Networked Embedded Sensing and Control: Workshop NESC'05: University of Notre Dame, USA, October 2005 Proceedings (Lecture Notes in Control and Information Sciences)

Lecture Notes in Control and Information Sciences Editors: M. Thoma · M. Morari 331 Panos J. Antsaklis Paulo Tabua...

Author: Panos J. Antsaklis | Paulo Tabuada

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Lecture Notes in Control and Information Sciences Editors: M. Thoma · M. Morari

331

Panos J. Antsaklis Paulo Tabuada (Eds.)

Networked Embedded Sensing and Control Workshop NESC'05: University of Notre Dame, USA October 2005 Proceedings With 96 Figures

Series Advisory Board F. Allg¨ower · P. Fleming · P. Kokotovic · A.B. Kurzhanski · H. Kwakernaak · A. Rantzer · J.N. Tsitsiklis

Editors Panos J. Antsaklis Paulo Tabuada 275 Fitzpatrick Hall Department of Electrical Engineering University of Notre Dame Notre Dame, IN 46556 USA

ISSN 0170-8643 ISBN-10 ISBN-13

3-540-32794-0 Springer Berlin Heidelberg New York 978-3-540-32794-3 Springer Berlin Heidelberg New York

Library of Congress Control Number: 2006921745 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, speciﬁcally the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microﬁlm or in other ways, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable to prosecution under German Copyright Law. Springer is a part of Springer Science+Business Media springer.com © Springer-Verlag Berlin Heidelberg 2006 Printed in Germany The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a speciﬁc statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Typesetting: Data conversion by editors. Final processing by PTP-Berlin Protago-TEX-Production GmbH, Germany Cover-Design: design & production GmbH, Heidelberg Printed on acid-free paper 89/3141/Yu - 5 4 3 2 1 0

To Melinda and Lily. Ao tinto do Alentejo e a` alheira de Mirandela.

Preface

This volume contains the proceedings of the Workshop on Networked Embedded Sensing and Control (NESC) held at the University of Notre Dame, Notre Dame, Indiana, on October 17-18 2005 (http://nesc.ee.nd.edu). Networked embedded control systems can be roughly described as collections of spatially distributed sensors, actuators and controllers whose behavior is coordinated through wired or wireless communication links. This integration between different technologies and scientiﬁc domains presents new and challenging fundamental problems underlying the foundations of this class of systems. The workshop acted both as a forum and as a catalyst for a dialogue between over 60 researchers working on diﬀerent aspects of networked systems, such as sensing and estimation, control, coding/communications and ad hoc networking. Diﬀerent research experiences and results were exchanged among researchers, while concrete steps towards establishing the scientiﬁc foundations and identifying the main scientiﬁc challenges were given. The workshop was organized as a single track event with 18 contributed papers over two days centered around four invited plenary lectures. The plenary lectures were ”Motion coordination for multi-agent networks” by Francesco Bullo, University of California at Santa Barbara; ”Embedded sensing and control: Applications and application requirements” by Tariq Samad, Honeywell; ”Control over communication networks: Impact of delays on performance” by Dawn Tilbury, University of Michigan; and ”The role of information theory in communication constrained control systems” by Sekhar Tatikonda, Yale University. The 18 contributed papers were presented in ﬁve sessions: Multi-Agent Control, Simulation and Implementation, Distributed Sensing, Filtering and Estimation, during the ﬁrst day; Control over Networks I and Control over Networks II, during the second day of the workshop.

VIII

Preface

What emerged from this workshop was the importance of exploring further the areas between the disciplines of systems and control, information theory, communication networks, distributed and collaborative control, and real-time systems. At the same time it became clear that being aware of application needs that help incorporate the right assumptions in our theoretical results, as well as of implementation issues that assist us in the crafting of our algorithms, is necessary. We are grateful to all invitees and contributors for making the workshop a success. We would also like to recognize the diligence and expertise of the Program Committee: John Baillieul, Boston University; Michael Branicky, Case Western Reserve University; Magnus Egerstedt, Georgia Institute of Technology; Nicola Elia, Iowa State University; Martin Haenggi, University of Notre Dame; Jo˜ ao Hespanha, University of California at Santa Barbara; George J. Pappas, University of Pennsylvania; Sekhar Tatikonda, Yale University; Dawn Tilbury, University of Michigan. We truly appreciate and acknowledge the generous support of our sponsors. The workshop was co-sponsored by the Oﬃce of Research of the University of Notre Dame, the Center for Applied Mathematics, the Department of Electrical Engineering, and the H.C. and E.A. Brosey Endowed Chair fund of the University of Notre Dame. It was also co-sponsored by the National Science Foundation EHS program and technically co-sponsored by the IEEE Control Systems Society. We do thank them all.

University of Notre Dame, Notre Dame, October 2005

Panos J. Antsaklis Paulo Tabuada

Contents

Part I Multi-agent Control Plenary Talk - Notes on Multi-agent Motion Coordination: Models and Algorithms Francesco Bullo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3

Discrete Time Kuramoto Models with Delay Benjamin I. Triplett, Daniel J. Klein, Kristi A. Morgansen . . . . . . . . . . .

9

Symmetries in the Coordinated Consensus Problem Ruggero Carli, Fabio Fagnani, Marco Focoso, Alberto Speranzon, Sandro Zampieri . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 On Communication Requirements for Multi-agent Consensus Seeking Lei Fang, Panos J. Antsaklis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 Applications of Connectivity Graph Processes in Networked Sensing and Control Abubakr Muhammad, Meng Ji, Magnus Egerstedt . . . . . . . . . . . . . . . . . . . . 69 Part II Simulation and Implementation Plenary Talk - Network-Embedded Sensing and Control: Applications and Application Requirements Tariq Samad . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 Simulation of Large-Scale Networked Control Systems Using GTSNetS ElMoustapha Ould-Ahmed-Vall, Bonnie S. Heck, George F. Riley . . . . . . . 87

X

Contents

neclab: The Network Embedded Control Lab Nicholas Kottenstette, Panos J. Antsaklis . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 Homogeneous Semantics Preserving Deployments of Heterogeneous Networks of Embedded Systems Aaron D. Ames, Alberto Sangiovanni-Vincentelli, Shankar Sastry . . . . . . 127 Part III Distributed Sensing, Filtering, and Estimation Distributed Kalman Filtering and Sensor Fusion in Sensor Networks Reza Olfati-Saber . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157 Belief Consensus and Distributed Hypothesis Testing in Sensor Networks Reza Olfati-Saber, Elisa Franco, Emilio Frazzoli, Jeﬀ S. Shamma . . . . . . 169 Distributed Evidence Filtering in Networked Embedded Systems Duminda Dewasurendra, Peter Bauer, Kamal Premaratne . . . . . . . . . . . . . 183 Part IV Control over Networks I Plenary Talk - Delays in Control over Communication Networks: Characterization, Impact, and Reduction Strategies Dawn Tilbury . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201 Anticipative and Non-anticipative Controller Design for Network Control Systems Payam Naghshtabrizi, Jo˜ ao P. Hespanha . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203 On Quantization and Delay Eﬀects in Nonlinear Control Systems Daniel Liberzon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219 Performance Evaluation for Model-Based Networked Control Systems Luis A. Montestruque, Panos J. Antsaklis . . . . . . . . . . . . . . . . . . . . . . . . . . . 231 Beating the Bounds on Stabilizing Data Rates in Networked Systems Yupeng Liang, Peter Bauer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251

Contents

XI

Disturbance Attenuation Bounds in the Presence of a Remote Preview Nuno C. Martins, Munther A. Dahleh, John C. Doyle . . . . . . . . . . . . . . . . 269 Part V Control over Networks II Plenary Talk - The Role of Information Theory in Communication Constrained Control Systems Sekhar Tatikonda . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289 Delay-Reliability Tradeoﬀs in Wireless Networked Control Systems Min Xie, Martin Haenggi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291 A Cross-Layer Approach to Energy Balancing in Wireless Sensor Networks Daniele Puccinelli, Emmanuel Sifakis, Martin Haenggi . . . . . . . . . . . . . . . 309 Distributed Control over Failing Channels C´edric Langbort, Vijay Gupta, Richard M. Murray . . . . . . . . . . . . . . . . . . . 325 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343

To Melinda and Lily. Ao tinto do Alentejo e a` alheira de Mirandela.

Preface

This volume contains the proceedings of the Workshop on Networked Embedded Sensing and Control (NESC) held at the University of Notre Dame, Notre Dame, Indiana, on October 17-18 2005 (http://nesc.ee.nd.edu). Networked embedded control systems can be roughly described as collections of spatially distributed sensors, actuators and controllers whose behavior is coordinated through wired or wireless communication links. This integration between different technologies and scientiﬁc domains presents new and challenging fundamental problems underlying the foundations of this class of systems. The workshop acted both as a forum and as a catalyst for a dialogue between over 60 researchers working on diﬀerent aspects of networked systems, such as sensing and estimation, control, coding/communications and ad hoc networking. Diﬀerent research experiences and results were exchanged among researchers, while concrete steps towards establishing the scientiﬁc foundations and identifying the main scientiﬁc challenges were given. The workshop was organized as a single track event with 18 contributed papers over two days centered around four invited plenary lectures. The plenary lectures were ”Motion coordination for multi-agent networks” by Francesco Bullo, University of California at Santa Barbara; ”Embedded sensing and control: Applications and application requirements” by Tariq Samad, Honeywell; ”Control over communication networks: Impact of delays on performance” by Dawn Tilbury, University of Michigan; and ”The role of information theory in communication constrained control systems” by Sekhar Tatikonda, Yale University. The 18 contributed papers were presented in ﬁve sessions: Multi-Agent Control, Simulation and Implementation, Distributed Sensing, Filtering and Estimation, during the ﬁrst day; Control over Networks I and Control over Networks II, during the second day of the workshop.

VIII

Preface

What emerged from this workshop was the importance of exploring further the areas between the disciplines of systems and control, information theory, communication networks, distributed and collaborative control, and real-time systems. At the same time it became clear that being aware of application needs that help incorporate the right assumptions in our theoretical results, as well as of implementation issues that assist us in the crafting of our algorithms, is necessary. We are grateful to all invitees and contributors for making the workshop a success. We would also like to recognize the diligence and expertise of the Program Committee: John Baillieul, Boston University; Michael Branicky, Case Western Reserve University; Magnus Egerstedt, Georgia Institute of Technology; Nicola Elia, Iowa State University; Martin Haenggi, University of Notre Dame; Jo˜ ao Hespanha, University of California at Santa Barbara; George J. Pappas, University of Pennsylvania; Sekhar Tatikonda, Yale University; Dawn Tilbury, University of Michigan. We truly appreciate and acknowledge the generous support of our sponsors. The workshop was co-sponsored by the Oﬃce of Research of the University of Notre Dame, the Center for Applied Mathematics, the Department of Electrical Engineering, and the H.C. and E.A. Brosey Endowed Chair fund of the University of Notre Dame. It was also co-sponsored by the National Science Foundation EHS program and technically co-sponsored by the IEEE Control Systems Society. We do thank them all.

University of Notre Dame, Notre Dame, October 2005

Panos J. Antsaklis Paulo Tabuada

Contents

Part I Multi-agent Control Plenary Talk - Notes on Multi-agent Motion Coordination: Models and Algorithms Francesco Bullo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3

Discrete Time Kuramoto Models with Delay Benjamin I. Triplett, Daniel J. Klein, Kristi A. Morgansen . . . . . . . . . . .

9

Symmetries in the Coordinated Consensus Problem Ruggero Carli, Fabio Fagnani, Marco Focoso, Alberto Speranzon, Sandro Zampieri . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 On Communication Requirements for Multi-agent Consensus Seeking Lei Fang, Panos J. Antsaklis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 Applications of Connectivity Graph Processes in Networked Sensing and Control Abubakr Muhammad, Meng Ji, Magnus Egerstedt . . . . . . . . . . . . . . . . . . . . 69 Part II Simulation and Implementation Plenary Talk - Network-Embedded Sensing and Control: Applications and Application Requirements Tariq Samad . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 Simulation of Large-Scale Networked Control Systems Using GTSNetS ElMoustapha Ould-Ahmed-Vall, Bonnie S. Heck, George F. Riley . . . . . . . 87

X

Contents

neclab: The Network Embedded Control Lab Nicholas Kottenstette, Panos J. Antsaklis . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 Homogeneous Semantics Preserving Deployments of Heterogeneous Networks of Embedded Systems Aaron D. Ames, Alberto Sangiovanni-Vincentelli, Shankar Sastry . . . . . . 127 Part III Distributed Sensing, Filtering, and Estimation Distributed Kalman Filtering and Sensor Fusion in Sensor Networks Reza Olfati-Saber . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157 Belief Consensus and Distributed Hypothesis Testing in Sensor Networks Reza Olfati-Saber, Elisa Franco, Emilio Frazzoli, Jeﬀ S. Shamma . . . . . . 169 Distributed Evidence Filtering in Networked Embedded Systems Duminda Dewasurendra, Peter Bauer, Kamal Premaratne . . . . . . . . . . . . . 183 Part IV Control over Networks I Plenary Talk - Delays in Control over Communication Networks: Characterization, Impact, and Reduction Strategies Dawn Tilbury . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201 Anticipative and Non-anticipative Controller Design for Network Control Systems Payam Naghshtabrizi, Jo˜ ao P. Hespanha . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203 On Quantization and Delay Eﬀects in Nonlinear Control Systems Daniel Liberzon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219 Performance Evaluation for Model-Based Networked Control Systems Luis A. Montestruque, Panos J. Antsaklis . . . . . . . . . . . . . . . . . . . . . . . . . . . 231 Beating the Bounds on Stabilizing Data Rates in Networked Systems Yupeng Liang, Peter Bauer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251

Contents

XI

Disturbance Attenuation Bounds in the Presence of a Remote Preview Nuno C. Martins, Munther A. Dahleh, John C. Doyle . . . . . . . . . . . . . . . . 269 Part V Control over Networks II Plenary Talk - The Role of Information Theory in Communication Constrained Control Systems Sekhar Tatikonda . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289 Delay-Reliability Tradeoﬀs in Wireless Networked Control Systems Min Xie, Martin Haenggi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291 A Cross-Layer Approach to Energy Balancing in Wireless Sensor Networks Daniele Puccinelli, Emmanuel Sifakis, Martin Haenggi . . . . . . . . . . . . . . . 309 Distributed Control over Failing Channels C´edric Langbort, Vijay Gupta, Richard M. Murray . . . . . . . . . . . . . . . . . . . 325 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343

Plenary Talk Notes on Multi-agent Motion Coordination: Models and Algorithms Francesco Bullo Mechanical Engineering Department Center for Control, Dynamical Systems and Computation University of California at Santa Barbara [email protected] http://motion.mee.ucsb.edu

Motion coordination is an extraordinary phenomenon in biological systems such as schools of ﬁsh and serves as a remarkable tool for man-made groups of robotic vehicles and active sensors. Although each individual agent has no global knowledge about the group as a whole or about the surrounding environment, complex coordinated behaviors emerge from local interactions. From a scientiﬁc point of view, the study of motion coordination poses novel challenges for systems and control theory. A comprehensive understanding of this phenomenon requires the joint ambitious study of mobility, communication, computation, and sensing aspects. In this brief document, we review some of our recent work on models and algorithms for coordinating the motion of multi-agent networks. In Section 1, we discuss models and classiﬁcations for multi-agent networks, i.e., groups of robotic agents that can sense, communicate and take local control actions. For these networks, we introduce basic notions of communication and control algorithms, coordination tasks and time complexity. Earlier eﬀorts in this direction are documented in [SY99, San01]; our treatment is inﬂuenced by [Lyn97a, JT92] and presented in detail in [MBCF05]. In Section 2, we discuss various basic algorithms for (i) rendezvous at a point and (ii) deployment over a given region. The proposed control and communication algorithms achieve these various coordination objectives requiring only spatially-distributed information or, in other words, single-hop communication. These rendezvous and deployment scenarios are treated extensively in [CMB04a, GCB05] and in [CMKB04a, CMB05a, CB04a], respectively. Early eﬀorts on related problems include [AOSY99, LH97]. The proposed models and examples shed some light on a novel class of control problems with insightful connections to the disciplines of distributed algorithms, geometric optimization, and algorithmic robotics.

P.J. Antsaklis, P. Tabuada (Eds.): Netw. Emb. Sens. and Cntrl., LNCIS 331, pp. 3–8, 2006. © Springer-Verlag Berlin Heidelberg 2006

4

F. Bullo

1 Robotic Networks and Complexity The global behavior of a robotic network arises from the combination of the local actions taken by its members. Each agent in the network can perform a few basic tasks such as sensing, communicating, processing information and moving according to it. The many ways in which these capabilities can be integrated make a robotic network a versatile and, at the same time, complex system. To understand the trade-oﬀs between performance, reliability and costs, it seems appropriate to propose a modeling framework where the execution of diﬀerent coordination algorithms can be appropriately formalized, analyzed and compared. We consider uniform networks of robotic agents deﬁned by a tuple S = (I, A, Ecmm ) consisting of a set of unique identiﬁers I = {1, . . . , N }, a collection of control systems A = {A[i] }i∈I , with A[i] = (X, U, X0 , f ), and a map Ecmm from X N to the subsets of I × I called the communication edge map. Here, (X, U, X0 , f ) is a control system with state space X ⊂ Rd , input space U , set of allowable initial states X0 ⊂ X, and system dynamics f : X ×U → X. An edge between two identiﬁers in Ecmm implies the ability of the corresponding two agents to exchange messages. A control and communication law for S consists of the sets: ¯ + , an increasing sequence of time instants, called com1. T = {t } ∈N ⊂ R munication schedule; 2. L, called the communication language, whose elements are called messages; 3. W , set of values of some logic variables w[i] ∈ W , i ∈ I, and W0 ⊆ W , subset of allowable initial values. These sets correspond to the capability of agents to allocate additional variables and store sensor or communication data; and the maps: 4. msg : T × X × W × I → L, called message-generation function; 5. stf : T × W × LN → W , called state-transition function; ¯ + × X × X × W × LN → U , called control function. 6. ctl : R To implement a control and communication law each agent performs the following sequence or cycle of actions. At each instant t ∈ T, each agent i communicates to each agent j such that (i, j) belongs to Ecmm (x[1] , . . . , x[N ] ). Each agent i sends a message computed by applying the message-generation function to the current values of t , x[i] and w[i] . After a negligible period of time, agent i resets the value of its logic variables w[i] by applying the statetransition function to the current value of w[i] , and to the messages y [i] (t ) received at t . Between communication instants, i.e., for t ∈ [t , t +1 ), agent i applies a control action computed by applying the control function to its state at the last sample time x[i] (t ), the current values of x[i] and w[i] , and to the messages y [i] (t ) received at t .

Notes on Multi-agent Motion Coordination

5

Some remarks are appropriate. In our present deﬁnition, all agents are identical and implement the same algorithm; in this sense the control and communication law is called uniform (or anonymous). If W = W0 = ∅, then the control and communication law is static (or memoryless) and no state-transition function is deﬁned. It is also possible for a law to be timeindependent if the three relevant maps do not depend on time. Finally, let us also remark that this is a synchronous model in which all agents share a common clock. Next, we establish the notion of coordination task and of task achievement by a robotic network. A (static) coordination task for a network S is a map T : X N → {true, false}. Additionally, let CC be a control and communication law for S. We say that CC achieves the task T if for all initial conditions [i] x0 ∈ X0 , the corresponding network evolution t → x(t) has the property that there exists T ∈ R+ such that T(x(t)) = true for t ≥ T . In control-theoretic terms, achieving a task means establishing a convergence or stability result. Beside this key objective, one might be interested in eﬃciency as measured by required communication service, required control energy or by speed of completion. We focus on the latter notion. The time complexity to achieve T with CC is TC(T, CC ) = sup

TC(T, CC , x0 ) | x0 ∈ X0N ,

where TC(T, CC , x0 ) = inf { | T(x(tk )) = true , ∀k ≥ }, and where t → (x(t)) is the evolution of (S, CC ) from x0 . The time complexity of T is TC(T) = inf {TC(T, CC )| CC achieves T} . Some ideas on how to deﬁne meaningful notions of communication complexity are discussed in [MBCF05]. In the following discussion, we describe certain coordination algorithms, which have been cast into this modeling framework and whose time complexity properties have been analyzed.

2 Example Algorithms and Tasks Key problems in motion coordination include the design of strategies for ﬂocking, motion planning, collision avoidance and others. Numerous such problems remain interesting open challenges as of today. For example, it is still not clear how to prescribe the agents’ motion in such a way as to achieve a generic prescribed geometric pattern; note that certain impossibility results are known [SY99]. Typically, coordination objectives are characterized via appropriate utility functions. We illustrate our approach by discussing two basic types of problems: rendezvous and deployment.

6

F. Bullo

Aggregation algorithms The rendezvous objective (also referred to as the gathering problem) is to achieve agreement over the location of the agents, that is, to steer each agent to a common location. An early reference on this problem is [AOSY99]. We consider two scenarios which diﬀer in the agents’ sensing/communication capabilities and the environment to which the agents belong. First [CMB04a], we consider the problem of rendezvous for agents equipped with range-limited communication in obstacle-free environments. In this case, each agent is capable of sensing its position in the Euclidean space Rd and can communicate it to any other robot within a given distance r. This communication service is modeled by the r-disk graph, in which two agents are neighbors if and only if their Euclidean distance is less than or equal to r. Second [GCB05], we consider visually-guided agents. Here the agents are assumed to belong to a nonconvex simple polygonal environment Q. Each agent can sense within line-of sight any other agent as well as sense the distance to the boundary of the environment. The relationship between the agents can be characterized by the so-called visibility graph: two agents are neighbors if and only if they are mutually visible to each other. In both scenarios, the rendezvous problem cannot be solved with distributed information unless the agents’ initial positions form a connected sensing/communication graph. Arguably, a good property of any rendezvous algorithm is that of maintaining connectivity between agents. This connectivitymaintenance objective is interesting on its own. It turns out that this objective can be achieved through local constraints on the agents’ motion. Motion constraint sets that maintain connectivity are designed in [AOSY99, GCB05] by exploiting the geometric properties of disk and visibility graphs. These discussions lead to the following algorithm that solves the rendezvous problems for both communication scenarios. The agents execute what we shall refer to as the Circumcenter Algorithm; here is an informal description. Each agent iteratively performs the following tasks: 1: acquire neighbors’ positions 2: compute connectivity constraint set 3: moves toward the circumcenter of the point set comprised of its neighbors and of itself, while remaining inside the connectivity constraint set. One can prove that, under technical conditions, the algorithm does achieve the rendezvous task in both scenarios; see [CMB04a, GCB05]. Additionally, when d = 1, it can be shown that the time complexity of this algorithm is Θ(N ); see [MBCF05].

Notes on Multi-agent Motion Coordination

7

Deployment algorithms The problem of deploying a group of agents over a given region of interest can be tackled with the following simple heuristic. Each agent iteratively performs the following tasks: 1: acquire neighbors’ positions 2: compute own dominance region 3: move towards the center of own dominance region This short description can be made accurate by specifying what notions of dominance region and of center are to be adopted. In what follows we mention two examples and refer to [CMKB04a, CMB05a, CB04a] for more details. First, we consider the area-coverage deployment problem in a convex polygonal environment. The objective is to maximize the area within close range of the mobile nodes. This models a scenario in which the nodes are equipped with some sensors that take measurements of some physical quantity in the environment, e.g., temperature or concentration. Assume that certain regions in the environment are more important than others and describe this by a density function φ. This problems leads to the coverage performance metric N

Have (p1 , . . . , pN ) =

min

Q i∈{1,...,N }

f ( q−pi ) φ(q)dq = i=1

Vi

f ( q−pi )φ(q)dq.

Here pi is the position of the ith node, f measures the performance of an individual sensor, and {V1 , . . . , VN } is the Voronoi partition of the nodes {p1 , . . . , pN }. If we assume that each node obeys a ﬁrst order dynamical behavior, then a simple gradient scheme can be easily implemented in a spatiallydistributed manner. Following the gradient of Have corresponds, in the previous algorithm, to deﬁning (1) the dominance regions to be the Voronoi cells generated by the agents, and (2) the center of a region to be the centroid of the region (if f (x) = x2 ). Because the closed-loop system is a gradient ﬂow for the cost function, performance is locally, continuously optimized. As a special case, when the environment is a segment and φ = 1, the time complexity of the algorithm can be shown to be O(N 3 log(N ε−1 )), where ε is a threshold value below which we consider the task accomplished; see [MBCF05]. Second, we consider the problem of deploying to maximize the likelihood of detecting a source. For example, consider devices equipped with acoustic sensors attempting to detect a sound-source (or, similarly, antennas detecting RF signals, or chemical sensors localizing a pollutant source). For a variety of criteria, when the source emits a known signal and the noise is Gaussian, we know that the optimal detection algorithm involves a matched ﬁlter, that detection performance is a function of signal-to-noise-ratio, and, in turn, that signal-to-noise ratio is inversely proportional to the sensor-source distance. In this case, the appropriate cost function is Hworst (p1 , . . . , pN ) = max

min

q∈Q i∈{1,...,N }

f ( q − pi ) = max f ( q − pi ), q∈Vi

8

F. Bullo

and a greedy motion coordination algorithm is for each node to move toward the circumcenter of its Voronoi cell. A detailed analysis [CB04a] shows that the detection likelihood is inversely proportional to the circumradius of each node’s Voronoi cell, and that, if the nodes follow this algorithm, then the detection likelihood increases monotonically as a function of time. Acknowledgments This document summarizes some results of joint work with Anurag Ganguli, Ketan Savla, Jorge Cort´es, Emilio Frazzoli, and Sonia Mart´ınez. The author thanks the organizers of the Workshop on Network Embedded Sensing and Control for the warm hospitality throughout the meeting and the opportunity to present this work. The author also gratefully acknowledges the partial support of ONR YIP Award N00014-03-1-0512, NSF SENSORS Award IIS0330008, and ARO MURI Award W911NF-05-1-0219.

Francesco Bullo received the Laurea degree ”summa cum laude” in Electrical Engineering from the University of Padova, Italy, in 1994, and the Ph.D. degree in Control and Dynamical Systems from the California Institute of Technology in 1999. From 1998 to 2004, he was an Assistant Professor with the Coordinated Science Laboratory at the University of Illinois at UrbanaChampaign. He is currently an Associate Professor with the Mechanical & Environmental Engineering Department at the University of California, Santa Barbara. His research interests include motion planning and coordination for autonomous vehicles, geometric control of mechanical systems, and distributed and adaptive algorithms. He is a recipient of the 2003 ONR Young Investigator Award. He is the coauthor (with Andrew D. Lewis) of the book ”Geometric Control of Mechanical Systems” (New York: Springer Verlag, 2004, 0-387-22195-6). He has published more than 70 papers in international journals, books, and refereed conferences. He serves on the Editorial Board of the ”IEEE Transactions on Automatic Control,” ”SIAM Journal of Control and Optimization,” and ”ESAIM: Control, Optimization, and the Calculus of Variations.” He also serves as Chairman of the IEEE Control Systems Society Technical Committee on Manufacturing, Automation, and Robotics Control (MARC).

Discrete Time Kuramoto Models with Delay Benjamin I. Triplett, Daniel J. Klein, and Kristi A. Morgansen Department of Aeronautics and Astronautics University of Washington Seattle, WA 98195-2400 {triplett,djklein,morgansen}@aa.washington.edu Summary. Motivated by the needs of multiagent systems in the presence of sensing and communication that is delayed, intermittent and asynchronous, we present here a discrete-time Kuramoto oscillator model that incorporates delays. Analytical results are derived for certain cases of heading-based stability of synchronized and balanced equilibrium sets both when delay is not present and when it is. In all cases, agents are assumed to be identical and fully connected, with constant and equal network delays. For complex cases, the analysis is supplemented with Monte Carlo simulations, and all results are demonstrated in simulation.

1 Introduction During the past several years, a great deal of research attention in the areas of dynamics and control theory has been directed to problems derived from applications in coordinated and multiagent systems. These applications appear in engineered systems (e.g. coordinated control of autonomous vehicles [BA98, BH98, CB04b, EH01, FM04b, Fer03, JLM03d, JK04, LF01]) as well as in natural systems (e.g. emergent patterns in groups of organisms [CDF+ 01, Gol96, GLR96, Hep97, MEK99, PEK99]). Analysis and control of systems of multiple vehicles or agents provides a great deal of challenge. The order of such systems tends to be large and can be dynamic (as members join and leave the group), the coupling between agents is mathematically nontrivial in most cases of interest (e.g. limited sensing range, dynamic communication), and in many cases allowance must be made for data transmission delay and clock asynchrony. Often, individual agent dynamics are taken to be linear integrators (either single or double). In general, such simpliﬁcations of the dynamics do not prevent application of the derived techniques to physical systems, and the mathematics of the resulting system are greatly simpliﬁed. However, a number of multiagent systems of interest, both engineered (e.g. aircraft, ground vehicles and underwater vehicles) and natural (e.g. ﬁsh, quadrapeds) are composed of

P.J. Antsaklis, P. Tabuada (Eds.): Netw. Emb. Sens. and Cntrl., LNCIS 331, pp. 9–23, 2006. © Springer-Verlag Berlin Heidelberg 2006

10

B.I. Triplett, D.J. Klein, and K.A. Morgansen

systems whose dynamics are inherently nonholonomic. The key features of the dynamics of the individual agents in such systems is that they have the ability to change their forward/backward velocity and the ability to change their turning radius, usually within ﬁnite bounds, but they do not have the ability to directly control lateral motion. The use of Frenet-Serret equations of motion has recently been shown to be a convenient tool for succinct description of such system dynamics [JK04]. By assuming the forward velocity to be constant and taking the turning radius to be a control function composed of relatively weighted spacing and alignment components, where the spacing occurs on a much slower time scale than the alignment, the group dynamics can be studied as a system of coupled oscillators. The study of such systems of oscillators has a well-established history in the physics and biology communities through the original work by Winfree [Win67] whose models were later reﬁned and formalized into the wellknown Kuramoto models and studies on collective synchronization (see e.g. [Str00b] for a survey). Oscillator models have recently begun to receive attention in the control theory community for stabilization of groups of autonomous agents [JMB04, SPL03]. Engineering studies have focused on stability results for continuous time versions of the model with either full or partial and dynamic connectivity of the agents. Here, we are interested in the eﬀects of delayed and discrete time interactions between the agents. Some extensions of the continuous time Kuramoto model to include delay have been considered in previous work [AV93, KPR97, YS99]. However, in engineered systems, for example in underwater applications, such delays often occur due to transmission time or due to shared communication media, and occur on a much slower time scale than the time scale for the agent dynamics. These resulting systems are thus best described with discrete time models with delayed interaction coupling. The contributions of the work here are an extension of the continuous time Kuramoto oscillator model to a discrete time version along with stability analysis for cases of systems both with and without delay. The presentation is organized as follows. For completeness, details of the connection between Frenet-Serret dynamics and continuous time oscillator models are given in the next section. In Section 3, stability results for discretized Kuramoto models with no delay are presented. Stability results for discretized systems with delay are given in Section 4. Simulation results are given in Section 5. We conclude with a discussion of ongoing and future work in Section 6.

2 Oscillator Models for Multivehicle Systems As presented in [JK04], planar vehicles with heading control can be described by the planar Frenet-Serret equations

Discrete Time Kuramoto Models with Delay

r˙ i = xi x˙ i = yi ui y˙ i = −xi ui

11

(1) i = 1...N

2

where r, x, y ∈ R , u ∈ R and θ is such that x=

cos(θ) sin(θ)

Here, r is the planar position of the center of mass of the vehicle, x is the unit tangent vector in the direction of motion, y is the unit vector normal to the direction of motion, and u is the controlled rate of angular motion, ˙ In [SPL03], this control is composed of a linear combination of alignu = θ. ment and spacing terms where the time scale of the spacing is assumed to be small enough relative to that of the alignment to allow the two to eﬀectively decouple. The form of the alignment control is taken to be that of a system of phase-coupled oscillators as in the Kuramoto model, written as u = θ˙i (t) = ωi −

K N

N

sin (θj (t) − θi (t))

(2)

j=1

where N is the number of agents in the system, ωi is the natural frequency of the ith agent, and θi is its corresponding phase. If the dynamics are represented relative to a rotating coordinate frame, then ωi = 0. As shown in [JMB04] and others, the sign of K determines whether the oscillators will align or anti-align their phases as the system evolves, and the magnitude of K determines the rate of convergence to steady state. The results from [JK04] are studied in the framework of phase coupled oscillators in [SPL03], where the topology of the equilibrium sets of systems like those in the prior work are analyzed and described.

3 The Discrete Time Kuramoto Model In many applications, networked multiagent control systems suﬀer from intermittent, delayed, and asynchronous communications and sensing. To accommodate such systems, in which the time scales of the network events are much slower than the underlying agent dynamics, we consider the Kuramoto model in synchronous discrete time with ﬁxed time step ΔT . Our goal in this section is to determine bounds on the quantity KΔT , where K is the control gain in the continuous model, for which the system will converge to either a state of synchrony in which all oscillators have the same phase, or a state of incoherence in which the phasor centroid is zero. The terms synchronized set and balanced set will be used respectively to describe these sets of states. Using a zero-order hold on the data to compute the control between updates at ﬁxed intervals ΔT , the discretized Kuramoto model is

12

B.I. Triplett, D.J. Klein, and K.A. Morgansen

θi (h + 1) = θi (h) −

KΔT N

sin (θj (h) − θi (h))

(3)

j

where we assume the natural frequencies of all oscillators in the model are zero. As with the continuous time system (2), this assumption corresponds to writing the system dynamics in a rotating coordinate frame before discretization. Stability analysis for the general system is quite challenging. Using linearization, we will show that the behavior of the discrete-time Kuramoto model without delay converges to the synchronized set when −2 < KΔT < 0, and to the balanced set for 0 < KΔT < 2, provided the initial conditions are in a local neighborhood of the equilibria. Further, in special cases, we will show that these results hold more strongly for any initial condition that is not an unstable equilibrium point for the respective case. Analysis for arbitrary numbers of vehicles in this more general case are not yet available, however, Monte Carlo simulations verify the extension. 3.1 The Two Oscillator Problem As the equilibria for this system are sets rather than discrete points, for full stability analysis we will apply LaSalle’s Invariance Principle rather than Lyapunov’s Stability Criteria. For reference, LaSalle’s Invariance Principle for discrete-time systems is stated as Proposition 1. (LaSalle’s Invariance Principle for Discrete Time Systems ¯ and [LaS86]) If V is a Lyapunov function on a linear space D, with closure D, the solution of the discrete time system is in D and bounded, then there is a number c such that the solution approaches M ∩V −1 (c) where M is the largest ¯ positively invariant set contained in the set E = {x ∈ Rm |ΔV = 0} ∩ D. To motivate the construction of the Lyapunov functions, we will refer to a parameters often used to study the synchrony of the continuous time system (2), the order parameter [Str00b], r(θ(h)) =

1 N

i

cos θi (h) sin θi (h)

.

Note that for all states in the synchronized set (identical headings), the order parameter has a value of r(θ) = 1, and for all states in the balanced set (opposing headings), r(θ) = 0. Based on this parameter, consider the following candidate Lyapunov function: V (θ(h)) ≡ =

1 N r(θ(h))2 2 1 2N

i

cos θi (h) sin θi (h)

2

.

Discrete Time Kuramoto Models with Delay

13

This function attains a maximum value of N2 and a minimum value of zero for all points on the synchronized and balanced sets, respectively. Using the deﬁnitions ΔV (x) ≡ V (f (x)) − V (x) and f (θi (h)) = θi (h + 1) as above, the ΔT time diﬀerence of V is   2 2 cos θi − KΔT sin (θ − θ ) j i 1 1 cos θ j N i   − ΔV (θ) = . sin θi 2N 2N sin θi − KΔT sin (θj − θi ) i

i

j

N

Every term in the above equation occurs at the same time, so the index, h, has been dropped for notational brevity. Choosing φi = θi − KΔT j sin (θj − θi ) , N the above equation becomes ΔV (θ) =

1 2N

2

cos φi sin φi

i

−

1 2N

i

cos θi sin θi

2

.

(4)

Employing the simpliﬁcation N i=1

cos θi sin θi

2

= cos2 θ1 + cos2 θ2 + . . . + cos2 θN + 2 cos θ1 cos θ2 + . . . + cos θN −1 cos θN + sin2 θ1 + sin2 θ2 + . . . + sin2 θN + 2 sin θ1 sin θ2 + . . . + sin θN −1 sin θN = N + 2 cos θ1 cos θ2 + . . . + cos θN −1 cos θN + sin θ1 sin θ2 + . . . + sin θN −1 sin θN = N + 2 cos (θ1 − θ2 ) + . . . + cos (θN −1 − θN ) N −1

N

cos (θi − θj ),

=N +2 i=1 j=i+1

(4) can be written as 1 ΔV (θ) = N

N −1

N

cos (φi − φj ) − cos (θi − θj ) .

(5)

i=1 j=i+1

Using the above, we can state the following. Theorem 1. For N = 2, the discrete time Kuramoto system (3) will converge to the balanced set, |θ1 − θ2 | = π, if 0 < KΔT < 2 and if the initial conditions satisfy θ1 (0) = θ2 (0), where (θ1 , θ2 ) ∈ T. Proof. We prove Theorem 1 by LaSalle’s Invariance Principle. The synchronized and balanced sets for N = 2 are S = {θ1 − θ2 | θ1 − θ2 = 0} and

14

B.I. Triplett, D.J. Klein, and K.A. Morgansen

B = {θ1 − θ2 | |θ1 − θ2 | = π}, respectively. Specify the torus minus the synchronized set as the domain, D = T − S, of the Lyapunov function V (θ). This choice of D contains no unstable equilibria. For N = 2, equation (5) becomes 1 cos (φ1 − φ2 ) − cos (θ1 − θ2 ) 2 1 KΔT = cos θ1 − θ2 − sin (θ2 − θ1 ) 2 2 KΔT + sin (θ1 − θ2 ) − cos (θ1 − θ2 ) 2 1 = cos (θ1 − θ2 + KΔT sin (θ1 − θ2 )) − cos (θ1 − θ2 ) . 2

ΔV (θ) =

(6)

Equation (6) only has zeros at B in D. Therefore, E is composed of only the desired equilibria points, so all of E is invariant, in which case M = E. To complete the proof we need to show that equation (6) is negative on D − E for the stated ranges of KΔT . We can then use the following inequality derived from equation (6) cos (θ + KΔT sin θ) < cos θ (7) where θ = θ1 − θ2 . The plot in Figure 1 shows the behavior of equation (7) which is dependent on the sign of K. Cosine is symmetric about zero so we can consider one half of a symmetric interval. For the domain 0 ≤ θ ≤ π cosine is a strictly decreasing function. Hence, for the inequality in question to hold, we must have that θ < θ + KΔT sin θ < 2π − θ. The left side of the inequality amounts to KΔT sin θ > 0 which is true as long as KΔT > 0 because sin θ is always positive on the interval in question. The right side of the inequality is more interesting. Here, we can discover the bounds on KΔT by taking the derivative of both sides with respect to θ, with the result KΔT < cos2 θ . Since the extreme values of cos θ on 0 ≤ θ ≤ π are ±1, we have the result that −2 < KΔT < 2. The ﬁnal result 0 < KΔT < 2 comes from taking the intersection of the two conditions on KΔT . Theorem 2. For N = 2, the discrete time Kuramoto system (3) will converge to the synchronized set, θ1 = θ2 , if −2 < KΔT < 0 and if the initial conditions satisfy |θ1 − θ2 | = π, where (θ1 , θ2 ) ∈ T. Proof. The proof uses similar analysis to Theorem 1. The main diﬀerences are the Lyapunov function switches signs, V = −V , and the domain is D = T−B. The largest invariant set is then M = S. For V to be negative on D − E, θ > θ + KΔT sin θ > 2π − θ which is true for all θ provided −2 < KΔT < 0.

cos(! + K T cos!)

cos!

Discrete Time Kuramoto Models with Delay

15

0 0.5 1

K T = 1.8 K T = 2.0 K T = 2.2

1.5

cos!

1.5

cos(! + K T cos!)

3

1 0.5

2

1

0

1

2

3

0

1

2

3

!

K T = 1.8 K T = 2.0 K T = 2.2

0 3

2

1

!

Fig. 1. Plot of cos (θ + KΔT sin θ) − cos θ demonstrating that the Lyapunov diﬀerence equation becomes positive for values of KΔT > 2 (near θ = ±π) and becomes negative for KΔT < −2 (near zero).

To generalize this result to an arbitrary numbers of oscillators, two points must be addressed. First, we need to show analytically that ΔV is negative for all points in D − E. This is a diﬃcult problem for arbitrary N with generic D. The second point is that we need to choose an initial set D which does not contain any unstable equilibria. It is important to note that the choice of D is signiﬁcantly diﬀerent for convergence to the synchronized and balanced sets. For example, any point on the balanced set is an unstable equilibrium point on the synchronized set, and vice versa. For convergence to the synchronized set, the choice of D = {θi | θi ∈ (γ − π/2, γ + π/2), i = 1, . . . , N } for any constant γ removes all unstable equilibria from the domain. This domain is necessary for analytical formality, but in practice, any initial conditions not precisely on an unstable equilibrium will work. For convergence to the balanced set, the choice of D for arbitrary N remains an open analytical problem, but again in practice this has not been a problem. 3.2 The N-Oscillator Problem The N -oscillator problem is more challenging. However, if we limit our attention to a smaller neighborhood about the synchronized set, we can prove a stability result for the N -oscillator problem by noting that an appropriate D is a region where all relative phases are small. We can also extend the N = 2

16

B.I. Triplett, D.J. Klein, and K.A. Morgansen

case to single equilibria in the balanced set by simply linearizing the system about the given equilibria. We therefore have the following result. Theorem 3. There is a non-zero region of attraction for which the system (3) of N phase-coupled oscillators will converge to the same phase provided that −2 < KΔT < 0. Proof. To begin, express the N -oscillator Lyapunov diﬀerence equation as follows   N −1 N N −1 N 1  ΔV (θ) = cos (θi − θj ) cos (φi − φj ) − N i=1 j=i+1 i=1 j=i+1  N −1 N N 1  KΔT = cos θi − θj − sin(θq − θi ) N i=1 j=i+1 N q=1  N N −1 N KΔT cos(θi − θj ) . sin(θq − θj ) − + N q=1 i=1 j=i+1 All equilibria in the synchronized set are given by θi = θ0 . Thus we can apply the small angle approximation for Δθij = θi − θj to get  N −1 N N KΔT 1  cos θi − θj − (θq − θi ) ΔV (θ) ≈ N i=1 j=i+1 N q=1  N N −1 N KΔT cos(θi − θj ) + (θq − θj ) − N q=1 i=1 j=i+1  N −1 N 1  cos (θi − θj − KΔT (θi − θj )) = N i=1 j=i+1  N −1

N

−

cos(θi − θj ) .

i=1 j=i+1

Each cosine in the sum is now only in terms of the diﬀerence between two phases, so we can compare term by term and use the results of Theorem 1. From continuity of the state space, it must hold that there exists an open neighborhood about the synchronized set where Δθ is small that does not contain any unstable equilibria. Using this neighborhood as D and applying LaSalle’s Invariance Principle completes the proof by noting that the synchronized set contains all of the positively invariant points in D.

Discrete Time Kuramoto Models with Delay

17

Similar analysis for the balanced set has not yet followed from this approach, although the result holds in simulation, so we provide the corresponding result as a conjecture and appeal to Monte Carlo simulations for the general case. Conjecture 1. There is a non-zero region of attraction for which the system of N phase-coupled oscillators will converge to the balanced set provided that 0 < KΔT < 2, and to the synchronous set provided that −2 < KΔT < 0. To show the stability of the N oscillator system with the most general region of attraction for the equilibria sets, we will use Monte Carlo techniques. To this end, we have run simulations of the system for many trials (typically on the order of 50000) over a range of values of KΔT . The minimum and maximum of ΔV for −3 < KΔT < 3 are shown as in Figure 2.

3

V Max V Min

N=7, Trials=50000

2

1 V 0

1

2

3

3

2

1

0 K T

1

2

3

Fig. 2. Monte Carlo simulation results for seven oscillators without delay. These results are used to support the notion that the stability bounds of KΔT are independent of N and are typical for any number of vehicles.

This, along with many other simulations not shown here, and at several values of N , supports the notion that the bounds of KΔT are independent of N . Speciﬁcally, there are three points worth noting in these results. First,

18

B.I. Triplett, D.J. Klein, and K.A. Morgansen

on −2 < KΔT < 0 and on 0 < KΔT < 2, the minimum and maximum of ΔV was respectively zero. This corresponds to initial conditions taken at equilibrium points. Second, for negative KΔT , ΔV could only be negative when KΔT < −2. This supports the notion that the system is guaranteed to converge to the synchronized set only for −2 < KΔT < 0, since the Lyapunov function for the synchronized set is the negative of V. Finally, for positive KΔT , the maximum value of ΔV was positive only for KΔT > 2, which supports the claim that the discrete time N -oscillator system is only guaranteed to converge to the balanced set for 0 < KΔT < 2.

4 Discrete Time Kuramoto Model with Time Delay The discrete-time Kuramoto model can be extended to allow for a class of delay with the formulation θi (h + 1) = θi (h) −

KΔT N

sin (θj (h − d) − θi (h)),

(8)

j=i

where we make the assumption that agent i always has current information about itself, and that information about all other agents is delayed by an integer number of time steps d ∈ N of duration ΔT . The zero-order hold is used to generate the control. Theorem 4. For any N and within some region of attraction, the discrete time Kuramoto model with any number of delay steps converges to alignment when − NN−1 < KΔT < 0. Proof. To prove stability for the N oscillator system with any integer time delay, we will use a diﬀerent approach from that of the non-delayed system analysis. We will linearize about the aligned equilibrium, which will allow us to write a linear matrix equation for the system. Stability is known from the location of the eigenvalues of the system matrix relative to the unit circle. Starting with equation (8), we can buﬀer the system and write it as     θ1 (h) θ1 (h − d)  θ2 (h)   θ2 (h − d)       ..   ..  .  .       θN (h)      = 0 I  θN (h − d)  , (9)  θ1 (h + 1)    A1 A2  θ1 (h)     θ2 (h + 1)   θ2 (h)      .  .  . . .  .  θN (h + 1)

θN (h)

where 0 is the N × N matrix of zeros, I is the N × N identity,

Discrete Time Kuramoto Models with Delay



0

  −KΔT   N  A1 =   ..  .  

−KΔT −KΔT N N

...

−KΔT N

...

−KΔT N

0

.. ...

.

19

 −KΔT N

−KΔT N

.. .

         

,

−KΔT N

0

N −1 N KΔT

... 1 +

N ×N

and A2 = diag 1 +

N −1 N KΔT

1+

N −1 N KΔT

.

For the system to be guaranteed to converge to the aligned state (i.e. Δθij = θi − θj , i, j ∈ {1 . . . N } satisﬁes the small angle approximation), we must have that the eigenvalues of the 2N × 2N system matrix above all lie within the unit circle. We ﬁnd that the eigenvalues of the system matrix are 1 1 + bc ± 2 2 1 1 λr = + bc ± 2 2

λ1,2 =

1 + 2bc + b2 c2 − 4(N − 1)b

(10)

1 + 2bc + b2 c2 + 4b,

(11)

where b = KΔT /N and c = N −1. The pairs of repeated eigenvalues, λr , occur N − 1 times, and all of the eigenvalues are real for any KΔT > 0 and N ≥ 2. For system stability, then, it is required that −1 ≤ λ1,2 ≤ 1 and −1 ≤ λr ≤ 1. The requirement for λ1,2 is satisﬁed when N−N −1 < KΔT < 0, and for λr when −2N N −2 < KΔT < 0. It is clear then that λ1,2 determine stability. The balanced set is more diﬃcult to work with, since the linearization for general N still involves cosines. A simple result for N = 2 and linearization about the balanced set is available and shows the same behavior as for the aligned set. Theorem 5. For N = 2 and within some region of attraction, the discrete time Kuramoto model with any number of delay steps converges to the balanced set when 0 < KΔT < 2. Proof. The N = 2 system with delay steps d, linearized about the balanced set, is described by      0 0 1 0 θ1 (h − d) θ1 (h)  θ2 (h)   0  θ2 (h − d) 0 0 1      (12) θ1 (h + 1) =  0 KΔT 1 − KΔT   θ1 (h)  . 0 2 2 KΔT KΔT θ2 (h + 1) θ2 (h) 0 0 1− 2 2 For the system to be guaranteed to converge for any θ1 , θ2 such that Δθ = θ1 − θ2 ≈ π we must have that the eigenvalues of the 4 × 4 system matrix

20

B.I. Triplett, D.J. Klein, and K.A. Morgansen

above all lie within the unit circle. We ﬁnd that the eigenvalues of the system matrix are √ KΔT 2 − KΔT ± K 2 ΔT 2 − 12KΔT + 4 . , 1, − 2 4 For the non-unity eigenvalues to be inside the unit circle, we require 0 < KΔT < 2. Note that one eigenvalue is identically unity. This eigenvalue corresponds to the fact that equilibrium on the balanced set can be about any angle, not speciﬁcally θi = 0.

5 Simulation Results To demonstrate the previously derived bounds on gains, we provide simulations of illustrative cases. Figure 3 shows simulation results for two vehicles with identical initial conditions when KΔT is within, and near the critical limits of, the asymptotically stable, marginally stable and unstable regions for synchronization with no delay. As expected, the vehicles quickly synchronize in the asymptotically stable case and do not converge in the marginally stable case. Note that the for unstable case of synchronization the oscillators quickly reach an equilibrium separation. Additionally, results for the synchronized 3.08 i

3.1 i

3.06

3.06

3.04 10 3.5 i

20 (a)

30

40

3.02

10

20 (b)

30

40

10

20 (d)

30

40

3.1

3.2

i

3.06 2.9 2.6

10

20 (c)

30

40

3.02

Fig. 3. Simulation of the discrete time Kuramoto model with two vehicles for (a) KΔT = −1.8, d = 0, (b) KΔT = −2.0, d = 0, (c) KΔT = −2.2, d = 0, and (d) KΔT = −1.8, d = 1.

Discrete Time Kuramoto Models with Delay

21

stability of two vehicles with one-step delay are shown in the ﬁnal panel of Figure 3. Again the derived results are veriﬁed. As indicated in the previous section, adding delay to the discrete-time Kuramoto model decreases the range of KΔT for which the system will converge. Analysis for small systems and Monte Carlo simulations for larger systems show that this result is dependent on the number of oscillators, N , but not the number of delay steps d > 0. The behavior of the system demonstrating these bounds is shown in Figure 4 with and without time delay for combinations of N = 4, N = 20, d = 0, d = 4, d = 8, and KΔT = −0.7. Given the results of the analysis, we expect that, for a choice of KΔT , increasing the delay for a constant number of vehicles will not aﬀect stability (Figure 4(a,b,c)), but that increasing the number of vehicles for a ﬁxed delay will lead to an unstable system (Figure 4(b,d)). Clearly these results validate both the analytical results for delay as well as the conjectures.

i

4

N = 4, d = 0

2 0

10

4

i

i

20

(a)

30

20

(c)

30

10

4

N = 4, d = 0

10

N = 4, d = 0

2 0

40

i

2 0

4

40

20

(b)

30

40

N = 4, d = 0

2 0

10

20

(d)

30

40

Fig. 4. Simulation of the Kuramoto model with KΔT = −1.4 and (a) N = 4 and d = 0, (b) N = 4 and d = 4, (c) N = 4 and d = 8, (d) N = 20 and d = 4. This simulation indicates that stability is independent from the delay, d, but dependent on the number of vehicles, N .

The connection between the theory of discrete time oscillators and the motivating multivehicle applications is illustrated in Figure 5. Here, a group of seven nonholonomic vehicles, modeled as (1) with control (3), is simulated with two diﬀerent values of KΔT . In Figure 5(a), 0 < KΔT < 2 so all vehicles align, whereas in 5(b) KΔT > 2 and the vehicles never reach alignment.

6 Conclusions and Directions of Future Work Motivated by the needs of multiagent systems in the presence of delayed, intermittent and asynchronous sensing and communication, we have presented

22

B.I. Triplett, D.J. Klein, and K.A. Morgansen K=0.03 T=1 N=7

(a)

K=0.5 T=5 N=7

(b)

Fig. 5. To illustrate the connection between the discrete time Kuramoto system and multiple vehicles, this plot shows N = 7 discrete time steering control vehicles with (a) a stable gain (KΔT < 2) and (b) an unstable gain (KΔT > 2).

here a discrete-time Kuramoto oscillator model. Analytical results have been shown for certain cases of stability of synchronized and balanced equilibria sets both when delay is not present and when it is. The nonlinearities of the problem make such analysis challenging, but the initial results presented here, both analytical and Monte Carlo, indicate that the phase coupled oscillator approach to multi-nonholonomic-vehicle systems with delay has potential. Several routes of investigation are being pursued beyond the stability analysis presented here. First, we would like to extend the models to systems with communication networks that are not complete. Such scenarios are more realistic in cases where data is obtained through sensors with limited range or with dynamic communication networks (e.g. [JLM03d]). With respect to the resulting delay from dynamic communication, one cannot usually assume that all data will be updated at the same time. In cases of a shared communication media, the data delays will be varied as in [GM05b, HM99]. As shown in continuous time systems, the pattern of communication has strong implications for stability. The combination of general dynamic communication systems with nonholonomic multivehicle models provides a rich context for future work. The applications here also have strong connections to graph theoretic techniques being applied to networked systems. In studies of continuous time systems, the convergence rate of the continuous time Kuramoto model has been shown to be dependent on λ2 (L), the second smallest eigenvalue of the graph Laplacian. This eigenvalues gives a measure of the connectedness of the communication network of the system [JMB04]. The communication network for our systems is described by a complete graph, meaning all agents are coupled to one another, which maximizes λ2 (L) for a given N . It has been shown that the maximum eigenvalue of the graph Laplacian determines convergence bounds for the linear average-consensus problem when delay is present in the

Discrete Time Kuramoto Models with Delay

23

communication network [KJH04, SM03b]. It is likely that similar mechanisms can be derived for the class of discrete time systems being explored here. In practice, multiple vehicle systems may have asynchronous communication dynamics. Asynchronous communication has been examined in the continuous time linear consensus problem [CMA05b], but the linear matrix tools are unavailable in the general phase coupled oscillator problem. The extension could be made to both the continuous and discrete time Kuramoto models for nonholonomic multivehicle control. In all, the use of Kuramoto oscillator models has provided fundamental results for use with continuous time, nonholonomic, multivehicle systems, and the extension to discrete time systems here shows initial promise and a potential for developing rigorous tools for systems with delays, asynchronicity, and dynamic communication systems is strong.

Acknowledgments This work was supported in part by NSF grant CMS-0238461 and in part by NSF grant BE-0313250.

Symmetries in the Coordinated Consensus Problem Ruggero Carli1 , Fabio Fagnani2 , Marco Focoso1 , Alberto Speranzon3 , and Sandro Zampieri1 1

2

3

Department of Information Engineering Universit` a di Padova Via Gradenigo 6/a, 35131 Padova, Italy {carlirug,zampi}@dei.unipd.it Dipartimento di Matematica Politecnico di Torino C.so Duca degli Abruzzi, 24,10129 Torino, Italy [email protected] Department of Signals, Sensors and Systems Royal Institute of Technology Osquldasv¨ ag 10, SE-10044 Stockholm, Sweden [email protected]

Summary. In this paper we consider a widely studied problem in the robotics and control communities, called consensus problem. The aim of the paper is to characterize the relationship between the amount of information exchanged by the vehicles and the speed of convergence to the consensus. Time-invariant communication graphs that exhibit particular symmetries are shown to yield slow convergence if the amount of information exchanged does not scale with the number of vehicles. On the other hand, we show that retaining symmetries in time-varying communication networks allows to increase the speed of convergence even in the presence of limited information exchange.

1 Introduction The design of coordination algorithms for multiple autonomous vehicles has recently attracted large attention in the control and robotics communities. This is mainly motivated by that multi-vehicle systems have application in many areas, such as coordinated ﬂocking of mobile vehicles [TJP03b, TJP03c], cooperative control of unmanned air and underwater vehicles [BLH01, BL02], multi-vehicle tracking with limited sensor information [MSJH04]. Typically the coordinating vehicles need to communicate data in order to execute a task. In particular they may need to agree on the value of certain coordination state variables. One expects that, in order to achieve coor-

P.J. Antsaklis, P. Tabuada (Eds.): Netw. Emb. Sens. and Cntrl., LNCIS 331, pp. 25–51, 2006. © Springer-Verlag Berlin Heidelberg 2006

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dination, the variables shared by the vehicles, converge to a common value, asymptotically. The problem of designing controllers that lead to such asymptotic coordination are called coordinated consensus problems, see for example [JLM03b, DD03, FM04a, OSM04a], and reference therein. The interest in these type of problems is not limited to the ﬁeld of mobile vehicles coordination but also involves problems of synchronization [Str00a, MdV02, LFM05]. The problem that mostly has been studied in the literature is the design of control strategies that lead to consensus when each vehicle shares its information with vehicles inside a neighborhood [JLM03b, TJP03b] and the communication network is time-varying [JLM03b, TJP03c, LMA04b]. Robustness to communication link failure [CMB04b] and the eﬀects of time delays [OSM04a] has also been considered recently. The consensus problem with time-invariant communication networks have been studied in [SBF05, FTBG05]. Randomly time-varying networks have also been analyzed in [HM04a]. In this paper we consider the consensus problem from a diﬀerent perspective. We are interested to characterize the relationship between the amount of information exchanged by the vehicles and the achievable control performance. More precisely, if we model the communication network by a directed graph, in which each arc represents information transmission from one vehicles to another one, we can expect that good control design methods have to yield better performance for graphs that are more connected. In other words, we model the communication eﬀort of each vehicle as the number of other vehicles it communicates with. In order to formally characterize the trade oﬀ between control performance and communication eﬀort we make the following assumption: the graph topology is independent of the relative position of the vehicles, and the vehicles are described by an elementary ﬁrst order model. The ﬁrst hypothesis, that could be realistic in networks of coupled oscillators [MdV02], it is certainly less plausible in applications involving mobile vehicles. Nevertheless a clear analysis of such simpliﬁed model seems to be a necessary prerequisite in order to better understand more realistic scenarios. The motivation of describing vehicles with an elementary model is that it allows a quite complete and clean analysis of the problem. The paper is organized as follows. In section 2 we formally deﬁne the consensus problem. In particular we restrict to linear state feedbacks. We then introduce an optimal control problem where the cost functional is related to the convergence rate to the barycenter of the initial position of the vehicles. Under some assumptions, described in section 3, it turns out that weighted directed graphs for which the adjacency matrix is doubly stochastic, are communication graphs that guarantee consensus. Such graphs can be interpreted as a Markov chain and the convergence rate can be related to the mixing rate of the chain [Beh99]. The problem turns out to be treatable for a class of time-invariant graphs with symmetries. In section 4 we introduce the class of Cayley graphs deﬁned on ﬁnite Abelian groups. Using tools for bounding the

Symmetries in the Coordinated Consensus Problem

27

mixing time of Markov chains deﬁned on groups [Beh99, SC04] and algebraic properties of ﬁnite Abelian groups we derive a bound on the convergence rate to the consensus. The bound is a function of the number of vehicles and the incoming arcs in each vertex of the communication graph, that is the total information each vehicle receives. The main result shows that imposing symmetries in the communication graph, and thus in the control structure, keeping bounded the number of incoming arcs in each vertex, makes the convergence slower as the number of vehicles increases. In section 5 we consider random solutions. In these strategies the communication graph is chosen randomly at each time step over a family of graphs with the constraint that the number of incoming arcs in each vertex is constant. A simple mean square analysis, shows that, in this way, we can improve the convergence rate obtained with time-invariant communication graphs. This holds true even if the random choice is restricted to families of graphs with symmetries. A similar analysis has been proposed in [HM04a] where however a diﬀerent model of randomly time varying communication graph was proposed and less neat results were obtained. In section 6 some computer simulations are reported.

2 Problem formulation Consider N > 1 vehicles whose dynamics are described by the following discrete time state equations x+ i = xi + ui

i = 1, . . . , N

where xi ∈ R is the state and represents the vehicle position, x+ i is the updated state and ui ∈ R is the control input. More compactly we can write x+ = x + u where x, u ∈ RN . The goal is to design a feedback control u = Kx,

K ∈ RN ×N

yielding the consensus, namely a control that asymptotically makes all the states xi converging to the same value. More precisely, our objective is to obtain a feedback matrix K such that the closed loop system x+ = (I + K)x , for any initial condition x(0) ∈ RN , satisﬁes lim x(t) = αv

t→∞

where v := (1, . . . , 1)T and where α is a scalar depending on x(0).

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Without further constraints the above problem is not particularly interesting because it admits completely trivial solutions. Indeed, if I + K is asymptotically stable (condition which can be trivially obtained by choosing, for instance, K = −I) then the rendezvous problem is clearly solved with α = 0 for all x(0). This is however an ineﬃcient solution since, in this way, nonzero initial states having equal components (in which the rendezvous has already occurred) would produce a useless control action driving all the states to zero. In the following we will impose the condition that the subspace generated by the vector v consists of all equilibrium points, and this happens if and only if Kv = 0 .

(1)

From now on, when we say that K solves the rendezvous problem, we will assume that condition (1) is veriﬁed. It is easy to see that in this way the rendezvous problem is solved if and only if the following three conditions hold: (A)the only eigenvalue of I + K on the unit circle is 1; (B)the eigenvalue 1 has algebraic multiplicity one (namely it is a simple root of the characteristic polynomial of I + K) and v is its eigenvector; (C)all the other eigenvalues are strictly inside the unit circle. It is clear that, in order to achieve this goal, it is necessary that the vehicles exchange their position. If no constraint is imposed on the amount of information exchanged, it is still quite easy to solve the above problem. In order to describe the information exchange associated to a speciﬁc feedback K it is useful to introduce certain graph theoretic ideas. To any feedback K we associate a directed graph GK with set of vertices {1, . . . , N } in which we add an arc from j to i whenever in the feedback matrix K the element Kij = 0, meaning that in the control of xi it is used the knowledge of xj . The graph GK is said to be the communication graph associated with K. Conversely, given any directed graph G with set of vertices {1, . . . , N }, we say that a feedback K is compatible with G if GK is a subgraph of G (we will use the notation GK ⊆ G). We will say that the rendezvous problem is solvable on a graph G if there exists a feedback K compatible with G and solving the rendezvous problem. From now on we will always assume that G contains all loops (i, i): each system has access to its own state. We would like to have a way to measure the performance of a given control scheme achieving rendezvous. The way to quantify this performance is by no means unique. Suppose we have deﬁned a cost functional R = R(K) to be minimized. We can then deﬁne RG = min{R(K) | GK ⊆ G} . We expect a meaningful cost functional to be sensitive to the amount information exchanged by the vehicles, in other words we would like RG to show

Symmetries in the Coordinated Consensus Problem

29

certain range of variation among all the possible communication graphs that can be considered. The simplest performance index is related to the speed of convergence towards the equilibrium point. Let P be any matrix such that P v = v and its spectrum (set of the eigenvalues) σ(P ) is contained in the closed disk centered in 0 and having radius 1. Deﬁne ρ(P ) =

1 if dim ker(P − I) > 1 max{|λ| : λ ∈ σ(P ) \ {1}} if dim ker(P − I) = 1 ,

(2)

we can then take R(K) = ρ(I + K). The index R(K) describes the exponential rate of convergence of x(t) towards the equilibrium point. However, such index does not show in general the desired sensitivity to the communication constraints. Indeed, if the graph G is described by 1 → 2 → · · · → N , and we consider the controller   −1 1 0 · · · 0 0 0 −1 1 · · · 0 0 .. .. .. . . .. .. K= . . . . . .  0 0 0 · · · −1 1 0 0 0 ··· 0 0 which fulﬁll condition (1), we obtain R(K) = 0. Thus, adding any communication edge will not lower this index. It is clear however that the above feedback have worse performance than others using a richer communication graph. This means that the spectral radius is not suﬃcient to highlight these diﬀerences. We then need to reﬁne the model. Since we are considering autonomous vehicles it seems reasonable to consider the cost in terms of control eﬀort, as for instance ∞

J(K) :=

||u(t)||2 .

t=0

Then the performance index consisting on the pair (ρ(I + K), J(K)) would better describe the problem. However this cost is hard to be analyzed. Therefore we consider a simpler index which is related to the previous one, namely 2

∞

u(t)

J (K) := t=0

∞

≤

||u(t)||2 .

t=0

Notice that, in our case, we have that J = ||x(∞) − x(0)||2 and so arg min{||x(∞) − x(0)||2 : x(∞) = αv , α ∈ R} =

1 T v x(0) v . N

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In this paper we will consider as performance index the pair (ρ(I +K), J (K)), which is relevant and treatable. Notice that all feedback strategies K producing rendezvous points that are the barycenter of the initial positions of the vehicles, namely such that lim x(t) =

t→∞

1 T v x(0) v , N

(3)

are all optimal with respect to the index J . This feedback maps are called consensus controls [OSM04a]. When K yields such a behavior, it will be called a barycentric controller. It is easy to see that K is a barycentric controller if and only if vT K = 0 . (4) Thus if we restrict to barycentric controllers that satisfy (1) then the performance index of interest is ρ(I + K). Moreover if we consider the displacement from the barycenter Δ(t) = x(t) −

1 T v x(0) v , N

it is immediate to check that, Δ(t) satisfy the same closed loop equation than x(t). In fact we have Δ(t + 1) = (I + K)Δ(t) . (5) Notice that the initial condition satisﬁes < Δ(0), v >= 0 .

(6)

Hence the asymptotic behavior of our rendezvous control problem can equivalently be studied by looking at the evolution (5) on the hyperplane characterized by the condition (6). The index ρ(I + K) seems, in this context, appropriate for analyzing the performance.

3 Stochastic and doubly stochastic matrices If we restrict to control laws K making I + K a nonnegative matrix, condition (1) imposes that I +K is a stochastic matrix. Since the spectral structure of such matrices is quite well known, this observation allows to understand easily what are the conditions on the graph that will ensure the solvability of the rendezvous problem. To exploit this we need to recall some notation and results on directed graphs (the reader can further refer to textbooks on graph theory such as [GR01a]). Fix a directed graph G with set of vertices V and set of arcs E ⊆ V ×V . The adjacency matrix A is a {0, 1} valued square matrix indexed by the elements in V deﬁned by (A)ij = 1 if and only (i, j) ∈ E. Deﬁne moreover the in-degree

Symmetries in the Coordinated Consensus Problem

31

of a a vertex i as deg(i) = j (A)ji . Vertices of in-degree equal to 0 are called sinks. A graph is called in-regular (of degree k) if each vertex has in-degree equal to k. A path in G consists of a sequences of vertices i1 i2 . . . . . . ir such that (i , i +1 ) ∈ E for every = 1, . . . , r − 1; i1 (resp. ir ) is said to be the initial (resp. terminal) vertex of the path. A vertex i is said to be connected to a vertex j if there exists a path with initial vertex i and terminal vertex j. A directed graph is said to be connected if given any pair of vertices i and j, at least one of the two is connected to the other. A directed graph is said to be irreducible if given any pair of vertices i and j, i is connected to j. Given any directed graph G we can consider its irreducible components, namely strongly connected subgraphs Gk with set of vertices Vk ⊆ V (for k = 1, . . . , s) and set of arcs Ek = E ∩ (Vk × Vk ) such that the sets Vk form a partition of V . The various components may have connections among each other. We deﬁne another directed graph TG with set of vertices {1, . . . , s} such that there is an arc from k1 to k2 if there is an arc in G from a vertex in Vk1 to a vertex in Vk2 . It can easily be shown that TG is always a union of disjoint trees. Standard results on stochastic matrices [Gan59, page 88 and 99] yield the following proposition. Proposition 1. Let G be a directed graph. The following conditions are equivalent: (i) The rendezvous problem is solvable on G. (ii) TG is connected and has only one sink vertex. Moreover if the above conditions are satisﬁed, any K such that I + K is stochastic, GK = G and Kii = −1 for every i ∈ VG is a possible solution. Among all possible solutions of the rendezvous problem, when the graph G satisﬁes the properties of Proposition 1, a particularly simple one can be written in terms of the adjacency matrix A of G. Consider indeed the matrix P:  (A)ji  deg(i) if deg(i) > 0 Pij =  0 if deg(i) = 0 then K = P − I solves the rendezvous problem. Notice the explicit form that the closed loop system assumes in this case: x+ i = xi +

1 deg(i)

(xj − xi ) .

(7)

j=i (j,i)∈E

If we restrict to control laws K making I +K a nonnegative matrix, conditions (1) and (4) are equivalent to the fact that I + K is doubly stochastic. This remark permits to obtain the following result (see [Gan59]). Proposition 2. Let G be a directed graph. The following conditions are equivalent:

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(i) The rendezvous problem on G admits a barycentric controller. (ii) G is irreducible Moreover if the above conditions are satisﬁed, any K such that I +K is doubly stochastic, GK = G and Kii = −1 for every i ∈ V is a possible solution. Notice that in the special case when the graph G is undirected, namely (i, j) ∈ E if and only if (j, i) ∈ E, it follows that we can ﬁnd solutions K to the rendezvous problem that are symmetric and that therefore are automatically doubly stochastic. One example is given by (7). We expect the spectral radius to be a meaningful cost functional when restricted to feedback controllers K such that I + K is doubly stochastic. More precisely we conjecture that, by taking ρds G = min{ρ(K) | K doubly stochastic, GK ⊆ G} , G1 ⊂ G2 implies that ρG1 > ρG2 . However we have been not able to prove this so far.

4 Symmetric Controllers The analysis of the rendezvous problem and the corresponding controller synthesis problem becomes more treatable if we limit our search to graphs G and matrices K exhibiting symmetries. We will show however that these symmetries yield rather poor performance in terms of convergence rate. In order to treat symmetries on a graph G in a general setting, we consider Cayley graphs deﬁned on Abelian groups. Let G be any ﬁnite Abelian group of order |G| = N , and let S be a subset of G which contains the zero. The Cayley graph G(G, S) is the directed graph with vertex set G and arc set EG(G,S) = {(g, h) : h − g ∈ S} . Notice that a Cayley graph is always in-regular: the in-degree of each vertex is equal to |S|. Notice also that irreducibility can be checked at an algebraic level: it is equivalent to the fact that the set S generates the group G which means that any element in G can be expressed as a ﬁnite sum of (not necessarily distinct) elements in S. If S is such that −S = S we say that S is inverseclosed. In this case the graph obtained is undirected. A Cayley graph supports (stochastic) matrices; to construct them it is suﬃcient to start from any function π : G → R such that π(g) = 0 if g ∈ S. Then deﬁne P by Pgh = π(g − h). Such a matrix will be called a Cayley matrix (adapted to the Cayley graph G(G, S)). We will also say that P is the Cayley matrix generated by the function π. To have candidate solutions to our rendezvous problem, we of course need something more. Notice that if it holds g π(g) = 1, then P satisﬁes both relations P v = v and v T P = v T . In the special case when π is a probability distribution (i.e. π(g) ≥ 0 for every

Symmetries in the Coordinated Consensus Problem

33

g) P is thus automatically a doubly stochastic matrix: such matrices P will be called Cayley stochastic matrices and for the rest of this section we will mostly work with them. Among the many possible choices of the probability distribution π, there is one which is particularly simple: π(g) = 1/|S| for every g ∈ S. In this case we have that P =

1 A, |S|

where A is the adjacency matrix of the Cayley graph G(G, S). Example 1. Let us consider the group ZN of integers modulo N and the Cayley graph G(ZN , S) where S = {−1, 0, 1}. Notice that in this case S is inverseclosed. Consider the uniform probability distribution π(0) = π(1) = π(−1) = 1/3 The corresponding Cayley stochastic matrix is given by   1/3 1/3 0 0 · · · 0 0 1/3 1/3 1/3 1/3 0 · · · 0 0 0 . P = .. .. .. . . .. .. .. .  .. .  . . . . . . 1/3 0

0 0 · · · 0 1/3 1/3

Notice that in this case we have two symmetries. The ﬁrst is that the graph is undirected and the second that the graph is circulant. These symmetries can be seen in the structure of the transition matrix P which, indeed, results both symmetric and circulant. The idea of considering Cayley graphs and Cayley stochastic matrices on Abelian groups is helpful in order to compute, or at least bound, the cost functional ρ(P ) deﬁned in (2). We can indeed consider the minimum ρCayley G of the spectral radius ρ(P ) as P varies among the stochastic Cayley matrices compatible with the given Cayley graph G. It will turn out that ρCayley can G ds be evaluated or estimated in many cases and clearly it holds ρCayley ≥ ρ G . G Before continuing we give some short background notions on groups characters and on harmonic analysis on groups, upon which the main results are built. 4.1 Cayley Stochastic Matrices on Finite Abelian Groups Let G be a ﬁnite Abelian group of order N , and let C ∗ be the multiplicative group of the nonzero complex numbers. A character on G is a group homomorphism χ : G → C ∗ (χ(g + h) = χ(g)χ(h) for all g, h ∈ G). Since we have that χ(g)N = χ(N g) = χ(0) = 1 , ∀g ∈ G

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it follows that χ takes values on the N th -roots of unity. The character χ0 (g) = 1 for every g ∈ G is called the trivial character. The set of all characters of the group G forms an Abelian group with respect to the pointwise multiplication: it is called the character group and ˆ The trivial character χ0 is the zero of G. ˆ It can be shown that denoted by G. ˆ is isomorphic to G, in particular its cardinality is N . If we consider the G vector space C G of all functions from G to C with the canonical Hermitian form 1 f1 (g)f2 (g) , < f1 , f2 >= N g∈G

ˆ is an orthonormal basis of C G . it follows that G The Fourier transform of a function f : G → C is deﬁned as ˆ → C , fˆ(χ) = fˆ : G

χ(−g)f (g) . g∈G

Example 2. Consider again the group ZN . The characters are given by 2π

χ (j) = ei N

j

,

j ∈ ZN ,

= 0, . . . , N − 1 .

The correspondence → χ yields an explicit isomorphism between ZN and ˆ N . Given any function f : ZN → C, its Fourier transform is given by Z fˆ(χ ) =

N −1

2π

f (j) e−i N

j

.

j=0

The cyclic case is instrumental to study characters for any ﬁnite Abelian group. Indeed it is a well known result in algebra [Hun74], which states that any ﬁnite Abelian group G is isomorphic to a ﬁnite direct sum of cyclic groups. In order to study characters of G we can therefore assume that G = ZN1 ⊕ · · · ⊕ ZNr . It can be shown [Beh99] that the characters of G are precisely the ˆ i for i = 1, . . . , r. maps (g1 , g2 , . . . , gr ) → χ1 (g1 )χ2 (g2 ) · · · χr (gr ) with χi ∈ G ˆN ⊕ · · · ⊕ Z ˆ N . Fix now a Cayley ˆ is (isomorphic to) Z In other terms, G 1 r graph on G and a Cayley matrix P generated by the function π : G → R. The spectral structure of P is very simple. To see this, ﬁrst notice that P can be interpreted as a linear function from C G to itself: simply considering, for f ∈ C G , (P f )(g) = h Pgh f (h). Notice that the trivial character χ0 ˆ corresponds to the vector v having all components equal to 1. For every χ ∈ G, it holds (P χ)(g) =

Pgh χ(h) = h∈G

π(g − h)χ(h) = h∈G

π(h)χ(g − h) = π ˆ (χ)χ(g) . h∈S

Hence, χ is an eigenfunction of P with eigenvalue π ˆ (χ). Since the characters form an orthonormal basis it follows that P is diagonalizable and its spectrum is given by

Symmetries in the Coordinated Consensus Problem

35

ˆ . σ(P ) = {ˆ π (χ) | χ ∈ G} We can think of characters as linear functions from C to C G : χ : z → zχ , and their adjoint as linear functionals on C |G| : χ∗ : f →< f, χ > . With this notation, χχ∗ is a linear function from C G to itself, projecting on the eigenspace generated by χ. In this way, P can be represented as π ˆ (χ)χχ∗ .

P = ˆ χ∈G

ˆ → C the matrix Conversely, it can easily be shown that given any θ : G θ(χ)χχ∗ ,

P = ˆ χ∈G

ˆ is a Cayley matrix generated by the Fourier transform π = θ. Suppose now P is the closed loop matrix of a system x+ = P x. The displacement from the barycenter can be represented as Δ = (I − χ0 χ∗0 )x . As we had already remarked Δ is governed by the same law (see equation (5)) Δ+ = P Δ . The initial condition Δ0 is characterized by < Δ0 , χ0 >. Notice that π ˆ (χ)t χ < Δ0 , χ > .

Δt = P t Δ 0 = ˆ χ∈G

Hence,

|ˆ π (χ)|t | < Δ0 , χ > |2 .

||Δt ||2 = ˆ χ∈G

This shows in a very simple way, in this case, the role of the spectral radius ρ(P ) = max |ˆ π (χ)| χ=χ0

in the convergence performance.

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4.2 The Spectral Radius for Stochastic Cayley Matrices The particular spectral structure of stochastic Cayley matrices allows to obtain asymptotic results on the behavior of the spectral radius ρ(P ) and therefore on the speed of convergence of the corresponding control scheme. Let us start with some examples. Example 3. Consider the group ZN and the Cayley graph G(ZN , S), where S = {0, 1}. Consider the probability distribution π on S described by π(0) = 1 − k , π(1) = k . where k ∈ [0, 1]. The Fourier transform of π is 2π

χ(−g)π(g) = 1 − k + ke−i N ,

ˆ (χ ) = π

= 1, . . . , N − 1 .

g∈S

In this case it can be shown that we have rendezvous stability if and only if 0 < k < 1 and that the rate of convergence is ρ(P ) =

2π

1 − k + ke−i N

max

1≤ ≤N −1

.

Hence, we have that ρCayley = min G k

The optimal

and k are ρCayley G

=

2π

max

1≤ ≤N −1

1 − k + ke−i N

.

= 1 and k = 1/2 yielding 1 1 + cos 2 2

2π N

1 2

1−

π2 1 2 N2

where the last approximation is meant for N → ∞. Example 4. Consider the group ZN and the Cayley graph G(ZN , S), where S = {−1, 0, 1, }. For the sake of simplicity we assume that N is even; very similar results can be obtained for odd N . Consider the probability distribution π on S described by π(0) = k0 , π(1) = k1 , π(−1) = k−1 . The Fourier transform of π is in this case given by 2π

2π

χ(−g)π(g) = k0 + k1 e−i N + k−1 ei N ,

π ˆ (χ ) =

= 1, . . . , N − 1 .

g∈S

We thus have ρCayley = G

min

max

(k0 ,1 ,k−1 ) 1≤ ≤N −1

2π

2π

k0 + k1 e−i N + k−1 ei N

.

Symmetries in the Coordinated Consensus Problem

37

Symmetry and convexity arguments allow to say that a minimum is for sure of the type k1 = k−1 . With this assumption the cost functional reduces to ρ(P ) = max k1

1 − 2k1 1 − cos

2π N

, |1 − 4k1 |

.

The minimum is achieved for k0 =

1 − cos 3 − cos

and we have ρCayley = G

2π N 2π N

,

k1 = k−1 =

1 + cos 3 − cos

2π N 2π N

1 3 − cos

1 − 2π 2

1 N2

2π N

(8)

where the last approximation is meant for N → +∞. Notice the asymptotic behavior of previous two examples: the case of communication exchange with two neighbors oﬀer a better performance. However, in both cases ρCayley → 1 for N → +∞. This fact is more general: if we keep G bounded the number of incoming arcs in a vertex, the spectral radius for Abelian stochastic Cayley matrices will always converge to 1. This is the content of our main result. Theorem 1. Let G be any ﬁnite Abelian group of order N and let S ⊆ G be a subset with |S| = ν + 1. Let π be a probability measure associated to the Cayley graph G(G, S). Then −2/ν ρCayley , G(G,S) ≥ 1 − CN

where C > 0 is a constant independent of G and S. Proof. See Appendix A. The consequence of theorem 1 we have that, if we consider any sequence of Cayley stochastic matrices PN adapted on Abelian Cayley graphs (GN , SN ) such that |GN | = N and |SN | = o(ln N ) then, necessarily, ρ(PN ) converges to 1. Notice that in Example 4 we have ν = 2 and an asymptotic behavior ρCayley 1 − 2π 2 N −2 while the lower bound of Theorem 1 is, in this case, G 2 −1 1 − 2π N . Can we achieve the bound performance? In other words, is the lower bound we have just found, tight? The following example shows that this is the case. Example 5. Consider the group ZνN and the Cayley graph G(ZνN , S), where S = {0, e1 , . . . , eν } where ej is the vector with all elements equal to 0 except a 1 in position j. Consider the probability distribution π on S described by

38

R. Carli et al.

π(0) = π(ei ) =

1 , ∀i = 1, . . . , ν . ν+1

The Fourier transform of π is

 χ(−g)π(g) =

π ˆ (χ 1 , . . . , χ ν ) = g∈S

where

j



ν

2π 1  1+ e−i N j  . ν+1 j=1

= 1, . . . , N − 1 , j = 1, . . . , ν. We thus have that for this graph ν

ρG =

2π 1 max ν+ e−i N ν + 1 1≤ j ≤N −1 j=1

j

.

It is easy to see that the above min-max is reached by j = 1 for every j = 1, . . . , ν or when ∃h ∈ {1, ν} such that h = 1 and for all j = h we have j = 0. This yields the value ρG =

2ν ν2 + 1 + cos 2 (ν + 1) (ν + 1)2

1 2

2π N

1−C

π2 π2 = 1 − C ν 2/ν 2 N (N )

where C is a constant. Notice we have exactly obtained the lower bound proven above.

5 Random Communication Graphs 5.1 Random Circulant Communication Graph A direct graph G = (V, E) is said to be a circulant directed graph if (i, j) ∈ E implies that (i + p mod N, j + p mod N ) ∈ E where p ∈ N. Observe that in a direct circulant graph each vertex has the same in-degree. In the following sometimes we will refer to the in-degree of the graph, meaning the in-degree of any of the vertices of the graph. Let G¯ the set of all circulant directed graphs G = (V, E) with in-degree ν +1 and such that the corresponding adjacency matrix A has aii = 0, ∀ 1 ≤ i ≤ N , meaning that, as we said perviously, each vehicle has access to its own state. In this strategy we suppose that at each time instant t the communication graph G(t) is chosen randomly from the set G¯ accordingly to a uniform distribution. This is equivalent to impose that the adjacency matrix A(t) of the communication graph G(t) is such that ν

A(t) = I +

Π αi (t)

i=1

where Π is the following circulant matrix [Dav79]

Symmetries in the Coordinated Consensus Problem

 0 1 0 ··· 0 0 1 ··· 0 0 0 ··· Π= . . . . .. .. .. . . 0 0 0 · · · 1 0 0 ···

0 0 0 .. .

0 0 0 .. .

39



0 1 00

and where {α1 (t)} , . . . , {αν (t)} are ν independent sequences of independent random variables uniformly distributed over the alphabet X = {1, 2, . . . , N }. We consider the following control law u(t) ν

u(t) =

ki Π αi (t)

k0 I +

x(t).

(9)

i=1

The close loop system then becomes ν

x(t + 1) =

(1 + k0 )I +

ki Π αi (t)

x(t)

(10)

i=1

The system (10) can be regarded as a Markov jump linear system [BCN04]. Notice that the state transition matrix in (10) is a circulant matrix and since Πv = v we have that the conditions (1) is satisﬁed. If we restrict to non-negative matrices then we have that the feedback gains k0 , . . . , kν are ν such that 1 + k0 , k1 , . . . , kν ≥ 0 and k0 + i=1 ki = 0. We conclude this section by observing that this strategy has an evident drawback from an implementation point of view : the same random choice, done at every time instance, needs to be known by all vehicles. A possible way to overcome this limitation is by using a predetermined pseudo-random sequence whose starting seed is known to everybody. 5.2 Random Communication Graph with Bounded In-Degree The strategy that we consider in this subsection is similar to the one presented in the previous subsection, but it overcomes the implementation issues. In this case we do not limit the time-varying communication graph G(t) to be circulant. We assume that each vehicle, besides knowing its own position, receives the state of ν vehicle chosen randomly and independently. Because of this it can happen that the resulting communication graph G(t) could have multiple arcs connecting the same pair of nodes. The feedback control in this case is ν

u(t) = k0 x(t) +

ki Ei (t)x(t)

(11)

i=1

where {Ei (t)}, i = 1, . . . , ν, are ν independent sequences of independent random processes taking value on the set of matrices

40

R. Carli et al. N ×N

Υ := E ∈ {0, 1}

: Ev = v

and equally distributed in such a set. Roughly speaking, the set Υ is constituted by all matrices with entries 0 or 1 that have exactly one element equal to 1 in each row. Since Ei (t)v = v, for all i = 1, . . . , ν and all t ≥ 0, we have that, as in the previous case, the in order for condition (1) to hold and to have a non-negative matric, the feedback gains k0 , . . . , kν must satisfy ν 1 + k0 , k1 , . . . , kν ≥ 0 and k0 + i=1 ki = 0. The close loop system becomes ν

ki Ei (t)x(t) .

x(t + 1) = (1 + k0 )I +

(12)

i=1

Notice that also the system (12) can be regarded as Markov jump linear system. 5.3 Convergence and Performance Analysis In order to study the asymptotic behavior of the two previous strategy, it is convenient to introduce the variable y(t) that is deﬁned in the following way. Consider 1 (13) Y = I − vv T N and let y(t) = Y x(t). (14) Notice that the component yi (t) of y(t), represents, by this deﬁnition, the displacement of xi (t) from the barycenter of the initial position of the vehicles, at the time instant t. Clearly we have that lim x(t) = αv

(15)

lim y(t) = 0 .

(16)

t→+∞

if and only if

t→+∞

Note that it holds both Y Π = ΠY and Y E = Y EY where E is any matrix in Υ . Then pre-multiplying (10) and (12) by Y we obtain ν

y(t + 1) =

(1 + k0 )I +

ki Π αi (t)

y(t)

(17)

i=1

if we consider the ﬁrst strategy and ν

y(t + 1) = F2 (t) =

ki Ei (t) y(t)

(1 + k0 )I +

(18)

i=1

In order to study the asymptotic properties of y(t) we consider E y(t) 2 where the expectation is taken over the set of the graph in which G(t) is chosen. We then have the following deﬁnition.

Symmetries in the Coordinated Consensus Problem

41

Deﬁnition 1 ([FL02b]). The jump linear Markov systems (17) and (18) are said to be asymptotically second moment stable if for any ﬁxed y(0) ∈ RN it holds (19) lim E y(t) 2 = 0. t→+∞

We can then state the two main results. Proposition 3. The system (17) is asymptotically second moment stable for any initial condition y(0) and for k0 = −ν/(1 + ν) and ki = 1/(1 + ν), 1 ≤ i ≤ ν. Moreover, for these values of ki we obtain the fasted convergence rate and we have that E

y(t)

2

1 1+ν

=

t

y(0) 2 .

(20)

Proof. See Appendix B. Proposition 4. The system (18) is asymptotically second moment stable for any initial condition y(0) and for k0 = −νN/(N + N ν − 1) and ki = N/(N + N ν − 1), 1 ≤ i ≤ ν. Moreover, for these values of ki we obtain the fastest convergence rate and we have that E

y(t)

2

=

N −1 N (1 + ν) − 1

t

y(0) 2 .

(21)

Proof. See Appendix C. Remark 1. Notice that the convergence rate obtained using the strategy with random circulant communication graphs it does not depend on N and ensures convergence even if ν = 1, which corresponds to the case when each vehicle receives the state of at most one another vehicle. The strategy with random communication graphs with in-degree bounded, attains the same converge rate of the ﬁrst only from N → +∞. However notice that both strategies have a better convergence rate than the one obtained using time-invariant communication graphs. This increase of performance has obtained by randomizing the choice over a pre-assigned family of graphs. Remark 2. Notice that by using random communication graphs with bounded in-degree the vehicles, in general, will not reach the consensus at the barycenter of the initial conﬁgurations, since vT

ν

ki Ei (t)

k0 I +

= vT

(22)

i=1

It is meaningful to study where the vehicles will reach consensus with respect to the barycenter of the initials conditions. In order to carry out this analysis we consider the mean square distance from the barycenter of the initials conditions namely we consider x(t)−v T x(0)v/N . We have the following result.

42

R. Carli et al.

Proposition 5. The mean square distance to the barycenter of the initial conﬁguration of the vehicle is bounded, namely, lim E (x(t) − bv)(x(t) − bv)T = α

t→∞

where α=

(1 − N )

1 x(0)T (I − Ω)x(0) N

ν 2 i=1 ki ν 2 1=1 ki − k0 N (k0

+ 2)

and where b = v T x(0)/N , and Ω = vv T /N and where the ki make the system (18) asymptotically second order stable. Proof. See Appendix D. Notice that when N → +∞ then the mean square distance to the νN barycenter tends to zero. If we use the control gains k0 = − N +N ν−1 and k1 = · · · = kν = N/(N + N ν − 1) then we have that lim E (x(t) − bv)(x(t) − bv)T =

t→∞

1 x(0)T (I − Ω)x(0) . N (N (1 + ν) − 1)

Notice that for ﬁxed N if ν grows the mean square distance to the barycenter becomes smaller.

6 Simulation Results In Figures 1-3, some computer simulation results are reported. The simulations show the time evolution of the state of n = 10 vehicles when they can exchange the state with at most one other vehicle, thus in this case we have ν = 1. The initial condition for all the three simulations is the same and the barycenter coordinate is 2.76. Figure 1 refers to the state evolution for time-invariant communication graph when the Cayley graph is the ring graph. Figure 2 refers to random circulant communication graph and Figure 3 refers to random communication graph with bounded in-degree. Notice that the random strategies, as expected, exhibit a faster rate convergence and that the random communication graph with bounded in-degree does not converge to the barycenter (in this case the consensus is reached at 0.71).

7 Conclusions In this paper we have analyzed the relationship between the communication graph and the convergence rate to the rendezvous point for a team of vehicles. Modelling the communication graph with a Cayley graph deﬁned on Abelian groups, namely a graph with symmetries, we have been able to bound the

Symmetries in the Coordinated Consensus Problem

43

10 8 6

Positions xi(t)

4 2 0 2 4 6 8

0

5

10 Time t

15

20

Fig. 1. Time-invariant Cayley graph with ν = 1. 10 8 6

Positions xi(t)

4 2 0 2 4 6 8

0

5

10 Time t

15

20

Fig. 2. Random circulant communication graph.

convergence rate. In particular we have proved that the convergence to the barycenter of the initial conﬁguration becomes slower and slower as the number of vehicles increases if the amount of information received by each vehicle remain constant. We have also considered some particular random strategies that consist in choosing randomly a communication graph in a predeﬁned family of graphs. In particular we have considered the circulant graphs and graphs with bounded in-degree. It turns out that choosing randomly over

44

R. Carli et al. 10 8 6

Positions xi(t)

4 2 0 2 4 6 8

0

5

10 Time t

15

20

Fig. 3. Random communication graph with bounded in-degree.

such family graphs we obtain higher performances then using time-invariant communication graphs. In [CFSZ] the analysis has been extended to random Cayley graphs and to communication graphs where the information is quantized.

Acknowledgments. Part of this research was supported by European Commission through the RECSYS project IST-2001-32515.

A Proof of Theorem 1 In order to prove theorem 1 we need the following technical lemma. Lemma 1. Let T = R/Z ∼ = [−1/2, 1/2[. Let 0 ≤ δ ≤ 1/2 and consider the hypercube V = [−δ, δ]k ⊆ Tk . For every Λ ⊆ Tk such that |Λ| ≥ δ −k , there exist x ¯1 , x ¯2 ∈ Λ with x ¯1 = x ¯2 such that x ¯1 − x ¯2 ∈ V . Proof. For any x ∈ T and δ > 0, deﬁne the following set L(x, δ) = [x, x + δ] + Z ⊆ T . Observe that for all y ∈ T, L(x, δ) + y = L(x+y, δ). Now let x ¯ = (¯ x1 , . . . , x ¯k ) ∈ Tk and deﬁne

Symmetries in the Coordinated Consensus Problem

45

k

L(¯ xi , δ) .

L(¯ x, δ) = i=1

Also in this case we observe that L(¯ x, δ)+ y¯ = L(¯ x + y¯, δ) for every y¯ ∈ Tk . Consider now the family of subsets {L(¯ x, δ), x ¯ ∈ Λ} . We claim that there exist x ¯1 and x ¯2 in Λ such that x ¯1 = x ¯2 and such that L(¯ x1 , δ) ∩ L(¯ x2 , δ) = ∅. Indeed, if not, we would have |Λ|δ k < 1 which contradicts our assumptions. Notice ﬁnally that L(¯ x1 , δ) ∩ L(¯ x2 , δ) = ∅ ⇔ L(0, δ) ∩ L(¯ x2 − x ¯1 , δ) = ∅ ⇔ x ¯2 − x ¯1 ∈ V .

We can now prove theorem 1. Proof. With no loss of generality we can assume that G = ZN1 ⊕ . . . ⊕ ZNr . Assume we have ﬁxed a probability distribution π concentrated on S. Let P be the corresponding stochastic Cayley matrix. It follows from previous considerations that the spectrum of P is given by N1 −1

σ(P ) =

Nr −1

... k1 =0

2π

2π

π(k1 , . . . , kr )ei N1 k1 1 · · · ei Nr kr

r

:

s

∈ ZNs ∀s = 1, . . . , r

kr =0

Denote by k¯j = (k1j , . . . , krj ), for j = 1, . . . , ν, the non-zero elements in S, and consider the subset r

Λ= i=1

r

ki1 i kiν i ,..., Ni Ni i=1

+ Zk |

s

∈ ZNs for 1 ≤ s ≤ r

⊆ Tν .

Let δ = ( i Ni )−1/ν and let V be the corresponding hypercube deﬁned as in Lemma 1. We claim that there exists ¯ = ( 1 , . . . r ) ∈ ZN1 × · · · × ZNr , ¯ = 0 such that r r kiν i ki1 i ,..., + Zk ∈ V . N N i i i=1 i=1 Indeed, if there exist two diﬀerent ¯ , ¯ ∈ ZN1 × · · · × ZNr such that r i=1

r

kiν i ki1 i ,..., Ni Ni i=1

+ Zν =

r i=1

r

kiν i ki1 i ,..., Ni Ni i=1

+ Zν ,

46

R. Carli et al.

then we have that, r

r

i=1

ki1 i kiν i ,..., Ni Ni i=1

+ Zν = 0 ,

where ¯ = ¯ − ¯ = 0 . On the other hand, if diﬀerent elements in ZN1 × · · · × ZNr always lead do distinct elements in Λ, then, |Λ| = i Ni = δ −ν . We can then apply Lemma 1 and conclude that there exist two diﬀerent ¯ , ¯ ∈ ZN × · · · × ZN such that 1 r r i=1

r

kiν i ki1 i ,..., Ni Ni i=1

Hence,

r

+ Zν −

r

ki1 i kiν i ,..., Ni Ni i=1

i=1

r

r i=1

kiν i ki1 i ,..., Ni Ni i=1

+ Zν ∈ V .

+ Zν ∈ V ,

where ¯ = ¯ − ¯ = 0 . Consider now the eigenvalue Nr −1

N1 −1 N2 −1

...

λ=

2π

π(k1 , . . . , kr )ei( N1 k1

kr =0 ν

k1 =0 k2 =0

2π

j

π(k1j , . . . , krj )ej( N1 k1

= π(0, . . . 0) +

2π 1+ N 2

2π 1+ N 2

k2j

k2

2π 2 +···+ Nr

2π 2 +···+ Nr

kr

krj

r)

r)

.

j=1

Its norm can be estimated as follows ν

|λ| ≥ π(0, . . . 0) +

π(k1j , . . . , krj ) cos

j=1 ν

≥ π(0, . . . 0) +

π(k1j , . . . , krj ) −

j=1

ν

2π j k N1 1

1

+

2π j k N2 2

π(k1j , . . . , krj )2π 2

j=1

2

+ ··· +

2π j k Nr r

r

1 1 ≥ 1 − 2π 2 2/ν N 2/ν N

and so we can conclude.

B Proof of Proposition 1. As previously we observe that E [ y(t) ] E[y(t)y T (t)]. We have that

2

= tr E y(t)y T (t) . Let P (t) =

Symmetries in the Coordinated Consensus Problem

P (t + 1) = E y(t + 1)y T (t + 1)  ν

= E  (1 + k0 )I +

Π αi (t) y(t)y T (t) (1 + k0 )I +

i=1

T

 

Π αi (t)

i=1 ν

=E E

ν

47

Π αi (t) y(t)×

(1 + k0 )I + i=1

ν

×y T (t) (1 + k0 )I +



T

| α1 (t), . . . , αν (t)

Π αi (t)

i=1

Since y(t) is independent from α1 (t), . . . , αν (t) we obtain  ν

P (t + 1) = E  (1 + k0 )I +

Π αi (t)

P (t) (1 + k0 )I +

i=1

+ (1 + k0 ) ν



ν

ki E(Π αi (t) ) P (t)+

i=1

ki E(Π −αi (t) ) P (t)+

i=1

j=1 j=i

ki kj E Π αi (t) P (t)E Π −αj (t) +

+

+

Π αi (t)



i=1

ν

ν

T

i=1

= (1 + k0 )2 P (t) + (1 + k0 ) ν

ν

ki2 E Π αi (t) P (t)Π −αi (t)

i=1

By straightforward calculations one can verify that E Π −α(t) = E Π α(t) = and tr P (t)vv T = 0. Hence

1 T N vv

E

y(t + 1)

2

= (1 + k0 )2 E

y(t)

2

ν

+

ki2 tr E Π αi (t) P (t)Π −αi (t)

i=1

Using the fact that tr (AB) = tr (BA) we can conclude that E

y(t + 1)

2

=

(1 + k0 )2 +

ν

ki2

E

y(t)

i=1

=

2

ν

(1 + k0 ) + i=1

t

ki2

y(0)

2

2

(23)

48

R. Carli et al.

Now it is easy to verify that  ν  ki2 | 1 + k0 , k1 , . . . , kν ≥ 0, min (1 + k0 )2 +  i=1

  1 kj = 0 =  1+ν j=0 ν

(24)

and that it is obtained by choosing k0 = −ν/(1 + ν) and kj = 1/(ν + 1) for all 1 ≤ j ≤ ν. With such a choice we have the convergence result (20)

C Proof of Proposition 2. 2

We observe that E [ y(t) ] = tr E y(t)y T (t) . Let P (t) = E[y(t)y T (t)]. We have that ν

P (t + 1) = E

(1 + k0 )y(t) +

ki Y Ei (t)y(t) × i=1 T

ν

× (1 + k0 )y(t) +

 

ki Y Ei (t)y(t) i=1

= (1 + k0 )2 P (t) + E (1 + k0 )y(t)y T (t)

ν

ki EiT (t)Y +

i=1 ν

Y Ei (t)y(t) y T (t)(1 + k0 ) +

+E  + E

i=1 ν

ki Y Ei (t)y(t)y T (t)

i=1

ν

 kj EjT (t)Y  .

j=1

Using the double expectation theorem and the fact that Ei (t) and P (t) are independent for any i = 1, . . . , r, we have that P (t + 1) = (1 + k0 )2 P (t) + (1 + k0 )P (t) E

ν

ki EiT (t)Y +

i=1 ν

+ (1 + k0 ) E  +Y E

ki Y Ei (t) P (t)+ i=1 ν

ν

(ki Ei (t))P (t) i=1

 (kj EjT (t)) Y.

j=1

Let Ω = vv T /N . Since E[Ei (t)] = Ω and since Y Ω = Ω Y = 0 we have that the ﬁrst two expectations in the previous equation are equal to zero. To compute the last expectation we need to distinguish two cases:

Symmetries in the Coordinated Consensus Problem

49

i = j : then Ei (t), EjT (t) and P (t) are all independent and thus the expectation factorizes. Two terms of the type Y Ω appear and thus for i = j the expectation is zero, i = j : then, since it can be veriﬁed by straightforward calculations that for any M ∈ RN ×N , 1 T (v M v)Ω + N

E Ei (t)M EiT (t) =

1 1 tr M − 2 v t M v I, N N

we have Y E ki Ei (t) P (t) ki EiT (t) Y = ki2 Y E Ei (t)P (t)EiT (t) Y =

ki2 T k2 Y v P (t) v ΩY + i tr (P (t)) Y N N ki2 T − 2 Y v P (t) vY N

The ﬁrst term of the previous equation is zero since Ω Y is zero. We thus obtain that P (t + 1) = (1 + k0 )2 P (t) + Now we consider E E

y(t + 1)

2

y(t) 2

1 N

ν

ki2 tr (P (t))Y −

i=1

1 N2

ν

ki2 Y v T P (t) vY.

i=1

= tr P (t), then we have

= (1 + k0 )2 E 1 − 2 N

y(t)

ν

2

+

1 N

ν

ki2 tr tr (P (t))Y

i=1

ki2 tr (v T P (t) vY ).

i=1

The term tr tr (P (t))Y = (N − 1) tr (P (t)) since tr (Y ) = N − 1 and the last term is zero since v T P (t) v =

N

N

(P (t))ij = 0 . i=1 j=1

We thus have the following diﬀerence equation E

y(t + 1)

2

=

Now it is easy to verify that   N −1 min (1 + k0 )2 + k,k1 ,...,kν  N

(1 + k0 )2 +

N −1 N

ν

ki2

E

y(t)

2

.

i=1

  1 kj = 1 = ki2 | 1 + k0 , k1 , . . . , kν ≥ 0,  1+ν j=1 i=1 ν

ν

(25)

50

R. Carli et al.

and that it is obtained by choosing k0 = −νN/(N (1 + ν) − 1) and kj = νN/(N (1 + ν)) for all 1 ≤ j ≤ ν. With such a choice we have the convergence result (21).

D Proof of Proposition 3. Let us deﬁne z(t) = x(t) − b v. It is not diﬃcult to prove that z(t) has the same close loop dynamics as the system in x(t), thus ν

z(t + 1) =

k i Ei

(1 + k0 )I +

z(t).

i=1

Let us consider P (t) = E[(x(t) − bv)(x(t) − bv)T ] = E[z(t)z T (t)]. Similar calculations as those done for the convergence rat yields P (t + 1) = (1 + k0 )2 P (t) + (1 + k0 )

ν

ki (P (t)Ω + ΩP (t)) + i=1

ν

1 T (v P (t) v)Ω + N

ki2

+ i=1

ν

1 1 tr P (t) − 2 v T P (t) v I + N N

ν

+

ki kj ΩP (t)Ω i=1 j=1 i=j

Let us deﬁne the following variables 1 tr P (t) N 1 s(t) = 2 v T P (t)v N

w(t) =

We want to compute the mean squared distance from the barycenter at steady state, namely we want to compute w(∞) := limt→∞ w(t). We have that   ν ν (1 + k0 )2 + i=1 ki2 −k0 (k0 + 2) − i=1 ki2 w(t + 1)  w(t) . = 1 ν 1 ν 2 2 s(t) s(t + 1) 1 − k k N i=1 i N i=1 i Σ

where the transition matrix Σ has eigenvalues λ1 = 1, since when the ν states agree, they do not move anymore and λ2 = k02 + NN−1 i=1 ki2 related to the convergence rate, which was computed before. The time evolution of w(t) and s(t) is thus given by

Symmetries in the Coordinated Consensus Problem

w(t) s(t)

51

= c1 λt1 a1 + c2 λt2 a2

where c1 , c2 are constants and a1 , a2 are the eigenvectors associated to λ1 and λ2 . At steady state the vector (w(∞), s(∞))T is aligned to the dominant eigenvector of Σ and thus w(∞) ≈ c1 . Simple calculations yield w(∞) = α where α=

(1 − N )

1 x(0)T (I − Ω) x(0) , N ν 2 i=1 ki ν 2 1=1 ki − k0 N (k0

+ 2)

.

On Communication Requirements for Multi-agent Consensus Seeking Lei Fang and Panos J. Antsaklis Department of Electrical Engineering University of Notre Dame Notre Dame, IN 46556 {lfang,antsaklis.1}@nd.edu Summary. Several consensus protocols have been proposed in the literature and their convergence properties studied via a variety of methods. In all these methods, the communication topologies play a key role in the convergence of consensus processes. The goal of this paper is two fold. First, we explore communication topologies, as implied by the communication assumptions, that lead to consensus among agents. For this, several important results in the literature are examined and the focus is on diﬀerent classes of communication assumptions being made, such as synchronism, connectivity, and direction of communication. In the latter part of this paper, we show that the conﬂuent iteration graph uniﬁes various communication assumptions and proves to be fundamental in understanding the convergence of consensus processes. In particular, based on asynchronous iteration methods for nonlinear paracontractions, we establish a new result which shows that consensus is reachable under directional, time-varying and asynchronous topologies with nonlinear protocols. This result extends the existing ones in the literature and have many potential applications.

1 Introduction In recent years, there has been growing interest in the coordinated control of multi-agent systems. One of the fundamental problems is the consensus seeking among agents, that is the convergence of the values of variables common among agents to the same constant value [BS03, JLM03c, LBF04, OSM04b, RB05a]. This need stems from the fact that in order for agents to coordinate their behaviors, they need to use some shared knowledge about variables such as direction, speed, time-to-target etc. This shared variable or information is a necessary condition for cooperation in multi-agent systems, as shown in [RBM04]. The challenge here is for the group to have a consistent view of the coordination variable in the presence of unreliable and dynamically changing communication topology with-

P.J. Antsaklis, P. Tabuada (Eds.): Netw. Emb. Sens. and Cntrl., LNCIS 331, pp. 53–67, 2006. © Springer-Verlag Berlin Heidelberg 2006

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L. Fang and P.J. Antsaklis

out global information exchange. For an extensive body of related work, see [BS03, BGP03, dCP04, HM04b, LW04, LBF04, RBA05, SW03, XB03]. The aforementioned consensus protocols all operate in a synchronized fashion since each agent’s decisions must be synchronized to a common clock shared by all other agents in the group. This synchronization requirement might not be natural in certain contexts. For example, the agreement of timeon-target in cooperative attack among a group of UAVs depends in turn on the timing of when to exchange and update the local information. This diﬃculty entails the consideration of the asynchronous consensus problem, where each agent updates on its own pace, and uses the most recently received (but possibly outdated) information from other agents. No assumption is made about the relative speeds and phases of diﬀerent clocks. Agents communicate solely by sending messages; however there is no guarantee of the time of delivery or even for a successful delivery. Under asynchronous communications, heterogeneous agents, time-varying communication delays and packet dropout can all be taken into account in the same framework. Nevertheless, the asynchronism can destroy convergence properties that the algorithm may possess when executed synchronously. In general, the analysis of asynchronous algorithms is more diﬃcult than that of their synchronous counterparts. We refer readers to [BT89, FS00, Koz03] for surveys on general theories of asynchronous systems. Work reported on the asynchronous consensus problem is relatively sparse compared to its synchronous counterparts. In [FA05], we introduced an asynchronous framework to study the consensus problems for discrete-time multi-agent systems with a ﬁxed communication topology under the spanning tree assumption (All the assumptions in this paragraph will be discussed in detail later). A distributed iterative procedure under the eventual update assumption was developed in [MSP+ 05] for calculating averages on asynchronous communication networks. The asynchronous consensus problem with zero time delay was studied in [CMA05a] where the union of the communication graphs is assumed to have a common root spanning tree. A nice overview of the asynchronous consensus problem is given in [BHOT05] where the authors link the consensus problem considered here to earlier work [TBA86]. For other related problems in asynchronous multi-agent systems, see [BL96, LMA04a, LPP02, VACJ98]. Asynchronism provides a new dimension to consensus problems and makes convergence harder to achieve. Under certain technical conditions, asynchronism is not detrimental to consensus seeking among agents. A natural question is what are the appropriate requirements on communication topologies to guarantee the convergence of consensus processes? In order to answer this question, we ﬁrst discuss the various assumptions on communication topologies commonly used in the literature and classify several of the existing consensus results by these communication assumptions. In Sect. 3, we prove the convergence of asynchronous consensus with zero time delay involving pseudocontractive mappings. This development is a generalization of the results of [CMA05a] and [XBL05b], and provides insight into why the choice be-

On Communication Requirements for Multi-agent Consensus Seeking

55

tween bidirectional and unidirectional communication assumptions can make the diﬀerence in establishing consensus convergence. In Sect. 4, we unify various communication assumptions using the conﬂuent iteration graph proposed in [Pot98]. Furthermore, a new convergence result for nonlinear protocols is developed based on the conﬂuent asynchronous iteration concept. This result contains some existing ones in [BHOT05] as special cases.

2 Preliminaries and Background 2.1 Deﬁnitions and Notations Let G = {V, E, A} be a weighted digraph (or direct graph) of order n with the set of nodes V = {v1 , v2 , . . . , vn }, set of edges E ⊆ V × V , and a weighted adjacency matrix A = [aij ] with nonnegative adjacency elements aij . The node indices belong to a ﬁnite index set I = {1, 2, . . . , n}. A directed edge of G is denoted by eij = (vi , vj ). For a digraph, eij ∈ E does not imply eji ∈ E. The adjacency elements associated with the edges of the graph are positive, i.e., aij > 0 if and only if eji ∈ E. Moreover, we assume aii = 0 for all i ∈ I. The set of neighbors of node vi is the set of all nodes which point to (communicate with) vi , denoted by Ni = {vj ∈ V : (vj , vi ) ∈ E}. A digraph G can be used to model the interaction topology among a group of agents, where every graph node corresponds to an agent and a directed edge eij represents a unidirectional information exchange link from vi to vj , that is, agent j can receive information from agent i. The interaction graph represents the communication pattern at certain time. The interaction graph is time-dependent since the information ﬂow among agents may be dynamically ¯ = {G1 , G2 , . . . , GM } denote the set of all possible interaction changing. Let G ¯ is assumed graphs deﬁned for a group of agents. Note that the cardinality of G to be ﬁnite. The union of a collection of graphs {Gi1 , Gi2 , . . . , Gim }, each with the vertex set V , is a graph G with the vertex set V and the edge set equal to the union of the edge sets of Gij , j = 1, . . . , m. A directed path in graph G is a sequence of edges ei1 i2 , ei2 i3 , ei3 i4 , · · · in that graph. Graph G is called strongly connected if there is a directed path from vi to vj and vj to vi between any pair of distinct vertices vi and vj . Vertex vi is said to be linked to vertex vj across a time interval if there exists a directed path from vi to vj in the union of interaction graphs in that interval. A directed tree is a directed graph where every node except the root has exactly one parent. A spanning tree of a directed graph is a tree formed by graph edges that connect all the vertices of the graph. The condition that a digraph contains a spanning tree is equivalent to the condition that there exists a node having a directed path to all other nodes. Let xi ∈ R, i ∈ I represent the state associated with agent i. A group of agents is said to achieve global consensus asymptotically if for any xi (0), i ∈ I, xi (t) − xj (t) → 0 as t → ∞ for each (i, j) ∈ I. Let 1 denote an n × 1

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column vector with all entries equal to 1. A matrix F ∈ Rn×n is nonnegative, F ≥ 0, if all its entries are nonnegative, and it is irreducible if and only if (I + |F |)n−1 > 0, where |F | denotes the matrix of absolute values of entries of F . Furthermore, if all its row sums are +1, F is said to be a (row) stochastic matrix. Let X ∗ be a nonempty closed convex subset of Rn , and let · be a norm on Rn . For any vector x ∈ Rn , y ∗ ∈ X ∗ is a projection vector of x onto X ∗ if x − y ∗ = miny∈X ∗ x − y . We use P (x) to denote an arbitrary but ﬁxed projection vector of x and dist(x, X ∗ ) to denote x − P (x) . Let T be an operator on Rn . It is paracontractive if Tx ≤ x

for all x ∈ Rn

(1)

and equality holds if and only if T x = x. An operator is nonexpansive (with respect to · and X ∗ ) if T x − x∗ ≤ x − x∗

for all x ∈ Rn , x∗ ∈ X ∗ ,

and pseudocontractive [SB01] (with respect to

·

(2)

and X ∗ ) if, in addition,

dist(T x, X ∗ ) < dist(x, X ∗ ) for all x ∈ / X ∗.

(3)

We use T to denote the set of all pseudocontractive operators. In the linear case, pseudocontractive operators are generalizations of paracontractive ones. But the converse is not true. Consider the following inequalities: T x − P (T x) ≤ T x − P (x) ≤ x − P (x) , for all x ∈ / X ∗.

(4)

Paracontractivity requires the second inequality to be strict, while pseudocontractivity requires any one of these two inequalities to be strict. Example 1. Let T ∈ Rn×n , X ∗ = {c1|c ∈ R}. Then T is pseudocontractive with respect to X ∗ and the inﬁnity norm · ∞ if and only if T 1 = 1 and for any x ∈ Rn such that mini xi < maxi xi , maxi (T x)i − mini (T x)i < maxi xi − mini xi . 2.2 Synchronous and Asynchronous Consensus Protocols We consider the following (synchronous) discrete-time consensus protocol [RB05a, OSM04b, Mor03] xi (t + 1) =

1 n j=1 aij (t)

n

aij (t)xj (t)

(5)

j=1

where t ∈ {0, 1, 2, · · · } is the discrete-time index, (i, j) ∈ I and aij (t) > 0 if information ﬂows from vj to vi at time t and zero otherwise, ∀j = i. The magnitude of aij (t) possibly represents time-varying relative conﬁdence of

On Communication Requirements for Multi-agent Consensus Seeking

57

agent i in the information state of agent j at time t or the relative reliabilities of information exchange links between agents i and j. We can rewrite (5) in a compact form x(t + 1) = F (t)x(t)

(6) a (t)

where x = [x1 , · · · , xn ]T , F (t) = [Fij (t)] with Fij (t) = Pn ij aij (t) , (i, j) ∈ I. j=1 An immediate observation is that the matrix F is a nonnegative stochastic matrix, which has an eigenvalue at 1 with the corresponding eigenvector equal to 1. The protocol (5) or (6) is synchronous in the sense that all the agents update their states at the same time using the latest values of neighbors’ states. The way described above to deﬁne F (t) in (6) is only one possible way among many others. In the following we assume that F (t) satisﬁes Assumption 1 below. Assumption 1 (Nontrivial interaction strength [BHOT05]) There exists a positive constant α such that (a) Fii (t) ≥ α, for all i, t. (b) Fij (t) ∈ {0} ∪ [α, 1], for all i, j, t. n (c) j=1 Fij (t) = 1, for all i, t. Now, in the asynchronous setting the order in which states of agents are updated is not ﬁxed and the selection of previous values of the states used in the updates is also not ﬁxed. Now let t0 < t1 < · · · < tn < · · · be the time instants when the state of the multi-agent system undergoes change. Let xi (k) denote the state of agent i at time tk . The index k is also called in the literature the event-based discrete time index. The dynamics of asynchronous systems can be written as xi (k + 1) =

n j=1

Fij (k)xj (sij (k)) if i ∈ I(k), xi (k) if i ∈ / I(k),

(7)

where sij (k) are nonnegative integers, I(k) are nonempty subsets of {1, · · · , n}. The initial states are speciﬁed by x(0) = x(−1) = · · · . Henceforth, we write the initial vector x(0) to abbreviate reference to this set of equal initial states. We refer to dij (k) = k − sij (k) as iteration delays and I(k) as updating sets. The following assumptions (called regularity assumptions) are usually made in the study of linear asynchronous linear systems. Assumption 2 (Partial asynchronism) (a) (Frequency of updating) The updating sets I(k) satisfy i+B

∃B ≥ 0,

I(k) = {1, · · · , n}, for all i. k=i

(8)

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(b) (Bounded-delay asynchronism) There exists a nonnegative integer B such that 0 ≤ k − sij (k) ≤ B < ∞, ∀ (i, j, k).

(9)

(c) sii (k) = k, for all i. Assumption 2(a) says that every agent should be updated at least once in any B + 1 iteration steps. Assumption 2(b) requires delays to be bounded by some constant B. Assumption 2(c) says that an agent generally has access to its own most recent value. Without loss of generality (but after renumbering in the original deﬁnition), we assume that I(k) is a singleton which contains a single element from {1, . . . , n}. Furthermore, if for all i there exists a nonnegative integer Bi such that for all j j+Bi

i∈

I(k),

(10)

k=j

we call I(k) an indexwise-regulated sequence [Pot98]. This condition expresses the fact that diﬀerent agents may have diﬀerent updating frequencies. 2.3 Other Communication Assumptions Let us review several important assumptions commonly used in the literature. Assumption 3 (Connectivity) (a) (Uniform strong connectivity) There ext+B ists a nonnegative integer B such that s=t G(s) is strongly connected for all t. (a’) (Uniform spanning tree) There exists a nonnegative integer B such that t+B s=t G(s) contains a spanning tree for all t. (b) (Nonuniform strong connectivity) t≥t0 G(t) is strongly connected for all t0 ≥ 0. (b’) (Nonuniform spanning tree) t≥t0 G(t) contains a spanning tree for all t0 ≥ 0. Assumption 4 (Direction of Communication) (a) (Bidirectional link) If eij ∈ G(t) then eji ∈ G(t). It implies that the updating matrix F (t) or F (k) is symmetric. (b) (Unidirectional link) eij ∈ G(t) does not imply eji ∈ G(t). In this case, the updating matrix F (t) or F (k) is not symmetric. Assumption 5 (Reversal link) (a) If (vi , vj ) ∈ G(t), then there exists some τ such that |t − τ | < B and (vj , vi ) ∈ G(τ ) [CMA05a, BHOT05].

On Communication Requirements for Multi-agent Consensus Seeking

59

(b) There is a nonnegative integer B such that for all t and all vi , vj ∈ V we have that if (vi , vj ) ∈ G(t) then vj is linked to vi across [t, t + B] [Mor04]. Assumptions 1-5 play diﬀerent roles in proving various consensus results and they are not necessarily independent from each other. Assumption 1 (Nontrivial interaction strength) and one of the items in Assumption 3 (Connectivity) are always necessary for the convergence of consensus protocols. They guarantee that any update by any agent has a lasting eﬀect on the states of all other agents. Assumption 2 (Partial asynchronism) describes a class of asynchronous systems. If Assumption 3(a) (Uniform strong connectivity) is satisﬁed, then Assumption 2(a) (Frequency of Updating) and Assumption 5(b) (Reversal link) are satisﬁed automatically. Assumption 4(a) (Bidirectional link) is a special case of Assumption 5 (Reversal link). Instead of requiring an instantaneous reversal link (vj , vi ) (T =0 for the bidirectional case) for the link (vi , vj ), we only need the reversal link (vj , vi ) to appear within a certain time window or just require vj links back to vi within a certain time window (The edge (vj , vi ) may not appear at all). 2.4 A Classiﬁcation of Consensus Results For better understanding of Assumptions 1-5, we categorize some of the existing consensus results in Table 1. Assumption 1 is omitted in the table since it is required in all the results listed. Table 1. A Categorization of the existing consensus results No.

Results

Synchronism

Connectivity

1 Th. 2 in [JLM03c] Sync. A3(a) 2 Prop. 2 in [Mor03] Sync. A3(b) 3 Prop. 1 in [Mor03] Sync. A3(a’) 4 Th. 3.10 in [RB05a] Sync. A3(a’) 5 Th. 1 [Mor04] Sync. A3(b) 6 Th. 2 in [FA05] A2(a),(b),(c) Fixed spanning tree 7 Th. 4 in [CMA05a] A2(a),(c) with Spanning trees with zero time delay the common root 8 Th. 1 in [BHOT05] A2(a),(c) with A3(a) zero time delay 9 Th. 3 in [BHOT05] A2(a),(b),(c) A3(a) 10 Th. 4 in [BHOT05] A2(a),(b),(c) A3(b) 11 Th. 5 in [BHOT05] A2(a),(b),(c) A3(b)

Direction Reversal link A4(a) A4(a) A4(b) A4(b) A4(b) A4(b) A4(b)

NA NA NA NA A5(b) NA NA

A4(b)

NA

A4(b) A4(a) A4(b)

NA NA A5(a)

Several comments are appropriate. First, asynchronous systems with zero time delay can be mapped to equivalent synchronous systems following the arguments in [FA05]. Therefore, Results No. 3 & 4 may be seen as a special

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case of Result No. 6. Second, Table 1 reveals the fact that the uniform connectivity is necessary under unidirectional communication but it is not necessary under bidirectional communication. The reason behind this fact is not at all clear from an intuitive perspective. In Sect. 3, we give an explanation to the diﬀerence between bidirectional and unidirectional communication from contractive operators’ point of view. Third, we utilize the iteration graph in Sect. 4.1 to unify various communication assumptions. Fourth, Result No. 9 considers the most general case among linear protocols; notice that all protocols in Table 1 are linear protocols. A nonlinear asynchronous protocol will be introduced in Sect. 4.2.

3 Bidirectional Versus Unidirectional Communication In this section, we investigate why diﬀerent assumptions on connectivity need to be imposed for bidirectional and unidirectional communication if consensus is to be achieved. Speciﬁcally, we see the consensus problem as a matrix iteration problem where the notions of paracontraction and pseudocontraction introduced in Sect. 2.1 are useful in proving convergence. For an easy exposition, we restrict ourselves to synchronous protocols with time-varying topologies and no time delays. That is, we consider the system updating equation x(t + 1) = F (t)x(t) where the matrix F (t) satisﬁes Assumption 1. It is easily deduced that F (t) is nonexpansive with respect to the vector norm · ∞ . In the bidirectional case, F (t) is also symmetric. From nonnegative matrix theory [HJ85], we know that the eigenvalues of F (t) lie in (−1, +1] for all t. It is now known that any symmetric matrix F the eigenvalues of which are in (−1, +1] is paracontracting with respect to Euclidian norm [BEN94]. However, F (t) is no longer paracontracting when its symmetry is lost. Example 2 (partly taken from [SB01]) (a) For the weight matrix   0.5 0.5 0 F1 =  0.25 0.5 0.25  , 0 0.5 0.5 induced from the simple communication topology as shown in Fig. 1(a), the norm and the set X ∗ are the same as in Example 1. For any x, P (x) = 0.5(maxi xi +mini xi )1. For x = [2, 2, 1]T , F1 x = (2, 1.75, 1.5)T , P (x) = 1.5·1, and P (F1 x) = 1.75·1. Thus the ﬁrst inequality in (4) is strict while the second one is an equality. So this operator is pseudocontractive, not paracontractive. (b) For an arbitrary F (t), there is no guarantee that it is pseudocontractive, e.g., the weight matrix

On Communication Requirements for Multi-agent Consensus Seeking





61

1 0 0 F2 =  0.5 0.5 0  , 0 0 1 which results from the communication topology in Fig. 1(b). For x = [2, 2, 1]T , F2 x = [2, 2, 1]T . Thus equalities hold throughout (4). F2 is not pseudocontractive but it is nonexpansive. 1

2

(a)

3

1

2

3

(b)

Fig. 1. Possible communication topologies for a three-agent system

An alternative proof for Result No. 2 was provided in [XBL05b] where the convergence of the paracontracting matrix products was explored. The following theorem in [EKN90] is the key to this development. Theorem 1. Suppose that a ﬁnite set of square matrices of same dimensions {F1 , · · · , Fr } are paracontracting. Let {I(t)}∞ t=0 , with 1 ≤ I(t) ≤ r, be a sequence of integers, and let J denote the set of all integers that appear inﬁnitely often in the sequence. Then for all x(0) ∈ Rn the sequence of vectors x(t+1) = FI(t) x(t) has a limit x∗ ∈ ∩i∈J H(Fi ), where H(F ) denotes the ﬁxed point subspace of a paracontracting matrix F , i.e., its eigenspace associated with the eigenvalue 1, H(F ) = {x|x ∈ Rn , F x = x}. Does there exist a result similar to Theorem 1 for pseudocontracting matrices? The answer is, fortunately, aﬃrmative. Theorem 2 ([SB01]). Let {Tt } be a sequence of nonexpansive operators (with respect to · and X ∗ ), and assume there exists a subsequence {Tti } which converges to T ∈ T . If T is pseudocontractive and uniformly Lipschitz continuous, then for any initial vector x(0), the sequence of vectors x(t + 1) = Tt x(t), t ≥ 0 converges to some x∗ ∈ X ∗ . Note that in our study X ∗ = {c1|c ∈ R}. We are thus one step away from proving Result No. 3 and it is exactly where Assumption A3(a) (the uniform connectivity) comes into play. Result No. 3 is an immediate result of Theorem 2 and the following lemma. Lemma 1. If Assumption A3(a) is satisﬁed, then there exists an integer B such that the matrix product F (t0 + B )F (t0 + B − 1) · · · F (t0 ) is pseudocontractive, with respective to · ∞ and X ∗ = {cl|c ∈ R}, for any t0 ≥ 0. To prove Lemma 1, we need the following technical results.

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Lemma 2 (Lemma 2 in [JLM03c]). Let m ≥ 2 be a positive integer and let F1 , F2 . . . , Fm be nonnegative n × n matrices. Suppose that the diagonal elements of all of the Fi are positive and let α and β denote the smallest and largest of these, respectively. Then Fm Fm−1 · · · F1 ≥

α2 2β

(m−1)

(Fm + Fm−1 + · · · + F1 ).

(11)

¯ Lemma 3 (Proposition 3.2 in [SB01]). Let x be a vector such that x < x with x = mini xi and x ¯ = maxi xi , F be an irreducible matrix with Fii > 0 for 1 ≤ i ≤ n, y = F x. Then the number of the elements in set {i|yi = x or yi = ¯}. x ¯} is at least one less than the number of the elements in {i|xi = x or xi = x Proof (of Lemma 1). Assume that there exists an inﬁnite sequence of contiguous, non-empty, bounded time-intervals [tij , tij+1 ), j ≥ 1, starting at ti1 , with the property that across each such interval, the union of the interaction graphs is strongly connected (Assumption A3(a)). Let Htj = F (tij+1 −1 ) · · · F (tij +1 )F (tij ). By Lemma 2 and the strong connectivity of the union of the interaction graphs, it follows that Htj is irreducible. It is easy to prove that Htj is nonexpansive with positive diagonal elements and Htj 1 = 1. Deﬁne H = Htn−1 Htn−2 · · · Ht1 and y = Hx. Applying Lemma 3 repeat¯} edly for (n − 1) times, we have at least one set of {i|yi = x} and {i|yi = x is empty. If, say, {i|yi = x} is empty, then yi > x for all 1 ≤ i ≤ n, and furthermore y − P (y) =

maxi yi − mini yi x ¯ − mini yi x ¯−x ≤ < = x − P (x) ; 2 2 2

therefore H is pseudocontractive. In other words, F (t0 + B )F (t0 + B − 1) · · · F (t0 ) is pseudocontractive for B = (n − 1)B. Let us brieﬂy summarize what we have presented in this section. In the bidirectional case, the weight matrices are always paracontracting. Theorem 1 can be applied directly to infer the convergence of the consensus processes. In the unidirectional case, the weight matrices are generally not pseudocontracting. In order to use Theorem 2, the uniform connectivity condition needs to be imposed so that the matrix products across certain time interval are pseudocontracting.

4 Iteration Graph and Nonlinear Asynchronous Consensus Protocols In the framework of [Pot98], the consensus problem is regarded as a special case of ﬁnding common ﬁxed points of a ﬁnite set of paracontracting multiple

On Communication Requirements for Multi-agent Consensus Seeking

63

point operators. That is, all the operators are deﬁned on (diﬀerent) products of Rn . To avoid the divergent phenomena, asynchronous iterations which fulﬁll certain coupling assumptions called conﬂuent are considered. Below we apply this theory of paracontractions and conﬂuence to derive a more general consensus result, which extends Result No. 9 by allowing nonlinear multiple point operators. For the purpose of self-containedness, we introduce several related deﬁnitions below. Let I be a set of indices, m ∈ N a ﬁxed number, and F = {F i |i ∈ I} be a pool of operators F i : Dmi ⊂ Rnmi → D, where mi ∈ {1, · · · , m}, ∀i ∈ I, and D ⊂ Rn is closed. Furthermore, let XO = {x(0), · · · , x(−M )} ⊂ D be a given set of vectors. Then, for sequences I = I(k) (k = 0, 1, . . . ) of elements in I, S = {s1 (k), . . . , smi(k) (k)}, k = 0, 1, . . . , of mi -tuple from N0 ∪ {−1, . . . , −M } with sl (k) ≤ k for all k ∈ N0 , l = 1, . . . , mi(k) , we study the asynchronous iteration given by x(k + 1) = F I(k) x(s1 (k)), . . . , x(smI(k) (k)) , k = 0, 1, . . .

(12)

An asynchronous iteration corresponding to F , starting with XO and deﬁned by I and S can be denoted by (F , XO , I, S). A ﬁxed point ξ of a multiple point operator F : Rnm → R is a vector ξ ∈ Rn which satisﬁes F (ξ, . . . , ξ) = ξ, and a common ﬁxed point of a pool is a ﬁxed point of all its operators in this sense. 4.1 Iteration Graph In essence, the communication assumptions deﬁne the coupling among agents or, more generally, the coupling of an iteration process. The existing assumptions often rely on interaction graphs to describe the “spatial” coupling among agents. However, ambiguity arises when asynchronism (e.g., delays) is allowed since the “temporal” coupling cannot be described directly. In the asynchronous setting, it is of importance to diﬀerentiate the same agent at diﬀerent time instants. To this end, we associate an iteration graph with the asynchronous iteration (F , XO , I, S). Every iteration, including initial vectors, gets a vertex, so the set of vertices is V = N0 ∪ {−1, . . . , −M }. A pair (k1 , k2 ) is an element of the set of edges E in the iteration graph (V, E), if and only if the k1 th iteration vector is used for the computation of the k2 th iteration vector. Below we illustrate the concept of iteration graph via an example. The interaction topologies of a three-agent system at diﬀerent time instants are shown in Fig. 2(a). It is easy to see that if the interaction pattern continues, Assumption 3(b) is satisﬁed. Let y(−1) = x1 (0), y(−2) = x2 (0), and y(−3) = x3 (0). At time k = 0, v2 communicates with v3 . By construction, we add a vertex 0 and an edge from vertex -2 to vertex 0 in the associated iteration graph, as shown in Fig. 2(b). Assume that v3 and v1 do not use its own past value. Therefore, we do not add an edge from vertex -3 to vertex 0 in the

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iteration graph. At time instant k = 1, v2 uses the value of v1 and its own past value to update its state, resulting in two edges in the iteration graph. Remark 1. An analogy to the iteration graph is the the reachability graph in the Petri net literature. The reachability graph is used for veriﬁcation and supervisory control and obtained sometimes via a method called unfolding that simpliﬁes the procedure [BFHJ03].

v2 v3

-3

0

-2

v2 v1

v3

-1

0

v2 v1

1

1

v3

2

2

3

v2 v1

v3 (a)

4

3

v2 v1

v3

4

v2 v1

v3

5

v1

5

(b) Fig. 2. Interaction topologies of an asynchronous system and its associated iteration graph.

Deﬁnition 1 (Conﬂuent asynchronous iteration [Pot98]). Let (F, XO, I, S) be an synchronous iteration. The iteration graph of (F , XO , I, S) is the digraph (V, E), whose vertices V are N0 ∪{−1, . . . , −M }, and whose edges E are given by (k, k0 ) ∈ E, iﬀ there is an 1 ≤ l ≤ mI(k0 −1) , such that sl (k0 − 1) = k. (F , XO , I, S) is called conﬂuent, if there are numbers n0 ∈ N, b ∈ N and a sequence bk (k = n0 , n0 + 1, . . . ) in N, such that for all k ≥ n0 the following is true: (i) For every vertex k0 ≥ k there is a directed path from bk to k0 in (V, E), (ii) k − bk ≤ B, (iii) S is regulated, (iv) for every i ∈ I there is a ci ∈ N so that for all k ≥ n0 there is a vertex wki in V , which is a successor of bk and a predecessor of bk+ci , and for which is I(wki − 1) = i. It is worth mentioning that when Assumptions 2(a)(b), 3(b), 4(b) are fulﬁlled, the associated asynchronous iteration is conﬂuent. Given an arbitrary iteration, there are simple ways to make its implementation conﬂuent [Pot98].

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On Communication Requirements for Multi-agent Consensus Seeking

Remark 2. As opposed to the original development in [CM69], here the whole vector is updated in every iteration step. Also, all components of vectors have the same delay. This does not impose a restriction since the vectors reduce to a scalar in our study. 4.2 Nonlinear Asynchronous Consensus Protocols Before introducing the main result of the paper, we need the following deﬁnition. Note that paracontracting operator in Deﬁnition 2(ii) corresponds to pseudocontracting operator as deﬁned in (3). (Without much confusion, the original deﬁnitions given in [Pot98] are followed for easy reference.) Deﬁnition 2. Let F be a pool of operations as in Deﬁnition 1, and X = (x1 , . . . , xmi ) an element of Rnmi . (i) If for all i ∈ I, X, Y ∈ Dmi and a norm

·

Gi (X) − Gi (Y ) < max xj − y j j

or Gi (X) − Gi (Y ) = xj − y j , ∀j ∈ {1, . . . , mi }, then F is called strictly nonexpansive on D. (ii) If for all i ∈ I, X ∈ Dmi and a norm · , F i is continuous on Dmi , then F is paracontracting on D, if for any ﬁxed point ξ ∈ Rn of F i , F i (X) − ξ < max xj − ξ j

or X = (x, . . . , x) and x is a ﬁxed point of F i . It is easy to see that every strictly nonexpansive pool is paracontracting. Moreover, an (e=extended)-paracontracting notion is introduced in [Pot98]. The (e)-paracontracting operators may be discontinuous, and have nonconvex sets of ﬁxed points. A simpliﬁed version of the main result in [Pot98] is now given. Theorem 3. Let F be a paracontracting pool on D ⊂ Rn , and assume that F has a common ﬁxed point ξ ∈ D. Then any conﬂuent asynchronous iterations (F , XO , I, S) converges to a common ﬁxed point of F . With the help of Theorem 3, Result No. 9 can be obtained by interpreting the diﬀerent rows of a stochastic matrix as multiple data operators. To see this, let fimi (j) (k) be for all i ∈ {1, . . . , n} the jth of mi nonzero entries in F (k)’s ith row, or let mi (j) = j, ∀j = 1, . . . , n, and mi = n, if this row is zero. Then the pool F = {F i |i = 1, . . . , Q} (Q, the total number of operators, is ﬁnite), deﬁned by F i (k) : Rmi → R, F i (k)(y 1 , . . . , y mi ) :=

mi

fimi (j) (k)y j , i = 1, . . . , Q

(13)

j=1

is strictly nonexpansive on all closed intervals D ⊂ R, if F i (1, . . . , 1) = 1.

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We are now ready to claim a new consensus result where F i is allowed to be nonlinear. To avoid confusion, we rewrite I(k) in (12) as pk below. Theorem 4. Consider the iteration xi (k + 1) = F i x1 (s1 (k), x2 (s2 (k)), . . . , xn (sn (k) .

(14) l

(i) Assume without loss of generality that the numbering of s (k), k = 0, 1, . . . , is chosen in such a manner that all components xl (sl (k)) in (14) themselves are updated at time sl (k), i.e., psi (k)−1 = i, ∀k ∈ N, i ∈ {1, . . . , n} with si (k) ≥ 1,

(15)

and, also w.l.o.g., that all initial vectors are multiples of 1. Deﬁne x(−k) := xk (0)1, ∀k = 1, . . . , n, and renumber in this way the elements of the sequences of sl (k), k = 0, 1, . . . , l = 1, . . . , n, for which sl (k) = 0. Then the asynchronous iteration (F , YO , I, S), given by s1 (k)), . . . , y(˜ smpk (k)) , k = 0, 1, . . . y(k + 1) := F pk y(˜

(16)

where F = {F pk |k = 0, 1, . . . } is paracontracting, I = pk , k = 0, 1, . . . , S = {˜ si (k)|k = 0, 1, . . . ; i = 1, . . . , mpk } is given by s˜i (k) := smpk (i) (k), ∀k ∈ N0 , i = 1, . . . , mpk ,

(17)

and YO by y(−l) := x1 (−l), l = 1, . . . , n, generates y(k + 1) = xpk (k + 1), ∀k ∈ N0 .

(18)

(ii) The pool F = F i : Rn → R|i ∈ {1, . . . , Q} is paracontracting and has a common ﬁxed point. Furthermore, there exists an agent i0 which updates its state using only a subset of the pool F . Every operator in this subset ∂F i is continuously diﬀerentiable in xi0 and ∂x = 0. Assume that si0 (k) = i0 max{k0 ≤ k|pk0 −1 = i0 } for all k > min{k0 ∈ N0 |pk0 = i0 } with pk = i0 . Then, under Assumptions 2(a)(b), 3(b), 4(b), the nonlinear protocol (16) or, equivalently, (14) guarantees asymptotic consensus. Proof. (i) follows by induction on k. Using the same argument as in the proof of Theorem 5(v) in [Pot98], it can be shown that the iteration (16) is conﬂuent. (ii) is then an immediate result of Theorem 3. Remark 3. In Fig. 2, the iteration graph is conﬂuent when the agent v2 always uses its own past value for updating. Suppose that no agents use their past values during the process (i.e., dashed edges no longer appear). After removing the dashed edges from Fig. 2(b), the iteration graph is no longer conﬂuent since there is no directed path from an odd-numbered vertex to an even-numbered vertex, and vice versa. This shows the necessity of existence of i0 in Theorem 4(ii).

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Theorem 4 is exact, rather than linearized and can be used to study multiagent systems with nonlinear couplings. Potential applications include distributed time synchronization and rendezvous of multi-robots with nonholonomic constraints.

5 Conclusions In this paper, we examined the diﬀerent assumptions made in the various consensus results in the literature so to better understand their roles in the convergence analysis of consensus protocols. A novel nonlinear asynchronous consensus result was also introduced using the theory of paracontracitons and conﬂuence. This result is more general than the existing ones and provides a powerful tool to study a wider range of applications. Many open problems remain; see [FAT05] for a detailed discussion.

Applications of Connectivity Graph Processes in Networked Sensing and Control Abubakr Muhammad, Meng Ji, and Magnus Egerstedt School of Electrical and Computer Engineering Georgia Institute of Technology, Atlanta, GA {abubakr,mengji,magnus}@ece.gatech.edu Summary. This paper concerns the problem of controlling mobile nodes in a network in such a way that the resulting graph-encoding of the inter-node information ﬂow exhibits certain desirable properties. As a sequence of such transitions between diﬀerent graphs occurs, the result is a graph process. In this paper, we not only characterize the reachability properties of these graph processes, but also put them to use in a number of applications, ranging from multi-agent formation control, to optimal collaborative beamforming in sensor networks.

1 Introduction As the complexity associated with the control design tasks for many modern engineering systems increases, strategies for managing the complexity have become vitally important. This is particularly true in the areas of networked and embedded control design, where the scale of the system renders classical design tools virtually impossible to employ. However, the problem of coordinating multiple mobile agents is one in which a ﬁnite representation of the conﬁguration space appears naturally, namely by using graph-theoretic models for describing the local interactions in the formation. In other words, graph-based models can serve as a bridge between the continuous and the discrete when trying to manage the design-complexity associated with formation control problems. Notable results along these lines have been presented in [RM03, Mes02, JLM03a, TJP03a, CMB05b]. The conclusion to be drawn from these research eﬀorts is that a number of questions can be answered in a natural way by abstracting away the continuous dynamics of the individual agents. Several terms such as link graphs, communication graphs and connectivity graphs have been used interchangeably in the literature for graphical models that capture the local limitations of sensing and communication in decentralized networked systems. In this paper, we give several applications of connectivity graphs and their dynamics: the so-called connectivity graph processes.

P.J. Antsaklis, P. Tabuada (Eds.): Netw. Emb. Sens. and Cntrl., LNCIS 331, pp. 69–82, 2006. © Springer-Verlag Berlin Heidelberg 2006

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The outline of this paper as follows. In Section 2 we provide various characterizations and computational tools that deal with the realization of connectivity graphs in their corresponding conﬁguration spaces, based on our work in [ME05, MEa]. These studies let us distinguish between valid and invalid graphical conﬁgurations and provide a computable way of determining valid transitions between various conﬁgurations, which is the topic of Section 3. Moreover, notions such as graph reachability and planning will be given a solid foundation. We moreover develop an optimal control framework, where the conﬁguration space is taken as the space of all connectivity graphs. Finally, Section 4 is devoted to the various applications of this framework. In particular, we study low-complexity formation planning for teams for mobile robots and collaborative beamforming in mobile sensor networks.

2 Formations, Connectivity Graphs, and Feasibility Graphs can model local interactions between agents, when individual agents are constrained by limited knowledge of other agents. In this section we summarize some previous results, found in [ME04a], of a graph theoretic nature for describing formations in which the primary limitation of perception for each agent is the limited range of its sensor. Suppose we have N such agents with identical dynamics evolving on R2 . Each agent is equipped with a range limited sensor by which it can sense the relative displacement of other agents. All agents are assumed to have identical sensor ranges δ. Let the position of each agent be xn ∈ R2 , and its dynamics be given by x˙ n = f (xn , un ), where un ∈ Rm is the control for agent n and f : R2 × Rm → R2 is a smooth vector ﬁeld. The conﬁguration space C N (R2 ) of the agent formation is made up of all ordered N -tuples in R2 , with the property that no two points coincide, i.e. C N (R2 ) = (R2 × R2 × . . . R2 ) − Δ, where Δ = {(x1 , x2 , . . . , xN ) : xi = xj for some i = j}. The evolution of the formation can be represented as a trajectory F : R+ → C N (R2 ), usually written as F (t)(x1 (t), x2 (t), . . . xN (t)) to signify time evolution. The spatial relationship between agents can be represented as a graph in which the vertices of the graph represent the agents, and existence of a pair of vertices on each edge tells us that the corresponding agents are within sensor range δ of each other. Let GN denote the space of all possible graphs that can be formed on N vertices V = {v1 , v2 , . . . , vN }. Then we can deﬁne a function ΦN : C N (R2 ) → GN , with ΦN (F (t)) = G(t), where G(t) = (V, E(t)) ∈ GN is the connectivity graph of the formation F (t). vi ∈ V represents agent i at position xi , and E(t) denotes the edges of the graph. eij (t) = eji (t) ∈ E(t) if and only if xi (t) − xj (t) ≤ δ, i = j. These graphs. The graphs are always undirected because the sensor ranges are identical. The motion of agents in a formation may result in the removal or addition of edges in the graph. Therefore G(t) is a dynamic structure. Lastly and most importantly, every graph in GN is not a connectivity graph. The last observation is not as obvious as the others, and

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it has been analyzed in detail in [ME04a]. A realization of a graph G ∈ GN is a formation F ∈ C N (R2 ), such that ΦN (F ) = G. An arbitrary graph G ∈ GN can therefore be realized as a connectivity graph in C N (R2 ) if Φ−1 N (G) is nonempty. We denote by GN,δ ⊆ GN , the space of all possible graphs on N agents with sensor range δ, that can be realized in C N (R2 ). In [abu05] we proved the following result. Theorem 1. GN,δ is a proper subspace of GN if and only if N ≥ 5. Formations can produce a wide variety of graphs for N vertices. This includes graphs that have disconnected subgraphs or totally disconnected graphs with no edges. However the problem of switching between diﬀerent formations or of ﬁnding interesting structures within a formation of sensor range limited agents can only be tackled if no sub-formation of agents is totally isolated from the rest of the formation. This means that the connectivity graph G(t) of the formation F (t) should always remain connected (in the sense of connected graphs) for all time t. In [ME05, MEa] we gave a detailed study of feasibility results using semideﬁnite programming methods and its relation to the Positivstellensatz for semialgebraic sets. In particular, we showed how to setup the feasibility of geometrical constraints in a possible graph as a linear matrix inequality (LMI) problem as follows. Recall that the connectivity graph (V, E) corresponding to the formation (x1 , x2 , · · · xN ) ∈ C N (R2 ) can be described by N (N − 1)/2 relations of the form δ 2 − (xi − xj )2 − (yi − yj )2 ≥ 0, if eij ∈ E, (xi − xj )2 + (yi − yj )2 − δ 2 > 0, if eij ∈ / E, where 1 ≤ i < j ≤ N and xi = (xi , yi ). Therefore the realization problem is equivalent to asking if there exist x1 , y1 , . . . , xN , yN such that these inequality constraints are satisﬁed. In [ME05] we showed that the non-feasibility problem is equivalent to asking if the set X = {x ∈ RM | xT Aij x ≥ 0, 1 ≤ i < j ≤ N, eij ∈ E, xT Blm x > 0, 1 ≤ l < m ≤ N, elm ∈ E} is empty for certain Aij and Blm matrices (See [ME05] for details). Since all semi-algebraic constraints on the set X are quadratic and Moreover, it was also shown that T Aij = ATij , Blm = Blm , we can use the celebrated S-procedure to transform the feasibility question into a linear matrix inequality (LMI) problem [BGFB94]. Theorem 2. Given symmetric n × n matrices {Ak }m k=0 , the following are equivalent: 1. The set {x ∈ Rn | xT A1 x ≥ 0, xT A2 x ≥ 0, · · · , xT Am x ≥ 0, xT A0 x ≥ 0, xT A0 x = 0} is empty. m 2. There exist non-negative scalars {λk }m k=1 such that −A0 − k=1 λk Ak ≥ 0. If a solution to this LMI exists then we say that we have a certiﬁcate of infeasibility. It was demonstrated how to use a standard LMI-software [Inc01] to

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eﬀectively solve a wide class of such feasibility problems. Once such infeasibility certiﬁcates have been obtained, they can be used in formation planning under constraints of communication and sensory perception.

3 Connectivity Graph Processes for Formation Planning As mentioned in Section 1, formation switching with limited global information is an important problem in multi-agent robotics. However, little work has been done so far that adequately addresses the problem of formation switching under limited range constraints. Therefore the ability to give exact certiﬁcates about what can and cannot be achieved under these constraints is a desirable result. Recall that the connectivity graph of the formation evolves over time as G(t) = (V, E(t)) = ΦN (x(t)). Under standard assumptions on the individual trajectories of agents, one gets a ﬁnite sequence of graphs {G0 , G1 , . . . , GM } for each ﬁnite interval of evolution [0, T ], where ΦN (x(0)) = G0 , Gi switches to Gi+1 at time ti , and ΦN (x(T )) = GM . We will often write this as G0 → G1 → . . . → GM and call such a sequence a connectivity graph process 1 . Such graph processes can be thought of as trajectories on the space GN,δ . In what follows we discuss the role of feasible, reachable and desirable sets when generating trajectories on this space. 3.1 Feasible Connectivity Graph Transitions The connectivity graph processes are generated through through the movement of individual nodes. For a connectivity graph Gj = (Vj , Ej ) = ΦN (x(tj )) let the nodes be partitioned as Vj = Vj0 ∪Vjm , where the movement of the nodes in Vjm facilitates the transition from Gj to the next graph Gj+1 and Vj0 is the set of nodes that are stationary. With the positions x0j = {xm (ti )}m∈Vj0 being ﬁxed, let Feas(Gj , Vjm , x0j ) ⊆ GN,δ be the set of all connected connectivity graphs that are feasible by an unconstrained placement of positions corresponding to Vjm in R2 . (We will often denote this set as Feas(Gj , Vjm ), when the the positions x0j are understood from context.) The set Feas(Gj , Vjm ) of feasible graph transitions can be computed using the semi-deﬁnite programming methods discussed above. It will be appropriate to explain the reason for keeping track of mobile and stationary nodes at each transition. In principle, it is possible to compute However, in order to manage the combinatorial growth in the number of possible graphs, it is desirable to let the transitions be generated by the movements of a small subset of nodes only. In fact, we will investigate the situation where er only move one node at a time. Let Vj0 = {1, . . . , k − 1, k + 1, . . . , N } and Vjm = {k}. It should be noted that the movement of node k can only result in the addition or deletion of edges that have node k as one of its vertices. Therefore the enumeration of 1

We borrow this term from Mesbahi [Mes02].

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the possible resulting graphs should count all possible combinations of such deletions and additions. This number can be easily seen to be 2N −1 for N nodes. Since we are also required to keep the graph connected at all times, this number is actually 2N −1 − 1, obtained after removing the graph in which node k has no edge with any other node. Now, we can use the S-procedure to evaluate whether each of the new graphs resulting from this enumeration is feasible. Since all nodes are ﬁxed except for xk = (x, y), the semi-algebraic set we need to check for non-feasibility is deﬁned by N − 1 polynomial inequalities over R[x, y]. Each of these inequalities has either of the following two forms, δ 2 − (x − xi )2 − (y − yi )2 ≥ 0, if eki ∈ E, (x − xi )2 + (y − yi )2 − δ 2 > 0, if eki ∈ / E. where 2 ≤ i ≤ N , E is the edge set of the new graph and we denote xi (tj ) by (xi , yi ) for i = k. This computation can be repeated for all N nodes so that we have a choice of N (2N −1 − 1) graphs. Each of the N − 1 inequalities can be written as either    −1 0 xi x   y  ≥ 0, if eik ∈ E, x y 1  0 −1 yi xi yi δ 2 − x2i − yi2 1 or

 x

y

1

Denoting by  −1 0 Ai =  0 −1 xi yi

1  0 −xi

0 1 −yi

  −xi x   y  > 0, if eik ∈ E. −yi 1 x2i + yi2 − δ 2

 xi , yi 2 2 2 δ − xi − y i



1 0 Bi =  −xi

0 1 −yi

 −xi , −yi 2 2 2 xi + y i − δ

and ignoring the lossy aspect of the S-procedure [BGFB94], we need to solve the LMI, λαi Aαi − λαj Bαj ≥ 0. −Aα1 − i=1,eαi k ∈E

j,eαj k ∈E

An example of such a calculation is given in Figure 1, where V 0 = {2, 3, 4, 5} and V m = {1}. The LMI control toolbox [Inc01] for MATLAB has been used to solve the LMI for each of these graphs in order to get the appropriate certiﬁcates. 3.2 Reachability and Connectivity Graph Processes Note that Feas(G0 , V0m ) does not depend on the actual movement of the individual nodes. In fact even if G ∈ Feas(G0 , V0m ), it does not necessarily mean

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Set of feasible transitions F eas(G0, {1}). 5

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Infeasible transitions. Fig. 1. Feasible and infeasible graphs by movement of node 1.

that there exists a trajectory by which G0 → G or even that G0 → Gf . . . G. We therefore need some notion of reachability on the space GN,δ . We say that a connectivity graph Gf is reachable from an initial graph G0 if there exists a connectivity graph process of ﬁnite length G0 → G1 → . . . Gf and a sequence of vertex-sets {Vkm } such that each Gk+1 ∈ Feas(Gk , Vkm ). If Vkm = V m at each transition, then every Gk ∈ Feas(G0 , V m ). (In particular, Gf ∈ Feas(G0 , V m ).) Consider all such G that are reachable from G0 with a ﬁxed Vm . We will denote this set by Reach(G0 , V m ). It is easy to see that Reach(G0 , V m ) ⊆ Feas(G0 , V m ). In the previous paragraphs, it was shown how to determine the membership for the set Feas(G0 , V m ). But, determining the membership for Reach(G0 , V m ) is not trivial. Under the assumption that individual nodes are globally controllable, a computational tool from algebraic geometry known as Cylindrical Algebraic Decomposition [bas] or CAD can be used for this purpose, as shown in [MEa, MEb]. For the semialgebraic set S deﬁned by the

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x4 xk (t0 )

x2

x1

xk (tf )

Fig. 2. Selection of Nash cells and the generation of trajectory in a graph process.

union of sensing discs of the stationary nodes V 0 , CAD provides a decomposition of S into so-called Nash cells. It has been proved in [MEb] that for two given connectivity graphs ΦN (x(0)) and Gf , there exists a ﬁnite connectivity graph process ΦN (x(0)) → G1 → . . . → Gf , and a corresponding trajectory x(t) ⊆ C N (R2 ), t ∈ [t0 , tf ] such that x(tf ) ∈ Φ−1 N (Gf ), if and only if there exists a ﬁnite collection of Nash cells in the CAD of S such that x(t) belongs to a cell in this collection for all t ∈ [t0 , tf ]. The construction used in proving this result result gives us the trajectories for the actual movement of the nodes. We omit the details of this proof for the sake of brevity. An example of CAD and the trajectory of the mobile node is depicted in Figure 2. 3.3 Global Objectives, Desirable Transitions and Optimality The whole purpose of a coordinated control strategy in a multi-agent system is to evolve towards the fulﬁlment of a global objective. This typically requires the minimization (or maximization) of a cost associated with each global conﬁguration. Viewed in this way, a planning strategy should basically be a search process over the conﬁguration space, evolving towards this optimum. If the global objective is fundamentally a function of the graphical abstraction of the formation, then it is better to perform this search over the space of graphs instead of the full conﬁguration space of the system. By introducing various graphical abstractions in the context of connectivity graphs, we have the right machinery to perform this kind of planning. In other words, we will associate a cost or score with each connectivity graph and then work towards minimizing it. Given Reach(G0 , V m ), a decision need to be taken regarding what Gf ∈ Reach(G0 , V m ) the system should switch to. For this we deﬁne a cost function Ψ : GN,δ → R and we choose the transition through Gf =

arg min

G∈Reach(G0 ,V m )

Ψ (G)

Here Ψ is analogous to a terminal cost in optimal control. If, in addition, we also take into account the cost associated with every transition in the graph

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process G0 → G1 → . . . → GM = Gf that takes us to Gf , then we would instead consider the minimization of the cost M −1

β(i)L(Gi , Gi+1 ),

J = Ψ (Gf ) + i=0

where L : GN,δ × GN,δ → R is the analogue of a discrete Lagrangian, β(i) are weighting constants, and Gi+1 ∈ Reach(Gi , V m ) at each step i. The Lagrangian lets us control the transient behavior of the system during the evolution of the graph process. As an example, let Gi = (Vi , Ei ) and deﬁne L(Gi , Gi+1 ) = |(Ei+1 \ Ei ) ∪ (Ei \ Ei+1 )|, where |.| gives the cardinality of a set. This Lagrangian is the symmetric difference of the sets of edges in the two graphs. Here, we penalize the addition or deletion of edges at each transition. The resulting connectivity graph process takes G0 to Gf with minimal structural changes. As another example, if we let L(Gi , Gi+1 ) = |Ei | − |Ei+1 |, then the desired Gf is generated while maximizing connectivity during the graph process.

4 Applications of Connectivity Graph Processes We now give some concrete applications of the connectivity graph processes. 4.1 Production of Low-Complexity Formations In [ME04b], we have presented a complexity measure for studying the structural complexity of robot formations. The structural complexity is based on the number of local interactions in the system due to perception and communication. When designing control strategies for distributed, multi-agent systems, it is vitally important that the number of prescribed local interactions is managed in a scalable manner. In other words, it should be possible to add new robots to the system without causing a signiﬁcant increase in the communication and computational burdens of the individual robots. On the other hand, an additional requirement when designing multi-agent coordination strategies should be that enough local interactions are present in order to ensure the proper execution of the task at hand. It turns out that the notion of structural complexity is the right measure to compare these conﬂicting requirements for the system. Therefore, it would be desirable to obtain graph processes that transform a formation with a high structural complexity to one with a lower complexity (and vice versa.) In [ME04b] we deﬁned the structural complexity of a connectivity graphs G as   deg(v ) j  deg(vi ) + C(G) = , d(vi , vj ) vi ∈V

vj ∈V,vi =vj

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where d : V × V → R+ is some distance function deﬁned between vertices. It was also observed that if G is a connected connectivity graph then the complexity of G is bounded above and below by C(δN ) ≤ C(G) ≤ C(KN ), where δN is the δ-chain on N vertices, and KN is the complete graph. The lowest complexity graphs or δ-chains (which are the line graphs or the Hamiltonian paths on all vertices), are important for formations that require minimal coordination. Therefore coming up with a planning mechanism to produce low-complexity formations from an arbitrary initial formation is a useful result in multi-agent coordination. Using the concepts from previous sections we deﬁne Ψ (G) = C(G) and L(Gi , Gi+1 ) = |(Ei+1 \ Ei ) ∪ (Ei \ Ei+1 )|, where Gi = (Vi , Ei ). In this way we tend to produce formations that are lower in complexity by generating a graph process that adds or deletes a small number of edges at each transition. The results of one such simulation has been shown in Figure 3. Here, the star-like graph in the upper left corner is the initial connectivity graph (with a higher structural complexity than a δ-chain on 5 vertices). The graph process evolves from left to right and then continues onto the lower rows in the same manner until it reaches the δ-chain in the lower right corner. The process involves transitions to various intermediate graphs. In this example, the mobile node is labelled as 1. As predicted by the CAD decomposition, it ﬁrst slides up to make an edge with node 2 and then rotates about node 2. It then passes by node 5 making various intermediate graphs, till it comes in the vicinity of node 4. Finally it makes a rotation about node 5 to form the δ-chain. Also note that due to the choice of the Lagrangian described above, the mobile node makes (or breaks) only a minimum number of edges at each graph transition. In this example, it can be seen that this number is always 1. Simulations for a relatively large number of nodes and more complex formations have also been done. It should be noted that the optimal trajectories thus obtained are only locally minimizing. A scheme for globally optimal behavior is currently under study. We are also working towards extending the number of mobile nodes by a decomposition of CAD into non-overlapping regions for each mobile node. 4.2 Collaborative Beamforming in Sensor Networks Another promising application of the framework presented in Section 3 is collaborative beamforming. In ad-hoc and wireless sensor networks, long range communication between clusters is always expensive due to limitations on power and communication channels. Collaborative beamforming is one way to solve this problem without dramatically increasing the complexity compared to non-collaborative strategies [OMPT, AC00]. With this method, a cluster of sensors synchronize their phases and collaboratively transmit or receive data in a distributed manner. By properly

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2 1

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designing the array factor, one can shape and steer the beam pattern in such a way that the array has either a high power concentration in the desired direction with little leakage (when transmitting), or a high gain in the direction of arrival (DOA) of the signal of interest with signiﬁcant attenuation in the direction of interference (when receiving). These properties enables Space-Division Multiplex Access (SDMA) among clusters. In most beamforming applications, the array geometry is assumed to be ﬁxed and the optimal beam pattern is formed by optimally weighing the signals received at individual nodes [VB88]. In this work, we further optimize the beam pattern by altering the geometry of the sensor array using connectivity graph processes.

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It should be mentioned that ﬁnding an optimal geometry is a diﬃcult design problem in array signal processing. Most designs favor a regular equispaced geometry such as linear, circular, spherical and rectangular grid arrays over random geometries[VB88]. If one follows the design philosophy in the array processing community, one would tend to drive all nodes to a regular geometry for obtaining better beam patterns. However, more exotic geometries have also been designed for particular applications. Moreover, the placement of nodes in a sensor network is not merely to optimize network communication, but also to maximize some beneﬁt associated with distributed sensing. Therefore, it seems beneﬁcial to optimize the geometries over some cumulative function of both the communication performance as well as the sensing performance of the network. We present this approach below. Let us ﬁrst study the beamforming for an arbitrary geometry. Following the standard notation in the array signal processing literature, we describe the positions of the individual nodes, signal and interference in polar coordinate system. The signal is located at a distance A and azimuthal angle φ0 while the interference is at an angle φi as shown in Fig. 4. The position xk = (xk , yk ) of the k-th node is given by (rk , θk ), where rk = x2k + yk2 and θk = tan−1 (yk /xk ). Given the position of the sensor array r = [r1 , r2 , . . . , rN ], θ = [θ1 , θ2 , . . . , θN ], we adopt the beamforming algorithm presented in [OMPT], where the gain in direction φ is given by the norm of the array factor θ F (φ|r, θ) = ej

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For known signal and interference directions, the objective of beamforming is to obtain high signal to interference ratio (SIR) and ﬁne resolution, i.e. we would like to keep the main lobe of the beam pattern as thin as possible while minimizing the power in the interference direction. Let ΔφH be the half power

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beam width (HPBW) of the main lobe as depicted in Figure 5. The power concentrations (accumulated gain) in the direction of signal and interference are respectively given by Ps (r, θ) =

φ0 +ΔφH /2 φ0 −ΔφH /2

|F (φ|r, θ)|2 dφ, Pi (r, θ) =

φi +ΔφH /2 φi −ΔφH /2

|F (φ|r, θ)|2 dφ.

We choose the following metric to evaluate the performance of a sensor array geometry. Ps (r, θ) = J (x), J (r, θ) Pi (r, θ)ΔφH where x is the sensor array geometry in cartesian coordinates. We wish to maximize the value of this metric over various geometries. To elaborate this point further, we give some example geometries and their respective beam patterns in Figure 7. Here, we assume that the signal direction φ0 is 0 degrees and the interference is coming at an azimuthal angle φi = 90 degrees. For comparison, the values of the metric have also been given on top of the beam patterns. Note that the linear array has a narrower beam but a large leakage in the interference direction. Similarly, the circular geometry in the bottom has low interference but a fat beam (i.e a large ΔφH ) in the direction of signal. The irregular patterns in the middle have a higher beneﬁt, although they lie in between the two extremes of beam width and interference nulliﬁcation. Moreover, as described in the above paragraphs, the metric to extremize may not be a function of the beamforming performance alone. Therefore it is reasonable to search over all possible geometries, rather than driving all nodes to a pre-determined regular geometry. Note that this metric may be diﬀerent for diﬀerent realizations of a particular connectivity graph. We make use of the cylindrical algebraic decomposition (CAD) algorithm for computing reachability to get a representative geometry r, θ for the connectivity graph G. 2 In this way we let J (x) = J (G), where ΦN (x) = G. If J (G) is some other performance metric associated with the function of the sensor network, then using the notation in Section 3, Ψ (G) = ν1 J (G) + ν2 J (G). Similarly we chose L(Gi , Gi+1 ) according to a desired transient behavior in the network. We give a snapshot from one such simulation for a particular choice of cost metric in Figure 7. Here, we have purposely chosen a small number of nodes and a relatively less relative displacement to demonstrate the value of the graph processes. The signal and interference directions are the same as in Figure 6. As the graph process evolves, notice the thinning of the main lobe in the signal direction. More pronounced is the attenuation in the interference direction, thus increasing the value of the metric at each transition.

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4.3 Other Applications In principle, the framework developed in Section 3 can be used for any application that required optimization over connectivity graphs. In a recent work [GM05a, dSGM05], it has been shown that the geometrical problem of determining coverage loss in a sensor network can be studied by looking at certain topological invariants of the topological spaces induced from the connectivity graph of the sensor network. This characterization associates a number with every connectivity graph that measures the number of coverage holes in the network. One can therefore use the connectivity graph processes to reduce the coverage holes in the network by setting up the appropriate terminal cost and

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the Lagrangian. A full investigation of this application (and many others) is a subject of current research.

Conclusions We have presented a generic framework for connectivity graph processes. The concepts of feasibility and reachability are useful for obtaining optimal trajectories on the space of connectivity graphs. These graphical abstractions are computable using the techniques of semi-deﬁnite programming and CAD, as demonstrated by various simulation results. This framework can be used for a wide range of applications. In particular, the problems of producing low complexity formations and collaborative beamforming has be studied by using connectivity graph processes.

Acknowledgments The authors wish to thank Mehran Mesbahi for discussing dynamic graphs and graph controllability concepts. This work was supported by the U.S. Army Research Oﬃce through the grant # 99838.

Plenary Talk Network-Embedded Sensing and Control: Applications and Application Requirements Tariq Samad Honeywell Labs Minneapolis, MN [email protected]

The ﬁrst part of my talk will cover some recent commercially successful applications of networked and embedded sensing and control solutions. These include applications in the process industries, home health care, and building management systems. In these products and solutions, technological advances in sensors, wireless, networks, and knowledge services have enabled new ways of solving outstanding societal and industry problems. Yet in many ways the current state-of-the-practice has just scratched the surface of technological possibility. Realizing the visions that many of us harbor for distributed sensor networks, however, will require mapping technical beneﬁts to economic value; the second part of the presentation discusses some of the complexities that arise in this process. Finally, I will present some recent research results in networked sensing for a military application–urban surveillance with networked UAVs. This research is still just that, but by discussing it in the context of key application requirements I hope to illustrate that an intimate connection between the two can be achieved.

Tariq Samad is a Corporate Fellow with Honeywell Automation and Control Solutions. He has been with various R&D organizations in Honeywell for 19 years, during which time he has been involved in a number of projects that have explored applications of intelligent systems and intelligent control technologies to domains such as autonomous aircraft, building and facility management, power system security, and process monitoring and control. He was the editor-in-chief of IEEE Control Systems Magazine from 1998 to 2003 and currently serves on the editorial boards of Control Engineering Practice and Neural Processing Letters. He has published about 100 articles in books, journals, and conference proceedings and he holds 11 patents with others pending.

P.J. Antsaklis, P. Tabuada (Eds.): Netw. Emb. Sens. and Cntrl., LNCIS 331, pp. 85–86, 2006. © Springer-Verlag Berlin Heidelberg 2006

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Tariq Samad

Dr. Samad is a Fellow of the IEEE and the recipient of an IEEE Third Millennium Medal and of a Distinguished Member Award from the IEEE Control Systems Society. He received a B.S. degree in Engineering and Applied Science from Yale University and M.S. and Ph.D. degrees in Electrical and Computer Engineering from Carnegie Mellon University.

Simulation of Large-Scale Networked Control Systems Using GTSNetS ElMoustapha Ould-Ahmed-Vall, Bonnie S. Heck, and George F. Riley School of Electrical and Computer Engineering Georgia Institute of Technology Atlanta, GA 30332-0250 {eouldahm,bonnie.heck,riley}@ece.gatech.edu

This paper presents a simulation framework for distributed control using wireless sensor networks. The proposed framework is an extension of the Georgia Tech Sensor Network Simulator (GTSNetS). The new features include models for diﬀerent elements of a control loop: an actuator class, a plant class and a user deﬁned controller application class. These models are integrated with the existing models for various components of a sensor network, diﬀerent network protocols, energy models and collaboration strategies. We present diﬀerent architectures for sensor networks in distributed control, depending on whether the sensing, the controller or the actuator elements of the control loop are distributed or centralized. Our tool allows the simultaneous study of the eﬀects of control and network strategies in a distributed control system. The simulation tool is demonstrated on a fault tolerance scheme. An existing distributed fault tolerance method that is applicable to sensor networks is expanded to allow nodes to learn about their failure probability and those of their neighboring nodes. Such an approach can be used to give diﬀerent weights to the information supplied by diﬀerent nodes depending on their conﬁdence level. The resulting fault tolerance scheme is implemented in a hierarchical networked control system and evaluated using the new simulation tool.

1 Introduction A wireless sensor network consists of a set of nodes powered by batteries and collaborating to perform sensing tasks in a given environment. It may contain one or more sink nodes (base stations) to collect sensed data and relay it to a central processing and storage system. A sensor node can be divided into three main functional units: a sensing unit, a communication unit and a computing unit.

P.J. Antsaklis, P. Tabuada (Eds.): Netw. Emb. Sens. and Cntrl., LNCIS 331, pp. 87–106, 2006. © Springer-Verlag Berlin Heidelberg 2006

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A networked control system consists of a set of control nodes having sensing, control and actuation capabilities and interacting using an overlapping network [BLP03]. In such a system, any of the three main control loop tasks of sensing, control and actuation can be performed in a distributed manner. The use of wireless sensor networks for distributed control oﬀers several beneﬁts. It allows cost reduction and eliminates the need for wiring. Wiring could become costly and diﬃcult in the case of a large number of control nodes needed for the sensing and control of a large process. However, the use of wireless sensor networks in a control system introduces new challenges. In fact, these networks can suﬀer unbounded delays and packet losses. There are several sensor network simulators available and widely used by the research community. There are also few simulation frameworks for networked control systems that have been proposed recently. However, we have not seen any simulator that combines the features of large-scale sensor networks with control applications. The co-simulation of control and network aspects of a networked control system allows the simultaneous study of the eﬀects of control issues and network characteristics on the system performance. The control design issues include stability, adaptability and fault tolerance. Network characteristics aﬀecting the control system performance include bandwidth, reliability and scalability. In this paper, we extend the Georgia Tech Sensor Network Simulator (GTSNetS) [OAVRHR05] to simulate distributed control using sensor networks. Several approaches for using sensor networks for distributed control are presented along with their implementations. One of the main features of our solution is its scalability. In fact, GTSNetS was shown to simulate networks of up to few hundred thousand nodes. The simulator is implemented in a modular way and the user is allowed to choose from diﬀerent architectural implementations. If a speciﬁc approach or algorithm is not available, the user can easily implement it by extending the simulator. This simulator is demonstrated using an existing Bayesian fault tolerance algorithm. The second major contribution of this paper consists of the extension of the Bayesian fault tolerance algorithm to adapt to dynamic failure rates. The enhanced algorithm allows nodes to learn dynamically about the operational conditions of their neighbors. Each node then gives diﬀerent weight factors (conﬁdence levels) to the information received from each of its neighbors. The weight factors are function of the failure probability of the speciﬁc neighbor. The probability of failure is computed from the reliability of the information received from the speciﬁc neighbor compared to other neighbors. The simulator facilitates the implementation and evaluation of the algorithm even for a network containing a large number of nodes. The remainder of the paper is as follows. Section 2 presents some of the related work. Section 3 gives an overview of GTSNetS. Section 4 presents the extension to GTSNetS to allow the simulation of networked control systems. Section 5 presents and compares the original and the enhanced fault tolerance algorithms. Section 6 simulates a distributed control scenario in a

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sensor network context with fault tolerance capabilities. Section 7 concludes the paper.

2 Related Work 2.1 Sensor Network Simulation Several sensor network simulators have been developed in recent years. Some of these simulators focus more on the network aspect, while others are more application speciﬁc. SensorSim [PSS00] was built upon the widely used ns2 [MF97] simulator. It provides models for diﬀerent parts of sensor network architecture: battery, sensors, radio, CPU, etc. It also models the power consumption of these components. A major feature of SensorSim is its ability to support hybrid simulation: integration between the simulator nodes and real sensor nodes collecting real data and running real applications. The SWAN [LPN+ 01] simulator is shown to be capable of simulating sensor networks of few tens of thousands of nodes. However, SWAN focuses more on the ad-hoc network aspect of sensor networks and less on the sensing function. TOSSF [PN02], developed by L. F. Perrone and D. M. Nicol of ISTS, is an adaptation of SWAN intended to simulate the execution of TinyOS applications at the source-level. It relaxes a major restriction of TOSSIM [LLWC03], the other TinyOS applications simulator. In fact, TOSSIM requires that all nodes within the simulated network run the same set of applications. However, these two simulators cannot be used to validate new applications for general use on sensor networks running any platform, since they are speciﬁcally tailored for TinyOS. In addition, these two simulators focus on the application behavior and do not necessarily capture all the aspects of a sensor network. SENS [SKA04] is platform-independent, unlike TOSSF and TOSSIM. A main feature of SENS is its detailed environment model. However, SENS lacks realistic energy models in that it assumes constant power consumption values in each operational mode. 2.2 Networked Control Simulation The simulation of networked control systems has received a growing amount of interest lately. This section discusses the two main simulation frameworks for networked control systems that have been proposed. These are: TrueTime, a Matlab-based simulation framework and an Agent/Plant extension to ns-2. TrueTime [HCA02] is based on Matlab/Simulink and allows the simulation of the temporal behavior of multi-tasking real-time kernels containing controller tasks. It proposes two event-driven Simulink blocks: a computer block and a network block. The computer block is used to simulate computer control activities, including task execution and scheduling for user-deﬁned threads and interrupt handlers. The network block is used to simulate the

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dynamics of a computer network using parameters such as message structure and message prioritizing function. However, this network block is not general enough to simulate various types of networks, especially sensor networks. It also suﬀers scalability problems. The widely used ns-2 was extended to allow simulation of the transmissions of plants and controllers in a networked control systems [BLP03]. The authors added a plant and an agent classes to ns-2. To the best of our knowledge, this solution is not yet interfaced with any of the ns-2-based sensor network simulators [PSS00]. In addition, it does not allow the simulation of large networked control systems.

3 GTSNetS Overview GTSNetS was built as an extension of GTNetS as a simulation framework for sensor networks. It inherits and beneﬁts from the basic design of GTNetS as discussed in [Ril03a, Ril03b]. By leveraging much of the existing capabilities of GTNetS, we were able to implement GTSNetS in a modular and eﬃcient manner, leading to the capability to simulate large-scale wireless sensor networks. GTSNetS was designed in such a way that it would not impose any architectural or design decisions on the user who wants to simulate a particular sensor network. We did not want GTSNetS to be a “do it yourself” type of simulator. Rather, the user can choose from various implemented alternatives: diﬀerent energy models, network protocols, applications and tracing options. Several diﬀerent methods for each of these choices are included in our baseline implementation. Should our models not be suﬃcient for a particular application, the existing models can easily be extended or replaced by the user. We consider a sensor node as composed of three main functional units: a sensing unit, a communication unit and a computing unit. A battery provides energy for these three units. In addition to regular sensor nodes, a sensor network can contain one or more sink nodes (base stations). These sink nodes interact with sensor nodes to collect sensed data and serve as a relay to the outside world. A sink node has a similar architecture to that of a regular node. The main diﬀerence is that it does not have a sensing unit. Because of the importance of lifetime in sensor networks, GTSNetS provides the capability to track the overall lifetime of a simulated network. It also measures the energy consumption of each one of the functional units and provides detailed statistics for each, allowing the user to study the effects of diﬀerent architectural choices on lifetime and energy consumption. Each one of the three units consuming energy (sensing unit, communication unit and computing unit) has several energy models to characterize its energy consumption. The implemented energy models vary in terms of complexity, accuracy and level of details. For example, for the sensing unit, three energy models have been implemented. The ﬁrst model is very simple and assumes a

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linear relationship between the consumed sensing energy and the size in bits of sensed data. The second sensing energy model takes into account the sensing range (in meters) in addition to the size of the sensed data to compute the sensing energy. The third model oﬀers greater details by dividing the sensing energy into two categories: the energy consumed by the ampliﬁers and the energy consumed by the analog to digital conversion process. This detailed model allows to better study the trade-oﬀ between energy and performance (e.g., accuracy of sensed data). Similar energy models are implemented for the computing and communication units. The user can choose from these models or add other energy models when necessary. The detailed description of GTSNetS implementation can be found in [OAVRHR05]. For the communication units, several sensor network speciﬁc network protocols are implemented. For example, as routing protocols we have implemented the directed diﬀusion and several geographical routing protocols. At the MAC layer, 802.11 and Bluetooth are also available. For more details, please refer to [OAVRHR05]. Other protocols for wireless ad hoc networks and wired networks, DSR, AODV, IP and TCP, are inherited from GTNetS [Ril03a, Ril03b]. The features of GTSNetS compared to other sensor network simulators, include: • A uniﬁed framework of existing energy models for the diﬀerent components of a sensor node. This allows the user to choose the energy model that best suits his needs • GTSNetS provides three models of accuracy of sensed data and allows for addition of new models. Modeling the accuracy of sensed data helps understanding the trade-oﬀ of quality versus lifetime • This simulator allows the user to choose among diﬀerent implemented alternatives: diﬀerent network protocols, diﬀerent types of applications, diﬀerent sensors, diﬀerent energy and accuracy models. New models, if needed, can be easily added. This makes GTSNetS very suitable for simulating sensor networks since such networks are application-dependent and their diversity cannot be represented in a single model • GTSNetS is currently, to the best of our knowledge, the most scalable simulator speciﬁcally designed for sensor networks. It can simulate networks of up to several hundred thousand nodes. The scalability is the result of active memory management and careful programming • Finally, GTSNetS can be used to collect detailed statistics about a speciﬁc sensor network at the functional unit level, the node level as well as at the network level

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4 Extension of GTSNetS to Simulate Distributed Control Using Sensor Networks Control is an important application area of networked embedded systems and sensor networks in particular. In fact, distributed control has been a standard for large-scale processes, such as industrial automation and mobile robotics. From a sensor network point of view, a sensor network can be used in several ways in a control system. 1. Distributed sensing with centralized control and actuation: This approach consists of using an entire sensor network as the sensing entity in a control system. This allows monitoring a large plant that cannot be covered using a single sensor. It also allows for fault tolerance since sensor nodes can be deployed to have several nodes covering each part of the plant. Information collected by the diﬀerent sensor nodes is fused and given as an input to the controller, which runs on a supervisor node outside of the sensor network. The controller generates a signal (action) that is applied to the plant by the actuator without involving the network. Figure 1 illustrates this approach. The arrows in the ﬁgure indicate the information ﬂow.

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Fig. 1. Distributed sensing with centralized control and actuation

2. Distributed sensing and actuation with centralized control: This approach is similar to the previous one. The main diﬀerence is that any corrective action on the plant is now applied by individual sensor nodes after receiving control commands from the controller. The controller is still run centrally at the supervisor node. Actuation messages are addressed either to all nodes in the network or to speciﬁc nodes as a function of their sensor readings. This could be the case, for example, when nodes in a speciﬁc region are reporting exceptionally high values. Figure 2 illustrates this approach.

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3. Distributed sensing, control and actuation: In this approach, the sensing, the control and the actuation are all performed inside the network. In this case, each control system acts as a sensor node. Each node can collect information about the plant, run control algorithms (act as a controller) and apply any necessary actions (actuator role). These nodes collaborate to control an entire system. However, each node monitors and actuates a speciﬁc plant that is a part of a larger system. In this case, each sensor node contains a controller, an actuator and a plant in addition to its normal components. In this approach, the sink node (base station) does not participate in the control process. It plays its traditional roles of collecting information, storing it and relaying it to the outside world when necessary. Figure 3 illustrates this approach.

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4. Hierarchical distributed sensing, control and actuation: This approach is similar to the previous one, except that a supervisor node can now aﬀect the parameters of the control algorithm executed on individual sensor nodes. It can, for example, change the reference value at individual nodes or load a new control algorithm depending on the current state of the plant as reported by the individual nodes. It can also change the parameters to adjust to changing network conditions. For example, if the energy level becomes low at some of the nodes, these nodes can be asked to sample at a lower rate. This approach is simulated later in this paper. Figure 4 illustrates this approach.

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Fig. 4. Hierarchical distributed sensing, control and actuation

To implement these diﬀerent control architectures, several classes were added to GTSNetS. Due the modularity of GTSNetS, the addition of these classes does not have any negative impact on the existing classes. It is facilitated by the possibility of code reuse in GTSNetS. The ﬁrst class (class Plant) models a plant. This class is derived from the real sensed object class that is already in GTSNetS. The main diﬀerence is that the plant can receive a command (signal) from an actuator. A second class implements the actuator (class Actuator). An actuator has a plant object associated with it. It can act on the plant in several manners depending on the commands received from the controller. Several methods of actuations are implemented. The user can choose from these methods or implement his own if necessary. The controller is modeled using an application class (class ApplicationSNController) derived from the sensor network application class. Each controller has a speciﬁc actuator object attached to it. This actuator object performs actions on the plant depending on the control commands received from the controller application.

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The diﬀerent control architectures are implemented by attaching some or all of these classes to individual nodes. To implement the ﬁrst architecture (distributed sensing with centralized control and actuation), we attach a plant object to each sensor node. However, since these nodes do not perform the controller and actuator roles, they each have a regular sensor network application. No actuator object is attached to the sensor nodes. A controller application is attached to the supervisor node. An actuator object is attached to this controller. This actuator acts on the plant object, which is sensed by the sensor nodes. In a similar way, the second architecture is implemented by attaching a controller application to the supervisor node. However, this controller application gives commands to actuators, which are attached to individual sensor nodes. Each actuator acts on the plant object attached to its sensor node. In the third architecture, a controller application is attached to each sensor node. This application commands an actuator that acts on a plant, which is attached to the same node. The sink node does not have any role in a fully distributed control application such as this case. The fourth architecture is implemented in a similar way to the third one, except that a controller application is now attached to the supervisor node. This application can modify the control algorithms or other parameters at the individual sensor nodes depending on the information they supply.

5 Fault Tolerance Algorithms A key requirement for the development of large-scale sensor networks is the availability of low cost sensor nodes operating on limited energy supplies. Such low cost nodes are expected to have relatively high failure rates. It is, therefore, important to develop fault tolerant mechanisms that can detect faulty nodes and take appropriate actions. A possible solution is to provide high redundancy to replace faulty nodes in a timely manner. However, the cost sensitivity and energy limitation of sensor networks make such an approach unsuitable [KPSV02]. Here, we focus on the problem of event region detection. Nodes are tasked to detect when a speciﬁc event is present within their sensing range. Such an event could be detected through the presence of a high concentration of a chemical substance [KI04]. Each node ﬁrst determines if its sensor reading indicates the presence of an event before sending this information to its neighbors or to a sink node. However, in case of failure the sensor can produce a false positive or a false negative. That is, a high reading indicating an event occurred when it did not or a low reading indicating the absence of event when one occurred. A distributed solution proposed by Krishnamachari et al. in [KI04] is presented in the subsection below. The simulations reported in [KI04] are

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duplicated in our simulator, which gives conﬁdence to the accuracy of our simulator. The algorithm in [KI04] assumes that all nodes in the network have the same failure rate and that this rate is known prior to the deployment. These are unrealistic assumptions, and we demonstrate through simulation that the algorithm introduces too many errors when such assumptions do not hold. We propose an enhanced version of the algorithm where nodes learn their failure rates as they operate by comparing their detection results with the ones of their neighbors. This mechanism is proved to reduce signiﬁcantly the number of introduced errors. 5.1 Distributed Bayesian Algorithm This solution presented in [KI04] considers a sensor reading of a high value as an indication of the presence of an event, while a low value is considered normal. It relies on the correlation between the node reading and the readings of its neighbors to detect faults and take them into account. The following binary variables are used to indicate if a node is in an event region (value 1) or in a normal region value (value 0): • Ti : indicates the real situation at the node (in an event region or not) • Si : indicates the situation as obtained from the sensor reading. It could be wrong in the case of failure • Ri : gives a Bayesian estimate of the real value of Ti using the Si values of the node and its neighbors It is assumed that all the nodes have the same uncorrelated and symmetric probability of failure, p: P (Si = 0|Ti = 1) = P (Si = 1|Ti = 0) = p

(1)

The binary values Si are obtained by placing a threshold on the reading of the sensor. The sensor reading when in an event region and when in a normal region are considered to have means of mf and mn , successively. The error term is modeled as a Gaussian distribution with mean 0 and standard deviation σ. In such a case, p is computed as follows using the tail probability of a Gaussian distribution: p = Q(

mf − mn ) 2σ

(2)

Assume each node has N neighbors. Deﬁne the evidence E(a, k) as the event that k of the N neighboring nodes report the same conclusion Si = a. Using the spatial correlation, we have: P (Ri = a|Ei (a, k)) =

k . N

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Each node can now estimate the value of Ri given the value of Si = a and Ei (a, k). This is given by: Paak = P (Ri = a|Si = a, Ei (a, k)) =

(1 − p)k (1 − p)k + p(N − k)

P (Ri = a|Si = a, Ei (a, k)) = 1 − Paak =

p(N − k) (1 − p)k + p(N − k)

(4) (5)

Three decision schemes are proposed: • Randomized: determine the values of Si , k and Paak ; generate a random number uε(0, 1); if u ≤ Paak , then set Ri = Si , else set Ri = Si • Threshold: a threshold θ is ﬁxed in advance; determine the values of Si , k and Paak ; if θ ≤ Paak , then set Ri = Si , else set Ri = Si • Optimal threshold: determine the values of Si , k; if k ≥ N2 , then set Ri = Si , else set Ri = Si It has been proved in [KI04] that the optimal value of θ in the threshold scheme is 1−p, which is equivalent to using the optimal threshold scheme. We, therefore, study only the randomized and the optimal threshold schemes. Several metrics have been developed to evaluate the performance of this Bayesian solution under diﬀerent settings. These metrics include: • Number of errors corrected: number of original sensor errors detected and corrected by the algorithm • Number of errors uncorrected: number of original sensor errors undetected and uncorrected by the algorithm • Reduction in errors: overall reduction in number of errors, taking into account the original errors and the ones introduced by the algorithm • Number of errors introduced by the solution: number of new errors introduced by the algorithm A full description of these metrics as well as their theoretical values can be found in [KI04]. 5.2 Simulation of the Bayesian Solution Using GTSNetS To simulate this particular solution, we only need to add one class (class ApplicationSNFToler). The modularity of GTSNetS allowed us to reuse without any change all the other modules: computing unit, communication unit, sensing unit containing a chemical sensor with the appropriate accuracy model and sensed object. We simulated a sensor network of 1024 nodes deployed in a region of 680 meters by 680 meters. Each two neighboring nodes are separated by 20 meters. The communication range was set to 23 meters. A parameter measuring the fault tolerance range was deﬁned in the ApplicationSN F T olerance. Nodes within this range are considered neighbors and taken into account by the fault

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tolerance algorithm to produce the prediction value Ri . The deﬁnition of this parameter disassociates the neighborhood region in which neighbors are taken into account by the algorithm from the communication range. This is a more realistic approach than the one used in [KI04]. This parameter was set such that each interior node has 4 neighbors taken into account by the algorithm. One source (sensed object) was placed at the lower left corner of the region of interest. The sensing range was set to 93 meters. These numeric values are only diﬀerent from the ones in [KI04] by a scaling factor. To reduce the size of the exchanged messages, it was decided to run an initial neighbors discovery phase prior to the execution of the fault tolerance algorithm. This avoids having to send node location along with every sensor reading message, which greatly reduces the sensor message size. This helps reducing the energy cost of the algorithm, but requires the existence of an identiﬁcation mechanism [OAVBRH05, OAVBHR05]. For the communication, every node communicates only with nodes in its fault tolerance range. In this speciﬁc simulation, this range is less than the communication range. Nodes, therefore, broadcast their messages to the neighboring nodes and there is no need for any routing protocol. When necessary, the user can use one of our implemented sensor network routing protocols, e.g., geographical routing or directed diﬀusion. Figures 5 and 6 give some of the performance metrics results for the optimal threshold and randomized schemes for various fault rates. This results were obtained by averaging over 1000 runs. We can see that both decision schemes correct a high percentage of original errors (about 90% for the optimal threshold and 75% for the randomized scheme at 10% failure rate). These graphs are in accordance with the ones reported in the original paper [KI04], which increases our conﬁdence in the correctness of GTSNetS. The simulations in [KI04] where conducted using Matlab and did not take into account many of the communication aspects and energy constraints present in sensor networks. Clearly, the optimal threshold scheme performs better than the randomized scheme. This is expected and is in accordance with the ﬁndings in [KI04]. However, the randomized scheme has the advantage of giving a level of conﬁdence in its decision to set Ri = Si or not. This conﬁdence level is given by Paak . This is not possible in the case of the optimal threshold scheme. 5.3 Enhancement of the Bayesian Fault Tolerance Algorithm The Bayesian algorithm assumes that all nodes in the network have the same failure probability p. It also assumes that this failure probability is known to the nodes prior to the deployment. These two assumptions are unrealistic. In fact, a node can become faulty with time either because of a lower energy level or because of changing environment conditions. This increases its fault rate. We can also have a heterogeneous sensor network with nodes that have diﬀerent operational capabilities and accuracy levels.

Simulation of Large-Scale Networked Control Systems Using GTSNetS Metrics for the original algorithm using the optimal threshold scheme 1

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For these reasons, we propose an enhancement to the algorithm under which nodes learn about their failure rates as they operate. This is done by comparing the sensor reading Si with the estimated value of Ri , obtained by taking into account the values reported by the neighbors, over time. We also give diﬀerent weights to information reported by nodes with diﬀerent fault rates. The reported information Sj from neighbor j is given a weight of 1 − pj , where pj is the current fault rate at node j sent along with its sensor reading. Several of the previous equations are modiﬁed for these dynamic failure rates. Suppose that k of the neighboring nodes obtained sensor decisions identical with the decision Si = a at the current node i. For simpliﬁcation of notation, assume that these nodes are numbered 1 to k and the rest of the neighbors are numbered k + 1 to N . We deﬁne Pea as the probability of the event Ri = a knowing that these ﬁrst k neighbors have each Sj = a. This k probability is constant and equal to N in the original algorithm. Its value is now given by: k (1 − pj ) (6) Pea = P (Ri = a|E(a, k)) = N1 1 (1 − pj ) Here pj is the current failure probability at the jth neighbor. Each node i keeps track of the current failure probability pj at each one of its neighbors, by looking at the number of times the reported value Sj by neighbor number j disagreed with the estimated value Ri . This avoids having to send this probability value along with every decision message, which saves the energy. The new value of Paak is given by: Paak = P (Ri = a|Si = a, E(a, k)) =

((1 − pi )Pea (1 − pi )Pea + pi (1 − Pea )

(7)

The parameter pi is the current failure probability of node i. The decision schemes are implemented as in the original algorithm using the new expression of Paak . Since the optimal threshold scheme does not use Paak to decide weather the node is faulty or not, we focus our attention on the randomization scheme. For each failure rate level p, as used in the original algorithm, we compute two adjacent probabilities to introduce variability as follows: p1 = p − 0.05

(8)

p2 = p + 0.05

(9)

This keeps the average failure rate unchanged at p in the overall network. Each sensor is assigned randomly one of these two probability levels. The node does not know in advance the failure probability of its sensor. It starts with a conservative estimate that is equal to the highest of the two probability levels (p2 ). A node i learns its real probability level as it operates by looking

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at the number of times its sensor decision Si disagreed with the estimate of the real situation Ri . This failure probability, pi , is update at every round. We compare the performance of the enhanced version of the algorithm with that of the original algorithm as well as with the original algorithm with prior knowledge of exact probability level p that is identical for all nodes. We obtain the performance by averaging over 1000 runs as in the previous section. We use the same network topology and conﬁguration parameters as in the previous section. Figure 7 gives the normalized number of corrected errors under the diﬀerent experimental settings. Clearly, we can see that the number of corrected errors remains relatively constant and does not suﬀer any degradation from the introduction of the assumptions of unknown and uneven fault rates. The only change is observed for very small probability levels (around 0.05), which is due to the ratio between p and the constant subtracted from (added to) it to produce p1 and p2 . Normalized number of corrected errors 0.85

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Figure 8 gives the normalized number of reduced errors under the diﬀerent experimental settings. We can see that the performance, as measured by the number of reduced errors, is worst for the original algorithm (for error rates of up to 15%) while the enhanced algorithm gives similar performance to the one for the case where the values of p are identical for all nodes and known prior to the deployment. The degradation in the performance of the original

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algorithm comes from the increased percentage of errors introduced by the algorithm. Normalized number of reduced errors 0.8

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Figure 9 gives the normalized number of errors introduced by the algorithm for diﬀerent p values. We can see that the performance, as observed before, beneﬁts from the introduction of the enhancement to the algorithm. The number of introduced errors is controlled and keeps close to the levels of the algorithm with prior knowledge of error rates that are even throughout the network. In conclusion of this section, we proved that prior knowledge of error rates is not necessary for the Bayesian fault tolerance algorithm to perform well and keep the level of errors under control. It is also clear that the assumption of even error rates is not necessary. Relaxing these two assumptions makes the algorithm much more realistic.

6 Hierarchical Distributed Control with Fault Tolerance In this section, we give an example of how our simulator can be used to study the behavior of a simple distributed hybrid control system. Each node is responsible for sensing, controlling and actuating a plant. A supervisor node is responsible for changing the parameters of the local controllers. For simplicity, we consider a single input single output proportional controller with gain K and measurement y. The plant is chosen as a ﬁrst order discrete

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time plant with pole at 1. A hybrid control is added by having the supervisor node modify the gain at a speciﬁc node if the local measurement y falls outside of a given Interval, [I1 , I2 . This is a simplistic case of hybrid control where the control is reconﬁgured depending on the operating region in which it lies. The node informs the supervisor node that it has changed the region by sending a message. The supervisor node sends back a message to update the K value at the speciﬁc node. A geographic routing scheme was used to communicate between the supervisor node and the sensor nodes. The location of the ﬁnal destination is contained in the message header. Each message is routed through the neighbor node that is closest to the ﬁnal destination among the neighbors that remain alive. Intermediate nodes route messages in a similar way. The enhanced fault tolerance algorithm, presented in Section 5, is incorporated in the control. A node that determines that it is faulty (by comparing its sensor reading with the readings of its neighbors) does not apply any control action and waits until the next reading. Each node is powered by a battery containing 2 joules at the beginning. Typical energy consumption parameters given in [BC02] were used for the energy models. Sensors collect data every 30 seconds. Each node broadcasts a heartbeat message every 100 seconds. These messages allow nodes to keep an updated list of neighbors that are still alive. This list is used in the geographical routing protocol. All nodes were, initially, conﬁgured with a fault

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probability p = 0.05. However, this probability is dynamically updated using the enhanced fault tolerance algorithm. The GTSNetS simulator can give results on the control performance at each node. In addition, it allows for an exploration of other network related parameters: communication protocols, energy consumption network-wide and at individual nodes, collaboration strategies, etc. Here, for example, we study the lifetime of the network as a function of the width of the operating range I = I2 − I1 for a network of 1024 nodes. The lifetime, here, is measured as the time from the starting of network operation till the network is considered dead. Here, the base station considers the network dead when more than ( 43 ) of its neighbors are dead. The death of a node is discovered when it no longer sends heartbeat messages. Figure 10 gives the network lifetime as a function of the tolerance interval for K = 0.5 initially. The lifetime increases with the width of the operating range. In fact, for larger operating ranges the sensor reading of individual nodes is less likely to fall outside of the range. This reduces the number of messages sent to the supervisor node, which allows its neighbors to live longer. Network lifetime as a function of width of the oprating range 850

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We also studied the percentage of sensor nodes remaining alive after the network death. In fact, since all nodes communicate with the supervisor node, the neighbors of this node are expected to die much sooner than most of other nodes. Figure 11 gives this percentage for an initial value of K = 0.5. We observe that this percentage is very high, as expected, and tends to increase with the width of the operating range. However, it remains relatively constant (be-

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tween 85 and 95% for most values) because the neighbors of the base station still die much sooner than the rest of the network even for large operating ranges. This problem could be overcome by increasing the network density near the supervisor node. Percent of ramaining nodes as a function of width of the operating range 95

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7 Conclusion We described the design and implementation of an extension of GTSNetS that allows the simulation of distributed control systems using sensor networks. This simulation tool can be used to study the combined eﬀects of network parameters and control strategies in a distributed control system. One of the main features of GTSNetS is its scalability to network size. In fact, it can be used to simulate networks of several hundred thousand nodes. Another important feature of GTSNetS is the fact that it is easily extensible by users to implement additional architectural designs and models. We also presented an extension of an existing Bayesian fault tolerance algorithm to adapt to dynamic fault rates. This extension relaxes two major assumptions of the original algorithm. Nodes are no longer assumed to know their failure rates prior to the deployment. A node estimates dynamically its failure rate by comparing its sensing values with those of its neighbors. It is also no longer assumed that all nodes have the same failure rate. Relaxing these two assumptions makes the algorithm much more realistic.

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Acknowledgments This work is supported in part by NSF under contract numbers ANI-9977544, ANI-0136969, ANI-0240477, ECS-0225417, CCR 0209179, and DARPA under contract number N66002-00-1-8934.

neclab: The Network Embedded Control Lab Nicholas Kottenstette and Panos J. Antsaklis Department of Electrical Engineering University of Notre Dame Notre Dame, IN 46556 {nkottens,antsaklis.1}@nd.edu Summary. The network embedded control lab, neclab, is a software environment designed to allow easy deployment of networked embedded control systems, in particular wireless networked embedded control systems (wnecs). A wnecs is a collection of interconnected plant sensors, digital controllers, and plant actuators which communicate with each other over wireless channels. In this paper neclab is introduced and explained using a simple ball and beam control application. We focus on wnecs which use the MICA2 Motes.

1 Introduction Typically, when a controls engineer needs to develop a new closed-loop control system she develops the control system in phases. The ﬁrst phase is to develop a mathematical model of the system and synthesize a controller. The second phase is to simulate the control system using tools such as MATLAB [mat]. In the third phase, using the results from the simulations, the engineer integrates sensors, actuators, remote data acquisition and control equipment into the system. This is done in order to acquire additional data and reﬁne the models in order to optimize the controller. When the third phase is complete, the engineer has optimized and deployed a robust control system. Systems with a higher degree of autonomy will also have fault detection and remote monitoring systems. Typically these digital control systems are developed using a dedicated data acquisition system attached to a cable interfaced to a computer running a real-time-control software, such as RTLinux [Yod]. For control systems in which a wired control system is not possible or desired, the available design tools for the engineer are limited at best. In this paper, a software environment is introduced called neclab, that is a software environment designed to allow easy deployment of networked embedded control systems, in particular wireless networked embedded control systems called wnecs. The components of neclab are presented in the following and described in terms of a classical control experiment, the ball and beam.

P.J. Antsaklis, P. Tabuada (Eds.): Netw. Emb. Sens. and Cntrl., LNCIS 331, pp. 107–125, 2006. © Springer-Verlag Berlin Heidelberg 2006

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Note that most of the tools currently available to aid the engineer develop software for wireless embedded systems are geared speciﬁcally for sensing. The majority uses Berkley’s TinyOS [Ber]. Note also that the majority of the TinyOS applications listed in [Ber], are not designed to be wirelessly reconﬁgurable. For example, one reconﬁgurable system which uses TinyOS is Harvard’s moteLab [WASW05], where each mote is connected to a dedicated programming board that is connected to an Ethernet cable. This is necessary in order for each mote to be reconﬁgured in order to use TinyOS. A reliable protocol, called Deluge, to enable wireless programming of TinyOS applications has been developed [HC04]. Deluge is currently part of the TinyOS development tree, and should be an integral part of the next stable release of TinyOS. We considered Deluge but in view of our sensor and control applications of interest we decided to work with an alternative to the TinyOS operating system called SOS [UCL]. SOS oﬀered an alternative working design for network reprogramming for the three following reasons. First the SOS operating system utilizes a basic kernel which should only have to be installed on the mote once. The second key element is that the SOS kernel supports small, typically one-twentieth the size of a TinyOS application, dynamically loadable modules over a network. Last, the SOS kernel supports a robust routing protocol, similar to MOAP [SHE03], to distribute modules over a wireless network. We built neclab, our networked embedded control system software environment using SOS. Speciﬁcally, neclab is a collection of software consisting of ﬁve main components. The ﬁrst component, build utilities, is a set of utilities designed to build and download all required software tools and libraries in order to use neclab. The second component, SOS, is an operating system developed by the Networked and Embedded Systems Lab (NESL) at UCLA. SOS is a highly modular operating system built around a message passing interface (MPI) which supports various processor architectures, including those on the MICA2 Motes. The third component, sos utilities, are the utilities to facilitate code development and deployment of SOS modules. The fourth component, necroot, is a ﬁle system structure and language designed to seamlessly interconnect individual motes for distributed control. The ﬁfth component, FreeMat utilities, are a set of utilities to facilitate wnecs design using FreeMat. FreeMat is a free interpreter similar to MATLAB but has two important advantages. First, FreeMat supplies a direct interface for C, C++, and FORTRAN code. Second, FreeMat has a built-in API for MPI similar to MatlabMPI [Kep]. neclab provides a mini-language built on facilities similar to those supported by UNIX. A modern UNIX OS supports facilities such as pipes, sockets and ﬁlters. neclab allows the engineer to develop a wnecs by designing control modules which can be interconnected using networking message pipes. A networking message pipe is an abstraction to pass arrays of structured binary data from one module to another over a network. A networking message type indicates how the data should be handled, for example, descriptors are used to

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indicate standard, error, routing, and control messages which are passed over a network; e.g. control messages are indicated by the control message type. Speciﬁcally, a networking message pipe is used to interconnect data ﬂows between networking sources, networking ﬁlters, and networking sinks. A networking source creates data which will be sent over a network to networking ﬁlters, and networking sinks. Similarly, a networking ﬁlter will receive data from either a networking source or another networking ﬁlter. The networking ﬁlter will proceed to optionally modify the data and send the new data to another networking ﬁlter or networking sink. A networking sink is where the network data ﬂow for a given route ends. In order to implement networking message pipes we will use the network message passing protocol provided by SOS. Like UNIX, SOS provides a way to run and halt programs which have a corresponding process id. These executable programs on SOS are known as modules. neclab provides an interface to pass networking conﬁguration messages to a module in order to conﬁgure and enable the network ﬂow of data between modules in the wnecs at run-time. Using these facilities we will demonstrate an implementation of a highly parallel wnecs in which a secondary controller is reconﬁgured, while the primary controller maintains a stable control-loop. Once reconﬁgured, the roles of the two controllers will be switched. Other, highlights will illustrate that a controls engineer can actually create concise routing tables by simply describing a wnecs with neclab. This is a natural result of wnecs in general. neclab’s use of SOS’s dynamic memory allocation services, easily allows for a system which enables a control engineer to work through the second and third phases of her design project. This highly conﬁgurable environment without wires would have been diﬃcult to implement with TinyOS since TinyOS does not support dynamic memory management. SOS on the other hand does. Other SOS features which neclab utilized are the ability to dynamically share functions and load executable ﬁles (modules) over a wireless channel while the SOS kernel is still running. SOS implies ﬂexibility, and as a result it was chosen as the core component to neclab. For a more detailed discussion on the advantages and diﬀerences of SOS as compared to other solutions, refer to [HKS+ 05]. neclab is not the ﬁrst project to utilize SOS. Other projects such as Yale’s XYZ Sensor Node project [LS05] and various projects at NESL are starting to use SOS such as the RAGOBOT [FLT+ 05]. In presenting neclab, we will illustrate its use by presenting a typical undergraduate control lab problem modiﬁed to be a wnecs. We will then generalize this problem, by describing a tutorial application, which a user can create if ﬁve MICA2 Motes and a programming board are available. As the tutorial application is described we will highlight the various components of neclab which highlight the many issues that have to be addressed in order to develop a robust wnecs.

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2 Problem Description Consider an undergraduate controls laboratory experiment that teaches a student how to control the position of a metal ball on a beam. The experiment uses a motor as an actuator, and two variable Wheatstone bridges for sensing. The bridges measure angular position of the motor, and the absolute position of the ball on the beam. In the ﬁrst laboratory experiment, the students are required to determine the actual model parameters of the ball and beam plant [Unib]. The next lab [Unia] teaches the student how to control the exact position of the ball on the beam. The student designs and implements the control system using MATLAB, Simulink [mat], and the Wincon server for real-time control [Qua]. The sensor inputs and outputs are accessible through the MultiQ board. We are going to replace this system using neclab, the MICA2 motes and a general purpose I/O boards developed for the MICAbot [MGM03]. Figure 1 illustrates such a system. With neclab installed on a host computer a MICA2 N gateway mote is typically accessed via a serial port. Figure 1 indicates that MICA2 N gateway is interconnected to a MIB510 programming board. See [Cro] for additional details on the MIB510. In order to control the ball and beam plant the following control loops (jobs) need to be implemented (spawned). First the MICA2 N actuator needs to reliably control the angular position α of the beam. This is achieved by controlling the angular position θl of the motor. The actuator will receive a desired angular position set-point and will control the motor. In networking terms, the MICA2 N actuator behaves as a networking sink for networking messages, and takes actions based on the messages sent to its control standard input. According to [Unia] the desired response time for controlling α should be around 0.5 seconds. This is a fairly aggressive target to meet for the low data rate wireless network control system. As a result, we have initially kept this control loop internal to the MICA2 N actuator. In order to do this we link this code statically to the kernel in order to guarantee a stable control loop on start-up. The second control loop involving the actual position of the ball, requires around a 4 second settling time, which is reasonable to implement over the wireless channel. This loop is accomplished by MICA2 N sensor sampling data from the ball position sensor output with the ATmega 128L built-in 10 bit A/D converter (see [Atm] for additional information on this chip’s features). The MICA2 N sensor behaves as a networking source for generating networking messages, sending its data along to MICA2 N controller-A and MICA2 N controller-B respectively. Depending on which controller is enabled, the enabled controller will behave as a networking ﬁlter by calculating an appropriate command to send to MICA2 N actuator based on the users desired set-point received. Figure 2, illustrates how this system can be implemented using SOS modules and messages which we will refer to as we discuss neclab.

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3 neclab Looking at ﬁgure 2, one can appreciate the number of distinct software components required for an engineer to obtain a working wnecs. In this ﬁgure the engineer has successfully built and installed kernels on ﬁve motes, loaded all the required modules on to the network, and created routing tables in order to create a stable closed loop controller to monitor and maintain the position of the ball. In order to use neclab the engineer must ﬁrst download the 1.x version of SOS to a unix based PC. The location SOS is installed will be referred to as SOSROOT; neclab will be in the SOSROOT/contrib/neclab directory which will be referred to as NECLABROOT. From there all the engineer needs to do is follow the instructions in the NECLABROOT/README SOS-1.X ﬁle. The ﬁrst task that neclab will do for the user is download, build and install all the necessary tools to build and install software on the MICA2 motes so to work with the FreeMat environment. Once the tools are installed, neclab will apply some minor patches in order to fully utilize the SOS software. From there the engineer should test and fully understand the example blink lab. The blink lab is discussed in Appendix A. 3.1 build utilities neclab has been designed so that a user does not require root access to build and install her tools. This oﬀers two distinct advantages, the ﬁrst being that the user can modify any appropriate component that is needed to maximize the performance of the system. For example, neclab actually downloads and allows the user to build an optimized BLAS (Basic Linear Algebra Subprograms) library using ATLAS (Automatically Tuned Linear Algebra Software) [WP05]. Second it provides all users with a consistent tool-kit eliminating potential software bugs associated with not using a consistent tool-chain. The key tool used is the build tool makeﬁles program which reads a conﬁguration ﬁle located in NECLABROOT/etc/build.conf and generates a custom makeﬁle for all the following tools: • • • • • • • • • • •

perl – key tool for various utilities in neclab [WOC00] avr-binutils – used for the MICA2 and MICAz motes [bin] avr-gcc – used for the MICA2 and MICAz motes [avrb] avr-libc – used for the MICA2 and MICAz motes [avra] ATLAS – used with FreeMat[WP05] FreeMat [fre] uisp – used to load an image on to the MICA2 and MICAz motes [uis] tcl – (Tool Command Language) required for the tk library [tcl] tk – graphical user interface toolkit library required for python [tcl] python – [pyt] used in conjunction with pexpect [pex] for automation SWIG – [swi] is a tool to connect programs written in C and C++ with high-level programming languages such as perl and python.

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The auto-generated makeﬁles are then installed in each corresponding NECLABROOT/src/build utilities/< tool > directory and invoked to download, build and install each < tool >. The installation directory is in NECLABROOT/tools/. As a result any user can build and use her own tools, without having to ask the system administrator for permission! 3.2 SOS, and sos utilities Referring back to Figure 2, on the local host PC (HOST PC), the engineer has just ﬁnished creating a wnecs using the neclab run tool provided by neclab. Each mote has a kernel; however, they do not have to be unique, as is clearly shown in Figure 2. For example, the MICA2 N gateway mote has the sosbase kernel which has a statically linked module which we will refer to by it’s process id SOSBASE PID. Other motes such as MICA2 N sensor, MICA2 N controller-A, and MICA2 N controller-B have a blank sos kernel with no statically linked modules. Finally the MICA2 N actuator has a custom kernel, custom sos, with statically linked modules ANG POS SEN PID (a module which manages the angular position sensor), and MOT ACT PID (a module which controls the angular position on the motor). The custom sos kernel was required to generate a pwm output in order to drive the H-Bridge on the MICAbot board. The remaining modules which are dynamically loaded and unloaded over the network, are either in an active or inactive state. When in an inactive state they do not consume any additional processor RAM but do take up program space in the ﬂash. In order to load and unload the modules the following programs are required. First, the sos server, sossrv, needs to be built, installed and started. neclab manages this with the makerules and neclab {sim,run} commands. The makerules ﬁrst manage building and installing sossrv into NECLABROOT/tools/bin. The neclab {sim,run} commands can be used either to begin a simulation of a wnecs with the neclab sim command or to run a wnecs using the neclab run command. Either command can be treated as equivalent for discussion. The neclab run command starts the sossrv and connects it to the MICA2 N gateway mote. Figure 2 indicates that sossrv is listening for clients on HOST PC via the loop-back interface on port 7915 while listening for incoming packets ﬁltered through the MICA2 N gateway attached to /dev/ttyS0. Next the neclab run tool creates an input ﬁfo and starts the SOS modd gw client. The modd gw client is an SOS application that runs natively on a PC, it provides a shell like interface in which user input can be redirected to the ﬁfo for automation. The modd gw client maintains a database ﬁle .mod version db local to where it is started. This database tracks the diﬀerent dynamic modules which are loaded and unloaded from the network. If another engineer chose to use the same motes in the lab, they will either need access to this ﬁle or re-install new kernels on all the motes. As a result neclab makes sure that the modd gw is started such that the database is located in a publicly accessible

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directory such as /tmp so others can access and run their own experiments. The neclab run tool then proceeds to build all the required network modules, load them on to the network, and establish the networking routes for the control system. Lastly, the closed loop control system is enabled and can be monitored and modiﬁed as necessary. Another tool neclab provides is the create module proto tool to generate a new module prototype for beginning development. neclab has built into its makerules a mechanisms to generate tags ﬁles to assist in tracking all the interrelationships that modules have with the kernel and other modules. Once the engineer has a satisfactory implementation she can use the appropriate options from the neclab run tool to easily rebuild and re-install new module images as necessary. These tools provide a stream-lined mechanism for simulating and generating wnecs. Building oﬀ of SOS’s novel technique of tracking module version numbers as they are dynamically loaded and unloaded. neclab has created a ﬁle-structure known as necroot to track, simulate, and develop wnecs. 3.3 necroot Modules are tracked by their process id, similar to a process id generated by the ps command on a unix machine. This id is actually used to direct messages which are passed on an SOS network. Every message passed on the SOS network has a corresponding destination process id and address. The process id ﬁeld; however, is only 8 bits, which supports only 255 unique process ids. Clearly more than 255 unique modules will be developed to be run on the SOS operating system, so there needs to be a clean way to address this limitation. SOS uses two ﬁles to deﬁne the association of a process id with a given module, mod pid.h and mod pid.c. The gen mod pid tool combined with makerules and the necroot design allow for dynamic generation of mod pid.h and mod pid.c for a corresponding lab project. In necroot the NECLABROOT/src/necroot/modules.conf provides a line by line list in which each entry consists of a full-path to a corresponding module and a short description of the module. Each item is delimited by a colon. The module process id is parsed directly from each modules.conf entry and added to mod pid.h. The corresponding description is then added to mod pid.c for simulation and debugging. Furthermore, static modules and modules to be loaded over the network are uniquely identiﬁed by grouping these modules between the < static >, < /static >, and < network >, < /network > tags respectively. The modules.conf is to serve as a global ﬁle, in it the key modules for neclab are listed. These entries include the moduled pc module (MOD D PC PID) and neclab’s conﬁguring module (CONFIGURING PID). The moduled pc module, is a statically linked module which runs on the modd gw client. The conﬁguring module provides an interface for creating networking sources, networking sinks and networking ﬁlters. It also provides the interface to conﬁgure modules and enable the control

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standard input/control standard output networking design referred to in the introduction. The remaining modules are identiﬁed in the engineer’s NECLABROOT/labs/ball beam lab/etc/modules.conf.local ﬁle. The next issue is to create a design which would allow a user to easily manage programming and tracking each corresponding mote’s kernel and module conﬁguration. This is solved by the construction of the NECLABROOT/src/necroot/etc/network.conf and the corresponding NECLABROOT/labs/ball beam lab/etc/network.conf.local ﬁles. Each entry follows nearly the same language structure as the build.conf ﬁle described in the build utilities section. The only diﬀerence instead of identifying a tool, each entry describes a mote and all the properties local to that mote. Using the build mote makeﬁles tool, a custom makeﬁle for each mote described in the network conﬁguration ﬁles is generated. Then the necroot directory structure is generated in the ball beam lab directory. Furthermore the engineer can use neclab run with the appropriate options to build and install the corresponding kernel on to her motes. For reference, a typical network.conf entry for a mote is as follows: <mote> SOS_GROUP = 13 ADDRESS = 2 NECLAB_PLATFORM = mica2 KERNEL_DIR = ${NECLABROOT}/src/patched_sos-1.x/sosbase X = 6 Y = -12 Z = 5 LOC_UNIT = UNIT_FEET TX_POWER = 255 #CHANNEL = ? has no effect for mica2

#$

The conﬁguration language is GNU Make combined with some XML-like tags to separate each mote entry. Although, not true for mobile motes, each nonmobile mote has a ﬁxed location. This information is typically used for geographic routing protocols, such as those implemented on Yale’s XYZ platform. SOS has built in the capability to track the location of each mote into its kernel; hence, the X, Y, Z, and LOC UNIT entries. The neclab project assisted in this design by introducing the Z dimension, a LOC UNIT to associate relative position with a given dimension, and a gps loc variable to track GPS location information. The gateway mote should typically assign itself a GPS location in order to assist with routing and interconnecting of other laboratories around the globe. Presently, gps loc maintains precision down to one second. A sample GPS entry, here at Notre Dame would have the following format (note the Z coordinate is an elevation relative to sea-level in feet). GPS_X_DIR = WEST GPS_X_DEG = 86

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GPS_X_MIN = 14 GPS_X_SEC = 20 GPS_Y_DIR = NORTH GPS_Y_DEG = 41 GPS_Y_MIN = 41 GPS_Y_SEC = 50 GPS_Z_UNIT = UNIT_FEET GPS_Z = 780 Note, each platform that SOS supports has a wireless radio setting unique to each mote. In the above sample the radio was set to full power. neclab is also working on formalizing the interface to modify the frequency and channel of the radio. As the comment notes, the MICA2 mote does not currently support the channel interface; however, the MICAz mote does. The other parameters such as the SOS GROUP id is used to ﬁlter radio packets out in the network that do not belong to that particular group. Each mote entry should have a unique, (SOS GROUP, ADDRESS) pair. Being able to group motes and designate diﬀerent channels, provides one way to allow multiple laboratories to interact and perform co-operative tasks. For example, the ball and beam lab, can be modiﬁed to simulate the transfer of one ball from one station to another, as might be done in transferring parts from one assembly cell to another. Furthermore, by building in the ability to group the motes and assign each group to a separate radio channel, we have created a mechanism to create groups which can co-operate without worrying about generating radio interference. This feature we plan to exploit with the MICAz motes when developing routing algorithms between groups. The routing will be implemented between the gateway motes of a group. The gateway mote will typically be attached to their respective HOST PC and the routing of packages will occur over the Internet. Finally, the issue of declaring the desired kernel is addressed with the KERNEL DIR entry. The KERNEL DIR entry provides the full path to the desired kernel to be loaded and built. Any modules that the engineer plans to statically link in her kernel should be addressed in her respective kernel makeﬁle. After the engineer has successfully built her newly created necroot ﬁle structure, the following root tree will result. [user@host_pc ball_beam_lab]$ find ./ -name "mote*" ./root/group13/mote1 ./root/group13/mote2 ./root/group13/mote3 ./root/group13/mote4 ./root/group13/mote5 ./root/group14/mote1 ./root/group14/mote2 ./root/group14/mote3

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./root/group14/mote4 ./root/group14/mote5 [user@host_pc ball_beam_lab]$ In each mote directory there is a module conﬁguration ﬁle, corresponding to every module identiﬁed in the corresponding modules.conf and modules.conf.local ﬁles. These are used in conjunction with the conﬁguring module to set module parameters and set up routes in the network. Looking in the mote1 directory for example the user will see the following additional ﬁles. [user@host_pc ball_beam_lab]$ ls -1 root/group13/mote1/*.mod root/group13/mote1/ball_beam_con.mod root/group13/mote1/ball_pos_sen.mod root/group13/mote1/configuring.mod root/group13/mote1/moduled_pc.mod root/group13/mote1/ang_pos_sen.mod root/group13/mote1/mot_act.mod [user@host_pc ball_beam_lab]$ These user editable ﬁles provide an interface in order to use the conﬁguring module to create routes. These routes can be conﬁgured using the command line or can optionally be declared in the routing.conf ﬁle. All lines starting with a # are comments. What the following segment of routing.conf should illustrate is that we have created a compact language to describe networking routes. The net source, net ﬁlter, and net sink commands are intended to emphasize that a module will be conﬁgured to be in one of those three states. The optional arguments such as –d{0,1} are to show that the engineer can describe up to two destinations and the behavior is determined by the respective –m{0,1} option which describes the networking message type for each destination. This allows an engineer to redirect control data ﬂow into error data ﬂow for monitoring for example. It should also be noted that although we are using the necroot ﬁle system to maintain and describe our routes, an ad-hoc routing module could be designed to utilize the conﬁguring module interface in order to build arbitrary networking routes based on some metric. This language can further be utilized to describe these routes using a graphical user interface. The ball beam lab’s etc/routing.conf ﬁle is as follows: #!/bin/sh #routing.conf #define some constants TIMER_REPEAT=0;TIMER_ONE_SHOT=1 SLOW_TIMER_REPEAT=2;SLOW_TIMER_ONE_SHOT=3 MSG_STANDARD_IO=35;MSG_ERROR_IO=36 MSG_CONTROL_IO=37;MSG_ROUTING_IO=38 cd $NECLABROOT/labs/ball_beam_lab/root/group13 #Main control-loop #MICA_2_N_sensor -> MICA_2_N_controller-A -> MICA_2_N_actuator

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#Secondary control-route #MICA_2_N_sensor -> MICA_2_N_controller-B #Configure the MICA_2_N_sensor (mote2/ball_pos_sen) routes net_source -m mote2/ball_pos_sen -t "$TIMER_REPEAT:1024" \ --d1=mote3/ball_beam_con --m1=$MSG_CONTROL_IO \ --d2=mote4/ball_beam_con --m2=$MSG_CONTROL_IO #Configure MICA_2_N_controller-A (mote3/ball_beam_con) routes net_filter -m mote3/ball_beam_con -t "$TIMER_REPEAT:2048" \ --d1=mote5/mot_act --m1=$MSG_CONTROL_IO \ --d2=mote1/moduled_pc --m2=$MSG_ERROR_IO #Configure MICA_2_N_controller-B (mote4/ball_beam_con) net_sink -m mote4/ball_beam_con -t "$TIMER_REPEAT:2048" #Configure MICA_2_N_actuator (mote5/mot_act) net_sink -m mote5/ball_beam_con -t "$TIMER_REPEAT:256" #Activate the routes $ net_enable {mote5/mot_act,mote4/ball_beam_con} net_enable {mote3/ball_beam_con,mote2/ball_pos_sen} As shown in Figure 2, three routes are clearly established. One route establishes the control loop from sensor to controller to actuator. A second route delivers sensor and controller debugging information back to a gateway mote. A third route enables a second controller to act as a slave. The ball-positionsensor module on mote2 is conﬁgured as a networking source, generating a control standard input output message to be sent to the ball-beam-controller module, on mote3 every second or 1024 counts. The ball-beam-controller on mote3 is conﬁgured as a networking ﬁlter. As a networking ﬁlter it handles control standard input output messages from mote2’s ball-position-sensor, computes the appropriate command and sends a control standard input output message to mote5’s motor-actuator module. Furthermore mote3’s ball-beamcontroller module will generate a debugging message destined for the gateway mote1’s non-resident moduled pc module every two seconds. By sending information to the gateway mote, neclab can support remote monitoring. The moduled pc module was arbitrarily chosen to illustrate that an arbitrary nonresident module id can be reserved in order to pass a message up to the sossrv program and a client can be designed to handle and display this message on the engineers HOST PC display. Terminating the control loop route as a networking sink, mote5’s motor-actuator, handles control standard input output messages from mote3’s ball-beam-controller and actuates the beam. The motor-actuator controls the angular position of the beam and requires a faster control loop of a quarter of a second; hence, the timer is set to 256 counts. 3.4 FreeMat utilities The FreeMat utilities will provide an interface for users familiar with MATLAB to appropriately modify conﬁguration parameters for neclab modules

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designed and written in C. These utilities are currently the least developed for neclab; however, the major design problems have been confronted. There were numerous obstacles which had to be overcome in order to develop these utilities. The ﬁrst accomplishment was getting FreeMat to build and install. Second was to integrate ATLAS, so that the engineer can have an optimized matrix library for simulation and development. Third was to develop the conﬁguring module to allow a higher level networking protocol to be implemented in order to interconnect modules in a manner similar to the MPI protocols identiﬁed in the introduction. Lastly, was identifying SWIG as a possible candidate to assist with generating shared libraries to allow us to interface our SOS modules with FreeMat. We have used SWIG to generate interface ﬁles and shared libraries for python which we have used to create our initial graphical user interface application. We have also used SWIG to interface to our conﬁguring module and do initial testing of the module. Although, FreeMat does provide a native interface for C and C++ programs, we feel that learning to eﬀectively use SWIG to handle interfacing with SOS will allow a more ﬂexible development environment in which users of neclab can use whatever high-level software tools they desire to use. The FreeMat utilities will allow an engineer to generate her own routing tables while allowing users to receive and monitor data from modules using the FreeMat client. The FreeMat client will connect to the sossrv and relay all data destined for the FREEMAT MOD PID. Setting parameters and displaying data should be transparent due to the conﬁguring interface provided by the combined conﬁguring module and the data ﬂash swapping interface provided by SOS. The conﬁguring interface provides all the necessary elements for a module to establish up to two routes, conﬁgure a timer and change up to 8 bytes of parameter data in the modules RAM. To handle larger data-sizes the user can either rely on the larger SOS RAM memory block of 128 bytes. The diﬃculty is that there are only four 128 byte RAM blocks available for the MICA2 and MICAz motes on SOS. The problem is further compounded in that the radio requires at least one 128 byte block to receive an SOS message over the network. In order to simultaneously send and receive a 128 byte message, a second 128 byte block of RAM needs to be allocated by the radio. This means that the user essentially has only two 128 large blocks of RAM available for allocation and they should be used for temporary operation such as being allocated to perform linear algebra routines. The second option is to dedicate a section of internal ﬂash memory for storing conﬁguration parameters and reading the conﬁguration parameters into the local stack during module run-time. This is a preferred option because swapping in 256 bytes of data into the stack should only require around a tenth of a millisecond. This is a feasible option as long as the engineer utilizes the co-operative scheduler provided by SOS, and avoids interrupt service routines, and nested function calls which require large amounts of the stacks memory. Being able to eﬀectively manage the RAM and stack will allow neclab to support a much larger design space. For example, by gaining the ability to

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conﬁgure up to 256 bytes of data, the engineer can begin to develop fourby-four full-state observers. The following sections of conﬁguring.h illustrate both the networking conﬁguration and control standard input interface. The networking conﬁguration is deﬁned by the conﬁguring message type. The control standard input is handled using the standard io message t and indicated by the MSG CONTROL IO message id. #define MSG_CONFIGURING MOD_MSG_START #define MSG_CONFIGURING_ENABLE (MOD_MSG_START + 1) #define MSG_CONFIGURING_DISABLE (MOD_MSG_START + 2) /* #define MSG_*_IO */ #define MSG_STANDARD_IO (MOD_MSG_START + 3) #define MSG_ERROR_IO (MOD_MSG_START + 4) #define MSG_CONTROL_IO (MOD_MSG_START + 5) #define MSG_ROUTING_IO (MOD_MSG_START + 6) /* #define MSG_*_IO */ #define CONFIGURING_SOURCE (1<<0) #define CONFIGURING_SINK (1<<1) #define CONFIGURING_FILTER (1<<2) #define CONFIGURING_ENABLED (1<<3) #define CONFIGURING_CONFIGURED (1<<4) #define CONFIGURING_RESERVED (1<<5) #define CONFIGURING_TIMER_0_BIT_0 (1<<6) #define CONFIGURING_TIMER_0_BIT_1 (1<<7) #define CONFIGURING_TIMER_0 0 #define SIZE_OF_MULTI_HOP_PACKET SOS_MSG_HEADER_SIZE #define SIZE_OF_MSG_SMALL_BLOCK (32) #define SIZE_OF_MSG_LARGE_BLOCK (128) typedef struct destination_type{ uint16_t daddr; /* Destination address */ sos_pid_t did; /* Destination pid */ uint8_t type; /* Message type to send */ } __attribute__ ((packed)) destination_t; #define CONFIGURING_N_DESTINATIONS 2 typedef struct configuring_type{ uint8_t flag; /* source, sink, etc. */ uint8_t size; /* Size of user data */ sos_pid_t pid; /* pid of user module */ uint8_t pad; /* pad for alignment */ /* destinations takes up to 8 bytes */ destination_t destinations[CONFIGURING_N_DESTINATIONS]; uint8_t data[8]; /* 8 bytes for user to store user specific data */

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} __attribute__ ((packed)) configuring_t;/* 20 bytes leaving 12 bytes for 6 function pointers in the app_state_t structure = 32 bytes = SIZE_OF_SMALL_BLOCKS for the avr processor */ typedef struct configuring_message_type{ int32_t interval; /* The ticks/counts you want to place for the timer if timer < 0 the timer is disabled by definition! */ configuring_t conf; } __attribute__ ((packed)) configuring_message_t; /* 24 byte packet leaving 8 bytes (8 required) for multi-hop routing worst case */ typedef enum {UINT8_T, INT8_T, UINT16_T, INT16_T, UINT32_T, INT32_T, FLOAT_T, USER_STRUCT_T} standard_io_types; typedef struct standard_io_message_type{ /* Engineer can send a vector of data_types */ uint8_t data_r; /* Engineer can send an array of structures as long as the size_of is specified */ uint8_t data_size_of; /* One of standard_io_types */ uint8_t data_type; /* sid of sender of standard io message */ sos_pid_t sid; uint8_t data[MAX_STANDARD_IO_LENGTH]; /*32-8-4=20*/ } __attribute__ ((packed)) standard_io_message_t; Refer to the neclab documentation for additional details.

4 Conclusion In developing neclab many challenging design issues were addressed and solved. neclab ﬁrst addressed the need to have a consistent tool chain, such as cross-compilers, for the MICA2 motes. This design problem was solved using neclab’s build utilities. The next design issue to be addressed was automating all the tasks associated with loading software on to the MICA2 motes. This was addressed by developing sos utilities. These utilities assisted in generating

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new modules, loading modules on to the network, building and installing new kernels on to the MICA2 motes. The next design issue was to create a mechanism to track the type of kernels and properties of each MICA2 mote on the network. This was accomplished, using the necroot design. Building on the tools developed from build utilities, the network.conf ﬁle provides a simple interface to manage each motes kernel, static-modules, radio conﬁguration, location, group, and address properties. Next by showing a well-designed conﬁguring module interface combined with necroot we demonstrated an eﬃcient way to establish routes using a routing.conf ﬁle. Finally, all these components allow us to build a set of FreeMat utilities to enable control-engineers to develop wnecs, using a MATLAB like interface. Using the routing.conf ﬁle, we also illustrated how control systems naturally create their own routes. The sensor is a networking source, creating a time-base for the system and generating messages to be routed to either a networking ﬁlter, or networking sink. The controller is a networking ﬁlter, receiving messages from the sensor, modifying the data and sending an appropriate message to either a networking ﬁlter, or networking sink. In the ball and beam example the controller sent a command message to the motor actuator. The motor actuator behaved as a networking sink, receiving the messages from either a networking source or networking ﬁlter and applied the appropriate input to the motor. We also showed in the ball and beam example how by utilizing a redundant controller we can accomplish the reprogramming on the ﬂy. Speciﬁcally, the second controller, initially conﬁgured as a networking sink, can be reconﬁgured with new control parameters and then switched to become a networking ﬁlter while conﬁguring the initial controller to be a networking sink. This shows that if a control system requires extensive time to reconﬁgure itself, the parallel architecture just described, can safely do so without having to risk having the system go unstable. This is a valuable feature, which may be used for control, reconﬁguration and fault diagnostics and identiﬁcation, switching control, etc. Working with the memory constrained MICA2 and MICAz motes, we identiﬁed a unique way to increase our module conﬁguration space. State-space control applications, in particular, those which require a state-observer, require large amounts of memory to describe a model of the system. In order to describe the system and make it re-conﬁgurable, we have identiﬁed that a section of internal ﬂash can be reserved for reading in module control parameters into the stack during module run-time. Using SOS with it’s co-operative scheduler, an engineer can wirelessly send new control parameters to the internal ﬂash section in order to reconﬁgure and tune a state-observer. The current implementation is slower, however, due to the increased memory read times from the internal ﬂash section. Although, neclab, is in its initial release, it has become a fairly advanced piece of software. One limitation, is related to the limited message routing infrastructure. Presently, neclab relies on the built in SOS message routing structure, which is quite advanced; however, conﬁguration messages can only

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be established if the gateway mote can access an individual node directly on the network. We are particularly interested in adding a more advanced routing interface, to enable multiple-hop routing for initial conﬁguration. We are going to be actively using neclab for designing wnecs, and hope the controlcommunity will ﬁnd neclab a useful tool for their own research.

Acknowledgment We would like to thank Simon Han from NESL, UCLA for inviting us to contribute to the SOS project and making neclab one of many projects which will make extensive use of the SOS operating system. Simon has provided valuable development, feedback and support for integrating neclab with SOS. We would also like to acknowledge the rest of the SOS team: Ram Kumar, Roy Shea, Andrew Barton-Sweeney, Eddie Kohler and Mani Srivastava for their collective feedback and support. Finally, we would like to acknowledge Joey Ernst for testing and assisting in the development of neclab.

Appendix A As shown in Figure 3, a basic network has been created and a blink module has been loaded and enabled on each MICA2 mote in order to control the blink rate of the yellow, green, and red LEDs. Each mote is a member of group 2, and each mote name corresponds to its address (mote1 has an address of 1). The blink module utilizes the conﬁguring module interface and a module speciﬁc LED structure. The LED structure contains the state of each LED, a basic clock structure for each LED, and a mask to selectively ﬁlter the state of a given LED. Mote1 serves as a networking source in which its timer is conﬁgured to repeat every 1024 counts or one second. Once conﬁgured, and enabled by a MSG CONFIGURING ENABLE message, mote1’s blink module will change the status (on/oﬀ) of the red LED every 4 seconds, green LED every 2 seconds, and the yellow LED every 8 seconds. Each time the led state changes a MSG STANDARD IO message is delivered to mote2’s blink module. Figure 3 indicates that mote1 currently has the yellow and green LEDs on. The message received by mote2 contains the new led state of mote1. Mote2, conﬁgured as a networking ﬁlter masks the received led state such that only the red and green led state will pass. As a result only mote2’s green LED is indicated as on. In this example, we chose to let the interval timer for mote2’s blink module update the new status of the led state on the physical blink display so there could be a delay up to half a second. This is done to illustrate that we can isolate the handling of asynchronous events with a synchronous task. Last, the ﬁltered led state of mote2 is delivered to mote3 as a MSG STANDARD IO message. Mote3, conﬁgured as a networking sink updates the newly received led state and displays the lit green LED with a lag of no greater than half a second.

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Note that modules on the network can be enabled/disabled by broadcasting a MSG CONFIGURING {ENABLE,DISABLE} message or motes can be enabled/disabled individually. Conﬁguring messages are delivered reliably to individual motes, using the SOS MSG RELIABLE interface. As a result, conﬁguring messages which are broadcast can not currently be acknowledged as being reliably sent; however, may still be deemed useful for a quick initial shutdown of a system.

Fig. 1. Ball and Beam Network Embedded Control System.

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Fig. 2. neclab block diagram for ball and beam control.

neclab: The Network Embedded Control Lab

Fig. 3. blink lab network routing diagram.

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Homogeneous Semantics Preserving Deployments of Heterogeneous Networks of Embedded Systems Aaron D. Ames, Alberto Sangiovanni-Vincentelli, and Shankar Sastry Center for Hybrid and Embedded Software Systems Department of Electrical Engineering and Computer Sciences University of California at Berkeley {adames,alberto,sastry}@eecs.berkeley.edu Summary. Tagged systems provide a denotational semantics for embedded systems. A heterogeneous network of embedded systems can be modeled mathematically by a network of tagged systems. Taking the heterogeneous composition of this network results in a single, homogeneous, tagged system. The question this paper addresses is: when is semantics (behavior) preserved by composition? To answer this question, we use the framework of category theory to reason about heterogeneous system composition and derive results that are as general as possible. In particular, we deﬁne the category of tagged systems, demonstrate that a network of tagged systems corresponds to a diagram in this category and prove that taking the composition of a network of tagged systems is equivalent to taking the limit of this diagram—thus composition is endowed with a universal property. Using this universality, we are able to derive veriﬁable necessary and suﬃcient conditions on when composition preserves semantics.

1 Introduction In an embedded system, diﬀerent components of the system evolve according to processes local to the speciﬁc components. Across the entire system, these typically heterogeneous processes may not be compatible, i.e., answering questions regarding the concurrency, timing and causality of the entire system—all of which are vital in the actual physical implementation of the system—can be challenging even if these questions can be answered for speciﬁc components. Denotational semantics provide a mathematical framework in which to study the behavior (signals, ﬂows, executions, traces) of embedded systems or networks thereof. This framework is naturally applicable to the study of heterogeneous networks of embedded systems since signals always can be compared, regardless of the speciﬁc model of computation from which they were produced.

P.J. Antsaklis, P. Tabuada (Eds.): Netw. Emb. Sens. and Cntrl., LNCIS 331, pp. 127–154, 2006. © Springer-Verlag Berlin Heidelberg 2006

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Tagged systems provide a denotational semantics for heterogeneous models of computation; they consist of a set of tags (a tag structure), variables and maps (behaviors) from the set of variables to the set of tags—hence, tagged systems are a speciﬁc case of the tagged signal model (cf. [LSV97]). A heterogeneous network of embedded systems, e.g., a network consisting of both synchronous and asynchronous systems, can be modeled by a network of tagged systems with heterogeneous tag structures communicating through mediator tagged systems. Benveniste et al. [BCC+ 04], [BCC+ 05] and [BCCSV03], introduced the notion of tagged systems and dealt with the issues we set forth in this paper; this work extends and generalizes the ideas introduced in these papers. Of course, there is a wealth of literature on semantics preservation in heterogeneous networks, cf. [BCG99], [Cas01], [LSV98], [PSVPC05], and [WN95], the last of which approaches the problem from a categorical prospective. A network of tagged systems can be implemented, or deployed, through heterogeneous parallel composition—obtained by taking the conjunction (intersection) of the behaviors that agree on the mediator tagged systems—which results in a single, homogeneous, tagged system. Thus, heterogeneous networks of tagged systems can be homogenized through the operation of composition. This paper addresses the question: When is semantics preserved by composition? That is, when is the homogeneous tagged system obtained by composing a heterogeneous network of tagged systems semantically identical to the original network? Understanding this question is essential to understanding when networks of (possibly synchronous) embedded systems can be implemented asynchronously while preserving the semantics of the original system. Since implementing asynchronous systems is often more eﬃcient (less overhead) when compared to the implementation of synchronous systems, deriving conditions on when this can be done while simultaneously preserving semantics would have many important implications. In this paper, taking a similar approach to Benveniste et al., we address the issue of semantics preservation. However, we use the formalism of category theory, i.e., we introduce the category of tagged systems: TagSys to obtain more general conditions for semantics preservation. We begin by considering a network of two tagged systems P1 and P2 communicating through a mediator tagged system M as described by the diagram: P1 → M ← P2 . The ﬁrst contribution of this paper is that we are able to show that the (classical notion of) heterogeneous composition of P1 and P2 over M, P1 M P2 , is given by the pullback (or ﬁbered product) of this diagram: P1

M P2

= P1 ×M P2 .

The importance of this result is that it implies that composition is endowed with a universal property; this universal property is fundamental in understanding when semantics is preserved. Consider the case when P1 and P2 have

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the same semantics, i.e., the same tag structure. Therefore, they always can communicate through the identity mediator tagged system, I, and the homogeneous composition of P1 and P2 , P1 P2 , is given by the pullback P1 ×I P2 of the diagram: P1 → I ← P2 . It is possible through this framework to give a precise statement of what it means to preserve semantics by composition over the mediator tagged system M: Semantics is preserved by composition if P1 P2 ≡ P1

M P2 .

Through the universality of the pullback, we are able to give veriﬁable necessary and suﬃcient conditions on semantics preservation. A corollary of our result is the suﬃcient conditions on semantics preservation established by Benveniste et al.. A network of tagged systems is given by an oriented graph Γ = (Q, E) together with a set of tagged systems P = {Pq }q∈Q communicating through a set of mediator tagged systems M = {Me }e∈E ; that is, for every e ∈ E, there is a diagram in TagSys of the form: Psource(e)

αe ✲ α Me ✛ e Ptarget(e) .

Equivalently, a network of tagged systems is given by a functor P(P,M,α) : HΓ → TagSys, where HΓ is a small category of a special form, termed an H-category, and obtained from Γ . As in the case of a network of two tagged systems, the heterogeneous composition of a network of tagged systems is given by the limit (a generalization of the pullback) of this diagram, MP

HΓ =← Lim −− (P(P,M,α) ),

and so composition again is deﬁned by a universal property. If all of the Pi s have the same semantics, then we can again consider the identity mediator I (which in this case is a set), and the homogeneous composition of this HΓ network is given by P = ← Lim −− (P(P,I,id) ). This formulation allows us to give a precise statement of when semantics is preserved: Semantics is preserved by composition if

MP

≡ P.

The universality of composition allows us to derive concrete necessary and sufﬁcient conditions on when semantics is preserved, indicating that this framework can produce results on semantics preservation that are both practical and veriﬁable. Although applications are not speciﬁcally discussed in this work, it is important to note that there are many practical implications of the results derived in this paper. Tagged systems were originally developed in order to better understand time-trigged architectures (cf. [Kop97]). Therefore, this theory is

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naturally applicable to such a framework; in fact, Benveniste et al. address locally time-trigged architectures and globally asynchronous locally synchronous architectures in [BCC+ 04], [BCC+ 05] and [BCCSV03]. Addressing compositionality issues in speciﬁc architectures utilizing the results presented in this paper is a promising area of future research. This paper is structured as follows: Section 2: Tag structures and tagged systems are introduced, along with the corresponding categories, Tag and TagSys, respectively. Section 3: Classical heterogeneous composition is reviewed, and it is shown that composing two tagged systems corresponds to taking the pullback of a speciﬁc diagram in TagSys. Section 4: The general composite of two tagged systems, communicating through a mediator tagged system, is deﬁned. The notion of semantics preservation for simple networks of this form is introduced, and necessary and suﬃcient conditions on semantics preservation are derived. Section 5: Networks of tag structures, tagged systems and behaviors are deﬁned. It is shown how to associated to these networks a functor and a small category, i.e., how these networks correspond to diagrams in the categories Tag, TagSys and Set, respectively. Section 6: A general method for universally composing networks of tagged systems is introduced. This is related to the composite of corresponding networks of tag structures and behaviors. Section 7: The notion of a semantics preserving deployment of a network of tagged systems is introduced. Necessary and suﬃcient conditions on semantics preservation are derived. Section 8: Concluding remarks are given.

2 A Categorical Formulation of Tagged Systems In this section, we begin by deﬁning the category of tag structures. This deﬁnition is used to understand how to associate a common tag structure to a pair of tag structures which can communicate through a mediator tag structure. Later, it will be seen how composing tagged systems mirrors this construction on tag structures. Before discussing composition, we ﬁrst must introduce the category of these systems. This category will be instrumental later in understanding how to form the heterogeneous composition of a network of tagged systems. Tag structures and the corresponding category. Fundamental to the notion of tagged systems is the notion of timing. This timing is encoded in a set of tags; these “tag” the occurrences of events, i.e., they index the events such that they are (partially) ordered. Hence, a set of tags or a tag structure is a partially ordered set T, with the partial order denoted by ≤. The category of tags, Tag, can be deﬁned as follows:

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Objects: Partially ordered sets, i.e., tag structures. Morphisms: Nondecreasing maps between sets ρ : T → T , i.e., if t ≤ t ∈ T then ρ(t) ≤ ρ(t ) ∈ T . Composition: The standard composition of maps between sets. Clearly, two objects in the category Tag are isomorphic if ρ : T → T is a bijection: there exists a ρ : T → T such that ρ ◦ ρ = idT and ρ ◦ ρ = idT . Note also that the terminal objects in the category Tag are just one point sets Ttriv := {∗} (called asynchronous tag structures), i.e., for all tag structures T there exists a unique morphism ρ : T → Ttriv deﬁned by ρ(t) ≡ ∗ which desynchronizes the tag structure. The synchronous tag structure is given by Tsync = N. Common tag structures. Tags are fundamental in understanding tagged systems in that morphisms of tag structures will induce morphisms of tagged systems. To better understand this, we will discuss an important operation on tag structures: the pullback (the pullback of elements in a category will be used extensively in the paper—see [Lan98] for a formal deﬁnition). Consider two tag structures T1 and T2 . We would like to ﬁnd a tag structure that is more general than T1 and T2 and has morphisms to both of these tag structures, i.e., we would like to ﬁnd a common tag structure for these two tag structures. To do this, ﬁrst consider the diagram in Tag: ρ1✲

T1

ρ2 T✛ T2 ,

(1)

where T is the mediator tag structure. We know that such a tag structure always exists since it always can be taken to be Ttriv (although this rarely is the wisest choice). We deﬁne the common tag structure to be the pullback of this diagram: T1 ×T T2 = {(t1 , t2 ) ∈ T1 × T2 : ρ1 (t1 ) = ρ(t2 )}.

(2)

The pullback is the desired common tag structure since it sits in a commutative diagram of the form: T1 ×T T2 π1

❄ T1

π✲ 2 T2 ρ2 ρ1✲ ❄ T

(3)

Moreover, that fact that the common tag structure is the pullback implies that for any other tag structure that displays the properties of a common tag structure, there exists a unique morphism from this tag structure to the common tag structure. More precisely, for any tag structure T such that the following diagram commutes:

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T ∃!

q2

✲ q1

T1 ×T T2 ✲

π1 ❄ T1

✲ ✲ T2 π2 ρ2 ❄ ρ1✲ T

(4)

there exists a unique morphism from T to T1 ×T T2 also making the diagram commute. This construction on tag structures both motivates and mirrors constructions that will be performed throughout this paper on tagged systems. To demonstrate this we must, as with tag structures, deﬁne tagged systems and the associated category.

σ(6, x)

E v e n ts

σ(2, x)

σ(1, x)

t1 t2

t3 t4 t5

t6

t7

t8

T a g s (tim in g )

Fig. 1. A graphical representation of a behavior of a tagged system.

Tagged systems. Following from [BCC+ 04], [BCC+ 05] and [BCCSV03] (although our notation slightly deviates from theirs), we deﬁne a tagged system. We then proceed to introduce the category of tagged systems. Let V be an underlying set of variables and D be the set of values that these variables can take, i.e., the domain of the variables. A tagged system is a tuple P = (V, T, Σ), where V is a ﬁnite subset of the underlying set of variables V, T is a tag structure, i.e., an object of Tag, and Σ is a set of maps: σ : N × V → T × D. Each of the elements of Σ, i.e., each of the maps σ, are referred to as V behaviors (or just behaviors when the variable set is understood). It is required that for each v ∈ V , the map δv (n) := π1 (σ(n, v)) : N → T is a morphism in Tag, that is, nondecreasing (and called a clock in [BCCSV03]).

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Remark 1. In deﬁning the set of behaviors of a tagged system, we made an explicit choice for the domain of the behaviors: N×V . This choice is motivated by the fact that the behaviors of a tagged system are signals generated by a computer, and hence discrete in nature. It is possible to consider other domains for the behaviors, e.g., R × V , without any signiﬁcate change to the theory introduced here. This indicates an interesting extension of this work to behavioral dynamical system theory (cf. [PW98, vdSJ02, vdSS01]). The category of tagged systems. We can use the formulation of tagged systems above in order to deﬁne the category of tagged systems, TagSys, as follows: Objects: Tagged systems P = (V, T, Σ). Morphisms: A morphism of tagged systems α : P = (V, T, Σ) → P = (V , T , Σ ) is a morphism (in the category of sets) of behaviors α : Σ → Σ . Composition: The standard composition of maps between sets. In other words, for α : P → P and α : P → P the composition of α : Σ → Σ and α : Σ → Σ is given by α ◦ α : Σ → Σ . From the deﬁnition of morphisms in the category TagSys, it follows that two tagged systems, P and P , are isomorphic, P ∼ = P , if and only if, to use the terminology from the literature, the two tagged systems are in bijective correspondence. A “forgetfull” functor. By the deﬁnition of the category TagSys, there is a fully faithful functor (cf. [Lan98]): U : TagSys → Set, where Set is the category of sets. This functor is deﬁned on objects and morphisms in TagSys as follows: for every diagram in TagSys of the form: α

P = (V, T, Σ) → P = (V , T , Σ ), the functor U is given by: α

α

U P = (V, T, Σ) → P = (V , T , Σ ) = Σ → Σ . When discussing composition, we often will blur the distinction between the categories TagSys and Set, i.e., we often will deﬁne the composition of a diagram of tagged systems by the behaviors of the composite system, and hence implicitly view it as an object of Set. In this case, we always will construct an object in TagSys with behaviors isomorphic to the corresponding object in Set. Induced morphisms of tag systems. Suppose that there is a morphism of tag structures ρ : T → T . Then there exists a tagged system P ρ together with

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an induced morphism of tagged systems (·)ρ : P → P ρ . First, if P = (V, T, Σ) we deﬁne P ρ = (V, T , Σ ρ ) where Σρ

:=

{σ ρ : N × V → T × D : σ ρ (n, v) = (ρ(t), d) iﬀ (t, d) = σ(n, v) for some σ ∈ Σ}.

That is, Σ ρ is deﬁned by replacing t with ρ(t) in the codomain of σ. With this deﬁnition of P ρ , we obtain a morphism ( · )ρ : P → P ρ , called the desynchronization morphism and deﬁned by, for each σ ∈ Σ, σ ρ (n, v) = (ρ(t), d)

def

⇔

σ(n, v) = (t, d).

Note that ( · )ρ is always surjective. Example 1. Consider the following synchronous tagged systems, P1 and P2 , deﬁned as follows: P1 P2

:= :=

(V1 = {x}, Tsync = N, Σ1 = {σ1 }), (V2 = {x, y}, Tsync = N, Σ2 = {σ2 , σ ˜2 }),

where σ1 (n, x)

:=

σ2 (n, v)

:=

σ ˜2 (n, v)

:=

(m(n), ), (m(n), ) (k(n), )

if if

v = x ∈ V2 , v = y ∈ V2

(m(n), ) (l(n), )

if if

v = x ∈ V2 . v = y ∈ V2

Here m(n), k(n) and l(n) are any strictly increasing functions with k(n) = l(n), and is a (single) arbitrary value in D. ρ For ρ : Tsync → Ttriv the desyncronization morphism, P1 ∼ = P1 because P1 ρ ρ consists of a single behavior. Since Σ2 = {σ2 , σ ˜2 }, Σ2 = {σ2 = σ ˜2ρ }, i.e., Σ2ρ consists of a single behavior. Therefore, P2 is not in bijective correspondence with P2ρ

3 Universal Heterogeneous Composition In this section, we discuss how to take the composition of a “simple” network of embedded systems. Given two tagged systems, P1 and P2 , we would like to form their composition, i.e., a single tagged system obtainable from these two tagged systems. We begin by reviewing the “standard” deﬁnition of composition, followed by a categorical reformulation of composition. We demonstrate that the composition of two tagged systems corresponds to the pullback of a speciﬁc diagram in the category of tagged systems. This will allow us later to generalize the notion of composition.

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Heterogeneous composition. Let P1 = (V1 , T1 , Σ1 ) and P2 = (V2 , T2 , Σ2 ) be two tagged systems. Consider a mediator tag structure T between the tag structures T1 and T2 , i.e., there exists a diagram in Tag: T1

ρ1✲

ρ2 T✛

T2

Recall that the common tag structure to T1 and T2 (relative to T) is given by the pullback of the above diagram in Tag (the ﬁbered product as deﬁned in (2)): T1 ×T T2 . We deﬁne the parallel composition of P1 and P2 over the mediator tag structure T by P1

T

P2 := (V1 ∪ V2 , T1 ×T T2 , Σ1 ∧T Σ2 ).

This notation for parallel composition is taken from [BCC+ 04]; the morphisms ρ1 and ρ2 are implicit in this notation. In the above deﬁnition, Σ1 ∧T Σ2 is given by the set of behaviors σ : N × (V1 ∪ V2 ) → (T1 ×T T2 ) × D such that the following condition holds: for all (n, v) ∈ N × (V1 ∪ V2 ), there exist unique σ1 ∈ Σ1 and σ2 ∈ Σ2 such that1 (i) σ(n, v) = ((t1 , t2 ), d)

⇔

(ii) (iii)

σ1 (n, v) = (t1 , d) if v ∈ V1 and σ2 (n, v) = (t2 , d) if v ∈ V2 (5) and σ1ρ1 (n, v) = σ2ρ2 (n, v) if v ∈ V1 ∩ V2 .

Since σ is uniquely determined by σ1 and σ2 , and vise-versa, we write σ = σ1 T σ2 . We pick this notation so as to be consistent with the literature, cf. [BCC+ 04], where a pair (σ1 , σ2 ) ∈ Σ1 × Σ2 is called uniﬁable when it satisﬁes condition (iii), and σ1 T σ2 is called the uniﬁcation of σ1 and σ2 . We will always assume that such a pair exists; in this case composition is well-deﬁned (Σ1 ∧T Σ2 is not the empty set). Universal heterogeneous composition. The common tag structure for the composition of two tagged systems is given by the pullback of a certain diagram. The natural question to ask is: can the composition of two tagged systems be realized as the pullback of a diagram of tagged systems of the form: P1 1

α✲ α2 1 M ✛ P2 ?

Note that conditions (i) and (ii) imply condition (iii); this follows from the fact that (t1 , t2 ) ∈ T1 ×T T2 , so ρ1 (t1 ) = ρ2 (t2 ) in condition (i) and (ii). Condition (iii) is stated for the sake of clarity.

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The importance of this question is that if the answer is yes, then the composition between two heterogeneous tagged systems is universal, i.e., deﬁned by a universal property. We then can ask when the composition of two tagged systems is the same as the composition of these tagged systems with diﬀerent tag structures, i.e., when semantics is preserved. It is possible to show that composition is in fact given by a universal property. In order to deﬁne composition universally, we must deﬁne the tagged system M in the above diagram. In this vein, and using the same notation as the above paragraph, deﬁne IT := (V1 ∩ V2 , T, Σ1 ∨T Σ2 ),

(6)

where T is a mediator tag structure (between T1 and T2 ) and Σ 1 ∨T Σ 2

:=

{σ : N × (V1 ∩ V2 ) → T × D : σ = σiρi |V1 ∩V2 , for σi ∈ Σi , i = 1 or 2}.(7)

Now for the tagged systems P1 and P2 there exist morphisms P1

ρ1 Res✲ Resρ2 1 IT ✛ 2 P2

(8)

deﬁned as follows: Resρi i (σi ) = σiρi |V1 ∩V2 : N × (V1 ∩ V2 ) → T × D for σi ∈ Σi , i = 1, 2. Clearly, such a morphism always exists. Note that IT is a mediator tagged system or channel between the tagged systems P1 and P2 ; IT “communicates” between P1 and P2 . In the case when T = T1 = T2 , I := IT is exactly the identity mediator tagged system or identity channel introduced in [BCC+ 04]. Theorem 1. Consider two tagged systems P1 = (V1 , T1 , Σ1 ) and P2 = (V2 , T2 , Σ2 ) with mediator tag structure T, i.e., suppose that there is a diagram in Tag: T1

ρ1✲

ρ2 T✛ T2 .

The parallel composition of P1 and P2 over this tag structure, P1 the pullback of the diagram: P1

T

P2 , is

ρ1 Res✲ Resρ2 1 IT ✛ 2 P 2

in the category of tagged systems, TagSys. Implications of Theorem 1. Before proving Theorem 1, we discuss some of the implications of this theorem.

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If we consider the following diagram in the category of sets, Set: Σ1

ρ1 Res✲ Resρ2 1 Σ 1 ∨T Σ 2 ✛ 2 Σ2 = U

ρ1 Res✲ Resρ2 1 IT ✛ 2 P 2 ,

P1

the pullback of this diagram is given by: Σ1 ×Σ1 ∨T Σ2 Σ2 = {(σ1 , σ2 ) ∈ Σ1 × Σ2 : Resρ11 (σ1 ) = Resρ22 (σ2 )}. It is important to note that the pullback of the above diagram (which is an object in Set) is related to—in fact, isomorphic to—the behavior of the tagged system P1 T P2 . More precisely, for the functor U : TagSys → Set, we have: Σ 1 × Σ1 ∨ T Σ2 Σ 2 ∼ = U (P1

T

P2 ) = Σ1 ∨T Σ2 .

(9)

This observation will allow us later to, justiﬁably, blur the distinction between pullbacks (and limits) in the categories TagSys and Set. In order to produce the bijection given in (9), ﬁrst note that there are projections deﬁned by: πi : Σ 1 ∧ T Σ 2 → Σ i

(10)

where for each σ1 T σ2 ∈ Σ1 ∧T Σ2 , πi (σ1 T σ2 ) := σi for i = 1, 2. Because of (5), it follows that any element σ ∈ Σ1 ∧T Σ2 can be written as σ = π1 (σ) T π2 (σ). Theorem 1 implies—by the universality of the pullback—that there is a bijection: (π1 , π2 ) : Σ1 ∧T Σ2 σ = σ1 T σ2

∼

−→ →

Σ 1 × Σ1 ∨ T Σ2 Σ 2 (σ1 = π1 (σ), σ2 = π2 (σ))

(11)

where the inverse of this map is the uniﬁcation operator: (·)

T

( · ) : Σ1 × Σ 1 ∨ T Σ 2 Σ 2 (σ1 , σ2 )

∼

−→ →

Σ 1 ∧T Σ2 σ1 T σ2

(12)

This completes the description of the bijection given in (9). Proof (of Theorem 1). From the deﬁnition of Σ1 ∧T Σ2 and Σ1 ∨T Σ2 , it follows that the following diagram in Set: Σ1 ∧T Σ 2 π2

❄ Σ2

π1 ρ2 Res✲ 2

✲ Σ1 Resρ11 ❄ Σ 1 ∨T Σ 2

commutes, which implies, by the deﬁnition of morphisms of tagged systems, that the following diagram in TagSys:

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P1

π1 ✲ P1

P2

T

Resρ11 Resρ22✲ ❄ IT

π2

❄ P2

commutes. Consider a tagged system Q such that the following diagram commutes: q1 ✲ P1

Q q2

❄ Resρ2 2 ✲ P2

We can deﬁne a morphism γ : Q → P1 γ(σ) = q1 (σ)

Resρ11 ❄ IT T

P2 by, for σ ∈ ΣQ , T

q1 (σ).

It follows that there is a commuting diagram: Q γ

q1

q2

✲ P1

T

P2

✲

π2 ❄ P2

π1 Resρ22

✲ ✲ P1

Resρ11 ❄ ✲ IT

Moreover, by replacing γ with any other morphism making the diagram commute, say γ˜ , it follows that for σ ∈ ΣQ γ˜ (σ) = π1 (˜ γ (σ))

T

π2 (˜ γ (σ)) = q1 (σ)

T

q2 (σ) = γ(σ).

So γ = γ˜ , i.e., γ is unique.

4 Equivalent Deployments of Tagged Systems Standard composition is just the pullback of a speciﬁc diagram in TagSys; this observation naturally allows us to generalize composition. To perform this generalization, we introduce the notion of a general mediator tagged system, M, and deﬁne composition to be the pullback of a diagram of the form: P1

α✲ α2 1 M ✛ P2

(13)

in TagSys. This process will be instrumental later in understanding how to take the composition of more general networks of embedded systems.

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We conclude this section by reviewing the deﬁnition of semantics preservation and giving necessary and suﬃcient conditions on when semantics is preserved. We apply these results to the special case of semantics preservation through desyncronization. Composition through mediation. Given the results of Theorem 1, we can develop a more intuitive notation for composition. Speciﬁcally, if T is the mediator tag structure, then we write P1 T P2 = P1 IT P2 (in the case when T1 = T2 = T and ρ1 = ρ2 = id in (1), we just write P1 P2 := P1 I P2 ). The mathematical reason for this is that P1 IT P2 is (isomorphic to) P1 ×IT P2 , i.e., the pullback of the diagram given in (8). The philosophical motivation for this notation is that the composition of P1 and P2 can be taken over general mediator tagged systems. In other words, the parallel composition of P1 and P2 over a general mediator tagged system M, denoted by P1 M P2 , is deﬁned to be the pullback of the diagram given in (13): P1

M P2

:= P1 ×M P2 .

This implies that if ΣM is the set of behaviors of M, then the set of behaviors for P1 M P2 is isomorphic to: Σ1 ×ΣM Σ2 = {(σ1 , σ2 ) ∈ Σ1 × Σ2 : α1 (σ1 ) = α2 (σ2 )}.

(14)

The explicit construction of P1 M P2 as a tagged system is not especially relevant, as we are interested only in its set of behaviors which must be isomorphic to the set of behaviors given in (14). That being said, this construction is a special case of a more general construction (given in Section 6) for which the construction of the tagged system is carried out. We only note that there are bijections (·)

M

( · ) : Σ1 ×ΣM Σ2

(π1 , π2 ) : ΣP1

M P2

∼

Σ P1

∼

Σ 1 × ΣM Σ 2

−→ −→

M P2

(15) (16)

which are generalizations of the uniﬁcation and projection maps given in (12) and (11), respectively. Speciﬁcation vs. deployment. Consider two tagged systems P1 and P2 with a mediator tag structure T. As in [BCC+ 05] (although with some generalization, since P1 and P2 are not assumed to have the same tag structure), we deﬁne the following semantics: Speciﬁcation Semantics: P1 IT P2 Deployment Semantics: P1 M P2 for some mediator tagged system M. The natural question to ask is when are the speciﬁcation semantics and the deployment semantics “equivalent.” Formally, and following from [BCC+ 05], we deﬁne a mediator M to be semantics preserving with respect to IT , denoted by

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P1

M P2

≡ P1

IT P2

(17)

if for all (σ1 , σ2 ) ∈ Σ1 × Σ2 , ∃ σ ∈ ΣP1

M P2

s.t. π1 (σ ) = σ1 and π2 (σ ) = σ2

∃ σ ∈ ΣP1

IT P2

s.t. π1 (σ) = σ1 and π2 (σ) = σ2 .

(18)

Utilizing Theorem 1, we have the following necessary and suﬃcient conditions on semantics preservation. Theorem 2. For two tagged systems P1 and P2 , P1

M P2

≡ P1

IT P2 ,

if and only if for all (σ1 , σ2 ) ∈ Σ1 × Σ2 : α1 (σ1 ) = α2 (σ2 )

⇔

Resρ11 (σ1 ) = Resρ22 (σ2 )

Proof. (Suﬃciency:) If (σ1 , σ2 ) ∈ Σ1 × Σ2 s.t. α1 (σ1 ) = α2 (σ2 ) by (14)

⇒

(σ1 , σ2 ) ∈ Σ1 ×ΣM Σ2

⇒

σ1

by (15)

by (18)

⇒

by (16)

⇒

by (14)

⇒

M

σ 2 ∈ Σ P1 π1 (σ1

∃ σ ∈ Σ P1

IT P 2

where σ2 ) = σ1 and π2 (σ1

M P2

M

M

σ2 ) = σ2

s.t. π1 (σ) = σ1 and π2 (σ) = σ2

(σ1 , σ2 ) ∈ Σ1 ×ΣIT Σ2

Resρ11 (σ1 ) = Resρ22 (σ2 ).

The converse direction proceeds in the same manner: if (σ1 , σ2 ) ∈ Σ1 × Σ2 s.t. Resρ11 (σ1 ) = Resρ22 (σ2 )

⇒

α1 (σ1 ) = α2 (σ2 ),

by (14), (15), (16), and (18). (Necessity:) We have the following implications: ∃ σ ∈ Σ P1

M P2

by (16)

s.t. π1 (σ ) = σ1 and π2 (σ ) = σ2

⇒

(σ1 , σ2 ) ∈ Σ1 ×ΣM Σ2

⇒ ⇒

α1 (σ1 ) = α2 (σ2 ) Resρ11 (σ1 ) = Resρ22 (σ2 )

⇒

(σ1 , σ2 ) ∈ Σ1 ×ΣIT Σ2

by (14)

by (14) by (15)

⇒

σ1

IT σ2 ∈ ΣP1 IT P2 and π1 (σ1 IT σ2 ) = σ1 and π2 (σ1

IT

σ2 ) = σ2 .

Homogeneous Semantics Preseving Deployments of Heterogeneous Networks

Therefore σ1 IT σ2 is the element of ΣP1 IT P2 such that π1 (σ1 and π2 (σ1 IT σ2 ) = σ2 , as desired. The other direction follows in the same way: ∃ σ ∈ ΣP1 ⇒

IT P2

σ1

IT

141

σ2 ) = σ1

s.t. π1 (σ) = σ1 and π2 (σ) = σ2

M σ2 ∈ ΣP1 M P2 and π1 (σ1 M σ2 ) = σ1 and π2 (σ1

M

σ2 ) = σ2 ,

by (14), (15) and (16). To demonstrate the power of Theorem 2, we prove the following theorem, which is a generalization of one of the two main theorems of [BCC+ 04]. Moveover, we show that the theorem in [BCC+ 04] is a corollary of this theorem; thus, our results are more general. First, we review the general set-up for this theorem. Desynchronization. Consider the case when P1 and P2 have the same tag structure, i.e., T1 = T2 = T. Consider a mediator tag structure T of T, i.e., suppose there exists a diagram in Tag: T

ρ✲ ρ T ✛

T.

In this case, we ask when the mediator tagged system IT is semantics preserving, i.e., when P1 P2 ≡ P 1

IT

P2 .

(19)

A very important example of when this framework is useful is in the desynchronization of tagged systems; in this case T = Ttriv = {∗}, and P1 IT P2 is the desynchronization of P1 and P2 . Using the notation of this paragraph, we have the following theorem and its corollary. Theorem 3. IT is semantics preserving w.r.t. I, P1 only if for all (σ1 , σ2 ) ∈ Σ1 × Σ2 : σ1ρ |V1 ∩V2 = σ2ρ |V1 ∩V2

⇒

IT

P2 ≡ P1 P2 , if and

σ1 |V1 ∩V2 = σ2 |V1 ∩V2 .

Proof. Note that by the deﬁnition of the desynchronization morphism ( · )ρ (and the fact that it is always surjective), it follows that σ1 |V1 ∩V2 = σ2 |V1 ∩V2

⇒

σ1ρ |V1 ∩V2 = σ2ρ |V1 ∩V2 .

Therefore, this result is a corollary of Theorem 2. Corollary 1. If Piρ is in bijection with Pi for i = 1, 2 and (P1 P2 )ρ = P1ρ P2ρ , then P1 IT P2 ≡ P1 P2 (IT is semantics preserving w.r.t. I).

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Proof. We need only show that the suppositions of the theorem imply for all (σ1 , σ2 ) ∈ Σ1 × Σ2 σ1ρ |V1 ∩V2 = σ2ρ |V1 ∩V2

⇒

σ1 |V1 ∩V2 = σ2 |V1 ∩V2 .

The result then follows from Theorem 3. To see that the desired implication holds, note that we have the following chain of implications: σ1ρ |V1 ∩V2 = σ2ρ |V1 ∩V2 ⇒ ⇒ ⇒ ⇒

(σ1 , σ2 ) (σ1ρ , σ2ρ ) σ1ρ IT σ2ρ ∃ σ ˜

∈ ∈ ∈ ∈

Σ1 ×ΣI Σ2 ∼ = ΣP1 IT P2 T Σ1ρ ×ΣI Σ2ρ ∼ = ΣP1ρ P2ρ T ρ (since (P1 P2 )ρ = P1ρ P2ρ ) ΣP 1 P2 Σ P1 P 2 s.t. σ ˜ ρ = σ1ρ IT σ2ρ .

Setting σ ˜i = πi (˜ σ ), the last of these implications implies that (˜ σ1ρ , σ ˜2ρ ) = ρ ρ ρ ρ ρ (σ1 , σ2 ) ∈ Σ1 ×ΣI Σ2 . Now, the fact that Pi is in bijection with Pi for T i = 1, 2 implies that σ ˜i = σi , or: (σ1 , σ2 ) = (˜ σ1 , σ ˜ 2 ) ∈ Σ 1 × Σ IT Σ 2

⇒

σ1 |V1 ∩V2 = σ2 |V1 ∩V2 .

Example 2. We would like to know semantics is preserved by desynchronization for the tagged systems given in Example 1, i.e., for ρ : Tsync → Ttriv the desyncronization morphism, is P1 P2 ≡ P1 ITtriv P2 ? First we apply the necessary and suﬃcient conditions given in Theorem 3. Since V1 ∩ V2 = {x}, σ1ρ (n, x) = σ2ρ (n, x) σ1ρ (n, x) = σ ˜2ρ (n, x)

⇒ ⇒

σ1 (n, x) = σ2 (n, x) σ1 (n, x) = σ ˜2 (n, x)

because σ1 (n, x) = σ2 (n, x) and σ1 (n, x) = σ ˜2 (n, x). Therefore, semantics is preserved. Note that Corollary 1 would not tell us whether semantics is preserved, because P2 is not in bijective correspondence with P2ρ , and so the conditions of the corollary do not hold. This demonstrates that Theorem 3 is a stronger result than Corollary 1.

5 Networks of Tagged Systems In this section, we introduce the notion of a network of tag structures, tagged systems and behaviors. Moreover, we are able to show that these objects correspond to diagrams in Tag, TagSys and Set, respectively. This observation will be fundamental in deﬁning composition for these networks.

Homogeneous Semantics Preseving Deployments of Heterogeneous Networks

143

Networks of tag structures. We begin by deﬁning a network of tag structures as in [BCC+ 05] (although we state the deﬁnition in a slightly diﬀerent manner). A network of tag structures is deﬁned to be a tuple (Γ, T, S, ρ), where • Γ is an oriented graph with Q the set of vertices and E the set of edges; for e = (i, j) ∈ E, denote the source of e by S(e) = i and the target of e by t(e) = j. • T = {Tq }q∈Q is a set of tag structures. • S = {Se }e∈E is a set of mediator tag structures, mediating between TS(e) and Tt(e) . • ρ = {(ρe , ρe )}e∈E is a set of pairs of morphisms in Tag, such that for every e ∈ E, there is the following diagram in Tag: TS(e)

ρe✲

ρ Se ✛ e Tt(e) .

Networks of tagged systems are deﬁned in an analogous manner. Networks of tagged systems. A network of tagged systems is deﬁned to be a tuple (Γ, P, M, α), where • Γ is an oriented graph. • P = {Pq }q∈Q is a set of tag structures. • M = {Me }e∈E is a set of mediator tagged systems, mediating between PS(e) and Pt(e) . • α = {(αe , αe )}e∈E is a set of pairs of morphisms in TagSys, such that for every e ∈ E, there is the following diagram in TagSys: αe α PS(e) ✲ Me ✛e Pt(e) . Suppose we have a network of tag structures (Γ, T, S, ρ) and a collection of tagged systems P = {Pq }q∈Q such that Pq has tag structure Tq . Then we can associate to this set of tagged systems a network of tagged systems, (Γ, P, IS , Resρ ) with: IS = {ISe }e∈E ,

ρ

e e )}e∈E Resρ = {(ResρS(e) , Rest(e)

ρ

e e where ISe is deﬁned as in (6), and ResρS(e) are deﬁned as in (8), and Rest(e) i.e., there is a diagram in TagSys of the form:

ρ

PS(e) for every e ∈ E.

e ResρS(e) Res e ✲ IS ✛ t(e) Pt(e) . e

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Networks of behaviors. We can deﬁne a network of behaviors from the network of tagged systems, (Γ, P, M, α), as a tuple: (Γ, ΣP , ΣM , α), where • • • •

Γ is an oriented graph. ΣP = {ΣPq }q∈Q , where ΣPq is the set of behaviors for Pq . ΣM = {ΣMe }e∈E , where ΣMe is the set of behaviors for Me . α = {(αe , αe )}e∈E is a set of pairs of morphisms in Set, such that for every e ∈ E, there is the following diagram in Set: αe α ΣPS(e) ✲ ΣMe ✛e ΣPt(e) .

The association of a network of behaviors from a network of tagged systems can be viewed categorically. For the network of tagged systems (Γ, P, M, α), the functor U : TagSys → Set yields the corresponding network of behaviors (Γ, ΣP , ΣM , α) because ΣP ΣM

= =

{ΣPq }q∈Q = {U (Pq )}q∈Q {ΣMe }e∈E = {U (Me )}e∈E

More generally, αe α ΣPS(e) ✲ ΣMe ✛e ΣPt(e) = U

αe α PS(e) ✲ Me ✛e Pt(e)

for every e ∈ E. H-cateogries. The goal is to deﬁne a network of tagged systems as a diagram in TagSys; to do this, we must ﬁrst deﬁne a speciﬁc type of small category termed an H-category2 and denoted by H. This is a small category in which every diagram has the form:3 •

✲ •✛

•

✲ •✛

•

···

•

✲ •✛

•

That is, an H-category has as its basic atomic unit a diagram of the form: ✲ •✛ • •, and any other diagram in this category must be obtainable by gluing such atomic units along the source of a morphism (and not the target). More formally, a small category, H, is an H-category if it satisﬁes the following two axioms: 2

3

The category H considered here actually is the opposite category to a H-small category, as deﬁned in [AS05]. The surprising thing about this is that these categories were considered in the context of hybrid systems; this suggests some interesting relationships between hybrid and embedded systems. ✲ denotes an arbitrary where • denotes an arbitrary object in H and morphism.

Homogeneous Semantics Preseving Deployments of Heterogeneous Networks

145

1. Every object in H is either the source of a non-identity morphism in H or the target of a non-identity morphism, but never both; i.e., for every diagram •

✲ •

✲ •

···

✲ •

•

✲

•

•

✛

in H, all but one morphism must be the identity (the longest chain of composable non-identity morphisms is of length one). 2. If an object in H is the target of a non-identity morphism, then it is the target of exactly two non-identity morphisms, i.e., for every diagram in H of the form • ✲ ✲ •· · · · · · · · ·•

either all of the morphisms are the identity or two and only two morphisms are not the identity. More on H-categories can be found in [AT] (where the H-categories deﬁned in [AT] are the opposite categories to the ones considered in this paper). We can associate to an oriented graph Γ an H-category HΓ by, for every edge in e ∈ E, deﬁning the following diagram S(e)

✲ e✛

t(e)

in HΓ . Of course, the identity morphisms must be added to each object in HΓ in order to complete the deﬁnition. More generally, the category of (oriented) H-categories can be formed, Hcat, and it can be demonstrated (cf. [AT]) that this category is isomorphic to the category of graphs: Hcat ∼ = Grph. Example 3. The following diagram shows an oriented cycle graph, Γ = Ck , and the associated H-category HCk :

k .. .

i−1

i+2

1

− e i✛ i

i+1

i ✲

✛ ei+1

✲

i−1

ei

✲ ei−1 ✲

ei i+1 ✛ 1

e2

3 .. .

H Ck

✛

i+

2

1

3 .. .

Ck

i + 2✛e

✲

✛

2 e ✲2

✛

1

ek ✛ ✛

k .. .

e k✲

e1 ✲

✲

e1

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A.D. Ames et al.

Diagrams as functors. A diagram in a category is just a collection of objects together with a collection of morphisms between these objects. Equivalently, a diagram in a category can be viewed as a functor from a small category to this category. This motivates an equivalent, arguably more simple and certainly more abstract, deﬁnition of a network of tag structures and a network of tagged systems. We deﬁne a network of tag structures and a network of tagged systems to be functors: T : H → Tag, P : H → TagSys, where H is an H-category. For example, if H is given by the diagram: • −→ • ←− •, then the network of tag structures given in (1) and the network of tagged systems given in (13) are deﬁned, respectively, by the functors: ρ1✲ ✛ρ2 T(• −→ • ←− •) = T1 T T2 , (20) P(• −→ • ←− •)

=

P1

α✲ α2 1 M ✛ P2 .

(21)

More generally, we can associate to a network of tag structures (Γ, T, S, ρ) and a network of tagged systems (Γ, P, M, α) functors: T(T,S,ρ) : HΓ → Tag,

P(P,M,α) : HΓ → TagSys,

where HΓ is the H-category associated with Γ , and T(T,S,ρ) and P(P,M,α) are deﬁned by: ρe ρ T(T,S,ρ) (S(e) ✲ e ✛ t(e)) := TS(e) ✲ Se ✛ e Tt(e) , (22) P(P,M,α) (S(e)

✲ e✛

t(e))

:=

αe α PS(e) ✲ Me ✛e Pt(e) , (23)

for every e ∈ E. If (Γ, P, M, α) is a network of tagged systems, and (Γ, ΣP , ΣM , α) is the associated network of behaviors, then there is a functor S(ΣP ,ΣM ,α) : HΓ → Set, where HΓ is the H-category associated with Γ , and S(ΣP ,ΣM ,α) is deﬁned to be the composite: P(P,M,α) ✲ TagSys U ✲ Set . HΓ In other words, S(ΣP ,ΣM ,α) is deﬁned by: S(ΣP ,ΣM ,α) (S(e)

✲ e✛

t(e))

= for every e ∈ E.

αe α ΣPS(e) ✲ ΣMe ✛e ΣPt(e)

:= U

αe α PS(e) ✲ Me ✛e Pt(e)

,

Homogeneous Semantics Preseving Deployments of Heterogeneous Networks

147

Example 4. Continuing Example 3, an example of a network of tagged systems associated to the H-category HCk is given in the following diagram: Me1 ✲

Pi+2

Pi−1

✲ α ei

i

αe ✲ i− 1

✛ αe

1 +

ei

Pi

✲ 1 α e i−

Pi+1

e2

✲ 1

αe

P3 .. .

✛ α e i+ 1

M e2

α

ek

✛ α

α e2

P2

P1

Pk .. .

P(HCk ) =

Mei+1

✲

1

k

✛α e

✛α

M ek ✛ αe

Mei−1

M ei

6 Universally Composing Networks of Tagged Systems In this section, we give a categorical formulation for composition. Since a network is just a diagram, the composition of a network is the limit of this diagram. To illustrate this concept, we ﬁrst consider a network of tag structures and demonstrate how the taking the composition of this network is consistent with the notion of a common tag structure as ﬁrst introduced in Section 2. We then discuss how these ideas can be generalized to networks of tagged systems. Finally, we explicitly relate the composite of a network of tagged systems with the composite of a network of behaviors. This relationship will be important when attempting to prove results relating to semantics preservation. Composing networks of tag structures. Recall that for two tag structures, T1 and T2 , communicating through a mediator tag structure, T, we obtained a single, common, tag structure that was unique up to isomorphism; the common tag structure, T1 ×T T2 , was the pullback of the diagram given in (1). But the pullback is just a special case of the limit (or inverse limit or projective limit; see [Lan98] for more details) of a functor, i.e., (•−→•←−•) (•−→•←−•) T1 T1 ×T T2 = Lim (T) = ← Lim ←−− −−

where T : (• −→ • ←− •) → Tag is deﬁned as in (20).

ρ1✲

ρ2 T✛

T2 ,

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Therefore, we can deﬁne a tag structure common to an entire network of tag structures by taking the limit of the corresponding diagram in Tag describing this network. If (Γ, T, S, ρ) is a network of tag structures and T(T,S,ρ) : HΓ → Tag is the corresponding functor and H-category, we deﬁne the common tag strucHΓ ture to be ← Lim −− (T(T,S,ρ) ), which because of the special structure of an Hcategory is given by:     HΓ Lim (T ) = (t ) ∈ T : ρ (t ) = ρ (t ), ∀ e ∈ E , q q∈Q q e (T,S,ρ) S(e) e t(e) ←−−   q∈Q

which corresponds to the common tag structure deﬁned in [BCC+ 05]. By the properties of the limit, we know that this is in fact the desired common tag structure since for every e ∈ E, we have a diagram of the form HΓ Lim ←−− (T(T,S,ρ) )

πS(e)

❄

TS(e)

πt(e) ✲ Tt(e) ρe ❄ ρe ✲ Se

which is a direct generalization of (3). Moreover, the limit is universal in the same sense as (4). Composing networks of tagged systems. As with networks of tag structures, we can consider the limit of a network of tagged systems (when viewed as a diagram)—this is the heterogeneous composition of the network. This is justiﬁed by the discussion in Section 4, where the composition of a network of two tagged systems, P1 and P2 , was deﬁned to be the pullback of these systems over the mediator tagged system, M: P1

M P2

(•−→•←−•) (•−→•←−•) = Lim (P) = ← Lim P1 ←−− −−

α✲ α2 1 M ✛ P2 ,

where P : (• −→ • ←− •) → Tag is deﬁned as in (20). This indicates a general, and universal, way of taking the composition of a network of tagged systems: through the limit. Consider a network of tagged systems (Γ, P, M, α), with the tagged systems Pq and Me given by Pq Me

= =

(Vq , Tq , Σq ), (Ve , Se , Σe ),

q ∈ Q, e ∈ E.

For the corresponding functor and H-category: P(P,M,α) : HΓ → TagSys

Homogeneous Semantics Preseving Deployments of Heterogeneous Networks

149

denote the heterogeneous composition of (Γ, P, M, α) by M P (to be consistent with the notation of [BCC+ 05]) and deﬁne it by (unlike [BCC+ 05])   MP

where Σ

MP

HΓ  := Lim ←−− (P(P,M,α) ) =

Vq , q∈Q

Tq , Σ

MP

,

(24)

q∈Q

is the set of behaviors σ :N×

Vq

→

q∈Q

Tq × D q∈Q

σ(n, v)

→

((tq )q∈Q , d)

such that the following conditions hold: for all (n, v) ∈ N × exists unique (σq )q∈Q ∈ q∈Q Σq such that4 for all e ∈ E (i )

Vq , there

σS(e) (n, v) = (tS(e) , d) if v ∈ VS(e) and σt(e) (n, v) = (tt(e) , d) if v ∈ Vt(e) and αe (σS(e) )(n, w) = αe (σt(e) )(n, w) ∀ w ∈ Ve .

(ii )

σ(n, v) = ((tq )q∈Q , d) ⇔

q∈Q

(iii )

Because σ ∈ Σ M P is uniquely determined by (σq )q∈Q ∈ the right-hand side of (25), we write (σq )q∈Q ∈ Σ

σ=

q∈Q

(25)

Σq satisfying

MP

M

for the corresponding element on the left-hand side of (25), and call it the uniﬁcation of (σq )q∈Q ∈ q∈Q Σq . Conversely, every element of Σ M P can be written as the uniﬁcation of an element of q∈Q Σq , so there are projection maps πq , q ∈ Q, given by πq : Σ σ=

MP

→

Σq

(σq )q∈Q

→

σq = πq (σ).

M

We can obtain a better understanding of the behaviors of the tagged system M P by considering the associated network of behaviors. Composing networks of behaviors. Let (Γ, P, M, α) be a network of tagged systems, (Γ, ΣP , ΣM , α) the associated network of behaviors, and S(ΣP ,ΣM ,α) = U ◦ P(P,M,α) : HΓ → Set, 4

Unlike (5), the third condition stated here is no longer redundant.

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the associated functor and H-category. Because of the special structure of an H-category, we can explicitly compute the limit of the functor S(ΣP ,ΣM ,α) ; it is given by HΓ Lim )= P ,ΣM ,α) ←−− (S(Σ   (σq )q∈Q ∈ 

Σq : αe (σS(e) ) = αe (σt(e) ),

∀ e∈E

q∈Q

  

(26) .

Now, there is a bijection: Σ The map from Σ

MP

MP

HΓ ∼ = Lim ←−− (S(ΣP ,ΣM ,α) ).

(27)

HΓ to Lim ←−− (S(ΣP ,ΣM ,α) ) is given by

(πq )q∈Q : Σ

MP

σ

∼

HΓ Lim ←−− (S(ΣP ,ΣM ,α) ) (πq (σ))q∈Q .

−→ →

(28)

The inverse of this map is given by the uniﬁcation operator generalized to the network case. That is M

HΓ (·):← Lim −− (S(ΣP ,ΣM ,α) )

(σq )q∈Q

∼

Σ

→

σ=

−→

(29)

MP

(σq )q∈Q , M

where σ is given as in the left-hand side of equation (25), which is well deHΓ ﬁned because of the deﬁnition of ← Lim −− (S(ΣP ,ΣM ,α) ), i.e., because an element HΓ (σq )q∈Q ∈ Lim ←−− (S(ΣP ,ΣM ,α) ) automatically satisﬁes the right-hand side of (25) by (26). Composition over identity mediator tagged systems. An especially interesting case is when the network of tagged systems is obtained from a network of tag structures (Γ, T, S, ρ), i.e., the network of tagged systems is given by (Γ, P, IS , Resρ ). Here we explicitly carry out the construction of IS P, and demonstrate how this yields the correct deﬁnition of IS P so as to be consistent with [BCC+ 05]. If Pq = (Vq , Tq , Σq ) for all q ∈ Q, then   IS P

HΓ  ρ =← Lim −− (P(P,IS ,Res ) ) =

q∈Q

HΓ Vq , Lim ←−− (T(T,S,ρ) ), Σ

where Σ IS P is deﬁned in the same way as Σ iﬁcations, i.e., Σ IS P is the set of behaviors

MP

IS P

,

with the appropriate mod-

Homogeneous Semantics Preseving Deployments of Heterogeneous Networks

σ :N×

Vq

→

q∈Q

σ(n, v)

→

HΓ Lim ←−− (T(T,S,ρ) ) × D ⊂

Tq × D q∈Q

((tq )q∈Q , d) ,

such that the following conditions hold: for all (n, v) ∈ N × exists unique (σq )q∈Q ∈ q∈Q Σq such that5 for all e ∈ E (i ) σ(n, v) = ((tq )q∈Q , d) ⇔

151

(ii ) (iii )

q∈Q

Vq , there

σS(e) (n, v) = (tS(e) , d) if v ∈ VS(e) and σt(e) (n, v) = (tt(e) , d) if v ∈ Vt(e) and ρe ρe σS(e) |VS(e) ∩Vt(e) = σt(e) |VS(e) ∩Vt(e) .

(30)

HΓ Note that the fact that σ ∈ Σ IS P takes values in Lim ←−− (T(T,S,ρ) ), rather than q∈Q Tq , is exactly because of condition (iii ). The conditions given in (30) demonstrate that our deﬁnition of Σ IS P is consistent with the one given in [BCC+ 05] (although our deﬁnition of Σ M P is more general that anything deﬁned in that paper). Moreover, (26), (27) and (30) imply that we have the following bijection

Σ

IS P

∼ = =

HΓ Lim (S(ΣP ,ΣIS ,Resρ ) ) (31) ← − −     ρ ρe e (σq )q∈Q ∈ Σq : σS(e) |VS(e) ∩Vt(e) = σt(e) |VS(e) ∩Vt(e) , ∀ e ∈ E   q∈Q

as deﬁned in (28) and (29). Here S(ΣP ,ΣIS ,Resρ ) = U ◦ P(P,IS ,Resρ ) : HΓ → Set is the functor corresponding to the network of behaviors (Γ, ΣP , ΣIS , Resρ ) obtained from the network of tagged systems (Γ, P, IS , Resρ ).

7 Semantics Preserving Deployments of Networks Using the framework established in this paper, we are able to introduce a general notion of semantics preservation. After this concept is introduced, we state the main result of this work: necessary and suﬃcient conditions for semantics preservation. We conclude this section by applying this result to the speciﬁc case of network desyncronization. Network speciﬁcation vs. network deployment. Generalizing the notion of speciﬁcation vs. deployment given in Section 4, we deﬁne the following semantics (using the notation of the previous paragraph): 5

Like (5) and unlike (25), the third condition stated here is again redundant because we are taking LimHΓ (T(T,S,ρ) ) as our tag structure. ←−−

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Network Speciﬁcation Semantics: Network Deployment Semantics:

IS P

MP

The set of mediator tagged systems M is said to be semantics preserving with respect to IS , denoted by M P ≡ IS P if for all (σq )q∈Q ∈

q∈Q

∃ σ ∈Σ ∃ σ∈Σ

Σq MP

IS P

s.t. πq (σ ) = σq

∀ q∈Q

s.t. πq (σ) = σq

∀ q ∈ Q.

(32)

We now are able to generalize the results given in Theorem 2 on semantics preservation to the networks of tagged systems case. Theorem 4. For the networks, (Γ, P, M, α) and (Γ, P, IS , Resρ ), MP

if and only if for all (σq )q∈Q ∈ αe (σS(e) ) = αe (σt(e) )

≡

q∈Q

IS P

Σq and all e ∈ E: ρ

e e (σt(e) ). ResρS(e) (σS(e) ) = Rest(e)

⇔

Proof. (Suﬃciency:) If Σq s.t. αe (σS(e) ) = αe (σt(e) )

(σq )q∈Q ∈

∀ e∈E

q∈Q by (26)

⇒

by (29)

⇒

by (32)

⇒

by (28)

⇒

by (31)

⇒

HΓ (σq )q∈Q ∈ Lim ←−− (S(ΣP ,ΣM ,α) ) M (σq )q∈Q ∈ Σ M P where πq ( M (σq )q∈Q ) = σq

∃ σ∈Σ

IS P

s.t. πq (σ) = σq

∀ q∈Q

∀ q∈Q

HΓ

ρ (σq )q∈Q ∈ Lim ←−− (S(ΣP ,ΣIS ,Res ) ) ρe ρe ResS(e) (σS(e) ) = Rest(e) (σt(e) ) ∀ e ∈ E.

The converse direction proceeds in the same manner: if ρ

(σq )q∈Q ∈ q∈Q by (29)

⇒

by (32)

⇒

by (28)

⇒

by (26)

⇒

e e (σt(e) ) Σq s.t. ResρS(e) (σS(e) ) = Rest(e)

IS (σq )q∈Q

∈Σ

πq

∃ σ ∈Σ

MP

where (σ ) q q∈Q = σq IS IS P

s.t. πq (σ ) = σq HΓ

(σq )q∈Q ∈ Lim ←−− (S(ΣP ,ΣM ,α) ) αe (σS(e) ) = αe (σt(e) ) ∀ e ∈ E.

∀ e∈E

∀ q∈Q

∀ q∈Q

Homogeneous Semantics Preseving Deployments of Heterogeneous Networks

153

(Necessity:) We have the following implications: ∃ σ ∈Σ

MP

by (28)

⇒

by (26)

⇒ ⇒

by (29)

⇒

s.t. πq (σ ) = σq

∀ q∈Q

HΓ (σq )q∈Q ∈ Lim ←−− (S(ΣP ,ΣM ,α) ) αe (σS(e) ) = αe (σt(e) ) ∀ e ∈ E ρe e ResρS(e) (σS(e) ) = Rest(e) (σt(e) ) ∀ e ∈ E IS (σq )q∈Q

∈Σ

πq

and (σ ) = σq q q∈Q IS IS P

∀ q ∈ Q. IS (σq )q∈Q

Therefore IS (σq )q∈Q is the element of Σ IS P such that πq σq for all q ∈ Q. The other direction follows in the same way: ∃ σ∈Σ

IS P

⇒

s.t. πq (σ) = σq

=

∀ q∈Q

M (σq )q∈Q ∈ Σ M P and πq ( M (σq )q∈Q )

= σq

∀ q ∈ Q,

by (26), (28) and (29), so M (σq )q∈Q is the element of Σ πq ( M (σq )q∈Q ) = σq for all q ∈ Q.

MP

such that

Network desyncronization. Let T and T be two tag structures and ρ : T → T be a morphism between these tag structures. By slight abuse of notation, let (Γ, T, T , ρ) denote the network of tag structures such that Tq = T for all q ∈ Q and Te = T , ρe = ρe = ρ for all e ∈ E; denote the corresponding network of tagged systems by (Γ, P, IT , Resρ ). Similarly, let (Γ, T, T, id) denote the network of tag structures with TS(e) = Tt(e) = Te = T for all e ∈ E, and with all morphisms of tag structures being the identity; denote the corresponding network of tagged systems by (Γ, P, I, Resid ). Therefore, this network consists of a set of tagged systems, all with the same tag structure, communicating through the identity tagged system. A special case in which this framework in interesting is when T = Ttriv = {∗}; in this case (Γ, P, IT , Resρ ) is the desynchronization of (Γ, P, I, Resid ). Utilizing the notation of Section 6, and generalizing the discussion on desynchronization given in this Section 4, we are interested in when HΓ HΓ P := Lim ←−− (P(T,T,id) ) ≡ Lim ←−− (P(T,T ,ρ) ) =:

IT

P.

In other words, we would like to know when IT is semantics preserving. The following corollary (of Theorem 4) says that this happens exactly when every element of IT is semantics preserving.

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Corollary 2. IT is semantics preserving, P ≡ (σq )q∈Q ∈ q∈Q Σq and all e ∈ E: ρ ρ σS(e) |VS(e) ∩Vt(e) = σt(e) |VS(e) ∩Vt(e)

⇒

IT

P, if and only if for all

σS(e) |VS(e) ∩Vt(e) = σt(e) |VS(e) ∩Vt(e) .

8 Conclusion We have presented a categorical approach to the analysis of heterogeneous composition of tagged systems. This approach has allowed us to derive necessary and suﬃcient conditions for semantics preservation in a heterogeneous network of tagged systems. This result can be considered a generalization of the approach of Benveniste et al. who introduced the notion of tagged systems and their heterogeneous composition and derived suﬃcient conditions for semantics preservation. The main theoretical tool utilized in this paper has been used in hybrid systems [AS05]. This is not surprising as hybrid systems are a special case of heterogeneous systems. To make this slightly more speciﬁc, a hybrid object over a category T is deﬁned to be a functor S : H → T, where T is any category. A network of tagged systems is just a hybrid object over the category TagSys, and a hybrid system is a hybrid object over the category Dyn. Therefore, if general results can be obtained for hybrid objects over categories (such as bisimulation relations, cf. [HTP03]) then they can be applied to networks of embedded systems. Conversely, the intuition gained in this paper on composing networks of tagged systems indicates a possible way of composing networks of dynamical systems: hybrid systems. Acknowledgement. The authors are indebted to Alessandro Pinto for his invaluable feedback. The second author is indebted to Albert Benveniste, Benoit Caillaud, Luca Carloni and Paul Caspi for the many discussions and the work that led to the deﬁnition of tagged systems and the issue of semantics preservation of heterogeneous networks of tagged systems. This research is supported by the National Science Foundation (NSF award number CCR0225610).

Distributed Kalman Filtering and Sensor Fusion in Sensor Networks Reza Olfati-Saber Thayer School of Engineering Dartmouth College Hanover, NH 03755. [email protected] Summary. The problem of distributed Kalman ﬁltering (DKF) for sensor networks is one of the most fundamental distributed estimation problems for scalable sensor fusion. This paper addresses the DKF problem by reducing it to two separate dynamic consensus problems in terms of weighted measurements and inverse-covariance matrices. These to data fusion problems are solved is a distributed way using lowpass and band-pass consensus ﬁlters. Consensus ﬁlters are distributed algorithms that allow calculation of average-consensus of time-varying signals. The stability properties of consensus ﬁlters is discussed in a companion CDC ’05 paper [OSS05b]. We show that a central Kalman ﬁlter for sensor networks can be decomposed into n micro-Kalman ﬁlters with inputs that are provided by two types of consensus ﬁlters. This network of micro-Kalman ﬁlters collectively are capable to provide an estimate of the state of the process (under observation) that is identical to the estimate obtained by a central Kalman ﬁlter given that all nodes agree on two central sums. Later, we demonstrate that our consensus ﬁlters can approximate these sums and that gives an approximate distributed Kalman ﬁltering algorithm. A detailed account of the computational and communication architecture of the algorithm is provided. Simulation results are presented for a sensor network with 200 nodes and more than 1000 links.

1 Introduction Sensor networks and intelligent arrays of micro-sensors have broad range of applications including information gathering and data fusion for modeling an environment, surveillance, active monitoring of forests & agricultural lands, health-care applications, collaborative information processing, and control of smart materials with embedded sensors [EGHK99, HKB99, MKPS01, CHZ02, ASSC02, FCGS02, CMKB04b, OFL04, MB04, CPX05, XBL05a, OSFFS05, FOSPPed]. The most fundamental distributed estimation problem for sensor networks is to develop a distributed algorithm [Lyn97b] for Kalman ﬁltering [AM79]. A

P.J. Antsaklis, P. Tabuada (Eds.): Netw. Emb. Sens. and Cntrl., LNCIS 331, pp. 157–167, 2006. © Springer-Verlag Berlin Heidelberg 2006

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scheme for approximate distributed Kalman ﬁltering (DKF) was proposed in [SOSM05b] based on reaching an average-consensus [OSM04c, SM03a, OS05c]. The work in [SOSM05b] only suggests a scalable scheme to tackle the DKF problem in a special case of full-information and does not contain the suﬃcient analytical results and distributed algorithms necessary to implement a distributed Kalman ﬁlter. This paper provides the essential distributed algorithms and analytical guarantees necessary to establish: a) The DKF problem can be reduced to two dynamic consensus problems regarding fusion of the measurements and covariance information and b) Solving the two dynamic consensus problems requires appropriate consensus ﬁlters (i.e. a low-pass ﬁlter and a band-pass ﬁlter). A detailed discussion of consensus ﬁlters that solve dynamic consensus problems and their stability properties is provided in [OSS05b, SOSM05a]. In particular, the low-pass consensus ﬁlter in [OSS05b] plays a crucial role in both data fusion problems in part b). The problem of decentralized Kalman ﬁltering was ﬁrst solved by Speyer [Spe79]1 in 1979. It was independently resolved by Rao, Durrant-Whyte, and Sheen in [RDWS93]. Both methods require a complete network with all-to-all links. This solution is not scalable for large-scale sensor networks due to its O(n2 ) communication complexity (n is the number of sensors/nodes). Thus, decentralized Kalman ﬁltering and distributed Kalman ﬁltering are two separate problems. In the latter one, each node only is allowed to communicate with its neighbors on a graph G that is connected but rather sparse. Consensus problems [OSM04c, SM03a] and their special cases have been the subject of intensive studies by several researchers [Mes05, OS05c, HM04c, Mor05, RB05b, XB04, XBL05a] in the context of formation control, selfalignment, and ﬂocking [OS05b] in networked dynamic systems. An in-depth comparison between the distributed Kalman ﬁlter introduced here and the existing decentralized fusion algorithms in [ZYZ+ 01, SR05, DW05, GDW94] is the subject of ongoing investigation. An outline of the paper is as follows: Section 2 provides some background on the information form of Kalman ﬁlter. Section 3 contains our ﬁrst main result on decomposition of a Kalman ﬁlter into n collaborative micro-Kalman ﬁlters with local communication. Consensus ﬁlters are described in Section 4. Simulation results for a sensor network with 200 nodes and over 1000 links are presented in Section 5. Finally, concluding remarks are made in Section 6.

2 Kalman Filter: Information Form Consider a sensor network with n sensors that are interconnected via an overlay network G (e.g. a connected undirected graph as shown in Fig. 1). 1

The original work by Speyer was brought up to the attention of the author by J. Shamma and has partially inﬂuenced the choice of the information form of the Kalman ﬁlter as well as the notation used in the paper.

Distributed Kalman Filtering and Sensor Fusion in Sensor Networks

159

Fig. 1. A sensor network with n = 200 nodes and l = 1074 links.

This section describes the so-called information form of the Kalman ﬁlter (IKF) according to [AM79, Spe79]. Let us describe the model of a process (e.g. a physical phenomenon or a moving object) and the sensing model of the IKF as follows: xk+1 zk

= =

Ak xk + Bk wk ; Hk x k + v k

x0

(1)

where zk ∈ Rnp represents the vector of p-dimensional measurements obtained via n sensors, wk and vk are white Gaussian noise (WGN), and x0 ∈ Rm denotes the initial state of the process that is a Gaussian random variable. Here is the information regarding the statistics of these variables: E(wk wl ) x0

= =

Qk δkl , E(vk vl ) = Rk δkl N (¯ x0 , P0 ).

(2) (3)

Given the measurements Zk = {z0 , z1 , . . . , zk }, the state estimates can be expressed as x ˆk Pk

= =

E(xk |Zk ), x ¯k = E(xk |Zk−1 ), Σk|k−1 , Mk = Σk|k

(4) (5)

where Σk|k−1 and Σk|k−1 denote the state covariance matrices and their inverses are known as the information matrices. Note that Σ0|−1 = P0 . Here are the Kalman ﬁlter iterations in the information form:

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Mk−1 Kk x ˆk Pk+1 x ¯k+1

= = = = =

Pk−1 + Hk Rk−1 Hk Mk Hk Rk−1 x ¯k + Kk (zk − Hk x ¯k ) Ak Mk Ak + Bk Qk Bk Ak x ˆk

(6) (7) (8) (9) (10)

3 Distributed Kalman Filter and Micro-KFs Our ﬁrst objective is to show how the information form of a central Kalman ﬁlter for a sensor network observing a process of dimension m with an npdimensional measurement vector zk can be equivalently expressed in consensus form using n micro-Kalman ﬁlters (µKF) with p-dimensional measurement vectors which are embedded in each sensor so that the network of microKalman ﬁlters collectively in a distributed way calculate the same state estimate x ˆ obtained via application of a central Kalman ﬁlter located at a sink node (e.g. for a moving object in a plane p = 2, m = 4 and n 1). Let us assume that there are n sensors with p × m measurement matrices Hi and sensing model: zi (k) = Hi x(k) + vi (k) Thus, deﬁning the central measurement, observation noise, and observation matrix as zc vc Hc

= = =

col(z1 , z2 , . . . , zn ), col(v1 , . . . , vn ), [H1 ; H2 ; · · · ; Hn ],

(11) (12) (13)

where Hc is a column block matrix. We get zc (k) = Hc x(k) + vc (k)

(14)

where the subscript “c” means “central”. Let Rc = diag(R1 , R2 , . . . , Rn ) denote the covariance of vc (i.e. we assume vi ’s are uncorrelated). We have2 M = (P + Hc Rc−1 Hc )−1 and

Kc = M Hc Rc−1 .

Thus, the state propagation equation can be expressed as 2

The iteration numbers are dropped whenever no confusions occur.

Distributed Kalman Filtering and Sensor Fusion in Sensor Networks

x ˆ

= =

x ¯ + Kc (zc − Hc x ¯) −1 x ¯ + M (Hc Rc zc − Hc Rc−1 Hc x ¯)

161

(15) (16)

Deﬁning the following m × m average inverse-covariance matrix S=

1 1 H R−1 Hc = n c c n

n i=1

Hi Ri−1 Hi

(17)

and the m-vector of average measurements yi = Hi Ri−1 zi , y =

1 n

n

yi ,

(18)

i=1

one gets the Kalman state update equation of a µKF as x ˆ=x ¯ + Mµ (y − S x ¯)

(19)

with a micro-Kalman gain of Mµ = nM , measurement consensus y, and inverse-covariance consensus value of S. The expression for Mµ can be stated as follows: Mµ = nM = ((nP )−1 + S)−1 . (20) Denoting Pµ = nP and Qµ = nQ, we obtain an update equation of dimension m × m for a µKF: (21) Pµ+ = AMµ A + BQµ B . Based on the above argument, we have the following decomposition theorem for Kalman ﬁltering in sensor networks: Theorem 1. (distributed Kalman ﬁlter) Consider a sensor network with n sensors and topology G that is a connected graph observing a process of dimension m using p ≤ m sensor measurements. Assume the nodes of the network solve two consensus problems that allow them to calculate average inversecovariance S and average measurements y at every iteration k. Then, every node of the network can calculate the state estimate x ˆ at iteration k using the update equations of its micro-Kalman ﬁlter (or µKF iterations) Mµ x ˆ + Pµ

= = =

(Pµ−1 + S)−1 , x ¯ + Mµ (y − S x ¯), AMµ A + BQµ B ,

(22) (23) (24)

x ¯+

=

Aˆ x.

(25)

This gives an estimate identical to the one obtained via a central Kalman ﬁlter. Remark 1. The gain Mµ of the micro-Kalman ﬁlter has O(m2 ) elements, whereas the Kalman gain K of the central Kalman ﬁlter has O(m2 n) elements. Thus, the calculations of the central KF require manipulation of large matrices which is not computationally feasible.

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Remark 2. We assume all nodes know n or solve a consensus problem to calculate n. This is necessary for calculation of Qµ = nQ. Considering that both S and y are time-varying quantities, one need to solve two dynamic consensus problems that allow asymptotic tracking of the values of S(k) and y(k) [SOSM05b]. The nature of these two dynamic consensus problem diﬀer in nature. Consensus in y(k) requires sensor fusion for noisy measurements yi that can be solved using a newly found distributed lowpass consensus ﬁlter given in [OSS05b]. The consensus regarding the inversecovariance matrices for calculation of S requires a band-pass consensus ﬁlter that will be described in the next section. Neither problems can be solved using a high-pass consensus ﬁlter alone. Based on the results in [OSS05b], the nodes of a network that uses a consensus ﬁlter only reach an ε-consensus (for non-static cases). Meaning that all agents reach a state that is in a closed-ball of radius ε 1 around the group decision value [OSS05b]. This means that practically every node calculates its approximate consensus values Sˆi and yˆi that all belong to small neighborhoods around S and y, respectively. This gives the following state and covariance update equations for the ith µKF: Mi

=

x ˆ

= = =

Pi+ + x ¯

(Pi−1 + Sˆi )−1 , x ¯ + Mi (ˆ yi − Sˆi x ¯), AMi A + BQµ B , Aˆ x,

(26) (27) (28) (29)

with Pi = nP . This is the perturbed version of the exact iterations of the µKF equation in Theorem 1. The convergence analysis of the collective dynamics of the perturbed µKF equations is the subject of future research.

4 Consensus Filters Theorem 1 does not amount to the solution of the DKF problem. So far, we have only managed to show that if two dynamic consensus problems in S and y are solved, then a distributed algorithm for Kalman ﬁltering in sensor networks exists. The crucial part of solving the DKF problem is solving its required dynamic consensus problems which have been addressed in [OSS05b] and partially in [SOSM05a]. We state the distributed algorithms for three consensus ﬁlters: a low-pass ﬁlter, a high-pass ﬁlter, and a resulting band-pass ﬁlter. Let us denote the adjacency and Laplacian matrix [GR01b] of G by A and L = diag(A1) − A, respectively. • Low-Pass Consensus Filter (CFlp , [OSS05b]): Let qi denote the mdimensional state of node i and ui denote the m-dimensional input of node i. Then, the following dynamic consensus algorithm

Distributed Kalman Filtering and Sensor Fusion in Sensor Networks

q˙i =

(qj − qi ) + j∈Ni

(uj − qi )

163

(30)

j∈Ni ∪{i}

that can be equivalently expressed as ˆ − Lu ˆ + (In + A)(u ˆ q˙ = −Lq − x)

(31)

ˆ = L ⊗ Im gives a low-pass with q = col(q1 , . . . , qn ), Aˆ = A ⊗ Im and L consensus ﬁlter with a MIMO transfer function ˆ −1 (In + A) ˆ Hlp (s) = [(s + 1)In + Aˆ + L)

(32)

from input u to output x. This ﬁlter is used for fusion of the measurements that calculates yˆi by applying the algorithm to Hi Ri−1 zi as the input of node i. • High-Pass Consensus Filter (CFhp , [OSS05b, SOSM05a]): Let pi denote the m-dimensional state of node i and ui denote the m-dimensional input of node i. Then, the following dynamic consensus algorithm (pj − pi ) + u˙ i

p˙ i =

(33)

j∈Ni

that can be equivalently expressed as e˙ p

= =

ˆ − Lu ˆ i −Le e+u

(34) (35)

ˆ = L ⊗ Im . This gives a high-pass consensus ﬁlter with an improper with L MIMO transfer function ˆ −1 s Hhp (s) = (sIn + L)

(36)

from input u to output x that becomes In as s → ∞. This ﬁlter apparently propagates high-frequency noise and by itself is inadequate for sensor fusion. • Band-Pass Consensus Filter (CFbp ): This distributed ﬁlter can be deﬁned as Hbp (s) = Hlp (s)Hhp (s) (37) that can be equivalently stated in the form of a dynamic consensus algorithm e˙ i pi

= =

q˙i

=

ˆ i − Lu ˆ i, −Le ei + ui ,

(38) (39)

(qj − qi ) + j∈Ni

(pj − qi ) j∈Ni ∪{i}

(40)

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with a state (ei , qi ) ∈ R2m , input ui , and output qi . This ﬁlter is used for inverse-covariance consensus that calculates Sˆi column-wise for node i by applying the ﬁlter on columns of Hi Ri−1 Hi as the inputs of node i. The matrix version of this ﬁlter can take Hi Ri−1 Hi as the input. Fig. 2 shows the architecture of each node of the sensor network for distributed Kalman ﬁltering. Note that consensus ﬁltering is performed with the same frequency as Kalman ﬁltering. This is a unique feature that completely distinguishes our algorithm with some related work in [SOSM05b, XBL05a].

Node i Sensor Data

Covariance Data

Low-Pass Consensus Filter

Band-Pass Consensus Filter

Micro Kalman Filter µKF) (µ

xˆ

(a)

(b)

Fig. 2. Node and network architecture for distributed Kalman ﬁltering: (a) architecture of consensus ﬁlters and µKF of a node and (b) communication patterns between low-pass/band-pass consensus ﬁlters of neighboring nodes.

5 Simulation Results In this section, we use our consensus ﬁlters jointly with the update equation of the micro-Kalman ﬁlter of each node to obtain an estimate of the position of a moving object in R2 that (approximately) goes in circles. The output matrix is Hi = I2 and the state of the process dynamics is 2-dimensional corresponding to the continuous-time system x˙ = A0 x + B0 w

Distributed Kalman Filtering and Sensor Fusion in Sensor Networks

with

0 1

A0 =

−1 0

165

, B0 = I2

The network has n = 200 sensors with a topology shown in Fig. 1. We use the following data: 1

Ri = 100(i 2 )I2 , Q = 25, P0 = I2 , x0 = (15, −10) . with a step-time of T = 0.02 (sec). Figs 3 and 4 and show the estimate obtained by nodes i = 100, 25. Apparently, the distributed and central Kalman ﬁlters provide almost identical estimates. Of course, the diﬀerence is in scalability of the DKF. In Fig. 5, the consecutive snapshots of estimates of all nodes are shown. The estimates appear as a cohesive set of particles that move around the location of the object.

40

Distributed vs. Cenralized Kalman Filter

Distributed Kalman Filter

20

x

x

20 0

0 −20 0

2

4

6 time (sec)

8

10

−20 0

12

20

2

4

6 time (sec)

8

10

12

2

4

6 time (sec)

8

10

12

20

y

y

0 0

−20 −40 0

2

4

6 time (sec)

(a)

8

10

12

−20 0

(b)

Fig. 3. Distributed position estimation for a moving object by node i = 100: (a) DKF vs. KF (DKF is the smooth curve in red) and (b) Distributed Kalman ﬁlter estimate (in red) vs. the actual position of the object (in blue).

6 Conclusions The importance of distributed Kalman ﬁltering (DKF) for sensor networks was discussed. We addressed the DKF problem by reducing it to two separate dynamic consensus problems in terms of weighted measurements and inverse-covariance matrices that can be viewed as two data fusion problems with diﬀerent natures. Both data fusion problems were solved is a distributed way using consensus ﬁlters. Consensus ﬁlters are distributed algorithms that allow calculation of average-consensus of time-varying signals. We employed a low-pass consensus ﬁlter for fusion of the measurements and a band-pass

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x 2

4

6 time (sec)

8

10

0

−50 0

12

50

50

0

0

−50 0

2

4

6 time (sec)

(a)

8

10

Distributed Kalman Filter

50

0

−50 0

y

Distributed vs. Cenralized Kalman Filter

y

x

50

12

−50 0

2

4

6 time (sec)

8

10

12

2

4

6 time (sec)

8

10

12

(b)

Fig. 4. Distributed position estimation for a moving object by node i = 25: (a) DKF vs. KF (DKF is the smooth curve in red) and (b) Distributed Kalman ﬁlter estimate (in red) vs. the actual position of the object (in blue).

consensus ﬁlter for fusion of the inverse-covariance matrices. Note that the stability properties of consensus ﬁlters is discussed in a companion paper [OSS05b]. We established that a central Kalman ﬁlter for sensor networks can be decomposed into n micro-Kalman ﬁlters with inputs that are provided by two consensus ﬁlters. This network of micro-Kalman ﬁlters was able to collaboratively provide an estimate of the state of the observed process. This estimate is identical to the estimate obtained by a central Kalman ﬁlter given that all nodes agree on two central sums. Consensus ﬁlters can approximate these sums and that gives an approximate distributed Kalman ﬁltering algorithm for sensor networks. Computational and communication architecture of the algorithm was discussed. Simulation results are presented for a sensor network with 200 nodes and 1074 links.

Acknowledgments I would like to thank Jeﬀ Shamma, Emilio Frazzoli, Jason Speyer, Demetri Spanos, Richard Murray, Elisa Franco, and Sonja Martinez for their valuable remarks. This research was supported in part by ARO under the grant #W911NF–04–1–0316, AFOSR MURI grant #F49620-02-1-0325, and NSF ITR grant #0325716.

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Fig. 5. Snapshots of the estimates of all nodes regarding the position of a moving object (a red dot).

Belief Consensus and Distributed Hypothesis Testing in Sensor Networks Reza Olfati-Saber1 , Elisa Franco2 , Emilio Frazzoli3 , and Jeﬀ S. Shamma3 1

2

3

Thayer School of Engineeing Dartmouth College Hanover, NH 03755 [email protected] Department of Electrical, Electronic, & Computer Engineering University of Trieste Trieste, Italy [email protected] Department of Mechanical & Aerospace Engineering University of California at Los Angeles Los Angeles, CA 90095 [email protected] [email protected]

Summary. In this paper, we address distributed hypothesis testing (DHT) in sensor networks and Bayesian networks using the average-consensus algorithm of OlfatiSaber & Murray. As a byproduct, we obtain a novel belief propagation algorithm called Belief Consensus. This algorithm works for connected networks with loops and arbitrary degree sequence. Belief consensus allows distributed computation of products of n beliefs (or conditional probabilities) that belong to n diﬀerent nodes of a network. This capability enables distributed hypothesis testing for a broad variety of applications. We show that this belief propagation admits a Lyapunov function that quantiﬁes the collective disbelief in the network. Belief consensus beneﬁts from scalability, robustness to link failures, convergence under variable topology, asynchronous features of average-consensus algorithm. Some connections between small-word networks and speed of convergence of belief consensus are discussed. A detailed example is provided for distributed detection of multi-target formations in a sensor network. The entire network is capable of reaching a common set of beliefs associated with correctness of diﬀerent hypotheses. We demonstrate that our DHT algorithm successfully identiﬁes a test formation in a network of sensors with self-constructed statistical models.

1 Introduction Sensor networks are comprised of wireless networks of low-cost sensors that can be embedded in an environment for data gathering and monitoring P.J. Antsaklis, P. Tabuada (Eds.): Netw. Emb. Sens. and Cntrl., LNCIS 331, pp. 169–182, 2006. © Springer-Verlag Berlin Heidelberg 2006

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purposes. Over the past decade, both sensor networks and ad-hoc networks of mobile agents have attracted many researchers from diverse backgrounds in computer science, robotics, signal processing, and control theory [RDWS93, EGHK99, HKB99, MKPS01, CHZ02, ASSC02, CMKB04b, OSS05a, OS05a, SOSM05b, XBL05a]. Information processing in sensor networks is closely related to traditional areas of research in systems and control such as multi-target tracking and sensor fusion back in 70’s and 80’s [BS78, Spe79, BSF88, Rei79, ZB93]. In a rather independent path, belief networks and graphical models were introduced in the pioneering work of Pearl [Pea88] that turned into a powerful tool for statistical learning. The connections between the two areas of decentralized sensor fusion & target tracking and decentralized computation (not distributed) on Bayesian networks has been investigated by Rao & DurrantWhyte [RDW93] and more recently by Alanyali et al. [ASSA04] and Saligrama et al. [SAS05]. An important issue that has been addressed is belief propagation in networks with loops (or cycles) [TJ02, YFW04, MVR05] which was not known initially [Pea88]. The objective stated in [MVR05] to average numbers over a network using belief propagation is equivalent in nature to solving average-consensus problem [OSM04c] in an asynchronous mode [HM04c]. Decentralized algorithms and distributed algorithms signiﬁcantly diﬀer in terms of their communication costs. In the decentralized algorithm proposed by Rao & Durrant-Whyte [RDW93] , every node is allowed to talk to all other nodes and therefore the network has O(n2 ) number of links. As a result, a decentralized algorithm is not scalable. In contrast, in our distributed algorithm for hypothesis testing, each node only talks to a few of its neighbors. The total number of links is either O(n), or O(n log(n)) and in either case the algorithm in scalable in n. Hence, the fundamental between a decentralized system and a distributed system is that an all-to-all communication (or fully connected) topology is admissible in a decentralized system whereas it is unacceptable for a distributed system. In this paper, our main objective is to solve the distributed hypothesis testing problem for sensors networks with applications to multi-target tracking and detection. The main features of interest are scalability and robustness of the algorithm to communication failures in the network. We demonstrate that distributed hypothesis testing can be posed as an average-consensus problem [OSM04c, SM03a] in terms of likelihood values of the nodes of the sensor network. As a byproduct of this algorithm, we obtain a novel belief propagation algorithm called belief consensus algorithm. Belief consensus works for loopy networks with arbitrary degree sequence. The convergence properties of this distributed algorithm is analyzed and a Lyapunov function is introduced that measures the disbelief in the network. The belief consensus algorithm allows distributed calculation of products of n probabilities that live in n different nodes. This enables distributed hypothesis testing that is scalable in comparison with a decentralized algorithm presented in [RDW93].

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To demonstrate the eﬀectiveness of belief consensus, we present a detailed example of distributed detection of multi-target formations of m ≥ 2 objects. Each sensor locally constructs statistical models of its observations that are later used for formation identiﬁcation/detection. The sensors execute the belief consensus algorithm to reach a common set of beliefs representing the probability of correctness of multiple hypotheses. The consensus vector of beliefs is used to collectively ﬁnd the most likely hypothesis (or formation) by the entire network of sensors. An outline of the paper is as follows: Section 2 discusses the connections between consensus problems and belief propagation in Bayesian networks and contains our main results. A Lyapunov function for convergence of belief consensus is given in Section 3. Distributed detection of multi-target formations is discussed in Section 3. The simulation results are presented in Section 5. Finally, concluding remarks are stated in Section 6.

2 Belief Consensus on Bayesian Networks Consider a sensor network with n sensors each making an observation z(i). Let Z = {z(1), . . . , z(n)} denote the set of observations/measurements of all sensors in the network. For example, each z(i) can be the position of m moving targets in a multi-target tracking framework. In this paper, we are interested in solving a multi-hypothesis testing problem for sensor networks using a distributed algorithm that primarily relies on the average-consensus algorithm [OSM04c]. Similar to the framework used by Rao & Durrant-Whyte [RDW93], calculation of posterior probabilities in a Bayesian network can be performed based on the following equation: n

p(ha |Z) = αp(ha )

p(z(i)|ha )

(1)

i=1

under the assumption of independence of conditional probabilities (α is a h normalization factor). The set of all hypotheses is H = {ha }na=1 . We are interested in distributed computation of the following quantity that depends on the measurements of n sensors with a network topology G = (V, E) that is an undirected graph: n

p(z(i)|ha ) =⇒ Q =

Q= i=1

1 1 log(Q) = n n

n

log(p(z(i)|ha ))

(2)

i=1

Following the terminology of Pearl [Pea88], let us refer to the conditional probability πi = p(z(i)|ha ) as the belief of agent i in the network. Deﬁning the likelihood of the belief of agent i as li := log(πi ). Following the terminology of Pearl [Pea88], let us call πi the belief of agent i in the network. We have Q =

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Ave{log(πi )}i∈V = Ave(l) and Q = exp(nQ ) = i πi where Ave(·) denotes the average l = (l1 , . . . , ln ) is the vector of likelihoods. In other words, our objective is to calculate the product of beliefs using a distributed algorithm. Let E ⊂ V × V be the set of edges of G. The set of neighbors of node i is deﬁned as Ni = {j ∈ V : (i, j) ∈ E}. The degree of node i is denoted by di = |Ni | (i.e. the number of neighbors of node i). The following averageconsensus algorithm calculates Q (and thus Q) in a distributed way on a network G: x˙ i = (xj − xi ), xi (0) = log(πi ) (3) j∈Ni

The discrete-time average-consensus algorithm can be expressed as xi (t + 1) = xi (t) + ε

(xj − xi ),

xi (0) = log(πi )

(4)

j∈Ni

for t = 0, 1, 2, . . . with 0 < ε < 1/Δ(G) that depends on the maximum node degree of the network Δ = maxi |Ni |. Based on the last iterative algorithm, the collective dynamics of consensus can be rewritten as x(t + 1) = (I − εL)x(t), or x(t + 1) = P x(t) (5) with P = I −εL where L is the graph Laplacian. By deﬁnition, P is a nonnegative matrix. Moreover, due to the fact that 1 = (1, . . . , 1)T is the left and right eigenvector of L (i. e. L1 = 0 and 1T L = 0), P becomes a doubly-stochastic nonnegative matrix. The inequality 0 < ε < 1/Δ(G) guarantees that P is a stable (or primitive) matrix based on Gershgorin’s disk theorem. In the following lemma, we show that all nodes asymptotically reach an agreement regarding Q (and therefore Q) if the network is connected. The solution exponentially converges to a consensus state with a speed that is greater or equal to µ2 = 1 − ελ2 with ε = 1/(1 + Δ). Lemma 1. Consider a connected network of agents with topology G. Suppose every agent applies the consensus algorithm in (4). Then, the state of all agents exponentially converges to x ¯(0) = ( i xi (0))/n with the least rate of µ2 = 1 − ελ2 and average-consensus is asymptotically achieved. Proof. Let x ¯ = Ave(x) and note that x ¯ is an invariant quantity. Set x ˜ i = xi − x ¯ and note that x ˜(t + 1) = P x ˜(t). By deﬁnition, x ˜ = 0 and thus x ˜ is i i orthogonal to 1 = (1, . . . , 1)T . This implies x ˜(t + 1) = P x ˜(t) ≤ µ2 x ˜(t) and as a result x ˜(t) exponentially converges to zero with a speed faster or equal to µ2 and limt→∞ xi (t) = x ¯(0) for all i. See [OSM04c] for more details.

Here is our main result on distributed computation of products of n conditional probabilities and derivation of a novel belief propagation algorithm:

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Theorem 1. Consider a connected (undirected) network of agents that exchange the likelihood of their beliefs li = log(πi ) according to the averageconsensus algorithm in (4). Then, the following statements hold: i) The agents are capable of asymptotically reaching a product-consensus ren garding the value of Q = ( i=1 πi ) with πi > 0, ∀i. ii) The belief consensus algorithm for agreement in πi ’s takes the form πi (t + 1) = πi (t)βi

πj (t)γ

(6)

j∈Ni

where βi = 1 − γdi > 0 for all i and γ satisﬁes 0 < γ < 1/Δ (di is the degree of node i and Δ = maxi di ). n iii) Q = i=1 πi is a invariant quantity along the solutions of (6) and the group consensus value is Q1/n (i.e. πi → Q1/n as t → ∞). Proof. Part i) follows directly from Lemma 1 and the identity Q = exp(nQ ) where Q = Ave(l) is the average of node likelihoods. To prove part ii), let us deﬁne πi = exp(xi ). Taking the exponential of both sides of the averageconsensus algorithm in (4), we obtain a belief consensus algorithm in the form  γ πi (t + 1) = πi (t) 

πj (t)/πi (t)

(7)

j∈Ni

by setting γ = ε. Noting that di = |Ni | and j∈Ni πiγ = πiγdi , equation (6) follows. Part iii) follows from the product-consensus algorithm in the form (7) and the fact that in an undirected network, the ratio πj /πi corresponding to edge (i, j) cancels out with the ratio πj /πi due to edge (j, i). Hence, one obtains γ  n n n πj (t) πi (t)  = πi (t), πi (t)  πi (t + 1) = π (t) πj (t) i=1 i<j i i=1 i=1 or Q(t + 1) = Q(t). Given that the product-consensus algorithm in (6) converges, in the limit πi = πj for all i, j, j = i and Q = πin which means asymptotically the belief of all agents converges to πi = Q1/n . According to Theorem 1, we have arrived at a novel belief propagation algorithm given in (6) for distributed computation of products of beliefs of all agents in a network. We call the following algorithm belief consensus4 : Algorithm 1: Belief Consensus is an iterative belief propagation algorithm in the following form 4

This is due the fact that all agents asymptotically reach the same belief.

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(πj /πi )γ .

πi ← πi

(8)

j∈Ni

Theorem 1 provides the convergece analysis of this algorithm. The belief consensus algorithm works for connected networks with (or without) loops and arbitrary degree sequences. A degree sequence for a graph is a vector (d1 , d2 , . . . , dn ) and an ”arbitrary degree sequence” means there are no restrictions on the degrees of all nodes to be equal or follow a particular pattern, or distribution. Traditionally, Pearl’s original belief propagation algorithm is applicable to trees (or graphs without loops). The work in [ASSA04, SAS05] does not suﬀer from this restriction. However, for networks of ﬁnite size, the algorithm produces meaningful results for regular networks in which the degree of all nodes are equal (i.e. d1 = . . . = dn ). The belief consensus algorithm in (8) does not suﬀer from either of the aforementioned restrictions. An important feature of Pearl’s belief propagation algorithm is that it is asynchronous. Based on recent results by Hatano & Mesbahi [HM04c] and Moreau [Mor05], asynchronous implementations of consensus algorithms converge. Therefore, the belief consensus algorithm in asynchronous mode converges as well. An advantage of equivalence between the average-consensus and belief consensus algorithms is that recently it was demonstrated by Olfati-Saber [OS05c] that quasi-random small-worlds [WS98] are ”ultrafast” in dynamic diﬀusion of information. In the sense that consensus algorithms converge multiple orders of magnitude faster than they do on lattice-type regular networks. Apparently, this provides a means of designing the information ﬂow of sensor networks in such a way that distributed inference and information processing can be carried out tremendously faster. Note that the belief propagation algorithm of Pearl is fairly general and applicable to broader class of learning problems as long as the graphical models are trees. In comparison, belief consensus applies speciﬁcally to hypothesis testing with a wide range of applications and arbitrary graph topology. The main contribution is not belief propagation in loopy graphs, rather introducing a novel belief propagation algorithm that is robust to a changing topology and has a guaranteed speed of convergence characterized by algebraic connectivity (or λ2 ) of the network. Remark 1. (eﬀects of uncertain network size) Suppose each node has an estimate n ˆ i of the size of the network (n) with an error δi , i.e. n ˆ i = n+δi . Assume δi /n 1. Then an estimate of the ith node of Q is ˆ i = exp(ˆ Q ni Q ) = exp(nQ )nˆ i /n = Qνi with limn→∞ νi = limn→∞ (1+ δni ) = 1. Therefore, for complex sensor networks ˆ i ≈ Q for all i. In case, δi = δ is equal for all nodes, then Q ˆ i = Q exp( δ Q ) = Q n Qc where c is a positive constant. This gives a new normalization constant β = cα in equation (1) and has no eﬀect on hypothesis testing.

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3 Lyapunov Function for Belief Consensus A unique advantage of the equivalence between average-consensus in likelihoods in (4) and belief consensus in (8) is that a Lyapunov function is available for convergence of the average-consensus algorithm in discrete-time that is called the disagreement function [OSM04c] and is deﬁned by W (x) = (x − x ¯1)T (x − x ¯1).

(9)

Given the change of coordinates xi = log(πi ), one obtains the following collective disbelief function ϕ(π) = exp(W (x)) ≥ 1

(10)

where π = (π1 , . . . , πn )T . The following theorem explicitly gives this function. Theorem 2. The disbelief function is the product of powers of beliefs with exponents that add up to zero, i.e. in the form ϕ(π) = i πiνi where νi = log(πi ) − (1/n) log(Q) is likelihood deviation of agent i. Furthermore, ϕ(π) takes its minimum value of 1 when all agents have the same belief. Proof. First, note that by deﬁnition of Q, written as

νi = 0. The function W can be

(log(πi ) − 1/n log(Q))2

(xi − x ¯ )2 =

W (x) =

i

i

i

Thus ϕ(π)

=

exp{(log(πi ) − 1/n log(Q))2 }

(11)

exp(νi log(πi /Q1/n ))

(12)

i

= i

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=

i

i

= i

πiνi

/Q(

P i

νi )/n

πiνi

(13) (14)

The last equality is due to i νi = 0. Since ϕ = exp(W ) and W ≥ 0, ϕ takes its minimum at 1 when all agents have the same belief, i.e. log(π) = 1/n log(Q) for all i. Remark 2. For two agents 1 and 2, the total disbelief can be stated as (π1 /π2 )(log(π2 )−log(π2 ))/2 .

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In Bayesian networks with variable topology5 , the disbelief function can be eﬀectively used as a common Lyapunov function in a similar way to the role of disagreement function in consensus on switching networks [OSM04c]. The feasibility of convergence analysis for belief consensus on switching Bayesian networks increases the hope for development of more general belief propagation algorithms for networks with variable interconnections. This framework could possibly have an instrumental role in understanding and modeling reallife biological and social networks with changing structural interactions.

4 Distributed Detection of Multi-target Formations We consider the problem of distributed detection of multi-target formations using a sensor network within a multi-target tracking framework. Each formation of the targets corresponds to a hypothesis ha . We consider nh = 4 types of (unperturbed) formations of targets and accordingly four hypotheses a = 1, . . . , nh . An example of the network of n = 20 sensors performing this distributed detection task is shown in Fig. 1. Remark 3. The term ”formation” here applies to an arrangement of physical objects. In complex social networks, formations represent intent or behavior of a group of agents. The framework discussed here is equally applicable to detection of intent of a group of adversarial agents in games or social systems. The application of the theoretical framework presented in this paper in social networks is the subject of ongoing research.

Fig. 1. A sensor network with n = 20 nodes.

5

To the best of our knowledge, there exist no results on belief propagation on networks with changing topologies.

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4.1 Sensing Model of Sensors Each sensor in a network with communication topology G observes a noisy set of measurements of the positions of m targets. The formations used in this paper all consists of m = 12 targets and are shown in Fig. 2. Let qj = (qj1 , qj2 )T ∈ R2 be the position of the jth target, then the ith sensor observes a noisy version of the position of all targets that is an m-vector z(i) = (y1 (i), . . . , ym (i)) in the form: yj (i) wj1 (i) wj2 (i)

= = =

qj + wj (i); ri cos(θi )); ri sin(θi ).

(15) (16) (17)

where wj (i) is the measurement noise of the ith sensor for observation of the jth target. The quantities ri and θi are random variables with uniform distribution over the intervals [0, r¯i ] and [−π, π].

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4.2 Feature Vectors for Multi-target Formation Detection We deﬁne a set of features for multi-target formations that are functions of the position of all targets in a formation but do not depend on ordering of the targets (i.e. the features are invariant under permutation of indices of the target). In addition. all the features are invariant with respect to rotation and

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translation of all targets. Let z be the observed position of m targets by some sensor (the index of the sensor is unimportant in this discussion). Deﬁne the radius of formation z as ρ(z) = max

j∈1,...,m

z(j) − z¯ ; z¯ =

1 m

m

z(j). j=1

Given the radius ρ(z), one can deﬁne a set of targets that are neighbors of each other using N (ρ) = {i, j : z(j) − z(i) ≤ ρ}. Moreover, let J = [z(1)|z(2)| · · · |z(m)] denote the 2 × m matrix of coordinates of points in the formation. Here are the ﬁve chosen features in the form x = f (z): m

x1

=

(

z(i) 2 )1/2

(18)

j=1

x2 x3 x4 x4

= = = =

λmax (JJ T )1/2 |N (ρ(z))| |N (2ρ(z))| |N (4ρ(z))|

(19) (20) (21) (22)

where | · | denotes the number of elements of a set. 4.3 Learning Statistical Models of Formations Every sensor represents a formation using its feature vector x = f (z) and based on nt = 100 observations (training data) creates Gaussian distribution of the features xk , k = 1, . . . , nf = 5. This can be done by measures mean and variance of the observed data as µk (i), σk2 (i) for sensors i = 1, . . . , n. An example of these distributions is shown in Fig. 3. 4.4 Multi-target Formation Detection In the detection stage, we would like to use a maximum a-posterior (MAP) estimator that ﬁrst calculates p(ha |Z) for a = 1, . . . , 4 using belief consensus algorithm and then uses the following recognition step: a∗ = arg max p(ha |Z). a

For doing so, ﬁrst we need to relate the statistics of the features of a formation to its prior probability p(zi |ha ). This can be greatly simpliﬁed by assuming that all features are statistically independent, i.e.

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p(z(i)|ha ) :=

p(xk (i)|ha )

(23)

k=1

where x(i) = f (z(i)) is the feature vector of the observation of the ith node. Regardless of the node index, we assume that each feature has a Gaussian distribution. As a result, the probability that a feature xk is in an interval of length δ 1 can be approximated as p(xk |ha ) ≈ δ ×

1 (xk − µk )2 exp − σk2 2πσk2

.

Since δ is a positive constant, it will be eventually absorbed in the normalization constant of posterior probability p(ha |Z) and therefore it can be dropped. Hence, we apply the following approximation for the kth feature calculated

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locally by every sensor according to sensor’s statistical model of mean and variance of nt previously observed data: p(xk |ha ) =

(xk − µk )2 1 exp − σk2 2πσk2

.

(24)

for the kth feature calculated locally by every sensor according to sensor’s statistical model of mean and variance of nt previously observed multi-target formations. 4.5 Belief Consensus Assuming that all nh formations are equally likely to occur, we obtain p(ha ) = 1/nh . Based on the procedure given in the preceding section, each sensor can calculate its nh -dimensional vector of beliefs πi (a) = p(zi |ha ). A MAP estimator only requires this knowledge that can be calculated either directly using belief consensus in (8), or indirectly using average-consensus algorithm in (4) (in terms of likelihood of beliefs).

5 Simulation Results Consider a network of n = 20 sensor that are distributed randomly in a square region as shown in Fig. 1. The objective is to detect a test formation after learning the statistical features of training data of nh = 4 formations shown in Fig. 2. Each sensor observes a randomly perturbed version of the formation of m = 12 targets that are spaced approximately at a distance of d = 2.The perturbation radius of sensors monotonically increases by their index as r¯i = 0.1 + .02i to create varying degrees of uncertainty. All sensors as well as the sensor network as a whole manage to correctly identify the test formation. h The sensors asymptotically reach a common set of beliefs {πi (a)}na=1 that allow them to choose the most likely hypothesis with maximum likelihood. The result of distributed detection of multi-target formations for a square formation (a = 3) are presented in Fig. 4. Apparently, the winning hypothesis is the one that its likelihood is negative but near zero (giving a relatively high common belief by all agents. We also tested whether a network of sensors that is trained with formations taht have m = 12 targets is capable of identifying formations with less or more targets. It turns out that the answer is aﬃrmitive for small changes in the number of targets.

6 Conclusions A novel belief propagation algorithm called belief consensus was introduced for distributed computation of products of beliefs of n nodes of a sensor network.

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200 0 −200

likelihoods

−400 −600 −800 −1000 −1200 −1400 −1600 0

20

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iterations

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Fig. 4. Evolution of likelihood of n = 20 nodes of a sensor network for four hypotheses based on belief consensus.

This computation is instrumental in distributed hypothesis testing in Bayesian networks. Belief consensus is a byproduct of the average-consensus algorithm in [OSM04c]. This belief propagation algorithm works for connected networks with loops/cycles and arbitrary degree sequence. A Lyapunov function was given for belief consensus in the form of collective disbelief of all agents. The disbelief function has a potentially important role in convergence analysis of belief consensus in Bayesian networks with variable topology. It was discussed that belief consensus (due to its connection to average-consensus) converges multiple orders of magnitude faster on quasi-random small-world networks compared to regular lattices or highly clustered small-worlds [OS05c]. A detailed example was provided for distributed detection of multi-target formations using a network of sensors. Belief consensus was used to reach a common set of beliefs associated with correctness of diﬀerent hypotheses. Then a MAP estimator was used to identify the formation/hypothesis with maximum likelihood. All agents could use a local MAP estimator after they reach a consensus with other agents regarding their set of beliefs. It was demonstrated that belief consensus successfully identiﬁes a test formation in a network of sensors with individually built statistical models of previously observed multitarget formations. Recently, belief consensus has been applied to distributed fault diagnosis in network embedded systems [FOSPPed].

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Acknowledgments This research was supported in part by ARO under the grant #W911NF– 04–1–0316, AFOSR MURI grant #F49620-02-1-0325, and NSF ITR grant #0325716. This work was performed while the ﬁrst author was a visiting scientist at the University of California, Los Angeles.

Distributed Evidence Filtering in Networked Embedded Systems Duminda Dewasurendra1 , Peter Bauer1 , and Kamal Premaratne2 1

2

Department of Electrical Engineering University of Notre Dame Notre Dame, IN {ddewasur, pbauer}@nd.edu Department of Electrical and Computer Engineering University of Miami Miami, FL [email protected]

Summary. Networked embedded sensor systems use the evidence gathered by spatially distributed heterogeneous sensor nodes possessing partial and diﬀerent scopes of expertise, to make inferences on the scenario under observation. For such system to make an accurate collective decision, the partial and incomplete evidence provided by nodes must be processed in a simple and straightforward manner during the information exchange and fusion process. To achieve these objectives, we present a novel, uniﬁed approach named “non-recursive evidence ﬁltering” based on the Dempster-Shafer (DS) formalism for evidence representation. It is capable of selectively fusing partial evidence in such a network to directly infer on events of interest such as threats occurring with a certain temporal distribution, while accommodating the varying reliability and accuracy of information sources. Certain restrictions on the coeﬃcients impose several challenges in the design of such evidence ﬁlters. We show that the gain of these evidence ﬁlters at frequency zero is always equal to one and all coeﬃcients must be non-negative. This suggests that arbitrary frequency shaping is not possible and a pure bandpass evidence ﬁlter is not realizable. A method to design a simple FIR evidence ﬁlter to detect periodic events of interest is presented. Multi-dimensional spatio-temporal evidence ﬁlters are discussed as a direct extension to the above, along with a low signature threat detection example that clearly illustrates the eﬀectiveness of the evidence ﬁltering concept in distributed networked embedded systems.

1 Introduction Networked embedded systems are playing an increasingly important role in today’s state-of-the-art distributed sensing applications. Many such applications use the signals generated by spatially distributed multiple heterogeneous sensor nodes over a period of time to infer on events of interest. No single node P.J. Antsaklis, P. Tabuada (Eds.): Netw. Emb. Sens. and Cntrl., LNCIS 331, pp. 183–198, 2006. © Springer-Verlag Berlin Heidelberg 2006

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has access to all the information and hence the inferences they provide can only be partial or incomplete. However the system as a whole is capable of collectively making accurate inferences or decisions on the scenario under observation by properly exchanging the data among the nodes [FJLJ03]. Due to the immense amount of raw signals acquired by the nodes, it is almost impossible to make a centralized decision by collecting all these raw data signals at a single node. Instead, a more practical approach is to preprocess raw sensor signals at individual nodes and exchange the processed information among the other nodes. Nevertheless, each node is an embedded system with only limited computational capacity, small range radio connectivity and limited battery life, making this a challenging task [DAW+ 04]. In such distributed networked embedded systems, because the preprocessed information or evidence is generated from heterogeneous sources that possess partial and diﬀerent scopes of expertise, the imperfections associated with these partial decisions must be appropriately modeled at each node if evidence combination is to be eﬀectively carried out over the network [P97, DJ01]. Moreover the reliability of each evidence source (sensor) can vary widely and, if not properly accommodated, these diﬀerences can have a signiﬁcant impact on the decisions being made [EKD+ 04]. New evidence with low reliability should not be given high signiﬁcance, priority or bandwidth [JEKP01]. In addition, an existing knowledge base in a particular node possesses a certain inertia and integrity [EKD+ 04, KDP05]. These need to be accounted for before new evidence from other nodes are used to update its current viewpoint. If the existing knowledge base consists of evidence gathered from highly reliable sources, or if it was constructed from a large number of sources over a long period of time, its high inertia has to be accommodated during the updating process. These issues are directly related to the burden placed on the network for information exchange during collective decision making. Hence, the overall success of the networked embedded sensing system will be determined in general by its ability to [JEKP01, EKD+ 04, RBA05, KDP05], • represent the essential information buried in the acquired raw sensor signals in a compact form, • accommodate the reliability, relative accuracy and the inertia of the evidence during information exchange over the network and in the fusion process, and • fuse partial evidence in a simple and straightforward manner to make direct inferences on the scene under observation. In this paper, we present a novel, uniﬁed approach named non-recursive evidence ﬁltering to achieve these objectives in a networked embedded system. Our approach is based on the Dempster-Shafer (DS) formalism for evidence representation [G76, SA98]. It is capable of selectively fusing partial evidence in such a network to directly infer on events of interest such as threats occur-

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ring with a certain temporal distribution, while accommodating the varying reliability and accuracy of information sources. The paper is organized as follows: Section 2 presents a brief review of DS theory. The concept of evidential ﬁltering is presented in Section 3 for the 1-D and spatio-temporal m-D case. The methods we have developed for designing FIR evidence ﬁlters is discussed in Section 4 followed by conclusions and future work.

2 Dempster Shafer Theory Evidence theory [G76] can be considered a generalization of Bayesian probability theory. It is capable of more easily accommodating the qualitative aspects of information during knowledge representation and reﬁnement under uncertainties [SA98, SAX01]. The underlying notions of this theory are brieﬂy discussed in this section. 2.1 Notation Reals are denoted via . We denote the total set of mutually exclusive and exhaustive propositions via Θ = {θ1 , . . . , θn }, viz., Θ is the corresponding frame of discernment (FOD) signifying the scope of ‘expertise’ and is the equivalent of the sample space of events in a Bayesian probability framework. A proposition θi represents the lowest level of discernible information. |Θ| denotes the cardinality of set Θ, and 2Θ denotes its power set. Propositions of interest are then of the form A ⊆ Θ generated from the power set 2Θ of Θ. We use A − B, A, B ⊆ Θ, to denote all propositions in A after removal of those propositions that may imply B, i.e., A − B = {θ : θ ∈ A, θ ∈ B}; A denotes Θ − A. The support for any proposition A is provided via a mass assignment m : 2Θ → [0, 1] where (i) m(∅) = 0. (ii) m(A) = 1. A⊆Θ

Those propositions in a frame Θ that possesses nonzero masses are called the focal elements of Θ and the set of all focal elements is denoted by F (Θ). For a given frame Θ, the triple {Θ, F , m} is referred to as its body of evidence (BOE). Given a BOE {Θ, F , m}, the belief assigned to A ⊆ Θ is Bel : 2Θ → [0, 1] where Bel(A) = m(B) (1) B⊆A

and the plausibility of A ⊆ Θ is P l : 2Θ → [0, 1] where P l(A) =

m(B). B∩A=∅

(2)

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The uncertainty associated with A, denoted by U n(A), can be deﬁned using the interval U n(A) = [Bel(A), P l(A)]. (3) The quantity m(A) measures the support assigned to proposition A only and to no smaller subset of A. However the belief assigned to any proposition is also committed to any other proposition it implies. In other words, belief of A must take into account the supports for all proper subsets of A as well. Hence Bel(A) represents the total support that can move into A without any ambiguity. When each focal set contains only one element, i.e., m(A) = 0, ∀|A| = 1, belief functions become probability functions. By assigning masses for composite propositions, it is quite straightforward to maintain uncertainty intervals for events using DS theory. This is a highly useful property within the context of networked embedded sensor systems where we need to quantify the uncertainty of possible events. Non-zero mass for a proposition A with |A| > 1 would represent the initial ambiguity among the events constituting A. For example, mass assignment m({a, b}) = 0.3 for the doubleton proposition A = {a, b} would represent the ‘ﬂoating’ mass which an embedded sensor node can commit to neither a nor b initially when the current evidence is not reﬁned enough to support each individual singleton a or b. With further evidence arriving at later times, this ﬂoating mass can move to support either singleton a or b. This leads to the direct quantiﬁcation of uncertainty on each event a and b using (3). In contrast to Bayesian probability theory, DS theory allows one to avoid casting knowledge to A if no evidence is available to directly support the event A. This is denoted by the relation Bel(A) + Bel(A) ≤ 1

(4)

The portion that is not assigned to A can be considered as the ignorance one has regarding the occurrence of A, and can be kept unassigned till we receive further evidence. Hence compared to probability theory, DS framework could be easily used in situations where a priori mass assignments are not feasible with the limited initial information available. In fact, [JR93] argues that the DS theory is far less reliant on prior information compared to Bayesian probability theory. By accommodating more evidence over time, the uncertainty interval may be reduced and one may be able to predict the occurrence or non-occurrence of A with a higher degree of certainty. These intuitive notions make DS theory an ideal candidate for evidence representation in distributed environments where a vast amount of information is processed over time in the presence of numerous types of uncertainties. When further evidence represented by perhaps a diﬀerent BOE spanning the same FOD is received from another sensor node, DS combination rule is employed to fuse the two, and to arrive at a combined BOE.

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Deﬁnition 1. Dempster’s Rule of Combination [G76]: The orthogonal sum of two mass assignments m1 : 2Θ → [0, 1] and m2 : 2Θ → [0, 1] deﬁned over the same FOD Θ is denoted by m1 ⊕ m2 : 2Θ → [0, 1] where m1 (C) · m2 (D) (m1 ⊕ m2 )(A) =

C∩D=A

m1 (C) · m2 (D)

1−

for C, D ∈ F (Θ),

(5)

C∩D=∅

provided the BOEs are non contradicting, i.e., m1 (C) · m2 (D) < 1.

(6)

C∩D=∅

Hence, the original DS formulation is limited to fusion among sources with identical FODs.

3 Evidence Filtering In our previous work [KDP05, EKD+ 04], we have introduced a novel DS theoretic evidence updating function to avoid certain problems associated with (5) and to even accommodate the combination of evidence generated from non-identical FODs. It is based on the premise that a knowledge base should only consider the portion of the incoming evidence that it is capable of discerning itself. We will consider the simpliﬁed case of identical FODs here. Suppose a node with a knowledge base denoted by the BOE1 {Θ, F1 , m1 } desires to update itself using new incoming evidence arriving from another node denoted by the BOE2 {Θ, F2 , m2 }, conditional to the occurrence of event A ⊆ Θ. Then the current knowledge base available in the ﬁrst node can be updated via : Deﬁnition 2. Conditional evidence combination function (CECF) [KDP05]: Consider the two BOEs {Θ, F1 , m1 } and {Θ, F2 , m2 }, and a given A ∈ F2 . Then for an arbitrary B ⊆ Θ1 , the updated belief Bel1 : 2Θ → [0, 1] and updated plausibility P l1 : 2Θ → [0, 1] are given by Bel(B)1 (k + 1) = αk · Bel(B)1 (k) + βk · Bel(B|A)2 (k) Pl(B)1 (k + 1) = αk · Pl(B)1 (k) + βk · Pl(B|A)2 (k)

(7)

where αk ≥ 0, βk ≥ 0 and αk + βk = 1. The constraints on positivity and sum are needed to ensure that the updated beliefs constitute a valid belief function. In eﬀect, the combined evidence is taken to be a linear combination of the available evidence and the new evidence conditioned to a proposition that has occurred. An interesting probabilistic interpretation and several intuitively appealing properties of the above updating strategy appear in [KDP05] along

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with various strategies for choosing the linear combination weights αk and βk . By properly choosing these coeﬃcients, one can easily accommodate for the reliability, relative accuracy and the inertia of the available evidence during the fusion process. The conditional in (7) is the Fagin-Halpern conditional: Deﬁnition 3. Conditional Belief and Plausibility [RJ91]: For the BOE {Θ, F , m}, A ∈ F2 and an arbitrary B ⊆ Θ, the conditional belief Bel(B|A) : 2Θ → [0, 1] and conditional plausibility P l(B|A) : 2Θ → [0, 1] of B given A are Bel(A ∩ B) Bel(A ∩ B) + Pl(A − B) Pl(A ∩ B) Pl(B|A) = Pl(A ∩ B) + Bel(A − B)

Bel(B|A) =

(8)

This can be considered a more natural extension of the Bayesian conditional notions. In this evidence updating strategy, the term Bel(B|A)2 (k) captures the portion of the incoming evidence conditioned to the current knowledge base while Bel(B)1 (k) captures the already available evidence. The updated belief is given by Bel(B)1 (k + 1) and similar notions hold for the plausibility. The subscripts 1 and 2 denote BOEs the evidence is generated from. Note that, these subscripts are discarded in the following discussion for simplicity. The index k in (7) denotes the temporal ordering of the evidence. Let k represent a discrete time index t = kT where T denotes the sampling time of the evidence at each node and t denotes the continuous time. Consider a sequence of new incoming evidence on B ⊆ Θ denoted by discrete time indices k = 0, 1, . . . , n − 1 with a ﬁxed T . Assume the initial belief is Bel(B)(0). Repeated application of (7) with ﬁxed α, β yields the diﬀerence equation Bel(B)(n) = αn · Bel(B)(0) + β ·

n−1

αi · Bel(B|A)(n − i − 1)

(9)

i=0

This can be interpreted as a 1-D, ﬁrst order linear ‘evidence ﬁlter’. If one assumes constant coeﬃcients α, β weighing the current and incoming evidence, then this type of ﬁlter can be characterized in the time domain by its impulse response hB (n) and in the spectral domain by its discrete time Z-transform HB (z). β z −1 (10) 1 − α z −1 The constant coeﬃcient α determine how much the past evidence is being weighed. An impulse response with a higher α value indicates that the past evidence already gathered is much more signiﬁcant compared to the incoming hB (n) = β · αn , n ≥ 0

⇐⇒

HB (z) =

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evidence. In case of lower α, the past evidence is weighed very little and accordingly the time horizon considered for decision making is very short. This interpretation of (7) gives useful insights to the frequency characteristic of the evidence (z = ejω ) such as detecting objects that are periodically occurring in the ﬁeld. A distinct advantage of this approach is that by using higher values of α, we can ensure that the past evidence is emphasized more, or else we can ﬁlter evidence over a very short time horizon using smaller values for α. Several versions of updated evidence can be computed in parallel, resulting in the notion of evidence ﬁlter banks. 3.1 Higher Order Evidence Filters This concept of evidence ﬁltering in time can be generalized to higher orders which could further enhance the evidence updating strategy. The most general form of (7) can be stated as the N -th order diﬀerence equation N

Bel(B)(k) =

N

αi · Bel(B)(k − i) + i=1

βj · Bel(B|A)(k − j)

(11)

j=1

where constant coeﬃcients αi , βi ≥ 0 and responds to the transfer function HB (z) =

1−

N i=1

αi +

N −j j=1 βj z N −i i=1 αi z

N j=1

βj = 1. This cor-

(12)

which is an N -th order IIR evidence ﬁlter. Hence, by properly choosing the coeﬃcients, we can directly obtain certain spectral characteristics of the evidence updating strategy which is highly useful in the context of distributed networked embedded systems. Note that this approach integrates the information fusion process with the detection of spectral characteristics of the events, while accommodating the key issues highlighted in Section 1 in a simple and straightforward manner. 3.2 Spatio-temporal Evidence Filtering Almost all real world applications of distributed networked embedded sensing take into consideration the occurrence of events of interest over both time and space. Hence, we further extend the above concept of temporal evidence ﬁltering to exploit spatio-temporal correlations of the collected evidence. The concept of evidence ﬁltering versus other forms of ﬁltering is illustrated in Figure 1. In eﬀect, an evidence ﬁlter adds one more dimension to the general spatio-temporal ﬁltering and explores the correlations among sensor signals of various modalities to improve and enhance the detection.

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Modality

n

mod 4

Spatio Temporal filter

Spatial filter

mod 3 mod 2 mod 1 mod 0

Temporal filter

t

Fig. 1. Concept of evidence ﬁltering

A general m-dimensional spatio-temporal evidence ﬁlter takes the form, Bel(B)(k1 , k2 , . . . , km ) =

...

αi1 ,i2 ,...,im Bel(B)(k1 − i1 , k2 − i2 , . . . , km − im )

+

...

βj1 ,j2 ...,jm Bel(B|A)(k1 − j1 , k2 − j2 , . . . , km − jm )

(13)

with indices k1 , k2 , . . . , km representing each dimension. The sum and positivity constraints for coeﬃcients hold true as before. For example, consider the 3-dimensional evidence ﬁlter consisting of 2-D space (indices n1 , n2 ) and 1-D time (index t) characterized by the diﬀerence equation Bel(B)(n1 , n2 , t) =

αi1 ,i2 ,τ Bel(B)(n1 − i1 , n2 − i2 , t − τ )

+

βj1 ,j2 ,τ Bel(B|A)(n1 − j1 , n2 − j2 , t − τ ).

(14)

This can be interpreted as an updating strategy that accounts for how the event A that occurs at times (t−τ ) and at locations (n1 −j1 , n2 −j2 ) contribute to the proposition B at time t and at location (n1 , n2 ). We note that, to compute the belief of B at time t and at location (n1 , n2 ), the beliefs of B computed at times (t − τ ) at at locations (n1 − i1 , n2 − i2 ) are required. Its spectral characteristics are given by the transfer function HB (z1 , z2 ) =

1−

N N −j1 −j2 j2 =1 βj1 ,j2 z1 z2 j1 =1 N N −i1 −i2 i1 =1 i2 =1 αi1 ,i2 z1 z2

.

(15)

The spatio-temporal evidence updating strategy in (14) requires certain causality conditions for the implementation of multi-dimensional recursion

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n2

n1 Fig. 2. First quadrant causality

[DR90]. The causality relationship in time is obvious (only past belief values contribute to the current belief). To support a 2-D recursion, ﬁrst quadrant causality must hold in the (n1 , n2 ) plane as illustrated in Figure 2 [JS89]. The arrows indicate the direction in which the locally generated belief values at each node need to be propagated. In order to avoid diﬃculties associated with the design of recursive m-D ﬁlters [DR90] in the presence of additional constraints, we will focus on the non-recursive case in Section 4.

4 Evidence Filter Design The sum and positivity constraints on the coeﬃcients in (11) impose several challenges in the evidence ﬁlter design. As a direct consequence we note that HB (ω)|ω=0 = 1, i.e., its DC gain is always equal to unity. Therefore, arbitrary frequency shaping is not possible and a pure bandpass that only detects event that occur at a certain repetition frequency is not realizable. We therefore investigate other methods of evidence ﬁlter design that rely on the output of a bank of ﬁlters. 4.1 FIR Evidence Filters In this section, we present the design of a bandpass FIR evidence ﬁlter with strictly positive coeﬃcients. Suppose we require an evidence ﬁlter to detect a certain type of periodically occurring threats in a region under surveillance. Assume synchronized network sampling and exchanging evidence with a sampling time of 4 seconds. To detect events on this particular type of threat

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dB

50

100

150

0

0.2

0.4

0.6

Normalized Frequency

0.8

1

Fig. 3. Spectral response of the original ﬁlter implementation

Coefficient Value

0.05

0

0.05

0

50

100

150

200

250

300

350

400

Fig. 4. Coeﬃcients of the original FIR ﬁlter implementation

occurring approximately at every 5 seconds, we desire a bandpass evidence ﬁlter centered around relative frequency 0.8. A length 400 FIR implementation of this ﬁlter using a Hamming window yields the spectral response shown in Figure 3 and its respective coeﬃcients are shown in Figure 4. To convert this into an evidence ﬁlter satisfying the sum and positivity constraints, we move the coeﬃcients to the positive region by adding a positive coeﬃcient shift. This can be characterized by adding a window function with length 400 and amplitude equal to the maximum negative coeﬃcient. Nevertheless, the addition of such a square window to the coeﬃcients would be equivalent to the addition of a sinc function in the frequency domain, thereby increasing the ripples in the ﬁlter frequency response. Hence, to minimize this eﬀect and maintain a minimum ripple ratio, we chose to add a Chebyshev window instead of a square window. A Chebyshev type window produces a small ripple ratio compared to many other window types. A Kaiser window

Distributed Evidence Filtering in Networked Embedded Systems 1 0.8 0.6 0.4 0.2 0

0

50

100

150

200

250

300

350

400

Fig. 5. Chebyshev window used to modify coeﬃcients

Coefficient value

15

x 10

3

10

5

0 0

50

100

150

200

250

300

350

400

Fig. 6. Modiﬁed Coeﬃcients after positive shift and scaling

0 20

dB

40 60 80 100 120 140

0

0.2

0.4 0.6 Normalized frequency

0.8

Fig. 7. Modiﬁed spectral response of the evidence ﬁlter

1

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would also be an attractive solution due to its parameterization. The one used in this design is shown in Figure 5 and is an approximation of the FIR impulse response envelope in Figure 4. The shifted coeﬃcients were scaled down such that they meet the sum constraint and the resulting coeﬃcients are shown in Figure 6. Such modiﬁcation of coeﬃcients inevitably alters the frequency response of the evidence ﬁlter as shown in Figure 7. Note that this resulting evidence ﬁlter has a narrow passband around frequency zero as expected, but otherwise the original ﬁlter characteristic has been preserved.

5 Low Signature Threat Detection In this section, we illustrate the multidimensional evidence ﬁltering concepts discussed in Section 3.2 using a simple 2-D example. Consider a vehicle traveling at a constant velocity along a straight path, as illustrated in 8. For simplicity, we consider only 3 modalities of detection assuming the vehicle consists of metal, and emits sound and light. Nodes ﬁtted with various types of sensors are placed along the path as part of a distributed networked embedded system for surveillance. In this scenario, the local objective is to detect the presence of a threat object (vehicle) characterized by acoustic(A), metallic(M), and light(L) properties traveling at a normalized speed of 4.

Fig. 8. Sensor nodes placed alongside vehicle path Table 1. Nodes equipped with sensor types A(acoustic), M(magnetic) and L(light) Distance to Node 12 14 17 21 23 25 30 32 36 38 39 46 50 A x x x x x x x M x x x x x x L x

Consider a path length of 50 and assume that the nodes equipped with sensors capable of sensing modalities A,M,L are placed along the path as shown in Table 1. It is important to note that the sensors capable of detecting this particular object have a highly sparse distribution along the path. We also note that, no single node that is capable of detecting all the three modalities is available.

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(a) 0.6 0.4 0.2 0 0.2

0

2

4

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6

(b)

8

10

12

8

10

12

8

10

12

0.6 0.4 0.2 0 0.2

(c)

0.6 0.4 0.2 0 0.2

time

Fig. 9. Raw sensor signals (a) Modality A from node at 12, (b) Modality M from node at 17, (c) Modality L from node at 39

0.2

Magnitude

0.15

0.1

0.05

0 0 20

Distance

40 60

9 10 7 8 5 6 4 3 Time 1 2

Fig. 10. Velocity ﬁlter impulse response

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Each sensor is modeled as generating a detection signal similar to that shown in Figure 9, cluttered by heavy noise. The amplitude and the width of the peak generated due to the vehicle passing close to a particular sensor along the path is dependant on various factors such as the sensitivity and detection radius of the sensor. In this simulation, we used approximate peak amplitudes 0.7, 0.8, 0.6 and detection radii 0.5, 0.5, 1 for modalities A,M,L respectively to generate the detection signals, since actual sensor data from a test setup was unavailable. The challenging task is to detect this low signature threat using the available few sensor signals. For this purpose, we employ the simple velocity ﬁlter shown in Figure 10. The diagonal impulse response shown here selectively ﬁlters the required velocity of 4 while the Hamming window shape minimizes the ripple eﬀects. The collection of all signals gathered on modality M and L are shown in Figures 11(a) and 11(b) respectively (Signals from modality A sensors not shown here). The response of the velocity ﬁlter shown in Figure 10 for the raw sensor signals from modalities M and L are shown in Figures 11(c) and 11(d) respectively. Figure 11(e) shows the response of the same ﬁlter with input as belief values calculated at each node. If a raw sensor signal passes a threshold of 0.5, the conditional belief Bel(B|A)2 (k) in (7) corresponding to the sensor modality is set to 1. These conditional beliefs were used in (7) to generate a BOE inside each node, based on their partial view of the scenario. Initial beliefs at each node for all three modalities were assumed to be zero and β was set to 1. The combined evidence from all nodes were processed together using the evidence ﬁlter given in Figure 10, as described in Section 3. Note that the signature of the threat object (vehicle) is clearly detected by using such basic evidence ﬁltering techniques as shown in 11(e), compared to the response from raw signal ﬁltering in 11(c) and 11(d).

6 Conclusion In this paper, we introduced the novel concept of evidence ﬁltering for distributed networked embedded sensor systems. Based on the DS evidential theory, it is capable of representing the essential information buried in the raw sensor signals as evidence and make direct inferences on the spectral characteristics of events of interest in a simple and straightforward manner. Spatio-temporal evidence ﬁlters were introduced as an extension to the general spatio-temporal ﬁlters and its generalization to the m-D case was brieﬂy discussed. A method to design a simple FIR evidence ﬁlter to detect periodic events of interest was also presented. The low signature threat detection example presented here clearly illustrates the eﬀectiveness of the evidence ﬁltering concept. Future work includes the development of design methods for IIR type evidence ﬁlters and extension of the concepts developed here to accommodate

Distributed Evidence Filtering in Networked Embedded Systems

(a) Raw signal - modality M

(b) Raw signal - modality L

(c) Response - modality M alone

(d) Response - modality L alone

(e) Response of evidence ﬁlter Fig. 11. 2D signal analysis

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heterogeneous sources. The 1-D evidence ﬁlter design presented in Section 4 can be extended to 2-D systems as shown in the example presented in Section 5.

7 Acknowledgements This material is based upon work supported by the National Science Foundation (NSF) under grants IIS-0325252 (ITR Medium) at University of Notre Dame and IIS-0325260 (ITR Medium) and IIS-0513702 at University of Miami.

Plenary Talk Delays in Control over Communication Networks: Characterization, Impact, and Reduction Strategies Dawn Tilbury Department of Mechanical Engineering University of Michigan [email protected]

The most notable trend in manufacturing over the past ﬁve years is probably the move towards networks at all levels of industrial systems. At lower levels in the factory infrastructure, these networks provide solutions with higher reliability, visibility and diagnosability, and enable capabilities such as distributed control, diagnostics, safety, and device interoperability. At higher levels, especially with the availability of internet services, networks enable factory-wide automated scheduling, control, and diagnostics; improve data storage and visibility; and open the door to e-manufacturing. The diagnostic and reconﬁgurability advantages of networked devices tempt control designers to implement feedback systems with networked communication of sensor and control signals. However, delays will inevitably occur, and it is well-known that time delays can degrade the performance of control systems. In this talk, we characterize the diﬀerent contributions of network delay for several types of popular industrial networks, including DeviceNet and Ethernet. Experimental results of the measured end-to-end delays are presented and analyzed. A qualitative characterization of the eﬀect of delays on performance is presented. Since the network delays are most signiﬁcant when the network traﬃc is high, we present several control strategies we have developed that can signiﬁcantly reduce the network traﬃc while maintaining closed-loop performance. To conclude, we will describe the testbed we have developed for networked control research and outline some future work in the area.

P.J. Antsaklis, P. Tabuada (Eds.): Netw. Emb. Sens. and Cntrl., LNCIS 331, pp. 201–202, 2006. © Springer-Verlag Berlin Heidelberg 2006

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D. Tilbury

Dawn M. Tilbury received the B.S. degree in Electrical Engineering, summa cum laude, from the University of Minnesota in 1989, and the M.S. and Ph.D. degrees in Electrical Engineering and Computer Sciences from the University of California, Berkeley, in 1992 and 1994, respectively. In 1995, she joined the faculty of the Mechanical Engineering Department at the University of Michigan, Ann Arbor, where she is currently an Associate Professor. She won the EDUCOM Medal (jointly with Professor William Messner of Carnegie Mellon University) in 1997 for her work on the web-based Control Tutorials for Matlab. An expanded version, Control Tutorials for Matlab and Simulink, was published by Addison-Wesley in 1999. She is co-author (with Joseph Hellerstein, Yixin Diao, and Sujay Parekh) of the book Feedback Control of Computing Systems. She received an NSF CAREER award in 1999, and is the 2001 recipient of the Donald P. Eckman Award of the American Automatic Control Council. She belongs to ASME, IEEE, and SWE, and is a member of the 2004-2005 class of the Defense Science Study Group (DSSG). Her research interests include distributed control of mechanical systems with network communication, logic control of manufacturing systems, and performance management and control of computing systems.

Anticipative and Non-anticipative Controller Design for Network Control Systems Payam Naghshtabrizi and Jo˜ ao P. Hespanha Department of Electrical and Computer Engineering Univ. of California at Santa Barbara Santa Barbara, CA 93106-9560 {payam,hespanha}@ece.ucsb.edu Summary. We propose a numerical procedure to design a linear output-feedback controller for a remote linear plant in which the loop is closed through a network. The controller stabilizes the plant in the presence of delays, sampling, and packet dropouts in the (sensor) measurement and actuation channels. We consider two types of control units: anticipative and non-anticipative. In both cases the closedloop system with delays, sampling, and packet dropouts can be modeled as delay diﬀerential equations. Our method of designing the controller parameters is based on the Lyapunov-Krasovskii theorem and a linear cone complementarity algorithm. Numerical examples show that the proposed design method is signiﬁcantly better than the existing ones.

1 Introduction Network Control Systems (NCSs) are spatially distributed systems in which the communication between plants, sensors, actuators, and controllers occurs through a shared band-limited digital communication network. Using network as a medium to connect spatially distributed elements of the system results in ﬂexible architectures and generally reduces wiring and maintenance cost, since there is no need for point to point wiring. Consequently, NCSs have been ﬁnding application in a broad range of areas such as mobile sensor networks, remote surgery, haptics collaboration over the Internet and unmanned aerial vehicles [HNX]. However, the use of a shared network, in contrast to using several dedicated independent connections, introduces new challenges. To transmit a continuous-time signal over a network, the signal must be appropriately sampled to be carried over a network as atomic units called packet. Hence there are some similarities between NCSs and sampled-data systems due to the sampling eﬀect. However, NCSs are signiﬁcantly diﬀerent from standard sampled-data systems since the delay in the feedback loop can be highly variable due to both access delay (i.e., the time it takes for a P.J. Antsaklis, P. Tabuada (Eds.): Netw. Emb. Sens. and Cntrl., LNCIS 331, pp. 203–218, 2006. © Springer-Verlag Berlin Heidelberg 2006

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shared network to accept data) and transmission delay (i.e., the time during which data are in transit inside the network). Both types of delays depend on highly variable network conditions such as congestion and channel quality. Since access and transmission delays have the same eﬀect with respect to the stability of NCSs, throughout the paper we use the term (NCS) delay to refer to access/transmission delay. Another signiﬁcant diﬀerence between NCSs and standard digital control is the possibility that data may be lost while in transit through the network. Typically, packet dropout results from transmission errors in physical network links (which is far more common in wireless than in wired networks) or from buﬀer overﬂows due to congestion. Long transmission delays sometimes result in out of order delivery, which can amount to a packet dropout if the receiver discards “outdated” arrivals. Reliable transmission protocols, such as TCP, guarantee the eventual delivery of packets. However, these protocols are not appropriate for NCSs since the re-transmission of old data is generally not very useful. We consider two types of control units: non-anticipative and anticipative. A non-anticipative control unit sends control updates at times ta , ∈ N, equal to {u : ∀ ∈ N}, where u is a single-value control command to be applied to the plant and held until the next control update arrives. An observerbased controller, constructs estimate x ˆ of the plant’s state x. These estimates are used to compute the updates u := −K x ˆ(ta ) sent through the actuation a channel to the actuators at times t , ∈ N suﬀering a (possibly variable) delay of τ a ≥ 0. In a loss-less network, the control signal u(t) is therefore updated according to u(t) = u = −K x ˆ(ta ),

∀t ∈ [ta + τ a , ta+1 + τ a+1 ),

∈ N.

An anticipative controller attempts to compensate the sampling and delay introduced by the actuation channel. For simplicity, we assume that the actuation channel is sampled with constant sampling interval ha = ta+1 − ta , ∀ ∈ N and that its delay is constant and equal to τ a = τ a , ∀ ∈ N. At each sampling time ta = ha , ∈ N the controller sends a time-varying control signal u (·) that should be used from the time ha + τ a at which it arrives until the time ( + 1)ha + τ a at which the next control update will arrive. This leads to u(t) = u (t),

∀t ∈ [ ha + τ a , ( + 1)ha + τ a ),

∈ N,

where u (t) is equal to −K x ˆ(t). However, the estimates x ˆ(·) needed in the interval [ ha + τ a , ( + 1)ha + τ a ) must be available at the transmission time ha , which requires the control unit to estimate the plant’s state up to ha + τ a time units into the future. [BPZ00a], [Nil98c] model NCSs as discrete-time systems. However, this approach requires the assumption that the total delay in the control loop

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is smaller than the sampling interval, otherwise the analysis becomes significantly more complex [LB00]. Alternatively NCSs can be modeled as delay diﬀerential equations (DDEs) for which the after mentioned restriction can be easily lifted [HNX]. An NCS with LTI plant model, anticipative or nonanticipative controller, with delays, sampling, and packet dropouts can be modeled as a DDE of the form ¯˙ (t) = A0 x x ¯(t) +

2

Ai x ¯ t − τi (t) , τi (t) ∈ [τi min , τi max , τ˙i (t) = 1 a.e. (1) i=1

The delay’s bounds τi min and τi max for i ∈ {1, 2}, are positive and depend on the sampling intervals, maximum number of consecutive packet dropouts, and upper and lower bounds on the delay in the measurement and the actuation channels. We ﬁnd suﬃcient conditions for the stability of (1), based on a Lyapunov-Krasovskii functional, in the form of matrix inequalities. For a given controller, the matrix inequalities turn out to be linear (LMIs). However, in design problems in which the controller parameters also appear as unknowns, the matrix inequalities are bilinear (BMIs) and non-convex. We propose a numerical method based on the linear cone complementarity algorithm introduced in [GOA97] to solve this problem. This method converts the feasibility of the original non-convex matrix inequalities into a sequence of the convex optimizations of a linear function subject to a set of LMIs, which can be eﬀectively solved by numerical packages such as MATLAB. Related work The stability of NCSs has received signiﬁcant attention in the literature. [ZB01] model NCSs with time-varying sampling intervals as hybrid systems and ﬁnd suﬃcient condition for stability. [MA03b] study the stability of modelbased NCSs in which the network delays are negligible and the measurement channel is loss-less and the controller is directly connected to the plant. The authors use an explicit model of the plant to produce an estimate of the plant’s state between transmission times which allows them to reduce the network usage. [LZA03] investigate the stability of NCSs with uncertain time delays and packet dropouts in the framework of switched systems. Signiﬁcant work has been devoted to ﬁnding upper bounds on transmission intervals, i.e., tk+1 − tk , ∀k ∈ N for which stability of the closed-loop is guaranteed. These upper bounds are sometimes called the maximum allowable transfer interval (MATI). [WYB99b, WYB01] ﬁnd MATI for linear and nonlinear systems. [NT04] study the input-output stability properties of general nonlinear NCSs using an argument based on small gain theorem. [YWCH04] design an observer-type output feedback controller to stabilize a plant through a network with admissible bounds on data packet losses and delays, by modeling NCSs as DDEs and using the Razumikhin theorem to prove the stability of DDEs. Our work is inspired by [YWCH04], but instead of the Razumikhin

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theorem, which generally leads to conservative designs, our analysis is based on a new descriptor system approach and the Lyapunov-Krasovskii functional proposed by [FS02a]. We refer the readers to [HNX] for a more through review of related literature. The results in this paper generalized those in [NH05] by considering variable intervals. Notations The notation is introduced as and when necessary. The set of all non-negative integers is denoted by N. Superscripts ’s’ and ’a’ are used to refer to the (sensor) measurement and the actuator channels and subscripts k and j are discrete indices taking values in N used for the measurement and the actuator channel events respectively. For example tsk and taj refer to the k-th and the jth sampling times in the measurement and the actuation channels respectively. This paper is organized as follows: In section 2 we introduce anticipative and non-anticipative control units. We show that the resulted closed-loop systems can be written as (1). In section 3, we provide a suﬃcient condition for asymptotic stability of (1) as a set of LMIs. In section 4 a numerical procedure is proposed to design a controller that stabilizes the plant for given admissible bound on delays, number of consecutive packet dropouts and sampling intervals. Then through examples we illustrate the use of our method.

2 Network Control Systems Modeling Fig. 1 shows an NCS consisting of a plant, actuators, sensors, and a control unit where the plant, actuators, and sensors are compound. The plant is LTI with state space model of the form x(t) = Ax(t) + Bu(t), ˙

y(t) = Cx(t),

Cont GH

(2)

Meas e

t

a

P

ul

yk N

uhl

q

uh

j

K

n

M

yhk H

g NM

Fig. 1. Two-channel feedback NCS with observer-based controller.

Anticipative and Non-anticipative Controller Design

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where x(t) ∈ Rn , u(t) ∈ Rl , y(t) ∈ Rm are the state, input, and output of the plant respectively. The measurements are sampled and sent at times tsk , k ∈ N. Assuming, for now, that there are no dropouts, the measurements {y(tsk ) : k ∈ N} are received by the control unit at times tsk + τks . These are used to construct an estimate of the plant state using s x ˆ˙ (t) = Aˆ x(t) + B u ˆ(t) + L y(tsk ) − C x ˆ(tsk ) , ∀t ∈ tsk + τks , tsk+1 + τk+1 , (3)

where u ˆ(t) is an estimate of the plant’s input at time t. Since u is constructed from data sent by the controller, in general we have u ˆ = u. We consider two types of control units: non-anticipative and anticipative. The two units defer on the construction of the control signal u. 2.1 Non-anticipative Control Unit Control signal The control unit sends control updates at times ta , ∈ N, equal to {−K x ˆ(ta ) : ∈ N}. In the absence of drops, these arrive at the plant a at times t +1 + τ a+1 , ∈ N, leading to u(t) = −K x ˆ(ta ),

∀t ∈ ta + τ a , ta+1 + τ a+1 .

Delay diﬀerential equation formulation

(4)

Deﬁning

τ s (t) := t − tsk ,

s , ∀t ∈ tsk + τks , tsk+1 + τk+1

τ a (t) := t − ta ,

∀t ∈ ta + τ a , ta+1 + τ a+1 ,

which we call them ﬁctitious delays (as opposed to the actual network delays τks and τja ), we can re-write (3) and (4) as 1 x ˆ˙ (t) = Aˆ x(t) + B u ˆ(t) + L y(t − τ s ) − Cx(t − τ s ) , u(t) = −K x ˆ(t − τ a ).

(5)

Note that s } , τ s ∈ min{τks }, max{tsk+1 − tsk + τk+1

∀k ∈ N , τ˙ s = 1 a.e.,

τ a ∈ min{τ a }, max{ta+1 − ta + τ a+1 } ,

∀k ∈ N , τ˙ a = 1 a.e.,

k

k

Fig. 2.a shows the evolution of τ s with respect to time where τks = τ¯s , ∀k ∈ N and constant sampling interval tsk+1 − tsk = hs , ∀k ∈ N. The derivative of τ s is almost always one, except at the times that new packets arrive to the controller side at tsk+1 + τ¯s , where τ s (the ﬁctitious delay) drops to τ¯s (the actual network delay). 1

For simplicity of notation through the paper we use τ s and τ a as τ s (t) and τ a (t) respectively.

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P. Naghshtabrizi and J.P. Hespanha r

r g

a

a

b

b

d

c

t

e m

c

d

e

f

t

n

s

Fig. 2. Evolution of τ with respect to time when (a) There is no packet dropout, (b) The packet sent at tsk is dropped.

Packet dropouts Packet dropouts in the measurement and actuation channels can be viewed as the delay that grow beyond the deﬁned bounds. If ms and ma consecutive packet dropouts happen in the measurement and actuation channels then s τ s ∈ min{τks }, max{tsk+ms +1 − tsk + τk+m s +1 } , k

k

τ a ∈ min{τ a }, max{ta+ma +1 − ta + τ a+ma +1 } . Fig. 2.b shows the situation that the measurement packet sent at t = tsk is dropped and as a result τ s grows up to 2hs + τ s . Closed-loop Deﬁning e := x − x ˆ, with regard to (2) and (5), the closed-loop can be written as x(t) ˙ A = e(t) ˙ 0

0 A

x(t) 0 + e(t) 0

0 −LC

xˆ(t) 0 + e(t) 0

LC −LC

x(t − τ s ) −BK + e(t − τ s ) 0

BK 0

x(t − τ a ) e(t − τ a )

0 0

xˆ(t − τ a ) . e(t − τ a )

or alternatively, as A xˆ˙ (t) = 0 e(t) ˙

0 A

xˆ(t − τ s ) −BK + e(t − τ s ) 0

2.2 Anticipative Control Unit Control signal For simplicity, we assume that the actuation channel is sampled with constant sampling interval ha = ta+1 − ta , ∀ ∈ N and that its delay is constant and equal to τ¯a = τ a , ∀ ∈ N. At each sampling time ta = ha , ∈ N the controller sends a time-varying control signal u (·) that should be used from the time ha + τ¯a at which it arrives until the time ( + 1)ha + τ¯a at which the next control update will arrive. This leads to u(t) = u (t),

∀t ∈ [ ha + τ¯a , ( + 1)ha + τ¯a ),

∈ N.

(6)

To stabilize (2), u (t) should be equal to −K x ˆ(t), where x ˆ(t) is an estimate of x(t). However, the estimates x ˆ(·) needed in the interval [ ha +¯ τ a , ( +1)ha +¯ τ a) must be available at the transmission time ha , which requires the control unit to estimate the plant’s state up to ha + τ¯a time units into the future.

Anticipative and Non-anticipative Controller Design

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Remark 1. Anticipative controllers send actuation signals to be used during time intervals of duration ha , therefore the sample and hold blocks in Fig. 1 should be understood in a broad sense. In practice, the sample block would send over the network some parametric form of the control signal u (·) (e.g., the coeﬃcients of a polynomial approximation to this signal). State predictor An estimate z(t) of x(t + ha + τ¯a ) is constructed as follows: z(t) ˙ = Az(t) + B u ˆ(t + ha + τ¯a ) + L yˆ(tsk ) − Cz(tsk − ha − τ¯a ) ,

(7)

s for ∀t ∈ [tsk + τks , tsk+1 + τk+1 ), k ∈ N. To compensate for the time varying delay and packet dropouts in the actuation channel, z would have to estimate x further into the future. Hence the assumptions of constant delay and loss-less actuation channel can be relaxed by predicting x further into the future. Control signal construction With such estimate available, the signal u (t) sent at times ta = ha , to be used in ha + τ¯a , ( + 1)ha + τ¯a , is then given by

u (t) = −Kz(t − ha − τ¯a ),

∀t ∈ [ ha + τ¯a , ( + 1)ha + τ¯a ),

∈ N,

(8)

which only requires the knowledge of z(.) in the interval t ∈ ( − 1)ha , ha , and therefore is available at the transmission time ha . Delay diﬀerential equation formulation Deﬁning s , ∀t ∈ tsk + τks , tsk+1 + τk+1

τ s (t) := t − tsk ,

and assuming that u ˆ = u, we conclude from (6),(7), and (8) that z(t) ˙ = (A − BK)z(t) + L y(t − τ s ) − Cz(t − ha − τ¯a − τ s ) , s

τ (t) ∈

min{τks }, max{tsk+1 k k

−

tsk

+

s } τk+1

,

∀k ∈ N,

(9)

s

τ˙ = 1 a.e.

Closed-loop Deﬁning e(t) := x(t + ha + τ¯a ) − z(t) and considering (2),(8), and (9), the closed-loop can be written as z(t) ˙ A − BK = e(t) ˙ 0

0 A

z(t) 0 + e(t) 0

LC −LC

z(t − ha − τ¯a − τ s ) . (10) e(t − ha − τ¯a − τ s )

3 Stability of Delay Diﬀerential Equations It was shown in section 2 that both types of control units result in the closedloop models of the following form x ¯˙ (t) = A0 x ¯(t) +

2

Ai x ¯ t − τi (t) , τi (t) ∈ [τi min , τi max , τ˙i (t) = 1 a.e. (11) i=1

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Until recently it was believed that the only available tool to study the stability of delay equations of the form (11) was the Razumikhin theorem. [FS02b] were able to use the Lyapunov-Krasovskii theorem to study the stability of system (11). In [FSR04b] they studied the stability of sampled-data systems with input delays as DDEs of the form (11) where τ1 ∈ [0, hs ). In sampleddata systems, at each sampling time delay drops to zero. However in NCSs as new information arrives the ﬁctitious delay drops to the actual network delay. The next theorem gives a suﬃcient condition for the asymptotic stability of (11) where the matrices Ai is 2n × 2n for i ∈ {0, 1, 2}. Theorem 1. The system (11) is asymptotically stable, if there exist 2n × 2n matrices P1 > 0, P2 , P3 , Si , Ri > 0 and 4n×4n matrices Z1i , Z2i and 2n×4n matrices Ti , i = 1, 2, that satisfy the following set of matrix inequalities 2 :   0 0 Ψ P A1 − T1 P A2 − T2   < 0,  (12a)  ∗ −S1 0 ∗ ∗ −S2 Ri ∗ Ri ∗

0

Ai P > 0, Z2i

Ti > 0, Z1i

∀i ∈ {1, 2},

(12b)

∀i ∈ {1, 2},

(12c)

where P := 2

Φ := i=1

Si 0

P1 P2

0 , P3

Ψ := P

0 A0

I 0 + −I A0

I −I

P + Φ (13)

T T 0 + τi min Z1i + (τi max − τi min )Z2i + i + i 0 0 τi max Ri

.

Proof of Theorem 1 See Appendix. Remark 2. The “triangular” structure of (10) is unique to the anticipate controller. With this type of controller, if we choose K so that A−BK is Hurwitz, asymptotic stability of the NCS is equivalent to the asymptotic stability of the (decoupled) dynamics of the error e(t) := x(t + ha + τ¯a ) − z(t), which is given by the following DDE e(t) ˙ = A e(t) − LC e t − τ (t) ,

t ≥ 0,

with τ ∈ [τmin , τmax ), τ˙ = 1, a.e., where s a ¯a }, τmin := min{τks + ha + τ¯a }, τmax := max{tsk+1+ms − tsk + τk+1+m s + h + τ k∈N

k∈N

and Theorem 1 provides the stability test with A0 = A and A1 = −LC. 2

Matrix entries denoted by ’*’ are implicitly deﬁned by the fact that the matrix is symmetric.

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Remark 3. Suppose that the matrices of the system Ω := A0 A1 A2 are not exactly known and that instead they are given as polytopic uncertainty given by N

N

f j Ωj ,

Ω∈

0 ≤ fj ≤ 1,

j=1

fj = 1 , j=1

where the N vertices of the polytope are described by Ωj := Aj0 A1j A2j . Stability of the system can be checked by solving the LMIs in Theorem 1 for each of the individual vertices and the same matrix variables.

4 Observer-Based Controller Design for NCSs When the controller parameters L and K are known, the system matrices Ai for i ∈ {0, 1, 2}, are constant and known. Hence (12a)-(12c) are in the form of LMIs and the stability of NCSs can be established easily. However when L and K are also variables to be deﬁned, since Ai contain unknowns, the matrix inequalities in Theorem 1 are in the form of BMIs. The next lemma taken from [dOCS00], plays a central role in transforming BMIs to a suitable form for a numerical algorithm we will develop. Lemma 1. Assume that Q is a symmetric matrix. There exists a symmetric matrix N > 0 such that J N U + U N J + Q < 0,

(14)

if and only if there exist symmetric matrices X, Y and a scalar α > 0 such that X = α2 Y −1 and U XU − Q ∗

J + αU Y

> 0.

(15)

The proof is obtained by Schur Lemma and the equation X = αN . Lemma 1 changes the matrix inequality (14), with products of unknowns J and N , into the inequality (15) in which such products are absent. However, (15) requires the non-convex constraint X = α2 Y −1 . Our goal now is to rewrite the matrix inequalities in Theorem 1 in the form of (14) and ultimately in the form of (15). It turns out that this form is suitable for developing a numerical procedure to compute K, L, and the other matrix variables in Theorem 1. Suppose that P2 > 0, P3 > 0, after matrix manipulations (12a) can be written as J0 (K, L) N U0 + U0 N J0 (K, L) + Q0 < 0, where   Γ −T1 −T2 A0 −I A1 A2 0  , J0 (K, L) = , Q0 =  ∗ −S1 A0 −I A1 A2 ∗ ∗ −S2 U0 =

I 0

0 I

0 0

0 , 0

N=

P2 0

0 > 0, P3

Γ =

0 P1

P1 + Φ. 0

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The matrix inequality (12b) can be written as Ji (K, L) N Ui + Ui N Ji (K, L) + Qi < 0, where Qi = −

Ri 0

0 , Z2i

Ji (K, L) = −

Ai Ai

0 0

0 , 0

Ui =

0 0

I 0

0 , I

for i ∈ {1, 2}. Theorem 1 can be read as the following theorem: Theorem 2. The anticipative or non-anticipative controller given in section 2 with the matrix variables K and L asymptotically stabilizes the plant (2) for given τi min and τi max , if there exist 2n×2n matrices P1 > 0, X1 > 0, X2 > 0, Y1 , Y2 , Si , Ri > 0, 4n × 4n Z1i , Z2i , 2n × 4n matrices Ti , n × 1 matrix L, 1 × n matrix K and α > 0 that satisfy the following matrix inequalities for i ∈ {1, 2}: U0 XU0 − Q0 ∗ Ui XUi − Qi ∗

J0 + αU0 > 0, Y

(16)

Ji + αUi >0 Y

(17)

Ri ∗

(18)

Ti > 0, Z1i

where X = α2 Y −1 ,

X=

X1 0

0 , X2

Y =

Y1 0

0 . Y2

Since A0 , A1 , A2 are linear functions of K and L, the matrix inequalities in (16) are in the form of LMIs also in the unknowns. However, the condition X = α2 Y −1 is not convex. In the next section we introduce a numerical procedure to solve this non-convex problem. Remark 4. A simple but conservative way to make the matrix inequalities in Theorem 1 suitable for controller synthesis for anticipative controllers with regard to Remark 2, consists of requiring that P2 > 0, P3 = ρP2 , for some positive constant ρ > 0 and making the (bijective) change of variables Y = P2 L, which transforms (12) into   YC − −T  Ψ ρY C < 0, ∗ −S R ∗

ρC Y − CY Z2

> 0,

R ∗

T > 0, Z1

with Ψ given by (13). This inequality is linear in the unknowns and can therefore be solved using eﬃcient numerical algorithms. The observer gain is found using L = P2−1 Y . This procedure introduces some conservativeness because it restricts P3 to be a scalar multiple of P2 .

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Numerical procedure We modify the complementarity linearization algorithm introduced in [GOA97] to construct a procedure to determine the feasibility of the matrix inequalities in Theorem 2 by solving a sequence of LMI’s as follows: 1. Pick α. 2. Find a feasible solution denoted by X0 , Y0 for the set of LMIs (16) and X I

I α−2 Y

≥0

(19)

3. Find the feasible solution denoted by Xj+1 , Yj+1 that solves the following problem Σj : min tr(Xj Y + XYj ) subject to (16), (19). 4. We choose satisfying (12a) and (12b) with N = α−1 X as the stopping criterion. If the matrix inequalities are satisﬁed, exit. Otherwise set j = j + 1 and go to step 3 if j < c (a preset number). The sequence Σj is monotonically decreasing, lower bounded by 8n × α−2 achieved at X = α2 Y −1 , if the LMIs in (16) are feasible with the constraint X = α2 Y −1 . It is numerically very diﬃcult to obtain the minimum such that the tr(Xj Yj+1 + Xj+1 Yj ) equals to 8n × α−2 . Instead we choose (12a) and (12b) with N = α−1 X as the stopping criterion. If after c times the stopping criterion is not satisﬁed we pick another α and we continue line search until the stopping criteria is satisﬁed. Examples Example 1. [BPZ00a, ZB01] Consider the following plant 0 x˙ 1 = 0 x˙ 2

1 −0.1

0 x1 + u, 0.1 x2

and assume that all the states are available for feedback. We assume there is no delay and packet dropout in the network and the state feedback K = 3.75 11.5 stabilizes the plant. [BPZ00a] model the system as a hybrid system and show that the closed-loop is stable as long as the sampling intervals are constant and equal to 4.5 × 10−4 . Later [ZB01] ﬁnd the less conservative upper bound 0.0593 for variable sampling intervals. Based on an exhaustive search the same authors ﬁnd that the ”true” upper bound is roughly 1.7194. The maximum variable sampling interval based on Theorem 1 is 0.8965.

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P. Naghshtabrizi and J.P. Hespanha 0.8 0.7

maximum NCS delay

0.6 0.5 0.4 0.3 0.2 0.1 0

0

1

2 3 4 Number of packet dropouts

5

6

Fig. 3. Number of consecutive packet dropouts versus the maximum of actual network delay. Results in our paper and [YWCH04] are shown by ∗ and respectively.

Suppose now that just the ﬁrst state is available for the feedback. Based on Theorem 2 and the proposed algorithm, we design a controller to stabilize the plant when the sampling interval of the measurement channel is constant and equal to hsk = 0.5, ∀k ∈ N. A non-anticipative control unit with K = 3.3348

9.9103 ,

L = 0.6772

0.1875 ,

stabilizes the plant for any actuation sampling intervals such that ha ≤ 0.7330, ∀ ∈ N. With the same measurement sampling interval, an anticipative control unit with K = 28.5347

83.8626 ,

L = 0.3518

0.0492 ,

stabilizes the plant for any actuation sampling intervals such that ha ≤ 0.976, ∀ ∈ N. As we expected the anticipative controller stabilizes the plant for larger sampling intervals. Example 2. [YWCH04] consider the following state space plant model −1.7 x˙ 1 = −1 x˙ 2

3.8 1.8

5 x1 + u, 2.01 x2

y(t) = 10.1

4.5

x1 , x2

with the controller directly connected to the actuator, which means that there is no delay, sampling, and packet dropouts in the actuation channel. [YWCH04] show that the controller with gains K = −.2115

2.5044 ,

L = 0.1043

0.0518 ,

Anticipative and Non-anticipative Controller Design

215

stabilizes the system for any ﬁctitious delay τ s in the interval [0, 0.3195]. Consequently as long as (tk+1+ms − tk ) + τks ≤ 0.3195 holds, the closed-loop system remains stable. Fig. 3 shows the number of consecutive packet dropouts versus the actual network delay such that the system remains stable in which hs := tk+1 − tk = .1s, ∀k ∈ N. Our result reveals a signiﬁcant improvement in comparison to [YWCH04]. For instance, when the measurement channel is loss-less (ms = 0, τi max − τi min = hs ), the LMIs in Theorem 2 are feasible up to τi max = 0.865 which means the closed-loop system with K = −1.7436

1.1409 ,

L = 0.0675

0.0267 ,

is stable for any τks ∈ [0, 0.765], ∀k ∈ N. If we assume a maximum number of consecutive packet dropouts ms = 6 (which corresponds to τi max − τi min = 7 × hs ), the LMIs are feasible up to τi max = 0.773 and the closed-loop system with K = −0.5310

0.1668 ,

L = 0.0564

0.0221 ,

is stable for any τks ∈ [0, 0.073], ∀k ∈ N. As expected, for a smaller number of consecutive packet dropouts, the system remains stable for larger τi max .

2.5

0.8 0.7

2 norm of the error

2

0.6 0.5

1.5

0.4 1

0.3 0.2

0.5

0.1 0

0

5

10

15

20 time

(a)

25

30

35

40

0

0

5

10

15

20

25

30

35

40

(b)

Fig. 4. Simulation results considering the nonlinear model for the inverted pendulum (a) L2 norm of state x, (b) L2 norm of error e = x(t + ha ) − z(t).

Example 3. [LCMA05] In this example we design an anticipative controller for an inverted pendulum. The linearized model is given by

216

P. Naghshtabrizi and J.P. Hespanha



0 0   x˙ = 1   0

0 0 0 3

(M + m)g l(m + 4M )

1 0 mg −3 m + 4M β+ε 3 l(m + 4M )

   0 0    0 0    β+ε  4α    u, x +  −4  m + 4M  m + 4M     −3α 0 l(m + 4M )

where x = X θ X˙ θ˙ ; M, m, l, g are the mass of the cart, the mass of the pendulum, half of the length of the pendulum, acceleration due to the gravity, α, β are motor speciﬁcations and ε is the viscous friction. The value of variables are given in [LCMA05], and ε = 2. The sampling interval of the sensor and measurement channels are equal to 0.05, and we assume τ a = 0, ∀ ∈ N. Our goal is to ﬁnd K and L such that the upper bound on the tolerable network delays in the measurement channel τks is maximized. The LMIs are feasible up to τimax = 0.195 when τi min = 0.05 (regarding (10), τi min = ha + τ¯a where we assumed τ¯a = 0). The observer gain L is obtained by solving the LMIs in Remark 4. The state feedback gain K is chosen such that A − BK is asymptotically stable. With 0.9706 −0.1100

−0.2082 −0.7815

0.0369 5.1597

K = −0.4802

−5.4503

−14.9865

L=

0.4441 , 26.4242 −2.9440 ,

the closed-loop system remains stable even in the presence of variable measurement channel delay smaller than 0.095 (note that τi max = ha +τks +(tsk+1 −tsk )). Fig. 4(a), Fig. 4(b) show the norm of x(t) and e(t) respectively when the pendulum’s non-linear model in [LCMA05] is used with the initial condition 0.1 0.1 0 0.15 .

5 Conclusions and Future Work We proposed two types of control units: non-anticipative and anticipative. NCSs with an LTI plant model and anticipative or non-anticipative controller can be modeled by a DDE such as (11). We found suﬃcient conditions for asymptotic stability of (11) in the form of matrix inequalities and presented a procedure to design output feedback control unit for NCSs. Our method shows signiﬁcant improvement in comparison to the existing results. We will extend our results to H∞ or H2 design. The problem of stabilizing the plant in the presence of saturation in the control loop will also be considered.

Anticipative and Non-anticipative Controller Design

217

Appendix Equation (11) can be written in as equivalent form [FS02b] 2

2

x(t) ˙ = y(t), −y(t) +

Ai x(t) − i=0

t

Ai

t−τi

i=1

y(s)ds = 0,

(20)

and the following Lyapunov-Krasovskii functional: 2

V (t) = x P1 x + i=1

0

t

−τi max t+θ

2

y (s)Ri y(s)dsdθ

+ i=1

t t−τi min

x (s)Si x(s)ds, (21)

where P1 > 0 . Diﬀerentiating the ﬁrst, second and third term of (21) with respect to t respectively gives x˙ , 0

2x P1 x(t) ˙ = 2˜ x (t)P 2

τi max y (t)Ri y(t) − i=1

(22a) t

t−τi max

y (τ )Ri y(τ )dτ ,

(22b)

2

x (t)Si x(t) − x (t − τi min )Si x(t − τi min ) ,

(22c)

i=1

where x ˜ = x(t)

y(t) . Substituting (20) into (22a), 2

dV (t) ≤x ˜ (t)Ψ˜ x ˜− x (t − τi min )Si x(t − τi min )− dt i=1 2

t t−τi max

i=1

y (τ )Ri y(τ )dτ + η,

where Ψ˜ = P

0

2 i=0

Ai

2

2˜ x (t)P

η= i=1

I + −I 0 Ai

t t−τi

0

2 i=0

Ai

I −I

2

P+ i=1

Si 0

0 , (23) τi max Ri

y(s)ds.

By Moon-park inequality [MPKL01], a bound on cross term, η, can be found as follows:

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P. Naghshtabrizi and J.P. Hespanha 2

t

η≤

y(s) x ˜(t)

t−τi min

i=1 2

t−τi min

+ i=1

t−τi

2

t

i=1

t−τi

= 2

y(s) x ˜(t)

i=1

t−τi

2

t

+2

t−τi min

i=1

Ti − 0 Ai P Z1i

Ri ∗

y(s) ds x ˜(t)

T˜i − 0 Ai P Z2i

y(s) ds x ˜(t)

y (s)Ri y(s)ds

t−τi min

+2

Ri ∗

y (s) T˜i − 0

y (s) Ti − 0

Ai P x ˜(t)ds Ai P x ˜(t)ds

2

+

x ˜ (t) τi min Z1i + (τi − τi min Z2i )˜ x(t), i=1

where Ri ∗ By choosing T˜i = 0 2

t

i=1

t−τi

η≤

Ti ≥ 0, Z1i

Ri ∗

T˜3 ≥ 0. Z2i

(24)

Ai P ,

y(s) Ri y(s)ds + 2x (t) Yi − 0

Ai P x ˜(t)

2

x (t − τi min ) Yi − 0

−2

Ai P x ˜(t)

i=1 2

˜(t). x ˜ (t) τi min Z1i + (τi max − τi min )Z2i x

+ i=1

Based on the Lyapunov-Krasovskii theorem, (11) is asymptotically stable if dV 2 for some ε > 0. Hence the system is asymptotically stable if dt ≤ −ε x (12a) holds. However any row and column of (12a) except the ﬁrst block row and column can be zero. The inequalities in (12b) and (12c) are in fact nonstrict. However for simplicity and since there is no numerical advantage we state them as strict inequality.

On Quantization and Delay Eﬀects in Nonlinear Control Systems Daniel Liberzon Coordinated Science Laboratory University of Illinois at Urbana-Champaign Urbana, IL 61801 [email protected] Summary. The purpose of this paper is to demonstrate that a uniﬁed study of quantization and delay eﬀects in nonlinear control systems is possible by merging the quantized feedback control methodology recently developed by the author and the small-gain approach to the analysis of functional diﬀerential equations with disturbances proposed earlier by Teel. We prove that under the action of a robustly stabilizing feedback controller in the presence of quantization and suﬃciently small delays, solutions of the closed-loop system starting in a given region remain bounded and eventually enter a smaller region. We present several versions of this result and show how it enables global asymptotic stabilization via a dynamic quantization strategy.

1 Introduction To be applicable in realistic situations, control theory must take into account communication constraints between the plant and the controller, such as those arising in networked embedded systems. Two most common phenomena relevant in this context are quantization and time delays. It is also important to be able to handle nonlinear dynamics. To the best of the author’s knowledge, the present paper is a ﬁrst step towards addressing these three aspects in a uniﬁed and systematic way. It is well known that a feedback law which globally asymptotically stabilizes a given system in the absence of quantization will in general fail to provide global asymptotic stability of the closed-loop system that arises in the presence of a quantizer with a ﬁnite number of values. One reason for this is saturation: if the quantized signal is outside the range of the quantizer, then the quantization error is large, and the control law designed for the ideal case of no quantization may lead to instability. Another reason is deterioration of performance near the equilibrium: as the diﬀerence between the current and the desired values of the state becomes small, higher precision is required,

P.J. Antsaklis, P. Tabuada (Eds.): Netw. Emb. Sens. and Cntrl., LNCIS 331, pp. 219–229, 2006. © Springer-Verlag Berlin Heidelberg 2006

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D. Liberzon

and so in the presence of quantization errors asymptotic convergence is typically lost. Due to these phenomena, instead of global asymptotic stability it is more reasonable to expect that solutions starting in a given region remain bounded and approach a smaller region. In [Lib03a] the author has developed a quantized feedback control methodology for nonlinear systems based on results of this type, under a suitable robust stabilization assumption imposed on the controller (see also [Lib03c] for further discussion and many references to prior work on quantized control). As we will see, this robustness of a stabilizing controller with respect to quantization errors (which is automatic in the linear case) plays a central role in the nonlinear results. The eﬀect of a suﬃciently small time delay on stability of a linear system can be studied by standard perturbation techniques based on Rouch´e’s theorem (see, e.g., [GKC03, HL01]). When the delay is large but known, it can in principle be attenuated by propagating the state forward from the measurement time. However, the case we are interested in is when the delay is not necessarily small and its value is not available to the controller. There is of course a rich literature on asymptotic stability of time-delayed systems [HL93, GKC03], but these results are not very suitable here because they involve strong assumptions on the system or on the delay, and asymptotic stability will in any case be destroyed by quantization. On the other hand, there are very few results on boundedness and ultimate boundedness for nonlinear control systems with (possibly large) delays. Notable exceptions are [Tee98, WP04, WB99a], and the present work is heavily based on [Tee98]. In that paper, Teel uses a small-gain approach to analyze the behavior of nonlinear feedback loops in the presence of time delays and external disturbances. Our main observation is that his ﬁndings are compatible with our recent results on quantization; by identifying external disturbances with quantization errors, we are able to naturally combine the two lines of work. For concreteness, we consider the setting where there is a processing delay in collecting state measurements from the sensors and/or computing the control signal, followed by quantization of the control signal before it is transmitted to the actuators. Other scenarios can be handled similarly, as explained at the end of the paper. Assuming that an upper bound on the initial state is available, our main result (Theorem 1 in Section 3) establishes an upper bound and a smaller ultimate bound on resulting closed-loop trajectories. Section 4 contains some interpretations and modiﬁcations of this result, and explains how global asymptotic stability can be recovered by using the dynamic quantization method of [BL00a, Lib03a]. In what follows, we will repeat some of the developments from [Tee98] in order to make the paper self-contained and also because we will need some estimates not explicitly written down in [Tee98]. We remark that the results of [Tee98] were also used in [TNK98] for a diﬀerent purpose, namely, to study the eﬀects of sampled-data control implementation. We take the delay to be ﬁxed for simplicity; however, everything readily generalizes to the case of a time-varying bounded delay (by simply replacing the value of the delay with

On Quantization and Delay Effects in Nonlinear Control Systems

221

its upper bound in all subsequent arguments). This latter type of delay is used for modeling sampled-data control systems (see, e.g., [FSR04a] and the references therein), thus such an extension could be useful for studying systems with sampled-data quantized feedback.

2 Notation and Preliminaries Consider the system

x˙ = f (x, u)

n

where x ∈ R is the state, u ∈ Rm is the control, and f : Rn × Rm → Rn is a C 1 function. The inputs to the system are subject to quantization. The (input) quantizer is a piecewise constant function q : Rm → Q, where Q is a ﬁnite subset of Rm . Following [Lib03a, Lib03c], we assume that there exist real numbers M > Δ > 0 such that the following condition holds: |u| ≤ M

⇒

|q(u) − u| ≤ Δ.

(1)

This condition gives a bound on the quantization error when the quantizer does not saturate. We will refer to M and Δ as the range and the error bound of the quantizer, respectively. We consider the one-parameter family of quantizers qµ (u) := µq

u , µ

µ > 0.

(2)

Here µ can be viewed as a “zoom” variable. This parameter is in general adjustable, but in this section we take µ to be ﬁxed. The range of the quantizer qµ is M µ and the error bound is Δµ. Assumed given is some nominal state feedback law u = k(x), which is C 1 and satisﬁes k(0) = 0. We take this feedback law to be stabilizing robustly with respect to actuator errors, in the following sense (see Section 4 for a discussion of this assumption and a way to relax it). Assumption 1 There exists a C 1 function V : Rn → R such that for some class K∞ functions1 α1 , α2 , α3 , ρ and for all x ∈ Rn and v ∈ Rm we have α1 (|x|) ≤ V (x) ≤ α2 (|x|) and |x| ≥ ρ(|v|) 1

⇒

∂V f (x, k(x) + v) ≤ −α3 (|x|). ∂x

(3) (4)

Recall that a function α : [0, ∞) → [0, ∞) is said to be of class K if it is continuous, strictly increasing, and α(0) = 0. If α is also unbounded, then it is said to be of class K∞ . A function β : [0, ∞) × [0, ∞) → [0, ∞) is said to be of class KL if β(·, t) is of class K for each ﬁxed t ≥ 0 and β(r, t) decreases to 0 as t → ∞ for each ﬁxed r ≥ 0. We will write α ∈ K∞ , β ∈ KL, etc.

222

D. Liberzon

We let the state measurements be subject to a ﬁxed delay τ > 0. This delay followed by quantization give the actual control law in the form u(t) = qµ (k(x(t − τ )))

(5)

and yields the closed-loop system x(t) = f (x(t), qµ (k(x(t − τ )))). ˙

(6)

We can equivalently rewrite this as x(t) ˙ = f x(t), k(x(t)) + θ(t) + e(t)

(7)

θ(t) := k(x(t − τ )) − k(x(t))

(8)

e(t) := qµ (k(x(t − τ ))) − k(x(t − τ )).

(9)

where and

To simplify notation, we will write a ∨ b for max{a, b}. Applying (4) with v = θ + e and deﬁning γ ∈ K∞ by γ(r) := ρ(2r), we have |x| ≥ γ(|θ| ∨ |e|)

⇒

∂V f (x, k(x) + θ + e) ≤ −α3 (|x|). ∂x

(10)

The quantity deﬁned in (8) can be expressed as θ(t) = −

t t−τ

k (x(s))f (x(s), k(x(s − τ )) + e(s))ds.

(11)

Substituting this into the system (7), we obtain a system with delay td := 2τ . For initial data x : [−td , 0] → Rn which is assumed to be given, this system has a unique maximal solution2 x(·). We adopt the following notation from [Tee98]: |xd (t)| := maxs∈[t−td ,t] |x(s)| for t ∈ [0, ∞), xd J := supt∈J |xd (t)| for a subinterval J of [0, ∞), and |ed (t)| and ed J are deﬁned similarly. In view of (11), for some γ1 , γ2 ∈ K∞ we have |θ(t)| ≤ τ γ1 (|xd (t)|) ∨ τ γ2 (|ed (t)|) (we are using the fact that f (0, k(0)) = 0, which follows from Assumption 1). Deﬁning γτ (r) := γ(τ γ1 (r)), γˆ (r) := γ(τ γ2 (r) ∨ r), and using (10), we have |x(t)| ≥ γτ (|xd (t)|) ∨ γˆ (|ed (t)|) ⇒ V˙ (t) ≤ −α3 (|x|). In light of (3), this implies 2

It is not hard to see that discontinuities of the control (5) do not aﬀect the existence of Carath´eodory solutions; cf. the last paragraph of Section 2.6 in [HL93].

On Quantization and Delay Effects in Nonlinear Control Systems

V (t) ≥ α2 ◦ γτ (|xd (t)|) ∨ α2 ◦ γˆ (|ed (t)|)

223

V˙ (t) ≤ −α3 (|x|)

⇒

hence by the standard comparison principle (cf. [Kha02, Chapter 4]) V (t) ≤ β(V (t0 ), t − t0 ) ∨ α2 ◦ γτ

xd

[t0 ,∞)

∨ α2 ◦ γˆ

ed

[t0 ,∞)

where β ∈ KL and β(r, 0) = r. Using (3) again, we have ˜ |x(t)| ≤ β(|x(t ˜x 0 )|, t − t0 ) ∨ γ

xd

[t0 ,∞)

∨ γ˜e

ed

[t0 ,∞)

(12)

˜ t) := α−1 (β(α2 (r), t)), γ˜x (r) := α−1 ◦ α2 ◦ γτ (r), and γ˜e (r) := where β(r, 1 1 −1 α1 ◦ α2 ◦ γˆ (r). The full expression for γ˜x (which depends on the delay τ ) is γ˜x (r) = α1−1 ◦ α2 ◦ γ(τ γ1 (r)).

(13)

Let us invoke the properties of the quantizer to upper-bound the quantization error e deﬁned in (9). Take κ to be some class K∞ function with the property that κ(r) ≥ max |k(x)| ∀r ≥ 0 |x|≤r

so that we have

|k(x)| ≤ κ(|x|)

∀ x.

Then (1) and (2) give |xd (t)| ≤ κ−1 (M µ)

⇒

|e(t)| ≤ Δµ.

(14)

3 Main Result Theorem 1. Let Assumption 1 hold. Assume that the initial data satisﬁes |xd (t0 )| ≤ E0

(15)

for some known E0 > 0. Choose a (small) ε > 0 and assume that for some Λ > 0 we have α1−1 ◦ α2 (E0 ) ∨ ε ∨ γ˜e (Δµ) < Λ < κ−1 (M µ).

(16)

Assume that the delay τ is small enough so that γ˜x (r) < r

∀ r ∈ (ε, Λ].

(17)

Then the solution of the closed-loop system (6) satisﬁes the bound xd

[t0 ,∞)

≤ α1−1 ◦ α2 (E0 ) ∨ ε ∨ γ˜e (Δµ)

(18)

and the ultimate bound xd for some T > 0.

[t0 +T,∞)

≤ ε ∨ γ˜e (Δµ)

(19)

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D. Liberzon

Proof. As in [Tee98], the main idea behind the proof is a small-gain argument combining (12) with the bound |xd (t)| ≤ |xd (t0 )| ·

(1 − sgn(t − td − t0 )) ∨ x 2

[t0 ,∞)

which gives xd

[t0 ,∞)

˜ ≤ |xd (t0 )| ∨ β(|x(t ˜x 0 )|, 0) ∨ γ

xd

∨ γ˜e

[t0 ,∞)

ed

.

[t0 ,∞)

˜ 0) = α−1 ◦ α2 (r) ≥ r. The condition (15) and the ﬁrst We know that β(r, 1 inequality in (16) imply that there exists some maximal interval [t0 , t¯) on which |xd (t)| < Λ. On this interval, using (14), (16), (17), and causality, we have the (slightly conservative) bound xd

[t0 ,t¯)

˜ d (t0 )|, 0) ∨ γ˜x ≤ β(|x ≤

α1−1

xd

∨ γ˜e

[t0 ,t¯)

ed

[t0 ,t¯)

◦ α2 (E0 ) ∨ ε ∨ γ˜e (Δµ) < Λ

Thus actually t¯ = ∞ and (18) is established. Next, denote the right-hand side of (18) by E and pick a ρ > 0 such that ˜ β(E, ρ) ≤ ε.

(20)

Using (12), we have xd

[t0 +td +ρ,∞)

≤ x

[t0 +ρ,∞)

˜ 0 , ρ) ∨ γ˜x ≤ β(E

xd

[t0 ,∞)

∨ γ˜e (Δµ)

≤ ε ∨ γ˜x (E) ∨ γ˜e (Δµ) From this, using (12) again but with t0 + td + ρ in place of t0 , we obtain xd

[t0 +2(td +ρ),∞)

≤ x

[t0 +td +2ρ,∞)

˜ ≤ β(|x(t ˜x 0 + td + ρ)|, ρ) ∨ γ

xd

[t0 +td +ρ,∞)

∨ γ˜e (Δµ)

˜ ∨ γ˜e (Δµ), ρ) ∨ β(˜ ˜ γx (E), ρ) ∨ γ˜x (ε ∨ γ˜e (Δµ)) ∨ γ˜ 2 (E) ∨ γ˜e (Δµ) ≤ β(ε x ≤ ε ∨ γ˜x2 (E) ∨ γ˜e (Δµ) in view of (16), (17), (20), and continuity of γ˜x at ε. There exists a positive integer n such that γ˜xn (E) ≤ ε ∨ γ˜e (Δµ). Repeating the above calculation, for T := n(td + ρ) we have the bound (19) and the proof is complete.

4 Discussion Hypotheses of Theorem 1 The only constraint placed on the quantizer in Theorem 1 is the hypothesis (16). It says that the range M of q should be large enough compared to

On Quantization and Delay Effects in Nonlinear Control Systems

225

the error bound Δ, so that the last term in (16) is larger than the ﬁrst one (then a suitable Λ automatically exists). A very similar condition is used in [Lib03a]. One direction for future work is to extend the result to “coarse” quantizers not satisfying such hypotheses (cf. [LH05], [Lib03c, Section 5.3.6]). The small-gain condition (17) is justiﬁed by the formula (13), which ensures that for every pair of numbers Λ > ε > 0 there exists a τ ∗ > 0 such that for all τ ∈ (0, τ ∗ ) we have γ˜x (r) < r

∀ r ∈ (ε, Λ].

The value of τ ∗ depends on the relative growth rate of the functions appearing on the right-hand size of (13) for small and large arguments. In particular, if both α1−1 ◦ α2 ◦ γ and γ1 have ﬁnite derivatives at 0, then for small enough τ the inequality (17) holds with ε = 0. In this case, the eﬀect of ε (called the oﬀset in [Tee98]) disappears, and the ultimate bound (19) depends on the quantizer’s error bound only. Also note that in the case of linear systems, α1−1 ◦ α2 ◦ γ and γ1 can be taken to be linear, and for small enough τ we have γ˜x (r) < r

∀ r > 0.

(21)

Assumption 1 says that the feedback law k provides input-to-state stability (ISS) with respect to the actuator error v, in the absence of delays (see [Son89, SW95]). This requirement is restrictive in general. However, it is shown in [Son89] that for globally asymptotically stabilizable systems aﬃne in controls, such a feedback always exists. (For linear systems and linear stabilizing feedback laws, such robustness with respect to actuator errors is of course automatic.) One way to proceed without Assumption 1 is as follows. Suppose that k just globally asymptotically stabilizes the system in the absence of actuator errors, so that instead of (4) we only have ∂V f (x, k(x)) ≤ −α3 (|x|). ∂x By virtue of [CT96, Lemma 1] or [Son90, Lemma 3.2], there exist a class K∞ function γ and a C 1 function G : Rn → GL(m, R), i.e., G(x) is an invertible ¯ e¯ ∈ Rm we have m × m matrix for each x, such that for all x ∈ Rn and θ, ¯ ∨ |¯ |x| ≥ γ(|θ| e|)

⇒

∂V α3 (|x|) f (x, k(x) + G(x)θ¯ + G(x)¯ e) ≤ − . (22) ∂x 2

To use this property, let us rewrite the system (7) as ¯ + G(x(t))¯ x(t) ˙ = f x(t), k(x(t)) + G(x(t))θ(t) e(t) where and

¯ := G−1 (x(t))θ(t) θ(t)

226

D. Liberzon

e¯(t) := G−1 (x(t))e(t). In view of (11), for some γ¯1 , γ¯2 , γ¯3 ∈ K∞ and a > 0 we have ¯ |θ(t)| ≤ τ γ¯1 (|xd (t)|) ∨ τ γ¯2 (|ed (t)|) and

|¯ e(t)| ≤ (¯ γ3 (|x(t)|) ∨ a)|e(t)|.

Using (22), we have |x(t)| ≥ γ τ γ¯1 (|xd (t)|) ∨ τ γ¯2 (|ed (t)|) ∨ γ¯3 (|x(t)|)|e(t)| ∨ a|e(t)|

⇒

α3 (|x|) V˙ (t) ≤ − . 2

As before, we obtain from this that −1 ¯ |x(t)| ≤ β(|x(t ¯1 ( xd 0 )|, t − t0 ) ∨ α1 ◦ α2 ◦ γ τ γ

∨ τ γ¯2 ( ed

[t0 ,∞) )

∨ γ¯3 ( x

[t0 ,∞) )

e

[t0 ,∞) )

[t0 ,∞)

∨a e

[t0 ,∞)

(23)

¯ 0) = α−1 ◦ α2 (r). Deﬁne where β¯ ∈ KL and β(r, 1 γ¯x (r) := α1−1 ◦ α2 ◦ γ τ γ¯1 (r) ∨ γ¯3 (r)Δµ and

γ¯e (r) := α1−1 ◦ α2 ◦ γ(τ γ¯2 (r) ∨ ar).

Then we have a counterpart of Theorem 1, without Assumption 1 and with γ¯x , γ¯e replacing γ˜x , γ˜e everywhere in the statement. The proof is exactly the same modulo this change of notation, using (23) instead of (12). The price to pay is that the function γ¯x depends on Δµ as well as on τ , so the modiﬁed small-gain condition requires not only the delay but also the error bound of the quantizer to be suﬃciently small. This can be seen particularly clearly in the special case of no delay (τ = 0); in this case we arrive at a result complementary to [Lib03a, Lemma 2] in that it applies to every globally asymptotically stabilizing feedback law but only when the quantizer has a suﬃciently small error bound.3

3

The argument we just gave actually establishes that global asymptotic stability under the zero input guarantees ISS with a given oﬀset on a given bounded region (“semiglobal practical ISS”) for suﬃciently small inputs. (A related result proved in [SW96] is that global asymptotic stability under the zero input implies ISS for suﬃciently small states and inputs.) This observation conﬁrms the essential role that ISS plays in our developments.

On Quantization and Delay Effects in Nonlinear Control Systems

227

Extensions In Theorem 1, the eﬀects of quantization manifest themselves in the need for the known initial bound E0 in (15) and in the strictly positive term γ ˜e (Δµ) in the ultimate bound (19). If the ultimate bound is strictly smaller than the initial bound, and if the quantization “zoom” parameter µ in (2) can be adjusted on-line, then both of these shortcomings can be removed using the method proposed in [BL00a, Lib03a]. First, the state measurement at time t0 gives us the value of x(t0 − τ ). We assume that no control was applied for t ∈ [t0 − τ, t0 ). If an upper bound τ ∗ on the delay is known and the system is forward complete under zero control, then we have an upper bound of the form (15). This is because for a forward complete system, the reachable set from a given initial condition in bounded time is bounded (see, e.g., [AS99]). An over-approximation of the reachable set can be used to actually compute such an upper bound (see [KV00, HST05] for some results on approximating reachable sets). Now we can select a value of µ large enough to satisfy (16) and start applying the control law (5) for t ≥ t0 . Let us assume for simplicity that (21) is satisﬁed, so that we can take ε = 0. Applying Theorem 1, we have from (19) that |xd (t0 + T )| ≤ γ˜e (Δµ). Next, at time t0 + T we want to select a smaller value of µ for which (16) holds with E0 replaced by this new bound (which depends on the previous value of µ). For this to be possible, we need to assume that α1−1 ◦ α2 ◦ γ˜e (Δµ) < κ−1 (M µ)

∀ µ > 0.

Repeating this “zooming-in” procedure, we recover global asymptotic stability. We mention an interesting small-gain interpretation of the above strategy, which was given in [NL05]. The closed-loop system can be viewed as a hybrid system with continuous state x and discrete state µ. After the “zooming-out” stage is completed, it can be shown that the x-subsystem is ISS with respect to µ with gain smaller than γˆ (Δ·), while the µ-subsystem is ISS with respect to x with gain γ˜e−1 /Δ = γˆ −1 ◦ α2−1 ◦ α1 /Δ. The composite gain is less than identity, and asymptotic stability follows from the nonlinear small-gain theorem. One important advantage of the small-gain viewpoint (which was also used to establish Theorem 1) is that it enables an immediate incorporation of external disturbances, under suitable assumptions. We refer the reader to [NL05] and [Tee98] for further details. We combined quantization and delays as in (5) just for concreteness; other scenarios can be handled similarly. Assume, for example, that the control takes the form u(t) = k(qµ (x(t − τ )))

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i.e., both the quantization and the delay4 aﬀect the state before the control is computed. The resulting closed-loop system can be written as x(t) ˙ = f x(t), k x(t) + θ(t) + e(t) where

θ(t) := x(t − τ ) − x(t)

and

e(t) := qµ (x(t − τ )) − x(t − τ ).

Assumption 1 needs to be modiﬁed by replacing (4) with |x| ≥ ρ(|v|)

⇒

∂V f (x, k(x + v)) ≤ −α3 (|x|). ∂x

This is the requirement of ISS with respect to measurement errors, which is restrictive even for systems aﬃne in controls (see the discussion and references in [Lib03a]). For γ(r) := ρ(2r) we have |x| ≥ γ(|θ| ∨ |e|)

⇒

∂V f (x, k(x + θ + e)) ≤ −α3 (|x|) ∂x

and we can proceed as before. The ISS assumption can again be relaxed at the expense of introducing a constraint on the error bound of the quantizer (see also footnote 3 above). A bound of the form (15) for some time greater than the initial time can be obtained by “zooming out” similarly to how it is done in [Lib03a], provided that an upper bound τ ∗ on the delay is known and the system is forward complete under zero control. Global asymptotic stability can then be achieved by “zooming in” as before. It is also not diﬃcult to extend the results to the case when quantization aﬀects both the state and the input (cf. [Lib03a, Remark 1]).

5 Conclusions The goal of this paper was to show how the eﬀects of quantization and time delays in nonlinear control systems can be treated in a uniﬁed manner by using Lyapunov functions and small-gain arguments. We proved that under the action of an input-to-state stabilizing feedback law in the presence of both quantization and small delays, solutions of the closed-loop system starting in a given region remain bounded and eventually enter a smaller region. Based on this result, global asymptotic stabilization can be achieved by employing a dynamic quantization scheme. These ﬁndings demonstrate that the quantized control algorithms proposed in our earlier work are inherently robust to time delays, which increases their potential usefulness for applications such as networked embedded systems. 4

Their order is not important in this case as long as the quantizer is ﬁxed and does not saturate.

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Acknowledgments The author would like to thank Dragan Neˇsi´c and Chaouki Abdallah for helpful pointers to the literature, and Andy Teel and Emilia Fridman for useful comments on an earlier draft. This work was supported by NSF ECS-0134115 CAR and DARPA/AFOSR MURI F49620-02-1-0325 Awards.

Performance Evaluation for Model-Based Networked Control Systems Luis A. Montestruque1 and Panos J. Antsaklis2 1

2

EmNet, LLC Granger, IN [email protected] Department of Electrical Engineering University of Notre Dame Notre Dame, IN [email protected]

Summary. The performance of a class of Model-Based Networked Control System (MB-NCS) is considered in this paper. A MB-NCS uses an explicit model of the plant to reduce the network bandwidth requirements. In particular, an Output Feedback MB-NCS is studied. After reviewing the stability results for this system and some lifting techniques basics, two performance measures related to the traditional H2 performance measure for LTI systems are computed. The ﬁrst H2 like performance measurement is called the Extended H2 norm of the system and is based on the norm of the impulse response of the MB-NCS at time zero. The second performance measure is called the Generalized H2 norm and it basically replaces the traditional trace norm by the Hilbert-Schmidt norm that is more appropriate for inﬁnite dimensional operators. The Generalized H2 norm also represents the average norm of the impulse response of the MB-NCS for impulse inputs applied at diﬀerent times. Examples show how both norms converge to the traditional H2 norm for continuous H2 systems. Finally, with the help of an alternate way of representing lifted parameters, the relationship between the optimal sampler and hold of a sampled data system and the structure of the Output Feedback MB-NCS is shown.

1 Introduction A networked control system (NCS) is a control system in which a data network is used as feedback media. NCS is an important area see for example [bib03] and [NE00c, YTS02, WYB99a]. Industrial control systems are increasingly using networks as media to interconnect the diﬀerent components. The use of networked control systems poses, though, some challenges. One of the main problems to be addressed when considering a networked control system is the size of bandwidth required by each subsystem. Since each control subsystem must share the same medium the reduction of the individual band-

P.J. Antsaklis, P. Tabuada (Eds.): Netw. Emb. Sens. and Cntrl., LNCIS 331, pp. 231–249, 2006. © Springer-Verlag Berlin Heidelberg 2006

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width is a major concern. Two ways of addressing this problem are: minimizing the frequency of transfer of information between the sensor and the controller/actuator; or compressing or reducing the size of the data transferred at each transaction. A shared characteristic among popular digital industrial networks are the small transport time and big overhead per packet, thus using fewer bits per packet has small impact over the overall bit rate. So reducing the rate at which packets are transmitted brings better beneﬁts than data compression in terms of bit rate used. The MB-NCS architecture makes explicit use of knowledge about the plant dynamics to enhance the performance of the system. Model-Based Networked Control Systems (MB-NCS) were introduced in [MA02a]. Consider the control of a continuous linear plant where the state estimated by a standard observer is sent to a linear controller/actuator via a network. In this case, the controller and observer uses an explicit model of the plant that approximates the plant dynamics and makes possible the stabilization of the plant even under slow network conditions. The controller makes use of a plant model, which is updated with the observer estimate, to reconstruct the actual plant state in between updates. The model state is then used to generate the control signal. The main idea is to perform the feedback by updating the model’s state using the observer estimated state of the plant. The rest of the time the control action is based on a plant model that is incorporated in the controller/actuator and is running open loop for a period of h seconds. Also a disturbance signal w and a performance or objective signal z are included in the setup. The control architecture is shown in Figure 1. The observer has as inputs the output and input of the plant. In the implementation, in order to acquire the input of the plant, which is at the other side of the communication link, the observer can have a version of the model and controller, and knowledge of the update time h. In this way, the output of the controller, that is the input of the plant, can be simultaneously and continuously generated at both ends of the feedback path with the only requirement being that the observer makes sure that the controller has been updated. This last requirement ensures that both the controller and the observer are synchronized. Handshaking protocols provided by most networks can be used to provide synchronization. The performance characterization of Networked Control Systems under diﬀerent conditions is of major concern. In this paper a class of networked control systems called Model-Based Networked Control Systems (MB-NCS) is considered. This control architecture uses an explicit model of the plant in order to reduce the network traﬃc while attempting to prevent excessive performance degradation. MB-NCS can successfully address several important control issues in an intuitive and transparent way. Necessary and sufﬁcient stability results have been reported for continuous and discrete linear time invariant systems with state feedback, output feedback, and with network-induced delays (see [MA02a, MA02b, MA03a]). Results for stochastic update-times have also been derived [MA04]. We have observed that the

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Fig. 1. MB-NCS with disturbance input and objective signal output.

stability of MB-NCS is, in general, a function of the update times, the difference between the dynamics of the plant and the dynamics of the plant model, and of the control law used, and we have quantiﬁed these relations. The performance of the MB-NCS can be studied using several techniques and considering diﬀerent scenarios. One promising technique is called Continuous Lifting [CF96, BPFT91, DL99]. Lifting is basically a transformation of a periodic system to a discrete LTI system. The main advantage of this approach is that most of the results available for LTI systems are readily applicable to the lifted system. The disadvantage is that the input and output spaces are inﬁnite dimensional and thus the parameters of the lifted system are operators and not matrices. New results in this area allow overcoming these diﬃculties [MRP99, MP99]. The next section brieﬂy introduces the lifting techniques used to derive the results contained in the paper. Then, an H2 like performance measure is introduced as the Extended H2 norm, here the interplay between the discrete and continuous nature of the system is observed in the calculation of the norm. Also, a way of calculating the Extended H2 norm using an auxiliary LTI discrete system is presented. The next section presents the Generalized H2 norm. This norm is important because it can also be related to the norm of a operator based transfer function. Again, a way of computing the norm using an auxiliary LTI discrete system is derived. The paper is ﬁnalized with a discussion of the techniques described in [MRP99, MP99] and their application to optimal control synthesis problems for data sampled systems. This allows

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to eﬃciently compute the optimal gains for the controller and observer. Not surprisingly it is shown that under certain conditions, the optimal gains for the H2 optimal observer and controller of the MB-NCS are equivalent to the optimal gains for the non-networked system. We will start by deﬁning the system dynamics: Plant Dynamics: x˙ = Ax + B1 w + B2 u z = C1 x + D12 u y = C2 x + D21 w + D22 u Observer Dynamics: ˆ 2 − LD ˆ 22 )u + Ly x ¯˙ = (Aˆ − LCˆ2 )¯ x + (B Model Dynamics: ˆx + B ˆ2 u x ˆ˙ = Aˆ Controller: u = Kx ˆ

(1)

The model state x ˆ is updated with the observer state x ¯ every h seconds. It can be shown that the system dynamics can be described by: Gzw :   x˙  x ¯˙  e˙

  A B2 K −B2 K x ˜ 22 K −Bˆ2 K − LD ˜ 22 K  x¯   LC2 Aˆ − LCˆ2 + Bˆ2 K + LD ˜ ˆ ˆ ˜ e LC2 LD22 K − LC2 A − LD22 K   B1 +  LD21 w LD21   x C1 D12 K −D12 K  x¯  , ∀t ∈ [tk , tk+1 ) e 

=

z

=

e

=

x¯ − xˆ = 0, t = tk+1

We will also use the following deﬁnitions:     x(t) B1 ¯(t) , BN =  LD21 , CN = C1 ϕ(t) =  x e(t) LD21  A B2 K ˆ 2 K + LD ˜ 22 K Λ =  LC2 Aˆ − LCˆ2 + B ˜ 22 K − LCˆ2 LC2 LD

(2)

D12 K

−D12 K

 −B2 K ˆ 2 K − LD ˜ 22 K  −B ˜ 22 K Aˆ − LD

(3)

Throughout this paper we will assume that the compensated model is stable and that the transportation delay is negligible (that is, the delay resulting from the use of a network is assumed to be zero, this assumption can be made for systems where the inverse of the delay is much smaller than the plant

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235

bandwidth). We will assume that the frequency at which the network updates the state in the controller is constant. The goal is to ﬁnd the smallest frequency at which the network must update the state in the controller, that is, an upper bound for h, the update time. A necessary and suﬃcient condition for stability of the output feedback MB-NCS is now presented. Theorem 1. The non-disturbed output feedback MB-NCS described by (1) is T x ¯T eT = globally exponentially stable around the solution xT  0 if I 0 0 I 0 0 and only if the eigenvalues of M =  0 I 0  eΛh  0 I 0  are 0 0 0 0 0 0 inside the unit circle. A detailed proof for Theorem 1 can be found in [MA02b]. Note that while there might be a maximum h at which the system is stable (known also as maximum transfer interval or MATI [WYB99a]), this does not imply that all smaller transfer times h result in stable systems. Some systems might exhibit alternating bands of stable and unstable transfer times. Before deﬁning the performance measures previously described a brief summary of the lifting technique is presented. As it was pointed out, lifting can transform a periodic linear system such as a MB-NCS into a discrete linear time invariant system with operator-valued parameters. These parameters are computed for a class of MB-NCS and used throughout the paper.

2 Continuous Lifting Technique We will give a brief introduction into the Lifting technique. We need to deﬁne two Hilbert spaces, the ﬁrst space is deﬁned as follows: L2 [0, h) =

u(t)/

h 0

uT (t)u(t)dt < ∞

(4)

The second Hilbert space of interest is formed by an inﬁnite sequence of L2 [0, h) spaces and is deﬁned: l2 (Z, L2 [0, h)) = l2 = =

T

[..., u−2 , u−1 , u0 , u1 , u2 , ...] /ui ∈ L2 [0, h),

+∞ −∞

h 0

uTj (t)uj (t)dt < ∞

(5) Now we can deﬁne the lifting operator L as mapping from L2e (L2 extended) to l2 : L : L2e → l2 , Lu(t) = [..., u−2 , u−1 , u0 , u1 , u2 , ...] where uk (τ ) = u(τ + kh), τ ∈ [0, h)

T

(6)

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L.A. Montestruque and P.J. Antsaklis

It can be shown that L preserves inner products and thus is norm preserving [BPFT91]. Since L is surjective, it is an isomorphism of Hilbert spaces. So, lifting basically transforms a continuous function into a discrete function where each element of the sequence is a continuous function restricted to [0, h). As an example of the application of this lifting technique we will compute the lifted parameters of a MB-NCS with output feedback previously presented (1). This system is clearly h periodic, and therefore we expect to get, after the lifting procedure, an LTI system of the form: ϕk+1 = Aϕk + B wk , z k = C ϕk + Dwk

(7)

To obtain the operators we “chop” the time response of the system described in (1) and evaluate at times kh. The lifted parameters can be calculated as: 

I A= 0 0

0 I 0

 0 0  eΛh , 0

C = CN eΛτ ,



I h Bw = 0  0 0 D w = CN

τ 0

0 I 0

 0 0  eΛ(h−s) BN w(s)ds 0

eΛ(τ −s) BN w(s)ds

(8) The new lifted system is now a LTI discrete system. Note the dimension of the state space is left unchanged, but the input and output spaces are now inﬁnite dimensional. Nevertheless, the new representation [MRP99, MP99] allows extending the results available for discrete LTI systems to the lifted domain. These tools have been traditionally used to analyze and synthesize sample and hold devices, and digital controllers. It is to be noted, though, that in this application the discrete part is embedded in the controller that doesn’t operate in the same way a typical sampled system does. Here, for instance, the controller gain operates over a continuous signal, as opposed to over a discrete signal as it is customary in sampled data systems.

3 An H2 Norm Characterization of a MB-NCS It is clear that, since the MB-NCS is h-periodic, there is no transfer function in the normal sense whose H2 norm can be calculated [BPFT91]. For LTI systems the H2 norm can be computed by obtaining the 2-norm of the impulse response of the system at t = t0 . We will extend this deﬁnition to specify an H2 norm, or more properly, to deﬁne an H2-like performance index [BPFT91]. We will call this performance index Extended H2 Norm. We will study the extended H2 norm of the MB-NCS with output feedback studied in the previous section and shown in Figure 1. The Extended H2 Norm is deﬁned as: 1/2

Gzw

xh2

=

Gzw δ (t0 ) ei i

2 2

(9)

Performance Evaluation for Model-Based Networked Control Systems

Theorem 2. The Extended H2 Norm, G T XBN BN T

xh2 , 1/2

237

of the Output Feedback MB-

where X is the solution of the NCS is given by G xh2 = trace discrete Lyapunov equation M (h) XM (h) − X + Wo (0, h) = 0 and Wo (0, h) T h T CN eΛt dt. is the observability Gramian computed as Wo (0, h) = 0 eΛ t CN Proof. We will compute the extended H2 norm of the system by obtaining the 2-norm of the objective signal z to an impulse input w = δ(t − t0 ). It can be shown that the response of the system to an input w = δ(t − t0 ) (assuming that the input dimension is one) is:  C1  D12 K  ϕ(t) −D12 K  I 0 eΛ(t−tk )  0 I 0 0  z(t)

ϕ(t)

=

=

(10) k  0 0  eΛh  0

B1

LD21

LD21

(11)

With x(t) x ¯(t) e(t) , A B2 K ˆ ˆ ˆ 2 K + LD ˜ 22 K  LC2 A − LC2 + B Λ= ˜ ˆ LC2 LD22 K − LC2 h = tk+1 − tk . ϕ(t) = 

 −B2 K ˆ 2 K − LD ˜ 22 K  , −B ˆ ˜ A − LD22 K

So we can compute the 2-norm of the output: z

2 2

where

= =

∞ t0 ∞ t0

z(t)T z(t)dt T BN M (h)T

k ΛT (t−tk )

e

 I 0 0 M (h) =  0 I 0  eΛh , BN = 0 0 0  C1   D CN = 12 K −D12 K

k

T CN CN eΛ(t−tk ) (M (h)) BN dt



B1

LD21

LD21

,

(12) It is easy to see that the norm of a system with more than one inputs can be obtained by taking the norm of the integral shown in (12). So at this point we can drop our assumption of working with a one dimension input system. We will concentrate now on the integral expression (12).

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Σ(h)

L.A. Montestruque and P.J. Antsaklis

=

∞ t0

=

T BN

=

T BN

k ΛT (t−tk )

T M (h)T BN ∞

i=0 ∞ i=0

ti+1 ti

e

M (h)T

M (h)T

i

k

T CN eΛ(t−tk ) (M (h)) BN dt CN

i ΛT (t−ti )

e

i

T CN CN eΛ(t−ti ) (M (h)) dt BN

Wo (0, h) (M (h))

i

BN (13)

where Wo (0, h) =

h 0

T

T eΛ t CN CN eΛt dt

Note that Wo (0, h) has the form of the observability Gramian. Also note the summation resembles the solution of a discrete Lyapunov equation. This Lyapunov equation can be expressed as: M (h)T XM (h) − X + Wo (0, h) = 0

(14)

In this equation we note that M (h) is a stable matrix if and only if the networked system is stable. Note that Wo (0, h) is a positive semi deﬁnite matrix. Under these conditions the solution X will be positive semi deﬁnite.♦ Note that the observability gramian can be factorized as Wo (0, h) = T h T T Caux Caux = 0 eΛ t CN CN eΛt dt. This allow us to compute the norm of the system as the norm of an equivalent discrete LTI system. h

T

T T Corollary 1. Deﬁne Caux Caux = 0 eΛ t CN CN eΛt dt and the auxiliary discrete system Gaux with parameters:Aaux = M (h) , Baux = BN , Caux , and Daux = 0 then the following holds:

Gzw

xh2

= Gaux

2

(15)

Example 1. We now present an example using a double integrator as the plant. 0 1 0.1 0.1 ; The plant dynamics are given by: A = ; B1 = 0 0 0.1 1 B2 = 0 1 ; C1 = ; C2 = ; D12 = 0.1; D21 = 0.1; D22 = 0. 0.1 0 −1 . A state estimator We will use the state feedback controller K = −2 20 100 is used to place the state observer eigenvalwith gain L = ues at −10. We will use a plant model with the following dynamics: Aˆ = 0.1634 0.8957 0.8764 ˆ2 = −0.1686 1.0563 , Cˆ2 = , B , −0.1072 −0.1801 0.1375 ˆ 22 = −0.1304. This model yields a stable NCS for update times up and D to approximately 7.5 sec. In Figure 2 we plot the extended H2 norm of the system as a function of the updates times. Note that as the update time of the MB-NCS approaches zero, the value of the norm approaches the norm of the non-networked compensated system. Also note that the optimal update time is around 0.8 sec, and it starts to

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239

0.045 0.04 0.035 0.03 2 xh2

0.0394

||Gzw||

0.025 0.02

0.015 0.01 0.005 0 0

1

2

3

4

5

6

update times h (sec)

Fig. 2. Extended H2 norm of the system as a function of the update times.

degrade as the update times become smaller. This pattern is repeated with other norms as shown in the next example.

4 A Generalized H2 Norm for MB-NCS In the previous section the Extended H2 Norm was introduced to study the performance of MB-NCS, this norm was deﬁned as the norm of the output of the system when a unit impulse at t = t0 is applied to the input. But since the MB-NCS is a time varying system it may seem inappropriate to apply this input only at t = t0 . By letting the input be δ (t − τ ) we arrive to an alternate deﬁnition. Since the system is h periodic we only need to consider τ ∈ [t0 , t0 +h). We will call this norm the Generalized H2 Norm and we deﬁne it as: Gzw

gh2

=

1 h

1/2

t0 +h t0

Gzw δ (t − i

2 τ ) ei 2

dτ

(16)

A detailed study of this norm can be found in [BPFT91]. Note that this norm evaluates the time average of the system response to the impulsive function applied at diﬀerent times. Another option for a generalized norm could have replaced the time average by the maximum over time. However, as it is shown later, there is relationship between the time averaged norm (16) and the norm of the operator-valued transfer function of Gzw that can be useful for frequency domain analysis. We will now show some relations arising from this norm. Let a continuous-time linear transformation G : L2 [0, h) → L2 [0, ∞) be deﬁned by:

240

L.A. Montestruque and P.J. Antsaklis t

(Gu) (t) =

0

g (t, τ ) u (τ ) dτ

(17)

Where g (t, τ ) is the impulse response of G. Let G be periodic and let its Hilbert-Schmidt norm G HS be deﬁned as: G

HS

h

=

0

1/2

∞

T

trace g (t, τ ) g (t, τ ) dtdτ

0

(18)

Then it is clear that: Gzw

gh2

1 = √ Gzw h

(19)

HS

Note the slight abuse of notation since originally Gzw was considered a transformation with domain L2 [0, ∞) while the Hilbert-Schmidt norm in (18) is deﬁned for transformations with domain on L2 [0, h). Now denote the lifted operator Gzw = LGzw L−1 with input-output relation given by the convolution:

zk =

k l=0

g k−l wl (20)

where

g k : L2 [0, h) → L2 [0, h) and

g k u (t) =

h 0

g (t + kh, τ ) u (τ ) dτ

The Hilbert-Schmidt operator for g k is given by:

gk

HS

=

h 0

h 0

1/2 T

trace g (t + kh, τ ) g (t + kh, τ ) dtdτ

(21)

Then it is easy to show that: Gzw

2 HS

∞

=

gk k=0

2 HS

= g

2 2

(22)

The last expression shows a relationship between the discrete lifted representation of the system and the Generalized H2 Norm. Finally we will show the relationship between the Generalized H2 Norm and the norm of an operatorvalued transfer function: ∞

g˜ (λ) = k=0

g k λk

(23)

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241

By deﬁning in a similar way the λ-transform for the input and output of the system we get: z˜ (λ) = g˜ (λ) w ˜ (λ)

(24)

Note that for every λ in their respective regions of convergence, w ˜ (λ) and z˜ (λ) are functions on [0, h); while g˜ (λ) is a Hilbert-Schmidt operator. Deﬁne the Hardy space H2 (D, HS) with operator-valued functions that are analytic in the open unit disc, boundary functions on ∂D, and with ﬁnite norm: g˜

2

2π

1 = 2π

0

g˜ e

jθ

1/2

2 HS

(25)

dθ

Note the norm in H2 (D, HS) is a generalization of the norm in H2 (D) by replacing the trace norm by the Hilbert-Schmidt norm. It can be shown that: Gzw

2 HS

= g

2 2

= g˜

2 2

(26)

We will now show how to calculate the Generalized H2 Norm of the Output Feedback MB-NCS. Deﬁne the auxiliary discrete LTI system: S

Gaux =

A Baux Caux 0

(27)

Where:  T Baux Baux =

h 0

T Caux Caux =

h 0

I  0 0 eΛ

T

τ

0 I 0

  I 0 T T Λ τ  0 0  eΛτ BN BN e 0 0

 0 0  dτ 0

0 I 0

T CN CN eΛτ dτ

(28)

Theorem 3. The Generalized H2 Norm, Gzw MB-NCS is given by Gzw

gh2

=

√1 h

D

2 HS

gh2 ,

of the Output Feedback

+ Gaux

2 2

1/2

.

Proof. The transfer function for Gzw can be written as: g˜ (λ) = D + C (˜ gt (λ)) B with g˜t (λ) =

A I I 0

(29)

Note that g˜t (λ) is a matrix-valued function and that g˜t (0) = 0 therefore the two functions on the right of (29) are orthogonal and:

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L.A. Montestruque and P.J. Antsaklis

h Gzw

2 gh2

= g˜ (λ)

2 2

= D

2 HS

+ C (˜ gt (λ)) B

2 2

(30)

The second norm on the right can be calculated as: C (˜ gt (λ)) B

2 2

=

1 2π

2π 0

C g˜t ejθ

By ﬁxing θ the integrand F = C g˜t ejθ B with impulse response:  I 0 f (t, τ ) = CN eΛt g˜t ejθ  0 I 0 0

2

B

HS

dθ

(31)

is a Hilbert-Schmidt operator  0 0  eΛ(h−τ ) BN 0

(32)

Then: F

2 HS

h h 0 0

= trace 

∗

f (t, τ ) f (t, τ ) dtdτ   I 0 0 T ∗ T h T Λ (h−τ )  0 I 0  g˜t ejθ Caux = trace  0 BN e Caux g˜t ejθ 0 0 0    I 0 0  0 I 0  eΛ(h−τ ) BN dτ  0 0 0      I 0 0 I 0 0 T h T Λ (h−τ )  0 I 0  dτ = trace  0  0 I 0  eΛ(h−τ ) BN BN e 0 0 0 0 0 0 g˜t ejθ T = trace Baux Baux g˜t e T = trace Baux g˜t ejθ

∗

jθ ∗

∗

T Caux Caux g˜t ejθ

T Caux Caux g˜t ejθ

T Caux Caux g˜t ejθ Baux

(33) So (31) can be calculated as the H2 norm of Caux g˜t ejθ Baux which corresponds to the H2 norm of Gaux .♦ To calculate the Generalize H2 Norm several calculations need to be done, among these are: 2

h t 0 0

T

T Λ τ T BN e CN CN eΛτ BN dτ dt HS    I 0 0 I 0 0 T T Baux Baux =  0 I 0  P22 P12  0 I 0  0 0 0 0 0 0 T −Λ BN BN P11 P12 = exp h T 0 P22 0 Λ T T M12 Caux Caux = M22 T M11 M12 CN −ΛT CN = exp h 0 M22 0 Λ

D

= trace 

(34)

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243

Note that in this particular case it was relatively easy to separate the inﬁnite dimensional components of the system from a ﬁnite dimensional core component. This is not always possible. In particular one might be tempted to apply the previous techniques to obtain a ﬁnite dimensional auxiliary discrete LTI that can be used to solve an H2 optimal control problem. The described separation technique can’t be carried out since the controller and observer gains operate over continuous signals. Nevertheless it will be shown later how to address an H2 optimal control problem using other techniques. Example 2. We now calculate the generalized H2 norm for the same system studied in the previous example. Some computational issues have to be addressed in order to do this. In particular the formulas given in (34) may yield inaccurate results because of scaling issues. In particular for the calculation of Baux and Caux the term (1,1) of the exponentials calculated may be too large in comparison with the other terms, this is because of the negative sign in front of the stable matrix Λ. Direct integration yields better results. Also the Cholesky factorization is usually only an approximation. The results though seem to represent reality in a reasonable way. This is veriﬁed by varying the tolerances in the integration algorithm and by measuring the error in the Cholesky factorization. The calculated generalized H2 norm is shown below for the same range of update times used for the previous example: 0.5 0.45 0.4

0.3

0.25

zw

||G ||

2 gh2

0.35

0.2 0.0394 0.15 0.1 0.05 0 0

1

2

3

4

5

6

update times h (sec)

Fig. 3. Generalized H2 norm of the system as a function of the update times.

In this example we see that the norm converges again to the non-networked H2 norm of the system as the update time goes to zero. The optimal update time in this case is around 1 sec, this is somewhat consistent with the previous example where the optimal update time is around 0.8 sec. Both examples coincide in that after the update time of 1 sec the performance starts to degrade.

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Since both performance measurements are deﬁned in a diﬀerent manner no real comparison can be made between them. It seems though that the Generalized H2 Norm is more appropriate since it considers the application of the impulse input at diﬀerent times. Also its link with a well-deﬁned operatorvalued transfer function makes it very attractive. The next section presents an alternate parameter representation that overcomes the inconveniencies of dealing with inﬁnite dimensional operators.

5 Optimal Controllers for MB-NCS In this section we address the issue of designing optimal controllers for MBNCS. We saw previously that lifting can transform a periodic system such as the MB-NCS into a discrete LTI system. Most results for the design of optimal controllers for discrete systems directly apply to the lifted system. Since the parameters of the lifted system are inﬁnite dimensional, computations using the integral representation given in (8) can be diﬃcult. This is evident when one considers operators such as

∗

I −D D

−1

, which appears for instance in

sampled data Hinf problems. To circumvent some of the problems associated with optimal control problems, an auxiliary discrete LTI system is obtained so that its optimal controller also optimizes the lifted system. The separation of the inﬁnite dimensionality from the problem is not always guaranteed. In particular we note that the controller for the auxiliary system works in the discrete time domain while the controller for the lifted system representing the MB-NCS in (8) works in continuous time. This means the controller has to be obtained using the lifted parameters directly. In this section we start by giving a brief summary of an alternative representation of the lifted parameters proposed by Mirkin & Palmor [MRP99, MP99]. This alternative representation allows performing complex computations using lifted parameters directly. Results on the computation of an optimal sampler, hold, and controller are shown and their equivalence with the components of the output feedback MB-NCS is shown. The representation of lifted parameters proposed by Mirkin & Palmor considers the lifted parameters as continuous LTI systems operating over a ﬁnite time interval. The main advantage of such representation lies in the possibility of simplifying operations over the parameters to algebraic manipulations over LTI systems with two-point boundary conditions. These manipulations can then be performed using well-know state-space machinery. Consider the following LTI system with two-point boundary conditions (STPBC):

Performance Evaluation for Model-Based Networked Control Systems

G:

x˙ (t) = Ax (t) + Bu (t) y (t) = Cx (t) + Du (t) Ωx (0) + Υ x (h) = 0

245

(35)

Here the square matrices Ω and Υ deﬁne the boundary conditions. In is said that the boundary conditions are well-posed if x (t) = 0 is the only solution to (35) when u (t) = 0. It can be veriﬁed that the STPBC G has well-posed boundary conditions if and only if the matrix: ΞG = Ω + Υ eAh

(36)

is non-singular. If the STPBC G has well-posed boundary conditions, then its response is uniquely determined by the input u(t) and is given as follows: y (t) = Du (t) +

h 0

KG (t, s) u (s) ds

(37)

where the kernel KG (t, s) is given by: KG (t, s) =

−1 CeAt ΞG Ωe−As B −1 −CeAt ΞG Υ eA(h−s) B

if 0 ≤ s ≤ t ≤ h if 0 ≤ t ≤ s ≤ h

(38)

We will use the following notation to represent (35): G=

A Ω C

Υ

B D

(39)

The following is a list of manipulations that are used to perform operations over STPBCs. 1) Adjoint System: G∗ =

−AT AT h T −T e Υ ΞG −B T

−T A Ω T ΞG e

T

h

CT DT

(40)

2) Similarity Transformation: (for T and S non singular) T GT −1 =

T AT −1 SΩT −1 CT −1

SΥ T −1

TB D

(41)

3) Addition: 

A1 0 G1 + G2 =  C1 4) Multiplication:

0 A2 C2

Ω1 0

0 Ω2

Υ1 0

0 Υ2

 B1 B2 D1 + D2 

(42)

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L.A. Montestruque and P.J. Antsaklis



A1 0 G1 G2 =  C1

B1 C 2 A2 D 1 C2

Ω1 0

0 Ω2

Υ1 0

 B1 D 2 B2 D1 D2 

0 Υ2

(43)

5) Inversion (exists if and only if det (D) = 0 and −1 det Ω + Υ e(A−BD C )h = 0) G−1 =

A − BD−1 C Ω −D−1 C

Υ

BD−1 D−1

,

(44)

This representation reduces the complexity of computing operators such −1

∗

as

I −D D

ξ=

. Using the integral representation of (8) one can get that −1

∗

I −D D

ω if and only if: h

ω (t) = ξ (t) +

t

T −Λ e BN

T

(t−s)

T CN CN

s 0

eΛ(s−τ ) BN ξ (τ ) dτ ds

(45)

It is not clear how to solve this equation. On the other hand using the alternative representation we note that: D=

Λ I CN

0

BN 0

(46)

Using the properties previously listed we obtain: ∗

I −D D

−1

=

I− 

Λ I CN

0

BN 0

T CN −ΛT CN T −BN BN Λ = T −BN 0

∗

Λ I CN 0 0

0 I

0

BN 0 I 0

−1

(47) 0 0

 0 BN I 

To be able to represent operators with ﬁnite dimension domains or ranges such as B and C two new operators are deﬁned. Given a number θ ∈ [0, h], the impulse operator Iθ transforms a vector η ∈ Rn into a modulated impulse as follows: ς = Iθ η ⇔ ς (t) = δ (t − θ) η

(48)

Also deﬁne the sample operator Iθ∗ , which transforms a continuous function ς ∈ Cn [0, h] into a vector η ∈ Rn as follows:

Performance Evaluation for Model-Based Networked Control Systems

η = Iθ∗ ς

⇔ η = ς (θ)

247

(49)

Note that the representation of Iθ∗ is as the adjoint of Iθ , even when this is not strictly true, it is easy to see that given an h ≥ θ the following equality holds: ς, Iθ η =

h 0

T

T

ς (τ ) (Iθ η) (τ ) dτ = ς (θ) η = Iθ∗ ς, η

(50)

The presented results allow to make eﬀective use of the impulse and sample operator. Namely the last two lemmas show how to absorb the impulse operators into an STPBC. Now let us present a result that links the solutions of the lifted algebraic discrete Riccati equation and the algebraic continuous Riccati equation for the continuous system for the H2 control problem. Lemma 1. Let the lifted algebraic discrete Riccati equation for the lifted system G = LGL−1 be as follows: T

∗

A XA − X + C C T

∗

∗

−1

T

∗

T

D C + B XA

D D + B XB

− A XB + C D

=0

(51)

and let the algebraic continuous Riccati equation for G be: AT X + XA + C T C − XB + C T D

−1

DT D

DT C + B T X = 0

(52)

then the conditions for existence of a unique stable solution for both Ricatti equations are equivalent, moreover if they exist, then X = X. This implies that in order to solve the optimal control problem we just need to solve the regular continuous Ricatti equation. We can for example obtain the optimal H2 state feedback “gain” given by ∗

∗

−1

F = − D D + B XB

∗

∗

D C + B XA .

It can be shown [MRP99, MP99] that: F =

A + BF I F

0

I 0

(53)

Here F is the H2 optimal control gain for the continuous system. Note that the expression in (53) exactly represents the dynamics of the actuator/controller for the state feedback MB-NCS when the modelling errors are zero and the feedback gain is the H2 optimal feedback gain. Finally, we present next a result that obtains the H2 optimal sampler, hold and controller.

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Lemma 2. Given the standard assumptions, when the hold device is given by (Hu) (kh + τ ) = φH (τ ) uk , ∀τ ∈ [0, h] and the sample device is given by h (Sy)k = 0 φS (τ ) y (kh − τ ) dτ , the H2 optimal hold, sampler, and discrete controller for the lifted system G = LGL−1 with   B1 B2 A G =  C1 0 D12  C2 D21 0 are as follows: Hold: Sampler: Controller: where:

φH (τ ) = F e(A+B2 F )τ φS (τ ) = −e(A+LC2 )τ L Θ I Kd = I 0 h Θ = e(A+B2 F )h + 0 e(A+LC2 )(h−τ ) LC2 e(A+B2 F )τ dτ

(54)

Remarks: Note that the H2 optimization problem solved in [MP99] is related to the Generalized H2 norm previously presented. That is, replacing the trace norm with the Hilbert-Schmidt norm. As it has been observed, there is a strong connection between the H2 optimal hold of a sampled system and the H2 optimal controller of the nonsampled system. As pointed out in [MP99] it is clear that the H2 optimal hold attempts to recreate the optimal control signal that would have been generated by the H2 optimal controller in the non-sampled case. That is, the H2 optimal hold calculated in [MP99] generates a control signal identical to the one generated by the non-sampled H2 optimal controller in the absence of noise and disturbances. Another connection exists between the H2 optimal sampler, hold, and discrete controller calculated in [MP99] and the output feedback MB-NCS. It is clear that when the modeling errors are zero and the gain is the optimal H2 gain, the optimal hold has the same dynamics as the controller/ actuator in the output feedback MB-NCS. The same equivalence can be shown between the combination of optimal sampler/discrete controller dynamics and the output feedback MB-NCS observer. The techniques shown here can be used to solve robust optimal control problems that consider the modeling error. This is possible due to the alternative representation that allows the extension of traditional optimal control synthesis techniques to be used with the inﬁnite dimensional parameters that appear in the lifted domain.

6 Conclusions The study of the performance of MB-NCS shows that a large portion of the available literature on sampled data systems cannot be directly applied to

Performance Evaluation for Model-Based Networked Control Systems

249

MB-NCS. Moreover, diﬀerent deﬁnitions of performance yield diﬀerent performance curves. For a constant controller and observer gains it was shown that the best transmission times are not necessarily the smallest ones. Using an alternate representation of the lifted parameters, a connection between the optimal hold, sampler, and discrete controller and the output feedback MBNCS was established. This representation opens a large new area of research in robust MB-NCS.

Beating the Bounds on Stabilizing Data Rates in Networked Systems Yupeng Liang and Peter Bauer Department of Electrical Engineering University of Notre Dame Notre Dame, IN 46556 {yliang1,pbauer}@nd.edu Summary. This paper applies simple, classical feedback control and quantization schemes to the problem of bit rate reduction for stabilizing unstable linear plants. In particular, state feedback, deadbeat control (pole placement at the origin), and standard ﬂoating point quantizers (with diﬀerentially encoded exponents) are used to reduce stabilizing data rate requirements to levels near or even below the theoretical bounds for certain classes of systems. First order systems and higher order controllable canonical forms are investigated. The robustness of the derived condition for stabilizing data rate with respect to system coeﬃcients is also addressed.

1 Introduction Over the last decade, embedded systems in general, and networked control systems in particular, have enjoyed tremendous attention. This is mainly due to the low cost of embedded devices and the wide availability of wireless (and wired) digital networks. One of the most interesting questions that has recently been answered in the area of networked control systems is the problem of minimal bit rate requirements in the feedback path for stabilizing an unstable system. In a linear time-invariant system, this bit rate was shown to be directly related to the unstable eigenvalues of the system. There are a large number of results that analyzed minimum bit rate requirements for linear time-invariant systems [FX03, TM04c, WCWC04, EM01a, WB99b, BL00b, FZ03, Lib03b, Tat00, LL04], time-variant systems [TM04c, Tat00, PBLP04, LB03, LB04a], nonlinear systems [LB04b], and stochastic systems [NE04, Tat00, TSM04]. All these results focused on the theoretical problem of establishing lower bounds for stabilizing data rates and were derived without introducing any compression schemes. In this paper, we explore the idea of using standard and simple methods of implementing quantization, feedback control and compression/encoding

P.J. Antsaklis, P. Tabuada (Eds.): Netw. Emb. Sens. and Cntrl., LNCIS 331, pp. 251–268, 2006. © Springer-Verlag Berlin Heidelberg 2006

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mechanisms to achieve or even beat the derived bounds for ﬁrst and higher order discrete time linear systems. The introduced system employs state feedback, ﬂoating point quantization, and a simple compression scheme of the exponent information by using diﬀerence encoding of two consecutively transmitted ﬂoating point values for stabilization. It is shown that, for the case of ﬁrst order systems, the theoretical bit rate bounds can always be achieved and sometimes violated in the average bit rate sense. This requires pole placement such that deadbeat control is achieved. Higher order controllable canonical systems are also investigated, and the obtained results are shown to be close to the theoretical bounds only for certain classes of systems that are theoretically characterized. Simulations show that our theoretically derived mantissa plus sign bit rate requirements are often conservative and the actual bit rates required are even lower. In certain cases, they indeed beat the theoretical minimum presented in previous literature. This is achieved by using a novel methodology developed in this paper, which minimizes the mantissa plus sign rate required for closed loop stabilization, and adopts an eﬃcient compression/encoding scheme for exponent information. This paper is organized as follows: In Section 2, we introduce two model components used in this analysis, and review the previous results about stabilizing data rate bounds. In section 3, we derive stabilizing mantissa plus sign rate bounds for ﬁrst order and higher order controllable canonical systems. Relationships between the obtained results and previous theoretical bounds are established and conservativeness properties are studied for certain classes of systems. Robustness of the derived results are also brieﬂy discussed. Practical system implementations approaching the theoretical bounds are proposed in Section 4. In particular, an eﬃcient compression/encoding scheme for exponent information and conditions on necessary stabilizing mantissa plus sign rate are presented. Additionally, the possibility of beating the theoretical bounds is discussed and simulation results highlighting such cases are provided. Finally, conclusions are oﬀered in Section 5.

2 Preliminaries and Models 2.1 Model Description Networked embedded systems (NES) implement feedback links over digital communication networks characterized by delays, quantization eﬀects, packet drop, synchronization errors, and data rate constraints. Good models capturing these eﬀects are important for the analysis of stability and performance properties of NES. In what follows, we will introduce two models that are used in the analysis: the quantization (data word) model and the system model. Some preliminary deﬁnitions are required. N refers to the set of integers and N0 denotes that of nonnegative integers. The set of real numbers is denoted as R, and that of the nonnegative real numbers is given by R0 . Last,

Beating the Bounds on Stabilizing Data Rates in Networked Systems

253

Fig. 1. Sector bound model for ﬂoating point quantization

the discrete time variable is represented as k ∈ N0 and the rational numbers are denoted by Q. 2.1.1. Quantization (Data Word) Model NES send quantized data over digital communication links in the feedforward and feedback paths. Three basic choices of quantization schemes are commonly used in practice: ﬁxed point, ﬂoating point and block ﬂoating point. It is known that ﬂoating point and block ﬂoating point schemes present a number of advantages over ﬁxed point schemes [MB95]. In particular, it was proven that the ﬁxed point quantization policy with a ﬁnite number of quantization levels cannot achieve an asymptotic closed-loop stability in the case of unstable plants [Del90a]. In contrast, ﬂoating and block ﬂoating point quantization oﬀer an approximately constant relative quantization error and allow for stabilization of unstable plants. We therefore select ﬂoating point quantization as the choice of quantization scheme. A ﬂoating point number is represented by x = (−1)s × m × 2e

(1)

where s ∈ {0, 1} is the sign of x, m ∈ [0.5, 1) is the mantissa of x and e = log2 |x| + 1 ∈ N is the exponent of x, with x denoting the largest integer less than or equal to the real number x.When x is stored using a total of l bits in ﬂoating point format, it consists of three parts: one bit for s, lm bits for m and le bits for e. Obviously, l = 1 + lm + le

(2)

Notice that here we have denoted lm as the number of mantissa bits not counting the 1 bit after the binary point that is always present to satisfy the normalization requirement 12 ≤ m < 1.

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Y. Liang and P. Bauer

Fig. 2. State feedback model with ﬂoating point quantization

A classical approach for the analysis of quantization eﬀects is to treat the quantization error as a nonlinearity and model it using the sector bound approach [FX03], as shown in Fig. 1. Then, for the case of rounding and inﬁnite dynamic exponent range with no overﬂow or underﬂow, the ﬂoating point quantizer can be deﬁned as Q : Rn −→ Qn such that Δ

Q(x(k)) = F lp(x(k)) = x(k)[I + E(k)] where x(k) ∈ Rn and

(3)

E(k) = [εij (k)]

with

[ε, ε), 0,

εij (k) =

∀i = j ∀i = j

ε = −ε = 2−(lm +1)

(4)

Therefore, the relative quantization error for each state xi (k) has a maximum absolute value ε. In particular, for Single Input Single Output (SISO) systems, the quantization error term E(k) reduces to a scalar, denoted as ε(k). Hence, (3) can be simpliﬁed as: Q(x(k)) = [1 + ε(k)]x(k). (5) 2.1.2. System model Fig. 2 depicts the state feedback model with a ﬂoating point quantizer in the feedback path. For simplicity, only SISO systems are considered in this paper. Furthermore, the sampling time is normalized to 1. The state space representation of the plant is given by the diﬀerence equations x(k + 1) y(k)

= =

Ax(k) + Bu(k) Cx(k)

(6)

where x ∈ Rn×1 , A ∈ Rn×n , B ∈ R1×n and u ∈ R. In addition, u(k) = r(k) + Q(K T · x(k))

(7)

Beating the Bounds on Stabilizing Data Rates in Networked Systems

255

where K ∈ R1×n . K T denotes the transpose of K. For zero external input cases, applying (5), (7) becomes u(k) = Q(K T · x(k)) = [1 + ε(k)] · K T · x(k)

(8)

where ε(k) denotes the scalar quantization error originated from quantizing K T · x(k). 2.2 Overview of Previous Results In recent years, many researchers [PBLP04, TM04c, NE04, WCWC04, Tat00, LL04, EM01a, WB99b, BL00b, FZ03, Lib03b] have analyzed the eﬀects of communication constraints on the stability property of networked control systems. Several theoretical lower bounds on stabilizing data rates have been obtained [PBLP04, TM04c, NE04, Tat00, LL04]. Some of them are summarized below. 2.2.1. Lower Bound on Average Stabilizing Data Rate When the stabilizing data rate is obtained in an average sense, it is given by [TM04c, NE04, Tat00] n

max(0, log2 |λi |)

R>

(9)

i=1

where λi are the eigenvalues of a discrete time system. 2.2.2. Lower bound on constant stabilizing data rate For the cases with constant rate requirement, the lower bound on stabilizing data rate is [TM04c] n

R>

max(0, log2 |λi | )

(10)

i=1

where, again, λi stands for the system eigenvalues. Furthermore, a tighter lower bound [LL04] is shown to be: n

max(1, |λi |)

R ≥ log2 i=1

(11)

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Y. Liang and P. Bauer

2.2.3. Deadbeat Cases with Mantissa Requirements and Time-Variant Delay State feedback and ﬂoating point quantization have been popularly used in practical implementations of NES. Hence, the stabilizing data rate issue in this particular type of system realization is worth exploring due to its simplicity and practical relevance. In addition, communication networks are usually characterized by various time variant delays. This is especially true in networks employing wireless links. This type of issues have been discussed in previous work [PBLP04]. A deadbeat controller was shown to result in a minimum required stabilizing mantissa rate, and corresponding lower bounds for the stabilizing mantissa length lm ∈ N0 were obtained. For ﬁrst order systems, the mantissa condition is given by: λm > Ti log2 |λ| − 1, ∀i ∈ N0 , lm ∈ N0 . (12) where Ti , i ∈ N0 represents the time-variant access delay modelled in [PBLP04] and λ stands for the system eigenvalue, which is assumed to be larger than 1 in magnitude. Furthermore, the expression for stabilizing mantissa plus sign rate rTi ∈ R0 was also derived [PBLP04], as follows: rTi =

lm + 1 > log2 |λ|. Ti

(13)

It can be seen that the stabilizing mantissa plus sign rate is not aﬀected by the time-variant access delay Ti present in the feedback path. Finally, an interesting case is worth emphasizing. Note that for an unstable system with 1 < |λ| < 2 and Ti = 1, according to (12), the lowest possible stabilizing mantissa length becomes 0, which is somewhat counterintuitive. In this case, zero mantissa bits except the ’1’ after the binary point are needed. Therefore, only the sign and exponent information needs to be transmitted.

3 Theoretical Results In this section, we extend the work in [PBLP04] by obtaining stabilizing data rate (sign, mantissa and exponent) bounds for systems depicted in Fig. 2, i.e. with a ﬂoating point quantizer on the feedback path but without considering time delays. We also investigate the relationship between the obtained bounds with previous results presented in Section 2.2. Interpretations and implications of the derived relationship are explored for diﬀerent types of systems, and the robustness of the obtained condition for stabilizing data rate with respect to system coeﬃcients is also addressed. Throughout this analysis, the external input is assumed to be zero and the sampling time is normalized to 1.

Beating the Bounds on Stabilizing Data Rates in Networked Systems

257

3.1 Deadbeat Systems with Mantissa Requirements In the following analysis, we focus on two types of deadbeat delay-free (no inherent time delay) systems: the ﬁrst order case and the n-th order SISO controllable canonical forms, respectively. 3.1.1 First Order Deadbeat System Using (5) and (6), the state diﬀerence equation for the ﬁrst order deadbeat delay-free control system is x(k + 1) = ax(k) + b · Q(K · x(k)) = −a · ε(k) · x(k)

(14)

by noting that a + bK = 0. In order to ensure the stability of the resulting system, the following inequality needs to be satisﬁed: |a · ε(k)| < 1

(15)

Applying (4), i.e. |ε(k)| ≤ 2−(lm +1) , we obtain |a| < 2lm +1

(16)

lm > log2 |a| − 1

(17)

which can be rearranged to:

It provides the condition for system coeﬃcient a to ensure closed-loop stability. It can also be reformulated in terms of the system eigenvalue λ as lm > log2 |λ| − 1

(18)

by recognizing the fact that λ = a. For delay-free systems, under the assumption of unit sampling time, the stabilizing mantissa plus sign rate is given by: rms = lm + 1

(19)

Speciﬁcally, for ﬁrst order systems, by applying (17), (18) to (19), we obtain the representation for rms as: rms > log2 |a| = log2 |λ|, rms ∈ N0 . 3.1.2 SISO Controllable Canonical System For SISO delay-free systems in controllable canonical form, we have:

(20)

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Y. Liang and P. Bauer

 A=

0 0 .. .

0 an

1 0 .. .

0 1 .. .

an−1

an−2

0

··· ··· ··· ··· ···

0



  0 0 , B = ... 0 1 a1 1 0 0 .. .

using (5) and (6), the system characteristic equation can be written as: n

xn (k)

=

ai xi (k − 1) + u(k − 1) i=1 n

n

ai xn (k − i))

ai xn (k − i) − Q(

= i=1

=

i=1

n

ai · xn (k − i)

−ε(k) ·

(21)

i=1

by applying the fact that xi (k − 1) = xn (k − i), i = 1, 2, · · · , n. Here, ε(k) ∈ [−2−(lm +1) , 2−(lm +1) ]. Note that the system represented in (21) is globally asymptotically stable [BMD93] if n

|ε(k)| ·

n

|ai | ≤ 2−(lm +1) ·

i=1

|ai | < 1

(22)

i=1

Therefore,

n

lm > log2

|ai | − 1

(23)

i=1

providing a lower bound for the mantissa length to ensure the overall system stability. Similar as in the ﬁrst order case, substituting (23) into (19), we obtain the stabilizing mantissa plus sign data rate, described as: n

rms > log2

|ai |, rms ∈ N0 .

(24)

i=1

3.2 Relationship to Bounds in Literature Below, we study the relationship between the rate bound (24) for deadbeat control with ﬂoating point quantization and the general bound (9), for the n-th order controllable canonical forms. Since ﬁrst order systems can be regarded as a special case of controllable canonical forms, the derived relationship can also be applied to ﬁrst order systems in a straightforward manner. First, we introduce some notations to be used in this section. λi , i = 1, 2, · · · , n stands for system eigenvalues. In particular, λuj , j = 1, 2, · · · , m

Beating the Bounds on Stabilizing Data Rates in Networked Systems

259

denote the unstable eigenvalues. ai , i = 1, 2, · · · , n denote the system coeﬃΔ n cients. Finally, we use Rmsb = log2 i=1 |ai | to represent the lower bound on Δ

m

stabilizing mantissa plus sign rate provided by (24), and Rrb = j=1 log2 |λuj | to describe the stabilizing rate bound introduced in (9). It can be seen from the following discussion, that Rmsb is always greater than or equal to the rate Rrb . For a controllable canonical form, the characteristic polynomial is given as: n

Δ

p(λ) =

(λ − λi ) = λn − a1 · λn−1 − · · · − an−1 · λ − an .

(25)

i=1

When an = 0, Rmsb can be simpliﬁed from (24) as: n

Rmsb

=

|ai |

log2 i=1

= =

a2 an−1 a1 | + | | + ··· + | |)] an an an a1 a2 an−1 log2 |an | + log2 (1 + | | + | | + · · · + | |). an an an

log2 [|an | · (1 + |

(26)

Δ

Now, denote Δ = log2 (1 + | aan1 | + | aan2 | + · · · + | an−1 an |). Since Δ ≥ 0, from (26), we conclude: n

n

log2 |λi |

λi | =

Rmsb ≥ log2 |an | = log2 | i=1

(27)

i=1

n

by noticing | i=1 λi | = |an |. This result implies that Rmsb can be more conservative than Rrb , depending on the value of Δ and system eigenvalue speciﬁcations. More detailed discussions are presented below. In the particular cases when an = 0, the above derivation becomes invalid. Relationships between the system coeﬃcients and eigenvalues need to be inspected for a comparison between Rmsb and Rrb . We begin our following discussion by interpreting the obtained relationship for diﬀerent classes of systems by investigating the conservativeness of Rmsb compared to Rrb . 3.2.1 Systems with Unstable ”Butterworth Type” Eigenvalue Distribution If ai = 0, i = 1, 2, · · · , n − 1 and |an | > 1, then the characteristic equation becomes λn +an = 0. Hence, the system eigenvalues occur on a circle centered at the origin with radius n |an | > 1 at equally spaced points, or in other words, distributed in a ”Butterworth type” fashion. Obviously, all the system eigenvalues are unstable and denoted by λui , i = 1, 2, · · · , n. In addition, Δ = 0. Therefore, applying (26), we obtain

260

Y. Liang and P. Bauer n

Rmsb = log2 |an | =

log2 |λui | = Rrb .

(28)

i=1

Hence, Rmsb coincides with Rrb in these systems and turns out to be nonconservative. Obviously, a ﬁrst order system with an unstable pole constitutes the simplest example of this type of system. 3.2.2 Systems with Repeating Unstable Eigenvalues For systems with only repeating unstable eigenvalues, the characteristic polynomial is given as p(λ) = (λ + β)n where |β| > 1. Therefore, Rmsb = nlog2 |β| + log2 (1 + | βan1 | + | βan2 | + · · · + | aβn−1 n |) according to (26). Because n β i = 0, we have Δ > 0. Therefore, in this situation, Rmsb is ai = i conservative compared to Rrb . 3.2.3 Stable Plants n

It is worth noting that while the condition i=1 |ai | < 1 ensures a stable n plant, i=1 |ai | > 1 does not ensure an unstable plant. Therefore, if one would choose to use feedback in the case of a stable plant, it may be necessary to use a non-zero rate for stabilization. (Of course, one can simply check the eigenvalues of the plant and then decide whether stabilization through feedback is necessary.) n Again, an interesting special case occurs for 1 ≤ i=1 |ai | < 2 (which can correspond to both, a stable or unstable plant): for this particular case, only exponent and sign information needs to be transmitted. 3.2.4 Robustness with Respect to System Coeﬃcients From (22), we know the condition for closed-loop system stability is n

|ai | < 2lm +1

(29)

i=1

where lm represents the stabilizing mantissa length. This property provides an important tool to check the required bit rate by inspection. It also sheds light on the robustness of bit rate requirements with respect to plant coeﬃcients. Fig. 3 shows regions of constant stabilizing mantissa rates as a function of plant coeﬃcients for the second order case. Regions of constant minimum mantissa rate requirements are concentric diamonds centered at the origin. n It should be noted that [BMD93], in the ﬁrst hyper-quadrant, i=1 |ai | ≥ 1 n describes unstable systems and i=1 |ai | < 1 describes stable systems.

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261

Fig. 3. System coeﬃcient set with stabilizing mantissa length speciﬁcation

4 Beating the Bounds: Possibility, Implementation, and Simulations In the previous work [LDB05], we investigated realizations of two diﬀerent system realizations for the purpose of approaching the theoretical stabilization data rate bounds, viz. (a) System Type I: systems with the controller co-located with the plant which is digital in nature. In this type of systems, we have access to the ﬂoating point exponent information extracted from the system internally to complete the stabilization of the overall system. In this way, we have saved feedback bandwidth a part of which would have to be devoted to transmitting exponent bits; (b) System Type II: systems with the controller geographically separated from the plant or the plant originates from a system which is analog in nature with states/outputs discretized for feedback over a network link. In this type of systems, the internal system information is not accessible any more, and, hence, both mantissa and exponent information of the quantized feedback data needs to be transmitted to achieve closed-loop stability. In this section, we focus on System Type II. We present the system implementation that minimizes the stabilizing data rate, and use simulation re-

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sults to validate that, in particular cases, one can indeed beat the theoretical bounds. 4.1 Exponent Compression/Encoding Scheme In order to minimize the stabilizing data rate, we propose an eﬃcient ”compression” scheme for exponent information, which needs to be transmitted over the feedback path for stabilization, as detailed below: 1) For the ﬁrst data packet, transmit complete exponent information (e). Store e at both the transmitter and receiver sides of the feedback network. 2) For the following data packets, store the exponent at the transmitter side and send bits representing its diﬀerence (Δe) with the previous one (P rev e) only (including the sign of Δe). 3) At the receiver side, recover the complete feedback exponent information using Δe and previously stored exponent (P rev e). Finally, store the recovered exponent as P rev e for the next round use. This compression scheme is especially eﬃcient for systems that are right at the boundary of stability, since exponent updates are then relatively rare. In addition to transmitting Δe instead of complete exponent bits, we adopt eﬃcient encoding schemes for Δe to further reduce the bits that need to be transmitted. Ideally, we propose an encoding method similar to Huﬀman coding [Lib03c]. In principle, we assign length-varying codes to diﬀerent values of Δe based on their probabilities. Speciﬁcally, we assign codes with less (more) bits to Δe of higher (lower) probabilities. As a result, the bits representing Δe information is minimized in an average sense. However, in some practical cases, implementation of this method is not feasible. One such example is when there are a large number of possible values for Δe, which may further be stochastic, it is impossible to assign a corresponding code to each possible Δe a prior. Even if it is possible to do so, most times this method generates a long processing delay to assign such probability based length-varying codes, which can profoundly degrade the system performance. To avoid the above drawbacks, in this paper, we adopt another coding methodology for Δe, which is easy to implement and reduces the bits that represent Δe information and need to be transmitted on the feedback path for stabilization, although it may not result in the strictly minimal feedback rate. In particular, the ﬁrst bit is ﬁxed as the sign bit of Δe, and the remaining bits are the absolute value of Δe represented in binary format, unless Δe = 0. In this case, no bits are transmitted in the feedback path, and it is acknowledged at the receiver side that, the exponent stays the same as the previous one. In fact, this turns out to be a most common case if the system is on the stability boundary. Finally, for the cases when Δe becomes an inﬁnite value, the binary bits are rounded up to a certain predeﬁned length before transmission.

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263

In essence, the above compression schemes are introducing memory into the system to reduce the information that needs to be sent at each sample time. In practice, this technique is commonly used in many applications. For example, it is used when streaming video over networks to reduce the required bit-rate while maintaining image quality. 4.2 Suﬃcient Mantissa Plus Sign Information for Stabilization As observed in section 3.2, Rmsb can coincide with Rrb in many cases. In addition, we have used conservative results to derive Rmsb in section 3.1, speciﬁcally, by employing sector bounds [FX03] and the stability condition in [BMD93]. Hence, we can expect that in certain cases the practically required stabilizing mantissa plus sign rate, denoted as Rms , can be lower than the theoretical bound Rmsb . In fact, many such examples have been identiﬁed in simulations. Some of them are presented in section 4.5 below. Nevertheless, a systematic way to determine the minimally required mantissa plus sign rate for stabilization is still open. Finally, it should be noted that the notation Rms here diﬀerentiates from the rms introduced in section 3, by representing the practically achievable stabilizing mantissa plus sign rate, while rms denotes a suﬃcient stabilizing mantissa plus sign rate above the bound Rmsb , which can be conservative and higher than Rms . 4.3 Implementation of Simulation In this section, we present the simulation implementation steps for System Type II: 1) Obtain the system state vector, and multiply it with the feedback gain vector required for pole placement. Using ﬂoating point quantization, obtain the mantissa, sign and exponent. The mantissa is rounded up to the minimum length (Lm ) suﬃcient to stabilize the overall system, assuming inﬁnite exponent range. Lm is identiﬁed using simulations, and its typical value is below (17) for ﬁrst order systems and (23) for higher order SISO systems, due to the conservative results in [FX03] and [BMD93] used in deriving these bounds, similar as discussed in section 4.2. Note that once Lm is determined for a system, it stays constant for all simulations. Similar to the relationship between Rms and rms discussed in section 4.2, Lm stands for the practically achievable minimum stabilizing mantissa length in simulations, while the notation lm used in section 3 and before denotes a suﬃcient stabilizing mantissa length satisfying the conditions speciﬁed in (17), (18) and (23) that can be conservative and higher than the practically required one, Lm . 2) Transmit the mantissa (Lm ) with the sign bit (1) completely. For the exponent part, use the compression scheme introduced in section 4.1, i.e. for the ﬁrst data packet, transmit complete exponent information (e). After that, store the exponent at the transmitter side and send information representing

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Y. Liang and P. Bauer

its diﬀerence with the previous exponent (Δe) only (including the sign of exponent diﬀerence). Δe is encoded in an eﬃcient way before transmission, as presented in section 4.1. In the case where two consecutive exponents are the same, no exponent information bit needs to be transmitted. 3) At the receiver end, recover the feedback states for stabilization using the sign, mantissa bits and information on the diﬀerence of the corresponding exponents. Note that under this scheme, we have assumed that the channel capacity is large enough to avoid saturation. According to the above implementation, the resulting stabilizing feedback data rate, denoted as Rstab , is given as Rstab = Rms + Re

(30)

The stabilizing mantissa plus sign rate Rms is given as Rms = Lm + 1

(31)

where Lm is deﬁned above, representing the minimum mantissa length that is suﬃcient for stabilization in simulations. The stabilizing exponent rate Re results from diﬀerence encoding of e. Using the second method presented in section 4.1, Re at each sampling time instance is given as  i=1 length(e) + 1; (32) Re = length(Δe) + 1; Δe = 0, i = 2, 3, · · ·  0; Δe = 0, i = 2, 3, · · · where i stands for the sampling time index and length(x) denotes the length of binary bits representing the absolute value of x. Note that the additional ’1’ bit per unit time is due to the sign bit of e or Δe. 4.4 Possibility of Beating the Theoretical Bounds As discussed in section 4.2, there may exist certain cases when the actually required stabilizing mantissa plus sign rate Rms can violate the theoretical bound Rmsb . If, furthermore, Rmsb coincides with Rrb , we have Rms lower than Rrb . In addition, if the stabilizing exponent rate Re is small enough, the overall stabilizing data rate Rstab given by (30) can indeed beat the theoretical bound Rrb provided by (9). This can be better illustrated using Fig. 4 and Fig. 5 below. Fig. 4 and Fig. 5 compare relationships between diﬀerent bounds and data rates, respectively. As can be observed, in the cases when Rmsb coincides with Rrb , due to the conservative nature of Rmsb , the practical mantissa plus sign rate Rms can actually be one (or more) bits per unit time less than Rmsb , and Rrb . In addition, if the exponent rate Re is small enough (say less than one

Beating the Bounds on Stabilizing Data Rates in Networked Systems

265

Fig. 4. Relationships between diﬀerent bounds

Fig. 5. Relationships between diﬀerent data rates when beating the bound (9)

bit per unit time), the stabilizing total data rate given by Rstab = Rms + Re can indeed beat the theoretical bound Rrb . In fact, such examples do exist, and are presented in section 4.5 below. Finally, it is important to note that the methodology presented in this work has diﬀerent assumptions than [TM04c] and the other results in NES literature. In particular, the theoretical bound Rrb was derived without considering any compression schemes, while this paper does adopt an eﬃcient exponent compression/encoding method. This paper, therefore, does not disprove the established theoretical bounds, but rather, using these theoretical results as a reference, it proposes a simple and standard system implementation that can actually stabilize the overall system with a feedback rate lower than the theoretical bound Rrb . This is invaluable for practical system design. 4.5 Simulation Results Based on the implementation method presented in section 4.3, we performed extensive simulations to obtain the practical minimum stabilization data rates for many ﬁrst and second order systems. In all simulations, feedback gains are chosen such that deadbeat control is achieved. Each investigated system was

266

Y. Liang and P. Bauer Table 1. Examples of ﬁrst order systems beating the theoretical bounds System pole(λ) Rrb (9)1 Rmsb (20)2 Rms 3 Re 4 Rstab 5 5000 −107

12.288 23.254

12.288 23.254

11 22

0.005 11.005 0.004 22.004

Table 2. Examples of second order systems beating the theoretical bounds System matrix(A) Rrb (9)1 Rmsb (24)6 Rms 3 Re 4 Rstab 5 „ « 0 1 8.966 9.229 8 0.009 8.009 „ 100 500 « 0 1 6.644 6.644 5 0.006 5.006 100 0

analyzed for random initial conditions over 100 simulations runs. The stability was observed in the sense of Lyaponov over all the simulation runs to determine the minimum required data rate to achieve stability. Each simulation run was of length 5000 time steps. The practically achieved minimum stabilizing mantissa plus sign rate Rms is speciﬁed by (31), and is constant for each particular system in all simulation runs. The stabilizing exponent rates for each of the 100 simulation runs of a particular system is determined by (32). These values are then averaged to render the stabilizing exponent rate Re for this system. It should be noted that for a given system, the diﬀerences between stabilizing exponent rates obtained in diﬀerent simulation runs are actually small. Finally, the minimally required stabilizing total data rate Rstab for each system is given by (30), i.e. produced by summing up the constant Rms and average value Re . Indeed, the resulting Rstab is obtained in an average sense rather than a strictly minimal value. One could stabilize the system using an even smaller Re under certain conditions, or, dynamically assign mantissa bits at a minimal level instead of keeping Rms constant for all time steps. However, both methods come at a cost of signiﬁcantly increasing implementation complexity, which need to be avoided in many situations. The value of this paper is to provide an easily implementable system realization that requires a stabilizing data rate close to or even lower than the theoretical bounds. In the following, we present simulation results for some ﬁrst and second order systems. −1 0 1 2 3 4

Stabilizing data rate bound (9) Stabilizing mantissa plus sign rate bound (20) Practical stabilizing mantissa plus sign rate (31) Stabilizing exponent rate (32) Stabilizing total data rate (30) Stabilizing mantissa plus sign rate bound (24)

Beating the Bounds on Stabilizing Data Rates in Networked Systems

267

4.5.1. First Order Case: For ﬁrst order unstable systems, the lower bound for stabilizing mantissa plus sign rate provided in (20) coincides with the stabilizing data rate bound in (9). However, as mentioned in section 4.2, (20) is sometimes conservative due to the conservative results in [FX03] and [BMD93]. We can therefore have cases when the practical stabilizing mantissa plus sign rate Rms is below the bounds in (20) and (9), and, hence, the resulting stabilizing data rate Rstab beats the theoretical bound (9) under the condition of a small exponent rate Re , according to the analysis in section 4.4. Two such examples are given in Table 1. One can see that the practical stabilizing data rates provided in the simulation results indeed beat the theoretical bound in (9) by more than 1 bits per unit time. 4.5.2. Second Order Case: For second order systems, (24) turns to be more conservative than (9) except for some particular cases, as discussed in section 3.2. However, similar results as the ﬁrst order systems can still be identiﬁed, as shown in Table 2. From this table, one can see the theoretical bounds are again violated by the stabilizing data rates achieved in simulations.

5 Conclusion This paper shows that the recently derived data rate bounds for stabilization of unstable linear time-invariant discrete time systems can be achieved and sometimes even violated with very simple controller, quantization and compression/encoding schemes. Using state feedback and ﬂoating point quantization with diﬀerence encoding of exponent information, it was shown that ﬁrst order systems can always be stabilized at approximately the data rate corresponding to the theoretical bound given by the logarithm of the unstable eigenvalue. In certain cases the stabilizing data rates were found to be well below the bound, which is due to the compression(encoding) of the exponent. It was proved that deadbeat control always allows for stabilizing mantissa plus sign rates that are exactly at the bound. Simulations showed that this mantissa plus sign rate can be conservative and that the required exponent rates were usually well under 1 bit per sample. In the case of higher order controllable canonical forms, the results were typically more conservative, even though classes of systems that yielded results close to or below the bounds were identiﬁed. Finally, the robustness of the mantissa rate with respect to system coeﬃcients was also investigated and resulted in a condition that can be decided by inspection of the plant coeﬃcients.

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Acknowledgement The authors gratefully acknowledge the support provided by the National Science Foundation (NSF) via Grant IIS 0325252 and the Center for Applied Mathematics, University of Notre Dame.

Disturbance Attenuation Bounds in the Presence of a Remote Preview Nuno C. Martins1 , Munther A. Dahleh2 , and John C. Doyle3 1

2

3

Electrical Engineering Department and the ISR University of Maryland College Park, MD 20742 [email protected] Department of Electrical Engineering and Computer Science and LIDS-MIT Cambridge, MA 02139 [email protected] Department of Electrical Engineering and CDS CALTECH Pasadena, CA 91125 [email protected]

Summary. We study the fundamental limits of disturbance attenuation of a networked control scheme, where a remote preview of the disturbance is available. The preview information is conveyed to the controller, via an encoder and a ﬁnite capacity channel. In this article, we present an example where we design a remote preview system by means of an additive, white and Gaussian channel. The example is followed by a summary of our recent results on general performance bounds, which we use to prove optimality of the design method.

1 Introduction If S(z) is the sensitivity transfer function [DFT92] of a single-input linear feedback loop, in discrete time, then Bode’s [Bod45] integral equation can be written as: π 1 log |S(ejω )|dω = log |λ| (1) 2π −π λ∈U P

where UP are the unstable poles of the open loop system [DFT92], which is assumed to be rational and strictly proper. By using feedback, one would expect that disturbance rejection can be improved. On the other hand, (1) quantiﬁes a fundamental limitation which says that disturbance rejection can be, at most, shaped in frequency. Due to its importance, Bode’s fundamental limitation has been extended to more general frameworks [SBG97] than the linear and time invariant one. The multi-dimensional version was provided in

P.J. Antsaklis, P. Tabuada (Eds.): Netw. Emb. Sens. and Cntrl., LNCIS 331, pp. 269–285, 2006. © Springer-Verlag Berlin Heidelberg 2006

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N.C. Martins, M.A. Dahleh, and J.C. Doyle

[FL88], while the time-varying case has been addressed in [Igl02] and certain non-linear systems have been analyzed in [ZI03, Igl01, SBKM99]. In recent publications, such as [MCF04] and references therein, the study of fundamental limitations generalizes to controllers with preview. Using an information theoretic formulation, Bode’s result was extended for feedback systems where the controller belongs to a general class [MD05a, MD04], which might include systems operating on a discrete or ﬁnite alphabet. The use of information theoretic quantities, which was ﬁrst suggested in [Igl01], also allows for the clear diﬀerentiation of the roles of causality and information ﬂow in the feedback loop [MD05a, MD04]. While causality is responsible for Bode’s fundamental limitation, information constraints in the feedback loop give rise to a new limitation[MD05a, MD04]. The work in [Eli04] also explores the connection between Bode’s integral formula and the ability to transmit information over a Gaussian channel, by means of linear and time invariant systems acting as encoders and decoders. Bode’s fundamental limitation is derived for a deterministic setting in [YGI+ ], under certain convergence conditions. 1.1 Problem Formulation It is well known that the use of disturbance previews may improve controller performance [JWPK+ 04]. In [TM05] one ﬁnds recent results in optimal preview control as well as a source of references to other related approaches. Recent results on fundamental limitations in the presence of reference preview are given in [CRHQ01, MCF04]. In this publication, we consider the diagram of Fig 1, where the controller has access to a remotely transmitted disturbance preview, represented as r. This scheme portrays a formulation where the disturbance results from a physical phenomenon, which must travel in space until it reaches the system. The travel time is represented as a delay of m units of time. At the same time, a remote preview signal r may be available to the controller, subject to information transmission/processing constraints at the remote preview system (RPS) block. We also adopt a Markovian model for the disturbance, where G is an auto-regressive shaping ﬁlter and w is the innovations process. Examples of remote preview systems can be found in animal life, such as the ones that use vision and hearing to perceive a future physical interaction. In these cases, the information/processing constraints arise from limited vision and hearing resolution as well as noise and limited information processing in the brain [Fit54]. Further examples are the information path of a heat-shock mechanism at the cellular level [ESKD+ 05] as well as navigation engineering systems. There are two extreme cases in this setup: the ﬁrst is when the disturbance can be fully transmitted4 , in that case the disturbance can be canceled by the 4

This would require a RPS block with inﬁnite capacity.

Disturbance Attenuation Bounds in the Presence of a Remote Preview

shaping ﬁlter

w✲

physical delay ✲

G ✲

-m

z

RPS

d✲

❧ +

r

✲

remote preview system

e

✲

✻

K

✛

271

✲

P y

❄

❧ +

q

✻

measurement noise Fig. 1. Structure of a Remote Preview System.

controller, and the second is the absence of remote preview information, which is the classical framework. In this paper, we study the situation in between, i.e., we consider that C > I∞ (r, d) > 0, where C is a ﬁnite positive constant, representing the Shannon capacity [CT91] of the RPS block, and I∞ (r, d) is the mutual information rate5 between the disturbance d and the remote preview signal r, in bits per unit of time. 1.2 Paper Organization The paper is organized as follows: Sections 1.3 and 1.4 introduce the notation and main deﬁnitions. The technical framework is given in section 2, where we also describe the measures of performance adopted in the paper. A motivation example is described in section 3, while the fundamental limitations, in their more general form, are stated in section 4. The conclusions are provided in section 5. Most of the Theorems for the general case are just stated and the proofs may be found in [MD05b]. 1.3 Notation The following notation is adopted: • Whenever it is clear from the context, we refer to a sequence {a(k)}∞ 0 of elements in Rn as a. A ﬁnite segment of a sequence a is indicated as def = ∅. . If kmax < kmin then akkmax = {a(k)}kkmax akkmax min min min • Random variables are represented using boldface letters, such as a. • If a(k) is a stochastic process, then we use a(k) to indicate a speciﬁc realization. Similar to the convention used for sequences, we may denote ∞ {a(k)}∞ 0 just as a and {a(k)}0 as a. A ﬁnite segment of a stochastic kmax process is indicated as akmin . • The probability density of a random variable a, if it exists, is denoted as pa . The conditional probability, given b, is indicated as pa|b . 5

This quantity is precisely deﬁned in section 1.4.

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• The expectation operator over a is written as E[a]. • We write log2 (.) simply as log(.). • We adopt the convention 0 log 0 = 0. 1.4 Basic Deﬁnitions of Information Theory In this section, we summarize the main deﬁnitions of Information Theory used throughout the paper. We adopt [Pin64], as a primary reference, because it considers general probabilistic spaces in a uniﬁed framework. The deﬁnitions and properties listed in this section hold under general assumptions, we refer to [Pin64] for further details. Deﬁnition 1. (from [Pin64] pp. 9 ) The mutual information I : (a; b) → R+ {∞}, between a and b, is given by Pa,b (Ei × Fj ) log

I(a; b) = sup ij

Pa,b (Ei × Fj ) Pa (Ei )Pb (Fj )

The supremum is taken over all partitions {Ei } of A and {Fj } of B, where A and B are the alphabets of a and b. The deﬁnition of conditional mutual information can be found in [CT91] or in [Pin64] (pp. 37). Notice that, in deﬁnition 1, A and B may be diﬀerent. Without loss of generality, we consider probability spaces which are countable or Rq , for some q. We also deﬁne the following quantities, denoted as diﬀerential entropy and conditional diﬀerential entropy, which are useful in the computation of I(·, ·), for certain cases relevant in this paper. Deﬁnition 2. If a is a random variable with alphabet A = Rq then we deﬁne the diﬀerential entropy of a as h(a) = −

Rq

pa (γ) log pa (γ)dγ

If b is a random variable with alphabet B = Rq then we deﬁne the conditional diﬀerential entropy of a given b as: h(a|b) = h(a, b) − h(b) = −

Rq

Rq

pa,b (γa , γb ) log pa|b (γa , γb )dγa dγb

(2)

If B is countable then h(a|b) is deﬁned as: h(a|b) = − γb ∈Sb

Rq

pa,b (γa , γb ) log pa|b (γa , γb )dγa

(3)

Likewise, the quantity h(a|b, c) is deﬁned by incorporating another sum if the alphabet of c is discrete, or an integral otherwise.

Disturbance Attenuation Bounds in the Presence of a Remote Preview

273

In order to simplify our notation, we also deﬁne the following quantities: Deﬁnition 3. (Information Rate) Let a and b be stochastic processes. The following is the deﬁnition of (mutual) information rate6 : I∞ (a; b) = lim sup N →∞

I(a0N −1 ; b0N −1 ) N

The use of information rates is motivated by its universality [CT91], i.e., it quantiﬁes the rate at which information can be reliably transmitted through an arbitrary communication medium. Deﬁnition 4. (Entropy Rate) For a given stochastic process, we also deﬁne entropy rate as: −1 ) h(aN m (4) h∞ (a) = lim sup N − m N →∞ where m is the time delay represented in Fig 5. In our formulation, em−1 and dm−1 may be deterministic or may take val0 0 ues on a countable set. Since the diﬀerential entropy for these variables is undeﬁned, we choose to deﬁne entropy rate as in (4). On the other hand, for any em−1 and dm−1 , there are no technical problems in using them with mutual 0 0 information or conditional entropy, such as I((em−1 , b); c) and h(b|c, em−1 ). 0 0 In the real world, all signals in Fig 5 should have some noise, with ﬁnite diﬀerential entropy, added to them. If that was the case then we could have deﬁned entropy rate in the standard way. We chose not to add noise everywhere, because it would complicate the paper and it would lead to the same results and conclusions. In this paper, we will refer to channels which are stochastic operators conforming to the following deﬁnition: Deﬁnition 5. (Channel) Let V and R be given input and output alphabets, along with a stochastic process, denoted as c, with alphabet C. In addition, consider a causal map f : C ∞ × V ∞ → R∞ . The pair (f, c) deﬁnes a channel. The following are examples of channels: • Additive white Gaussian channel: V = R = C = R, c is an i.i.d. white Gaussian sequence with unit variance and f (c, v)(k) = c(k) + v(k). • Binary symmetric channel: V = R = C = {0, 1}, c is an i.i.d sequence satisfying P(c(k) = 1) = pe and f (c, v)(k) = c(k) +mod2 v(k) For any given channel speciﬁed by (f, c), the supremum of the rates, at which information can be reliably transmitted, is a fundamental quantity denoted as capacity [CT91]. The formal deﬁnition of capacity, denoted by C, can be found in [CT91], for which the following holds: sup I∞ (v; f (v, c)) ≤ C pv

6

(5)

Throughout the paper, for simplicity, we refer to mutual information rate simply as information rate.

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1.5 Spectral Properties of Asymptotically Stationary Stochastic Processes We adopt the following deﬁnition of asymptotic power spectral density. Deﬁnition 6. A given zero mean real stochastic process a is asymptotically stationary if the following limit exists for every γ ∈ N: ¯ a (γ) def R = lim E[(a(k + γ) − E[a(k + γ)])(a(k) − E[a(k)])] k→∞

(6)

We also use (6) to deﬁne the following asymptotic power spectral density: Fˆa (ω) =

∞

¯ a (k)e−jωk R

(7)

k=−∞

1.6 Elementary Properties of h and I. Below, we provide a short list of properties of diﬀerential entropy and mutual information. A more complete list is included in the Appendix, at the point where the more complex proofs are derived. Consider that a and b are given random variables and that the alphabet of a is A = Rq , while B, the alphabet of b, may be a direct product between Rq and a countable set7 . Under such assumptions, the following holds [CT91]: • I(a; b) = I(b; a) = h(a) − h(a|b) • for any arbitrary function φ, h(a|b) = h(a + φ(b)|b) ≤ h(a) , where equality holds if a and b are independent. • (Data Processing Inequality) I(a; b) ≥ I(φ(a); θ(b)), where equality holds if φ and θ are injective. The above properties hold if we replace entropy by entropy rate and mutual information by information rate. If a is a stationary process8 then [CT91]: h∞ (a) ≤

1 4π

π −π

log(2πeFˆa (ω))dω

(8)

where equality holds if a is Gaussian.

7

8

Notice that if b takes values in a direct product between Rq and a countable ¯ c) where b ¯ takes values in Rq and c has set, then we can represent it as b = (b, a countable alphabet. Using such decomposition one can use our deﬁnitions of conditional diﬀerential entropy to perform computations As we show in [MD05b], this inequality also holds for asymptotically stationary processes.

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2 Technical Framework and Assumptions Regarding the general scheme of Fig 1, the following assumptions are made: • w is a scalar (w(k) ∈ R), unit variance, identically and independently distributed stochastic process. For each k, w(k) is distributed according to a density pw , satisfying |h∞ (w(k))| < ∞. • G is an all-pole stable ﬁlter of the form: G(z) =

α

1−

p m=1

am z −m

(9)

where p ≥ 1, ai and α > 0 are given. We chose this form of G, as a way to model the disturbance d, not only because it is convenient that G−1 is well deﬁned and causal, but also because there exists a very large class of power spectral densities that can be arbitrarily well approximated by |G(ejω )|2 [Pri83]. In addition, we assume that G has zero initial conditions. • given n, P is a single input plant with state x(k) ∈ Rn , which satisﬁes the following state-space equation: x(k + 1) =

xu (k + 1) Au = xs (k + 1) 0

b 0 x(k) + u e(k) bs As

(10)

y(k) = Cx(k), |λi (Au )| ≥ 1, |λi (As )| < 1 and k ≥ 0 The state partitions xu and xs represent the unstable and stable openloop dynamics, respectively. In addition, the initial state x(0) is a random variable satisfying |h(xu (0))| < ∞. • q, w, x(0) and c are mutually independent, where c represents the channel noise according to deﬁnition 5. • the measurement noise q is such that the following holds: I(x(0); um−1 )<∞ 0 meaning that the controller does not have access to an exact description of the initial state of the plant, for k ∈ {0, . . . , m − 1}. Since um−1 = em−1 , 0 0 m−1 we could have equivalently required that I(x(0); e0 ) < ∞. has a probability • e is a scalar (e(k) ∈ R) stochastic process, for which ekkmax min density function for every ﬁnite kmin , kmax ≥ m. 2.1 Performance Measures Using Entropy Rates and Asymptotic Power Spectra In section 4, which comprises our most general results, we characterize limits of performance by means of a lower bound to the diﬀerence h∞ (e) − h∞ (d). In standard texts, such as [PP02, CT91], the entropy rate of a given stochastic process is interpreted as a measure of randomness or power. We use the

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entropy rate to gage performance, not only because it is technically convenient, by allowing us to derive inequalities involving the information rate at the remote preview system (RPS) block, but also because it is a fundamental quantity which can be related to other, more common, measures of performance. The following is a list containing a few inequalities, which relate the entropy rate with other performance measures9 : • for a general stochastic process a, the following holds: lim sup V ar(a(k)) ≥ k→∞

1 2h∞ (a) 2 2πe

• if a is asymptotically stationary with an integrable power spectral density Fˆa (ω) then the following holds: h∞ (a) ≤

1 4π

π −π

log 2πeFˆa (ω)

Deﬁnition of a sensitivity-like function Deﬁnition 7. If the stochastic process e, represented in Fig 1, is asymptotically stationary10 then we deﬁne the following sensitivity-like function: def

Sd,e (ω) = lim sup σq →0

Fˆe (ω) Fˆd (ω)

(11)

The following is a list of remarks regarding Sd,e (ω): • if the RPS block is absent in Fig 1 and K is linear and time-invariant then Sd,e (ω) is the absolute value of the standard sensitivity function [DFT92]. • we deﬁne Sd,e (ω) in the limit, as σq goes to zero, because we want Sd,e (ω) to capture the eﬀect of the RPS block in disturbance attenuation. In general, the existence of measurement noise, with σq > 0, would add an extra term on Fˆe (ω) that could mask the beneﬁcial eﬀect of the RPS block. • in general, if we allow K to be non-linear, or if the RPS block is present, then Sd,e will depend on G. Consequently, the fundamental limitations expressed as a function of Sd,e should be interpreted as point-wise in the space of disturbances, in contrast with the no preview, linear and time-invariant case, for which Sd,e is uniform over the set of disturbances. Limitations in terms of the ratio represented by Sd,e should be interpreted as follows: once we have a spectral model of the disturbance, say Fˆd , then limitations in Sd,e translate immediately to limitations in Fˆe . Clearly, for 9 10

The ﬁrst fact follow from standard results in [CT91]. Since the last fact does not follow immediately from [CT91], we refer the reader to a proof in [MD05b]. When we refer to e as asymptotically stationary, we mean that e is asymptotically stationary for all σq > 0.

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each spectral model of the disturbance, Sd,e gives as much information about Fˆe as a classic sensitivity function would. As a consequence, our results show, for any given disturban! ce spectrum, that there are limits of attenuation and that certain spectra Fˆe are not attainable. It has been suggested in previous publications [DS92] that such point-wise property, instead of the uniformity characteristic of induced measures, is needed in the study of feedback loops with general non-linear controllers. In section 4, we derive inequalities in terms of Sd,e , which hold regardless of K, G and the RPS block. • in our formulation, d is asymptotically stationary, but e may not be. If e is not asymptotically stationary then Sd,e (ω) is undeﬁned. We have not tried to attribute a frequency domain interpretation based on other non-stationary notions, such as wavelets or evolutionary power spectral densities. In the absence of stationarity, we resort directly to entropy rates.

3 Motivation Case Study: A Remote Preview System Based on an Additive White Gaussian Channel In this section, we consider a particular case of the scheme of Fig 1, where the RPS block is constructed by means of an encoder and an additive white Gaussian channel, as depicted in Fig 2. We minimize the log-sensitivity integral, by searching over linear and time-invariant E and K, and show that the optimum explicitly depends on the capacity of the channel. Since such solution is derived for the linear and time-invariant case, we end the section by asking the question: “how good is this particular selection of E and K?”. The answer relies on the results of section 4, where we derive universal bounds that hold for arbitrary channels, controllers and encoders, which might be non-linear and time-varying and operating on arbitrary alphabets. This example is intended for motivation only and the general results are stated and proved in section 4 and in the Appendix.

w✲

✲

G ✲

E

-m

z

v✲

❧ +

c

✻

d✲ r

✲

❧ + ✻ u

e

K

✲ ✛

✲

P y

❄

❧ +

q

✻

Fig. 2. Structure of a Remote Preview System (RPS), using an additive white Gaussian channel.

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N.C. Martins, M.A. Dahleh, and J.C. Doyle

3.1 Technical Framework for the Case-Study For the purposes of this example, we consider the following: • w(k) is unit variance, zero mean, white and Gaussian. • the channel is speciﬁed by r = v + c, where c(k) is white, zero mean and Gaussian with variance σc2 > 0. • P is stabilizable by means of a stable, linear and time invariant controller H(z). • q is i.i.d. zero mean and Gaussian with variance σq2 > 0. 3.2 Optimal Design of a Linear and Time-Invariant Remote Preview Control Scheme In this example, we will derive an optimal choice of linear and time-invariant E and K, for the scheme of Fig 2.

w✲

✲

G ✲

-m

z

d✲

˜

e

d ❧✲ ❧ ✲+ +

ˆ −d v ❧✲ r E ✲+ D H ✻ c ✻

✻

✛

✲

P y

❄

❧ +

q

✻

Fig. 3. Complete scheme, comprising the feedback loop and the Remote Preview System, using an additive white Gaussian channel.

We wish to ﬁnd a scheme that achieves the following minimum: 1 E,K∈ LTI 2π min

π −π

log Sd,e (ω)dω

(12)

under the following power constraint, at the output of E: lim sup V ar(v(k)) ≤ 1 k→∞

(13)

where the minimum is taken over linear and time-invariant (LTI) E and K, while Sd,e is the sensitivity of deﬁnition 7. We have decided to investigate (12), because we wanted to determine by how much Bode’s equality can be relaxed in the presence of an AWGN channel, along with linear and time-invariant K and E. Since K is assumed linear, we choose to design the preview system and the feedback controller separately, as shown in Fig 3. The preview system is composed by an encoder E and a decoder D, while H is responsible for

Disturbance Attenuation Bounds in the Presence of a Remote Preview

279

stabilization and performance. For this scheme, the overall controller K results from the additive contribution of D and H, according to: K(r, y) = Dr + Hy

(14)

At the end of section 3.4, we explain why our solution is optimal. In addition, we show that the optimal solution is not unique because the cost function is invariant under any LTI, stable and stabilizing H. Such degree of freedom is useful because it allows H to be designed to meet additional speciﬁcations. 3.3 First Step: Design of E and D As a ﬁrst step, we choose to determine the following minimum: π

1 E,D∈ LTI 2π min

−π

log Sd,d˜ (ω)dω

(15)

under the following power constraint, at the output of E: lim sup V ar(v(k)) ≤ 1 k→∞

(16)

where Sd,d˜ is a particular case of the sensitivity of deﬁnition 7, which is given by: 1 1 2 Fˆd˜ (ω) Fˆ˜ (ω) 2 def Sd,d˜ (ω) = (17) = d jω |G(e )| Fˆd (ω) From a functional point of view, E and D of Fig 3 should generate the best ˆ so that the eﬀects of d can be reduced by means of disturbance possible d ˜ = d− cancellation (subtraction). The disturbance residual is indicated as d ˆ d. Our approach consists in designing E as a whitening ﬁlter, that recovers w from d, and constructing D as the mean-square optimal estimate of w, followed by a ﬁlter which replicates the eﬀect of G, as well as a matching delay. The resulting E and D are given by:

w✲

✲

G ✲

E

-m

z

v✲

❧ +

c

✻

d✲ r

✲

˜ d ❧ ✲ + ✻ ˆ −d D

Fig. 4. Structure of the forward loop used to study the properties of the optimal design of E and D, in the presence of an additive white Gaussian channel.

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N.C. Martins, M.A. Dahleh, and J.C. Doyle

E = G−1 D = −G

(18)

z −m 1 + σc2

(19)

In order to gage the quality of our design, we focus on the forward loop depicted in Fig 4. The following is the resulting asymptotic power spectral ˜ density of d: |G(ejω )|2 (20) Fˆd˜ (ω) = 1 + σ12 c

From (20), we also ﬁnd that: log Fˆd˜ (ω) = 2 log |G(ejω )| − C

(21)

where C = 21 log(1 + σ12 ) is the capacity of the remote preview channel11 . c Clearly, (21) indicates that our scheme makes use of the channel in reducing ˜ The larger the capacity C the better the overall power-spectral density of d. the disturbance reduction. Using (21), we can also derive the following equality: 1 2π

π −π

log Sd,d˜ (ω)dω = −C

(22)

At this point, the natural question to ask is: “Assuming that Sd,d˜ is well deﬁned12 , can we do better than (22), by using some other choice of nonlinear and time-varying E and D?”. The answer to the question is no, i.e., the following holds, under the power constraint (16): ˜ is asymptotically stationary ) ∀E, D ( such that d π 1 ˜ (ω)dω ≥ −C 2π −π log Sd,d

(23)

In particular, we ﬁnd that our choices (18)-(19) are optimal and that the minimum in (15) is equal to −C. The rigorous statement of (23) is given in Theorem 2, for the more general situation where feedback is also present. From Theorem 2, we also conclude that (23) holds regardless of E, D and the channel. Below, we sketch a simple proof of (23) as a way to illustrate the basic principle, which, due to its simplicity, can be explained in just a few steps. In order to make it short, we will assume familiarity with the properties listed in 11 12

This formula gives the capacity of an additive white Gaussian channel [CT91], with noise power σc2 and input power constraint σv2 ≤ 1. ˜ is asympSince d is asymptotically stationary, Sd,d˜ is well deﬁned provided that d totically stationary. Without asymptotic stationarity, we resort directly to entropy rates to deﬁne measures of performance. The complete analysis is provided in section 4.

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section 1.6. Since this result is proved in detail and generality in the Appendix, the reader has the option to skip directly to section 3.4. Sketch of a proof of (23). By expressing mutual information as a diﬀerence of diﬀerential entropies [CT91], we get: ˆ = I∞ (d, d) ˆ ≤C h∞ (d) − h∞ (d|d) (24) where the last inequality results from the data-processing inequality, i.e., ˆ ≤ I∞ (v, r) and from I∞ (v, r) ≤ C, which holds for any channel. I∞ (d, d) Using basic properties of h, we arrive at: ˆ = h∞ (d − d| ˆ d) ˆ ≤ h∞ (d − d) ˆ = h∞ (d) ˜ h∞ (d|d)

(25)

Together with (24), (25) leads to the following entropy rate inequality, which is valid regardless of E, D and the channel: ˜ ≥ h∞ (d) − C h∞ (d)

(26)

˜ so that Fˆd and S ˜ are well deﬁned, Assuming asymptotic stationarity of d, d,d we have13 : π ˜ ≤ 1 log 2πeFˆd˜ (ω) dω (27) h∞ (d) 4π −π Since d is Gaussian, the following holds: h∞ (d) =

1 4π

π −π

log 2πeFˆd (ω) dω

(28)

In summary, substituting (27) and (28) into (26), we obtain: 1 2π

π −π

log

Fˆd˜ (ω) Fˆd (ω)

1 2

dω ≥ −C

(29)

which holds regardless of E, D and the channel. 3.4 Optimal Scheme in the Presence of a Linear Feedback Controller We go back and focus on the original problem (12), by re-writing it as: 1 E,D,H∈ LTI 2π min

π −π

log Sd,e (ω)dω

(30)

under the following power constraint, at the output of E: lim sup V ar(v(k)) ≤ 1 k→∞

13

(31)

This inequality is proved in [MD05b], under asymptotic stationarity assumptions. The standard proof, assuming stationarity, is given in [CT91]

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N.C. Martins, M.A. Dahleh, and J.C. Doyle

At this point, we investigate how our choices (18)-(19) will aﬀect the complete scheme of Fig 3, in the presence of an arbitrary stable and stabilizing linear and time-invariant controller H. In terms of Fig 2, if we substitute (18)-(19) in (14), we obtain the following: z −m 1 + σc2

K(r, y) = −

Gr + Hy

(32)

Standard computations lead to the following asymptotic power spectral density: Fˆ˜ (ω) + |H(ejω )|2 σq2 (33) Fˆe (ω) = d |1 − P (ejω )H(ejω )|2 Using deﬁnition 7, (17) and (33), we arrive at the following: Sd,e (ω) = lim sup σq →0

Sd,d˜ (ω) Fˆe (ω) = ˆ |1 − P (ejω )H(ejω )| Fd (ω)

(34)

By making use of (34), we can compute the following integral: π −π

log Sd,e (ω)dω

=

π −π

+

log π

−π

1 |1 − P (ejω )H(ejω )|

log Sd,d˜ (ω)dω

dω (35)

which, by means of Bode’s equality [DFT92] and (22) applied to the right hand side, can be written as: 1 2π

n

π −π

log Sd,e (ω)dω =

max{0, log |λi (A)|} − C

(36)

i=1

From (23) and (35), we conclude that (36) attains the optimal value of (30). 3.5 Motivation for Studying the General Case If C = 0, i.e. in the absence of a RPS block, then (36) is identical to Bode’s integral equation. Otherwise, if C > 0 then the identity (36) shows that our choices (18)-(19) make good use of the channel because they relax Bode’s integral by a factor of −C. If we allow plants P which are not stabilizable by means of a stable, linear and time-invariant H then (35) indicates that the following inequality still holds for any linear and time invariant E and K: 1 2π

π −π

n

max{0, log(|λi (A)|)} − C

log Sd,e (ω)dω ≥ i=1

(37)

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283

where we used the fact that the ﬁrst term on the right hand side of (35) will also account for the unstable poles of H, if any. This motivated us to ask the following questions, which are addressed in section 4: • adopting the general assumptions of section 2 plus asymptotic stationarity of e, can we beat (37) by allowing a larger class of E and K, such as nonlinear and operating over arbitrary alphabets? Notice that by allowing non-linear K, we also consider the case where K cannot be written as an additive contribution from r and y, such as in (32). • what if, beyond the extensions of the previous question, we allow arbitrary channels, can (37) be beaten? • for the case where e may not be asymptotically stationary, general E, K and arbitrary channels, can we extend (37) by means of another inequality involving entropy rates ? In section 4, we show that the answers to the above questions are no, no and yes, respectively.

4 Fundamental Limitations for the General Case In this section, we present performance bounds for the scheme14 of Fig 5. In such case, the remote preview system is constructed by means of an arbitrary channel and a general encoder. The proofs of the Theorems may be found in [MD05b]. In the rest of the paper, we adopt the following assumptions: • (A1) E and K are causal operators deﬁned in the appropriate spaces, i.e., the output of E must belong to the channel input alphabet, which might be discrete or continuous. Similarly, the output of the channel must be deﬁned in the alphabet of r, at the input of K (see Fig 5). • (A2) (Feedback well-posedness) we assume that the feedback system is well-posed, i.e., that there exists a causal operator J such that the following is well deﬁned: ∀k ≥ 0, u(k) = J(x(0), r, d, q)(k) (38) 4.1 Derivation of a General Bound Involving Entropy Rates As we have discussed in section 2.1, we use h∞ (e) − h∞ (d) as a performance measure for the most general case, where we do not require e to be asymptotically stationary. The following Theorem provides an universal lower bound for h∞ (e) − h∞ (d) as a function of the unstable poles of P and the capacity 14

Although our results are valid for the general scheme of Fig 1, for simplicity, we consider the concrete scenario depicted in Fig 5.

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N.C. Martins, M.A. Dahleh, and J.C. Doyle

w✲

✲

G ✲

E

-m

z

d✲

v r ✲ Channel ✲

❧ + ✻ u

e

K

✲ ✛

✲

P y

❄

❧ +

q

✻

Fig. 5. Structure of a remote preview system, using a general communication channel.

of the remote preview channel. All of the remaining results in this section are, in one way or another, consequences of such universal lower bound. Theorem 1. [MD05b] Consider the feedback interconnection represented in Fig 5. In addition, assume that the state of P satisﬁes supk E [(x(k ))T x(k)] < ∞. For any encoder E and controller K, satisfying (A1) and (A2), the following is true: n

h∞ (e) − h∞ (d) ≥

max{0, log |λi (A)|} − C

(39)

i=1

where C represents the capacity [CT91] of the remote preview channel. The proof of Theorem 1 can be found in [MD05b]. In addition, [MD05b] comprises not only all the details of the proof, but it also contains preliminary results, which clarify aspects such as the role of causality and stability. As shown in [MD05b], the eﬀect of stability is related to the results in [TM04a, YB03, NE00b]. The aforementioned references address the problem of control under ﬁnite-rate constraints, but their proofs regarding the minimum stabilizing rate hold in general. An example for a control design method for such class of problems is given in [SDS04]. 4.2 Expressing Performance Limitations by Means of Asymptotic Power Spectral Densities: An Extension to Bode’s Integral Formula Under asymptotic stationary assumptions, Theorem 2 ascribes a frequency domain interpretation to Theorem 1. Theorem 2. [MD05b] Consider the feedback interconnection represented in Fig 5. In addition, assume that the state of P satisﬁes supk E[x [ T (k)x(k)] < ∞. If the encoder E and the controller K are such that (A1)-(A2) are satisﬁed and e is asymptotically stationary then the following is true: 1 2π

π −π

log

√

n

2πeSd,e (ω) dω ≥

max{0, log |λi (A)|} − C + h∞ (w) (40) i=1

Disturbance Attenuation Bounds in the Presence of a Remote Preview

285

where C represents the capacity [CT91] of the RPS channel. In addition, if w is Gaussian then (40) is given by: 1 2π

π −π

n

max{0, log |λi (A)|} − C

log Sd,e (ω)dω ≥

(41)

i=1

Remark 1. Now, we can conﬁrm our assertion, at the end of section 3.5. Any other choice of non-linear E and K, in the example of section 3, still has to satisfy (41) and (37).

5 Conclusions In this paper, we have analyzed an example of a networked control scheme, where a ﬁnite horizon preview of the disturbance is remotely available to the controller. In our results, the capacity of the preview communication system is an essential quantity. Under asymptotic stationary assumptions, we have shown, for a given ﬁnite capacity, that Bode’s integral formula can be extended. We have illustrated how some general bounds, derived in [MD05b], might be achieved.

Acknowledgments The authors would like to thank Ola Ayaso (MIT), Jorge Goncalves (Cambridge U., U.K.) and Mustafa Khammash (UCSB) for providing references to examples in Biology. We also would like to express our gratitude to Prakash Narayan (U. Maryland) for carefully reading our ﬁrst manuscript, to Sanjoy Mitter (MIT) for interesting suggestions and to Jie Chen (U California Riverside) for pointing us to a few references in Preview Control. This work was sponsored by the UCLA, MURI project title: “Cooperative Control of Distributed Autonomous Vehicles in Adversarial Environments”, award: 0205-G-CB222. Nuno C. Martins was supported by the Portuguese Foundation for Science and Technology and the European Social Fund, PRAXIS SFRH/BPD/19008/2004/JS74.

Plenary Talk The Role of Information Theory in Communication Constrained Control Systems Sekhar Tatikonda Department of Electrical Engineering Yale University [email protected]

Communication is an important component of distributed, networked, and cooperative control systems. In most current systems the communication aspects are designed separately from the control aspects. Recently there has been an increasing interest in studying control systems employing multiple sensors and actuators that are geographically distributed. For these systems this design separation is no longer tenable. Hence there is a need for a uniﬁed view of control and communication. In this talk we discuss how tools from information theory can be used to develop such a view. Speciﬁcally we apply source coding, channel coding, binning, and coding with side-information techniques to a variety of control with communication problems including settings with non-traditional information patterns, multiple sensors, and multiple agents.

Sekhar Tatikonda is currently an assistant professor of electrical engineering at Yale University. He received his PhD degree in EECS from MIT in 2000. He was a postdoctoral fellow in EECS at UC-Berkeley from 2000-2002. His research interests span topics in communications, information theory, control, and machine learning. He has made contributions in the areas of control with communication constraints, feedback in communication channels, and iterative message-passing algorithms.

P.J. Antsaklis, P. Tabuada (Eds.): Netw. Emb. Sens. and Cntrl., LNCIS 331, p. 289, 2006. © Springer-Verlag Berlin Heidelberg 2006

Delay-Reliability Tradeoﬀs in Wireless Networked Control Systems Min Xie and Martin Haenggi Department of Electrical Engineering University of Notre Dame Notre Dame, IN 46556 {mxie,[email protected]} Summary. Networked control systems (NCS) require the data to be communicated timely and reliably. However, random transmission errors incurred by wireless channels make it diﬃcult to achieve these qualities simultaneously. Therefore, a tradeoﬀ between reliability and latency exists in NCSs with wireless channels. Previous work on NCSs usually assumed a ﬁxed transmission delay, which implied that the failed packet will be discarded without retransmission, and thus reliability is reduced. When the channel errors are severe, the NCS cannot aﬀord the resulting packet loss. In this paper, a delay-bounded (DB) packet-dropping strategy is associated with automatic repeat request (ARQ) at the link layer such that the failed packets will be retransmitted unless they are out-of-date. On the one hand, the packet delay is controlled to be below the predetermined delay bound. On the other hand, reliability will be improved because failed packets are given more retransmission opportunities. This paper investigates the tradeoﬀ between packet delay and packet loss rate with DB strategies. Due to the multihop topology of the NCS, medium access control (MAC) schemes are needed to schedule the transmission of multiple nodes. Moreover, spatial reuse should be taken into account to improve the network throughput. In this paper, two MAC schemes are considered, m-phase time division multiple access (TDMA) and slotted ALOHA. They are compared for diﬀerent sampling rates and delay bounds. TDMA outperforms ALOHA in terms of both end-to-end (e2e) delay and loss rate when the channel reception probability is above 0.5 and/or traﬃc is heavy. However, ALOHA shows a self-regulating ability in that the node eﬀective transmit probability depends only on the sampling rate and channel reception probability, but is essentially independent of the ALOHA-dependent parameters. Then, for light traﬃc, a simple ALOHA with transmit probability 1 is preferred over TDMA in NCSs. The derived relationship between the sampling rate, the e2e delay (or delay bound), and the packet loss rate is accurate and realistic and can be used in NCSs for more accurate performance analyses.

P.J. Antsaklis, P. Tabuada (Eds.): Netw. Emb. Sens. and Cntrl., LNCIS 331, pp. 291–308, 2006. © Springer-Verlag Berlin Heidelberg 2006

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1 Introduction Networked sensing and control systems require real-time and reliable data communication. In the wireless environment, due to the random channel errors, it is diﬃcult, if not impossible, to guarantee hard delay bounds with full reliability. Many applications of networked control systems (NCS) are delaysensitive, but they can tolerate a small amount of data loss. Therefore, it is suﬃcient to provide NCSs a balanced guarantee between the delay and the loss rate. Previous work on the QoS in NCS usually assumed that the packet transmission delay is ﬁxed [LMT01, LG03, LG05, BPZ00b]. In the practical wireless environment, this assumption hardly holds since the network-induced delay is random. In order to verify the assumption, in many previous works the packets are allowed only one transmission attempt. In [LG03, LG05], the wireless channel is modeled by a Bernoulli process with a success probability ps . The failed packets are immediately discarded without any retransmission attempts. In this case, the packet delay is a constant equal to one time slot. But the network is rather unreliable. Reliability completely depends on the channel parameter ps . If ps 1, the packet loss rate pL = 1 − ps will be too large to be tolerated. On the other hand, in the fully reliable network, in order to guarantee 100% reliability, the failed packets will be retransmitted until they are received successfully. Then, the resulting delay is tightly controlled by the channel parameter ps . Similarly, as ps 1, the delay will be very long and cannot be tolerated by real-time applications. Meanwhile, more energy will be wasted to retransmit packets that are outdated and thus useless for the controller. In addition, from the perspective of network stability, the traﬃc rate (or the data sampling rate) is constrained to be smaller than the channel reception probability ps . In an interference-limited network, ps at the hotspot nodes are very small. Then, only light traﬃc can be accommodated by such networks. A feasible solution to balance latency and reliability is to drop a small percentage of packets. A simple strategy is “ﬁnite buﬀer” (FB) [Zor02]. If the buﬀer is full, some packets will be thrown out. To guarantee a hard delay bound, consider a bounded delay (BD) dropping strategy [Zor02], in which the failed packets will be retransmitted until they are received correctly or their delay exceed a delay bound B. The maximum packet delay is guaranteed to be B. The dropping strategy is associated with the node scheduling algorithm to determine which packets are eliminated. For instance, the NCS applications always prefer the newly packets over the old packets. Therefore, using priority scheduling (high priority to new packets) or Last-Come-First-Serve (LCFS) scheduling, the old packets will be dropped to yield the buﬀer space for new packets. With respect to real-time NCS applications, the BD strategy is employed in this paper. The NCS using the BD scheme is referred to as Delay Bounded (DB)-network. Compared to the fully reliable network, the DB network has

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several advantages. First, network stability is not an issue. In case that the traﬃc load is too heavy to be accommodated by the channel, some packets will be discarded, and the network can be ﬁnally self-stabilized. Hence, a traﬃc rate higher than the channel reception probability ps is allowed in the DB network. Second, less energy is wasted to retransmit packets that will be dropped eventually (refer these packets to as “marked” packets). As a matter of fact, given the multihop topology, the sooner the marked packets are dropped, the better. Third, in interference-limited networks, as the overall traﬃc load decreases, the channel reception probability ps will be enhanced to admit more traﬃc. This feature is particularly helpful for NCS applications. At the cost of discarding outdated packets, more recent packets will be transmitted to the controller. The disadvantage caused by the BD strategy is unreliability, which is measured by the packet loss rate pL in this paper. However, with coding or network control packets, the transmission errors can be combated and reliability will be improved. Using B and pL as the maximum delay and reliability and plugging into the controller [LG03, LG05], a more accurate lower bound on the NCS performance can be obtained. The system model for a NCS is outlined in Fig. 1(a), where the source (e.g., sensor) data are transmitted over multiple wireless hops to the controller. Node 1 to N are relays. The data generated at the source node is timecritical, e.g., periodic data used for updating controller output. A loop exists between the source, the controller and plant. The set of communication links is modeled as a tandem queueing network (Fig. 1(b)). Given the multihop topology, multiple nodes in tandem compete for transmission opportunities. Then MAC schemes are needed to schedule the node transmission order to avoid collision and take advantage of spatial reuse. If packets are of ﬁxed length, the objective is to analyze the discrete-time tandem queueing network controlled by wireless MAC schemes. Plant

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1.1 Related Work The analysis of multihop MAC schemes in wireless networks essentially involves two issues, the wireless MAC scheme and the tandem queueing system. Previous work usually analyzed these two issues separately. The analysis on tandem networks focuses on the queueing delay, without consideration of the MAC-dependent access and backoﬀ delay, and the retransmission delay induced by collision. For example, in [PP92, PR98], the delay performance is derived assuming that the node transmits immediately if the server is free, and the transmission is always successful. In [Leh96, Leh97], real-time queueing theory is proposed to explore real-time Jackson networks with Poisson arrivals and exponential servers for heavy traﬃc. Poisson arrival and exponential service possess the memoryless property, which signiﬁcantly simpliﬁes the analysis from the perspective of queueing theory. However, most traﬃc models are not as memoryless as Poisson and cause more diﬃculties in the analysis, particularly when associated with MAC schemes like TDMA. Moreover, the traﬃc load in NCS is generally not heavy. The assumption of heavy traﬃc in [Leh96, Leh97] leads to an underestimation of the network performance. On the other hand, the analysis of MAC schemes concentrates on only the access and transmission delay while ignoring the queueing delay. For simplicity, it is usually assumed that the network has a single-hop topology, an inﬁnite number of nodes, and a single packet at each node such that no queueing delay is incurred to the packet. More importantly, the traﬃc ﬂow is generated in an unpractical way that every node always has packets to transmit whenever it is given a transmission opportunity. In [Fer77, YY03] study the access delay of Bernoulli and Poisson arrivals. In [BTR92], the queueing delay was claimed to be derived with a special MAC scheme that allows one node to completely transmit all its packets as long as it captures the channel. For a speciﬁc node, the so-called queueing delay actually includes only access delay and transmission delay. In the wireless environment, the analysis of MAC schemes additionally considers the impact of wireless channel characteristics. In [ZP94, AB87], a capture model is used to calculate the average packet transmission delay of ALOHA and CSMA for Rayleigh fading channels. [FM03] proposes optimum scheduling schemes for a line sensory network to minimize the end-to-end (e2e) transmission delay. With respect to the BD scheme, its eﬀect upon a single node is discussed in [LC04]. In [SS99], various TDMA schemes combined with the BD scheme are studied in the single-hop scenario. This paper extends to the multihop scenario and studies the e2e performance. 1.2 Our Contributions This paper investigates the tradeoﬀ between the end-to-end (e2) delay, reliability, and the sampling rate in multihop NCS. The main contribution is to

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jointly study the MAC scheme, the dropping strategy and the tandem network. The network performance like delay and packet loss rate is determined by several factors, including traﬃc, the routing protocol, the channel characteristics, and the MAC scheme. Emphasizing on the MAC scheme, we focus on a regular line network as shown in Fig. 1(b), which disposes of the routing and inter-ﬂow interference problem. The obtained performance provides an upper bound for general two-dimensional networks because 1) the interﬂow interference is zero; 2) networks with equal node distances achieve better performance than those with unequal or random node distances. Two MAC schemes are studied, m-phase TDMA and slotted ALOHA. In the former, every node is allocated to transmit once in m time slots, and nodes m hops apart can transmit simultaneously. In the latter, the node independently transmits with transmit probability pm . The wireless channel is characterized by its reception probability ps . The sampled source data (generated by the source in Fig. 1(a)) is modeled as constant bit rate (CBR). The CBR traﬃc ﬂow is not only easy to be generated but also more practical than the traﬃc models used in previous work for MAC schemes, where the traﬃc load is assumed to be so heavy that the node is always busy transmitting. In practice, this heavy traﬃc assumption leads to an unstable network. There is no restriction on the node scheduling algorithm. To simplify the analysis, we assume First-Come-First-Serve (FCFS). Other scheduling algorithms like LCFS and priority scheduling can be chosen to better serve the NCS application demand. We will show later that in the TDMA mode, the CBR arrival results in a non-Markovian queueing system, which substantially complicates the analysis. Despite of the enormous queueing theory literature, it is still diﬃcult to track the transient behavior of non-Markovian systems [Tri01], while the BD strategy is implemented based on the system transient behavior. Even if the source is deterministic and smooth, calculating the delay distribution in a time-dependent BD strategy is still a challenge. In addition, non-Poisson arrival causes correlations between the delays and queue lengths at individual nodes. Closed-form solutions exist only for some special networks, like Jackson networks, which do not include the network in this paper. In one word, accurate analyses are almost impossible in a multihop network with a long path. Therefore, we start the analysis with the ﬁrst node, then investigate the network performance through simulation results. Combining MAC with the BD strategy, we compare DB-TDMA and DBALOHA in terms of the delay and the packet loss rate. The DB-MAC networks are also compared with their non-dropping counterparts to exhibit the advantage of the BD strategy. Since the traﬃc intensity is not necessarily as heavy as close to 1, ALOHA possesses a self-regulating property, which does not exist in the heavy traﬃc assumption. That is, although ALOHA attempts to control the packets transmission with the transmit probability pm , the node actually transmits with a probability pt , which is independent of pm .

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This paper is organized as follows. Section 2 describes the system model and presents some results for the non-dropping MAC schemes. Then, TDMA and ALOHA are studied in Section 3. The self-regulating property of ALOHA is specially discussed. Their performance are compared through a set of simulation results in Section 4. The paper is concluded with Section 5.

2 System Model The set of communication links in NCS (Fig. 1(a)) is modeled as a tandem queueing network, which is composed of a source node (node 0), N relay nodes, and the unique destination node (node N + 1, also referred to as the sink node). A CBR ﬂow of ﬁxed-length packets is generated at the source node. Relay nodes do not generate traﬃc. Time is divided into time slots. One time slot corresponds to the transmission time of one packet. The sampling rate of CBR traﬃc is 1/r, r ∈ N. Given the channel reception probability ps , the channel is modeled as a Bernoulli process with parameter ps . The e2e delay bound is B. From the perspective of energy eﬃciency, it is not recommended to drop the packet only when its e2e delay exceeds B, which often happens when the packet reaches the last few nodes to the sink. The longer the route the packet traverses, the more energy is wasted if the packet will be eventually dropped. So, a local BD strategy is preferred. In order to determine how the local BD strategy is implemented, we ﬁrst review the cumulated delay distribution in non-dropping tandem networks. In [XH05], a fully reliable tandem queueing network is studied. CBR Trafﬁc is transformed to correlated and bursty through the error-prone wireless channels. Even with correlation, the e2e delay is approximately linear with the number of nodes with respect to both the delay mean and delay variance, as conﬁrmed by simulation results in Fig. 2. Then, it is reasonable to uniformly allocate B among nodes based on their relative distances to the source node [Pil01], i.e., the local delay bound Di is set to be Di = iB/N = iD (i ∈ [1, N ], Di ∈ N). Packets are dropped at node i if their cumulated delay exceeds Di . Intuitively, if a node experiences a delay at node i longer than Di , then it is highly possible that it has delay longer than B at the ﬁnal node N . Parameters of interest include the cumulated delay di and the packet loss rate piL at node i (1 ≤ i ≤ N ). As proved in [XH05], the e2e delay mean of ALOHA is about ps /(1 − ps ) times than that of TDMA. The gap is even larger for the delay variance as shown in Fig. 2(b). However, in the wireless multihop network, TDMA is not feasible to be implemented, and simple MAC schemes like ALOHA are more desirable, even though TDMA substantially outperforms ALOHA in terms of both throughput and delay. The BD strategy is a solution to reduce the performance gap between TDMA and ALOHA.

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Note that in the DB network, the packets are dropped according to their delays. Conventional queueing theory keeps track of the buﬀer size and cannot capture the packet dropping event [Zor02]. So, we use a delay model [LC04], in which the system state is denoted by the packet delay such that the delaydependent packet dropping event can be directly depicted through the system state.

3 Delay Bounded Wireless Multihop Networks 3.1 m-Phase TDMA The m-phase TDMA scheduler takes advantage of spatial reuse so that nodes i, m + i, 2m + i, . . . (1 ≤ i ≤ m) can transmit simultaneously. A transmission can be either a transmission of a new packet or a retransmission of a failed packet. Instead of being divided into time slots, now the time is divided into frames of m time slots. For a node, the beginning of a frame is the beginning of the time slot allocated to this node. The transmission rate is 1/m, and the transmission is successful with probability ps . To guarantee system stability, r > m. For heavy traﬃc, we assume m < r < 2m. At the frame level, the service time is geometric with ps . We start with the ﬁrst node since it determines the traﬃc pattern of all subsequent nodes. At the frame level, the interarrival time 1 < r/m < 2 is not an integer even though it is a constant. As a matter of fact, the number of packet arrivals in one frame jumps between 0 and 1, depending on the arrival pattern of all previous r − 1 frames. This dependence makes the resulting queueing system more complicated than the G/M/1 system, where the interarrival times are identical and independent. Hence, a standard Markov chain cannot be established as usual to keep track of the buﬀer size. Instead, we resort to the delay model

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that denotes the delay of the Head of Line (HOL) packet as the system state [LC04]. The system state is the waiting time of the HOL packet in terms of time slots, but the state transitions happen at the frame boundaries. Since the waiting time is the diﬀerence between the present time and the packet arrival time, the state value might be negative when the queue is empty and the next packet arrival does not happen. The absolute value of the negative state represents the remaining time till the next packet arrival. With a constant interarrival time r, the transition probability matrix P = {Pij } is:  0 ≤ i ≤ D − m, j = i − Δ,  ps qs 0 ≤ i ≤ D − m, j = i + m, Pij = (1)  1 D − m < i ≤ D, j = i − Δ or i < 0, j = i + m, where Δ := r − m > 0. At frame t, let the HOL packet be packet k and its waiting time wk (t). If the transmission is successful, packet k departs at frame t, and the subsequent packet k + 1 becomes the HOL packet at frame t + 1. The waiting time of packet k + 1 at frame t is wk+1 (t) = wk (t) − r. It increases by m up to wk+1 (t + 1) = wk (t) − r + m = wk (t) − Δ at frame t + 1. Therefore, the system state transit from wk (t) to wk (t) − Δ with probability ps . If wk (t) < Δ, packet k is the last packet in the buﬀer and the buﬀer becomes empty after its transmission. Then, the system transits to a negative state i = −(Δ − wk (t)) < 0. For m < r < 2m, the server idle time does not exceed one frame. Then, there must be a packet arrival during frame t+1. This new packet may arrive in the middle of frame t + 1 and cannot be transmitted immediately. The waiting time to access the channel is m + i > 0. Then the negative state i transits to a positive state m + i with probability 1. If the transmission is failed and wk (t) ≤ D − m, the HOL packet remains at the buﬀer and will be retransmitted after one frame. Its delay increases by m up to wk (t + 1) = wk (t) + m ≤ D with probability qs . If wk (t) > D − m, this HOL packet k will experience a delay greater than D after one frame and be discarded (maybe in the middle of the frame). Then, at the beginning of frame t + 1, packet k + 1 becomes the HOL packet with a delay wk+1 (t + 1) = wk (t)−Δ. Recall that if the transmission is successful, the positive state wk (t) transits to state wk (t) − Δ, as well. In other words, if wk (t) > D − m, the system state always transits to wk (t) − Δ with probability 1, regardless of whether the transmission is successful or failed. The steady-state probability distribution{πi } can be obtained either iteratively or by using mathematical tools to solve π = πP. For a critical case Δ = 1, {πi } is derived in terms of π0 as follows (qs 1 − ps ):  Ki π0 1 + q k i≤D−m (−qs pm s s ) g(k)  pis k=1 (2) πi = D−m  qs

πj j=i−m

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where

i − km + 1 i − km i+1 − p s , Ki = m+1 k k If the HOL packet is transmitted successfully, its delay at the ﬁrst node is wk (t) plus one time slot for transmission. Therefore, the delay distribution {di } is completely determined by the probabilities of non-negative states, πi−1 di = . (3) j≥0 πj g(k) =

In the m-phase TDMA network, the node transmits at the frame boundaries. A packet may be dropped in the middle of a frame if it experiences a delay greater than D − m at the beginning of this frame and fails to be transmitted. The packet dropping probability is qs

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3.2 Slotted ALOHA In slotted ALOHA, each node independently transmits with probability pm . Note that pm represents the node transmission opportunity. The node actually transmits only if it is given a transmission opportunity and it has packets to transmit, which depends on its buﬀer occupancy. Traﬃc and the channel model are the same as in TDMA. Again, we start with the ﬁrst node, which is observed at the time slot level. Given the success probability ps , a packet departs the node if and only if the node is scheduled to transmit and the transmission is successful, with a probability a = ps pm . Otherwise, the packet is retained in the buﬀer or discarded. The service time is geometric with a. The system state is the waiting time of the HOL packet. The state transition probabilities are  1 − a i ∈ [0, D), j = i + 1 i ∈ [0, D), j = i − r + 1 Pij = a (8)  1 i < 0, j = i + 1 or i = D, j = D − r + 1. At slot t, assume that the HOL packet is packet k with delay wk (t). If wk (t) < D and the packet successfully departs the node with probability a, the next packet becomes the HOL packet at slot t+1 with delay wk+1 (t+1) = wk+1 (t)+ 1 = wk (t) − r + 1. Otherwise, if the packet fails to depart the node with probability (1 − a), it remains as the HOL packet with its delay increased by one. If wk (t) = D, either the packet is transmitted successfully or not, it has to be deleted from the buﬀer since its delay exceeds the bound D at slot t + 1. In this case, the system state transits from D to D − r + 1 with probability 1. The negative states indicate an empty buﬀer and the system is waiting for the next new packet arrival. The delay distribution {di } is calculated based on {πi } and (3) like in TDMA. Since the packet dropping possibly occurs at the time slot boundaries, the packet dropping probability is (1)

pL =

(1 − a)πD = r(1 − ps pm )πD . λ

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i+r−1

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i
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where k = max{0, i}. (10) holds for {di }, as well. Diﬀerent from the TDMA network, (10) exhibits that di is jointly determined by r − 1 consecutive states above i, which results in a forward iteration. The pmf essentially consists of three sections, [1, D − 2r + 2], [D − 2r + 3, D − r + 1], and [D − r + 2, D + 1]. A general setting is D > 2(r − 1), so that the pmf contains all three sections. Note that the node transmits with probability pm when its buﬀer is nonempty. In other words, the eﬀective transmit probability is pt = PB pm , where

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PB is the node busy probability. In previous work, the traﬃc load is so heavy that PB = 1. Then the eﬀective transmit probability pt is identical to the transmit probability pm so that the performance of ALOHA networks can be optimized by manipulating pm . However, if the traﬃc load is light and PB 11 , which is highly possible in ALOHA networks, simply optimizing pm does not necessarily lead to optimization of the network performance. As a matter of fact, based on queueing theory, the busy probability of node i is PBi = λi /(ps pm ), where λi is the arrival rate to node i. As the delay bound D goes to inﬁnity, it is easy to show that λi = 1/r for all i and thus PB = 1/(rps pm ). Naturally, the eﬀective transmit probability pt = 1/(rps ) depends only on the traﬃc rate 1/r and the channel reception probability ps , and is completely independent of pm . Since the network performance is essentially determined by the eﬀective transmit probability pt , this observation implies that the ALOHA parameter pm does not contribute to the change of the network performance. In the DB-ALOHA network, due to packet dropping, (i) (i) the arrival rate λi to node i is a function of the loss rate pL , λi = λi−1 (1−pL . (i) The packet loss rate pL depends on the delay bound D, the service rate a = 1/(ps pm ) and the arrival rate λi−1 . Through intuition, pt is not completely independent of pm . However, in most cases the loss rate is required to be relatively small. Then PBi ≈ 1/(rps pm ), and it is reasonable to say that pt is independent of pm . From the analysis in [XH05] we can see that the longer delay of ALOHA is mostly caused by the longer access delay, which is proportional to 1/pm . When the traﬃc load is light, a large pm will lead to a small access delay and signiﬁcantly improve the e2e delay. In this sense, to optimize the delay performance of ALOHA, pm should be chosen based on the traﬃc load, which has been ignored in previous work because of the heavy traﬃc assumption. For example, in [LG04], the throughput is maximized by pm = 1/N without consideration that the node does not need contend for transmission opportunities when its buﬀer is empty.

4 Simulation Results A set of extensive simulation results are provided to expose the performance of the DB-TDMA and DB-ALOHA network. First of all, the fully reliable (non-dropping) TDMA network is compared with the DB-TDMA network in Fig. 3. We set m = 3 and ps = 0.8 for both networks. For the DB network, we additionally set the interarrival time r = 4 and D = 4, which results in an e2e dropping probability pL ≈ 0.20. It implies that 20% packets will be discarded and the throughput is (1 − 0.2)/r = 1/5. In the non-dropping network, all generated packets will be successfully delivered to the sink and the throughput is exactly the traﬃc rate 1/r. Accordingly, for the non-dropping network, we 1

PB is essentially a function of pm .

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Fig. 3. Comparison of the non-dropping and DB-TDMA network with m = 3, N = 9, ps = 0.8

With the BD strategy, the e2e delay decreases. A substantial improvement is particularly reﬂected on the delay variance that is reduced by 60%. On the one hand, simply reducing the traﬃc load at the ﬁrst node does not improve the network performance signiﬁcantly. On the other hand, although introducing redundant packets does increase the traﬃc load, it enhances the delay performance. In addition, the lost packets can be compensated for by the redundant packets, which ensures reliability. In this sense, the BD strategy is very helpful to achieve a good balance between latency and reliability. 4.1 m-Phase TDMA Network Throughout this section, the m-phase TDMA network is assumed to have m = 3, r = 4, N = 8. Compared to the pmf of the non-dropping TDMA network, the pmf of the cumulated delays from node 0 to node i (1 ≤ i ≤ N ) is truncated based on B/D (Fig. 4) . For D > 2m, the pmf of the cumulated delay is scaled. In Fig. 5, the e2e delay mean and variance are shown for ps = 0.8, D = 10. The delay mean approximately linearly increases with the number of nodes. In comparison with the non-dropping TDMA network (D = ∞), the delay mean is reduced by 40% and the delay variance by 75%. The resulting e2e packet loss rate pL = 0.0414 ( as listed in Table 4.1) is acceptable when the aforementioned packet-level coding scheme is applied to combat the network unreliability. The dropping probability is to be traded oﬀ against the delay. A smaller D (i) results in a smaller delay and a higher local dropping probability pL at node (i) i, which is shown in Fig. 6(a). As D increases, pL decreases more slowly. It

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Fig. 4. pmf of the cumulated delay in the TDMA network with m = 3, r = 4, pr = 0.8, N = 8 Table 1. E2e dropping probabilities of the TDMA network with m = 3, r = 4, N = 8 D 1 5 10 15 20 0.7 0.9615 0.3742 0.2183 0.1678 0.1431 0.75 0.9331 0.2621 0.1229 0.0803 0.0589 0.8 0.8880 0.1550 0.0414 0.0142 0.0058 ps

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implies that the major packet loss occurs at the ﬁrst few nodes of the chain. This property is desirable since the downstream nodes do not need to spend energy to transmit packets that will ﬁnally be discarded. For large D, the per-node dropping probability asymptotically converges to zero. Fig. 7 demonstrates the eﬀects of D. Both the delay mean and the delay variance are nearly linear in D, particularly when D is small. Unsurprisingly, the e2e dropping probability pL asymptotically decreases with D. For large D, say D > 10, the decrease of pL becomes very slow. Thus, simply increasing D does not help to improve reliability, but does harm the delay performance. There may exist an optimal D to achieve the best balance between the delay and the packet loss. Unlike in fully reliable networks [XH05], the delay perfor-

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mance is not severely deteriorated by the drop of ps (Fig. 8). Moreover, as long as D is suﬃciently large, even if the traditional stability condition does not hold, the resulting e2e dropping probability is so moderate that both the data latency and reliability are guaranteed. For instance, considering the critical case ps = m/r = 0.75, for D ≥ 20, the packet loss pL ≤ 0.05 is negligible. For small D like D = 10 and pL ≤ 0.13, it is not diﬃcult to introduce redundant packets for reliability. 80

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4.2 Slotted ALOHA This section discusses the performance of the DB-ALOHA. To compare with TDMA, we set pm = 1/m and m = 3, r = 4, N = 8. Diﬀerent from the TDMA system, the pmf tail is both truncated and twisted by applying the BD scheme (Fig. 9). But the delay central moments and the dropping probability behave similarly as in TDMA. Speciﬁcally, the delay mean and variance

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linearly increase with the number of nodes (Fig. 5(a) andFig. 5(b)), and the (i) per-node dropping probability pL diminishes with the node index i and the ﬁrst node experiences the maximum packet loss (Fig. 5(c) and Fig. 6(b)). The e2e dropping probability pL is listed in Table 2. 0.12

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Like the TDMA network, both the delay mean and variance are approximately linear with D (Fig. 10(a) and Fig. 10(b)). The dropping probability pL (Fig. 10(c)) sharply decreases with small D. However, for D suﬃciently

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large, say D ≥ 30, the decreasing speed is decelerated and pL eventually converges to zero. Apparently, a larger D is needed for pL to reach zero. The impact of ps is displayed in Fig. 11. Again, the drop of ps causes a very small diﬀerence in the delay mean and variance, but results in an increase of the per-node packet dropping rate. Moreover, the per-node dropping probability asymptotically converges to zero. 1

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4.3 Comparison In the fully reliable network, the TDMA network substantially outperforms the ALOHA network in terms of the delay (Fig. 2). In the DB network, the performance gap between TDMA and ALOHA becomes fairly small. As shown in Fig. 5, the diﬀerence of the delay mean between the TDMA and ALOHA network decreases from 300% (reliable) to 20% (DB); while the delay variance diﬀerence changes from 750% (reliable) to 10% (DB). The main performance degradation caused by the random access is the dropping probability. For (i) ALOHA, the local dropping probability pL at node i is almost ﬁve times (i) than that of TDMA. Moreover, pL of ALOHA converges to zero more slowly

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than TDMA. However, if packet-level coding is applied to introduce redundant packets to compensate for the dropped packets, ALOHA is is a feasible MAC scheme that achieves a good delay performance. As a tradeoﬀ, when the gap in pL is reduced, the gap in the delay moments will be increased. The pair (B, pL ) can be used in the controller design to optimize the NCS performance as shown in [Nil98a, LG04]. With nonzero packet loss pL and the maximum packet delay B, the controller system can be written as a Markovian Jump Linear System (MJLS). Optimizing the MJLS system can optimize the NCS network with MAC schemes.

5 Conclusions This paper aims to provide a more accurate measurement to optimize NCS networks with MAC schemes. Previous work usually assumed the wireless channels as a constant-delay, which is not practical. We derive the e2e delay and packet loss probability of DB multihop wireless networks for two MAC schemes, m-phase TDMA and probabilistic slotted ALOHA. These parameters can be used by the controller to evaluate the performance. The e2e delay mean and variance are approximately linear with the number of nodes. The (i) local dropping probabilities pL asymptotically converge to zero. A moderate delay bound B is suﬃcient to guarantee a small packet loss and thus achieve a good balance between reliability and latency. Compared to fully reliable networks, the e2e delay of DB networks becomes less sensitive to the channel reception probability ps . This improvement is desirable since the network performance is not expected to rapid ﬂuctuate with ps , which basically cannot be controlled. Besides, with the BD strategy, the delay performance gap between TDMA and ALOHA is reduced. Due to the implementation complexity and overhead, TDMA is less favored than ALOHA. But ALOHA has poor delay performance. With the reduced performance gap, ALOHA becomes more practical. We also show that ALOHA possesses the self-regulating property. For light traﬃc, unlike previous work [LG04], we ﬁnd that a large transmit probability pm does not degenerate the network performance as for heavy traﬃc. Previously, pm = 1/N (N is the number of nodes) is thought to optimize the throughput. As the network enlarges, pm becomes very small and the resulting delay is long. From the perspective of the delay, a large pm is preferred, particularly when it does not decrease the throughput substantially. This paper considers a FCFS node scheduling for analysis tractability. Other scheduling algorithms that favor the newly arriving packets like LCFS and priority scheduling, may be more desirable for NCS applications. In the future work we will discuss how NCSs perform when associating with the BD strategy and other scheduling algorithms.

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Acknowledgment The authors would like to thank the support of the Center for Applied Mathematics (CAM) Fellowship of the University of Notre Dame and the partial support of NSF (grant ECS03-29766).

A Cross-Layer Approach to Energy Balancing in Wireless Sensor Networks Daniele Puccinelli, Emmanuel Sifakis, and Martin Haenggi Department of Electrical Engineering University of Notre Dame Notre Dame, IN {dpuccine,esifakis,mhaenggi}@nd.edu Summary. We consider a many-to-one real-time sensor network where sensing nodes are to deliver their measurements to a base station under a time constraint and with the overall target of minimizing the energy consumption at the sensing nodes. In wireless sensor networks, the unreliability of the links and the limitations of all resources bring considerable complications to routing. Even in the presence of static nodes, the channel conditions vary because of multipath fading eﬀects due to the motion of people or objects in the environment, which modify the patterns of radio wave reﬂections. Also, sensing nodes are typically battery-powered, and ongoing maintenance may not be possible: the progressive reduction of the available energy needs to be factored in. The quality of the links and the remaining energy in the nodes are the primary factors that shape the network graph; link quality may be measured directly by most radios, whereas residual energy is related to the node battery voltage, which may be measured and fed into the microcontroller. These quantities may be used to form a cost function for the selection of the most eﬃcient route. Moreover, the presence of a time constraint requires the network to favor routes over a short number of hops (a.k.a. the long-hop approach, in the sense that a small number of long hops is used) in order to minimize delay. Hop number information may be incorporated into the cost function to bias route selection toward minimum-delay routes. Thus, a crosslayer cost function is obtained, which includes raw hardware information (remaining energy), physical layer data (channel quality), and a routing layer metric (number of hops). A route selection scheme based on these principles intrinsically performs node energy control for the extension of the lifetime of the individual nodes and for the achievement of energy balancing in the network; intuitively, the long-hop approach permits the time-sharing of the critical area among more nodes. A novel, practical algorithm based on these principles is proposed with the constraints of the currently available hardware platforms in mind. Its beneﬁts are investigated with the help of computer simulation and are illustrated with an actual hardware implementation using Berkeley motes.

P.J. Antsaklis, P. Tabuada (Eds.): Netw. Emb. Sens. and Cntrl., LNCIS 331, pp. 309–324, 2006. © Springer-Verlag Berlin Heidelberg 2006

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1 Introduction A large portion of wireless sensor network research over the past few years has concentrated on routing [AKK04]. In this extensive body of work, the most common metric used for the assessment of routing quality is the number of hops in a route [JM96]. However, nodes in wireless sensor networks are typically static, and link quality has to be taken into account to avoid routing over lossy links. In the literature on ad hoc networks, various metrics are suggested in order to take packet loss into account. An interesting example is ETX (Expected Transmission Count) [CABM03], which uses perlink measurements of packet loss ratios in both directions of each wireless link to ﬁnd the high throughput paths which require the smallest number of transmissions to get a packet to destination. Since sensor nodes have heavy energy constraints, the battery capacity of individual nodes should also be taken into account. Battery-aware routing algorithms [SWR98] typically try to extend the lifetime of the network by distributing the transmission paths among nodes with greater remaining battery resources. The key observation is that minimum-energy routes can often unfairly penalize a subset of the nodes. Most eﬀorts in this direction are targeted toward ad hoc networks, and are often not portable to wireless sensor networks, which are the focus of the present work. The distinguishing features of wireless sensor networks make routing much more challenging. Wireless sensor networks often need to self-organize and operate in an untethered fashion. Sensor nodes are usually static, but location information is often unavailable, and geographic routing is thus not an option. Typically, constraints in terms of energy, processing, and storage capacities are extremely tight: resources need to be carefully managed, and lifetime extension of the sensing nodes is a major concern. The adoption of a many-to-one traﬃc scheme is the primary cause of energy unbalance leading to premature discontinuation of node activity. If the base station is not located within the reach of a source, a multihop scheme needs to be adopted, and other nodes are used as relays to guarantee a connection from that source to the base station. However, if many sources are in that situation, the nodes directly within the reach of the base station are located on most forwarding paths, have an increased workload, and their lifetime is likely to be shortened. In the special case of one-to-one traﬃc, schemes that ﬁnd and rely on optimal routing paths also cause energy unbalancing, as they unevenly distribute the workload across the network and shorten the lifetime of the nodes along the optimal path. In [SR02], the occasional use of suboptimal routes is suggested as a countermeasure; this solution is shown to yield a signiﬁcant lifetime improvement with respect to methods based on optimal routes alone. An eﬃcient attempt to balance the energy distribution in the network is necessarily reliant upon some form of monitoring of the quality of the links which allows an assessment of connectivity. The remaining battery power is also of interest, since we aim at having a reasonably uniform energy balancing.

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Our main goal in this paper is to investigate the beneﬁts of a joint metric which considers hop count, link quality, and remaining battery power. We look at the routing problem in wireless sensor networks from a cross-layer perspective, by introducing an algorithm to build a routing tree by means of a heuristic metric encompassing physical layer information such as link strength and battery power as well as routing layer information (number of hops). The scenario is as follows: in a wireless sensor network comprising of N severely energy-constrained (battery-operated) lower-end sensing nodes, we intend to use fairly eﬃcient routes for node lifetime extension; hence, we propose a strategy with energy-balancing guarantees. We assess its validity and beneﬁts by means of simulation, and describe a hardware implementation with the most recent generation of Berkeley motes, MICAz. The remainder of the paper is organized as follows. Section 2 describes our algorithm and illustrates it with examples obtained with a custom network simulator. Section 3 presents an evaluation of the performance of our scheme obtained by means of extensive simulation, and Section 4 details the hardware implementation. Section 5 includes some closing remarks.

2 A Routing Scheme with Energy-Balancing Guarantees From a dynamic programming perspective, the problem of routing from a source s to a destination node d corresponds to deﬁning a policy at a generic node ri for the choice of a relay ri+1 in the direction of d given the list of nodes s, r0 , r1 , ..., ri−1 . Routing over many short hops (e.g., nearest-neighbor routing) enjoys a lot of support. From an energy-consumption viewpoint, it is claimed that dividing a hop over a distance d into n short hops yields an energy beneﬁt of nα−1 , where α is the path loss exponent [Rap01]. Unfortunately, this comes from the use of oversimpliﬁed link models such as the disk model [Gil61]: a transmission can either fail or succeed depending on whether the distance is larger or smaller than the so-called transmission radius. This model totally ignores fading and is therefore inaccurate for a realistic modeling of the wireless medium [Rap01]. It is shown in [Hae04] that for a block fading channel multihop routing does not oﬀer any energy beneﬁts for α=2. Routing over fewer (but longer) hops carries indeed many advantages [HP]; from our point of view, the most interesting are the reduction of energy consumption at the sensing nodes, the achievement of a better energy balancing, the more aggressive exploitation of sleep modi, and the lack of route maintenance overhead. If the sensing nodes only occasionally need to act as relays, they can sleep longer and only consume energy to make their own data available. With these ideas in mind, we propose a lightweight multihop routing scheme with energy-balancing guarantees. Our scheme is an example of ﬂat routing, in the sense that all nodes are assigned similar roles (hierarchies are not deﬁned). It can be considered dynamically proactive, since all routes are computed prior to being used and periodically updated.

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In a network of N nodes (N − 1 sensing nodes and a base station), the generation of the routing tree is performed as follows: • Every sensing node sends a test packet to the base station; due to the unreliability of wireless links, the base station is only able to receive a fraction of such test packets. • The base station uses the received test packets to measure the quality of the links to the sensing nodes, and feeds this information back to them. • The sensing nodes that receive a reply from the base station broadcast a route setup packet to advertise their nearest-neighbor status as well as the quality of their link to the base and their battery voltage. Route setup packets may be seen as pointers toward the direction of the base station. • Other sensing nodes that receive these packets generate and broadcast similar packets to advertise that they are two hops away from the base station. • The information travels upstream from the base station into the network until all nodes know their depth and the tree is fully deﬁned. The route setup is targeted at downstream communication, and its eﬀectiveness is reliant upon link symmetry. In environments where asymmetric links are abundant, link quality estimation from reverse link quality information often does not work and handshakes between nodes are necessary.

Table 1. List of symbols. Symbol Meaning Ai li→k Mi,k Li Di vi Vi l[i, k]

Address of node i. Quality of the link between node i and node k. Metric of the route found at node i going through k. min(li→j , Lj ), Mi,j = max0≤k≤N −1 Mi,k , i = k. LBS 1. Depth of node i. Battery voltage of node i. min(vi , Vk ), Mi,k = max0≤j≤N −1 Mi,j , i = j. VBS 1. min(li→k , Lk ) (k-th entry in the table internal to node i).

We integrate a route selection scheme into the tree formation procedure by adding the following operations: • Each sensing node maintains a table indicating the nodes it can reach, the quality of the links to such nodes, their battery voltage, and their depth. All this information can be inferred by the route setup packets mentioned above. In particular, the depth can be inferred by using a counter initialized to 0 in the setup packets sent by the base station. The nearest neighbors of the base station will thus set it to 1, their nearest neighbors will increment it to 2, and so forth.

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Table 2. Setup packet sent by the base station. Abs Lbs Vbs Dbs bs 1

1

0

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Vk

Dk

min(lk→j , Lj ) min(vk , Vj ) Dj + 1

Table 4. Structure of the internal table for node i, which is able to communicate with nodes k and j. AL

V D

M

k min(li→k , Lk ) Vk Dk +1 Mi,k j min(li→j , Lj ) Vj Dj +1 Mi,j

• Sensing node i at depth Di receives a setup packet from node k at depth Dk ; the structure of setup packets is shown in Table 3. In the remainder of the paper, we assume the quantities L and V to be normalized as to be comprised within the interval [0, 1]. If Dk ≥ Di , the setup packet from node k is discarded (it does not point toward the base station). If Dk < Di , on the other hand, the k-th entry in the table internal to node i is processed as displayed in Table 4. It should be observed that the base station does not need to maintain an internal table. The rationale behind these operations is to keep track of possible routes to the base station and be able to order them on the basis of a joint metric favoring good links, relays with abundant energy resources, and low number of hops. By keeping track of the minimum link quality and the lowest voltage in the route, bottlenecks may be identiﬁed. • In node i, a metric is computed for each entry in the internal table. For instance, a possible metric for the k-th entry is given by Mi,k =

min(li→k , Lk ) + Vk +

1 Dk +1

, (1) 3 where k is a downstream nearest neighbor of i. This metric favors energy balancing: the number of hops from node i to the base station through

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node k is given by Dk + 1, and using its inverse in the metric results into a larger metric for routes over a small number of hops. Node i operates the decision to route over the node whose entry in the table has the largest metric. We will use this metric in the numeric examples. This procedure is to be repeated periodically for dynamic route maintenance and update (in case of topology changes, e.g. due to mobility or to the death of one or more nodes). The eﬀectiveness of this approach is reliant upon the existence of a solid MAC scheme minimizing in-network interference.

Fig. 1. An example of a sensor network with 5 nodes. Node 5 represents the base station. The numbers near the links indicate their quality, whereas the numbers below the nodes represent the battery voltage.

We will now show a simple example of a network with N =5 nodes (shown in Figure 1) in order to clarify how the algorithm works. • Nodes 1 through 4 send a test packet to node 5. Due to the features of the wireless channel, node 5 can only communicate with nodes 3 and 4: the test packets sent by the other nodes are not received by node 5. • Node 5 sends a setup packet to 3 and 4; this packet is shown in Table 5.

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Table 5. Setup packet sent by node 5. A 5 L5 V 5 D 5 5

1 1 0

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(2)

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On the basis of the internal table, the packet shown in table 6 is sent. Since there is only one possible route (node 5 is the only entry in the internal table), L4 = l[4, 5]. • Node 3 ﬁlls the row pertaining to node 5 in its internal table as follows: l[3, 5] = min(l3→5 , L5 ) = 0.4

(6)

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V3

D3

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• Node 2 ﬁlls the rows pertaining to nodes 3 and 4 on the basis of the received setup packets. The row for node 3 is ﬁlled as follows: l[2, 3] = min(l2→3 , L3 ) = 0.4

(10)

V3 = 0.2

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(12)

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(13)

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The route with the highest metric, namely the one related to entry 4, is selected, L2 = l[2, 4], and the test packet shown in Table 8 is sent out. • Only nodes 1, 3, and 4 receive the test packet from 2. Nodes 3 and 4 do not process them, as they come from a higher-depth node.

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• Node 1 ﬁlls the row pertaining to node 2 in its internal table as follows: l[1, 2] = min(l1→2 , L2 ) = 0.1

(18)

V2 = 0.6

(19)

since node 2 has chosen route (2, 4, 5), and node 4 (v4 = 0.6) is the bottleneck of route (1, 2, 4, 5) in terms of battery voltage. D2 + 1 = 3

(20)

0.1 + 0.6 + 0.3 = 0.33. (21) 3 • Node 1 is thus able to reconstruct the route to node 5: the ﬁrst step is node 2, the only entry in the table internal to node 1. Node 2 knows that packets need to be routed over node 4, whose entry has the best metric. Node 4 is a nearest neighbor of the base station, so the chosen route is (1, 2, 4, 5). M1,2 =

3 Performance Analysis We have used a custom network simulator to assess the validity of our routing scheme in a scenario with N =100 nodes uniformly randomly deployed in a 30m × 30m environment. The source and destination are placed at the opposite ends of one of the diagonals of the square deployment area. We will show that the use of our proposed joint metric can lead to a signiﬁcant gain in terms of energy balancing with respect to approaches that only consider link quality. A one-to-one scenario with one node trying to communicate to the base station is simulated. The simulator evaluates link quality on the basis of a multipath channel model with additive Gaussian noise. Our simulator initially ﬁnds a total of 72 routes: 8 over 3 hops, and 64 over 4 hops. Routes with more hops are discarded. Figure 2 shows the performance of a routing scheme using bottleneck link quality as the only metric for route selection. The route with the best bottleneck link quality is chosen, which does not at all guarantee energy-balancing: each chosen route is perused until its relaying nodes are fully depleted. The use of a joint metric between node k and node 0 along the route (k, k − 1, ..., 1, 0) obtained as Mk =

min(Lk→k−1 , ..., L1→0 ) + min(Vk−1 , ..., V0 ) + 3

1 Dk

(22)

dramatically improves energy balancing: Figure 3 shows that the death of the ﬁrst node is delayed from step 12 to step 118. The impact of this result can be better appreciated by considering that a wireless sensor network typically

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D. Puccinelli, E. Sifakis, and M. Haenggi 80

Number of 3−hop Routes Number of 4−hop Routes

70 60 50 40 30 20 10 0 0

50

100 Step

150

200

Fig. 2. The number of 3-hop and 4-hop routes steadily decreases as the batteries of the nodes are depleted: a routing scheme based on bottleneck link quality does not ensure a proper energy balancing.

follows a many-to-one traﬃc pattern: keeping as many nodes alive for as long as possible is extremely important for the sensing capabilities of the network. Other metrics are also possible. A stronger emphasis may be placed on either energy balancing or link quality by using a weighted average of the type Mk = α min(Lk→k−1 , ..., L1→0 ) + β min(Vk−1 , ..., V0 ) + (1 − α − β)

1 . Dk

(23)

Figure 4 shows the performance of our scheme with a weight assignment of the form α=1/6 and β=1/2, which places an even stronger emphasis on energy balancing, and further delays the death of the ﬁrst node (from step 118 to step 131). With both this weighted metric and the unweighted bottleneck metric, a gain of an order of magnitude can be achieved with respect to the scheme based on bottleneck link quality. These gains are of course upper bounds, as in our simulation we assume that the energy consumption is negligible if the nodes are not actively relaying packets. These upper bounds can be approached with an aggressive use of sleep modes and low-power listening techniques which reduce the receive energy, which is normally comparable to the transmit energy. We have seen the advantages of a joint metric, but the analysis above is based on the idea of using bottleneck quantities. Figure 5 shows that in-

A Cross-Layer Approach to Energy Balancing in Wireless Sensor Networks 80

319

Number of 3−hop Routes Number of 4−hop Routes

70 60 50 40 30 20 10 0 0

50

100 Step

150

200

Fig. 3. Our joint metric ensures better energy balancing: routes start to become unusable after 118 steps.

corporating average route link quality and average route battery voltage (as opposed to minima) into the selection metric fails to provide energy balancing and leads to disadvantages such as the premature loss of shorter routes. Bottlenecks should not be ignored: if a K-hop route includes K − 1 good hops and a bottleneck, it will have a good average but will also prematurely cease to exist if selected.

4 Hardware Implementation We implemented our energy-aware routing scheme in hardware using the low power MICAz platform. MICAz represents the latest generation of Berkeley motes, and is commercialized by Crossbow. It is built around an ATMega128L microprocessor, and features a CC2420 802.15.4-compliant ZigBee-ready radio. IEEE 802.15.4 is a standard for low-rate, wireless personal area networks which provides speciﬁcation for the physical and the MAC layer. At the physical layer, it deﬁnes a low-power spread spectrum radio operating at 2.4GHz with a bit rate of 250kb per second. ZigBee is a collection of high level communication protocols built on top of the IEEE 802.15.4 MAC layer. We implemented the algorithm with the operating system TinyOS [HSW+ 00]; our code has a reasonably small footprint (about 10KB of ROM occupancy, and less than 500B of RAM occupancy).

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D. Puccinelli, E. Sifakis, and M. Haenggi 80

Number of 3−hop Routes Number of 4−hop Routes

70 60 50 40 30 20 10 0 0

50

100 Step

150

200

Fig. 4. The use of weights within the metric can lead to even better energy balancing results.

We placed 10 MICAz motes in a lab environment in an arrangement shown in Figure 6. This network is to deliver a continuous stream of packets at a rate of 46 packets/min from node 1 to node 10 over a duration of 105 hours and 20 minutes (the time that was needed for the radio of the source node to become unusable due to the overly low voltage). Although the nodes themselves are static, people moving in the lab cause a certain amount of fading. Fresh batteries were used, the transmit power is -15dBm for both control and application traﬃc, the MAC scheme is standard CSMA/CA, and no acknowledgments are used to makes links reliable. In this implementation, unused nodes sleep 66% of the time, and low power listening modes are not used. The lifetime gain in this experiment can therefore be expected to be fairly far from the upper bounds, but can still provide us with a valid proof of concept. In the 105h experiment, 290720 packets were transmitted and 256313 were successfully received at the BS; this corresponds to a packet loss of about 12%. Given that there were no retransmissions at all and the transmit power was relatively small, this loss is quite acceptable. Among all the routes found, 9% were single-hop, 89% were two-hop, and 1.8% three-hop. The mean path loss over a distance of about 8m and an obstructed line-of-sight path prevents packets from being received at a transmit power of -15dBm. So the fact that single-hop routes exist indicates that the algorithm exploits positive fading states, i.e., is opportunistic, thereby allowing all relay nodes to sleep for some

A Cross-Layer Approach to Energy Balancing in Wireless Sensor Networks 80

321

Number of 3−hop Routes Number of 4−hop Routes

70 60 50 40 30 20 10 0 0

50

100 Step

150

200

Fig. 5. The use of average quantities (as opposed to bottleneck quantities) leads to the premature loss of 3-hop routes.

7m 7 8 4m

4

6 5

3 2

Source 1

9 10 Base Station

All nodes elevated by 1m, except: Node 1: on floor Node 8: 1.6m

Fig. 6. Setup of the 10-node experiment in an indoor environment.

time. More precisely, one-hop routes dominate between hours 58 and 64; in this time lapse, the room was empty, and the particular arrangement of the furniture created a pattern of reﬂections that placed the destination in a good fading spot with respect to the source, thereby allowing one-hop communication. As soon as people entered the room again, the particular arrangement that had created this situation was modiﬁed, and the source was no longer able to exploit static fading to reach the destination directly.

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Remaining Battery Voltage

3.2 3 2.8 Source → 2.6 2.4 2.2 2 1.8 0

20

40

60 Time [hours]

80

100

Fig. 7. Battery discharge curves for the 10 nodes.

4

8

x 10

← Node 2

7

Packets Relayed

6

← Node 5

5 4 ← Node 7

3

← Node 8

2

← Node 1 ← Node 3

1

← Node 9

0 0

← Node 6 ← Node 4

20

40

60 80 Time [hours]

100

120

Fig. 8. Cumulated number of packets relayed by each node.

With short-hop routing, the discharge curves of the nodes would all lie below the one for the source node in Figure 7, since all the relays would

A Cross-Layer Approach to Energy Balancing in Wireless Sensor Networks

Number of Control Packets

500 400

323

0.5 nodes/m2 1 node/m2 2 nodes/m2

300 200 100 0 0

50

100 150 Number of Nodes

200

Fig. 9. Number of control packets as the network size increases.

not only transmit each packet but also receive each packet, and the receive energy is substantial. The use of sleep modes is certainly not aggressive in the algorithm used; by increasing the length of the sleeping periods, the high gains shown by our simulations can be approached. This is easier in networks with a high number of nodes, as the chances of losing connectivity because of the excessive use of sleep modes are reduced. This shows that our scheme has very interesting scalability properties; our long-hop approach enhances such properties, in the sense that with as few long hops as possible the volume of routing traﬃc is minimized. The number of control packets for route discovery M is related to the number of nodes by the relation N − 1 + 2n1 + i=2 ni , where N is the number of nodes, M is the maximum depth in the routing tree, and ni is the number of nodes which can reach the base station in i hops. In fact: 1) N − 1 nodes send a test packet to the base station; 2) only n1 nodes can actually reach it, and the base station thus sends n1 feedback packets to such nodes; 3) these n1 nodes broadcast one route setup packet each; 4) so do their n2 nearest neighbors; 5) this goes on until the nM nearest neighbors at M hops from the base station are reached. Figure 9 shows that the number of control packets as predicted by our simulator for three diﬀerent node densities scales approximately linearly with the number of nodes, which is again promising for scalability.

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5 Closing Remarks The main contribution of the present work is the introduction of a new approach to route selection in wireless sensor networks. With the constraints of real hardware in mind, we suggest the use of a joint metric as part of a cross-layer approach to achieve energy-balancing. Simulation results show a gain of up to one order of magnitude in node lifetime extension with respect to routing schemes based on link quality. A successful hardware implementation with MICAz motes is indicative of the lightweight nature of our scheme. Control applications would greatly beneﬁt from a scheme that relies on a reduced number of hops because of the inherent delay beneﬁt, and the dramatic extension of network lifetime provided by our routing strategy is extremely appealing.

Acknowledgements The support of NSF (grants ECS 03-29766 and CAREER CNS 04-47869) is gratefully appreciated.

Distributed Control over Failing Channels C´edric Langbort, Vijay Gupta, and Richard M. Murray Division of Engineering & Applied Sciences California Institute of Technology 1200 E. California Blvd., Pasadena CA 91125 {clangbort,gupta,murray}@ist.caltech.edu Summary. We give suﬃcient convex conditions for the existence of a distributed controller, with the same interconnection structure as the plant, when the latter is composed of heterogeneous subsystems that communicate with their neighbors over packet-dropping channels. Our linear matrix inequalities generalize previous results for the case of ideal communication links and, although conservative, are typically far more tractable for large-scale systems than recently obtained suﬃcient and necessary conditions.

1 Introduction Ever since the early works reported in [Bel74], the ﬁeld of Distributed Control has mostly been concerned with the design of controllers with particular geometrical structures or information patterns for large-scale systems involving multiple cooperating and/or coupled agents. However, in cases where this coupling corresponds to actual communication as, e.g., in a team of vehicles broadcasting their positions to teammates over a wireless network, the constraints experienced by such systems go beyond the mere speciﬁcation of “which subsystem can communicate with which” and communication limitations should also be taken into account. This latter point, i.e., the analysis of the role of such degradations as quantization, packet drops and random delays, has received relatively little attention in the structured control community, apart from notable exceptions like [RL04, Vou00], that included propagation delays in their theory. The ﬁeld of Control over Networks, on the other hand, has focused on understanding the role of these limitations. In particular, researchers have considered control problems with non-ideal communication links between the sensor, actuator or controller and the plant. Beginning with the seminal work of Delchamps [Del90b], quantization issues have been studied, among others, by Wong & Brockett [WB99c], Nair & Evans [NE00a], Tatikonda & Mitter [TM04b] and so on. Eﬀects of packet-drops on stability and performance have

P.J. Antsaklis, P. Tabuada (Eds.): Netw. Emb. Sens. and Cntrl., LNCIS 331, pp. 325–342, 2006. © Springer-Verlag Berlin Heidelberg 2006

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Fig. 1. A schematic representation of the control architecture. Communication links are only present between the lighter subsystems (the controller’s) if they are between the darker ones (the plant’s). Both links fail according to the same random process.

also been addressed by e.g., Nilsson [Nil98b], Sinopoli et al. [SSF+ 04], AzimiSadjadi [AS03], Gupta et al. [GSHM05]... Unfortunately, all these results, derived in the context of a single centralized plant, cannot be applied directly to large-scale distributed control systems. Our goal, in this paper, is to initiate a merger of the tools of Control over Networks and Distributed Control to arrive at a theory of Networked Control Systems that can handle the eﬀects and implications of both spatial structure and limited communication on the control design problem. We restrict our attention to networks of possibly diﬀerent Linear Time Invariant (LTI) systems, where coupling variables are exchanged between units over non-ideal communication links that can fail, and hence, drop packets stochastically. We consider several possible models for these failures, but assume that enough bits are allocated to each coupling variable and that the error-correcting code operating at the output of each link is eﬃcient enough for each packet containing the value of a variable to be either destroyed or transmitted without error. This is the so-called packet erasure model of a channel. For such systems, we consider both the stability and performance analysis and control synthesis problem. The obtained controller has the same structure as the plant, i.e., that it is itself an interconnected system with the same communication links as the plant, as illustrated in Figure 1. Such an architecture is appropriate in cases where it is too costly or impossible to create a speciﬁc network for communication between controllers, but some information still needs to be exchanged to obtain satisfactory performance. Our work relies on similar dissipativity arguments as those used in [LCD04] and the results take the form of a set of Linear Matrix Inequalities (LMIs), with a particular structure that can be exploited for eﬃcient numerical resolution as was done, e.g., in [LXDB04].

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327

After explaining the main elements of our approach and discussing some robustness and conservatism issues for the simpliﬁed case of a single failing communication link in Section 2, we present our analysis and synthesis results for general interconnections in Section 3. Finally, a numerical example is given in Section 4, which illustrates the trade-oﬀs and computational beneﬁts of our distributed control conditions.

2 The Case of Two Interconnected Subsystems We ﬁrst focus on the simplest type of interconnected system: that consisting of two (diﬀerent) subsystems with one communication link between them. This is done so as to introduce the main ideas at play in the paper, while avoiding unnecessary notational complications. The general case will be addressed in Section 3. Each subsystem is LTI and represented in state-space by xi (k + 1)=Ai xi (k) + Bi vi (k) wi (k)=Ci xi (k) + Di vi (k), i = 1, 2.

(1) (2)

xi is the state of subsystem i and takes its values in Rmi , while vi and wi respectively stand for the input received from and sent to subsystem i’s neighbor. We will assume that both signals take their values in Rni . To account for the fact that information exchanged between subsystems can be dropped randomly at each time-step, we assume the following interconnection relation: v1 (k) v2 (k)

= δ(k)

w2 (k) for all k ≥ 0, w1 (k)

(3)

where δ(k) is a stochastic process taking values in {0, 1} at each time k. Implicit in equation (3) is the fact that n1 = n2 . In the remainder of this section, we will denote both dimensions by n. This way of representing failing channels as random structured perturbations is close in spirit to the approach taken in [EM01b]. In contrast with this reference, however, we will exploit the particular structure of this ‘perturbation’ to obtain a controller with the same interconnection topology as the plant. Also, we will consider the following two models of failure processes, which correspond to various degrees of knowledge about the properties of the network. Deﬁnition 1. We will say that a property of the interconnected system holds in – the independent failure model: if it holds when δ(k), k ≥ 0 are i.i.d random variables, taking value 1 with a given probability p and 0 with probability (1 − p).

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the arbitrary failure model: if it holds for δ being any time-inhomogeneous Markovian process.

Relations (1) and (3) do not specify the interconnected system entirely, as the various signals involved may not exist or be unique. We thus need to introduce the following notion of well-posedness. Deﬁnition 2. The interconnection of subsystems (1) with communication link (3) is nominally well-posed if and only if matrix I−

0 D2 D1 0

is invertible.

When an interconnected system is nominally well-posed, states x1 and x2 satisfy a set of jump diﬀerence equations, namely x1 (k + 1) x (k) = A(δ(k)) 1 , x2 (k + 1) x2 (k)

(4)

where A is a rational matrix function of δ, A(δ) =

B1 0 A1 0 + 0 B2 0 A2

I −δ

0 D2 D1 0

−1

δ

0 C2 , C1 0

and their ﬁrst and second moment are well-deﬁned at all time k ≥ 0. 2.1 Distributed analysis We are interested in testing stability of the subsystems’ interconnection and, ultimately, in designing distributed stabilizing controllers for them. We will use the notion of second moment stability, which has several equivalent deﬁnitions [Lop]. We chose to follow that of [FL02a, SS05], namely Deﬁnition 3. We say that a well-posed interconnected system with communication link (3) is stable when lim EΔ(k−1)

k→∞

x1 (k) x2 (k)

xt1 (k)xt2 (k)

=0

where expectation is taken with respect to Δ(k − 1) = {δ(1), . . . , δ(n − 1)} . Using results of [FL02a] and [Lib03d], we can give the following analysis conditions. Proposition 1. Assume the interconnected system is nominally well-posed. Then it is stable – with independent failures, if and only if there exists P > 0 such that (1 − p) A(0)t P A(0) + p A(1)t P A(1) < P.

(5)

Distributed Control over Failing Channels

329

– with arbitrary failures, if there exists P > 0 such that A(0)t P A(0)

(6) (7)

While condition (5) is both necessary and suﬃcient for stability with independent failures, LMIs (6) are only suﬃcient. In fact, these latter conditions are already only suﬃcient for stability under arbitrary deterministic switching. Our reason for resorting to these conservative conditions is that, as we will see, the set of controllers ensuring that they are satisﬁed in a closed-loop system is convex. However, in order to obtain distributed controllers, we need to relax these analysis tests even further and formulate LMIs involving each separate subsystem. Proposition 2. Assume subsystems (1) are such that D1 = D2 = 0. Then their interconnection with communication link (3) is nominally well-posed. It is stable with independent failures, if there exist P1 , P2 > 0 and symmetric matrices S1 and S2 such that t

Ati Pi Ai − Pi Ati Pi Bi Ci 0 < Bit Pi Ai Bit Pi Bi 0I

Si

Ci 0 0I

for i = 1, 2,

S2 = −E(p)t S1 E(p)

(8) (9)

0 √1p I . where E(p) := √ pI 0 Proof. Let D := diag i=1,2

Ati Pi Ai − Pi Ati Pi Bi . Bit Pi Ai Bit Pi Bi

From Equation (8) we then get that t     t   I0 I 0 I 0 I0  0 C2   0 C2  00 00        (1 − p)  0I  D 0I  + p  0 I  D  0 I  00 00 C1 0 C1 0 <

C1 0 0 C2

t

pS1 + (1 − p)

I0 00

+pE(1)S2 E(1) + (1 − p)E(1)

t

S1 I0 00

(10)

I0 00

t

S2

I0 E(1) 00

C1 0 . 0 C2

Because of relation (9), the term in parentheses on the right-hand side of (10) vanishes while the left-hand side is equal to (1 − p) A(0)t

P 0 P 0 P1 0 A(1) − 1 A(0) + p A(1)t 1 0 P2 0 P2 0 P2

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because D1 = D2 = 0. Thus, we have established that condition (5) holds for P 0 P = 1 . 0 P2 Conditions (8-9) can be seen as dissipativity results in the sense of [Wil72]: LMIs (8) require each subsystem to be dissipative with respect to supply rate matrix Si , while equation (9) imposes that the interconnection is neutral on average for these supply rates. The same notions can be used and further constraints added to the supply rates to obtain a hierarchy of increasingly conservative stability analysis conditions for interconnections with failing channels, when less and less information is available regarding the failure process. For example, we can treat an intermediate failure model where packet-drops are independent from each other at each time step but there is some uncertainty regarding the probability of failure. Proposition 3. Let the interconnection of subsystems (1) and interconnection (3) be nominally well-posed and assume that conditions (8-9) are satisﬁed for p = p and with Si =:

Xi Yi , Xi < 0, and Zi > 0 for i=1, 2. Yit Zi

The the system is stable for all failures of the link described by a random process {δ(k)}∞ k=0 such that (i) δ(k) and δ(l) are independent for k = l, (ii)for all k ≥ 0, p(k) := Prob(δ(k) = 1) < p . Finally, we can give the following result for the arbitrary failure model. Proposition 4. The interconnection of subsystems (1) with communication link (3) is nominally well-posed and stable with arbitrary failures if the following LMIs are satisﬁed Ati Pi Ai − Pi Ati Pi Bi Ci D i < Bit Pi Ai Bit Pi Bi 0 I

t

Si

Ci D i 0 I

for i = 1, 2,

S2 = −E(1)t S1 E(1) where, in addition, matrix Si partitioned according to wi and vi , satisfy Si =:

Xi Yi , Xi < 0 for all i = 1, 2. Yit Zi

(11) (12)

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Proof. Well-posedness follows from classical quadratic separation arguments that we give for completeness. Because of (11), Di I

t

Si

Di > 0 for i = 1, 2. I

But, since S2 = −E(1)t S1 E(1), I D2 which implies that Im

t

S1

I <0 D2

I D1 ∩ Im = {0} D2 I

and, in turn, proves nominal well-posedness. −1 0 D2 0 C2 For stability, we ﬁrst let M := I − . Then, using the D1 0 C1 0 notations of Proposition 2 and the fact that M11 = D2 M21 , M22 = D1 M12 , M12 = C2 + D2 M22 , M21 = C1 + D1 M11 , we can write A(0)t

C1t X1 C1 P 0 0 P1 0 < A(0) − 1 0 P2 0 C2t X2 C2 0 P2

and A(1)t

M11 M12 P 0 P1 0 < A(1) − 1 M21 M22 0 P2 0 P2

t

S2 + E(1)t S1 E(1)

M11 M12 . M21 M22

which proves stability, according to Proposition 2.

In the next section, we apply these results to closed-loop systems and derive controller synthesis conditions. 2.2 Distributed controller synthesis We now assume that each subsystem has an additional control input ui and measured output yi and is given in state-space by xi (k + 1)=Ai xi (k) + Bi vi (k) + Biu ui (k) wi (k)=Ci xi (k) + Di vi (k) + Diu ui (k) yi (k)=Ciy xi (k) + Diy vi (k)

(13) (14) (15)

for all i = 1, 2, k ≥ 0. Also, the interconnected system is assumed to be nominally well-posed. We want to design another interconnected system, the controller, with subsystems

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xki (k + 1)=Aki xki (k) + Bik vik (k) + Biy yi (k) k wik (k)=Cik xki (k) + Dik vik (k) + Diy yi (k) k ui (k)=Ciu k xi (k) + Diu k vi (k) + Diyu yi (k)

(16) (17) (18)

and the same communication link as the plant, such that the closed-loop system is nominally well-posed and stable. When we say that the controller “uses the same communication links as the plant”, we mean that its interconnection relation is v1k (k) w2k (k) = δ(k) for all k ≥ 0, k v2 (k) w1k (k) for the same switching signal δ as the plant. The loops between controller’s and plant’s subsystems are closed by eliminating ui and yi according to equations (13) and (16). Thus we assume that there is no communication link between a plant’s subsystem and its corresponding controlling unit, which is justiﬁed in practice if they are co-located. The obtained closed-loop system is an interconnected system itself, with subsystems’ state-space matrices Aci Bic Cic Dic being aﬃne in the ith controller’s subsystem’s data Θi , and interconnection relation given by      w1 (k) 0 In 0 0 v1 (k) In 0 0 0  w2 (k) v2 (k)   v1k (k) = δ(k)  0 0 0 Ink  w1k (k) for all k ≥ 0. 0 0 I nk 0 v2k (k) w2k (k) Applying analysis conditions of Section 2.1 to the closed-loop system for the arbitrary failure model, we obtain the following Proposition 5. The closed-loop system is well-posed and stable with arbitrary failures, if there exist P1c , P2c > 0 and symmetric matrices S1c and S2c such that (Aci )t Pic Aci − Pic (Aci )t Pic Bic Cic Dic c t c c c t c c < (Bi ) Pi Ai (Bi ) Pi Bi 0 I

t

Sic

Cic Dic 0 I

S2c = −E(1)t S1c E(1) Sic =:

Xic Yic (Yic )t Zic

, Xic < 0.

for i = 1, 2,

(19) (20) (21)

Because the closed-loop subsystems’ matrices are aﬃne functions of the controller’s data Θi , conditions (19) are not jointly convex in Pic , Sic and these data. The following theorem shows that they are nevertheless equivalent to a set of LMIs.

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Theorem 1. There exist matrices such that stability conditions (19)-(20) (21) are satisﬁed in closed-loop if and only if there exist symmetric matrices ˜ g, X ˜ g and matrices Y g , Y˜ g such that the following P1g , P2g , P˜1g , P˜2g , X1g , X2g , X 1 2 LMIs are satisﬁed: Pig > 0, Xig < 0,

Pig I ≥ 0 for all i = 1, 2 I P˜ig −Xig I ˜ g ≥ 0 for all i = 1, 2 I −X i



 t  g  A1 B1 0 0 A1 B1 P1 0 g      1 0 0  1 t  I 0   0 −P1   I 0  (NX (NX )  )<0 g g    C1 D 1 0 0 −X1 −Y C1 D1  0 0 −(Y g )t X2g 0 I 0 I t  g    P2 0 0 0 A2 B2 A2 B2 g    2   0 0  2 t  I 0   0 −P2   I 0  (NX )<0 )  (NX g g t    C2 D2 0 0 −X2 (Y ) C2 D 2  0 I 0 I 0 0 Y g X1g   t  ˜ g  0 0 P1 0 −I 0 −I 0 ˜g 0  t t   t t 0  1 t  A1 B1   0 −P1   A1 B1  (N 1˜ ) > 0 (NX ˜)  g g ˜ −Y˜   0 −I  X 0 −I   0 0 −X 1 ˜g ˜ C1t D1t C1t D1t 0 0 −(Y g )t X 2  t  ˜ g   0 0 P2 0 −I 0 −I 0 ˜g 0  t t  t t  0  2 t  A2 B2   0 −P2   A2 B2  (N 2˜ ) > 0 (NX ˜)  g g t    ˜ ˜ 0 −I 0 0 −X2 (Y )  0 −I  X t t ˜g C2 D 2 C2t D2t 0 0 Y˜ g X 1

(22)

(23)

(24)

(25)

y y u t u t i i and NX where the columns of NX ˜ span the null-space of Ci Di and (Bi ) (Di ) respectively, for all i. The corresponding controller can be computed once a feasible point is known and its subsystems share three times as many signals as the plant’s subsystems, i.e., nk = 3 n.

Proof. The equivalence between (19)-(20)-(21) and the LMIs of Theorem 1 is proved in a similar way as in [LCD04], where distributed stabilizing controllers are constructed for systems with ideal channels. One starts with the bilinear matrix inequalities (19) and tries to obtain equivalent convex conditions using the Elimination Lemma of [Sch01]. Of crucial importance for this purpose is the fact that the inertia condition in

S1c 0 0 S2c

= (n + nk , 0, n + nk )

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C. Langbort, V. Gupta, and R.M. Murray

holds. This is implied by condition (20). It is also to guarantee this property that we need to choose nk = 3n when reconstructing the controller from the LMI conditions of Theorem 1.

3 General Multi-link Interconnections We now consider distributed control design for interconnections of N > 2 subsystems, where all communication links can fail separately. In addition to internal stability, we ask that the H∞ –norm of the closed-loop system be less than one. Being able to synthesize such suboptimal controllers will allow us to explore the inﬂuence of decentralization on performance in Section 4. The dynamics of each subsystem is described by xi (k + 1)=Ai xi (k) + Bi vi (k) + Biu ui (k) + Bid di (k) wi (k)=Ci xi (k) + Di vi (k) + Diu ui (k) + Did di (k)

+Diyd di (k) yi (k)=Ciy xi (k) + Diy vi (k) + zi (k)=Ciz xi (k) + Diz vi (k) + Dizu ui (k) + Dizd di (k),

(26) (27) (28) (29)

where di and zi are the exogenous disturbance and performance output of the ith subsystem, respectively. In (26), vi and wi represent the aggregate signals exchanged between subsystem i and its neighbors i.e., t t t t vit = vi1 . . . viN ; wit = wi1 . . . wiN

each vij and wij belonging to Rnij (nij = nji ) and corresponding to signals shared by subsystem i and j. When subsystem i and j do not communicate, we simply let nij = 0. As in Section 2, interconnection links between two subsystems are described by vij (k) wji (k) = δij (k) for all k ≥ 0, 1 ≤ i ≤ j ≤ N (30) vji (k) wij (k) for a stochastic process δij taking values in {0, 1}. In the remainder of this section, we will restrict ourselves to the arbitrary failure model and assume that each δij is a switching signal of the form presented in Section 2. The concept of well-posedness, which guarantees that all signals are well-deﬁned irrespective of the system’s failure mode, generalizes naturally to this case. Deﬁnition 4. We say that a system described by (26) and (30) is well-posed if, for any value of δij = δji in {0, 1} (1 ≤ i ≤ N, 1 ≤ j ≤ N ), the only vectors {vij }1≤i≤N , {wij }1≤i≤N satisfying 1≤j≤N

1≤j≤N

vij = δij wji and are zero.

wi Di ∈ Im vi I

for all i, j

Distributed Control over Failing Channels

335

Finally, following [SS05], we deﬁne the H∞ –norm of the interconnected system (26)-(30) as ∞ t k=0 EΔ(k−1) z(k) z(k) (31) sup sup ∞ t δ(0) d=0 k=0 EΔ(k−1) d(k) d(k) where, in (31), we have let t (k) and dt (k) = dt1 (k). . .dtN (k) z t (k) = z1t (k). . .zN

for all k ≥ 0 and assumed xi (0) = 0 for all i = 1...N . When this norm is strictly less than one, we say that the system is contractive. With these deﬁnitions in place, we can state the counterpart of Proposition 4 for general interconnections. Proposition 6. The interconnection of subsystems (26) (with dimui = dimyi = 0 for all i) with communication link (30) is well-posed and stable and contractive with arbitrary failures, if there exist symmetric matrices Pi > 0 and Xij < 0 for all i, j = 1...N and matrices Yij for all i ≥ j, with Yii skewsymmetric, such that, for all i = 1...N  t   0 0 0 0 Pi 0 Ai Bi Bid Ai Bi Bid   I 0 0   0 −Pi 0  0 0 0  I 0 0      Ci Di Dd   0 0 Zi11 Zi12 0 0   Ci Di Dd  i   i     0 I 0   0 0 Z 12 t Z 22 0 0   0 I 0  < 0 i i      C z Dz Dzd   0 0 0 0 I 0  Ciz Diz Dizd  i i i 0 0 I 0 0 I 0 0 0 0 0−I 

where

(32)

Zi11 := − diag Xij 1≤j≤N

Zi22 := diag (Xji ) 1≤j≤N

Zi12 := diag −diag Yij , diag Yjit . 1≤j≤i

i<j≤N

Proof. We will only give the proof of well-posedness. From LMIs (32) we obtain, in particular, that N

wij vij

t

Sij wij vij ≥ 0

(33)

j=1

for all i = 1...L and all wij , vij such that wi vi ∈ Im Di I , with equality only if all wij ’s and vij ’s are zero. In (33), we have let

(34)

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C. Langbort, V. Gupta, and R.M. Murray



Sij :=

Xij Yij ifj ≤ i,  Yijt −Xji 

Xij −Yjit ifj > i. Yji −Xji

Now assume that, in addition to relation (34), vij = δij wji for all i, j and let F := {(i, j)|δij = 0} be the set of failing communication links and E := {(i, j)|δij = 1} its complementary set in the set of non-oriented edges of the complete graph over N vertices, with self-loops. We can them rewrite (33) as t

t wij Xij wij + (i,j)∈F

wij wji Sij wij wji ≥ 0 (i,j)∈E

for all i. Summing this latter expression over i and using the deﬁnition of Sij , we get      i

(i,j)∈F



t Xij wij  + wij ≤0

i

t wij wji Sij wij wji  ≥ 0.

(i,j)∈E =0 t

Hence, wij = 0 for all (i, j) ∈ F and, in turn, (i,j)∈E wij wji Sij wij wji = 0 for all i. As a result, we have equality in inequality (33), which implies wij = vij = 0 for all (i, j) ∈ E as well and proves well-posedness.

The proof of stability and contractiveness amounts to showing that (32) implies the Bounded Real Lemma conditions of [SS03]. The steps are very similar to those used in the proof of Proposition 4 and are thus omitted. Nevertheless, we would like to draw the reader’s attention to two points: 1. Analysis conditions of Proposition 6 can handle interconnections with selfloops (i.e. signals of the form vii and wii ). This will prove useful when we treat a practical example in Section 4. 2. When LMIs (32) are satisﬁed, stability and contractiveness are guaranteed irrespective of whether diﬀerent links fail independently of each other or not. This is because these conditions require each failing link to decrease the stored energy xti Pi xi of its adjacent subsystems, no matter what mode the other channels are in. In contrast, the multi-link counterpart of the probability-dependent condition given in Proposition 2 would not have this property, since every link would only decrease subsystem’s stored energy on average, which requires to know the joint failure probability and to account for all global failure modes.

Distributed Control over Failing Channels

337

Probability-dependent stability tests can be given for general interconnections when Di = 0 for all subsystems (as we did in Proposition 2). However, since we do not know how to ensure this property on a closedloop system and we ultimately want to formulate synthesis conditions, we decided to focus on the arbitrary failure model for the multi-link case. Our distributed control synthesis theorem also has a counterpart for general interconnections. Theorem 2. There exists a distributed controller, with the same structure as the plant, such that the closed-loop system is well-posed, stable and contractive g , with arbitrary failures, if there exist symmetric matrices Pig > 0, P˜ig > 0, Xij g g g g g ˜ for all i, j = 1...N and matrices Y , Y˜ for i ≥ j, with Y , Y˜ skewX ij ij ij ii ii symmetric such that g I −Xij Pig I ≥ 0, g ˜g ≥ 0 ˜ I −X I Pi ij



i NX

where

(35)

  0 0 Ai Bi Bid 0 0  I 0 0    12 d  0 0 Z i i g g   C i D i D i  NX <0 t 22  0 I 0  0 0 Z   i g g 0 I 0  Ciz Diz Dizd  0 0 I 0 0−I (36)    t P˜ g 0 0 0 0 0 i −I 0 0  g 0 0 0 0   At B t B d t   0 −P˜i    i  i i ˜ 11 ˜ 12 0 0    0  Z 0 0 Z −I 0 i i t     i g g NX   t t  ˜  Ci Dit Did   0 0 Z˜ 12 t Z˜ 22 0 0      i i g g   0 0 −I     0 0 0 0 I 0 z t z t zd t (Ci ) (Di ) Di 0 0 0 0 0−I (37)   −I 0 0  At B t B d t   i  i i  0 −I 0    i × t <0 t N  Ci Dit Did  X˜    0 0 −I  t t t (Ciz ) (Diz ) Dizd

t  g 0 Pi 0 Ai Bi Bid  I 0 0   0 −Pig 0    0 Zi11 t  Ci Di Did   0     0 I 0   0 0 Z 12    i C z Dz Dzd   0 0 0 i i i 0 0 I 0 0 0

Zi11

g

Zi12

0 0

g ; Zi22 = − diag Xij 1≤j≤N

g

g

g = diag Xji 1≤j≤N

= diag −diag Yijg , diag Yjig 1≤j≤i

i<j≤N

t

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C. Langbort, V. Gupta, and R.M. Murray

Z˜i11

g

Z˜i12

g ˜ ij ; = − diag X 1≤j≤N

g

Z˜i22

g

= diag

1≤j≤N

= diag −diag Y˜ijg , diag 1≤j≤i

i<j≤N

Y˜jig

g ˜ ji X t

.

It is interesting to compare Theorem 2 with the recent results of Seiler & Sengupta [SS05], regarding H∞ optimal control of networked systems with stochastic packet-losses (with failures modeled as time-homogeneous Markovian processes with known transition probability matrix). While they were able to derive necessary and suﬃcient LMI conditions for the existence of a stabilizing controller in this case, the latter was centralized and, more importantly, the conditions involved one matrix variable for each of the possible failure modes. In particular, if all links can fail independently at each time step, the conditions of [SS05] have O(m2 (N 2 + 2E )) scalar variables, where we have let m := max mi ; E := Card{(i, j)|nij = 0}. i

The LMIs of Theorem 2, on the other hand, only require O(N m2 +En2 ) (with n = maxi,j nij ) such variables, irrespective of the number of modes. The price we pay for this increased tractability is a high degree of conservatism for the Markovian failure model, as the controller we obtain guarantees a given performance level for arbitrary failures and the conditions of Proposition 6 are only suﬃcient for this case. Nevertheless, our controller has the same interconnection structure as the plant.

4 A Power Network Example In this section, we apply our tools to a model of power network where each transmission line fails stochastically. The goal is to control each power plant individually to ensure stability and performance of the whole network, in the face of such failures. Our model is adapted from [CL04]. The dynamics of the ith load-driving generator is given by xi (k + 1)=Ri xi (k) + Li Ii (k) + Biu ui (k) Vi (k)=Ki xi (k)

(38) (39)

where, at each time k, Ii (k) ∈ R2 (respectively Vi (k)) contains the real and imaginary part of the current (respectively voltage) deviations from a chosen operating point, ui (k) ∈ R is the control torque applied to regulate the generator and xi (k) ∈ R2 is the state of the generator, corresponding to the deviations from the rotor’s angular velocity’s and angle’s reference proﬁle. Each generator is connected to a load with admittance matrix Yii ∈ R2×2 and to other generators through a transmission line of stochastically-varying

Distributed Control over Failing Channels

339

Fig. 2. (a)A power network with N = 3 generators ; (b) The corresponding model including multiple self-loops. See text for details.

admittance δij (k)Yij ∈ R2×2 , j = 1...N , as depicted in Figure 2(a). In keeping with our previous model of arbitrary link failures, we assume that processes δij are Markovian and take values in {0, 1} but that their characteristics are unknown. As already explained at the end of Proposition 6, assumptions regarding whether two lines fail independently of each other are irrelevant for our purposes. Using Kirchhoﬀ’s laws, we derive that Ii (k) = Yii Vi (k) + δ(k)

Yij (Vi (k) − Vj (k)) j=i

and, in turn, that δij (k)Yij (Ki xi (k)−Kj xj (k))+Biu ui (40)

xi (k+1) = (Ri +Li Yii Ki )xi +Li j=i

for all i = 1...N . Introducing a disturbance di , on the control torque, a measured output yi = xi and a performance output zit = xti uti , for each generator, equations (40) can be put in the form of (26), with Ai := (Ri + Li Yii Ki ); Bi := Li . . . Li ; Bid := Biu 4(N −1)

and

 Y1i  ..   .    Y(i−1)i   Ki x i .  wij = −Yij Ki xi for i = j ; wii =   Y(i+1)i   .   ..  YN i 

340

C. Langbort, V. Gupta, and R.M. Murray

It is then apparent that signals vii and wii do not satisfy an interconnection relation of the form (30). Instead, they satisfy vii (k) =

diag

1≤j≤N,j=i

(δij (k)I) wii (k) for all k > 0.

(41)

There are two elements that make relation (41) diﬀerent from that used in our deﬁnition of interconnected systems in Section 3. 1. First, at each instant k, wii (k) and vii (k) are related through a blockdiagonal matrix, instead of a diagonal one. This can be easily ﬁxed by partitioning both signals according to the block-structure of the matrix as t t (i−1) (i+1) t 1 t N t , wii = wii . . . wii wii . . . wii t vii =

t

(i−1)

1 vii . . . vii

t

(i+1)

vii

t

N . . . vii

t

,

j j and treating each relation between wii and vii separately as j j j vii (k) = δii (k)I wii (k) for all k ≥ 0.

(42)

In so doing, we introduce multiple edges between subsystems (in fact multiple self-loops) and, although the interconnection graph thus obtained is not of the form presented before, it is straightforward to extend our tools to this class by simply relabeling all these new edges. 2. Second, each one of these new self-loop fails according to the same random variable as an other edge, i.e., j δii (k) = δij (k) for all k ≥ 0, 1 ≤ i = j ≤ N.

As explained earlier, this correlation between diﬀerent links’ failures is easily handle by the conditions of Proposition 6. In order to numerically investigate the conservatism of our methods, we applied the distributed analysis conditions of Section 2 to the model (40) with N = 2 interconnected generators, and compared the best guaranteed performance to the exact H∞ -norm of the system, computed using the Bounded Real Lemma of [SS03]. The generator characteristics are as follows: Ri =

1 20

00

, Li =

Biu = −10 , Ki =

10 , 00 00 , 0−1

for all i. The load matrices Yii and Yij were generated at random. We re-scaled the system’s inputs and outputs until the LMIs of Proposition 2 and [SS03]

Distributed Control over Failing Channels

341

120

best guaranteed performance

100

80

60

40

20

0 0.1

0.2

0.3

0.4

0.5 0.6 0.7 probability of failure

0.8

0.9

1

Fig. 3. H∞ -norm and best performance upper-bound guaranteed by conditions (11( for two randomly generated power networks with N = 2 generators, as probability of link failure is varied. Note that the bounds produced by (11) are probabilityindependent. The two types of lines correspond to two diﬀerent random systems.

were within one percent of not being feasible. We thus obtained a tight upperbound both on the best closed-loop performance guaranteed by our method and on the H∞ -norm. Typical results are plotted in Figure 3. Then, we also applied the synthesis methods of Section 3 to the model (40), this time with N = 10 generators interconnected over the complete graph (i.e. nij = 2 for all i, j = 1...10). If all links can fail independently, the system has 245 diﬀerent modes, which makes the conditions of [SS05] impractical. In fact, our semi-deﬁnite program solver (SeDuMi) ran out of memory when we tried to solve the corresponding LMIs. In contrast, the same solver, on the same machine, had no diﬃculty solving LMIs (32) and, for the test-case where Yii =

i 1 − 10 i − 10 1

; Yij =

1−1 11

for i = j,

gave rise to a distributed controller with a guaranteed upper-bound on closedloop performance of γ = 358.4. This shows that, even if one is not interested in a distributed controller, there is a computational advantage in using the suﬃcient synthesis conditions of Section 3.

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C. Langbort, V. Gupta, and R.M. Murray

5 Conclusion and Future Work We have extended pre-existing tools for the synthesis of distributed controllers to include tolerance to communication failures between the various components of an interconnected system. In so doing, we have taken a ﬁrst step towards bridging the gap between the theories of Distributed Control and Control over Networks. There are several directions in which the results of the present paper could be deepened. For example, staying in the realm of perfect erasure channels, it would be nice to be able to tackle particular models of link failures (such as the popular Gilbert-Elliott model) less conservatively than we have done, and design distributed controllers which can guarantee better performance in these cases. Modifying our tools to include quantization eﬀects and/or other type of degradations would also be of interest.

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Author Index

Bauer, Peter, 183, 251 Bullo, Francesco, 3

Martins, Nuno C., 269 Montestruque, Luis A., 231 Morgansen, Kristi A. , 9 Muhammad, Abubakr, 69 Murray, Richard M., 325

Carli, Ruggero, 25

Naghshtabrizi, Payam, 203

Dahleh, Munther A., 269 Dewasurendra, Duminda, 183 Doyle, John C., 269

Olfati-Saber, Reza, 157, 169 Ould-Ahmed-Vall, ElMoustapha, 87

Ames, Aaron D., 127 Antsaklis, Panos J., 53, 107, 231

Egerstedt, Magnus, 69 Fagnani, Fabio, 25 Fang, Lei, 53 Focoso, Marco, 25 Franco, Elisa, 169 Frazzoli, Emilio, 169 Gupta, Vijay, 325 Haenggi, Martin, 291, 309 Heck, Bonnie S., 87 Hespanha, Jo˜ao P., 203 Ji, Meng, 69 Klein, Daniel J. , 9 Kottenstette, Nicholas, 107 Langbort, C´edric, 325 Liang, Yupeng, 251 Liberzon, Daniel, 219

Premaratne, Kamal, 183 Puccinelli, Daniele, 309 Riley, George F., 87 Samad, Tariq, 85 Sangiovanni-Vincentelli, Alberto, 127 Sastry, Shankar, 127 Shamma, Jeﬀ S., 169 Sifakis, Emmanuel, 309 Speranzon, Alberto, 25 Tatikonda, Sekhar, 289 Tilbury, Dawn, 201 Triplett, Benjamin I. , 9 Xie, Min, 291 Zampieri, Sandro, 25

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