Discrete-time Sliding Mode Control: A Multirate Output Feedback Approach (Lecture Notes in Control and Information Sciences)

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

323

B. Bandyopadhyay
S. Janardhanan

Discrete-time Sliding Mode Control A Multirate Output Feedback Approach With 68 Figures

Series Advisory Board

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

Authors Prof. B. Bandyopadhyay S. Janardhanan Interdisciplinary Programme in Systems and Control Engineering Mumbai-400 076 India

ISSN 0170-8643 ISBN-10 ISBN-13

3-540-28140-1 Springer Berlin Heidelberg New York 978-3-540-28140-5 Springer Berlin Heidelberg New York

Library of Congress Control Number: 2005931593 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-Verlag. Violations are liable to prosecution under German Copyright Law. Springer is a part of Springer Science+Business Media springeronline.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 authors. 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

M¯ atr Devo Bhav¯ ah, Pitr Devo Bhav¯ ah, ¯ arya Devo Bhav¯ Ach¯ ah, ( Mother is a diety, Father is a diety, Teacher is a diety)

Dedicated to our parents and teachers who made us capable enough to write this book

P7eface

Sliding mode control is a simple and yet robust control technique. In case of sliding mode control, the system states are made to confine to a selected subset of the state space so as to achieve some desirable dynamics. Traditionally, a relay-based control has been used for this purpose and had its roots in the variable structure system philosophy. Developed in the erstwhile Soviet Union, the concept was pioneered by Vadim Utkin. With the incerasing use of computers and discrete-time samplers in controller implementation in the recent past, discrete-time systems and computer based control have become topics that have a lot of potential in them. This had opened up the field of sliding mode control of discrete-time systems. Many researchers; W. B. Gao, E. Misawa, A. Bartoszewicz, K. Furuta, C. Milosavljevic, to cite a few had worked in this field. However, much of the work had been concentrated on state feedback based control. But, it is of common knowledge that only the system output is available for the controller design. More often than not, the system output is not coincident with the system state. This leads to the requirement of output feedback based sliding mode control strategies. The existing literature on output feedback sliding mode control is either very restrictive, by being applicable to only a specific class of systems, even when one looks at the control of LTI systems alone. A wider class of systems can be controlled, if one adopts dynamic sliding mode controllers. However, the system complexity is increased in the process. This is the motivation of this monograph : An output feedback sliding mode control philosophy which can be applied to almost all controllable and observable systems, while at the same time being simple enough as not to tax the computer too much. We found the answer in the synergy of the multirate output sampling concept and the concept of discrete-time sliding mode control. This work would have been incomplete had it not been for the kind cooperation and help from many. Particularly, we needed much help from our associates Vishvjit K. Thakar, Vitthal S. Bandal and T. C. Manjunath in find-

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Preface

ing appropriate applications to complete the final chapter of the monograph. We would like to use the oppurtunity to thank them.

Mumbai, May 2005

Bijnan Bandyopadhyay Janardhanan Sivaramakrishnan

Con2en26

1

2

3

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Sliding Mode Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.1 Variable Structure Systems . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.2 Continuous-time Sliding Mode . . . . . . . . . . . . . . . . . . . . . . 1.1.3 Continuous-time Sliding Mode Control . . . . . . . . . . . . . . . 1.1.4 Equivalent Control and the Reaching Law Approach . . 1.1.5 Discrete-time Sliding Mode Control . . . . . . . . . . . . . . . . . . 1.2 Multirate Output Feedback . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.1 Multirate Output to State Relationship . . . . . . . . . . . . . . 1.2.2 Advantage of Multirate Output Sampling over Discrete-time Observer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Motivation for Multirate Output Feedback based Discrete-time Sliding Mode Control . . . . . . . . . . . . . . . . . . . . . . . . Switching Function based Multirate Output Feedback Sliding Mode Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Quasi-Sliding Mode Control in Deterministic Systems . . . . . . . . 2.1.1 Illustrative Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Multirate Output Feedback based Quasi-Sliding Mode Control for Uncertain Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Multirate Sampled Output to State Relationship in Uncertain Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.2 Review of State Feedback based QSM control of Uncertain Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.3 Output Feedback Sliding Mode Control Algorithm based on New Reaching Law . . . . . . . . . . . . . . . . . . . . . . . . 2.2.4 Illustrative Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 1 2 5 9 10 11 13 14 15 17 17 18 19 20 20 22 24

Multirate Output Feedback based Discrete-time Sliding Mode in LTI Systems with Uncertainty . . . . . . . . . . . . . . . . . . . . 27 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

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3.1.1 Relaxation of Quasi-sliding mode criterion . . . . . . . . . . . . 3.2 Multirate Output Feedback based Discrete-time Sliding Mode Control for Uncertain Systems with Matched Uncertainty 3.2.1 A Brief Review on State Feedback based DSMC Control Strategy for Matched Uncertain Systems . . . . . . 3.2.2 Multirate Output Feedback Control Algorithm . . . . . . . . 3.2.3 Numerical Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.4 Comparison with State Feedback based Control . . . . . . . 3.3 Multirate Output Feedback based Discrete-time Sliding Mode Control of LTI Systems with Unmatched Uncertainty . . . 3.3.1 Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.2 Multirate Output Feedback based Control Law . . . . . . . . 3.3.3 Illustrative Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.4 Comparative Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.5 Analysis of Simulation Results . . . . . . . . . . . . . . . . . . . . . . 3.4 Multirate Output Feedback based Integral Sliding Mode in Discrete-time Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.1 Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.2 State Based Discrete-Integral Sliding Mode . . . . . . . . . . . 3.4.3 Multirate Output Feedback based DISMC . . . . . . . . . . . . 3.4.4 Illustrative Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

Multirate Output Feedback based Discrete-time Quasi-Sliding Mode Control of Time-Delay Systems . . . . . . . 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 The Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1 Discretization of a Time Delay System . . . . . . . . . . . . . . . 4.3 Design of Sliding Surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Multirate Sampling of Time-Delay Systems . . . . . . . . . . . . . . . . . 4.4.1 Contribution of the Disturbance Term during Multirate Sampling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.2 Relationship between State and Multirate Output in Time-Delay Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Quasi-Sliding Mode Control Algorithm for Form 1 Systems . . . 4.5.1 Multirate Output to State Relationship in Time-Delay Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.2 Multirate Output Feedback based Sliding Mode Control Algorithm for Form 1 Systems . . . . . . . . . . . . . . . 4.5.3 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6 Quasi-Sliding Mode Control Algorithms for Form 2 Systems . . 4.6.1 State based Control Algorithm . . . . . . . . . . . . . . . . . . . . . . 4.6.2 Multirate Output Feedback Discrete-time Sliding Mode Control Algorithm for Form 2 Systems . . . . . . . . . 4.6.3 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.7 Quasi-Sliding Mode Control Algorithm for Form 3 Systems . . .

27 28 28 29 31 34 35 35 36 38 39 44 44 44 45 46 48 51 51 51 52 53 55 55 56 57 57 58 59 60 60 61 62 63

Contents

4.7.1 Multirate Output Feedback based Control Algorithm . . 4.7.2 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.8 Discrete-time Sliding Mode Control of Form 4 Systems . . . . . . . 4.8.1 Reaching Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.8.2 Multirate Output Feedback based Discrete time Sliding Mode Control Law for Time-Delay Systems with Uncertainty . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.8.3 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.8.4 Performance in System without Disturbance . . . . . . . . . . 5

6

Multirate Output Feedback Sliding Mode for Special Classes of Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Multirate Output Feedback Discrete-time Sliding Mode Control based Tracking Controller for Nonminimum Phase Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.2 Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.3 Two-part Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.4 Determination of the Nominal Zero Dynamics Trajectory ( 2,0 ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.5 Multirate Output Feedback Based Tracking Control . . . 5.1.6 Numerical Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Multirate Output Feedback based Quasi-Sliding Mode for a Class of Nonlinear Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.2 Background : Finitely Discretizable Systems . . . . . . . . . . 5.2.3 Multirate Output Sampling in Nonlinear Systems . . . . . 5.2.4 Discrete-time Sliding Mode Control for Nonlinear Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.5 Illustrative Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Multirate Output Feedback Discrete-time Sliding Mode Control based with Prescribed (Rd Ud ) Sliding Sector . . . . . . . . . 5.3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.2 Discrete-time VSC with Rd Ud Sliding Sector . . . . . . . . . . 5.3.3 Multirate Output Feedback DSMC Controller for Rd Ud Sliding Sector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

XI

63 65 66 66 67 68 70 71 71 71 72 72 74 76 77 81 81 81 83 85 85 93 93 93 99

Discrete-time Terminal Sliding Mode: Concept . . . . . . . . . . . . 105 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 6.2 Continuous-time Terminal Sliding Mode Control . . . . . . . . . . . . . 105 6.3 Discretization of TSM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 6.4 Discrete-time Terminal Sliding Mode Control . . . . . . . . . . . . . . . 108 6.4.1 DTSM Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 6.4.2 Illustrative Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 6.5 Multirate Output Feedback based DTSM Algorithms . . . . . . . . 112

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6.5.1 Multirate Output Feedback based DTSM Algorithms for LTI Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 6.5.2 Multirate Output Feedback based Discrete-time Terminal Sliding Mode in Output Feedback Linearizable Nonlinear Systems . . . . . . . . . . . . . . . . . . . . . 117 7

Applications of Multirate Output Feedback Discrete-time Sliding Mode Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 7.1 Position Control of Permanent Magnet DC Stepper Motor . . . . 121 7.1.1 Stepper Motor Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 7.1.2 Discrete Time Sliding Mode Control Using Multirate Output Feedback: Regulator Case . . . . . . . . . . . . . . . . . . . 124 7.1.3 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 7.2 Power System Stabilizer Design using Multirate Output Feedback Sliding Mode Control . . . . . . . . . . . . . . . . . . . . . . . . . . . 128 7.2.1 Power System Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128 7.2.2 Controller Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . 130 7.2.3 Nonlinear Simulation Study . . . . . . . . . . . . . . . . . . . . . . . . . 132 7.3 Vibration Control of Smart Structure using Multirate Output Feedback based Sliding Mode Control . . . . . . . . . . . . . . . . . . . . . . 132 7.3.1 Modeling of a Smart Cantilever beam . . . . . . . . . . . . . . . . 133 7.3.2 Multirate Output Feedback based Sliding Mode Controller design for Vibration Control . . . . . . . . . . . . . . . 134

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

Li62 of Figw7e6

1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 2.1 2.2

3.1 3.2

Asymptotic Stability of VSS with Lyapunov Stable Systems . . . 3 Asymptotic Stability of VSS with Unstable Constituent Systems 4 An illustration of sliding mode in variable structure systems . . . 5 Order based switching scheme in multi-input systems . . . . . . . . . 6 Eventual sliding mode switching scheme for multi-input systems 7 Illustration depicting the state velocity vector for a system with discontinuous right-hand side . . . . . . . . . . . . . . . . . . . . . . . . . . 8 Illustration of Multirate Output Feedback based Control Philosophy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 Comparative plots of error norms in state computation using multirate sampling and Luenberger observer . . . . . . . . . . . . . . . . . 15 Quasi-Sliding Mode Control based on Multirate Output Feedback : (a) State responses (b) Sliding function (c) Control input (d) Phase portrait . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 Quasi-Sliding Mode Control based on Multirate Output Feedback for Uncertain System : (a) State responses (b) Sliding function (c) Control input (d) Phase portrait . . . . . . 25

Plot of v(µ) with constant disturbance component . . . . . . . . . . . Closed loop response of the system with constant disturbance component . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Plot of v(µ) with varying disturbance component . . . . . . . . . . . . . 3.4 Closed loop response of the system with varying disturbance component . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 The mechanical system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6 Plot of Sliding Function vs (µ) using state feedback . . . . . . . . . . . 3.7 Evolution of the System states using state feedback . . . . . . . . . . . 3.8 Plot of the state feedback based control inputs . . . . . . . . . . . . . . . 3.9 Plot of Sliding Functions v(µ) using multirate output feedback . 3.10 Evolution of the System states using multirate output feedback

32 32 33 34 38 40 41 41 42 43

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List of Figures

3.11 Plot of the control inputs using multirate output feedback . . . . . 43 3.12 The phase plots of the actual system and expected system . . . . . 48 3.13 The control input applied to the system . . . . . . . . . . . . . . . . . . . . . 49 4.1 4.2 4.3 4.4 4.5 4.6

Plots for Systems of Form 1 : a. Time Response of 1 , b. Time Response of 2 , c. Input Profile, d. Profile of the sliding function v(µ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . System Plots for State Delay and Retarded Input Channel : a. Time Response of 1 , b. Time Response of 2 , c. Input Profile, d. Profile of the sliding function v(µ) . . . . . . . . . . . . . . . . . Plots for Systems with Output Delay : a. Time Response of 1 , b. Time Response of 2 , c. Input Profile, d. Profile of the sliding function v(µ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Disturbance Example 1 : (a). State Response (b). Phase plot (c). System input (d). Sliding function . . . . . . . . . . . . . . . . . . . . . . Disturbance Example 2 : (a). State Response (b). Phase plot (c). System input (d). Sliding function . . . . . . . . . . . . . . . . . . . . . . System without Disturbance : (a). State Response (b). Phase plot (c). System input (d). Sliding function . . . . . . . . . . . . . . . . .

59 62 65 68 69 70

5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8

Comparative plot of the system output and the reference signal 78 Plot of the control input . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 Plot of the tracking error in the system output . . . . . . . . . . . . . . . 79 Plot of the bounded zero dynamics of the system . . . . . . . . . . . . . 80 Response of System States. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 Phase Portrait of the System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 Evolution of Sliding Surfaces and Control Inputs. . . . . . . . . . . . . . 92 A representative third order system with simplified Rd Ud − sliding sector with two stable modes . . . . . . . . . . . . . . . . . . . . . . . . 96 5.9 A representative third order system with simplified Rd Ud − sliding sector with one stable mode . . . . . . . . . . . . . . . . . . . . . . . . 97 5.10 Comparative Plots of the State, Input and Lyapunov function a (µ) of the system with and without initial state estimation error. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 5.11 Comparative Linear controller responses of the State, Input and Lyapunov function a (µ) of the system with and without initial state estimation error. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 6.1 6.2 6.3 6.4

Plot for various possible solutions . . . . . . . . . . . . . . . . . . . . . . . . . . 107 Response of System States in Discrete-time Terminal Sliding Mode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 Discrete-time Terminal Sliding Mode Control Input . . . . . . . . . . . 111 Response of system outputs with DTSM in linear system using control law (6.24) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

List of Figures

6.5 6.6 6.7 6.8 6.9 6.10 6.11 6.12 6.13 7.1 7.2 7.3 7.4 7.5 7.6 7.7 7.8 7.9 7.10 7.11 7.12 7.13 7.14 7.15

XV

Profile of control inputs with DTSM in linear system using control law (6.24) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 Profile of sliding functions with DTSM in linear system using control law (6.24) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 Response of system outputs with DTSM in linear system using control law (6.25) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 Profile of control inputs with DTSM in linear system using control law (6.25) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 Profile of sliding functions with DTSM in linear system using control law (6.25) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 State trajectories of MROF based DTSM controlled feedback linearizable nonlinear system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 Output samples of MROF based DTSM controlled feedback linearizable nonlinear system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 Control input profile of MROF based DTSM controlled feedback linearizable nonlinear system . . . . . . . . . . . . . . . . . . . . . . 120 Sliding function plot of MROF based DTSM controlled feedback linearizable nonlinear system . . . . . . . . . . . . . . . . . . . . . . 120 Response of system states (a)Direct axis current (b) Quadrature axis current (c) Angular velocity (d) Angular position . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 Response of winding currents and voltages (a) Current in winding A (b) Current in winding B (c) Voltage in winding A (d) Voltage in winding B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 Response of the switching planes (a)switching plane 1 (b) switching plane 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 Single Machine Infinite Bus System . . . . . . . . . . . . . . . . . . . . . . . . . 128 Block diagram of Single Machine Infinite Bus System . . . . . . . . . 129 Block diagram of a Power System with PSS . . . . . . . . . . . . . . . . . . 130 Response of Slip to MOF-SMC PSS when fault is applied at 1 sec . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 Response of delta to MOF-SMC PSS when fault is applied at 1 sec . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 Profile of PSS output to MOF-SMC PSS when fault is applied at 1 sec . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132 A flexible cantilever beam / smart beam . . . . . . . . . . . . . . . . . . . . 133 A smart structure beam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 System Response to an impulse excitation . . . . . . . . . . . . . . . . . . . 136 Generated control input profile for an impulse excitation . . . . . . 136 System Response to a sinusoidal disturbance . . . . . . . . . . . . . . . . . 137 Generated control input profile for a sinusoidal disturbance . . . . 137

Nomencla2w7e

Abbreviations DSM Discrete-time Sliding Mode DSMC Discrete-time Sliding Mode Control DTSM Discrete-time Terminal Sliding Mode LTI Linear Time-Invariant MIMO Multi Input Multi Output MROF Multirate Output Feedback QSM Quasi-Sliding Mode QSMB Quasi-Sliding Mode Band QSMC Quasi-Sliding Mode Control RP Representative Point SISO Single Input single output SMC Sliding Mode Control TSM Terminal Sliding Mode VSS Variable Structure System List of Symbols General notation for the matrix transpose operation (•)T R The field of real numbers The vector space of vectors of length ν with real entries Rn ˜ η(µ) Disturbance vector in an LTI system State matrix in continuous-time model of time delay LTI system A0 Delayed-State matrix in continuous-time model of time delay LTI sysA1 tem Input matrix in continuous-time model of time delay LTI system B0 Delayed-Input matrix in continuous-time model of time delay LTI sysB1 tem C Output matrix of discrete-time LTI model Sliding function parameter eT Terminal sliding function parameter eTt Error between system and reference state vectors gx State matrix in discrete-time model of LTI system with time delay G0

Nomenclature

G1 I0 I1 Jd (•) µu µx µy ∂ P φ q θ v X Xpre  a

k

x f H Hτ d

S

Sτ W zu zx zy

XVIII

Delayed-state matrix in discrete-time model of LTI system with time delay Input matrix in discrete-time model of LTI system with time delay Delayed-input matrix in discrete-time model of LTI system with time delay Identity function Ratio of input delay to input sampling time Ratio of state delay to input sampling time Ratio of output delay to input sampling time Number of inputs of an LTI system model Ratio of input and output sampling rates in a multirate system Number of states of an LTI system model Number of outputs of an LTI system model Reference signal Sliding function General notation for a φ × φ transformation matrix Reference preview time of tracking controller for nonminumum phase system Control input of a dynamical system Lyapunov function System state of a dynamical system System output of a dynamical system Multirate sampled system output Shuffle product operator Width of the quasi-sliding mode band Input matrix of discrete-time LTI model sampled at interval of ∆ sec Input matrix of discrete-time LTI model sampled at interval of z sec Sliding sector in a discrete-time LTI system System state matrix of discrete-time LTI model sampled at interval of ∆ sec System state matrix of discrete-time LTI model sampled at interval of z sec Nonlinear system representation in continuous time Delay in input in LTI system with input delay Delay in state in LTI system with state delay Delay in output in LTI system with output delay

1 In27odwc2ion

In this chapter, a brief introduction to the concepts of sliding mode control and multirate output feedback would be given. The chapter would describe and distinguish the concepts of variable structure control and sliding mode control and would clarify the concept of discrete-time sliding mode and discrete-time sliding mode control. The chapter would also explain the concept of multirate output sampling and the relationship between the system states and the output samples for linear time invariant systems.

1?1 Sliding Mode Con27ol 1.1.1 Variable Structure Systems The underlying idea of sliding mode control is variable structure control. In variable structure control, the structure of the control input is changed in accordance to the system states. This, in turn would result in dynamics that were not realizable with any of the constituent control structures working alone. This property of variable structure control may be illustrated by the following examples [73]. Example 1 Consider the second order system ¨ = −Σ having the structures defined by Σ = ,12 and Σ = ,22 , with ,12 κ ,22 . The resultant dynamics in both cases would be concentric ellipses in the phase

B. Bandyopadhyay, S. Janardhanan: Discrete-time Sliding Mode Contr., LNCIS 323, pp. 1–16, 2006.

© Springer-Verlag Berlin Heidelberg 2006

2

1 Introduction

plane as given in Figs. (1.1(a) and 1.1(b)). Hence, the system is not asymptotically stable. However, the system can be made to be asymptotically stable by changing the dynamics at the co-ordinate axes with the switching logic Σ=

,12 if ,22 if

˙ κ0 ˙ ≤0

It can be seen in this example that even though none of the systems were asymptotically stable, the application of VSS has rendered the composite system to be asymptotically stable as illustrated in Fig. (1.1(c)) Example 2 Now consider the second order unstable system represented by the dynamics ¨−˙ +Σ =0

(1.1)

with the feedback being either of Σ = ±,ε , κ 0. Both the systems are unstable. The only stable motion present in the system is along one of the eigenvector corresponding to Σ = −,. If switching occurs along this line and along = 0 with the switching law, Σ=

,ε if v κ 0 εv = e + ˙ −,ε if v ≤ 0

 e = −ω = − + 2

2 +, 4

the resulting variable stucture system (VSS) would now be asymptotically stable as seen in Fig. (1.2(c)), in spite of both the constituent systems being unstable(Figs. (1.2(a) and 1.2(b))). Thus, it is seen that the concept of variable structure systems may be applied to design a controller of varying structure to bring out dynamics in the system that are not realizable by the use of any single control structure. 1.1.2 Continuous-time Sliding Mode In Example 2 in Section 1.1.1, the parameter e is chosen as equal to the stable eigenvalue of one of the constituent systems. This led to a switching behaviour that rendered the system stable. However, in this case, the performance of the system is very sensitive to system parameters. Now, consider the case wherein 2

the system (1.1) is switched along the line e2 + ˙ = 0ε 0 p e2 p − 2ξ + ξ4 + ,. In this case, the resultant dynamics would be different. The system states would now approach the line e2 + ˙ = 0 on the phase plane and would

1.1 Sliding Mode Control

3

.

X I

X

.

X

II

X

(a)

(b)

.

X II

I

X

I

II

(c) Fig. 1.1. Asymptotic Stability of VSS with Lyapunov Stable Systems

stay on it. The state would then asymptotically approach the origin. This is illustrated in Fig. (1.3).

4

1 Introduction .

.

X

X

II

I

X

X

(b)

(a) .

I

X

II

X

I

II

(c) Fig. 1.2. Asymptotic Stability of VSS with Unstable Constituent Systems

It can be seen in the figure that in the vicinity of the line v = e2 + ˙ = 0, the vector i is in a direction towards v = 0. Mathematically, this can be represented as [73] lim v˙ p 0ε lim− v˙ κ 0

(1.2)

vv˙ p 0

(1.3)

s→0+

or more concisely as

s→0

This is termed as the reaching condition. A more stringent criterion is the h-reachability condition [20] which is given as vv˙ p −h|v|ε h κ 0

1.1 Sliding Mode Control

5

.

I

X

II

c2x+dx/dt=0

X

II

I

Fig. 1.3. An illustration of sliding mode in variable structure systems

Hence, any system trajectory close to the line would converge to the line and thereafter remain on the line v = 0. It can be said that the trajectory slides along the line. This motion is termed as sliding mode. Further, when the states are confined to v = 0, system dynamical equation would depend solely on e and not on the value of  or Σ . Hence, the dynamics of the system would be robust against parameter variations. Formally, sliding mode may be defined as follows : Definition 1.1 (Sliding Mode). Sliding motion or sliding mode may be defined as the evolution of the state trajectory of a system confined to a specified non-trivial sub-manifold of the state space with stable dynamics. 1.1.3 Continuous-time Sliding Mode Control Let us now consider a input affine dynamical system ˙ = i ( ε y) + j( ε y)(y)

(1.4)

where i and j are φ-dimensional continuous functions in ε  and y. is an φ - dimensional column vector and  is a ∂ - dimensional function. Let us also assume the existence of a ∂ - vector of φ − 1 dimensional manifolds represented as v( ε y) = v1 ( ε y) v2 ( ε y) · · · vm ( ε y)

T

=0

(1.5)

where, vi ( ε y) are continuous and differentiable functions of and y. It is assumed, without loss of generality, that the system confined to each of the sliding manifolds, vi = 0ε < = 1ε 2ε · · · ε ∂ is stable, atleast in the lyapunov sense.

6

1 Introduction

The aim of the control input (y) is to bring the system states onto the intersection of the chosen manifolds in a finite time and then move the system states towards the origin of the state space. Due to the multiplicity of the sliding manifolds in a multi-input system, this approach to the intersection may occur in one of the following manners. • The system representative point (RP) can approach any one of the manifolds first and slide along it till it approaches the intersection of the first and a second manifold. This trend would continue until it approaches the (φ − ∂)-dimensional intersection of all manifolds denoted as v = 0(refer Fig. (1.4)). Again, the order of approach of the sliding manifolds may or may not be fixed. The switching scheme being termed as fixed order switching scheme or free order switching scheme accordingly. • Alternatively, the RP can approach and ’hit’ the (φ−∂)-dimensional intersection directly, the path crossing any of the individual sliding manifolds but generally not staying in any one of them.(refer Fig. (1.5)). This type of a switching scheme is called as eventual sliding mode switching scheme. A detailed study of the various possible ways of approach is dealt in [31].

X2

X1 X3

Fig. 1.4. Order based switching scheme in multi-input systems

In both the aforementioned cases and also in case of sliding mode control of single input systems the state trajectory is required to reach on to the sliding manifold within a finite amount of time for sliding motion to start and exist. Unlike asymptotic convergence, this added requirement of finite time puts on additional conditions on the sliding mode controller.

7

1.1 Sliding Mode Control

X3

X2

X1

Fig. 1.5. Eventual sliding mode switching scheme for multi-input systems

Discontinuity of Control Signal Analysing Eqn. (1.2), it may be seen that the function v˙ would have a discontinuity at v = 0. Using Eqn. (1.4), the value of v˙ may be obtained as v˙ =

rv (i ( ε y) + j( ε y)(y)) ρ r

Due to the continuous nature of iε j and the differentiable nature of v, it can be concluded that in order for v˙ to be discontinuous, the control input (y) needs to be discontinuous. Thus, the control input (y) that could bring sliding mode in a system would be of the form (y) =

+ ( ε y) with v( ε y) κ 0ε − ( ε y) with v( ε y) p 0ρ

(1.6)

where + and − are continuous functions with + = − . Discontinuous Equations and Continuation Method As a control engineer, one would not only be interested in the behaviour of the system outside the sliding manifold but also on the manifold. The control structure suggested in (1.6) defines the state velocites as ˙=

i + = i ( ) + j( )+ with v( ε y) κ 0ε i − = i ( ) + j( )− with v( ε y) p 0ρ

(1.7)

But, the motion on the sliding surface is not clearly defined. The equations do not give a clear indication of the state velocity for v( ε y) = 0. A method to resolved this problem, termed as continuation method [22] has been elaborated in [74]

8

1 Introduction

x

2

C

Vector − f in point C 0

Vector f , phase velocity in sliding mode Vector + f in point C

Discontinuity Surface

x1

Fig. 1.6. Illustration depicting the state velocity vector for a system with discontinuous right-hand side

At each point on the discontinuity surface, the velocity vector determining the solution belongs to a minimal convex closed set containing all the values of i ( ) + j( ) when covers the entire f-neighbourhood of the point under consideration(with f tending to zero). For continuation of a solution on the sliding manifold, Filippov’s method [22] gives the following result : To determine the velocity of i 0 in sliding mode, at each point on the sliding manifold, the velocity i − and i + should be plotted and their ends connected. In this way, a minimal convex hull is obtained. Since, by definition sliding mode occurs on the discontinuity surface, the state velocity vector of that motion lies on a plane tangential to this surface and therefore its end is the intersection point of the tangential plane and the straight line connecting the ends of vectors i + and i − (refer Fig. (1.6)). Thus on the sliding manifold, the velocity vector is of the form ˙ = i 0 ( ε y)ε i 0 = .i + + (1 − .) i − ε 0 ≤ . ≤ 1ρ Calculating . using the fact that during sliding motion v˙ = 0, gives the state velocity as

1.1 Sliding Mode Control

˙=

rv − i r rv − (i − i + ) r

i+ −

rv + i r rv − (i − i + ) r

i −ρ

9

(1.8)

1.1.4 Equivalent Control and the Reaching Law Approach Equivalent control constitutes an equivalent input which, when exciting the system, produces the motion of the system on the sliding surface whenever the system is on the surface [14]. Suppose, the system trajectory intersects the sliding surface at time 1 , and a sliding mode exists. The existence of sliding mode implies that for all n 1 , the system trajectory would satisfy v( ( )) = 0 and hence v( ˙ (y)) = 0. Thus, the equivalent control that maintains the sliding mode is the input eq satisfying v˙ =

rv rv rv + i ( ε y) + j( ε y)eq = 0ρ ry r r

Assuming that the matrix may be calculated as eq = −

(1.9)

rv j( ε y) is non-singular, the equivalent control r

rv j( ε y) r

−1

rv rv + i ( ε y) ρ ry r

(1.10)

However, the equivalent control is only effective once the state trajectory hits the sliding manifold. A formal control algorithm, possibly variable structure, has to be formulated to bring the system states on to the sliding manifold. One of the approaches of sliding mode controller design in a general dynamical system is the so called reaching law approach. In the reaching law approach, the dynamics of the switching function are directly expressed. Then the sliding function dynamics can be expressed with a general structure v˙ = −Tis (v) − Lsgn(v)

(1.11)

where, T and L are positive definite matrices of appropriate dimensions and is (v) is such that is,i (v)vi κ 0ε ∀vi = 0. Some of the possible dynamics are shown below [31]: 1. The constant rate reaching law : v˙ = −Lsgn(v)ρ

(1.12)

2. The constant plus proportional rate reaching law : v˙ = −Tv − Lsgn(v)ρ

(1.13)

10

1 Introduction

3. The power-rate reaching law: v˙ i = −µi |vi |α، ε 0 p ,i p 1ρ

(1.14)

A control input can now be constructed using (1.11) for the system (1.4) (y) = −

rv j( ε y) r

−1

rv rv + i ( ε y) + Tis (v) + Lsgn(v) ρ (1.15) ry r

It is worthy to note at this juncture that the reaching law based control in (1.15) would become the equivalent control (1.10) when the system state is on the sliding manifold. 1.1.5 Discrete-time Sliding Mode Control In the recent years, research has been carried out in the field of discrete-time sliding mode control(DSMC). DSMC is the discrete-time counterpart of the continuous-time sliding mode control discussed in the earlier sections. In the case of discrete-time sliding mode control, the measurement and control signal application are performed only at after regular intervals of time and the control signal is held constants in between these instants. A discretetime extension of the reaching law approach [31] was proposed by Gao et al [27]. The reaching law in this case would be of the form v(µ + 1) − v(µ) = −Tz v(µ) − Lz sgn(v(µ))ρ where z is the sampling interval of the discrete-time system. A reaching law based discrete-time control law has been derived in [27], for an LTI system (µ + 1) = Sτ (µ) + Hτ (µ)

(1.16)

and a stable sliding surface v(µ) = eT (µ) = 0, to be of the form (µ) = − eT Hτ

−1

eT Sτ − eT + Tz eT

(µ) + >z sgn(v(µ))

(1.17)

However, the control law (1.17) can only bring a quasi-sliding modr and −1 would introduce a chattering of amplitude (2J − Tz ) >z into the system. An important property of discrete-time systems is that the control signal is computed and varied only at the sampling instants. This makes discrete-time control inherently discontinuous. Hence, unlike the case of continuous sliding mode control, a discrete-time sliding mode control law need not necessarily be of variable structure or need to have explicit discontinuity. Control laws based on this concept have also been developed [4, 7]. These control algorithms try to satisfy the condition v(µ + <) = 0 for some < ≥ 1ρ

11

1.2 Multirate Output Feedback

The resultant control law, in the case of < = 1, using the algorithm proposed in [4] would be of the form (µ) = − eT Hτ

−1

eT Sτ

(µ)ρ

(1.18)

It can be seen here that the control is now no longer of variable structure. However, it is a sliding mode control technique. Hence, it can be seen that though sliding mode control started from variable structure control, during the course of its development, the concept of sliding mode became independent of VSC.

1?v Mwl2i7a2e Ow2pw2 Feedback Multirate Output Feedback (MROF) is the concept of sampling the control input and sensor output of a system at different rates. It was found that multirate output feedback can guarantee closed loop stability, a feature not assured by static output feedback [69] while retaining the structural simplicity of static output feedback. Much research has been performed in this field [9, 29, 44, 45, 52]. In multirate output feedback, the control input [9, 45] or the sensor output [29] is sampled at a faster rate than the other. In this book, the term multirate output feedback is used to refer the situation wherein the system output is sampled at a faster rate as compared to the control input. It was found that state feedback based control laws of any structure may be realized by the use of multirate output feedback, by representing the system states in terms of the past control inputs and multirate sampled system output [3, 36, 62]. The book deals with the application of multirate output feedback concept to discrete-time output feedback controller design. Further references to multirate output feedback imply that the system output is sampled at a faster rate as compared to the control input. Consider the ∂-input, q-output, φ-th order continuous time LTI system ˙ = A + Bε =C ρ

(1.19)

Let the system sampled at a sampling interval of z sec be represented as ((µ + 1)z ) = Sτ (µz ) + Hτ (µz )ε (µz ) = C (µz )ρ

(1.20) (1.21)

Let the control input  be applied with a sampling interval of z sec and the system output is sampled with a faster sampling period of ∆ = z xP sec, where P is an integer greater than or equal to the observability index of the system. Let the system sampled at the ∆ interval be represented using the triplet (Sε Hε C). It is assumed, without loss of generality that the pair (Sτ ε Hτ ) is controllable and the pair (Sε C) is observable

12

1 Introduction

Definition 1.2 (Observability Index). The observability index of a system (Sε Hε C) is the smallest positive integer π such that     C C  CS    CS     = Rank  . Rank  .     ..    .. ν ν−1 CS CS Now consider the ∆ system ((µ + 1)∆) = S (µ∆) + H (µ∆)ε (µ∆) = C (µ∆)ρ Hence,

(µz + ∆) = S (µz ) + H (µz )

Using the fact that  is unchanged in the interval z ≤ y p (µ + 1)z , the z system state dynamics may be constructed from the ∆ system dynamics in the following manner. (µz + ∆) = S (µz ) + H (µz )ε (µz + 2∆) = S (µz + ∆) + H (µz ) = S2 (µz ) + (S + J) H (µz )ε .. .. . . ((µ + 1)z − ∆) = SN −1 (µz ) +

N −2

Si H (µz )ε

i=0

((µ + 1)z ) = SN (µz ) +

N −1

Si H (µz )ρ

(1.22)

i=0

Comparing the state equations (1.20) and (1.22), it can be inferred that SN = S τ N −1

S i H = Hτ ρ

i=0

Further, if the past P multirate-sampled system outputs are represented as   ((µ − 1)z )  ((µ − 1)z + ∆)    (1.23)  k = .  ..  (µz − ∆)

and the notation µ is used to represent µz , for brevity, then the multirate output sampled system dynamics can be represented as

1.2 Multirate Output Feedback

(µ + 1) = Sτ (µ) + Hτ (µ)ε k+1 = C0 (µ) + D0 (µ)ε where,





C  CS  2  C0 =  CS  .. .

CSN −1



0   CH      ε D0 =  C (S + J) H   ..  . C

N −2 i=0

13

(1.24) (1.25)     ρ  

(1.26)

Si H

1.2.1 Multirate Output to State Relationship The state (µ) can be expressed in terms of system outputs input (µ) using (1.25) as [36] (µ) = C0T C0

−1

C0T (

k+1

− D0 (µ))

k+1

and control (1.27)

Remark 1.3 (Invertibility of C0T C0 ). The C0 , of dimension qP × P , would be of full rank (of rank φ) due to the assumption of the observability of the system and P ≥ π. Thus, the matrix C0T C0 would be a φ × φ matrix of rank φ. Hence, it would be invertible. Substituting the value of (µ + 1) can be derived as

(µ) from (1.27) in (1.24), and expression for

(µ + 1) = My

k+1

+ Mu (µ)ε

where My = Sτ C0T C0

−1

C0T ε

−1 C0T C0

Mu = Hτ − S τ

(1.28) C0T D0 ε

or equivalently, the state (µ) can be expressed using the past multirate output samples k and the immediate past control input (µ − 1) as (µ) = My

k

+ Mu (µ − 1)ρ

(1.29)

Thus, using the relation (1.29), any control of the form (µ) = iu ( (µ)) can be realized using past input and output samples as (µ) = iu (My

k

+ Mu (µ − 1)) ρ

An illustration of this multirate control philosophy is given in Fig. (1.7).

14

1 Introduction u(k)

τ

Unit Delay

u(k−1)

System

τ/Ν y(t)

State

[]

y

Computation

k

Output Stack

Controller

Fig. 1.7. Illustration of Multirate Output Feedback based Control Philosophy

1.2.2 Advantage of Multirate Output Sampling over Discrete-time Observer Multirate output feedback does not make use of the ’present’ outputs or input, thus providing for a time-delay required for control law implementation in practice. Further, it should be noted here that the multirate output feedback technique is different from an observer based technique one important aspect. The system states are computed exactly after just one sampling interval as opposed to a theoretically infinite time taken by an observer. In case of multirate output feedback, the error between the computed state and the actual state of the system goes to zero once a multirate sampled output measurement is available, i.e, after ‫ ܈‬sec. However, in case of an observer, the error between the estimated and the actual system state decreases asymptotically, but generally goes to zero only as time approaches infinity. Even the best designed Luenberger observer can assure zero error only after s sampling instants. This is illustrated in the following numerical example. Consider the discrete-time system (µ + 1) =

0ρ4 1 −0ρ7 0ρ7

(µ) = 1 0ρ5

(µ) +

0 (µ) 1

(µ)

For this system a discrete-time Luenberger observer is designed as ˆ(µ + 1) =

0ρ4 1 0 0ρ0906 ˆ(µ) + (µ) + −0ρ7 0ρ7 1 −0ρ5811

(µ) − 1 0ρ5 ˆ(µ)

15

1.3 Motivation for MROF-DSMC

So that the error dynamics are stable. For the same system, the state may be computed from multirate output samples as mof (µ)

=

−0ρ2279 1ρ0596 −1ρ5241 1ρ3907

k

+

1.6

−0ρ0923 (µ − 1) 0ρ8789

Using Multirate Output Feedback Using Luenberger Observer

1.4

Error Norm (|x−x

est

|)

1.2

1

0.8

0.6

0.4

0.2

0

−0.2

0

5

10

15

Time Samples

20

25

30

Fig. 1.8. Comparative plots of error norms in state computation using multirate sampling and Luenberger observer

The error norms i.e., (µ) − ˆ(µ) and (µ) − mof (µ) , are plotted in Fig. (1.8). It can be clearly seen here that the multirate output sampling based state computation is much more efficient than a Luenberger observer. It converges in only one sampling interval, which is not possible with a Luenberger observer.

1?u Mo2ixa2ion fo7 Mwl2i7a2e Ow2pw2 Feedback ba6ed Di6c7e2e-2ime Sliding Mode Con27ol The use of digital computers and samplers in the control circuitry in the recent years, has made the use of discrete-time system representation more justifiable for controller design than continuous-time representation. In the recent years,

16

1 Introduction

few researchers have worked on discrete-time sliding mode controller design [7,24,27]. In discrete-time sliding mode control, the control input is calculated once in every sampling interval and is held constant during this period. Due to the finite sampling frequency, it may happen in discrete-time sliding mode, that the system states are unable to move along the sliding surface. They may move about the surface, thus giving a sliding-like mode or Quasi-sliding motion (QSM). The aforementioned sliding mode control strategies are based on state feedback. Since all the systems states may not be available for measurement in most systems, such control strategies have a problem from the implementation point of view. This prompted the development of output feedback sliding mode control strategies [18,48,83]. However, these control strategies also have certain shortcomings. Sliding mode control strategies based on static output feedback may not exist for control for all controllable and observable linear systems, whereas dynamic controllers would increase the complexity of the system. In the following chapters, discrete-time sliding mode control strategies based on multirate output feedback [29] are discussed. In this technique, the system output is sampled at a rate faster than the control input. As seen in Section 1.2, any state feedback based control algorithm can be converted to an output feedback based control algorithm by the use of multirate output sampling. Consequently, the control algorithm is based on output feedback and at the same-time, is applicable to all controllable and observable systems. Thus, it has the advantages of both state feedback and output feedback control philosophies.

v Syi2ching Fwnc2ion ba6ed Mwl2i7a2e Ow2pw2 Feedback Sliding Mode Con27ol

v?1 Qwa6i-Sliding Mode Con27ol in De2e7mini62ic S062em6 In [27], a reaching law approach was proposed for the quasi-sliding mode control of discrete-time LTI systems of form (1.20). The reaching law approach aims at satisfying the condition v(µ + 1) − v(µ) = −τz v(µ) − >z sgn(v(µ))ε

(2.1)

where, v(µ) = eT (µ) is the sliding function, τ and > are controller parameters satisfying the relationship 1 − τz κ 0ε > κ 0. Remark 2.1 (The Quasi Sliding Mode Band). The quasi-sliding motion has been defined in [27] as a motion in which the system states approaches monotonically to the vicinity of the sliding surface v(µ) = 0 and on reaching a band termed as the quasi-sliding mode band (QSMB), it moves about the sliding surface in a zigzagging motion, crossing and re-crossing the sliding surface in subsequent time steps, with the magnitude of the zig-zag motion being within the band in subsequent time steps. The magnitude, f, of the QSMB can be computed by solving (2.1) for v(µ) = f and v(µ + 1) = −v(µ). [6] −2f = −τz f − >zρ Thus giving f=

>z ρ 2 − τz

The state feedback based control law that satisfies the reaching condition (2.1) can be derived to be [27]

B. Bandyopadhyay, S. Janardhanan: Discrete-time Sliding Mode Contr., LNCIS 323, pp. 17–25, 2006. © Springer-Verlag Berlin Heidelberg 2006

2 Switching function based MROF-SMC

18

(µ) = − eT Hτ

−1

eT Sτ − eT + τz eT

(µ) + >z sgn(eT (µ)) ρ

(2.2)

As mentioned in Section 1.2, any state feedback based control algorithm may be converted to an output feedback based control algorithm by the use of multirate output feedback concept. Thus, by substituting for (µ) in (2.2) from (1.29), the multirate output feedback based quasi-sliding mode control law can be derived to be (µ) = Fy

k

+ Fu (µ − 1)

− e T Hτ

−1

(2.3)

>z sgn eT My

k

+ eT Mu (µ − 1) ε

where Fy = − e T Hτ

−1

eT Sτ − eT + τz eT My ε

−1

eT Sτ − eT + τz eT Mu ε Fu = − e T Hτ Mu = Hτ − M y D 0 ε My = Sτ C0T C0

−1

C0T ρ

2.1.1 Illustrative Example Consider the second order LTI system sampled at an interval of z = 0ρ1 sec as (µ + 1) =

01 −1 1

(µ) = 1 0

(µ) +

0 (µ)ε 1

(2.4)

(µ)ρ

A stable sliding surface is designed as v(µ) = eT (µ) = 0ε eT = −0ρ8 1 . The observability index of the system is 2. Hence, choosing P = 2 and the controller parameters as τ = 1ε > = 0ρ1, the multirate output feedback based quasi-sliding mode controller can be derived to be (µ) =

−1ρ68 1ρ7

T

−0ρ01sgn

k

+ 0ρ96(µ − 1) −1ρ2 0ρ35

T k

+ 1ρ05(µ − 1) ρ

The system responses are shown in Fig. (2.1). The width of the quasisliding mode band with the multirate output feedback sliding mode controller (2.3) is the same as with the state feedback based controller (2.2). For the example considered here, the quasi-sliding mode band width would turn out τ = 0ρ0053. to be f = 2−qτ

19

System States

1

Sliding Function (s(k))

2.2 MROF-QSMC in Uncertain Systems x 1 x2

0.5 0

0

−0.2

−0.5

−0.4

−1

−0.6

−1.5 −2

0.2

0

1

2

3

Time (sec)

4

5

−0.8

0

2

3

Time (sec)

4

5

(b)

(a)

0.5

0

0

−0.5

−0.5

x

2

0.5

Control input

1

−1

−1

−1.5

−1.5

−2

0

1

2

3

Time (sec)

4

5

−2 −2

−1

(c)

x1

0

1

(d)

Fig. 2.1. Quasi-Sliding Mode Control based on Multirate Output Feedback : (a) State responses (b) Sliding function (c) Control input (d) Phase portrait

Remark 2.2 (Generation of Initial Control). The initial control (0) cannot be derived from multirate output feedback based algorithms as the system output is not known before y = 0. This problem can be circumvented by generating the initial control alone based on an assumed initial state 0 by using the control law (2.2)

v?v Mwl2i7a2e Ow2pw2 Feedback ba6ed Qwa6i-Sliding Mode Con27ol fo7 Unce72ain S062em6 Consider the system (1.20), but with an added uncertainty in the state equation. ˜ (µ + 1) = Sτ (µ) + Hτ (µ) + Dτ η(µ)ε

(2.5)

(µ) = C (µ)ρ ˜ is the disturbance vector representing the combined effect of unmodeled η(µ) dynamics and external disturbances affecting the system. It is assumed here

20

2 Switching function based MROF-SMC

˜ that η(µ) is bounded and is a disturbance signal that satisfies the matching condition as imposed in [19]. Let the system representation for a sampling interval of ∆ sec, assum˜ remains unchanged during each z ing that the disturbance component η(µ) interval, be ˜ (µz + (m + 1)∆) = S (µz + m∆) + H (µz ) + Dη(µ)

(2.6)

(µz + m∆) = C (µz + m∆) Using the matrix relationships similar to (1.22), the value of D may be computed in terms of the z -system parameters as −1

N −1

S

D=

i

Dτ ρ

(2.7)

i=0

2.2.1 Multirate Sampled Output to State Relationship in Uncertain Systems Proceeding on similar lines as with the case without uncertainty in Section 1.2, the multirate output samples of the uncertain system can be related to the system states, input and disturbance vectors as [40] k+1

where

˜ = C0 (µ) + D0 (µ) + Cd η(µ) 

0  CD  Cd =  .. . C

(2.8)



N −2 i=0

  ρ  Si D

From (2.8) and (2.5), the system state (µ) can be represented as a function of the past multirate output samples, past control and disturbance signals as (µ) = My where

k

˜ − 1)γ + Mu (µ − 1) + Md h(.

(2.9)

Nd = Dτ − Ny Cd u

2.2.2 Review of State Feedback based QSM control of Uncertain Systems ˜ is bounded, it is Consider the system (2.5). Since, it is assumed that η(µ) T ˜ correct to assume that η(µ) = e Dτ η(µ) will also be bounded. Let the bounds be ηl ≤ η(µ) ≤ ηu ρ

21

2.2 MROF-QSMC in Uncertain Systems

Let us also define the mean and spread of η(µ) as ηl + ηu ε 2 ηu − ηl ρ fd = 2

(2.10)

η0 =

(2.11)

A reaching law approach was proposed in [27] for the quasi-sliding mode control of systems of the form (2.5). The control is made to satisfy the reaching condition v(µ + 1) − v(µ) = −τz v(µ) − >z sgn(v(µ)) + η(µ) − η0 − fd sgn(v(µ))ρ (2.12) Thus, the sign of the increment in v(µ) is made to be always in opposite sense to that of v(µ), irrespective of the value of the disturbance factor η(µ). A control law that satisfies the reaching law (2.12) can be computed to be (µ) = − eT Hτ

−1

− e T Hτ

−1

eT Sτ − eT + τz eT

(µ)

(2.13)

(η0 + (fd + >z ) sgn(v(µ))) ρ

The bound on the Quasi-sliding mode band In order to satisfy the crossing-recrossing condition of quasi-sliding mode, sgn(v(µ + 2)) = −sgn(v(µ + 1)) = sgn(v(µ))ρ On the other hand, taking into account the reaching law (2.12), we have v(µ + 2) = (1 − τz )v(µ + 1) − >z sgn(v(µ + 1)) +η(µ + 1) − η0 − fd sgn(v(µ + 1)) 2

= sgn(v(µ)) (1 − τz ) |v(µ)| + τz >z + fd τz +(1 − τz ) (η(µ) − η0 ) + (η(µ + 1) − η0 ) ρ

(2.14)

˜ Due to be boundedness of η(µ), the value of |η(m) − η0 | ≤ fd . Hence, for the assurance of quasi-sliding mode, i.e, for sgn(v(µ+2) = sgn(v(µ)) for arbitrarily small magnitude of v(µ), the following condition has to be satisfied. [6] τz (>z + fd ) κ (2 − τz )fd ρ Hence, imposing the constraint on the controller parameters as τz >z κ fd ρ 2(1 − τz )

(2.15)

2 Switching function based MROF-SMC

22

Further, using the fact that while the system is in sliding mode sgn(v(µ+1)) = −sgn(v(µ))ε v(µ) = 0, the bound on the quasi sliding mode band may be calculated from (2.12) as f ≤ 2fd + >zρ The state feedback based control law (2.13) can be converted to output feedback based control law by substituting for (µ) from (2.9). However, due ˜ to the presence of the uncertainty Md η(µ), the control would not be implementable. Hence, for achieving a multirate output feedback based control algorithm, the control law has to redesigned from a modified reaching law suited for output feedback. 2.2.3 Output Feedback Sliding Mode Control Algorithm based on New Reaching Law Consider the reaching law v(µ + 1) − v(µ) = −τz v(µ) − >z sgn(v(µ))

(2.16)

+j(µ − 1) + η(µ) − η0 − j0 − (fd + fg )sgn(v(µ)) where, ˜ = j(µ) ≤ ju ε jl ≤ eT Sτ − eT + τz eT Md η(µ) jl + ju j0 = ε 2 ju − jl ρ fg = 2 Now, a state feedback based control law satisfying the reaching law (2.16) can be formulated as (µ) = − eT Hτ

eT Sτ − eT + τz eT

(µ)

(2.17)

T

− e Hτ (η0 + j0 + (fd + fg )sgn(v(µ)) − j(µ − 1)) ρ The control law (2.17) has an uncertain component j(µ − 1). However, if the multirate output feedback control is attempted by replacing the state vector (µ) from (2.9), the uncertain components in (2.9) and (2.17) cancel out to give a multirate output feedback based quasi sliding mode control algorithm that does not have uncertainty components. (µ) = Fy −

k + Fu (µ − 1) eT Hτ (η0 + j0 +

(2.18) (fd + fg + >z )sgn(v(µ))) ρ

2.2 MROF-QSMC in Uncertain Systems

23

Quasi Sliding Mode Bound in Multirate Output Feedback Control Following the same technique employed for the state feedback based control algorithm, the width of the quasi-sliding mode bound may be computed to be fy ≤ 2 (fd + fg ) + >z

(2.19)

with the condition on the controller parameters τz >z κ (fd + fg ) 2(1 − τz ) to ensure quasi sliding motion. Thus, it increases the bound on the quasisliding mode band in comparison to the state feedback based quasi-sliding mode control algorithm. Additional Constraint on Controller Parameters The reason for the control algorithm in (2.13) not being exactly realized, is the presence of the uncertainty term Md i (µ) in the output to state relationship (2.9). Thus, the state and hence the sliding function v(µ) cannot be exactly computed. However, the sign of v(µ) is essential for generation of the control (2.18). Since, only the sign is essential, but not the exact value, sgn(v(µ)) is replaced with sgn(¯ v(µ)), where v¯(µ) is computed using the formula v¯(µ) = eT (My

k

+ Mu (µ − 1)) + ν0 ε

(2.20)

where, ˜ ≤ νu ε νl ≤ eT Md η(µ) νu + ν l ε ν0 = 2 νu − ν l fl = ρ 2 v(µ)| κ fl . However, when v(µ)) whenever |¯ The value of sgn(v(µ)) = sgn(¯ the value of |¯ v(µ)| p fl , the sign of v(µ) cannot be determined accurately as v(µ) is in the range of v¯(µ) ± fl . Hence, in order to assure quasi sliding mode band in spite of this uncertainty, the width of the quasi-sliding mode band should be such that it encompasses this ambiguous band of |v(µ)| ≤ 2fl . Thus, it gives an additional constraint on the controller parameters as 2 (fd + fg ) + >z κ 2fl ρ

(2.21)

24

2 Switching function based MROF-SMC

2.2.4 Illustrative Example Consider the system in (2.4) with an additional uncertainty. (µ + 1) =

01 −1 1 +

(µ) +

0 (µ) 1

0 sin(0ρ1µ)ε 0ρ01

(µ) = 1 0

(2.22)

(µ)ρ

Choosing the sliding surface v(µ) = −0ρ8 1 (µ) = 0 and P = 2, the various controller parameters can be computed to be η0 = 0ε fd = 0ρ01ε j0 = 0ε fg = 0ρ0086ε ν0 = 0ε fl = 0ρ0105ε and the value of > satisfying the condition for recrossing, for τ = 2 is calculated as 2 (fd + fg ) (1 − τz ) > = 1ρ63 κ = 1ρ49ρ τz 2 This gives a quasi-sliding mode bound of fy = >z + fd + fg = 0ρ182ρ The resultant control is of the form (µ) =

−1ρ56 1ρ66

T

−0ρ182sgn

k

+ 0ρ857(µ − 1) −1ρ2 0ρ35

T k

+ 1ρ05(µ − 1) ρ

The simulation results can be seen in Fig. (2.2). It can be noted here that though the actual quasi-sliding mode band is of much less width as compared to the bound fy . This is because the bound is calculated for the worst case scenario of disturbance which may seldom persist for a long time in an uncertain system.

25

2.2 MROF-QSMC in Uncertain Systems

0.5

δ

0.2

x1 x2

y

0

Sliding Function

System States

1

−δy

−0.2

0

−0.4

−0.5 −1

−0.6

0

1

2

3

Time (secs)

4

5

−0.8

0

1

4

5

0.1 0

0.2

−0.1

0

−0.2

2

0.4

x

Control Input

3

(b)

(a)

0.6

−0.2

−0.3

−0.4

−0.4

−0.6

−0.5

−0.8

2

Time (secs)

0

1

2

3

Time (secs) (c)

4

5

−0.6 −1

−0.5

0

x1

0.5

1

(d)

Fig. 2.2. Quasi-Sliding Mode Control based on Multirate Output Feedback for Uncertain System : (a) State responses (b) Sliding function (c) Control input (d) Phase portrait

u Mwl2i7a2e Ow2pw2 Feedback ba6ed Di6c7e2e-2ime Sliding Mode in LTI S062em6 yi2h Unce72ain20

u?1 In27odwc2ion This chapter introduces various techniques of multirate output feedback based discrete-time sliding mode control of LTI systems with uncertainty. Systems satisfying the so-called matching condition [19] would be studied first. Linear systems that do not satisfy this condition would be handled next in the chapter. Finally the chapter would present a discrete-time version of the integral sliding-mode [76] control technique. This technique can be used to impart robustness to a controller structure. The chattering phenomenon exists, and is unavoidable in the discrete-time sliding mode control strategies discussed in the previous sections. This is due to the fact that a switching function is used in the control. Since, discretetime control is inherently discontinuous, the discontinuity occurring at each input sampling instant, an explicit discontinuity may not be required for the existence of sliding mode in discrete-time systems. 3.1.1 Relaxation of Quasi-sliding mode criterion In [7], it was suggested that the condition of quasi-sliding mode may be relaxed to the following criterion. Definition 3.1. We call the quasi-sliding mode in the f vicinity of the sliding hyperplane v(µ) = eT (µ) = 0 a motion of the system such that |v(µ)| ≤ f

(3.1)

where the positive constant f is called the quasi-sliding-mode band width. This definition is essentially different from the one proposed in [27], since it does not require the system state to cross the sliding plane in each successive control step. Consequently, we eliminate chattering (i.e., after the transient

B. Bandyopadhyay, S. Janardhanan: Discrete-time Sliding Mode Contr., LNCIS 323, pp. 27–49, 2006. © Springer-Verlag Berlin Heidelberg 2006

28

3 MROF-DSMC in Uncertain Systems

period, the system state and its output do not change in each successive control step) and achieve an essential reduction of the control effort and improved quality of the quasi-sliding mode control. Definition 3.2. We say that the system (2.5) satisfies the reaching condition of the quasi-sliding mode in the f vicinity of the sliding surface if and only if for any µ ≥ 0 the following condition is satisfied. |v(µ + <)| ≤ fε ∀< κ 0

(3.2)

where f is a positive constant.

u?v Mwl2i7a2e Ow2pw2 Feedback ba6ed Di6c7e2e-2ime Sliding Mode Con27ol fo7 Unce72ain S062em6 yi2h Ma2ched Unce72ain20 3.2.1 A Brief Review on State Feedback based DSMC Control Strategy for Matched Uncertain Systems Bartoszewicz proposed a discrete-time sliding mode control strategy that guarantees finite time convergence of the state trajectory to the sliding line [7]. It also avoids chattering by not using a switching function. This method is briefly discussed in the following. Consider a discrete-time φ-th order system sampled with sampling time z sec ˜ (µ + 1) = Sτ (µ) + Hτ (µ) + η(µ) (µ) = C (µ)

(3.3)

˜ p 0) = 0. Sτ ε Hτ ε C are matrices of appropriate It is assumed that η(µ dimensions with (Sτ ε Hτ ) being controllable and (Sτ ε C) being observable. Bartoszewicz proposed a new reaching law v(µ + 1) = η(µ) − η0 + vd (µ + 1)

(3.4)

where vd (µ) is a priori known function such that the following apply. • If |v(0)| κ 2fd ε then vd (0) = v(0) vd (µ)ρvd (0) ≥ 0ε for any µ ≥ 0 vd (µ) = 0ε for any µ ≥ µ ∗ |vd (µ + 1)| p |vd (µ)| − 2fd ε for any µ p µ ∗ • Otherwise, i.e., if |v(0)| ≤ 2fd ε then vd (µ) = 0 for any µ ≥ 0ρ

(3.5)

3.2 MROF-DSMC for Matched Uncertainty

29

The positive integer µ ∗ is chosen by the designer to achieve a good trade off between faster convergence and magnitude of the control ρ One possible definition of vd (µ) when |v(0)| κ 2fd is µ∗ − µ v(0)ε µ = 0ε 1ε · · · ε µ ∗ ε µ∗ |v(0)| ρ µ∗ p 2fd

vd (µ) =

The control law which satisfies the reaching law in Eqn. (3.4) can be computed by using Eqn. (3.3) as (µ) = − eT Hτ

−1

eT Sτ (µ) + η0 − vd (µ + 1) ρ

The control law so designed guarantees that for any µ ≥ µ ∗ ε the system states satisfy the inequality |v(µ)| = |η(µ − 1) − η0 | ≤ fd Hence, the states of the system settle within a quasi-sliding mode band whose width is less than half of the width of the band achieved by the control law proposed in [27]. 3.2.2 Multirate Output Feedback Control Algorithm Consider the system described by Eqns. (2.5 and 2.8). We define a new variable g(µ) as ˜ g(µ) = eT Sτ Md η(µ)ρ Since the disturbance is bounded we have gl ≤ g(µ) ≤ gu ρ Let us define the average value of g(µ) and the maximum deviation of g(µ) from this value as g0 = 0ρ5 (gu + gl ) ε fe = 0ρ5 (gu − gl ) respectively. Now, let us consider a new reaching law [35] for output feedback sliding mode for the system in (3.3) as v(µ + 1) = η(µ) − η0 + g(µ − 1) − g0 + vd (µ + 1) where vd (µ + 1) is a a priori known function which satisfies the conditions in (3.5) for the bound of (fd + fe ) instead of fd . We consider vd (µ) to be of the form

3 MROF-DSMC in Uncertain Systems

30

µ∗ − µ v(0)ε µ = 0ε 1ε · · · ε µ ∗ ε µ∗ |v(0)| ρ µ∗ p 2 (fd + fe ) vd (µ) = 0ε µ κ µ ∗ vd (µ) =

Thus, ˜ eT Sτ (µ) + Hτ (µ) + η(µ) = η(µ) − η0 + g(µ − 1) − g0 + vd (µ + 1) eT Sτ (µ) + eT Hτ (µ) + η(µ) = η(µ) − η0 + g(µ − 1) − g0 + vd (µ + 1) eT Sτ (µ) + eT Hτ (µ) = −η0 + g(µ − 1) − g0 + vd (µ + 1) Now, the control input can be derived as (µ) = − eT Hτ

−1

eT Sτ (µ) + η0 − g(µ − 1) + g0 − vd (µ + 1)

(3.6)

Using Eqn. (2.9), (µ) = − eT Hτ

−1

− e T Hτ

−1

= − e T Hτ

−1

− e T Hτ

−1

= − e T Hτ

−1

e T S τ My

k

˜ − 1) + Mu (µ − 1) + Md η(µ

(3.7)

(η0 − g(µ − 1) + g0 − vd (µ + 1)) e T S τ My

k

+ eT Sτ Mu (µ − 1)

(g(µ − 1) − g(µ − 1) + η0 + g0 − vd (µ + 1)) e T S τ My

k

+ eT Sτ Mu (µ − 1) + η0 + g0 − vd (µ + 1)

Hence, the control input can be computed using the past output samples and the immediate past input signal. But, at µ = 0ε there are no past outputs for use in control, hence (0) is obtained by ignoring g(µ − 1) and g0 (as we expect no disturbance before µ = 0 to affect the system) from Eqn. (3.6) and assuming an initial state (0) to obtain (0) = − eT Hτ

−1

eT Sτ (0) + η0 − vd (1) ρ

Proof of Convergence When the control input derived from Eqn. (3.7) is applied to the system, the system obeys the reaching law, v(µ + 1) = η(µ) − η0 + g(µ − 1) − g0 + vd (µ + 1) or equivalently v(µ) = η(µ − 1) − η0 + g(µ − 2) − g0 + vd (µ) for µ κ max(µ ∗ ε 2)ε vd (µ) = 0 and therefore

3.2 MROF-DSMC for Matched Uncertainty

31

v(µ) = η(µ − 1) − η0 + g(µ − 2) − g0 ε thus |v(µ)| = |η(µ − 1) − η0 + g(µ − 2) − g0 | ≤ |η(µ − 1) − η0 | + |g(µ − 2) − g0 | = fd + fe |v(µ)| ≤ fd + fe giving a sliding mode band of fy ≤ fd + fe . 3.2.3 Numerical Example Example 1 Consider the system cited in [7] (µ + 1) =

1 1 0 0ρ5

(µ) = 1 0

(µ) +

0 0 (µ) + ε 1 1

(µ)ρ

The sliding line is chosen as eT = 1 1 ρ Computation of the parameters gives −0ρ707 0ρ707 0ρ293 −0ρ707 0 ε Mu = ε Md = −0ρ854 0ρ854 1ρ146 −0ρ854 1 ηu = ηl = 1ε gu = gl = 1ρ5ε fd = fe = 0ε µ ∗ = 15

My =

For an initial condition (0) = 1000 0 control input is derived to be (µ) = 1ρ987 −2ρ987

k

T

ε using Eqn. (3.7), the sliding mode

− 2ρ013(µ − 1) + vd (µ + 1) − 2ρ5

The simulation results are shown in Figs. (3.1, 3.2). Fig. (3.1) shows the evolution of the variable v(µ)ρ It can be seen that the state trajectory converges to the sliding line in finite time after µ ∗ = 15 sampling instants. Also there is a discrete-time sliding mode motion without chatter. The output response of the system is given in Fig. (3.2). It is evident here that the output converges without exhibiting undesirable chatter. The system responses are exactly the same as obtained through state feedback in [7]. Example 2 Consider the discrete-time LTI system (µ + 1) =

0 1 0ρ4 −0ρ3

(µ) = 1 0

(µ)

(µ) +

0 0 (µ) + 1 sin(µx2) exp(−µx5)

32

3 MROF-DSMC in Uncertain Systems 1000 900 800

Sliding Function, s(k)

700 600 500 400 300 200 100 0 0

10

20

30

40

50

60

Time samples, k

70

80

90

100

Fig. 3.1. Plot of s(k) with constant disturbance component 1000 900 800

System Output, y(k)

700 600 500 400 300 200 100 0 0

10

20

30

40

50

60

Time Samples, k

70

80

90

100

Fig. 3.2. Closed loop response of the system with constant disturbance component

33

3.2 MROF-DSMC for Matched Uncertainty

The sliding line is chosen as eT = −0ρ4 1 ρ Computation of the parameters gives 5ρ407 −6ρ94 2ρ814 7ρ94 0 ε Mu = ε Md = −1ρ222 2ρ082 0ρ156 −2ρ082 1 ηu = 0ρ564ε ηl = −0ρ162ε fd = 0ρ363ε

My =

gu = 0ρ1131ε gl = −0ρ395ε fe = 0ρ254 µ∗ = 6

0

Sliding Function, s(k)

−0.5

Proposed Algorithm Bartoszewicz Algorithm

−1

−1.5 −2

−2.5 −3

−3.5 −4

0

5

10

15

20

25

30

Time Samples, k

35

40

45

50

Fig. 3.3. Plot of s(k) with varying disturbance component

For an initial condition (0) = 10 0 control input is derived to be (µ) = −3ρ019 4ρ233

k

T

ε using Eqn. (3.7), the sliding mode

− 1ρ017(µ − 1) + vd (µ + 1) − 0ρ06

The simulation results for a time-varying disturbance are shown in Figs. (3.3, 3.4). Fig. (3.3) shows the evolution of the sliding function v(µ). The plot compares the performance of the multirate output feedback based controller with the state feedback based control algorithm in [7]. It can be seen that the state trajectory converges to the quasi-sliding mode band. It is also noticeable that in this particular case, the output feedback based controller gives a better performance as compared to the state feedback based controller in [7]. The quasi-sliding mode band width is lesser when the multirate controller is used. The output response of the system is given in Fig. (3.4).

34

3 MROF-DSMC in Uncertain Systems 10

Proposed Algorithm Bartoszewicz Algorithm

8

System Output, y(k)

6

4

2

0

−2

−4

0

5

10

15

20

25

30

Time Samples, k

35

40

45

50

Fig. 3.4. Closed loop response of the system with varying disturbance component

3.2.4 Comparison with State Feedback based Control The control algorithm suggests an increase in the upper bound of quasi-sliding mode band width to fd + fe from fd as proposed by a state feedback control based on [7]. This is the trade-off for not being able to use the entire state information of the system. However, the actual width of the sliding mode band may be appreciably less in particular cases. Since, the discrete-time sliding mode band is computed for the worst case scenario, i.e., the magnitudes of η(µ) and g(µ −1) adding up, there is a possibility that in some of the cases, the disturbance vector can be such that η(µ) and g(µ − 1) are of opposite sense (as in the case of Example 2). In such a case, the multirate output feedback based control would perform better than a state feedback based control. However, this cannot be generalized. In most cases, the state feedback based algorithm would give a better performance. If this algorithm is applied to a deterministic systems or systems with a constant disturbance, i.e., fd = fe = 0, by choosing the value of µ ∗ to be any suitable positive integer, the system states would converge exactly to the sliding surface after µ ∗ time samples without any chatter and effectively reducing the sliding mode band to zero. In this case, both the state feedback based control and the output feedback based control systems behave in the same manner.

35

3.3 MROF-DSMC for Unmatched Uncertainty

u?u Mwl2i7a2e Ow2pw2 Feedback ba6ed Di6c7e2e-2ime Sliding Mode Con27ol of LTI S062em6 yi2h Unma2ched Unce72ain20 3.3.1 Problem Statement Consider the discrete-time system representation for a sampling rate of z sec ˜ (µ + 1) = Sτ (µ) + Hτ (µ) + Dd η(µ)

(3.8)

(µ) = C (µ) where, ∈ Rn ε  ∈ Rm ε ∈ Rp and w ∈ Rq , with the matrices being of appropriate dimensions. The matching condition rank(Hτ ) = rank([Hτ |Dd ]) is ˜ not necessarily satisfied. η(µ) is the bounded disturbance vector. T Let v(µ) = e (µ) = 0ε v ∈ Rm ε e ∈ Rn×m ε eT Hτ = 0 be the stable sliding manifold. Further, it is also assumed that the bounded disturbance is such that ˜ ≤ ηu ηl ≤ η(µ) = eT Dd η(µ) where, the inequality ηl = ηl,1 ηl,2 · · · ηl.m

T

≤ η(µ) = η1 (µ) η2 (µ) · · · ηm (µ)

T

implies that ηl,i ≤ ηi ε < = 1ε 2ε · · · ε ∂. The mean and spread of η(µ) is defined as (ηl + ηu ) (ηu − ηl ) η0 = ε fd = (3.9) 2 2 The aim is to achieve quasi-sliding mode for the system in (3.8). A state feedback based approach to solve this problem has been proposed in [70]. However, as mentioned in the introduction, states are not always available for measurement. So, output feedback control becomes mandatory in most of the cases. Hence, the problem statement now becomes the quasi-sliding mode control of (3.8) by using the system output only. It has been mentioned in [70], that a static output feedback based solution of the problem does not exist. In this section, a multirate output feedback strategy is considered for solving the problem. Using the multirate output sampling technique, an approximation of the state (µ) can be computed using the past system input and output samples as (3.10) ¯(µ) = My k + Mu (µ − 1) + ν0 and the approximate sliding function v¯(µ) can be defined as v¯(µ) = eT ¯(µ) = eT My

k

+ eT Mu (µ − 1) + eT ν0

(3.11)

36

3 MROF-DSMC in Uncertain Systems

3.3.2 Multirate Output Feedback based Control Law Theorem 3.3. The multirate output feedback based control law (µ) = − eT Hτ

−1

eT (Sτ − J) (¯(µ)) + Lisat (¯ v(µ)ε s) + η0

(3.12)

where, ¯ and v¯ are obtained from Eqns. (3.10 and 3.11) respectively, and satisfying the conditions isat (¯ v(µ)ε s) = sat sat

s¯1 (k) φ1

sat

s¯2 (k) φ2

· · · sat

s¯ (k) φ

T

v¯i (µ) vi | κ s i sgn(¯ vi )ε |¯ (3.13) = s¯، ε |¯ v i | ≤ si si φ، (•)i indicates the <-th component of the vector (•) unless indicated otherwise L = diag (L1 ε L2 ε · · · ε Lm ) ε Li κ (fα + fβ + fd )i ˜ ≤ ,u ,l ≤ ,(µ) = eT Sτ Md η(µ) (,l + ,u ) (,u − ,l ) = e T S τ ν0 ε f α = ,0 = 2 2 ˜ ≤ δu δl ≤ δ(µ) = eT Md η(µ) (δu − δl ) (δl + δu ) = e T ν0 ε f β = δ0 = 2 2 0 p 2diag(s) − L − diag (fα + fβ + fd )

(3.14) (3.15) (3.16) (3.17) (3.18) (3.19)

would achieve quasi-sliding mode for the discrete-time system (3.8). i.e, ∀ (0)ε ∃µ ∗ such that (µ) ∈ B ⊂ Bε ∀µ κ µ ∗ . Proof. Let us define the quasi-sliding mode band B = { |¯ v(µ) ≤ s} Now consider the lyapunov function ai (µ) = v¯2i (µ). |¯ vi (µ)| decreases monotonically if ai (µ + 1) p ai (µ)ε i.e., [∆¯ v(µ) + 2¯ v(µ)] ∆¯ v(µ) p 0ε ∀¯ v(µ) = 0

(3.20)

where ∆¯ vi (µ) = v¯i (µ + 1) − v¯i (µ). Using (3.12-3.19), ∆¯ v(µ) is obtained as ∆¯ v(µ) = v¯(µ + 1) − v¯(µ) ˜ + ν0 − ¯(µ) − ν0 + Md η(µ ˜ − 1) = eT ¯(µ + 1) − Md η(µ) ˜ − 1) − η(µ) ˜ = eT ( (µ + 1) − (µ)) + eT Md η(µ ˜ − (µ) + eT Md η(µ ˜ − 1) − η(µ) ˜ = eT Sτ (µ) + Hτ (µ) + Dd η(µ) = eT (Sτ − J) (µ) − eT (Sτ − J) (¯(µ)) + Lisat (¯ v(µ)ε s) + η0

37

3.3 MROF-DSMC for Unmatched Uncertainty

˜ + eT Md η(µ ˜ − 1) − η(µ) ˜ +eT Dd η(µ) = eT (Sτ − J) (µ) − eT (Sτ − J)

˜ − 1) + ν0 (µ) − Md η(µ

˜ + eT Md η(µ ˜ − 1) − η(µ) ˜ −Lisat (¯ v(µ)ε s) − η0 + eT Dd η(µ) ˜ − 1) − eT Sτ ν0 + eT ν0 − eT Md η(µ) ˜ + (η(µ) − η0 ) = eT Sτ Md η(µ −Lisat (¯ v(µ)ε s)

∆¯ vi (µ) = (,(µ − 1) − ,0 )i + (δ0 − δ(µ))i + (η(µ) − η0 )i v¯i (µ) −Li sat si

(3.21)

For v¯i (µ) κ si , let Li = (fα + fβ + fd )i + i (µ)ε i (µ) κ 0, then ∆¯ vi (µ) = (,(µ − 1) − ,0 )i + (δ0 − δ(µ))i + (η(µ) − η0 )i − Li ≤ (fα + fβ + fd − L)i = −i (µ) p 0 [2¯ v(µ) + ∆¯ v(µ)] ≥ 2s − L − fα − fβ − fd κ 2s − 2s = 0 =⇒ [2¯ vi (µ) + ∆¯ vi (µ)] ≥ 0 vi (µ) p 0 and [2¯ vi (µ) + ∆¯ vi (µ)] κ 0. Hence, for all v¯i (µ) κ si ε we have ∆¯ Hence, using (3.20), it can be said that the lyapunov function ai (µ) decreases monotonically and decreases at least by an amount of ∆amin,i = (2si − Li − (fα,i + fβ,i + fd,i )) (Li − (fα,i + fβ,i + fd,i )) (3.22) A similar argument can be given for the case of v¯i (µ) p −si . These conclude that B is attractive. i.e, for ∀ ∈ x Bε ∃µs such that ¯(µs ) ∈ B. vi (µ)| ≤ si ε, let v¯i (µ) = >i (µ)si ε |>i (µ)| ≤ 1. From, Eqns. (3.11 Next, for |¯ and 3.21), Li v¯i (µ) si + (,(µ − 1) − ,0 − δ(µ) + δ0 + η(µ) − η0 )i Li vi (µ + 1) = 1 − v¯(µ) + (,(µ − 1) − ,0 + η(µ) − η0 )i si s i − Li |¯ vi (µ)| + |,i (µ − 1) − ,0,i | + |ηi (µ) − η0,i | =⇒ |v(µ + 1)| ≤ si ≤ |si − Li | + fα,i + fd,i v¯i (µ + 1) =

1−

Now, consider the case si κ Li , then |vi (µ + 1)| ≤ si + (fα,i + fd,i − Li ) |vi (µ + 1)| p si − fβ,i and if si p Li , then

38

3 MROF-DSMC in Uncertain Systems

|vi (µ + 1)| ≤ (fα,i + fd,i + Li ) − si |vi (µ + 1)| p 2si − fβ,i − si = si − fβ,i and finally if si = Li , then using (3.14 and 3.19), it can be shown that si κ fα,i + fβ,i + fdi =⇒ |vi (µ + 1)| ≤ fα,i + fdi p si − fβ,i Thus, it can be said that ∀ (µ) such that v¯i (µ) ≤ si , we can assure that |vi (µ + 1)| p si − fβ,i . Therefore, for all (µ) ∈ B |¯ vi (µ + 1) + δi (µ) − δ0,i | p si − fβ,i |¯ vi (µ + 1)| − |δi (µ) − δ0,i | p si − fβ,i |¯ vi (µ + 1)| p si Therefore, it can be said that the band B is positively invariant. Thus, using (3.22), it can be concluded that for all (µ) ∈ Rn ε ∃µ ∗ v¯2i (0) i=1,···,m ∆amin,i

µ ∗ = max

(3.23)

such that ∀µ κ µ ∗ ε (µ) ∈ Bε B = { (µ)||v(µ)| p s − fβ } ⊂ B. Hence, it is proved that a quasi-sliding mode is achieved in the system using multirate output feedback control.  3.3.3 Illustrative Example The validity and effectiveness of the multirate output feedback based control strategy is analysed and compared with the state feedback based algorithm proposed in [70] using the following example. Consider the mechanical system T shown in Fig. (3.5). Let ∂ = 1, µ = 2, d = 3, = τ1 τ˙1 τ2 τ˙2 τ3 τ˙3 , q1

q2

q3

b

f u1

m

m k

m k

Fig. 3.5. The mechanical system

u2

3.3 MROF-DSMC for Unmatched Uncertainty

39

T

 = 1 2 . Let us assume that only the positions are measurable. Then the continuous time system representation would be ˙ (y) = Ac (y) + Bc (y) + Dc i (y) (y) = C (y)       0 00 01 0 0 0 0 0 1 0  −2 0 2 0 0 0        0 0 0  0 0 0 1 0 0      ε Dc  ε Bc =  Ac =    1ε   0 0  2 0 −4 −3 2 3  0 0 0  0 0 0 0 0 1 0 01 0 0 2 3 −2 −3   100000 C = 0 0 1 0 0 0 000010 The system input being sampled at z = 0ρ2 sec and is held constant during ˜ the period using a zero order hold circuit. Let the disturbance η(µ) affecting ˜ = [−15 + sin(3ρ5σµz )] the discrete-time system representation be η(µ) 3.3.4 Comparative Study State Feedback based Control Using the state feedback based control algorithm proposed in [70], a control signal was generated such that the quasi-sliding mode band is minimum. The resultant state feedback based control uses a single sliding function vs (µ) = e1 (µ) and has a structure vs (µ) ss G = diag (e1 Hτ,1 ε e1 Hτ,2 ε · · · ε e1 Hτ,m ) Hτ = Hτ,1 Hτ,2 · · · Hτ,m

(µ) = −G−1 N (µ) − G−1 Ls sat

(3.24)

T

N = [.1 |.2 | · · · |.m ] ε .i ∈ Rn Ls = L 1 L2 · · · Lm m

T

Li ∈ R

.Ti = e1 (Sτ − J)

i=0 m

Li = LΣ = k + 2z >ε > κ 0 i=0

ss κ k + z > k = max (|η(µ)|)

(3.25)

40

3 MROF-DSMC in Uncertain Systems

For, the example considered the following control achieves the minimum possible quasi-sliding mode band of width ss = 0ρ7530 for a disturbance magnitude of k = 0ρ7529. e1 = −0ρ1394 −0ρ0055 −0ρ9432 −0ρ1389 −0ρ1600 −0ρ2146 T 19ρ5764 0  8ρ1366 0    −20ρ4114 0  (µ) − 160ρ7151 sat (µ) = −    0 3ρ8933 2ρ1002    0 0ρ0805  0 0ρ8162 

v(µ) 0ρ7530

(3.26)

The simulation results for the control law proposed in [70] are shown below in Figs. (3.6-3.8). 1

0

Sliding Function ss(k)

−1

−2

−3

−4

−5

−6

0

10

20

30

40

50

60

Sampling Instants (k)

70

80

90

100

Fig. 3.6. Plot of Sliding Function st (k) using state feedback

Multirate Output Feedback based Control Law The system output is sampled at a rate twice that of the input sampling rate. i.e., ∆ = 0ρ1 sec. The sliding surfaces are chosen in a manner such that one of them is the sliding surface designed for the state feedback. This was done for

3.3 MROF-DSMC for Unmatched Uncertainty 40

q1 q 2 q 3

30

Displacements (m)

20

10

0

−10

−20

−30

0

20

40

60

Sampling Instants (k)

80

100

120

Fig. 3.7. Evolution of the System states using state feedback 200

u1 u2

150

Control Inputs (N)

100

50

0

−50

−100

−150

0

10

20

30

40

50

60

Sampling Instants (k)

70

80

90

Fig. 3.8. Plot of the state feedback based control inputs

100

41

42

3 MROF-DSMC in Uncertain Systems

the comparison purpose. The second surface is so chosen as to stabilize the dynamics of τ1 . −0ρ1394 −0ρ0055 −0ρ9432 −0ρ1389 −0ρ1600 −0ρ2146 0ρ7071 0ρ7071 0 0 0 0

eT =

Using the control algorithm in Eqns.(3.12-3.19), the control signal is computed to be T  1ρ9808 −2ρ0787  −0ρ7256 −0ρ7146     −1ρ9616 2ρ1573   ¯(µ)  (µ) =    −0ρ1579 −3ρ8780   −0ρ0192 −0ρ0787  −0ρ0287 −0ρ8134 −0ρ0046 −0ρ1684 sat 2ρ9319 0ρ0162

+

v¯1 (µ)x0ρ1444 v¯2 (µ)x0ρ0172

+

0ρ1441 15ρ5899

The simulation results with the multiple sliding function based multirate output feedback control are presented in Figs. (3.9-3.11) 3

s1 s2

2

Sliding Function s(k)

1

0

−1

−2

−3

−4

−5

−6

0

20

40

60

80

100

120

Sampling Instants (k)

140

160

180

200

Fig. 3.9. Plot of Sliding Functions s(k) using multirate output feedback

3.3 MROF-DSMC for Unmatched Uncertainty 4

q 1 q2 q3

3.5 3

Displacements (m)

2.5 2 1.5 1 0.5 0 −0.5 −1

0

50

100

150

200

Sampling Instants (k)

250

Fig. 3.10. Evolution of the System states using multirate output feedback 20

u1 u2

Control Inputs (N)

15

10

5

0

−5

−10

0

20

40

60

80

100

120

Sampling Instants (k)

140

160

180

200

Fig. 3.11. Plot of the control inputs using multirate output feedback

43

44

3 MROF-DSMC in Uncertain Systems

3.3.5 Analysis of Simulation Results It can be seen from Fig. (3.9) that the quasi-sliding mode band width by T using the multirate output feedback algorithm is s = [0ρ1444 0ρ0172] , which is much lesser than that achieved by using the state feedback based control [70] as is observed in Fig. (3.6). It is worthy to note here that the same sliding function v1 = vs = e1 (µ) settles to a very tight quasi-sliding band when the multirate output feedback based control law is used. This performance is however not achieved despite the use of entire state information with the algorithm (3.24). The system states in Fig. (3.10) also settle within a much tighter band than in Fig. (3.7).Further, the multirate output feedback based control required to achieve the quasi-sliding mode, as shown in Fig. (3.11) are also of much lesser magnitude as compared to the control algorithm in (3.24) as in Fig. (3.8). Hence, or the example considered, the multirate output feedback based control algorithm is clearly able to achieve a much better performance than the state feedback based algorithm proposed in [70] even when less information is available. The main reason for this improvement in performance is that the multirate output feedback algorithm takes advantage of any bias in the disturbance signal whereas the latter algorithm does not. Thus, in case of disturbance signals with a large amplitude but little variation(as in the chosen example , the mutlirate output feedback algorithm performs much better. Further, the multirate output feedback based algorithm uses the available freedom of multiple sliding surfaces to achieve better performance than that acheivable by a single sliding surface. However, in case of systems wherein the disturbance is actually unbiased, the output feedback control algorithm would result in a quasi-sliding mode band of width fα + fd which is slightly more that the band resulting from state feedback i.e., fd . But, this is an expected behaviour

u?4 Mwl2i7a2e Ow2pw2 Feedback ba6ed In2eg7al Sliding Mode in Di6c7e2e-2ime S062em6 3.4.1 Problem Statement Consider the LTI system discretized with a sampling interval of z sec, ˜ (µ + 1) = Sτ (µ) + Hτ (µ) + Dτ η(µ) (µ) = C (µ) with matching condition satisfied.i.e. there exists a ηu (µ) such that, ˜ = Hτ ηu (µ) Dτ η(µ)

(3.27) (3.28)

3.4 MROF based Integral Sliding Mode in Discrete-time Systems

45

˜ Let the state be φ-vector, input be ∂-dimensional, disturbance vector η(µ) be of dimension φd and the output be a q vector. The analysis is restricted to systems wherein φd ≤ q ≤ φ. The aim is to design a control (µ) such that system (3.27) would have a response of the deterministic system (µ + 1) = Sτ (µ) + Hτ 0 (µ)

(3.29)

inspite of the uncertainty in the system, with both systems (3.27) and (3.29) starting at the same initial condition. The input 0 (µ) is the control that would drive the system in a desired manner. The point to be noted is that this is not a tracking problem as both systems start from the same initial condition and have almost the same dynamics. 3.4.2 State Based Discrete-Integral Sliding Mode Let the control that achieves the above task be of two parts as [75, 76] (µ) = 0 (µ) + 1 (µ)

(3.30)

where 0 (µ) is the ideal control and 1 (µ) is used to reject the perturbations. Consider the sliding function w(µ) = v(µ) + ‫(܈‬µ)

(3.31)

where, v(µ) = 0 is a function representing a stable manifold, designed based on system states in a manner similar to ordinary sliding mode. The second term ‫(܈‬µ) induces the integral effect. The aim is to design a control such that while the system is in sliding motion, the control 1 (µ) has an ’equivalent value of 1eq (µ) = −ηu (µ), which rejects the disturbance. The value of ‫(܈‬µ) is determined such that the sliding mode starts from initial time itself, i.e, w(0) = 0. During the sliding mode, w(µ + 1) − w(µ) = 0ε v(µ + 1) − v(µ) + ‫(܈‬µ + 1) − ‫(܈‬µ) = 0ρ or eT (Sτ − J) (µ) + eT Hτ (0 (µ) − ηu (µ)) + eT Hτ ηu (µ) = 0 +‫(܈‬µ + 1) − ‫(܈‬µ) Thus giving the equations ‫(܈‬µ + 1) = −eT (Sτ − J) (µ) + eT Hτ 0 (µ) + ‫(܈‬µ)ε ‫(܈‬0) = −v(0)ρ

(3.32) (3.33)

46

3 MROF-DSMC in Uncertain Systems

The incremental control 1 (µ) is computed in the following manner. Consider the equation w(µ + 1) − w(µ) = v(µ + 1) − v(µ) + ‫(܈‬µ + 1) − ‫(܈‬µ)ε ˜ = eT Hτ 1 (µ) + eT Dτ η(µ)ρ An ideal control that would bring w(µ + 1) = 0 would be of the form 1 (µ) = − eT Hτ

−1

˜ w(µ) − eT Dτ η(µ) ρ

˜ However, η(µ) is an unknown factor. Hence, this control cannot be imple˜ ˜ mented. Instead, if it is assumed that η(µ) is a slowly varying signal, η(µ) ˜ may replaced by the estimate of η(µ − 1) in the control law [67]. Now, denot˜ − 1) and η(µ) ¯ as the estimate of η(µ), the control law ing η(µ) = eT Dτ η(µ becomes 1 (µ) = − eT Hτ

−1

¯ w(µ) − η(µ) ε

(3.34)

¯ is computed using the following where, the estimation of the disturbance η(µ) algorithm. ¯ = w(µ) − w(µ − 1) − eT Hτ 1 (µ − 1)ε η(µ) ¯ = 0ρ η(0)

(3.35) (3.36)

Thus the final control law is of the form (µ) = 0 (µ) + 1 (µ) T

1 (µ) = − e Hτ T

w(µ) = e ‫(܈‬µ) ‫(܈‬0) ¯ η(µ) ¯ η(0)

−1

¯ w(µ) − η(µ)

(µ) + ‫(܈‬µ)

(3.37) (3.38) (3.39)

T

= −e ((Sτ − J) (µ − 1) + Hτ 0 ( (µ − 1))) + ‫(܈‬µ − 1) (3.40) = −eT (0) (3.41) = w(µ) − w(µ − 1) − eT Hτ 1 (µ − 1) =0

(3.42) (3.43)

3.4.3 Multirate Output Feedback based DISMC As already mentioned before, the system states are not always available for measurement. Hence, the control law (3.37-3.43) cannot be implemented directly in all cases. In this section, we devise a method of computing the control input (µ) using the past system outputs and control input signals. The term w(µ) in the control, is in essence an integrator which integrates the error in the actual and the expected system response. The control is to be designed so that this integrated error is nullified. Hence, it is imperative that

3.4 MROF based Integral Sliding Mode in Discrete-time Systems

47

there be no error in the computation of v(µ) as any error in the computation would be integrated and would increase with time. The task of computing v(µ) can be accomplished by using the concept of a disturbance estimator [77] along with the usual multirate output feedback technique [40]. State Computation and Disturbance Estimation in Multirate Sampled Systems Consider the system (3.27) with input being sampled at z sec and the output being sampled at ∆ sec then the multirate sampled system has the form ˜ (µ + 1) = Sτ (µ) + Hτ (µ) + Dτ η(µ)ε ˜ k+1 = C0 (µ) + D0 (µ) + Cd η(µ)ε Thus,

(µ) = RTR ˜ η(µ)

where

−1

RT (

k+1

− D0 (µ)) ε

(3.44) (3.45)

(3.46)

R = C0 Cd ρ

T ˜ Substituting this value of (µ) η(µ) in the state equation (3.44) and shifting the variable µ, the expression for (µ) in terms of k and (µ − 1) can be computed as

(µ) = T R T R

−1

RT

k

+ Hτ − T R T R

−1

R T D0 (µ − 1) (3.47)

T = Sτ Dτ

Design of Multirate Output Feedback based DISMC Using the computed value of (µ) from (3.47) in the control law (3.37-3.41) ¯ from Eqn. (3.46), the discreteand estimating the disturbance component η(µ) time integral sliding mode controller can be formulated as a multirate output feedback based controller of the following structure. (µ) = T R T R

−1

RT

k

+ Hτ − T R T R

−1

R T D0 (µ − 1)ε (3.48)

(µ) = 0 (µ) + 1 (µ)ε T

1 (µ) = − e Hτ T

−1

¯ w(µ) − η(µ) ε

w(µ) = e (µ) + ‫(܈‬µ)ε ‫(܈‬µ) = −eT ((Sτ − J) (µ − 1) + Hτ 0 (µ)) + ‫(܈‬µ − 1)ε

(3.49) (3.50) (3.51) (3.52)

48

3 MROF-DSMC in Uncertain Systems

‫(܈‬0) = −eT (0) ¯ = eT Dτ η(µ ˜ − 1) η(µ)

(3.53)

= eT Dτ 0n ×n Jn ×n ¯ = 0ρ η(0)

RTR

−1

RT (

k

− D0 (µ − 1)) ε (3.54) (3.55)

3.4.4 Illustrative Example Consider the LTI system (µ + 1) =

11 −1 2

(µ) = 1 0

(µ) +

0 0 ˜ η(µ) (µ) + −1 1

(µ)

˜ = g−k/5 sin(µx10). The system The system has a disturbance function η(µ) needs to behave as an unperturbed system with the nominal control 0 (µ) = F (µ), where F = −0ρ89 1ρ3 . The resultant plots are shown in Figs. (3.12, 3.13). It can be seen from the phase plots in Fig. (3.12) that the constructed multirate output feedback control law is able to make the perturbed system behave in a manner similar to an unperturbed feedback system. The plot of the applied control is given in Fig. (3.13). 1

System Response Target Response

0.8

0.6

0.4

State x

2

0.2

0

−0.2

−0.4

−0.6

−0.8 −1.5

−1

−0.5

0

0.5

1

State x1

1.5

2

2.5

3

Fig. 3.12. The phase plots of the actual system and expected system

3.4 MROF based Integral Sliding Mode in Discrete-time Systems 3

2

1

Control Input

0

−1

−2

−3

−4

−5

−6

0

50

100

150

200

250

Samples

300

350

400

Fig. 3.13. The control input applied to the system

450

500

49

4 Mwl2i7a2e Ow2pw2 Feedback ba6ed Di6c7e2e-2ime Qwa6i-Sliding Mode Con27ol of Time-Dela0 S062em6

4?1 In27odwc2ion In recent years, the study of sliding mode control for systems with time delay in its state dynamics has received consideration. The delays present in a system can be broadly categorized into three categories. The input delay, state delay and output delay. Input delay is caused by the transmission of a control signal over a long distance. State delay is due to transmission or transport delay among interacting elements in a dynamic system. The output delay is to delay due to sensors. A lumped function based approach is adopted in [50] to design a sliding mode control law for systems with state delay. An added uncertainty is handled in [13]. In [78], a control law has been proposed for systems where delay is present in both state and input. A state feedback discrete-time sliding mode controller for time-delay system has also been investigated in [54]. In [58], an observer based sliding mode algorithm has been proposed. The algorithm handles systems with computational delay. This chapter introduces multirate output feedback based sliding mode control algorithms for control of both discrete-time LTI deterministic systems with time delay and time-delay systems with uncertainty [34]. The algorithms for the deterministic systems are based on the reaching law approach proposed in [27] for systems with delay in either input, state or the measurement channel. An algorithm has also been discussed for control of time-delay systems with uncertainty in the state equation. This algorithm is based on the uncertainty based reaching law approach discussed in Section 3.2 for uncertain systems without time delay.

4?v The P7oblem S2a2emen2 Consider the continuous-time system representation ˜ ˙ (y) = A0 (y) + A1 (y − zx ) + B0 (y) + B1 (y − zu ) + η(y)

(4.1)

(y) = C (y − zy )

B. Bandyopadhyay, S. Janardhanan: Discrete-time Sliding Mode Contr., LNCIS 323, pp. 51–70, 2006. © Springer-Verlag Berlin Heidelberg 2006

4 MROF based QSMC in Time-Delay Systems

52

˜ is where, the delay terms zx ε zu ε zy ≥ 0 and the disturbance vector η(y) bounded. The information of the states (y) is known for y ∈ [−zx ε 0] and the input for y ∈ [−zu ε 0]. The aim is to design a multirate output feedback based sliding mode controller for systems with the above structure. Various combinations of the aforementioned delays and disturbance would be considered and control algorithms are derived for each type of system. The following assumptions are made in relation to the delay and disturbance terms in this chapter. It is assumed that the state, input and output delays in the system, whenever occurring, are in one of the following forms [38] Form 1 : B0 = 0ε η = 0ε zy = 0ε

(4.2)

Form 2 : B0 = 0ε η = 0ε ε zy = 0ε zx κ zu ε Form 3 : B0 = 0ε η = 0ε zy p {zx ε zu }ε

(4.3) (4.4)

Form 4 : B0 = 0ε η = 0ε η ∈ u(B0 )ε zy = 0

(4.5)

It is assumed here, without loss of generality that the pair (A0 ε B0 ) is controllable and (A0 ε C) is observable. 4.2.1 Discretization of a Time Delay System A continuous-time system of structure represented in (4.1), when discretized at a sampling time of z sec, with z such that zx = µx zε zu = µu zε zy = µy z where {µx ε µu ε µy } ∈ W, W denoting the set of whole numbers, would yield the following best-approximated discrete-time model [2, 11]. (µ + 1) = G0 (z ) (µ) + G1 (z ) (µ − µx ) + I0 (z )(µ) + I1 (z )(µ − µu ) +Dτ ηd (µ)ε (4.6) (µ) = C (µ − µy )ε where G0 (z ) = gA0 τ ε G1 (z ) = I0 (z ) = I1 (z ) = Dτ ηd (µ) =

τ

0

0

τ

τ

0

gA0 (τ −ω) A1 ηψε gA0 (τ −ω) B0 ηψε gA0 (τ −ω) B1 ηψε

(k+1)τ kτ

(4.7)

˜ gA0 ((k+1)τ −ω) Dt η(ψ)ηψρ

˜ Remark 4.1. It is obvious from the equations that the boundedness of η(y) would ensure the boundedness of ηd (µ). This can be proved in the following manner.

4.3 Design of Sliding Surface

53

˜ ˜ is bounded, there exists a vector ηmax such that |η(y)| p ηmax Proof. Since η(y) ˜ ˜ element-wise, where |η(y)| denotes the element-wise absolute value of η(y). From Eqn. (4.7), the value of ηd (µ) can be bounded as |ηd (µ)| = ≤ ≤ ≤

(k+1)τ kτ (k+1)τ kτ (k+1)τ kτ τ 0

˜ gA0 ((k+1)τ −ω) η(ψ)ηψ ˜ ηψ gA0 ((k+1)τ −ω) η(ψ) gA0 ((k+1)τ −ω) ηmax ηψ

gA0 (τ −ω) ηψηmax

≤ |G0 (z )ηmax | Remark 4.2. It is to be noted here that the exact discretization of a time-delay system is impossible to be represented in a closed form [68]. This is due to the fact that the actual contribution of the delayed state in the z system is (k+1)τ kτ

gA0 ((k+1)τ −ω) A1 (ψ − zx ) ηψ

and it is approximated to G1 (z ) (µ − µx ) =

τ 0

gA0 (τ −ω) A1 ηψ

(µ − µx )

for the purpose of obtaining a closed form representation. In fact, the actual contribution cannot be represented as a multiple of (µ − µx ) or a finite linear combination of (<) ε < ∈ Z.

4?u De6ign of Sliding Sw7face The first step in the design of the required DSMC control algorithm would be to design a sliding surface v = eT from the continuous time representation (4.1). A method for constructing the sliding surface for state-delay systems has been given in [50]. The method can be extended for a class of systems with both state and input delay. The brief description of the method is as follows. Construct a transformation Xn such that T Xn B0 = 0 B0,T

T

T ε Xn B1 = 0 B1,T

T

(4.8)

4 MROF based QSMC in Time-Delay Systems

54

where B0,T is an invertible ∂ × ∂ matrix and is it assumed that there exists an ∂ × ∂ matrix Xm such that B0 Xm = B1 ρ T = Xn ε then Let ‫܈ = ܈‬1T ‫܈‬2T ‫˙܈‬1 (y) = A11 ‫܈‬1 (y) + A12 ‫܈‬2 (y) +A¯11 ‫܈‬1 (y − zx ) + A¯12 ‫܈‬2 (y − zx ) ε ‫˙܈‬2 (y) = A21 ‫܈‬1 (y) + A22 ‫܈‬2 (y) +A¯21 ‫܈‬1 (y − zx ) + A¯22 ‫܈‬2 (y − zx )

(4.9) (4.10)

+B0,T  (y) + B1,T  (y − zu ) ε where, Xn A0 Xn−1 =

A¯ A11 A12 ε Xn A1 Xn−1 = ¯11 A21 A21 A22

A¯12 A¯22

Let the sliding surface to be designed be of the form v (y) = L‫܈‬1 (y) + ‫܈‬2 (y) ρ Therefore, the system (4.9) is converted to ‫˙܈‬1 (y) = (A11 − A12 L) ‫܈‬1 (y) + A¯11 − A¯12 L ‫܈‬1 (y − zx ) ρ

(4.11)

This system is asymptotically stable if the roots of the characteristic equation q (v) = det vJn−m − A11 + A12 L − A¯11 − A¯12 L gτx s = 0

(4.12)

are in the open left half of the complex plane. Eqn. (4.12) has infinitely many solutions for zx κ 0ρ Hence, it is impractical to check for its solutions. A frequency sweeping based approach to calculate appropriate L is discussed in [12]. A more practical Lyapunov function based approach has been discussed in [66]. Here, we consider the positive-definite candidate Lyapunov function a (yε ‫܈‬1 (y) ε ‫܈‬1 (y − zx )) = ‫܈‬1T (y) R ‫܈‬1 (y) +

t

t−τx

‫܈‬1 () T‫܈‬1 () η

where R is the positive definite symmetric solution of the Riccati equation A11 R + R AT11 − R A12 U−1 AT12 R = −T with T and U as some positive-semi-definite and positive-definite matrices respectively. The negative definiteness of a˙ and hence closed loop system stability is assured [66] if L is chosen as L = U−1 A12 Rε 0 p A12 U−1 AT12 − A¯11 − A¯12 U−1 AT12 R T−1 A¯11 − A¯12 U−1 AT12 R ρ

(4.13)

The sliding function v can thus be constructed as v (y) = eT (y) = L Jm Xn (y) ρ The value of L is chosen so that that DSMC be feasible.

τ 0

(4.14)

eT gA0 (τ −ω) B0 ηψ is nonsingular in order

4.4 Multirate Sampling of Time-Delay Systems

55

4?4 Mwl2i7a2e Sampling of Time-Dela0 S062em6 4.4.1 Contribution of the Disturbance Term during Multirate Sampling Consider the system states in (4.6) observed at a sampling interval ∆ = z xP , where P is chosen to be an integer greater than the observability index of the system. The system input (µ) is applied with a sampling interval of z sec. Let us assume here that the disturbance vector ηd (µ) present in the z system in (4.6) manifests itself in the ∆ system dynamics as η (µ). For simplicity, it is assumed that η (µ) remains constant over the z interval µz ≤ y p (µ + 1)z , the ∆ - system states can be computed to be (µz + ∆) = G0 (∆) (µz ) + G1 (∆) (µz − zx ) (4.15) +I0 (∆)  (µz ) + I1 (∆)  (µz − zu ) +η (µ)ε (µz + 2∆) = G0 (∆) (µz + ∆) + G1 (∆) (µz − zx + ∆) (4.16) +I0 (∆)  (µz + ∆) + I1 (∆)  (µz − zu + ∆) + η (µ)ε Substituting for (µz + ∆) from (4.15) into (4.16), (µz + 2∆) = G20 (∆) (µz ) +

1

G0 (∆)1−i G1 (∆) (µz − zx + <∆)

i=0

+ (G0 (∆) + J) (I0 (∆)(µz ) + I1 (∆)(µz − zu )) + (G0 (∆) + J) η (µ) = function( ε ) + (G0 (∆) + J) η (µ) ε

(4.17)

An expression for (µz + 2∆) can also be obtained as (µz + 2∆) = G0 (2∆) (µz ) + G1 (2∆) (µz − zx ) +I0 (2∆)  (µz ) + I1 (2∆)  (µz − zu ) + η2 (µ)ε where, η2 (µ) is the contribution of the disturbance term. From (4.17), it is easily deducible that η2 (µ) = (G0 (∆) + J) η (µ).Thus, (µz + 2∆) = G0 (2∆) (µz ) + G1 (2∆) (µz − zx ) +I0 (2∆)  (µz ) + I1 (2∆)  (µz − zu ) (G0 (∆) + J) η (µ)ε Proceeding in similar lines, ((µ + 1) z ) = G0 (P ∆) (µz ) + G1 (P ∆) (µz − zx ) +I0 (P ∆)  (µz ) + I1 (P ∆)  (µz − zu ) N −1

+ i=0

Gi0 (∆) η (µ)ρ

(4.18)

56

4 MROF based QSMC in Time-Delay Systems

Now, using the properties of LTI systems and comparing (4.18) with (4.6), the following relationship can be arrived at N −1

η (µ) =

−1

Gi0 (z )

ηd (µ)ρ

i=0

4.4.2 Relationship between State and Multirate Output in Time-Delay Systems Consider the case where the system output is being sampled after every ∆ sec and the input is applied with an interval of z sec., then we can obtain the relationship between the system states (µ) and the lifted output k+k +1 , by using the relationship (µ) = C (µ − µy ) as (µz + zy ) = C (µz ) ε (µz + zy + ∆) = C (µz + ∆) = C (G0 (∆) (µz ) + G1 (∆) (µz − µx z ) +I0 (∆)  (µz ) + I1 (∆)  (µz − µu z ) + η (µ)) ε .. . ((µ + µy + 1)z − ∆) = CG0 ((P − 1) ∆) (µz ) +G1 ((P − 1) ∆) (µz − µx z ) +C (I0 ((P − 1) ∆)  (µz )) N −2

+CI1 ((P − 1) ∆)  (µz − µu z ) + C

Gi0 (∆)η (µ)ρ

i=0

Thus, representing (µz ) as (µ), we have, k+k +1

C0T

k+1+k

= C0 (µ) + C1 (µ − µx ) +D0  (µ) + D1  (µ − µu ) + Cd ηd (µ)ε

(4.19)

= C0T (C0 (µ) + C1 (µ − µx )) +C0T (D0  (µ) + D1  (µ − µu ) + Cd ηd (µ)) ε −1

(µ) = C0T C0 C0T k+1+k − C1 (µ − µx ) − (D0  (µ) + D1  (µ − µu ) + Cd ηd (µ))) ρ

(4.20)

where,  k

  = .  ..

 ((µ − 1) z ) ((µ − 1) z + ∆)     (µz − ∆)

(4.21)

57

4.5 Quasi-Sliding Mode Control Algorithm for Form 1 Systems













0 C    CG1 (∆)  CG0 (∆)      CG1 (2∆)   CG0 (2∆)  C0 =    ε C1 =     ..  ..   . . CG1 (z − ∆) CG0 (z − ∆)  0    CI1 (∆)      CI1 (2∆)   ε D1 =      ..   . CI0 (z − ∆) CI1 (z − ∆)   0  C −1  N −1   CG0 (∆) + C  i Cd =  G0 (∆)    ..  i=0 . N −2 C i=0 Gi0 (∆) 

0  CI0 (∆)   D0 =  CI0 (2∆)  .. .

and the values of G0 ε G1 ε I0 ε I1 can be computed using the Eqns. (4.7).

4?5 Qwa6i-Sliding Mode Con27ol Algo7i2hm fo7 Fo7m 1 S062em6 Consider the system (4.1) with zy = 0 and η(y) = 0. The system has no sensor delay nor does it have any disturbance or uncertainty in it. Such systems mostly occur in process control where the large transportation lags between different system components cause state delays and input delays in the system [38]. Substituting zy = 0 and η(y) = 0 into the discrete-time system representation (4.6) would give the z system model as (µ + 1) = G0 (z ) (µ) + G1 (z ) (µ − µx ) + I0 (z ) (µ)

(4.22)

+I1 (z ) (µ − µu ) (µ) = C (µ) 4.5.1 Multirate Output to State Relationship in Time-Delay Systems For the system described above, the multirate output can be calculated in terms of the system states using (4.19) as k+1

= C0 (µ) + C1 (µ − µx ) + D0 (µ) + D1 (µ − µu )

4 MROF based QSMC in Time-Delay Systems

58

from which the value of (µ) can be obtained in terms of the multirate output as (µ) = C0T C0

−1

C0T (

k+1

− C1 (µ − µx ) − D0 (µ) − D1 (µ − µu ))

Substituting this value of (µ) in the state equation (4.22) and (4.21), an expression for the system state (µ + 1) can be obtained in terms of k+1 and the past state and control inputs as (µ + 1) = My

k+1

+ Mxĝ (µ − µx ) + Mu0 (µ) + Muĝ (µ − µu )

or equivalently (µ) = My

k

+ Mxĝ (µ − µx − 1) + Mu0 (µ − 1) + Muĝ (µ − µu − 1) (4.23)

where My = G0 (z ) C0T C0

−1

C0T

Mxĝ = G1 (z ) − G0 (z ) C0T C0

−1

C0T C1

Mu0 = I0 (z ) − G0 (z ) C0T C0

−1

C0T D0

Muĝ = I1 (z ) − G0 (z ) C0T C0

−1

C0T D1

4.5.2 Multirate Output Feedback based Sliding Mode Control Algorithm for Form 1 Systems Consider the reaching law proposed in [27]. v(µ + 1) − v(µ) = −τz v(µ) − >z sgn(v(µ))ρ

(4.24)

The state based control law that would satisfy the above said reaching condition for the system (4.22) can be derived to be (µ) = − eT I0 (z ) eT I0 (z )

−1

−1

τz eT − eT + eT G0 (z )

(µ)

eT G1 (z ) (µ − µx )eT I1 (z )(µ − µu ) ρ

(4.25)

Substituting the value of (µ) from (4.23) in (4.25), the multirate output feedback based discrete-time sliding mode control law for systems of Form 1 can be derived to be  (µ) = − eT I0 (z ) T

− e I0 (z ) − eT I0 (z ) T

− e I0 (z )

−1 T

e (τz + G0 (z ) − J) My

e G1 (z ) (µ − µx )

−1 T

e (τz + G0 (z ) − J) Mxĝ (µ − µx − 1)

−1 T

e (τz + G0 (z ) − J) Mu0  (µ − 1)

− eT I0 (z )

−1 T

− eT I0 (z )

−1 T

T

− e I0 (z )

k

−1 T

e I1 (z )  (µ − µu ) e (τz + G0 (z ) − J) Muĝ  (µ − µu − 1)

−1

>z sgn eT (µ) ρ

(4.26)

4.5 Quasi-Sliding Mode Control Algorithm for Form 1 Systems

59

Generation of Initial Control For µ = 0, the information about (−µx − 1) and (−µu − 1) is not available. Hence, the control law (4.26) cannot be applied directly. For this case, the control law (0) is computed using (4.25) for the given data of state and input for y p 0. 4.5.3 Simulation Results Consider the system (4.1) with A0 =

−1 1 −1 0ρ4 ε A1 = 0 −2 0 −1

B0 =

0ρ6 0ρ6 ρ ε B1 = 1ρ0 1ρ0

(4.27)

C= 10 and zx = 0ρ8 secε zu = 0ρ4 secε z = 0ρ2 sec. Using the design procedure described in Section 4.3, the sliding surface can be designed as v( ) = [ −ρ6 1 ] = 4

3

3

2

2

State x

State x

1

2 1 0

0

−1

−1 −2

1

0

10

20

30

Time in sec

40

−2

50

0

10

40

50

40

50

0.02

Sliding Function

3 2

Input

30

(b)

(a)

4

0

−0.02

1

−0.04

0

−0.06

−1

−0.08

−2 −3

20

Time in sec

0

10

20

30

Time in sec (c)

40

50

−0.1

−0.12

0

10

20

30

Time in sec (d)

Fig. 4.1. Plots for Systems of Form 1 : a. Time Response of x1 , b. Time Response of x2 , c. Input Profile, d. Profile of the sliding function s(k)

60

4 MROF based QSMC in Time-Delay Systems

0. The multirate output feedback QSMC law is designed for P = 4 and ∆ = z xP using the technique in Section 4.5. The reaching law parameters τ and > are chosen as τ = 0ρ005ε > = 0ρ01. The initial data is given as b(y) = [3ρ2ε 2]T ε y ∈ [−0ρ8ε 0] and (y) = 3ε y ∈ [−0ε 4ε 0] The simulation results are shown in Fig. (4.1). Figs. (4.1(a) and 4.1(b)) give the state responses of the system under the multirate output feedback based sliding mode control algorithm for the Form 1 systems. Fig. (4.1(c)) shows the input profile generated by the algorithm. The profile of the sliding function v(µ) is shown in Fig. (4.1(d)). The convergence of the time-delay system to quasi-sliding mode is clearly visible here.

4?3 Qwa6i-Sliding Mode Con27ol Algo7i2hm6 fo7 Fo7m v S062em6 Deterministic systems without output delay, and with only a retarded input affecting the system, fall into the category of Form 2 systems as said in (4.3). The additional criterion for such systems is that they should be controllable with B1 as the input matrix, i.e., the pair (G0 (z )ε I1 (z )) must be controllable. They would have the discrete-time representation (µ + 1) = G0 (z ) (µ) + G1 (z ) (µ − µx ) + I1 (z ) (µ − µu )ε

(4.28)

(µ) = C (µ)ρ Since the system input (µ) would affect the state only after µu sampling instants, the control law (µ) is to be designed so as to control the future state (µ + µu ). 4.6.1 State based Control Algorithm Consider the reaching law in (4.24), the control can be derived in similar lines to (4.25) as (µ − µu ) = − eT I1 (z )

−1

− eT I1 (z )

−1

− eT I1 (z )

−1

τz eT − eT + eT G0 (z )

(µ)

eT G1 (z ) (µ − µx ) >z sgn(v(µ))

or equivalently (µ) = − eT I1 (z ) T

− e I1 (z ) − eT I1 (z )

−1

τz eT − eT + eT G0 (z )

−1

T

−1

(µ + µu )

e G1 (z ) (µ − µx + µu ) >z sgn(v(µ + µu ))

(4.29)

61

4.6 Quasi-Sliding Mode Control Algorithms for Form 2 Systems

But, the future state (µ + µu ) is not known by measurement. Hence, it has to be computed through extrapolation using the state equation of (4.28). (µ + µu ) = G0k (z ) (µ) +

k −1

Gi0 (z )G1 (z ) (µ − µx + µu − < − 1)(4.30)

i=0 k −1

+

Gi0 (z )I1 (z )(µ − < − 1)

i=0

(4.31) Remark 4.3. From (4.29), it can be seen that, the contribution of the retarded state term is eT G1 (z ) (µ − µx + µu ). Therefore, in order for the algorithm to function, we have the condition µ − µx + µu ≤ µ − 1, equivalently µx κ µu . Thus, the algorithm has the restriction that If the system has state delay, then the delay in the input channel must be less than that in the state. The restriction does not apply if the system does not have a state delay. Further, it can be seen that the control is predictive in nature. The control (µ) would satisfy the reaching law for v(µ+µu ), and hence the reaching phase cannot be expected before µ = µu in systems with retarded input channel. Substituting the value of (µ + µu ) from (4.30) into the control law (4.29), the state based control algorithm can be derived to be (µ) = − eT I1 (z )

−1

− eT I1 (z )

−1

− eT I1 (z )

−1

τz eT − eT + eT G0 (z ) G0k

(µ)

(4.32)

>z sgn(v(µ + µu )) (τz − J) eT

k −1

Gi0 (z )G1 (z ) (µ − µx + µu − < − 1)

i=0 T

− e I1 (z )

−1

T

T

T

k −1

τz e − e + e G0 (z )

Gi0 (z )I1 (z )(µ − < − 1)

i=0

− eT I1 (z )

−1 T

k

e

Gi0 (z )G1 (z ) (µ − µx + µu − < − 1)

i=0

4.6.2 Multirate Output Feedback Discrete-time Sliding Mode Control Algorithm for Form 2 Systems Proceeding in a manner similar to Section 4.5.2, substituting for (µ) in terms of the past system outputs and past state and input information, the multirate output feedback based control law can be derived as (µ) = − eT I1 (z )

−1

− eT I1 (z )

−1

τz eT − eT + eT G0 (z ) Gk0 (z )My >z sgn(v(µ + µu ))

k

4 MROF based QSMC in Time-Delay Systems

62

− eT I1 (z )

−1

− eT I1 (z )

−1

τz eT − eT + eT G0 (z ) G0k (z )Muĝ (µ − µx − 1)(4.33) (τz − J) eT

k −1

Gi0 (z )G1 (z ) (µ − µx + µu − < − 1)

i=0

− eT I1 (z )

−1

τz eT − eT + eT G0 (z )

k −1

Gi0 (z )I1 (z )(µ − < − 1)

i=0

− eT I1 (z )

−1 T

k

Gi0 (z )G1 (z ) (µ − µx + µu − < − 1)

e

i=0

2

2

1 0

2

0

State x

State x

1

4

−1

−2

−2

−4 −6

−3 0

10

20

30

Time in sec

40

−4

50

0

10

(a)

2

0.1

Sliding Function

0.2

0

Input

30

40

50

30

40

50

(b)

4

−2

0

−0.1

−4

−0.2

−6

−0.3

−8 −10

20

Time in sec

−0.4

0

10

20

30

Time in sec (c)

40

50

−0.5

0

10

20

Time in sec (d)

Fig. 4.2. System Plots for State Delay and Retarded Input Channel : a. Time Response of x1 , b. Time Response of x2 , c. Input Profile, d. Profile of the sliding function s(k)

4.6.3 Simulation Results Consider the system in the numerical example (4.27) with B0 = 0. The system is now affected by retarded input alone. A control is designed for this system using the control law discussed in (4.33) with τ = 0ρ005ε > = 0ρ01. The simulation results are presented in Fig. (4.2). The control input (y) = 6ε y ∈

4.7 Quasi-Sliding Mode Control Algorithm for Form 3 Systems

63

[−0ρ4ε 0]. It can be seen from Fig. (4.2.d) that the system converges to the sliding surface v(µ) = 0 very quickly.

4?t Qwa6i-Sliding Mode Con27ol Algo7i2hm fo7 Fo7m u S062em6 Systems with output or sensor delays are also very common in practical application. Many mechanical systems and thermal processes have sensor delays. A system with delays in state (due to transportation lag), input (due to actuator lag) and system output would have a discrete-time representation (µ + 1) = G0 (z ) (µ) + G1 (z ) (µ − µx ) + I0 (z )(µ) +I1 (z )(µ − µu )ε (µ) = C (µ − µy )ρ

(4.34)

4.7.1 Multirate Output Feedback based Control Algorithm The state equation of (4.34) is similar to that of (4.22). Therefore,the state based control law is also the same for both systems. It would be described by (4.25). However, since the output of the system is delayed in nature, the control law needs to be predictive in nature. Using (4.20), the retarded state (µ − µy ) can be expressed in terms of the multirate output as (µ − µy ) = My

k + Mx ĝ

(µ − µx − µy − 1) + Mu0 (µ − µy − 1) + Muĝ (µ − µu − 1) (4.35)

Remark 4.4. It can be inferred from (4.35) that for µ = 1 the value of (−µx − µy ) and (−µu − µy ) need to be available for the computation of the state vector (µ − µy ). Thus, it can be said that for systems with output delay more information is needed for the control implementation. The algorithm would need prior information of (µ)ε µ ∈ [−µx − µy ε 0] and (µ)ε µ ∈ [−µu − µy ε 0]. As with the case of Form 2 systems, the controller needs to be predictive in Form 3 systems also. In order to determine (µ) from the information available, the following equation may be used. k −1

k

Gi0 (z )G1 (z ) (µx,y,i )

(µ) = G0 (z ) (µ − µy ) + i=0 k −1

Gi0 (z )I0 (z )(µ + µy − < − 1)

+ i=0 k −1

Gi0 (z )I1 (z )(µ − µu + µy − < − 1)

+ i=0

64

4 MROF based QSMC in Time-Delay Systems k

= G0 (My

k

+ Mxĝ (µ − µx − µy − 1) + Mu0 (µ − µy − 1))

k −1

Gi0 (z )G1 (z ) (µ − µx + µy − < − 1)

+ i=0 k −1

Gi0 (z )I0 (z )(µ + µy − < − 1)

+ i=0 k −1

Gi0 (z )I1 (z )(µ − µu + µy − < − 1)

+ i=0 k

µx,y,i

+G0 (z )Muĝ (µ − µu − µy − 1) = µ − µx − µy − < − 1

Proceeding in a manner similar to that adopted in Section 4.6.2, the output feedback based discrete-time sliding mode control algorithm for systems with output delay can be expressed as follows. (µ) = − eT I0 (z )

−1

− eT I0 (z )

−1

T

− e I0 (z ) − eT I0 (z )

k

iG (z )G0 (z ) (My

k iG (z )G0 −1 k iG (z )G0 −1

k

+ Mxĝ (µx,y,i ))

(z )Mu0 (µ − µy − 1)

(4.36)

(z )Muĝ (µ − µu − µy − 1)

k −1

Gi0 (z )I0 (z )(µ + µy − < − 1)

iG (z ) i=0

− eT I0 (z )

−1

k −1

Gi0 (z )G1 (z ) (µ − µx + µy − < − 1)

(τz − J) eT i=0

− eT I0 (z )

−1

k −1

(τz − J) eT

Gi0 (z )I1 (z )(µ − µu + µy − < − 1) i=0

− eT I0 (z )

−1 T

k

Gi0 (z )G1 (z ) (µ − µx + µy − < − 1)

e

i=0

− eT I0 (z )

−1 T

k

Gi0 (z )I1 (z )(µ − µu − µy − 1)

e

i=0 T

− e I0 (z ) T

T

−1

>z sgn(v(µ))ε

T

iG (z ) = τz e − e + e G0 (z ) ρ

(4.37)

Remark 4.5. Similar to systems of Form 2, in systems with retarded output the delay in the output channel must be lesser than that in the state and input channels, i.e., µ y p µx ε µy p µu whenever state or input delay exists in the system.

65

4.7 Quasi-Sliding Mode Control Algorithm for Form 3 Systems

Remark 4.6. The technique of Form 2 and Form 3 systems can be combined to design a output feedback DSMC controller for systems with a retarded input (B0 = 0) and output delay. The controller structure is much similar to (4.36), but now would have a prediction horizon of µy + µu instead of µy . This means that the µy in the exponents and the summation terms of Eqn. (4.36) would now have to be replaced by µy + µu . The restriction on the delays in such systems would be µ x κ µu κ µy whenever a state delay exists in the system.

4

2 1

2

State x

State x

1

2 0

0

−1

−2 −4

−2

0

10

20

30

Time in sec

40

−3

50

0

10

40

50

30

40

50

0.05

Sliding Function

2 1

Input

30

(b)

(a)

3

0

−0.05

0

−1 −2

−0.1

−0.15

−3 −4

20

Time in sec

0

10

20

30

Time in sec (c)

40

50

−0.2

0

10

20

Time in sec (d)

Fig. 4.3. Plots for Systems with Output Delay : a. Time Response of x1 , b. Time Response of x2 , c. Input Profile, d. Profile of the sliding function s(k)

4.7.2 Simulation Results Consider the system in (4.27) with an output delay zy = 0ρ2 sec. An output feedback based DSMC control is designed for this system using the control law (4.36). The simulation results are shown in Fig. (4.3). Figs. (4.3(a) and 4.3(b)) give the state response of the system under the multirate output feedback control control algorithm for Form 3 systems. Fig.

66

4 MROF based QSMC in Time-Delay Systems

(4.3(c)) shows the input profile generated by the algorithm. The profile of the sliding function v(µ) is shown in Fig. (4.3(d)). The convergence of the time-delay system to quasi-sliding mode is clearly visible here.

4?8 Di6c7e2e-2ime Sliding Mode Con27ol of Fo7m 4 S062em6 Most practical systems have uncertainty in them due to unmodelled dynamics or disturbances affecting the systems. Time delay systems with uncertainty fall into the category of Form 4 systems [42]. They can be represented in discrete form as (µ + 1) = G0 (z ) (µ) + G1 (z ) (µ − µx ) + I0 (z )(µ) +I1 (z )(µ − µu ) + ηd (µ) (µ) = C (µ)

(4.38)

For this system the multirate output to system state relationship can be derived using Eqns. (4.20 and 4.21) as (µ) = My k + Mx (µ − µx − 1) + Mu0 (µ − 1) +Mu  (µ − µu − 1) + Md ηd (µ − 1)ρ where

Md = J − G0 (z ) C0T C0

−1

C0T Cd ρ

(4.39)

(4.40)

In order that the control be feasible and finite, it is assumed that the disturbance parameter ηd (µ) is bounded so that ηl ≤ eT ηd (µ) = η (µ) ≤ ηu ε gl ≤ eT G0 (z ) Md ηd (µ) = g˜ (µ) ≤ gu and we also define the mean value and spread of the above functions as (ηl + ηu ) (ηu − ηl ) ε fd = ε 2 2 (gl + gu ) (gu − gl ) ε fe = ρ g0 = 2 2

η0 =

4.8.1 Reaching Law Consider the reaching law designed in Section 3.2 based on that state feedback based control proposed in [7]. v (µ + 1) = vd (µ + 1) + η (µ) − η0 + g˜ (µ − 1) − g0 described in Section 3.2.

(4.41)

67

4.8 Discrete-time Sliding Mode Control of Form 4 Systems

4.8.2 Multirate Output Feedback based Discrete time Sliding Mode Control Law for Time-Delay Systems with Uncertainty From the reaching law (4.41), eT (µ + 1) = eT G0 (z ) (µ) + eT G1 (z ) (µ − µx ) +eT I0 (z )  (µ) + eT I1 (z )  (µ − µu ) + η (µ) = vd (µ + 1) + η (µ) + g˜ (µ − 1) − η0 − g0 ρ Therefore, to satisfy the reaching law, the control input  (µ) can be computed to be  (µ) = − eT I0 (z ) T

− e I0 (z ) + eT I0 (z )

−1 T

e (G0 (z ) (µ) + G1 (z ) (µ − µx ))

(4.42)

−1 T

e I1 (z )  (µ − µu )

−1

(vd (µ + 1) + g˜ (µ − 1))

T

− e I0 (z ) (η0 + g0 ) ρ Now substituting for (µ) in terms of the multirate sampled output control law can be obtained to be  (µ) = − eT I0 (z )

−1 T

− eT I0 (z )

−1 T

T

− e I0 (z ) − eT I0 (z ) T

− e I0 (z ) − eT I0 (z )

e G0 (z ) My

k,

the

(4.43)

k

e G1 (z ) (µ − µx )

−1 T

e G0 (z ) Mx

(µ − µx − 1)

−1 T

e G0 (z ) Mu0  (µ − 1)

−1 T

e I1 (z )  (µ − µu )

−1 T

e G0 (z ) Mu  (µ − µu − 1)

T

+ e I0 (z ) (vd (µ + 1) − η0 − g0 ) ρ Generation of Initial Control For µ = 0ε the information about (−µx − 1) and  (−µu − 1) is not available. Hence, the control law (4.43) cannot be directly applied. For this case, the control law  (0) is computed using (4.42) for a given initial state and data of initial delayed state and input and assuming g˜ (µ − 1)−g0 = 0. The assumption is due to the obvious reason that to generate  (0) the output information is not used and hence contribution of the disturbance vector in the system output need not be considered in constructing it. Since the same reaching law is used as in Section 3.2, the width of the quasi-sliding mode band would also remain the same, i.e., |v(µ)| p fd + fe .

4 MROF based QSMC in Time-Delay Systems

68

System States

10

5 0

0

State x1 State x2

−5

2

−10

X

−10

−20

−15

−30 −40

−20 0

10

20

Time, sec

30

−25 −40

40

−30

−20

−10

0

10

(b)1

(a)

10

0.5

Sliding Function

0

0

−0.5

Input

−10 −20

−1

−1.5

−30 −40

X

0

10

20

Time, sec

30

40

−2

−2.5

0

10

20

Time, sec

30

40

(d)

(c)

Fig. 4.4. Disturbance Example 1 : (a). State Response (b). Phase plot (c). System input (d). Sliding function

4.8.3 Simulation Results Example 1 For the system (4.27), the simulations were carried out with the following system and controller parameters. η(y) =

1ρ2 −0.4t g sin(2y) 2 T

b(0) = 4ρ43 0ρ42 (y) = 3ε y ∈ [−0ρ4ε 0) ε µ ∗ = 20ρ The simulation results are shown in Fig. (4.4). Fig. (4.4(a)) gives the state response of the system under the designed output feedback control algorithm. The phase plot of the system is shown in Fig. (4.4(b)). It can be clearly seen here that the system enters into a quasi-sliding mode and is confined to the vicinity of the sliding surface. Fig. (4.4(c)) shows the input profile generated by the algorithm. It can be seen that the control input is oscillatory. This oscillatory behavior is due to the presence of both (µ − µu ) and (µ) in the system input channel. The profile of the sliding function v(µ) is shown in

69

4.8 Discrete-time Sliding Mode Control of Form 4 Systems

10

5

0

0 −5

−10 −20

2

State x1 State x2

−10

X

System States

Fig. (4.4(d)). The convergence of the time-delay system to quasi-sliding mode is clearly visible here too. The decaying oscillations observed in the sliding function is due to the oscillatory nature of the disturbance considered.

−15

−30 −40

−20 0

10

20

Time,sec

30

−25 −40

40

−30

(a)

−10

0

X1

10

(b) 0.5

0

0

Sliding Function

10

−0.5

Input

−10 −20

−1

−1.5

−30 −40

−20

0

10

20

Time, sec (c)

30

40

−2

−2.5

0

10

20

Time, sec

30

40

(d)

Fig. 4.5. Disturbance Example 2 : (a). State Response (b). Phase plot (c). System input (d). Sliding function

Example 2 The disturbance signal considered in the previous example was one which decayed with time. Here the control law is tested against a sustained disturbance T η(y) = 1ρ2 2 sin(2y) and the simulation results are shown in Fig(4.5). For the disturbance defined above, the bounds were found to be η0 = g0 = 0, fd = 0ρ1783 and fe = 0ρ3822. It can be seen in Fig. (4.5(a)) that the system states converge to a band around origin. In Fig. (4.5(b)), the sliding mode behavior of the system can be observed in the phase portrait. The plot of the control input is shown in Fig. (4.5(c)). Fig. (4.5(d)) shows the time response of the sliding function v(µ). It can be seen here that the quasi-sliding mode band is well within the estimated bound of 0ρ5605. Thus, it may be said that the performance of the system may be much better than that suggested by the bound on quasi-sliding mode band. The sustained oscillations observed in Figs. (4.5(c) and 4.5(d)) is due to the sustained oscillations in the disturbance.

4 MROF based QSMC in Time-Delay Systems

70

4.8.4 Performance in System without Disturbance

10

5

0

0 −5

−10 −20

2

State x1 State x2

−10

X

System States

The case of the time-delayed system without disturbance with the application of the control law in Eqn. (4.43) is considered here. The results are shown in Fig. (4.6). It can be seen in Fig. (4.6(a)) that the system states converge to the origin. Fig. (4.6(d)) shows the time response of the sliding function v(µ). It can be seen here that the states converge to the sliding surface in finite time and without chatter.

−15

−30 −40

−20 0

10

20

Time (sec)

30

−25 −40

40

−30

X

−10

0

10

1

(a)

(b) 0.5

0

0

Sliding Function

10

−0.5

Input

−10 −20

−1

−1.5

−30 −40

−20

0

10

20

Time (sec) (c)

30

40

−2

−2.5

0

10

20

Time (sec)

30

40

(d)

Fig. 4.6. System without Disturbance : (a). State Response (b). Phase plot (c). System input (d). Sliding function

5 Mwl2i7a2e Ow2pw2 Feedback Sliding Mode fo7 Special Cla66e6 of S062em6

5?1 Mwl2i7a2e Ow2pw2 Feedback Di6c7e2e-2ime Sliding Mode Con27ol ba6ed T7acking Con27olle7 fo7 Nonminimwm Pha6e S062em6 5.1.1 Introduction The tracking problem of systems is one that has come under a lot of investigation. The initial studies were on the transfer function models of the system, based on internal model control [23], which used the inverse dynamics of the system to design a tracking controller. The problem with this simple logic arouse in cases wherein the system was nonminimum phase, i.e., had unstable zeros. In such a case, the inverse dynamics were unstable, which generated a closed loop system that may be stable , but was not internally stable [1]. Further study on the topic brought into light numerous techniques that handled the tracking problem, for both linear [5, 43] and nonlinear systems [16, 17, 32]. The research on unstable zero dynamics was also extended to discrete-system representations [26, 59]. In recent times, the concept of sliding mode control [21, 73] is being used for the tracking of nonminimum phase systems [5, 26, 43, 59] for its inherent insensitivity to plant parameter variations. A sliding mode tracking control has been proposed in [43] that uses a ’two part’ control for continuous-time systems with unstable zero dynamics. A reference preview based predictivetype of control has been discussed in [59] to handle unstable zeroes in discretetime representations. Most of the aforementioned control strategies fall into one of the two categories. They are either state feedback based strategies or are based on dynamic output feedback. As already mentioned in the introduction, it may not always be possible to implement a state feedback based control law as all the states may not be available for measurement.

B. Bandyopadhyay, S. Janardhanan: Discrete-time Sliding Mode Contr., LNCIS 323, pp. 71–103, 2006.

© Springer-Verlag Berlin Heidelberg 2006

72

5 Multirate Output Feedback Sliding Mode for Special Classes of Systems

This section discusses a multirate output feedback sliding mode control based algorithm for tracking control of a SISO nonminimum phase discretetime LTI system representations [39]. 5.1.2 Problem Statement Consider the stable LTI discrete time system obtained for a sampling time of z as v (µ + 1) = Sτ v (µ) + Hτ  (µ) ε

(5.1)

(µ) = Cv (µ) ρ v ∈ Rn ε  ∈ Rε ∈ R The system output has a relative degree of ν ≥ 1 and therefore can also be represented in the following manner 1 (µ

+ 1) = A11

2 (µ

+ 1) = A21 1 (µ) + A22 2 (µ)ε (µ) = 1 0 · · · 0 1 (µ)ε 1

∈ Rl ε

1 (µ)

2

+ A12

2 (µ)

+ B1 (µ)ε

(5.2)

∈ Rn−l

where the ν-th order subsystem for 1 is in the normal. The system (5.1) is stable and which may be nonminimum phase and is required to follow the reference signal θ which is the output of the ν-th order system described by the stable and bounded dynamic equations ‫( ܈‬µ + 1) = R ‫( ܈‬µ)

(5.3)

‫܈‬i (µ + 1) = ‫܈‬i+1 (µ) ε < = 1ε 2ε · · · ε ν − 1 θ (µ) = 1 0 · · · 0 ‫( ܈‬µ) ‫ ∈ ܈‬Rl ε θ ∈ R q≤ν≤φ The relationship between the system representations (5.1) and (5.2) is given by an invertible transformation 1 (µ) 2 (µ)

= X v(µ)

(5.4)

The problem of designing a tracking controller for the above system with zero dynamics may be solved in the following manner. 5.1.3 Two-part Control A method was proposed in [43] to tackle the tracking problem in case of continuous-time system representations using state feedback. This section introduces an output feedback based method to tackle the tracking problem in discrete-time systems.

5.1 MROF-based Nonminimum Phase System Tracking

73

Let us assume that the system output is following the reference trajectory then the system error dynamics would be ∆ 1 (µ) = ∆ 1 (µ + 1) =

1 (µ)

− ‫(܈‬µ) = 0 1 (µ + 1) − ‫(܈‬µ + 1) = 0

= A11 1 (µ) + A12 2 (µ) + B1 (µ) − R ‫(܈‬µ) = A11 ‫(܈‬µ) + A12 2,0 (µ) + B1 (µ) − R ‫(܈‬µ) = 0ε where 2,0 (µ) is the nominal trajectory followed by 2 (µ) when the state 1 is exactly following its reference ‫(܈‬µ). Thus, the nominal input 0 (µ) to maintain the state on the reference trajectory can be computed as 0 (µ) = − B1T B1

−1

B1T (A11 − R ) ‫(܈‬µ)

− B1T B1

−1

B1T A12

2,0 (µ)ρ

(5.5)

The mismatch dynamics of the system can now be expressed as ∆ ∆

1 (µ

+ 1) = A11 ∆ 2 (µ + 1) = A21 ∆

1 (µ)

+ A12 ∆ 1 (µ) + A22 ∆

2 (µ)

+ B1 ∆(µ)ε 2 (µ)ρ

(5.6)

The incremental input ∆(µ) is the input that needs to be pumped to the system so that the mismatch dynamics are stabilized. This can be set as ∆ ∆

∆(µ) = F

1 (µ)

2 (µ)

+ ksgn(v(µ))ρ

∆(µ) can be obtained by using reaching law based sliding mode control for the system of the the error dynamics. The sliding function v(µ) is constructed as v(µ) = ∆ 1 (µ) + µ∆ 2 (µ)ρ (5.7) Thus, the control input to the system would be (µ) = 0 (µ) + ∆(µ) = − B1T B1

−1

B1T (A11 − R ) ‫(܈‬µ)

− B1T B1

−1

B1T A12

+kρsgn(v(µ))ρ 1 (µ) =F − 2 (µ)

where

d

2,0 (µ)

+F

∆ ∆

1 (µ)

2 (µ)

d (µ)

−

B1T B1

−1

B1T (A11 − R ) ‫(܈‬µ)

−

B1T B1

−1

B1T A12

2,0 (µ)

+ kv<jφ (v(µ)) ρ

is the nominal state vector represented as

(5.8)

74

5 Multirate Output Feedback Sliding Mode for Special Classes of Systems d (µ)

= ‫(܈‬µ)T

T T 2,0

ρ

Now, a control can be designed to enable the φth order system to track a given νth order reference signal. However, this needs the knowledge of the nominal trajectories of the system. The nominal trajectory of the state 1 can be easily inferred to be the reference state ‫܈‬ρ But, the nominal trajectory of the zero dynamics 2 cannot be inferred directly from the reference. A method to accomplish this proposed in [59] is used here. Remark 5.1. It is worth to note here that the controller can also be designed using the approach in section 3.2 instead of the switching function based sliding mode control approach. 5.1.4 Determination of the Nominal Zero Dynamics Trajectory (x2,0 ) The Infinite Horizon Method Consider the system (5.2) to be moving along the nominal trajectory with the nominal input being applied to the system ( the incremental input would be equal to zero as there is no error in the tracking). Then, the system dynamics can be represented as ‫( ܈‬µ + 1) = Rρ‫(܈‬µ)ε 2,0 (µ + 1) = A21 ‫( ܈‬µ) + A22

2,0 (µ)ρ

(5.9)

A point to be noted at this juncture would be that since the system is assumed to be in the normal from, the structure of ‫(܈‬µ) can be said to be as ‫(܈‬µ) = θ(µ) θ(µ + 1) · · · θ(µ + ν − 1)

T

= θk,l ρ

(5.10)

It can also be seen here clearly that the zero dynamics are not determinable from the reference. A method has been suggested in [59] to overcome this difficulty. We define a transformation X2 such that   As 0 0  0 Am 0  = X2 A22 X2−1 ε 0 0 Au  s 

2 m 2 u 2

= X2

2ε

where As has all strictly stable eigenvalues of A22 ε Am has the marginally stable eigenvalues and Au has all unstable eigenvalues of A22 . Applying this transformation to the zero dynamics equation in (5.9), we get

5.1 MROF-based Nonminimum Phase System Tracking s 2 (µ m 2 (µ u 2 (µ

+ 1) = As s2 (µ) + A21,s ‫(܈‬µ)ε + 1) = Am m 2 (µ) + A21,m ‫(܈‬µ)ε + 1) = Au

u 2 (µ)

+ A21,u ‫(܈‬µ)ρ

75

(5.11) (5.12) (5.13)

If the values of s2 and m 2 are known for some initial instant µi ε then their further values on the nominal trajectory can be obtained by simple simulation using (5.11 and 5.12). But, as the eigenvalues of Au are unstable this method cannot be used to obtain the solution of (5.13). It would give an unbounded trajectory for u2 . However, if the final value of u2 is known, the solution of (5.13) can be obtained by solving it backwards in time. Thus, assuming that the zero dynamics are unactuated initially, i.e., 2 (−∞) = 0 and that the reference trajectory is zero for time instants µ p 0, we have the solution for (5.11, 5.12 and 5.13) as s 2 (µ)

k−1

=

As(k−i) (A21,s ‫))<(܈‬

(5.14)

(k−i) Am (A21,m ‫))<(܈‬

(5.15)

Au−(i+1−k) (A21,u ‫))<(܈‬

(5.16)

i=0 m 2 (µ)

u 2 (µ)

k−1

= i=0 ∞

= i=k

Restrictions on the Reference Signal It should be noted here that since the reference is bounded, the system in (5.11) has a bounded response. However, the same cannot be commented about the system in (5.12). This is unity for all values of µ − < and therefore is because the norm of Ak−i m the weightage of ‫(܈‬µ − <) does not diminish as (µ − <) → ∞. Hence, it imposes a condition that the reference signal must be such that it stabilizes the marginally stable zero dynamics. With the generated zero dynamics from (5.14, 5.15 and 5.16), the desired (nominal) state can be constructed as    ‫(܈‬µ) s   ‫(܈‬µ) 2 (µ) =  (5.17) d (µ) =  −1  m   (µ) (µ) X2 2,0 2 u 2 (µ) Substituting this desired state into the control input function in (5.8) would result in the desired input for tracking of the nonminimum phase system. From the above equations, it can be seen that the controller so developed is predictive. The computation of the stable and marginally stable zero dynamics would require the desired output ν steps in advance. However, the computation of the unstable part depends on the total future of the desired output trajectory, which would not be a computationally elegant technique.

76

5 Multirate Output Feedback Sliding Mode for Special Classes of Systems

An alternative would be a trade off between the accuracy of the desired state u 2 and a bounded preview of the reference trajectory. This mode of controller design is handled in the next section. Bounded Preview Method The case where the preview time is limited is considered here. We will assume that the number of time steps for which the reference signal is known is limited to Xpre ≥ νρ It can be concluded with the help of ( 5.10) that the bounded preview time does not affect the requirements of (5.14 and 5.15). But, this is not the case with (5.16). Setting all unknown terms of of ‫ = )<(܈‬0ε ∀< κ (µ + Xpre ), we would come to the following estimate of u2 (µ)ρ T

ˆu2 (µ)

ĝ

Au−(i+1) (A21,u ‫(܈‬µ + <)) ρ

=

(5.18)

i=0

The error of the above estimate can be found to be ∞

Au−(i+1) (A21,u ‫(܈‬µ + <)) ρ

gpre (µ) = − i=T

(5.19)

ĝ

By the boundedness of the reference signal and the stable eigenvalues of the matrix A−1 u ε the above error signal is bounded as well. The error in the estimation of state can be found as gx (µ) = ˆd (µ) − d (µ)     0   0  gx (µ) =  ρ  X2−1  0 gpre (µ) The error is the state is driven by the bounded error gpre and hence would be bounded as well. Since, the estimation error does not affect the state ‫ˆ܈‬, there would be no error in the output tracking. 5.1.5 Multirate Output Feedback Based Tracking Control As pointed out in the introduction, all the system states may not be available for measurement. Therefore, the control law in (5.8) should be realized using only the system outputs and past inputs. This can be accomplished in the following manner. [39] Consider the system (5.1). Let the system representation for a sampling time of ∆ = z xPε P κ π, where π is the observability index of the system, be v (µ + 1) = Sv (µ) + H  (µ) (µ) = Cv (µ)

(5.20)

5.1 MROF-based Nonminimum Phase System Tracking

77

then a multirate output feedback representation of the system, with the output sampled at an interval ∆ and an input sampling interval z , would be v (µ + 1) = Sτ v (µ) + Hτ  (µ) ε k+1 = C0 v(µ) + D0 (µ)ε

(5.21)

where k+1 ε C0 ε D0 have the definitions described in (1.23) and (1.26). The value of v(µ) can be computed in terms of k and (µ) using (1.29) as v(µ) = My

k

+ Mu (µ − 1)ρ

(5.22)

The overall control input given to the system can be computed from (5.8) using (5.4, 5.10, 5.17 and 5.22) as (µ) = F X My

k

+ F X Mu (µ − 1)

−

B1T B1

−F

d (µ)

−1

−

B1T (A11 − R ) θk,l B1T B1

−1

B1T A12

2,0 (µ)

+ksgn(v(µ)) where

 2,0 (µ)

= X2−1  

  = X2−1  =



s 2 (µ) m  2 (µ) u (µ) 2

 = X2−1 

d (µ)

(5.23)



k−1 (k−i) (A21,s ‫))<(܈‬ i=0 As  k−1 (k−i)  (A21,m ‫))<(܈‬ A m i=0 T ĝ −(i+1) (A21,u ‫(܈‬µ + <)) i=0 Au  k−1 (k−i) (A21,s θi,l ) i=0 As  k−1 (k−i)  (A21,m θi,l ) i=0 Am T ĝ −(i+1) (A θ )) A u 21,u k+i,l i=0

θk,l 2,0 (µ)

v(µ) = eT X v(µ) = eT X (My

k

+ Mu (µ − 1)) ρ

5.1.6 Numerical Example Consider the discrete-time system with zero dynamics as     0 1 0 0 v (µ + 1) =  −ρ7 1ρ5 −0ρ1  v (µ) +  1   (µ) ε 0ρ2 0ρ1 1ρ05 0

78

5 Multirate Output Feedback Sliding Mode for Special Classes of Systems 3

Reference Signal System Output

2

Amplitude

1

0

−1

−2

−3

0

100

200

300

400

500

Sampling Instants (k)

600

Fig. 5.1. Comparative plot of the system output and the reference signal

(µ) = 1 0 0 v(µ) v(0) = 1 0 0

T

The system dynamics can be split up as 1

(µ + 1) =

0 1 −0ρ7 1ρ5 +

2

1

(µ) +

0 −0ρ1

2

(µ)

0  (µ) 1

(µ + 1) = 0ρ2 0ρ1

1

(µ) + 1ρ05

2

(µ)

From the state equations it can be observed that the system has zero dynamics and the zero dynamics is unstable. The system is required to track the signal ‫(܈‬µ + 1) =

0 1 ‫(܈‬µ) −0ρ9775 1ρ524

θ(µ) = 1 0 ‫(܈‬µ) ‫(܈‬0) = −1 0

T

The sliding line for the error dynamics is designed as

5.1 MROF-based Nonminimum Phase System Tracking 0.3 0.2 0.1

Control Input (u)

0 −0.1 −0.2 −0.3 −0.4 −0.5 −0.6 −0.7

0

50

100

150

200

250

300

Time Samples (k)

350

400

450

500

400

450

500

Fig. 5.2. Plot of the control input 2 1.5 1

Tracking Error

0.5 0

−0.5 −1

−1.5 −2 −2.5 −3

0

50

100

150

200

250

300

Time Samples (k)

350

Fig. 5.3. Plot of the tracking error in the system output

79

80

5 Multirate Output Feedback Sliding Mode for Special Classes of Systems 0.5 0 −0.5

Zero Dynamics (X3)

−1 −1.5 −2

−2.5 −3 −3.5 −4 −4.5

0

100

200

300

Time Samples (k)

400

500

600

Fig. 5.4. Plot of the bounded zero dynamics of the system

v (µ) = −0ρ6623 1ρ0 0ρ1230 ∆ (µ) and the bounded preview time Xpre is chosen as 10 samples. The control is designed as 

T −5ρ156  (µ) =  11ρ112  −5ρ985

T 0ρ1456  0ρ25  k − 0ρ0693

+1ρ616(µ − 1) − −0ρ01

2,0



0ρ2755 −0ρ324

T

d

(µ)

θ (µ) θ (µ + 1)

− 0ρ01ρsgn (v (µ)) ρ

The simulation results are presented here. It can be seen from Fig. (5.1) that the system is able to track the reference signal. The tracking error due to the usage of bounded preview is found to be of negligible magnitude, as shown in Fig. (5.3). The control input generated by the multirate output feedback algorithm is shown in Fig. (5.2). From Fig. (5.4), it can be seen that the output feedback control algorithm not only tracks the signal but also keeps the zero dynamics bounded.

5.2 MROF-QSMC for Finitely Discretizable Nonlinear Systems

81

5?v Mwl2i7a2e Ow2pw2 Feedback ba6ed Qwa6i-Sliding Mode fo7 a Cla66 of Nonlinea7 S062em6 5.2.1 Introduction The application of sliding mode control strategy to nonlinear system representations has received considerable attention in the recent years [48, 60, 65, 84]. A model predictive sliding mode control algorithm for continuous-time nonlinear systems has been proposed in [84]. In [60], a dynamic sliding surface based adaptive algorithm has been proposed for sliding mode in discrete-time nonlinear systems of a specific structure. But, the above said algorithms are state feedback based. As discussed in the introductory chapter, all the system states are not always available for measurement. Hence, the system output should be used for control purposes. Output feedback based sliding mode control laws have been proposed in [48,65]. However, these control laws require the sliding surface to be a function of the system outputs. It is not always possible to represent a stable sliding surface in terms of the system output, which restricts the scope of possible sliding manifolds. Even in case of a output feedback based sliding surface is successfully constructed, it cannot ensure that the system as a whole would always be stabilized [72]. This section presents a method for the multirate output feedback based discrete-time sliding mode control of a class of nonlinear systems by using the concept of finite discretizability [10]. It uses the reaching law concept proposed in [27]. 5.2.2 Background: Finitely Discretizable Systems Definition Let = ( 1 ε · · · ε n ) be the local coordinates for an open neighborhood of τ, defined as Yq ⊂ N . where N is a real analytical connected φ-dimensional manifold. Consider the locally-controllable and observable nonlinear system of the form m

i (y)bi ( (y))

W : ˙ (y) =

(5.24)

i=1

(y) = j ( (y)) where b1 ε · · · ε bm are real analytical vector fields on Yq ,  = (1 ε · · · ε m ) ∈ Rm and j : Rn → Rp is a polynomial function of . The solution of (5.24) corresponding to a constant control (y) =  ¯, for m  ¯ b and J is the y ≥ 0, is denoted (exp yβ ) (Jd )|x(0) , where β = d i=1 i i identity function.

82

5 Multirate Output Feedback Sliding Mode for Special Classes of Systems

Definition 5.2. [10] The nonlinear system W is said to be finitely discretizable at the order π ≥ 1 if the solution of (5.24), corresponding to a constant control (y) =  ¯ for y ≥ 0 is a polynomial of degree π − 1 in y, ∀y ≥ 0ε ∀¯  ∈ Rm and (y0 ) ∈ Yq i.e., (y + y0 ) = (exp yβ ) (Jd )|x(t0 ) = (Jd )|x(t0 ) + yβ (Jd )|x(t0 ) + · · · +

(5.25)

yν−1 β (ν−1) (Jd )|x(t0 ) (π − 1)!

∀y ≥ 0ε ∀¯  ∈ Rm ε ∀ (0) ∈ Yq ρ  In other words, one has β ν+µ (Jd )|x(t0 ) = 0ε ∀. κ 0ρ

(5.26)

Thus, if the system is discretized at a sampling interval of z sec, the discrete-time representation would be ((µ + 1)z ) = (exp yβ ) (Jd )|x(kτ ) = (Jd )|x(kτ ) + yβ (Jd )|x(kτ ) + · · · +

(5.27)

ν−1

y β (ν−1) (Jd )|x(kτ ) (π − 1)!

Remark 5.3. The system is considered to be driftless only for the convenience of notation. All the definitions can be extended to drift systems by adding, say, m+1 ≡ 1 Sufficient Condition of Finite Discretization We will use the capital letter J = (<1 ε · · · ε <m ) to denote multi-indices with <µ ∈ Nε . ≤ ∂. We also define |J| = <1 + · · · + <m

i b I = b1i1 · · · bm i1 i I x b = b1 x · · · xbm ∂Y where bβ = ∂Y ∂x b, ∂x representing the Jacobian matrix [47] and “ x” denotes the shuffle product inductively defined on the length as follows

bxJd = Jd xb = b b i xβ j = b b i−1 xβ j + β b i xβ j−1 ρ The shuffle product is associative and commutative.

5.2 MROF-QSMC for Finitely Discretizable Nonlinear Systems

83

Definition 5.4. Consider Rn with the coordinates = ( 1 ε · · · ε n ). A dilation is a map ft : R+ ×Rn → Rn is of the form ft (λ( )) = λ (yr1 1 ε · · · ε yr n ), where λ : Rn → Rn is a polynomial in and we assume that θi ∈ Nε < ≤ φε θi ≤  θi+1 . Definition 5.5. A polynomial λ : Rn → Rn is homogeneous of degree m ∈ Z with respect to a dilation ft if ft∗ = λ ◦ ft = yj λ. Let I be the algebra of real polynomial functions in ( 1 ε · · · ε n ), we define Ij = {λ ∈ Iε ft∗ λ = yj λ} and set Ij = {0}ε ∀m p 0ε then I = ⊕j≥0 Ij . We denote by Rj the set of all  polynomials homogeneous of degree ≤ m i.e., Rj = ⊕jl=0 Il Definition 5.6. A polynomial vector field b is said to be homogeneous of degree −v ∈ Z with respect to a dilation ft if ∗

(ft ) b (λ) = ys ft∗ (b (λ)) ε λ ∈ I or equivalently b(λ) ∈ Ij−s if λ ∈ Ij .



Theorem 5.7. Let b1 ε · · · ε bm be real analytical vector fields, with polynomial coefficients, homogeneous of degree −1 with respect to the dilation ft ( ) = ( r11 ε · · · ε rn ), then W is finitely discretizable at most of the order  θn + 1. Proof. [10] Let the identity function Jd = (Jd,1 ε · · · ε Jd,n ), with Jd,i ( ) = i , then Jd,i ( ) is a polynomial function homogeneous of degree θi with respect to ft . By virtue of definitions 5.5 and 5.6, it is immediate that b I Jd,i ( ) ∈ Rr، −|I| . Thus, if |J| κ θn = sup θi , then b I Jd,i ( ) = 0ε ∀< ∈ {1ε · · · ε }. In particular, since b xI Jd,i ( ) = 0, for |J| κ θn ε ∀< ∈ {1ε · · · ε φ}. Hence, any family F of polynomial vector fields homogeneous of degree −1 generates a finite discretized of order at most θn + 1. 5.2.3 Multirate Output Sampling in Nonlinear Systems Consider the nonlinear system (5.24). Let W be controllable, observable, and finitely discretizable. Let the system input is given with a sampling interval of z sec and the outputs i are sampled at intervals ∆i = z xPi ε Pi ∈ Nε < = 1ε · · · ε q. It can then be shown that the system states can be expressed as a function of past Pi samples of outputs i and immediate past control input [41]. Proof. Since = j( ) is a polynomial function in , using the result that the finite discretization property is preserved under polynomial transformation [10], it can be said that would also be finitely discretizable. i (y + z )ε < ∈ Nε < ≤ q would therefore be of the form i (y0

+ z) =

i (y0 )

+z

(1) i (y0 )

+ ··· +

z N، −1 (Pi − 1)!

(N، −1) (y0 ) i

(5.28)

84

5 Multirate Output Feedback Sliding Mode for Special Classes of Systems

¯ε ∀y ∈ [y0 ε y0 + z ). Due to the assumption for some Pi ∈ Nε {zε y} ∈ R+ ε (y) =  that the system W is observable, the system states can be expressed as (y0 ) = i

ε ˙ε

(2)

ε···ε

(Nmax )−1

ε ¯

(5.29)

Pmax = sup Pi i=1,···,p

where Pi is the highest order derivative of i appearing in the nonlinear continuous-time observer. Now if the system input  is applied and held constant for every z sec interval and each of the system outputs i is sampled at (k) (k) a rate ∆i = Nτ، , then using (5.28), and using i to denote i (µz ) i (µz ) i (µz

=

+ ∆i ) =

(0) i ε (0) i

+ ∆i

(1) i

+ ··· +

، −1 ∆N i (Pi − 1)!

(N، −1) ε i

.. . i

((µ + 1)z − ∆i ) =

(5.30) (0) i

+ ··· +

((Pi − 1)∆i )N، −1 (Pi − 1)!

(N، −1) ρ i

The left hand side of the above set of Pi equations constitute the multirate output samples for i . The equations are independent in the Pi variables (N، −1) (1) and hence the output derivatives can be obtained by solviε i ε · · · ε i ing (5.30).     (0) i (µz ) i   (1)  i (µz + ∆i )   i   −1    (5.31) = A   . ..  i  ..     . (N، −1) i ((µ + 1)z − ∆i ) i t=kτ   1 0 ··· 0 i −1   ∆، ،  1 ∆i ··· (N، −1)!)   (5.32) Ai =  . ρ .. ..   .. . .   (i، −1) ،) 1 (Pi − 1)∆i · · · ((N، −1)∆ (N، −1)! This along with the observability condition in (5.29) and the discrete state equation (5.27) would mean that one can derive the system state information at y = (µ + 1)z by measuring the outputs i for the period y ∈ [µzε (µ + 1)z ) ¯ constant with a sampling interval ∆i respectively and holding the input  =  during the same period. Hence, the observability of the continuous time system along with the finite discretization property ensures that a multirate output sampling interval of Nτ، for each output i for the input sampling interval of z is a sufficient condition for the discrete-time observability of the nonlinear system.

85

5.2 MROF-QSMC for Finitely Discretizable Nonlinear Systems

5.2.4 Discrete-time Sliding Mode Control for Nonlinear Systems State based Control Using a strategy similar to that discussed in [27], we first design stable sliding surfaces vi (y) = 0ε < = 1ε · · · ε ∂ by finding the relationship between the states so that a chosen candidate Lyapunov function a ( ) has a˙ ( ) p 0. Since, the system stability is conserved on discretization, the same sliding surfaces vi (µz ) = vi (µ) = 0 would also be stable for the discrete-time system representation. For the reminder of the section, the notation (µ) is used instead of (µz ) for brevity. Now applying the reaching condition vi (µ + 1) − vi (µ) = −τi z vi (µ) − >z sgn(vi (µ)) and substituting the value of (µ + 1) from the discrete system representation (5.27), one can solve for i (µ) and obtain the control inputs that would guide the system along the chosen sliding surfaces. Multirate Output Feedback Control As discussed in the introduction, the above algorithm may not be always implementable because all the states may not be measurable, or even physical variables. It has already been shown in Section 5.2.3 that the observability and finite discretizability of the system ensures that each of the system states can be represented as a function of the past Pi multirate samples of the output and the past input i (µ − 1) [41]. Thus, the state based control derived in Section. 5.2.4 can now be easily translated to one that is based on past output samples and the immediate past control signal, whenever the finitely discretizable system is observable in continuous time by using the procedure described in Section 5.2.3. 5.2.5 Illustrative Example The above said multirate output feedback based discrete-time sliding mode control technique has been illustrated in the following example. Consider the following continuous time system representation defined in the manifold Yp : 1 κ −1ε { 2 ε 3 } ∈ R2 ˙ 1 = 1 ˙ 2 = 2 ˙3 = 1 1 2

=

(5.33) 2

+

2

1 3

The system has vector fields b1 = r1 ε b2 = r2 and the drift vector field β = ( 1 2 + 2 ) r3 , where ri denotes the partial derivative with respect to

86

5 Multirate Output Feedback Sliding Mode for Special Classes of Systems

It can be verified that relative to the dilation ft = y 1 ε y 2 ε y3 3 , the vector fields are homogeneous of degree −1. The verification for b1 has been shown here. i.

∗

(ft ) b1 ( ) = b1 (ft ( )) = b 1 y 1 ε y 2 ε y3 3 r = y 1 ε y 2 ε y3 3 r 1 = (yε 0ε 0) ρ r ft (b1 ( )) = ft ( 1ε 2ε 3) r 1 = ft ((1ε 0ε 0)) 0 = ft 1 ε 0ε 0 = (y

1)

0

ε 0ε 0

= (1ε 0ε 0) ρ Hence, it can be seen ∗

(ft ) b1 ( ) = y1 (ft∗ (b1 ( ))) Therefore, the vector field b1 is homogeneous of degree −1 with respect to the dilation ft . The property can be easily verified for the other vector fields also. Therefore, the system (5.33) is finitely discretizable. For a sampling time of z sec the discrete-time representation can be given as (µ + 1) = 2 (µ + 1) = 3 (µ + 1) = 1

(µ) + z 1 (µ) ε 2 (µ) + z 2 (µ) ε 3 (µ) + z ( 1 (µ) 2 (µ) + z2 + ( 2 (µ)1 (µ) + (1 + 2 z3 + 1 (µ)2 (µ)ρ 3

(5.34)

1

2 (µ)) 1 (µ)) 2 (µ))

Design of Sliding Surfaces The system is a multi-input system and hence requires the design of two sliding surfaces. Dividing the system (5.33) into two coupled sub-systems with states ( 1 ) and ( 2 ε 3 ), it can be observed that the only possible sliding surface for the former system would be (5.35) v1 = 1 = 0 and in order to obtain the sliding surface for the second subsystem we use the x2 candidate Lyapunov function a = 23 , which would give

87

5.2 MROF-QSMC for Finitely Discretizable Nonlinear Systems

a˙ =

3

(

1 2

+

2)

and thus a stable sliding surface for this sub-system would be v2 =

3

+

1 2

+

2

(5.36)

=0

Multirate Output Sampling based Nonlinear Observer From the discrete model (5.34), it can be said that if the outputs have multiplicities as P1 = 2ε P2 = 4, then it would be a sufficient condition for the system states to be computable through multirate output sampling. However, by choosing P1 = 1ε P2 = 2, i.e., ∆1 = zε ∆ = ∆2 = τ2 , the discrete-time observer can be derived as 1 (µ)

=

11 (µ)

+ z 1 (µ − 1)ε i1 (µ) 1 ε 2 (µ) = 3∆ (2 11 (µ) + 1 (µ − 1) ∆ + 2) i2 (µ) 1 ρ 3 (µ) = 3 (2 11 (µ) + 1 (µ − 1) ∆ + 2)

(5.37) (5.38) (5.39)

where i1 (µ) = 6 ( 22 (µ) − 21 (µ)) + 9∆2 2 (µ − 1) ( 11 (µ) + 1) +4∆3 1 (µ − 1) 2 (µ − 1) ε 2 i2 (µ) = 6 11 (µ) + 1 ∆2 2 (µ − 1) + 12 11 (µ) 22 (µ) +12

11 3

(µ) ∆3 1 (µ − 1) 2 (µ − 1) + 12

22

(µ)

+12∆ 1 (µ − 1) 2 (µ − 1) + 12 ( 11 (µ) + 1) ∆2 2 (µ − 1) −6 ( 11 (µ) + 1) 21 (µ) + 31 (µ − 1) ∆ (4 22 (µ) − 3 21 (µ)) −4∆2 2 (µ − 1) 21 (µ − 1) ∆2 − 3 ε 11 (µ)

=

21 (µ) = 22 (µ) =

1 (µ

− 1)ε

2 (µ − 1)ε 2 (µz − ∆)ρ

Thus, the system states can be derived using the past Pi multirate output samples and the immediate past control signals. Remark 5.8. It is to be noted here that during the estimation of the states 2 (µ) and 3 (µ) a singularity would occur whenever (2

11 (µ)

+ 1 (µ − 1) ∆ + 2) = 0ρ

Therefore, the control signal 1 should be computed in such a manner that this condition is avoided.

5 Multirate Output Feedback Sliding Mode for Special Classes of Systems

88

Controller Design Computation of 1 (µ) Using the Gao’s [27] reaching law for v1 (µ), the control signal 1 (µ) can be derived as v1 (µ + 1) − v1 (µ) = −τ1 z − >1 z sgn(v1 (µ)) z 1 (µ) = −τ1 z − >1 z sgn(v1 (µ)) 1 (µ) = −τ1 − >1 sgn(v1 (µ))

(5.40)

with the restrictions on τ1 ε >1 as τ1 ε >1 κ 0ε

(5.41)

1 − τ1 z κ 0ρ

(5.42)

This control would ensure that the state 1 (µ) converges monotonically to within the quasi-sliding mode band of width given by f1 = The condition

>1 z ρ 2 − τ1 z

(5.43)

f1 p 1

(5.44)

is imposed so that the sliding mode control 1 has a quasi-sliding mode band completely inside Yp . If the value of  (µ) is substituted from (5.40) into (2 11 (µ) + 1 (µ − 1) ∆ + 2) and then equated to zero, we get the disallowed state as follows. 1. For

1

κ 0ε (

1

(µ) + 1) +

1

(µ) 1 −

∆ (−τ1 2

1

(µ) − >1 )

= 0ε

∆ >1 − 1 ε 2 (>1 ∆ − 2) ρ 1 (µ) = (2 − τ1 ∆)

∆ τ1 2

=

Since the Gao’s reaching law stipulates 1 − τ1 ∆ κ 0ε the denominator would always be positive, thus if it is ensured that (>1 ∆ − 2) p 0ε the above case can be completely ignored.

(5.45)

5.2 MROF-QSMC for Finitely Discretizable Nonlinear Systems

2. For

1

89

p 0ε (

1

(µ) + 1) +

1

(µ) 1 −

∆ (−τ1 2

∆ τ1 2

1

(µ) + >1 )

= 0ε

∆ >1 + 1 ε 2 (∆>1 + 2) ρ =− (2 − ∆τ1 ) =−

d1

This case can also be ignored provided it is ensured that the disallowed states falls outside the manifold Yp . That is by imposing the condition (∆>1 + 2) κ 1ρ (2 − ∆τ1 )

(5.46)

Since the initial state 1 (0) would be inside the manifold Yp and the control 1 would take it monotonically to a band of width f1 p 1, the disallowed state 1 (µ) = d1 would not be encountered. 3. And the special case of 1 (µ) = 0ε In this case, the system becomes of a reduced order and hence it is the observer that has to be modified (and not the control input, which would obviously be 1 (<) = 0ε < ≥ µ). The new discrete state equations would be 2

(µ + 1) =

2

(µ) + z 2 (µ) ε

3

(µ + 1) =

3

(µ) + z

Hence, in this case,

(2,3)

2

(µ) +

z2 2 (µ) ρ 2

(µ + 1) can be estimated as

(µ) + 32 2 (µ) ∆2 ε ∆ 2 3 (µ + 1) = 2 22 (µ) − 21 (µ) + 2 (µ) ∆ ρ 2

(µ + 1) =

22

(µ) −

21

Computation of 2 (µ) Similar to 1 (µ), the control input 2 (µ) can be computed in the following manner. v2 (µ + 1) − v2 (µ) = −τ2 z v2 (µ) − >2 z sgn (v2 (µ)) ε ( 2 (µ)1 (µ)

1 (µ) 2 (µ)

+ 2 (µ) z

+

1 (µ)

+2 (µ)

2 (µ))

+

+

z (1 + 2

z2 +z 3

z +1 2 1 (µ))

1 (µ)

= −τ2 (v2 (µ)) − >2 sgn (v2 (µ)) ρ

5 Multirate Output Feedback Sliding Mode for Special Classes of Systems

90

2 (µ) = −

(τ2 v2 (µ) + >2 sgn (v2 (µ))) ( − i3 (µ) τ 2

2 (µ)1 (µ) ε i3 (µ) z2 3z +z i3 (µ) = 1 (µ) + ∆ + 2 3

−

1 (µ) 2 (µ)

+ i3 (µ)

2 (µ))

+1

(5.47) 1 (µ)

with the inequality conditions τ2 ε > 2 κ 0 1 − τ2 z κ 0

(5.48) (5.49)

Here too, there would be a singularity encountered in the computation of the control 2 whenever the denominator vanishes. In this case the disallowed state would be 1. If

1

κ0 0=z

d2

2. If

1

=

1 (µ) τ2 3

z (1 + 2

+

+ z >1 − ∆ τ2 3

+ z τ1

+

z (1 + 2

3∆ −

1 (µ))

+

z2 +z 3

(−τ1

1 (µ)

− >1 )

+

z2 + 1 (−τ1 3

1 (µ)

+ >1 )

ρ

p0 0=z

d3

=

−

1 (µ)

τ2 3

3∆ −

1 (µ))

+ 1 >1 + ∆ τ2 3

+ 1 τ1

ρ

Both these cases would be avoided if τ1 and >1 are chosen such that z2 + z >1 − ∆ p 0ε 3 z2 + z τ1 p 0ρ 3∆ − 3

(5.50) (5.51)

In this case, 1 = 0 does not cause any singularity in 2 (µ) When the states in control law (5.40, 5.47) are substituted from the nonlinear multirate observer constructed in (5.37), the control law would now be translated into one based on multirate output feedback.

5.2 MROF-QSMC for Finitely Discretizable Nonlinear Systems

91

Simulation Study A simulation of the response of the system (5.33) under the designed control, was studied. The control inputs 1 and 2 were designed according to (5.40) and (5.47). The sampling time was chosen as z = 0ρ1 sec, and the controller parameters were chosen as τ1 = τ2 = 2ε >1 = >2 = 0ρ1 so as to satisfy the inequality conditions in equations (5.41, 5.42, 5.44-5.46, 5.48-5.51). 5

x 1 x2 x 3

4

System States

3

2

1

0

−1

0

5

10

Time (sec)

15

Fig. 5.5. Response of System States. T

The simulation results for b(0) = 2ρ5 5 0 are shown in Figs. (5.5-5.7). Fig. (5.5) gives the time-response of the system states when the designed control is applied to the system. The phase portrait of the system is shown in Fig. (5.6). The evolution of the sliding surfaces v1 and v2 and the plots of the control inputs are given in Fig. (5.7). It can be seen from the plots (Fig. (5.7)) that the sliding surfaces decrease monotonically in magnitude to within the quasi-sliding mode band. The response of the system states and the applied control inputs are also found to be satisfactory.

5 Multirate Output Feedback Sliding Mode for Special Classes of Systems

1 0.8

X3

0.6 0.4 0.2 0

2

5

4

3

2

1

1

0

0

X1

X

2

Fig. 5.6. Phase Portrait of the System 20

Sliding Function s

Sliding Function s

1

2

2.5 2

1.5 1 0.5 0

−0.5

0

5

10

Time (sec)

10 5 0

0

5

10

15

0

5

10

15

Time (sec)

10

Control Input u

2

0

−1

0

−10

−2 −3

−20

−4

−30

−5 −6

15

−5

15

1

Control Input u1

92

0

5

10

Time (sec)

15

−40

Time (sec)

Fig. 5.7. Evolution of Sliding Surfaces and Control Inputs.

5.3 MROF-DSMC based on Pe Re Sliding Sector

93

5?u Mwl2i7a2e Ow2pw2 Feedback Di6c7e2e-2ime Sliding Mode Con27ol ba6ed yi2h P7e6c7ibed .PdRdq Sliding Sec2o7 5.3.1 Introduction As mentioned in the introduction, the concept of discrete-time sliding mode control is being researched in detail in the recent years. Using the Lyapunov function based approach proposed in [24], Furuta and Pan [25] proposed a sliding sector approach, wherein the control input is zero inside the designed sector. This method used the inherent stable modes of the system as hidden control inputs, thus taking the error dynamics to the origin with control at minimal instants. The algorithm discussed in this section attempts to derive an multirate output feedback version of the DSMC algorithm presented in [25]. 5.3.2 Discrete-time VSC with Pd Rd Sliding Sector The concept of Rd Ud sliding sector was proposed by Furuta and Pan in [25]. The controller is so designed such that a given Lyapunov function continues to decrease in the state space with the derivative less than the specified negative value. Inside the sector, the Lyapunov function decreases for zero input with specified velocity. And outside the sector, the control law is used. Consider the continuous-time system representation ˙ = A + Bε =C ρ

(5.52)

Let the discrete-time system representation of this system sampled at an interval of z sec be (µ + 1) = Sτ (µ) + Hτ  (µ) ε (µ) = C (µ) ε

(5.53)

where (µ) ∈ Rn and  (µ) ∈ R are the state and input vectors respectively, Sτ and Hτ are constant matrices of appropriate dimensions, and the pair (Sτ ε Hτ ) is assumed to be controllable and (Sτ ε C) is observable. Discrete Pd Rd Sliding Sector Definition 5.9. The Rd −norm · fined as (µ)

P

=

T

P

of the discrete-time system state is de-

(µ)Rd (µ)

1/2

ε (µ) ∈ Rn ε

where Rd is an φ × φ positive-definite symmetric matrix.

(5.54) 

5 Multirate Output Feedback Sliding Mode for Special Classes of Systems

94

Definition 5.10. A discrete-time Rd Ud sliding sector is defined as d

=

(µ) |

T

STτ Rd Sτ

Qd =

(µ)Qd (µ) ≤ 0 ε

(5.55)

− Rd + U d

where Rd is an φ × φ positive-definite symmetric matrix, Ud is an φ × φ posT CR ε CR ∈ Rl×n ε ν ≥ 1ε and itive semi-definite symmetric matrix, Ud = CR (Sτ ε CR ) is an observable pair. It should be noted here that such a sector does not exist in systems with no stable modes.  Inside the Rd Ud − sliding sector, the Rd −norm decreases with zero input because ∆M (µ) = M (µ + 1) − M (µ) = T (µ) STτ Rd Sτ − Rd ≤−

T

(5.56) (µ)

(µ) Ud (µ)

where the candidate Lyapunov function M (µ) is equal to the square of the Rd −norm, i.e., M (µ) =

(µ)

2 P

=

T

(µ)Rd (µ) κ 0ε

(5.57)

n

∀ (µ) ∈ R ε (µ) = 0ρ It should be reiterated here that for the existence of the Rd Ud sliding sector the open loop system must have at least one stable mode. Theorem 5.11. For plant (5.53), the Rd Ud −sliding sector defined in (5.55) exists for any positive-definite symmetric matrix Rd and any positive semidefinite symmetric matrix Ud described in Definition 5.10, and can be rewritten as (5.58) (µ) |v2 (µ) ≤ f 2 (µ) ε (µ) ∈ Rn d = where v2 (µ) = 2

f (µ) =

T

(µ) Rd1 (µ) ε

(5.59)

T

(µ) Rd2 (µ) ε

(5.60)

and Rd1 and Rd2 are φ × φ positive-semi-definite symmetric matrices Proof. Denote



Qd = STτ Rd Sτ − Rd + Ud

Then the Rd Ud −sliding sector defined in (5.55) is determined by T

(µ)Qd (µ) ≤ 0ρ

For the matrix Qd , there exists a real orthogonal matrix Y ∈ Rn×n such that

5.3 MROF-DSMC based on Pe Re Sliding Sector

95

Y T Qd Y = η
= { (µ)| |v(µ)| ≤ f (µ) ε (µ) ∈ Rn }

(5.61)

where the linear function v(µ) and the square root f (µ) of the quadratic function f 2 (µ) are respectively determined by v (µ) = Vd (µ)ε Vd ∈ U1×n ε f (µ) =

T

(5.62)

(µ) ∆d (µ)ε n×n

∆d ∈ R

(5.63)

ε ∆d ≥ 0 (∆d = 0) 

To design the simplified Rd Ud −sliding sector the following procedure is suggested in [25]. • Choose Rd as the solution of the discrete-time Riccati equation Rd = T + STτ Rd Sτ −STτ Rd Hτ 1 + HτT Rd Hτ

(5.64) −1

HτT Rd Sτ ε

where T ∈ Rn×n is a positive-semi-definite symmetric matrix.

96

5 Multirate Output Feedback Sliding Mode for Special Classes of Systems

• Choose HτT Rd Sτ

Vd =

1 + HτT Rd Hτ

ε

(5.65)

∆d = T − Ud ε ∆d ≥ 0ε ∆d = 0ρ

(5.66)

A illustration of the simplified Rd Ud -sliding sector, for a third order system has been given in Figs. (5.8 and 5.9).

s(k)= 0

s(k)= δ(k)

1

s(k)= −δ(k)

x3

0.5

0

−0.5

−1 −1 −0.5

1 0

x1

0.5 0

0.5

−0.5 1

−1

x2

Fig. 5.8. A representative third order system with simplified Pe Re − sliding sector with two stable modes

The proof for the existence of Rd Ud sliding sector when the terms in (5.61) are chosen as in Eqns. (5.64-5.66) can be shown as follows. Proof. Using Eqns. (5.57 and 5.64), it can be seen that the following holds when the input (µ) is zero. ∆M(µ) = =

T T

(µ)STτ Rd Hτ 1 + HτT qd Hτ

−1

HτT Rd Sτ (µ) −

T

(µ)T (µ)

(µ)STτ Rd Hτ 1 + HτT qd Hτ

−1

HτT Rd Sτ (µ) −

T

(µ)∆d (µ)

T

− (µ)Ud (µ) = v2 (µ) − f 2 (µ) − T (µ)Ud (µ) ≤ − T (µ)Ud (µ)ε ∀ (µ) ∈ d ρ Therefore, (5.61) defines a simplified Rd Ud -sliding sector.



5.3 MROF-DSMC based on Pe Re Sliding Sector

97

2

1.5

|s(k)|=δ(k)

|s(k)|=0

1

x3

0.5

0

−0.5

−1

−1.5

−2 1

x2

0

−1

−1

0

1

x1

Fig. 5.9. A representative third order system with simplified Pe Re − sliding sector with one stable mode

Remark 5.13. For simplicity choose a constant θ (0 p θ p 1) and let ∆d = θT and Ud = (1 − θ) Tρ Discrete-time State Feedback based VS Controller with Pd Rd Sliding Sector A discrete-time state feedback based VS controller with the Rd Ud − sliding sector (5.61) was designed in [25] for the controllable system (5.53) as  (µ) =

0ε (µ) ∈ 1 (µ) (µ) ∈ x

d d

1 (µ) = −d−1 (F (µ) + Ld sgn (dv(µ)) f(µ)) √ 1+λ |d| 0 p Ld ≤ min 1ε λ

(5.67)

98

5 Multirate Output Feedback Sliding Mode for Special Classes of Systems

Ld2 Ud κ STτ VdT Vd Sτ d = Vd Hτ ε λ = HτT Rd Hτ ε F = Vd Sτ ρ This control would render the system (5.53) to be quadratically stable, with the Lyapunov function a (µ) =

T

(µ)Rd (µ) + v2 (µ)

decreasing monotonically in the entire state space, if d is invertible. The proof for this claim can be given as follows [25]. Proof. Inside the Rd Ud -sliding sector, |v(µ)| ≤ f(µ) and the VS control input (µ) is equal to zero. In this case, the Rd -norm (µ) P decreases because ∆M(µ) = M(µ + 1) − M(µ) = v2 (µ) − f 2 (µ) − ≤−

T

(µ)Ud (µ) ≤ 0ε ∀ (µ) ∈

T

(µ)Ud (µ)

dρ

At the same time, the Lyapunov function a (µ) also decreases since ∆a (µ) = a (µ + 1) − a (µ) =

T

(µ) STτ Rd Sτ − Rd

(µ) + v2 (µ + 1) − v2 (µ)

= v2 (µ + 1) − f 2 (µ) − T (µ)Ud (µ) = T (µ) F T F − Ud (µ) − T (µ)Ud (µ) p−

T

(µ)Ud (µ)ε ∀ (µ) ∈

dρ

Outside the Rd Ud -sliding sector, |v(µ)| κ f(µ) and the discrete-time VS control input is determined by (µ) = −d−1 (F (µ) + Ld sgn (dv(µ)) f(µ)) with which the following holds : 2

v2 (µ + 1) = (Vd (µ + 1)) = (Vd Sτ (µ) + Vd Hτ (µ))

2

= Ld2 f 2 (µ) p Ld2 v2 (µ) ≤ v2 (µ)ε ∀ (µ) ∈ x dρ Thus, the system state will move toward the inside of the Rd Ud -sliding sector and will move inside it in a finite number of steps if Ld is small enough. In this case, the Rd -norm | (µ)|P may increase, but the Lyapunov function a (µ) keeps decreasing because ∆a (µ) =

T

(µ) STτ Rd Sτ − Rd

+v2 (µ + 1) − v2 (µ)

√

(µ) + 2

T

(µ)STτ Rd Hτ (µ) + HτT Rd Hτ 2 (µ)

1 + λ v(µ)(µ) + λ2 (µ) − T (µ)Ud (µ) √ = Ld2 − 1 f 2 (µ) + 2 1 + λ v(µ)(µ) + λ2 (µ) − T (µ)Ud (µ) = v2 (µ) − f 2 (µ) + 2

5.3 MROF-DSMC based on Pe Re Sliding Sector

99

√

1 + λ v(µ)(µ) + λ2 (µ) − T (µ)Ud (µ) √ = −2d−1 1 + λ v(µ) (F (µ) + Ld sgn (dv(µ)) f(µ)) + λd−2

≤2

2

× (F (µ) + Ld sgn (dv(µ)) f(µ)) − T (µ)Ud (µ) √ = −2|d|−1 1 + λ |v(µ)| − λ|d|−1 Ld f(µ) × (Ld f(µ) + sgn (dv(µ)) F (µ)) −λd−2 Ld2 f 2 (µ) − p

T

(µ)Ud (µ)ε ∀ ∈ x

T

(µ)F T F (µ) −

T

(µ)Ud (µ)

d

Therefore, the Lyapunov function a (µ) decreases in both the inside and the outside of the Rd Ud -sliding sector, i.e., it decreases in the entire state space, which means that the VS control law (5.67) results in a quadratically stable discrete-time VS control system.  5.3.3 Multirate Output Feedback DSMC Controller for Pd Rd Sliding Sector This section presents an approach to achieve the Rd Ud sliding sector behaviour discussed in Section 5.3.2 by using multirate output feedback technique discussed in Section 1.2. This would make the control law more practical as all the states of the system are not always available for measurement whereas the system outputs are always measurable. In fact, only a measurable quantity can be assigned as a system output. General Multirate Output Feedback based DSMC with Pd Rd Sliding Sector The multirate output feedback version of DSMC with Rd Ud -sliding sector can be designed by first designing the state feedback control law using the technique discussed in [25], then combining the result with the output feedback methodology discussed earlier. Thus, the control law can be designed in the following manner [37]. First it is needed to define the sliding sector d in terms of the system outputs and past inputs. The sliding sector is defined as d

= { (µ)| |v (µ)| ≤ f (µ)} ε = (µ)|v2 (µ) ≤ f 2 (µ) ε =

(µ)|

Substituting the value of be rewritten as

T

(µ) VdT Vd (µ) ≤

T

(µ) ∆d (µ) ρ

(µ) from Eqn. (1.27), the sliding sector can now

100

5 Multirate Output Feedback Sliding Mode for Special Classes of Systems

= { (µ)| (µ) ≤ 0} ε  (µ) = kT uyy k + 2uuy k  (µ − 1) + uuu 2 (µ − 1) ε d

(5.68)

uyy = FyT VdT Vd − ∆d Fy ε

uuy = FuT VdT Vd − ∆d Fy ε uuu = FuT VdT Vd − ∆d Fu ρ As shown in (5.67), the control input (µ) is zero inside input outside d is

d

and the control

(µ) = −d−1 (F (µ) + Ld sgn (dv(µ)) f(µ)) = −d−1 F (µ) − sgn(dv(µ))d−1 Ld f 2 (µ) = −d−1 F i ( k ε ε µ) −sgn (dVd i ( k ε ε µ)) d−1 Ld f (µ) ε

(5.69)

i T ( k ε ε µ) ∆d i ( k ε ε µ)ε

f (µ) =

i ( k ε ε µ) = Fy

k

+ Fu  (µ − 1) ρ

Thus, the discrete time sliding mode control law with Rd Ud sliding sector can be rewritten in terms of the system outputs and past inputs as (µ) =

0  (µ) ≤ 0ε 2 (µ)  (µ) κ 0ε

(5.70)

where 2 = −d−1 F i ( k ε ε µ) − d−1 (sgn (dv (µ)) Ld f (µ)) ε  (µ) = kT uyy k + 2uuy k  (µ − 1) + uuu 2 (µ − 1) ε v (µ) = Vd i ( k ε ε µ) ε f (µ) =

i T ( k ε ε µ) ∆d i ( k ε ε µ)ε

i ( k ε ε µ) = Fy

k

+ Fu  (µ − 1) ρ

Thus, the control input  is a nonlinear function of the system output and past input signals. Generation of Initial Control Input (0) The control methodology uses past signals to generate the present control input. But, at µ = 0ε there would be no past information available. Hence, the initial control signal (0) is computed by using the formulation given in (5.67) by assuming the value of the initial state b0 as b0,est ρ Since, the error in initial state estimation does not propagate in this control technique, the error b0 − b0,est would be rectified at µ = 1 itself.

5.3 MROF-DSMC based on Pe Re Sliding Sector

101

Numerical Example Consider the continuous time system b˙ =

0 1 0 b+  0ρ5 −2ρ5 1

(5.71)

= 11 b The system is discretized at a sampling interval of z = 0ρ1 sec and the output is sampled at an interval ∆ = 0ρ025 sec ρ The value of P is chosen to be 4. Choosing T as J2 and Ud = ∆d = 0ρ5T; Rd ,the solution of the discrete-time Riccati equation (5.64) is obtained as Rd =

49ρ2137 16ρ1838 16ρ1838 7ρ9354

and Vd = 1ρ6368 0ρ7267 ρ The values of Fy and Fu are found to be Fy =

−5ρ5819 −1ρ5735 2ρ1915 5ρ7291 5ρ1126 1ρ6037 −1ρ6921 −4ρ7887

Fu =

−0ρ4696 0ρ4711

The simulation results for Ld = 0ρ15 and for an initial state of b0 = T b0,est = 1 0 are given in Fig. (5.10). Figure (5.10) shows the comparative plots for the states 1 and 2 (in (a) and (b)), the profile of the control input  and also the monotonic decay of the Lyapunov function a (µ) ρ The figure also shows a comparative graph of the T same when the initial state is ‘wrongly’ assumed to be b0,est = 0ρ8 0ρ2 ρ (Both the estimates lead to the same system output.) It can be seen in the figure that the multirate output feedback control law is able to stabilize the system and make the Lyapunov function a decrease monotonically. The control input need not be persistent to achieve this. Moreover, even in case of an error in the estimation of the initial state, the control law ensures convergence of the state trajectories to those obtained without estimation error. Special Case: Linear Control It can be noted that the control signal is nonlinear in due to the presence of the term f (µ). But, if T is chosen to be a positive-semi-definite matrix of unity rank then ∆d would also be a positive-semi-definite matrix of unity rank. Thus it can be represented as [37] T C∆ ε C∆ ∈ R1×n ρ ∆d = C∆

(5.72)

5 Multirate Output Feedback Sliding Mode for Special Classes of Systems

102

0 0.8

−0.5

0.6

1

−1 x2

x

0.4

−1.5

0.2

−2

0 −0.2

0

1

2

Time in Sec

3

−2.5

4

0

1

2

Time in Sec

50

Lyapunov Function V

Control Input u

4

(b)

(a) 5 0

without Initial State Error with Initial State Error

40

−5

30

−10

20

−15

10

−20 −25

3

0

1

2

Time in Sec

3

4

0

0

1

2

Time in Sec

3

4

(d)

(c)

Fig. 5.10. Comparative Plots of the State, Input and Lyapunov function V (k) of the system with and without initial state estimation error.

Therefore, f (µ) can now be simplified as f (µ) =

T

(µ) ∆d (µ)

=

T

TC (µ) C∆ ∆ (µ)

=

2

(C∆ (µ)) = |(C∆ (µ))| ρ

(5.73)

Substituting this value of f (µ) in the control formulation in (5.67), the control law can be represented as (µ) =

0  (µ) ≤ 0ε 3 (µ)  (µ) κ 0ε

where 3 = −d−1 F i ( k ε ε µ) − d−1 Ld C∆ sgn (j (µ)) i ( k ε ε µ) ε j (µ) = di T ( k ε ε µ) VdT C∆ i ( k ε ε µ) ε and the sliding function (µ) can be expressed as  (µ) = sgn (Vd i ( k ε ε µ)) Vd i ( k ε ε µ) −sgn (C∆ i ( k ε ε µ)) C∆ i ( k ε ε µ) ρ

(5.74)

5.3 MROF-DSMC based on Pe Re Sliding Sector

103

Thus, for a special choice of Tε the control  can be made linear in  (µ − 1).

k

and

Numerical Example For the system in (5.71), when Tε Uε ∆d and Ld are chosen as 22 ε U = ∆d = 0ρ5Tε 22

T=

C∆ = 1 1 ε Ld = 0ρ15ρ The resultant control is linear in both k and  (µ − 1) ρ The simulation results T for an initial condition b0 = 1 0 ε with the initial state approximation b0,est = b0 and for an erroneous b0,est = 0ρ8 0ρ2 1

T

are given in Fig. (5.11).

0

0.8

−0.5

2

−1

x

x1

0.6 0.4

−1.5

0.2 −2

Without Initial State Error With Initial State Error

0 0

1

2

Time in Sec

3

4

Lyapunov Function V

Control Input u

0

1

0

1

2

3

4

2

3

4

Time in Sec

50

0

40

−5

30

−10

20

−15

10

−20 −25

−2.5

0

1

2

Time in Sec

3

4

0

Time in Sec

Fig. 5.11. Comparative Linear controller responses of the State, Input and Lyapunov function V (k) of the system with and without initial state estimation error.

3 Di6c7e2e-2ime Te7minal Sliding Mod: Concep2

3?1 In27odwc2ion The behavior of the system in sliding mode is determined by the sliding manifold upon which the system states are constrained. Sliding hyperplanes, being the most generally used manifolds guarantee asymptotic stability of the system in the sliding mode. This would mean that the system states converge to the origin, but generally at an infinite time. Terminal sliding mode (TSM) control [55] aims at designing a sliding mode control strategy that would guarantee a finite time convergence to the origin. This is accomplished by using a nonlinear sliding line which results in finite time convergence [30]. The concept of terminal sliding mode in continuous time systems has been studied in detail [56, 79–81]. The discrete-time equivalent of the continuous time terminal sliding mode control would be investigated in this chapter.

3?v Con2inwow6-2ime Te7minal Sliding Mode Con27ol Definition 6.1. Terminal sliding mode control is the method of achieving finite-time convergence by using the concept of sliding mode control. Consider the following SISO continuous-time system representation ˙ i = i+1 ε < = 1ε 2ε · · · ε φ − 1 ˙ n = c( ) + d( )

(6.1)

T

where, = [ 1 ε 2 ε · · · ε n ] ∈ Rn . The system is assumed to be controllable. Hence, 0 ∈ x d( ). A terminal sliding mode control can be designed for the system (6.1) by designing a series of φ − 1 nested nonlinear sliding manifolds [82] as

B. Bandyopadhyay, S. Janardhanan: Discrete-time Sliding Mode Contr., LNCIS 323, pp. 105–120, 2006. © Springer-Verlag Berlin Heidelberg 2006

106

6 Discrete-time Terminal Sliding Mode: Concept

v1 = v2 = .. . vn−1 =

v˙ 0 + ,1 vγ01 + δ1 vρ01 ε v˙ 1 + ,2 vγ12 + δ2 vρ12 ε .. . ρ −1 γ −1 + δn−1 vn−2 ε v˙ n−2 + ,n−1 vn−2

(6.2) (6.3) (6.4)

where, v0 = 1 , ,i κ 0ε δi κ 0 and ki and ui are rational numbers satisfying 0 p ki p 1ε ui κ ki ε < = 1ε · · · ε φ − 1. It can be easily shown [82] that if vn−1 is made zero in finite time by applying an appropriate control, then vn−2 will also become zero in finite time, so will vn−3 ε · · · ε v0 in that order. Thus, 1 = v0 = 0 becomes in finite time. The control that would achieve this can be given as  = −d−1 ( ) c( ) +

n−2 k=0

γ

,k+1 vk

+1

ρ

+ δk+1 vk

+1

(6.5)

−L vjφ (vn−2 ) ε L κ 0ρ

3?u Di6c7e2i a2ion of TSM In case of sliding mode control based on a linear sliding hyperplane, the asymptotic stability found in continuous time system is also found in discrete-time sliding mode control [4]. However, the finite time convergence property assured by the nonlinear sliding manifold, in continuous time systems, is not preserved when the same concept is extended to discrete-time systems. This can be explained by the following example. The stability of an autonomous system is preserved on discretization. Hence, a sliding manifold that assures stability in continuous time is also stable when the system is discretized. Consider the case of a second order system. ˙ 1 = 2ε ˙ 2 = c( ) + d( )ρ

(6.6) (6.7)

For this system, the sliding manifold for terminal sliding mode would be of the form, v = 2 + , γ1 + δ ρ1 = 0ρ (6.8) For a sampling time of z sec, the discrete representation of the system (6.6) and the sliding manifold (6.8) can be given as 1 (µ

+ 1) = 1 (µ) + z 2 (µ)ε (µ + 1) = cτ ( (µ)ε (µ)ε z )ε 2 v(µ) =

2 (µ)

+,

γ 1 (µ)

+δ

(6.9) ρ 1 (µ)ρ

(6.10)

107

6.3 Discretization of TSM

where, cτ ( (µ)ε (µ)ε z ) is the exact discretization [10] of the differential equation (6.7). Assume that there exists a discrete-time control strategy (µ) = τ (µ) such that it brings all ∈ R2 onto the sliding manifold v(µ) = 0. If the closed loop system has to have finite-time convergence, then there exists a µ ∗ p ∞ such that (µ) = 0ε ∀µ κ µ ∗ . Hence, 1 (µ ∗ + 1) = 0 and v(µ ∗ ) = 0 as (µ) belongs to the sliding manifold. Thus, 2 (µ

∗

)+,

∗ ∗ 1 (µ ) + z 2 (µ ) γ ∗ ρ ∗ 1 (µ ) + δ 1 (µ )

x2+20.1x0.3 +0.3x0.5 =0 and x1+20x2=0 1 1

60

100

(a)

40

x2

2

x

0

−20

−2

0

x

x2+2x0.5 +3x51=0 1

2

4

−100 −4

and x1+20x2=0

100

(c)

−2

0

x

1 x2+40x0.3 +5x31=0 and 1

2

0 −50

4

x1+20x2=0 (d)

50

x2

2

x

(b)

0

1

50

−100 −2

x2+17.1x0.3 +2x3.1 =0 and x1+20x2=0 1 1

−50

−40

100

(6.11) (6.12)

50

20

−60 −4

= 0ε = 0ρ

0 −50

−1

0

x

1

2

−100 −4

−2

1

0

x

2

4

1

Fig. 6.1. Plot for various possible solutions

The various possible graphs for Eqn. (6.11) and Eqn. (6.12) are given in Fig. (6.1). The plot in Fig. (6.1(a)) shows the graphs for 0 p kε u p 1. It can be seen that the system of equations would then have three solutions in the real domain. Figs. (6.1(b),6.1(c) and 6.1(d)) show the graphs for cases with 0 p k ≤ 1 p u. It can be seen that the cases have five, three and one solutions respectively. The graphs in Fig. (6.1) represent all the possible ‘configurations’ that the combination of Eqns. (6.11, 6.12) may take. Thus, it may be inferred that for all ,ε δ κ 0ε Eqns. (6.11, 6.12) have exactly three real solutions, when

108

6 Discrete-time Terminal Sliding Mode: Concept

0 p k p u ≤ 1 and at most five solutions when 0 p k p 1 p u. One of the roots would be at = 0 and the others in the second and fourth quadrant respectively with the form (∓z s ε ± s ), where s is the real and positive solution(s) of the equation 1 − ,z γ

γ−1

− δz ρ

ρ−1

=0

(6.13)

solved for . Thus, a finite time convergence is possible with the nonlinear sliding line approach only if the control is so designed such that it can exactly reach one of the points (∓z s ε ± s ). Otherwise, the system states would converge towards origin along the sliding manifold (6.12). However, the origin is not reached in a finite number of iterations as the system is not crossing (∓z s ε ± s ). Neither can the system go arbitrarily close to the origin due to the sampled behavior of the system. Thus, the system states would settle into an oscillatory behavior of nature (µ + 1) = − (µ), and satisfying Eqn. (6.12) for maintaining the sliding mode behavior. The system would oscillate (∓z u ε ± u ), where u are the solutions of z γ γ−1 z ρ ρ−1 1−, −δ = 0ρ (6.14) 2 2 Therefore, it may be concluded that the use of a nonlinear sliding line in discrete-time systems does not guarantee terminal sliding mode. The problem of terminal sliding mode control in discrete-time systems should be handled in a manner independent of the continuous-time terminal sliding mode philosophy.

3?4 Di6c7e2e-2ime Te7minal Sliding Mode Con27ol Any discrete-time time-invariant system may be represented as (µ + 1) = F ( (µ)ε (µ))ε

(6.15)

(µ) = I( (µ))ρ (6.16) Define the discrete-time terminal sliding manifold as Sd = { (µ)|v( (µ)) = 0}

(6.17)

where v( (µ)) is a real-valued function. If the system dynamics confined to the discrete-time terminal sliding manifold be (µ + 1) = Fc ( (µ))

(6.18)

the finite time convergence property can be expressed in discrete-time as (µ + µd ) = Fck ( (µ))ε µd ∈ Z+ ε µd p ∞

(6.19)

6.4 Discrete-time Terminal Sliding Mode Control

109

for all (µ) ∈ Sd . This condition would mean that Fc (ρ) is a nilpotent function with index µd . Therefore, discrete-time terminal sliding mode may be achieved of a sliding manifold is designed such that the system dynamics confined to the manifold is nilpotent. A formalized discrete-time terminal sliding mode (DTSM) control algorithm may thus be derived in the following manner. 6.4.1 DTSM Algorithm 1. Using an appropriate diffeomorphism ‫(܈‬µ) = Σ ( (µ)), transform the system (6.15) into a Brunowsky-like canonical form

‫((܈‬

‫܈‬i (µ + 1) = ‫܈‬i+1 (µ)ε (6.20) < = 1ε 2ε · · · ε h1 − 2ε h1 ε · · · ε h1 + h2 − 3ε · · · ε φ − ∂ − 1 ، (µ + 1) = ‫(܈‬n−m+i) (µ)ε < = 1ε 2ε · · · ε ∂ 㴀=1 η㴀 )−i) ‫܈‬j (µ + 1) = cd,j (‫(܈‬µ)) + dd,j (‫(܈‬µ))(µ)ε m = φ − ∂ + 1ε · · · ε φ

where,

 dd,n−m+1 (‫(܈‬µ))   .. dd =   . 

dd,n (‫(܈‬µ))

is an invertible ∂ × ∂ for all ‫(܈‬µ) and hi represents the controllability m index of system with respect to the <-th input i (µ). Thus, i=1 hi = φ. 2. Design the sliding manifold for the transformed system as v(µ) = ‫܈‬n−m+1 (µ) · · · ‫܈‬n (µ)

T

ρ

(6.21)

The dynamics constrained to the manifold v(µ) = 0 would then be of the form ‫(܈‬µ + 1) = Ad ‫(܈‬µ)ε where ‫(܈‬µ) = ‫܈‬1 (µ) ‫܈‬2 (µ) · · · ‫܈‬n (µ)

T

ρ

and Ad is a nilpotent matrix with a maximum possible index of φ. 3. Design the sliding mode control signal so as to have v(µ + 1) = 0 [4]. Thus, −1 (µ) = −d−1 d (‫(܈‬µ))cd (‫(܈‬µ)) = −dd (Σ ( (µ)))cd (Σ ( (µ)))ρ

(6.22)

Thus, for any ∈ Rn , v(µ + 1) = 0 and ‫(܈‬µ + φ) = 0 due to nilpotency of Ad . Thus, (µ + φ) = Σ −1 (‫(܈‬µ + φ)) = 0, i.e, discrete-time terminal sliding mode is achieved.

110

6 Discrete-time Terminal Sliding Mode: Concept

It should be noted here that terminal sliding mode in discrete-time system is possible if and only if there exists a diffeomorphism ‫(܈‬µ) = Σ ( (µ)) such that the system dynamics in ‫ ܈‬co-ordinate system is of the form (6.20). This would ensure that the control signal is always bounded. An interesting observation can be drawn from the discrete-time terminal sliding mode controller. In case of TSM in continuous time systems, a nonlinear sliding manifold and a switching function based control is required to bring out finite time convergence. However, in case of discrete-time systems, when the system is linear or is a nonlinear system in Brunowsky canonical form, a linear control and a linear sliding manifold is sufficient. No explicit switching needs to be incorporated in the control. 6.4.2 Illustrative Example The discrete time terminal sliding mode control algorithm may be illustrated through the following numerical example. Consider the system     2 2 (µ) + ix (µ) 1 (µ + 1) ε  2 (µ + 1)  =  ix (µ) 4 2 1 (µ) + 2 2 (µ)ix (µ) + ix (µ) 3 (µ + 1) where

ix (µ) =

3 (µ)

Using the transformation 

  ‫܈‬1 (µ)  ‫܈‬2 (µ)  =  ‫܈‬3 (µ)

−

2 1 (µ)

3 (µ)

− (µ) − 1 2 (µ)

+ (µ)ρ 

2 1 (µ) 2  2 (µ)

the system is transformed into     ‫܈‬1 (µ + 1) ‫܈‬2 (µ)  ‫܈‬2 (µ + 1)  =  ‫܈‬3 (µ) ρ ‫܈‬3 (µ + 1) ‫܈‬1 (µ) + (µ) The sliding manifold is designed as ‫܈‬3 (µ) =

2 (µ)

= 0ρ

The discrete-time terminal sliding mode control law for this system can be formulated as (µ) = −‫܈‬1 (µ) = −

3 (µ)

+

2 1 (µ)ρ

It is worthy to note that the control that induces terminal sliding mode in the system is linear in the ‫ ܈‬co-ordinates. The simulation results for (0) = T 232 are shown in Figs. (6.2, 6.3). It can be observed that the system states converge to the origin in a finite number of steps (three to be precise, in the example).

111

6.4 Discrete-time Terminal Sliding Mode Control

3.5

3

2.5

System States

2

1.5

1

0.5

0

−0.5

−1

0

1

2

3

Sampling Instant

4

5

6

Fig. 6.2. Response of System States in Discrete-time Terminal Sliding Mode 8

6

Control Input

4

2

0

−2

−4

0

0.5

1

1.5

2

2.5

3

Sampling Instant

3.5

4

4.5

Fig. 6.3. Discrete-time Terminal Sliding Mode Control Input

5

112

6 Discrete-time Terminal Sliding Mode: Concept

Remark 6.2. In step 3 of the discrete-time terminal sliding mode algorithm the reaching law v(µ + 1) = 0 may be replaced by the a priori known function based reaching law v(µ + 1) = vd (µ + 1) as proposed by Bartoszewicz in [7]. This would give a control law of the form (µ) = −d−1 d (Σ ( (µ)))cd (Σ ( (µ))) + vd (µ + 1)

(6.23)

where, vd is the a priori known function satisfying the constraints • vd (0) = v(0). • |vd (µ + 1) ≤ |vd (µ)| for all µ ≥ 0 • vd (µ) = 0 for all µ ≥ µ ∗ , where µ ∗ is an appropriately chosen time sample.

3?5 Mwl2i7a2e Ow2pw2 Feedback ba6ed DTSM Algo7i2hm6 6.5.1 Multirate Output Feedback based DTSM Algorithms for LTI Systems It has been shown in the previous sections that for a LTI system, discrete terminal sliding mode can always be achieved by choosing a sliding surface such that the reduced order system has all its poles at the origin. Thus, for a sliding surface designed by the above philosophy, denoted as eTt , the control law that would satisfy the reaching conditions mentioned above can be represented as a multirate output feedback based terminal sliding mode control law in the following manner. • For a LTI system of the form (1.20), by using (1.29) and the control law (6.22) in order to satisfy the reaching condition v(µ + 1) = 0, the control law would be [3] (µ) = − eTt Hτ

−1 T et Sτ

(My

k

+ Mu (µ − 1)) ρ

(6.24)

• Another multirate output feedback based control algorithm can also be designed to satisfy the reaching law, v(µ + 1) = vd (µ + 1), proposed by Bartoszewicz [7]. Using the control algorithm discussed in Section 3.2.1, the control input can be formulated as [40] (µ) = − eTt Hτ

−1 T et Sτ

(My

k

+ Mu (µ − 1) − vd (µ + 1)) ρ

(6.25)

Illustrative Example Consider the controllable and observable MIMO LTI system [28] ˙ = A + Bε =C ρ

(6.26)

6.5 Multirate Output Feedback based DTSM Algorithms

with

113

 −16ρ031 −3ρ75 0ρ059 0 3ρ067 6ρ92 −1ρ928  −7ρ23 −6ρ137 −0ρ2 0 −5ρ342 −10ρ932 22ρ862      −0ρ447 −0ρ1590 0 0ρ029 −0ρ721 −0ρ564    0 −4 −0ρ497 0 0 A =  −0ρ066 −0ρ014 ε   1ρ338 0ρ677 −0ρ002 0 −0ρ255 −0ρ607 −0ρ402    6ρ087 0ρ613 0ρ01 0 −0ρ358 −9ρ112 9ρ222  7ρ028 2ρ181 −0ρ003 0 −0ρ478 9ρ222 −16ρ737 

−14ρ88 6ρ74 0 0 1ρ788 10ρ76 −3ρ778 −0ρ8 − 28 0 0 0ρ914 −8ρ935 20ρ007   1000000 0 1 0 0 0 0 0  C= 0 0 1 0 0 0 0ρ 0001000

B=

T

ε

The system is sampled with an input sampling interval of z = 1 sec and an output sampling interval of ∆ = z x2 = 0ρ5 sec. Using (6.24), the control input is derived to be 

0ρ0006  0ρ0003   −0ρ1957   −0ρ4681 (µ) =   0ρ8247   0ρ1477   0ρ1970 3ρ4615

T −0ρ0007 0ρ0028   −1ρ7316   0ρ0741   −3ρ5754   −0ρ2570   3ρ3802  −0ρ5261

k

+

0ρ3177 0ρ1673 (µ − 1)ρ −2ρ2748 −1ρ4771

The simulation results are shown in Figs. (6.4-6.6). It can be seen in Fig. (6.6) that the sliding functions go to zero in one sampling instant and Fig. (6.4) shows all the system outputs also moving to zero in finite time. The control input is shown in Fig. (6.5). Though it is not plotted here, all the system states also reach the origin after a finite number of iterations. Sometimes it may happen that the control law (6.24) demands many abrupt changes in the input magnitude to achieve finite time convergence. A relatively smoother control can be obtained by using the control law (6.25), by relaxing the time of convergence to the sliding surface. For the system (6.26), a multirate output feedback based terminal sliding mode control algorithm can be designed, using (6.25), for a multivariate value T of µ ∗ = 10 15 , as

114

6 Discrete-time Terminal Sliding Mode: Concept 0.3

y 1 y 2 y 3 y

0.25

4

0.2

System Outputs

0.15

0.1

0.05

0

−0.05

−0.1

−0.15

0

1

2

3

4

5

Time (secs)

6

7

8

9

10

Fig. 6.4. Response of system outputs with DTSM in linear system using control law (6.24) 0.02

Control Input u

1

0.01 0

−0.01 −0.02 −0.03 −0.04

0

1

2

3

4

0

1

2

3

4

5

6

7

8

9

10

5

6

7

8

9

10

Time (secs)

0.1

Control Input u

2

0.05 0

−0.05 −0.1

−0.15 −0.2

−0.25

Time (secs)

Fig. 6.5. Profile of control inputs with DTSM in linear system using control law (6.24)

115

6.5 Multirate Output Feedback based DTSM Algorithms 0.2

s1 s 2

0.1

Sliding Function

0

−0.1

−0.2

−0.3

−0.4

−0.5

−0.6

0

1

2

3

4

5

Time (secs)

6

7

8

9

10

Fig. 6.6. Profile of sliding functions with DTSM in linear system using control law (6.24) 0.5

y1 y 2 y 3 y4

0.4

0.3

System Outputs

0.2

0.1

0

−0.1

−0.2

−0.3

−0.4

0

5

10

Time (secs)

15

20

25

Fig. 6.7. Response of system outputs with DTSM in linear system using control law (6.25)

116

6 Discrete-time Terminal Sliding Mode: Concept 0.04

Control Input u

1

0.03 0.02 0.01 0

−0.01 −0.02

0

5

10

0

5

10

Time (secs)

15

20

25

15

20

25

0.2

Control Input u2

0.1 0

−0.1 −0.2 −0.3 −0.4 −0.5

Time (secs)

Fig. 6.8. Profile of control inputs with DTSM in linear system using control law (6.25) 0.2

s 1 s 2

0.1

Sliding Function

0

−0.1

−0.2

−0.3

−0.4

−0.5

−0.6

0

5

10

Time (secs)

15

20

25

Fig. 6.9. Profile of sliding functions with DTSM in linear system using control law (6.25)

6.5 Multirate Output Feedback based DTSM Algorithms



0ρ0006  0ρ0003   −0ρ1957   −0ρ4681 (µ) =   0ρ8247   0ρ1477   0ρ1970 3ρ4615 +

117

T

−0ρ0007 0ρ0028   −1ρ7316   0ρ0741   −3ρ5754   −0ρ2570   3ρ3802  −0ρ5261

k

+

0ρ3177 0ρ1673 (µ − 1) −2ρ2748 −1ρ4771

1 0ρ15 v (µ + 1)ρ 0 1 d

The simulation results are shown in Figs. (6.7-6.9). It can be seen here in Fig. (6.8) that the control input is much less abrupt as compared to Fig. (6.5). 6.5.2 Multirate Output Feedback based Discrete-time Terminal Sliding Mode in Output Feedback Linearizable Nonlinear Systems Feedback linearizable discrete-time nonlinear systems can be equivalently represented as linear systems in appropriate state and input co-ordinates [33,51,53]. The relationship in state and the multirate output may be used to design a multirate output feedback based controller for feedback linearizable systems [63]. A multirate output feedback based discrete-time terminal sliding mode controller can be designed for feedback linearizable nonlinear systems using the following procedure. 1. Find the linearized model of the system (6.15) in appropriate co-ordinates by using the concepts of feedback linearization [33,53] to get a linear model of form ‫(܈‬µ + 1) = Sτ ‫(܈‬µ) + Hτ (µ)ε (µ) = C‫(܈‬µ)ε

(6.27) (6.28)

with the transformation as ‫(܈‬µ) = Σ ( (µ))ε v(µ) = Σu ( (µ)ε (µ))ρ 2. Design the multirate output feedback terminal sliding mode control for the linear system (6.27) using one of the control algorithms (6.24, 6.25) discussed in Section 6.5.1. 3. Since state co-ordinate transformation does not affect the control law, a transformation back to the original co-ordinate frame is unnecessary in this case. However, whenever input co-ordinate transformation is also

118

6 Discrete-time Terminal Sliding Mode: Concept

involved, a re-translation into the  co-ordinate is necessary using the transformation (µ) = Σu−1 Σ −1 (My

k

+ Mu v(µ − 1)) ε v(µ) ρ

The technique is illustrated here with the help of a numerical example. Illustrative Example Consider the system    1 (µ + 1)  2 (µ + 1)  =  3 (µ + 1) (µ) =

2 (µ)

2 + 3 (µ) + (µ) + 3 (µ) 3 (µ) + (µ) + 3 (µ)  ε 2 1 (µ) − 2 (µ)

3 (µ)ρ

Using a co-ordinate transformation    ‫܈‬1 (µ)  ‫܈‬2 (µ)  =  ‫܈‬3 (µ)



3 (µ) 2  − 1 2 (µ) ε 2 (µ)

v(µ) = (µ) + 3 (µ) we have the system represented in the (‫܈‬ε v) co-ordinates as        ‫܈‬1 (µ + 1) ‫܈‬1 (µ) 010 0  ‫܈‬2 (µ + 1)  =  0 0 1   ‫܈‬2 (µ)  +  0  v(µ) 100 ‫܈‬3 (µ + 1) 1 ‫܈‬3 (µ)   ‫܈‬1 (µ) (µ) = 1 0 0  ‫܈‬2 (µ)  ρ ‫܈‬3 (µ) The sliding surface that would ensure discrete-time terminal sliding mode for this system is v(µ) = eTt ‫(܈‬µ) = 0 0 1 ‫(܈‬µ) = 0ρ The control law can then be formulated using (6.24) as v(µ) = −1 2ρ5321 −2ρ5321

k

− 0ρ052v(µ − 1)ρ

From v(µ), the value of the actual control input (µ) can be calculated using the formula (µ) =

1 6

3

108v(µ) + 12 12 + 81v 2 (µ) −

3

8 108v(µ) + 12 12 + 81v 2 (µ) T

The simulation results for an initial condition of b0 = 1 2 3 are presented in Figs. (6.10-6.13). It can be seen that even in this case, the system states converge to the origin in a finite number of sampling instants.

119

6.5 Multirate Output Feedback based DTSM Algorithms 4

X1 X2 X

3

3

System State Responses

2

1

0

−1

−2

−3

−4

0

1

2

3

Normalized Time

4

5

6

Fig. 6.10. State trajectories of MROF based DTSM controlled feedback linearizable nonlinear system 4

3

System Output Samples

2

1

0

−1

−2

−3

−4

0

1

2

3

Normalized Time

4

5

6

Fig. 6.11. Output samples of MROF based DTSM controlled feedback linearizable nonlinear system

120

6 Discrete-time Terminal Sliding Mode: Concept 1.5

1

Control Input

0.5

0

−0.5

−1

−1.5

0

1

2

3

Normalized Time

4

5

6

Fig. 6.12. Control input profile of MROF based DTSM controlled feedback linearizable nonlinear system 2.5

2

Sliding Function

1.5

1

0.5

0

−0.5

0

0.5

1

1.5

2

2.5

3

Normalized Time

3.5

4

4.5

5

Fig. 6.13. Sliding function plot of MROF based DTSM controlled feedback linearizable nonlinear system

t Applica2ion6 of Mwl2i7a2e Ow2pw2 Feedback Di6c7e2e-2ime Sliding Mode Con27ol

In this chapter, the applicability of the concept of multirate output feedback based discrete-time sliding mode control, will be analysed for a few practical systems.

t?1 Po6i2ion Con27ol of Pe7manen2 Magne2 DC S2eppe7 Mo2o7 The stepper motor is an electromechanical system used in incremental motion control which converts digital input signal into angular motion of the rotor. The rotor rotates a fixed step depending on its construction. The small step motion can be obtained using micro stepping. It finds application in precise position control. Accurate positioning at high stepping rate is possible with closed loop control. In the existing literature various methods for closed loop control of stepper motor are proposed and implementation results have been presented. In [85] a method has been proposed for position control of permanent magnet stepper motor using exact linearization. Another important method for model based control using exact linearization has been proposed in [8] where a nonlinear observer has been used for speed estimation. In [87] a flatness based static and dynamic sliding mode control for control of stepper motor has been proposed. Recently a state feedback based discrete sliding mode control [71] for control of stepper motor has been investigated. 7.1.1 Stepper Motor Model The stepper motor is designed to rotate in steps as its windings get excited by digital input pulses. The speed of rotation can be adjusted by adjusting the delay between the applied input pulses. It has a stator and rotor. For permanent magnet stepper motor, rotor consists of permanent magnet. Depending upon the stator and rotor teeth step angle can be decided. There

B. Bandyopadhyay, S. Janardhanan: Discrete-time Sliding Mode Contr., LNCIS 323, pp. 121–139, 2006. © Springer-Verlag Berlin Heidelberg 2006

122

7 Applications of MROF-DSMC

are windings on the stator pole which are energized in a specific sequence to produce stepping motion. The windings of the most permanent magnet stepper motors are bipolar. In this type of arrangement two sets of windings are wound in each pole opposing each other. Hence, bidirectional field is achieved by passing current in one direction in either of the two coil [49]. Such windings simplifies the drive requirements. For winding, constructional other modeling details [46] [49] provides excellent references. The motor model given in [8], [86] is η
[va − U
=

= ψρ

(7.1)

Where va and vb are the phase voltages,
=

e v(Prθ ) v<φ(Pr ) −v<φ(Pr ) e v(Pr )

va ε vb

(7.3)

where
7.1 Position Control of Permanent Magnet DC Stepper Motor

η
123

(7.4)

Discrete State Space Model of the Stepper Motor Discrete model of the stepper motor is obtained by discretizing the system of Eqn.(7.4) with sampling time z . The discrete time state space representation of the system can be given as 1 (µ

+ 1) = (1 − µ1 ) 2 (µ + 1) = (1 − µ1 ) 3 (µ

+ 1) = µ3 (µ + 1) = µ6 4

1 (µ)

+ µ5 2 (µ) − µ5

2 (µ)

+ (1 − µ4 ) (µ) + 4 (µ)ε 3

2 (µ) 3 (µ)

+ 1 (µ) 1 (µ) 3 (µ) − µ2 3 (µ) + 2 (µ)ε 3 (µ)ε

(7.5)

where 1 (µ)

=
Now, it is possible to define linearizing outputs for the system as 1 (µ)

= 2 (µ) = of

1 (µ) 4 (µ)ρ

All the system state variables and control inputs can be expressed in terms 1 , 2 and their higher order differences in the following manner. 1 (µ)

=

1 (µ)ε

+ 2) − 2 (µ + 1) (1 − µ4)( 2 (µ + 1) − − µ3 µ6 µ3 µ6 2 (µ + 1) − 2 (µ) ε 3 (µ) = µ6 4 (µ) = 2 (µ)ρ 2 (µ)

.

=

2 (µ

2 (µ))

ε

(7.6)

7 Applications of MROF-DSMC

124

Linearized Model of the Stepper Motor We get the linear state space model after simplifying the equations (7.5 by using auxillary input v1 and v2 defined as [71] 1 (1 (µ) + µ5 2 (µ) 3 (µ) − µ6 µ3 v2 (µ) = 2 2 (µ) − (µ5 1 (µ) + µ2 ) µ6

v1 (µ) =

1 (µ)) 3 (µ)

− µ4

2 (µ)

+

µ42 µ3

3 (µ)

which would give the linear state space model as 1 (µ

+ 1) =

1 (µ)

+ µ6 v1 (µ)ε

2 (µ

+ 1) = (1 + µ4 )

3 (µ

+ 1) = µ3

µ42 µ62 v2 (µ)ε 3 (µ) + µ3 µ3 2 (µ) + (1 − µ4 ) 3 (µ)ε

4 (µ

+ 1) = µ6

3 (µ)

2 (µ)

+

−

4 (µ)ρ

(7.7)

The above system can be expressed as (µ + 1) = Sτ (µ) + Hτ v(µ)

(7.8)

(µ) = C (µ) where



10 0  0 1 + µ4 −k42 k3 Sτ =  0 µ 1 − µ4 3 00 µ6 C=

  µ6 0 0 k62   0  0 k3 ε H z = 0 0 0 1 0 0

  ε 

1000 ρ 0001

7.1.2 Discrete Time Sliding Mode Control Using Multirate Output Feedback: Regulator Case Discrete Time State Feedback Sliding Mode Control: Background Results The auxiliary quasi-sliding mode control v(µ) can be obtained using the control law (2.2) as v(µ) = − eT Hτ

−1

eT Sτ − eT + τz eT

(µ) + >z sgn(eT (µ)) ρ

(7.9)

The actual control (µ) can be derived from v(µ) for the stepper motor as

7.1 Position Control of Permanent Magnet DC Stepper Motor

1 (µ) = v1 (µ)µ6 + µ1 1 (µ) − µ5 2 (µ) 3 (µ)ε µ2 µ2 2 (µ) = v2 (µ) 6 + (µ1 + µ4 ) 2 (µ) + µ2 3 (µ) − 4 µ3 µ3

125

(7.10) 3 (µ)

+ µ5

1 (µ) 3 (µ)ρ

This is the state feedback sliding mode control for the nonlinear model of the permanent magnet stepper motor. Discrete-time Multirate Output Feedback Sliding Mode Control for PMDC Stepper Motor The auxiliary control v(µ) can be calculated in terms of output samples and past control using Eqns. (2.3) and the actual control (µ) is calculated using Eqns. (7.10) as 1 (µ) = µ6 v1 (µ) + µ1 (q1 k + τ1 v1 (µ − 1)) −µ5 (q2 k + τ2 v1 (µ − 1))(q3 k + τ3 v1 (µ − 1)) µ2 2 (µ) = 6 v2 (µ) + (µ1 + µ4 )(q2 k + τ2 v2 (µ − 1)) µ3 µ2 +(µ2 − 4 )(q3 k + τ3 v2 (µ − 1)) µ3 +µ5 (q1 k + τ1 v2 (µ − 1))(q3 k + τ3 v2 (µ − 1)) where, qi and τi represent the <-th row of the matrices My and Mu as defined in Eqn. (1.28). However, in case of the PMDC stepper motor k for this particular case is defined as 

k

 − z)  2 (µz − z )   =  2 (µz − z + ∆)  2 (µz − z + 2∆) 1 (µz

(7.11) and the definitions of the matrices C0 and D0 of system (7.8) are changed in appropriate manner from the value defined in Eqn. (1.26). Using the algorithm, simulation is carried out to observe the performance of the stepper motor. The simulation results are presented in next section. 7.1.3 Simulation Results Using the motor parameter as given in [86] and shown in Table 7.1 simulation is carried out to observe the performance of the stepper motor using the

126

7 Applications of MROF-DSMC −3

5

0.1

iq,amps

id,amps

0.15

0.05

0

5

t (a)

10

−10

15

0

t 2 (b)

1

3

0.3

θ,radians

ω,rad/sec

1

0.5

0

−0.5

0

−5

0 −0.05

x 10

0.2 0.1 0

0

2

t

4

−0.1

6

0

5

10

(c)

t 15 (d)

20

0.15

0.15

0.1

0.1

ib,amps

ia,amps

Fig. 7.1. Response of system states (a)Direct axis current (b) Quadrature axis current (c) Angular velocity (d) Angular position

0.05

0.05

0 −0.05

0

0

1

2

t3 (a)

4

5

−0.05

4

t

6

8

2 1.5

1

Vb,Volts

Va,Volts

2

(b)

2

0

1 0.5 0

−1 −2

0

−0.5 0

2

t 4 (c)

6

−1

0

1

2

t3 (d)

4

5

Fig. 7.2. Response of winding currents and voltages (a) Current in winding A (b) Current in winding B (c) Voltage in winding A (d) Voltage in winding B

7.1 Position Control of Permanent Magnet DC Stepper Motor

127

−6

x 10 switching plane 1

5 0 −5 −10 −15 4.1

4.2

4.3

4.4

4.5

4.6

4.7

4.8

4.9

5

time

−6

switching plane 2

x 10 2 0 −2 −4 −6

4.74

4.76

4.78

4.8

4.82

4.84

4.86

4.88

4.9

4.92

time

Fig. 7.3. Response of the switching planes (a)switching plane 1 (b) switching plane 2

algorithm based on output samples. The parameters for the controller are selected so as to satisfy conditions for reaching law and multirate output feedback technique. These parameters are z = 0ρ1ε γ = 0ρ0002ε τ = 2ε P = 3 and ∆ = 0ρ033 sec. Table 7.1. Motor Parameters R L km J B Nr

19.1388Ω 40mH 0.1349N M /A 4.1295 ∗ 10−4 kg.m2 0.0013N m/rad/sec 50

The objective of the control is to bring the states of the system from any initial position to zero. Fig.7.1 shows the response of the system states. It can be seen that the position of the motor reaches to zero from any initial position without oscillation. Fig. 7.2 shows the response of the winding current and winding voltages. The

128

7 Applications of MROF-DSMC

response of the switching function is shown in Fig. 7.3 which is as expected for the reaching law based control.

t?v Poye7 S062em S2abili e7 De6ign w6ing Mwl2i7a2e Ow2pw2 Feedback Sliding Mode Con27ol 7.2.1 Power System Modeling Consider a single machine infinite bus system shown in Fig. 7.4. For simplicity, we assume a synchronous machine represented by model 1.0 neglecting damper windings both in the η and τ axes. Also, the armature resistance of the machine is neglected and the excitation system represented by a single time-constant system [61]

Re

Xe

G

Eb Fig. 7.4. Single Machine Infinite Bus System

The rotor mechanical equations are ηf = ψB (Vm − Vmo ) ε ηy ηVm = −DVm + Xm − Xe ε 2I ηy Xe = Eqo
q

−

d

(7.12) (7.13)
(7.14)

Linearizing Eqns. (7.12),(7.13) and (7.14) and by taking Laplace transform, we obtain ψB ψB ∆Vm = ∆ψε v v 1 [∆Xm − ∆Xe − D∆Vm ] ρ = 2Iv

∆f = ∆Vm

(7.15) (7.16)

129

7.2 Power System Stabilizer Design using MROF-SMC

K1 ∆ Te1

∆Τm +

−

Σ ∆ Te2

∆Sm

1 2Hs

∆δ

ωB s K4

K2

K5

∆Εq _

K5

Σ

1 + s Td0 K3

Σ

1 + s TE

∆Εfd

∆ vref

_

KE

+

+

K6 Fig. 7.5. Block diagram of Single Machine Infinite Bus System

System Representation The block diagram of the system, consisting of the representation of the rotor swing equation, flux decay and excitation system is shown in Fig. 7.5. Here the damping term in the swing equation is neglected for convenience. The slip Vm of machine is taken as the system output. The transfer function of the system is approximated as ∆Vm =

1 ∆Xm − 2Iv

K2 K5 K6

Xdo vx (L6 LE ) + 1

+ L1

f ε

(7.17)

assuming ∆Xm to be small, ∆Vm = −

1 2Iv

K2 K5 K6

Xdo vx (L6 LE ) + 1

+ L1

 fρ

(7.18)

State Space Model of Single Machine System From the block diagram shown in Fig.7.5, the following state space equations for the entire system can be derived using Heffron-Phillip’s model: [61], [15] ˙ = A + B ( aref +

as ) ε

(7.19)

7 Applications of MROF-DSMC

130

T m

Te Mech. System

S

V V

ref

AVR

E

Network

δ m

Eq

i i d q

fd

Field Winding

s

PSS Fig. 7.6. Block diagram of a Power System with PSS

where 

=  f ∆Vm ∆Eq ∆Ef d  0 ψB 0 K2 D  − K1 − 2H − 2H  2H A =  − K4 0 − T 1K  T −

 Kγ K5

Tγ

BT = 0 0 0

0

Kγ Tγ

−

 3 Kγ K6

Tγ

ε 0 0 − T1



− T1γ

   ε 

(7.20)

ε

C= 1000 ρ The damping term D is included in the swing equation. The eigenvalues of the matrix should lie in left half of the s-plane for the system to be stable. The effect of various parameters (for example LE and XE ) can be examined from eigenvalue analysis. It is to be noted that the elements of A are dependent on the operating condition. 7.2.2 Controller Implementation A multirate output feedback based controller is designed for the linearized system (7.20), taking the machine slip as the plant output. The sampling time is taken as f = 0ρ05 sec and ∆ = 0ρ0125 sec. The multirate output feedback

7.2 Power System Stabilizer Design using MROF-SMC

131

Generator 1

0.02

Multirate OFB SMC PSS Classical PSS

0.015 0.01 0.005

Slip

0 −0.005 −0.01 −0.015 −0.02 −0.025

0

2

4

Time in sec.

6

8

10

Fig. 7.7. Response of Slip to MOF-SMC PSS when fault is applied at 1 sec Generator 1

2.2

Multirate OFB SMC PSS Classical PSS

2 1.8 1.6

Delta

1.4 1.2 1 0.8 0.6 0.4 0.2

0

2

4

Time in sec.

6

8

10

Fig. 7.8. Response of delta to MOF-SMC PSS when fault is applied at 1 sec

132

7 Applications of MROF-DSMC Generator 1

0.1

Control input Multirate OFB SMC PSS

0.08 0.06

Control input

0.04 0.02 0 −0.02 −0.04 −0.06 −0.08 −0.1

0

2

4

Time in sec.

6

8

10

Fig. 7.9. Profile of PSS output to MOF-SMC PSS when fault is applied at 1 sec

sliding mode control algorithm designed in (2.3) is used for this purpose. The designed control is applied to the true nonlinear model of the system constructed in VJN Y MJP L
(refer Fig. (7.6)). 7.2.3 Nonlinear Simulation Study The nonlinear simulation results are given in Figs. (7.7, 7.8 and 7.9). The results show that with classical PSS or without a PSS, the power transfer capability is limited in case of small signal disturbances in the power system. With the multirate output feedback sliding mode control PSS, the power transfer capability of the power system is extended and also the controller damps out the rotor oscillations in 5 − 6 sec as compared to 10 − 15 sec in case of a classical PSS as seen in Fig. (7.7).

t?u Vib7a2ion Con27ol of Sma72 S27wc2w7e w6ing Mwl2i7a2e Ow2pw2 Feedback ba6ed Sliding Mode Con27ol Vibration control of any system is always a formidable challenge for a system designer. Active control of vibrations relieves a designer from strengthening the structure from dynamic forces and the structure itself from extra weight and cost. The need for intelligent structures such as smart structures arises

7.3 Vibration Control of Smart Structure using MROF-SMC

133

from the high performance requirements of such structural members in numerous applications. Intelligent structures are those which incorporate actuators and sensors that are highly integrated into the structure and have structural functionality, as well as highly integrated control logic, signal conditioning and power amplification electronics. An active vibration control system consists of an actuator, controller, sensor and the system or the plant (beam), which is to be controlled. When an external force is applied to the beam as shown in Fig. (7.10), it is subjected to vibrations. These vibrations should be suppressed. F2

Z

M2

M1 w 1θ 1 F1

F2 Z

w 2θ 2 Lb

M2

M1 w 1θ 1 F1

w 2θ 2 Lb

Fig. 7.10. A flexible cantilever beam / smart beam

7.3.1 Modeling of a Smart Cantilever beam The dynamic equation of the smart structure is obtained by using both the regular beam elements and the piezoelectric beam elements by considering the first two dominant vibratory modes. The mass and stiffness matrices of the smart system includes the sensor / actuator mass and stiffness and the bonding between the sensor / actuator layer and the master structure. The cable capacitance between the sensor and signal-conditioning device is considered negligible and the temperature effects are neglected. The state space model for the smart structure with sensor / actuator pair at the fixed end of the beam is developed using the Finite Element Method [64] and is diagrammatically shown in Fig. (7.11). The dynamic equation of the smart structure is obtained by using both the regular beam elements and the piezoelectric beam elements. The mass and stiffness matrices of the dynamic equation of the smart structure includes the sensor / actuator mass and stiffness. The dynamic equation of motion of the smart structure developed using the FEM method and the Euler-Bernoulli theory is given by (7.21) M¨ τ + Cτ˙ + Kτ = fext + fcntrl where Mε C and K represent the mass, damping, stiffness matrices of the smart beam and ,fext and fcntrl represent the external and control forces acting on the beam. The transformation of (7.21) to the generalized co-ordinate frame(τ = Tj), in order to reduce the system to represent the first two modes would yield the new set of equations represented as

134

7 Applications of MROF-DSMC

Actuator Input z Actuator 1

1

2

3

4

X

Sensor 1

Sensor Output Fig. 7.11. A smart structure beam ∗ ∗ M∗ j¨ + C∗ j˙ + K∗ j = fext + fcntrl

(7.22)

with the matrices representing the beam coefficients in the generalized coordinate frame. From (7.22) state space representation of the system may be generated as     ˙1 1  2  ˙2  0 0 J  +  =  (7.23)  ˙3  −M∗−1 K∗ −M∗−1 C∗  3  M∗−1 TT h ˙4 4 +

0 M∗−1 TT fext   1

 3

 = 0p  

2 4

where, the system states following manner

i

are related to the generalized co-ordinates in the 1 2

= jε

3 4

= jε ˙

The system output is the strain experienced by the piezo-electric sensor. 7.3.2 Multirate Output Feedback based Sliding Mode Controller design for Vibration Control The vibration control of a smart cantilever beam requires a control law that is calpable of handling uncertainty. Thus, the multirate output feedback discrete-

7.3 Vibration Control of Smart Structure using MROF-SMC

135

time sliding mode control strategy [40], based on the Bartoszewicz control law [7], as discussed in Section 3.2 is used for the vibration control purpose [57]. The spectifications of the smart cantilever beam used are as follows : Cantilever beam Piezoelectric Sensor Length 0ρ3 m 0ρ075 m Breadth 0ρ03 m 0ρ03 m Thickness 5 × 10−4 m 3ρ5 × 10−4 m Density 8030 Ljx∂3 7700 Ljx∂3 11 Young’s Modulus 1ρ931 × 10 R c 6ρ8 × 1010 R c Simulation Results Two different situations are considered for the study here. The first case study is the excitation of the system by an impulse signal. The system responses and the generated control input are shown in Figs. (7.12 and 7.13) respectively. It can be noticed in Fig. (7.12) that the uncontrolled system takes much longer time to damp out the oscillations as compared to the system with the designed sliding mode control input. In the second case study, the system is subjected to a sustained sinusoidal disturbance of frequency 40 I‫܈‬. The responses and control input profile are shown in Figs. (7.14 and 7.15). The system output plots in (7.14) suggest that in the uncontrolled system the higher modes of the system get excited due to the introduction of the sinusoidal signal. This would cause vibrations in the smart structure and, if left unattended can lead to wear and tear of the system. But, when a sliding mode control is applied to the system, this high frequency component can be eliminated from the system response, thus shielding the beam from much wear and tear. However, the low frequency component of the disturbance, which is affecting the system directly due to the excitation, cannot be eliminated from the response. It is also worthy to note from the plot of the control input in Fig. (7.15) that the control algorithm generates an oscillatory control signal to compensate for the oscillatory behaviour of the disturbance.

7 Applications of MROF-DSMC 0.03

With Control Without Control

0.02

Sensor Output

0.01

0

−0.01

−0.02

−0.03

0

0.1

0.2

0.3

0.4

Time in Secs

0.5

0.6

0.7

0.8

Fig. 7.12. System Response to an impulse excitation 0.14

0.12

0.1

Control Input

136

0.08

0.06

0.04

0.02

0

−0.02

0

0.2

0.4

0.6

0.8

1

1.2

Time in Secs

1.4

1.6

1.8

2

Fig. 7.13. Generated control input profile for an impulse excitation

7.3 Vibration Control of Smart Structure using MROF-SMC −3

2.5

x 10

With Control Without Control

2 1.5

Sensor Output

1 0.5 0

−0.5 −1 −1.5 −2 −2.5

0

0.1

0.2

0.3

0.4

Time in Secs

0.5

0.6

0.7

0.8

Fig. 7.14. System Response to a sinusoidal disturbance −3

12

x 10

10

Control Input

8

6

4

2

0

−2

0

0.2

0.4

0.6

0.8

1

1.2

Time in Secs

1.4

1.6

1.8

2

Fig. 7.15. Generated control input profile for a sinusoidal disturbance

137

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Indez

Pe Re sliding sector 93 Pe norm 93 definition 94 discrete-time sliding mode 93 existence 96 quasi sliding mode 97 simplified 96 definition 95 design 95–96 illustration 96 VS controller 97 multirate output feedback 99, 101 continuation method

7–9

discrete-time sliding mode 10, 16 Pe Re sliding sector 93 inherent discontinuity 10 integral multirate 46, 47 reaching law 46 state feedback 45 multirate output feedback 29, 40, 134 control 30, 36 reaching law 29 state feedback 28, 39 control 29 reaching law 28 terminal sliding mode 105, 108 control 109 definition 108 multirate output feedback 112 nilpotent suraface 109

sliding surface 109 state feedback 112 time delay systems 66, 70 control 67 uncertain 67 uncertain 27, 134 comparison 34, 39–42, 70 integral 44 matched 28 time delay 67 unmatched 35 multirate output 11 feedback 13, 85 Pe Re sliding sector 93 advantage 14 applications 38, 121, 128, 130, 132, 134 comparison 34, 70 constraints 23 feedback linearizable systems 117 initial control 19, 30, 59, 67, 100 motivation 15 nonlinear observer 87 nonlinear systems 117, 121, 125, 128, 130 reaching law 22, 29 sliding mode band 23, 31, 36 terminal sliding mode 112 time delay 51, 58, 61, 67 uncertain 19, 22, 27, 35, 67 variable structure control 99, 101 sampling 11–13 disturbance estimation 47

146

Index nonlinear observer 83–84 nonlinear systems 83, 87 output-state relationship 13, 56 time delay 55 uncertain 20, 55

nonlinear systems feedback linearizable 117 feedback linearizable systems terminal sliding mode 117 finitely discretizable systems 81 condition 82 definition 81 dialtion 83 homogeneous 83 multirate sampling 83 shuffle product 82 modeling stepper motor 121 multirate observer 83–84 sufficient condition 84 observer 84 quasi sliding mode 81, 85 state feedback 85 sliding surface 86 lyapunov function 85 stepper motor 121 nonminimum phase systems 72 nominal zero dynamics trajectory 74 bounded preview method 76 infinite horizon method 74 preview time 76 tracking 71 multirate output feedback 76 nominal trajectory 73 restrictions 75 two part control 72 zero dynamics 72 power system modeling 128 single machine 129 state space 129 single machine 129 control 130 stabilizer 130 multirate output feedback 130

128,

simulation 132 single machine 132 quasi sliding mode 17 band 17 bound 21 control 18 Pe Re sliding sector 97 multirate output feedback 18, 125 state feedback 17, 18, 124 uncertain 19–21 criterion 17 relaxation 27 nonlinear systems 81, 85 lyapunov function 85 state feedback 85 stepper motor 121, 124 nonminimum phase systems 71 two part control 72 reaching law 17, 21, 85 time delay systems 51 form 1 57 form 2 60 form 3 63 sliding mode 5 continuous-time 2, 5 control discontinuity 7 equivalent 9 reaching law 9 terminal 105 discrete-time 10 inherent discontinuity 10 function 4 reaching condition 4 switching scheme 6 terminal definition 105 smart structure 133 construction 133 modeling 133 Euler-Bernoulli theory 133 FEM method 133 state space 134 vibration control 132 multirate output feedback 132 stepper motor 121 control

Index multirate output feedback state feedback 124 D-Q transformation 122 differential flatness 123 model 121, 122 continuous time 122 discrete-time 123, 124 flat outputs 123 linearized 124 PMDC 121 quasi-sliding mode 124

125

terminal sliding mode continuous time control 106 definition 105 discretization 106 sliding surface 105 discrete-time 105, 108 control 109 definition 108 multirate output feedback 112 nilpotent surface 109 sliding surface 109 state feedback 112 discrete-time systems nonlinear systems 117 finite time convergence 105 time delay input 51, 52

147

output 51, 52 state 51, 52 uncertain 52 time delay systems 51 discretization 52 form 1 52, 57 multirate 58 form 2 52, 60 multirate output feedback 61 state feedback 60 form 3 52, 63 multirate output feedback 63 form 4 52, 66 multirate 67 multirate disturbance 55 output-state relation 56–57 multirate sampling 55 representation 51 sliding surface design 53 condition 54 uncertain 67 unstable zero dynamics see nonminimum phase systems72 variable structure control 2, 97 multirate output feedback systems 1

99, 101

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