DISCRETE-TIME STOCHASTIC SLIDING MODE CONTROL USING FUNCTIONAL OBSERVATION
SATNESH SINGH
DEPARTMENT OF ELECTRICAL ENGINEERING INDIAN INSTITUTE OF TECHNOLOGY DELHI
MAY 2019
© Indian Institute of Technology Delhi (IITD), New Delhi, 2019
DISCRETE-TIME STOCHASTIC SLIDING MODE CONTROL USING FUNCTIONAL OBSERVATION
by
SATNESH SINGH
DEPARTMENT OF ELECTRICAL ENGINEERING Submitted
in fulfilment of the requirements of the degree of Doctor of Philosophy to the
INDIAN INSTITUTE OF TECHNOLOGY DELHI
MAY 2019
CERTIFICATE
This is to certify that the thesis entitled“Discrete-Time Stochastic Sliding Mode Control Using Functional Observation”, submitted by Satnesh Singh to the Indian Institute of Technology Delhi, for the award of the degree of Doctor of Philosophy in Electrical Engineering, is a record of the bonafide research work carried out by him under my supervision and guidance. The thesis has reached the standards fulfilling the requirements of the regulations relating to the award of the degree.
The results contained in this thesis have not been submitted either in part or in full to any other University or Institute for the award of any degree or diploma to the best of my knowledge.
Dr. S. Janardhanan Department of Electrical Engineering, Indian Institute of Technology Delhi.
(Supervisor)
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ACKNOWLEDGEMENTS
First of all, I would like to express my heartfelt thanks to my research supervisor Dr. S.
Janardhanan. Without his consistent support, guidance and patience, the work presented in this thesis could not have been possible. His technical acumen, precise suggestions and timely discussions are wholeheartedly appreciated. I also favour to thank my student research committee members Prof. I. N. Kar, Prof. S. D. Joshi and Prof. S. Dharmaraja for their inspirational suggestions and critical comments time to time during the entire period of research work. I thank Prof. Bijnan Bandyopadhyay, Systems and Control Engineering, IIT Bombay for giving me an opportunity to attend IEEE fall school on modern sliding mode control in Oct. 2016. Lectures delivered by eminent faculty during the training helped me throughout my PhD work. I am also very thankful to my M.Tech supervisor Dr. B. B. Sharma for kindling my curiosity in control systems. It was his encouragement that I was strong-willed to do a PhD in the first spot.
I would always be grateful to the faculty and staff members of the Electrical Engineering Department of IIT Delhi for their assistance and support. Notably, Mr. Virender Singh needs a special mention here for arranging all the computing facilities to support the research work.
I could not have accomplished this work without the help of several individuals. I would like to encourage all those people who have contributed to the completion of my thesis. Especially my lab mates Koena Mukherjee, Madan Mohan Rayguru, Sumit Kumar Jha, Venkat Bokka, Niraj Choudhary and Abhilash Patel, deserve special mention for their technical inputs and creating ambience for the research.
On a personal note, I express heartfelt gratitude to my family members, especially my elder brother Upendra Singh, and parents for their support and constant motivation throughout the work. Finally, I thank The Omnipotent who conferred me with his elegance and strength
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throughout the research work.
I am inadequate to express my gratitude in words, however as once a wise man wrote, “I stopped explaining myself when I realised people only understand from their level of percep- tion”.
Satnesh Singh
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ABSTRACT
Sliding mode control (SMC) is one of the most efficient robust methods for controlling systems with uncertainty. SMC methods provide some advantages such as easy and straightforward realisation, quick response and low sensitivity to uncertainties in the model of the system and disturbances. The target of this thesis is to provide a comprehensive investigation of functional state estimation based SMC for discrete-time stochastic systems and propounds an idea of discrete-time stochastic SMC. SMC for discrete-time stochastic systems with bounded disturbances is proposed in this work. Subsequently, SMC is designed for the discrete-time stochastic system such that the states will lie within the specified band. Furthermore, this result has been extended for the incomplete state information. In that case, states are estimated by the Kalman filter approach and SMC is designed.
A functional observer based SMC is designed for discrete-time stochastic systems. The control signal is calculated by a functional observer method. In many cases, the disturbances in the system may not always satisfy the matching condition. Therefore, a functional observer based SMC is designed for discrete-time stochastic systems with unmatched uncertainty. A disturbance dependent based sliding surface method is proposed to ameliorate the effect of unmatched uncertainty in the stochastic system. Next, SMC design using functional state estimation is proposed for parametric uncertain discrete-time stochastic systems. A sufficient condition of stability is proposed based on Gershgorin disc theorem, which provides the estimate of the eigenvalues location of the matrix.
Most of engineering systems have an inherent time-delay such as sensor measurements and communication delays. This thesis also proposes SMC method for linear discrete-time delayed stochastic systems. Subsequently, the stability and convergence analysis of the proposed method are provided. Furthermore, SMC of a delayed stochastic system for incomplete state information has also been considered, where states are estimated by the Kalman filter approach. Moreover,
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a functional observer based SMC method for the discrete-time delayed stochastic system is proposed. Therefore, SMC has been calculated by the functional observer approach. Finally, functional observer-based state feedback and SMC law are compared graphically as well as numerically.
Further, the previous result is extended for the case where parametric uncertainty are present in the system and the state delay matrix. Functional observer-based SMC is developed for uncertain state-delayed systems. Subsequently, the SMC method for discrete-time delayed stochastic systems is investigated. Finally, SMC has been calculated by using a functional observer approach. A sufficient condition of stability is proposed based on Gershgorin disc theorem, which provides the estimate of the eigenvalues location of the given matrix. Finally, in the last chapter, a summary of the thesis contributions is presented and future directions of research are identified.
Keywords: Sliding mode control, Functional observers, Time delay, Discrete-time systems, Stochastic systems.
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सार
ससस्टम को नियंत्रित करिे के सिए स्िाइडडंग मोड कंट्रोि (एसएससी) सबसे अधिक ई-रोगी मजबूत तरीकों में से एक है अनिश्चितता के साथ। एसएमसी विधियां कुछ फायदे प्रदाि करती हैं जैसे कक आसाि और सरि अहसास, त्िररत प्रनतकिया और ससस्टम के मॉडि में अनिश्चितताओं के प्रनत कम संिेदिशीिता और गड़बड़ी। इस थीससस का िक्ष्य एक व्यापक जांि प्रदाि करिा है असतत समय स्टोकेश्स्टक प्रणासियों के सिए कायाात्मक राज्य का अिुमाि एसएमसी
आिाररत है और प्रॉप्स ए असतत समय स्टोकेश्स्टक एसएमसी का वििार। बाउंड के साथ असतत समय स्टोकेश्स्टक ससस्टम के सिए एसएमसी इस काया में गड़बड़ी प्रस्तावित है। इसके बाद, एसएमसी असतत समय के सिए डडजाइि
ककया गया है स्टोकेश्स्टक प्रणािी ऐसी है कक राज्य विसशष्ट एड बैंड के भीतर श्स्थत होंगे। इसके अिािा, यह अपूणा
राज्य सूििा के सिए पररणाम बढाया गया है। उस श्स्थनत में, राज्यों का अिुमाि है किमि क़िल्टर दृश्ष्टकोण और एसएमसीद्िारा डडजाइि ककया गया है।
एक कायाात्मक पयािेक्षक आिाररत एसएमसी असतत समय स्टोकेश्स्टक ससस्टम के सिए डडजाइि ककया गया है।
नियंिण संकेत एक कायाात्मक पयािेक्षक विधि द्िारा गणिा की जाती है। कई मामिों में, गड़बड़ी ससस्टम में हमेशा
समिाि श्स्थनत को संतुष्ट िहीं ककया जा सकता है। इससिए, एक कायाात्मक पयािेक्षक आिाररत एसएमसी को बेजोड़
अनिश्चितता के साथ असतत समय स्टोकेश्स्टक ससस्टम के सिए डडजाइि ककया गया है। ए अशांनत पर आिाररत स्िाइडडंग सतह विधि के प्रभाि को संशोधित करिे का प्रस्ताि है स्टोकेश्स्टक प्रणािी में बेजोड़ अनिश्चितता। अगिा, कायाात्मक राज्य का उपयोग करके एसएमसी डडजाइि अिुमाि पैरामीट्रट्रक अनिश्चित असतत समय स्टोकेश्स्टक ससस्टम के सिए प्रस्तावित है। एक पयााप्त Gershgorin डडस्क प्रमेय के आिार पर श्स्थरता की श्स्थनत प्रस्तावित है, जो अिुमाि प्रदाि करती है मैट्रट्रक्स के आइगेन्वॅल्यूसस्थाि।
अधिकांश इंजीनियररंग प्रणासियों में एक अंतनिाट्रहत समय-देरी होती है जैसे सेंसर माप और संिार में देरी। यह थीससस भी रैखिक असतत समय देरी के सिए एसएमसी विधि का प्रस्ताि है स्टोकेश्स्टक ससस्टम। इसके बाद, प्रस्तावित विधि
की श्स्थरता और असभसरण विचिेषण ट्रदए गए है। इसके अिािा, अपूणा राज्य सूििा के सिए एक वििंत्रबत स्टोिश्स्टक प्रणािी का एसएमसी यह भी मािा जाता है, जहां राज्यों का अिुमाि किमि िेटर दृश्ष्टकोण से है। इसके अिािा, असतत समय देरी िािे स्टोकेश्स्टक ससस्टम के सिए एक कायाात्मक पयािेक्षक आिाररत एसएमसी विधि है का प्रस्ताि
रिा। इससिए, एसएमसी की गणिा कायाात्मक पयािेक्षक दृश्ष्टकोण द्िारा की गई है। आखिरकार, कायाात्मक पयािेक्षक- आिाररत राज्य प्रनतकिया और एसएमसी कािूि की तुििा रेिांकि के साथ-साथ की जाती है संख्यािुसार।
इसकेअलावा, पिछलेिरिणामकोउसमामलेकेललएबढायाजाताहैजहाांलसस्टममेंिैिामीट्रिकअनिश्चितताऔििाज्यपवलांब मैट्रिक्समौजूदहैं।कायाात्मक प्रेक्षक आिाररत एसएमसी अनिश्चित के सिए विकससत ककया गया है राज्य-वििंत्रबत ससस्टम। इसके बाद, असतत समय के सिए एसएमसी विधि स्टोकेश्स्टक प्रणासियों में देरी हुई जांि की जाती है। अंत में, एक कायाात्मक पयािेक्षक दृश्ष्टकोण का उपयोग करके एसएमसी की गणिा की गई है। Gershgorin डडस्क प्रमेय के आिार पर श्स्थरता की पयााप्त श्स्थनत प्रस्तावित है, जो प्रदाि करती है ट्रदए गए मैट्रट्रक्स के आइगेन्वॅल्यूसस्थाि
का अिुमाि। अंत में, आखिरी अध्याय में, ए थीससस योगदाि का सारांश प्रस्तुत ककया गया है और अिुसंिाि की
भविष्य की ट्रदशाएं आइडी एड हैं।
कीिडा: स्िाइडडंग मोड नियंिण, कायाात्मक पयािेक्षक, समय की देरी, असतत समय प्रणािी, स्टोकेश्स्टक ससस्टम।
Contents
Certificate i
Acknowledgements iii
Abstract v
List of Figures xiii
List of Tables xvii
Nomenclature xix
1 Introduction 1
1.1 Motivation . . . 3
1.2 Contributions . . . 4
1.3 Structure of the Thesis . . . 5
2 Preliminary Concepts 7 2.1 Sliding mode Control . . . 7
2.1.1 Sliding mode Control for Continuous-time Systems . . . 8
2.1.2 Discrete-time Sliding Mode Control . . . 11
2.1.3 Discrete-time Sliding Mode Control for Stochastic Systems . . . 14
2.2 Observers . . . 21
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2.2.1 State Observation . . . 23
2.2.2 Observer based SMC for discrete-time systems . . . 24
2.2.3 Observer based SMC for discrete-time stochastic systems . . . 24
2.3 Functional Observers . . . 28
2.3.1 Linear Functional Observer . . . 30
2.3.2 Main advantages of functional observer . . . 32
3 Design of SMC for Discrete-time Stochastic Systems with Bounded Distur- bances 33 3.1 Introduction . . . 33
3.2 Problem Formulation . . . 34
3.3 SMC for Stochastic Systems in Presence of Bounded Disturbances . . . 35
3.3.1 SMC for Stochastic Systems with Complete State Information in the Presence of Bounded Disturbances . . . 35
3.3.2 SMC for Stochastic Systems with Incomplete State Information in the Presence of Bounded Disturbances . . . 35
3.4 Stability Analysis . . . 37
3.4.1 Complete State Information Case . . . 37
3.4.2 Incomplete State Information Case . . . 37
3.5 Simulation Results . . . 38
3.6 Conclusion . . . 40
4 Functional Observer based SMC for Discrete-time Stochastic Systems 41 4.1 Introduction . . . 41
4.2 Problem Formulation . . . 42
4.3 Sliding Function and Controller Design . . . 42
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4.3.1 Design of Sliding Function . . . 42
4.3.2 Synthesis of Control Law . . . 43
4.4 Linear Functional Observer based SMC Design . . . 44
4.5 Simulation Results . . . 49
4.6 Conclusion . . . 54
5 Functional Observer based SMC for Discrete-time Stochastic Systems with Unmatched Uncertainty 55 5.1 Introduction . . . 55
5.2 Problem Formulation . . . 56
5.3 Sliding Function and Controller Design . . . 57
5.3.1 Sliding Function Design . . . 57
5.3.2 Augmented System . . . 60
5.3.3 Controller Design . . . 60
5.4 Linear Functional Observer based SMC Design . . . 61
5.5 Simulation Example and Results . . . 65
5.5.1 Comparative Study . . . 68
5.6 Conclusion . . . 70
6 Stochastic SMC for Parametric Uncertain Systems using Functional Ob- server 71 6.1 Introduction . . . 71
6.2 Problem Formulation . . . 72
6.3 Sliding Function and Controller Design . . . 73
6.3.1 Sliding Function Design . . . 73
6.3.2 Design of Controller . . . 76
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6.4 Design of Functional Observer based SMC . . . 77
6.5 Simulation Results . . . 80
6.6 Conclusion . . . 81
7 Functional Observer based SMC for Discrete-Time Delayed Stochastic Sys- tems 83 7.1 Introduction . . . 83
7.2 System Description and Problem Formulation . . . 85
7.3 Design of DSMC for Stochastic Systems with Complete State Information . . . . 86
7.3.1 Sliding Function Design . . . 86
7.3.2 Discrete-time sliding mode control Design . . . 89
7.4 Design of DSMC for Stochastic Systems with Incomplete State Information . . . 90
7.4.1 Controller Design Strategy . . . 91
7.4.2 Simulation Results . . . 93
7.5 Functional Observer based SMC Design for Time Delayed Stochastic Systems . . 99
7.5.1 General Case . . . 99
7.5.2 Functional Observer without Internal Delay . . . 105
7.6 Simulation Example and Results . . . 106
7.6.1 Comparative Study . . . 107
7.6.2 Functional observer based SMC without internal delay . . . 112
7.7 Conclusions . . . 114
8 Functional Observer based SMC for Parametric Uncertain Discrete-Time Delayed Stochastic Systems 115 8.1 Introduction . . . 115
8.2 Problem Formulation . . . 116
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8.3 Design of Sliding Function and Controller . . . 117
8.3.1 Design of Sliding Function . . . 118
8.3.2 Design of DSMC . . . 123
8.4 Design of Functional Observer based SMC for Parametric Uncertain Time-delay Systems . . . 124
8.4.1 General Case . . . 124
8.4.2 Internal Delay Free Observer . . . 126
8.5 Simulation Example and Results . . . 127
8.5.1 General Case . . . 127
8.5.2 Internal Delay Free Observer Case . . . 130
8.6 Conclusions . . . 132
9 Conclusions and Future Directions of Work 133 9.1 Conclusions . . . 133
9.2 Future Directions of Work . . . 134
Bibliography 137
xi
List of Figures
2.1 Filippov method to analyze sliding mode . . . 10
2.2 The reaching phase and sliding phase in sliding mode . . . 11
2.3 Simulation responses of complete state information:(a)and (b) state variables (c) control input and (d) sliding function. . . 21
2.4 Simulation responses of incomplete state information:(a)and (b) state variables (c) control input and (d) sliding function. . . 29
3.1 Complete state information in the presence of bounded disturbances: (a)-(c) time response of the states . . . 38
3.2 Complete state information in presence of bounded disturbances (a) Control Input and (b) Sliding Function . . . 39
3.3 Incomplete state information in presence of bounded disturbances (a) Control Input and (b) Sliding Function . . . 40
4.1 Evolution of control input u1(k) . . . 51
4.2 Evolution of control input u2(k) . . . 52
4.3 Evolution of estimation error eu1(k) . . . 52
4.6 Evolution of sliding functions2(k) . . . 52
4.4 Evolution of estimation error eu2(k) . . . 53
4.5 Evolution of sliding functions1(k) . . . 53
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5.1 Response for (a) control input u1(k) and (b) control input u2(k) with presence
and absence of unmatched uncertainty. . . 68
5.2 Response for (a) estimation error eu1(k) and (b) estimation error eu2(k) with presence and absence of unmatched uncertainty. . . 69
5.3 Response for (a) sliding function s1(k) and (b) sliding function s2(k) with pres- ence and absence of unmatched uncertainty. . . 69
6.1 Evolution of control input u(k). . . 80
6.2 Evolution of estimation error eu(k). . . 81
6.3 Evolution of sliding functions(k) . . . 81
7.1 Response for (a) state x1(k) and its state estimation ˆx1(k), (b) state x2(k) and its state estimation ˆx2(k), (c) statex3(k) and its state estimation ˆx3(k). . . 94
7.2 Complete state information (a) Control Inputu1(k) and (b) Sliding Function s1(k) 95 7.3 Complete state information (a) Control Inputu2(k) and (b) Sliding Function s2(k) 96 7.4 Incomplete state information (a) : Control Inputu1(k) and (b) : Sliding Function s1(k) . . . 97
7.5 Incomplete state information (a) : Control Inputu2(k) and (b) : Sliding Function s2(k) . . . 98
7.6 Evolution of Control Inputu(k) . . . 109
7.7 Evolution of estimation error eu(k) . . . 110
7.8 Evolution of Sliding Function s(k) . . . 110
7.9 Evolution of Control Inputu(k) without internal delay . . . 112
7.10 Evolution of estimation error eu(k) without internal delay . . . 113
7.11 Evolution of Sliding Function s(k) without internal delay . . . 113
8.1 Evolution of control input u(k) . . . 129
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8.2 Evolution of estimation error eu(k) . . . 129
8.3 Evolution of sliding functions(k) . . . 130
8.4 Evolution of control input u(k) with delay free case . . . 131
8.5 Evolution of estimation error eu(k) with delay free case . . . 131
8.6 Evolution of sliding functions(k) with delay free case . . . 132
xv
List of Tables
2.1 Distinguished observer and order of observers . . . 32 3.1 Parameters in simulation results . . . 39
7.1 Performance Comparison of Different Control Strategies . . . 111
xvii
Nomenclature
Acronyms Description
CSMC Continuous-time Sliding Mode Control
DSM Discrete-time Sliding Mode
DSMC Discrete-time Sliding Mode Control
QSM Quasi Sliding Mode
QSMB Quasi Sliding Mode Band
KF Kalman Filter
LTI Linear Time Invariant
LFO Linear Functional Observers
LQR Linear Quadratic Regulator
LMI Linear Matrix Inequality
SFC State Feedback Control
SMB Sliding Mode Band
SMC Sliding Mode Control
DARE Discrete-time Algebraic Riccati Equation
a.s. Almost sure
p.d.s. Positive definite symmetric
VSS Variable Structure Systems
VSC Variable Structure Control
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Continued from previous page
Acronyms Description
VSCS Variable Structure Control Systems
List of Symbols Description
(.)T General notation for the matrix transpose operation
R The field of real numbers
Rn The n-dimensional real vector space
Z The set of integer numbers
⊗ Kronecker product
A State matrix in discrete-time system
Ad State matrix in discrete-time system with time delay
C Output matrix in discrete-time system
B Input matrix of control in discrete-time system
Γ Process noise matrix in discrete-time system
F Unknown input matrix of the system
G Measurement noise matrix in discrete-time system
d The disturbance vector effect on the sliding function
d0 The mean of disturbance bounds du and dl
dl Lower bound of the disturbance d
du Upper bound of the disturbance d
P Probability measure
Ω Sample space
F Set of events
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List of Symbols Description
In An n×n identity matrix
0n An n×n zero matrix
u Control input
θ Regulating the approaching speed
E(.) Expectation operator
E{.|.} Conditional expectation
k Variable denoting time instant
kx Constant known delay in state matrix
m Number of inputs in system
n Number of states in system
p Number of outputs of system
w Vector of process noise
v Vector of measurement noise
Q Matrix of process noise co-variance
R Matrix of measurement noise co-variance
||.|| Euclidean norm, the norm of a vector
q Order of functional observer
s Sliding function
Sµc Sliding mode band
τ >0 Sampling time
κ, ψ Tuning parameters
∀ For all
Designating the completion of proof
→ Tends to
xxi
List of Symbols Description
, Defined as
⊂ Proper subset of
∩ Intersection
∈ Belongs to
C Complex plane
δ Performance index
ρ(X) Range space of the matrix X
x State of a dynamical system
ˆ
x Estimated system state of a dynamical system
y System output of a dynamical system
Υ Kalman gain
c Sliding function parameter
K Sliding function gain matrix
V Lyapunov function
U General notation for an n×n transformation matrix
L Functional gain matrix
M, J, H, E Unknown functional observer matrices
Md, Jd, Ed Unknown functional observer matrices with time delay
xxii
