Discrete-Time Stochastic Sliding Mode Control Using Functional Observation

129 Pages·2020·2.515 MB·English

by  Satnesh Singh, S. Janardhanan

Lecture Notes in Control and Information Sciences 483
Satnesh Singh
S. Janardhanan
Discrete-Time
Stochastic
Sliding Mode
Control Using
Functional
Observation
Lecture Notes in Control and Information
Sciences
Volume 483
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Satnesh Singh S. Janardhanan
(cid:129)
Discrete-Time Stochastic
Sliding Mode Control Using
Functional Observation
123
SatneshSingh S. Janardhanan
Department ofElectrical Engineering Department ofElectrical Engineering
Indian Institute of Technology Delhi Indian Institute of Technology Delhi
NewDelhi, India NewDelhi, India
ISSN 0170-8643 ISSN 1610-7411 (electronic)
Lecture Notesin Control andInformation Sciences
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Preface
Many control applications that are encountered in practice require the design of a
suitable controller to make a given system behave in a specified manner. For this
purpose, the system under consideration is modelled mathematically, but a gap
always lies between the original model and its mathematical model [1]. These
mismatches along with the unknown external disturbances affect the system’s
performance.Totackletheeffectofexternaldisturbancesandparametervariations,
many robust control techniques have been developed. One of the most effective
such techniques is variable structure control (VSC); in VSC, the structure of a
closed loop is changed according to some decision rule; this rule is called the
switching function [2, 3]. VSC provides a precise solution to the problem of
maintaining stability and consistent performance in the face of bounded distur-
bances. VSC theory was first proposed in 1959 [4] and has been extensively
developed since then with the invention of high-speed control devices. But due to
the trouble of implementation in high-speed switching, it did not attain much
popularity initially.
Slidingmodecontrol(SMC)hasbeenextensivelyrecognizedasarobustcontrol
strategy for its ability to make a control system very sturdy, which yields the
complete elimination of external disturbances satisfying the matching conditions
[5, 6]. In this control technique, a control input is designed such that the state
trajectory of the system reaches a prescribed manifold in finite time and thereafter
remainsonitinspiteofthepresenceofuncertaintiesinthesystem.Theprescribed
manifold is called a sliding manifold, the motion of the state trajectory on the
sliding manifold is known as the sliding mode or sliding motion, and the corre-
sponding control is called SMC. Since the state trajectory stays on the sliding
manifold in the sliding mode, the sliding manifold alone stipulates the behaviour
of the closed-loop system in the sliding mode. Therefore, the sliding manifold can
beusedtospecifythedesiredperformanceofthesystemunderconsideration.Later,
Utkin [6] presented a review paper on VSC using sliding modes that led to a
renewalofinterestinthisarea.Toestablishandmaintaintheslidingmode,control
v
vi Preface
is designed such that the state trajectory is always directed towards a sliding
manifold. To satisfy this constraint, SMC utilises the idea of VSC. Many practical
applications of SMC have been reported in the control literature such as flight
control, robotic manipulator’s and servo systems [7].
Traditionally, a high-frequency switching control action is used to force the
system dynamics to slide along the sliding manifold. Thus, high-frequency
switching is an inherent characteristic of SMC, which results in invariance of the
systemstateinthefaceofuncertainties.However,thehigh-frequencycomponentin
the control input leads to the problem of undesirable high-frequency vibration,
calledchattering,intheclosed-loopsystem.Manymethodshavebeendevelopedto
mitigate the effect of chattering [8, 9].
Usually, a switching type of control is used to achieve a change in controller
structure. Since an increasing number of modern control systems are implemented
by computers, the study in the discrete-time domain, i.e., discrete-time SMC, has
been an important topic in the SMC literature. However, it has been realized that
directly applying continuous-time SMC algorithms for discrete-time systems will
lead to many problems, such as sample/holds effects, large chattering amplitude,
discretizationerrors,oreveninstability.Tocopewiththeaforementionedproblems,
theideaofdiscrete-timeSMC(DSMC)hasbeenintroduced.Thus,theSMCdesign
for discrete-time representation of a system is more reasonable than a
continuous-time system in digitally controlled systems. As a result, there is now
significantinterestandresearchinSMCfordiscrete-timesystems,andanumberof
discrete-time sliding mode (DSM) techniques have been developed [5, 10–13].
An essential property of a discrete-time system is that the control signal is
computed and varied only at sampling instants, which makes discrete-time control
inherently discontinuous. Hence, unlike the case of continuous SMC, the control
law need not necessarily be of variable structure or have an explicit discontinuity.
This method involves the design of a sliding surface that generates a stable
reduced-ordermotionandthedesignofasuitablecontrollawtoforceaclose-loop
responseofasystemtoaslidingsurfaceandtomaintainitsubsequently.Here,the
system states move about the sliding manifold, but are inadequate to stay on it,
whence the terminology quasi-sliding mode (QSM) [14]. In other words, the tra-
jectories of discrete-time systems may notremain on a predesigned sliding surface
becauseofthelimitationinthesamplingrate(thesamplingratecannotbeinfinite).
However, the state trajectories may be able to remain within a boundary layer
aroundtheslidingsurfacecalledaquasi-slidingmodeband(QSMB).Afewstudies
have also proved that the chattering phenomenon in DSMC vanishes if the dis-
continuous control part is eliminated from the feedback control law. The control
law in the absence of an explicit discontinuous component is then called a linear
controllaw[3,5,11,13].Fromtheabove,itcanbeconcludedthattheDSMCcan
ensuretheboundednessofthetrajectoriesinsideaQSMB,evenwithouttheuseofa
variablestructurecontrolstrategy.Thispropertycanstillbeensuredinthepresence
ofmatcheduncertaintyinthesystemdynamics.Fromtheaboveobservations,itcan
be inferred that the use of a switching function in the control law may not neces-
sarily enhance the performance [15].
Preface vii
Motivation
Discrete-time stochastic systems are predominant in numerous applications, and
manysuccessfulattemptshavebeenmadetoaddresstherobuststabilisationofsuch
systems [16–18]. Most of the studies in SMC available in the literature do not
consider the presence of stochastic noise in the systems. However, it has been
noticed that many real-world systems and a natural process may be disturbed by
various noises such as process and measurement noise. This means that stochastic
system representations are more aligned with reality. Therefore, it is crucial to
extend the SMC theory to stochastic systems [19].
IntheframeworkofDSMCforstochasticsystems,onlyafewworksareavailablein
theliterature[20,21].AmethodofSMCfordiscrete-timestochasticsystemshasbeen
designed. However, in contrast with sliding function sðkÞ design of discrete-time
system and discrete-time stochastic systems is always different in nature. Sliding
function design in stochastic systems is always probabilistic. Hence, the idea and
definition of DSM cannot be applied directly in discrete-time stochastic systems.
The design of a controller for each control problem uses either a state feedback
controller or an output feedback controller depending on the available means of
measurement [22, 23]. Traditionally, SMC was developed in an environment in
whichallthestatesofthesystemareavailable.Thisisnotaveryrealisticsituation
for practical problems and has motivated the development of functional
observer-based SMC.
In this sense, this book intends to develop functional observer-based robust
control strategies for stochastic systems. The use of a functional observer reduces
the observer order substantially, and sliding mode control addresses the issue of
robustness of the controller [24–26]. To the best of the authors’ knowledge, the
proposed methodology has not been previously applied to discrete-time stochastic
systems. Motivated by the above observations, this book explores the problem of
designing functional observer-based SMC for discrete-time stochastic systems
explicitly. This book attempts to fill such gaps in the SMC literature.
The Book
The prime contributions of this book are the sliding function design for various
categoriesoflineartime-invariant(LTI)systemsanditscontrolapplications,which
are summarised in brief as follows:
1. SMC for discrete-time stochastic systems with bounded disturbances is
designed.Subsequently,anSMCcontrollawisdesignedforastochasticsystem
such that the states will lie within the specified band. Further, this result has
been extended to the case of incomplete information, in which case, states are
estimated by the Kalman filter approach and SMC is designed when the state
information is not available for the systems states.
viii Preface
2. Afunctionalobserver-basedSMCisdesignedforlineardiscrete-timestochastic
systems. Sliding function, stability, and convergence analysis are given for the
stochastic system. Existence conditions and stability analysis of a functional
observer are provided. Finally, the controller is calculated by a functional
observer method. This leads to a nonswitching type of SMC.
3. A functional observer-based SMC is designed for discrete-time stochastic sys-
tems in the presence of unmatched uncertainty. A state- and disturbance-
dependent sliding function method is proposed to reduce the effect of unmat-
ched uncertainty in the stochastic system. Finally, SMC is calculated by a
functional observer method.
4. Next, DSMC is designed for parametric uncertain stochastic systems. SMC
design using a functional observer is proposed for parametric uncertain
discrete-time stochastic systems. SMC is calculated by a functional observer
method. To mitigate the side effect of the parameters’ uncertainty on the esti-
mationoferrordynamics,asufficientconditiononstabilityisproposedbasedon
Gershgorin’s circle theorem.
5. An SMC method is proposed for discrete-time delayed stochastic systems.
Stability and convergence analysis of the proposed method are provided.
Furthermore, DSMC of a delayed stochastic system for incomplete state
informationhasalsobeenconsidered,wherestatesareestimatedbytheKalman
filter approach. A functional observer-based SMC method for discrete-time
delayed stochastic systems isproposed.Therefore,SMC has been estimated by
the functional observer approach. Finally, functional the observer-based state
feedback and SMC law are compared graphically as well as numerically.
6. Next, functional observer-based SMC is developed for state time-delayed
stochasticsystems,inthepresenceofparameteruncertaintiesinthestateandin
the delayed state matrix. Finally, SMC has been calculated using a functional
observerapproach.Tomitigatethesideeffectoftheparameters’suncertaintyon
the estimation error dynamics, a sufficient condition on stability is proposed
based on Gershgorin’s circle theorem. A simulation example is considered to
emphasize the functional observer-based SMC design.
Theaimofthisworkistobridgethegapbetweenthediscrete-timeslidingmode
andthediscrete-timestochasticslidingmodebybringinginmanyconceptsthatare
well defined in the former domain into the latter domain using the functional
observer. It is written in a manner such that graduate students interested in sliding
mode control, and particularly the discrete-time variety, will be able to grasp the
difference in the design philosophy of continuous and discrete sliding modes, and
wehopethatitwillpavethewayforfutureresearchintheareaofapplication-based
discrete-time sliding mode control.
New Delhi, India Satnesh Singh
August 2019 S. Janardhanan
Preface ix
AcknowledgementsToourparents,teachers,andfamily,fortheknowledgetheyimparted,and
thesupportgiventous,withoutwhichwewouldnothavebeencapableofwritingthisbook.
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