Table Of ContentStudies in Systems, Decision and Control 139
Bijnan Bandyopadhyay
Abhisek K. Behera
Event-Triggered
Sliding Mode
Control
A New Approach to Control System
Design
Studies in Systems, Decision and Control
Volume 139
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Bijnan Bandyopadhyay Abhisek K. Behera
(cid:129)
Event-Triggered Sliding
Mode Control
A New Approach to Control System Design
123
Bijnan Bandyopadhyay Abhisek K.Behera
Systems andControl Engineering Systems andControl Engineering
Indian Institute of Technology Bombay Indian Institute of Technology Bombay
Mumbai,Maharashtra Mumbai,Maharashtra
India India
ISSN 2198-4182 ISSN 2198-4190 (electronic)
Studies in Systems,DecisionandControl
ISBN978-3-319-74218-2 ISBN978-3-319-74219-9 (eBook)
https://doi.org/10.1007/978-3-319-74219-9
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Dedicated to my Ph.D. guide, late Dr. S. S.
Lamba, former Professor, IIT Delhi
Bijnan Bandyopadhyay
Dedicated to my parents and gurus
Abhisek K. Behera
Preface
Variable structure systems using sliding mode control (SMC) was originated in
USSRinthelatefiftiestostabilizetheuncertaindynamicalsystemswithrelayasa
feedback control law. It gained popularity outside USSR only after the late sev-
enties due to a survey paper in English by Prof. Vadim I. Utkin. Since then, it has
become now a well-established robust control technique to deal with the uncer-
tainties intheplant,andachievingthesystemstability.Avastnumberofscientific
publications and the practical applications of this control technique have made
SMC as an important area in the control literatures.
The attention on design of SMC in discrete-time domain was paid by many
researchers soon after the importance of microprocessor and computer/processors
are realized in control applications in the early eighties. The first and important
observationinthediscrete-timedesignisthatnoexactslidingmodeisachievedas
in the continuous-time counterpart. A new notion of sliding mode is introduced
which is known a quasi-sliding mode (QSM). This has led to the development of
discrete-time SMC as an important area in SMC due to its practical importance.
Many designapproacheshavebeen proposed toimprovetheperformance ofSMC
for the sampled-data system.
In this monograph, a new approach to design SMC is presented using a novel
implementation strategy, namely event-triggering. In this strategy, the control is
updatedwheneveracertainstabilizingconditionisviolated,andhence,thesystem
stability is always maintained. Due to the need-based control strategy, it finds a
major application in spatially distributed control systems to reduce the communi-
cation among different sensor and actuator ends. So, the resources of the systems
are optimally used. The event-triggering-based design of SMC not only gives the
robustperformancebutalsoensuresminimaluseofresourcesinthecontrolsystem.
This monograph presents the recent results on event-triggered SMC for robust
stabilization of dynamical systems. In the first part of the monograph, the prelim-
inariesonsampled-datasystemswithanintroductiontoevent-triggered controlare
presented to familiarize the readers the event-triggering-based design of control
law. In addition to this, a brief introduction to SMC and its design are also dis-
cussed. Then, the design of event-triggered SMC for both linear and nonlinear
vii
viii Preface
systemsisgiveninChaps.2and3.Theevent-triggeredSMCispresentedforlinear
systems guaranteeing the semi-global and global stability in Chap. 2. However,
only local stabilization result is discussed for the nonlinear systems, which is
presented in detail in Chap. 3. In the event-triggered control, the state trajectory is
continuously measured to evaluate the triggering condition which may not be
practical in all applications. So, Chaps. 4 and 5 present few variants of
event-triggered control, namely self-triggered and discrete event-triggered control,
respectively. In the self-triggering strategy, the triggering strategy is developed
without using the continuous state measurements. On the other hand, the periodic
state measurements are used in discrete event-triggered control and the control is
updatedwhenitisviolatedatsomeperiodicinstants.Inrecenttime,therehasbeen
aconsiderableamountofinterestinstabilityofquantizedcontrolsystem.Thefinal
chapter presents the design of event-triggered SMC with quantized state mea-
surements. It is our belief that the material of the monograph would serve its
purpose and explore the new challenges on the topic.
This work would have not been completed without the support and encour-
agementfrommanyofourfriendsandcolleagues.Theauthorswouldliketothank
IndianInstituteofTechnologyBombayforprovidingtheconduciveenvironmentto
carry out the research reported in the monograph. Finally, we extend our gratitude
to our family for their love, support and understanding throughout the process of
this endeavour.
Mumbai, India Bijnan Bandyopadhyay
December 2017 Abhisek K. Behera
Contents
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Computer-Controlled Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.1.1 Basic Architecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.1.2 Design Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 Event-Triggered Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.2.1 Preliminary Idea . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.2.2 Stability of Event-Triggered Systems . . . . . . . . . . . . . . . . 8
1.2.3 Need of Event-Triggered Control . . . . . . . . . . . . . . . . . . . 11
1.3 Sliding Mode: An Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.3.1 Dynamics During Sliding Mode . . . . . . . . . . . . . . . . . . . . 14
1.3.2 Design of Sliding Mode Control. . . . . . . . . . . . . . . . . . . . 17
1.4 Discrete-Time Sliding Mode . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
1.4.1 Switching-Based Reaching Law . . . . . . . . . . . . . . . . . . . . 22
1.4.2 Switching-Free Reaching Law . . . . . . . . . . . . . . . . . . . . . 23
1.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
1.6 Notes and References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2 Event-Triggered Sliding Mode Control for Linear Systems . . . . . . . 27
2.1 System Description. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.2 Event-Triggered Sliding Mode Control . . . . . . . . . . . . . . . . . . . . 28
2.2.1 Stability of Sliding Motion. . . . . . . . . . . . . . . . . . . . . . . . 32
2.2.2 Stability of Event-Triggered System . . . . . . . . . . . . . . . . . 33
2.2.3 Simulation Results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
2.3 Global Event-Triggered Sliding Mode Control . . . . . . . . . . . . . . . 37
2.3.1 Global Event-Triggering Rule. . . . . . . . . . . . . . . . . . . . . . 37
2.3.2 Design of Sliding Mode Control. . . . . . . . . . . . . . . . . . . . 38
2.3.3 Global Stability of Event-Triggered System. . . . . . . . . . . . 41
2.3.4 Simulation Results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
ix
x Contents
2.4 Event-Triggered Sliding Mode Control for Multivariable
Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
2.4.1 Event-Triggered Design of SMC . . . . . . . . . . . . . . . . . . . 45
2.4.2 Event-Triggering Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
2.4.3 Simulation Results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
2.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
2.6 Notes and References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
3 Event-Triggered Sliding Mode Control for Nonlinear Systems. . . . . 55
3.1 System Description. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
3.1.1 Design of Sliding Mode Control. . . . . . . . . . . . . . . . . . . . 56
3.2 Event-Triggered Sliding Mode Control . . . . . . . . . . . . . . . . . . . . 57
3.2.1 Stability of Sliding Motion. . . . . . . . . . . . . . . . . . . . . . . . 60
3.3 Event-Triggering Strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
3.3.1 Design of Event-Triggering Scheme with Constraints . . . . 63
3.4 Event-Triggering with Delay. . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
3.5 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
3.5.1 Without Delay. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
3.5.2 With Delay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
3.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
3.7 Notes and References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
4 Self-Triggered Sliding Mode Control for Linear Systems . . . . . . . . . 77
4.1 System Description. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
4.2 Self-Triggering Scheme Without Delay . . . . . . . . . . . . . . . . . . . . 78
4.2.1 Simulation Results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
4.3 Self-Triggering Scheme with Delay . . . . . . . . . . . . . . . . . . . . . . . 81
4.3.1 Design of Self-Triggered Sliding Mode Control. . . . . . . . . 85
4.3.2 Simulation Results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
4.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
4.5 Notes and References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
5 Discrete Event-Triggered Sliding Mode Control for Linear
Systems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
5.1 System Description. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
5.2 Discrete-Time Sliding Mode . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
5.2.1 Bartoszewicz’s Reaching Law . . . . . . . . . . . . . . . . . . . . . 96
5.2.2 Design of Discrete-Time Sliding Mode Control. . . . . . . . . 97
5.3 Discrete Event-Triggered Sliding Mode Control: State
Feedback Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
5.3.1 Event-Triggered Bartoszewicz’s Reaching Law . . . . . . . . . 99
