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

Lecture Notes in Control and Information Sciences Editors: M. Thoma · M. Morari 323 B. Bandyopadhyay S. Janardhana...

Author: B. Bandyopadhyay | S. Janardhanan

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

323

B. Bandyopadhyay S. Janardhanan

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

Series Advisory Board

F. Allgöwer · P. Fleming · P. Kokotovic · A.B. Kurzhanski · H. Kwakernaak · A. Rantzer · J.N. Tsitsiklis

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

ISSN 0170-8643 ISBN-10 ISBN-13

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

Library of Congress Control Number: 2005931593 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in other ways, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable to prosecution under German Copyright Law. Springer is a part of Springer Science+Business Media springeronline.com © Springer-Verlag Berlin Heidelberg 2006 Printed in Germany The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Typesetting: Data conversion by authors. Final processing by PTP-Berlin Protago-TEX-Production GmbH, Germany Cover-Design: design & production GmbH, Heidelberg Printed on acid-free paper 89/3141/Yu - 5 4 3 2 1 0

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

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

P7eface

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

VIII

Preface

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

Mumbai, May 2005

Bijnan Bandyopadhyay Janardhanan Sivaramakrishnan

Con2en26

1

2

3

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

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

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

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Contents

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

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

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

Contents

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

6

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

XI

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

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

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Contents

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

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

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

Li62 of Figw7e6

1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 2.1 2.2

3.1 3.2

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

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

32 32 33 34 38 40 41 41 42 43

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List of Figures

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

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

59 62 65 68 69 70

5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8

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

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

List of Figures

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

XV

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

Nomencla2w7e

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

Nomenclature

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

k

x f H Hτ d

S

Sτ W zu zx zy

XVIII

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

1 In27odwc2ion

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

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

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

© Springer-Verlag Berlin Heidelberg 2006

2

1 Introduction

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

,12 if ,22 if

˙ κ0 ˙ ≤0

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

(1.1)

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

,ε if v κ 0 εv = e + ˙ −,ε if v ≤ 0

e = −ω = − + 2

2 +, 4

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

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

1.1 Sliding Mode Control

3

.

X I

X

.

X

II

X

(a)

(b)

.

X II

I

X

I

II

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

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

4

1 Introduction .

.

X

X

II

I

X

X

(b)

(a) .

I

X

II

X

I

II

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

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

(1.2)

vv˙ p 0

(1.3)

s→0+

or more concisely as

s→0

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


5

.

I

X

II

c2x+dx/dt=0

X

II

I

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

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

(1.4)

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

T

=0

(1.5)

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

6

1 Introduction

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

X2

X1 X3

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

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

7


X3

X2

X1

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

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

rv (i ( ε y) + j( ε y)(y)) ρ r

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

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

(1.6)

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

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

(1.7)

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

8

1 Introduction

x

2

C

Vector − f in point C 0

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

Discontinuity Surface

x1

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

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


˙=

rv − i r rv − (i − i + ) r

i+ −

rv + i r rv − (i − i + ) r

i −ρ

9

(1.8)

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

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

Assuming that the matrix may be calculated as eq = −

(1.9)

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

rv j( ε y) r

−1

rv rv + i ( ε y) ρ ry r

(1.10)

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

(1.11)

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

(1.12)

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

(1.13)

10

1 Introduction

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

(1.14)

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

rv j( ε y) r

−1

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

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

(1.16)

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

−1

eT Sτ − eT + Tz eT

(µ) + >z sgn(v(µ))

(1.17)

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

(2.1)

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

>z ρ 2 − τz

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

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

2 Switching function based MROF-SMC

18

(µ) = − eT Hτ

−1

eT Sτ − eT + τz eT

(µ) + >z sgn(eT (µ)) ρ

(2.2)

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

k

+ Fu (µ − 1)

− e T Hτ

−1

(2.3)

>z sgn eT My

k

+ eT Mu (µ − 1) ε

where Fy = − e T Hτ

−1

eT Sτ − eT + τz eT My ε

−1

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

−1

C0T ρ

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

01 −1 1

(µ) = 1 0

(µ) +

0 (µ)ε 1

(2.4)

(µ)ρ

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

−1ρ68 1ρ7

T

−0ρ01sgn

k

+ 0ρ96(µ − 1) −1ρ2 0ρ35

T k

+ 1ρ05(µ − 1) ρ

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

19

System States

1

Sliding Function (s(k))

2.2 MROF-QSMC in Uncertain Systems x 1 x2

0.5 0

0

−0.2

−0.5

−0.4

−1

−0.6

−1.5 −2

0.2

0

1

2

3

Time (sec)

4

5

−0.8

0

2

3

Time (sec)

4

5

(b)

(a)

0.5

0

0

−0.5

−0.5

x

2

0.5

Control input

1

−1

−1

−1.5

−1.5

−2

0

1

2

3

Time (sec)

4

5

−2 −2

−1

(c)

x1

0

1

(d)

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

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

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

(2.5)

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

20


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

(2.6)

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

N −1

S

D=

i

Dτ ρ

(2.7)

i=0

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

where

˜ = C0 (µ) + D0 (µ) + Cd η(µ) 

0  CD  Cd =  .. . C

(2.8)



N −2 i=0

  ρ  Si D

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

k

˜ − 1)γ + Mu (µ − 1) + Md h(.

(2.9)

Nd = Dτ − Ny Cd u

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

21

2.2 MROF-QSMC in Uncertain Systems

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

(2.10)

η0 =

(2.11)

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

−1

− e T Hτ

−1


(µ)

(2.13)

(η0 + (fd + >z ) sgn(v(µ))) ρ

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

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

(2.14)

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

(2.15)


22

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

(2.16)

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


(µ)

(2.17)

T

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

k + Fu (µ − 1) eT Hτ (η0 + j0 +

(2.18) (fd + fg + >z )sgn(v(µ))) ρ


23

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

(2.19)

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

k

+ Mu (µ − 1)) + ν0 ε

(2.20)

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

(2.21)

24


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

01 −1 1 +

(µ) +

0 (µ) 1

0 sin(0ρ1µ)ε 0ρ01

(µ) = 1 0

(2.22)

(µ)ρ

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

−1ρ56 1ρ66

T

−0ρ182sgn

k

+ 0ρ857(µ − 1) −1ρ2 0ρ35

T k

+ 1ρ05(µ − 1) ρ

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

25


0.5

δ

0.2

x1 x2

y

0

Sliding Function

System States

1

−δy

−0.2

0

−0.4

−0.5 −1

−0.6

0

1

2

3

Time (secs)

4

5

−0.8

0

1

4

5

0.1 0

0.2

−0.1

0

−0.2

2

0.4

x

Control Input

3

(b)

(a)

0.6

−0.2

−0.3

−0.4

−0.4

−0.6

−0.5

−0.8

2

Time (secs)

0

1

2

3

Time (secs) (c)

4

5

−0.6 −1

−0.5

0

x1

0.5

1

(d)

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

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

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

(3.1)

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


28

3 MROF-DSMC in Uncertain Systems

period, the system state and its output do not change in each successive control step) and achieve an essential reduction of the control effort and improved quality of the quasi-sliding mode control. Definition 3.2. We say that the system (2.5) satisfies the reaching condition of the quasi-sliding mode in the f vicinity of the sliding surface if and only if for any µ ≥ 0 the following condition is satisfied. |v(µ + i (µ)| ≤ 1. From, Eqns. (3.11 Next, for |¯ and 3.21), Li v¯i (µ) si + (,(µ − 1) − ,0 − δ(µ) + δ0 + η(µ) − η0 )i Li vi (µ + 1) = 1 − v¯(µ) + (,(µ − 1) − ,0 + η(µ) − η0 )i si s i − Li |¯ vi (µ)| + |,i (µ − 1) − ,0,i | + |ηi (µ) − η0,i | =⇒ |v(µ + 1)| ≤ si ≤ |si − Li | + fα,i + fd,i v¯i (µ + 1) =

1−

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

38


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

µ ∗ = max

(3.23)

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

q2

q3

b

f u1

m

m k

m k

Fig. 3.5. The mechanical system

u2

3.3 MROF-DSMC for Unmatched Uncertainty

39

T

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

(µ) = −G−1 N (µ) − G−1 Ls sat

(3.24)

T

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

T

Li ∈ R

.Ti = e1 (Sτ − J)

i=0 m

Li = LΣ = k + 2z >ε > κ 0 i=0

ss κ k + z > k = max (|η(µ)|)

(3.25)

40


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

v(µ) 0ρ7530

(3.26)

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

0

Sliding Function ss(k)

−1

−2

−3

−4

−5

−6

0

10

20

30

40

50

60

Sampling Instants (k)

70

80

90

100

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

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

3.3 MROF-DSMC for Unmatched Uncertainty 40

q1 q 2 q 3

30

Displacements (m)

20

10

0

−10

−20

−30

0

20

40

60


80

100

120

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

u1 u2

150

Control Inputs (N)

100

50

0

−50

−100

−150

0

10

20

30

40

50

60


70

80

90

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

100

41

42


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

eT =

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

+

v¯1 (µ)x0ρ1444 v¯2 (µ)x0ρ0172

+

0ρ1441 15ρ5899

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

s1 s2

2

Sliding Function s(k)

1

0

−1

−2

−3

−4

−5

−6

0

20

40

60

80

100

120


140

160

180

200

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

3.3 MROF-DSMC for Unmatched Uncertainty 4

q 1 q2 q3

3.5 3

Displacements (m)

2.5 2 1.5 1 0.5 0 −0.5 −1

0

50

100

150

200


250

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

u1 u2

Control Inputs (N)

15

10

5

0

−5

−10

0

20

40

60

80

100

120


140

160

180

200

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

43

44


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

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

(3.27) (3.28)

3.4 MROF based Integral Sliding Mode in Discrete-time Systems

45

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

(3.29)

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

(3.30)

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

(3.31)

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

(3.32) (3.33)

46


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

−1

˜ w(µ) − eT Dτ η(µ) ρ

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

−1

¯ w(µ) − η(µ) ε

(3.34)

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

(3.35) (3.36)

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

1 (µ) = − e Hτ T

w(µ) = e ‫(܈‬µ) ‫(܈‬0) ¯ η(µ) ¯ η(0)

−1

¯ w(µ) − η(µ)

(µ) + ‫(܈‬µ)

(3.37) (3.38) (3.39)

T

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

(3.42) (3.43)

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

3.4 MROF based Integral Sliding Mode in Discrete-time Systems

47

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

(µ) = RTR ˜ η(µ)

where

−1

RT (

k+1

− D0 (µ)) ε

(3.44) (3.45)

(3.46)

R = C0 Cd ρ

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

(µ) = T R T R

−1

RT

k

+ Hτ − T R T R

−1

R T D0 (µ − 1) (3.47)

T = Sτ Dτ

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

−1

RT

k

+ Hτ − T R T R

−1

R T D0 (µ − 1)ε (3.48)

(µ) = 0 (µ) + 1 (µ)ε T

1 (µ) = − e Hτ T

−1

¯ w(µ) − η(µ) ε

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

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

48


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

(3.53)

= eT Dτ 0n ×n Jn ×n ¯ = 0ρ η(0)

RTR

−1

RT (

k

− D0 (µ − 1)) ε (3.54) (3.55)

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

11 −1 2

(µ) = 1 0

(µ) +

0 0 ˜ η(µ) (µ) + −1 1

(µ)

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

System Response Target Response

0.8

0.6

0.4

State x

2

0.2

0

−0.2

−0.4

−0.6

−0.8 −1.5

−1

−0.5

0

0.5

1

State x1

1.5

2

2.5

3

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

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

2

1

Control Input

0

−1

−2

−3

−4

−5

−6

0

50

100

150

200

250

Samples

300

350

400

Fig. 3.13. The control input applied to the system

450

500

49

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

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

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

(4.1)

(y) = C (y − zy )


4 MROF based QSMC in Time-Delay Systems

52

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

(4.2)

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

(4.3) (4.4)

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

(4.5)

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

τ

0

0

τ

τ

0

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

(k+1)τ kτ

(4.7)

˜ gA0 ((k+1)τ −ω) Dt η(ψ)ηψρ

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

4.3 Design of Sliding Surface

53

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

(k+1)τ kτ (k+1)τ kτ (k+1)τ kτ τ 0

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

gA0 (τ −ω) ηψηmax

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

gA0 ((k+1)τ −ω) A1 (ψ − zx ) ηψ

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

τ 0

gA0 (τ −ω) A1 ηψ

(µ − µx )

for the purpose of obtaining a closed form representation. In fact, the actual contribution cannot be represented as a multiple of (µ − µx ) or a finite linear combination of (z sgn eT (µ) ρ

(4.26)

4.5 Quasi-Sliding Mode Control Algorithm for Form 1 Systems

59

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

−1 1 −1 0ρ4 ε A1 = 0 −2 0 −1

B0 =

0ρ6 0ρ6 ρ ε B1 = 1ρ0 1ρ0

(4.27)

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

3

3

2

2

State x

State x

1

2 1 0

0

−1

−1 −2

1

0

10

20

30

Time in sec

40

−2

50

0

10

40

50

40

50

0.02

Sliding Function

3 2

Input

30

(b)

(a)

4

0

−0.02

1

−0.04

0

−0.06

−1

−0.08

−2 −3

20

Time in sec

0

10

20

30

Time in sec (c)

40

50

−0.1

−0.12

0

10

20

30

Time in sec (d)

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

60


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

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

(4.28)

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

−1

− eT I1 (z )

−1

− eT I1 (z )

−1

τz eT − eT + eT G0 (z )

(µ)

eT G1 (z ) (µ − µx ) >z sgn(v(µ))

or equivalently (µ) = − eT I1 (z ) T

− e I1 (z ) − eT I1 (z )

−1


−1

T

−1

(µ + µu )

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

(4.29)

61

4.6 Quasi-Sliding Mode Control Algorithms for Form 2 Systems

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

k −1

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

i=0 k −1

+

Gi0 (z )I1 (z )(µ − < − 1)

i=0

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

−1

− eT I1 (z )

−1

− eT I1 (z )

−1

τz eT − eT + eT G0 (z ) G0k

(µ)

(4.32)

>z sgn(v(µ + µu )) (τz − J) eT

k −1

Gi0 (z )G1 (z ) (µ − µx + µu − < − 1)

i=0 T

− e I1 (z )

−1

T

T

T

k −1

τz e − e + e G0 (z )

Gi0 (z )I1 (z )(µ − < − 1)

i=0

− eT I1 (z )

−1 T

k

e

Gi0 (z )G1 (z ) (µ − µx + µu − < − 1)

i=0

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

−1

− eT I1 (z )

−1

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

k


62

− eT I1 (z )

−1

− eT I1 (z )

−1

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

k −1

Gi0 (z )G1 (z ) (µ − µx + µu − < − 1)

i=0

− eT I1 (z )

−1


k −1

Gi0 (z )I1 (z )(µ − < − 1)

i=0

− eT I1 (z )

−1 T

k

Gi0 (z )G1 (z ) (µ − µx + µu − < − 1)

e

i=0

2

2

1 0

2

0

State x

State x

1

4

−1

−2

−2

−4 −6

−3 0

10

20

30

Time in sec

40

−4

50

0

10

(a)

2

0.1

Sliding Function

0.2

0

Input

30

40

50

30

40

50

(b)

4

−2

0

−0.1

−4

−0.2

−6

−0.3

−8 −10

20

Time in sec

−0.4

0

10

20

30

Time in sec (c)

40

50

−0.5

0

10

20

Time in sec (d)

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

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


63

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

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

(4.34)

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

k + Mx ĝ

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

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

k

Gi0 (z )G1 (z ) (µx,y,i )

(µ) = G0 (z ) (µ − µy ) + i=0 k −1

Gi0 (z )I0 (z )(µ + µy − < − 1)

+ i=0 k −1

Gi0 (z )I1 (z )(µ − µu + µy − < − 1)

+ i=0

64

4 MROF based QSMC in Time-Delay Systems k

= G0 (My

k

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

k −1

Gi0 (z )G1 (z ) (µ − µx + µy − < − 1)

+ i=0 k −1

Gi0 (z )I0 (z )(µ + µy − < − 1)

+ i=0 k −1

Gi0 (z )I1 (z )(µ − µu + µy − < − 1)

+ i=0 k

µx,y,i

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

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

−1

− eT I0 (z )

−1

T

− e I0 (z ) − eT I0 (z )

k

iG (z )G0 (z ) (My

k iG (z )G0 −1 k iG (z )G0 −1

k

+ Mxĝ (µx,y,i ))

(z )Mu0 (µ − µy − 1)

(4.36)

(z )Muĝ (µ − µu − µy − 1)

k −1

Gi0 (z )I0 (z )(µ + µy − < − 1)

iG (z ) i=0

− eT I0 (z )

−1

k −1

Gi0 (z )G1 (z ) (µ − µx + µy − < − 1)

(τz − J) eT i=0

− eT I0 (z )

−1

k −1

(τz − J) eT

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

− eT I0 (z )

−1 T

k

Gi0 (z )G1 (z ) (µ − µx + µy − < − 1)

e

i=0

− eT I0 (z )

−1 T

k

Gi0 (z )I1 (z )(µ − µu − µy − 1)

e

i=0 T

− e I0 (z ) T

T

−1

>z sgn(v(µ))ε

T

iG (z ) = τz e − e + e G0 (z ) ρ

(4.37)

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

65


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

4

2 1

2

State x

State x

1

2 0

0

−1

−2 −4

−2

0

10

20

30

Time in sec

40

−3

50

0

10

40

50

30

40

50

0.05

Sliding Function

2 1

Input

30

(b)

(a)

3

0

−0.05

0

−1 −2

−0.1

−0.15

−3 −4

20

Time in sec

0

10

20

30

Time in sec (c)

40

50

−0.2

0

10

20

Time in sec (d)

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

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

66


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

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

(4.38)

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

Md = J − G0 (z ) C0T C0

−1

C0T Cd ρ

(4.39)

(4.40)

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

η0 =

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

(4.41)

67

4.8 Discrete-time Sliding Mode Control of Form 4 Systems

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

− e I0 (z ) + eT I0 (z )

−1 T

e (G0 (z ) (µ) + G1 (z ) (µ − µx ))

(4.42)

−1 T

e I1 (z ) (µ − µu )

−1

(vd (µ + 1) + g˜ (µ − 1))

T

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

−1 T

− eT I0 (z )

−1 T

T

− e I0 (z ) − eT I0 (z ) T

− e I0 (z ) − eT I0 (z )

e G0 (z ) My

k,

the

(4.43)

k

e G1 (z ) (µ − µx )

−1 T

e G0 (z ) Mx

(µ − µx − 1)

−1 T

e G0 (z ) Mu0 (µ − 1)

−1 T

e I1 (z ) (µ − µu )

−1 T

e G0 (z ) Mu (µ − µu − 1)

T

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


68

System States

10

5 0

0

State x1 State x2

−5

2

−10

X

−10

−20

−15

−30 −40

−20 0

10

20

Time, sec

30

−25 −40

40

−30

−20

−10

0

10

(b)1

(a)

10

0.5

Sliding Function

0

0

−0.5

Input

−10 −20

−1

−1.5

−30 −40

X

0

10

20

Time, sec

30

40

−2

−2.5

0

10

20

Time, sec

30

40

(d)

(c)

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

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

1ρ2 −0.4t g sin(2y) 2 T

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

69

4.8 Discrete-time Sliding Mode Control of Form 4 Systems

10

5

0

0 −5

−10 −20

2

State x1 State x2

−10

X

System States

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

−15

−30 −40

−20 0

10

20

Time,sec

30

−25 −40

40

−30

(a)

−10

0

X1

10

(b) 0.5

0

0

Sliding Function

10

−0.5

Input

−10 −20

−1

−1.5

−30 −40

−20

0

10

20

Time, sec (c)

30

40

−2

−2.5

0

10

20

Time, sec

30

40

(d)

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

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


70

4.8.4 Performance in System without Disturbance

10

5

0

0 −5

−10 −20

2

State x1 State x2

−10

X

System States

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

−15

−30 −40

−20 0

10

20

Time (sec)

30

−25 −40

40

−30

X

−10

0

10

1

(a)

(b) 0.5

0

0

Sliding Function

10

−0.5

Input

−10 −20

−1

−1.5

−30 −40

−20

0

10

20

Time (sec) (c)

30

40

−2

−2.5

0

10

20

Time (sec)

30

40

(d)

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

5 Mwl2i7a2e Ow2pw2 Feedback Sliding Mode fo7 Special Cla66e6 of S062em6

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

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

© Springer-Verlag Berlin Heidelberg 2006

72

5 Multirate Output Feedback Sliding Mode for Special Classes of Systems

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

(5.1)

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

+ 1) = A11

2 (µ

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

∈ Rl ε

1 (µ)

2

+ A12

2 (µ)

+ B1 (µ)ε

(5.2)

∈ Rn−l

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

(5.3)

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

= X v(µ)

(5.4)

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

5.1 MROF-based Nonminimum Phase System Tracking

73

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

1 (µ)

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

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

−1

B1T (A11 − R ) ‫(܈‬µ)

− B1T B1

−1

B1T A12

2,0 (µ)ρ

(5.5)

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

1 (µ

+ 1) = A11 ∆ 2 (µ + 1) = A21 ∆

1 (µ)

+ A12 ∆ 1 (µ) + A22 ∆

2 (µ)

+ B1 ∆(µ)ε 2 (µ)ρ

(5.6)

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

∆(µ) = F

1 (µ)

2 (µ)

+ ksgn(v(µ))ρ

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

−1

B1T (A11 − R ) ‫(܈‬µ)

− B1T B1

−1

B1T A12

+kρsgn(v(µ))ρ 1 (µ) =F − 2 (µ)

where

d

2,0 (µ)

+F

∆ ∆

1 (µ)

2 (µ)

d (µ)

−

B1T B1

−1

B1T (A11 − R ) ‫(܈‬µ)

−

B1T B1

−1

B1T A12

2,0 (µ)

+ kv<jφ (v(µ)) ρ

is the nominal state vector represented as

(5.8)

74

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

= ‫(܈‬µ)T

T T 2,0

ρ

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

2,0 (µ)ρ

(5.9)

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

T

= θk,l ρ

(5.10)

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

2 m 2 u 2

= X2

2ε

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

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

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

u 2 (µ)

+ A21,u ‫(܈‬µ)ρ

75

(5.11) (5.12) (5.13)

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

k−1

=

As(k−i) (A21,s ‫))1 + 2) ρ =− (2 − ∆τ1 ) =−

d1

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

(5.46)

Since the initial state 1 (0) would be inside the manifold Yp and the control 1 would take it monotonically to a band of width f1 p 1, the disallowed state 1 (µ) = d1 would not be encountered. 3. And the special case of 1 (µ) = 0ε In this case, the system becomes of a reduced order and hence it is the observer that has to be modified (and not the control input, which would obviously be 1 (2 z sgn (v2 (µ)) ε ( 2 (µ)1 (µ)

1 (µ) 2 (µ)

+ 2 (µ) z

+

1 (µ)

+2 (µ)

2 (µ))

+

+

z (1 + 2

z2 +z 3

z +1 2 1 (µ))

1 (µ)

= −τ2 (v2 (µ)) − >2 sgn (v2 (µ)) ρ


90

2 (µ) = −

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

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

−

1 (µ) 2 (µ)

+ i3 (µ)

2 (µ))

+1

(5.47) 1 (µ)

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

(5.48) (5.49)

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

1

κ0 0=z

d2

2. If

1

=

1 (µ) τ2 3

z (1 + 2

+

+ z >1 − ∆ τ2 3

+ z τ1

+

z (1 + 2

3∆ −

1 (µ))

+

z2 +z 3

(−τ1

1 (µ)

− >1 )

+

z2 + 1 (−τ1 3

1 (µ)

+ >1 )

ρ

p0 0=z

d3

=

−

1 (µ)

τ2 3

3∆ −

1 (µ))

+ 1 >1 + ∆ τ2 3

+ 1 τ1

ρ

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

(5.50) (5.51)

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

5.2 MROF-QSMC for Finitely Discretizable Nonlinear Systems

91

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

x 1 x2 x 3

4

System States

3

2

1

0

−1

0

5

10

Time (sec)

15

Fig. 5.5. Response of System States. T

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


1 0.8

X3

0.6 0.4 0.2 0

2

5

4

3

2

1

1

0

0

X1

X

2

Fig. 5.6. Phase Portrait of the System 20

Sliding Function s

Sliding Function s

1

2

2.5 2

1.5 1 0.5 0

−0.5

0

5

10

Time (sec)

10 5 0

0

5

10

15

0

5

10

15

Time (sec)

10

Control Input u

2

0

−1

0

−10

−2 −3

−20

−4

−30

−5 −6

15

−5

15

1

Control Input u1

92

0

5

10

Time (sec)

15

−40

Time (sec)

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

5.3 MROF-DSMC based on Pe Re Sliding Sector

93

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

(5.52)

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

(5.53)

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

P

=

T

P

of the discrete-time system state is de-

(µ)Rd (µ)

1/2

ε (µ) ∈ Rn ε

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

(5.54)


94

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

=

(µ) |

T

STτ Rd Sτ

Qd =

(µ)Qd (µ) ≤ 0 ε

(5.55)

− Rd + U d

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

T

(5.56) (µ)

(µ) Ud (µ)

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

(µ)

2 P

=

T

(µ)Rd (µ) κ 0ε

(5.57)

n

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

f (µ) =

T

(µ) Rd1 (µ) ε

(5.59)

T

(µ) Rd2 (µ) ε

(5.60)

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

Qd = STτ Rd Sτ − Rd + Ud

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

(µ)Qd (µ) ≤ 0ρ

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

5.3 MROF-DSMC based on Pe Re Sliding Sector

95

Y T Qd Y = η

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