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

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 Advisor Professor, Ph.D Sung Hyun, Kim School of Electrical Engineering, University of Ulsan Professor Committee Ph.D Sung Hwan, Kim School of Electrical Engineering, University of Ulsan Professor Committee Ph.D Jong Pil Yun Korean Institute of Industrial Technology The Dissertation was officially defended in the Master Dissertation Defense Presentation held by the School of Electrical Engineering, University of Ulsan on November 28, 2016 There, the Dissertation named ”Stabilization of semi-Markovian jump systems with uncertain probability intensities and its extension to quantized control” was successfully defended by graduate student Nguyen Ngoc Hoai An VITA 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 University of Ulsan, South Korea under the supervisor of Professor Kim Sung Hyun Since then, she has conducted researches in Embedded Control Eystem Laboratory ACKNOWLEDGEMENTS 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 of his research and for his supervisor which helped me complete this work I am greatly indebted to him for his full support, constant encouragement and advice both in technical and non-technical matters I would also like to thank the other members of my M.E supervisory committee for many useful interactions and for contributing their broad perspective in redefining the ideas in this dissertation I am grateful to my friends, my lab mate of Embedded Control System Laboratory (ECSL), University of Ulsan (UOU), for their friendship, enthusiastic help and cheerfulness during my stay in South Korea The financial support of the BK21 and BK21+ programs is also gratefully acknowledged Last, but certainly not the least, I would like to thank my family for their spirit 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 ABSTRACT Stabilization of semi-Markovian jump systems with uncertain probability intensities and its extension to quantized control by Nguyen Ngoc Hoai An Advisor: Prof Kim Sung Hyun This dissertation concentrates on the issue of stability analysis and control synthesis for semi-Markovian jump systems (S-MJSs) with uncertain probability intensities Here, to construct a more applicable transition model for S-MJSs, the probability intensities are taken to be uncertain, and this property is totally reflected in the stabilization condition via a relaxation process established on the basis of time-varying transition rates Moreover, an extension of the proposed approach is made to tackle the quantized control problem of SMJSs, where the infinitesimal operator of a stochastic Lyapunov function is clearly discussed with consideration of input quantization errors Simulation examples show the effectiveness of the proposed method Notation 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 squared △ matrix Q For N+ s = {1, 2, , s},   Q11 Q1s [ ] [ ] △ △   , Qi i∈N+s = [Q1 Q2 Qs ], Qij i,j∈N+s =   Qs1 Qss     Q Q Q 12 13 1s Q1  0 Q23 Q2s     [ ]D △ Q2  △   [ ]U  Qi i∈N+s = , Qij i,j∈N+s =  ,      Q(s−1)s  Qs 0 where Qi and Qij 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 Acknowledgements Abstract Notation INTRODUCTION 1.1 Motivation 1.2 Previous works 1.3 Research Contribution 7 8 STOCHASTIC STABILITY ANALYSIS 2.1 Introduction 2.2 System description 2.3 Stochastic Stability Analysis 2.4 Conclusion RELAXED STOCHASTIC STABILITY 3.1 Introduction 3.2 Probability intensity analysis 3.3 Relaxed Stochastic Stability Analysis 3.4 Conclusion ANALYSIS CONTROL DESIGN 4.1 Introduction 4.2 Control design 4.3 Extension on input quantization 4.4 Conclusion error control 10 10 10 11 12 13 13 13 14 15 16 16 16 17 18 Table of Contents SIMULATION RESULTS 19 5.1 Example 19 5.2 Example 19 SUMMARY OF CONTRIBUTIONS AND FURTHER WORKS 22 6.1 Introduction 22 6.2 Summary of Contributions 22 6.3 Future Research Directions 23 Publication 23 Reference 24 Page Chapter INTRODUCTION 1.1 Motivation Over the past few decades, considerable attention has been paid to Markovian jump systems (MJSs) since such systems are suitable for representing a class of dynamic systems subject to random abrupt variations In addition to the growing interest from their representation ability, MJSs have been widely applied in many practical applications, such as manufacturing systems, aircraft control, target tracking, robotics, networked control systems, solar receiver control, and power systems (see [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12] and references therein) Following this trend, numerous investigations are underway to deal with the issue of stability analysis and control synthesis for MJSs with complete/incomplete knowledge of transition probabilities in the framework of filter and control design problems: [13, 14, 15, 16] with a complete description of transition rates, and [17, 18, 19, 20, 21, 22] without a complete description Generally, in MJSs, the sojourn-time is given as a random variable characterized by the continuous exponential probability distribution, which tends to make the transition rates time-invariant due to the memoryless property of the probability distribution The thing to be noticed 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 jump systems (S-MJSs) to overcome the limitation of this memoryless property Chapter 1: INTRODUCTION 1.2 Previous works As reported in [26, 27, 28], the mode transition of S-MJSs is driven by a continuous stochastic process governed by the nonexponential sojourn-time distribution, 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 of practical 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) semiMarkovian jump parameters, which was extended to the state estimation and sliding mode control by [33] Besides, [27] considered the Weibuill distribution for the stability analysis of S-MJSs and introduced a sojourn-time partition technique to make the derived stability criterion less conservative Continuing this, [32] applied the sojourn-time partition technique to the design of H∞ state-feedback control for S-MJSs with time-varying delays After that, another partition technique of dividing the range of transition rates was proposed by [31] to derive the stability and stabilization conditions of S-MJSs with norm-bounded uncertainties Most recently, [35] designed a reliable mixed passive and H∞ filter for semi-Markov jump delayed systems with randomly occurring uncertainties and sensor failures Also, [28] considered semi-Markovian switching and random measurement while designing a sliding mode control for networked control systems (NCSs) Based on the above observations, it can be found that their key issue mainly lies in finding more applicable transition models for S-MJSs, capable of a broad range of cases In this light, one needs to explore the impacts of uncertain probability intensities in the study of S-MJSs, and then provide a relaxed stability criterion absorbing the property of the resultant time-varying transition rates However, until now, there have been almost no studies that intensively establish a kind of relaxation process corresponding to the stabilization problem of S-MJSs with uncertain probability intensities 1.3 Research Contribution This dissertation addresses the issue of stability analysis and control synthesis for S-MJSs with uncertain probability intensities One of the main Page Chapter 3: RELAXED STOCHASTIC STABILITY ANALYSIS given as follows: for all i and j (j ̸= i) ∈ N+ s, ( ( ) ) ( ( ) ) β β h i h i βi βi −1 and gi (h) = βi h exp − , Gi (h) = − exp − αi αi αi (3.1) which leads to πij (h) = qij πi (h) = qij gi (h) βi = qij βi hβi −1 − Gi (h) αi (3.2) As a special case, let βi = Then, we can represent MJSs from (3.2), that is, the transition rate πij (h) can be reduced to an h-independent value as follows: πij (h) = qij πi (h) = qij /αi Accordingly, it can be claimed that (3.2) expresses a more generalized transition model, compared to the case of MJSs Remark As shown in (3.2), the transition rate πij (h) is time-varying and depends on the probability intensity qij 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 πi (h) and qij , respectively, as follows: πi,1 ≤ πi (h) ≤ πi,2 and qij,1 ≤ qij ≤ qij,2 Then, from πij (h) = qij πi (h), the bounds of πij (h) are decided as follows: πij,1 ≤ πij (h) ≤ πij,2 , where { { qij,1 · πi,1 , if j ̸= i qij,2 · πi,2 , if j ̸= i πij,1 = and πij,2 = − πi,2 , otherwise − πi,1 , otherwise In accordance with Remark 1, an auxiliary constraint can be established as (2) follows: Π(h) ∈ SΠ , where { } (2) △ SΠ = [πij ]i,j∈Ns+ πij,1 ≤ πij ≤ πij,2 , ∀i, j ∈ N+ s The following section presents the stochastic stability condition for S-MJSs (1) ∩ (2) (2.4) with transition rates boundaries Π(h) ∈ SΠ SΠ 3.3 Relaxed Stochastic Stability Analysis Lemma Suppose that there exists Pi > 0, for all i ∈ N+ s , such that ∩ (2) (1) > Qi (h), Π(h) ∈ SΠ SΠ , ∀i ∈ N+ (3.3) s, Page 14 Chapter 3: RELAXED STOCHASTIC STABILITY ANALYSIS ∑ △ where Qi (h) = He(Pi Ai ) + sj=1 πij (h)Pj and πij (h) ∈ [πij,1 , πij,2 ] Then, (1) ∩ (2) S-MJSs (2.4) with SΠ SΠ are stochastically stable However, it is worth noticing that solving (3.3) of Lemma 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 (1) ∩ (2) stochastic stability condition for (2.4) with Π(h) ∈ SΠ SΠ { } Theorem Suppose that there exists matrices Gi , Sij , Xij , Yij i,j∈N+s ∈ Rnx ×nx { } and symmetric matrices Pi > i∈N+s ∈ Rnx ×nx such that ] [ (1, 1) (1, 2) , ∀i ∈ N+ (3.4) 0> s, (∗) (2, 2) ≤ He(Xij ), ∀i, j ∈ N+ s, + ≤ He(Yij ), ∀i, j ∈ Ns , (3.5) (3.6) where µij |j̸=i = 1, µij |j=i = −1, ( ) ∑ ( ) (1, 1) = He Pi Ai − He πij,1 πij,2 Xij , [ j∈N+ s ] , (1, 2) = Pj + Gi + (πij,1 + πij,2 )Xij + µij Yij j∈N+ s [ ( ) ([ )]D ]U (2, 2) = He Sia − Xia + He S + S + ia ib a,b∈Ns + a∈Ns (1) Then, S-MJSs (2.4) with Π(h) ∈ SΠ ∩ (2) SΠ are stochastically stable PROOF The proof of Theorem is shown in my full Dissertation 3.4 Conclusion To sum up, this Chapter is successful in using relaxation technique to transform the infinite stability condition into the finite and solvable LMI form The Theorem proven in this Chapter is so important that the following generalized case of S-MJSs analysis will be inherited and developed In the next Chapter, the control design is conducted to generalize the stability condition for S-MJSs via relaxation technique Page 15 Chapter CONTROL DESIGN 4.1 Introduction This chapter covers the control synthesis on stochastically stability condition for the S-MJSs The control input is considered in the S-MJSs system and the final stability condition as proven in Chapter is then adapted to the current system model Again, in this chapter, the relaxation technique places a key role to successfully convert final stability conditions into the finite LMI form Further, as an extension, the condition of input quantization error is reflected such that the input-quantized S-MJSs is stochastically stable following to Theorem in Chapter 4.2 Control design Let us consider the following mode-dependent state-feedback control law: u(t) = Fi x(t), (4.1) ( ) △ where Fi = F ζ(t) = i Thereby, the resultant closed-loop system under (2.1) and (4.1) is given by: x(t) ˙ = Ai x(t) + Bi u(t) = (Ai + Bi Fi )x(t) = A¯i x(t) (4.2) The following theorem provides a ∩ relaxed stochastic stabilization condition (1) (2) for S-MJSs (4.2) with Π(h) ∈ SΠ SΠ 16 Chapter 4: CONTROL DESIGN { } { Theorem Suppose that there exist matrices F¯i i∈N+s ∈ Rnx ×nx and Gi , { } } Sij , Xij , Yij , Qij i,j∈N+s ∈ Rnx ×nx , and symmetric matrices P¯i > i∈N+s ∈ Rnx ×nx such that ] [ (1, 1) (1, 2) 0> , ∀i ∈ N+ (4.3) s, (∗) (2, 2) ≤ He(Xij ), ∀i, j ∈ N+ s, + ≤ He(Yij ), ∀i, j ∈ Ns , [ ] Qij P¯i 0≤ , ∀i, j ̸= i ∈ N+ s, (∗) P¯j (4.4) (4.5) (4.6) where ϵij |j̸=i = 1, ϵij |j=i = 0, µij |j̸=i = 1, µij |j=i = −1, ( ) ∑ ( ) (1, 1) = He Ai P¯i + Bi F¯i − He πij,1 πij,2 Xij , [( j∈N+ s ] ) 1 (1, 2) = ϵij Qij + (1 − ϵij )P¯i + Gi + (πij,1 + πij,2 )Xij + µij Yij , 2 j∈N+ s [ ( ([ ) )]D ]U (2, 2) = He Sia − Xia + He Sia + Sib a,b∈N+s + a∈Ns (1) Then, the closed-loop system (4.2) with Π(h) ∈ SΠ stable, where Fi = F¯i P¯i−1 ∩ (2) SΠ is stochastically PROOF The proof of Theorem is shown in my full Dissertation 4.3 Extension on input quantization error control Hereafter, as a practical extension of the proposed approach, we consider the following input-quantized S-MJSs: ( ) x(t) ˙ = Ai x(t) + Bi q u(t) , (4.7) where q(•) stands for quantization operator with the quantization ( a uniform ) ( ) ( ) level δ > 0, i.e., q u(t) = δ · round u(t)/δ Here, note that q u(t) = u(t) + φ(t), where the kth element of the quantization error φ(t) satisfies δ φk (t) ≤ , ∀k ∈ N+ nu Page 17 Chapter 4: CONTROL DESIGN Thus, (4.7) can be rewritten as ( ) x(t) ˙ = Ai x(t) + Bi u(t) + φ(t) , ( ) where φ(t) = q u(t) − u(t) is known Continuously, as a mode-dependent state-feedback law, we adopt u(t) = Fi x(t) + νi (t) (4.8) where νi (t) ∈ Rnu is the input quantization error Then, the resultant closedloop system is described as ( ) x(t) ˙ = A¯i x(t) + Bi νi (t) + φ(t) (4.9) The following theorem provides a relaxed stochastic stabilization condition for S-MJSs (4.9) with input quantization error Theorem Let νi,k (i.e., the kth element of νi ) be given as follows: ( ( ) ( )) + νi,k (t) = −δ · sgn si,k (t) · max 0, sgn sTi (t)φ(t) , ∀i ∈ N+ s , k ∈ Nnu , (4.10) the kth element of si (t) where si (t) = BiT Pi x(t) ∈ Rnu and{ si,k { } (t) denotes nx ×nx ¯ and Gi , Sij , Xij , Yij , Suppose that there exist matrices Fi i∈N+s ∈ R } } { nx ×nx ¯ Qij i,j∈N+s ∈ R , and symmetric matrices Pi > i∈N+s ∈ Rnx ×nx such (1) ∩ (2) that (4.3)–(4.6) hold Then, the closed-loop system (4.9) with Π(h) ∈ SΠ SΠ −1 ¯ ¯ is stochastically stable, where Fi = Fi Pi PROOF The proof of Theorem is shown in my full Dissertation 4.4 Conclusion In conclusion, the stability control synthesis of S-MJSs is successfully proven and defined Therefore, the stability condition in Theorem built on relaxation technique is efficient and applicable for the generalized S-MJSs As an extension on the stabilization analysis, the consideration on quantization error show the sufficient condition of input quantization error such that S-MJSs is stable in Theorem Page 18 Chapter SIMULATION RESULTS 5.1 Example 5.2 Example Figure shows the behavior of the state response for the mode transition generated according to (α1 , β1 ) = (0.5, 2.0), (α2 , β2 ) = (1.0, 2.0), and (α3 , β3 ) = (1.5, 2.0) Here, the control input νi (t) is designed in accordance with (4.10), and the quantization level is assumed to be δ = 0.1 From Fig 6.3, it can be seen that the states converge from the initial-state condition (1, −1, 0.5) to the origin as time increases Consequently, Theorem provides a suitable mode-dependent control for the S-MJSs with input quantization errors as well as uncertain probability intensities 19 Chapter 5: SIMULATION RESULTS (a) (c) (b) 1.8 G1(h) 1.6 0.9 g1(h) g3(h) 0.8 0.6 0.7 (α3,β3) = (1.5,2.0) (α2,β2) = (1.0,2.0) (α1,β1) = (0.5,2.0) 0.8 1.2 0.6 0.4 0.6 0.5 0.4 0.3 G (h) 0.4 0.2 g2(h) 0.2 0.2 0 G3(h) 0.8 1.4 0.1 h 0 h 0 h Figure 5.1: (solid line) Cumulative distribution functions Gi (h) and (dashdotted line) probability distribution functions gi (h) mode ζ(t) (a) 10 15 20 25 30 time (sec) (b) (c) 100 x1(t) x2(t) −2 −4 −6 80 numbers control u(t) state x(t) 10 60 40 time (sec) −10 10 10 15 15 20 time (sec) 20 25 25 20 30 30 0 Figure 5.2: (a) mode evolution, (b) state response and control input, and (c) Monte Carlo simulation results Page 20 10 15 20 settling time (sec) 25 Chapter 5: SIMULATION RESULTS x1(t) x2(t) −2 mode ζ(t) state x(t) x3(t) −4 −6 −8 2 −12 10 time (sec) −10 time (sec) Figure 5.3: State response and mode evolution Page 21 10 Chapter SUMMARY OF CONTRIBUTIONS AND FURTHER WORKS 6.1 Introduction This chapter is to summarize the main contributions of this dissertation and to discuss further works In particular, a brief summary of these results is presented in the next section whereas concerning future research directions are discussed in Section 6.3 6.2 Summary of Contributions • The issue of stability analysis for S-MJSs with uncertain probability intensities has been addressed in this dissertation • Here, the boundary constraints of probability intensities have been totally reflected in the stabilization condition via a relaxation process established on the basis of time-varying transition rates • This dissertation also solves the stabilization control synthesis for SMJSs with complete description • Furthermore, as an extension, the quantized control problem of S-MJSs has been addressed herein 22 Chapter 6: SUMMARY OF CONTRIBUTIONS AND FURTHER WORKS • Through simulation examples, the effectiveness of the proposed method has been shown 6.3 Future Research Directions Before finalizing the thesis, we mention here with some brief remarks on extension works in the future as following: • This dissertation is successful in uncertain probability intensity analysis for the S-MJSs Therefore, we are inspired to conduct research on “Robust H∞ output-feedback control of semi-Markovian jump systems with uncertain probability intensities” where the H∞ output feed back control will be the new light in this future research • Incomplete transition description is always a mysterious and challenging to jump systems analysis Consequently, we are also desired to study about “Mixed passive and H∞ control for nonhomogeneous Markovian jump systems with actuator saturation and incomplete transition description” Page 23 Publication [1] Kim Sung Hyun, and Ngoc Hoai An Nguyen.“Stochastic stability analysis of semi-Markovian jump linear systems via a relaxation technique for time-varying transition rates.” Control, Automation and Systems (ICCAS), 2015 15th International Conference on IEEE, 2015 [2] Ngoc Hoai An Nguyen, Kim Sung Hyun and Jun Choi.“Stabilization of semi-Markovian jump systems with uncertain probability intensities and its extension to quantized control.” Mathematical Problems in Engineering, 2016 (accepted) [3] Kim Tong Hyun, Kim Sung Hyun and Ngoc Hoai An Nguyen “Speed Control of DC Motors with time-varying load torque via H2 Control Scheme.” Institute of Control, Robotics and Systems, 2015 30th ICROS Annual Conference ICROS, 2015 24 Bibliography [1] E.K Boukas, Manufacturing systems: LMI approach, IEEE Trans Autom Control 51 (2006) 1014-1018 [2] F Martinelli, Optimality of a two-threshold feedback control for a manufacturing system with a production dependent failure rate, IEEE Trans Autom Control 52 (2007) 1937-1942 [3] B.L Stevens, F.L Lewis, Aircraft modeling, dynamics and control, Wiley, New York (1991) [4] X.R Li, V P Jilkov, A survey of maneuvering target tracking: multiplemodel methods, IEEE Trans Aerosp Electron Syst 41 (2005) 1255-1321 [5] J.K Mills, A A 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Ngoc Hoai An Nguyen, Kim Sung Hyun and Jun Choi. Stabilization of semi- Markovian jump systems with uncertain probability intensities and its extension to quantized control. ” Mathematical Problems... School of Electrical Engineering, University of Ulsan on November 28, 2016 There, the Dissertation named Stabilization of semi- Markovian jump systems with uncertain probability intensities and its. .. mode control for semi- Markovian jump systems with mismatched uncertainties, Automatica 51 (2015) 385-393 [34] F Li, L Wu, P Shi, Stochastic stability of semi- Markovian jump systems with mode-dependent