Stabilization of Semi-Markovian Jump Systems with Uncertain Probability Intensities and Its Extension to Quantized Control

Abstract

This paper 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 S-MJSs, where the infinitesimal operator of a stochastic Lyapunov function is clearly discussed with consideration of input quantization errors.

[ProQuest: [...] denotes non US-ASCII text; see PDF]

Ngoc Hoai An Nguyen 1 and Sung Hyun Kim 1 and Jun Choi 2

Academic Editor:Haitao Zhang

1, School of Electrical Engineering, University of Ulsan, Daehak-ro 93, Nam-Gu, Ulsan 680-749, Republic of Korea

2, Korea Institute of Industrial Technology, Jongga-ro 55, Jung-gu, Ulsan 44413, Republic of Korea

Received 19 July 2016; Accepted 1 November 2016

This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction

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 [1-9] 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: [10-13] with a complete description of transition rates and [14-20] 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 [21-23]). Thus, another interesting topic has recently been studied in semi-Markovian jump systems (S-MJSs) to overcome the limitation of this memoryless property.

As reported in [24-26], 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 [21, 22, 25, 27-29] and references therein). Of them, the first attempt to overcome the limits of MJSs was made by [21, 22] for the stability analysis of systems with phase-type (PH) semi-Markovian jump parameters, which was extended to the state estimation and sliding mode control by [29]. Besides, [25] considered the Weibull 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, [28] 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 [27] to derive the stability and stabilization conditions of S-MJSs with norm-bounded uncertainties. Most recently, [30] designed a reliable mixed passive and _H∞ filter for semi-Markov jump delayed systems with randomly occurring uncertainties and sensor failures. Also, [26] 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.

This paper addresses the issue of stability analysis and control synthesis for S-MJSs with uncertain probability intensities. One of our main contributions is to discover more reliable and scalable transition models for S-MJSs on the basis of their time-varying and boundary properties. To this end, this paper provides a valuable theoretical approach of constructing practical transition models for S-MJSs (1) by taking into account uncertain probability intensities and (2) by reflecting their available bounds in the transition rate description. Further, in a different manner from other works, all constraints on time-varying transition rates are totally incorporated into the stabilization condition via a relaxation process established on the basis of time-varying transition rates. Here, it is worth noticing that our relaxation process is developed in such a way that all possible slack variables can be included therein. In contrast to other works, our relaxation process plays a key role in obtaining a finite and solvable set of linear matrix inequalities (LMIs) from parameterized matrix inequalities (PLMIs) arising from uncertain probability intensities. On the other hand, the quantization module that converts real-valued measurement signals into piecewise constant ones has been commonly used to implement a variety of networked control systems over wired or wireless communications (see [31, 32]). Especially among optical wireless communications, the visible light communication can be applied as a data communication channel to transmit the control input to the S-MJSs under consideration. Thus, as an extension, this paper tackles the quantized control problem of S-MJSs, where the infinitesimal operator of a stochastic Lyapunov function is clearly discussed with consideration on input quantization errors. In addition, this paper proposes a method for reducing the influence of input quantization errors in the control of S-MJSs, which is also one of our main contributions. Finally, simulation examples show the effectiveness of the proposed method.

Notation . The notations X≥Y and X-Y means that X-Y is positive semidefinite and positive definite, respectively. In symmetric block matrices, ([low *]) is used as an ellipsis for terms induced by symmetry. For any square matrix Q, He[Q]=Q+^QT . For ^Ns+ [triangle, =]{1,2,...,s}, [figure omitted; refer to PDF] 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.

2. System Description

Let us consider the following continuous-time semi-Markovian jump linear systems (S-MJSs): [figure omitted; refer to PDF] where x(t)∈^R_nx and u(t)∈^R_nu denote the state and the control input, respectively. Here, (ζ(t),t≥0) denotes a continuous-time semi-Markov process that takes values in the finite space ^Ns+ and further has the mode transition probabilities: [figure omitted; refer to PDF] where _{limh[arrow right]0} [...](o(h)/h)=0 and _πij (h) denotes the transition rate from mode i to mode j at time t+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 ∏(h)[triangle, =]_{(πij (h))i,j∈^Ns+} belongs to the following set: [figure omitted; refer to PDF] Before going ahead, for later convenience, we define the system matrix for the ith mode as (_Ai ,_Bi )[triangle, =](A(_ζk =i), B(_ζk =i)), and set _∏i (h)[triangle, =]^{(_πi1 (h) [...] _πis (h))T} =^{(_πij (h))j∈Ns+ T} . Also, to deal with the stability analysis problem in such a stochastic setting, we consider the following definition.

Definition 1.

An S-MJS (2) with u(t)=0 is stochastically stable if its solution is such that, for any initial condition _x0 and _ζ0 , [figure omitted; refer to PDF]

3. Stochastic Stability Analysis

First of all, let us consider (2) with u(t)≡0: [figure omitted; refer to PDF] The following lemma presents the stochastic stability condition for (6) with ∏(h)∈^SΠ(1) .

Lemma 2.

Suppose that there exists _Pi >0, for all i∈^Ns+ , such that [figure omitted; refer to PDF] where _Qi (h)[triangle, =]He(_PiAi )+^∑j=1s_πij (h)_Pj . Then, S-MJSs (6) with ∏(h)∈^SΠ(1) are stochastically stable.

Proof.

Let us consider a stochastic Lyapunov function candidate of the following form: [figure omitted; refer to PDF] where P(ζ(t)) is taken to be positive definite. Then, the infinitesimal generator ∇ of the stochastic process (x(t),ζ(t),t≥0) acting on V(t) is given by [figure omitted; refer to PDF] where _pij (t,Δ)=Pr(ζ(t+Δ)=j|"ζ(t)=i), [figure omitted; refer to PDF] Here, the probabilities _pij (t,Δ) and _pii (t,Δ) are described as [figure omitted; refer to PDF] where _Gi (h) denotes the cumulative distribution function of the sojourn-time h at mode i and _qij stands for the probability intensity such that ^{∑j=1, j≠is} [...]_qij =1 and _qii =-1. Especially in (11), the probability intensity _qij is taken to have uncertainties. Meanwhile, from (6), it follows that x(t+Δ)=(_Ai Δ+I)x(t), for Δ[arrow right]0, which leads to [figure omitted; refer to PDF] That is, from (11) and (12), it is clear that [figure omitted; refer to PDF] Thus, using (14), _Ψ1 (t) and _Ψ2 (t) become [figure omitted; refer to PDF] That is, ∇V(t) becomes [figure omitted; refer to PDF] where _πi (h)=_gi (h)/1-_Gi (h) denotes the transition rate of the system jumping from mode i. As a result, by defining _πij (h)=_qijπi (h), for j≠i, and _πii (h)=-^{∑j=1, j≠is}_πij (h)=-_πi (h), we have [figure omitted; refer to PDF] In what follows, by the generalized Dynkin's formula [33], it is clear that [figure omitted; refer to PDF] which results in [figure omitted; refer to PDF] Thus, from (7), it follows that [figure omitted; refer to PDF] Finally, by Definition 1, the proof can be completed.

In this paper, 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 _Gi (h) and probability distribution function _gi (h) are given as follows: for all i and j(j≠i)∈^Ns+ , [figure omitted; refer to PDF] which leads to [figure omitted; refer to PDF] As a special case, let _βi =1. Then, we can represent MJSs from (22); that is, the transition rate _πij (h) can be reduced to an h-independent value as follows: _πij (h)=_qijπi (h)=_qij /_αi . Accordingly, it can be claimed that (22) expresses a more generalized transition model, compared to the case of MJSs.

Remark 3.

As shown in (22), 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 (7), 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)=_qijπi (h), the bounds of _πij (h) are decided as follows: _πij,1 <=_πij (h)<=_πij,2 , where [figure omitted; refer to PDF]

In accordance with Remark 3, an auxiliary constraint can be established as follows: ∏(h)∈^SΠ(2) , where [figure omitted; refer to PDF] The following lemma presents the stochastic stability condition for S-MJSs (6) with ∏(h)∈^SΠ(1) ∩^SΠ(2) .

Lemma 4.

Suppose that there exists _Pi >0, for all i∈^Ns+ , such that [figure omitted; refer to PDF] where _Qi (h)[triangle, =]He(_PiAi )+^∑j=1s_πij (h)_Pj and _πij (h)∈[_πij,1 ,_πij,2 ]. Then, S-MJSs (6) with ^SΠ(1) ∩^SΠ(2) are stochastically stable.

However, it is worth noticing that solving (25) of Lemma 4 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 (25). To this end, the following theorem provides a relaxed stochastic stability condition for (6) with ∏(h)∈^SΠ(1) ∩^SΠ(2) .

Theorem 5.

Suppose that there exists matrices _{(Gi ,Sij ,Xij ,Yij )i,j∈^Ns+} ∈^{R_nx ×_nx} and symmetric matrices _{(Pi >0)i∈^Ns+} ∈^{R_nx ×_nx} such that [figure omitted; refer to PDF] where _μij|j≠i =1, _μij|j=i =-1, [figure omitted; refer to PDF] Then, S-MJSs (6) with ∏(h)∈^SΠ(1) ∩^SΠ(2) are stochastically stable.

Proof.

From the first constraint ∏(h)∈^SΠ(1) , we can obtain, under (28), [figure omitted; refer to PDF] Further, under (27), the second constraint ∏(h)∈^SΠ(2) provides [figure omitted; refer to PDF] Here, note that (30)-(32) can be converted, respectively, into [figure omitted; refer to PDF] where ^{(1,1)[low *]} =-_∑j∈^Ns+ He(_{πij,1πij,2Xij} ) and ^{(2,2)[low *]} =^{(He(_Sia ))a∈Ns+ D} +He(^{(_Sia +_Sib )a,b∈Ns+ U} ). Likewise, according to the form of (33)-(35), we can rewrite _Qi (h)<0 in Lemma 4 as follows: [figure omitted; refer to PDF] As a result, by combining (36) with (33)-(35) through the S-procedure [34], the stochastic stability condition (25) is given by [figure omitted; refer to PDF] where (1,1), (1,2), and (2,2) are given in the body of Theorem 5. Therefore, as (26) implies (37), the proof can be completed.

4. Control Design

Let us consider the following mode-dependent state-feedback control law: [figure omitted; refer to PDF] where _Fi [triangle, =]F(ζ(t)=i). Thereby, the resultant closed-loop system under (2) and (38) is given by [figure omitted; refer to PDF] The following theorem provides a relaxed stochastic stabilization condition for S-MJSs (39) with ∏(h)∈^SΠ(1) ∩^SΠ(2) .

Theorem 6.

Suppose that there exist matrices _{(F-i )i∈^Ns+} ∈^{R_nx ×_nx} and _{(Gi ,Sij ,Xij ,Yij ,Qij )i,j∈^Ns+} ∈^{R_nx ×_nx} and symmetric matrices _{(P-i >0)i∈^Ns+} ∈^{R_nx ×_nx} such that [figure omitted; refer to PDF] where _{[...]ij|j≠i} =1, _[...]ij|j=i =0, _μij|j≠i =1, _μij|j=i =-1, [figure omitted; refer to PDF] Then, closed-loop system (39) with ∏(h)∈^SΠ(1) ∩^SΠ(2) is stochastically stable, where _Fi =_F-i^{P-i -1} .

Proof.

In light of (25), the stabilization condition of (39) is given by [figure omitted; refer to PDF] Also, pre- and postmultiplying (45) by _P-i [triangle, =]^Pi-1 yields [figure omitted; refer to PDF] where _F-i [triangle, =]_FiP-i . Here, by employing _Qij such that (43) holds (i.e., _Qij ≥_P-iPjP-i ), we can convert (46) into [figure omitted; refer to PDF] Thereupon, _Q-i (h) is factorized as follows: [figure omitted; refer to PDF] where ^{(1,2)[low *]} =_{((1/2)[...]ijQij +(1/2)(1-[...]ij )P-i )j∈^Ns+} .

Next, as in the proof of Theorem 5, the constraint ∏(h)∈^SΠ(1) ∩^SΠ(2) can be represented as [figure omitted; refer to PDF] where ^{(1,1)[low *]} =-_∑j∈^Ns+ He(_{πij,1πij,2Xij} ) and ^{(2,2)[low *]} =^{(He(_Sia ))a∈Ns+ D} +He(^{(_Sia +_Sib )a,b∈Ns+ U} ). As a result, by the S-procedure, combining (48) with (49) results in [figure omitted; refer to PDF] where (1,1), (1,2), and (2,2) are given in the Theorem 6. Therefore, as (40) implies (50), the proof can be completed.

Hereafter, as a practical extension of the proposed approach, we consider the following input-quantized S-MJSs: [figure omitted; refer to PDF] where q(*) stands for a uniform quantization operator with the quantization level δ>0; that is, q(u(t))=δ·round(u(t)/δ). Here, note that q(u(t))=u(t)+[straight phi](t), where the kth element of the quantization error [straight phi](t) satisfies [figure omitted; refer to PDF] Thus, (51) can be rewritten as [figure omitted; refer to PDF] where [straight phi](t)=q(u(t))-u(t) is known. Continuously, as a mode-dependent state-feedback law, we adopt [figure omitted; refer to PDF] Then, the resultant closed-loop system is described as [figure omitted; refer to PDF] The following theorem provides a relaxed stochastic stabilization condition for S-MJSs (55) with input quantization error.

Theorem 7.

Let _νi,k (i.e., the kth element of _νi ) be given as follows: [figure omitted; refer to PDF] where _si (t)=^BiT_Pi x(t)∈^R_nu and _si,k (t) denotes the kth element of _si (t). Suppose that there exist matrices _{(F-i )i∈^Ns+} ∈^{R_nx ×_nx} and _{(Gi ,Sij ,Xij ,Yij ,Qij )i,j∈^Ns+} ∈^{R_nx ×_nx} and symmetric matrices _{(P-i >0)i∈^Ns+} ∈^{R_nx ×_nx} such that (40)-(43) hold. Then, the closed-loop system (55) with ∏(h)∈^SΠ(1) ∩^SΠ(2) is stochastically stable, where _Fi =_F-i^{P-i -1} .

Proof.

From (55), it follows that x(t+Δ)=(_A-i Δ+I)x(t)+_Bi Δ(_νi (t)+[straight phi](t)), for Δ[arrow right]0, which yields [figure omitted; refer to PDF] Thus, it is given that [figure omitted; refer to PDF] Then, based on (11) and (12), we have [figure omitted; refer to PDF] which leads to [figure omitted; refer to PDF] Hence, for ^siT (t)[straight phi](t)<=0, letting _νi (t)≡0 implies [figure omitted; refer to PDF] Further, for ^siT (t)[straight phi](t)>0, letting _νi,k (t)=-δ·sgn[...](_si,k (t)) yields [figure omitted; refer to PDF] That is, (56) allows (61) to be reduced to (62) for all ^siT (t)[straight phi](t). In what follows, we need to derive the stabilization condition from (62), which is omitted herein because it is in line with the proof of Theorem 6.

5. Numerical Examples

Example 1.

Consider the following system with three different modes: for x0 =(3.0,-5.0) and ζ0 =3, [figure omitted; refer to PDF] Here, the Weibull distribution for the sojourn-time is set by (α1 ,β1 )=(0.5,2.0), (α2 ,β2 )=(1.0,2.0), and (α3 ,β3 )=(1.5,2.0) (see Figure 1). Thereby, a reasonable interval h∈[h0 ,h1 ] can be obtained such that ∫h0h1gi (h)dh≥0.99, from which the lower and upper bounds (πi,1 ,πi,2 ) of πi (h) can be also found. As a result, by considering the uncertain probability intensities such that 0.3<=qij <=0.7, for all i and j, the following matrices such that πij (h)∈(πij,1 ,πij,2 ) are established from Remark 3: [figure omitted; refer to PDF] Figure 2(a) shows the mode evolution generated from the stochastic setting. Besides, from Theorem 6, the following control gains are obtained: [figure omitted; refer to PDF] Figure 2(b) shows the behavior of the state response coupled with the mode transition depicted in Figures 2(a) and 2(c) which presents the Monte Carlo simulation result of the settling time for 5000 different mode transition configurations, where its mean value and standard deviation are 12.2979 and 3.126, respectively. From Figure 2(b), it can be seen that the states converge from the initial condition (3.0,-5.0) to the equilibrium point as time increases. Consequently, Theorem 6 provides an applicable method for deriving a relaxed stabilization condition for S-MJSs with uncertain probability intensities.

Figure 1: Cumulative distribution functions _Gi (h) (solid line) and probability distribution functions _gi (h) (dash-dotted line).

(a) [figure omitted; refer to PDF]

(b) [figure omitted; refer to PDF]

(c) [figure omitted; refer to PDF]

Figure 2: (a) Mode evolution, (b) state response and control input, and (c) Monte Carlo simulation results.

(a) [figure omitted; refer to PDF]

(b) [figure omitted; refer to PDF]

(c) [figure omitted; refer to PDF]

Example 2.

Consider the following system representing an inverted pendulum: for _x0 =(1,-1,0.5) and _ζ0 =3: [figure omitted; refer to PDF] where _x1 (t) is the angle of the inverted pendulum, _x2 (t) is the angular velocity, _x3 (t) is the input current, u(t) is the control input voltage, g is the acceleration due to gravity, m and l are the mass and length, respectively, _kb and _km are the back emf constant and motor torque constant, respectively, _La and R(ζ(t)) are the inductance and resistance of the DC motor, respectively, and N is the gear ratio. Here, we set _La =1, g=9.8, l=1, m=1, N=10, _Km =_Kb =0.1, and [figure omitted; refer to PDF] Then, the linearized model of (67) is given by [figure omitted; refer to PDF] Besides, the matrices _{(πij,1 )ij∈^N3+} and _{(πij,2 )ij∈^N3+} such that _πij (h)∈[_πij,1 ,_πij,2 ] are taken to be the same as in Example 1. Thereupon, Theorem 7 provides the following control gains: [figure omitted; refer to PDF] Figure 3 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 (56), and the quantization level is assumed to be δ=0.1. From Figure 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 7 provides a suitable mode-dependent control for the S-MJSs with input quantization errors as well as uncertain probability intensities.

Figure 3: State response and mode evolution.

[figure omitted; refer to PDF]

6. Concluding Remarks

The issue of stability analysis and control synthesis for S-MJSs with uncertain probability intensities has been addressed in this paper. 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. Furthermore, as an extension, the quantized control problem of S-MJSs has been addressed herein. Through simulation examples, the effectiveness of the proposed method has been shown.

Acknowledgments

This work was supported by the National Research Foundation of Korea Grant funded by the Korean Government (NRF-2015R1A1A1A05001131).

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