# Quantized State-Feedback Stabilization for Delayed Markovian Jump Linear Systems with Generally Incomplete Transition Rates

Authors: Yanbo Li; Peng Zhang; Yonggui Kao; Hamid Reza Karimi
Journal: Abstract and Applied Analysis
 (2014)

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## Abstract

This paper is concerned with the robust quantized state-feedback controller design problem for a class
of continuous-time Markovian jump linear uncertain systems with general uncertain transition rates and
input quantization. The uncertainties under consideration emerge in both system parameters and mode
transition rates. This new uncertain model is more general than the existing ones and can be applicable
to more practical situations because each transition rate can be completely unknown or only its estimate
value is known. Based on linear matrix inequalities, the quantized state-feedback controller is formulated
to ensure the closed-loop system is stable in mean square. Finally, a numerical example is presented to
verify the validity of the developed theoretical results.


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## Body

## 1. Introduction

Markovian jump systems have been serving as popular tools for analyzing plants subjected to random abrupt changes, such as random component failures, abrupt environment changes, disturbance, and changes in the interconnections of subsystems; see [1–11] and the references therein. In practice, transition rates are difficult to precisely estimate the transition rates. Therefore, developing the analysis and synthesis method for MJS with uncertain transition rates is of great importance. Bounded uncertain transition rates (BUTRs) are introduced when the precise values of the transition rates are unknown, but their bounds (upper bounds and lower bounds) are known; see, for example, [12–14]. However, in some practical cases, to obtain the bound of every TR is difficult or even impossible. Zhang et al. [15–18] proposed another description for the uncertain TRs, which is partly unknown TRs (PUTRs). In this description, some of the TRs can be unknown. This model also simulates researchers’ interests (see, e.g., [19–22]). In this PUTR model, every transition rate is either exactly known or completely unknown, which may be too restrictive in many practical situations. Therefore, Guo and Wang [23] proposed generally uncertain TRs to deal with a more practical situation where the transition rates can be completely unknown or only its bound is known.On the other hand, in many modern engineering practices, information processing devices, such as analog-to-digital and digital-to-analog converters, have been widely used and brought about some advantages, such as lower cost, reduced weight and power, and simple installation and maintenance. However, server deterioration of system performance or even system instability may also be induced. Signal quantization should be fully considered in such cases. Nowadays, the feedback stabilization problem is considered by utilizing dynamic quantizers [24–26] and static quantizers [27–36]. The stabilization problem for single-input discrete Markov jump linear systems via mode dependent quantized state feedback is addressed by Xiao et al. in [37], but the transition rates are assumed to be completely known. Ye et al. [38, 39] considered H2 control of Markov jump linear systems with unknown transition rates and input quantization. To the best of our knowledge, no result has been presented for control design of delayed continuous-time Markov jump linear uncertain systems with generally unknown transition rates and input signal quantization.In this paper, robust quantized state-feedback stabilization for delayed Markovian jump linear systems with generally uncertain transition rates is addressed. In Section2, the considered systems are formulated and the purposes of the paper are stated. In Section 3, the main results are derived via linear matrix inequalities. The structure of the controller consists of two parts. The nonlinear part is provided to eliminate the effect of input quantization. The linear part is obtained by solving LMIs to deal with model uncertainties and unknown transition rates. Section 4 concludes the paper.Notation. In this paper, Rn and Rn×m denote the n-dimensional Euclidean space and the set of all n×m real matrices, respectively. N+ represents the set of positive integers. The notation P>0(P≥0) means that P is a real symmetric and positive-definite (semi-positive-definite) matrix. For notation (Ω,F,P),Ω represents the space, F is the σ-algebra of subsets of the sample space, and P is the probability measure on F. E{·} stands for the mathematical expectation. Matrices, if their dimensions are not explicitly stated, are assumed to be compatible for algebraic operations.


## 2. Problem Formulation

Consider the following stochastic system with Markovian jump parameters, defined on a complete probability space(Ω,F,P):
(1)x˙(t)=A(rt)x(t)+Ad(rt)x(t-τ)+(B(rt)+ΔB(rt))q(u(t)),t≥0,x0=x(0),r0=r(0),
where x(t)∈Rn is the system state. The mode jumping process {rt,t≥0} is a right-continuous Markov process on the probability space taking values in a finite state space S={1,2,…,s} with the mode transition probabilities:
(2)Pr{rt+Δ=j∣rt=i}={πijΔ+o(Δ),ifi≠j,1+πijΔ+o(Δ),ifi=j,
where Δ>0, lim⁡Δ→0(o(Δ)/Δ)=0, and πij≥0(i,j∈S,j≠i) is the TR from mode i at time t to mode j at time t+Δ, and
(3)πii=-∑j=1,j≠isπij.
for each i∈S.The mode TR matrixΠ≜(πij) is considered to be generally uncertain. For instance, the TR matrix for system (1) with s operation modes may be expressed as
(4)[π^11+Δ11?π^13+Δ13⋯???π^23+Δ23⋯π^2s+Δ2s⋮⋮⋮⋱⋮?π^s2+Δs2?⋯π^ss+Δss],
where π^ij and Δij∈[-δij,δij](δij≥0) represent the estimate value and estimate error of the uncertain TR πij, respectively, where π^ij and δij are known. “?’’ represents the complete unknown TR, which means its estimate value π^ij and estimate error bound are unknown. For notational clarity, for all i∈S, the set Ui denotes Ui=Uki∪Uuki with Uki≜{j:theestimatevalueofλijisknownforj∈S}, Uuki≜{j:theestimatevalueofλijisunknownforj∈S}. Moreover, if Uki≠Ø, it is further described as Uki={k1i,k2i,…,kmi}, where kmi∈N+ represent the mth bound-known element with the index kmi in the ith row of matrix Π. According to the properties of the TRs (e.g.,λij≥0(∀i,j∉S,i≠j)andλii=-∑j=1,j≠iN‍λij), we assume that the known estimate values of the TRs are well defined. That is, the following assumptions hold.Assumption 1.
IfUki=S, then λ^ij-δij≥0(∀j∈S,j≠i),λ^ii=-∑j=1,j≠iNλ^ij≤0, and δii=∑j=1,j≠iNδij>0.Assumption 2.
IfUki≠S,andi∈Uki, then λ^ij-δij≥0(∀j∈Uki,j≠i),λ^ii+δii≤0 and ∑j∈Ukiλ^ij≤0.Assumption 3.
IfUki≠S and i∉Uki, then λ^ij-δij≥0(∀j∈Uki).Remark 4.
The above assumption is reasonable, since it is the direct result from the properties of the TRs(5)(e.g.,πij≥0(∀i,j∉S,i≠j),πii=-∑j=1,j≠isπij).
The above description about uncertain TRs is more general than either the BUTR or PUTR models. To show this, we rewrite the two uncertain models as follows: BUTR model (see [12–14]):
(6)[π^11+Δ11π^12+Δ12⋯π^1s+Δ1sπ^21+Δ21π^22+Δ22⋯π^2s+Δ2s⋮⋮⋱⋮π^s1+Δs1π^s2+Δs2⋯π^ss+Δss]
with π^ij-δij≥0(∀j∈S,j≠i), π^ii=-∑j=1,j≠isπ^ij,andδii=-∑j=1,j≠isδij. PUTR model (see [15–22]):
(7)[π11?π13⋯???π23⋯π2s⋮⋮⋮⋱⋮?πs2?⋯πss].
Obviously, if UkiØ,∀i∈S, the GUTR model (4) will reduce to the BUTR model (6); if δij=0,∀i∈S,∀j∈Uki, the GUTR model (4) will reduce to the PUTR model (7). The GUTR model (4) is more general than the other two models; therefore, it is more practicable.For convenience, the system matricesAi≜A(rt=i), Adi≜Ad(rt=i), and Bi≜B(rt=i),ΔBi≜ΔB(rt=i),i∈S, are known matrix functions of the Markovian process. Then system (1) can be described by
(8)x˙(t)=Aix(t)+Adix(t-τ)+(Bi+ΔBi)q(u(t)),t≥0,x0=x(0),r0=r(0).The following assumptions are assumed to be valid.Assumption 5.
Δ
B
i
=
B
i
M
i
Ξ
i
(
t
)
F
i and |MiΞi(t)Fi|∞≤ψi, where Mi and Fi are known constant matrices with appropriate dimensions, Ξi(t) is time-varying uncertain matrix satisfying Ξi(t)ΞiT(t)≤I, and parameter ψi satisfies 0≤ψi<1.In addition, the quantizerq(·) is defined by an operator function round(·) which rounds to the nearest integer; that is,
(9)q(u(t))=μ·round(u(t)μ),
where μ(>0) is called a quantizing level of the quantizer. In computer-based control systems, the value of μ depends on the sampling accuracy and is known a priori. q(·) is the uniform quantizer with the fixed level μ. Define eμ=q(u(t))-u(t); since each component of eμ is bounded by the half of the quantizing level μ, we have |eμ|∞≤μ/2.The objective of this paper is to design a state-feedback control law(10)u(t)=Kix+uic,Ki=K(rt),whenrt=i
such that the resulting closed-loop system is stochastically stable. The nonlinear part of the controller uic is designed against the effect of signal quantization, and the linear part Kix is proposed to deal with model uncertainties and unknown transition rates.Lemma 6 (Petersen, 1987).
Given a symmetric matrixΠ and matrices M, N with appropriate dimensions, then Π+MF(t)N+NTFT(t)MT<0 for all F(t) satisfying FT(t)F(t)≤I, if and only if there exists a scalar ε>0 such that the following inequality holds:
(11)Π+εMMT+ε-1NTN<0.Lemma 7 (14).
Given any real numberε and any matrix Q, the matrix inequality
(12)ε(Q+QT)≤ε2T+QT-1QT
holds for any matrix T>0.


## 3. Stochastic Stability Analysis

The goal of this section is to develop an analysis result of stability for system (1) with general uncertain TRs.Theorem 8.
Consider that uncertain Markovian jump system (8) with a GUTR matrix (4) is stochastically stable if there exist matrices Pi>0(i∈S), Tij>0(i∉Uki,j∈Uki), Vijl>0(i,j∈Uki,Uuki≠Ø,l∈Uuki),Wij>0(i,j∈Uki,Uuki=Ø), and Q>0 such that the following LMIs are feasible for i=1,2,…,s.
Ifi∉Uki,
(13)[Γ11PiAdiPik1i-Pi⋯Pikmi-Pi*-Q0⋯0**-Tik1i⋯0***⋱⋮****-Tikmi]<0(14)Pi-Pj≥0,∀j∈Uuki,j≠i.
Ifi∈Uki and Uuki≠Ø,foronel∈Uuki,
(15)[Θ11PiAdiPk1i-Pl⋯Pkmi-Pl*-Q0⋯0**-Vik1il⋯0***⋱⋮****-Vikmil]<0.
Ifi∈Uki and Uuki=Ø,(16)[Ψ11PiAdiP1-Pi⋯Pi-1-PiPi+1-Pi⋯Ps-Pi*-Q0⋯00⋯0**-Wi1⋯00⋯0***⋱⋮⋮⋮⋮****-Wi(i-1)0⋮⋮*****-Wi(i+1)⋮⋮******⋱⋮*******-Wis]<0,where
(17)Γ11=AiTPi+PiAi+KiTBiTPi+PiBiKi+σiPiBiMiMiTBiTPi+1σiKiTFiTFiKi+∑j∈Uki‍π^ij(Pj-Pi)+∑j∈Uki‍δij24Tij+QΘ11=AiTPi+PiAi+KiTBiTPi+PiBiKi+σiPiBiMiMiTBiTPi+1σiKiTFiTFiKi+∑j∈Uki‍π^ij(Pj-Pl)+∑j∈Uki‍δij24Vijl+QΨ11=AiTPi+PiAi+KiTBiTPi+PiBiKi+σiPiBiMiMiTBiTPi+1σiKiTFiTFiKi+∑j∈S,j≠i‍π^ij(Pj-Pi)+∑j∈S,j≠i‍δij24Wij+Q.
Then the controller designed as
(18)u(t)=Kix+uic
can drive the state trajectory to the origin asymptotically, where Ki=YiXi-1,Pi=Xi-1, and uic=(1+ψ)u/(2(1-ψ))sign⁡(xTPiBi).Proof.
Take the Lyapunov function candidateV=xTPix+∫t-τt‍xT(α)Qx(α)dα; then along the system trajectory of plant (8), the weak infinitesimal operator Jax[·] of the process x(t),rt,t≥0, for plant (8) at the point t,x,i is as follows:
(19)Jax[V]=x˙TPix+xTPix˙+xT∑j=1N‍λijPjx=[+(Bi+ΔBi)(Kix+uic+eμ)]Aix+Adix(t-τ)+(Bi+ΔBi)(Kix+uic+eμ)]TPix+xTPi[+(Bi+ΔBi)(Kix+uic+eμ)]Aix+Adix(t-τ)nl+(Bi+ΔBi)(Kix+uic+eμ)]+xT∑j=1N‍λijPjx=xT[AiTPi+PiAi+Pi(Bi+ΔBi)Kinl+KiT(Bi+ΔBi)TPi+Q]x+xT(t-τ)AdiTPix+xTPiAdix(t-τ)-xT(t-τ)Qx(t-τ)+2xTPi(Bi+ΔBi)(uic+eμ)+xT∑j=1N‍λijPjx.
According to Assumption 2, uic in Theorem 8, and Lemma 6, one can obtain that
(20)2xTPi(Bi+ΔBi)(uic+eμ)=2xTPiBiuic+2xTPiΔBi(uic+eμ)+2xTPiBieμ=2xTPiBiuic+2xTPiBiMiΞi(t)Fi(uic+eμ)+2xTPiBieμ≤2xTPiBiuic+2|xTPiBi|1|MiΞi(t)Fi|∞(|uic|∞+μ2)+2|xTPiBi|1μ2=2xTPiBiuic+2|xTPiBiBi|1ψ(|uic|∞+μ2)+|xTPiBi|1μ≤2xTPiBiuic+(1+ψ)|xTPiBi|1μ+2ψ|xTPiBi|1|uic|∞=-(1+ψ)μ1-ψ|xTPiBi|1+(1+ψ)|xTPiBi|1μ+ψ(1+ψ)μ1-ψ|xTPiBi|1=0.
It follows from (19) and (20) that
(21)Jax[V]≤xT[AiTPi+PiAi+Pi(Bi+ΔBi)Kinl+KiT(Bi+ΔBi)TPi+Q]x+xT(t-τ)AdiTPix+xTPiAdix(t-τ)-xT(t-τ)Qx(t-τ)+xT∑j=1N‍λijPjx.
Since ΔBi=BiMiΞi(t)Fi, we introduce
(22)Φi≜xT[AiTPi+PiAi+KiTBiTPi+PiBiKinl+PiBiMiΞi(t)FiKi+KiTFiTΞiT(t)MiTBiTPi+Q]x+xT(t-τ)AdiTPix+xTPiAdix(t-τ)-xT(t-τ)Qx(t-τ).
Then, the above inequality can be rewritten as
(23)Jax[V]≤Φi+xT[∑j=1N‍λijPj]x.
Three cases should be considered.
Case
I
(
i
∉
U
k
i
). In this case, note that ∑j∈Uuki,j≠iπij=-πii-∑j∈Ukiπij and πij≥0,j∈Uuki,j≠i; then from (23), we have
(24)Jax[V]≤Φi+xT[∑j∈Uki‍πijPj+πiiPi+∑j∈Uuki,j≠i‍πijPi]x=Φi+xT[∑j∈Uki‍πijPj+πiiPinll+(-πii-∑j∈Uki‍πij)Pi]x=Φi+xT[∑j∈Uki‍πij(Pj-Pi)]x=Φi+xT[∑j∈Uki‍(π^ij+Δij)(Pj-Pi)]x=Φi+xT[∑j∈Uki‍π^ij(Pj-Pi)+∑j∈Uki‍Δij(Pj-Pi)]x.
There, note that πij=π^ij+Δij for j∈Uki.
On the other hand, in view of Lemma7, we have
(25)∑j∈Uki‍Δij(Pj-Pi)=∑j∈Uki‍[12Δij(Pj-Pi)+12Δij(Pj-Pi)]≤∑j∈Uki‍[(12Δij)2Tij+(Pj-Pi)Tij-1(Pj-Pi)]≤∑j∈Uki‍[δij24Tij+(Pj-Pi)Tij-1(Pj-Pi)].
From (24) and (25), we have
(26)Jax[V]≤Φi+xT×{∑j∈Uki‍π^ij(Pj-Pi)+∑j∈Uki‍[δij24Tij+(Pj-Pi)Tij-1(Pj-Pi)]}x.
Hence, Jax[V]<0 holds if
(27)H=[A11PiAdi*-Q]<0,
where
(28)A11=AiTPi+PiAi+KiTBiTPi+PiBiKi+PiBiMiΞi(t)FiKi+KiTFiTΞiT(t)MiTBiTPi+Q+∑j∈Uki‍π^ij(Pj-Pi)+∑j∈Uki‍[δij24Tij+(Pj-Pi)Tij-1(Pj-Pi)].
Let
(29)A-11=AiTPi+PiAi+KiTBiTPi+PiBiKi+Q+∑j∈Uki‍π^ij(Pj-Pi)+∑j∈Uki‍[δij24Tij+(Pj-Pi)Tij-1(Pj-Pi)];
then we have
(30)H=[A-11PiAdi*-Q]+[PiBiMi0]Ξi(t)[FiKi0]+[FiKi0]TΞiT(t)[PiBiMi0]T<0,
which is equivalent to (13) by Schur complement.
Case
II
(
i
∈
U
k
i
and
U
u
k
i
≠
Ø
). There must be an l∈Uuki such that P(l)-P(j)≥0, ∀j∈Uuki. We define
(31)Jax[V]≤Φi+xT[∑j∈Uki‍λijP(j)+∑j∈Uuki‍λijP(l)]x=Φi+xT[∑j∈Uki‍πijPj-(∑j∈Uki‍πij)Pl]x=Φi+xT[∑j∈Uki‍πij(Pj-Pl)]x=Φi+xT[∑j∈Uki‍(π^ij+Δij)(Pj-Pl)]x=Φi+xT[∑j∈Uki‍π^ij(Pj-Pl)+∑j∈Uki‍Δij(Pj-Pl)]x.
By using Lemma 7 again, we have
(32)∑j∈Uki‍Δij(Pj-Pl)=∑j∈Uki‍[12Δij(Pj-Pl)+12Δij(Pj-Pl)]≤∑j∈Uki‍[(12Δij)2Vijl+(Pj-Pl)Vijl-1(Pj-Pl)]≤∑j∈Uki‍[δij24Vijl+(Pj-Pl)Vijl-1(Pj-Pl)].
From (31) and (32), we have
(33)Jax[V]≤Φi+xT×{∑j∈Uki‍π^ij(Pj-Pl)+∑j∈Uki‍[δij24Vijl+(Pj-Pl)Vijl-1(Pj-Pl)]}x.
Hence, Jax[V]<0 holds if
(34)A=[A11PiAdi*-Q]<0,
where
(35)A11=AiTPi+PiAi+KiTBiTPi+PiBiKi+PiBiMiΞi(t)FiKi+KiTFiTΞiT(t)MiTBiTPi+Q+∑j∈Uki‍π^ij(Pj-Pl)+∑j∈Uki‍[δij24Vijl+(Pj-Pl)Vijl-1(Pj-Pl)].
Let
(36)A-11=AiTPi+PiAi+KiTBiTPi+PiBiKi+Q+∑j∈Uki‍π^ij(Pj-Pl)+∑j∈Uki‍[δij24Vijl+(Pj-Pl)Vijl-1(Pj-Pl)];
then we have
(37)A=[A-11PiAdi*-Q]+[PiBiMi0]Ξi(t)[FiKi0]+[FiKi0]TΞiT(t)[PiBiMi0]T<0,
which is equivalent to (14) by Schur complement.
Case
III
(
i
∈
U
k
i
and
U
u
k
i
=
Ø
)
. In this case,
(38)Jax[V]≤Φi+xT[∑j=1,j≠iN‍λijP(j)+λiiP(i)]x=Φi+xT[∑j∈Uki‍πijPj-(∑j∈Uki‍πij)Pi]x=Φi+xT[∑j∈Uki‍πij(Pj-Pi)]x=Φi+xT[∑j∈Uki‍(π^ij+Δij)(Pj-Pi)]x=Φi+xT[∑j∈Uki‍π^ij(Pj-Pi)+∑j∈Uki‍Δij(Pj-Pi)]x.
In view of Lemma 7, we have
(39)∑j=1,j≠is‍Δij(Pj-Pi)=∑j=1,j≠is‍[12Δij(Pj-Pi)+12Δij(Pj-Pi)]≤∑j=1,j≠is‍[(12Δij)2Wij+(Pj-Pi)Wij-1(Pj-Pi)]≤∑j=1,j≠is‍[δij24Wij+(Pj-Pi)Wij-1(Pj-Pi)].
From (38) and (39), we have
(40)Jax[V]≤Φi+xT×{∑j=1,j≠is‍π^ij(Pj-Pi)nll+∑j=1,j≠is‍[δij24Wij+(Pj-Pi)nlllll×Wij-1(Pj-Pi)[δij24Wij]{∑j=1,j≠is‍π^ij(Pj-Pi)}x.
Hence, Jax[V]<0 holds if
(41)R=[R11PiAdi*-Q]<0,
where
(42)R11=AiTPi+PiAi+KiTBiTPi+PiBiKi+PiBiMiΞi(t)FiKi+KiTFiTΞiT(t)MiTBiTPi+Q+∑j∈Uki‍π^ij(Pj-Pi)+∑j∈Uki‍[δij24Wij+(Pj-Pi)Wij-1(Pj-Pi)].
Let
(43)R-11=AiTPi+PiAi+KiTBiTPi+PiBiKi+Q+∑j∈Uki‍π^ij(Pj-Pi)+∑j∈Uki‍[δij24Wij+(Pj-Pi)Wij-1(Pj-Pi)];
then we have
(44)R=[R-11PiAdi*-Q]+[PiBiMi0]Ξi(t)[FiKi0]+[FiKi0]TΞiT(t)[PiBiMi0]T<0,
which is equivalent to (15) by Schur complement. The proof is completed.Remark 9.
As mentioned in Section2, if, Uuki=Ø∀i∈S, then GUTR matrix reduces to BUTR one. Similarly, if, δij=0∀j∈Uki,∀i∈S, GUTR matrix reduces to PUTR one. Therefore, Theorem 8 can also be applicable to the MJSs with BUTRs or PUTRs. Because the BUTR methods require the estimate of every rate to be known, such methods cannot be applied to the GUTR model. By replacing the uncertain TRs with unknown ones, the generally uncertain TR matrix can become partly known TR matrix, so that the PUTR methods can be applied to GUTR model. However, such methods are inevitably conservative for GUTRs since the information of the TRs’ estimates cannot be utilized.


## 4. Conclusions

The stability problems for a class of Markovian jump linear systems with generally uncertain transition rates are investigated in this paper. The considered systems are more general than the systems with bounded uncertain transition rates or partly unknown transition rates, which can be viewed as two special cases of the systems we tackled here. The LMI-based stochastic stability condition for the underlying systems is derived. There are some possible directions to extend the proposed model and method. Tracking control and fault-tolerant control are two important research areas due to their wide application in the practical systems [40–46]. However, for the tracking control and fault-tolerant control, no research has focused on the case of general uncertain transition rates. Therefore, it is worth further extending the proposed method to deal with these problems.



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*Source: 961925-2014-05-26.xml*