Automotive Model Predictive Control: Models, Methods and Applications (Lecture Notes in Control and Information Sciences, Volume 402)

Lecture Notes in Control and Information Sciences 402 Editors: M. Thoma, F. Allgöwer, M. Morari Luigi del Re, Frank A...

Author: Luigi Del Re | Frank Allgöwer | Luigi Glielmo | Carlos Guardiola | Ilya Kolmanovsky

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Lecture Notes in Control and Information Sciences 402 Editors: M. Thoma, F. Allgöwer, M. Morari

Luigi del Re, Frank Allgöwer, Luigi Glielmo, Carlos Guardiola, and Ilya Kolmanovsky (Eds.)

Automotive Model Predictive Control Models, Methods and Applications

ABC

Series Advisory Board P. Fleming, P. Kokotovic, A.B. Kurzhanski, H. Kwakernaak, A. Rantzer, J.N. Tsitsiklis

Editors Luigi del Re

Carlos Guardiola

Institute for Design and Control of Mechatronical Systems Johannes Kepler University Linz Altenbergerstr. 69 4040 Linz

Universidad Politécnica de Valencia (UPV) Departamento de Máquinas y Motores Térmicos Camino de Vera, s/n. 46022 Valencia (Spain)

Frank Allgöwer

Ilya Kolmanovsky

Institute for Systems Theory and Automatic Control, University of Stuttgart Pfaffenwaldring 9 70550 Stuttgart

Technical Leader, Powertrain Control R&A Ford Research and Adv. Engineering Ford Motor Company 2101 Village Road Dearborn, MI 48124

Luigi Glielmo Università del Sannio in Benevento Facoltà di Ingegneria Corso Garibaldi 107, 82100 Benevento Ordinario di Automatica

ISBN 978-1-84996-070-0

e-ISBN 978-1-84996-071-7

DOI 10.1007/978-1-84996-071-7 Lecture Notes in Control and Information Sciences Library of Congress Control Number: 2009943555 c

ISSN 0170-8643

2010 Springer-Verlag Berlin Heidelberg

MATLAB and Simulink are registered trademarks of The MathWorks, Inc., 3 Apple Hill Drive, Natick, MA 01760-2098, U.S.A. http://www.mathworks.com 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 any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable for prosecution under the German Copyright Law. 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. Typeset & Cover Design: Scientiﬁc Publishing Services Pvt. Ltd., Chennai, India. Printed in acid-free paper 543210 springer.com

Preface

Automotive control has developed over the decades from an auxiliary technology to a key element without which the actual performances, emission, safety and consumption targets could not be met. Accordingly, automotive control has been increasing its authority and responsibility – at the price of complexity and diﬃcult tuning. The progressive evolution has been mainly led by speciﬁc applications and short term targets, with the consequence that automotive control is to a very large extent more heuristic than systematic. Product requirements are still increasing and new challenges are coming from potentially huge markets like India and China, and against this background there is wide consensus both in the industry and academia that the current state is not satisfactory. Model-based control could be an approach to improve performance while reducing development and tuning times and possibly costs. Model predictive control is a kind of model-based control design approach which has experienced a growing success since the middle of the 1980s for “slow” complex plants, in particular of the chemical and process industry. In the last decades, several developments have allowed using these methods also for “fast” systems and this has supported a growing interest in its use also for automotive applications, with several promising results reported. Still there is no consensus on whether model predictive control with its high requirements on model quality and on computational power is a sensible choice for automotive control. Are the successful applications exceptions or are they paving the way for its wider use? To discuss these questions, a group of researchers with very diﬀerent backgrounds has gathered in Feldkirchen near Linz (Austria) on February 9-10, 2009, in a workshop organized by the Austrian Center of Competence in Mechatronics (ACCM), and under the auspices of the Johannes Kepler University in Linz. The 25 participants from 11 countries were chosen to represent diﬀerent aspects of the topic, from combustion to numerics. Of course, they do not represent but a small part of the corresponding scientiﬁc communities, but a larger meeting would have made the fruitful discussion more diﬃcult.

VI

Preface

The contributions in this book are revised versions of most presentations given in this workshop and reﬂect the interdisciplinary nature of the topic. The book starts with a general presentation aimed at setting the stage, the attention of the interplay ﬁve contributions on diﬀerent modeling approaches follow, the last one being a bridge to the control part. Three presentations address mainly methodological issues, followed by application examples. Neither the workshop nor this collection of contributions would have been possible without the support of several people (in particular of Daniel Alberer, Daniela Hummer and Peter Ortner). Thanks are due also to the reviewers of the single chapters who have done an important and essential work.

Hosting Organisation Austrian Center of Competence in Mechatronics, Linz Austria Johannes Kepler University Linz, Austria

Program Committee Frank Allg¨ ower Luigi del Re Luigi Glielmo Carlos Guardiola Ilya Kolmanovsky

Universit¨ at Stuttgart, Germany Johannes Kepler Universit¨ at Linz, Austria Univerisit´ a del Sannio, Italy Universidad Politecnica de Valencia, Spain Ford Research and Advanced Engeneering, USA

Organizing Committee Daniel Alberer Daniela Hummer Peter Ortner

Johannes Kepler University Linz, Austria Johannes Kepler University Linz, Austria Johannes Kepler University Linz, Austria

Referees ¨ J. Angeby F. Allg¨ ower M. Alamir A. Amstutz J.P. Arr`egle F. Borrelli L. del Re S. Di Cairano M. Diehl L. Eriksson

P. Falcone L. Glielmo C. Guardiola D. Hrovat T.A. Johansen R. Johansson E. Kerrigan B. Koch I. Kolmanovsky D. Limon

P. Moulin G. Naus A. Ohata J.G.V. Ortiz P. Ortner N. Peters R. Scattolini G. Stewart P. Tunest˚ al

Contents

1

Chances and Challenges in Automotive Predictive Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Luigi del Re, Peter Ortner, Daniel Alberer 1.1 Introduction: The Rationale . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Alternatives for Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.1 First Principles Models . . . . . . . . . . . . . . . . . . . . . . . . 1.2.2 Data-only Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.3 Advanced Use of Data . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Alternatives for Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.1 Basic Algorithmic Approaches . . . . . . . . . . . . . . . . . . 1.3.2 Coping with Nonlinearity . . . . . . . . . . . . . . . . . . . . . . 1.4 Chances: State and Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 4 5 6 7 9 9 12 15 19 19

Part I: Models 2

On Board NOx Prediction in Diesel Engines: A Physical Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Jean Arr`egle, J. Javier L´ opez, Carlos Guardiola, Christelle Monin 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Main Physical/Chemical Mechanisms of NOx Formation/Destruction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 NOx Re-burning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.2 NOx Formation in LTC Conditions . . . . . . . . . . . . . . 2.3 Mechanisms and Model Sensitivity . . . . . . . . . . . . . . . . . . . . . 2.3.1 Structure of Physically-based NOx Models . . . . . . . 2.3.2 Flame Temperature Determination . . . . . . . . . . . . . . 2.4 Input Parameters Accuracy . . . . . . . . . . . . . . . . . . . . . . . . . . . .

25

25 26 27 29 30 30 31 33

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2.4.1 2.4.2

Intake Air Mass Flow Rate Accuracy . . . . . . . . . . . . Air + EGR Mixture Temperature and Oxygen Fraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

4

Mean Value Engine Models Applied to Control System Design and Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Pierre Olivier Calendini, Stefan Breuer 3.1 State of the Art Mean Value Engine Model . . . . . . . . . . . . . . 3.2 System Model Structure as a Response to the Requirements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 Bond Graph Applied to Mean Value Engine Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.2 Naturally Aspirated and Turbocharged Engine in Bond Graph Structure . . . . . . . . . . . . . . . . . . . . . . 3.3 Basic Blocs for Building Mean Value Models . . . . . . . . . . . . . 3.3.1 The Volume Bloc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.2 The Gas Exchange Bloc . . . . . . . . . . . . . . . . . . . . . . . 3.3.3 Heat Exchange Models . . . . . . . . . . . . . . . . . . . . . . . . 3.3.4 Combustion Model Possibilities . . . . . . . . . . . . . . . . . 3.3.5 Environment Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Application Example: Choice of an Air Loop Control Strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.1 Implementation of the Robustness Simulation . . . . 3.4.2 Results of the Robustness Simulations . . . . . . . . . . . 3.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Physical Modeling of Turbocharged Engines and Parameter Identiﬁcation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Lars Eriksson, Johan Wahlstr¨ om, Markus Klein 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 MVEM Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1 Library Development . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.2 Building Blocks: Component Models . . . . . . . . . . . . 4.2.3 The Engine Cylinders: Flow, Temperature, and Torque . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.4 Implementation Examples . . . . . . . . . . . . . . . . . . . . . . 4.3 Modeling of a Diesel Engine with EGR/VGT . . . . . . . . . . . . 4.3.1 Experimental Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.2 Minimum Number of States . . . . . . . . . . . . . . . . . . . . 4.3.3 Model Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Gray-Box Models and Identiﬁcation . . . . . . . . . . . . . . . . . . . . 4.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

33 33 34 35 37 38 39 39 40 42 42 43 43 43 44 45 47 49 50 51 53 53 54 54 55 59 60 60 61 64 64 65 68 69

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5

6

Dynamic Engine Emission Models . . . . . . . . . . . . . . . . . . . . . . . Markus Hirsch, Klaus Oppenauer, Luigi del Re 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Data-based Model Identiﬁcation . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Mean Value Emission Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.1 Input Selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.2 Model Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.3 Parameter Identiﬁcation . . . . . . . . . . . . . . . . . . . . . . . 5.3.4 Regressor Selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.5 Realization and Results . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Crank Angle Based Emission Model . . . . . . . . . . . . . . . . . . . . 5.4.1 Workﬂow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.2 1-zone Process Calculation . . . . . . . . . . . . . . . . . . . . . 5.4.3 2-zone Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.4 Emission Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.5 Model Development and Veriﬁcation . . . . . . . . . . . . 5.5 Data for Identiﬁcation: Input Design . . . . . . . . . . . . . . . . . . . . 5.6 Limitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

IX

73 73 75 76 76 77 77 78 78 79 79 80 80 81 82 82 85 85 86

Modeling and Model-based Control of Homogeneous Charge Compression Ignition (HCCI) Engine Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 Rolf Johansson, Per Tunest˚ al, Anders Widd 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 6.2 HCCI Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 6.2.1 Fuel Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 6.2.2 Auto-ignition Modeling . . . . . . . . . . . . . . . . . . . . . . . . 93 6.2.3 Thermal Modeling and Auto-ignition . . . . . . . . . . . . 93 6.3 Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 6.3.1 Model Predictive Control . . . . . . . . . . . . . . . . . . . . . . 98 6.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

Part II: Methods 7

An Overview of Nonlinear Model Predictive Control . . . . Lalo Magni, Riccardo Scattolini 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Problem Formulation and State-feedback NMPC Control Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.1 Feasibility and Stability in Nominal Conditions . . . 7.2.2 The Robustness Problem . . . . . . . . . . . . . . . . . . . . . . . 7.3 Output Feedback and Tracking . . . . . . . . . . . . . . . . . . . . . . . .

107 107 108 109 109 111

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7.3.1 Output Feedback . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.2 Tracking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4 Implementation Problems and Alternative Approaches . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

9

Optimal Control Using Pontryagin’s Maximum Principle and Dynamic Programming . . . . . . . . . . . . . . . . . . . . Bart Saerens, Moritz Diehl, Eric Van den Bulck 8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 Optimal Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2.1 Pontryagin’s Maximum Principle . . . . . . . . . . . . . . . . 8.2.2 Dynamic Programming . . . . . . . . . . . . . . . . . . . . . . . . 8.3 Vehicle and Powertrain Model . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.1 Vehicle and Driveline Model . . . . . . . . . . . . . . . . . . . . 8.3.2 Engine Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4 Minumum-fuel Acceleration with the Maximum Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.5 Minumum-fuel Acceleration with Dynamic Programming . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.6 Discussion of the Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.6.1 Comparison between the Maximum Principle and Dynamic Programming . . . . . . . . . . . . . . . . . . . . 8.6.2 Comparison with Other Research . . . . . . . . . . . . . . . 8.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . On the Use of Parameterized NMPC in Real-time Automotive Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mazen Alamir, Andr´e Murilo, Rachid Amari, Paolina Tona, Richard F¨ urhapter, Peter Ortner 9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2 The Parameterized NMPC: Deﬁnitions and Notation . . . . . 9.3 Example 1: Diesel Engine Air Path Control . . . . . . . . . . . . . . 9.4 Example 2: Automated Manual Transmission Control . . . . . 9.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

111 112 113 115 119 119 120 123 124 124 125 125 129 132 134 135 135 136 137 139

140 141 142 145 148 148

Part III: Applications 10 An Application of MPC Starting Automotive Spark Ignition Engine in SICE Benchmark Problem . . . . . . . . . . . . Akira Ohata, Masaki Yamakita 10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2 Control Design Strategy in MBD . . . . . . . . . . . . . . . . . . . . . . . 10.3 Benchmark Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

153 153 159 161

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10.4 Application of MPC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165 10.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169 11 Model Predictive Control of Partially Premixed Combustion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Per Tunest˚ al, Magnus Lewander 11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3 PPC Deﬁnition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.4 Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.4.1 Control Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.5 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.5.1 Response to EGR Disturbance . . . . . . . . . . . . . . . . . . 11.5.2 Response to Load Changes . . . . . . . . . . . . . . . . . . . . . 11.5.3 Response to Speed Changes . . . . . . . . . . . . . . . . . . . . 11.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 Model Predictive Powertrain Control: An Application to Idle Speed Regulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Stefano Di Cairano, Diana Yanakiev, Alberto Bemporad, Ilya Kolmanovsky, Davor Hrovat 12.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.2 Engine Model for Idle Speed Control . . . . . . . . . . . . . . . . . . . . 12.3 Control-oriented Model and Controller Design . . . . . . . . . . . 12.4 Controller Synthesis and Reﬁnement . . . . . . . . . . . . . . . . . . . . 12.4.1 Feedback Law Synthesis and Functional Assessment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.4.2 Prediction Model Reﬁnement . . . . . . . . . . . . . . . . . . . 12.5 Experimental Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 On Low Complexity Predictive Approaches to Control of Autonomous Vehicles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Paolo Falcone, Francesco Borrelli, Eric H. Tseng, Davor Hrovat 13.1 Introduction to Autonomous Guidance Systems . . . . . . . . . . 13.2 Vehicle Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.3 Low Complexity Predictive Path Following . . . . . . . . . . . . . . 13.3.1 Two Levels Autonomous Path Following . . . . . . . . . 13.3.2 Single Level Autonomous Path Following . . . . . . . . 13.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

171 171 172 172 174 174 175 175 177 177 179 180 181 183

183 184 185 189 189 190 191 193 193 195

195 198 201 201 204 206

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13.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209 14 Toward a Systematic Design for Turbocharged Engine Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Greg Stewart, Francesco Borrelli, Jaroslav Pekar, David Germann, Daniel Pachner, Dejan Kihas 14.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.2 Engine Control Requirements . . . . . . . . . . . . . . . . . . . . . . . . . . 14.2.1 Steady-state Engine Calibration . . . . . . . . . . . . . . . . 14.2.2 Control Functional Development . . . . . . . . . . . . . . . . 14.2.3 Functional Testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.2.4 Software Development . . . . . . . . . . . . . . . . . . . . . . . . . 14.2.5 Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.2.6 Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.2.7 Certiﬁcation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.2.8 Release and Post-release Support . . . . . . . . . . . . . . . 14.2.9 Iteration Loops . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.3 Modeling and Control for Turbocharged Engines . . . . . . . . . 14.3.1 Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.4 Model Predictive Control and Computational Complexity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.4.1 Explicit Predictive Control . . . . . . . . . . . . . . . . . . . . . 14.4.2 On the Complexity of Explicit MPC Control Laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.5 Summary and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 An Integrated LTV-MPC Lateral Vehicle Dynamics Control: Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Giovanni Palmieri, Osvaldo Barbarisi, Stefano Scala, Luigi Glielmo 15.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.2 Full Vehicle Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.3 Lateral Vehicle Dynamic Control Strategy . . . . . . . . . . . . . . . 15.3.1 Reference Signals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.3.2 Estimation of Tire Variables . . . . . . . . . . . . . . . . . . . . 15.3.3 Supervisor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.3.4 Model Predictive Control . . . . . . . . . . . . . . . . . . . . . . 15.3.5 An Alternative 2PI Regulator . . . . . . . . . . . . . . . . . . 15.4 A Reduced Model for Slip Control . . . . . . . . . . . . . . . . . . . . . . 15.5 A Slip Control Strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.5.1 Feedback Action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.6 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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16 MIMO Model Predictive Control for Integral Gas Engines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ¨ Jakob Angeby, Matthias Huschenbett, Daniel Alberer 16.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.2 System Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.3 Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.4 Model Predictive Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.5 Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.5.1 Objective Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.5.2 Model Derivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.6 Model Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.7 Real-time MPC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.8 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.9 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 A Model Predictive Control Approach to Design a Parameterized Adaptive Cruise Control . . . . . . . . . . . . . . . . . Gerrit J.L. Naus, Jeroen Ploeg, M.J.G. Van de Molengraft, W.P.M.H. Heemels, Maarten Steinbuch 17.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.2 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.2.1 Quantiﬁcation Measures . . . . . . . . . . . . . . . . . . . . . . . 17.2.2 Parameterization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.3 Model Predictive Control Problem Setup . . . . . . . . . . . . . . . . 17.3.1 Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.3.2 Control Objectives and Constraints . . . . . . . . . . . . . 17.3.3 Control Problem / Cost Criterion Formulation . . . . 17.4 Controller Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.4.1 Parameterization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.4.2 Implementation Issues . . . . . . . . . . . . . . . . . . . . . . . . . 17.4.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.5 Conclusions and Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Author Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285

Chapter 1

Chances and Challenges in Automotive Predictive Control Luigi del Re, Peter Ortner, and Daniel Alberer

Abstract. Recent years have witnessed an increased interest in model predictive control (MPC) for fast applications. At the same time, requirements on engines and vehicles in terms of emissions, consumption and safety have experienced a similar increase. MPC seems a suitable method to exploit the potentials of modern concepts and to fulfill the automotive requirements since most of them can be stated in the form of a constrained multi input multi output optimal control problem and MPC provides an approximate solution of this class of problems. In this introductory chapter, we analyze the rationale, the chances and the challenges of this approach. This chapter does not intend to review all the literature, but to give a flavor of the challenges and chances offered by this approach.

1.1 Introduction: The Rationale Most automotive control problems can be stated in terms of an optimal multiobjective problem. In most automotive cases the system to be controlled shows complex, interconnected dynamics, and has bounded inputs [6, 31, 33, 49, 55] – either physically bounded, like the opening of a throttle which is limited to a certain range, or constrained by purpose, like the position of a variable turbocharger vane, which must be kept in a given range depending on the actual operating point to avoid a damage of the turbocharger. As an example, let us consider the reference engine control problem to be solved for the emission certification of a passenger vehicle. It requires the vehicle to follow a given speed profile as in Figure 1.1 [32] under normalized conditions, while keeping the regulated components of the exhaust emissions in average below a given threshold defined by law. Of course, besides these mandatory requirements, also customers’ expectations must be met, and this leads to further requirements Luigi del Re, Peter Ortner, and Daniel Alberer Institute for Design and Control of Mechatronical Systems, Johannes Kepler Universit¨at Linz e-mail: {luigi.delre,peter.ortner,daniel.alberer}@jku.at L. del Re et al. (Eds.): Automotive Model Predictive Control, LNCIS 402, pp. 1–22. c Springer-Verlag Berlin Heidelberg 2010 springerlink.com

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Fig. 1.1 New European driving cycle (NEDC)

e.g. in terms of drivability or fuel consumption. All this describes a classical multiobjective problem, as several partly conflicting targets have to be optimized using a limited set of choices (e.g. injection times and quantities, exhaust gas recirculation (EGR)-valve position, turbocharger vane position, etc.). In practice, usually one of the target quantities is chosen as the minimization target (usually the fuel consumption) while the others are restated as constraints (secondary targets). An optimal performance under these conditions is achieved by solving the finite horizon optimal control problem described by min

u(t)∈U ∀t∈[t0 ,te ]

te

J = ψ (t, x, u)dt t0

⎧ T (t) = T (x(t), u(t)) = Tre f (t) ± Δ T (t) ⎪ ⎪ ⎪ ⎪ u j, min ≤ u j (t) ≤ u j, max j = 1..m ⎨ s.t. x˙ = f (x, u) ⎪ te ⎪ ⎪ 1 ⎪ ⎩ te −t0 φi (t, x, u)dt ≤ Φi i = 1..n

(1.1)

t0

where T is the engine torque, Tre f is the torque reference trajectory implicitly defined by the vehicle parameters and the speed profile, Δ T (t) the torque tolerance corresponding to the speed tolerance, J is the main target, ψ is the instantaneous value of the cost function (typically the fuel consumption), t0 and te are the time limits of the test cycle, and u j,min and u j,max the actuator bounds. x is the vector of states which are relevant for modeling the torque and emissions. φi (t) are the regulated quantities, typically emissions (e.g. NOx ), with their admissible cumulative limit over the test length expressed by Φi . Additionally, pointwise-in-time

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Fig. 1.2 Basic idea of the MPC operation

Fig. 1.3 General MPC control loop (from [20])

constraints on emissions (e.g. visible smoke) could be included in the problem formulation. Many other automotive control problems can also be formulated in terms of a suitable optimal problem. For instance, adaptive cruise control [9], or the stabilization of a vehicle along the trajectory – the lateral dynamic stabilization – can be described as an optimal tracking problem with constraints on the inputs and on the outputs. In the latter case, for example, the system inputs are typically the steering angle and the traction and braking torques on the wheels while the outputs are the vehicle translatory and rotatory coordinates. The driver will typically choose the steering angle and indicate by the gas pedal position his traction torque request, the constraints result e.g. from the maximum transmissible cornering forces between tire and road. Notice that these quantities are usually unknown (as the load distribution in the car and especially the tire and road conditions are not known). Then the optimal problem can be stated for instance in following terms: given an initial vehicle speed, a driver torque request and a steering angle, determine the values of the traction and braking forces over the next samples to minimize the difference between applied and requested torque while keeping the cornering forces below the (unknown) limit. Solving directly such optimization problems is usually not practically possible, even though attempts have been made in this direction for a very long time (e.g. see

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[51] for an early example). There are a number of reasons for this, among which the lack of a suitable model, and the excessive computational load for the actual electronic control units (ECUs), especially in view of the high dimensional and mostly nonconvex problems arising from the complex and nonlinear nature of most applications. MPC is a technique first developed in the 1980s [22] for the MIMO control of complex linear plants with bounded inputs and constrained variables and at first successful for systems with slow dynamics. It has become a standard in some technical fields like refineries and has been extended over the years to many more applications and faster systems (see e.g. [20] and [44]). MPC does not solve the general optimal control problem, but yields an approximate “receding horizon” solution. This approximate solution is in many cases close to the real optimal solution at each time instant. The receding horizon optimal sequence of control inputs to the plant is computed over a limited number of steps (Figure 1.2) so to minimize a cost function under constraints, but only the values corresponding to the next sample time are used. The computation is repeated at every sample while taking into account the actual change of the state of the plant [45], thus leading to a closed loop control. Figure 1.3 shows a generic structure of the implementation of a MPC with three core blocks: a system model, a dynamic optimizer and an evaluation block, including the cost function and the constraints on the inputs, outputs and states. For linear systems, such a framework exists and has been used already for a long time. However, this is not yet the case for a nonlinear framework, as automotive applications would mostly need. Indeed, even though impressive progress has been achieved over the last decades, it is not sensible to expect dynamic optimizer to be able to produce reliable results with any model, and the art – the real challenge – consists in finding sufficiently complex models to capture the main behavior of nonlinear plants while being still suitable for online finite horizon optimization. In the following, we shall shortly review the options for modeling and for online optimization and then highlight both their possibilities and risks. By this, it will become evident that, as expected, no single solution is available but the choice will be dictated by the different degrees of requirements and of physical insight of the specific task.

1.2 Alternatives for Modeling MPC is to a large extent a feedforward approach, and so depends strongly on model quality – drawbacks of using poor models for feedforward are well known (see e.g. [13]), whose main message can be resumed stating that, if the model information is too poor, it might be better not to use it at all. The problem with most automotive problems is that they are, by their nature, open-loop; nobody measures the cornering forces and the tire-road contact coefficients or the emissions in a production vehicle. As a consequence, the question becomes not on whether to use feedforward or not, but on how to derive a sufficiently precise model for it without too much effort.

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As automotive control belongs to the technical fields in which experiments are usually possible, and even normal, the model designer has the full choice of modeling options, ranging from first principle models to purely data-based ones.

1.2.1 First Principles Models Automotive first principle models range from precise computational (distributed) fluid dynamics (CFD) models, in which the flows and possibly combustion is modeled precisely, to mostly lumped models for vehicle control in multi-body languages like ADAMS, partly produced directly from 3D drawings produced e.g. in CATIA. In the case of engines, models may be complex combustion models down to mean value models commonly used in engine control simulation [23]. First principle models have always been the first choice of designers as they allow physical insight in the systems and experimenting with design changes. In the academic automotive community, there is still an impressive interest in using them and using the feedback properties of most control systems to compensate their shortcomings. These result from a common characteristic of all first principle models is that they essentially contain the information known a priori, i.e., what the modeler is aware of at the start of the project. While this is unavoidable in some cases, e.g., when the system does not yet exist, it bears the enormous disadvantage that all deviations from expectation will not included. As a consequence, this approach will work with systems which are relatively easy to model, like the basic vehicle dynamics. Modeling the dynamics of a vehicle can be performed for instance using the “bicycle model” (see Figure 1.4) which assumes a stiff connection between two wheels which have very few degrees of freedom and essentially one unknown coefficients – the tire-road contact. Based on these simple assumptions the dynamics can be set up and used for studies and simulations, sometimes with surprisingly good results. The industrial community has mostly a different approach, especially in the case of engine control. Control candidates with high parameter numbers are set up, and tuned experimentally with an enormous effort, yielding in practice an approximated inverse model and is used to design a reliable, yet utterly heuristic controller. Industrial software exists, like AVL’s Cameo, to support this process. There is a consensus that something else, in particular model-based control, would be better, but it is not available. The main reason for this is that first principle models tend to be too simple to describe the real systems to the necessary degree of precision necessary to meet the increasing performance requirements. Not surprisingly, model-based design works better in fields like traction control, which relies on partly well known models with relatively lax requirements (e.g. braking force may shortly lead to block the wheel, but not continuously) while they fail in cases like emissions, for which hardly any online models exist and for which a single wrong transient may jeopardize the whole test.

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Fig. 1.4 Simplified vehicle “bicycle mode” [16]:, a,b represents the distance of the front and rear wheel to center of gravity, Fl f the front wheel longitudinal force, Fc f the front wheel lateral (cornering) force, Fl r the rear wheel longitudinal force, Fcr the rear wheel lateral force, I the car inertia, m the car mass, δr the rear wheel steering angle, δ f the front wheel steering angel, α the tire slip angle, vl f the longitudinal velocity of the front wheel, and vc f the lateral velocity of the front wheel

This essential difference in modeling is mirrored in the practical development of the models for MPC and successful applications: while control of longitudinal dynamics as in [52], or the lateral approaches as in [17] can be built upon first principle models with few unknown parameters, the same is not true for many applications of engine control. The applications have been concentrated essentially on the air path control [42, 46], whereas the mostly unknown emission models are not included in the control formulation. Even though first principle models are always the most complete information source, frequently they cannot be used for control design, as they are too complex for on-line use, they must be approximated. So, a natural question could be: why bother for a good first principle model and not look directly for a good approximation?

1.2.2 Data-only Models The natural alternative to first principle models are data-based models, which typically fix a parameterized candidate model without reference to the physics of the real plant and requires large quantities of data in order to compute the optimal estimation of the parameters. The main drawbacks of purely data-based models are, of course, the need of data, but also the frequently unknown extrapolation properties: as in all nonlinear identification problems, the choice of the excitation is critical, because actually all moments of the exciting signal are critical [26]. For some cases, as emissions, the candidate model structure will in general not contain the true model, therefore system identification becomes essentially an approximation tool. Against this background, there has been much work in this

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Fig. 1.5 The measured and estimated NOx values using an LPV model

field with universal approximations, either as artificial neural networks (ANN) [11, 28, 29, 50] or in other forms like multi-linear models. Unfortunately, simple mapping approaches, like artificial neural networks may fail and produce not only wrong extrapolations but even wrong interpolations. Also “practical” approximators, e.g. linear parameter varying (LPV) methods [31, 56] as well as simple structures which can be easily included in the optimization like the one used by [46] have been used in the quest for suitable models, i.e., for those who capture the main nonlinearities while remaining sufficiently simple. The different model quality achievable for different quantities can be easily seen in the comparison of the two following figures concerning an LPV approximation of NOx emissions [57] (Figure 1.5) and the same method used for the prediction of air path behavior [58] (Figure 1.6). Not surprisingly LPV identification works fine for the air path, but is not able to cover to nonlinearities in the emission modeling, as the underlying system is much more complex [21], in particular for soot [35, 36, 47, 53]. Moreover, the dimension of data-based models increases significantly (in comparison to a linear approach).

1.2.3 Advanced Use of Data Against this background, there is a rationale for combining both approaches so to get simple and high performing models. There are at least two ways to do this, the first one, the so called gray-box model, uses physical understanding to describe explicitly aspects of the models which can be described analytically and combines them

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Fig. 1.6 A validation result of the mass air flow (MAF) model

with data-based models. So [18] has developed methods for estimation of torque which include physical and databased methods respectively, the same approach has recently been used by [24] to get emission models. For a different example see e.g. [40]. Figure 1.7 shows a simple example of a gray-box model – the temperature dependency is treated using a physically motivated function, while the main production of emissions is performed by data-based approaches. A second possibility consists in looking for global patterns in the observed data which can be expressed analytically (with few parameters) and thus infer not only the parameters of a given model but also its structure from the data. Figure 1.8 shows the results for the case of opacity prediction using a model derived from the data using a genetic programming (GP) approach [59] to define its structure [4]. The resulting models tend to be complex and to yield different structures according to the data used for identification.

Fig. 1.7 Example of an emission gray-box model: all the input quantities are ECU variables, from [24]

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Chances and Challenges in Automotive Predictive Control

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16 target estimated

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opacity (%)

10 8 6 4 2 0 −2

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Fig. 1.8 Evaluation (on a part of the NEDC data) of models produced by GP for the smoke emissions

In this context, it is interesting to note [25] that the real critical issue is not the method used for the identification, not even the class of functions, but the specific data.

1.3 Alternatives for Optimization The second critical block in Figure 1.3 is the dynamic optimizer. Its performance is critical both for the performance of the closed loop as well as for its online implementability, especially for fast systems. In the automotive applications three approaches have been gaining importance over the last years, which can be roughly classified as explicit MPC, online optimization with guaranteed computational times, and optimization using simplified problems. All these three methods have been tested in real automotive applications, moreover all three have been used on the same test bench for the air path control of a Diesel engine.

1.3.1 Basic Algorithmic Approaches 1.3.1.1

Explicit MPC

This explicit MPC formulation is based on the idea that the active constraints define regions of an extended state space. Inside each of these regions, MPC behaves like an affine state feedback control law which can be computed a priori [7].

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Fig. 1.9 Section of the polyhedral partition of MPC 1

Storing all these control laws off-line means that at every instant a linear controller can be used, with a low computational effort. Thus, the real online effort consists in finding the active region, i.e., to select the proper controller. This effort can be substantial, but it is possible to show that it corresponds to looking for the active constraint set in the standard QP algorithm. Therefore, the final numerical effort will be significantly lower as in the online case where the QP algorithm should also compute the solution to the optimal equation to the specific constraint set. This method has become very popular also due to the availability of a toolbox1. Figure 1.9 shows a typical partition of the state space corresponding to the active constraint set for a specific problem [42]. 1.3.1.2

Online MPC and NMPC

Nonlinear optimization has been an important research topic for a long time. Consequently, many different approaches have been developed to tackle the problem, among them hypothesis on the expected changes of the active constraints in the next future. In this direction a recent contribution [19] has yielded an algorithm which converges to the correct solution, but allows also an intermediate stop after a fixed time without substantial problems (see Figure 1.10). The advantage of this method is that it allows a fixed time suboptimal and feasible solution of the QP solution. This can also be embedded into a more complicated nonlinear structure, in which for instance the values of the state transition matrix 1

http://www.dii.unisi.it/hybrid/toolbox/ http://control.ee.ethz.ch/∼ mpt/

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Chances and Challenges in Automotive Predictive Control

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Fig. 1.10 Homotopy paths (solid) from one QP to the next with limited number of active set changes.

over the prediction horizon are computed according initial prediction, as done for instance in [41]. 1.3.1.3

Recasting the Problem

In many cases the consideration of the control horizon as done usually may be misleading. Consider for instance the injection in the engine: it is not possible to give a continuous input signal over a given time, but it is only possible to inject a given quantity at a given time. Therefore, the proper optimization problem should determine the time and quantity of the injections and not the time profile. This would simplify the problem from finding the optimal value of the input u for the next steps to find the optimal value u at the optimal time instant tk and with a reduction of dimensions of the search space. The same principle can be applied in many different situations retaining the “main” degrees of freedom while simplifying the overall structure and the parameter number. Against this background, several ideas have been developed, in particular those by [1]. This approach which is mainly used in NMPC is the parameterized MPC approach. This approach falls in the category of fully nonlinear solutions like [1, 14, 39]. Moreover, the parameterized approach enables a low dimensional online optimization problem to be derived that can be solved using simple and therefore potentially certified solution. This last feature is of great importance when talking about solutions that have to be adopted in industrial large production units context. Indeed, in such context, implementing dedicated algorithms may be incompatible with automotive company certification requirements. The control parametrization approach amounts to choose the candidate piecewise open loop control profiles within a class of control profiles that are defined by a low dimensional vector of parameters. Figure 1.11 shows a possible parametrization of the control input F(.) over a time interval [0 t f ]. This parametrization is used in the definition of the predictive

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Fig. 1.11 Parametrization of control input [2]

control formulation. F(.) is defined by one parameter only, namely its asymptotic value F f [2].

1.3.2 Coping with Nonlinearity The use of the methods presented above is not limited to linear systems. For instance, the explicit approach is applicable to NL systems (see [30]). However, as linearity makes the computation of the solution much easier, there is a strong interest in reducing nonlinear problems to linear ones. A first possibility consists in noting that some nonlinearities are not really there. Indeed, many nonlinear plants are actually linear plants in a “wrong” coordinate system and some of them can be transformed into linear systems with different constraints using e.g. a differential geometric approach [12]. If the nonlinear plant is output linearizable [27], it is possible to transform the original problem into a new one with state depending constraints. While this may not necessarily be an improvement, it is also possible in some cases to approximate these constraints by time varying ones. Multi-linear models are much more general and a very popular choice. Multilinearity can be achieved in time domain (apparently leading a linear time varying structure, as used e.g., in [38], or in the state space, using several linear models switched according to the operating point as used for instance in [42]. Using this kind of models in MPC may pose some small additional problems (e.g. in terms of continuity of the state estimation), but the complexity of the whole system does not become significantly larger. For systems which really exhibit switching behavior, more general hybrid models can be used, which consist of continuous and discrete dynamics and switches according to Figure 1.12. Hybrid models can be derived in very different ways, see e.g., [48] for a recent review).

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Fig. 1.12 Hybrid system

Fig. 1.13 left: Engine brake torque (dashed line: desired value, solid line: response of the nonlinear model), right: Regime of combustion

To use hybrid models in an MPC framework dynamic programming and multiparametric programming are combined to obtain the optimal state-feedback control law for hybrid systems. This method has been used for real applications in the automotive areas of DISC engines and traction control as e.g. presented in [6, 8]. Figure 1.13 and 1.14 show the results of a hybrid MPC controlling a nonlinear DISC engine simulator [6]. The hybrid prediction model is obtained by linearizing and discretizing the nonlinear plant model and implemented via the hybrid system description language (HYSDEL2 ). The key feature of the approach is the simultaneous manipulation of discrete and continuous control variables within a receding horizon optimal control framework. It has been shown that the transient response of the engine can be shaped as desired and constraints imposed by the engine feasible operating range can be enforced. Another possibility which has already been tested for automotive applications, consists in keeping system and constraints linear but using a nonlinear cost function. It can be shown that under some conditions [10], solving this optimization problem is equivalent to solving the linear optimal problem for a Wiener model, 2

http://www.dii.unisi.it/ hybrid/DISCengine.html

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Fig. 1.14 left: Air-to-fuel Ratio (dashed line: desired value, solid line: response of the nonlinear model, dash-dotted lines: constraints), right: Spark timing (solid line: response of the nonlinear model, dash-dotted lines: constraints k1

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a common nonlinear model consisting of a linear model followed by a static nonlinearity, the main limit being on the slopes of the static nonlinearity. Figure 1.15 shows a possible output nonlinearity. In this case the optimization of the nonlinear setup can be substituted by a standard but extended linear QP including some slack variables. There are other approaches for nonlinear MPC (e.g. neural network based [60]) but in order not to overload this chapter we shall not discuss them further here.

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1.4 Chances: State and Outlook The former discussion was aimed at showing that MPC can provide a solution, but we still have to discuss whether it provides new chances. An answer to this question can be provided by Figure 1.17 which compares the tracking results of fresh air (MAF) and boost pressure (or manifold absolute pressure (MAP)) in a Diesel engine using an explicit MPC based on identified multi-linear models with the production ECU control, both driven by the same reference profiles. MAF 800 measured MPC setpoint measured ECU

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Fig. 1.17 Comparison of the MPC and conventional ECU fresh air and boost pressure tracking during a NEDC cycle from [42]

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It also shows the use of the corresponding actuators, the exhaust recirculation valve (EGR) and the variable geometry turbocharger vane (VGT). Despite the fact that multi-linear models offer only an approximation of the real behavior of the air path, MPC shows a better performance as it reacts much faster and has a more intensive use of the actuators than the production control. This is due to the fact that MPC takes explicitly in account the cross couplings and the limits, while the standard ECU control tries to compensate the coupling via additional feed forward elements. Especially in the transients the MPC can go faster since also the physical bounds of the actuator valves are included as constraints in the MPC law and can therefore be fully exploited. MPC can do this without the need of an antiwindup mechanism, which is needed for integral acting controllers that are tuned to meet the actuator saturations. Notice also that the setpoint value for MAF is not reachable, and of course neither MPC nor the conventional approach will reach it. A better dynamic behavior, however, is not enough to achieve a better performance in terms of emissions or consumption. However, performing an optimization of the setpoints, much better results can be obtained, as shown in figure 1.18. The corresponding price in terms of actuator activity is summarized in the table 1.1, the advantages in table 1.2. Table 1.1 Increase in actuator activity (from [1])

ECU MPC Optimization

VGT activity

EGR activity

100 % 122.3 % 122.3 %

100 % 192.5 % 248.2 %

Table 1.2 Emissions under MPC compared to production ECU for a warm NEDC NOx Opacity Total HC CO high concentration CO low concentration CO2

-10 % -54 % +4 % +46 % +16 % -2 %

Further improvements can be obtained using nonlinear approaches. For example, figure 1.20 shows the application of a linear MPC based on an asymmetric cost function including slack variables to regulate the NOx emissions in a stationary gas engine. The results claim that the use of an asymmetric cost function can significantly improve the desired control performance. Using online optimization still yields a better performance. Figure 1.21 depicts a comparison of a standard controller on a passenger car Diesel engine and the

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result of an online-optimization during a load transient on an engine testbed. The objective for the optimization was to maintain the torque response while reducing the emission peaks of smoke emissions and NOx by shaping the trajectories of the fuel injection (qtot in the figure), the EGR valve position (Xegr) and the guide vane position of the turbocharger (Xvgt). The procedure was implemented on a dynamical engine-testbed, whereas during the optimization the testbed was used for function evaluation. For more details see [3].

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Fast model predictive control is a fast expanding area, and at least two commercial implementations of engine MPC have been announced (by Honeywell and H¨orbiger Control).

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Fig. 1.21 Comparison of standard and optimized system outputs (left) and system inputs (right)

1.5 Conclusions MPC is by far not the only approach suited for model-based engine control (see e.g., [5, 15, 37, 53, 54]) and MPC has significant drawbacks in terms of computational and/or memory loads, non-intuitive tuning and robustness properties (see e.g., [34]). Still MPC is able to handle constraints, complex interconnected dynamics and in particular offers a systematic design procedure. Since most of the advantages of MPC rely on the model, a well-designed feedforward signal may perform similar, but is more calibration intensive and cannot be designed adaptively, where one could think of an adaptive model used in an online MPC to account for ageing, production spread and similar phenomena. This, however, would rule out the explicit MPC version since the controller is only computed offline, and adaptation must be online. The real bottleneck lies in the modeling: we still need better tools to achieve on-line identification of sufficiently precise models for design and operation. Once these become available, MPC could turn out to be a standard design method for automotive control.

Acknowledgment The authors would like to thank the ACCM for their financial support as well as Richard Fuerhapter, Markus Hirsch and Xiaoming Wang for their support.

References [1] Alamir, M.: Stabilization of Nonlinear System Using Receding-Horizon Control Schemes: A parametrized approach for Fast Systems. LNCIS. Springer, London (2006) [2] Alamir, M., Sheibat-Othman, N., Othman, S.: Constrained nonlinear predictive control for maximizing production in polymerization processes. IEEE Transactions on Control Systems Technology 15, 315–323 (2007)

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[3] Alberer, D., del Re, L.: Optimization of the transient diesel engine operation. In: Proceedings of the 9th International Conference on Engines and Vehicles, Capri, Naples, Italy (2009) [4] Alberer, D., del Re, L., Winkler, S., Langthaler, P.: Virtual sensor design of particulate and nitric oxide emissions in a DI diesel engine. In: Proceedings of the ICE 2005, 7th International Conference on Engines for Automobile, Capri, Italy (September 2005) [5] Ammann, M., Fekete, N.P., Guzzella, L., Glattfeder, A.H.: Model based control of the VGT and EGR in turbocharged common-rail diesel engine: Theory and passenger car implementation. SAE Transactions 112, 527–538 (2003) [6] Bemporad, A., Giorgetti, N., Kolmanovsky, I.V., Hrovat, D.: A hybrid system approach to modeling and optimal control of DISC engines. In: 41th IEEE Conf. on Decision and Control, pp. 1582–1587 (2002) [7] Bemporad, A., Morari, M., Dua, V., Pistikopoulos, E.N.: The explicit linear quadratic regulator for constrained systems. Automatica 38, 3–20 (2002) [8] Borrelli, F., et al.: An MPC/hybrid system approach to traction control. IEEE Trans. Contr. Systems Technology 14(3), 541–552 (2006) [9] Corona, D., Necoara, I., De Schutter, B., van den Boom, T.: Robust hybrid MPC applied to the design of an adaptive cruise controller for a road vehicle, San Diego, California, pp. 1721–1726 (2006) [10] Alberer, D., Kirchsteiger, H., del Re, L., Ferreau, H.J., Diehl, M.: Receding horizon optimal control of wiener systems by application of an asymmetric cost function. In: IFAC Workshop on Control Applications of Optimisation, Jyv¨askyl¨a, Finnland, May 6-8 (2009) [11] Lichtenth¨aler, D., Ayeb, M., Theuerkauf, H.J., Winsel, T.: Improving real-time SI engine models by integration of neural approximators, SAE Technical Paper Series, Paper No. 1999-01-1164 (1999) [12] del Re, L.: Hybrid MPC for minimum phase nonlinear plants. In: Tagungsband Third European Control Conference (1995) [13] Devasia, S.: Should model-based inverse inputs be used as feedforward under plant uncertainty? IEEE Transactions on Automatic Control 47(11), 1865–1871 (2002) [14] Diehl, M., Bock, H.B., Schl¨oder, J.P.: A real-time iteration scheme for nonlinear optimization in optimal feedback control. Siam Journal on Control and Optimization 43, 1714–1736 (2005) [15] St¨olting, E., Seebode, J., Gratzke, R., Behnk, K.: Emissionsgef¨uhrtes motormanagement f¨ur nutzfahrzeuganwendungen. In: MTZ December (2008) [16] Falcone, P., Borelli, F., Asgari, J., Tseng, H.E., Hrovat, D.: Predictive active steering control for autonomous vehicle systems. IEEE Transactions on Control Systems Technology 15 (2007) [17] Falcone, P., Borelli, F., Asgari, J., He, T., Hrovat, D.: A model predictive control approach for combined braking and steering in autonomous vehicles. In: Proceedings of the IEEE Conference on Control and Automation (2007) [18] Falcone, P., de Gennaro, M.C., Fiengo, G., Glielmo, L., Santini, S., Langthaler, P.: Torque generation model for diesel engine. In: 42nd IEEE Conference on Decision and Control, Hawaii, USA, December 2003, pp. 1771–1776 (2003) [19] Ferreau, H.J., Bock, H.G., Diehl, M.: An online active set strategy for fast solution of parametric quadratic programs with applications to predictive engine control. Diploma Thesis, University of Heidelberg (2006) [20] Findeisen, R., Alg¨ower, F.: An introduction to nonlinear model predictive control. In: 21st Benelux Meeting on Systems and Control (2002)

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[21] Wang, G., Li, G., Liu, Y., Chen, L., Zhang, X., Lu, J.: A Developed Model for Emission Prediction. SAE Paper No. 1999-01-0233 (1999) [22] Garcia, C.E., Morshedi, A.M.: Quadratic programming solution of dynamic matrix control QDMC. Chemical Engineering Communications 46, 73–87 (1986) [23] Heywood, J.: Internal Combustion Engine Fundamentals (1988) [24] Hirsch, M., Alberer, D., del Re, L.: Grey-box control oriented emissions models. In: 17th IFAC World Congress, Seoul, Korea (July 2008) [25] Hirsch, M., del Re, L.: Adapted D-optimal experimental design for transient emission models of diesel engines. In: Proceedings of the 9th International Conference on Engines and Vehicles, Capri, Naples, Italy (2009) [26] Hjalmarsson, H., Martensson, J.: Optimal input design for identification of non-linear systems: Learning from the linear case. In: Proceedings of the 2007 American Control Conference, New York City (July 2007) [27] Isidori, A.: Nonlinear Control Systems, 2nd edn. (1997) [28] Galindo, J., Luj´an, J.M., Serrano, J.R., Hern´andez, L.: Combustion simulation of turbocharger hsdi diesel engines during transient operation using neural networks. Applied Thermal Engineering 25, 877–898 (2005) [29] Desantes, J.M., Lopez, J.J., Garcia, J.M., Hernandez, L.: Application of neural networks for prediction and optimization of exhaust emissions in a h.d. diesel engine. SAE-Paper No. 2002-01-1144 (2002) [30] Johansen, T.A.: Approximate explicit receding horizon control of constrained nonlinear systems. Automatica 40, 293–300 (2004) [31] Jung, M.: Mean-value modelling and robust constrol of the airpath of a turbocharged diesel engine. PhD Thesis,University of Cambridge (2003) [32] Klingenberg, H.: Automobile Exhaust Emission Testing. Springer, Heidelberg (1996) [33] Langthaler, P.: Model Predictive Control of a diesel engine airpath. PhD Thesis, Johannes Kepler Universit¨at (2007) [34] Lazar, M., Heemels, W.P.M.H., Munoz de la Pena, D., Alamo, T.: Further results on Robust MPC using Linear Matrix Inequalities. In: Assessment and Future Directions of Nonlinear Model Predictive Control. LNCIS. Springer, Heidelberg (2009) [35] Li, X., Wallace, J.S.: A phenomenological model for soot formation and oxidation in direct-injection diesel engines. SAE Paper No. 952428 (1995) [36] Costa, M., Merola, S., and Vaglieco, B.M.: Mulitdimensional modelling and spectroscopic analysis of the soot formation process in a diesel engine. 2002-01-2161 (2002) [37] Hafner, M., Sch¨uller, M., Nelles, O., Isermann, R.: Fast neural networks for diesel engine control design (2000) [38] Barbarisi, O., Palmieri, G., Scala, S., Glielmo, L.: European Journal of Control 15(3-4) (2009) [39] Ohtsuka, T.: A continuation/GMRES method for fast computation of nonlinear receding-horizon control. Automatica 40, 563–574 (2004) [40] Oppenauer, K., del Re, L.: Hybrid emission models. In: Proceedings of the 9th International Conference on Engines and Vehicles, Capri, Naples, Italy (2009) [41] Ortner, P., Bergmann, R., Ferreau, H.J., del Re, L.: Nonlinear model predictive control of a diesel engine airpath. In: IFAC Workshop on Control Applications of Optimisation, Agora, Finland (2009) [42] Ortner, P., del Re, L.: Predicitve control of a diesel engine air path. IEEE Transactions on Automatic Control 15, 449–456 (2007) [43] Previdi, F., Lovera, M.: Identification of non-linear parametrically varying models using separable least squares. International Journal of Control 77, 1382–1392 (2004)

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[44] Qin, S.J., Badgwell, T.A.: A survey of industrial model predictive control technology. Control Engineering Practice 11, 733–764 (2003) [45] Rawlings, J.B.: Tutorial overview of model predictve control. IEEE Control Systems Magazine (2000) [46] R¨uckert, J., Richert, F., Schloßer, A., Abel, D., Herrmann, O.E., Pfeifer, A., Pischinger, S.: Ein modellgest¨utzter pr¨adiktiver ansatz zur regelung von ladedruck und agr-rate beim nutzfahrzeug-dieselmotor. In: Steuerung und Regelung von Fahrzeugen und Motoren - AUTOREG 2004, vol. 1828, pp. 131–141 (2004) [47] Lee, S., Shin, D., Lee, J., Sung, N.: Soot emission form a direct injection diesel engine. Paper No. 2004-01-0927 (2004) [48] Paoletti, S., Juloski, A.L., Ferrari-Trecate, G., Vidal, R.: Identification of hybrid systems: a tutorial. European Journal of Control 513(2-3), 242–260 (2007) [49] Stewart, G., Borelli, F.: A model predictive control framework for industrial turbodiesel engine control. In: IEEE Conference on Decision and Control (2008) [50] Winsel, T., Ayeb, M., Lichtenth¨aler, D., Theuerkauf, H.J.: A neural estimator for cylinder pressure and engine torque. SAE Technical Paper Series, Paper No. 1999-01-1165 (1999) [51] Tennant, J.A., Rao, H.S., Powell, J.D.: Engine characterization and optimal control. In: IEEE Conference on Decision and Control including the Symposium on Adaptive Processes (1979) [52] Terwen, S., Back, M., Krebs, V.: Predictive powertrain control for heavy duty truck. In: Proceedings of the IFAC Symposium on Advances in Automotive Control, p. 39 (2004) [53] Tree, D., Svensson, K.I.: Soot processes in compression ignition engines. Progress in Energy and Combustion Science 33, 272–309 (2007) [54] van Nieuwstadt, M.J., Moraal, P.E., Kolmanovsky, I.V., Stefanopoulou, A., Wood, P., Criddle, M.: Decentralized and multivariable designs for EGR-VGT control of a diesel engine. In: IFAC Workshop on Advances in Automotive Control, Mohican State Park, OH (1998) [55] Vasak, M., Baotic, M., Morari, M., Petrovic, I., Peric, N.: Constrained optimal control of an electronic throttle. International Journal of Control 79(5), 465–478 (2006) [56] Wei, X.: Advanced LPV techniques for diesel engines. PhD Thesis, Johannes Kepler Universit¨at (2006) [57] Wei, X., del Re, L., Langthaler, P.: LPV dynamical models of diesel engine nox emission. In: First IFAC Symposium on Advances in Automotive Control, University of Salerno, Salerno, Italy, April 2004, pp. 262–267 (2004) [58] Wei, X., del Re, L., Lihua, L.: Air path identification of diesel engines by lpv techniques for gain scheduled control. Mathematical and Computer Modelling of Dynamical Systems 14(6), 495–513 (2008) [59] Winkler, S., Hirsch, M., Affenzeller, M., del Re, L., Wagner, S.: Virtual sensors for emissions of a diesel engine produced by evolutionary system identification. In: Proceedings of the 12th International Conference on Computer Aided System Theory (2009) [60] Yu-Jia, Z., Ding-Li, Y.: A neural network model based MPC of engine AFR with singledimensional optimization. In: Liu, D., Fei, S., Hou, Z.-G., Zhang, H., Sun, C. (eds.) ISNN 2007. LNCS, vol. 4491, pp. 339–348. Springer, Heidelberg (2007)

Chapter 2

On Board NOx Prediction in Diesel Engines: A Physical Approach Jean Arr`egle, J. Javier L´opez, Carlos Guardiola, and Christelle Monin

Abstract. For pollutant emissions predictive physical modeling in diesel engines, three key points have to be taken into account: • the suitability of the physico-chemical mechanisms to describe the important processes in the desired range of operating conditions; • the sensitivity of these mechanisms and the corresponding models to errors in input parameters; and • the accuracy when determining the input parameters. This chapter takes the NOx formation/destruction process modeling as an example to illustrate the most important aspects of this kind of modeling.

2.1 Introduction The on-board quantification of engine-out pollutant emissions seems to be a key point for the management of the engine itself, and more specifically of the aftertreatment system. This quantification can or could be performed either by direct measurement of the considered species, or by prediction from physical or empirical models based on a more or less important number of measurements of engine operating parameters. It is important to take into account three main aspects when evaluating the potential of a given technique or methodology for pollutant prediction: 1. The model of formation/destruction, with three key elements: • the main physico-chemical mechanisms involved; • the way these mechanisms are modeled, ranging from parameterized maps to detailed physico-chemical models of the diesel flame; and Jean Arr`egle, J. Javier L´opez, Carlos Guardiola, and Christelle Monin CMT-Motores T´ermicos, Universidad Polit´ecnica de Valencia, Camino de Vera s/n 46022 Valencia, Spain e-mail: {arregle,jolosan3,carguaga,chrismon}@mot.upv.es L. del Re et al. (Eds.): Automotive Model Predictive Control, LNCIS 402, pp. 25–36. c Springer-Verlag Berlin Heidelberg 2010 springerlink.com

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• the input parameters to these models, which should be measured or estimated on-board. 2. The sensitivity of the final result to the errors in the measurement or in the determination of these input parameters. Frequently, this sensitivity depends much less on the complexity of the model used than on the selection of the input parameters to be measured so as to characterize the engine operating conditions. 3. The accuracy when determining the input parameters. Obviously, this accuracy depends on the sensor characteristics and its drift because of fouling and/or ageing during the engine lifetime. But another important aspect is the representativeness of the measurement. A clear example is the determination of the temperature or the oxygen mass fraction of the air+EGR mixture in the intake manifold: because of the frequent heterogeneity of this mixture and the pulsed character of the flow, the measurement can be far away from the mean values of the flow. The prediction of NOx production in a diesel engine can be a good example to illustrate all those ideas. On the one hand, a clear benefit on the management of after-treatment systems (e.g. NOx traps) can be achieved and, on the other hand, the NOx formation/destruction process in a diesel flame can be modeled with a high level of accuracy.

2.2 Main Physical/Chemical Mechanisms of NOx Formation/Destruction The experimental and theoretical studies about the fundamentals of the combustion process showed the existence of four mechanisms of NOx formation [10, 11, 14, 15], which are summarized in Table 2.1. In standard diesel combustion conditions it is often assumed [1, 2, 7, 16] that nearly all NOx are formed via the Zeldovich thermal mechanism [8, 17]. In fact, in the majority of diesel combustion predictive models Table 2.1 NO formation mechanisms. Formation mechanism N2 + O ↔ NO + N N + O2 ↔ NO + O N + OH ↔ NO + H (extended mechanism) Prompt NO CH + N2 ↔ HCN + N (Fenimore) HCN → CN → NCO → NH → N → NO “via N2 O” NO N2 + O + M → N2 O + M (main reac- N2 O + O → 2 NO tions) N2 O + O → N 2 + O 2 Fuel NO no N in diesel fuel Thermal NO (Zeldovich)

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(either 0D, 1D or 3D models) only this mechanism is taken into account. However, in some particular operating conditions other NOx formation mechanisms can not be neglected anymore.

2.2.1 NOx Re-burning In a diesel diffusion flame, the reaction zone separates the region inside the flame, with fuel and burned gases, from the region outside the flame, with oxygen and burned gases. Because of the lack of oxygen and the presence of fuel and high temperatures, the region inside the flame is a reductive atmosphere, which is incompatible with NOx formation. Thus NOx are formed in the reaction zone and the region outside the flame, with oxygen and temperature levels still high enough. An important question is the following: what happens with the formed NOx if they are re-entrained inside the reductive atmosphere inside the flame (as sketched in Figure 2.1)? In some modern power plants the NOx re-burning process is used to remove NOx by reduction in presence of HCCO, an intermediate formed in fuel-rich regions. In the context of diesel combustion, where the characteristic times are much shorter, some experimental and theoretical studies showed that the NOx re-burning mechanism has to be taken into account [4, 12, 13]. In a first study performed in a single-cylinder engine, a certain amount of artificial NOx was injected in the intake manifold and the NOx content in the exhaust manifold was measured after the combustion process; results are shown in Figure 2.2. If no NOx is injected in the intake manifold, the NOx measured in the exhaust correspond to those formed in the cylinder. As far as the amount of NOx injected in the

Fig. 2.1 Schema of the process of NOx re-entrainment into the flame

Fig. 2.2 Experimental illustration of the NOx destruction process in a diesel engine

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Fig. 2.3 Comparison of the NOx destruction process with and without the re-burning mechanism

intake manifold increases, it is observed that the amount of NOx in the exhaust increases less than expected, being even smaller than the amount injected in the intake in some cases. Consequently, it is clear that in some combustion conditions a NOx destruction process exist in the cylinder of a diesel engine. This NOx destruction process can be, somehow, coherent with the Zeldovich thermal mechanism. However, a theoretical study of chemical kinetics analysis shows that the NOx reduction is quite low if only the Zeldovich mechanism is taken into account, as shown in Figure 2.3. On the contrary, if the NOx re-burning mechanism used in power plants is added, the NOx reduction observed is much more significant even if the characteristic combustion time is very small (some tens of μ s). This re-burning mechanism is particularly important when the combustion temperature is relatively high and when a big amount of NOx is re-entrained inside the diffusion flame. Both conditions are possible at the same time at engine operating points inside the medium/high load range.

Fig. 2.4 NOx prediction with and without NOx reduction by re-burning mechanism

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The example shown in Figure 2.4, where both experimental and modeling (1D prediction model [3]) results are plotted together, show that it is difficult to properly predict NOx at high load operation points without a re-burning mechanism. In the four analyzed operating points, the only changed parameter is the injection duration (consequently, the injected fuel mass). Experimentally it is observed that with the highest load the NOx content in the exhaust manifold is even lower compared to a case with lower load. This behavior can not be reproduced by modeling if the NOx re-burning reduction mechanism is not integrated in the model.

2.2.2 NOx Formation in LTC Conditions The search of lower and lower NOx emissions pushes the evolution of diesel engines towards low temperature combustion processes, with extremely high EGR rates. In such conditions a chemical kinetics theoretical analysis shows that the “prompt” and “via N2 O” formation mechanisms are not negligible anymore [1, 9, 16].

Fig. 2.5 Comparison between estimated NOx calculated with thermal Zeldovich mechanism and total NOx (thermal + prompt + via N2 O). (Flame temp. from 1900 to 2900 K, YO2 from 0.07 to 0.23, relative A/F from 0.5 to 1)

Obviously, the NOx formation rate for each mechanism is very changeable and depends mainly on the local conditions of temperature, oxygen fraction and fuel fraction. However, it is possible to find a common behavior along the whole operating range (either classical or LTC) existing in a diesel engine. Figure 2.5 shows, for a wide range of combustion conditions, the error that is introduced when NOx emissions are calculated using the Zeldovich thermal mechanism alone. It is observed that for high local NOx values in the flame (diesel combustion without EGR) the error is very small. For intermediate values, up to 100 ppm (current diesel combustion with EGR), the error ranges between 10 to 20%. However, for LTC the error can be much higher. For this reason it seems important that diesel combustion models to be applied to engines operating with high EGR rates include the “prompt” and “via N2 O” mechanisms of NOx formation. There are two ways to achieve that:

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• performing the complete chemical kinetic calculation of these mechanisms, what would increase considerably the computing time compared to the Zeldovich model alone; and • estimating the error performed when only the Zeldovich thermal mechanism is used and introducing a correction in the results. In this case the computing time is kept practically the same as when using the Zeldovich mechanism alone, which is an attractive solution for on-board prediction models. Summarizing, these results show that even if the Zeldovich thermal mechanism is the most important in the whole range of operating conditions in a diesel engine, a good prediction of NOx emissions can be obtained only if it is taken into account (Figure 2.6): • the destruction process (re-burning) of the NOx re-entrained into the high temperature diffusion flame (point at medium-high load with low EGR rates); and • the “prompt” and “via N2O” formation mechanisms, usually negligible but predominant at conditions with very high EGR rates (with low temperature flames).

Fig. 2.6 Main mechanisms involved depending on the operating conditions

2.3 Mechanisms and Model Sensitivity The NOx prediction accuracy does not depend only on the suitability of the physicochemical mechanism considered, but also on the model sensitivity to input parameters errors.

2.3.1 Structure of Physically-based NOx Models Many different possible ways to describe the combustion process in a diesel engine can be used so as to predict the amount of formed NOx. From the simplest to the most complex (top to bottom in Figure 5.8), it can be mentioned: • Gray box models based on a series of semi-empirical relationships. They can perform an appropriate prediction once they are tuned, and with very low requirements in calculation power and time. However, it is nearly impossible to

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Fig. 2.7 Structure of physically-based NOx models

extrapolate their predictions out of the operating range where they have been tuned or in case that the engine undergoes any modification; • Combination of simple thermodynamical models for the estimation of the adiabatic flame temperature and semi-empirical NOx formation models. They are an evolution of the previous case that slightly improves their prediction capability in case of engine modifications; • Combination of a thermodynamical model, a 1D description of the mixing and combustion process and a description of the NOx formation/destruction chemical kinetics. They imply a significant advance compared to the previous models. The key point is to determine the instantaneous local conditions at and near the flame so as to reproduce the NOx kinetic-chemical evolution. Their predictive character allows the extrapolation to conditions not necessarily used for the calibration of the model, with different types of combustion processes; and • Complete discretisation of the atomisation/mixing/combustion/NOx formation processes with a 3D CFD code. It is the evolution of the previous case, with an improved capacity of detailed description of the reproduced processes. The cost of this enlarged potential is a huge requirement in calculation power and time, which are a priori incompatible with on-board operation. It is very important to underline that independently of the way of describing and discretising the diesel flame so as to model these mechanisms, and also independently of the complexity of the model, the key point when predicting NOx is the determination of the flame temperature.

2.3.2 Flame Temperature Determination A first way to determine the flame temperature on-board is described in Figure 2.8. Combining the value of some engine operating parameters registered classically in the ECU with some additional information of in-cylinder pressure, the following magnitudes can be determined:

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A/F ratio; in-cylinder mass; EGR rate and in-cylinder oxygen fraction; and evolution of in-cylinder mean temperature.

From this information, it is possible to determine the flame temperature and the amount of formed NOx in a more or less detailed and precise way.

Fig. 2.8 Path for flame temperature and NOx prediction from ECU parameters and measured in-cylinder pressure

This way to determine the flame temperature will be the same either if a gray box or a 3D CFD model is used. A sensitivity study [5] allowed showing the important amplification of the error that occurs during the determination of the flame temperature and its consequences over the NOx prediction. The model sensitivity varies, obviously, with the studied operating conditions. However, in general it can be observed in Figure 2.9 that an error of a relatively small variation in an input parameter can translate into a considerably big error in the final NOx prediction. For example, an error/variation of +5% in intake temperature (Tint) has as a consequence an error between 60 to 100% in NOx prediction. Taking into consideration more reasonable initial errors, it is obtained that a variation of ±1% in each of the input parameters translates into an important error up to ±33% in final NOx prediction. It is important to underline that this extremely high sensitivity is a direct consequence of the involved physico-chemical mechanisms, and it is independent of the type of model used to describe these mechanisms. This shows the intrinsic difficulty of defining the mass and the oxygen fraction of the air + EGR + residual mixture in the cylinder, as well as the exponential dependency of NOx emissions as a function of flame temperature. A possible way to improve the situation is to change the way the EGR rate and the in-cylinder oxygen fraction is determined. Moving from the standard method, based on the measurement of the air mass flow rate and the intake temperature and pressure, to a method where the oxygen fraction is directly measured by an oxygen sensor, the sensitivity to the input errors is slightly reduced. However, a variation in ±1% in the value of each input parameter translates into a still important error up to 21% in final NOx prediction.

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Fig. 2.9 NOx prediction sensitivity to the model input parameters

2.4 Input Parameters Accuracy In the previous section it was underlined that the calculation of the flame temperature and the NOx emissions is extremely sensitive to errors and imprecisions in model input parameters independently of the model used. The most critical parameters are: • Air mass flow rate; • Air + EGR mixture oxygen mass fraction; and • Air + EGR mixture temperature. The two following examples help to illustrate the accuracy of the measurement of these parameters in current engines.

2.4.1 Intake Air Mass Flow Rate Accuracy In current diesel engines the air mass flow rate is usually measured by a hot wire/film anemometer placed before the turbocompressor. This element is relatively sensitive to possible drifts along the lifetime of the engine. Besides, despite its location it can be affected by the pressure waves coming from the engine. Figure 2.10 shows the error introduced by the flowmeter placed in an engine compared to a reference measurement performed upstream, in a position completely isolated of the pulsations. It is observed that the error can be very important at some specific engine speeds. But even in the rest of the engine speed range the scattering is not negligible. A global mean error of 5% can be estimated, which is considerably higher than the 1% assumed for the sensitivity analysis.

2.4.2 Air + EGR Mixture Temperature and Oxygen Fraction Temperature and oxygen sensors are very precise nowadays. However, in many current engines the air + EGR mixture is not homogeneous. Besides provoking a

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Fig. 2.10 Intake mass air flow rate measurement error in a current engine

Fig. 2.11 Dispersion in the oxygen mass fraction measurement depending on the sensor location

cylinder-to-cylinder composition variation, this fact introduces a difficulty in how to define a representative value of temperature and oxygen fraction of the mixture. This heterogeneity is caused by the pulsating character of the air and EGR flows and to the geometry of the intake ducts. Figure 2.11 shows that depending on the location of the oxygen sensor the measured value varies in ±4%. Also in this case, the magnitude of the error is much higher than the 1% assumed for the sensitivity analysis introduced in the previous section.

2.5 Conclusions The examples presented for the particular case of NOx emission prediction models in diesel engines show the convenience of finding an appropriate equilibrium between three fundamental aspects: 1. The suitability of the physico-chemical mechanisms used. This aspect is very important if a pollutant emission prediction model wants to be used in a wide range of operating conditions. This is a key aspect in modern diesel engines, because

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thanks to their extremely high flexibility they can change their combustion mode when changing their speed or load. Evidently, the involved main mechanism can not be the same in a standard diffusion combustion, or in a low temperature diffusion combustion, or in a partially or completely premixed combustion... The trend to use fuels with modified characteristics, particularly biofuels, should also be taken into account when defining the physico-chemical mechanisms used in the models. For these new combustion modes and fuels, important efforts in basic comprehension studies are still required so as to identify, characterize and quantify the important mechanisms. 2. The complexity of the model used to discretise the combustion process and to apply the selected physico-chemical mechanisms. 3. The accuracy of the model input parameters. Because of the extremely high sensitivity of some mechanisms to errors in the parameters defining the in-cylinder conditions at intake valve closing, this point uses to be a critical point. For this reason, an important stage when elaborating an on-board pollutant emissions prediction model is the search of: • the improvement of the accuracy of some sensors; • the optimization of the design of some engine parts, as for example the intake manifold, so as to make easier and to improve the operation of the sensors by avoiding pulsations, inhomogeneities, noise; • using pollutant emission measuring systems, even if they are not accurate, so as to have a feedback and to validate and/or calibrate the rest of sensors; and • redundant systems so as to evaluate on-line the accuracy of the sensors. Among these three fundamental aspects, the most critical ones are, with no doubts, the first and the last. This allows selecting among the different model type choices, the configuration optimizing the compromise between accuracy and computational requirements for the concerned application.

References [1] Amneus, P., Gauss, F., Kraft, M., Vressner, A., Johansson, B.: NOx and N2 O formation in HCCI engines. SAE Paper 2005-01-0126 (2005) [2] Andersson, M., Johansson, M., Hultqvist, A.: A real time NOx model for conventional and partially premixed diesel combustion. SAE Paper 2006-01-0195 (2006) [3] Arr`egle, J., L´opez, J.J., Mart´ın, J., Mochol´ı, E.: Development of a mixing and combustion zero-dimensional model for diesel engines. SAE Paper 2006-01-1382 (2006) [4] Payri, F., Arr`egle, J., L´opez, J.J., Mochol´ı, E.: Diesel NOx modeling with a reduction mechanism for the initial NOx coming from EGR or re-entrained burned gases. SAE Paper 2008-01-1188 (2008) [5] Arr`egle, J., L´opez, J.J., Guardiola, C., Monin, C.: Sensitivity study of a NOx estimation model for on board applications. SAE Paper 2008-01-0640 (2008) [6] Chikahisa, T., Konno, M., Murayama, T.: Analysis of NO formation characteristics and control concepts in diesel engines from NO reaction kinetic consideration. SAE Paper 950215 (1995)

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[7] Easley, W.L., Mellor, A.M., Plee, S.L.: NO formation and decomposition models for DI diesel engines. SAE Paper 2000-01-0582 (2000) [8] Fenimore, C.P.: Formation of nitric oxide in premixed hydrocarbon flames. In: 13th Symposium International of Combustion, pp. 373–379 (1971) [9] Hern´andez, J.J., P´erez, C.J., Sanz-Argent, J.: Role of the chemical kinetics on modeling NOx emissions in diesel engines. Energy Fuels 22(1), 262–272 (2008) [10] Heywood, J.B.: Internal combustion engines fundamentals. McGraw-Hill, New York (1988) [11] Hill, S.C., Smoot, D.L.: Modelling of nitrogen oxides formation and destruction in combustion systems. Progress in Energy and Combustion Science 26, 417–458 (2000) [12] Kidoguchi, Y., Miwa, K., Mohammadi, A.: Reduction mechanism of NOx in rich and high turbulence diesel combustion. In: 5th COMODIA Symposium, vol. 5(1), pp. 108– 114 (2001) [13] Kidoguchi, Y., Noge, H., Miwa, K.: DeNOx mechanism caused by thermal cracking hydrocarbons in stratified rich zone during diesel Combustion. In: COMODIA 2004, vol. 6, pp. 73–80 (2004) [14] Lavoie, G.A., Heywood, J.B., Keck, J.C.: Experimental and theoretical study of nitric oxide formation in internal combustion engines. Combustion Science and Technology 1, 313–326 (1970) [15] Turns, S.R.: An introduction to combustion. In: Concepts and applications, McGrawHill, New York (1996) [16] Yoshikawa, T., Reitz, R.D.: Development of an improved NOx reaction mechanism for low temperature diesel combustion modeling. SAE Paper 2008-01-2413 (2008) [17] Zeldovich, Y.B.: The oxidation of nitrogen in combustion and explosions. Acta Physicochemica, USSR 21, 577–628 (1946)

Chapter 3

Mean Value Engine Models Applied to Control System Design and Validation Pierre Olivier Calendini and Stefan Breuer

Abstract. The importance of simulation in power train and combustion engine development is undisputed today, but the search for the most efficient use of simulation in the development cycle is still ongoing. In parallel available computing power and the number of tools are increasing. The choice of the right tool has significant impact on the development process. In this chapter some insight will be given into a process relying on the mean value model approach for control strategy development and validation regarding the development of a 1.6-litres 4-cylindre direct injection diesel engine respecting Euro 5 legislation. To reach the performance and emission targets in a cost effective way, simulation was used early in the project to study different concepts of air loop control architecture and then was employed for the control function development of the air loop control. The present study addresses the main components of the mean value model, the air loop, the degree of refinement in regards to combustion and the actuator dynamics. Two variants of a new control concept have been studied with considerable refinement. To position the predicted performance they were compared to the conventional Euro 4 approach. To obtain significant simulation results the work was concentrated on specific engine life situations. The comparatively fast execution of mean value models has been exploited to realize statistical computations on the robustness of the controlled system. The main targets of control development are a high degree of precision but also a robust behavior in real life. To asses the performance of the control approaches under study, the main sources of disturbances du to series production were identified and than this knowledge was used to build a mean engine model that allowed to represent their impact on the engine. This allowed distinguishing the control approaches not only on their capacity to perform well on a nominal engine but their ability to cope with the large spectrum of engine behavior to be expected in series production. The obtained results allowed for an intelligent choice of the control strategy, choice which has been confirmed by experimental work. Pierre Olivier Calendini and Stefan Breuer PSA Peugeot Citro¨en 18, rue des Fauvelles 92252 La Garenne-Colombes e-mail: [email protected],[email protected] L. del Re et al. (Eds.): Automotive Model Predictive Control, LNCIS 402, pp. 37–52. c Springer-Verlag Berlin Heidelberg 2010 springerlink.com

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3.1 State of the Art Mean Value Engine Model Use of simulation tools is more and more widespread today. The field of system modeling is rather busy, with impulses coming from a lot of different areas as control function development but also from system design. The challenge of today is no longer in creating a precise representation of the reality, but to deploy a known and efficient methodology to get the right answers in time with the project schedule. Precision is still necessary but no longer a major hurdle. Contrary to dedicated tools, mean value models are most valuable when sticking to the major influences in the system but including its sub-systems. For studies of particular details other tools as 1-D to 3-D calculations are more suitable especially if very accurate precision is required. Engine simulation with mean value models dates back to Euro 3 engine development. The first deployment was on air loop control for turbo-charged direct injection diesel engines. The successful use in Euro 4 diesel engine development made clear that there was still a lot of potential to be unleashed. The number of topics accessible for mean value models has reached beyond engine control, and diesel engines. They include today the larger domain of the power train as a whole. The modeling bases have been clarified. The model is defined in response to the ever more stringent development requirements: • represent the essential system behavior, especially predict critical phenomena as response time and sensitivity but also include all major external influences; • development work today is iterative with a tight schedule not only in software or component design; • system behavior has an impact for growing number of engineers in engine development, not all accustomed to simulation work; • as schedules and budgets shrink, more and more questions are to be treated in simulation; and • concepts and control strategies get more and more complex, so validation plays an essential role in today’s engineers tasks. In response to these requirements, simulation has to evolve. The model complexity has to be adapted as good as possible to the required task. The set-up time for simulation needs to be low, as well as the computation time, while still delivering correct and precise answers. No longer an expert tool known only to the numerics specialist and the control engineer, has the model had to be usable after a short training period. This means the model development has to focus more on usability and common language with the client departments. In the same time, system simulation now includes new sub systems, touching engineers who on one hand bring in their component or sub system models, and on the other hand are less familiar with control loops and engine operating points. With those subsystems, new physical phenomena have to be described. A rising topic is validation, with its need for so-called environment models to stimulate control units for testing purposes. This specific model application requires real time capability.

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3.2 System Model Structure as a Response to the Requirements To improve model setup time and reliability, system models need to be decomposed into smaller, easier to handle elements. The main challenge to enable model element exchange is interface design. Be it to have polymorphism on combustion models or be it to integrate externally supplied sub-system models, as for example actuator models or heat exchanger models. This allows creating one model structure and realizing all tasks with adapted models from this “model family”. The interface definitions have a significant impact on all simulation activity. Being to light brings the risk of later modifications, breaking the model “family” and inducing unexpected work. Being too restrictive creates bulk and eventually may harm computation performance. The interfaces are also directly linked to the model structure. They imply which physical phenomena is described and where it is located. As a simple example, the location of the crank-shaft torque balance, in the engine or in the clutch? The solution used is a practical application of the Bond Graph theory [4, 5, and 6]. Experience has shown that this solution, when properly realized, has a number of advantages: • • • •

easy exchange between different physical domains, due to conservation of power; well suited for computation when used in an integral way; implies a structure of alternating flow and effort elements; and allows for simple structures as well as complex systems as the standardized interfaces allow grouping of sub-systems as long as the boundaries are identical this is particularly interesting for polymorphism.

Some difficulties had to be overcome. The Bond Graph theory is not part of typical engineer formation, and the point that the approach is not intuitive at first sight. Still a major challenge is to convince all contributors to adopt this formalism as it obliges them to combine their models to get a result. As an example, the before mentioned torque balance can be used. Typically several approaches are possible. The chosen one is to unite component torque with its inertia, and to tie speed and angle together, based on the bond graph approach described below. This has the advantage that torques and inertias can be summed and divided together, reducing the risk to consider a component only partially. To ensure coherence between models of different origins, SI units are compulsory. This gives the following interface definition: Table 3.1 Torque balance interface definition Value Torque and inertia Angle and speed

unit [Nm, kg m2 ] [rad, rad/s]

orientation in out

3.2.1 Bond Graph Applied to Mean Value Engine Models While several definitions are possible for standardizing the interfaces, several evolutions led to the table below. This choice is indirectly including another simulation

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Table 3.2 Torque balance interface definition Physical domain Mechanics (translational) Mechanics (rotational) thermodynamics heat hydraulics electrics chemistry

Effort/State units Speed [m/s]

Flow units Force [N]

Rotational speed [rad/s] Torque [Nm] Mass and enthalpy flow [Pa, K, kg] [kg/s; J/s] Temperature [K] enthalpy flow [J/s] Pressure [Pa] Volume Flow [m3/s] Tension [V] Courant [A] Free energy [J.mol] Molar flow [mol/s]

Applications Actuators, vehicle movement, ... Crankshaft, gears, ... Gas flow, pneumatics Intercooler, engine cooling, ... Pneumatic actuators, ... Electric actuators Oxygen sensors, (combustion),

hypothesis. Acoustics concerning pressure wave propagation are not represented, as this would imply to include a lot more detail on duct design. So cooling circuit models are simple thermal models, and air path acoustics are not directly modeled. This simplification has proven possible as the trend towards engine downsizing has made turbocharging more commonplace. Modeling of turbocharged engine has less need to take into account acoustics. For naturally aspirated engine this is not true and a specific solution had to be found. The above mentioned table is surely not the only solution possible for a bond graph respecting interface definition, as it is also not complete. The listed physical domains have appeared at least once in a project. The link between this table and its underlying theoretical aspect is to be a guideline and standard for building the more practical interface definitions of the models elementary blocs.

3.2.2 Naturally Aspirated and Turbocharged Engine in Bond Graph Structure Building up mean engine models once the interfaces have been defined is a rather easy task. If the constraint of alternating flow and effort bloc is respected, the basic structure can be assembled quickly. Both naturally aspirated and turbocharged engines rely on similar modeling for combustion, aspiration, torque generation, and fuel system modeling. This is simplified due to the use of elementary blocs available in a library created and maintained by the simulation group. Another shared element are the environment and the test scenarios. This part of the model structure enables the resulting engine model to be run in different life situations, on the engine test bench, on the power-train bench or in the car. The common parts of the elements describe the air composition, ambient pressure and temperature, as well as known engine life situation as NEDC cycle or performance test. These elements are the

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Fig. 3.1 Structure of a naturally aspirated engine

same independent of the engine, while respecting the same interface definition as the other elements. The naturally aspirated engine presents few volume blocs, keeping the numbers of integrators low. This simplifies the model but representing the aspiration of the engine is a major difficulty, which can only be solved today by additional information, either from other engines, test results or more detailed and specific acoustics computations. This allows including the effects of pressure waves in the intake and exhausting ducts. Nevertheless such models are in successful use today. This type of engine has gotten lots of attention from a system model point of view. This can be explained by several reasons, all of which imply more or less directly the complexity of this engine concept. A turbocharged engine requires quite some more sensors and actuators as well as governing control functions.

Fig. 3.2 Structure of a turbocharged engine

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The naturally aspirated engine presents few volume blocs (or nodes, in green), keeping the numbers of integrators low. This simplifies the model but representing the aspiration of the engine is a major difficulty, which can only be solved today by additional information, either from other engines, test results or more detailed and specific acoustics computations. This allows including the effects of pressure waves in the intake and exhausting ducts. Nevertheless such models are in successful use today. This type of engine has gotten lots of attention from a system model point of view. This can be explained by several reasons, all of which imply more or less directly the complexity of this engine concept. A turbocharged engine requires quite some more sensors and actuators as well as governing control functions.

3.3 Basic Blocs for Building Mean Value Models Mean value models are composed of different types of basic blocs. The most common ones in use for representing the air loop are those describing volumes, gas exchange and heat exchange.

3.3.1 The Volume Bloc The volume bloc computes an effort in the bond graph reference frame. As many as necessary flow elements can be connected to one volume element. It computes pressure, temperature and mass inside a homogeneous volume. To obtain these values, the bloc computes the mass balance resulting in the currently contained mass.

mvol = minit

∑ i

dmupstream vol dmdownstream vol + dt dt dt i i

(3.1)

The energy balance feeds the energy variation into the internal energy equation that computes the temperature. dhvol =∑ dt i

dhupstream vol dQexchange dQreaction dhdownstream vol + + + (3.2) dt dt dt dt i i enthal py f lows

heat f lows

With temperature and mass fixed, we can compute the pressure with the ideal gas equation. This variant of the volume bloc is dedicated to fixed volumes as for example tubes, exhaust manifolds and others. To represent a combustion chamber or a pneumatic actuator, a variable volume is needed. It requires a little more complex set of equations, but the basic equations are strictly the same.

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3.3.2 The Gas Exchange Bloc This bloc determines the exchange between exactly two volume blocs. It represents an ideally short connection, where a pressure drop occurs. The reasoning is the pressure difference between two volumes drives the flow in-between. Depending on the situation the choice is between two possibilities, the Bernoulli equation: dm Δp = S · ρ ·Cd 2 · (3.3) dt ρ Or the Barr´e de St. Venant equation:

κ1 p dm pin = Cd · S · ρin · √ · dt pin rin · Tin

κ −1 2·κ κ p · 1− · κ −1 pin

κ +1

2·(κ −1) √ dm pin 2 = Cd · S · ρin · √ · κ· dt κ +1 rin · Tin

(3.4)

(3.5)

The later allows for supersonic situations (eq. 3.5) (inlet and exhaust valves for example) or closing actuators. On all flow elements a decision has to be made with respect to the associated heat exchange. The theory for both equations requires the hypothesis of infinitesimally small distances.

3.3.3 Heat Exchange Models The process of heat transfer can be modeled in various ways, for reasonable first estimations one can rely on the semi-physical Nusselt laws that are available for standardized configurations of heat exchange. Nu =

α ·L λ

(3.6)

The difficulty is to know the conditions beforehand and to choose the right set of values. In any case the heat exchange depends on the temperatures of the two elements exchanging heat. dQ = α · S · (T2 − T1 ) (3.7) dt In the model structure the definition of heat exchanges may be rather challenging as in reality all physical elements present take part. The identification of all major contributors is crucial. This relies on the given task and the required precision.

3.3.4 Combustion Model Possibilities One distinguishes two distinct families in combustion models, time based and crankangle based models. The major difference is the continuous computation in the first

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case and the recurring computation in the later. Continuous computation is far less computationally intensive as there is no need to determine precisely timed events as inlet and outlet valve closing and opening or ignition timing. Time based models are either more or less simple maps, based on test results or more detailed calculations. This gives no real prediction possibility. To increase this prediction potential, introducing some basic physics and indexing tables to non dimensional parameters can improve results quite a lot. Below an example result obtained on a 1.6 liter turbocharged direct injection diesel engine:

Fig. 3.3 Gas temperature [ ˚ C] over engine speed [rpm] inside the Exhaust manifold at full load (simulation: yellow triangle)

Crank angle based models allow to access more of the internal variables of the engine, but require more information to set-up correctly. This approach has been rather widely published, some often used models are Wiebe [1], Chmela [2] or Hiroyasu [3] but many others exist or are derived variants of the former. Not to be forgotten is that a crank angle based model also requires more detail on the heat exchange and the fuel system. This increases notably the complexity of the resulting model with more time required to identify all necessary parameters. Nevertheless progress is ongoing and rising computation power allows today to use the crankangle based combustion models in real time applications for validation of control functions linked to individual cylinder combustion control.

3.3.5 Environment Model Using simulation means always also fixing boundary conditions and defining test scenarios, especially for system models. Many test cases are shared, as are most boundary conditions, running tests on the official homologation cycles (for example NEDC) is one common case as are performance tests (example : 1000m from standing start), but also engine tests as full load bench tests. An efficient system model

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Fig. 3.4 Gas pressure [mBar] over engine speed [rpm] inside the Exhaust manifold at full load (simulation: yellow triangle)

should cover all the standard needs while enable easy implementation of additional test scenarios.

3.4 Application Example: Choice of an Air Loop Control Strategy The potential of mean value engine models for system development can be demonstrated on the following example of control strategy development. During the development of a Euro 5 turbocharged Diesel engine, the question of the right choice of air loop control appeared as soon as the engine concept was defined. The air function needs to fulfill several requirements especially to supply the right amounts of air and EGR to the combustion chamber at any time during engine operation. The main challenge is to obtain good transient behavior, on three items, fast response, precise realization of set points and robustness, operating under different boundary conditions. At this state of development three possibilities were considered: • reuse of the existing Euro 4 control, essentially map based; • model-based approach, sensor based; and • model-based approach with virtual sensors. To assess the performance of each solution, the corresponding function had to be implemented and integrated into a power train system model that allowed realizing different test cases. Those test cases had to be chosen with respect to the possibilities of the strategy. The test cases should allow for each control to show its full potential and create equal and comparable conditions for all three. The test case should also contain a challenging situation for the control, by having a considerable impact on the realization of the combustion requirements by the air loop. Experience quickly led to the choice of the high load part of the NEDC. The

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Fig. 3.5 Elements used in the example

acceleration from 100 to 120 km/h needs the highest torque and requires precise mixture control to ensure respect of emission legislation. With regard to the engine model, an existing model of a 1.6l turbocharged direct injection diesel engine was adapted to the task. It makes use of in-house libraries R Simulink. The different control strategies were implemented, based on MATLAB while the engine model was updated to the engines Euro 5 definition. The performance and precision tests quickly showed that all three control strategies could be tuned to obtain these requirements for a nominal engine. So the question of robustness became the main issue. 140 set speed realised speed

120

100

80

60

40

20

0

-20

0

200

Fig. 3.6 NEDC speed trace

400

600

800

1000

1200

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Fig. 3.7 Torque, speed and boost pressure during the 100 to 120km/h acceleration

3.4.1 Implementation of the Robustness Simulation The main input to this approach came from the constraints of series production. Any industrial process results in more or less varying end-products and engines are no exception. Any component differs more or less with respect to its specified tolerance interval. This is true for directly measurable parameters as well as for functional requirements (example: efficiencies). So one task is to identify all impacting parameters and to check how they are represented in the mean value model. Then the variation has to be translated into the definition used in the model. 0.35 0.3

p rob ability [ ]

0.25 0.2 0.15 0.1 0.05

0 50

55

60

65 70 75 80 intercooler efficiency [%]

85

Fig. 3.8 Varying intercooler efficiency on one operating point

90

95

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The following 23 parameters were identified as being functionally relevant for the performance of the air loop, and thus included in the study: • Turbocharger – Pressure drop – Efficiency • Air duct pressure drops – – – – –

Air filter Intercooler Throttle EGR circuit Exhaust pipe

• Heat exchange – Intercooler efficiency – EGR cooler efficiency • Engine – Exhaust manifold temperature – Injected fuel mass – Volumetric efficiency • Environment – Ambient temperature – Ambient pressure – Coolant temperature • Sensor precision – – – –

Pressures Position feedback Temperatures Air flow

Depending on the information available for each parameter, either absolute or relative variations needed to be implemented in the different parts of the engine model. As interdependence of the parameters could not be excluded beforehand, a variant of the Monte Carlo method was chosen. Random sample engines were created based on the collected parameter data. As the simulations are computationally intensive, the choice was made to start with about 1000 samples, and with an extension pushed to 1300 totally. These numbers are below the theoretical level to obtain good statistical correlation but going from 1000 to 1300 samples showed no significant evolution in the results.

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Fig. 3.9 Implementation of absolute and relative variations in Simulink

3.4.2 Results of the Robustness Simulations One advantage of using a complete mean value engine model to asses the robustness of a control strategy is the easy access to all functional parameters of the engine. Not only the usual parameters can be considered but all simulated components (for example actuator positions). To determine the performance of the control strategies the values for realized boost pressure and mass air flow were observed: A reduction of the area between the outermost lines indicates a better robustness against the varying sample engines. The main conclusions are rather obvious. There is a significant gain in robustness possible. The use of a model-based approach can be justified, and the study translates the expected improvement between a virtual sensor to a real one. These results oriented the further work towards the model-based solution. Further work on power train test benches confirmed the observed behavior. A secondary result was a classification by level of importance of every parameter with respect to the functional variables observed. This was obtained in linking input variation to result variation. This classification leads to improved specification of the associated systems or components.

Fig. 3.10 Boost pressure comparison between Euro 4 (left) against model-based approach with virtual sensor (right)

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Fig. 3.11 Mass air flow comparison between Euro 4 (left) against model-based approach with virtual sensor (right)

Fig. 3.12 Boost pressure comparison between model-based approach with virtual sensor (left) against real sensor (right)

Fig. 3.13 Mass air flow comparison between model-based approach with virtual sensor (left) against real sensor (right)

3.5 Conclusions Mean value models are today a common tool for system development. They allow for a good compromise between precision and a reasonable delay. They even make possible approaches as shown on robustness that are technically or economically impossible. They are most useful in situations where complex systems are concerned. The use of mean value model increases the knowledge about the system early in

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the development phase, and allows for faster assessments of concepts and choices. These gains can be converted as needed into an improved development process, acting either or simultaneously on: • • • •

costs (less or better use of bench tests); quality (check system behavior in rarely considered situations); delay (more confidence in concepts allows for earlier decisions); and performance (the choice of the best compromise for different requirements).

The importance of mean value engine models will grow as the power-train gets more complex, especially if considering hybrid power-trains. This implies connecting to physical domains that were up to now not really considered.

Definitions m

mass

h

enthalpy

p

pressure

T

temperature

i

indices

in

upstream condition

ρ

density

α

heat exchange coefficient

S

square section

L

characteristic length

Cd

discharge coefficient

Q

heat flow

Nu Nusselt number

κ

ratio of specific heats

λ

thermal conductivity

References [1] Wiebe, I.I.: Brennverlauf und Kreisprozess von Verbrennungsmotoren. VEB Verlag Technik, Berlin (1970) [2] Chmela, F.G., Orthaber, G.C.: Rate of Heat Release Prediction for Direct Injection Diesel Engines Based on Purely Mixing Controlled Combustion, SAE 1999-01-0186 (1999) [3] Hiroyasu, H., Kadota, T.: Models for Combustion and Formation of Nitric Oxide and Soot in DI Diesel Engines; SAE 760129 (1976)

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[4] Paynter, H.M.: Analysis and design of engineering systems. MIT Press, Cambridge (1961) [5] Karnopp, D.C., Rosenberg, R.C.: System dynamics: A unified approach. John Willey and sons, New York (1975) [6] Thoma, J.U.: Simulation by bondgraphs. Springer, Berlin (1990)

Chapter 4

Physical Modeling of Turbocharged Engines and Parameter Identification Lars Eriksson, Johan Wahlstr¨om, and Markus Klein

Abstract. The common theme in this chapter is physical modeling of engines and the subjects touch three topics in nonlinear engine models and parameter identification. First, a modeling methodology is described. It focuses on the gas and energy flows in engines and covers turbocharged engines. Examples are given where the methodology has been successfully applied, covering naturally aspirated engines and both single and dual stage turbocharged engines. Second, the modeling with the emphasis on models for EGR/VGT equipped diesel engine. The aim is to describe models that capture the essential dynamics and nonlinear behaviors and that are relatively small so that they can be utilized in model predictive control algorithms. Special emphasis is on the selection of the states. The third and last topic is related to parameter identification in gray-box models. A common issue is that parameters with physical interpretation often receive values that lie outside their admissible range during the identification. Regularization is discussed as a solution and methods for choosing the regularization parameter are described and highlighted.

4.1 Introduction Modeling is a standard tool in the design and analysis of control systems and this chapter touches on three topics related to physical and semi-physical modeling. The first topic introduces a general component-based framework for developing Mean Value Engine Models (MVEM) of engines. Applying this approach produces accurate models that have many states and are therefore well suited for controller validation and observer design. Next the application of diesel engines with Exhaust Gas Recirculation (EGR) and Variable Geometry Turbine (VGT) is discussed in more detail. These systems Lars Eriksson, Johan Wahlstr¨om, and Markus Klein Vehicular Systems, Dept. of Electrical Engineering, Link¨oping University, SE-581 83 Link¨oping e-mail: [email protected],[email protected] L. del Re et al. (Eds.): Automotive Model Predictive Control, LNCIS 402, pp. 53–71. c Springer-Verlag Berlin Heidelberg 2010 springerlink.com

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exhibit non-trivial behaviors that are challenging to control and MPC based designs are thus interesting for this application. In particular these engines: • are multiple input and multiple output systems, with strong couplings between the channels; • have strong non-linear behaviors with sign reversals in some of the channels; and • have classical non-minimum phase behavior in some of the channels. The aim is to develop a model that; describes this MIMO system from actuators to sensors, has a minimum set of equations and components, and captures the important system properties listed above. Special emphasis is put on maintaining minimality which is important when the aim is a model that can be used in an MPC framework. The last issue, to be discussed, is parameter identification in non-linear models, which is an important topic in the automotive area. A frequently utilized modeling approach is to use gray-box models where first principles are used to capture the essential properties of the system and then complement this with tuning parameters, e.g. component efficiencies, that are used to fine tune the model to measurement data. This approach often gives either over-parameterized models, or parameters that are barely identifiable, with large parameter uncertainties as a result. Applying standard system identification methods for identifying the tuning parameters, e.g. prediction error methods, frequently results in unphysical values on the tuning parameters. These issues will be discussed together with a remedy based on regularization.

4.2 MVEM Modeling Combustion engines are complex systems and the modeling thus covers several domains, i.e. thermo- and fluid-dynamical, chemical, mechanical, and electrical, when compiling a complete system model. Furthermore the level of detail for the models also vary, all in all giving an abundance of available engine models, and a selection of the appropriate model must be made when considering the application for which the model is built. Here the focus is on models that are related to analysis and control of the gas flows (air, exhaust, and EGR) in engines and this brings the selection to the model family that is called Mean Value Engine Models (MVEM). Models and components in this family have been around for a long time but the term MVEM was coined in [1] and there is now a rich literature in this area, see e.g. [2, 3, 4, 5, 6, 7, 8, 9] for some few examples. Much use has been made of this model family, for example system analysis and controller validation, and it has been an important cornerstone in the model-based control of engines, see e.g. [10, 11, 12, 13, 14, 15, 16, 17, 18] for some examples.

4.2.1 Library Development One important issue to consider when working with models and modeling in a large organization is to enable reuse of models and components. To achieve this the

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models and subsystems must be designed according to design rules. These design rules can be supported by modeling tools such as for example AMESIM [19] or Modelica [20, 21, 22] but can also come from an agreed upon standard, describing rules for the design of components and most importantly their interfaces. This is R done in MATLAB /Simulink, which has become a widespread tool in the industry. In addition to the interfaces the development of a library also involves the definition of the subsystems, and this is the next topic. When analyzing engines and their models, it becomes apparent that there is a set of components that frequently occur and these are therefore natural building blocks that are well suited for generic components in a library. The modeling methodology is to divide the system into components and then defining boundaries and interactions between components with the aid of physics and thermodynamics. The components are arranged according to a scheme where control volumes (CV) are placed in series with flow governing components such as for example compressor, engine, or flow restrictions. These flow governing components are here collectively named flow components (FC). Control volumes have the mass and energy conservation equations and the flow components determine the transport of mass and energy. Examples of control volumes are those where mass is collected: intake manifold, exhaust manifold, all the sections of the pipes between components including the inlets and outlets of the upstream and downstream components respectively. Examples of flow components in a turbo charged engine are: air filter, compressor, intercooler, throttle, engine, turbine, catalyst, exhaust system. To exemplify this methodology the first part of the intake system of a turbo charged engine is Air filter - pipe - compressor - pipe - intercooler FC - CV FC - CV FC With this component view on the modeling it is easy to develop and maintain a library with a set of generic components, and also allow for extensions.

4.2.2 Building Blocks: Component Models With the division into components, given above, the component models have to be developed but there are also design choices with the interfaces. The main equations in the control volumes are mass and energy balances and therefore the natural choice

Fig. 4.1 Basic structure of a gas flow model for a combustion engine, showing the flow component and the control volumes. The arrows show the causality of the system, where the control volumes have pressure and temperature state information and the flow components use this information to determine the mass transport and energy flow (usually implemented using the temperature)

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is to have the mass and energy flows given by the restrictions. Furthermore it is also beneficial to base the model equations on measurable quantities, such as mass flow, pressure, and temperature, since the models can then easily be tuned and validated. Consequently it is natural to select the pressure and the temperature as state variables for the control volumes and the mass flow and temperature of the flowing fluid as the transported properties in the flow components, see Figure 4.1 for an example of how the information flows between the components. In the following sections the components will be discussed separately and the equations are presented using the upstream and downstream properties (the notation follows the flow direction) that will be denoted with subscripts us and ds. 4.2.2.1

Control Volumes

Control volumes are usually modeled using either a simple mass balance and the ideal gas state equation or using both the mass balance and energy balance together with the ideal gas state equation. With the first approach the differential equation for the pressure p in the volume V becomes d p RT = (m˙ us − m˙ ds ) dt V

(4.1)

where R is the ideal gas constant, m˙ denotes mass flow and the subscripts us and ds represent the in- and out-flows respectively. In general this model does not fulfill the energy equation, but it gives good agreement with measured pressures and it is simple to implement and tune to dynamic measurement data from engines. The other choice is to use both mass- and energy-balance, and with pressure p and temperature T as states this results in the following set of equations m= = =

dT dt dp dt

1 m cv

pV RT

m˙ us cv (Tus − T ) + R(Tusm˙ us − T m˙ ds ) + Q˙ RT dT ˙ us − m˙ ds ) + mR V (m V dt

(4.2)

where cv is the specific heat at constant volume and Q˙ is the heat transfer. The temperature of the out-flow from this component is the same as the temperature in the control volume, i.e. Tds = T . See [4] for a longer discussion and comparison. A further extension is to also consider multi-component gases, e.g. separating air and exhaust gases. In such a case the mass- and energy-conservation equations are used but the mass conservation includes each gas specie. Depending on the accuracy of the gas models the resulting equations can become more or less involved. 4.2.2.2

Flow Components

Several components that are placed in the air path of the engine have pressure losses over them e.g. air filter, intercooler, catalyst, exhaust system, and pipe bends. These pressure losses are well described by the equation for incompressible and turbulent flow, see for example [23], where the pressure drop has a quadratic dependence

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on the mass flow, pus − pds = C f r RpTusus · m˙ 2 , where pus is upstream pressure, pds is downstream pressure, Tus is upstream temperature, and C f r is a component parameter usually tuned to data. To fit into the modeling framework the equation is rewritten so that it returns the mass flow m˙ as function of pressure and temperature ⎧ pus (pus −pds ) ⎨ pus − pds ≥ plin C f r Tus m˙ = (4.3) pus p√ us −pds ⎩ otherwise C f r Tus p lin

Another aspect that is important to take into account is that the function should fulfill the Lipschitz condition to guarantee that there exists a solution to the ODE [24]. Therefore the region close to zero has a linear region around pus − pds ≤ plin . This extension is also motivated by the physics since the flow is laminar for low flow velocities and when the flow velocity increases it becomes turbulent and this gives a transition, see [25] for a discussion and another transition model. Applications and validations of these component models are found in for example [6, 26, 9]. 4.2.2.3

Actuator Valves

Actuator valves are frequently used to control flows, e.g. throttle, EGR-valve, wastegate, and compressor bypass. These have higher pressure losses and flow velocities and require that models for compressible flows are used. The standard model is pus m(u, ˙ pus , pds , Tus ) = √ Ae (u)Ψ (Πth ) (4.4) R Tus

γ

γ −1 γ +1 2 2γ p 2 ds Ψ (Πth ) = Πth γ − Πth γ , Πth = max( , ) γ −1 pus γ + 1 c

where γ = cvp and Ae (u) is the effective area as a function of the control input u (see Appendix C in [27] for a derivation of the model). This model should also be extended so that it fulfills the Lipschitz condition at ppds = 1 to avoid numerical us problems and stiffness for large control valve openings and for low flows. A Note on Flow Temperatures: in most of the components that utilize (4.3) or (4.4) there is no production or consumption of neither heat nor work, and changes in potential- and kinetic energy over the component can normally be neglected. As a result the temperature after the component becomes equal to that before, i.e. Tds = Tus . However some components do affect the temperature, e.g. intercooler and exhaust system. For such systems separate temperature models are used for Tds , complementing the flow models. See [5, 6] for some examples. 4.2.2.4

Compressor and Turbine Components

Compressors and turbines also influence the flows and pressures and are modeled in a similar manner as the flow components above, with the rotational speed ωtc as an

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additional input. Complementing the flow model there is also an efficiency model and in generic form they are all expressed as m˙ c = fc,m˙ (pus , pds , Tus , ωtc )

m˙ t = ft,m˙ (pus , pds , Tus , ωtc )

(4.5)

ηc = fc,η (pus , pds , Tus , ωtc )

ηt = ft,η (pus , pds , Tus , ωtc )

(4.6)

where the functions fx,x (·) are determined from compressor and turbine maps, like the ones displayed in Figure 4.2. The graphical maps are generated from manufacturer provided tables. These tables contain experimental data with flows, pressure ratios, and efficiencies and they often only cover a portion of the operating region around the design point, see e.g. [28, 29] for more information about the turbo maps. There is no generally accepted model for the functions fx,x (·) and the most frequent options are to use either parameterized functions [3, 29, 30, 9], that can be more or less complex, or to use the provided manufacturer data directly as lookup tables. When implementing models for turbo chargers it is important to account for the corrected quantities, that the maps are using in their representation, and to provide extrapolation capabilities outside the range covered by the turbo maps. Compressors and turbines also influence the temperature of the gas. The temperatures are determined with the help of the efficiencies, that are defined as

ηc =

γc −1

γc pds −1 pus , Tds Tus − 1

ηt =

1 − TTds us γtγ−1 t 1 − ppds us

(4.7)

where γc and γt are ratios of specific heat for the compressor and turbine respectively. Solving these for Tds and inserting the efficiency models (4.6) gives the temperatures out of the compressor and turbine.

Fig. 4.2 Graphical representations of the compressor (left) and turbine (right) maps. These maps provide the connection between the mass flow, pressure, and rotational speed of the devices as well as the efficiency of the device

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In turbocharged engines the rotational dynamics of the turbocharger is a dominating dynamics and thus has a significant influence on the transient response. This is modeled by the mechanical coupling between the compressor and turbine, using the power balance between them and Newton’s second law. The power consumption of the compressor and power production of the turbine are Pc = m˙ c c p,c (Tds − Tus ),

Pt = m˙ t c p,t (Tus − Tds )

(4.8)

where c p,c and c p,t are specific heats at constant pressure for the compressor and turbine. These are used when describing the rotational dynamics of the turbocharger d ωtc 1 = (Pt ηm − Pc ) dt Jtc ωtc

(4.9)

where Jtc is the mass moment of inertia in the turbo. In most turbo maps the mechanical efficiency ηm is included in the turbine efficiency and then ηm = 1 should be inserted in (4.9). It is of practical importance to consider that (4.9) is singular when ωtc = 0. This can be handled by specifying a lower limit on the turbine speed ωtc,min , which is not a big loss of generality since the turbocharger only gives a significant influence when it is rotating with high speed.

4.2.3 The Engine Cylinders: Flow, Temperature, and Torque The last component to be considered in this overview is the engine block with the cylinders, and here it is the mass flow and torque production that are of interest. The mass flow into the cylinders pulsates with the opening and closing of the valves to the individual cylinders. Pulsations from the cylinders can also be seen in other variables, see e.g. the exhaust pressure in Figure 4.6. However when modeling the flow in the framework of MVEM these time varying flows are averaged over one or more cycles, giving a mean value of the flow. This viewpoint is in fact the basis for calling this model family mean value engine models. This averaging is sufficient for most control purposes where for example for fuel control the main interest is in how much air is inducted into the cylinder during one stroke. The cylinder flow depends largely on the density in the intake manifold ρ = RpTimim and engine speed N, or more specifically the rate with which the engine displaces the volume, i.e. Vnd rN where Vd is the engine displacement and nr is the number of revolutions per stroke, nr = 2 for a four stroke engine. The flow is frequently modeled with the aid of the volumetric efficiency ηvol , that is a parameter that encapsulates how efficient the engine induction process is, see [27] for a longer discussion. The mass flow into the engine block is given by m˙ ac = ηvol (pim , N)

pim Vd N R Tim 2

where N is given in rps. The volumetric efficiency depends on many parameters and the most important ones are intake manifold pressure as well as engine speed,

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ηvol (pim , N). It is often described either by a map or a parametrized function. Other effects such as charge cooling by fuel evaporation and exhaust back pressure dependence can also be included, see e.g. [27, 31]. Complex engines with variable valve actuation require that these effects are also included in the ηvol (·) model. The gases that exit the cylinder have high temperature and are used to drive the turbine, which thus requires a submodel for the engine downstream temperature. This model is frequently described with the aid of an energy balance over the engine, stating that the waste energy from the engine cycle will determine the temperature increase over the engine Tds = Tus + f (m˙ ac , N, λ , θign , . . .)

(4.10)

where mass flow, engine speed, air-to-fuel ratio λ , and ignition timing θign are important factors to account for. 4.2.3.1

Engine Torque

Engine torque is also an important output from the engine since it is the driving force for the vehicle or other loads connected to the engine. A frequently used approach is to model the torque based on the work production and consumption. A starting point is the chemical energy that the fuel carries into the engine during a stroke m f qLHV . This is converted to work during the high pressure part of the cycle with an efficiency ηig (λ , θign , . . .) that depends on different operating conditions. Then there is a loss of work during the gas exchange phase, called pumping work Wpump(pi , pe , N) and finally there is friction work W f ric (N). Which gives the following torque model M=

1 m f qLHV ηig (λ , θign , . . .) − Wpump(pi , pe , N) − W f ric (N) nr 2 π

(4.11)

Examples of sub models for this generic model can be found in [7, 27, 32].

4.2.4 Implementation Examples Based on the ideas presented above it is possible to implement a model library with reusable components. Several implementations, based on this modeling methodology, have been used in different research projects [26, 33, 34]. Figure 4.3 shows four application examples that serve the purpose of illustrating the modularity of the modeling approach.

4.3 Modeling of a Diesel Engine with EGR/VGT The emission limits for heavy duty trucks are constantly being reduced. To fulfill the legislated requirements, technologies like Exhaust Gas Recirculation (EGR) systems and Variable Geometry Turbochargers (VGT) are introduced, see Figure 4.4. These systems, from actuators to sensors, are strongly coupled

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NA Simple

NA

TCSI

TSTCSI

61

Fig. 4.3 Four Simulink examples of models for SI engines, showing the modularity of the modeling methodology. A simple model for a Naturally Aspirated (NA) engine, a little more complex model for an NA engine, a Turbo Charged SI (TCSI) engine, and a to Two-stage TCSI (TSTCSI) engine

multiple-input multiple-output systems that have non-minimum phase behaviors, overshoots, and sign reversals [35, 36, 37, 38]. For this system a control oriented mean value engine model is developed. As described in Section 4.1, the aim is to develop a model that has a minimum number of states and captures the system properties above. The minimum number of states is investigated by analyzing engine experiments, modeling methodology, and system simulations.

4.3.1 Experimental Data Important system properties such as non-minimum phase behaviors, overshoots, and sign reversals have been investigated by performing steps in EGR and VGT position in an engine test cell. Steps in VGT position are performed in Figure 4.5 and Figure 4.6. Figure 4.5 shows that there is a non-minimum phase behavior in the channel VGT to compressor flow Wc . Figure 4.6 shows a step response at another operating point with a higher load and where the initial VGT position is more closed compared to Figure 4.5. The result is that there is a sign reversal in the DC-gain in the channel VGT to compressor flow Wc and that the non-minimum phase behavior becomes an overshoot instead for this channel. Further, steps in EGR position are performed in [38] showing a non-minimum phase behavior in the channel EGR position to intake manifold pressure. The fundamental physical explanation of these system properties is that the system consists of two dynamic effects that interact: a fast pressure dynamics in the manifolds and a slow turbocharger dynamics. These two dynamic effects often work

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Fig. 4.4 Illustration of a diesel engine with VGT and EGR, showing the fresh gas side with compressor and intercooler on the right and hot gas side with the turbine and EGR valve to the left

Fig. 4.5 Responses to steps in VGT position showing a non-minimum phase behavior in the channel VGT to compressor flow Wc . Operating point: 40 % load, engine speed ne =1200 rpm, and EGR position uegr =100 %

against each other which results in the system properties above. These two dynamics can be seen in Figure 4.6 where the exhaust manifold pressure pem has a fast dynamics in the beginning of the step response and where the turbocharger speed nt has a slow dynamics during the complete step response. Actuator dynamics are analyzed by performing several steps of different sizes in the EGR and VGT positions, covering the range of the actuators. The steps are then normalized to unit steps according to Figure 4.7 showing that the VGT actuator can be described by a linear first order system with a time delay. Further, several step responses for the EGR actuator have overshoots and they can be described by

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Fig. 4.6 A response to a step in VGT position showing a sign reversal in the DC-gain and an overshoot in the channel VGT to compressor flow Wc . Operating point: 50 % load, engine speed ne =1200 rpm, and EGR position uegr =100 %

Fig. 4.7 Step responses for the actuator dynamics showing that the VGT actuator can be described by a linear first order system with a time delay and that the EGR actuator can be described by a second order system with an overshoot and a time delay

a second order system with a time delay. The sum of the time delay and the time constant for the actuators are approximately equal to half the time constant for the intake manifold pressure dynamics. Consequently, it is important to consider the actuator dynamics in order to describe the complete system dynamics.

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Fig. 4.8 A model structure of the EGR/VGT diesel engine with flow components (FC) and control volumes (CV) that follows the modeling methodology in Section 4.2.1

4.3.2 Minimum Number of States The minimum number of states for a mean value engine model that describes the gas flow dynamics in Figure 4.5 and Figure 4.6 are three. This can be understood as follows. It requires one state to describe the slow dominating turbocharger dynamics. To describe the fast pressure dynamics, control volumes are used according to the modeling methodology in Section 4.2.1. The question is: what is the minimum number of control volumes required? The answer to this question depends on which flow components must be modeled and how they are related to each other. To model the turbocharger dynamics, flow components for the compressor and the turbine have to be used. To model the flow through the EGR-valve and the engine two additional flow components must be used. These four flow components are arranged according to Figure 4.8. Following the modeling methodology in Section 4.2.1, there must be a control volume between each flow component leading to that the minimum number of control volumes are two and they are arranged according to Figure 4.8. Each control volume is modeled using one pressure state according to (4.1) leading to that the total number of states are three: intake and exhaust manifold pressure and turbocharger speed. An example of a third order model that captures the system dynamics in Figure 4.5 and Figure 4.6 is the well known model described in [39].

4.3.3 Model Extensions The previous section investigates the minimum number of states required to describe the gas flow dynamics. However, in Section 4.3.1 it is mentioned that it is important also to model the actuator dynamics. According to the step responses in Figure 4.7, the VGT actuator can be modeled using one time delay and one state, and the EGR actuator can be modeled using one time delay and two states. Further, it is important to model the oxygen concentration in the EGR gas due to that this concentration affects the oxygen fuel ratio in the cylinder. This effect is considered by adding a state for the oxygen concentration in each control volume. Many models in the literature, that approximately have the same complexity as the model described here, use three states for each control volume in order to describe the temperature dynamics [39, 35, 40]. However, the model described here uses only two states for each manifold: pressure and oxygen mass fraction. Model

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extensions are investigated in [38] showing that inclusion of temperature states has only minor effects on the dynamic behavior. Furthermore, the pressure drop over the intercooler is not modeled since this pressure drop has only small effects on the dynamic behavior. However, this pressure drop has large effects on the stationary values, but these effects do not improve the complete engine model [38]. In summary, to model the gas flow dynamics, actuator dynamics, and oxygen concentration dynamics, the minimum number of states required is eight: intake and exhaust manifold pressure and oxygen concentration, turbocharger speed, and three states for the actuators. An example of a eighth order model is proposed and validated in [38] that shows that the proposed model captures the system dynamics seen in Figure 4.5 to Figure 4.7.

4.4 Gray-Box Models and Identification This third, and last section, discusses the important issue of parameter identification in gray box models. A normal situation when modeling automotive systems is that there is a desire to have a physical interpretation of the model components and the parameters. However, in most cases it is not possible to determine all model parameters x from physics, geometry or other kinds of system knowledge and it is then necessary to determine them from experimental input and output data Z N . Parameters are frequently determined with the help of prediction error methods [41], where the model is used together with the experimental data to determine a prediction error εi (x, Z N ) for each sample. The best parameters are then selected as those that minimize the prediction error and where the sum of the squared residuals over all N samples is the most frequently used criterion. Thus the following quadratic criterion WN (x, Z N ) =

1 N 1 2 ∑ 2 εi (x, Z N ) N i=1

(4.12)

is used and a minimization of WN (x, Z N ) is performed to determine the parameters x. Standard solvers are usually applied to solve this problem and the MarquardtLevenberg algorithm (see e.g. [42, 43]) is frequently employed. There are some challenges associated with the parameter identification of semiphysical models, for example the model can have identifiability problems or model errors can influence the interpretation of the parameters. Gray-box modeling with physical model components can result in models with many unknown parameters. A consequence of this is that the model structure can have problems with the identifiability of some parameters. Even if the parameters are identifiable some parameters can have a very small effect on WN , and this is manifested in that the optimization problem is not well posed. In particular the hessian of WN (x, Z N ), with respect to x, can be ill conditioned and the solver may require many optimization steps before it converges or it might not converge before the maximum number of iterations have been reached.

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Another but related problem is a consequence of model errors. All models are approximations of the real world and when a model is used to describe a complex system simplifications are usually made which results in a model error. When the parameters are identified this will give a bias in the parameters when the parameters are adjusted to give the best fit to the data and thus cover for the unmodeled effects. It is also common that the parameter values that are received as the solution to (4.12) lie far outside the region where they have a physical interpretation. One example is discussed in [44] where a wall temperature that should be above 100◦ C ended up around -200◦C. Loosely speaking, when the optimization problem (4.12) is solved for the parameters x, there is no account taken for the fact that the parameters should have a physical meaning. The parameters are thus freely adjusted to the value that best fits the data. When the model has a low sensitivity to one or several parameters these can easily be moved far outside their normal range due to model errors or noise.

Regularization A frequently used remedy for the problems outlined above is to use regularization techniques [41]. This is achieved by including nominal parameter values x# for the d parameters, e.g. obtained through prior knowledge. In addition a (diagonal) regularization matrix δ ∈ Rd×d is added that assigns how much we trust the nominal values (the prior). Then the following augmented problem is solved WN (x, x# , Z N , δ ) =

1 N 1 ∑ 2 epsilon2i (Z N , x) + (x − x#)T δ (x − x# ) N i=1

(4.13)

This balances between nominal parameters and data and if δ is selected positive definite this can also improve the condition number of the hessian of WN and restrains the parameters. It can also be seen as a generalization of the frequently used Levenberg-Marquardt method [42] that employs a diagonal matrix to ensure positive definiteness of the hessian in the optimization steps. After this remedy one problem still remains, and that is the selection of δ . Here a starting point comes from the statistics. If we make the following assignment

δ = δx diag(δ j ),

δ j = 1, . . . , d,

δj =

1 2N σ j 2

and if the following conditions are fulfilled; the measurement noise is white and Gaussian and the parameter uncertainty is also Gaussian, then the solution to (4.13) will be the maximum aposteriori (MAP) estimate. Thus the following approach is appealing from an engineering perspective. Do the best possible job on finding values for x# and δ j then look at δx , and this will be done in the next section. However, before proceeding it is also worth to mention that other methods, for supporting the parameter estimation, have been discussed in the literature. One is to add inequality constraints on the parameters to ensure that the stay within given bounds. With

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Fig. 4.9 An example of Hansen’s L-curve (solid) that balances prior knowledge with data. It is presented in log-log scale with the x-axis showing the RMSE of the prediction error and the y-axis showing the difference between the parameter prior and the estimated prior, but scaled with Łδ . Figure from [46]

identifiability problems for some parameters this often leads to that one of more of the parameters end up on their bounds. Both regularization and inequality constraints can be employed on the same problem. The L-curve One way to balance between data and prior knowledge can be done with the aid of the (Hansen’s) L-curve [45], see Figure 4.9. To generate the curve the problem (4.13) is solved for several values of δx . The x-axis shows the RMSE of the prediction error, which is large when δx is large since the parameters are then determined essentially by the prior. The y-axis shows the difference between the parame1 ter prior and the estimated parameter, scaled with Lδ = diag(δ j ) 2 , which increases when the δx is decreased and the parameters leave the prior to be determined by

Fig. 4.10 Two examples of Hansen’s L-curve. In the left plots simulation data is used with a prior that has 5% error in the parameters and then model can thus describe the data, in this case there is a sharp corner in the L-curve. In the right plots experimental data is used and the model is thus an approximation. See [46] for more details

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the data. When δx decreases there is a sharp transition where the residual error ε falls while the nominal parameter deviation x# remains basically constant. For small values of δx , RMSE(ε ) is approximately the same as the noise level. Due to that the upper part of the graph resembles the letter “L”, the curve is called an L-curve [45]. The noise level and the parameter deviation x# − xt are known in simulations, and are therefore included in Figure 4.9 as dashed dotted lines. The cross-over point of these two lines is close to the corner of the L. It is therefore natural to expect that the corresponding δx is a good compromise between data fitting and penalizing the parameter deviation [47]. Typically there is a wide range of δx : s corresponding to the points on the Lcurve near its corner. Therefore, the location of the corner should be found by some numerical optimization routine, and not by visual inspection [45, 47], especially if one wants to automatize the search procedure. Three methods for automating the search procedure are investigated in [46] and these are • M2:1 Millers a priori choice rule [48]; • M2:2 Morozov’s discrepancy principle [49]; and • M2:3 Maximum curvature of the L-curve [45] with extensions to the basic method. where the methods are named in the same way as in [46]. The performance of these methods are investigated for cylinder pressure models both on simulated and measured data, see Figure 4.10. On simulated data the methods perform very well, i.e. when the model structure contains the true model. On experimental data, i.e. when the model structure does not necessarily contain the true model, there are some occasions when the corner isn’t sharp or when there are several elbows in the transition region. But with these caveats there is still a good support for the parameter estimation provided by the regularization and corner detection. With the regularization the user can choose between two cases of the parameter uncertainty, either all δ are equal (in a normalized sense) or they are chosen individually. These options were investigated in different cases and it was not clear that the second case was better in all cases but on average it was the best choice [46]. It requires more effort to decide upon the uncertainty for each parameter, but pays off in better estimates that are more robust to a false nominal parameter value. Once a choice of parameter uncertainty has been done, no user interaction is needed in the parameter identification. Utilizing L-curve and corner detection methods thus gives a systematic/automatic method for determining δx .

4.5 Conclusions Three different aspects of control oriented modeling based on the physics have been addressed and they are summarized below. The first topic is component based modeling and library design for validation general purpose simulation. By analyzing the engine components and their constituting equations together with insight into the component topology of engines it is possible to design a model library with high

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degree of reusability. The strength is demonstrated by the multitude of applications that have been successfully addressed with the design method. Building tailored minimal models for a particular application is the second topic. A diesel engine with EGR and VGT is used as example and the necessary system dynamics is studied with respect to its ability to capture non linear effects from actuator to sensor response. A three state model, with intake and exhaust manifold pressures as well as turbocharger speed as states, is the minimum size that captures the essential dynamics and non-linearities. When more effects are needed for EGR and λ -control gas composition and actuator dynamics are those that influence the dynamics in those particular applications. Additional components or states for the temperature do not have a significant influence on the essential dynamics. Parameter identification in gray-box models is the last topic and regularization is studied as a support. Mixing prior information and measurement data improves the convergence of the optimization problem and helps the optimization to provide physically reasonable parameter values. An application of Hansen’s L-curve and corner detection methods is studied and it gives a systematic/automatic method for determining the regularization parameter. Acknowledgements. This work has been supported by Swedish Energy Agency and the Strategic Research Center MOVIII, funded by the Swedish Foundation for Strategic Research, SSF.

References [1] Hendricks, E.: A compact, comprehensive model of large turbocharged, two-stroke diesel engines, SAE Paper 861190 (1986) [2] Hendricks, E., Sorenson, S.C.: Mean Value Modelling of Spark Ignition Engines, SAE Paper 900616 (1990) [3] M¨uller, M., Hendricks, E., Sorenson, S.C.: Mean Value Modelling of Turbocharged Spark Ignition Engines. SAE SP-1330 Modeling of SI and Diesel Engines (SAE Technical Paper 980784), pp. 125–145 (1998) [4] Hendricks, E.: Isothermal vs. adiabatic mean value SI engine models. In: 3rd IFAC Workshop, Advances in Automotive Control, Preprints, Karlsruhe, Germany, March 2001, pp. 373–378 (2001) [5] Eriksson, L.: Mean value models for exhaust system temperatures. SAE Transactions, Journal of Engines, 2002-01-0374 111(3) (September 2002) [6] Eriksson, L., Nielsen, L., Brug˚ard, J., Bergstr¨om, J., Pettersson, F., Andersson, P.: Modeling and simulation of a turbo charged SI engine. Annual Reviews in Control 26(1), 129–137 (2002) [7] Guzzella, L., Onder, C.H.: Introduction to Modeling and Control of Internal Combustion Engine Systems. Springer, Heidelberg (2004) [8] Kiencke, U., Nilsen, L.: Automotive Control Systems – For Engine, Driveline, and Vehicle, 2nd edn. Springer, Heidelberg (2005) [9] Eriksson, L.: Modeling and control of turbocharged SI and DI engines. Oil & Gas Science and Technology - Rev. IFP 62(4), 523–538 (2007)

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[10] Aquino, C.F.: Transient A/F control characteristics of the 6 liter central fuel injection engine. SAE Paper 810494 (1981) [11] Hendricks, E., Sorensen, S.C.: SI engine controls and mean value engine modeling, SAE Paper 910258 (1991) [12] Ault, B.A., Jones, V.K., Powell, J.D., Franklin, G.F.: Adaptive air-fuel ratio control of a spark-ignition engine. SAE SP-1029 (SAE Paper 940373), pp. 109–118 (1994) [13] Turin, R.C., Geering, H.P.: Model-based adaptive fuel control in an SI engine. SAE SP-1029 (SAE Paper 940374), pp. 119–128 (1994) [14] Bidan, P., Boverie, S., Chaumerliac, V.: Nonlinear control of a spark-ignition engine. IEEE Transactions on Control System Technology 3(1), 4–13 (1995) [15] Jensen, P.B., Olsen, M.B., Poulsen, J., Hendricks, E., Fons, M., Jepsen, C.: A new family of nonlinear observers for SI engine air/fuel ratio control. SAE Paper 970615, pp. 91–101 (1997) [16] Guzzella, L., Amstuz, A.: Control of diesel engines. IEEE Control Systems (special issue on powertrain control) 18(5) (October 1998) [17] Andersson, P., Eriksson, L.: Mean-value observer for a turbocharged SI-engine. In: IFAC Symposium on Advances in Automotive Control, Universty of Salerno, Italy, April 19-23, pp. 146–151 (2004) [18] Deur, J., Ivanovic, V., Pavkovi, D., Jansz, M.: Identification and speed control of SI engine for idle operating mode. Number SAE Technical Paper 2004-01-0898 (2004) [19] Le Berr, F., Miche, M., Colin, G., Le Solliec, G., Lafossas, F.: Modelling of a turbocharged SI engine with variable camshaft timing for engine control purposes. SAE Technical paper 2006-01-3264 (2006) [20] Eriksson, L.: VehProLib - Vehicle propulsion library, library development issues. In: Proceedings of the 3rd International Modelica Conference, Link¨oping, November 3-4 (2003) [21] Tiller, M.: Introduction to Physical Modeling With Modelica. Kluwer Academic Publishers, Dordrecht (2001) [22] Batteh, J., Tiller, M., Newman, C.: Simulation of engine systems in modelica. In: Proceedings of the 3rd International Modelica Conference, pp. 139–148 (2003) [23] Massey, B.: Mechanics of Fluids. Stanley Thornes, 7th edn. (1998) [24] Hairer, E., Wanner, G.: Solving Ordinary Differential Equations I: Nonstiff Problems, 2nd revised edn. Springer, Heidelberg (1987) [25] Ellman, A., Pich´e, R.: A two regime orifice flow formula for numerical simulation. Journal of Dynamic Systems, Measurement, and Control 121(4), 721–724 (1999) [26] Andersson, P.: Air Charge Estimation in Turbocharged Spark Ignition Engines. PhD thesis, Link¨opings Universitet (December 2005) [27] Heywood, J.B.: Internal Combustion Engine Fundamentals. McGraw-Hill series in mechanical engineering. McGraw-Hill, New York (1988) [28] Watson, N., Janota, M.S.: Turbocharging the Internal Combustion Engine. The Macmillan Press Ltd., Basingstoke (1982) [29] Moraal, P., Kolmanovsky, I.: Turbocharger Modeling for Automotive Control Applications. SAE Technical Paper 1999-01-0908, pp. 309–322 (1999) [30] Sorenson, S.C., Hendrick, E., Magnusson, S., Bertelsen, A.: Compact and accurate turbocharger modeling for engine control. In: Electronic Engine Controls 2005 (SP1975), number SAE Technical Paper 2005-01-1942 (2005) [31] Andersson, P., Eriksson, L.: Cylinder air charge estimator in turbocharged SI-engines. SAE Technical Paper 2004-01-1366 (2004)

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[32] Nilsson, Y., Eriksson, L., Gunnarsson, M.: Torque modeling for optimising fuel economy in variable compression engines. International Journal of Modeling, Identification and Control (IJMIC) 3(3) (2008) [33] Leufven, O., Eriksson, L.: Time to surge concept and surge control for acceleration performance. In: IFAC World Congress, Seoul, Korea (2008) [34] Thomasson, A., Eriksson, L., Leufven, O., Andersson, P.: Wastegate actuator modeling and model-based boost pressure control. Submitted to E-COSM 2009, IFAC Workshop on Engine and Powertrain Control, Simulation, and Modeling (2009) [35] Kolmanovsky, I.V., Stefanopoulou, A.G., Moraal, P.E., van Nieuwstadt, M.: Issues in modeling and control of intake flow in variable geometry turbocharged engines. In: Proceedings of 18th IFIP Conference on System Modeling and Optimization, Detroit (July 1997) [36] Jung, M.: Mean-Value Modelling and Robust Control of the Airpath of a Turbocharged Diesel Engine. PhD thesis, University of Cambridge (2003) [37] Winge Vigild, C.: The Internal Combustion Engine – Modeling, Estimation and Control Issues. PhD thesis, Technical University of Denmark (2001) [38] Wahlstr¨om, J.: Control of EGR and VGT for Emission Control and Pumping Work Minimization in Diesel Engines. PhD thesis, Link¨oping University (2009) [39] Jankovic, M., Jankovic, M., Kolmanovsky, I.: Constructive lyapunov control design for turbocharged diesel engines. In: IEEE Transactions on Control Systems Technology (2000) [40] Stefanopoulou, A.B., Kolmanovsky, I., Freudenberg, J.S.: Control of variable geometry turbocharged diesel engines for reduced emissions. IEEE Transactions on Control System Technology 8(4), 733–745 (2000) [41] Ljung, L.: System Identification Theory for the User, 2nd edn. Prentice Hall, Englewood Cliffs (1999) [42] Gill, P.E., Murray, W., Wright, M.H.: Practical Optimization. Academic Press, London (1981) ˚ Numerical Methods for Least Squares Problems. SIAM, Philadelphia (1996) [43] Bj¨ork, A.: [44] Eriksson. L.: Requirements for and a systematic method for identifying heat-release model parameters. Modeling of SI and Diesel Engines (SP-1330), SAE Technical Paper 980626, pp. 19–30 (1998) [45] Hansen, P.C., O’Leary, D.P.: The use of the L-curve in the regularization of discrete ill-posed problems. SIAM Journal on Scientific Computing 14(6), 1487–1503 (1993) [46] Klein, M.: Single-Zone Cylinder Pressure Modeling and Estimation for Heat Release Analysis of SI Engines. PhD thesis, Link¨opings Universitet (September 2007) ˜ [47] Engl, H.W., Hanke, M., Neubauer, A.: Regularization of inverse problems. Kluwer Academic Publishers, London (1996) [48] Miller, K.: Least squares methods for ill-posed problems with a prescribed bound. SIAM Journal on Mathematical Analysis 1(1), 52–74 (1970) [49] Morozov, V.A.: Methods for solving incorrectly posed problems. Springer, New York (1984)

Chapter 5

Dynamic Engine Emission Models Markus Hirsch, Klaus Oppenauer, and Luigi del Re

Abstract. The classical trade off between nitrogen oxides (NOx ) and particulate matters (PM) is still one of the key topics for Diesel engine developers. This article gives an overview about models for these emissions usable for online engine and exhaust after treatment control, offline optimization and virtual sensors for monitoring. Two different ways for obtaining such models are presented in detail: first we present a black-box data-based mean value model which estimates engine raw emissions from quantities available in the engine control unit. Second a gray-box model is shown in which also physical equations are used to describe emission formation over crank angle with the measured cylinder pressure as main input.

5.1 Introduction Nitrogen Oxides (NOx ) (a combination of nitrogen oxide NO and nitrogen dioxide NO2 ) and particulate matter (PM) are the primary and for manufactures most demanding pollutants in Diesel engines. Limits can be met by in cylinder measures like adjustment of injection and exhaust gas recirculation (EGR) or otherwise by aftertreatment devices like Selective Catalytic Reduction (SCR), NOx storage catalyst systems (NSC) or by Diesel particulate filters (DPF). To optimize the operation of these devices either on or offline, models for the engine raw emission are essential [14]. These models should be valid for steady state operating points but also for transient changes, as here absolute values can become very high (cp. Figure 5.1) and highest potential of reduction is there. Markus Hirsch and Luigi del Re Institute for Design and Control of Mechatronical Systems, JKU Linz, Austria e-mail: {markus.hirsch,luigi.delre}@jku.at Klaus Oppenauer JKU Linz, Austria, permanent address: BMW Motoren GmbH, Hinterbergerstrae 2, A-4400 Steyr e-mail: [email protected] L. del Re et al. (Eds.): Automotive Model Predictive Control, LNCIS 402, pp. 73–87. c Springer-Verlag Berlin Heidelberg 2010 springerlink.com

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Fig. 5.1 Emissions during transients are significant

The combustion itself is a consistently running process where emissions are formed. Therefore a model taking into account each combustion process would be predestinated for good representation. Even so, such detailed (in general crank angle based) models of the combustion are often not required as only a mean emission value, which represents averaged emissions during the combustion process of all cylinders, is necessary. Figure 5.2 shows different possible boundaries for models of engine operation including the emissions: if the whole engine is seen as a blackbox system, only two input quantities, accelerometer position given by the driver and external load, a result from the vehicle mass, translation, road resistance and acceleration or a test bench dynamometer, are inputs for an entire engine model. In such a case engine speed, temperature, etc. are other outputs besides emissions. In this article we are interested only in models relating the emission outputs with the inputs coming from the ECU, which can be used for control and optimization applications, therefore an approach close to the combustion process is reasonable. Here many engine states like speed, airpath states etc. present possible inputs for the emission model which results in a higher accuracy. Before development of two emission

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Fig. 5.2 Different levels for emission models

models – a crank angle based model and a mean value model (levels of treatment are shown in Figure 5.2) – is presented, some general aspects on data-based model identification are explained.

5.2 Data-based Model Identification Although the main processes involved in NOx and PM formation (see [7, 27]) are theoretically well understood, first principles models (e.g. [13, 26]) describing these processes in CRDI Diesel engines, are not able to provide control oriented models with sufficient accuracy, as most information, e.g. on the exact local concentrations and temperatures in the combustion chamber, is not available. Therefore, the main challenge in dynamical emission modeling consists in finding models that represent the highly nonlinear process of emission formation with a sufficient accuracy, but still can be parameterized using only data available on a production engine. This can be done by data-based model identification where parameters of a model which should relate input quantities u (here available engine states) to output quantities y, emissions, are defined by measurement data. This relationship is usually a mathematical expression defined by a in general nonlinear function f containing parameters given in the vector θ and a some kind of model error ε (cp. (5.1)). y = f (u, θ ) + ε

(5.1)

The art of model identification is characterized by finding the optimal θ within a given range Θ that way a criterion J depending on the model error ε becomes minimal for the data set of length n used for this optimization. For further information on data-based identification procedures see[17].

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θ = arg min J (yi − f (u, θ )i ) i ∈ [1 . . . n] θ ∈Θ

(5.2)

Depending how the structure of f is chosen, data-based models, can be divided into different groups: on one side of the spectrum, there are so-called white-box models (e.g. [19]) where the structure of f is defined by basic chemical and physical equations and only (some) parameters are adapted and identified respectively to represent the emissions of a specific engine. On the other side of the spectrum, so called black-box models assume no special structure of f at all but the correct structure should be approximated by a general function class. Many model classes here offer sufficient approximation properties. Very popular classes for black-box emission models are artificial neural networks (ANN) (e.g. [4]), models applying genetic algorithms (e.g. [3]) as well as models where the nonlinearity is handled by a polynomial structure (e.g. [9]. In between these two extremes there are different shades of gray-box models; here some physical knowledge is included in the structure of f and other terms have to be identified from data. In this article two different ways for obtaining data-based emission models are presented: first we present a black-box mean value model defining engine raw emissions by in the engine control unit given quantities. Second we show a gray-box model where the model structure is given by first principle equations and parameters are optimized to describe the emission formation over crank angle with the measured cylinder pressure as main input.

5.3 Mean Value Emission Model Here no first principles based structure of f in (5.1) was assumed, but the structure of f was chosen very flexible that way general approximations could be done.

5.3.1 Input Selection For building up a data-based model first input quantities have to be defined. Here, all available ECU values were taken as possible inputs. As the number of possible input candidates is very high and many quantities are equivalent, an initial preselection was done by process know-how and data-driven statistical input selection [12]. A result of this process was that for standard operating conditions (heated up engine and standard ECU control), quantities of total injected fuel mass per cycle (qtot ) and engine speed (N) in combination with some quantities describing the airpath states (e.g. manifold absolute pressure (MAP) and oxygen concentration of the exhaust gas (O2exh )) are sufficient (see Table 5.1). Engine temperatures have an influence on emissions, but as variations during normal conditions are relatively small this input could be excluded. However, if the cold start phase should be included in the model, this temperature, best represented by the oil temperature, has to be included in the model too. One possibility therefore would be a temperature depended scaling factor (see [9]) or including this temperatures into the identification process.

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Table 5.1 Input variables of the mean value model Input Channel Unit

Description

u1 u2 u3 u4

total injected fuel mass per cycle engine speed manifold absolute pressure volume concentration of oxygen in the exhaust

qtot N MAP O2exh

mg/cyc rpm hPa %

5.3.2 Model Structure Eq. (5.3) presents the chosen structure of f , a polynom of maximum order l with q = 4 input channels offering good approximation properties. f = θ0 +

l

q

∑ ∑

θi1 ...im

m=1 i1 ...im =1 i1 ≤···≤im

m

∏ ui p

(5.3)

p=1

Though the emission model defined here should include dynamic properties too, a steady state formulation is sufficient, as most relevant dynamics can be excluded by taking airpath states of MAP and O2exh instead of airpath inputs, opening level of the EGR valve and VGT position, as model inputs. Otherwise airpath dynamics could be included by adding delayed values of the inputs and the output into the set of channels defining the present output, obtaining a polynomial NARX structure model (see [20]). To reduce the polynomial degree l of (5.3) and therefor to obtain a preferable “more linear” model, a power transformation of the output (cp. [1]) can be useful. Hence, not the nominal output is defined by f but a nonlinear transformation (e.g. the logarithm of the nominal output). Eq. (5.3) is linear in parameters, therefore (5.1) can be written in the form: y = θ T · u˜ + ε

(5.4)

θ is the parameter vector which has to be identified and u˜ the regressor vector containing the elements of the polynom. In matrix notation this can be defined for a whole time series: Y = θ T · U˜ + E (5.5) U˜ is the data matrix containing n regression vectors for every sample and Y the output vector, while E is the unknown error vector.

5.3.3 Parameter Identification For the identification process a cost function J (cp. (5.2)), summing up square errors, was chosen. Very common therefore is the matrix notation.

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J =

n

∑

yk − θ T · u˜k

2

T Y − θ T · U˜ = Y − θ T · U˜

(5.6)

k=1

Finding the best parameter vector θ implies minimizing this function and can be done explicitly by an ordinary least square estimation: −1 T θ = U˜ T U˜ U˜ Y = = M −1 U˜ T Y

(5.7)

M here defines the information matrix composed of the input correlations.

5.3.4 Regressor Selection U˜ can contain, depending on the assumed polynomial order l, a high number of regresors for which parameters have to be identified. According to (5.7) and due to noise and measurement errors, there will always be a solutions for all these parameters whether their regressor is necessary and meaningful or not. Therefore, a regressor selection can be included to the identification algorithm. Here by means of statistical tests (e.g. T-test) the significance of each regressor is computed. If some regressors are not significant the worst is neglected and identification procedure is started again without this parameter. This is done iteratively until only significant parameters are defining the model. More on this backward selection algorithm and alternative procedures can be found in [6]).

5.3.5 Realization and Results The engine used to conduct the experiments was a production 2 liter 4 cylinder CRDI Diesel engine on an dynamical test bench. The target values for engine parameters qtot , N, MAP and O2exh have been specified by a design of experiment strategy for maximizing the information content of the measurements (further detail in chapter 5.5). NOx emission acquisition was done with a Cambustion fNOx400 system. This, on chemiluminescence based, fast measurement device holds rising times less then 15 ms. PM analysis were done by the AVL439 opacimeter. For defining the parameters of the models 180 points have been measured. This means that for the identification of each parameter (polynomial order l of the model was set to 2) 12 points were measured. Each of these 180 points was hold 5 seconds, which results in a total measurement time of 20 minutes, including measurements for definition of the boundaries for save operating conditions. After measurements were done, parameters of the models defining NOx and PM (represented by opacity) respectively have been identified according to (5.7), applying also regressor selection. Figure 5.3 shows the model validation on part of a standard driving cycle, also run on the test bench using standard ECU control.

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Fig. 5.3 Validation of the mean value model on a standard driving cycle

5.4 Crank Angle Based Emission Model To get deeper information how the emission is formed and oxidized during the combustion process a crank angle (CA) based model could be used. Such a model has a smaller timescale to look on the process, compared to a mean value model. A main advantage of a CA based model is the ability to extrapolate. Changed emissions caused by changed combustion can be predicted because the main input is the measured cylinder pressure of the mentioned test engine. This quantity inherently takes into account combustion changes and is useful for emission prediction in transient conditions. Previous use of CA based emission models have only focused on steady state operating modes because of the complexity and the range of validity, but actually there is no physical restriction not to use these models during transients too. In this chapter the workflow of a possible crank angle based emission model is described. The model starts at inlet valve closing (IVC) and calculates emissions over crank angle until exhaust valve opening (EVO). At EVO the model values should fit to values defined by mean value models as well as measured emissions in the tailpipe.

5.4.1 Workflow The described workflow is shown in Figure 5.4. It starts with a classical 1-zone thermodynamic process calculation depending mainly on the measured cylinder pressure over crank angle. Some starting values for the gas compositions, assumptions for the wall heat release and crank geometry values are also necessary. In this first step mean values over the whole combustion chamber are calculated over crank

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Fig. 5.4 Model workflow for CA based emission models

angle. The next step is to divide the combustion chamber into two zones, the burned and the unburned zone. This is essential because the 1-zone mean temperature is too low for an adequate NO model. At least the emissions are derived with the statevariables calculated in further steps.

5.4.2 1-zone Process Calculation A process calculation according to [21] is used which is based on the following 3 equations: • mass conservation; (5.8) • energy conservation; and (5.9) • ideal gas equation (5.10) After some modifications and simplifications 3 linear explicit coupled differential equations can be obtained. dm dϕ

=

dmB dϕ

dQB dϕ

= dT dϕ

1 dQB HU d ϕ

=

→

dλ dϕ

B = −λ mB01+mB dm dϕ

=

(

)

(5.8)

dQ 1+ R1 δδ Tu + ddϕp VR δδ Tu +m δδ up + d ϕW m·λ δu 1+ Hu + HT δδTu + HU (mBO +mB ) δ λ U U

p ddVϕ

dm p· ddVϕ +V · ddϕp −m·T · ddR ϕ −R·T · d ϕ m·R

(5.9) (5.10)

5.4.3 2-zone Model Based on the burn rate calculated in the 1-zone model a post 2-zone model is used to derive temperature and λ in the burned and unburned zone. The empirical 2zone model published by Heider and Woschni [5] is extended with a parallel mixing flow depending on global turbulence according to [15]. Advantage of the empirical energy transport formulation is that the energy equations for both zones are not required. The temperature difference between the two zones is described in equation (5.11) with a crank angle depending B-value and a fixed A-value analog to [5]. The B-value in equation (5.12) is mainly a function of the pressure difference between

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Fig. 5.5 Results 1-zone and 2-zone calculation

the measured pressure and a calculated pressure without combustion. A data based description for the A-value developed for a modern common rail Diesel engine is shown in equation (5.13). The β -parameters in equation (5.13) have to be defined by data-based identification by least square optimization similar to (5.7). Results for one operating point are shown in Figure 5.5. Tburned − Tunburned = B (ϕ ) · A

B (ϕ ) =

ϕEVO ϕCS

A = β1

β2

1.2+(λ −1.2)

(5.11)

ϕ

(p−pnoC )·mburned d ϕ − ϕCS (p−pnoC )·mburned d ϕ ϕEVO ϕCS (p−pnoC )·mburned d ϕ

XSwirl 100%

2.2

XSwirl +β3 1− 100%

+ β4

XEGR 100%

+ β5

(5.12) XEGR 100%

2 (5.13)

5.4.4 Emission Models Based on the 1- and 2-zone results the emissions over crank angle are calculated. 5.4.4.1

Nitrogen Oxide Model

The extended Zeldovich mechanism [28] is used to calculate the NO emissions. A simplified equation [24] and reaction rates analog to (5.14) are used. 1− d [NO] = 2k1f [N2 ] [O] dt 1+

[NO] 2

f f k k 1 2 [N ][O ] 2 2 b k1 k2b b k1 [NO] f f k2 [O2 ]+k3 [OH]

(5.14)

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Particulate Matter Model

A PM model with formation and oxidation term is used, similar to [8] and [25]. The pre exponential terms AFormation and AOxidation again are identified by measurement data. dmPMOxidation dt 6113 dmPMFormation dmB 1.8 − Tburned = AFormation dt p e dt dmPMOxidation − 4.6 = AOxidation mPM nOH p0.4 e Tburned dt dmPM dt

=

dmPMForamtion dt

−

(5.15)

5.4.5 Model Development and Verification Model development and verification was done with conventional tailpipe exhaust gas analyzers and an optical global 2 color spectroscopy over crank angle at the 2 liter 4 cylinder CRDI test engine. The global 2 color spectroscopy determines burning soot temperature by measuring the intensity of the radiation at two wavelengths over crank angle. With the Hottel and Broughten analogy [11] and assumptions it is possible to derive the soot concentration in the cylinder. Detailed information about the 2 color spectroscopy can be found in [15, 16, 18, 23]. An example is shown in Figure 5.6.

Fig. 5.6 The compared modeled and measured emissions

5.5 Data for Identification: Input Design The idea of input design is optimizing the input sequence u (t) that way it excites the system preferably and makes identification of each parameter separable in some sense though measurement time is minimal. Random signals (e.g. white noise or

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Fig. 5.7 Detail of the optimized input sequence for identification of the mean mean value model

pseudo random binary signals e.g. [2]) are known as good candidates to this end, especially for linear systems. Indeed, studies like [16] have confirmed that simultaneous and independent random excitations present the best input signals. Unfortunately, this occurs only after a rather long time (in theory infinity), because it is essentially a blind excitation, all possible components are excited. However, if something more is known on the system to be identified, this information can be used to tailor the signal to it. In the case of the functions presented here structure of the function f is assumed, for the gray-box model see eq. (5.13) and (5.15) respectively and for the black box model see eq. (5.3), therefore inputs can be optimized for the structure.

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Fig. 5.8 Validation of the emission model on a part of the NEDC

The task of input design is to enforce a consistent parameter estimation. An essential requirement for the explicit least square identification is the invertibility of the information matrix M (see (5.7)), where correlations of the inputs u are included: M = U˜ T U˜ (5.16) Hence, input sequences should be composed that way an inversion can be done easily and effects of limited numerical resolution and appearing noise, always critical for inversion, can be minimized. Therefore optimal input design algorithms define input signals so that way they maximize a given criterion J (M), usually related to the condition of M, a common one being the D-criterion, defined as J (M) = 1n det (M) (for some alternative proposals see [22]). Especially for the identification of nonlinear systems, input design becomes essential in order that all regressors are exciting and cross-correlation become minimal. By advanced strategies also nonuniform boundaries, limiting the safe operating range (e.g. not all combination of MAP and O2exh are reasonable in terms of reachability, ignition as well as engine limits) can be included (cp. [10]). Figure 5.7 shows details of the input sequence used for the identification of the mean value model presented in chapter 5.3. Figure 5.8 shows the performance of the emission estimation in two test cases.

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5.6 Limitations A model will always represent just an approximation of the real system. Only effects which have been considered during the modeling and identification process can be included. Hence, emission models presented in this article represent standard operating conditions quite well but cannot be used as some constant assumed states like fuel composition, engine aging etc. change. Such effects just can be seen by physical emission sensors but here emission models can be useful for monitoring and fault detection. Especially for nonlinear models extrapolation to regions where no data was available for identification can be very critical as system behavior can be completely different there. For the emission model this implies that measured real emission can become completely different as combustion characteristics change. The acceptance of a model should thus be guided by “usefulness” rather than “truth” and model limitations always to be kept in mind.

5.7 Summary This article presented possibilities for defining data-based emission model for Diesel engines. Depending on the point of view, emissions can be defined either crank angle based or cumulatively by mean value models. Ideas of data-based parameter identification were presented in a very general way and possibilities for the structure of the regression function were addressed. The presented gray-box CA based emission model is mainly based on formulations given by first principles equations at which some unknown parameters are defined by data based identifications. Contrary to the fixed structure here, the approach for the mean value model was a more general one, as model structure was chosen very general in terms of an universal approximator. Robustness given by the physical based fixed structure for the gray-box model, can here partly be obtained by statistical tests for regressor selection. Results show that models representing (NOx ) perform very well. Even so, more attention should be drawn to models for PM as here good results are harder to obtain. Important for the parameter identification is data used therefore. In particular for nonlinear identification data has to excite all appearing regressors independently. In order to obtain adequate input sequences of minimal measurement duration optimal input design strategies are essential.

Definitions for CA Based Model A

Empirical value which describes maximum temperature between burned and unburned zone

AFormation Pre exponential parameter for PM calculation formation term AOxidation Pre exponential parameter for PM calculation oxidation term

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HU

Lower heating value

f ki kib

Reaction rates forward (NO)

m

Total in-cylinder gas mass

mB

Fuel mass burned

mburned

Total gas mass in burned zone

Reaction rates backward(NO)

munburned Total gas mass in unburned zone mPM

In-cylinder PM mass

p

Cylinder pressure

pnoC

Cylinder pressure without combustion

QB

Fuel energy burned

QW

Wall heat energy

R

Gas constant

T

Mean temperature (1-zone)

Tburned

Temperature burned zone

Tunburned Temperature unburned zone u

Inner energy

V

Cylinder volume

XEGR

Percent EGR rate extern

V

Percent swirl valve position

λ

Mean relative air/fuel ratio (1-zone)

λburned

Relative air/fuel ratio in the burned zone

λunburned Relative air/fuel ratio in the unburned zone ϕ

Crank angle

[.]

Concentration

References [1] Box, G.E.P., Cox, D.R.: An analysis of transformation. Journal of the Royal Statistical Society 26(2), 211–252 (1964) [2] Chen, J., Yu, C.: Optimal input design using generalized binary sequence. Automatica 33(11), 2081–2084 (1997) [3] del Re, L., Langthaler, P., Furtmueller, C., Winkler, S., Affenzeller, M.: Nox virtual sensor based on structure identification and global optimization. SAE, Paper 2005-010050 (2005) [4] Hafner, M., Schueller, M., Nelles, O., Isermann, R.: Fast neural networks for diesel engine control. Control Engineering Practice 8, 1211–1221 (2000)

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[5] Heider, G., Woschni, G., Zeillinger, K.: 2-zonene rechenmodell zur vorausberechnung der no-emissionen. MTZ 59, 770–775 (1998) [6] Henig, C.: University of Hamburg, Department of Mathematics (2004) [7] Heywood, J.B.: McGraw-Hill series in mechanical engineering (1988) [8] Hiroyasu, H., Kadota, T.: Development and use of a spray combustion modelling to predict diesel engine efficiency and pollutant emission. The Japan Society of Mechanical Engineers 26, 569–575 (1983) [9] Hirsch, M., Alberer, D., del Re, L. [10] Hirsch, M., del Re, L.: Adapted d-optimal experimental design for transient emission models of diesel engines. SAE, Paper 2009-01-0621 (2009) [11] Hottel, H.G., Broughten, F.P.: Determination of true temperature and total radiation form luminous gas flames. Industrial and Engineering Chemistry: Analytical Edition 4(2), 166–174 (1932) [12] Johnson, R.A., Wicher, D.W.: Applied multivariate statistical anlaysis. Prentice-Hall, Englewood Cliffs (1998) [13] Jung, D., Assanis, D.: Multi-zone di diesel spray combustion model for cycle simulations studies of engine performance and emissions. SAE, Paper 2001-01-1246 (2001) [14] Karagiorgis, S., Glover, K., Collins, N.: Control challenges in automotive engine management. European Journal of Control 13, 92–104 (2007) [15] Kozuch, P.: University Stuttgart (2004) [16] Li, X., Wallace, J.S.: In-cylinder measurement of temperatures and soot concentration using the two-color method. SAE, Paper 950848 (1995) [17] Ljung, L.: PTR Prentice Hall infromation and system science series. Prentice Hall PTR, Englewood Cliffs (1999) [18] Mohammad, I.S., Borman, G.L.: Measurement of soot and flame flame temperature along three directions in the cylinder of direct injection diesel. SAE, Paper 910728 (1991) [19] Ouenou-Gamo, S., Ouladsine, M., Rachid, A.: Measurement and prediction of diesel engine exhaust emissions. ISA Transactions 37, 135–140 (1998) [20] Piroddi, L., Spinelli, W.: An identification algorithm for polynomial narx models on simulation error minimization. International Journal of Control 76(17), 1767–1781 (2003) [21] Pischinger, R., Klell, M., Sams, T.: PTR Prentice Hall infromation and system science series. Springer, Heidelberg (2002) [22] Pukelsheim, F.: SIAM - classics in applied mathematics. SIAM, Philadelphia (2006) [23] Schubinger, R.: ETH Zurich (2001) [24] Urlaub, A.: Springer, Heidelberg (1989) [25] Vanhaelst, R.: University Magdeburg (2003) [26] Wang, G., Li, G., Liu, Y., Chen, L., Zhang, X., Lu, J.: A developed model for emission prediction of a diesel engine. SAE, Paper 1999-01-0233 (1999) [27] Warnatz, J., Maas, U., Dibble, R.W.: Springer, Berlin (1999) [28] Zeldovich, Y.B.: The oxidation of nitrogen in combustion ans explosions. Acta Physiochemica, URSS 21 (1946)

Chapter 6

Modeling and Model-based Control of Homogeneous Charge Compression Ignition (HCCI) Engine Dynamics Rolf Johansson, Per Tunest˚al, and Anders Widd

Abstract. The Homogeneous Charge Compression Ignition (HCCI) principle holds promise to increase efficiency and to reduce emissions from internal combustion engines. As HCCI combustion lacks direct ignition timing control and auto-ignition depends on the operating condition, control of auto-ignition is necessary. Since auto-ignition of a homogeneous mixture is very sensitive to operating conditions, a fast combustion phasing control is necessary for reliable operation. To this purpose, HCCI modeling and model-based control with experimental validation were studied. A six-cylinder heavy-duty HCCI engine was controlled on a cycle-to-cycle basis in real time using a variety of sensors, actuators and control structures for control of the HCCI combustion in comparison. The controllers were based on linearizations of a previously presented physical, nonlinear, model of HCCI including cylinder wall temperature dynamics. The control signals were the inlet air temperature and the inlet valve closing. A system for fast thermal management was installed and controlled using mid-ranging control. The resulting control performance was experimentally evaluated in terms of response time and steady-state output variance. For a given operating point, a comparable decrease in steady-state output variance was obtained either by introducing a disturbance model or by changing linearization point. Additionally, the robustness towards disturbances was investigated.

6.1 Introduction The motivation for studying the homogeneous charge compression ignition (HCCI) engine principle is the promise of low levels of exhaust emissions with regards to Per Tunest˚al Department of Energy Sciences, Div. Combustion Engines, Lund University, PO Box 118, SE22100 Lund Sweden e-mail: [email protected] Rolf Johansson and Anders Widd Department of Automatic Control, Lund University, PO Box 118, SE22100 Lund Sweden e-mail: {Rolf.Johansson,Anders.Widd}@control.lth.se L. del Re et al. (Eds.): Automotive Model Predictive Control, LNCIS 402, pp. 89–104. c Springer-Verlag Berlin Heidelberg 2010 springerlink.com

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NOx , while still retaining an acceptable overall efficiency [17]. Pioneering efforts towards this new engine principle, also called controlled auto-ignition (CAI), were reported in [16, 23, 32, 43, 58]. Depending on the purpose, modeling of HCCI engine dynamics may exhibit different complexity and format such as: • multi-zone models including chemical kinetics to simulate engine operation in a large operating range; • multidimensional CFD for optimization of fuel injection and combustion chamber design; and • single-zone reduced-order dynamic models (for model-based control). A significant challenge with HCCI is the control of the combustion phasing, this is essential in order to control the load, to obtain low fuel consumption and emissions. For closed-loop control of the combustion phasing, feedback signals are necessary and in-cylinder pressure feedback is, perhaps, the most straightforward approach. In practice, the crank angle α of 50% burnt fuel (CA50 or α50 or θ50 ) has proved to be a reliable indicator of on-going combustion [6, 42]. In closed-loop control of an HCCI engine, several means to actuate the combustion phasing have been tested –e.g., dual fuels [42], variable valve actuation (VVA) [1, 8], variable compression ratio [17, 29], and thermal management [30, 39]. For control design purposes appropriate models and system variables useful for feedback control are needed. Previously, it was shown that physical modeling and system identification can be used to obtain low-complexity models of the HCCI dynamics [59, 7, 52]. For closed-loop HCCI engine operation, it was reported that the combustion phasing can be stabilized by means of a PID controller [42]; LQG control [59]; and MPC control [8]. A fast and robust control of α50 appears to be necessary in order to stabilize HCCI engine control. It is also desirable that the load, peak cylinder pressure, peak rate of cylinder pressure and emissions are controlled simultaneously. This is a multiinput multi-output (MIMO) control problem where the controller has to be able to handle constraints on several variables. In a comparison among several control methods, it will be demonstrated that Model Predictive Control (MPC) control could be used with favorable properties [4, 36]. All of the actuators suggested have control constraints and MPC has the benefit of explicitly taking the constraints into account. Whereas monitoring of α50 or other methods to sense on-going combustion for feedback control of an HCCI engine all rely on pressure sensors, these sensors may be expensive. One candidate to replace pressure sensors is the use of electronic conductive properties for the reaction zone [26]. This phenomenon is called ion current for which no expensive sensor is needed. Ion current has been successfully used in closed-loop control of SI engines [22]. The basic principle of ion current sensing is that a voltage is applied over an electrode gap inserted into the gas volume (combustion chamber) [26]. The common belief so far has been that ion current levels are not measurable for the highly diluted HCCI combustion. However, a recent study shows that it is not the dilution level in itself but the actual fuel/air equivalence ratio φ which is an important factor for the signal level [24, 59].

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In this chapter, we will report new modeling and experimental results on HCCI control, complementing our previously published results on control of a six-cylinder heavy-duty engine, evaluating a variety of control methods (MPC and PID) and actuators (VVA, dual fuel), and experimental results on HCCI control of a singlecylinder heavy-duty engine evaluating a variety of sensors (in-cylinder pressure, ion current) [7, 8, 9]. The purpose of this chapter is to provide an overview of state of the Art HCCI engine modeling with particular attention to control-oriented modeling. The structure of the chapter is the following: An overview of HCCI modeling is given with particular emphasis on modeling suitable for model-based control, followed by a model-based control description, discussion and conclusions.

6.2 HCCI Modeling There are two often used methods to obtain models of HCCI engine dynamics suitable for control; physical modeling [52] and modeling by means of system identification [7, 8]. Physical modeling based on conservation laws and chemical kinetics has attractive intuitive component-based features but suffers from complexity issues with adverse effects in application. Whereas system identification has proved to be a very effective modeling tool for prototyping, it may provide results hard to interpret from a physical point of of view. The purpose of modeling has an obvious influence on focus and the complexity of modeling [21, 34]. As modeling and simulation may easily become too detailed and computationally expensive to serve purposes of model-based control, low-complexity models and reduced-order models become relevant. A minimum requirement of physical modeling is explanation of the nature of the in-cylinder pressure traces where adiabatic compression combines with fuel-dependent autoignition [10, 25, 44]. In previous work, modeling choices involve aspects of chemical kinetics, cycle-to-cycle coupling, in-cylinder concentrations of reactants, wall temperature dynamics, pressure dynamics, and auto-ignition timing. Modeling details fall into categories of single-zone models, multi-zone models, multidimensional computational fluid dynamics (CFD) models, sometimes combined with exosystem simulation on the form of stochastic disturbances, load modeling, sensor modeling. Both physical aspects and operational aspects require attention. Shaver et al. singled out six distinct stages in modeling of HCCI engine operation—i.e., induction, compression, combustion, expansion, exhaust and residence in the exhaust manifold [46, 53, 54]. As for stable operation, combustion phasing control design requires appropriate models and system output variables usable for feedback control. Recently, mode-transition operation and control of Diesel-HCCI and SI-HCCI engines and other hybrid control aspects have received attention [55].

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6.2.1 Fuel Modeling The necessity of developing a practical iso-octane mechanism for HCCI engines was presented after various different experiments and currently available mechanisms for iso-octane oxidation being reviewed and the performance of these mechanisms applied to experiments relevant to HCCI engines being analyzed [49, 40]. A skeletal mechanism including 38 species and 69 reactions was developed, which could predict satisfactorily ignition timing, burn rate and the emissions of HC, CO and NOx for HCCI multi-dimensional modeling [40]. Comparisons with various experiment data including shock tube, rapid compression machine, jet stirred reactor and HCCI engine indicate good performance of this mechanism over wide ranges of temperature, pressure and equivalence ratio, especially at high pressure and lean equivalence ratio conditions. By applying the skeletal mechanism to a single-zone model of an HCCI engine, it was found that the results were substantially identical with those from the detailed mechanism developed by Curran et al. [20] but the computing time was reduced greatly [40]. A model for the auto-ignition of hydrocarbons applicable to 3D internal combustion engine calculations was proposed [18]. The limits of classical methods using an auto-ignition delay are investigated when cool flame phenomena are present. A method based on tabulated reaction rates was presented to capture the early heat release induced by low temperature combustion. Cool flame ignition delay when present and cool flame fuel consumption are also tabulated. The reaction rate, fuel consumption, and cool flame ignition delay tables were built a priori from complex chemistry calculations. The reaction rates, which directly depend on instantaneous changes of thermodynamic conditions, were then integrated during the 3D engine calculation. The model is first validated through comparisons with complex chemistry calculations in constant and variable volume configurations where good agreement was found. The model was applied both to a Diesel computation with spray injection and residual gases, and to a Diesel-HCCI configuration. Comparisons with experimental results showed that the auto-ignition essential features were well reproduced in these cases [18]. The combination of CFD computations with detailed chemistry leads to excessive computation times, and is not achievable with current computer capabilities. A reduced chemical model for n−heptane is described, in view of its implementation into a CFD simulation code [38]. Firstly, the reduction process to get to the 61-step mechanism is detailed and then the 26-step mechanism is described; this further reduction is carried out under various conditions that include a range of interest in engine applications. Validation work in reference to the original detailed mechanism and two reduced mechanisms was published in the literature, focusing on the prediction of ignition delay times under constant as well as variable volume conditions [38]. A good and accurate reproduction of both ignition delay times and heat release was reported to be reached with the 26-step model [38]. Despite the rapid combustion typically experienced in HCCI, components in fuel mixtures do not ignite in unison or burn equally. In experiments and modeling of blends of diethyl ether (DEE) and ethanol, the DEE led combustion and proceeded

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further toward completion, as indicated by 14 C isotope tracing [37]. A numerical model of HCCI combustion of DEE and ethanol mixtures supports the isotopic findings. Although both approaches lacked information on incompletely combusted intermediates plentiful in HCCI emissions, the numerical model and 14 C tracing data agreed within the limitations of the single-zone model. Despite the fact that DEE is more reactive than ethanol in HCCI engines, they were sufficiently similar and prevented incidence of a large elongation of energy release or significant reduction in inlet temperature required for light-off, both desired effects for the combustion event. This finding suggests that, in general, HCCI combustion of fuel blends may have preferential combustion of some of the blend components [37].

6.2.2 Auto-ignition Modeling Whereas HCCI engines have been shown to have higher thermal efficiencies and lower NOx and soot emissions than spark ignition engines, the HCCI engines experience very large heat release rates which can cause too rapid an increase in pressure. One method of reducing the maximum heat release rate is to introduce thermal inhomogeneities, thereby spreading the heat release over several crank angle degrees [56]. Direct numerical simulations (DNS) showed that both ignition fronts and deflagration-like fronts may be present in systems with such inhomogeneities [19]. Here, an enthalpy-based flamelet model was presented and applied to four cases of varying initial temperature variance. This model used a mean scalar dissipation rate to model mixing between regions of higher and lower enthalpies. The predicted heat release rates agree well with the heat release rates of the four DNS cases. The model was shown to be capable of capturing the combustion characteristics for the case in which combustion occurs primarily in the form of spontaneous ignition fronts, for the case dominated by deflagration-type burning, and for the mixed mode cases. The enthalpy-based flamelet model shows considerably improved agreement with the DNS results over the popular multi-zone model, particularly, where both deflagrative and spontaneous ignition are occurring, that is, where diffusive transport is important [19]. Another fuel model is the Shell model [28]. Further contributions on auto-ignition modeling can be found in [48].

6.2.3 Thermal Modeling and Auto-ignition HCCI combustion is often achieved without a completely homogeneous mixture. In order to derive a control-relevant model, however, we might firstly proceed by assuming that the mixture is homogeneous, thus allowing a single-zone cylinder model [5]. Such assumptions may be justified by laser-diagnostic measurements in our experimental set-up [47]. To reproduce the effects relevant for combustion phasing control it is required that the auto-ignition model captures the effects on ignition delay (induction time) of varying species concentrations, temperature trace, and fuel quality. Several alternative approaches are possible for modeling the instant of autoignition for fuels. High-complexity models, e.g., (Primary Reference Fuels (PRF),

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857 species, 3,606 reactions, CHEMKIN/LLNL) [57], have been used to model complete combustion. In addition to ignition prediction, such models are also aimed at describing intermediate species and end product composition. Reduced chemical kinetics models, e.g., (PRF fuels, 32 species, 55 reactions, CHEMKIN) [61], have also been proposed, where reactions with little influence on the combustion have been identified and removed. For simulation of multi-cycle scenarios it is necessary to keep the model complexity low in order to arrive at reasonable simulation times. An attractive and widespread alternative is to use the Shell model [28], which is a lumped chemical kinetics model using only five representative species in eight generic reactions. This model is aimed at prediction of auto-ignition rather than describing the complete combustion process. Compression ignition delay may also be described by empirical correlations, such as the knock integral condition ti dt t=0

τ

=1

(6.1)

where ti is the instant of ignition and τ is the estimated ignition time (ignition delay) at the instantaneous pressure and temperature conditions at time t, often described by Arrhenius type expressions [31, 50]. A drawback is that dependence on species concentrations is normally not regarded. An integral condition with concentration dependence was used in [51, 52] in a similar study for propane fuel, where also auto-ignition models based on very simple reaction mechanisms were evaluated. Alternatives to physical or physics-based models are to use system identification to obtain models or to use empirical look-up tables. The latter gives insufficient physical insight, and require substantial efforts to calibrate. In this work, the Arrhenius model was chosen to describe the process of auto-ignition. A static model is then used to describe the major part of the actual combustion and corresponding heat release. The result from the Shell model was compared to results from an integrated Arrhenius rate threshold model and the Planet mechanism model [2, 3]. To the purpose of detailed treatise, modeling of the cylinder, auto-ignition, integrated Arrhenius threshold, combustion, and heat transfer are now provided: Cylinder Gas Model: First Law of Thermodynamics The cylinder gas dynamics are described by conservation laws such as the the first law of thermodynamics

δ QHR = (1 +

cv cv )pdV + V dp + δ QHT R R

(6.2)

where p is the cylinder pressure, V the volume, R is the gas constant, and c p [kJ/mol·K], cv = c p − R the specific heat capacities. The time derivatives of QHR and QHT denote rates of heat released by the combustion process and heat flowing from the wall, respectively.

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Gas Properties The gas is described as a mixture of dry air and fuel, and the combustion products are nitrogen, carbondioxide and water. Specific heat for each species i is described by NASA polynomial approximations of JANAF data c p,i (T ) =

Ru Mi

5

∑ ai, j T j−3

(6.3)

j=1

where Mi is the molar mass of species i and T is the cylinder temperature [13, 27]. The mixture specific heat is then c p (T ) =

1 ni Mi c p,i (T ) n∑ i

(6.4)

where ni is the mole of species i and n is the molar substance amount contained in the cylinder. Shell Auto-ignition Model The Shell auto-ignition model for hydrocarbon fuels [28], Ca Hb , is based on a general eight-step chain-branching reaction scheme with lumped species: the hydrocar¯ intermediate species Q, and the chain branching agent B. bon fuel RH, radicals R, kq

RH + O2 −→ 2R¯

(initiation)

k

p R¯ −→ R¯ + products and heat (propagation cycle)

f k

1 p R¯ −→ R¯ + B (propagation forming B)

f k

2 p R¯ + Q −→ R¯ + B (propagation forming B)

f3 k p

R¯ −→ out (linear termination) f4 k p

R¯ −→ R¯ + Q (propagation forming Q) kt

2R¯ −→ out (quadratic termination) kb

B −→ 2R¯ (degenerate branching)

(6.5) (6.6) (6.7) (6.8) (6.9) (6.10) (6.11) (6.12)

Auto-ignition is described by integrating the time variations of species concentrations from the beginning of the compression stroke. ¯ d[R] ¯ 2 − f3 k p [R] ¯ = 2 kq [RH][O2 ] + kb[B] − kt [R] dt d[B] ¯ + f2 k p [Q][R] ¯ − kb[B] = f1 k p [R] dt

(6.13) (6.14)

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d[Q] ¯ − f2 k p [Q][R] ¯ = f4 k p [R] dt d[O2 ] ¯ = −gk p[R] dt

(6.15) (6.16)

¯ Q, and B are not considered in thermodynamic computations for the The species R, gas mixture. The stoichiometry is approximated by assuming a constant CO/CO2 ratio, ν , for the complete combustion process, with oxygen consumption g = 2[a(1 − ν ) + b/4]/b mole per cycle. The heat release from combustion is given by dQHR ¯ = k p qV [R] dt

(6.17)

where q is the exothermicity per cycle for the regarded fuel. The propagation rate coefficient is described as kp = (

1 1 1 + )−1 + k p,1 [O2 ] k p,2 k p,1 [RH]

(6.18)

To capture dependence of induction periods on fuel and air concentrations the terms f1 , f3 , and f4 are expressed as fi = fi◦ [O2 ]xi [RH]yi

(6.19)

Rate coefficients and rate parameters ki and fi◦ are then described by Arrhenius rate coefficients −Ei −Ei , fi◦ = Ai exp (6.20) ki = Ai exp Ru T Ru T We use the acronym FuelMEP to denote the mean effective pressure calculated from the quantity of fuel injected. Calibrated parameters for a number of fuels, including a set of Primary Reference Fuels (PRF), are found in the literature [28]. PRFx is a mixture of n-heptane and iso-octane, where the octane number x is defined as the volume percentage of iso-octane. Parameters for PRF90 were used in the simulations. Auto-ignition can be defined as the crank angle where a specific percentage of the total fuel energy has been released. Although somewhat arbitrary a choice, a standard threshold is θ01 , corresponding to 1% fuel burnt. Integrated Arrhenius Rate Threshold The Arrhenius form can be used to determine the rate coefficient describing a singlestep reaction between two molecules [63]. The single-step rate integral condition is based on the knock integral with θ

Kth =

θIVC

1/τ · d θ /w

[

mol ] dm3

1/τ = A exp(−Ea /(Ru T ))[Fuel]a [O2 ]b

(6.21) [

mol ] dm3 · s

(6.22)

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where θ is the crank angle and θIVC is the crank angle of the inlet valve closure. The integral condition describes a generalized reaction of fuel and oxygen and this is an extreme simplification of the large number of reactions that take place during combustion. The empirical parameters A [(mol/dm3 )1−a−b /s], Ea [J/kg], a, b, and Kth [mol/dm3] are determined from experiments. Values for n-heptane and iso-octane from [63] were used in the comparison below with A = 4.65 · 1011 [(mol/dm3 )1−a−b/s], Ea = 15.1 [J/kg], a = 0.25, b = 1.5, Kth = 1.6 ·105 [mol/dm3]. Auto-ignition was defined as the crank angle where the integral condition has reached the threshold Kth . Sensitivity analysis of integrated Arrhenius rate thresholding was made by Chiang and Stefanoupolou [15]. Combustion When auto-ignition is detected by the Shell model or the Integrated Arrhenius Rate Threshold, the completion of combustion is described by a Wiebe function [66]. θ − θ0 m+1 xb (θ ) = 1 − exp −a( (6.23) ) Δθ where xb denotes the mass fraction burnt, θ is the crank angle, θ0 start of combustion, Δ θ is the total duration, and a and m adjustable parameters that fix the shape of the curve. The heat release is computed from the rate of xb and the higher heating value of the fuel. Heat Transfer Heat is transfered by convection and radiation between in-cylinder gases and cylinder head, valves, cylinder walls, and piston during the engine cycle. In this case the radiation is neglected. This problem is very complex, but a standard solution is to use the Newton law for external heat transfer dQW = hc AW (T − TW ) dt

(6.24)

where QW is the heat transfer by conduction, AW the wall area, TW the wall temperature, and the heat-transfer coefficient hc given by the Nusselt-Reynolds relation by Woschni [67] hc = 3.26B−0.2 p0.8 T −0.55 (2.28S p)0.8 (6.25) where S p is the mean piston speed and B is the bore. A modified correlation for HCCI was presented in [12]; hnew (t) = αscaling L(t)−0.2 p(t)0.8 T (t)−0.73 vtuned (t)0.8

(6.26)

where L(t) is the characteristic length and vtuned (t) is given by vtuned (t) = C1 S¯ p +

C2Vd Tr (p − pmot) 6prVr

(6.27)

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In the model used in the experiments, the cylinder wall temperature was updated using a low-complexity model assuming the steady-state temperature distribution within the wall. A conductive flow through the wall and a convective flow between the gas charge and the wall were modeled [64]. Similar assumptions were made in [11]. When the engine is run with low levels of residuals, as was the case in the experiments, the cylinder wall temperature represents the main cycle-to-cycle dynamics of the engine. The residual coupling was modeled similar to [54], where the thermal capacity and the initial temperature of the charge is given by a weighted average of the properties of the inlet air and that of the residuals.

6.3 Experiments Detailed reviews of experimental set-up and conditions are given in [8, 9, 11]. A cycle-resolved model of HCCI presented in [65] was used to design model predictive controllers. The controlled output was the crank angle of 50 % burnt fuel (here denoted θ50 ). The control signals were the inlet air temperature and the crank angle of inlet valve closing. A fast thermal management system was used to obtain fast intake temperature actuation.

6.3.1 Model Predictive Control The Model Predictive Control (MPC) strategy employed in the experiments was centered around the minimization of the cost function Hp

Hu −1

i=1

i=0

J(k) = ∑ Y (i|k) + where

∑

U (i|k)

Y (i|k) = ||y(k ˆ + i|k) − r(k + i|k)||2Q, ˆ + i|k)||2R U (i|k) = ||Δ u(k

(6.28)

(6.29)

and y(k ˆ + i|k) is the predicted output at time k + i given a measurement at time k, Δ u(k ˆ + 1|k) is the predicted change in control signal, and r(k + i|k) is the reference value at time k + i. The parameters H p and Hu define the length of the prediction horizon and the control horizon. At each sample, the cost function in (6.28) is minimized by determining a sequence of changes to the control signal Δ u(k + i|k), i = 0 . . . Hu − 1, subject to the constraints ymin ≤ y(k) ≤ ymax

(6.30)

umin ≤ u(k) ≤ umax Δ umin ≤ Δ u(k) ≤ Δ umax

(6.31) (6.32)

for all k. The first step of the optimal sequence is then applied to the plant and the optimization is repeated in the next step yielding a new optimal sequence [36].

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Fig. 6.1 Results of consecutive set-point changes for θ50 , θIVC , TIn vs. cycle index (upper graphs) and response to disturbances θ50 , θIVC , TIn , IMEPn vs. cycle index (lower graphs) [64]

As witnessed by Fig 6.1, successful model-based control was accomplished both for setpoint tracking and disturbance rejection.

6.4 Conclusions In addition to aspects of modeling related to thermodynamics, chemical combustion kinetics, and engine operation, careful attention is required for control-oriented

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combustion modeling and the interactions among dynamics, control, thermodynamics and chemical combustion properties. Modeling of engine-load transients as well as thermal transients also belong to this important domain of modeling (Figure 6.1). Progress in this area is important and necessary for successful and robust control such as model-predictive control. Within the project a cycle-resolved, physicsbased, model of HCCI has been developed. The model includes a low-complexity model of the cylinder wall temperature dynamics in order to capture the relevant time-scales of transient HCCI when only small amounts of hot residuals are trapped in the cylinder. The temperature evolution of the gas charge is modeled as isentropic compression and expansion with three heat transfer events during each cycle. As compared to previous results, our model including wall heat dynamics and concentration dependence was successful as a basis for model-predictive control with setpoint tracking as well as disturbance rejection. Recently, research focused on design and evaluation of model predictive controllers based on linearizations of the model. The considered control signals were the inlet valve closing and the intake temperature. Simulations were used for the initial control design and the resulting controller was tested experimentally. The control performance was evaluated in terms of response time to set-point changes and the resulting output variance. It was found that a comparable decrease in the output variance in some operating points could be achieved either by introducing a disturbance model or by changing linearization. All tested set-point changes were accomplished within 20 engine cycles or less. Only minor changes to the intake temperature were required for moderate changes. The closed-loop system showed good robustness towards disturbances in engine speed, injected fuel energy, and the amount of recycled exhaust gases.

Acknowledgements This research was supported by the Competence Center of Combustion Processes (KCFP), Project Closed-Loop Combustion Control, Swedish Energy Administration (Ref. 22485-1). The authors would like to thank Johan Bengtsson, Daniel Blom, Kent Ekholm, Maria Karlsson, Petter Strandh for cooperation on HCCI modeling and control.

References ˚ [1] Agrell, F., Angstr¨ om, H.E., Eriksson, B., Wikander, J.: Transient control of HCCI through combined intake and exhaust valve actuation. SAE Technical Papers 200301-3172 (2003) [2] Ahmed, S., Tzeuch, T., Amn´eus, P., Blurock, E., Soyhan, H., Mauss, F.: PLANET D4 report, The European Community (2003) [3] Amn´eus, P.: Homogeneous Ignition—Chemical Kinetic Studies for IC-Engine Applications. PhD thesis, Div. Combustion Physics, Lund University, Lund, Sweden (2002)

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[4] Bemporad, A., Borelli, F., Morari, M.: Model predictive control based on linear programming-the explicit solution. IEEE Trans. Automatic Control 47, 1974–1985 (2002) [5] Bengtsson, J., G¨afvert, M., Strandh, P.: Modeling of HCCI engine combustion for control analysis. In: Proc. Conf. Decision and Control (CDC 2004), Bahamas (December 2004) [6] Bengtsson, J., Strandh, P., Johansson, R., Tunest˚al, P., Johansson, B.: Closed-loop combustion control of homogeneous charge compression ignition (HCCI) engines dynamics. Int. J. Adaptive Control and Signal Processing 18, 167–179 (2004) [7] Bengtsson, J., Strandh, P., Johansson, R., Tunest˚al, P., Johansson, B.: System identification of homogenous charge compression ignition (HCCI) engine dynamics. In: IFAC Symp. Advances in Automotive Control (AAC 2004), Salerno, Italy, April 19-23 (2004) [8] Bengtsson, J., Strandh, P., Johansson, R., Tunest˚al, P., Johansson, B.: Hybrid control of homogeneous charge compression ignition (HCCI) engine dynamics. Int. J. Control 79(5), 422–448 (2006) [9] Bengtsson, J., Strandh, P., Johansson, R., Tunest˚al, P., Johansson, B.: Hybrid modelling of homogeneous charge compression ignition (HCCI) engine dynamic—A survey. International Journal of Control 80(11), 1814–1848 (2007) [10] Bitar, E.Y., Schock, H.J., Oppenheim, A.K.: Model for control of combustion in a piston engine. SAE Technical Paper 2006-01-0401 (2006) [11] Blom, D., Karlsson, M., Ekholm, K., Tunest˚al, P., Johansson, R.: HCCI engine modeling and control using conservation principles. SAE World Congress, SAE Technical Paper 2008-01-0789, Detroit, MI (April 2008) [12] Chang, J., G¨uralp, O., Filipi, Z., Assanis, D., Kuo, T.W., Najt, P., Rask, R.: New heat transfer correlation for an HCCI engine derived from measurements of instantaneous surface heat flux. SAE Technical Papers (2004-01-2996) (2004) [13] Chase Jr., M.W., Davies, C.A., Davies Jr., J.R., Fulrip, D.J., McDonald, R.A., Syverud, A.N.: JANAF Thermochemical tables. J. Physical and Chemical Reference Data 14(Suppl. 1) (1985) [14] Chiang, C.J., Stefanopoulou, A.G., Jankovic, M.: Nonlinear observer-based control of load transitions in homogeneous charge compression ignition engines. IEEE Trans. Control Systems Technology 15(3), 438–448 (2007) [15] Chiang, C.J., Stefanopoulou, A.G.: Sensitivity analysis of combustion timing and duration of homogeneous charge compression ignition (HCCI) engines. In: Proc. 2006 Am. Control Conf., Minneapolis, MN, USA (June 2006) [16] Christensen, M., Einewall, P., Johansson, B.: Homogeneous charge compression ignition (HCCI) using isooctane, ethanol and natural gas—A comparison with sparkignition operation, SAE Technical Paper 972874 (1997) [17] Christensen, M., Hultqvist, A., Johansson, B.: Demonstrating the multi-fuel capability of a homogeneous charge compression ignition engine with variable compression ratio. SAE Technical Papers 1999-01-3679 (1999) [18] Colin, O., Pires da Cruz, A., Jay, S.: Detailed chemistry-based auto-ignition model including low temperature phenomena applied to 3-D engine calculations. Proc. Combustion Institute 30, 2649–2656 (2005) [19] Cook, D.J., Pitsch, H., Chen, J.H., Hawkes, E.R.: Flamelet-based modeling of autoignition with thermal inhomogeneities for application to HCCI engines. Proc. Combustion Institute 31, 2903–2911 (2007)

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[20] Curran, H.J., Gaffuri, P., Pitz, W.J., Westbrook, C.K.: A comprehensive modeling study of isooctane oxidation. Combustion and Flame 129, 253–280 (2002) [21] Egeland, O., Gravdahl, J.T.: Modeling and Simulation For Automatic Control. Marine Cybernetics, Trondheim, Norway (2002) [22] Eriksson, L., Nielsen, L., Glavenius, M.: Closed loop ignition control by ionization current interpretation. SAE Technical Papers 970854 (1997) [23] Fieweger, K., Blumenthal, R., Adomeit, G.: Self-ignition of s.i. engine model fuels: A shock tube investigation at high pressure. Combustion and Flame 109, 599–619 (1997) [24] Franke, A.: Characterization of an Electrical Sensor for Combustion Diagnostics. PhD thesis, ISRN LUTFD/TFCP–80–SE, Department of Physics, Lund University, Lund, Sweden (2002) [25] Gavillet, G.G., Maxson, J.A., Oppenheim, A.K.: Thermodynamic and thermochemical aspects of combustion in premixed charge engines revisited. SAE Technical Paper 930432, 20 (1993) [26] Gillbrand, P., Johansson, H., Nytomt, J.: Method and apparatus for detecting ion current in an internal combustion engine ignition system. Technical report, U.S. Patent No. 4,648,367 (1987) [27] Gordon, S., McBride, B.J.: Computer Program for Calculation of Complex Chemical Equilibrium Compositions and Applications: I. Analysis; II. Users Manual and Program Description. NASA Reference Publication 1311, October 1994 (June 1996) [28] Halstead, M.P., Kirsch, L.J., Quinn, C.P.: The auto-ignition of hydrocarbon fuels at high temperatures and pressures—Fitting of a mathematical model. Combustion and Flame 30, 45–60 (1977) [29] Haraldsson, G., Tunest˚al, P., Johansson, B., Hyvonen, J.: HCCI combustion phasing with closed-loop combustion control using variable compression ratio in a multicylinder engine. SAE Transactions 112(4), 1233–1245 (2003) (also JSAE 20030126) [30] Haraldsson, G., Tunest˚al, P., Johansson, B., Hyvonen, J.: HCCI closed-loop combustion control using fast thermal management. SAE Transactions 113(3), 599–610 (2004) [31] Heywood, J.B.: Internal Combustion Engine Fundamentals. McGraw-Hill, New York (1988) [32] Hultqvist, A., Christensen, M., Johansson, B., Franke, A., Richter, M., Ald´en, M.: A study of the homogeneous charge compression ignition combustion process by chemoluminescence imaging, SAE Technical Paper 1999-01-3680 (1999) [33] Ishibashi, Y., Asai, M.: Improving the exhaust emissions of two-stroke engines by applying the activated radical combustion. SAE Technical Papers 960742 (1996) [34] Johansson, R.: System Modeling and Identification. Prentice Hall, Englewood Cliffs (1993) [35] Johansson, R., Rantzer, A.: Nonlinear and Hybrid Systems in Automotive Control. Springer, London (2003) [36] Maciejowski, J.: Predictive Control with Constraints. Prentice Hall, Pearson Education, England (2002) [37] Mack, J.H., Flowers, D.L., Buchholz, B.A., Dibble, R.W.: Investigation of HCCI combustion of diethyl ether and ethanol mixtures using carbon 14 tracing and numerical simulations. Proc. Combustion Institute 30, 2693–2700 (2005) [38] Maroteaux, F., Noel, L.: Development of a reduced n−heptane oxidation mechanism for HCCI combustion modeling. Combustion and Flame 146, 246–267 (2006) [39] Martinez-Frias, J., Aceves, S.M., Flowers, D.L., Smith, J.R., Dibble, R.W.: HCCI engine control by thermal management. SAE Technical Papers 2000-01-2869 (2000)

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[40] Jia, M., Xie, M.: A chemical kinetics model of iso-octane oxidation for HCCI engines. Fuel, 2593–2604 (2006) [41] Olsson, J.O., Tunest˚al, P., Haraldsson, G., Johansson, B.: A turbo charged dual fuel HCCI engine. SAE Technical Papers 2001-01-1896 (2001) [42] Olsson, J.O., Tunest˚al, P., Johansson, B.: Closed-loop control of an HCCI engine. SAE Technical Papers 2001-01-1031 (2001) [43] Onishi, S., Hong Jo, S., Shoda, K., Do Jo, P., Kato, S.: Active thermo-atmosphere combustion (ATAC)—A new combustion process for internal combustion engines. SAE Technical Papers 790501 (1979) [44] Oppenheim, A.K., Kuhl, A.L., Packard, A.K., Hedrick, J.K., Johnson, W.P.: Model and control of heat release in engines. SAE Technical Paper SAE 960601, Engine Combustion and Flow Diagnostics, SAE SP-1157, pp. 15–23 (1996) [45] Rausen, D.J., Stefanopoulou, A.G., Kang, J.M., Eng, J.A., Kuo, T.W.: A mean-value model for control of homogeneous charge compression ignition (HCCI) engines. In: Proc. 2004 Am. Control Conf. (ACC 2004), Boston, Massachusetts, USA (July 2004) [46] Ravi, N., Roelle, M.J., Jungkunz, A.F., Gerdes, J.C.: A physically based two-state model for controlling exhaust recompression HCCI in gasoline engines. In: Proc. IMECE 2006, Chicago, IL, USA (November 2006) [47] Richter, M., Franke, A., Engstr¨om, J., Hultqvist, A., Johansson, B., Ald´en, M.: The influence of charge inhomogeneity on the HCCI combustion process. SAE Technical Papers 2000-01-2868 (2000) [48] Sankarana, R., Hong, G.I., Hawkes, E.R., Chen, J.H.: The effects of non-uniform temperature distribution on the ignition of a lean homogeneous hydrogen-air mixture. Proc. Combustion Institute 30, 875–882 (2005) [49] Semenov, N.N.: Chain Reactions. Goskhimtekhizdat, Leningrad (1934); transl.: Chemical Kinetics and Chain Reactions. Oxford University Press, Oxford (1935) [50] Shahbakhti, M., Koch, R.: Control oriented modeling of combustion phasing for an HCCI engine. In: Proc. 2007 Am. Control Conf., New York City, NY, USA (July 2007) [51] Shaver, G.M., Gerdes, J.C., Jain, P., Caton, P.A., Edwards, C.F.: Modeling for control of HCCI engines. In: Proc. American Control Conference, Denver, CO, pp. 749–754 (2003) [52] Shaver, G.M., Gerdes, J.C., Roelle, M.: Physics-based closed-loop control of phasing, peak pressure and work output in HCCI engines utilizing variable valve actuation. In: Proc. American Control Conference, Boston, MA, pp. 150–155 (2004) [53] Shaver, G.M., Roelle, M., Gerdes, J.C.: Decoupled control of combustion timing and work output residual-affected HCCI engines. In: Proc. 2005 American Control Conf., Portland, OR, June 8-10, pp. 3871–3876 (2005) [54] Shaver, G.M., Roelle, M., Gerdes, J.C.: Two-input two-output control model of HCCI engines. In: Proc. 2006 American Control Conf., Minneapolis, MN, June 14-16, pp. 472–477 (2006) [55] Shaver, G.M., Roelle, M.J., Gerdes, J.C.: Modeling cycle-to-cycle dynamics and mode transition in HCCI engines with variable valve actuation. Control Engineering Practice 14, 213–222 (2006) [56] Sj¨oberg, M., Dec, J.E.: Comparing enhanced natural thermal stratification against retarded combustion phasing for smoothing of HCCI heat-release rates. SAE Technical Papers 2004-01-2994 (2004) [57] Sj¨oberg, M., Dec, J.E.: Combined effects of fuel-type and engine speed on intake temperature requirements and completeness of bulge-gas reactions for HCCI combustion. SAE Technical Papers 2003-01-3173 (2003)

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[58] Stockinger, M., Sch¨apert¨ons, H., Kuhlmann, P.: Versuche an einem gemischansaugenden Verbrennungsmotor mit Selbstz¨undung. MTZ 53, 80–85 (1992) [59] Strandh, P., Bengtsson, J., Christensen, M., Johansson, R., Vressner, A., Tunest˚al, P., Johansson, B.: Ion current sensing for HCCI combustion feedback. SAE Technical Papers 2003-01-3216 (2003) [60] Suzuki, H., Koike, N., Ishii, H., Odaka, M.: Exhaust purification of Diesel engines by homogeneous charge with compression ignition. SAE Technical Papers 970315 (1997) [61] Tanaka, S., Ayala, F., Keck, J.C.: A reduced chemical kinetic model for HCCI combustion of primary reference fuels in a rapid compression machine. Combustion and Flame 133, 467–481 (2003) [62] Thring, R.H.: Homogeneous-charge compression-ignition (HCCI) engines. SAE Technical Papers 892068 (1989) [63] Turns, S.R.: An Introduction to Combustion—Concepts and Applications, 2nd edn. McGraw-Hill, Singapore (2000) [64] Widd, A., Ekholm, K., Tunest˚al, P., Johansson, R.: Experimental evaluation of predictive combustion phasing control in an HCCI engine using fast thermal management and VVA. In: Proc. 18th IEEE Conf. Control Applications, St. Petersburg, Russia (July 2009) [65] Widd, A., Tunest˚al, P., Johansson, R.: Physical modeling and control of homogeneous charge compression ignition (HCCI) engines. In: Proc. 47th IEEE Conf. Decision and Control, Cancun, Mexico, December 2008, pp. 5615–5620 (2008) [66] Wiebe, I.I.: Brennverlauf und Kreisprozessrechnung. VEB Verlag Technik, Berlin (1970) [67] Woschni, G.: A universally applicable equation for instantaneous heat transfer coefficient in the internal combustion engine. SAE Technical Papers 670931 (1967)

Chapter 7

An Overview of Nonlinear Model Predictive Control Lalo Magni and Riccardo Scattolini

Abstract. This chapter reviews some of the main approaches, results and open problems in Nonlinear Model Predictive Control. The style of the presentation is maintained at a high level, reducing to the minimum the mathematical details.

7.1 Introduction This chapter provides a critical overview of Nonlinear Model Predictive Control (NMPC), focusing both on theoretical results and on open issues still to be solved for a widespread use of NMPC in real-world applications. Section 2 introduces a general NMPC formulation, i.e. the system to be controlled, the constraints on the input and the state variables, the optimization problem, the implementation of the state feedback control law through the Receding Horizon principle. Then, the attention is placed on the main stability results nowadays available. It is also discussed how to modify the basic formulation to guarantee the robustness properties of the control system with respect to exogenous or state dependent disturbances. In Section 3 the output feedback and tracking problems are discussed and the few theoretical results nowadays available are reviewed. Finally, in Section 4 two main practical issues are addressed, namely the computational burden associated to the optimization problem to be solved on line and the need to dispose of a reliable nonlinear plant model. Two possible solutions of these problems are discussed. The first one consists of resorting Linear Parameter Varying models, eventually estimated from input-output data. The second is based on the use of specific plant representations, such as NARX, Volterra, Wiener and Hammerstein models. Lalo Magni Universit´a di Pavia, Italy e-mail: [email protected] Riccardo Scattolini Politecnico di Milano, Italy e-mail: [email protected] L. del Re et al. (Eds.): Automotive Model Predictive Control, LNCIS 402, pp. 107–117. c Springer-Verlag Berlin Heidelberg 2010 springerlink.com

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The style of the presentation is maintained at a high, descriptive, level by reducing to the minimum the mathematical details, focusing on problems and possible solutions, and providing a non-exhaustive list of references to relevant papers.

7.2 Problem Formulation and State-feedback NMPC Control Law The system under control is assumed to be described by the discrete-time model x(k + 1) = f (x(k), u(k)), k ≥ t, x(t) = x

(7.1)

where k is the discrete time index, x(k) ∈ Rn and u(k) ∈ Rm are the state and input variables, respectively, and f (·, ·), with f (0, 0) = 0, is a continuous function of its arguments. The state and control variables are required to fulfill the following constraints x ∈ X, u ∈ U (7.2) where X and U are compact subsets of Rn and Rm , respectively, both containing the origin as an interior point. For system (7.1) a stabilizing “auxiliary” control law u = κ f (x)

(7.3)

is assumed to be known. Let X f ⊆ X be a positively invariant set for the closedloop system (7.1), (7.3) which is a domain of attraction of the origin and such that x ∈ X f implies κ f (x(k)) ∈ U, k ≥ t. In order to enhance the performance provided by the auxiliary control law and to enlarge its positive invariant set X f , an NMPC algorithm can be derived as follows. First, let ut1 ,t2 := [u(t1 )u(t1 + 1) . . . u(t2 )], t2 ≥ t1 , then, denoting by N a positive integer representing the prediction horizon, l(·, ·) a stage cost and V f (·) a terminal penalty, define the following Finite Horizon Optimal Control Problem (FHOCP) min J(x, ut,t+N−1 , N) =

ut,t+N−1

t+N−1

∑

l(x(k), u(k)) + V f (x(t + N))

(7.4)

k=t

subject to (i) the state dynamics (7.1) with x(t) = x, (ii) the constraints (7.2), k ∈ [t,t + N − 1], (iii) the terminal state constraint x(t + N) ∈ X f . o At every time instant, the optimal control sequence ut,t+N−1 is computed by solving the FHOCP with x = x(t); then, according to the Receding Horizon approach it o (x) where uo (x) is the first column of uo is set κ MPC (x) = ut,t t,t t,t+N−1 . This implicitly defines the NMPC control law u = κ MPC (x)

(7.5)

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7.2.1 Feasibility and Stability in Nominal Conditions The first problem to be considered in the design of an MPC algorithm concerns feasibility, i.e. the property guaranteeing that, given a feasible solution of FHOCP at a generic time instant k, a feasible solution exists also at the next time instant k + 1. In general, the state and input constraints (7.2) together with the terminal state constraint x(t + N) ∈ X f are such that a feasible solution of FHOCP exists only in a set X MPC of the state space. Once feasibility is achieved, the stability of the origin of the closed-loop system (7.1), (7.5) must also be guaranteed. In order to deal with feasibility and stability, many different choices of the design parameters, i.e. the auxiliary control law (7.3), the stage cost l(x, u), the terminal set X f and the terminal cost V f have been proposed in the literature, see e.g. [7, 13, 18, 21, 20, 32, 37, 45, 41] and the survey paper [42]. In view of these results, the nominal stability issue can be considered solved, at least in nominal conditions, by taking the optimal o cost V (x, N) = J(x, ut,t+N−1 , N) as a Lyapunov function for the closed-loop system. Notably, V (x, N) can even be not continuous, in view of the possible discontinuity of the NMPC control law (7.5) caused by the constraints (7.2) and of the terminal state constraint. However, the stability of the origin can still be proven by means of the results reported in [26].

7.2.2 The Robustness Problem Stability in nominal conditions can not be sufficient to guarantee a safe use of NMPC algorithms in most practical cases. For example, model uncertainties, parameter variations, external disturbances, or the use of dynamic observers to reconstruct the value of unmeasurable states can strongly influence the closed-loop stability and performance properties. For this reason, in recent years many efforts have been devoted to the development of robust NMPC methods. In this framework, the system under control is usually assumed to be described by x(k + 1) = f (x(k), u(k), w(k)), k ≥ t, x(t) = x

(7.6)

where the disturbance w ∈ Rq can model a wide number of the uncertainties above described; as such it can be assumed to be either a state and control dependent term, i.e. w(k) = fw (x(k), u(k)) (7.7) fw (·, ·) being a suitable function, or an external bounded signal, i.e. w∈W

(7.8)

where W ⊆ Rq is a known compact set containing the origin. The presence of the disturbance w strongly impacts on feasibility. In fact, even though at a generic time instant k the optimization problem is feasible, i.e. x(k) ∈ X MPC , the effect of the disturbance could bring the state outside the feasibility region in the next time instants. To deal with this problem, in the design of robust NMPC

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algorithms a common approach consists of considering the a-priori worst possible effect of the future disturbances and guaranteeing that, event in this case, feasibility is not lost. In other words, a min-max optimization problem must be stated and solved, where the maximization is associated to the effect of the disturbance over the considered prediction horizon, while the minimization of the selected cost function must be performed with respect to the future control actions given by the control law u(k + i) = uop(k + i) + κi(x(k + i)), i = 0, ..., N − 1

(7.9)

where uop(k + i), i = 1, ..., N are open-loop terms to be computed through the optimization problem, while the functions κi (x) are closed-loop terms which can be either time-invariant and selected a-priori or whose parameters can be optimized on-line. The algorithms proposed in the literature for linear and nonlinear systems can be roughly classified as follows. • Methods where the maximization problem is solved off-line and the design parameters are suitably modified to guarantee feasibility. In these approaches, the basic idea is to include in the problem formulation some additional constraints on the states x!(k + i), i = 1, ..., N predicted over the considered horizon. Specifically, ! + i) ⊂ X chosen so that, the predicted state x!(k + i) is forced to belong to a set X(k for any feasible disturbance sequence w(k + j), j = 0, ..., i − 1, the real state still ! + i), i = 1, ..., N forms a “tube” where the belongs to X . The sequence of sets X(k predicted state is forced to remain. In addition, the auxiliary control law is usually chosen so that the terminal set X f is a robustly positive invariant set for the corresponding closed-loop system. Once the “tube” has been computed and the additional constraints on the future states have been included into the problem formulation, minimization of the selected cost function can be performed online by resorting to the Receding Horizon principle. In the context of MPC for nonlinear systems, [29] describes an algorithm where only the open-loop terms uop(k + i) (7.9) are considered, while fixed closed-loop functions κi (x(k + i)) are used in [51]. • Methods where the whole min-max problem is solved on-line. In this case, the control law (7.9) is usually made only by parametrized state-feedback control laws κi (x). The computational burden turns out to be high, but less restrictive constraints must be a-priori imposed on the state evolution over the prediction horizon. An example of application of this approach is described in [35]. Once the feasibility problem has been solved, the stability issue must be considered. In case of persistent disturbances, it is not possible to require the asymptotic stability of the origin, but only “practical stability”, i.e. convergence of the state trajectory to a robust positively invariant set containing the origin. The size of this set obviously depends on the worst feasible disturbance. Asymptotic stability of the origin can however be obtained for state dependent disturbances, provided that a suitable H∞ -type cost function is used in the optimization problem, see e.g. [35]. Concerning stability, in recent years it has been shown that the concept of Input to State Stability (ISS), see e.g. [24, 53], is the most appropriate tool for the analysis and synthesis of robust NMPC algorithms, see [36, 50].

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Finally it is worth noting that considering the effect of the most critical disturbance leads to solutions which are often very conservative. In many practical cases, one could accept a small probability of unfulfillment of the control design requirements in exchange for better performances on the average. In view of this consideration, a recent trend consists of deriving NMPC algorithms where the state and input constraints are required to be fulfilled in a probabilistic sense, see [47, 55, 49].

7.3 Output Feedback and Tracking In this section, the output feedback and tracking problems and the corresponding theoretical results nowadays available are reviewed.

7.3.1 Output Feedback In most of the engineering problems, the state variables can not be measured, while only output variables y(k) = g(x(k), u(k)) (7.10) with y ∈ R p , are available for control. In these cases, it is mandatory to resort to a state observer which computes a state estimate on the basis of the measures of the inputs u and outputs y of the system. For linear systems, it is well know that the joint use of a linear stabilizing state feedback control law and a stable state observer guarantees the overall stability of the closed-loop system in view of the so-called “separation principle”, see e.g. [1]. Unfortunately, this argument does not hold for nonlinear systems, and a local analysis based on linearization, see e.g. [25], can be made only in a neighborhood of the origin when the system can be well represented by its linearized model. For this reason, the output feedback problem for nonlinear systems is much more difficult than in the linear case. Even more significantly, the separation principle can not be automatically applied to feedback schemes with an MPC controller even when the system under control is linear. In fact, as already noted, the MPC control law is in general nonlinear due to the state and input constraints. The main approaches proposed so far to the output feedback control problem with NMPC can be classified as follows. • A first approach consists of considering the state estimation error as a perturbation term acting on the system controlled by a stabilizing state feedback control law. Provided that this error is asymptotically vanishing and the state feedback control law enjoys some strong stability properties, such as the exponential stability, the stability of the overall closed-loop system can be stated in view of the inherent robustness guaranteed by the control law. Following this approach, a sort of separation principle for discrete-time nonlinear systems has been presented in [34] where regional results are derived, i.e. stability is guaranteed in a region containing the origin. This allows one to combine well known nonlinear observers, such as the popular Extended Kalman Filter, with a stabilizing

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state feedback control law, for example designed with NMPC. In the context of NMPC, a similar approach has been also adopted in [52]. • In other methods, the design of the control law is performed by explicitly considering the problem of state reconstruction in the design of the feedback regulator. By considering continuous-time systems, in [44] it has been proposed to synthesize a robust stabilizing state-feedback control law based on the knowledge of a bound on the observer error. In [17] the approach taken is to to synthesize a high gain observer (see [25]) “sufficiently fast” with respect to the dynamics of the state feedback controlled system. In so doing, semiglobal practical stability can be achieved, i.e. for any initial condition in any compact set contained in X MPC the system state asymptotically reaches any small set containing the origin.

7.3.2 Tracking In the tracking problem, the output y is required to follow a given reference signal yo . When the system under control is linear and yo is the output of an unforced, unstable exosystem (for example it is a step, a ramp, a diverging exponential), according to classical results in linear control theory, asymptotic zero error regulation is achieved by resorting to the internal model principle, which basically states that the dynamics of the exosystem must be included into the regulator, see [11]. In the case of neutrally stable exosystems (linear systems with eigenvalues on the stability boundary) and nonlinear plants, the extension of the internal model principle has been described in [23], where it has been shown that the proper structure of the controller is made by the parallel connection of a copy of the exosystem and a stabilizing regulator. The design of this stabilizing regulator with NMPC has been discussed in [33]. A simpler, yet significant problem concerns the tracking of references which can be assumed constant and equal to yo , at least beyond a given time window of length nr steps in the future. In this case, it is possible to compute an equilibrium pair (x, u) for system (7.1) such that x = f (x, u)

(7.11)

yo = g(x, u)

(7.12)

and define δ x(k) = x(k) − x, δ u(k) = u(k) − u. Then, the system under control can be written as

δ x(k + 1) = f!(δ x(k), δ u(k)), k ≥ t, δ x(t) = δ"x

(7.13)

and for this new system the NMPC problem can be reformulated (with an obvious meaning of symbols) as a regulation problem by choosing the performance index to be minimized as J(δ"x, δ ut,t+N−1 , N) =

t+N−1

∑

k=t

l(δ x(k), δ u(k)) + V f (δ x(t + N))

(7.14)

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In many practical applications, a reasonable assumption is that the reference signal is piecewise constant, that is its value changes from time to time. In these cases, the above procedure suffers from three main drawbacks, which must be considered in the development of an efficient NMPC algorithm: When yo changes, one has to recompute the corresponding equilibrium and to reformulate the optimization problem. An algorithm which avoids to recompute the pair (x, u) through the use of a dynamic NMPC control law also when the system state is available has been proposed in [38]. An alternative solution has been described in [16] where a pseudolinearization method is adopted to derive a model predictive control strategy. In so doing, an invertible state variable change and a state feedback law are introduced such that the transformed and controlled closed-loop system has the same (constant) linearization independent of the constant reference signal. However, the computation of the pseudolinearization transformation and of the feedback law may be rather cumbersome. The feasibility problem must be considered with care. In fact, when the reference changes it is not possible to guarantee a-priori that a feasible input sequence exists even if the target equilibrium (x, u) is inside X × U. A possible solution is to move from the current equilibrium to the target one by passing through a number of intermediate equilibria chosen to satisfy the feasibility requirement. In a different context, a similar approach has been considered in [28] to design gain scheduling regulators. In the case of a plant-model mismatch, or when unknown disturbances affect the system, a steady-state error can occur. To guarantee an asymptotic null error, two possible approaches can be followed. The first one consists of estimating an “equivalent disturbance term” responsible of this error and compensate it in the control law, see e.g. [48] where the approach is discussed in detail for linear systems. The second solution is based on the insertion of an integral action in the regulator (recall the internal model principle). In many popular linear MPC methods, such as GPC [9] or DMC [10] the integrator is placed in front of the process, so that the control move computed by the algorithm is the control variation Δ u(k) = u(k) − u(k − 1). However, for systems described in state space form, this approach requires the use of a state observer even when the state itself is measurable, otherwise any model uncertainty could lead to closed loop instability, see [31]. For this reason, it appears preferable to place the integrators on the error components, as suggested in the NMPC algorithms described in [33, 39]. In this way, it is also possible to easily consider nonsquare systems with more inputs than outputs and to include the minimum number of integrators.

7.4 Implementation Problems and Alternative Approaches In addition to the points raised in the previous sections, a number of further problems must be solved for an efficient implementation of NMPC. Among them, it is possible to recall:

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• the need to compute the solution of a difficult (nonconvex) optimization problem within the prescribed sampling time. This can be done either by resorting to an efficient optimization procedure, see e.g. [3, 6, 14], or by reformulating the problem in a different and simpler context; and • the availability of a reliable state space model of the system under control. Notably, the development of a physical plant model-based on first principles is often a prohibitive task and requires the knowledge of many uncertain parameters. This is one of the motivations of the success of many popular MPC algorithms (such as GPC or DMC) based on linear models obtained from simple experiments or through black-box identification procedures. In order to partially solve these limitations, two main approaches have been followed. The first is based on the use of specific classes of linear models to simplify the underlying optimization problem. The second consists of resorting to nonlinear models with a prescribed structure and whose parameters can again be estimated from process data. In the first approach, the plant is described by a Linear Parameter Varying (LPV) model suitable for the definition of computationally tractable MPC algorithms. In LPV systems, the matrices defining the model can be made to depend, through suitable continuous or discontinuous functions, either on an external signal or on the current state value. In this second case, the models are also often called PieceWise Affine (PWA), and a number estimation techniques are available, see e.g. [15]. With reference to LPV models, in [2, 30] it is proposed to define a family of linear models, each one describing the plant local behavior at a given operating regime. The global LPV model is then made by the collection of these local models and is used to derive a scheduled MPC algorithm. Gain scheduling MPC regulators have also been developed in [54] and [8], where the LPV model is obtained by means of an embedding procedure. Concerning PWA models, stabilizing MPC algorithms have recently been presented in e.g. [27, 43]. As for the second approach, in [12] a NMPC algorithm has been developed for Nonlinear AutoRegressive eXogenous (NARX) models. Notably, NARX models can be estimated from plant data and can be reformulated in terms of a nonminimal state space description where the state is made by past input and output values, so that an observer is not required. Hammerstein and Wiener models. i.e. linear models with a static nonlinearity at the input or at the output, have been used to obtain NMPC algorithms, see e.g. [19, 22] (Hammerstein models), [4], [46] (Wiener models), [5] (Wiener and Hammerstein). Also for these systems many identification procedures are available. Moreover, the static nonlinearities can be compensated for through inversion or by transforming them in polytopic descriptions. Finally, the most natural extension of linear impulse response models, that is the so-called Volterra models have been used for NMPC in e.g. [40]. Acknowledgments. The research of R. Scattolini has received funding from European Community through FP7/2007-2013 under grant agreement n. 223854 (“Hierarchical and Distributed Model Predictive Control of Large Scale Systems”, HD-MPC project).

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References [1] Anderson, B.D.O., Moore, J.B.: Optimal Control: Linear Quadratic Methods. PrenticeHall, Englewood Cliffs (1990) [2] Arkun, Y., Banerjee, A., Lakshmanan, N.M.: Self scheduling MPC using LPV models. In: Berber, R., Kravaris, C. (eds.) Nonlinear Model Based Control. NATO ASI Series. Kluwer Academic Publishers, Dordrecht (1998) [3] Biegler, L.T.: Efficient solution of dynamic optimization and NMPC problems. In: Allg¨ower, F., Zheng, A. (eds.) Nonlinear Predictive Control, Progress in Systems Theory Series. Birkh¨auser, Basel (2000) [4] Bloemen, H.H.J., Chou, C.T., van den Boom, T.J.J., Verdult, V., Verhaegen, M., Backx, T.C.: Wiener model identification and predictive control for dual composition control of a distillation column. Journal of Process Control 11, 601–620 (2001) [5] Bloemen, H.H.J., van den Boom, T.J.J., Verbruggen, H.B.: Model-based predictive control for Hammerstein-Wiener systems. International Journal of Control 74, 482–495 (2001) [6] Bock, H.G., Diehl, M., Leineweber, D.B., Schl¨oder, J.P.: A direct multiple shooting method for real-time optimization of nonlinear DAE processes. In: Allg¨ower, F., Zheng, A. (eds.) Nonlinear Predictive Control, Progress in Systems Theory Series. Birkh¨auser, Basel (2000) [7] Chen, H., Allg¨ower, F.: A quasi-infinite horizon nonlinear model predictive control scheme with guaranteed stability. Automatica 34, 1205–1217 (1998) [8] Chisci, L., Falugi, P., Zappa, G.: Gain-scheduling MPC of nonlinear systems. Int. J. of Robust and Nonlinear Control 13, 295–308 (2003) [9] Clarke, D.W., Mothadi, C., Tuffs, P.S.: Generalized predictive control- part I and II. Automatica 23, 137–160 (1987) [10] Cutler, C.R., Ramaker, B.C.: Dynamic matrix control - a computer control algorithm. In: Proceedings Joint Automatic Control Conference, San Francisco, CA (1980) [11] Davison, E.J.: The robust control of a servomechanism problem for linear timeinvariant multivariable systems. IEEE Trans. on Automatic Control 21, 25–34 (1976) [12] De Nicolao, G., Magni, L., Scattolini, R.: Stabilizing predictive control of nonlinear ARX models. Automatica 33, 1691–1697 (1997) [13] De Nicolao, G., Magni, L., Scattolini, R.: Stabilizing receding-horizon control of nonlinear time-varying systems. IEEE Trans. on Automatic Control AC-43, 1030–1036 (1998) [14] Diehl, M., Bock, H., Schl¨oder, J., Findeisen, R., Nagy, Y., Allg¨ower, F.: Real-time optimization and nonlinear model predictive control of processes governed by differentialalgebraic equations. Journal of Process Control 12, 577–585 (2002) [15] Ferrari-Trecate, G., Musselli, M., Liberati, D., Morari, M.: A clustering technique for the identification of piecewise affine systems. Automatica 39, 205–217 (2003) [16] Findeisen, R., Chen, H., Allg¨ower, F.: Nonlinear predictive control for setpoint familes. In: Proc. Amer. Contr. Conf., pp. 260–264 (2000) [17] Findeisen, R., Imsland, L., Allg¨ower, F., Foss, B.A.: Output feedback stabilization of constrained systems with nonlinear predictive control. International Journal of Robust and Nonlinear Control 13, 211–228 (2003) [18] Fontes, F.A.C.C.: A general framework to design stabilizing nonlinear model predictive controllers. Systems & Control Letters 42, 127–143 (2001) [19] Fruzzetti, K.P., Palazoglu, A., McDonald, K.A.: Nonlinear model predictive control using Hammerstein models. Journal of Process Control 1, 31–41 (1999)

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[20] Grimm, G., Messina, M.J., Tuna, S.E., Teel, A.R.: Model predictive control: for want of a local control Lyapunov function, all is not lost. IEEE Transactions on Automatic Control 50, 546–558 (2005) [21] Gyurkovics, E., Elaiw, A.M.: Stabilization of sampled-data nonlinear systems by receding horizon contyrol via discrete-time approximations. Automatica 40, 2017–2028 (2004) [22] Harnischmacher, G., Marquardt, W.: Nonlinear model predictive control of multivariable processes using block-structured models. Control Engineering Practice 15, 1238– 1256 (2007) [23] Isidori, A.: Nonlinear Control Systems, 3rd edn. Springer, London (1995) [24] Jiang, Z.-P., Wang, Y.: Input-to-state stability for discrete-time nonlinear systems. Automatica 37, 857–869 (2001) [25] Khalil, H.K.: Nonlinear systems. Prentice Hall, Englewood Cliffs (2002) [26] Lazar, M.: Model predictive control of hybrid systems: Stability and robustness. PhD thesis, Eindhoven University of Technology, The Netherlands (2006) [27] Lazar, M., Heemels, W.P.M.H., Weiland, S., Bemporad, A.: Stabilizing model predictive control of hybrid systems. IEEE Trans. on Automatic Control 31, 1813–1818 (2006) [28] Leonessa, A., Haddad, W.H., Chellaboina, V.: Nonlinear system stabilization via hierarchical switching control. IEEE Trans. on Automatic Control 46, 17–28 (2001) [29] Limon, D., Alamo, T., Camacho, E.F.: Input-to-state stable MPC for constrained discrete-time nonlinear systems with bounded additive uncertainties. In: Proc. of the 41st IEEE Conference on Decision and Control, Las Vegas, NV, USA, pp. 4619–4624 (2002) [30] Lu, Y., Arkun, Y.: A scheduling quasi min-max model predictive control algorithm for nonlinear systems. Journal of Process Control 12, 589–604 (2002) [31] Magni, L.: On robust tracking with nonlinear model predictive control. International Journal of Control 75, 399–407 (2002) [32] Magni, L., De Nicolao, G., Magnani, L., Scattolini, R.: A stabilizing model-based predictive control for nonlinear systems. Automatica 37, 1351–1362 (2001) [33] Magni, L., De Nicolao, G., Scattolini, R.: Output feedback and tracking of nonlinear systems with model predictive control. Automatica 37, 1601–1607 (2001) [34] Magni, L., De Nicolao, G., Scattolini, R.: On the stabilization of nonlinear discretetime systems with output feedback. International Journal of Robust and Nonlinear Control 14, 1379–1391 (2004) [35] Magni, L., De Nicolao, G., Scattolini, R., Allgower, F.: Robust model predictive control of nonlinear discrete-time systems. International Journal of Robust and Nonlinear control 13(3-4), 229–246 (2003) [36] Magni, L., Raimondo, D.M., Scattolini, R.: Regional input-to-state stability for nonlinear model predictive control. IEEE Trans. on Automatic Control 51, 1548–1553 (2006) [37] Magni, L., Scattolini, R.: Model predictive control of continuous-time nonlinear systems with piecewise constant control. IEEE Trans. on Automatic Control 49, 900–906 (2004) [38] Magni, L., Scattolini, R.: On the solution of the tracking problem for non-linear systems with MPC. International Journal of Systems Science 36, 477–484 (2005) [39] Magni, L., Scattolini, R.: Tracking of non-square nonlinear continuous time systems with piecewise constant model predictive control. Journal of Process Control 17, 631– 640 (2007)

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[40] Maner, B.R., Doyle III, F.J., Ogunnaike, B.A., Pearson, R.K.: Nonlinear model predictive control of a simulated multivariable polymerization reactor using second-order volterra models. Automatica 32, 1285–1301 (1996) [41] Mayne, D.Q., Michalska, H.: Receding horizon control of nonlinear systems. IEEE Trans. on Automatic Control 35, 814–824 (1990) [42] Mayne, D.Q., Rawlings, J.B., Rao, C.V., Scokaert, P.O.M.: Constrained model predictive control: Stability and optimality. Automatica 36, 789–814 (2000) [43] Mayne, D.Q., Rakovic, V.: Model predictive control of constrained piecewise affine discrete-time systems. Int. Journal of Robust and Nonlinear Control 13, 261–279 (2003) [44] Michalska, H., Mayne, D.Q.: Moving horizon observers and observer-based control. IEEE Trans. on Automatic Control 40, 995–1006 (1995) [45] Mhaskar, P., El-Farra, N.H., Christofides, P.D.: Uniting bounded control and MPC for stabilization of constrained linear systems. Automatica 40, 101–110 (2004) [46] Norquay, S.J., Palazoglu, A., Romagnoli, J.A.: Model predictive control based on Wiener models. Chemical Engineering Science (1998) [47] Pala, D., Magni, L., Scattolini, R.: Stochastic model predictive control of constrained linear systems with additive uncertainty. In: Proc. of the European Control Conference, Budapest, Hungary (2009) [48] Pannocchia, G., Bemporad, A.: Combined design of disturbance model and observer for offset-free model predictive control. IEEE Trans. on Automatic Control 52, 1048– 1053 (2007) [49] Primbs, J.A., Sung, C.H.: Stochastic receding horizon control of constrained linear systems with state and control multiplicative noise. IEEE Trans. on Automatic Control 54(2), 221–230 (2007) [50] Raimondo, D.M., Limon, D., Lazar, M., Magni, L., Camacho, E.F.: Min-max model predictive control of nonlinear systems: a unifying overview on stability. European Journal of Control 15, 5–21 (2009) [51] Rakovic, S.V., Teel, A.R., Mayne, D.Q., Astolfi, A.: Simple robust control invariant tubes for some classes of nonlinear discrete time systems. In: Proc. of the 45th IEEE Conf. on Decision and Control, pp. 6397–6402 (2006) [52] Scokaert, P.O.M., Rawlings, J.B., Meadows, E.S.: Discrete-time stability with perturbations: Application to model predictive control. Automatica 33, 463–470 (1997) [53] Sontag, E.D.: Smooth stabilization implies coprime factorization. IEEE Trans. on Automatic Control 34, 435–443 (1989) [54] Wan, Z., Kothare, M.V.: Efficient scheduled stabilizing model predictive control for constrained nonlinear systems. Int. J. of Robust and Nonlinear Control 13, 331–346 (2003) [55] Yan, J., Bitmead, R.R.: Incorporating state estimation into model predictive control and its application to network traffic control. Automatica 41, 595–604 (2005)

Chapter 8

Optimal Control Using Pontryagin’s Maximum Principle and Dynamic Programming Bart Saerens, Moritz Diehl, and Eric Van den Bulck

Abstract. This chapter describes the application of Pontryagin’s Maximum Principle and Dynamic Programming for vehicle driving with minimum fuel consumption. The focus is on minimum-fuel accelerations. For the fuel consumption modeling, a six-parameter polynomial approximation is proposed. With the Maximum Principle, this consumption model yields optimal accelerations with a linearly decreasing acceleration as a function of the velocity. This linear acceleration behavior is also observed in real traffic situations by other researchers. Dynamic Programming is implemented with a backward recursion on a specially chosen distance grid. This grid enables the calculation of realistic gear shifting behaviour during vehicle accelerations. Gear shifting dynamics are taken into account.

8.1 Introduction Transport is known to be the fastest rising source of CO2 and other pollutant emissions. Environmental issues, the dependency on oil and the increasing fuel prices trigger the interest of lowering the fuel consumption of vehicles. Four distinct methods to achieve this, exist: 1. fuel: by using alternative fuels; 2. vehicle: by improving vehicle technology; lowering rolling friction, air resistance and vehicle weight, and increasing powertrain efficiency; 3. road: by adapting the infrastructure and traffic management; and 4. driver: by changing the driving behavior. Bart Saerens and Eric Van den Bulck Department of Mechanical Engineering, Katholieke Universiteit Leuven, Celestijnenlaan 300, B-3001 Heverlee, Belgium e-mail: {bart.saerens,eric.vandenbulck}@mech.kuleuven.be Moritz Diehl Optimization in Engineering Center (ESAT/SCD), Katholieke Universiteit Leuven, Kasteelpark Arenberg 10, B-3001 Heverlee, Belgium e-mail: [email protected] L. del Re et al. (Eds.): Automotive Model Predictive Control, LNCIS 402, pp. 119–138. c Springer-Verlag Berlin Heidelberg 2010 springerlink.com

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This chapter focusses on the latter. Numerous studies show the influence of the driving style on fuel consumption [17, 12, 11]. Research predicts that by adapting the driving style, fuel consumption can be reduced by 15 to 25 % [16, 36]. Thanks to the increasing environmental awareness of drivers, minumum-fuel driving1 has become more and more popular. Ecodriving2 is a set of rules for a fuel efficient driving style. Several studies show that ecodriving can reduce fuel consumption on average between 15 and 25 % in short term periods. On long term periods, old driving habits will re-emerge and consumption savings are between 5 and 8 % [13]. Ecodriving tools are also emerging (e.g. gear selection indicators, forcefeedback accelerator pedals, FEST [34],...). These minumum-fuel driving aids can only reach their maximum potential if they are based on optimal control algorithms that minimize the fuel consumption. This chapter discusses the use of optimal control as a basis for minumum-fuel vehicle driving. Only conventional powertrains are considered (no hybrid or CVT) and the main focus is on accelerations. Section 8.2 gives an overview of different methods for optimal control and gives some more details on Pontryagin’s Maximum Principle and Dynamic Programming. Since optimal control is model-based, Sect. 8.3 discusses the vehicle and powertrain modeling. There will be a focus on the fuel consumption modeling, which is essential for minumum-fuel driving. Based on the described model, Sects. 8.4 and 8.5 apply optimal control for vehicle accelerations. In Sect. 8.4 this is done with Pontryagin’s Maximum Principle, which gives an analytical insight in the problem, yet is rather complex. Section 8.5 uses Dynamic Programming. With this simple method, gear shifting can be included. Section 8.6 compares the results of the Maximum Principle and Dynamic Programming and also compares them with results from other research. Section 8.7 concludes the chapter.

8.2 Optimal Control Optimal control3 uses dynamic system models to minimize an objective function, with multiple constraints taken into account. Consider the following simplified optimal control problem in ordinary differential equations (ODE): min

x(.), u(.),te

te 0

L(x(t), u(t)) dt + E(x(te )),

(8.1a)

subject to x(0) − x0 = 0, x(t) ˙ − f (x(t), u(t)) = 0, h(x(t), u(t)) 0, r(x(te )) = 0. 1 2 3

∀t ∈ [0, te ], ∀t ∈ [0, te ],

(initial value) (ODE model)

(8.1b) (8.1c)

(path constraints) (8.1d) (terminal constraints) (8.1e)

Driving with an adapted driving style in order to save fuel. www.ecodrive.org Also referred to as dynamic optimization.

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The problem is visualized in Figure 8.1. The goal is to minimize an objective function (8.1a), that can consist of a Mayer term E and a Lagrange term L, subject to certain constraints. The horizon length te can be fixed or free for optimization. The above formulation is by far not the most general. Unnecessary notational overhead is avoided by omitting e.g. differential algebraic equations (DAE), multi-phase motions, or coupled multipoint constraints, which are, however, all treatable by selected optimal control methods. Generally speaking, there are three basic approaches to address optimal control problems: (a) indirect methods, (b) Dynamic Programming and (c) direct methods, cf. the top row of Figure 8.2. (a)Indirect methods use the necessary conditions of optimality of the infinite problem to derive a boundary value problem (BVP) in ordinary differential equations (ODE), as e.g. described in [9]. This BVP must numerically be solved, and the approach is often sketched as “first optimize, then discretize”. The class of indirect methods encompasses also the well known calculus of variations and the EulerLagrange differential equations, and Pontryagin’s Maximum Principle [28]. The numerical solution of the BVP is mostly performed by shooting techniques or by collocation. The two major drawbacks are that the underlying differential equations are often difficult to solve due to strong nonlinearity and instability, and that changes in the control structure, i.e. the sequence of arcs where different constraints are active, are difficult to handle: they usually require a completely new problem setup. Moreover, on so called singular arcs, higher index DAE arise which necessitate specialized solution techniques. (b)Dynamic Programming [3, 5] uses the principle of optimality of subarcs to compute recursively a feedback control for all times t and all x0 . In the continuous time case, this leads to the Hamilton-Jacobi-Bellman (HJB) equation, a partial differential equation (PDE) in state space. Methods to numerically compute solution approximations exist, e.g. [24] but the approach severely suffers from Bellman’s “curse of dimensionality” and is restricted to small state dimensions, like the application in this paper. 6

path constraints h(x, u) 0 states x(t)

initial value x0

r terminal

constraint r(x(te )) = 0

r p

controls u(t)

0

Fig. 8.1 Simplified optimal control problem

t

pte

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Optimal Control

PP

PP

PP

PP

PP

Dynamic Programming (Hamilton-Jacobi-Bellman Equation): Tabulation in State Space

Indirect Methods (Pontryagin Maximum Principle): Solve Boundary Value Problem

Direct Methods: Transform into Nonlinear Program (NLP)

Single Shooting: Only discretized controls in NLP (sequential)

Collocation: Discretized controls and states in NLP (simultaneous)

Multiple Shooting: Controls and node start values in NLP (simultaneous)

( ( ( ( (( ( ( ( (((( ( ( (( (((

Fig. 8.2 Overview of numerical methods for optimal control

(c)Direct methods [6, 7, 23, 31] transform the original infinite optimal control problem into a finite dimensional nonlinear programming problem (NLP). This NLP is then solved by variants of state of the art numerical optimization methods, and the approach is therefore often sketched as “first discretize, then optimize”. One of the most important advantages of direct compared to indirect methods is that they can easily treat inequality constraints, like the inequality path constraints in the formulation above. This is because structural changes in the active constraints during the optimization procedure are treated by well developed NLP methods that can deal with inequality constraints and active set changes. All direct methods are based on a finite dimensional parameterization of the control trajectory, but differ in the way the state trajectory is handled, cf. the bottom row of Figure 8.2. For solution of constrained optimal control problems in real world applications, direct methods are nowadays by far the most widespread and successfully used techniques. However, for rather simple optimal control problems, indirect methods and Dynamic Programming still have their value. The analytical nature of Pontryagin’s Maximum Principle enables one to get a physical insight in the control problem. Dynamic Programming is very easy to implement and can easily handle all sorts of constraints and dynamics. Therefore, this paper will use the latter two methods

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to solve the minumum-fuel vehicle driving problem. The remainder of this Section will briefly discuss these two methods.

8.2.1 Pontryagin’s Maximum Principle This indirect method can solve rather complex control problems, yet for illustration, a very simple one is considered: min

x(.), u(.),te

te 0

L (x(t), u(t), t) dt,

(8.2a)

subject to: x(0) − x0 = 0, x(t) ˙ − f (x(t), u(t)) = 0,

∀t ∈ [0, te ],

(initial value) (ODE model)

h(u(t)) 0,

∀t ∈ [0, te ],

(control constraints) (8.2d)

x(te ) − xe = 0,

(8.2b) (8.2c)

(terminal constraints) (8.2e)

with L the Lagrange objective function, x the state variables and u the controls. The controls can be constrained, for simplicity the state variables are considered unconstrained. To solve the optimal control problem with the Maximum Principle [9], the Hamiltonian H is defined: H (x(t), u(t), λ (t), t) ≡ L (x(t), u(t), t) + λ T [ f (x(t), u(t), t)] .

(8.3)

Here, λ are the adjoint variables that augment the system. The dynamics of these variables are given by:

∂H ∗ (x (t), u∗ (t), λ ∗ (t), t) . λ˙ ∗ (t) = − ∂x

(8.4)

The superscript ∗ denotes optimality. Optimal control is achieved by choosing controls u∗ that minimize the Hamiltonian at each time instant, for given state and adjoint variables: H (x∗ (t), u∗ (t), λ ∗ (t), t) H (x∗ (t), u(t), λ ∗ (t), t) ,

h(u∗ (t)) 0.

(8.5)

=M (t)

Thus, the optimal control problem is translated to a boundary value problem with dynamics (8.2c, 8.4). Constraints on the state variables are given by equations (8.2b, 8.2e), contraints on the controls by equation (8.2d). Possible terminal constraints on the adjoint variables are given by the following equations:

λi∗ (te ) δ xe,i = 0 and H (x∗ (te ), u∗ (te ), λ ∗ (te ), te ) δ te = 0,

(8.6)

where δ xe,i = 0 if xe,i is contrained and δ te = 0 if δ te is constrained. Thus when a state variable xi has no terminal constraint, the according adjoint variables λi has

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one. The actual optimization is done analytically and one has to find the initial values for the adjoint variables λ (0), so that possible terminal constraints are met.

8.2.2 Dynamic Programming Dynamic Programming uses discrete system models. Consider the following optimal control problem: min s, q

N−1 li (si , qi ) ∑i=0

+ E (sN ) ,

(8.7a)

subject to s0 − x0 = 0, si+1 − fi (si , qi ) = 0, hi (si , qi ) 0,

i = 0, . . . , N − 1, i = 0, . . . , N,

r (sN ) = 0.

(initial value)

(8.7b)

(discrete system) (path constraints)

(8.7c) (8.7d)

(terminal constraints)

(8.7e)

s are the discretized state variables and q the discretized controls. Dynamic Programming can easily get rid of inequality constraints hi and r by giving infinite costs li (s, q) or E(s) to infeasible points (s, u). The basis of Dynamic Programming is a backward iteration. For k = N − 1, N − 2, . . .: Jk (s) = min lk (s, q) + Jk+1 ( fk (s, q)), q

(8.8)

=J˜k (s, q)

starting with: JN (s) = E(sN ).

(8.9)

Based on J˜k , one can obtain feedback laws for k = 0, 1, . . . , N − 1: q∗k = arg min lk (s, q) + Jk+1 ( fk (s, q)) . q

(8.10)

=J˜k (s, q)

For given initial values s0 , one can thus obtain the optimal trajectories of sk and qk by the closed-loop system: sk+1 = fk (sk , q∗k (sk )) .

(8.11)

This is a forward recursion yielding q0 , . . . , qN−1 .

8.3 Vehicle and Powertrain Model This Section covers the modeling of the vehicle and the powertrain. Only longitudinal dynamics are considered. The powertrain consists of a combustion engine and a driveline (clutch, gearbox and differential).

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8.3.1 Vehicle and Driveline Model The total load force Fl [N] on the wheels of the vehicle is given by: dv Fl = λ M dt inertia

v2 Scd ρa 2

+

+

aerodynamic resistance

cr Mg cos φ + Mg sin φ , rolling friction

(8.12)

road slope

where M [kg] is the mass of the vehicle, λ [-] takes the rotational inertia of the driveline into account, S [m2 ] is the frontal surface of the vehicle, cd [-] the drag coefficient of the vehicle, ρa [kg/m3 ] the density of air, v [m/s] the velocity of the vehicle, cr [-] the rolling friction coefficient, g = 9.81 m/s2 the gravity acceleration and φ [rad] the slope of the road. The driveline converts the force on the wheels of the vehicle into a torque T [Nm] on the engine shaft. This is modeled by means of algebraic relations: Fl v = ω T ηm ,

(8.13)

where ω [rad/s] is the engine rotation speed and ηm [-] the power transmission efficiency, which is considered constant. Further: v = Rw

ω , id ig

(8.14)

where Rw [m] is the effective rolling radius of the wheels, id [-] the reduction ratio of the differential and ig [-] the reduction ratio of the gearbox. Table 8.1 shows the parameters of the vehicle and driveline model, for a mediumsize passenger car. Table 8.1 Parameters of the vehicle and driveline model M

1365

kg

λ

1.089

-

Rw

0.316

m

id

4.31

-

ig [3.32, 2, 1.36, 1.01, 0.82]

-

ηm

0.95

-

cr

0.015

-

Scd

0.65

m2

ρa

1.2

kg/m3

8.3.2 Engine Model The engine is modeled as a black box. The engine speed n [rpm] or ω [rad/s] is the state variable, the engine brake torque T [Nm] (torque at the engine shaft) is the

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control and the fuel mass flow rate m˙ f [kg/s] is the output. Another possibility is to use, instead of n and T , the vehicle velocity v [m/s], acceleration a [m/s2 ], and the gear i [-] [21, 29]. Since in this paper we want to minimize the fuel consumption of a vehicle, the modeling of the consumption is a very important issue. A first method is to use mapped data, or piecewise affine approximations of this data [21, 4]. Although this method is very accurate, it has some disadvantages: it takes a lot of time to perform engine tests to obtain detailed consumption maps if they are not provided by the engine manufacturer and this method is not appropriate for indirect optimal control, where one prefers simple analytical expressions. Polynomials are a good candidate for such an analytical expression and they are commonly used [33, 27, 32, 10, 35, 22]. This paper will also use a polynomial approximation of the fuel consumption. The goal is to make this model as simple as possible, while still being useful. A first condition for an appropriate polynomial function can be derived using Pontryagin’s Maximum Principle. Consider the simplified minumum-fuel problem with a generic driveline: te

min

ω (.), T (.),te 0

m˙ f (ω (t), T (t)) dt,

subject to: d ω (t) T (t) − l(ω ) = , dt I Tmin (ω (t)) T (t) Tmax (ω (t)),

(8.15a)

∀t ∈ [0, te ],

(8.15b) (8.15c)

with l(ω ) [Nm] a load function and I [kg·m2 ] the rotational inertia. A generic fuel consumption polynomial is proposed: m˙ f (ω , T ) = m˙ f0 (ω ) + esfc(ω , T ) ω T,

(8.16)

with m˙ f0 [kg/s] the fuel mass flow rate at zero torque and esfc [kg/J] the “extra specific fuel consumption”. Using the Maximum Principle, the Hamiltonian is given by:

λ λ H = m˙ f0 (ω ) − l(ω ) + + esfc(ω , T ) ω T. (8.17) I I The optimal torque T ∗ , is the torque that minimizes H . If the esfc is independent of T , then T ∗ = Tmin or T ∗ = Tmax , depending on the sign of λI + esfc(ω ) ω . This yields that an acceleration with a vehicle should be “full throttle”, which is not a realistic result [30]. Therefore, the first condition for an appropriate fuel consumption polynomial is that the esfc is a function of the torque T . Furthermore, a second consideration about the fuel consumption should be made. Talking about minimizing fuel consumption, most researchers have the specific fuel consumption sfc = ωm˙Tf [kg/J] in mind. Although when comparing the fuel consumption of cars, the mileage is the most relevant. Since a car always has the task to travel a certain distance, the fuel consumption per traveled distance is the primary objective. Considering a fixed gear, this translates into a consumption per rotation

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Table 8.2 Parameters of the engine model a1 1.1046 × 10−5

kg/rad

−7.7511 × 10−8

kg·s/rad2 )

a3 1.6958 × 10−10

kg·s2 /rad3 )

a4

1.7363 × 10−8

kg/(rad·Nm)

a5

6.4277 × 10−11

kg·s/(rad2 ·Nm)

a6

1.6088 × 10−10

kg/(rad·Nm2 )

b1

1.5545

Nm·s/rad

b2

−4.8907 × 10−3

Nm·s2 /rad2

b3

4.0442 × 10−6

Nm·s3 /rad3

a2

cpr [kg/r] of the engine. In order to model the consumption per traveled distance of a car to a certain degree of accuracy, the cpr should at least be a quadratic function of the engine speed. Therefore, the fuel mass flow rate should be at least cubic in ω (or n). Taken the two previous considerations into account, a polynomial approximation of the fuel consumption is proposed: m˙ f = a1 ω + a2ω 2 + a3 ω 3 + a4ω T + a5ω 2 T + a6ω T 2 .

(8.18)

This model is identified with measurements on an engine dynamometer4, Table 8.2 shows the parameters. Note that these parameters do not model the idle consumption very well, however this is of secondary importance when accelerating a vehicle.

Fig. 8.3 Fuel mass flow rate m˙ f [g/s]

4

1.6 l gasoline engine, 81 kW, 150 Nm.

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Fig. 8.4 Consumption per rotation cpr [mg/r]

Fig. 8.5 Specific fuel consumption sfc [g/kWh]

The maximum torque Tmax is a function of ω : Tmax (ω ) = b1 ω + b2 ω 2 + b3ω 3 .

(8.19)

The parameters are given in Table 8.2. The minimum torque Tmin is negative and is determined by the friction of the engine and driveline, the vehicle dynamics and the brakes. Since accelerations only use positive torques, the lower torque limit will not be considered in this paper. Figures 8.3, 8.4 and 8.5 show the fuel mass flow rate, the consumption per rotation and the specific fuel consumption of the engine respectively, as modeled by (8.18).

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8.4 Minumum-fuel Acceleration with the Maximum Principle In this Section, we solve the minumum-fuel vehicle driving problem with Pontryagin’s Maximum Principle. This approach has already been used by Schwarzkopf and Leipnik [32] and by Stoicescu [33]. Stoicescu uses a 4-parameter fuel consumption polynomial approximation, with the esfc independent of T and indeed obtains bangoff-bang controls. Schwarzkopf and Leipnik use a better 9-parameter polynomial approximation and obtain realistic results for selected driving loads. This Section uses the fuel consumption model (8.18). Consider the following minumum-fuel vehicle acceleration problem in a fixed gear: te

min

ω (.),T (.),te 0

m˙ f (ω (t), T (t)) dt,

(8.20a)

subject to: ω (0) − ω0 = 0,

(8.20b)

ω (te ) − ωc = 0, d ω (t) T (t) − l(ω ) − = 0, dt I θ (0) = 0, θ (te ) − θe = 0, d θ (t) − ω (t) = 0, dt Tmin (ω (t)) T (t) Tmax (ω (t)),

(8.20c) ∀t ∈ [0,te ],

(8.20d) (8.20e) (8.20f)

∀t ∈ [0,te ],

(8.20g)

∀t ∈ [0,te ].

(8.20h)

The aim is to accelerate a vehicle from a initial velocity v0 to a cruising velocity vc , with minimum fuel consumption mf , on a level road (φ = 0). In a fixed gear, v0 and vc correspond to certain engine rotation speeds ω0 and ωc , according to (8.14). The dynamics of the vehicle (8.20d), are given by (8.12–8.14): l(ω ) = Scd ρa and: I=

Rw id ig

λM ηm

3

ω2 cr MgRw + , 2ηm id ig ηm

Rw id ig

(8.21)

2 .

(8.22)

A second state θ [rad], the position of the engine shaft, is added. In a fixed gear, this corresponds to the distance s [m/s] traveled by the car: s = iRwig θ . d The optimal control problem yields the following Hamiltonian:

λ1 λ1 H = − (ω ) + λ2ω + m˙ f0 (ω ) + + esfc(ω , T ) ω T. (8.23) I I

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The dynamics of the augmented system are given by: d λ1 = − ∂∂H ω , dt d λ2 = − ∂∂H θ = 0. dt

(8.24a) (8.24b)

The optimal torque T ∗ minimizes the Hamiltonian. Some special conditions follow from the Maximum Principle [9]: dM • if Tmin T ∗ Tmax , then: ∂∂ H T ∗ = 0 and dt = 0; • if the total amount of rotations θe is free, then: λ2 = 0; and • if the total time te is free, then: M (te ) = 0.

The Maximum Principle translates the control problem into a boundary value problem: one should determine λ1 (0) and λ2 such that the terminal constraints are met. First consider the control problem that a vehicle should travel a substantially long distance (e.g. se = 1 km), with no time constrain given (te is free) and we omit the end velocity constraint (8.20c). It is logical (and can also be proven), that the vehicle will accelerate to an optimal cruising velocity vc,opt where the fuel consumption per traveled distance is minimal. The vehicle will cruise at this velocity until the desired distance se is traveled. Now assume an acceleration to this cruising velocity, or corresponding ωopt (this is the minimum-fuel vehicle acceleration). Since there is no time constraint: Mopt = 0. Furthermore: Topt = l(ωopt ). From ∂H ∂ T = 0 one can obtain λ1,opt , and from (8.23) λ2,opt . With this information the solution can be calculated with a backward simulation of the adjoint system from vopt until v = v0 . This backward simulation avoids numerical iterations to calculate the initial values of λ and allows much faster calculations. Note that at v = vopt : ddtλ1 = 0 and the system is stuck in a steady state. In order to make the backward simulation

Fig. 8.6 Optimal accelerations with the Maximum Principle, in the cpr-map

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Fig. 8.7 Optimal accelerations evolve exponentially towards the cruising velocity

Fig. 8.8 Linear behavior of optimal accelerations, the start of the accelerations is not linear due to the torque limits of the engine

work, it should start from v˜ = vopt − ε , with ε small. The method explained above is still valid since the optimal torque is continuous and therefore T ∗ Tmax and M = 0 at v. ˜ If the cruising velocity to attain is not the optimal cruising velocity vc = vc,opt , there are two possible cases: 1. vc < vc,opt The principle of optimality yields that the minimum-fuel acceleration to vc is the initial part of the minimum-fuel acceleration to vc,opt , upto v = vc ; and

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2. vc > vc,opt Since a cruising velocity vc > vc,opt is not optimal, an optimal acceleration will never reach vc . In order to do so, one has to adapt the fuel consumption model such that vc = vc,opt . For certain driving conditions, cruising at vc,opt can be considered too slow (e.g. highway). The will to cruise at higher velocities thus comes indirectly from a time constraint. In the maximum principle, this is enforced by the Hamiltonian H . This yields that the fuel consumption model should be adapted as follows: m˙ f = m˙ f − Mc , (8.25) such that the adapted cpr is minimal at vc . In this equation Mc is a kind of pseudoHamiltonian. Figure 8.6 shows two optimal accelerations in the cpr-map from 35 km/h, to the optimal cruising velocity 67 km/h and to a cruising velocity of 90 km/h. The cruising torque Tc [Nm] is the necessary engine torque for driving at a constant velocity. Figures 8.7 and 8.8 show that these trajectories yield an asymptotical and exponential evolution towards the cruising velocity and a linear decreasing acceleration as a function of the velocity.

8.5 Minumum-fuel Acceleration with Dynamic Programming In this Section we solve the same minumum-fuel acceleration problem as Section 8.4, with Dynamic Programming and gear shifting. This approach has been used before. Hooker et al. [20, 21] use forward dynamic programming on a time grid, both with instantaneous gear shifts and with shift dynamics. Using a time grid, one needs the distance as an extra state variable. This makes the dynamic programming slow, due to the “curse of dimensionality” [3]. Therefore, Monastyrsky and Golownykh [26], and Hellstr¨om [19] use a distance grid, so that there is only one state variable left: the vehicle velocity. The problem here is that at slow velocities, one distance increment takes a lot of time compared to higher velocities. This gives problems for optimizing gear shifting in an acceleration. This paper uses normal backward Dynamic Programming, with a specially adapted distance grid and gear shifting with shift dynamics. This original solution technique is the following. Let se be the total traveled distance and Δ tshift the time needed for gear shifting (around 0.3 to 0.5 s). First, a linear acceleration as a function of the distance from v0 to ve is assumed. Then, the distance is divided in K pieces Δ sk , such that every Δ sk has a duration Δ tk = σ Δ tshift , longer than the time required for gear shifting (σ > 1):

Δ sk = vk−1 σ tshift ,

k = 1, 2, . . . K.

(8.26)

The control problem is equivalent to (8.20), integrated over the distance s instead of t, and discretized:

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K−1

min v, a, i

∑ mf,k (vk , vk+1 , ik , ik+1 , Δ sk+1 ),

(8.27a)

k=0

subject to: v0 − v(0) = 0,

(8.27b)

vK − v(te ) = 0.

(8.27c)

The calculation of the fuel consumption for every distance increment k is as follows: • the fuel consumption mf,k (vk , vk+1 , ik , ik+1 , Δ sk+1 ) [kg] over a distance Δ sk+1 , from velocity vk to vk+1 , in a fixed gear ik = ik+1 : mf,k = m˙ f (n¯ k , T¯k ) with n¯ k and T¯k from v¯k , (v

−v )v¯

d v¯ dt

Δ sk+1 , v¯k

and ik according to (8.12–8.14), where v¯k =

(8.28) vk +vk+1 2

k k and ddtv¯k = k+1 ; Δ sk+1 • if a gear shift occurs, it is assumed to be at the beginning of the distance increment Δ s. The powertrain doesn’t deliver torque during the shifting, the vehicle is coasting and no fuel consumption is assumed (idle consumption can be taken into account). Thus, at the beginning of Δ sk+1 , the car will decelerate from vk to vk = vk + a¯shifttshift , with a¯shift from (8.12) with v = vk and Fl = 0.

v +v

The car will have traveled a distance Δ sk+1,shift = k 2 k tshift . After the shifting, the car has to drive the remainder of Δ sk+1 in gear ik+1 . The fuel consumption mf,k vk , vk+1 , ik+1 , ik+1 , Δ sk+1 − Δ sk+1,shift in this remaining part, is as explained above; • if no shift dynamics are considered, the whole distance Δ sk+1 is driven in gear ik+1 . The search space for v is discretized with Δ v. The minumum-fuel problem is then solved recursively as described in Section 8.2.2: Jk (v) = min mf (vk , vk+1 , ik , ik+1 , Δ sk+1 ) + Jk+1 (vk+1 ). v, i

(8.29)

To speed up the calculation for an acceleration, no decelerations or downshifting is considered. Since the original assumption of the linear acceleration as a function of the distance was wrong, one can redefine the distance grid Δ s with the obtained solution and do the Dynamic Programming again to become a better solution. Figures 8.9 and 8.10 show the results for an acceleration from 5 km/h to the optimal cruising velocity of 67 km/h, in 1 km. The exponential behavior can be seen again. Gear shifting happens very fast at low engine speeds. A comparison is made between Dynamic Programming with instantaneous gear shifting and shifting with shift dynamics. The influence of the dynamics is rather small: without dynamics a distance of 1 km is traveled in 64.05 s with a fuel consumption of 55.08 g, with dynamics this is done in 64.47 s and 55.26 g. The only substantial difference between

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Fig. 8.9 Optimal acceleration with Dynamic Programming and gear shifting

Fig. 8.10 Optimal acceleration with Dynamic Programming and gear shifting

the two is the upshift from 4th to 5th gear, as shown in figure 8.10. The shown results are based on the following parameters: Δ tshift = 0.3 s, σ = 1.5 and the velocity has been discretized with Δ v = 1 km/h.

8.6 Discussion of the Results This Section discusses the results obtained in Sections 8.4 and 8.5. The two approaches are compared with each other and then with other research.

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Fig. 8.11 Comparison of results with Dynamic Programming and Pontryagin’s Maximum Principle

8.6.1 Comparison between the Maximum Principle and Dynamic Programming Figure 8.11 shows an acceleration from 35 km/h to the optimal cruising speed of 67 km/h over 850 m. If se > 850 m, the optimal velocity trajectory will start with the same acceleration over 850 m and then the car will cruise the remaining distance at 67 km/h. The minumum-fuel acceleration is calculated with both Pontryagin’s Maximum Principle and Dynamic Programming in 5th gear. The results are fairly similar: 42.2 g in 51.2 s versus 42.6 g in 50.9 s. Although Dynamic Programming and the Maximum Principle yield almost the same results, one can favor one above the other. Dynamic Programming has the advantage that it is very easy to implement and can easily take into account gear shifting. It can also use mapped engine data. Pontryagin’s Maximum Principle on the other hand, gives more precise solutions and the calculations are faster then Dynamic Programming. The calculation of a minimum-fuel acceleration with the Maximum Principle takes around 1 s on a normal computer, calculation with Dynamic Programming around 1 minute.

8.6.2 Comparison with Other Research The solution of a minumum-fuel acceleration with Pontryagin’s Maximum Principle yields accelerations that linearly decrease with the velocity, as shown in Figure 8.8. This is caused by the specific structure of the proposed fuel consumption model (8.18). This linear acceleration behavior has also been observed with real vehicles in normal traffic situations [1, 2, 8, 14, 18, 25], so that one can conclude that this is

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normal driving behavior. This is important for the application in driving assistance systems, taken the driver acceptance into account. A linear vehicle acceleration can be modeled with: a = α − β v.

(8.30)

In the most recent acceleration study, Bonneson [8] identifies α ± 2 m/s2 and β ± 0.12 s−1 for normal passenger cars. It is not clear to what velocity this acceleration takes place. A fast comparison with the results presented in this paper for an acceleration to the optimal velocity (α = 1.6 and β = 0.09), shows that the average driver accelerates to fast. Note that this might not be a representative comparison, since the calculations in this paper are only done for one specific vehicle. El-Shawarbi et al. [15] measures the effect of acceleration levels on the fuel consumption. Although the test vehicle has a much bigger engine then the one modeled in this paper, they have a very similar fuel consumption per traveled distance. Three acceleration types are defined: aggressive, normal and mild. For an acceleration from standstill over 1.4 km, the mild acceleration (to 104 km/h) consumes the least amount of fuel. The same acceleration starting from 35 km/h would take around 1300 m, this is almost the same result as obtained in this paper.

8.7 Conclusions This paper discusses the problem of minumum-fuel vehicle driving, accelerations in particular. This problem is solved with optimal control. The different methods for optimal control are discussed briefly and two methods (Pontryagin’s Maximum Principle and Dynamic Programming) more in detail. A polynomial fuel consumption model is favored over a detailed consumption map. This kind of model can be used for every optimal control method and is relatively easy to obtain. To avoid bang-off-bang control and in order to have a decent mileage model of the vehicle, a six-parameter fuel consumption polynomial approximation is proposed. With Pontryagin’s Maximum Principle, this consumption model yields minumum-fuel accelerations that decrease linearly as a function of the vehicle velocity. This linear acceleration behavior is also observed in real traffic situations. Although the six-parameter polynomial model might not be the most precise consumption model, it can be very interesting for implementation in driving assistance systems if one considers driver acceptance. For solving the boundary value problem with the Maximum Principle, a backward simulation is proposed. This method avoids shooting or collocations methods and allows fast calculations. The minumum-fuel acceleration problem is also addressed with Dynamic Programming. This method can easily include gear shifting. A backward recursion with an adaptive distance grid and dynamic gear shifting is proposed and implemented. This approach yields almost the same results as with the Maximum Principle.

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References [1] Beakey, J.: Acceleration and Deceleration Characteristics of Private Passenger Vehicles. In: Proc. 18th Annual Meeting Part I: Highway Research Board, Washington DC, USA, pp. 81–89 (1938) [2] Bellis, W.R.: Capacity of Traffic Signals and Traffic Signal Timing. Highway Research Board Bulletin 271, 45–67 (1960) [3] Bellman, R.E.: Dynamic Programming. University Press, Princeton (1957) [4] Bemporad, A., Borodani, P., Mannelli, M.: Hybrid control of an automotive robotized gearbox for reduction of consumptions and emissions. In: Maler, O., Pnueli, A. (eds.) HSCC 2003. LNCS, vol. 2623, pp. 81–96. Springer, Heidelberg (2003) [5] Bertsekas, D.: Dynamic Programming and Optimal Control, 3rd edn., vol. 2. Athena Scientific, Belmont (2007) [6] Biegler, L.T.: Solution of dynamic optimization problems by successive quadratic programming and orthogonal collocation. Computers and Chemical Engineering 8, 243– 248 (1984) [7] Bock, H.G., Plitt, K.J.: A multiple shooting algorithm for direct solution of optimal control problems. In: Proc. 9th IFAC World Congress Budapest, pp. 243–247. Pergamon Press, Oxford (1984) [8] Bonneson, J.A.: Modeling Queued Driver Behavior at Signalized Junctions. Transportation Research Record 1365, 99–107 (1992) [9] Bryson, A.E., Ho, Y.-C.: Applied Optimal Control. Wiley, New York (1975) [10] Chang, D.J., Morlok, E.K.: Vehicle Speed Profiles to Minimize Work and Fuel Consumption. J. Transportation Engineering 131(3), 173–192 (2005) [11] Chang, M.F., Evans, L., Herman, R., Wasielewski, P.: The Influence of Vehicle Characteristics, Driver Behavior, and Ambient Temperature on Gasoline Consumption in Urban Traffic. Technical Report GMR-1950, General Motors Research Laboratories (1976) [12] Chang, M.F., Herman, R.: Driver Response to Different Driving Instructions: Effect on Speed, Acceleration and Fuel Consumption. Technical Report GMR-3090, General Motors Research Laboratories (1979) [13] CIECA. Internal project on Eco-driving in category B driver training & the driving test. Final report, CIECA, November 5 (2007) [14] Dockerty, A.: Accelerations of Queue Leaders from Stop Lines. Traffic Engineering and Control 8(3), 150–155 (1966) [15] El-Shawarby, I., Ahn, K., Rakha, H.: Comparative field evaluation of vehicle cruise speed and acceleration level impacts on hot stabilized emissions. Transportation Research Part D: Transport and Environment 10(1), 13–30 (2005) [16] Evans, L.: Driver Behavior Effects on Fuel Consumption in Urban Driving. Human Factors 21, 389–398 (1979) [17] Evans, L., Herman, R.: Automobile Fuel Economy on Fixed Urban Driving Schedules. Transportation Science 12, 137–152 (1978) [18] Gillespie, T.D.: Start-Up Accelerations of Heavy Trucks on Grades. Transportation Research Record 1052, 107–112 (1986) [19] Hellstr¨om, E.: Explicit use of road topography for model predictive cruise control in heavy trucks. Master’s thesis, Vehicular Systems, Dept. of Electrical Engineering at Link¨opings universitet (2005); Reg nr: LiTH-ISY-EX–05/3660–SE [20] Hooker, J.N.: Optimal Driving for Single-Vehicle Fuel Economy. Transportation Research Part A: General 22(3), 183–201 (1988)

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[21] Hooker, J.N., Rose, A.B., Roberts, G.F.: Optimal Control of Automobiles for Fuel Economy. Transportation Science 17(2), 146–167 (1983) [22] Jahns, G., Forster, K.J., Hellickson, M.: Computer simulation of diesel engine performance. Trans. American Society of Agricultural Engineers 33(3), 764–770 (1990) [23] Kraft, D.: On converting optimal control problems into nonlinear programming problems. In: Schittkowski, K. (ed.) Computational Mathematical Programming. NATO ASI, vol. F15, pp. 261–280. Springer, Heidelberg (1985) [24] Lions, P.L.: Generalized Solutions of Hamilton-Jacobi Equations. Pittman (1982) [25] Long, G.: Acceleration Characteristics of Starting Vehicles. Transportation Research Record 1737, 58–70 (2000) [26] Monastyrsky, V.V., Golownykh, I.M.: Rapid Computation of Optimal Control for Vehicles. Transportation Research Part B: Methodological 27(3), 219–227 (1993) [27] Nielsen, L., Kiencke, U.: Automotive control systems. Springer, Berlin (2000) [28] Pontryagin, L.S., Boltyanski, V.G., Gamkrelidze, R.V., Miscenko, E.F.: The Mathematical Theory of Optimal Processes. Wiley, Chichester (1962) [29] Post, K., Kent, J.H., Tomlin, J., Carruthers, N.: Vehicle characterization and fuel consumption prediction using maps and power demand models. Int. J. Vehicle Design 6(1), 72–92 (1985) [30] Saerens, B., Vandersteen, J., Persoons, T., Swevers, J., Diehl, M., Van den Bulck, E.: Minimization of the fuel consumption of a gasoline engine using dynamic optimization. Applied Energy 86(9), 1582–1588 (2009) [31] Sargent, R.W.H., Sullivan, G.R.: The development of an efficient optimal control package. In: Stoer, J. (ed.) Proc. 8th IFIP Conf. Optimization Techniques. Springer, Heidelberg (1977) [32] Schwarzkopf, A.B., Leipnik, R.B.: Control of Highway Vehicles for Minimum Fuel Consumption over Varying Terrain. Transportation Research 11, 279–286 (1977) [33] Stoicescu, A.P.: On Fuel-optimal Velocity Control of a Motor Vehicle. Int. J. Vehicle Design 16(2-3), 229–256 (1995) [34] van der Voort, M., Dougherty, M.S., van Maarseveen, M.: A prototype fuel-efficiency support tool. Transportation Research Part C: Emerging Technologies 9, 279–296 (2001) [35] Wang, G., Zoerb, G.C.: Determination of optimal working points for diesel engines. Trans. American Society of Agricultural Engineers 32(5), 1519–1522 (1989) [36] Waters, M.H.L., Laker, I.B.: Research on Fuel Conservation for Cars. Technical Report 921, Transport and Road Research Laboratory, Crowthorne, England (1980)

Chapter 9

On the Use of Parameterized NMPC in Real-time Automotive Control Mazen Alamir, Andr´e Murilo, Rachid Amari, Paolina Tona, Richard F¨urhapter, and Peter Ortner

Abstract. Automotive control applications are very challenging due to the presence of constraints, nonlinearities and the restricted amount of computation time and embedded facilities. Nevertheless, the need for optimal trade-off and efficient coupling between the available constrained actuators makes Nonlinear Model Predictive Control (NMPC) conceptually appealing. From a practical point of view however, this control strategy, at least in its basic form, involves heavy computations that are often incompatible with fast and embedded applications. Addressing this issue is becoming an active research topics in the worldwide NMPC community. The recent years witnessed an increasing amount of dedicated theories, implementation hints and software. The Control Parametrization Approach (CPA) is one option to address the problem. The present chapter positions this approach in the layout of existing alternatives, underlines its advantages and weaknesses. Moreover, its efficiency is shown through two real-world examples from the automotive industry, namely: • the control of a diesel engine air path; and • the Automated Manual Transmission (AMT)-control problem. In the first example, the CPA is applied to the BMW M47TUE Diesel engine available at Johannes Kepler University, Linz while in the second, a real world S MART Mazen Alamir and Andr´e Murilo Gipsa-lab, CNRS-University of Grenoble. BP 46, Domaine Universitaire, 38400 Saint-Martin d’H`eres, France e-mail: [email protected], [email protected] Rachid Amari and Paolina Tona IFP Powertrain engineering, BP3, 69390, Vernaison, France e-mail: {paolino.tona,rachid.amari}@ifp.fr Richard F¨urhapter and Peter Ortner Johannes Kepler University, Linz, Austria e-mail: {richard.fuerhapter,peter.ortner}@jku.at L. del Re et al. (Eds.): Automotive Model Predictive Control, LNCIS 402, pp. 139–149. c Springer-Verlag Berlin Heidelberg 2010 springerlink.com

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hybrid demo car available at IFP is used. It is shown that for both examples, a suitably designed CPA can be used to solve the corresponding constrained problem while requiring few milliseconds of computation time per sampling period.

9.1 Introduction The last decade witnessed an increasingly rich literature concerning the way NMPC schemes have to be adapted to fit the real-time requirements when applied to fast systems. Among many possible classifications, a straightforward one consists in splitting the approaches into two main categories: the first amounts to bringing the problem into the linearly hybrid world while the second keeps handling the nonlinear representation of the systems. Many approaches fall in the first category such as the explicit off-line feedback computation approach based on the Piece-Wise Affine (PWA) approximation [19], the Linear Parameter Varying (LPV) approach [10] using dynamic linearization and the recently developed active set approach for on-line solution of MPC for PWA models [11, 12]. Roughly speaking, these approaches address the real-time requirement by replacing the original problem by a new and generally different one which possesses a highly structured form that lends itself to efficient computations1. Although these linearization-like approaches attempt to solve a modified problem that can be quite different from the original one, they encountered and still have a huge popularity. This is based on the belief that the complete solution of the original nonlinear constrained problem would be intractable anyway within the available computation time. This difficulty, for a long time considered as insuperable, inspired the idea of distributing the optimization process over the system lifetime [1]. Several approaches emerged that implement this simple idea including the multiple shooting real-time iteration [8], the Continuation/GMRES (Generalize Minimum Residual) based differential approach [18] and the Control Parametrization Approach (CPA) [2]. The philosophy that underlines the CPA lies in the following ideas: 1. open-loop control profiles showing very simple time structures, when used in a receding-horizon framework, generally lead to very rich closed-loop control profiles that correspond to a small drop in the overall resulting optimality [4]; 2. although the global optimum of the NMPC cost function corresponding to a low dimensional parametrization is necessarily higher than that of a classical trivial piecewise constant parametrization, for a well designed parametrization however, it is more likely that the former would be easier to achieve than the latter due to the difference in the problem complexity. More clearly, in a constrained computational context, the suboptimal solution of a simple optimization problem may be better than the suboptimal solution of a far more complex one; and 1

It is shown hereafter through the Automated Manual Transmission example that even in the case where the problem can be put in a constrained Quadratic Programming (QP) form, the real-time requirements make it necessary to resort to some kind of dedicated parametrization.

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3. classical piece-wise constant control parametrization in which all the control values are degrees of freedom result in unnecessarily high dimensional optimization problems. This is a fortiori true in the case where even the state values along the future system’s trajectory are taken as degrees of freedom. Although these high dimensional problems are highly structured and can therefore be quite efficiently handled by dedicated algorithms, they still need incompressible high demanding preparation steps. Moreover, the dedicated softwares and memory storage needed for such problems are generally incompatible with embedded capacities. This is particularly true in the automotive applications. There are many other advantages of the CPA such as its ability to explicitly exploit the model structure that is generally strong in electromechanical systems and the possibility to use control parametrization that takes into account the constraints of the problem at the very definition of the parametrization. For a more detailed presentation of these issues, the reader can refer to [2] and the related references. This chapter is organized as follows: first, some definitions and notations describing the control parametrization approach are introduced in Section 9.2. Then two automotive control examples are proposed to illustrate the efficiency of the approach, namely, the diesel engine air path control problem (Section 9.3) and the automated manual transmission control problem (Section 9.4). Finally, the paper ends with a discussion and some concluding remarks.

9.2 The Parameterized NMPC: Definitions and Notation Let us consider a time-invariant dynamic model given in the following general form: x(t) = X(t, x0 , u)

;

x ∈ Rn

;

u ∈ U[0,T ]

;

t≤T

(9.1)

where x(t) is the state at instant t ≤ T when initialized at (0, x0 ) and under the control profile u defined on [0, T ]. Let τ > 0 denote some sampling period such that T = N τ . Each map U pwc : P × Rn → UN defines on [0, T ] a parameterized piecewise constant (p.w.c) control profile (with parameters in P × Rn ⊂ Rn p × Rn ) such that: u(t) = u(k) (p, x);

t ∈ [tk−1 ,tk ];

tk = kτ

U pwc (p, x) := (u(1) (p, x) . . . u(N) (p, x)) ∈ UN The state trajectory under the p.w.c control profile U pwc (p, x0 ) is denoted hereafter by X(·, x0 , p). More generally, using a straightforward abuse of notation, for each sampling instant t j = jτ ( j ∈ N), the notation X (t, x(t j ), p) denotes the state trajectory of the model at instant t j + t under the p.w.c control profile defined by U pwc (p, x(t j )) over the time interval [t j ,t j + T ]. Recall that the NMPC strategy relies on the solution at each decision instant t j of an optimization problem of the form # $ under C(p, x(t j )) ≤ 0 (9.2) p(x(t ˆ j )) := arg min J(p, x(t j )) p∈P

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where J(p, x(t j )) is some cost function defined on the system trajectory starting from the initial condition (t j , x(t j )) under the p.w.c control profile defined by U pwc (p, x(t j )). The condition C(p, x(t j )) ≤ 0 gathers all the problem constraints defined on the same trajectory including possible final constraints on the state. Classical NMPC formulation states that once a solution p(x(t ˆ j )) is obtained, the first control in the corresponding optimal sequence U pwc (p, x(t j )), namely K(x(t j )) := ˆ u(1) ( p(x(t j )), x(t j )) is applied to the system during the sampling period [t j ,t j+1 ]. This clearly results in the sampled-time state feedback law defined by: K := u(1) ( p(·), ˆ ·) : Rn → U

(9.3)

When a system with fast dynamic is considered however, only a finite number q ∈ N of iterations of some optimization process S can be performed during the sampling period [t j−1 ,t j ]. This lead to the following extended dynamic closed-loop system: x(t j+1 ) = X (τ , x(t j ), p(t j )) p(t j+1 ) = S q p+ (t j ), x(t j )

(9.4) (9.5)

where S q denotes q successive iterations of S starting from the initial guess p+ (t j ) which is related to p(t j ) to guarantee (if possible) the translatability property (see [2] for more details). The stability of the extended system (9.4)-(9.5) heavily depends on the performance of the optimizer S , the number of iterations q and the quality of the model (see [3] for more details).

9.3 Example 1: Diesel Engine Air Path Control Compared to standard gasoline engines, Diesel engines show better torque characteristics at low speed and reduced fuel consumption. However, the major drawback in such engines is the emission of oxides of nitrogen (NOx) and particulate matter (PM). To this respect, the intake manifold air pressure (MAP) and the mass air flow (MAF) of the engine play an important role. More precisely, it has been shown that if the set-points for these two variables are chosen correctly, a precise tracking of these variables leads to an efficient combustion process that corresponds to a low level of emission [16, 20]. The control inputs are the exhaust gas recirculation (EGR) and the variable geometry turbocharger (VGT) valves. It is shown hereafter that a simple low-dimensional parametrization of the control profile leads to a solution that is real-time implementable while explicitly addressing the constraints on both the control input and its derivative. Moreover, the resulting controller may work as a black box system, making the solution a priori compatible with any possible future improved model. An identified model has been obtained [19] based on real world test bench2. Results showed that the MAF and the MAP are basically influenced by the control inputs EGR, VGT, and by two measured disturbances, namely fuel injection 2

BMW M47TUE diesel engine, at Johannes Kepler University Linz.

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Fig. 9.1 Experimental results of the real-time parameterized NMPC under a part of the NEDC reference, using Np =1.5s, λ =1, b=5 and 30 iterations. The values of MAP and MAF are relative ones. Except in the end, where model uncertainties are very high, the CPA provides a good tracking performance for MAF and MAP

and speed engine. The identified model, sampled at 50 ms, shows the following structure: x+ = [A(u, w)]x + [B]u + [G]w y = [C(u, w)]x where x ∈ Rn is the state (n=8), y ∈ Rm is the vector of measured and regulated output (m=2), namely, the MAF and the MAP, w ∈ R2 is the measured disturbance vector (namely, the fuel injection and the engine speed) and u ∈ R2 is the control input representing the position (in %) of the EGR and VGT valves. The following constraints are imposed on the controlled inputs: uc + u ∈ [umin , umax ]

δ u ∈ [−δmax , +δmax ] where δ u = u(t j+1 ) − u(t j ) and uc is the central value around which the model is identified. The control problem is to design an output feedback that forces the regulated output y to approximately track some desired set-point yd ∈ Rm . Therefore, the

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controller needs the states that affect the evolution of the regulated output to be reconstructed. For this purpose, a Moving Horizon Observer (MHO) is designed [for more details, the reader can refer to [17]]. Having the state estimation x(t ˆ j ) at hand, the CPA consists basically in the computation of the steady state control u∗ and the definition of a temporal parametrization that structurally meets the constraints. The steady state control u∗ and the corresponding stationary state x∗ can be calculated by solving a simple optimization problem. More precisely given the measured vector w and the desired value yd , The steady state control is computed by solving the following two-dimensional optimization problem: u∗ (w, yd ) := arg min yc (ud , w) − yd 2 ud ∈U

;

ud ∈ [umin , umax ]

(9.6)

yc (ud , w) = C(ud , w)[In − A(ud , w)]−1 · [B.ud + G.w] ∗

∗

∗

−1

x (u , w) = [In − A(u , w)]

(9.7)

∗

· [B.u + G.w]

(9.8)

Based on the steady state control u∗ (w, yd ), the following temporal parametrization can be defined: ∗ −λ .i·τ max −uc u(i) (p, x(t (9.9) ˆ j )) = Satuumin + α2 (p).e−b·λ .i·τ −uc u + α1 (p).e where the αi ’s are solutions of the following p-dependent linear system of equations: u∗ + α1 (p) + α2 (p) = u(t j ) (e

−λ ·τ

− 1) · α1(p) + (e

−b·λ ·τ

(9.10) − 1) · α2(p) = pδmax

;

p ∈ [−1, +1] (9.11) 2

where λ > 0, b ∈ N are fixed parameters. The two components of p ∈ [−1, +1]2 are the only remaining degrees of freedom that have to be optimized on-line (together with the online computation of u∗ (w, yd )). Moreover, the continuity of the control sequence can be guaranteed in (9.10) and the constraints on the derivative are met in (9.11). Based on the above notation, the cost function used to compute the best control parameter p is defined as follows: N

J(p, x(t ˆ j )) =

ˆ j ), p) − Y ∗ (i, y(t j ), yd )2 + ρ · X (N, x(t ˆ j ), p) − x∗ (u∗ , w) ∑ Y (i, x(t

i=1

Y ∗ (i, yd ) = yd + e−3τ .i/tr · [y(t j ) − yd ] where Y ∗ (i, y(t j ), yd ) is a filtered version for the set points, tr the desired settling time of the closed loop and ρ a weighting term on the terminal state. For more details on the optimisation process and the time needed for its on-line solution, see [17]. Let us note however that given the small size of the optimization problems, several simple non-smooth optimization algorithms (Powell, simplex, etc.) have been tested without noticeable differences in the results.

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Experimental validation on the real world Diesel engine is shown on Figure 9.1. The real time platform used is a 480MHz Autobox-dSPACE system. The routines R environment and 50 ms as were developed in C language using the MATLAB sampling period. A part of the New European Driving Cycle (NEDC) is used as validation scenario and results are compared to the performance obtained via the existing Engine Control Unit (ECU). One can note that during the time interval [200 − 250] s, the NMPC controller outperforms the ECU while it gives roughly the same performance elsewhere. For completeness, note that an integrator was added in the control design in order to eliminate the offset error [17]. Recall that the proposed scheme can be easily used with more elaborated and fully nonlinear models without noticeable increase in complexity or computation time. Preliminary results in that direction are very promising and will be reported in future communications.

9.4 Example 2: Automated Manual Transmission Control Automated Manual Transmission (AMT) technology combines the fuel efficiency of manual transmission with the smooth operation of an automatic transmission. It operates similarly to a manual transmission except that it does not require direct clutch actuation or gear shifting by the driver. Transmission actuators are computercontrolled and embedded control strategies are in charge of ensuring smooth clutch engagement and gear shifting. AMT control has attracted considerable attention in the last recent years [14, 7, 13, 15, 9, 5] and a detailed state of the art discussion is clearly beyond the scope of this contribution. However, it is relevant for the present contribution to mention that real-time implementation was a major issue in all the contributions that involved optimal control-like design. A detailed recent study, [15] concluded that due to its large computational cost, explicit MPC such as the one used in [7] is not suitable yet for this type of problems. The same statement applies to the constrained finite horizon LQ scheme proposed in [14] because of the classical parametrization being used. Finally, the NMPC clutch engagement strategy proposed in [9] involves an initial open-loop phase using look-up table since prediction horizon longer than 0.6 seconds would be incompatible with real-time requirements. In what follows, it is shown how a parameterized approach enables a unified (start-up and gear shifting) and completely closed-loop real-time NMPC that explicitly handles the problem constraints to be obtained. However, because of the lack of space, only a sketch of the solution is proposed as the detailed description and results can be found in [5, 6]. In particular, the presentation concentrates on the start-up mode while the proposed solution [5, 6] covers all the gearing configurations. The system can be described by the following simplified model: Je ω˙ e = Te − sign(ωsl ) · Tc (xc ) (9.12) (9.13) Jc + Je (ig , id ) ω˙ c = sign(ωsl ) · Tc (xc ) − g(ωc , θcw , ωw , ig , id ) ω c Jw ω˙ w = (ig id ) · g(ωc , θcw , ωw , ig , id ) − TL (ωw ) ; θ˙cw = − ωw (9.14) ig id

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where the J’s, T ’s, ω ’s and θ ’s represent inertias, torques, angular velocities and angles respectively. The indices e, c, m, t, w and L refer to engine, clutch, mainshaft, transmission, wheel and load respectively. ig and id are the gear ratio and the differential ratio while Jeq is the equivalent inertia. Finally, ωsl = ωe − ωc is the slip speed and θcw = θc − θw . The control objective is to: 1) guarantee a smooth clutch engagement by forcing ωsl to track a dynamically generated reference ωslre f (·) that reaches 0 after some transient time t f . This duration t f must faithfully reflect the actual driver torque demand expressed through the accelerator pedal position X pedal (transparency). re f 2) regulate ωe around some reference value ωe = max ωe0 , T −1 (Ted (X pedal , ωe )) where ωe0 is the idle speed set-point, T (ωe ) is the maximum torque at speed ωe while Ted is the desired torque as interpreted from some static map given the pedal position X pedal and the engine speed ωe . These control objectives have to be fulfilled while meeting saturation constraints on the control inputs (Te ∈ [Temin , Temax (ωe )], Tc ∈ [Tcmin , Tcmax (ωe )]) and their rates of change (T˙e ∈ [T˙emin , T˙emax ], T˙c ∈ [T˙cmin , T˙cmax ]). In order to design a real-time implementable MPC, the model (9.12)-(9.14) is simplified as follows: Je ω˙ e = u1 − sign(ωsl ) · u2 + δe Jc + Je (ig , id ) ω˙ c = sign(ωsl ) · u2 − δc

(9.15) (9.16)

where u = (Tesp , Tcsp )T is the vector of set-points fed to the low level torque controllers while δe and δc gather all model mismatches and/or tracking errors including the unknown torque TL . The MPC described hereafter uses estimated version δˆe and δˆc of δe and δc obtained through a Kalman-like observer using the measurement u, ωe and ωc . Given a prediction horizon N p · τ , the parameterized solution involves a scalar parameter p that is used to define a quadratic cost in the p.w.c control profile u according to % % Np % % ωsl (k + i) − ωslre f (k + i, p) % % Ω (u, p, ω (k)) := ∑ % where (9.17) % re f % % ω (k + i) − ω (k + i) e e i=1 Q

ωslre f (k + i, p) =

1 − i/p ωsl (k) (1 + λ · i/p)2

(9.18)

Referring to the notation of Section 9.2, the following definitions are used: J(p, ω ) = |p − t f (X pedal )| ; U pwc (p, ω ) = arg min Ω (u, p, ω ) (9.19) u & ' & ' C(p, ω ) ≤ 0 ⇔ U pwc (p, ω ) meets the constraints (9.20) where t f (X pedal ) is the clutch engagement time associated to the pedal position X pedal as computed from a look-up table while U pwc (p, ω ) is the analytically given

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Fig. 9.2 Experimental validation of the real-time parameterized control strategy on the IFPS MART car AMT. The central plot shows a typical evaluation of the tasks computation time. The MPC-related computation time never exceeds 0.35 ms that is reached during saturation phase where the number of iterations needed for the dichotomy is around 21 iterations. Note that during these period, the scalar parameter increases drastically in order to meet the actuators constraints.

unconstrained minimum of the quadratic cost (9.17). More precisely, p must be chosen as close as possible to t f while resulting in an admissible control profile. It can be proved that this problem is feasible over the region of interest and can be solved by simple dichotomy. Figure 9.2 shows experimental results of a start-up manoeuver using the parameterized MPC. Note the saturation on the applied torques as well as the evolution of the parameter p that reached 47 during the saturation phase. The central plot shows a typical evaluation of the processor computational load. The software has been run in a multi-task configuration including two tasks: a secondary task (task2) corresponding to the MPC algorithm, computed every 50ms and a main task (task1), including the rest of the powertrain control modules, computed every 1ms. The MPC strategy takes approximately 0.05 ms when only one iteration is needed. Whereas during constraint saturation this time increases to a mean of approximately 0.28 ms (≈ 15 iterations) with a maximum of 0.35 ms (≈ 21 iterations). R The control software is written in MATLAB /Simulink and run in real-time on a xPC Target environment hosted on a micro-PC with a 1GHz Pentium III and 256 Mb of RAM of which only 0.3 Mb are used by the engine control software.

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9.5 Conclusion In this paper, the parametric approach to NMPC implementation is described and two automotive control problems are used to illustrate its efficiency. The main goal of this paper is to underline the fact that generic formulations are rarely compatible with available real-time and embedded computational facilities. However, dedicated and even oversimplified settings may lead to remarkably nice results thanks to the receding-horizon principle’s efficiency in recovering closed-loop optimality.

References [1] Alamir, M.: Nonlinear receding horizon sub-optimal guidance law for minimum interception time problem. Control Engineering Practice 9(1), 107–116 (2001) [2] Alamir, M.: Stabilization of Nonlinear System Using Receding-Horizon Control Schemes: A parametrized approach for Fast Systems. LNCIS. Springer, London (2006) [3] Alamir, M.: A Framework for Monitoring Control Updating Period in Real-Time NMPC. In: Assessement and future direction in NMPC. LNCIS. Springer, Heidelberg (2008) [4] Alamir, M., Marchand, N.: Constrained minimum-time-oriented feedback control for the stabilization of nonholonomic systems in chained form. Journal Optimization Theory with Applications 118(2), 229–244 (2003) [5] Amari, R., Alamir, M., Tona, P.: Unified mpc strategy for idle-speed control, vehicle start-up and gearing applied to an automated manual transmission. In: Proceedings of the 17th IFAC World Congress, Seoul, South Korea (July 2008) [6] Amari, R., Tona, P., Alamir, M.: Experimental evaluation of a hybrid mpc strategy for vehicle start-up with an automated manual transmission. In: Proceedings of the European Control Conference (ECC 2009), Budapest (2009) [7] Bemporad, A., Borrelli, F., Glielmo, L., Vasca, F.: Hybrid control of dry clutch engagement. In: Proceedings of the European Control Conference, Porto, Portugal (October 2001) [8] Diehl, M., Bock, H.G., Schlo¨oder, J.P.: A real-time iteration scheme for nonlinear optimization in optimal feedback control. SIAM Journal on Control and Optimization 43, 1714–1736 (2005) [9] Dolcini, P.J., Canudas de Wit, C., Bechart, H.: Observer-based optimal control of dry clutch engagement. Oil & Gas Science Technology 62(4), 615–621 (2007) [10] Falcone, P., Borrelli, F., Tseng, H.E., Asgari, J., Hrovat, D.: Linear time varying model predictive control and its application to active steering systems: Stability analysis and experimental validation. International Journal of Robust and Nonlinear Control 18(8), 862–975 (2008) [11] Ferreau, H.J., Bock, H.G., Diehl, M.: An online active set strategy for fast parametric quadratic programming in mpc applications. In: Proceedings of the IFAc Workshop on Nonlinear Model Predictive Control For Fast Systems, Grenoble (October 2006) [12] Ferreau, H.J., Ortner, P., Langthaler, P., del Re, L., Diehl, M.: Predictive control of a real-world diesel engine using an extended online active set strategy. Annual Reviews in Control 31, 293–301 (2007) [13] Glielmo, L., Iannelli, L., Vacca, V., Vasca, F.: Grearshift control for automated manual transmission. IEEE/ASME Transactions on Mechatronics 11(1), 17–26 (2006)

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[14] Glielmo, L., Vasca, F.: Optimal control of dry clutch engagement. In: Transmission and Driveline Symposium, SAE 2000 World Congress, Detroit, Michigan (March 2000) [15] Van Der Heijden, A.C., Serrarens, A.F.A., Camlibel, M.K., Nijmeijer, H.: Hybrid optimal control of dry clutch engagement. International Journal of Control (October 2007) [16] Johnson, T.V.: Diesel emission control in review. SAE, Paper 2001-01-0184 (2001) [17] Murilo, A., Alamir, M., Ortner, P.: Fast NMPC Scheme for a Diesel Engine Air Path. In: Proceedings of the European Control Conference (ECC 2009), Budapest (2009) [18] Ohtsuka, T.: A continuation/gmres method for fast computation of nonlinear recedinghorizon control. Automatica 40, 563–574 (2004) [19] Ortner, P., del Re, L.: Predictive control of a diesel engine air path. IEEE Transactions on Control Systems Technology 15(3), 449–456 (2007) [20] van Nieuwstadt, M.J., Kolmanovsky, I.V., Moraal, P.E.: Coordinated egr-vgtcontrol for diesel engines: An experimental comparison. SAE, Paper 2000-01-0266 (2000)

Chapter 10

An Application of MPC Starting Automotive Spark Ignition Engine in SICE Benchmark Problem Akira Ohata and Masaki Yamakita

Abstract. Research Committee on Advanced Powertrain Control Theory in Society of Instrument and Control Engineers (SICE) provided a benchmark control design problem on a V6 automotive spark ignition engine simulation that has strong nonlinearity and discrete event features. Challengers have to start the engine and regulate the engine speed at 650rpm within 1.5s after the ignition by their designed controllers actuating the spark advance and the fuel injector of each cylinder as well as the throttle valve. The background is that systematic control design methodologies are challenged in Model-based Development (MBD) that has been expected to resolve the serious complexity issue of automotive control system developments. MPC is a candidate of the recommended control design methodologies. (Generalized Predictive Control) GPC was studied in this chapter because it can be embeddable on production ECUs with the limited execution speed and the ROM/RAM memory sizes. GPC with the optimized feedforward control succeeded to satisfy the benchmark problem requirements. However, lots of efforts were necessary to design the feedforward and the local linear models, therefore, it is necessary to further continue to investigate NMPC to get the potential benefits of MPC.

10.1 Introduction The requirements of CO2 reduction and clean exhaust gas have given great impacts to the automotive engines. Those require coordinated control among engine, transmission and other automotive control systems as well as navigation system. That makes engine control very sophisticated and complex. Moreover, the requirements from the durability, the quality and reducing development time have been becoming Akira Ohata Toyota Motor Corporation e-mail: [email protected] Masaki Yamakita Tokyo Institute of Technology L. del Re et al. (Eds.): Automotive Model Predictive Control, LNCIS 402, pp. 153–170. c Springer-Verlag Berlin Heidelberg 2010 springerlink.com

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validation

Virtual World

*SILS, PILS

combining

controller model

plant model Rapid Prototyping ECU

*HILS

plant hardware vehicle

actuators

Functional Spec.

control design

=

=

Functional Spec.

controller

sensors

software

hardware

combining

Real World

validation

*PILS: Processor In the Loop Simulation *SILS: Software In the Loop Simulation *HILS: Hardwar e In the Loop Simulation

Fig. 10.1 Concept of Model-based Development

very strong. This situation has made control system developments extremely complex. It is considered that to manage the complexity can be the key of the future success in the automotive industry. Model-based Development (MBD) is highly expected to resolve the complexity issue of control system developments. The concept is shown in Figure 10.1. A control system in the actual world consists of the controlled hardware and the electronic controller. After the required validation tests, the developed control system is put into the production. The structure combining the controlled hardware and the Electronic Control Unit (ECU) in the actual world is modeled in the virtual world. Therefore, there are the controlled hardware and the controller models in virtual world. The closed loop system is validated through the required simulation tests. This is called Software In the Loop Simulation (SILS). An interesting thing is that there are two links between the actual and virtual world. One is Rapid Prototyping ECU in which an actual hardware is controlled with the virtual controller. That means the control logic works on a general purpose PC with the sufficient execution speed and the memory sizes. The other link is Hardware In the Loop Simulation (HILS) in which a real time hardware model is controlled with the actual controller. That can make debugging the developed controller efficient because of easy recreation of observed bugs. Moreover, some actuators can be put into the closed loop when the models don’t exist and the accuracies are hardly guaranteed. The key of MBD is to use the hardware and controller models as the functional specifications. In the ECU development, embedded codes can be generated form the controller model because the specification is sufficiently accurate. In the hardware development, the hardware must be corrected if some differences are detected. That may be unacceptable for many hardware developers because they tend to have

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System Development

Requirements and constraints analysis SystemValidation Engine System combination Requirements and constraints analysis

Validation

Hardware design Parts Partsdrawing drawing& Plant Model

Eng.+Virtual ECU Virtual Eng+ECU Combination

Verification

parts

Hardware parts

Prototyping parts

Engine Hardware Development

Validation

Combination Software module

Requirements and constraints analysis Control design

Verification

Model Specification

Software modules embedded code development

Control Development

Fig. 10.2 Concurrent MBD Process

doubts about model accuracies and hardly admit that they are wrong. However, they would produce trifle errors that may cause tough and urgent iterated works in the latter part of the development. The early detection of produced defects can considerably increase the productivity of the development and the quality of the control system because the effects of overlooked defect may be spread to many portions. The comparison between the product and the specification is a basic method to detect the error and the model can make it very accurate and efficient. Figure 10.2 shows the development process in MBD. It consists of three V cycles. The process starts from the system requirements and constraints analysis. The system design is performed according to the result. Control and hardware designs are done also according to the requirements and constraints published from the system design. The outputs of each design are the models as the functional specifications of the components. The following processes are prototyping hardware parts and embedded codes developments. The behaviors of the outputs are compared with the models in the verification processes. Thus, defects are detected immediately. These components are combined and the hardware system and ECU are constructed. The closed loop system validations are performed with Rapid Prototyping ECU and HILS. Here, validation means to guarantee the correctness of the developed products. It can be expected that the quality of the actual closed loop system is sufficiently high because defects can be detected and fixed immediately. The final validation is performed and the development is completed. The purpose of this MBD process is continuously improving the process with the visualization of the process quality and removing iterated works as much as possible with the accurate verifications and the validations. Of course, MBD can make works faster but removing iterated works crossing relevant divisions and companies are especially effective because they are costly and require lots of additional works.

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However, many technologies are required to realize the MBD process. We have categorized them into the seven technical areas as follows: • • • • • • •

A: plant modeling; B: control design; C: calibration; D: verification/validation; E: model execution; F: model/data management; and G: process management.

Each technical area consists of seven sub-technical areas and a sub-technical area R Automotive Advisory consists of seven sub-sub-technical areas. Japan MATLAB Board (JMAAB) defined similar diagrams that are obtained from the website of JMAAB. MBD requires many technologies as shown in the diagrams. Therefore, the collaboration among industries and universities is highly desired because a company can’t cover all of them only by itself. The activities of control system development are categorized into Plant Modeling, Control Design, Calibration and Verification & Validation (V&V). Required services for the activities have to be prepared as shown in Figure 10.3. All required models, data and services should be timely provided to control development activities. However, it is very difficult to gather them because they are usually distributed and not well managed. Therefore, we need the system in which newly developed models and data are managed as well as the existing models and data. For the purpose, MBD framework is defined as shown in Figure 10.3. In this chapter, only Control design area is described. The sub-technical areas and the sub-sub technical areas are defined in the list below. As mentioned above, JMAAB defined the similar diagram which can be also obtained from their website. 1. Control Design Methodologies: Model Driven Control, Hybrid System Control Design, State Estimation, Adaptive & Learning Control, Large Scale Control System, Auto-code Generation, Optimization with Constraints 2. Control Simplification: Definition of Simplicity, Simulation and Identification, Symbolic Manipulation, Function Approximation, Relaxation of Constraints, Order and Parameter Reductions, Piecewise Linear System 3. Control Calibration: Test Facility, Automated Measurements, Design of Experiments, Calibration Process, Identification Criteria, Optimization, Model Identification 4. Control Specification: Style Guidelines, Design Specification, Data Dictionary, Requirements Specification, I/O Interface Specification, Communication Specification, Documents 5. Control Evaluation: Test Data Generation, Test Scenario Database, Verification & Validation, Evaluation Metrics, Automated Evaluation, Comparison with Experiments, Model Execution

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6. Control Design Process Definition: Requirement Analysis, Scheduling, Reporting, Standardization, Process Integration, Requirement Tracker, Set of Process Definitions 7. Model/Data management: Configuration Management, Linkage with Legacy Codes, Control Architecture, Data Exchange, Version Management, Model Differencing, Meta Modeling of Model and Data

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Fig. 10.6 Workflow in MBD (Approximated recreation of the relationship between the inputs and the outputs)

The above list implies that a designed control must be simplified. The memory size and the execution speed of production ECU are highly limited because of the cost reason. Generally, derived control by an advanced control requires the high execution speed and the big ROM/RAM memory sizes. Therefore, it is expected that the control can be simplified from the original control through the order reduction, piecewise linearization and function approximations. Some portions of the control may be implemented as the combination of maps and simple functions that can make the implementation easy. Figure 10.4 shows how important simplification is in MBD. We can have a complex model at the beginning of the development. A complex model derives the complex control generally. There are two major technologies, symbolic and numerical manipulations, for the simplification. Control design with Simplified model can derive a simpler control than the one derived from the original model. But, the control may be still complex for the implementation. In the case, the control must be simplified. Figure 10.5 shows the current workflow for the implementation of the designed control. In the current workflow, the source codes are developed according to the specification. The embedded codes are obtained through the compilation. The process is the exact recreation between the inputs and the outputs of the controller from the one defined with the source codes. Next, calibration in which the control parameters are determined by experiments is performed. Iterated works are required when some defects are found in the followed verification and validation process. On the other hand, the specification must be executable in MBD and prototyping embedded codes are automatically generated. Thus, calibration is possible at this stage. The difference from the current workflow is “Math Expression” of the control model. Hopefully, embedded codes are generated from it. At this process,

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“Math Expression” is simplified so that the defined relationship between inputs and the outputs should be same. That is called “Exact Simplification”. This process can remove redundancies of the control model. Next, the relationship is recreated with the acceptable error. Thus, more simple embedded codes can be expected than the original.

10.2 Control Design Strategy in MBD Many engineers are working on a control system development. Systematic ways are essential for such the collaborated works. Currently, heuristic control design is the mainstream in the automotive industry. However, it takes much time to train such skilful engineers. Hopefully, the required control is systematically derived from the plant model in MBD as shown in Figure 10.1. That is the reason why Model Driven Control (MDC) is placed into Control Design Methodologies. The program code and the GUI diagram are considered as the plant model if they describe the relationships between the inputs and the outputs of the target controlled hardware. However, there is no general way to derive the required control from a complex nonlinear model with the time delays, the state/input constraints and the discrete event features. They highly affect the control performance and recommended methods of control design in MBD must deal with them. MPC is an attractive control design for automotive control developers because of the following capabilities: 1. it can deal with the time delay; 2. it can deal with the range and speed constraints of actuator and also state constraints; 3. it can construct the feedforward control; 4. it can allow the near constraint boundary operation of the constraints; and 5. it can give intuitively understandable plant behaviors. However, it may require time and memory consuming calculation. That is a serious concern for automotive control. Figure 10.7 shows the trends of the CPU speed and the ROM size. The figure says that the CPU speed may reach 400MHz and the ROM size may reach around 2Mbytes in 2008. This figure was made in 2004 and the prediction was a little bit optimistic. We knew that the maximum speed of single core CPU is 200MHz and expected the usage of cache memory or multi-core. The actual direction can be multi-core. We need to be watching the future increase of the CPU speed to capture the trend in the new situation. The prediction of the memory size may be almost right. The increases don’t mean that we will be able to have enough resources for installing advanced controls because the number of the actuators has been increasing. Thus, the limitation of the calculation speed and memory sizes will be still problem near future. Generally, the controlled object is modeled by dx = fm (x, u) , dt

x ∈ Rn , u ∈ Rm .

(10.1)

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Fig. 10.7 Trends of CPU speed and memory size for engine control

It is considered that almost all control problems can be reduced to the minimization of the const function described by T

V (x, u) = 0

l (x, u) dt + F (x (T )) ,

(10.2)

where l (x, u) > 0 and u ∈ U. U is the admissible set of the input. The solution is given through the Hamilton-Jacobi-Bellman equation with the boundary condition V ∗ (x, T ) = F (x) as follows:

∂ ∗ ∂ V (x, u) = min H x, u, V ∗ (x, u) . (10.3) u∈U ∂t ∂x Hamiltonian H in equation (10.3) is expressed by H (x, u, λ ) = l (x, u) + λ f (x, u) . The input u∗ (x) minimizing H is given by

∂ ∗ ∗ u (x) = arg min H x, u, V (x, u) . u∈U ∂x

(10.4)

(10.5)

However, the equation (10.3) can’t be solved generally. MPC avoids the difficulty by calculating parameterized discrete time inputs from the current time to the specified horizon. MPC uses the cost function VMPC (k) usually expressed by Hp

VMPC (k) =

∑

i=Hw

Hu

ˆz (k + i |k ) − r (k + i |k )2Q + ∑ Δ uˆ (k + i |k )2R , i=0

(10.6)

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where the time is expressed by t = T k, (k = 0, 1, 2, · · ·), T is the sampling period, zˆ is the predicted controlled variable vector, uˆ is the predicted input vector, H p is the prediction horizon, Hw is the window parameter, Hu is the control horizon, Q is positive semi-definite matrix and Ris positive definite matrix. Here, Hw can be related with the time delay from the input and the output. To minimize the cost function (10.6), time consuming numerical calculation is necessary when a model is complex and the input constraints are used. Therefore, there are two ways basically. One is that an obtained control is simplified. For the purpose, the relationship between the inputs and the outputs is recreated within the acceptable error by the simple control model. However, the impact analysis of the approximation error may be not easy. The other way is that the original plant model is simplified and the constraints are relaxed or ignored. However, the advantages of MPC may be lost. In order to apply MPC to complex nonlinear systems with constraints, the references [13], [14] and [15] may be consulted.

10.3 Benchmark Problem SICE (Society of Instrument and Control Engineers) provided a V6 spark ignition engine model and a control design problem for benchmarking control theories in 2006. Nine Japanese universities wrestled with the problem and succeeded to start the engine. Figure 10.8 shows the model consisting of the engine, the starter and the controller blocks on SIMULINK. They can’t change the inputs and the outputs of the controller block. Figure 10.9 shows the inside of the engine block. The Challengers could freely look into the insides of the blocks. However, no document was provided and the challengers had to derive the model equations by their selves if they want. The model is an instantaneous one and the cylinder pressures are calculated. The combustible air-fuel ratio is defined from 9 to 20. The model is constructed with first principles (conservation lows) without any adjustments by using experimental data rpm

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in order that the challengers can guess the engine behavior with physical considerations. Therefore, it isn’t accurately coincident with actual engine behaviors although qualitative behaviors are well captured. The inputs and the outputs of the controller block are as follows: x˙ (t) = fengine (x (t) , u1 (t) , u2 (k) , u3 (k))

(10.7)

y (t) = [y1 (t) , y2 (t), y3 (t)] T

(10.8)

Inputs: y1 (t) : engine speed y2 (t) : crank angle of the cylinder #1 y3 (t) : air flow rate at the throttle valve Outputs: u1 (t) : throttle valve angle u2 (k) = [ fi1 , fi2 , fi3 , fi4 , fi5 , fi6 ] :spark advances for the cylinders u3 (k) = [Sa1 , Sa2 , Si3 , Sa4 , Sa5 , Sa6 ] : fuel injections for the cylinders. The requirements of the control design are as follows. The closed loop system should be asymptotically stable: 1. 2. 3. 4.

the engine speed should reach 650 ± 50rpm within 1.5s; the overshoot of the engine speed after ignition should be sufficiently suppressed; the engine speed should be recovered in response to disturbances; and chattering shouldn’t appear.

In addition to the above requirements, the robustness is also required. We specify variances of the following items to check the robustness: 1. initial crank angle; 2. friction torque;

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3. starter motor speed; and 4. fuel evaporation characteristics (Normal and slow evaporations). Figure 10.10 shows the engine speed excursion when the spark advances are constant and the air fuel ratio is controlled at the stoichiometric. The engine speed rapidly rises, the big overshoot appears and the engine speed converges around 650rpm with slow-moving fluctuation attenuating gradually. The most difficult point of this problem is that the engine behavior drastically changes between firing and misfiring as shown in Figure 10.11. Therefore, the AF ?

Air mass LL : lower limit Fuel mass

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Fig. 10.13 Structure of the control

challengers must avoid misfiring. The next difficulty is that the first ignition easily releases the sufficient amount of energy to make the overshoot. This means that the feedforward control is essential at starting the engine. Figure 10.12 shows an example of control result also provided for reference. To avoid misfiring, air-fuel ratio must be kept within the combustible air-fuel ratio range. For the purpose, the air charge estimation is important. Negligently, the engine model can be copied in the controller block for the purpose. The fuel injection control can be derived from the inverse model of the fuel behavior model. By using air charge estimation and the fuel injection control, the engine can start.

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After that, the engine speed can be controlled with the throttle valve and the spark advance controls. The spark advance controller adopts a PID control to regulate the engine speed at 650rpm and the throttle valve angle is controlled so that the spark advance gives the best torque. However, the above control design can’t suppress the overshoot. Thus, the throttle angle at the beginning should be sufficiently small to reduce the air charge at the first ignition. However, the throttle angle can’t keep engine running. Therefore, the throttle angle must be opened after the engine running. Finally, the spark advance feedback is designed to stabilize the engine speed. The control structure is shown in Figure 10.13. This method is a corner-cutting control design. However, it simply shows that this problem has a solution at least. It also indicates that a required control can be obtained if the engine models can be accurately simplified and the above control can be simplified.

10.4 Application of MPC In this study, we adopted the strategy that the developed controller should be embedded on a production ECU. Thus, we decided to apply Generalized Predictive Control (GPC) to the local linear models. However as mentioned in the chapter 3, the first firing may cause the overshoot and the nonlinearity is extremely strong and the characteristic rapidly changes from every cycle to cycle when staring the engine. In this situation, feedback controls can’t work effectively. Therefore, we have to apply a scheduled feedforward control to suppress the overshoot sufficiently and activate a feedback control after the change becomes slow. In order to determine the scheduled feedforward control, the sequence of the fuel injections, the spark ignitions and throttle angle during 30 strokes were optimized with Particle Swarm Optimization (PSO)[10, 11]. The reason why we chose PSO was that the optimization parameter space was huge, 30 cycles × 13 inputs, and the engine model was a very complex nonlinear model. The following cost function was applied to the input sequence optimization. 2 + Q2 ymaxy −yd d k=1 2 2 30 30 u −u u1k −u0 1 1 1k 1k−1 +Q3 ∑ 30 + Q4 ∑ 30 u˙1 max u0 30

J = Q1 ∑

1 30

k=1

y1k −yd yd

2

(10.9)

k=1

But, the optimization was interrupted by the conditions of misfiring. Therefore, the fuel injection control was developed in advance of PSO. First, an observer was designed to estimate the air charge based on a simple dynamic air model that was a mean value model. Next, a simple discrete time fuel model was derived and the inverse model was constructed. The fuel injection control consisted of these two. Under the injection control, the sequence of the spark advances and the throttle valve angle were optimized by PSO. The weights from Q1 to Q4 had to be determined by a trial and error method. Figure 10.16 shows the excursion of the engine speed with the optimized input sequence shown in Figure 10.15. The designed control

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succeeded to start the engine, remove the overshoot and control the engine speed at 650rpm rapidly. It is better than the example provided. Only the feedforward control isn’t robust and it is required to design the feedback control. The local ARX models are given by JIT (Just In Time) modeling method [9]. JIT can avoid the troublesome modeling effort but derive the required linear model from the stored simulation data. A query of the input and the output sequence determines the best ARX model corresponding to the operation condition L expressed by (10.10) ΦL q−1 y (k) = ΨL q−1 u (k − 1) .

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The database of input and output data sets is created in advance by using provided simulation results. The query chose neighbor sets and the local ARX model is identified by using the data. The error was defined by e (k) = y (k) − yr (k). Here, −1 q = Δ yr(k) = 0. Equation (10.10) is transferred to equation (10.11) with Δ Φ L AL q−1 , ΨL q−1 = BL q−1 and white noise ξ (k): AL q−1 e (k) = BL q−1 Δ u (k − 1) + ξ (k) .

(10.11)

The following cost function is expressed by J = Eˆ (k)T Λe Eˆ (k) + Δ U (k)T Λu Δ U (k) .

(10.12)

Where Eˆ (k) = [eˆ (k + N1 ) , eˆ (k + N1 + 1) , · · · , eˆ (k + N2 )]T

Δ U (k) = [Δ u (k) , Δ u (k + 1) , · · · , Δ u (k + Nu − 1)]T Λe (k) = diag([λe (N1 ) , λe (N1 + 1) , · · · , λe (N2 )]) Λu (k) = diag([λu (1) , λe (2) , · · · , λe (Nu )]T ). Here, the following Diophantine’s equations are introduced: T q−1 = AL q−1 E j q−1 + q− j F q−1 E j q−1 BL q−1 = T q−1 G j q−1 + q− j S q−1 . Equations (10.11), (16.4) and (10.14) derive the following equation: −1 E j q−1 e (k + j) = G j q Δ u (k + j − 1) + h j (k) + ξ (k + j) T (q−1 ) where h j (k) =

Fj (q−1 ) T (q−1 )

e (k) +

S j (q−1 ) T (q−1 )

(10.13)

(10.14)

(10.15)

Δ u (k − 1).

Therefore, the output prediction is expressed as follows: Eˆ (k) = G j Δ U (k) + H (k) .

(10.16)

H (k) = [h (k) , k (k + 1) , · · · , h (k + Nu − 1)]T .

(10.17)

Where H (k) is defined by

To obtain the control minimizing the cost function (10.12), the equation (10.16) is substituted into the equation (10.11) and the partial differentiation with respect to Δ U (k) gives the following equation: −1 T Δ U (k) = − GTj Λe G j + Λu G Λe H (k) .

(10.18)

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The deference Δ u (k) of the input is chosen from the equation (10.18) and the control at k is finally obtained by u (k) = u (k − 1) + Δ u (k) .

(10.19)

To evaluate the robustness of this control, farther simulations are performed with changing the initial crank angle, the engine friction loss and the fuel evaporation characteristic. A typical result is shown in Figure 10.16. The figure shows the sufficient robustness. Misfiring often interrupted the input sequence optimization. Actually, even the first ignition couldn’t be succeeded without the realization of the combustible airfuel ratio. Therefore, the fuel injection control had to be developed first in order to avoid misfiring. For the purpose, the air charge observer is derived from a simple air charge model and the inverse model of the injected fuel behavior is effective to keep the combustible air-fuel ratio. The next important observation was that the first ignition easily caused the engine speed overshoot. Therefore, it was obvious that the engine speed feedback was too slow to suppress the overshoot and the feedforward with the throttle and the spark advance were necessary. The feedforward design with the optimized input sequences was necessary. The activation timing of the feedback based on GPC has to be decided. The nonlinearity of the engine highly depends on the engine speed. Therefore, the linear model around the target engine speed can be effective when the feedforward succeeded to regulate the engine speed very close the target speed. However, that looses the robustness. The feedforward design may be simplified and more consideration and evaluations can be necessary. The GPC is not memory consuming and the most memory consuming portion is to obtain the local linear model. Actually, this control design is quite similar to the current design in automotive industry. However, the benefit of MPC became hardly visible. Nonlinear MPC must be tried if we want to get the full 800

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advantages of MPC. The scheduled feedforward can be derived with MPC concept and the control design can be more unified way.

10.5 Summary The automotive industry has encountered the complexity issue of control system developments. Model-based Development (MBD) is expected to resolve the issue. Control Design is the central technology in MBD and systematic control design is highly required because many engineers are collaborating in automotive control system developments. In order to investigate the possible control designs, SICE provided a benchmark problem with a V6 spark ignition engine model. The challengers had to start the engine model and stabilize the engine speed within 1.5s. MPC is a candidate of recommended control design in MBD. However, it needs the high execution speed and the big memory sizes of controller. We deeply considered the balance between the embeddable control and the expected systematic control design through the benchmark problem. We have obtained the following results of MPC applied to the benchmark problem. 1. We took the strategy that GPC embeddable on production ECUs was given the first priority and the obtained benefits were evaluated. 2. The developed control succeeded to satisfy the requirements of the benchmark problem. However, the following heuristic works were necessary: • the feedforward design based on the inputs sequence optimized by PSO; • the local linear models construction with JIT modeling; and • the adjustment of when the feedback of GPC starts. 3. The constraint of embeddable MPC makes the potential advantages hardly visible because it was difficult to derive the required control systematically and heuristic works were required. 4. We have to continue to investigate NMPC if it can efficiently derive the feedforward that was derived by PSO in this benchmark trial. 5. We will have to devise the methodology installing NMPC on production ECUs or adopt the control simplification. For the comparison with other methods, the papers in MoA28 “Benchmark for Engine Cold Start” in IFAC World Congress 2008 are good references [12].

References [1] Maciejowski, J.M.: Predictive Control with Constraints. Pearson Education Limited, London (2002) [2] Camacho, E.F., Bordons, C.: Model Predictive Control in the Process Industry. In: Advances in Industrial Control. Springer, Berlin (1995) [3] Clarke, D.W.: Application of Generalized Predictive Control to Industrial Process. IEEE Control System Magazine 122, 49–55 (1988)

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[4] Kaneda, Y., Yamakita, M.: An Application of JIT Modeling and Control with a Nonlinear Local Model to IPMC Artificial Muscle Actuatirs. In: Proc. of SICE 2005 (2005) [5] Heywood, J.B.: Internal Combustion Engine Fundamentals. McGraw-Hill, New York (1998) [6] Kiecke, U., Nielsen, L.: Automotive Control Systems for Engine, Driveline and Vehicle, 2nd edn. Springer, Heidelberg (2004) [7] Guzzela, L., Onder, C.H.: Introduction to Modeling and Control of Internal Combustion Engine Systems, pp. 178–179. Springer, Heidelberg (2004) [8] Akaike, H.: Fitting Autoregressive Model for Prediction. Annals of the Institute of Statistical Mathematics 21, 243–247 (1969) [9] Stenman, A., Gustsfsson, F., Ljung, L.: Just In Time Models for Dynamical Systems. In: Proc. 35th Conf. Decision and Control, pp. 1115–1120 (1996) [10] Clerc, M.: Particle Swarm Optimization, ISTE (2006) [11] Kennedy, J., Ebeerhart, R.: Particle Swarm Optimization. In: Proc. of IEEE Int. Conf. on Neural Networks, Piscataway, NJ, pp. 1942–1948 (1995) [12] MoA28.1 – MoA28.5, in IFAC WC 228, session MoA28, Benchmark for Engine Cold Start [13] Ohtsuka, T.: A continuation/GMRES method for fast computation of nonlinear receding horizon control. Automatica 40, 563–574 (2004) [14] Murayama, A., Yamakita, M.: Speed Control of Vehicles with Variable Valve Lift Engine by Nonlinear MPC. In: Proc. of ICROS-SICE 2009 (2009) [15] Canale, M., Fagiano, L., Milanese, M.: Set Membership approximation theory for fast implementation of Model Predictive Control laws 45, 45–54 (2009)

Chapter 11

Model Predictive Control of Partially Premixed Combustion Per Tunest˚al and Magnus Lewander

Abstract. Partially premixed combustion is a compression ignited combustion strategy where high exhaust gas recirculation (EGR) levels in combination with early or late injection timing result in a prolonged ignition delay yielding a more premixed charge than with conventional diesel combustion. With this concept it is possible to get low smoke and NOx emissions simultaneously. Accurate control of injection timing and injection duration is however necessary in order to achieve this favorable mode of combustion. This chapter presents a method for controlled PPC operation. The approach is to control the time between end of injection and start of combustion which if positive yields sufficient premixing. Model Predictive Control was used to control the engine which was modeled using System Identification. The results show that it is possible to assure PPC operation in the presence of both speed/load transients and EGR disturbances.

11.1 Introduction Partially premixed combustion (PPC) is a promising combustion concept which combines low smoke and NOx emissions with higher controllability than with homogeneous charge compression ignition (HCCI). PPC occurs when the ignition delay is long enough for fuel and air to mix prior to combustion. A prolonged ignition delay can be achieved by advanced or retarded injection timing in combination with high EGR levels [4, 5, 6, 8]. The EGR lowers the combustion temperature which leads to a decrease in NOx while the enhanced mixing yields lower smoke levels. With even more EGR it is possible to obtain rich low temperature Diesel Per Tunest˚al Lund University, P.O. Box 118, 221 00 Lund, Sweden e-mail: [email protected] Magnus Lewander Lund University, P.O. Box 118, 221 00 Lund, Sweden e-mail: [email protected] L. del Re et al. (Eds.): Automotive Model Predictive Control, LNCIS 402, pp. 171–181. c Springer-Verlag Berlin Heidelberg 2010 springerlink.com

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combustion. This approach is simulated in [1] and is shown to produce low smoke and NOx emissions. The penalty in combustion efficiency is however substantial so the approach taken in this chapter is with lean combustion. It is of interest to find a way of operating the engine with enough ignition delay to get a large degree of premixing while maintaining sufficient control of engine load and combustion phasing. The above mentioned studies have presented PPC for steady state conditions mainly applied to single-cylinder engines. This work is focused on how to achieve PPC in a multi cylinder engine by applying cylinder individual, cycle to cycle control with the fuel injection system as actuator. It is desired to be able to control PPC, i.e. the level of premixing, at different engine speeds, EGR levels, and loads to make the combustion system versatile. Since load is highly dependent on the amount of fuel injected, load control needs to be integrated into the concept. Additionally, the combustion timing must be kept within a certain region to prevent excessive pressure rise rates and to maintain engine efficiency. Keeping an exast combustion timing is however not necessary. There are also constraints in the injection equipment that need to be considered. Given these requirements a suitable control method is model predictive control (MPC) which has good multiple input multiple output (MIMO) properties and takes constraints into account explicitly [7, 3]. In the presented approach constraints are introduced both for actuator saturation regarding injection timing and for combustion phasing in order to ensure high efficiency. In this study system identification was used to model the dynamics between fuel injection and combustion characteristics such as mean effective pressure, combustion phasing and mixing period, i.e. delay between end of injection (EOI) and start of combustion (SOC). The system identification results in a linear time invariant model which allows implementation of explicit MPC.

11.2 Experimental Setup The engine that was used in these experiments is a Volvo diesel engine with specification as in Table 11.1. It has a short route EGR system and a variable geometry turbine (VGT) which is used to get the desired inlet pressure. The VGT also generates exhaust back pressure which, together with the inlet pressure and the EGR-throttle position, determines the EGR level.

11.3 PPC Definition To make sure that the engine runs in PPC mode a measurable system output that is closely related to the combustion character is needed. It has previously been shown that mixing period (MP) [6] provides such information. Additionally it is measured every cycle and suitable for feedback control. MP is defined as the crank angle delay between end of injection and the crank angle of 10% accumulated heat release (CA10). CA10 is thought of as a practical

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Table 11.1 Engine Specifications Operated Cylinders Displaced Volume Bore Stroke Connecting Rod Length Number of Valves Compression Ratio Fuel Supply

6 2.13 l 131 mm 158 mm 267.5 mm 24 16 DI

measure of the start of combustion and it is chosen since it occurs sufficiently early in the combustion event but is more robust than for example CA1. If MP is positive it means that all the fuel was injected before the combustion onset which promotes premixed combustion and defines PPC mode in this study. The relation between MP and the combustion character can be studied in Figure 11.1 which was first published in [6]. Points with positive MP show premixed combustion while points with negative MP include a significant part of diffusion combustion.

Fig. 11.1 Mixing period and the corresponding heat release rate (HRR) [6]

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11.4 Control Since an operating point is defined by engine speed and load, error free set-point tracking of indicated mean effective pressure (IMEP) is crucial. The engine should run in PPC mode when possible so control of MP is also very important. MP and CA50 are highly dependent on both start of injection (SOI) and ignition delay (ID) while IMEP depends almost solely on ID. It is possible to control MP or CA50 to arbitrary set-points given a specific ID but not simultaneously. Hence, it was decided that an MP that assures PPC should be prioritized before set-point tracking of CA50. CA50 has an influence on both efficiency and combustion noise but near the optimum timing the dependencies are quite week. The resulting strategy was to apply MPC to keep CA50 in a predefined region by setting constraints that allow early as well as conventional combustion timings while IMEP and MP are controlled to set-point values. The set-point value for MP was set to zero which should give sufficient air/fuel premixing if maintained. MPC has previously been successfully applied to engine control problems and an early example is presented in [2].

11.4.1 Control Design The tuning parameters for the MPC used for PPC control were chosen as follows. The control signal, SOI, was constrained to be no earlier than 17 crank angle degrees

Fig. 11.2 Response for a rapid change in EGR at 1200 revolutions per minute (rpm)

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(CAD) before top dead center (BTDC) in order to maintain sufficient start of injection pressure. The rate by which the control signals could change was constrained which made the system more robust. On the output side, CA50 was constrained to be in the interval [-4, 12] in order to prevent over-advanced or over-retarded combustion. The input constraints were hard while the output constraints were soft. The output weights were selected such that control errors in IMEP and MP were almost equally penalized while errors in CA50 had a very low penalty which had the effect that CA50 did not follow any reference. This approach was applied to emulate constrained but uncontrolled output functionality. A prediction horizon of 10 combustion cycles and a control horizon of 3 combustion cycles proved to be suitable. To include integral action, a disturbance observer approach was used instead of adding integrating states which would complicate the choice of weights in the control design. Integral action was used on IMEP and MP but not on CA50.

11.5 Results Controller performance was evaluated with respect to three kinds of disturbances: set-point changes of IMEP; step changes of EGR level; and ramp changes of engine speed. The results are presented in the Sections 11.5.1–11.5.3.

11.5.1 Response to EGR Disturbance In Figure 11.2 a rapid decrease in EGR level is applied to the controlled system. Since the EGR level can not be correctly measured or estimated during this change

Fig. 11.3 PPC development in one cylinder for a rapid change in EGR level

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Fig. 11.4 Step response for changes in load at 1500 rpm

the presented EGR level is changed from the initial value to the final value in one engine cycle while it takes a couple of cycles in reality. In this case the controller manages to keep both IMEP and MP close to their setpoints. It is observed that the difference in CA50 between the 40% EGR case and the 30% EGR case is large both in terms of the mean value and the spread between the cylinders. The heat release rate (HRR) for Cylinder 1 during the EGR change is presented in Figure 11.3. The change in combustion timing is very clear as well as the change in shape. The figure shows that the controller manages to make the engine run in PPC mode during the decrease in EGR. All HRRs show the characteristic low

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Fig. 11.5 PPC for different cylinders versus conventional Diesel combustion

temperature reactions part and then pre-mixed combustion where the low EGR case shows some pressure oscillations following an earlier combustion with higher HRR.

11.5.2 Response to Load Changes Figure 11.4 shows the results for changes in load at 1500 rpm. The controller follows the set-point in IMEP while the mixing period is slightly negative for the highest load for all cylinders except Cylinders 4 and 5. The reason for this behavior is saturation of SOI and the shorter injection duration needed for Cylinders 4 and 5. In addition to the physical explanation Figure 11.4 shows the importance of cylinder individual control. When the load is lowered the controller manages to keep MP at its set-point again. CA50 is close to the constraint and violates it slightly for Cylinders 5 and 6 when the load is changed. Figure 11.5 illustrates how the controller achieves PPC operation in different ways for three cylinders in Cycle 50 from the experiments presented above. A comparison is made to combustion achieved with conventional timings at the same load. Studying the shape of the HRR it seems like Cylinder 1 is running with less EGR or higher compression than Cylinder 4 which has longer combustion duration and later timing. All HRRs show evidence of low temperature reactions which are characteristic for PPC.

11.5.3 Response to Speed Changes Figure 11.6 shows the response when changing the engine speed from 1200 rpm to 1800 rpm. The step change between 1200 and 1400 rpm is taken care of very

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Fig. 11.6 PPC development in one cylinder for a rapid change in EGR level

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Fig. 11.7 PPC development in one cylinder for a rapid change in engine speed from 1200 rpm to 1400 rpm

quickly by the controller. However, a dip in MP follows a couple of cycles later due to the translation of ID from crank angle to time. Since the engine speed is sampled once per second, the speed used in the translation may be subject to a delay of up to one second. With engine speed measurement this disturbance would have been avoided. Up to 1600 rpm the controller performs well but then it is not possible to keep MP at 0 or IMEP at 7 bar due to physical limitations. The HRR during the step change in speed from 1200 rpm to 1500 rpm is shown in Figure 11.7. Between Cycles 94 and 96 major changes are observed. The system stabilizes again between Cycles 100 and 110. The controller maintains the engine in PPC operation for every cycle during the change in engine speed.

11.6 Discussion The presented approach is able to maintain PPC within physical limitations. This means that when the load gets too high MP will become negative and thus a certain degree of diffusion combustion will take place. The controller will however ensure that the smallest possible MP is maintained and thus minimize the diffusion part of the combustion. The strategy to control MP to a set-point and merely constrain CA50 to an acceptable interval comes from the fact that it is impossible to control MP and combustion timing independently with a single injection setup. For each load the combination of EGR level, inlet pressure and temperature will define the feasible CA50-MP space. The presented control method is thus not to be seen as a standalone engine controller but requires a supervisory controller that sets the stage in terms of load reference, EGR level, inlet pressure and temperature. The presented

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MPC controller will then control the fuel injection timing and duration to ensure PPC, if possible.

11.7 Conclusions Model predictive control has been used to control partially premixed combustion in a six-cylinder heavy duty engine and the following conclusions can be drawn. PPC operation can be assured by feedback control of the mixing period, i.e. the time between end of injection and start of combustion. By keeping MP at zero the air/fuel premixing is sufficient to yield PPC. A linear 3rd order state-space model with start of injection and injection duration as inputs and CA50, IMEP and MP as outputs was found using system identification. This model was validated for different speeds, loads, air/fuel ratios and EGR levels and managed to capture the general trends and dynamics of the system. Model Predictive Control is a suitable control method since it has good MIMO properties and takes constraints into account explicitly. This makes it possible to keep CA50 in a region of permitted combustion timings without controlling it to a set-point which is necessary in order to be able to control MP and IMEP freely. The results show that the controller performs well up to the physical limitations. It is able to do reference tracking for both IMEP and MP and disturbances in EGR and engine speed are compensated for.

Abbreviations ATDC After Top Dead Center BTDC Before Top Dead Center CAx Crank Angle of x % accumulated heat release CAD Crank Angle Degrees EGR Exhaust Gas Recirculation HCCI Homogeneous Charge Compression Ignition HRR Heat Release Rate ID Injection Duration IMEP Indicated Mean Effective Pressure MIMO Multiple Input Multiple Output MP Mixing Period MPC Model Predictive Control NOx Oxides of nitrogen PPC Partially Premixed Combustion rpm Revolutions Per Minute SOI Start Of Injection VGT Variable Geometry Turbine

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References [1] Akihama, K., et al.: Mechanism of the Smokeless Rich Diesel Combustion by Reducing Temperature. SAE Transactions, Society of Automotive Engineers 110(3), 648–662 (2001) [2] Bengtsson, J., Strandh, P., Johansson, R., Tunest˚al, P., Johansson, B.: Model Predictive Control of Homogeneous Charge Compression Ignition (HCCI) Engine Dynamics. In: Proceedings of the 2006 IEEE International Conference on Control Applications. IEEE, Los Alamitos (2006) [3] Camacho, E.F., Bordons, C.: Model Predictive Control, 2nd printing edn. Springer, Heidelberg (2007) [4] Kimura, S., et al.: New Combustion Concept for Ultra Clean and High-Efficiency Small DI Diesel Engines. SAE Transactions, Society of Automotive Engineers 108(3), 2128– 2137 (1999) [5] Kimura, S., et al.: Ultra-Clean Combustion Technology Combining a Low-Temperature and Premixed Combustion Concept for Meeting Future Emission Standards. SAE Transactions, Society of Automotive Engineers 110(4), 239–248 (2001) [6] Lewander, M., Ekholm, K., Johansson, B.: Tunest˚al, P., Milovanovic, N., Keeler, N., Harcombe, T., Bergstrand, P.: Investigation of the Combustion Characteristics with Focus on Partially Premixed Combustion in a Heavy Duty Engine. SAE Technical Paper 200801-1658. Society of Automotive Engineers (2008) [7] Maciejowski, J.M.: Predictive Control with Constraints. Pearson Education, Essex (2002) [8] Musculus, M.P.B.: Multiple Simultaneous Optical Diagnostic Imaging of Early-Injection Low-Temperature Combustion in a Heavy-Duty Diesel Engine. SAE Transactions, Society of Automotive Engineers 115(3), 83–110 (2007)

Chapter 12

Model Predictive Powertrain Control: An Application to Idle Speed Regulation Stefano Di Cairano, Diana Yanakiev, Alberto Bemporad, Ilya Kolmanovsky, and Davor Hrovat

Abstract. Model Predictive Control (MPC) can enable powertrain systems to satisfy more stringent vehicle requirements. To illustrate this, we consider an application of MPC to idle speed regulation in spark ignition engines. Improved idle speed regulation can translate into improved fuel economy, while improper control can lead to engine stalls. From a control point of view, idle speed regulation is challenging, since the plant is subject to time delay and constraints. In this chapter, we first obtain a control-oriented model where ancillary states are added to account for delay and performance specifications. Then the MPC optimization problem is defined. The MPC feedback law is synthesized as a piecewise affine function, suitable for implementation in automotive microcontrollers. The obtained design has been extensively tested in a vehicle under different operating conditions. Finally, we show how competing requirements can be met by a switched MPC controller.

12.1 Introduction As the requirements for automotive vehicles become more stringent and customers expect improved driveability and better fuel economy, advanced powertrain control strategies become an appealing tool to be exploited in meeting these objectives. In this chapter we consider the use of Model Predictive Control (MPC) [14] as a control strategy to achieve high controller performance. MPC presents several advantages with respect to standard control strategies, such as the capability of explicitly enforcing constraints and the optimization of a user defined performance objective, while maintaining the inherent robustness of feedback control. Explicit model Stefano Di Cairano, Diana Yanakiev, Ilya Kolmanovsky, and Davor Hrovat Powertrain Control R&A, Ford Motor Company, Dearborn, MI e-mail: {sdicaira,dyanakie,ikolmano,dhrovat}@ford.com Alberto Bemporad Dipartimento di Ingegneria dell’Informazione, Universit`a di Siena, Italy e-mail: [email protected] L. del Re et al. (Eds.): Automotive Model Predictive Control, LNCIS 402, pp. 183–194. c Springer-Verlag Berlin Heidelberg 2010 springerlink.com

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predictive control techniques [2] can be used to synthesize the controller in the form of a piecewise affine function, which is feasible for implementation microcontrollers that have limited memory and computational power. In this chapter we demonstrate the design, synthesis, and experimental validation of a model predictive controller for Idle Speed Control (ISC) [10, 8, 9]. ISC is of continuing importance in spark ignited engines, since a higher performance controller may allow for lower engine idle speed or for operating with lower spark reserve with no risk of engine stalls. Lower engine idle speed or lower spark reserve reduce fuel consumption [8]. Several techniques have been considered to control idle speed, including state feedback [10], speed gradient with backstepping [11], observer-based control [3, 6], output feedback stabilization [7], hybrid controllers [1], and adaptive control [15], see also the references therein. In this chapter we design a controller that regulates engine speed at the indicated idle setpoint by controlling the airflow in the engine manifold via an electronic throttle. The challenges are the delay between the change in the commanded airflow and the change in the engine torque, and the constraints on the system input (limits on the throttle position) and output (allowed range for the engine speed error). The chapter is organized as follows. In Section 12.2 we introduce a nonlinear engine model used for idle speed control and for simulation, and in Section 12.3 the MPC controller, based on a linearized model, is designed. After the controller is tested in closed-loop with the nonlinear model, the complexity is assessed and the control law is synthesized in Section 12.4. Experimental validation is discussed in Section 12.5. The conclusions are summarized in Section 12.6

12.2 Engine Model for Idle Speed Control Engine rotational dynamics are based on Newton’s second law, 1 30 N˙ = (Me − ML ), J π

(12.1)

where N is the engine speed measured in rpm, Me [Nm] is the net engine brake torque, and ML [Nm] is the load torque applied to the crankshaft. In port-fuel injection engines, the torque cannot be instantaneously changed, and engine torque production dynamics have to be considered, Me (t) = Me,δ (t − td ) − Mfr (t) − Mpmp(t),

(12.2)

where Me,δ (t − td ) [Nm] is the indicated torque delayed by the intake-to-torque production time delay, td [s], Mfr (t) are the engine friction torque losses, and Mpmp (t) are the pumping torque losses. In Equation 12.2, constant spark retard is assumed, while near idle Mfr (t) + Mpmp (t) can be approximated as constant. The delay td is about 360 degrees of the crankshaft revolution, td (t) =

60 . N(t)

(12.3)

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Additional dynamics are associated with manifold filling. Under constant gas temperature assumption, the intake manifold pressure dynamics are p˙im =

RTim (Wth − Wcyl ), Vim

(12.4)

where Wth [kg/s] is the mass flow rate through the electronic throttle and Wcyl [kg/s] is the mean value of the mass flow rate into the engine cylinders. In (12.4), Tim [K] is the intake manifold temperature, Vim [m3 ] is the intake manifold volume and R is the ideal gas constant in kJ/kg/K. Assuming stoichiometric air-to-fuel ratio, the engine indicated torque is approximately proportional to the cylinder air charge, mcyl [kg], Me,δ (t) = γ1 mcyl (t) = γ1

Wcyl (t) . N(t)

(12.5)

where γ1 is an engine dependent parameter. The cylinder flow is a function of the intake manifold pressure, pim [Pa] and engine speed, N, Wcyl (t) =

γ2 pim (t)N(t) + γ0 , γ1

(12.6)

and γ0 , γ2 are constant parameters. Near idle, the flow through the throttle is choked and one can approximate Wth (t) = γ3 ϑ (t), (12.7) where ϑ [deg] is the throttle position and γ3 is an engine dependent constant. From γ1 (12.5), (12.6) it follows that Me,δ (t) = γ2 pim (t) + N(t) γ0 . Differentiating this and using 12.4, (12.6), and (12.7) RTim N RTim γ0 · γ1 ˙ M + γ2 γ3 ϑ − 2 N. M˙ e,δ = −γ2 Vim γ1 e,δ Vim N Neglecting the (typically small) term

γ0 ·γ1 ˙ N N2

(12.8)

it follows that

RTim N RTim M + γ2 γ3 ϑ . M˙ e,δ = −γ2 V1 γ1 e,δ Vim

(12.9)

Thus, the complete engine model near idle is defined by 12.1, 12.2, 12.3 and 12.9.

12.3 Control-oriented Model and Controller Design From Section 12.2 it follows that the engine dynamics are described by a second order system 12.1, 12.2, 12.9 with time delay 12.3. Around idle they can be conveniently approximated as a second order system subject to input delay [9], Yp (s) = G(s)e−std U(s),

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¯ and N¯ is the where td can be approximated as constant by 12.3 where N(t) = N, idle speed setpoint. Model 12.10 represents the deviations of engine speed from ¯ in response to variations in throttle position idle speed setpoint, y p (t) = N(t) − N, form nominal position, u(t) = ϑ (t) − ϑ¯ , where ϑ¯ is the nominal throttle position to compensate for Mfr (t) + Mpmp(t) and for the nominal load torque M¯ L at idle. Model 12.10 has the advantage of being easily identifiable from both simulation and/or experimental data. The coefficients in 12.10 can be computed in the time domain from step-response data. In particular, we can identify the linearized model from an available high-fidelity nonlinear simulation model-based on 12.1, 12.2, 12.3 and 12.9 where the coefficients are obtained from experimental data. Although direct identification from data is possible, we prefer to use a nonlinear simulation model first, because it allows for testing the closed-loop behavior in simulation. Through simulations we obtain a qualitative understanding of the effects of the tuning knobs on the closed-loop behavior much more rapidly than by experiments. This understanding is fundamental to reduce the calibration time in the vehicle, so that the overall controller calibration procedure cost is reduced. Figure 12.1 shows the results of the identification where the parameters in 12.10 are obtained from the relations L ≈ exp(−πζ (1 − ζ 2 )−1/2 ), ωd = ω (1 − ζ 2 )1/2 , tM = πωd−1 , and where L is the normalized overshoot, tM is the oscillation period and ωd is the damped system frequency. The evolution of the engine speed as predicted ˆ = y p (t)+ N, ¯ is compared to the engine speed evolution by the linearized model, N(t) ˆ − for the nonlinear model, N(t), in Figure 12.1. The plot of the error εN (t) = N(t) N(t) indicates that a satisfactory fit is obtained. The identified transfer function G(s) is then converted to discrete-time state space form with sampling Ts = 30ms, x f (k + 1) = A f x f (k) + Buδ (k), y f (k) = C f x f (k),

x f ∈ R2 ,

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where uδ is the delay-free input. For idle speed N¯ = 600rpm, td = 100ms≈ 4Ts . The discrete-time model of a signal u(·) delayed by nδ ∈ Z0+ steps is x δ ∈ Rn δ ,

xδ (k + 1) = Aδ xδ (k) + Bδ u(k), uδ (k) = Cδ xδ (k), ( 0 ··· 0 ) Aδ = In −1 ... , δ 0

(12.12) Bδ = [1 0 · · · 0] , T

Cδ = [0 · · · 0 1].

By cascading 12.11 and 12.12, the complete discrete-time model of 12.10 is obtained, x x p (k + 1) = A p x p (k) + B pu(k), x p = xδf , x p ∈ R6 , (12.13a) (12.13b) y p (k) = C p x p (k). The plant model 12.13 has to be extended with ancillary states that are used to enforce specifications. In idle speed control, steady-state errors due to changes in load torque caused, for instance, by power steering or air-conditioning system, or due to errors in the scheduled airflow feedforward have to be removed. In order to achieve offset free constant disturbance rejection one can introduce integral action [12] by adding the dynamics q(k + 1) = q(k) + Ts y p (k), where q ∈ R is the discrete-time integral of the output. When integral action is added to 12.13, the resulting model is x(k + 1) = Ax(k) + Bu(k), y(k) = Cx(k), $ # # A 0 A = TsCpp 1 , B =

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(12.14a) (12.14b)

.

(12.14c)

Model 12.14 will be used as the MPC prediction model. In order to complete the specification of the plant, the constraints on systems inputs and states have to be defined. Limits on the engine speed, in rpm, are (conservatively) defined to avoid engine stalls and engine flares, −150 ≤ y p ≤ 150 .

(12.15)

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(12.16)

where uFF is the scheduled feedforward term in nominal conditions at idle, and ideally uFF = ϑ¯ . Note that the full (feedforward+feedback) throttle input is ϑ (t) = uFF + u(t). Finally, the cost function for the finite horizon optimal control problem in the MPC strategy is specified. Since a non-zero input u can be required at steady state to reject disturbances, weighting the input increment in the cost function is more suitable,

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h−1 J y(k), u(k), u(k − 1) = ∑ (y(i|k) − ry )T Q(y(i|k) − ry ) + Δ u(i|k)SΔ u(i|k) i=0

(12.17) where h ∈ Z is the prediction horizon, y(k) = y(0|k), . . . , y(h − 1|k) and u(k) = + u(0|k), . . . , u(h − 1|k) are the output and input sequences predicted at step k, respectively, ry is the output setpoint, Δ u(i|k) = u(i|k) − u(i|k − 1), and u(−1|k) = u(k − 1). Here, the weight matrices Q and S are both positive definite. Collecting 12.14, 12.15, 12.16, 12.17, the optimal control problem is formulated as h−1

min ρσ 2 + ∑ (y(i|k) − ry )T Q(y(i|k) − ry ) + Δ u(i|k)SΔ u(i|k) (12.18a)

σ ,u(k)

i=0

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(12.18b)

y(i|k) = Cx(i|k), i = 0, . . . , h − 1, umin ≤ u(i|k) ≤ umax , i = 0, . . . , h − 1,

(12.18c) (12.18d)

ymin − σ 1 ≤ y(i|k) ≤ ymax + σ 1, i = 0, . . . , hc − 1,

(12.18e)

σ ≥ 0, u(i|k) = u(hu − 1|k), i = hu , . . . , h − 1, u(−1|k) = u(k − 1), x(0|k) = x(k), ˆ

(12.18f) (12.18g)

where 1 is a vector entirely composed of 1, x(k) ˆ is the state estimate at time k, hc ≤ h is the constraint horizon, and hu ≤ h is the control horizon. For the idle speed MPC controller, in 12.18, −150 150 , ymax = , umin = −uFF , umax = 8 − uFF , ymin = −∞ +∞ and 12.18e enforce soft output constraints. Soft constraints can be violated at the price of a penalty, modelled by the optimization variable σ ∈ R0+ , weighted by the positive constant ρ , which must be at least two orders of magnitude larger than the other weights. Soft output constraints are used to avoid 12.18 becoming unfeasible due to large unmeasured disturbances. At each step k ∈ Z0+ , the MPC controller reads the measurements and computes the state estimate x(k), ˆ then solves the quadratic programming problem 12.18 to obtain the optimal input sequence u∗ (k), and applies the input u(k) = u∗ (0|k). For the idle speed controller, a Kalman filter has been used for state estimation. Closed-loop simulation tests were performed in Simulink with the MPC in closed-loop with the nonlinear engine simulation model 12.1, 12.2, 12.3 and 12.9. In the simulations, N¯ = 600rpm, ϑ¯ = 2.09, h = 30, hu = 2, hc = 3, ry = [0 0]T . The feedforward is uFF = 2.5, slightly different from ϑ¯ in order to test the regulation capabilities of the controller and the rejection of feedforward input errors. At t = 0s the controller has to regulate the system to 600rpm. Then, a disturbance rejection test is performed by introducing an unmeasured torque disturbance load M˜ L = 20Nm, simulating, for instance, the power steering motor load, starting at t = 7s, and ending at t = 22s.

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Fig. 12.2 Simulation of the MPC idle speed controller in closed-loop with the nonlinear engine model

Figure 12.2 shows the resulting closed-loop behavior for two different tunings of the cost function weights. In Figure 2(a) the closed-loop dynamics exhibit overshoot, while in Figure 2(b) they are overdamped.

12.4 Controller Synthesis and Refinement The controller designed in Section 12.3 needs to undergo further steps before being tested on the vehicle. First, it must be synthesized in a way that allows implementation in the automotive microcontrollers. Then, it must be refined to work properly with an experimental engine.

12.4.1 Feedback Law Synthesis and Functional Assessment Solving the optimal control problem 12.18 in conventional automotive microcontrollers may still be too computationally demanding. Also, the CPU operations and the memory requirements of the optimization algorithms are difficult to predict. To overcome these difficulties, the MPC controller is explicitly synthesized [2]. The explicit MPC control law of problem 12.18 is a piecewise affine state feedback, u(k) = ϕMPC (x(k), ry (k), u(k − 1)), where ϕMPC (x, ry , u−1 ) = Fix x + Fir ry + Fiu u−1 + gi if (x, ry , u−1 ) ∈ Pi , i = 1, . . . , s, and {Pi }si=1 is a polyhedral partition of the statereference-previous input space. For the idle speed control optimization problem 12.18, horz = 30, horzc = 3, horzu = 2, and as a consequence there are 4 input constraints and 6 output constraints enforced along the horizon. The obtained explicit controller is composed of 35 regions, and the parameters in function ϕ (·) are 7 states, 2 references, 1 previous input. The worst case number of operations for both region search and command computation that have to be executed at each control cycle is less than 5000, which amount to less than 2 · 105 operations per second. Hence, according to [13], the explicit controller uses less than 1% of the CPU in the worst case. It shall be noted that at idle the microcontroller is, in general, underloaded, due to the low rate of

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engine event triggered tasks. The controller will require less than 6KB of data memory storage. Finally, since the explicit feedback law is computed, local stability of the closedloop system can be simply evaluated. For the controller shown in Figure 2(b) the largest eigenvalue absolute value is λ max = 0.97, confirming that the closed-loop system is locally stable. Piecewise quadratic Lyapunov functions can be constructed using LMI to confirm global stability.

12.4.2 Prediction Model Refinement The controller designed in Section 12.3 is based on a simulation model. Although this allows for controller design, evaluation, and functional assessment without the need of experimental data, the controller benefits from a refinement to efficiently operate with a specific engine. The refinement involves two steps: (i) the controloriented engine model 12.10 is identified again, this time using experimental data, and, (ii) the weights of the cost function 12.17 are re-tuned. The results of step (i) in the experimental V6 4.6L engine are shown in Figure 12.3, where the identified engine model is validated with respect to the experimental data. The model error is larger than the one in Figure 12.1. However, since MPC is a feedback strategy, the resulting model uncertainty does not present an issue. The experimental data can be used to evaluate the measurements and process noise, and the modeling errors. This allows to appropriately tune the Kalman filter for estimating the state of model 12.14 from available measurements, the engine speed in our case.

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12.5 Experimental Validation The refined controller was experimentally validated in nominal and non-nominal conditions. A nominal test involving power-steering torque load rejection at nominal idle (N¯ = 600rpm) with the transmission in Neutral is shown in Figure 12.4. The MPC controller (solid line) is compared with a tuned baseline PID-based controller. The MPC significantly outperforms the baseline controller. The main reason is the explicit constraint enforcement, and the delay model, which allows to implement a more aggressive control action (by increasing the output regulation weight Q) without losing stability.

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The controller performance was also evaluated at non-nominal conditions, with the transmission in Drive, a decreased desired idle speed (N¯ = 525rpm), and, as a consequence, a larger delay. Also, due to the transmission load, the engine dynamics are slightly different. In the test (see Figure 12.5) two disturbances are introduced and then removed almost simultaneously, power-steering load and air conditioning load. Due to the recognized large increase in the load, the scheduled setpoint for idle speed changes during the experiment. The MPC controller outperforms again the baseline controller, both in terms of disturbance rejection and time-varying reference tracking. Finally, in Figure 12.6 a cold-start test is shown, where the idle speed setpoint is initially very high (about 1250rpm), and then slowly approaches the nominal value (600rpm). The feedback controller is activated when the setpoint reaches 1000rpm. During the tests, several disturbances are introduced, including power steering load, air conditioning load, and multiple repetitions of these. Even in this test, the controller shows a very satisfactory performance. The tests in Figures 12.4–12.6 show that the MPC achieves higher performance by acting more aggressively on the control input. As a consequence, the controller

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may be unnecessarily sensitive around the setpoint due to measurement noise. In order to maintain high performance load rejection and reduce the sensitivity around the setpoint, two MPC controller can be scheduled, ϕ1 (x(k), ry (k), u(k − 1)) with low sensitivity, and ϕ2 (x(k), ry (k), u(k − 1)) with aggressive action. In particular, ϕ1 can be tuned to have the same sensitivity as the baseline controller by using the techniques presented in [4]. A switched control law is applied, u(k) = if |y p | < y then ϕ1 (x(k), ry (k), u(k − 1)) else ϕ2 (x(k), ry (k), u(k − 1)), (12.19)

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where y is the switching threshold that is selected by analyzing the measurement and the process noise. The results for such a controller in a test similar to the one in Figure 12.4 are shown in Figure 12.7, where we see that the MPC still outperforms the baseline, but now around the setpoint the controller action is more cautious.

12.6 Conclusions We have presented the design, synthesis, and validation of a model predictive controller for idle speed regulation. The MPC controller provides higher performance in terms of disturbance rejection and convergence rate than a tuned baseline PIDbased controller, and it is synthesized in a form that is suitable for implementation in standard automotive microcontrollers. The predictive features of the MPC strategy better account for intake-to-torque production time-delay, while performance objective and soft output constraints result in faster, nonlinear compensation of disturbances than the conventional baseline controller. As we have shown for the idle speed regulation problem, the MPC solution is well within the capability of automotive microcontrollers, with a CPU load smaller than 1%. An idle speed MPC controller commanding both throttle and spark delay is under investigation, and preliminary results are shown in [5].

References [1] Balluchi, A., Di Natale, F., Sangiovanni-Vincentelli, A., van Schuppen, J.: Synthesis for idle speed control of an automotive engine. In: Alur, R., Pappas, G.J. (eds.) HSCC 2004. LNCS, vol. 2993, pp. 80–94. Springer, Heidelberg (2004) [2] Bemporad, A., Morari, M., Dua, V., Pistikopoulos, E.: The explicit linear quadratic regulator for constrained systems. Automatica 38(1), 3–20 (2002)

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[3] Bengea, S., Li, X., De Carlo, R.: Combined controller-observer design for uncertain time delay systems with applications to engine idle speed control. J. of Dynamic Sys., Meas. and Control 126, 772–780 (2004) [4] Di Cairano, S., Bemporad, A.: Model predictive controller matching: Can MPC enjoy small signal properties of my favorite linear controller? In: Proc. European Control Conf. Budapest, Hungary (2009) (to appear) [5] Di Cairano, S., Yanakiev, D., Bemporad, A., Kolmanovsky, I., Hrovat, D.: An MPC design flow for automotive control and applications to idle speed regulation, Cancun, Mexico, pp. 5686–5691 (2008) [6] Gibson, A., Kolmanovsky, I., Hrovat, D.: Application of disturbance observers to automotive engine idle speed control for fuel economy improvement. In: Proc. American Contr. Conf., Minneapolis, MN, pp. 1197–1202 (2006) [7] Glielmo, L., Santini, S., Cascella, I.: Idle speed control through output feedback stabilization for finite time delay systems. In: Proc. American Contr. Conf., Chicago, IL, pp. 45–49 (2000) [8] Guzzella, L., Onder, C.: Introduction to modeling and control of internal combustion engine systems. Springer, Heidelberg (2004) [9] Hrovat, D., Sun, J.: Models and control methodologies for IC engine idle speed control design. Control Engineering Practice 5(8), 1093–1100 (1997) [10] Kiencke, U., Nielsen, L.: Automotive control systems for engine, driveline, and vehicle. Springer, Heidelberg (2000) [11] Kolmanovsky, I., Yanakiev, D.: Speed gradient control of nonlinear systems and its applications to automotive engine control. Journal of SICE 47(3) (2008) [12] Kwakernaak, H., Sivan, R.: Linear optimal control systems. Wiley-Interscience, New York (1972) [13] Magner, S., Cooper, S., Jankovic, M.: Engine control for multiple combustion optimization devices. In: Proc. SAE World Congress, Detroit, MI, Paper 26–21–0003 (2006) [14] Rawlings, J.: Tutorial overview of model predictive control. IEEE Control Systems Magazine, 38–52 (2000) [15] Yildiz, Y., Annaswamy, A., Yanakiev, D., Kolmanovsky, I.: Adaptive idle speed control for internal combustion engines. In: Proc. American Contr. Conf., New York, NY, pp. 3700–3705 (2007)

Chapter 13

On Low Complexity Predictive Approaches to Control of Autonomous Vehicles Paolo Falcone, Francesco Borrelli, Eric H. Tseng, and Davor Hrovat

Abstract. In this chapter we present low complexity predictive approaches to the control of autonomous vehicles. A general hierarchical architecture for fully autonomous vehicle guidance systems is presented together with a review of two control design paradigms. Our review emphasizes the trade off between performance and computational complexity at different control levels of the architecture. In particular, experimental results are presented, showing that if the controller at the lower level is properly designed, then it can handle system nonlinearities and model uncertainties even if those are not taken into account at the higher level.

13.1 Introduction to Autonomous Guidance Systems The architecture in Figure 13.1, borrowed from the aerospace field [15, 21, 22, 4], describes the main elements of an autonomous vehicle guidance system and it is composed of four modules: the trajectory/mode generator, the trajectory/mode replanning, the low-level control system, and the vehicle and the environmental model. The trajectory/mode planning module pre-computes off-line the vehicle trajectory together with the timing and conditions for operation mode change. In the Paolo Falcone Chalmers University of Technology, Department of Signals and Systems, G¨oteborg, SE-41296, Sweden e-mail: [email protected] Francesco Borrelli University of California at Berkeley, Department of Mechanical Engineering, Berkeley, CA 94720-1740, USA e-mail: [email protected] Eric H. Tseng and Davor Hrovat Ford Research Laboratories, Research and Innovation Center, 2101 Village Road, Dearborn, MI, 48121, USA e-mail: {htseng,dhrovat}@ford.com L. del Re et al. (Eds.): Automotive Model Predictive Control, LNCIS 402, pp. 195–210. c Springer-Verlag Berlin Heidelberg 2010 springerlink.com

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aerospace field, examples of operation mode selection include aeroshell parachute deployment or heatshield release. In the automotive filed this could include switching between two or more types of energy source (i.e., gas, electricity, hydrogen) or (in a very futuristic scenario) morphing between different vehicle shapes. The trajectory and the mode of operation computed off-line can be recomputed on-line during the drive by the trajectory/mode replanning module based on current measurements, at fixed points or on the occurrence of certain events (such as tracking errors exceeding certain bounds, hardware failure, excessive wind, the presence of an obstacle). The low-level control system commands the vehicle actuators such as front and rear steering angles, four brakes, engine torque, active differential and active suspensions based on sensor measurements, states and parameters estimations and reference commands coming from the trajectory/mode replanning module. Such reference commands can include lateral and longitudinal positions, pitch, yaw and roll rates. The low-level control system objective is to keep the vehicle as close as possible to the currently planned trajectory despite measurement noise, unmodeled dynamics, parametric uncertainties and sudden changes on vehicle and road conditions which are not (or not yet) taken into account by the trajectory replanner. In particular, when a vehicle is operating near its stability limit, these additional noises, disturbances and uncertainties must be considered, possibly through detecting the vehicle’s internal state, and compensated for. For example, if rear tires saturates, a skillful driver would switch his/her steering input from the usual steering command for trajectory following to a counter-steering one for stabilizing the vehicle. We remark that the scheme in Figure 13.1 is an oversimplified scheme and that additional hierarchical levels could be present both in the trajectory/mode replanning module and in the low-level control system module. The union of the first three modules is often referred to as Guidance and Navigation Control System (GNC system). Typically the trajectory replanner and the low-level control system modules do not share the same information on environment and vehicle. For instance, the replanning algorithms can use information coming from cameras or radars which may not

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be used at the lower level. Also, typically, the frequency at which the trajectory replanning module is executed is lower than the one of the lower level control system. The design of both modules makes use of vehicle and environment models with different levels of detail. The fidelity of the dynamical model used for the design of the two modules is dictated, among many factors, by a performance/computational resource compromise and in the literature there is no accepted standard on this. Studies on GNC algorithms vary in (i) the focus (trajectory replanner and/or the low-level control system) (ii) the type of vehicle dynamical model used, (iii) the type of control design used, and (iv) inputs and sensors selection. In [14] the trajectory replanner module is based on a receding horizon control design. The planning problem is formulated as a constrained optimization problem minimizing a weighted sum of arrival time, steering and acceleration control efforts. The vehicle model is a simple rear-centered kinematic model with acceleration, speed, steering, steering rate and rollover constraints are imposed. The lower level uses two separated PIDs to control longitudinal and lateral dynamics. The longitudinal controller acts on throttle and brakes while the lateral controls on the steering angle. The main ideas underlying the planner used in [14] are presented in [17], where a Nonlinear Trajectory Generation (NTG) algorithm is proposed. The GNC architecture in [20] is similar to [14]. The trajectory planning task is posed as a constrained optimization problem. The cost function penalizes obstacles collision, distance form the pre-computed offline trajectory and the lateral offset from the current trajectory. At the lower level, a PI controller acts on brakes and throttle to control the longitudinal dynamics. A simple nonlinear controller, instead, is used to control the lateral dynamics through the steering angle. No details about the vehicle dynamical model used [20] is disclosed. In [19] a design paradigm similar to [14] is used to design a receding horizon optimal control, combined with a real/time trajectory generation and optimization, for a flight control application. Intuitively, the higher the fidelity of the model used by the trajectory replanner is, the easier the job for the lower level control algorithm becomes. At the same time, more performing and robust lower level controllers are required in order to compensate for tire forces saturation due to external disturbances and/or model uncertainties (e.g., road friction coefficient). The two extreme cases are described next. On highly uneven road surfaces (e.g., snowy roads with icy spots) the low level controller has to frequently compensate for tire forces saturations due to alternate adherence conditions. On the other hand, if the road surface is uniform (e.g., dry asphalt), even a simple low level controller could easily follow a feasible trajectory computed by the trajectory replanner. In this chapter, we review two predictive approaches to the design of autonomous path following algorithms via front steering and braking. In the first approach, the trajectory replanner at the upper level is based on an oversimplified vehicle model, i.e. a point-mass model with bounds on the acceleration computed as function of the road friction coefficient. The low level controller is designed based on a linear vehicle model without accounting for tire forces limitations. For instance in [12] a predictive controller and in [25] a simple proportional feedback law are used, both based on linear vehicle models. In the second approach, the trajectory replanner is

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omitted and the low level controller is designed based on detailed vehicle models including nonlinear tire characteristics. The chapter is organized as follows: in Section 13.1 we present a hierarchical architecture for fully autonomous vehicles. In Section 13.2, we briefly summarize the most relevant vehicle model nonlinearities for control design. In Section 13.3, we review two control design approaches to autonomous path following, derived from the architecture presented in Section 13.1. Section 17.4.3 shows experimental results obtained with one of the presented approaches, while Section 13.5 closes the chapter with final remarks.

13.2 Vehicle Modeling Vehicle and tire modeling has been thoroughly studied over the past decades [18, 13]. In [6, 7] we have distilled the extensive literature in order to derive simplified vehicle models to be used for designing autonomous path following algorithms. Figure 13.2 sketches the considered vehicle model. The following notation is used: y˙ and x˙ are the lateral and longitudinal vehicle velocities in the body frame, respectively, ψ is the vehicle orientation (yaw angle) in the inertial frame, ψ˙ is the yaw rate, Y and X are the lateral and longitudinal position of the vehicle in the inertial frame, respectively, δ f is the front steering angle, α f is the front tire slip angle (i.e., the angle between the tire velocity vector v and the tire longitudinal axis), Fc f and Fl f are the front lateral (or cornering) and longitudinal tire forces, respectively. By replacing the second subscript f with r, the variables for the rear axles are defined

Fig. 13.2 The simplified vehicle dynamical model

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similarly. The planar motion of the vehicle in an inertial frame, subject to lateral, longitudinal and yaw dynamics, is described through a tenth order nonlinear model. The detailed model equations are not reported here and can be found in [6, 11]. The nonlinear dynamics can be compactly written as follows:

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where the state and input vectors are ξ = [y, ˙ x, ˙ ψ , ψ˙ , Y, X , ω f ,l , ω f ,r , ωr,l , ωr,r ] and u = [δ f , Tb f ,l , Tb f ,r , Tbr,l , Tbr,r ], respectively, with ω the wheels’ angular speeds and Tb the braking torques, and μ = [μ f ,l , μ f ,r , μr,l , μr,r ] is the vector of friction coefficients at the four wheels. The function f μ in (13.1) is a complex nonlinear function of states and inputs. In particular the right hand side of (13.1) is computed based on the cornering and longitudinal tire forces Fc and Fl , respectively, showed in Figure 13.2 for each wheel. The complexity and the prediction capability of the vehicle model (13.1) are strongly dependant on the model used for computing the forces Fc and Fl . Accurate physical modeling of tire forces is quite involving and several models have been proposed in literature over the past two decades. An exhaustive review of existing tire models can be found in [23]. Next we focus on properties of the nonlinear tire force characteristics, which are the most important from a vehicle control perspective. For the sake of simplicity, in this paper we consider a Pacejka tire model [1], used in [3, 7, 6] for autonomous path following problems via combined steering and braking. Different models, based on complex friction models [5], might be considered. The tire model in [1] is a semi-empirical static nonlinear model that takes into consideration the interaction between the longitudinal and the cornering forces in combined braking/driving and steering. In particular, the tire forces are computed through two nonlinear functions as follows: Fc = fc (α , s, μ , Fz ),

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with rw the wheel radius and vl the wheel longitudinal velocity, μ is the road friction coefficient and Fz is the normal force. We point out that the tire slip angle α and the slip ratio s are nonlinear functions of the system states. In Figure 13.3 the longitudinal and lateral tire forces in combined braking/driving and cornering are shown. In particular in Figure 3(a) the longitudinal force Fl is plotted versus the slip ratio for μ = 0.9 (corresponding to dry asphalt) for different values of the tire slip angle α . In Figure 3(b), the lateral force Fc is plotted versus

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the tire slip angle for different values of the slip ratio. In Figure 13.4 the lateral force Fc is plotted versus the longitudinal force Fl for different values of the tire slip angle α . We observe that, for all α , the curves are enclosed in the thicker outer ellipse according to the inequality AFl2 + BFc2 ≤ μ Fz , (13.4) where the product μ Fz is the maximum tire force that can be generated and the parameters A and B depend on the tire contact patch geometry and affect the maximum lateral and longitudinal forces. We highlight the following key properties of the tire force characteristics shown in Figures 13.3 and 13.4; 1. in pure cornering maneuvers, i.e., at zero slip ratio, the modulus of the the lateral tire force starts from zero, increases within the linear region, reaches a peak equal to μ |Fz | and then decreases to a constant value with increasing slip angle. Analogously, in pure braking/driving, i.e., at zero tire slip angle, the longitudinal force depends on the slip ratio in a similar way; 2. in combined steering and braking/driving both lateral and longitudinal forces are affected by the tire slip angle and the slip ratio. In particular, as the modulus of the slip ratio increases, the slope of the lateral tire force characteristic and the maximum achievable force decrease. Analogously, as the modulus of the tire slip angle increases, the slope of the longitudinal tire force characteristic and the maximum achievable force decrease; and 3. the resultant of the longitudinal and lateral tire forces is upper bounded by the force μ |Fz |.

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Remark 1. Using such complex nonlinear tire models is necessary in order to correctly model a vehicle operating close to the boundary of the linearity region where maximum forces are achieved. Remark 2. Different simplified models can be found in literature based on both nonlinear and linearized tire forces characteristics [18, 13].

13.3 Low Complexity Predictive Path Following Next we review two low complexity predictive approaches to autonomous path following, derived from the general architecture in Figure 13.1. The first approach, presented in [12], includes both an upper level trajectory replanner and a low level predictive control algorithm minimizing the vehicle deviation from the replanned trajectory. The second consists of a controller at low level only, minimizing the deviation from the off-line pre-computed trajectory. In both approaches, the objective is to limit the computational complexity of the predictive control problems, while accounting for the tire force limitations highlighted in Section 13.2.

13.3.1 Two Levels Autonomous Path Following In this Section we present a two levels design paradigm based on a receding horizon trajectory replanner and a low level vehicle controller. The idea underlying the considered approach is to account for the limited tire forces in the predictive trajectory replanner, executed at lower frequency than the low level controller. The latter, instead, is simpler and based on linearized tire characteristics. In principle,

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the low level control should be able to keep the vehicle within the linearity region of the tire characteristic (see Figure 13.3), while following the resulting replanned path. Therefore, any simple low level vehicle controller, based on linearized tire models, could be used to follow the replanned path. In particular, in this paper the trajectory replanner is based on a simple point mass model with lateral acceleration constraints. Every sampling time, the trajectory is replanned over a future horizon subject to path curvature constraints. Such constraints are derived from lateral acceleration bounds, set by the road friction coefficient. A Model Predictive Control (MPC)-based low level steering controller can be used as in [12] to follow the replanned path. Alternatively, the simpler steering feedback controller in [25] could be used. This is a feedback proportional controller with a feedforward term, depending on the deviation of a look-ahead point from the desired path. The point-mass model introduced in this Section describes in a simple way the motion of a vehicle along a prescribed path. We assume that the path is defined in the inertial frame XY through a sequence of pairs (X (t),Y (t)), t ≥ 0. We make use of the following simplifying assumption: Assumption 1. The vehicle acceleration is bounded by the constant μ g, where μ is the road friction coefficient and g is the gravitational acceleration. Under the Assumption 1 the motion of the vehicle along the path is described by the following system of constrained differential equations: v˙x (t) = ax (t),

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where vx and vy are the velocities tangential and normal to the path, ax is the longitudinal vehicle acceleration, c(t) is the path curvature at time t and is computed as follows: ¨ X˙ (t)Y¨ (t) − Y˙ (t)X(t) c(t) = 3 . X˙ 2 (t) + Y˙ 2 (t) 2

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According to the Assumption 1 the system (13.5)-(13.6) is subject to the following constraint: a2x (t) + v4x (t)c2 (t) ≤ μ g, ∀t ≥ 0. (13.7) Remark 3. The constraint (13.7) on the longitudinal and lateral accelerations stems from the limitation of the maximum force transmission to the road surface [13], i.e., point 3 in Section 13.2. Next we show how the trajectory replanning task can be formulated as a receding horizon problem. We discretize the model (13.5) and equation (13.6) with the sampling time T . We assume that at each sampling time t a desired off-line precomputed trajectory is provided over a finite time horizon of N steps, with N ∈ Z+ ,

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in terms of a sequence of pairs (Xir ,Yir ), with i = t + 1, . . . ,t + N. Consider the following cost function: Jrepl (At , Xt#, Yt , vdes , Xt r , Yt r ) = $ r 2 r 2 = ∑t+N i=t+1 α1 (Xi − Xi ) + (Yi − Yi ) # $ 2 α2 vxi,t − vdes + α3 a2xi,t , + ∑t+N−1 i=t

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where At = [axt,t , . . . , axt,t+N−1 ], Xt = [Xt+1 , . . . , Xt+N ], Yt = [Yt+1 , . . . ,Yt+N ], vdes is r , . . . , X r ], Y r = [Y r , . . . ,Y r ], v the desired vehicle velocity, Xt r = [Xt+1 xi,t = t t+N t+N t+1 vx (iT ) and axi,t = ax (iT ), with i ≥ 0. In the cost function (13.8) the first summand penalizes the deviation of the replanned path (Xi ,Yi ) from the desired path (Xir ,Yir ) , i = t + 1, . . .,t + N, the second summand penalizes the deviation of the vehicle longitudinal velocity from the desired velocity vdes , while the third summand penalizes the longitudinal acceleration. These three summands are weighted through the tuning parameters α1 , α2 and α3 . At each time t = kT Nt , with Nt ∈ Z+ , N ≥ Nt and k ∈ Z+ , we solve the following optimization problem: min

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where amax is the maximum longitudinal acceleration. Remark 4. For the sake of readability, in the equation (13.9b) the dependency of ck,t on Xk , Xk−1 , Xk−2 ,Yk ,Yk−1 ,Yk−2 is dropped, i.e., ck,t is used instead of ck,t (Xk , Xk−1 , Xk−2 ,Yk ,Yk−1 ,Yk−2 ). Once a solution At ∗ , Xt ∗ , Yt ∗ of problem (13.9) has been found, the first Nt samples of the sequences Xt ∗ and Yt ∗ are provided to the low-level control as reference trajectory, i.e., replanned path. At time t + Nt T the problem (13.9) is solved over a shifted horizon based on new measurements of vehicle longitudinal speed and lateral and longitudinal positions in the inertial frame. Remark 5. We have shown how a trajectory replanner can be designed based on a very simple vehicle model, i.e., a point mass model. In order to account for tire force limitations, constraints on the lateral accelerations have been added. In principle any vehicle model could be used at this level. For instance a bicycle model [16] or a two tracks model [13], including tire nonlinear models, could be used.

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The trajectory replanner presented in this Section can be complemented by a lower level vehicle control. This can be designed based on either a simplified linear vehicle model as in [25] or a more complex model, based on nonlinear tire characteristics, as in [7, 6]. Nevertheless, we remark that advanced low level controllers, based on nonlinear tire characteristics, might be required in order to compensate for disturbances (e.g., sudden changes of road friction, wind gusts) and model nonlinearities. For instance, as shown next in Section 17.4.3, a low level control, designed based on nonlinear tire characteristics, can effectively compensate for tire force saturation occurring in high speed manoeuvres on low friction surfaces.

13.3.2 Single Level Autonomous Path Following In this Section we present an approach to autonomous path following derived from the architecture in Figure 13.1 and based on a low level control only. In particular, the control objective in this low level control is to minimize the deviation from the off-line precomputed path. The path following problem is formulated as a receding horizon problem. The control algorithms used in this paper are based on the design approaches presented in [7] and [9], next briefly summarized for the sake of clarity. We consider a discrete time version of the vehicle model (13.1) with a sampling time Ts :

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where H p and Hc are the prediction and control horizons respectively, Δ U T (t) = T = [ψ , Y ]. Q, R and S are the weighting [Δ u(t), . . . , Δ u(t + Hc − 1)] and ηre re f re f f matrices of appropriate dimensions. The cost function (13.12) is minimized in receding horizon, subject to the vehicle dynamics (13.10) and constraints on input u(t) and input changes Δ u(t). In partic∗ , . . . , Δ u∗ ular, we denote by Δ Ut ∗ [Δ ut,t t+Hc −1,t ] the sequence of optimal steering and braking commands computed at time t by minimizing (13.12), for the current observed states ξ (t), the previous input u(t − 1) and the given vector of friction

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coefficients μ (t), subject to the vehicle dynamics (13.10), input and state constraints. Then, the first sample of Δ Ut ∗ is used to compute the optimal steering and braking commands and the resulting state feedback control law is ∗ u(t, ξ (t)) = u(t − 1) + Δ ut,t (t, ξ (t)).

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The state feedback control law (13.13) is obtained as a solution of a Non Linear Program (NLP) with 5Hc optimization variables, 10H p equality nonlinear constraints (i.e., the model dynamics) and 20Hc linear constraints (i.e., constraints on inputs and inputs rate of change). The real-time solution of this nonlinear program would require significant computation infrastructures. In [7, 8, 6], we have proposed predictive path following algorithms based on repeatedly updated linear vehicle models. In particular, every time step the cost function (13.12) is minimized subject to the dynamics (13.10), linearized around the current state and inputs, and input and input rate constraints. In [2] the authors apply the concept of Hybrid Parameter-varying Model Predictive Control (HPV-MPC) autonomous vehicle steering. In particular, the receding horizon control problem is based on a prediction model with non-constant, parameter-varying system matrices. The authors investigates side wind rejection and autonomous double lane change via active front steering scenarios. As an alternative approach, one can locally linearize the optimization problem, and compute local explicit solution to multiparametric quadratic programs, as proposed in [24]. In particular, the method uses quadratic approximations to the nonlinear cost and linear approximations to the constraints equations. This allows to approximate the solution of the NLP with a set of quadratic programs (QP), for which explicit equivalent solution functions can be calculated by means of multiparametric quadratic programs (mp-QP). The advantage is that these approximations can be computed off-line as an explicit, piecewise linear function of the state. The online computational burden is reduced to evaluation of a simple piecewise linear look-up table. A different approach to formulate simplified low level predictive control problems was used in [9], where we presented a low complexity autonomous path following algorithm via combined steering and braking, based on a simplified vehicle model derived from (13.1) and based on the following set of simplifications Simplification 1. At front and rear axles, the left and right wheels are lumped in a single wheel. Simplification 2. Wheels dynamics are much faster than yaw, lateral and longitudinal dynamics and are thus neglected. Simplification 3. Braking application induces only yaw moment without longitudinal and/or lateral force changes. Simplification 4. In the lateral tire force calculation the tire slip ratio is assumed to be zero, i.e., Fc = fc (α , 0, μ , Fz ). Simplification 5. The steering and braking effects on vehicle speed are negligible.

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Remark 6. Since braking is used for yaw stabilization only, minimum and singlesided usage of brakes is expected. The Simplifications 1 to 5 are therefore deemed reasonable. By the Simplifications 1-5, the model (13.1) reduces to a 6-th order nonlinear dynamical model and can be compactly written as follows:

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where the state and input vectors are ξ = [y, ˙ x, ˙ ψ , ψ˙ , Y, X ] and u = [δ f , M], respectively, with M the yaw moment induced by braking, and μ = [μ f , μr ] is the vector of friction coefficients at the front and rear axles. The minimization of the cost function (13.12), subject to the discrete time version of (13.14), input and input rate constraints, reduces to an NLP with 2Hc optimization variables, 6H p equality nonlinear constraints and 8Hc linear constraints. An outer braking logic, as in [9], then computes the braking torques at the four wheels from the yaw moment M.

13.4 Results Simulation results of a two level autonomous guidance system, designed as in Section 13.3, have been presented in [12]. There, the receding horizon trajectory replanner is complemented with an MPC low level control based on linearized vehicle models. The union of the two systems is compared against a complex low level MPC controller (the trajectory replanning level is omitted in this case), based on time-varying linearized vehicle models and additional stabilizing constraints originating from the tire nonlinearities. Simulation results show that the second approach is able to perform as well as the first, at the price of additional constraints and longer prediction horizons (i.e., longer preview). On the other hand, as shown next, the low level controller used in the second approach is able to perform complex countersteering manoeuvres in order to compensate for tire forces saturation occurring in high speed experiments on low friction surfaces. Such remarkable results have been obtained in simulations and confirmed by experimental tests. Next we show experimental results of two combined braking and steering low level controllers. These have been designed according to two different low complexity design paradigm. The first, next referred to as Controller A, is based on the design approach presented in [7, 8, 6], where a quadratic program is solved in receding horizon, based on linearized vehicle dynamics updated every time step. Additional state and input constraints are added in order to force the vehicle to operate in the linear region of the tire force characteristics. The second controller, next referred to as Controller B, has been designed based on the approach presented in [9]. In particular, a Nonlinear MPC controller is designed based on a simplified nonlinear vehicle model derived from (13.1), where the state vector is defined as for (13.14) and the input vector is u = [δ f , Fbl , Fbr ], with Fbl , Fbr the braking forces at the left

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and right side of the vehicle. Fbl and Fbr are then distributed to the front and rear axle through a simple braking logic described in [9]. Both controllers have been implemented in real time on a dSpace rapid prototyping platform (see [7] for a description of the experimental setup) on a mid-size passenger car, where other vehicle control systems, like Anti lock Braking System (ABS) and yaw control systems, were disabled. In the considered testing scenario, the vehicle enters the double lane change on a slippery surface, with a given initial forward speed. In both Controllers A and B the road friction coefficient is assumed known. In particular, an estimator is assumed to exist, online adjusting the friction coefficient. In the experimental results showed next, the friction coefficient

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By comparing Figures 6(b) and 13.8, we observe that in both experiments the front and rear vehicle tire slip angles exceed the bounds of the tire linearity region. Nevertheless, both controllers are able to steer the vehicle back to the linear region of the tire characteristics. We point out that such complex countersteering manoeuvres are achieved in Controller A, thanks to ad hoc state and input constraints derived from the nonlinear tire characteristics, and in Controller B, thanks to the use of nonlinear tire characteristics.

13.5 Conclusions In this paper, we have reviewed two design paradigms for predictive control of autonomous vehicles, derived from a general hierarchical architecture for fully autonomous vehicle guidance systems. The focus of this paper has been on the complexity of the two main components of such architecture, that is, the trajectory replanning and the low level controller. In the two considered design paradigms the low level controller is designed based on linearized and detailed vehicle models, respectively. We have presented experimental results showing that the use of detailed vehicle models in the design of the low level control enables complex countersteering manoeuvres, compensating tire nonlinearities and model uncertainties.

References [1] Bakker, E., Nyborg, L., Pacejka, H.B.: Tyre modeling for use in vehicle dynamics studies. SAE paper # 870421 (1987) [2] Besselmann, T., Morari, M.: Hybrid Parameter-Varying MPC for Autonomous Vehicle Steering. European Journal of Control 14(5), 418–431 (2008) [3] Borrelli, F., Falcone, P., Keviczky, T., Asgari, J., Hrovat, D.: MPC-based approach to active steering for autonomous vehicle systems. Int. J. Vehicle Autonomous Systems 3(2/3/4), 265–291 (2005) [4] Calhoun, P.C., Queen, E.M.: Entry vehicle control system design for the mars smart lander. In: AIAA Atmospheric Flight Mechanics Conference, AIAA 2002-4504 (2002) ˚ om, K.J., Lischinsky, P.: A new model for control [5] Canudas de Wit, C., Olsson, H., Astr¨ of systems with friction. IEEE Trans. Automatic Control 40(3), 419–425 (1995) [6] Falcone, P.: Nonlinear Model Predictive Control for Autonomous Vehicles. PhD thesis, Universit´a del Sannio, Dipartimento di Ingegneria, Piazza Roma 21, 82100, Benevento, Italy (June 2007) [7] Falcone, P., Borrelli, F., Asgari, J., Tseng, H.E., Hrovat, D.: Predictive active steering control for autonomous vehicle systems. IEEE Trans. on Control System Technology 15(3) (2007) [8] Falcone, P., Borrelli, F., Asgari, J., Tseng, H.E., Hrovat, D.: Linear time varying model predictive control and its application to active steering systems: Stability analysis and experimental validation. International Journal of Robust and Nonlinear Control 18, 862–875 (2008) [9] Falcone, P., Borrelli, F., Asgari, J., Tseng, H.E., Hrovat, D.: Mpc-based yaw and lateral stabilization via active front steering and braking. Vehicle System Dynamics 46, 611– 628 (2008)

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[10] Falcone, P., Borrelli, F., Asgari, J., Tseng, H.E., Hrovat, D.: Experimental validation of integrated steering and braking model predictive control. Accepted for publication in Int. J. Vehicle Autonomous Systems (2009) [11] Falcone, P., Borrelli, F., Asgari, J., Tseng, H.E., Hrovat, D.: Integrated braking and steering model predictive control approach in autonomous vehicles. In: Fifth IFAC Symposium on Advances of Automotive Control (2007) [12] Falcone, P., Borrelli, F., Asgari, J., Tseng, H.E., Hrovat, D.: A hierarchical model predictive control framework for autonomous ground vehicles. In: Proc. American Contr. Conf., Seattle, Washington (2008) [13] Kiencke, U., Nielsen, L.: Automotive Control Systems. Springer, Heidelberg (2000) [14] Foote, T.B., Cremean, L.B., Gillula, J.H., Hines, G.H., Kogan, D., Kriechbaum, K.L., Lamb, J.C., Leibs, J., Lindzey, L., Rasmussen, C.E., Stewart, A.D., Burdick, J.W., Murray, R.M.: Alice: An information-rich autonomous vehicle for high-speed desert navigation. Submitted to Journal of Field Robotics (2006) [15] Lu, W.M., Bayard, D.S.: Guidance control for mars atmospheric entry: Adaptivity and robustness. Technical report, JPL Technical Report (1997) [16] Margolis, D.L., Asgari, J., Multipurpose, J.: models of vehicle dynamics for controller design. SAE Technical Papers (1991) [17] Milam, M.B.: Real-Time Optimal Trajectory Generation for Constrained Dynamical Systems. PhD thesis, California Institute of Technology, Pasadena, CA (2003) [18] Pacejka, H.B.: Tire and Vehicle Dynamics. In: Automotive Engineering, 2nd edn. Society of Automotive Engineers, Inc. (2006) [19] Hauser, J., Murray, R.M., Jadbabaie, A., Miliam, M.B., Petit, N., Dunbar, W.B., Franz, R.: Online control customization via optimization-based control. In: Samad, T., Balas, G. (eds.) Software-Enabled Control: Information Technology for Dynamical Systems. IEEE Press, Los Alamitos (2003) [20] Montemerlo, M., Thrun, S., Dahlkamp, H., Stavens, D., Aron, A., Diebel, J., Fong, P., Gale, J., Halpenny, M., Hoffmann, G., Lau, K., Oakley, C., Palatucci, M., Pratt, V., Stang, P., Strohband, S., Dupont, C., Jendrossek, L.E., Koelen, C., Markey, C., Rummel, C., van Niekerk, J., Jensen, E., Alessandrini, P., Bradski, G., Davies, B., Ettinger, S., Kaehler, A., Nefian, A., Mahoney, P.: Stanley, the robot that won the darpa grand challenge. Journal of Field Robotics (2006) (Accepted for publication) [21] Smith, R.S., Bayard, D.S., Farless, D.L., Mease, K.D.: Aeromaneuvering in the martian atmosphere: Simulation-based analyses. AIAA Journal Spacecraft and Rockets 37(1), 139–142 (2000) [22] Smith, R., Boussalis, D., Hadaegh, F.: Closed-loop aeromaneuvering for a mars precision landing. In: NASA University Research Centers Tech. Conf., Albuquerque, NM, February 1997, pp. 942–947 (1997) [23] Svedenius, J.: Tire Modeling and Friction Estimation. PhD thesis, Department of Automatic Control, Lund University, Lund, Sweden (April 2007) [24] Tøndel, P., Johansen, T.A.: Control allocation for yaw stabilization in automotive vehicles using multiparametric nonlinear programming. In: Proc. American Contr. Conf. (2005) [25] Tseng, H.E., Asgari, J., Hrovat, D., Van Der Jagt, P., Cherry, A., Neads, S.: Evasive maneuvers with a steering robot. Vehicle System Dynamics 43(3), 197–214 (2005)

Chapter 14

Toward a Systematic Design for Turbocharged Engine Control Greg Stewart, Francesco Borrelli, Jaroslav Pekar, David Germann, Daniel Pachner, and Dejan Kihas

Abstract. The efficient development of high performance control is becoming more important and more challenging with ever tightening emissions legislation and increasingly complex engines. Many traditional industrial control design techniques have difficulty in addressing multivariable interactions among subsystems and are becoming a bottleneck in terms of development time. In this article we explore the requirements imposed on control design from a variety of sources: the physics of the engine, the embedded software limitations, the existing software hierarchy, and standard industrial control development processes. Decisions regarding the introduction of any new control paradigm must consider balancing this diverse set of requirements. In this context we then provide an overview of our work in developing a systematic approach to the design of optimal multivariable control for air handling in turbocharged engines.

14.1 Introduction The goal of this chapter is to present the emerging problems facing development of control for increasingly complex turbocharged engines and to discuss potential solutions. In turbocharged diesel engines ever-tightening emissions legislation drives the incorporation of new sensors [27, 33], actuators, and subsystems such as multistage turbochargers, complex exhaust gas recirculation (EGR) topologies [31], and aftertreatment devices (such as selective catalytic reduction [26, 32], lean NOx traps, and diesel particulate filters [21]). These changes are all introduced in the context of Greg Stewart, David Germann, and Dejan Kihas Honeywell Automation and Control Solutions, 500 Brooksbank Avenue, North Vancouver, BC, Canada, V7J 3S4 Francesco Borrelli Department of Mechanical Engineering, University of California, Berkeley, CA 94720 Jaroslav Pekar and Daniel Pachner Honeywell Prague Laboratory, V Parku 2326/18, 148 00 Prague 4, Czech Republic L. del Re et al. (Eds.): Automotive Model Predictive Control, LNCIS 402, pp. 211–230. c Springer-Verlag Berlin Heidelberg 2010 springerlink.com

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difficult analysis and decisions regarding the tradeoff of development and product cost, reliability, fuel economy, drivability, and emissions. The rising complexity of engines and the demand of tighter performance is increasing the complexity of the control functionality that is required to manage the engine. It is widely recognized that the development of the control is becoming a bottleneck in the development of engines and systematic approaches to developing performant controls would be welcome provided they mitigate these burdens. Typically most engine makers require control design techniques that provide some combination of improved closed-loop performance and reduced development effort. The relative importance of each consideration depends to a large extent on the business model of the particular control system developer. In this chapter we will discuss a wide view of the problem of developing such a systematic approach with the goal of integration into industrial practice. In so doing we need to consider the interaction of many practical issues; including engine physics and its resulting nonlinearities and multivariable interactions, the desired closed-loop performance, engine variability due to production dispersion and ageing, the restrictions in CPU time and memory due to the embedded electronic control unit (ECU) platform, the hierarchical software structure into which the final control function must integrate, the existing development process for engine control, and the range of personnel with whom any new process must interact. The scope of a solution to such a problem is large, but is aided by the significant body of previous research – both academic and industrial – in the domain. The requirements associated with engine modeling and closed-loop control performance are considered in [28, 20, 15, 34, 16] and references therein. As will be introduced below, our approach includes a model predictive control (MPC) component and previous MPC engine control examples may be found in [23, 17]. The interplay of control design and computational restrictions1 (CPU and memory) were noted in [17] where a fully nonlinear MPC (NMPC) is proposed which demonstrates improved performance with respect to linear state feedback and inputoutput linearization approaches. As pointed out in [17], the main drawback of most NMPC techniques comes from the fact that they typically require far more computational power than is available on modern automotive ECUs. To address these issues we have implemented several practical simplifications and have separately explored the question of online implementation of MPC in depth. This approach has allowed us to successfully implement a multivariable MPC controller in a production ECU [29]. Selected aspects of the underlying techniques are outlined in Section 14.4 below. Section 14.2 presents the requirements to be met by control design, Section 14.3 describes how the proposed modeling and model predictive control techniques are suited to address these requirements, Section 14.4 presents some more detailed results on one of the many technical aspects that are needed to be overcome in order to achieve an industrial quality systematic approach to control design. 1

More detailed discussions of the very important subject of requirements and management of engine control software design and its innovations may be found in references such as [25, 30].

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14.2 Engine Control Requirements In this Section we present the specific requirements that must be addressed by any engine control design process. These are discussed with reference to a high-level description of a typical engine control design process which has evolved over many years to address the industrial need of creating controllers for highly nonlinear engines which are to be hosted on ECUs and deployed across a fleet of thousands to millions of vehicles that may stay in active service as long as 20 years. As will be seen below, the support of all phases of this process requires a wide range of skills that include an understanding of engine physics, performance specifications (including emissions legislation), embedded software issues, and calibration and postrelease support. Typically these activities involve separate skills and groups within a company and it is important that any proposed technology change anticipates and addresses all of these areas. While specific engine control design processes will vary site to site, many similarities exist and a high level outline of a typical process is illustrated in Figure 14.1 which is similar to those described in references such as [25].

14.2.1 Steady-state Engine Calibration Also known as “base-mapping” in this pre-control phase an engine is calibrated in order to produce the actuator positions at a coarse grid of engine speed and load points such that certain desired steady-state criteria are traded off. This step is often performed by executing a design of experiment that sweeps the relevant engine actuators at a selection of speed and load points while recording the engine’s performance in terms of fuel consumption and emissions. Upon completion of the experiment the desired engine steady-state operating points (outputs and actuator positions as a function of exogenous variables such as speed and injected fuel quantity) are determined by optimizing with respect to requirements imposed by legislated drive cycle limits. This is a heavily experimental stage and requires much insight into the engines’ behavior.

14.2.2 Control Functional Development In this step control engineers decide on the partitioning of the engine functionality and, where needed, propose and configure new control functions with the goal of approximately delivering the steady-state engine calibration while simultaneously providing enough flexibility to meet the certification and drivability requirements that are evaluated over transient driving conditions. This stage is often performed with the aid of rapid prototyping tools that support the porting of high-level code R or Simulink into an ECU bypass system. Commercially availsuch as MATLAB able examples include [12, 2, 22]. Figure 14.2 is helpful for understanding the environment into which a controller must be designed. An engine is a highly nonlinear plant (see for example

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Fig. 14.1 Simplified illustration of the engine control development process

[28, 20, 15, 16, 19]) and in most cases requires the development of nonlinear control strategies. For example, when considering the response of airside parameters such as compressor flow and boost pressure to the variable geometry turbine (VGT) actuator, the control designer must address a steady-state gain that changes sign and the fact that the nonlinearities are a function of the engine speed and fuel injection quantity and thus may change more quickly than the dynamics of the air loop which are often dominated by the turbocharger inertia [20, 29]. In addition one must often consider the fact that multiple engine configurations may be required to be addressed by relatively minor configuration changes to a single control strategy. Figure 14.2 illustrates the hierarchical software environment into which the developed controller function must be integrated. An individual subcontroller is typically configured to be responsible for an engine subsystem and a given subcontroller may be responsible for fuel injection control, aftertreatment control, exhaust gas recirculation (EGR) control, or turbocharger (wastegate or VGT) control. The illustrated higher level function is responsible for engine monitoring and distribution of signals to the subcontrollers which typically includes information about special

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modes (cold start, regeneration of aftertreatment, etc.), setpoints, feedforward actuator values, and time varying constraints for actuators and engine states. Actuator constraints typically include the enforcement of minimum and maximum bounds on actuator positions and a common example of a state constraint is the imposition of an upper bound on the turbocharger speed in order to prevent damage to the turbocharger wheels. In our work we are concentrating on developing techniques that enable the user design multivariable controllers that may be substituted for one or more subcontrollers, initially focusing on unifying the control of the air induction subsystems and coordinating their interaction with fuel injection and aftertreatment subcontrollers. Multivariable interactions among subsystems have typically been neglected in traditional engine control design, but are rapidly becoming crucial to consider as performance requirements become more stringent due to legislation and the emergence of aftertreatment devices that are most effective in certain operating windows of temperature, flow, and composition of the exhaust gas. The techniques and tools required in the “control functional development” stage are those that enable the developer to systematically create models and design controllers for integration into the existing control hierarchy that will provide acceptable performance for a wide variety of engine configurations and performance specifications.

Fig. 14.2 Illustration of relative position of subcontroller algorithms within the ECU software hierarchy. The symbols yi (t) and u j (t) represent the ith and jth subsets of sensor and actuator information respectively. The symbols f k (t) denote the information transferred to the kth subcontroller and may include setpoints, feedforward actuator values, and time varying constraints for actuators and engine states. The techniques discussed in this chapter consider controller design for the replacement of one or more subcontrollers with an optimal multivariable controller

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14.2.3 Functional Testing In this phase the developing control functionality may be progressively tested in simulation, in the engine test cell, in vehicle, and over a range of ambient conditions of temperature and altitude. This step is typically performed iteratively with the control functional development and the developer must evaluate how well the designed control function interacts with the engine and any other existing control loops to deliver the desired performance.

14.2.4 Software Development This phase may be performed in-house by the engine maker or may involve a third party software provider. The software developers receive a functional specification from the above step and then specify, code, and test the desired function into the embedded software environment which may include the reformulation of the code to use fixed-point arithmetic. In this phase a key requirement is to respect the memory and processor limitations of the ECU. Furthermore, as this phase is potentially long and expensive, wherever possible one prefers to use control algorithm structures with the flexibility to enable control changes without requiring re-entry into the software development phase. The duration and expense of this phase has motivated the development of autocoding tools and techniques which enable the conversion of high-level languages R and Simulink into production-quality embedded code. Such such as MATLAB tools have begun to make inroads into various production applications [30].

14.2.5 Integration The engine developer then integrates the developed code which may be tested in a simulated engine environment before proceeding to testing and debugging in the test cell and vehicle.

14.2.6 Calibration This is a detailed phase in which the numerous free parameters of the engine controller are tuned or calibrated to provide the desired steady-state and transient performance. This phase requires that controller tuning tools be efficient, intuitive, and well-integrated into the existing calibration environment. This phase also includes all of the calibration required for the diagnostics portions of the software which are indirectly related to the control functionality.

14.2.7 Certification Refers to the set of tasks related to preparing, documenting and performing the certification testing that is relevant to the vehicle class and geographical region in which it will be used.

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14.2.8 Release and Post-release Support Once the software and control have been accepted, then the engine or vehicle is released to market. Subsequent control calibration or other fixes may be required in the post-release support of the product. Since post-release revision to the control is limited and expensive, the controllers designed and calibrated throughout the above steps must perform well over a fleet of engines each with differences in the manufactured components, and must maintain system performance over the lifetime as the engine components age and change in different ways.

14.2.9 Iteration Loops Figure 14.1 illustrates the above steps along with some standard iteration paths. The “short-path functional iteration” between functional testing and control functional development has been made relatively rapid by modern rapid prototyping systems. It is these iterations in which the major decisions regarding overall controller structure are made and thus it is critical that the developer be provided tools that direct or otherwise assist control structure decisions to be made systematically. In this stage it is noted that the designer must design a controller to work with the engine physics of its designated task, and also to interact appropriately with “neighboring” control functions. In [29], the hierarchical nature of engine control is described and it is pointed out that any developed control function must be integrated into this hierarchy where the supervisory levels deliver signals including setpoints, actuator feedforward, time-varying constraints for actuators or engine states to the lower-level components of the control hierarchy. The illustrated “software iterations” on the lower-right portion of the diagram indicates any rework that is required for fixing bugs or other errors in the software coding. The “long-path functional iteration” illustrated on the left side of the diagram indicates the required iteration path when controller code is found to be unacceptable at the functional level at a later stage of development. Since this path contains the entire software development cycle, it may incur a cost of several months of development time2 using existing development tools and processes, and is thus considered undesirable in the majority of cases. The above discussion has presented a high-level description of the engine control development process from which it is clear that several key requirements to be met by any control technology include: 1. to maintain performance of the family of highly nonlinear and multivariable engines that lie within the uncertainty bounds of a production fleet; 2. integration into the hierarchical control structure (including time varying constraints); 3. to fit within the tight processor and memory requirements for implementation in an ECU; and 2

The time and effort incurred in the “long-path functional iteration” is sometimes reduced by the use of autocoding tools such as [30] in the Software Development phase.

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4. to provide a systematic, efficient development process to facilitate software development and enable a reduced calibration burden.

14.3 Modeling and Control for Turbocharged Engines Section 14.1 discussed the trends and pressures facing the development of modern engine control while Section 14.2 overviewed the scope of the requirements for any strategy – new or old. In this Section we discuss the merits of using optimal model predictive control technology to address these issues. In light of the above requirements, MPC shows several strong advantages: • MPC is ideally suited for handling the signals from the higher level of the hierarchy illustrated in Figure 14.2. Particularly the setpoints and time varying input and output constraints [29]; • MPC has a general algorithmic structure which can cover many different control problems (e.g. unconstrained or constrained, single or multivariable) without requiring software structure changes; and • the tuning of MPC (while not quite as straightforward as is often claimed in MPC literature) can be made intuitive with the appropriate software tools to allow practical usage by people with a wide variety of technical backgrounds and interests [14]. Figures 14.3 and 14.4 contain representative examples of the on-engine results achieved using model predictive control on two very different engine control problems. Application details may be found in [29]. Figure 14.3 is an illustration of the simultaneous control of measured EGR flow and engine-out NOx concentration in a heavy-duty diesel engine using the EGR valve and variable cylinder valve as actuators. Such a control configuration could have application in coordinated engine-aftertreatment control. Figure 14.4 represents a more standard control problem where the intake manifold pressure and the compressor flow are simultaneously controlled to respective setpoints using the VGT vanes and EGR valve while the engine traverses the indicated transient in engine speed and injected fuel quantity. On the other hand, we will have to consider and address the challenges faced by MPC to achieve the desired reduction in development time and effort. These include: • modeling: one is obliged to develop an efficient and reliable process to generate the control oriented models that are required by all advanced control techniques; • computation: MPC – and especially nonlinear MPC – typically require too much computing power and memory for implementation on a modern ECU (see for example [17]). This topic is discussed in some detail in Section 14.4 below; and • usability: for deployment, advanced control techniques must be developed to the point of industrial quality, such that they do not require such detailed knowledge of MPC or modeling that disqualifies all but a small group of specialists for their use.

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Fig. 14.3 Simultaneous control of EGR flow and engine-out NOx concentration using the EGR valve and variable cylinder valve actuators in a heavy duty diesel engine. The setpoints are indicated by the dashed blue lines and the sensor measurements in red in the upper two subplots. The actuator constraints are illustrated as the broken magenta lines and the actuator positions as the solid green lines in the lower two subplots. The axes have been scaled for deidentification considerations

14.3.1 Modeling It is almost accepted as a truism that model-based control design techniques are used in order to cut down on the development time. A better phrasing of that statement may be that one should take care to ensure that the modeling results in a reduction of the overall the control development effort. Traditional, non-model-based approaches to engine control design have an ironic advantage of avoiding a potentially onerous and difficult modeling process. Of course traditional approaches are acknowledged to have serious limitations in the complexity of control problem they can address and furthermore engine models have many benefits beyond their use in control synthesis (see for example [11]). The models we develop in the course of our design are highly nonlinear and the reliable identification of their free parameters is a serious challenge, the detailed description of which is outside the scope of this chapter. We present a high level overview of the key issues here and will publish the technical details elsewhere.

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Fig. 14.4 Simultaneous control of boost pressure and compressor flow using the VGT and EGR valve actuators in a small diesel engine. The MPC was implemented on a production ECU (Motorola MPC555). In the upper two subplots the setpoints are represented by the solid blue lines and the measured sensor signals by the dashed green lines. The actuator constraints are constant at 5% and 95% respectively. The axes have been scaled for deidentification considerations

A practical modeling approach needs to be configurable to a wide variety of engines including single and multistage turbochargers, low and high pressure EGR, various actuators such as valves, throttles, wastegates, VGT vanes, variable valve actuation, and various sensor selections and locations on the engine. With an eye on their intended use, one must implicitly trade off model complexity and precision since low order models are preferable for model-based control design. Figure 14.5 illustrates two different engine layouts which have been built from our library of components. The dynamic response is governed by the states associated with the intake and exhaust manifolds and also the turbocharger speed(s). In more complicated engine layouts, such as the illustrated multiple turbocharger example, correspondingly more states are required to include the associated dynamics.

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Fig. 14.5 Example engine layouts for a standard single-stage turbocharged and a series turbocharged engine both with high-pressure exhaust gas recirculation (EGR)

Next the configured model must be calibrated to the engine such that it matches the true nonlinear and dynamic input-output response across all operating conditions. The model identification is typically required to work with some mix of steady-state and transient data (see e.g. [16]). To address this nonlinear model identification problem we take a two-step approach. First the individual components (listed above and illustrated in Figure 14.5) are fit one by one to component maps (if available) and recorded engine data. In practice, the overall model quality achieved from an assembled collection of components is typically insufficient for capturing the required input-output behavior of the engine and the second step of the model identification is performed by executing a global nonlinear optimal fitting of the available parameters. In practice, this step typically results in a dramatic improvement in model accuracy and a representative example is illustrated in Figure 14.6. On a technical level the model identification is further complicated by the fact that turbocharged engine models have inherent feedback paths via the EGR and the turbocharger shaft. Still more feedback paths are often added during standard

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Fig. 14.6 Representative accuracy plots obtained from modeling a two-stage turbocharged engine with the proposed medium-fidelity (7-11 dynamic states in this example) nonlinear control oriented model. Steady-state data from the full range of engine speeds and loads is represented. The blue circles indicate the model results when each component has been fit locally and the red squares illustrate the improved model results following the developed optimization-based simultaneous fitting of all model parameters. The straight lines indicate error bounds of 5%, 10%, and 15% with respect to the measured engine data

modeling procedures [20, 16]. Thus the sensitivity and even stability of the model with respect to its tuning parameters must be treated with due care to preserve the desired match to the true engine response. This modeling process results in a continuous-time nonlinear model in which f and h denote the state update and output functions, respectively: x(t) ˙ = f (x, u, v),

y(t) = h(x, u, v)

(14.1)

where y represent the controlled variables, the array u represents the actuator setpoints to be computed by the controller, and v represents the exogenous inputs to the system. The content of these variables is problem specific and configurable. The controlled variables y may include some subset of boost pressure, compressor air flow, EGR flow, turbocharger speed, engine-out NOx, exhaust temperature, etc. The actuators u may include variable geometry turbine (VGT) vanes, variable

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(cylinder) valve actuators (VVA), EGR valve, intake or exhaust throttles. The exogenous inputs v will typically include engine speed and fuel injection quantity as a baseline and may be further refined by coolant temperature, and ambient pressure and temperature, etc. The following Section will use the nonlinear model (14.1) as input to an MPC design and will discuss the simplifications required for practical controller synthesis.

14.4 Model Predictive Control and Computational Complexity This section presents the control methodology used and possible approaches to its real-time implementation.

14.4.1 Explicit Predictive Control We consider a piecewise affine (PWA) discrete-time approximation of the system dynamics (14.1) x(k + 1) = Aσ x(k) + Bσ u(k) + Bvσ v(k) + Bwσ w(k) + fσ v w y(k) = Cσ x(k) + Cσ v(k) + Cσ w(k) + gσ for

x u v w

(14.2)

(k) ∈ Cσ

where x(k), u(k), y(k), v(k), w(k) are the state, input, output, measured and unmeasured disturbances, respectively at time kTs where Ts is the sampling time, fσ and gσ are constant vectors. The natural number σ (k) ∈ {1, 2, . . . , M} is the operating point at time kTs and it is a function of inputs u(k), states x(k) and disturbances v(k). The set {Cσ }M i=1 is a polyhedral partition of the state, input and measured disturbance set. System (14.2) is subject to the following time varying constraint on inputs and outputs for all k ≥ 0. u(k) ∈ U (umin (k), umax (k)), y(k) ∈ Y (ymin (k), ymax (k))

(14.3)

where U (umin (k), umax (k)) and Y (ymin (k), ymax (k)) are polyhedra for all k ≥ 0. Consider the problem of letting the output of system (14.2) track a given reference yre f ,k while satisfying input and output constraints (14.3). Assume that estimates/measurements of the state x(k) and disturbances v(k) are available at the current time k and consider the following cost function Jk (x, v, w, Δ U, ε ) := yHp +k − yre f ,Hp +k )||P2 + k+H −1 R 2 + ∑t=k p (yt − yre f ,t )Q 2 + δ ut 2 + ρε

(14.4)

where vM 2 = v Mv. Then, the finite time optimal control problem is solved at time k,

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min

Δ UHc ,ε

Jk (x(k), v(k), w(k), Δ UHc , ε )

(14.5a)

s.t. xt+1 = Aσ xt + Bσ ut + Bvσ vt + Bwσ wt + fσ wt+1 = Awσ wt + Buσ ut yt = Cσ xt + Cσv vt + Cσw wt + gσ x if uv ∈ Cσ

(14.5b) (14.5c)

w k

t = k, . . . , k + H p − 1

ut = δ ut + ut−1 ut ∈ U (umin (k), umax (k)),

(14.5d)

yt ∈ Y (ymin (k), ymax (k)) ⊕ ε t = k, . . . , k + Hc − 1, ε > 0

(14.5e)

xk+Hp ∈ X f δ ut = 0, t = k + Hc , . . . , k + H p − 1

(14.5f) (14.5g)

vt = v(t − 1), t = k + 1, . . ., k + H p uk−1 = u(k − 1)

(14.5h) (14.5i)

xk = x(k), wk = w(k), vk = v(k)

(14.5j)

where the column vector Δ UHc := [δ uk , . . . , δ uHc −1 ] and ε are the optimization vectors, H p and Hc denote the output prediction horizon and the control horizon and X f is the terminal region. Note that the subscript notation is used to distinguish between the variables of the optimization problem (14.4)-(14.5) and the state, input, disturbances and outputs of the system model (14.2). Let Δ UH∗c = {δ u∗k , . . . , δ u∗k+Hc −1 } and ε ∗ be the optimal solution of (14.4)-(14.5) at time k. Then, the first sample of UH∗c (obtained from Δ UH∗c and u(k − 1)) is applied to the system: u(k) = u∗k .

(14.6)

The optimization (14.4)-(14.5) is repeated at time k + 1, based on the new state xk+1 = x(k + 1), measured disturbances vk+1 = v(k + 1), additive unmeasured disturbances wk+1 = w(k + 1), input and output constraints, yielding a moving or receding horizon control strategy. In (14.4) we assume that Q = Q 0, R = R 0, P 0. In problem (14.4)-(14.5) the following assumptions are used A1 H p > Hc and the control signal is assumed constant for all Hc ≤ k ≤ H p . This allows the reduction of the computational complexity of the MPC scheme; A2 the exogenous disturbance v is assumed constant over the horizon. If PWA prediction models for v(k) are available they could be included in the MPC formulation (14.4)-(14.5); A3 the region σ is constant over the horizon (14.5c); and

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A4 soft Constraints on outputs, i.e. Y (ymin (k), ymax (k)) ⊕ ε := {y|y + ε ∈ Y (ymin (k), ymax (k))}. Remark 1. Assumption (A3) basically implies that for any given time we simply implement a linear MPC for one member of the set of linear systems. Ideally the assumption should be removed in order to predict switches between affine dynamics over the horizon H p . This would improve both performance and attractivity region of the closed loop system. Nevertheless, we have been forced to use assumption (A3) by the current limitations of automotive ECUs. In fact, by removing (A3), problem (14.4)-(14.5) becomes a mixed integer quadratic program (MIQP) whose explicit solution [7] requires more floating point operations for its evaluations and more memory for its storage. The optimization problem (14.4)-(14.5) can be recast as a quadratic program (QP) . min

Δ UHc ,ε

1 2 Δ UHc Hσ Δ UHc

+ Hε ,σ ε 2 + φ (k) Fσ Δ UHc

subj. to Gσ Δ UHc ≤ Wσ + Eσ φ (k)

(14.7)

where

φ (k) := [x(k) u(k − 1) v(k) w(k) yre f ,k · · · · · · yre f ,k+Hp umin (k) umax (k) ymin (k) ymax (k)] and φ (k) ∈ Rn p . Problem (14.7) is a multiparametric quadratic program that can be solved by using the algorithm presented in [5]. Once the multiparametric problem (14.7) has been solved, the solution Δ UH∗c = Δ UH∗c (φ (k)) of problem (14.4)(14.5) and therefore u∗ (k) = u∗ (x(k)) is available explicitly as a function of the set of parameters φ (k) for all φ (k) ∈ X0 . X0 ⊆ Rn p is the set of initial parameters φ (0) for which the optimal control problem (14.4)-(14.5) is feasible. The following result [5] establishes the analytical properties of the optimal control law and of the value function. Theorem 1. [5] The control law δ u∗ (k) = fσ (φ (k)), fσ : Rn p → Rm , obtained as a solution of (14.7) is continuous and piecewise affine on polyhedra fσ (φ ) = Fσi φ + giσ if φ ∈ CRiσ , i = 1, . . . , Nσr

(14.8)

where the polyhedral sets CRiσ = {φ ∈ Rn p |Hσi φ ≤ Kσi }, i = 1, . . . , N r are a partition of the feasible polyhedron X0 . As discussed in [5] the implicit form (14.7) and the explicit form (14.8) are equal, and therefore the stability, feasibility, and performance properties are automatically inherited by the piecewise affine control law (14.8). Clearly, the explicit form (14.8) has the advantage of being easier to implement, M lookup tables, one for each operating point σ , are uploaded on the ECU and at each time step k the MPC resorts to selecting the current operating point σ , searching for the region CRσj containing the

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current vectors of parameters φ (k) and implementing the corresponding controller Fσj φ (k) + gσj . The following lists some of the major practical issues which have been encountered while implementing the proposed MPC on an automotive ECU. In this chapter we will focus on only the final listed issue. • state estimation; estimation of the state x(k) and w(k) of system (14.2) is a nontrivial task. We have used a bank of M Kalman filters which run in parallel in order to smoothen the estimation during switch between different operating points; • time-varying actuator and state constraints; the proposed formulation has been designed to address constraints that vary arbitrarily as a function of time and thus independently of the state variables or gain scheduling parameters. • constraint satisfaction under steady state disturbances; for a range of steady-state exogenous disturbance v(k) = v¯ the reference yre f might be infeasible for the given input and output constraints and thus not be trackable. At steady-state, the objective function will be composed of two terms with conflicting objectives: satisfy the constraints (ρε 2 ) and track yre f . In addition, model uncertainty and high ρ weights (usually used to strictly enforce soft constraints) can lead to oscillating behavior and poor performance. A possible solution to this problem based on computing regions of attraction and switching tuning has been given in [29]; and • limited ECU memory; we had to modify the explicit implementation in order to be able to run the MPC (14.4) to (14.6) in an industrial ECU (even for short horizons). We have used the Karush-Kuhn-Tucker conditions which lead to (14.8) in order to reduce the memory required for storing Fσ and gσ in (14.8). More details can be found in [8]. In the next Section we will provide more details on the last steps. The interested reader is refereed to [29] for a thorough treatment of the aforementioned topics.

14.4.2 On the Complexity of Explicit MPC Control Laws The efficient solution of the optimization problem (14.7) depends on the problem properties and on the hardware platform. In [9, 10] we have compared the computational time and storage demand associated to (i) active-set QPs for solving (14.7) and to (ii) the evaluation of the explicit solution (14.8). In particular we have shown that there might be alternative ways for solving the optimization problem (14.7) which are more efficient than evaluating the explicit linear MPC (14.8). However, the corresponding state-feedback controllers do not have the nice piecewise affine closed-form solution as the explicit solution presented in [5, 4]. Next we provide a brief overview of the main results and refer the reader to [9, 10] for a more formal and detailed discussion. The evaluation of the explicit solution (14.8) in its simplest form would require: (i) the storage of the list of polyhedral regions and of the corresponding affine control laws, (ii) a sequential search through the list of polyhedra for the i-th polyhedron that contains the current state in order to implement the i-th control law. Since verifying if a point φ belongs to a critical region means to verify primal and dual

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Fig. 14.7 Comparison of various algorithms for solving QP (14.9)

conditions, then the on-line search for the polyhedron containing φ can be compared to the main steps of a QP solver. In fact in [9] we have shown that an active set QP solver requires more operations at each iteration than an explicit solver. This is obtained at the price of increased memory requirement. In fact, for the evaluation of (14.8) the polyhedral partition and the gains have to be stored which, in general, largely surpass the memory required for an active set QP (simply the matrices of the QP (14.9)). The main idea behind alternative approaches can be simply explained as follows. Rewrite the optimization problem (14.7) for a fixed σ compactly as [5] min z

1 2 z Hz

subj. to Gz z ≤ bz (φ )

(14.9)

where z is the optimization variable, and bz (φ ) is an affine function of φ . Let I be the set of constraint indices. Consider a subset A of the constraints index A ⊆ I . Given a matrix M, MA denotes the submatrix of M consisting of the rows indexed by A . Consider the solution z∗ (φ ) and λ ∗ (φ ) when the set A is active at the optimum [9]: z∗ = Hz−1 Gz,A (Gz,A H −1 Gz,A )−1 bz,A (φ ) = TA bA (φ ) λ ∗ = −(Gz,A Hz−1 Gz,A )−1 bz,A (φ ) = SA bA (φ )

(14.10)

The alternative class of algorithms presented in [9, 10], computes and stores the matrices SA and TA off-line for all optimal sets of active constraints A . Then, the online steps are: (1) compute bz (φ ), verify duality conditions by computing λ ∗ from (14.10), compute the optimizer candidate z∗ by using (14.10) and verify primal feasibility conditions Gz z∗ ≤ bz (φ ). In conclusion, one can identify three classes of algorithms as illustrated in Figure 14.7: the explicit solvers (upper-left in the figure), the QP solvers (lower-right in the figure) and hybrid solvers which trade-off memory and computation in a different

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way and which, for certain classes of problems, can be more efficient than both QP solvers and explicit solvers [9, 10]. Remark 2. We remark that there exist other very efficient approaches appeared in the literature for solving predictive control problems for linear and PWA systems [18, 3, 6, 24, 13]. The comparison of our approach to other approaches would be problem dependent and requires the simultaneous analysis of several issues such as speed of computation, storage demand and real time code verifiability. This is an involved study and as such is outside of the scope of this chapter. We refer the reader to [1] for a good review of explicit predictive control.

14.5 Summary and Conclusions The demand for systematic and efficient techniques for the development, calibration, and deployment of control algorithms is undisputed in modern industrial engine design. The key issues are the achieved closed-loop performance and also the time, effort, and expense that are required to achieve it. Within the context of a standard industrial engine control development process we surveyed the set of control requirements at a high level, including: • to maintain performance of the family of highly nonlinear and multivariable engines that lie within the uncertainty bounds of a production fleet; • to integrate into the hierarchical control structure (including time varying constraints); • to fit within the tight processor and memory requirements for implementation in an ECU; and • to provide a systematic, efficient development process to facilitate software development and enable a reduced calibration burden. With respect to these requirements we discussed our recent work in attempting to address this set of requirements in the form of a systematic process and set of software tools that allow the user to configure a model using components from a library, to automatically and robustly fit this model to engine and component data, to use this model in the synthesis of both the feedforward and feedback components of a multivariable control strategy. We selected model predictive control (MPC) as the underlying technology due to its ability to address the multivariable interactions among subsystems, its ability to be integrated into the existing control software hierarchy found in industrial electronic control units (ECUs) – including its straightforward accommodation of time varying input and output constraints, and its general algorithmic structure which can cover many control problems without requiring software structure changes. We next overviewed some of the issues involved when considering implementing MPC for a nonlinear plant within the limited resources of a modern ECU, particularly paying attention to balancing the tradeoff between memory and processor usage. We presented examples of the overviewed control being used to control the air handling on two very different engine applications.

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Acknowledgements This work would not have been possible without the support of Honeywell.

References [1] Alessio, A., Bemporad, A.: A survey on explicit model predictive control. In: Allgower, F., Raimondo, D.M., Magni, L. (eds.) Nonlinear Model Predictive Control: Towards New Challenging Applications. LNCIS, vol. 384, pp. 345–369. Springer, Heidelberg (2009) [2] ETAS ASCET (2009), http://www.etas.com [3] Baoti´c, M., Borrelli, F., Bemporad, A., Morari, M.: Efficient on-line computation of constrained optimal control. SIAM Journal on Control and Optimization 47, 2470– 2489 (2008) [4] Bemporad, A., Borrelli, F., Morari, M.: Min-max Control of Constrained Uncertain Discrete-Time Linear Systems. IEEE Transactions on Automatic Control 48(9), 1600– 1606 (2003) [5] Bemporad, A., Morari, M., Dua, V., Pistikopoulos, E.N.: The explicit linear quadratic regulator for constrained systems. Automatica 38(1), 3–20 (2002) [6] Biegler, L.T., Zavala, V.M.: Large-scale nonlinear programming using ipopt: An integrating framework for enterprise-wide dynamic optimization. Computers & Chemical Engineering 33(3), 575–582 (2009); Selected Papers from the 17th European Symposium on Computer Aided Process Engineering held in Bucharest, Romania (May 2007) [7] Borrelli, F.: Constrained Optimal Control of Linear and Hybrid Systems. LNCIS, vol. 290. Springer, Heidelberg (2003) [8] Borrelli, F., Baotic, M., Pekar, J., Stewart, G.: On the Complexity of Explicit MPC Algorithms. Technical Report (August 2008), http://www.me.berkeley.edu/˜frborrel/pub.php [9] Borrelli, F., Baotic, M., Pekar, J., Stewart, G.: On the complexity of explicit MPC laws. In: European Control Conference (August 2009) [10] Borrelli, F., Baotic, M., Pekar, J., Stewart, G.: On the complexity of explicit mpc laws. Technical report, Mechanical Eng. Department, UC Berkeley, USA (March 2009) [11] Calendini, P.O., Breuer, S.: Mean value models. In: Workshop on Automotive Predictive Control: Models, Methods and Applications, Linz, Austria (2009) [12] dSPACE GmbH (2006), http://www.dspace.com [13] Ferreau, H.J., Bock, H.G., Diehl, M.: An online active set strategy to overcome the limitations of explicit mpc. International Journal of Robust and Nonlinear Control 18, 816–830 (2008) [14] Gheorghe, C., Lahouaoula, A., Backstrom, J., Baker, P.: Multivariable CD control of a large linerboard machine utilizing multiple multivariable MPC controllers. In: Proceedings of PaperCon 2009, St Louis, USA (2009) [15] Guzzella, L., Amstutz, A.: Control of diesel engines. IEEE Control Systems Magazine 18(2), 53–71 (1998) [16] Guzzella, L., Onder, C.H.: Introduction to Modeling and Control of Internal Combustion Engines. Springer, Heidelberg (2004) [17] Herceg, M., Raff, T., Findeisen, R., Allgower, F.: Nonlinear model predictive control of a turbocharged diesel engine. In: Proceedings of 2006 IEEE Conference on Control Applications, pp. 2766–2771 (2006)

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[18] Jones, C., Grieder, P., Rakovi´c, S.: A logarithmic-time solution to the point location problem for closed-form linear MPC. In: IFAC World Congress, Prague, Czech Republic (2009) [19] Jung, M., Glover, K.: Control-oriented linear parameter-varying modelling of a turbocharged diesel engine. In: Proceedings of 2003 IEEE Conference on Control Applications, pp. 155–160 (2003) [20] Kolmanovsky, I.V., Stefanopoulou, A.G., Moraal, P.E., van Nieuwstadt, M.: Issues in modeling and control of intake flow in variable geometry turbocharged engines. In: 18th IFIP Conference on System Modelling and Optimization (1997) [21] Masoudi, M., Konstandopoulos, A., Nikitidis, M.S., Skaperdas, E., Zarvalis, D., Kladopoulou, E., Altiparmakis, C.: Validation of a model and development of a simulator for predicting the pressure drop of diesel particulate filters. SAE Technical Paper Series, 2001-01-0911 (2001) [22] ATI No-Hooks OnTarget (2009), www.accuratetechnologies.com [23] Ortner, P., del Re, L.: Predictive control of a diesel engine air path. IEEE Transactions on Control Systems Technology 15(3), 449–456 (2007) [24] Santos, L.O., Afonso, P.A.F.N.A., Castro, J.A.A.M., Oliveira, N.M.C., Biegler, L.T.: On-line implementation of nonlinear MPC: an experimental case study. Control Engineering Practice 9(8), 847–857 (2001) [25] Schauffele, J., Zurawka, T.: Automotive Software Engineering: Principles, Processes, Methods, and Tools. In: SAE International, Warrendale, PA (2005) [26] Sch¨ar, C.M.: Control of a Selective Catalytic Reduction Process. PhD thesis, Diss. ETH Nr. 15221, Measurement and Control Laboratory, ETH Zurich, Switzerland (2003) [27] Schilling, A., Alfierir, E., Amstutz, A., Guzzella, L.: Emissions-controlled diesel engines. MTZ - Motortechnische Zeitschrift 68(11), 27–31 (2007) [28] Stefanopoulou, A.G., Kolmanovsky, I., Freudenberg, J.S.: Control of variable geometry turbocharged diesel engines for reduced emissions. IEEE Trans. Contr. Syst. Technol. 8(4), 733–745 (2000) [29] Stewart, G.E., Borrelli, F.: A model predictive control framework for industrial turbodiesel engine control. In: Proc. 47th IEEE Conf. on Decision and Control, Cancun, Mexico, pp. 5704–5711 (2008) [30] Thate, J.M., Kendrick, L.E., Nadarajah, S.: Caterpillar automatic code generation. SAE 2004-01-0894 (2004) [31] van Aken, M., Willems, F., de Jong, D.-J.: Appliance of high EGR rates with a short and long route EGR system on a heavy duty diesel engine. SAE Technical Paper Series, 2007-01-0906 (2007) [32] van Helden, R., Verbeek, R., Willems, F., van der Welle, R.: Optimization of urea SCR deNOx systems for HDdiesel applications. SAE Technical Paper Series, 2004-01-0154 (2004) [33] Wang, D.Y., Yao, S., Shost, M., Yoo, J.H., Cabush, D., Racine, D., Cloudt, R., Willems., F.: Ammonia sensor for closed-loop SCR control. SAE 2008-01-0919 (2008) [34] Wei, X., del Re, L.: Gain scheduled H-infinity control for air path systems of diesel engines using LPV techniques. IEEE Transactions on Control Systems Technology 15(3), 406–415 (2007)

Chapter 15

An Integrated LTV-MPC Lateral Vehicle Dynamics Control: Simulation Results Giovanni Palmieri, Osvaldo Barbarisi , Stefano Scala, and Luigi Glielmo

Abstract. In this work we present the integration of a Linear-time-varying Modelpredictive-control (LTV-MPC), designed to stabilize a vehicle during sudden lane change or excessive entry-speed in curve, with a slip controller that converts the desired longitudinal tire force variation to pressure variation in the brake system. The lateral controller is designed using a 3DOF vehicle model taking into account both yaw rate and side slip angle of vehicle while the slip controller is a gain scheduled proportional controller with feedforward action. The performances are validated through simulation: in particular, the authors use a proprietary simulator calibrated on an oversteering sport commercial car and commercial simulator calibrated on a standard light car. Simulation results show the benefits of the control methodology in that very effective steering manoeuvres can be obtained as a result of this feedback policy while satisfying input constraints and show the importance of the introduction of inputs constraints in the control strategy design.

15.1 Introduction Nowadays a great number of control systems are present in a car, each of them being designed to control a particular part of the vehicle: engine, clutch, gearbox, active and semiactive suspension control, power steering, electronic differential, etc. Giovanni Palmieri and Luigi Glielmo Dipartimento di Ingegneria - Universit`a degli Studi del Sannio, Benevento, Italy e-mail: {palmieri,glielmo}@unisannio.it Stefano Scala ELASIS S.C.p.A, Via Ex Aeroporto, 80038 Pomigliano D’Arco (NA), Italy e-mail: [email protected] Osvaldo Barbarisi Dipartimento di Ingegneria - Universit`a degli Studi del Sannio, Benevento, Italy. Currently he works at MENEA Medjimurje Energy Agency Ltd, Croatia e-mail: [email protected] L. del Re et al. (Eds.): Automotive Model Predictive Control, LNCIS 402, pp. 231–255. c Springer-Verlag Berlin Heidelberg 2010 springerlink.com

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Several of these systems contribute to improve performances, comfort and car safety in different working conditions, the most critical systems being those involving stability and safety of the car: emergency braking, cornering stability control, steering or fast manoeuvres to avoid obstacles and collisions with other vehicles. The development of new and safer strategies for vehicle control requires a deep knowledge of the dynamic behavior of the vehicle and of all subsystems it is composed of. It is possible, then, to properly integrate all subsystem actions and optimize their cooperation toward such objectives as minimum braking space, cornering, roadholding, and so on. Obviously, proper modeling is essential to achieve the above goal. In this work, we illustrate a model-based lateral stability control integrated with a slip controller. The goal of lateral controllers is clearly illustrated in Figure 15.1 where the lower curve shows the nominal trajectory that the vehicle would track when the driver imposes a steering input in presence of dry road with a high tire-road friction coefficient. If the friction coefficient is small or the vehicle speed is too high, then the vehicle may be unable to track the nominal trajectory and may follow the trajectory of larger radius as shown in the upper curve of the figure. Then, one of the goals of the lateral control system is the yaw velocity of the vehicle to track as much as possible the nominal motion expected by the driver, as shown by the middle curve in Figure 15.1 [1]. In the last years, many works have underlined the importance to include the side slip angle information into the control strategy. The side slip angle, as shown in Figure 15.1, is formed by the longitudinal axis of the vehicle frame and the direction

Fig. 15.1 Step steering with VDC

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of velocity vector [2]; when, over low friction surfaces, it grows too large, the capability of tires to generate lateral forces is reduced which significantly compromises the performance of the vehicle control system. Then, the second goal of the control action is the reduction of side slip angle. In order to achieve both goals, it is possible to manipulate three groups of variables: the front (and seldomly, rear) steering angle, the braking force (through the so called differential braking) and the engine torque, usually employed together with the former two. Active steering is used to make little corrections on the steering wheel angle profile which the driver imposes [3], or to totally control the tires on unmanned vehicles [4, 5]. Differential braking is the method usually employed in vehicle stability control systems [6, 7, 8, 9, 10] which uses a suitable split of the braking forces among the tires to apply a moment to the vehicle. Other controllers integrate the differential braking technique with the engine torque. There are different configurations of these controller types depending on the possibility offered by the driveline of distributing the engine torque among the wheels, e.g. active differential on the front wheels or on all the wheels [6, 11, 12, 13]. Some differential braking strategies are illustrated with a hierarchical control scheme in [14, 15], where several uncoupled SISO controllers with a supervisor strategy are proposed for guaranteeing stable and robust behavior. In [16, 17, 5] the lateral controller was a model-based MIMO system. In particular in [16, 5] the control model was based on a complete four-wheel model in body and inertial frame (GPS positioning was available) and the input is the wheel steering angle, while in [17] the model is the full vehicle model in body frame and the control inputs are the variation of longitudinal forces on each tires. Those strategies have been designed by using Linear Time Varying (LTV) model predictive control (MPC) techniques. The main contribution of this work is the integration of the control strategy presented in [18] with a slip controller, which drives the pressure of the brake cylinder in the brake system, as showed in Figure 15.2. The Lateral Controller regulates the yaw rate and the vehicle side slip angle by acting on the longitudinal force of tires. The slip controller regulates the brake pressure to achieve the desired longitudinal force of the tire. Since wheel dynamics are much faster than vehicle body dynamics, the design of lateral control neglects wheel dynamics. However the LTVMPCcontrollerenablesamoreprecise design which takes into account the constraints on tire adherence and the velocity of the braking system. This issue is illustrated by comparing the LTV-MPC controller with a PI-based controller which does not explicitly takes into account those constraints. The work is structured as follows: in Section 15.2 we describe the full vehicle model used to develop the MPC control strategy and the slip controller; in Section 15.3 we summarize the lateral control strategy presented in [18] and one implementation of classical parallel connection of two SISO controllers in anti-windup configuration; in Section 15.4 we illustrate the classical slip dynamic model used to design the slip controller, described in Section 15.5; in Section 15.6 we illustrate the results obtained through Elasis – Centro Ricerche Fiat (CRF) proprietary software package simulating an oversteering rear traction sport commercial car and the

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Supervisor Reference signal

ψ&ref

βref

v

e β~

~

β

Observer

ψ&

eψ&

MPC/2PI Lateral controller

Slip Controller

pBCij

Vehicle

ψ& , ω ij , δ ay,ax

Fig. 15.2 General control scheme R software simulating an understeering light car. results obtained through Carsim In the last Section we present conclusions and the future work.

15.2 Full Vehicle Model The double tracks model of the vehicle is depicted in Figure 15.3; the dynamics of the vehicle are described in details in [7] and reported in Equations 15.1a–15.1d for the sake of completeness. The state variables are the side slip angle of vehicle β , the yaw rate ψ˙ and the rotational speed ωi j of each wheel. (In the following we will often omit the indices i, j to indicate specific wheel quantities when this does not cause confusion and simplifies notation.) There are two different input types: 1. the wheel turn angle δ is a “disturbance input”, i.e., it is not a controlled variable, since the drive imposes it; and 2. the contribution of engine torque on each wheel Teng and the braking torque TB on each wheel, instead, are given by the controller. ⎤ ⎤ ⎡ f 1 v, β , ψ˙ , δ , FLi j , FSi j β˙ ⎣ ψ¨ ⎦ = ⎣ f2 v, β , ψ˙ , δ , FLi j , FSi j ⎦ , i = {F, R}, j = {L, R} (15.1a) ω˙ i j f3 FLi j , Tengi j , TBi j 1 {(FLFL + FLFR ) sin(β + δ ) − (FSFL + FSFR ) cos(β − δ ) f 1 v, β , ψ˙ , δ , FLi j = Mv + (FLRL + FLRR ) sin β − (FSFR + FLFL ) cos β } − ψ˙ , (15.1b) 1 f2 v, β , ψ˙ , δ , FLi j = {la (FSFL − FSFR ) cos δ − la (FLFL − FLFR ) sin δ + lb (FSRL + FSRR ) Jz + lc (FSFL − FSFR ) cos δ +lc (FLFL − FLFR ) sin δ + lc (FLRR − FLRL )} , (15.1c) 1 −rFLi j − TBi j + Tengi j . (15.1d) f 3 FLi j , Tengi j , TBi j = Jw ⎡

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vwFL

y

Fy RL

vw y

Fl RL

FxRL

c

FcRL

Fy RR

FxFL

FcFL

vFR Fy FR

Fl RR

x FL

FL

Fl FL

\

D FL vw

Fy FL

D FR

x

a

b FxRR

GF FxFR

FcRR

FcFR

Fl FR

Fig. 15.3 Double track vehicle model

15.2.0.1

Lateral and Longitudinal Forces

The interaction of forces between the wheel and the ground are described by the semi-empirical Pacejka model which we simplify by assuming that, for each wheel, the longitudinal force (resp., the lateral force) depends only on the longitudinal slip (resp., only on the side slip angle), i.e. [19]: fL (s; μ , Fz ) = f˜L (s, α , μ , Fz ) α =0 , (15.2a) fS (α ; μ , Fz ) = f˜S (s, α , μ , Fz ) , (15.2b) s=0

where f˜L and f˜S are the full (i.e., combined) Pacejka’s formulas for longitudinal and lateral force, α is the wheel side slip angle, i.e. the angle between the longitudinal wheel axis and the velocity vector vw . In the following we will often omit the dependence on μ and Fz for the sake of simplicity. The functions fL (pure longitudinal, i.e., driving/braking) and fS (pure lateral, i.e., cornering) are odd w.r.t. s and α respectively. Finally, the tire stiffness coefficients are defined as [7] c (α ; μ , Fz ) :=

∂ fS (α ; μ , Fz ) . ∂α

(15.3)

15.3 Lateral Vehicle Dynamic Control Strategy The algorithm we propose here assumes the following variables to be directly measured: δ , ψ˙ , ωi j , ay and ax ; the variables β and v, instead, have to be estimated on the basis of the above mentioned measurements (details in [7, 2, 20]). The control

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scheme is depicted in Figure 15.2. Given the vehicle velocity v, a proper safety bound to guarantee tires adherence for the side slip angle β is computed; on the basis of v and the steering wheel angle δ we compute a reference yaw rate signal ψ˙ ref (i.e., the yaw rate “desired” by the driver). When either the difference between the actual yaw rate and the reference yaw rate exceeds a safety range or the vehicle side slip angle exceeds the bound, the Supervisor block enables the Control block to compute the braking torques to be applied independently to each wheel.

15.3.1 Reference Signals 15.3.1.1

Reference Yaw Rate

To compute the reference yaw rate signal we use the relation between a given steering angle and the corresponding yaw rate at steady state for a bicycle model, sometimes called Ackermann yaw rate, given by [2, p. 231] vx δ, ψ˙ Ack := l 1 + v2x /v2ch

(15.4)

where l = la + lb . We can then compute a bound to take into account the physical limitation of the vehicle; in particular, it is possible, following [7] to compute

ψ˙ max =

aymax , vx

(15.5)

where aymax is the nominal maximum lateral acceleration. Now, we define ψ˙ b as

ψ˙ Ack (15.6) ψ˙ b = ψ˙ max sat ψ˙ max where “sat” denotes the saturation function. The signal ψ˙ b is then smoothed by a unit gain low pass filter whose poles are obtained by the linearization around β = 0 and ψ˙ = 0 of the bicycle model: W (p) :=

a p2 + bp + a

,

(15.7)

with cF cR l 2 + mv2 (cR lb − cF la ) , Jz mvx Jz + mla2 cF + Jz + mlb2 cR , b(vx ) = Jz mvx

a(vx ) =

where, for compact notation, we define cF = cFL + cFR and cR = cRL + cRR. Hence, the yaw rate reference signal is given by

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ψ˙ ref := W (p) ψ˙ b . 15.3.1.2

237

(15.8)

Bounds on the Vehicle Side Slip Angle

Following [7, p. 322] we require β! to be limited in the interval [−βmax , βmax ] where ⎧ k1 −k2 3 k1 −k2 2 ⎪ ⎨ 2 v3 vx − 3 v2 vx + k1 if vx < vch , ch ch βmax := (15.9) ⎪ ⎩k if vx ≥ vch . 2 The previous formula yields a smoothed version of the maximum side slip angle, as shown in Figure 15.4. Reasonable values for parameters k1 and k2 are 10π /180 and 3π /180 respectively.

Fig. 15.4 Dependence of maximum vehicle body side slip angle on speed

When β!(t) ∈ [−βmax , βmax ] no feedback is activated; when β!(t) > βmax (resp. β!(t) < −βmax ) a regulation level βref = βmax (resp. βref = −βmax ) is activated. Thus we define the error variable ⎧ ⎪ ⎨ β! + βmax when β˜ < −βmax , (15.10) eβ! := 0 when β!(t) ∈ [−βmax , βmax ], ⎪ ⎩! ˜ β − βmax when β > βmax .

15.3.2 Estimation of Tire Variables 15.3.2.1

Tire Side Slip Angle

The tire side slip angle α is the angle between the wheel velocity vector vw and the longitudinal axis of the wheel itself (see Figure 15.3). It results:

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αFL = − δ + atan

v sin β˜ + la ψ˙ v cos β˜ − lc ψ˙

la ψ˙ , −δ + β˜ + v

v sin β˜ + la ψ˙ la ψ˙ , −δ + β˜ + αFR = − δ + atan v v cos β˜ + lc ψ˙ v sin β˜ − lb ψ˙ lb ψ˙ , β˜ − αRL =atan ˜ v v cos β − lc ψ˙ lb ψ˙ v sin β˜ − lb ψ˙ , β˜ − αRR =atan v v cos β˜ + lc ψ˙

(15.11a) (15.11b) (15.11c) (15.11d)

where we suppose cos β˜ 1, sin(β˜ ) β˜ and v |lc ψ˙ |. 15.3.2.2

Longitudinal Slip Ratio

The longitudinal slip ratio is defined as [19] si j :=

r ω i j − v wi j , max{rωi j ; vwxi j }

(15.12)

where r is the actual rolling radius of the tire, ω is the angular speed of the tire (which is measured) and vwx is the linear speed of the tire center along the longitudinal axis of the wheel (which is estimated, e.g. vwxRR = vx + lc ψ˙ ). 15.3.2.3

Vertical Tire Forces Computation

As it is shown in [7], by neglecting suspension dynamics of the vehicle, the four forces of contact to road of the tires coincide with the vertical forces Fz . By balancing the torques at the points of contact between tires and the surface of the road, one obtains [7, p. 306] 1 h lb h 1 lb FzFL ∼ mg − max + may , = 2 la + lb 2 la + lb la + lb lc 1 h lb h 1 lb mg − max − may , FzFR ∼ = 2 la + lb 2 la + lb la + lb lc ∼ 1 la mg + 1 h max + la h may , FzRL = 2 la + lb 2 la + lb la + lb lc 1 1 la h l h a mg + max − may . FzRR ∼ = 2 la + lb 2 la + lb la + lb lc 15.3.2.4

(15.13a) (15.13b) (15.13c) (15.13d)

Estimation of Tire Forces

Once the slips s and α and the vertical tire forces are estimated, we use the simplified Paceijka’s formulas (15.2) to compute the tire force FL and FS . We assume knowledge of μ .

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Fig. 15.5 Lateral and longitudinal forces of the tire inscribed in the ellipsoid of adherence. We draw negative longitudinal forces because we deal with braking

15.3.2.5

Bounds on the Control Inputs

If we represent the longitudinal and lateral forces that the road can exert on the tire as in Figure 15.5, they belong to an area depending on the conditions of road μ and the vertical force Fz . We assume (but this can be actually justified on the basis of the combined Pacejka’s formulas) that the area can be approximated by the ellipsoid (see Figure 15.5)

ε FL2 + FS2 = r2 ,

(15.14)

. 2 where r := FSLIM and ε := FSLIM FLLIM ; FLLIM and FSLIM represent the maximum longitudinal force and lateral force respectively, defined as FLLIM (μ , Fz ) := max fL (s; μ , Fz ) ,

(15.15a)

FSLIM (μ , Fz ) := max fS (α ; μ , Fz ) .

(15.15b)

s

α

Given the actual side force FS , the maximum achievable longitudinal force that avoids the unstable slipping of the wheel is given by

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FLmax

/ # $ 1 2 r − (FS )2 . := ε

(15.16)

Finally we point out that, since we are considering the braking action of the tires, rather than the acceleration action, the longitudinal forces FL will be negative (see Figure 15.5) and the “maximum” braking forces corresponding to a side force FS will be −FLmax .

15.3.3 Supervisor The lateral dynamics controller is activated when either the error on yaw rate eψ˙ or the error on side slip angle eβ! exceed certain respective activations thresholds. The controller is deactivated when both eψ˙ and eβ! are within those thresholds for a period Tdel . Figure 15.6 illustrates the statechart of the corresponding automa.

>e

\

@ >

~ ! e\on OR eE~ ! eEon

@ YesVDC

NoVDC

>e

\

on \

@ >

~ ! e OR eE~ ! eEon

@

Wait

Normal

>e

enable=0

\

@

>

@

off e\off AND eE~ eE~ / t 0

t t0 t Tdel

t

enable=1

Fig. 15.6 Statechart of Supervisor. The default state is NoVDC where the boolean enable signal is set to low and the controller is disabled. When one of the two errors signals is outside the bounds, state YesVDC is activated (the boolean enable signal becomes high). The state YesVDC has two child-states: Normal and Wait. The first is the default state, while Wait is activated when both errors are inside the thresholds; if this condition holds for a time t − t0 > Tdel , where t0 is the time when the controller enters in Wait state, then the NoVDC is activated

The thresholds on yaw rate error depends on the vehicle speed. Since the vehicle responds to the steering wheel angle in different manners as the vehicle speed changes, we chose to shape the yaw rate error activation threshold eon ψ˙ and the deacoff tivation threshold eψ˙ as the following 2vx /vch eON eon ˙ , ψ˙ (vx ) = 1 + v2x /v2ch ψ

(15.17a)

on eoff ψ˙ (vx ) =ξ eψ˙ (vx ),

(15.17b)

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where eON ψ˙ > 0 is a calibration parameter. The reader can see that (15.17a) is obtained from (15.4) by setting δ = 1 and dividing by vch /(2l) (which is the maximum yaw rate for δ = 1 attained at vx = vch ); ξ is a calibration parameter with ξ ∈ (0, 1), tipically ξ = 0.75. The side slip angle error activation threshold eon ! has been chosen constant, since β

in any condition the side slip angle should be small; the deactivation threshold eoff = ξ eon . β˜ β˜ The delay Tdel on the deactivation was introduced after observing that, without it, when the steering wheel angle changes direction abruptly, the controller is disabled (for a very short time) since also the errors on yaw rate and side slip angle change their signs. Figure 15.7 shows an example of enabling signal.

Fig. 15.7 Example of enabling signal: (a) side slip angle error eβ˜ with its thresholds, (b) yaw rate errors eψ˙ with its thresholds, and (c) enable signal

15.3.4 Model Predictive Control The model presented in Section 15.2 is nonlinear. To reduce the computational load rather than working on the model (15.1a), we first compute the linearization of the submodel described by the first two Equations of 15.1a around the current sub-state (β , ψ˙ ). We remind that fS (α + Δ α ) ∼ = fS (α ) + c (α ) Δ α . Further, in our control model we consider as input the variation Δ FL of the tire braking (i.e., negative) forces constrained to −FLmax ≤ FL + Δ FL ≤ 0

(15.18)

Δ FLmin ≤ Δ FL ≤ Δ FLmax , Δ F˙Lmin ≤ Δ F˙L ≤ Δ F˙Lmax ,

(15.19a)

so that

(15.19b)

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where Δ FLmin := −FLmax − FL and Δ FLmax := −FL . Inequalities 15.19a and 15.19b represent physical bounds: (15.19a) describes the tire forces domain in a particular working point, and (15.19b) represents the slew-rate of the braking system. The control goal is to maintain the errors eβ! and eψ˙ close to zero. The linearized model around the working point (v, β!, δ , FL ) is ⎡ ⎤ Δ FL ⎢Δ F FL ⎥ LFR ⎥ !, ψ˙ , δ , FL , FS , β e˙ = A v, β!, δ , FL e + B v, β!, δ ⎢ + d v, ij ij ⎣ Δ FLRL ⎦ Δ FLRR # $T a11 a12 ! where e = eβ! , eψ˙ and A v, β , δ , FL = with a21 a22

(15.20)

1 cF cR (cos δ + δ sin δ ) − − (FLFR + FLFL ) cos δ + mv m mv 1 cF cF − (FLRR + FLRL ) − 2 β! sin δ − 2 la ψ˙ sin δ , mv mv mv l c la cF b R a12 = − 2 cos δ + β! sin β! + 2 − 1, mv mv la cF lb cR a21 = − cos δ + + (cFR sin δ − cFL sin δ ) lc , Jz Jz l 2 cR l 2 cF a22 = − a cos δ − b + (cFR sin δ − cFL sin δ ) lc ; Jz v Jz v

a11 = −

and B v, β!, δ =

(

sin δ −β! cos δ sin δ −β! cos δ mv mv la sin δ −lc cos δ la sin δ +lc cos δ Jz Jz

) β! β! − mv − mv , lc − Jlcz Jz

⎡ ⎤ f1 v, β!, ψ˙ , δ , FLi j , FSi j ⎦. d v, β!, ψ˙ , δ , FLi j , FSi j = ⎣ f2 v, β!, ψ˙ , δ , FLi j , FSi j The model and its constraints are then discretized by using backward Euler’s method with sampling time Ts , thus obtaining xk+1 = AT (k)xk + BT (k)uk + dT (k)

(15.21)

Δ F Δ F Δ F Δ F where u = |t=kTs is the controlinput, AT (k) = I + L L L L k FL FR RL RR A v, β!, δ , FL |t=kTs Ts is the system matrix, BT (k) = B v, β!, δ |t=kTs Ts is the input matrix, dT (k) = d v, β!, ψ˙ , δ , FLi j , FSi j |t=kTs Ts .

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We now consider a prediction horizon H p Ts , for some integer H p , in which the following approximations hold: AT =AT (k) ∼ = AT (k + 1) ∼ = ... ∼ = AT (k + H p − 1), BT =BT (k) ∼ = BT (k + 1) ∼ = ... ∼ = BT (k + H p − 1),

(15.22b)

d T =dT (k) ∼ = dT (k + 1) ∼ = ... ∼ = dT (k + H p − 1).

(15.22c)

(15.22a)

We also consider a control horizon Hc Ts , for some integer Hc ≤ H p , and define Uk = uk ... uk+Hc −1 . At each time k we solve the following optimization problem: Uk∗ = arg min Uk

Hp −1

∑

xTk+h+1 Q xk+h+1 + uTk+h R uk+h

(15.23a)

h=0

subject to xk+h+1 = AT xk+h + BT uk+h + d T , e (kT ) xk = β s , eψ˙ (kTs ) uk+h+1 = uk+h ≤ uk+h ≤ u

min

≤ uk+h − uk+h−1 ≤ Δ u

u

Δu

∀h ≥ Hc − 1,

min

max

(15.23b) (15.23c) (15.23d)

,

(15.23e) max

,

(15.23f)

where u−1 = 0. Inequality 15.23e is related to (15.19a) and inequality 15.23f is related to (15.19b). Since the cost function 15.23a is quadratic and the constraints 15.23b–15.23f are linear, the optimization problem (15.23) is convex and can be solved with an efficient quadratic programming (QP) solver (see [20] for details). We denote by Uk∗ = [u∗k , . . . , u∗k+Hc −1 ] the sequence of optimal braking torques computed at time k by solving problem (15.23) from the current observed state xk . Then the first element u∗k of Uk∗ is actually applied to the system at time k. Even though the MPC algorithm is developed and implemented in discrete time, it is notationally convenient to employ a continuous time description in the remaining part of the work. To do so we denote: u(t)MPC = [hold(t; Δ Fk , Ts ))/Ts

(15.24)

where, with some abuse of notation, hold(t; uk , Ts )) = uk

for t ∈ [kTs , (k + 1)Ts).

(15.25)

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15.3.5 An Alternative 2PI Regulator As a comparison term with the LTV-MPC controller, we present here a classical controller scheme composed of two parallel SISO controller: a Proportional-Integrator (PI) regulator with anti-windup on the yaw rate error eψ˙ and a Proportional (P) regulator on the side slip angle error eβ˜ . The output of the controller is the variation of yaw torque necessary to stabilize the vehicle:

Δ Myaw = kPψ˙ eψ˙ + kIψ˙

t 0

eψ˙ (τ ) d τ + kPβ eβ .

(15.26)

The maximum variation of yaw torque is limited by physics constraints on tires as shown in paragraph 15.3.2.4, so the anti-windup action keeps the torque Myaw inside its limits,

Δ Mmax = − (lc cos δ + la sin δ ) Δ FLmax − lc Δ FLmax , FL RL Δ Mmin = (lc cos δ

− la sin δ ) Δ FLmax + lc Δ FLmax , FR RR

(15.27a) (15.27b)

corresponding to the maximal variations of brake force at each wheel. The variation of yaw torque is made by left wheels when Δ M > 0, by right wheels when Δ M < 0. Then, Δ M is distributed between front and rear wheel according to maximal longitudinal Δ FLmax enforced at each wheel (see equation 15.18), ij

ΔM < 0

ΔM > 0

0

Δ M Δ F max +LFL Δ F max

Δ MLFL = Δ MLRL = Δ MLF R = Δ MLRR =

Δ F max

LFL

LRL

FL

RL

Δ F max Δ M Δ F max +LRL Δ FLmax L

0 Δ FLmax FR Δ FLmax +Δ FLmax Δ M FR max RR Δ FL RR Δ FLmax +Δ FLmax Δ M FR

(15.28)

0 0

RR

The corresponding values of desired longitudinal tire forces are

Δ MLFL , lc cos δ + la sin δ Δ MLFR = , lc cos δ + la sin δ 1 = Δ MLFL , lc 1 = Δ MLFL . lc

Δ FLFL =

(15.29a)

Δ FLFR

(15.29b)

Δ FLRL Δ FLRR

(15.29c) (15.29d)

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15.4 A Reduced Model for Slip Control The desired variation of longitudinal forces is achieved by the braking system. The slip controller computes the desired longitudinal slip corresponding to the desired longitudinal force and computes the braking force that has to be applied. For control design we will consider the third equation of (1a) and, thanks to the fast time response of the wheel, we can neglect the other dynamics by considering as constant the yaw rate, vehicle side sleep angle and the vehicle velocity during the time interval of slip control intervention. With this assumption the model of angular wheel velocity becomes Jw ω˙ = −r fL (s) + Teng − TB ,

(15.30)

and the derivative of the slip ratio can be written as s˙ =

rω˙ v . max{v2 , (rω )2 }

(15.31)

We distinguish two cases: Braking: When v ≥ rω , then s ∈ [−1, 0] and Equation 15.31 yields s˙ =

rω˙ , v

so that Equation 15.30 becomes

r s˙ = −r fL (s) + Teng − TB . vJw Traction:

(15.32)

(15.33)

When rω ≥ v then s ∈ [0, 1] and Equation 15.31 can be written as s˙ =

rω˙ v rω˙ (1 − s)2 = 2 2 r ω v

(15.34)

so that (15.30) becomes s˙ =

r vJw

−r fL (s) + Teng − TB (1 − s)2 .

(15.35)

r fL (s) = Teng − TB

(15.36)

Notice that the roots of

are the equilibrium points s¯ of both (15.33) and (15.35) with s ∈ [−1, 0] and s ∈ [0, 1], respectively. Such equilibrium points may or may not exist according to value of the RHS of Equation 15.36, and there may be one or two equilibrium points. If ∂ fL (s) ¯ /∂ s is positive, s is asymptotically stable so that we define the stable interval as the interval of s where the above derivative is positive.

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τ ff

−r

uMPC

∫

FLref

sref

sL

es

W

τ fb

∫

FL

μ

TB

wheel

s

FZ

Fig. 15.8 Slip control scheme. For the sake of simplicity we included into the “wheel” block the algorithm related to the estimation of slip ratio s

One of the goals of vehicle dynamic control is to keep the longitudinal slip s inside the stable interval where typically the longitudinal slip is close to zero and (1 − s)2 ∼ = 1. Hence, in order to design the longitudinal slip controller, we consider the model

r s˙ = −r fL (s) + Teng − TB . (15.37) vJw Finally, since in this work we only deal with the braking phase with accelerator pedal released and a high gear, we neglect Teng and use the model

r (−r fL (s) − TB ) . (15.38) s˙ = vJw

15.5 A Slip Control Strategy Goal of the slip controller is to obtain the desired longitudinal force FLref (t) := FLref (0) +

t 0

uMPC (τ ) d τ

(15.39)

by applying a braking torque TB . The desired slip sref is obtained through inversion of Pacejka’s formula, computed for values of slip inside the stable interval: sref = fL−1 (FLref ).

(15.40)

The control action on the wheel is given by T˙B = τ f f + τ f b = T˙B f f + T˙B f b .

(15.41)

The slip control action, then, is the sum of a feedforward action τ f f and a feedback action τ f b . The complete control scheme is depicted in Figure 15.8. The feedforward control law is given by

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T˙B f f = τ f f = −ruMPC , TB f f = −rFLref + const = −rFLref ,

(15.42)

where uMPC is the control input obtained from (15.24). In the Equation 15.42 we ignore the const value because the wheel slip and the longitudinal forces are continuously estimated so that, using (15.40), we can write r fL (sref ) + TB f f = 0.

(15.43)

15.5.1 Feedback Action Let us define the slip error as es = sref − s. Now, taking into account (15.39), one has d −1 # ref $ f FL (t) dt L ∂ −1 # ref $ ˙ ref FL (t) FL f = ∂F L

L ∂ uMPC , = 1/ fL sref ∂s

s˙ref =

so that the dynamic equation of error can be written as

∂ ref r ref e˙s = s˙ − s˙ = 1/ fL s (−r fL (s) − TB ) . uMPC − ∂s vJw

(15.44)

The first term of RHS of Equation 15.44 (of order of 10−5 in our simulations), can be neglected w.r.t. to the second one (of order 102 ). Thus, Equation 15.44 becomes

r e˙s = (r fL (s) + TB) vJw

∂ r ∼ r fL sref − r fL sref es + TB f f + TB f b = vJw ∂s

∂ ref r −r fL s es + TB f b = vJw ∂s = −(a1 a2 )es + a1TB f b , (15.45) where the term r fL sref + TB f f is zero in view of (15.43) and a1 =

r , vJw

a2 = r

∂ fL (sref ), ∂s

T˙B f b = τ f b ,

(15.46)

with a2 > 0 if sref is in the stable interval. The transfer function V (p; sref ), between feedback control input τ f b and slip ratio error es , is given by

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V (p; sref ) =

a1 . p (p + a1a2 )

(15.47)

We use a proportional regulator whose gain depends on a1 and a2

τ f b = −k p es .

(15.48)

The transfer function of the closed loop system Vo (s) is given by Vo (p) =

a1 k p . p2 + a1 a2 p + a1 k p

(15.49)

The bandwidth of (15.49) has to be kept at least one order of magnitude less than the sampling frequency of the regulator fs (here fs = 1000[Hz]). So we choose the parameters of regulator by imposing a1 k p =

2π fs 10

2 (15.50)

so that 1 kp = a1

2π fs 10

2 .

(15.51)

Equation 15.49 becomes Vo (p) =

2π 2 10 f s 2 . 2 p + a1a2 p + 210π fs

(15.52)

Simulations show that the above closed loop transfer function depends mildly on the operating conditions: vehicle speed v, vertical force Fz and friction coefficient μ .

15.6 Simulation Results To test our control strategy we have chosen a manoeuver known as the ATI90 in dry asphalt condition (μ = 1). In this manoeuver the driver: 1. turns the steering wheel from −90deg to +90deg; 2. decides the initial speed (in our simulations, 120 km/h for the oversteering car and 90 km/h for the understeering car ); and 3. releases the accelerator pedal when the manoeuvre starts. We tested our strategy on an oversteering sport car simulated through an ELASISCRF (Fiat group) proprietary simulator and an understeering light car simulated R The tuning parameters are: through Carsim . 1. sampling time Ts = 20 [ms]; 2. the prediction horizon H p = 5;

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Fig. 15.9 Simulation without control strategy for the oversteering car: (a) longitudinal velocity, (b) lateral acceleration, (c) yaw rate and reference yaw rate, (d) steering wheel angle, and (e) side slip angle. Notice that when the value of β exceeds 80, the simulation can be considered finished

3. the control horizon Hc = 3; 4. the control weight for the side slip angle β : q1 =10 for the oversteering car and q1 =5 for the light car; 5. the control weight for yaw rate ψ˙ : q2 =1 for the oversteering car and q2 =0.7 for the light car; 6. the control weight matrix R for the variation of longitudinal forces:

10−11 |maxFzij | 10−11 |maxFzij | 10−11 |maxFzij | 10−11 |maxFzij | for the oversteering , , , |FzFL | |FzFR | |FzRL | |FzRR | −8 10 |maxFzij | 10−8 |maxFzij | 10−8 |maxFzij | 10−8 |maxFzij | car and diag for the light car, , |Fz | , |Fz | , |Fz | |FzFL | FR RL RR

diag

where the |Fzmax | is the maximum normal forces of four wheels; and 7. the deactivation time Tdel = 0.12 [s], the side slip error activation threshold eon ! = β

on 0.5 [deg] and deactivation threshold is eoff ! = 0.75e ! .

β

β

The choice of the horizons length is a compromise between computational load and necessary information for the prediction model. We present results and performance at 140 [km/h] for the oversteering sport car and at 90 [km/h] for the understeering light car; both cars, when performing this

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Fig. 15.10 Simulation with control strategy @140 km/h, vehicle variables: (a) longitudinal velocity, (b) lateral acceleration, (c) yaw rate and reference yaw rate, (d) steering wheel angle, and (e) side slip. angle

Fig. 15.11 Simulation with control strategy @140 km/h, control variables from top: (a) steering wheel angle, (b) MPC control input (variation of longitudinal tire force), (c) output of slip controller (oil pressure in the braking system), and (d) activation signal of control

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Fig. 15.12 Comparison between LTV-MPC strategy (solid line) and 2PI strategy (dashed), vehicle variables: (a) steering wheel angle, (b) side slip angles and reference side slip angle, (c) yaw rates and reference yaw rate, (d) longitudinal speeds

manoeuvre at 100 [km/h] (oversteering sport car) and 90 [km/h] (understeering light car), lose stability. This is illustrated in Figure 15.9 for the oversteering sport car, where one can notice that β exceeds 80; this behavior is due to the oversteering characteristics, the rear traction, and the excitation of nonlinear dynamics. In Figure 15.9 we show the value of 1. 2. 3. 4. 5.

longitudinal speed vx in km/h; lateral acceleration ay in g s; car yaw rate (solid line) and reference yaw rate (dashed line) in deg/s; steering wheel angle SWA in deg; and side slip angle β in deg.

In Figure 15.10 it is clear that during the manoeuvre the longitudinal speed decreases smoothly to about 90km/h, the lateral acceleration reaches the maximum value when the steering angle is constant and the yaw rate tracking is satisfactory. There is no significant high frequency chattering, thus guaranteeing the driver’s comfort. Notice too that the maximum value of the sideslip angle is 4. We wish to point out that the peak of the yaw rate signal toward the end of the second steering change, which depends on the sudden change of steering and subsequent loss of authority of tires on one side, is not immediately followed by new action on the other side.

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Fig. 15.13 Comparison of control variables between LTV-MPC strategy (solid line) and 2PI strategy (dashed line), performed @90 km/h on the understeering light car: (a) oil pressure of braking system for front-left wheel, (b) oil pressure of braking system for front-right wheel, (c) oil pressure of braking system for rear-left wheel, and (d) oil pressure of braking system for rear-right wheel

In Figure 15.11 we show the control action. The second plot depicts the variation of longitudinal forces. These have to be inside their physical limits, which depend on load transfer and normal force on each wheels, so that when the car is curving on the right, the left forces increase and the right forces are approximately zero, and viceversa on a left curve. In other words tires with a larger instantaneous vertical load have a greater authority. In the third plot the final control input (the pressure of brake cylinder) are depicted. The slew rate of each signal depends on the limitation of input variation (see 15.19b). These signals are computed only when the controller is enabled; the enabling signal can be seen in the last plot of Figure 15.11. In Figure 15.12 the same variables of Figure 15.10 for the manoeuvre with 90[km/h] longitudinal entry speed are shown for the understeering light car comparing the LTV-MPC and the 2PI controller. During this manoeuvre the tires work close to their saturation zone. In 2PI simulation it is possible to see some oscillations of yaw rate though the tracking of reference side slip angle is better. This is due to LTV-MPC being more conservative than 2PI controller regarding the control effort bounds and its slew rate, as can be perceived in Figure 15.13. There the limitation due to brake system is more apparent and the slope of braking pressures underlines the control difficulty to stabilize the vehicle during sudden changes of steering. The

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use of braking system is on the whole stronger with 2PI than LTV-MPC which, furthermore, penalizes the longitudinal speed.

15.7 Conclusions In this chapter a novel integrated vehicle dynamics control has been presented that utilizes differential braking. It is based on an MPC force controller cascaded with a slip controller. The simulations were performed on an oversteering sport car using an ELASIS-CRF proprietary simulator and on an understeering light car in the R dataset. The results showed that the use of a MIMO controller yields Carsim encouraging results. The strategy will be hardware-in-the-loop tested before being implemented on test car. The next steps of our work, apart the above-mentioned experimental tests, will be the introduction of on-line estimation of the friction coefficient μ and the use of engine torque in the control strategy.

Definitions (·)i j the subscript i stands for F (front) or R (rear), the subscript j stands for L (left) or R (right), with respect to the body axis system depicted in Figure 15.3 αi j wheel side slip angle β vehicle side slip angle β! estimated β δ wheel turn angle μi j tire-road friction coefficient ωi j angular speed of the tire ψ˙ yaw rate ψ˙ ref reference yaw rate eψ˙ error between yaw rate and reference yaw rate eβ˙ error between estimated vehicle side slip angle and vehicle reference side slip angle ax vehicle longitudinal acceleration ay vehicle lateral acceleration ci j tire cornering stiffness h distance from center of gravity (CoG) to the road la longitudinal distance from CoG to the front axle lb longitudinal distance from CoG to the rear axle 2lc track width M mass of the vehicle ri j rolling radius of the tire si j longitudinal slip ratio v vehicle velocity

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characteristic speed wheel velocity vector of the tire center velocity of the vehicle along the x direction velocity of the vehicle along the y direction longitudinal force on the tire lateral force on the tire vertical force on the tire yaw inertia moment of vehicle around the z–axis inertia of wheel around the y–axis the contribution of engine torque on each wheel braking torque

References [1] Piyabongkarn, D., Rajamani, R., Grogg, J., Lew, J.: Development and experimental evaluation of a slip angle estimator for vehicle stability control. IEEE Transactions on Control Systems Technology 17(1), 78–88 (2009) [2] Rajamani, R.: Vehicle Dynamics and Control. Springer, New York (2005) [3] Ackermann, J., B¨unte, T., Odenthal, D.: Advantages of active steering for vehicle dynamics control (1999) [4] Borrelli, F., Falcone, P., Keviczky, T., Asgari, J., Hrovat, D.: Mpc-based approach to active steering for autonomous vehicle systems. International Journal of Vehicle Autonomous Systems 3(2-3), 265–291 (2005) [5] Falcone, P., Borrelli, F., Asgari, J., Tseng, H.E., Hrovat, D.: Predictive active steering control for autonomous vehicle systems. IEEE Transactions on Control Systems Technology 15(3), 566–580 (2007) [6] Piyabongkarn, D., Lew, J., Grogg, J., Kyle, R.: Stability-enhanced traction and yaw control using electronic limited slip differential, Vehicle Dynamics and Simulation, no. SAE technical papers 2006-01-1016 (April 2006) [7] Kienke, U., Nielsen, L.: Automotive Control Systems. Springer, Heidelberg (2000) [8] Daily, R., Bevly, D.: The use of GPS for vehicle stability control systems. IEEE Transactions on Industrial Electronics 51(2), 270–277 (2004) [9] Anwar, S.: Predictive yaw stability control of a brake-by-wire equipped vehicle via eddy current braking. In: American Control Conference, 2007, July 9-13, pp. 2308– 2313 (2007) [10] Ackermann, J.: Robust yaw damping of cars with front and rear wheel steering. In: Proceedings of the 31st Conference on Decision and Control, Tucson-Arizona, December 1992, pp. 2586–2590 (1992) [11] Zhao, S., Li, Y., Zheng, L., Lu, S.: Vehicle lateral stability control based on sliding mode control. In: Proc. IEEE International Conference on Automation and Logistics, August 2007, pp. 638–642 (2007) [12] Ouladsine, M., Shraim, H., Fridman, L., Noura, H.: Vehicle parameter estimation and stability enhancement using the principles of sliding mode. In: Proc. American Control Conference ACC 2007, July 9-13, pp. 5224–5229 (2007)

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[13] Kolesz´ar, P., Trencs´eni, B., Palkovics, L.: Development of an electronic stability program completed with steering intervention for heavy duty vehicles. In: Proceedings of the IEEE International Symposium on Industrial Electronics, June 2005, vol. 1, pp. 20–23 (2005) [14] Dongshin Kim, K.K., Lee, W., Hwang, I.: Development of mando esp (electronic stability program). In: Proceeding of the 2003 SAE World Congress. SAE 2003-10-0101 (March 2004) [15] van Zanten, A.T., Erthadt, R., Pfaff, G.: VDC, the vehicle dynamics control of Bosch. In: Proceeding of the International Congress and Exposition, March 1995. SAE950759, pp. 9–26 (1995) [16] Borrelli, F., Bemporad, A., Fodor, M., Rovat, D.: An MPC/hybrid system approach to traction control. IEEE Transactions on Control Systems Technology 14(3), 541–552 (2006) [17] Palmieri, G., Falcone, P., Tseng, H.E., Glielmo, L.: A preliminary study on the effects of roll dynamics in predictive vehicle stability control. In: Proc. 47th IEEE Conference on Decision and Control CDC 2008, December 9-11, pp. 5354–5359 (2008) [18] Barbarisi, O., Palmieri, G., Scala, S., Glielmo, L.: Dynamically constrained differential braking for yaw rate control and side slip control. In: Proc. European Control Conference ECC 2009 (August 2009) [19] Pacejka, H.: Tire and Vehicle Dynamics. SAE International (2006) [20] Barbarisi, O., Palmieri, G., Scala, S., Glielmo, L.: LTV-MPC for yaw rate control and side slip control with dynamically constrained differential braking. European Journal of Control (2009) (to appear)

Chapter 16

MIMO Model Predictive Control for Integral Gas Engines ¨ Jakob Angeby, Matthias Huschenbett, and Daniel Alberer

Abstract. The legal requirement of NOx emission reduction from legacy gas engines used in compressor stations asks for an improved engine control. A gas engine is a MIMO system with strong coupling, the inputs and outputs being limited by physical constraints and customer requirements. The engines drive compressors that change the load at time instants known in advance and the load change pattern can be modeled. A MIMO online linear model predictive controller (MPC) with the objective of keeping the fuel/air ratio and the engine speed constant was applied and compared to the standard SISO PID controls. The tracking of the fuel/air ratio during the transients was improved up to 80% when using the MPC approach which is sufficient to meet the up-coming emission legislation.

16.1 Introduction Abatement of NOx emissions is a main concern in the automotive industry and has been given considerable attention in the research community. However, there are other internal combustion engine applications that require considerable attention, like ship transportation and stationary plants. In the paper stationary gas engines in compressor stations are considered. Such engines are used in the US gas pipeline network to compensate the pressure losses and to adjust the flow according to the customer consumption. They are often built as integral engines whose crank shaft directly drives the compressor pistons, and flow regulation is usually performed in a discrete way, switching between clearance pockets of different size [2] at approximately constant speed. Due to the large installation cost, engines are changed very ¨ Jakob Angeby and Matthias Huschenbett Hoerbiger Engine Solutions e-mail: {jakob.aengeby,matthias.huschenbett}@hoerbiger.com Daniel Alberer Institute for Design and Control of Mechatronical Systems, Johannes Kepler University Linz, Austria e-mail: [email protected] L. del Re et al. (Eds.): Automotive Model Predictive Control, LNCIS 402, pp. 257–272. c Springer-Verlag Berlin Heidelberg 2010 springerlink.com

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seldom, and most installations include ”legacy“ engines, which were not designed for the actual emission levels. Recently, new NOx targets have been fixed which should be fulfilled without substantial modifications of the engines. The lean burn engines under consideration run mainly under static conditions. The most economic solution for emission abatement on the considered engines is to operate them at very lean fuel/air ratios without an additional aftertreatment. However, sudden changes of the compressor work load lead to unacceptable changes of the fuel/air ratio (φ ) and so of NOx . The compressor load changes can be modeled using physics. The compression volume is changed through switching in and out a number of ”pockets“ which are cavities of fixed volumes that are part of the compressor clearance volume. When a pocket is switched in/out the compressor clearance volume is increased/decreased and as a consequence the load is decreased/increased. Since many pocket changes occur in several steps – which may include opening and closing of large and small pockets, and these changes are not synchronized – the integral engine goes through a series of different states before reaching the intended value [1]. On a control relevant time scale this evolution of the clearance volume can be represented as a continuous load change on the engine crank shaft which is known both in shape and timing [4]. The MIMO nature of the plant, the existence of hard bounds on the manipulated variables and soft bounds on the outputs speak for a model predictive approach. Furthermore, MPC provides the ability to incorporate preview information of the externally requested compressor load torque. The parameter changes in the plant would call for a jump parameter approach [7]. However, as the parameter jumps can be described using specific load step models which describe their effects on the torque as if they were load changes, the model predictive control framework can be used instead of more complex algorithms (as e.g. in [6]).

16.2 System Description The considered system consists of a turbocharged 2 stroke reciprocating gas engine with counter-flow scavenging and a reciprocating compressor both acting on the same crankshaft, also called “integral engine”. For the current analysis a Clark TLA 6 was used, with 6 combustion and 3 compressor cylinders. Typically the engine is operated in a narrow range (see Table 16.1). The engine runs on very lean fuel mixtures. In order to achieve a good ignition of the lean fuel mixture, each cylinder has two pre combustion chambers, where fuel Table 16.1 Engine operating condition

Engine speed [rpm] Power [hp] Relative fuel/air ratio φ [-]

Nominal value 300 2000 0.59

Operating range 280-300 1700-2000 0.55-0.62

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Fig. 16.1 Integral Engine

is injected such that a locally rich fuel mixture is formed and ignited by spark plugs. The flames that shoot out from the pre chambers then in turn ignite the fuel in the main combustion chamber. The integral engine has four main controls: • fuel: the speed is controlled by the injected fuel amount (through the fuel rail pressure); • air: φ is controlled with the turbocharger waste gate; • ignition timing: the ignition timing is usually kept constant; and • load: the load is adjusted with the compressor by adding or removing discrete amounts of compressor clearances (pockets). The pockets can be seen in Figure 16.1, where 2 small pockets are mounted in “Vshape” at each compressor cylinder and one big at the front. In standard operation the pockets are used to maintain almost constant engine speed by adapting the load

Fig. 16.2 Pocket loading chart

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Fig. 16.3 Engine speed and φ during LS 8-7 and standard control

Fig. 16.4 Standard vs. model based control structure

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to the slowly changing pipeline conditions. The considered integral engine has 6 small (FCV-1 to FCV-6 in Figure 16.2) and 3 big pockets (FCV-7 to FCV-9 in Figure 16.2) which results in 28 load stages, also called load steps (LS). LS28 indicates the minimum load (all pockets open) and LS1 the maximum. The LS can only be passed sequentially, as shown in Figure 16.2. For example, in the LS change 8 to 7 (8-7) six small pockets must be opened and a large one closed. Since this procedure is not perfectly synchronized, substantial load deviations arise. The consequences can be seen in Figure 16.3 (under the existing standard control action). Notice that once a LS transition is initialized the resulting transient load torque trajectory cannot be influenced any more. The standard control configuration implemented on a PLC (process logic control) is shown in Figure 16.4. The controllers are usually simple SISO PIDs, although in particular φ has a strong coupling with the injected fuel amount. This leads to the assumption, that a MIMO model based control, which inherently takes into account the couplings in the system, should be able to increase the control performance. The model based approach replaces the cascaded standard structure, which is sketched in Figure 16.4, too.

16.3 Problem Statement The objective is to keep NOx below a given threshold while keeping the engine speed within specified limits under varying load conditions. There are no catalytic after-treatment devices applied to the engines under consideration and therefore the engines are operated lean (see Figure 16.5) in order to keep the NOx at an acceptable level. The relation between φ and NOx is known, hence φ can be used as an indirect measurement of NOx which simplifies the control problem. The main control target

Fig. 16.5 NOx as a function of φ

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Fig. 16.6 System description

is therefore defined as to keep φ at a pre-defined set point. The secondary objective is to keep the engine speed at the desired reference value. The system for control design is illustrated in Figure 16.6. The combustion cylinders and the compressor act on the same crankshaft. Unfortunately, the torque cannot be measured directly, only the effect can be measured through the engine speed. GFC (governor fuel command) represents the valve position of the fuel supply, i.e. with GFC the injection pressure and consequently the injected fuel amount can be controlled. WG (waste gate) is the opening position of the turbocharger turbine bypass and LS (load step) sets the pocket combination. The system outputs are engine speed and φ , which is calculated on the base of charge air pressure, temperature, scavenging efficiency and injected fuel amount. There are some control constraints. Although engine speed is not the main objective, a maximum speed limit may not be trespassed. The air manifold (AMP) pressure must stay within a certain range, too. The maximum value is related to physical constraints and moreover due to a decreasing efficiency of the turbocharger at low pressure, the air system can become unstable (counter flow scavenging relies on the pressure difference between input and exhaust port), consequently there is also a lower bound for AMP.

16.4 Model Predictive Control Model-based predictive control (MPC) provides the optimal solution subject to constraints [3]. For the present work online linear model predictive control was applied, where a quadratic problem is solved in every time instant to determine the optimal

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system inputs. In the following, the basic principles are briefly described and in the Sections below the application to the integral engine is presented. One can formulate the control problem as an optimization problem with constraints. Indeed, consider the following receding horizon formulation Ny −1

min U

∑

T T et+k|t Q et+k|t + δ ut+k|t R δ ut+k|t

(16.1)

k=0

subject to xt+k+1|t = Axt+k|t + But+k|t yt+k = Cxt+k|t umin ≤ ut+k ≤ umax δ umin ≤ δ ut+k ≤ δ umax

(16.2)

where Q and R are positive definite matrices of suitable dimensions, xt+k|t ∈ ℜn is the estimated value of the state at the time t + k as predicted at the time t, yt+k|t the corresponding output, and et+k|t = yre f ,t+k|t − yt+k|t is the trackΔ

ing error. U = [δ ut , ..., δ ut+Nu −1 ] ∈ ℜmNu is a vector containing the Nu future changes of the m-dimensional system input considered in the optimization problem, usually called control horizon, with ut+k = ut+k−1 + δ ut+k . Ny is the number of considered prediction values of the outputs (the prediction horizon). For more details see [3]. MPC solves this problem at each time step t by substituting xt+k|t = j Ak xt + ∑k−1 j=0 A But+k−1− j and recasting the problem as a quadratic program (QP) &

$ ' # T yT FU xtT ut−1 re f ,t U ⎡ ⎤ xt s.t. GU ≤ W + E ⎣ ut−1 ⎦ = W + E ξ yre f ,t min

1 T 2 U HU +

(16.3)

where H, F, G, W and E can easily be obtained from the plant model and Q, R. In (16.3), ξ is an extended state vector which contains the actual state, the last values of the manipulated variables and the reference values. As the optimization problem (16.3) depends on the current state xt , the implementation of MPC requires an online solution of a QP at each time step. As mentioned before, problems in the form of (16.3) can be solved efficiently by numerical QP solvers, e.g. using the online active set strategy as proposed in [5] for the MPC context.

16.5 Implementation In the next Subsections the adaptation of the integral engine task to the MPC framework is presented.

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16.5.1 Objective Function The control target is to track given set points of φ and engine speed and keep the control effort low. 2 2 J = c1 · φre f − φ + c2 · RPMre f − RPM + c3 · δ u2 . (16.4) Hereby, the weighting factor c1 was chosen much higher than c2 and c3 to enforce the tracking of φ . Notice that c1 and c2 of (16.4) equal the main diagonal elements of Q in (16.1) and c3 equals R.

16.5.2 Model Derivation The MPC formulation (16.1) and (16.2) is based on a linear model of the plant to be controlled, the engine. The compressor asserts a load on the engine fly wheel and is modeled separately. The MIMO engine model was chosen as a gray-box consisting of a number of simple interconnected single input single output (SISO) models that are easy to determine from straight forward measurements. The approach is motivated by known physical relationships and experience. The compressor model, however, is slightly more complicated due to the series of discrete non-linear

Fig. 16.7 Calculated compressor load torque for LS7-8 (top figure) and LS8-7 (bottom figure)

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Fig. 16.8 Gray-box control model scheme

compressor model changes that occur when the pockets are switched in and out. Figure 16.1 shows the calculated compressor torque (based on indicated pressure measurement) during LS7-8 and LS8-7. During LS8-7 the load is decreased by 45% for a short moment, whereas the steady state values differ by 5% only. This is explained by that as a change from one level to another is requested, then a number of compressor pockets are switched in and out, respectively. Since these actions are not synchronized, the load can be changed significantly during the transient. The sequence of torque changes (levels and timings) were identified from data recordings and stored in look up tables. The hereby identified compressor load change pattern is treated as a known change that shall be compensated for by the MPC such that the impact on the objective function (16.1) is minimized, i.e. it is used as a feed forward to the controller. Figure 16.8 depicts the gray-box scheme of the control model, where several dynamics subsystems are combined with affine stationary blocks. The fuel-to-torque path for instance was captured by a first order lag element with time delay and a constant DC gain. Furthermore, the engine speed impact on the mechanical injection system was taken into account in the block Combustion, too. The turbocharger was modeled as a second order MISO system and the crankshaft was assumed to be an ideal integrator. In a separated identification procedure first the DC gains were determined with steady state measurements and then dynamical data – suitable pseudo random binary input signals for WG and GFC were applied – was used to identify the dynamical parts. The input delay times were detected by single actuator steps and finally the subsystems were combined to an overall MIMO state space control model. Figure 16.9 presents validation results of the overall model on dynamical data, whereas the main system behavior is captured with a sufficient accuracy.

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Fig. 16.9 Validation of the gray-box model for the fuel flow, the engine speed (RPM), the fuel manifold pressure (FMP), the air manifold pressure (AMP) and the relative fuel/air ratio (PHI)

16.6 Model Extensions To achieve offset-free tracking additional integrator states ek were added as error model for every output1 xk u xk+1 A0 B 1 B2 · · k = + 0 I ek ek+1 0 0 vk (16.5) xk Yk = C I · ek T

where uk = [ GFCk W Gk ] and vk denotes the compressor torque that in this modeling approach is viewed as an input. For the MPC application to follow, however, vk is used as a measured disturbance and incorporated in the state vector. 1

Notice that in the the final implementation the error states ek are determined by a Kalman estimator under the assumption of a white noise input.

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⎤ ⎡ ⎤ ⎡ ⎤ ⎤ ⎡ xk xk+1 A 0 B2 B1 ⎣ ek+1 ⎦ = ⎣ 0 I 0 ⎦ · ⎣ ek ⎦ + ⎣ 0 ⎦ · uk 0⎡0 I ⎤ 0 vk+1 vk . x k Yk = C I 0 · ⎣ ek ⎦ vk

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⎡

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Future information of the compressor load pattern vk can easily be incorporated in the model by introducing a fictive “time delay” in the model representation that allows us to “see” load changes before they will affect the engine, hereby enabling the controller to take the necessary actions to minimize (16.1). Accordingly, a specified “delay time” is added to the compressor torque signal, which has to be taken into account in the model. This system model augmentation can be seen as a future measured disturbance representation, ⎤ ⎡ ⎤ ⎡ xk A 0 B2 0 ... 0 0 ⎥ ⎢ ⎥ ⎡ ⎤ ⎢ ⎥ ⎡ ⎤ ⎢ 0 I 0 0 ... 0 0 ⎥ ⎢ ek B1 ⎢ 0 0 0 1 0 0 0 ⎥ ⎢ v1,k ⎥ xk+1 ⎥ ⎢ ⎥ ⎢0 ⎥ ⎢ ⎢ ⎥ ⎢ ⎥ ⎥ ⎢ ⎣ ek+1 ⎦ = 0 0 0 0 1 0 0 · v2,k ⎥ ⎢ ⎥ + ⎣ ... ⎦ · uk ⎢ ⎢ ⎥ ⎥ ⎢ Vk+1 ⎥ ⎢ 0 0 0 0 0 ... 0 ⎥ ⎢ ... 0 (16.7) ⎣ 0 0 0 0 0 0 1 ⎦ ⎣ vN−1,k ⎦ 0 0 0 0 0 0 1 ⎡ vN,k ⎤ xk Yk = C I 0 · ⎣ ek ⎦ Vk where (v1,k , v2,k , . . . , vN−1,k , vN,k ) is the a priori known compressor load. Through this model formulation we are able to “see” the expected compressor load N sampling instants ahead. If a load step will take place within this future measured (compressor) “residing disturbance prediction horizon” of N sampling periods ahead, then the model will sense this and the MPC will optimize the engine performance according to (16.3). Clearly, N is a design variable and, just as the control and prediction horizons, subject to a trade off between computational burden and performance. As a rule of thumb, N shall be chosen large enough to allow for the engine control system (turbo, waste gate, fuelling system) to “prepare” itself for load step to come, but not larger due to the cost in computational burden that comes with the increase in model dimension.

16.7 Real-time MPC The QP condensing, which is required to prepare the problem and make it suitable R The real time minimization required for a QP-solver, was done in MATLAB . during execution of the MPC was done using the software package qpOASES. A Kalman filter was designed for the control model (16.5) and furthermore tuned on the plant to have a fast state estimation and satisfactory disturbance filtering. The

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Fig. 16.10 Hoerbiger Advanced Engine Controller (AEC)

real time application was implemented on a Hoerbiger Advanced Engine Controller (AEC) system with a control horizon of 5, a prediction horizon of 300 and a sampling period of 0.1s. The MPC is enabled in nominal engine operation only (e.g. engine startup is handled by a different controller). In case of repeated feasibility problems of the QP solver or in case of a poor control performance (which might be caused by an abnormal system behavior) a special safety routine becomes active and MPC is deactivated.

16.8 Results In the following the MPC implementation is compared to the standard control during load changes. LOAD DECREASE Figure 16.11 and Figure 16.12 show a comparison of PLC and MPC performance for LSLS 14-15 which is a load decrease. The MPC was implemented with 10s future measured disturbance information (from the compressor model). A comparison between the objective function values corresponding to the PLC (PID) and the MPC, respectively, gives (The cost function C(.) refers to a relative comparison of the sum of squared deviations to the reference in a time window from 10s before the LS change to 30s after it.): • C(PHIMPC) = 78.9% of PID cost function; • C(RPMMPC) = 88.7% of PID cost function; and • C(NOxMPC) = 88.5% of PID cost function. The objective function shows good improvement, as compared to an already good performance using the PLC. During a load decrease the fuel is decreased until the new (lower) power level is reached. A load decrease, however, is not as challenging as an increase. Then the controller must suddenly increase the power which calls for rich fuel mixtures before the engine has settled at the new load level.

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Fig. 16.11 Comparison of φ and engine speed during a load decrease

Fig. 16.12 Comparison of NOx emissions during a load decrease

LOAD INCREASE Figure 16.13 shows a comparison between the PLC (PID) and the MPC at LS 15-14 which is a load increase. The same design parameters were used as in Figure 16.11. The load step begins at t = 40s. It is clearly seen in Figure 16.13 that the GFC is increased before the real LS change occurs, hereby boosting the turbo and “preparing” the engine for the load step soon to come. A comparison between the objective function values corresponding to the PLC (PID) and the MPC, respectively, gives: • C(PHIMPC) = 20.4% of PID cost function; • C(RPMMPC) = 84.7% of PID cost function; and • C(NOxMPC) = 3.9% of PID cost function. The objective functions indicate a significant improvement. This is confirmed by measurements of the NOx as shown in Figure 16.15.

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Fig. 16.13 Comparison of φ and engine speed during a load increase

Fig. 16.14 GFC actuation during a load increase

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Fig. 16.15 Comparison of NOx emissions during a load increase

Using the MPC, over 80% reduction in φ -variance from the set point can be achieved with significantly reduced emissions as a consequence.

16.9 Conclusions A linear model-based predictive controller was successfully applied to an integral gas engine. The effects due to the nonlinear and time dependent switching of the pockets were modeled as a torque change on the crankshaft and used a future measured reference. The inclusion of the hereby future measured disturbance information in MPC allows the controller to set the actuators prior to the real load step change, whereas the impact of the known load change is reduced. The introduced controller architecture (MPC + future measured disturbance) reduced the deviations in the objective function up to 80%, hereby enabling continued use of the installed legacy integral engine fleet without tripping the emission regulations by an upgrade of the engine control system.

Acknowledgment The authors want to thank the Pipeline Research Council for the financial support and in particular the team at the SoCalGas compressor station in Newberry Springs for the assistance running the tests at the TLA 6 engine.

References [1] Emission Reduction & Stability Improvement by Model Based Predictive Control of Legacy Gas Engines, Dallas, Texas (2007) [2] Bloch, H.P., Hoefner, J.J.: Reciprocating Compressors: Operation and Maintenance. Gulf Professional Publishing (1996) [3] Camacho, E.F., Bordons, C.: Model Predicitve Control, 2nd edn. Springer, London (2004) [4] del Re, L., Alberer, D., Ranzmaier, M., Huschenbett, M.: Mimo model predictive control for integral gas engines under switching disturbances, San Antonio, Texas, USA (2008)

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[5] Bock, H.G., Ferreau, H.J., Diehl, M.: An online active set strategy to overcome the limitations of explicit mpc. International Journal of Robust and Nonlinear Control 18, 816– 830 (2008) [6] Lam, S., Davison, E.J.: Control of plants with changing dynamics using switching model predictive control, Montreal, QC, Canada (1979) [7] Sworder, D., Robinson, V.: Feedback regulators for jump parameter systems with state and control dependent transition rates. IEEE Transactions on Automatic Control 18, 355–360 (1973)

Chapter 17

A Model Predictive Control Approach to Design a Parameterized Adaptive Cruise Control Gerrit J.L. Naus, Jeroen Ploeg, M.J.G. Van de Molengraft, W.P.M.H. Heemels, and Maarten Steinbuch

Abstract. The combination of different desirable characteristics and situationdependent behavior cause the design of adaptive cruise control (ACC) systems to be time consuming and tedious. This chapter presents a systematic approach for the design and tuning of an ACC, based on model predictive control (MPC). A unique feature of the synthesized ACC is its parameterization in terms of the key characteristics safety, comfort and fuel economy. This makes it easy and intuitive to tune, even for nonexperts in (MPC) control, such as the driver. The effectiveness of the design approach is demonstrated using simulations for some relevant traffic scenarios.

17.1 Introduction Adaptive cruise control (ACC) is an extension of the classic cruise control (CC), which is a widespread functionality in modern vehicles. Starting in the late 1990s with luxury passenger cars, ACC functionality is now available in a number of commercial passenger cars as well as trucks. The objective of CC is to control the longitudinal vehicle velocity by tracking a desired velocity determined by the driver. Only the throttle is used as an actuator. ACC extends CC functionality by automatically adapting the velocity if there is a preceding vehicle, using the throttle as well as the brake system. Commonly, a radar is used to detect preceding vehicles, measuring the distance xr and the relative velocity vr between the vehicles. Hence, besides Gerrit J.L. Naus, M.J.G. Van de Molengraft, W.P.M.H. Heemels, and Maarten Steinbuch Eindhoven University of Technology, P.O.box 513, 5600 MB, Eindhoven, The Netherlands e-mail: {g.j.l.naus,m.j.g.v.d.molengraft,w.p.m.h.heemels}@tue.nl, [email protected] Jeroen Ploeg TNO Automotive, P.O.box 756, 5708 HN, Helmond, The Netherlands e-mail: [email protected] L. del Re et al. (Eds.): Automotive Model Predictive Control, LNCIS 402, pp. 273–284. c Springer-Verlag Berlin Heidelberg 2010 springerlink.com

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vh,ah

radar beam

target vehicle vt

xr ,vr

Fig. 17.1 The ACC-equipped host vehicle, driving with velocity vh and acceleration ah , automatically follows a preceding target vehicle, driving with velocity vt

CC functionality, ACC enables also automatic following of a predecessor. In Figure 17.1, a schematic representation of the working principle of ACC is shown. Considering the automatic-following functionality, ACC systems typically consist of two parts: a vehicle-independent part and a vehicle-dependent part [13]. In Figure 17.2, a schematic representation of the ACC control loop is shown. The vehicle-independent part determines a desired acceleration/deceleration profile for the vehicle. The vehicle-dependent part ensures tracking of this profile via actuation of the throttle and the brake system through uth and ubr , respectively. The latter part can thus be regarded as a controller for the longitudinal vehicle acceleration. As every vehicle has different dynamics, this part is vehicle dependent. The distance xr and relative velocity vr = vt − vh with respect to the preceding vehicle are measured using a radar. ACC system xr ,vr vehicle ah,d vehicle uth host vehicle radar ah ,vh vh indep- ah depen- ubr endent dent

Fig. 17.2 Schematic representation of an ACC control loop

Focusing on the vehicle-independent part, the primary control objective is to ensure following of a preceding vehicle. Considering the corresponding driving behavior, ACC systems are generally designed to have specific key characteristics, such as safety, comfort, fuel economy, traffic-flow efficiency and minimizing emissions [18]. In general, however, these characteristics typically impose contradictory control objectives and constraints, complicating the controller design. For instance, to ensure safe following, the system should be agile, requiring high acceleration and deceleration levels, which is not desirable considering comfort or fuel economy [10]. To account for different characteristics, a weighted optimization can be employed. For example, a model predictive control (MPC) approach can be adopted, which also facilitates constraint satisfaction [3, 9]. Besides the contradictory desirable characteristics, driver acceptance of the system requires ACC behavior to mimic human driving behavior to some extent [4]. Apart from the fact that human driving behavior is driver specific and time varying, it is also situation dependent [10, 19, 20]. The desired situation dependency of the

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designs give rise to many tuning variables, which makes the design and tuning time consuming and error prone. In this chapter, the design of an ACC is presented, accounting for the contradictory characteristics and the many tuning variables. The focus is on the design of the vehicle-independent part of the automatic-following functionality. The contribution is the design of an ACC, which is parameterized by the key characteristics safety, comfort and fuel economy, with at most one tuning variable for each characteristic. The setting of the ACC can then easily be changed, possibly even by the driver. The organization of this chapter is as follows. The problem formulation and the setup are presented in Sections 17.2 and 17.3. The parameterization and results are discussed in Section 17.4. Finally, conclusions and an outlook on future work are given.

17.2 Problem Formulation The problem formulation involves parameterization of the ACC, based on the chosen key characteristics.

17.2.1 Quantification Measures In this research, safety, comfort and fuel economy are chosen as the key characteristics of the desired behavior of an ACC. Considering safety, however, we note that the ACC is not a safety system such as an emergency braking or a collision avoidance system. ACC is primarily a comfort system that incorporates safety in the sense that appropriate driving actions within surrounding traffic are guaranteed. To enable quantification of the key characteristics, desirable properties of these characteristics, so-called quantification measures, have to be defined. Typically, the safety of a traffic situation increases for increasing inter-vehicle distance and decreasing relative velocity. Hence, regarding safety, the inter-vehicle distance and the relative velocity will be used as quantifications measures [11]. Regarding comfort, the (peak) acceleration and (peak) jerk levels will be used as quantification measures [8, 16]. Concerning fuel consumption, especially the average velocity and the deceleration time are important measures [6, 15]. Both measures are influenced by the acceleration and deceleration levels. Hence, regarding fuel economy, these levels will be used as quantification measures.

17.2.2 Parameterization This research presents the design of a parameterized ACC, with, in the end, only a few design parameters, that are directly related to the key characteristics of the behavior of the ACC. The limited number of intuitive tuning variables enables quick and easy adaptation of the ACC to different desirable driving behavior. Importantly, these variables can also be used by nonexperts in (MPC) control, like the average

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driver, to change the behavior of the ACC system to the driver’s own wishes. Enabling the driver to set these variables, really makes the ACC easily adjustable. Correspondingly, the design parameters Ps , Pc and Pf are defined, indicating to what extent the driving behavior of an ACC-controlled vehicle is either safe, comfortable or fuel economic, with Ps ∈ [0, 1], Pc ∈ [0, 1] and Pf ∈ [0, 1]. Incorporating Ps , Pc and Pf in the controller design yields a parameterized ACC, i.e., ACC(Ps , Pc , Pf ), with Ps , Pc and Pf as tuning variables directly related to the behavior of the ACC. The systematic approach presented here, makes it possible to redesign the system relatively easy, and reduces the amount of time-consuming and error-prone trialand-error techniques in the design. Although, focus lies here on safety, comfort and fuel economy, the approach is general and can be adopted for any characteristics, e.g., traffic flow efficiency or minimizing emissions.

17.3 Model Predictive Control Problem Setup In this section, the control problem formulation is discussed.

17.3.1 Modeling A model predictive control (MPC) synthesis is adopted to design the ACC. The MPC synthesis requires a model of the relevant dynamics to use as a prediction model. Consider again the control structure as presented in Figure 17.2. Focusing on the design of the vehicle-independent control part, the model should cover the longitudinal host vehicle dynamics, the vehicle-dependent control part and the longitudinal relative dynamics, which are measured by the radar. Assuming that the vehicle-dependent control part ensures perfect tracking of the desired acceleration ah,d (t), the internal vehicle dynamics and the vehicle-dependent control part together can be modeled by a single integrator, relating the host vehicle velocity vh (t) and the (desired) acceleration ah (t) = ah,d (t). The continuous-time equations, modeling the dynamics, are given by: ⎧ t ⎨ xr (t) = xr (0) + 0 vr (τ )d τ vr (t) = vr (0) + 0t ar (τ )d τ (17.1) ⎩ vh (t) = vh (0) + 0t ah (τ )d τ where xr (t) the relative position, vr (t) = vt (t) − vh(t) the relative velocity, ar (t) = at (t) − ah (t) the relative acceleration, vh (t) the host vehicle velocity, and ah (t) the host vehicle acceleration at time t ∈ R+ . The values of xr (t) and vr (t) are measured by the radar and measurements of vh (t) and ah (t) are available. As the acceleration of the target vehicle at (t) is unknown, it is, for now as a nominal case, assumed to be zero in the MPC prediction model, yielding ar (t) = −ah (t). In the end, at (t) acts as a disturbance on the closed loop system. MPC is commonly designed and implemented in the discrete-time domain. Therefore, the continuous-time equations (17.1) are converted into a discrete-time

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model via exact discretization with sample time Ts , and using a zero-order-hold assumption on ah (t). The signals are considered at the sampling times t = k Ts where k ∈ N represents the discrete time steps: x(k + 1) = Ax(k) + Bah (k)

k∈N

(17.2)

where x(k) = (xr (k), vr (k), vh (k))T , with some slight abuse of notation, and ⎞ ⎛ 1 2⎞ ⎛ 1 Ts 0 − 2 Ts A = ⎝ 0 1 0 ⎠ , B = ⎝ −Ts ⎠ (17.3) 0 0 1 Ts Considering the control structure as presented in Figure 17.2, the host vehicle acceleration ah (k) = ah,d (k) can be regarded as the control input. Furthermore, as all states of x(k) are measured, the output equation becomes y(k) = x(k), k ∈ N, yielding: 6 x(k + 1) = Ax(k) + Bu(k) M: k∈N (17.4) y(k) = x(k) with u(k) = ah (k) and A and B as defined in (17.3). Finally, the input-output model M (17.4) is converted into an increment inputoutput (IIO) model Me [7]. This enforces integral behavior, i.e., enabling a nonzero control output u(k) for zero error e(k), thus providing the possibility to prevent steady state errors in, for example, the following distance. The IIO model is given by: 6 xe (k + 1) = Ae xe (k) + Be δ u(k) Me : k∈N (17.5a) ye (k) = xe (k) where xe (k) = (xT (k), u(k − 1))T is the new state vector, δ u(k) = u(k) − u(k − 1) the new control input, and ⎞ ⎛ ⎛ ⎞ 0 1 Ts 0 − 12 Ts2 ⎜ 0 1 0 −Ts ⎟ ⎜0⎟ ⎟ ⎜ ⎟ (17.5b) Ae = ⎜ ⎝ 0 0 1 Ts ⎠ , Be = ⎝ 0 ⎠ 1 0 0 0 1 are the new model matrices. The model Me (17.5) will be used as the MPC prediction model for the vehicle-independent control part in the remainder of this chapter.

17.3.2 Control Objectives and Constraints Typically, the primary control objective of an ACC amounts to following a target vehicle at a desired distance xr,d (k). Often, a so-called desired time headway thw,d is used to define this desired distance, yielding

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xr,d (k) = xr,0 + vh (k)thw,d

(17.6)

with xr,0 a constant representing the desired distance at standstill, and the desired time headway thw,d a measure for the time it takes to reach the current position of the preceding vehicle if the host vehicle continues to drive with its current velocity, i.e., for constant vh (k). Correspondingly, the tracking error at discrete time k ∈ N is defined as e(k) = xr,d (k) − xr (k). Hence, the primary control objective, denoted as O1 , comes down to minimizing the absolute tracking error |e(k)|, k ∈ N. Besides the primary control objective O1 , several secondary objectives as well as constraints, related to the key characteristics safety, comfort and fuel economy, have to be included. These secondary objectives and constraints are based on the quantification measures discussed in Section 17.2.1: the absolute value of the relative velocity |vr (k)| and the peak values of the host vehicle acceleration |ah (k)| and the jerk, which will be denoted by | jh (k)|, should be kept small. Furthermore, the relative position should always be positive, i.e., xr (k) > 0, and the absolute values of the acceleration of the host vehicle |ah (k)| and the absolute value of the jerk | jh (k)| are constrained. The constraints on the acceleration and the jerk are given by ah,min = −3.0 m s−2 [17], ah,max (vh (k)) = ah,0 − α vh (k), and | jh (k)| ≤ jh,max , where jh,max , ah,0 and α are appropriately chosen positive constants. The parameter α will allow to decrease ah,max for increasing vh (k). The IIO model accommodates the constraint on | jh (k)|, using the variation in the control output δ u(k) as a measure for the jerk jh (k). Correspondingly, the constraint on the jerk is transformed into |δ u(k)| ≤ jh,max . Summarizing, the constraints are given by: ⎧ 0 < xr (k) ⎨ C : ah,min ≤ u(k) ≤ ah,max (vh (k)) k ∈ N (17.7) ⎩ |δ u(k)| ≤ jh,max where u(k) = ah,d (k) = ah (k).

17.3.3 Control Problem / Cost Criterion Formulation As we use MPC, a cost criterion J, which is minimized over a prediction horizon Ny , has to be defined. The future system states are predicted using the model Me (17.5) and the current state xe (k|k) := xe (k) at discrete time step k as initial condition. This yields the predicted states xe (k + n|k) and the predicted tracking error e(k + n|k), n = 0, 1, . . . , Ny for a selected input sequence δ U(k|k) = T δ u(k|k), . . . , δ u(k + Ny − 1|k) , starting at discrete time step k. Based on the prediction of the future system states, the minimization problem yields an optimal control sequence, subject to constraints (17.7) on the inputs and outputs. The cost criterion is typically formulated as a linear or as a quadratic criterion. To solve the corresponding optimization problem results in a linear program (LP) or a quadratic program (QP). Finding the solution of an LP is less computationally demanding than the corresponding solution of a QP, although this can also be

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done efficiently. The tuning of linear formulations, however, suffers from practical drawbacks, which explains why MPC is often formulated using a quadratic criterion [7, 14]. We will use the quadratic criterion: J(δ U(k|k), xe (k)) =

y ξ T (k + n|k) Q ξ (k + n|k) + ∑n=1 Nu −1 + ∑n=0 δ uT (k + n) R δ u(k + n)

N

(17.8)

with ξ (k + n|k) (e(k + n|k), vr (k + n|k), ah (k + n|k))T a column vector incorporating the primary and secondary control objectives, with ah (k + n|k) = u(k + n|k), and Q = diag(Qe , Qvr , Qah ) and R = Qjh the weights on the tracking error and the secondary control objectives. Furthermore, Ny and Nu denote the output and the control horizon, respectively, where Nu ≤ Ny . Moreover, for Nu ≤ n < Ny the control signal is kept constant, i.e., δ u(k + n|k) = 0 for Nu ≤ n < Ny . Finally, u(k + n|k) = u(k + n − 1|k) + δ u(k + n|k), for n ≥ 0. Given a full measurement of the state xe (k) of the model Me (17.5) at the current time k, the MPC optimization problem at time k is formulated as minimize J(δ U(k|k), xe (k)) δ U(k|k)

(17.9)

subject to the dynamics Me (17.5) the constraints C (17.7) The controller will be implemented in a receding horizon manner, meaning that at every time step k, an optimal future input sequence δ U∗ (k|k) = (δ u∗ (k|k), . . . , δ u∗ (k + Ny − 1|k))T is computed in the sense of the minimization problem (17.9). The first component of this vector, δ u∗ (k|k), is used to compute the new optimal control output u∗ (k) = u(k − 1) + δ u∗(k|k). This u∗ (k) is applied to the system, i.e., u(k) = u∗ (k), after which the optimization (17.9) is performed again for the updated measured state xe (k + 1) = (xT (k + 1), u(k))T .

17.4 Controller Design The final controller design, the implementation, and simulation results are presented in this section.

17.4.1 Parameterization The MPC controller design incorporates all quantification measures regarding safety and comfort. This yields a significant number of MPC tuning parameters, given by the desired time headway thw,d , the constraints on the acceleration and jerk, ah,min , ah,max and jh,max , respectively, the weights Q = diag(Qe , Qvr , Qah ) and R = Qjh , and the control and prediction horizons Nu and Ny .

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Using affine relationships between the MPC tuning parameters on the one hand and the ACC design parameters for safety, Ps , for comfort, Pc , and for fuel economy, Pf , on the other hand, the MPC tuning parameters are explicitly related to the key characteristics safety, comfort, and fuel economy. In this way, the MPC tuning parameters are all determined as a function of these three essential design parameters. The setting of these design parameters indicates to what extent the driving behavior is either safe, comfortable or fuel economic, with Ps ∈ [0, 1], Pc ∈ [0, 1] and Pf ∈ [0, 1]. Due to space limitations, we will not discuss in detail how these affine relationships are actually constructed, see [12]. In this specific case in which we considered comfort, safety and fuel economy as key characteristics, it can be assumed that the key characteristics are complementary: the design of the relationships between the MPC tuning parameters and the ACC design parameters indicates a decrease in comfort of the driving for increasing safety, and vice versa. For example, small acceleration and jerk peak values, indicating a high level of comfort, induce a long time to steady state, which is not desirable regarding safety. Furthermore, the quantification measures chosen to indicate comfort, are similar to those indicating fuel economy. Consequently, in this case, a single parameter P ∈ [0, 1] results: P = Pc ,

Ps = 1 − P,

Pf = P,

P ∈ [0, 1]

(17.10)

If other characteristics would be considered in the design, typically more design parameters would remain in the end. Parameterization of the ACC with safety, comfort and fuel economy amounts to incorporating in the original optimization problem (17.9) the relationships between the MPC tuning parameters, thw,d , ah,min , ah,max , jh,max , Q, R, Nu and Ny , and the design parameters, Ps , Pc and Pf , accounting for (17.10). This yields minimize J(P, δ U(k|k), xe (k)) δ U(k|k)

(17.11)

subject to the dynamics Me (17.5) the constraints C (17.7) where C = C (P) as a result of the parameterization. Changing the behavior of the ACC system comes down to adjusting P. Allowing the driver to change P ∈ [0, 1], enables him to influence the behavior of the controller focusing on either safe, or comfortable and fuel economic driving, depending on the driver’s own desire.

17.4.2 Implementation Issues The total controller design is implemented via the Multi Parametric Toolbox [5]. Online solving of the optimization problem (17.9) at each time step yields an implicit solution. Solving (17.9) as a multi-parametric quadratic program (mpQP) with parameter vector xe enables an explicit form of the solution by offline optimization. The resulting explicit controller inherits all stability and performance properties of

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the implicit controller and has the form of a piecewise affine (PWA) state feedback law [1, 2]. Solving the mpQP, provides a set Xf ⊆ Rnx , with nx the dimension of xe , of states for which the constrained optimization problem (17.9) is feasible. Since the control law is given by a PWA state feedback law, the feasible set Xf is partitioned into R polyhedral regions Ri , i = 1, . . . , R, such that Xf =

R 9

Ri

(17.12)

i=1

where intRi ∩ intRj = 0, / for i = 1, . . . , R, j = 1, . . . , R and i = j. At time step k, the optimal input δ u∗ (k|k) is then given by

δ u∗ (k|k) = Fi xe (k) + fi ,

for xe (k) ∈ Ri ,

i = 1, . . . , R

(17.13)

To compute the control input at discrete time step k ∈ N, (17.13) has to be evaluated. Regarding the explicit solution, the most time-consuming part is determination of the region Ri that contains xe (k). However, online tuning is prohibited by the offline optimization. As a solution, one might store various explicit controllers for a finite number of values P ∈ n/N for n = 0, 1, 2, . . . , N. For implementation of the implicit controller, solving an optimization in every time step is required. Hence, the computational demand depends on the available solver, which is not desirable targeting industrial acceptance. However, P can be changed online in a continuous manner. Depending on the system requirements, one may adopt either solution.

17.4.3 Results

acc. [m s−2] speed [km/h] distance [m]

To illustrate the influence of varying P ∈ [0, 1], simulations are performed for some relevant traffic scenarios using an explicit controller. For a finite number of values N = 10 for P, the number of regions in the explicit ACC laws ranges from 110 to 120. In Figures 17.3 and 17.4 the results of the approach of a vehicle at standstill 200 100 0 40 30 20 10 0 0.5

−0.5 −1.5 −2.5 10 13

20

30

40

50

60

70

80

time [s]

Fig. 17.3 The distance xr , the host vehicle velocity vh and the acceleration ah , corresponding to the approach of a vehicle at standstill. The solid black, the dashed black and the solid gray lines show the results for increasing P ∈ {0.2, 0.5, 0.8}

acc. [m s−2] speed [km/h] distance [m]

282

G.J.L. Naus et al.

45 35 25 15 5 80 70 60 50 40 30 2.5 0.5

−1.5 −3.5 15

20

25

30

35

40

time [s]

Fig. 17.4 The distance xr , the host vehicle velocity vh and the acceleration ah , corresponding to a negative cut in. The solid black, the dashed black and solid gray lines represent the results for increasing P ∈ {0.2, 0.5, 0.8}, respectively. The narrow black line in the middle figure represents the velocity of the target vehicle vt

and a negative cut-in scenario for different settings P ∈ {0.2, 0.5, 0.8} are shown, showing the proper working of the parameterization. By changing the setting of the design parameter P ∈ [0, 1], the behavior of the ACC system changes, with respect to the comfort, the safety and the fuel economy of the resulting driving action. In Figure 17.3, the results of the approach of a vehicle at standstill are shown. At 13 s, the vehicle at standstill is detected by the radar, which has a range of 180 m. Before that, no vehicle is detected. The following behavior of the ACC system becomes more comfortable as well as more fuel economic for increasing P. Firstly, the deceleration peaks decrease, and secondly, the total deceleration time increases. As a result, the average velocity decreases, which is beneficial regarding fuel economy. A negative cut in scenario involves the cut in of a vehicle driving with a velocity vt (k) < vh (k) at an inter-vehicle distance xr (k) < xr,d (k), see Figure 17.4: at 20 s, a vehicle cuts in 30 m in front of the host vehicle with a velocity of 50 km h−1, while the host vehicle is driving at 80 km h−1. Before that, no preceding vehicle is detected and, hence, no distance is measured. From a safety point of view, direct reaction and substantial braking are required, disregarding the setting of P, i.e., comfort or fueleconomy-related measures. The results in Figure 17.4 indeed show this behavior, indicating that safe behavior is guaranteed for any value of P. Furthermore, the results show that for decreasing P the desired steady state distance increases, which is a result of the parameterization thw,d = thw,d (P), and is desirable regarding safety.

17.5 Conclusions and Future Work In this chapter, a systematic procedure to design an ACC is presented, which is directly parameterized by the key characteristics safety, comfort and fuel economy of the ACC behavior. The goal of the parameterization of the ACC is to reduce the time it takes to tune the system and to enable the tuning for nonexperts in (MPC) control, such as the driver. This requires that the tuning should be simple and intuitive with only a few design parameters that are directly related to the key characteristics of

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the ACC. To this end, the corresponding design parameters Ps , Pc and Pf are defined. Due to the generality of the approach, other characteristics can be straightforwardly incorporated in the design, using the same systematic design procedure. The approach is based on (explicit) MPC. The parameterized ACC is obtained by carefully mapping the many tuning parameters of the MPC setup to the three design parameters Ps , Pc and Pf only, which, in this specific case, could be united in one design parameter P. Simulations have shown the proper functioning of the parameterized ACC for some relevant traffic scenarios. Changing the behavior of the system by changing the setting of the design parameter P, has proven to work in a desired manner. Future research will focus on experiments, and on extending the two-vehicle model to multiple vehicles. Taking vehicle-to-vehicle communication into account too, allows for the design of so-called cooperative ACC (CACC) systems. The communication provides additional information concerning the surrounding traffic in addition to the radar data, which can be very beneficial to the system behavior.

References [1] Bemporad, A., Morari, M., Dua, V., Pistikopoulos, E.: The explicit linear quadratic regulator for constrained systems. Automatica 38(1), 3–20 (2002) [2] Bemporad, A., Heemels, W., De Schutter, B.: On hybrid systems and closed-loop MPC systems. IEEE Trans. Automatic Control 47(5), 863–869 (2002) [3] Corona, D., De Schutter, B.: Adaptive Cruise Control for a SMART car: A comparison benchmark for MPC-PWA control methods. Trans. on Control Syst. Technol. 16(2), 365–372 (2008) [4] Driel, C.V., Hoedemaeker, M., Arem, B.V.: Impacts of a congestion assistant on driving behaviour and acceptance using a driving simulator. J. Transportation Res. F10, 139– 152 (2007) [5] Kvasnica, M., Grieder, P., Baotic, M., Christophersen, F.: Multi-Parametric Toolbox (2006) [6] Latteman, F., Neiss, K., Connolly, T.: The predictive cruise control - a system to reduce fuel consumption of heavy-duty trucks. SAE Technical Pap. Ser. (2004-01-2616) (2004) [7] Maciejowski, J.: Predictive control with constraints. Prentice Hall - Pearson Educ. Ltd. (2002) [8] Martinez, J.J., De Wit, C.C.: A safe longitudinal control for Adaptive Cruise Control and Stop-and-Go scenarios. IEEE Transaction on Control Syst. Technology 15(2), 246– 258 (2007) [9] M¨obus, R., Baoti´c, M., Morari, M.: Multi-object Adaptive Cruise Control. In: Maler, O., Pnueli, A. (eds.) HSCC 2003. LNCS, vol. 2623, pp. 359–374. Springer, Heidelberg (2003) [10] Moon, S., Yi, K.: Human driving data-based design of a vehicle Adaptive Cruise Control algorithm. Vehicle Syst. Dynamics 46(8), 661–690 (2008) [11] Naus, G.J.L., Van den Bleek, R.P.A.M., Ploeg, J., Scheepers, B., Van de Molengraft, M.J.G., Steinbuch, M.: Explicit MPC design and performance evaluation of an ACC Stop-&-Go. In: Proc. of the 2008 Am. Control Conf., Seattle, WA, USA, pp. 224–229 (2008)

284

G.J.L. Naus et al.

[12] Naus, G.J.L., Ploeg, J., Van de Molengraft, M.J.G., Heemels, W.P.M.H., Steinbuch, M.: Design and implementation of parameterized adaptive cruise control: An explicit model predictive control approach. Submitted for journal publication (2009) [13] Prestl, W., Sauer, T., Steinle, J., Tschernoster, O.: The BMW Active Cruise Control ACC. SAE Technical Pap. Ser (2000-01-0344) (2000) [14] Rao, C., Rawlings, J.: Linear programming and Model Predictive Control. J. Process Control 10(2), 283–289 (2000) [15] Saerens, B., Vandersteen, J., Persoons, T., Swevers, J., Diehl, M., den Bulck, E.V.: Minimization of the fuel consumption of a gasoline engine using dynamic optimization. J. Appl. Energy (2009) (accepted) [16] International Organization for Standardization: Mechanical vibration and shock - Evaluation of human exposure to whole-body vibration. NEN-ISO 2631 (1997) [17] International Organization for Standardization: Transport information and control systems - Adaptive Cruise Control systems - performance requirements and test procedures. NEN-ISO 15622 (2002) [18] Vahidi, A., Eskandarian, A.: Research advances in intelligent Collision Avoidance and Adaptive Cruise Control. IEEE Transactions on Intell. Transportation Syst. 4(3), 143– 153 (2003) [19] Widmann, G., Daniels, M., Hamilton, L., Humm, L., Riley, B., Schiffmann, J., Schnelker, D., Wishon, W.: Comparison of lidar-based and radar-based Adaptive Cruise Control systems. SAE Technical Pap. Ser. (2000-01-0345) (2000) [20] Zhang, J., Ioannou, P.: Integrated roadway / Adaptive Cruise Control system: Safety, performance, environmental and near term deployment considerations. Calif. PATH Research Rep. (UCB-ITS-PRR-2004-32) (2004)

Author Index

Lewander, Magnus 171 L´ opez, J. Javier 25

Alamir, Mazen 139 Alberer, Daniel 1, 257 Amari, Rachid 139 ¨ Angeby, Jakob 257 Arr`egle, Jean 25

Magni, Lalo 107 Monin, Christelle 25 Murilo, Andr´e 139

Barbarisi, Osvaldo 231 Bemporad, Alberto 183 Borrelli, Francesco 195, 211 Breuer, Stefan 37 Calendini, Pierre Olivier

37

del Re, Luigi 1, 73 Di Cairano, Stefano 183 Diehl, Moritz 119 Eriksson, Lars

273

Ohata, Akira 153 Oppenauer, Klaus 73 Ortner, Peter 1, 139 Pachner, Daniel 211 Palmieri, Giovanni 231 Pekar, Jaroslav 211 Ploeg, Jeroen 273

53

Falcone, Paolo 195 F¨ urhapter, Richard 139 Germann, David 211 Glielmo, Luigi 231 Guardiola, Carlos 25 Heemels, W.P.M.H. 273 Hirsch, Markus 73 Hrovat, Davor 183, 195 Huschenbett, Matthias 257 Johansson, Rolf

Naus, Gerrit J.L.

89

Kihas, Dejan 211 Klein, Markus 53 Kolmanovsky, Ilya 183

Saerens, Bart 119 Scala, Stefano 231 Scattolini, Riccardo 107 Steinbuch, Maarten 273 Stewart, Greg 211 Tona, Paolina 139 Tseng, Eric H. 195 Tunest˚ al, Per 89, 171 Van de Molengraft, M.J.G. Van den Bulck, Eric 119 Wahlstr¨ om, Johan 53 Widd, Anders 89 Yamakita, Masaki 153 Yanakiev, Diana 183

273

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Automotive Model Predictive Control: Models, Methods and Applications (Lecture Notes in Control and Information Sciences, Volume 402)

Nonlinear Model Predictive Control: Towards New Challenging Applications (Lecture Notes in Control and Information Sciences)

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