Stabilization of semi markovian jump systems with uncertain probability intensities and its extension to quantized control

REPUBLIC OF KOREA UNIVERSITY OF ULSANSchool of Electrical Engineering Stabilization of semi-Markovian jump systems with uncertain probability intensities and its extension to quantized c

REPUBLIC OF KOREA UNIVERSITY OF ULSAN

(School of Electrical Engineering)

Stabilization of semi-Markovian jump systems with uncertain probability intensities and its extension to quantized control

SUMMARY OF DISSERTATION

for the Degree of

MASTER OF ENGINEERING

(Electrical Engineering)

NGUYEN NGOC HOAI AN

November 2016

The dissertation is officially defended at

UNIVERSITY OF ULSAN, Ulsan City, South Korea

Nguyen Ngoc Hoai An was born in Da Nang City, Viet Nam on June

22, 1990 She received the B.E degree (2013) in Electrical Engineering from

Da Nang University of Technology, Da Nang City, Viet Nam

In March 2015, she began working full time towards her M.E at sity of Ulsan, South Korea under the supervisor of Professor Kim Sung Hyun.Since then, she has conducted researches in Embedded Control Eystem Lab-oratory

I would like to express my sincere gratitude to my dissertation advisor,Professor Kim Sung Hyun, for giving me a lot of opportunities to be part ofhis research and for his supervisor which helped me complete this work I

am greatly indebted to him for his full support, constant encouragement andadvice both in technical and non-technical matters

I would also like to thank the other members of my M.E supervisorycommittee for many useful interactions and for contributing their broad per-spective in redefining the ideas in this dissertation

I am grateful to my friends, my lab mate of Embedded Control SystemLaboratory (ECSL), University of Ulsan (UOU), for their friendship, enthu-siastic help and cheerfulness during my stay in South Korea

The financial support of the BK21 and BK21+ programs is also gratefullyacknowledged

Last, but certainly not the least, I would like to thank my family for theirspirit support and encouragement during the course of my studies

Especially, my parents and my younger brother have always believed in

me and have supported every endeavor of mine

Stabilization of semi-Markovian jump

systems with uncertain probability intensities and its extension to quantized control

Here, to construct a more applicable transition model for S-MJSs, theprobability intensities are taken to be uncertain, and this property is totallyreflected in the stabilization condition via a relaxation process established

on the basis of time-varying transition rates Moreover, an extension of theproposed approach is made to tackle the quantized control problem of S-MJSs, where the infinitesimal operator of a stochastic Lyapunov function isclearly discussed with consideration of input quantization errors Simulationexamples show the effectiveness of the proposed method

3

Some following notations are used in this dissertation

The notation X ≥ Y and X − Y means that X − Y is positive semi-definite

and positive definite, respectively In symmetric block matrices, (∗) is used

as an ellipsis for terms induced by symmetry For any square matrix Q,

He[Q] = Q + QT where QT denotes the transpose matrix of the squaredmatrix Q For N+

where Q i and Q ij denote real submatrices with appropriate dimensions or

scalar values The notation E[•] denotes the mathematical expectation, and

diag(•) stands for a block-diagonal matrix The notation λmax(•) denotes the

maximum eigenvalue of the argument, and exp(•) indicates the exponential

distribution

Table of contents

Vita 1

Acknowledgements 2

Abstract 3

Notation 4

1 INTRODUCTION 7 1.1 Motivation 7

1.2 Previous works 8

1.3 Research Contribution 8

2 STOCHASTIC STABILITY ANALYSIS 10 2.1 Introduction 10

2.2 System description 10

2.3 Stochastic Stability Analysis 11

2.4 Conclusion 12

3 RELAXED STOCHASTIC STABILITY ANALYSIS 13 3.1 Introduction 13

3.2 Probability intensity analysis 13

3.3 Relaxed Stochastic Stability Analysis 14

3.4 Conclusion 15

4 CONTROL DESIGN 16 4.1 Introduction 16

4.2 Control design 16

4.3 Extension on input quantization error control 17

4.4 Conclusion 18

5

Table of Contents

5.1 Example 1 195.2 Example 2 19

6 SUMMARY OF CONTRIBUTIONS AND FURTHER WORKS 22

6.1 Introduction 226.2 Summary of Contributions 226.3 Future Research Directions 23

Marko-a complete description GenerMarko-ally, in MJSs, the sojourn-time is given Marko-as

a random variable characterized by the continuous exponential probabilitydistribution, which tends to make the transition rates time-invariant due tothe memoryless property of the probability distribution The thing to benoticed here is that the use of constant transition rates plays a limited role

in representing a wide range of application systems (see [23, 24, 25]) Thus,another interesting topic has recently been studied in semi-Markovian jumpsystems (S-MJSs) to overcome the limitation of this memoryless property

7

Chapter 1: INTRODUCTION

1.2 Previous works

As reported in [26, 27, 28], the mode transition of S-MJSs is driven by acontinuous stochastic process governed by the nonexponential sojourn-timedistribution, which leads to the appearance of time-varying transition rates.Thus, it has been well recognized that S-MJSs are more general than MJSs

in real situations Further, with this growing recognition, various problems

on S-MJSs have been widely studied for successful utilization of a variety ofpractical applications (see [23, 24, 27, 31, 32, 33, 34] and references therein)

Of them, the first attempt to overcome the limits of MJSs was made by[23, 24] for the stability analysis of systems with phase-type (PH) semi-Markovian jump parameters, which was extended to the state estimationand sliding mode control by [33] Besides, [27] considered the Weibuill dis-tribution for the stability analysis of S-MJSs and introduced a sojourn-timepartition technique to make the derived stability criterion less conservative.Continuing this, [32] applied the sojourn-time partition technique to the de-sign of H ∞ state-feedback control for S-MJSs with time-varying delays Af-

ter that, another partition technique of dividing the range of transition rateswas proposed by [31] to derive the stability and stabilization conditions ofS-MJSs with norm-bounded uncertainties Most recently, [35] designed a re-liable mixed passive and H ∞ filter for semi-Markov jump delayed systems

with randomly occurring uncertainties and sensor failures Also, [28] ered semi-Markovian switching and random measurement while designing asliding mode control for networked control systems (NCSs) Based on theabove observations, it can be found that their key issue mainly lies in findingmore applicable transition models for S-MJSs, capable of a broad range ofcases In this light, one needs to explore the impacts of uncertain probabilityintensities in the study of S-MJSs, and then provide a relaxed stability cri-terion absorbing the property of the resultant time-varying transition rates.However, until now, there have been almost no studies that intensively estab-lish a kind of relaxation process corresponding to the stabilization problem

consid-of S-MJSs with uncertain probability intensities

1.3 Research Contribution

This dissertation addresses the issue of stability analysis and control thesis for S-MJSs with uncertain probability intensities One of the main

syn-Chapter 1: INTRODUCTION

contributions is to discover more reliable and scalable transition models forS-MJSs on the basis of their time-varying and boundary properties To thisend, this dissertation provides a valuable theoretical approach of constructing

a practical transition models for S-MJSs 1) by taking into account uncertainprobability intensities, and 2) by reflecting their available bounds in the tran-sition rate description Further, in a different manner from other works, allconstraints on time-varying transition rates are totally incorporated into thestabilization condition via a relaxation process established on the basis oftime-varying transition rates Here, it is worth noticing that the relaxationprocess is developed in such a way that all possible slack variables can beincluded therein In contrast to other works, the relaxation process plays akey role in obtaining a finite and solvable set of linear matrix inequalities(LMIs) from parameterized matrix inequalities (PLMIs) arising from uncer-tain probability intensities On the other hand, the quantization modulethat converts real-valued measurement signals into piecewise constant oneshas been commonly used to implement a variety of networked control sys-tems over wired or wireless communications (see [29, 30]) Especially, amongoptical wireless communications, the visible light communication can be ap-plied as a data communication channel to transmit the control input to theS-MJSs under consideration Thus, as an extension, this dissertation tacklesthe quantized control problem of S-MJSs, where the infinitesimal operator of

a stochastic Lyapunov function is clearly discussed with consideration on put quantization errors In addition, this dissertation proposes a method forreducing the influence of input quantization errors in the control of S-MJSs,which is also one of main contributions Finally, simulation examples showthe effectiveness of the proposed method

in-Page 9

process that takes values in the finite space N+

s and further has the modetransition probabilities:

Chapter 2: STOCHASTIC STABILITY ANALYSIS

where limh →0 (o(h)/h) = 0 and π ij (h) denotes the transition rate from mode

i to mode j at time t + h and π ii (h) = −∑s

j=1,j ̸=i π ij (h) Further, h indicates the sojourn-time elapsed when the system stays at mode i from the last jump (i.e., h is set to 0 when the system jumps) In particular, the transition rate matrix Π(h)=△ [

Before going ahead, for later convenience, we define the system matrix for the

ith mode as (A i , B i) = (A(ζ △ k = i), B(ζ k = i)), and set Π i (h)=△ [

Definition 1 An S-MJS (2.1) with u(t) = 0 is stochastically stable if its

solution is such that for any initial condition x0 and ζ0,

2.3 Stochastic Stability Analysis

First of all, let us consider (2.1) with u(t) ≡ 0:

Π are stochastically stable.

PROOF The proof of Lemma 1 is shown in my full Dissertation.

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Chapter 2: STOCHASTIC STABILITY ANALYSIS

2.4 Conclusion

In this Chapter, the Lemma 1 and its Proof is given to reveal the original

stochastically stability condition for the S-MJSs with u(t) = 0 In the next

Chapter, the probability intensity analysis will be represented and provenand lead to another important constraints on stability condition Those newconstraints are accompanied by the original condition construct the Lemma 2which is the final stability condition for the S-MJSs in that case After that,relaxation technique will be shown its role in converting the final stabilitycondition

sufficient constraints in the final stability condition for S-MJSs with u(t) =

0 Technically, those stability conditions are revised to the finite LMI-formthanks to the relaxation technique, which is the most important technique

in this dissertation

3.2 Probability intensity analysis

In our research, as a model of probability distribution for the sojourn-time

h ≥ 0, we utilize the Weibull distribution with shape parameter β > 0

and scale parameter α > 0, since such a distribution has been witnessed as

an appropriate choice for representing the stochastic behavior of practical

systems In other words, to represent the probability distribution of h, its cumulative function G i (h) and probability distribution function g i (h) are

13

Chapter 3: RELAXED STOCHASTIC STABILITY ANALYSIS

given as follows: for all i and j (j ̸= i) ∈ N+

π ij (h) = q ij π i (h) = q ij g i (h)

1− G i (h) = q ij

β i

α β i i

h β i −1 . (3.2)

As a special case, let β i = 1 Then, we can represent MJSs from (3.2), that

is, the transition rate π ij (h) can be reduced to an h-independent value as follows: π ij (h) = q ij π i (h) = q ij /α i Accordingly, it can be claimed that (3.2)expresses a more generalized transition model, compared to the case of MJSs

Remark 1 As shown in (3.2), the transition rate π ij (h) is time-varying and

depends on the probability intensity q ij Thus, to derive a finite number of solvable conditions from (2.5), there is a need to consider the lower and upper bounds of both π i (h) and q ij , respectively, as follows: π i,1 ≤ π i (h) ≤ π i,2 and

q ij,1 ≤ q ij ≤ q ij,2 Then, from π ij (h) = q ij π i (h), the bounds of π ij (h) are

decided as follows: π ij,1 ≤ π ij (h) ≤ π ij,2 , where

The following section presents the stochastic stability condition for S-MJSs

(2.4) with transition rates boundaries Π(h) ∈ S(1)

Π

∩

S(2)

Π

3.3 Relaxed Stochastic Stability Analysis

Lemma 2 Suppose that there exists P i > 0, for all i ∈ N+

Chapter 3: RELAXED STOCHASTIC STABILITY ANALYSIS

Π are stochastically stable.

However, it is worth noticing that solving (3.3) of Lemma 2 is still equivalent

to solving an infinite number of LMIs, which is an extremely difficult problem.Thus, it is necessary to find a finite number of solvable LMI-based conditions

from (3.3) To this end, the following theorem provides a sufficient relaxed stochastic stability condition for (2.4) with Π(h) ∈ S(1)

Π are stochastically stable.

PROOF The proof of Theorem 1 is shown in my full Dissertation

3.4 Conclusion

To sum up, this Chapter is successful in using relaxation technique to form the infinite stability condition into the finite and solvable LMI form.The Theorem 1 proven in this Chapter 3 is so important that the follow-ing generalized case of S-MJSs analysis will be inherited and developed Inthe next Chapter, the control design is conducted to generalize the stabilitycondition for S-MJSs via relaxation technique

trans-Page 15

a key role to successfully convert final stability conditions into the finite LMIform Further, as an extension, the condition of input quantization error

is reflected such that the input-quantized S-MJSs is stochastically stablefollowing to Theorem in Chapter 3

Thereby, the resultant closed-loop system under

(2.1) and (4.1) is given by:

˙x(t) = A i x(t) + B i u(t) = (A i + B i F i )x(t) = ¯ A i x(t). (4.2)The following theorem provides a relaxed stochastic stabilization condition

Chapter 4: CONTROL DESIGN

Theorem 2 Suppose that there exist matrices {¯

PROOF The proof of Theorem 2 is shown in my full Dissertation

4.3 Extension on input quantization error

Chapter 4: CONTROL DESIGN

Thus, (4.7) can be rewritten as

− u(t) is known Continuously, as a mode-dependent

state-feedback law, we adopt

Theorem 3 Let ν i,k (i.e., the kth element of ν i ) be given as follows:

where s i (t) = B i T P i x(t) ∈ R n u and s i,k (t) denotes the kth element of s i (t).

Suppose that there exist matrices {¯

is stochastically stable, where F i = ¯F i P¯i −1

PROOF The proof of Theorem 3 is shown in my full Dissertation.

4.4 Conclusion

In conclusion, the stability control synthesis of S-MJSs is successfully provenand defined Therefore, the stability condition in Theorem 2 built on relax-ation technique is efficient and applicable for the generalized S-MJSs As anextension on the stabilization analysis, the consideration on quantization er-ror show the sufficient condition of input quantization error such that S-MJSs

is stable in Theorem 3