This paper focuses on the stability issue of discrete-time networked control systems with random Markovian delays and uncertain transition probabilities, wherein the random
time delays exist in the sensor-to-controller and controller-to-actuator. The resulting closed-loop system is modeled as a discrete-time Markovian delays system governed by two Markov chains. Using Lyapunov stability theory, a result is established on the Markovian structure and ensured that the closed-loop system is stochastically stable. A simulation example illustrates the validity and feasibility of the results.
1. Introduction
Networked control systems (NCS) find many successful applications in power grids, manufacturing plants, vehicles, aircrafts, spacecrafts, remote surgery, and so on [1]. Compared with the traditional control systems, the use of the communication networks brings many advantages such as low cost, reduced weight, and simple installation and maintenance, as well as high efficiency, flexibility, and reliability. However, inserting communication networks into feedback control loops has also resulted in several interesting and challenging issues, such as packet dropouts [2], time delays [3–10], quantization [11], time-varying transmission intervals [12], distributed synchronization [13], or some of the constraints considered simultaneously [14–17], which make the analysis and design of NCS complex. These imperfections block the way of harvesting reliable NCS by implementing existing control techniques [18]. To overcome these drawbacks, significant attention has been paid to the NCS research ranging from system identification and stability analysis to controller and filter designs. See the survey papers [1, 19, 20] and the references therein.
The network-induced time delays are known to be the major challenges in NCS, which may be potential causes for the deteriorating performance or instability of NCS. Consequently, numerous works have been conducted on the time-delay issue in the past years. For example, in [21], the mixed H2/H∞ control issue of NCS with random time delays has been investigated based on Markovian jump linear systems method. In [9], the stability problem of NCS with uncertain time-varying delays has been investigated. The stability and stabilization of NCS with random time delay usually use Markovian jump linear systems (MJLS) approach, and, recently, many significant achievements have been obtained for MJLS in [22–26]. However, most of the approaches for NCS based on Markovian jump systems framework assumed that the Markovian transition probabilities are known a priori, which severely limit the utility of the Markov model. Furthermore, such assumption may not hold true especially in the case where networked control is applied to the remote plants. Recently, the H∞ filter problem for a class of uncertain Markovian jumping systems with bounded transition probabilities has been investigated in [27], but the well-established results cannot be directly used to NCS. To the best of the authors’ knowledge, up to now, very limited efforts have been devoted to studying the system with uncertain transition probability matrices for NCS, which motivates our investigation.
In this paper, we address the analysis and design of NCS with random time delays modeled by Markov chains in forward sensor-to-controller (S-C) and feedback controller-to-actuator (C-A) communication links and with the uncertain transition probability matrices. The main contributions of this paper are highlighted as follows. (i) A model is proposed for NCS with random Markovian delays and uncertain transition probability matrices. (ii) The system modeled will be more generalized and avoid the ideal assumption that the transition probabilities are known a priori. (iii) New criteria for stability are obtained based on a Lyapunov approach. Finally, a numerical example is provided to demonstrate the effectiveness of the proposed control scheme for NCS with random time delays and uncertain transition probability matrices.
The remainder of this paper is organized as follows. A model with Markovian delays and uncertain transition probabilities is obtained in Section 2. The main results are obtained based on a Lyapunov approach and the linear matrix inequalities technique in Section 3. Section 4 presents the simulation results. Finally, the conclusions are provided in Section 5.
Notations. Matrices are assumed to have appropriate dimensions. ℝn and ℝn×m denote the n-dimensional Euclidean space and the set of all n×m real matrices, respectively. The notations A>0 (A<0) indicate that A is a real symmetric positive (negative) definite matrix. I and 0 denote the identity matrix and the zero matrix with appropriate dimensions, respectively. Superscripts “T” and “-1” stand for the matrix transposition and the matrix inverse, respectively. 𝔼[·] stands for the mathematical expectation and diag{A,B} stands for a block-diagonal matrix of A and B. I and 0 denote the identity matrix and zero matrix with appropriate dimensions, respectively. sym{A} denotes the expression A+AT, and * means symmetric terms in symmetric entries.
2. NCS Model
The framework of networked control systems is depicted in Figure 1. The plant, sensor, controller, and actuator are spatially distributed and closed through a network. Random time delays exist in both of S-C and C-A.
Diagram of a NCS with time delays.
The plant is described by the following discrete-time linear time-invariant plant model:
(1)x(k+1)=Ax(k)+Bu(k),
where x(k)∈ℝn is the system state vector and u(k)∈ℝm is the control input. A and B are known real constant matrices with appropriate dimensions.
For this system, we will consider a state feedback controller as follows:
(2)u¯(k)=Kx¯(k),
where K is the state feedback controller gain.
Random S-C and C-A time delays are d(k) and τ(k), respectively. d(k) and τ(k) are assumed be bounded; that is, 0≤d_≤d(k)≤d¯, 0≤τ_≤τ(k)≤τ¯, where d_=min{d(k)}, d¯=max{d(k)}, τ_=min{τ(k)}, and τ¯=max{τ(k)}. One way to model delays d(k) and τ(k) is by using the finite-state Markov chains presented in [21]. The main advantages of the Markov model considering the dependence between delays are that the current time delays in real networks delays are frequently related to the previous delays. In this paper d(k) and τ(k) are modeled as two homogeneous Markov chains.
By substituting controller (2) to plant (1), we obtain a closed-loop system as follows:
(3)x(k+1)=Ax(k)+BKx(k-ηk),
where ηk=τ(k)+d(k-τ(k)).
In system (3), {d(k),k∈ℤ} and {τ(k),k∈ℤ} are two finite state discrete-time homogeneous Markov chains with values in the finite sets S1={0,…,s1} and S2={0,…,s2} with the uncertain transition probability matrices π^ and λ^. π^={π^ij} and λ^={λ^nm} denote the uncertain transition probability matrices of Markov chain d(k) and τ(k), respectively, with probabilities π^ij and λ^nm, which are defined by
(4)Pr{d(k+1)=j∣d(k)=i}=π^ij,Pr{τ(k+1)=n∣τ(k)=m}=λ^mn,
where Pr{d^0=i}=π^i≥0, Pr{τ^0=m}=λ^m≥0 and ∑j=0,j≠is1π^ij=1-π^ii, ∑n=0,n≠ms2λ^mn=1-λ^mm for all {i,j}∈S1 and {m,n}∈S2. The transition probability matrices π^≜[π^ij] and π^≜[λ^mn] are unknown a priori but belong to the following bounded compact set:
(5)π^=π+Δπ,λ^=λ+Δλ,
where π≜[πij] (i,j∈S1) and λ≜[λmn] (m,n∈S2) are known constant matrices. Δπ≜[Δπij] (i,j∈S1) and Δλ≜[Δλij] (i,j∈S2) denote the uncertainty in the transition probability matrices, where Δπ and Δλ satisfy
(6)∑j=0,j≠is1Δπij=-Δπij(i,j∈S1),∑n=0,n≠ms2Δλmn=-Δλmn(m,n∈S2),
where 0≤|Δπij|≤ɛij, 0≤|Δλmn|≤ɛmn, and ɛij and ɛmn are the known small scalar for all (i,j∈S1) and (m,n∈S2), respectively.
Remark 1.
Closed-loop system (3) is a linear system with the Markovian delays d(k) and τ(k), which describe the behavior of the S-C and C-A random time delays, and with the uncertain transition probabilities.
Remark 2.
The uncertain transition probabilities π^ and λ^ contain the certain terms π and λ, and the uncertain terms Δπ and Δλ, respectively. The uncertain terms Δπ and Δλ are bounded, and the sums of the elements in each row are zeros.
3. Stability Analysis and Controller Design
By applying a Lyapunov approach and a linear matrix inequality technique, this section provides sufficient conditions for the stochastic stability and the synthesis of state feedback controller design of the system (3).
Definition 3 (see [21]).
The closed-loop system (3) is said to be stochastically stable if, for every finite x0=x(0), initial mode d0=d(0)∈S1 and τ0=τ(0)∈S2, there exists a finite 𝒲>0 such that
(7)𝔼{∑k=0∞∥x(k)∥2∣x0,d0,τ0}<x0T𝒲x0.
Theorem 4.
For the system (3), random but bounded scalars d(k)∈[d_d¯] and τ(k)∈[τ_τ¯]. If, for each mode {i,j}∈S1 and {m,n}∈S2 and matrices Pi,m>0, Q1>0, Q2>0, Q3>0, R1>0, and R2>0, ℳs=[ℳ1sℳ2sℳ3s] and K exist that satisfy the following matrix inequalities:
(8)Γi,m=[-R1-10Ξ1*-R2-1Ξ2**Ξ3]<0,
where
(9)Ξ1=[t¯(A-I)0t¯BK00],Ξ2=[(t¯-t_)(A-I)0(t¯-t_)BK00],Ξ3=✠+symℳ¯TΩ,✠=[✠11P¯i,mR100*✠22000**✠33R1+R2R2***✠440****✠55],✠11=P¯i,m-Pi,m-R1+Q1+Q2+(t¯-t_+1)Q3,✠22=P¯i,m,✠33=-Q3-2R1-2R2,✠44=-Q1-R1-R2,✠55=-Q2-R2,ℳ¯=[ℳ1sℳ2sℳ3s00],Ω=[A-I-IBK00],P¯i,m=∑j=0s1∑n=0s2π^ijλ^mnPj,n,t¯=d¯+τ¯,t_=d_+τ_,
and π^ij and λ^mn are defined in (4) and (5).
Then the closed-loop system (3) is stochastically stable.
Proof.
For the closed-loop system (3), the stochastic Lyapunov functional candidate is constructed as follows:
(10)V(k)=V1(k)+V2(k)+V3(k)+V4(k),
with
(11)V1(k)=x(k)TP(d(k),τ(k))x(k),V2(k)=∑l=k-t¯k-1x(l)TQ1x(l)+∑l=k-t_k-1x(l)TQ2x(l),V3(k)=∑θ=-t¯+2-t_+1∑l=k+θ-1k-1x(l)TQ3x(l)+∑l=k-ηkk-1x(l)TQ3x(l),V4(k)=∑θ=-t¯+10∑l=k+θ-1k-1t¯δ(l)TR1δ(l)+∑θ=-t¯+1-t_∑l=k+θ-1k-1(t¯-t_)δ(l)TR2δ(l),
where P(d(k), τ(k))>0, Q1>0, Q2>0, Q3>0, R1>0, and R2>0.
Let δ(l)=x(l+1)-x(l), noting that x(k+1)=Ax(k)+BKx(k-ηk). Then 0=(A-I)x(k)-δ(k)+BKx(k-ηk). For simplicity, we will use the following notations: ζ(k)=[x(k)Tδ(k)Tx(k-ηk)T]T. Then, for any weighting matrices ℳs with compatible dimensions (and let ℳs=[ℳ1sℳ2sℳ3s]), we have 2ζ(k)TℳsT((A-I)x(k)-δ(k)+BKx(k-ηk))=0. Along the trajectory of the solution of the closed-loop system (3), we obtain
(12)𝔼[ΔV1(k)]=𝔼{[x(k)+δ(k)]TP¯i,m[x(k)+δ(k)]}-x(k)TPi,mx(k)+2ζ(k)TℳsT×((A-I)x(k)+BK(k-ηk)-δ(k)),(13)𝔼[ΔV2(k)]=x(k)T(Q1+Q2)x(k)-x(k-t¯)TQ1x(k-t¯)-x(k-t_)TQ2x(k-t_),(14)𝔼[ΔV3(k)]≤(t¯-t_+1)x(k)TQ3x(k)-x(k-ηk)TQ3x(k-ηk),(15)𝔼[ΔV4(k)]=t¯2δ(k)TR1δ(k)+(t¯-t_)2δ(k)TR2δ(k)-∑l=k-t¯k-1t¯δ(l)TR1δ(l)-∑l=k-t¯k-t_-1(t¯-t_)δ(l)TR2δ(l).
By Jensen’s inequality, we can obtain
(16)-∑l=k-t¯k-1t¯δ(l)TR1δ(l)=-(∑l=k-t¯k-ηk-1+∑l=k-ηkk-1)(t¯-ηk+ηk)δ(l)TR1δ(l)≤-((t¯-ηk)∑l=k-t¯k-ηk-1δT(l)R1δ(l)+ηk∑l=k-ηkk-1δT(l)R1δ(l))≤-((∑l=k-t¯k-ηk-1δ(l))TR1(∑l=k-t¯k-ηk-1δ(l))ddddd+(∑l=k-ηkk-1δ(l))TR1(∑l=k-ηkk-1δ(l)))=-(x(k-ηk)-x(k-t¯))TR1(x(k-ηk)-x(k-t¯))-(x(k)-x(k-ηk))TR1(x(k)-x(k-ηk)).
Similarly, we have
(17)-∑l=k-t¯k-t_-1(t¯-t_)δ(l)TR2δ(l)≤-(∑l=k-t¯k-ηk-1δ(l))TR2(∑l=k-t¯k-ηk-1δ(l))+(∑l=k-ηkk-t_-1δ(l))TR2(∑l=k-ηkk-t_-1δ(l))=-(x(k-ηk)-x(k-t¯))TR2(x(k-ηk)-x(k-t¯))-(x(k-t_)-x(k-ηk))TR2(x(k-t_)-x(k-ηk)).
By substituting (16) and (17) to (15) and then combining (12), (13), and (14), we have
(18)𝔼[ΔV]≤ξ(k)T{Ξ3+[t¯(A-I)0t¯BK00]Tsssssssssss×R1[t¯(A-I)0t¯BK00]dddddddd+[(t¯-t_)(A-I)0(t¯-t_)BK00]Tdddddddd×R2[(t¯-t_)(A-I)0(t¯-t_)BK00][t¯(A-I)0t¯BK00]T}ξ(k)=ξ(k)TΓi,mξ(k),
where ξ(k)=[ζ(k)Tx(k-t¯)Tx(k-t_)T]T. By using the Schur complement, (8) guarantees that Γi,m<0. Therefore,
(19)𝔼[ΔV]≤-λmin(-Γi,m)ξ(k)Tξ(k)≤-ηx(k)Tx(k),
where λmin(-Γi,m) denotes the minimal eigenvalue of -Γi,m and η=inf{λmin(-Γi,m)}. From (19), it follows that, for any t>0,
(20)𝔼[V(k+1)]-𝔼[V(0)]≤-η∑k=0t𝔼[x(k)Tx(k)].
Furthermore
(21)∑k=0t𝔼[x(k)Tx(k)]≤1η𝔼[V(0)].
By taking t→∞ as the limit, we obtain
(22)∑k=0∞𝔼[x(k)Tx(k)]≤1η𝔼[V(0)]=1ηx0TP(d0,τ0)x0<∞.
According to Definition 3, the closed-loop system (3) exhibits stochastic stability for all uncertain transition probability matrices.
Theorem 4 gives a sufficient condition for the stochastic stability of the system (3). However, it should be noted that the controller gain K cannot be obtained according to the condition in (8) because of the nonlinear terms R1-1 and R2-1 and the uncertain terms Δπ and Δλ. To handle this problem, the equivalent LMI conditions are given as follows.
Before proceeding further, we provide the following lemma that will play a significant role in processing the uncertainty terms Δπ and Δλ of uncertain transition probability matrices π^ and λ^.
Lemma 5 (see [28]).
For any vectors of a,b∈ℛn and positive matrix Z∈ℛnZ,nZ, the following holds:
(23)2aTb≤aTZa+bTZ-1b.
Theorem 6.
For the system (3), the random but bounded scalars d(k)∈[d_d¯] and τ(k)∈[τ_τ¯]. If, for each mode {i,j}∈S1 and {m,n}∈S2, the tuning parameters φ1>0 and φ2>0, the scalars ɛij>0 and ɛnm>0, and matrices P^i,m>0, X>0, Q^1>0, Q^2>0, Q^3>0, R^1>0, R^2>0, Zi>0, Zm>0, and Zj,n>0, and Y exist such that
(24)[Θ1100Θ14*Θ220Θ24**Θ33Θ34***Θ44]<0,
where
(25)Θ11=-
sym
{X}+R^1,Θ22=-
sym
{X}+R^2,Θ33=diag{-Z^m,-Z^i,-Z^j,n},Θ14=[t¯(AX-X)0t¯BY00],Θ24=[(t¯-t_)(AX-X)0(t¯-t_)BY00],Θ34=[Ψ1Ψ1000Ψ2Ψ2000Ψ3Ψ3000],Z^m=diag{Zm,…,Zm}︸s1(s2-1),Z^i=diag{Zi,…,Zi}︸(s1-1)s2,Z^j,n=diag{Zj,n,…,Zj,n}︸s1s2,Ψ1=[πi1ΔP^1,m⋯πis1ΔP^s1,m],Ψ2=[λm1ΔP^i,1⋯λms2ΔP^i,s2],Ψ3=[P^1,1⋯P^s1,s2],ΔP^j,m=[P^j,1-P^j,m⋯P^j,m-1-P^j,mP^j,m+1-P^j,m⋯P^j,s2-P^j,m],ΔP^i,n=[P^1,n-P^j,n⋯P^i-1,n-P^j,nP^i+1,n-P^j,n⋯P^s1,n-P^j,n],Θ44=
sym
{ΠTΩ^}+[✠^11ϑ1⊛ϑ2R^100*ϑ12⊛000**ϑ22✠^33ϑ2(R^1+R^2)ϑ2R^2***✠^440****✠^55],✠^11=⊛-P^i,m-R^1+Q^1+Q^2+(t¯-t_+1)Q^3,✠^33=-Q^3-2R^1-2R^2,✠^44=-Q^1-R^1-R^2,✠^55=-Q^2-R^2,⊛=∑j=0s1∑n=0s2(πijλmnP^j,n+(ɛijɛmn)24Zj,n)+∑j=0s1∑n=0,n≠ms2πijɛmn24Zm+∑j=0,j≠is1∑n=0s2λmnɛij24Zi,Π=[III00],Ω^=[AX-X-ϑ1Xϑ2BY00].
Then the closed-loop system (3) is stochastically stable and the controller u¯(k)=Kx¯(k)=YX-1x¯(k) is a state feedback controller of the system (3).
Proof.
Let Δ¯i=diag{I,I,Xi,m,ϑ1Xi,m,ϑ2Xi,m,Xi,m,Xi,m}, ℳ1s-1=Xi,m, ℳ2s-1=ϑ1Xi,m, and ℳ3s-1=ϑ2Xi,m, where ϑ1>0 and ϑ2>0 are known tuning parameters. We restrict Xi,m to be the same for all {i,m} (namely, Xi,m=X) and give the notations as
(26)P^i,m=XTPi,mX,P¯^i,m=XTP¯i,mX,R^1=XTR1X,R^2=XTR2X,Q^1=XTQ2X,Q^2=XTQ2X,Q^3=XTQ3X.
Pre- and postmultiplying Δ¯iT and Δ¯i to (8), respectively, we have
(27)[-XR^1-1XT0Ξ^1*-XR^2-1XTΞ^2**Ξ^3]<0,
where
(28)Ξ^1=[t¯(AX-X)0t¯BY00],Ξ^2=[(t¯-t_)(AX-X)0(t¯-t_)BY00],Ξ^3=✠~+symΠTΩ^,✠~=[✠~11ϑ1P¯^i,mϑ2R^100*ϑ12P¯^i,m000**ϑ22✠^33ϑ2(R^1+R^2)ϑ2R^2***✠^440****✠^55],✠~11=P¯^i,m-P^i,m-R^1+Q^1+Q^2+(t¯-t_+1)Q^3,P¯i,m=∑j=0s1∑n=0s2π^ijλ^mnP^j,n,
where t¯ and t_ are defined in Theorem 4 and ✠^33, ✠^44, and ✠^55 are defined in Theorem 6.
According to the assumption on uncertain transition probabilities π^ and λ^ and the fact that ∑j=0,j≠is1Δπij=-Δπii and ∑n=0,n≠ms2Δλmn=-Δλmm, one has
(29)P¯^i,m=∑j=0s1∑n=0s2π^ijλ^mnP^j,n=∑j=0s1∑n=0s2(πij+Δπij)(λmn+Δλmn)P^j,n=∑j=0s1∑n=0s2πijλmnP^j,n+∑j=0s1∑n=0s2πijΔλmnP^j,n+∑j=0s1∑n=0s2ΔπijλmnP^j,n+∑j=0s1∑n=0s2ΔπijΔλmnP^j,n.
Note that
(30)∑j=0s1∑n=0s2πijΔλmnP^j,n=∑j=0s1πij(∑n=0,n≠ms2ΔλmnP^j,n+ΔλmmP^j,m)=∑j=0s1πij∑n=0,n≠ms2Δλmn(P^j,n-P^j,m).
By Lemma 5 and the fact that |Δλmn|≤ɛmn, we have
(31)∑j=0s1∑n=0s2πijΔλmnP^j,n≤∑j=0s1πij∑n=0,n≠ms2(14ɛmn2Zm+(P^j,n-P^j,m)Tddddddddddddddd×Zm-1(P^j,n-P^j,m)14).
Similarly, we have
(32)∑j=0s1∑n=0s2ΔπijλmnP^j,n≤∑j=0,j≠is1∑n=0s2λmnsssssssssssss×(14ɛij2Zi+(P^j,n-P^i,n)TZi-1(P^j,n-P^i,n)).∑j=0s1∑n=0s2ΔπijΔλmnP^j,n≤∑j=0s1∑n=0s2(14ɛij2ɛmn2Zj,n+P^j,nTZj,n-1P^j,n).
Note that, for any matrix X, we have XW-1XT≥sym{X}-W for W=R^1 and W=R^2. Combining (29), (31), and (32) and by the Schur complement, (24) can be yielded easily from (27); this completes the proof of Theorem 6.
4. Numerical Example
In this section, we illustrate our results through an example. We apply the results in Theorem 6 to a simple inverted pendulum system [5] shown in Figure 2, which is a two-order unstable system. The state variables are [φφ˙]T, where φ is the angular position of the pendulum. The parameters used are m=0.1 kg and L=1 m, without friction surfaces. The sampling time is Ts=0.05 s. The plant matrices are given by
(33)A=[1.01230.05020.49201.0123],B=[0.01250.5020].
A simple inverted pendulum.
We assume that the stochastic Markovian jumping S-C delay d(k)∈{0,1} and C-A delay τ(k)∈{0,1,2} and their uncertain transition probability matrices are given as follows:
(34)π=[0.40.60.70.3],λ=[0.40.30.30.20.50.30.40.20.4],Δπ=[0.02-0.02-0.010.01],Δλ=[0.03-0.030-0.020.010.01-0.030.020.01].
The eigenvalues of A are 1.1695 and 0.8551. Therefore, the discrete-time model is unstable.
Figures 3 and 4 show part of the simulation of the Markov chains mode. The initial conditions are as follows: d(0)=0, τ(0)=0, and x(0)=[0.1-0.1]T. By Theorem 6, when ɛij=0.02, ɛmn=0.03, ϑ1=0.09, and ϑ2=12, we can obtain the gain matrix K of state feedback controller (2) which is constructed as
(35)K=YX-1=[-0.1046-0.1177][0.1636-0.39230.39231.4211]-1=[-2.4757-0.7662].
Values of the S-C delay d(k).
Values of the C-A delay τ(k).
The state trajectories of the system (3) are shown in Figure 5, where two curves represent state trajectories under the controller gains K. Figure 5 also indicates that the system (3) is stochastically stable.
State trajectories under K.
Remark 7.
In this example, the uncertain transition probabilities are given as a discrete probability distribution function. When the uncertain transition probability is given as a continuous probability distribution function, we can use the H2 norm of the continuous probability distribution function as the upper bound to simulation.
5. Conclusions
The state feedback stabilization problem for a class of NCS with the S-C and C-A random time delays is investigated in this paper. The resulting closed-loop NCS is modeled as a linear system with uncertain Markovian transition probabilities. New sufficient conditions on stochastic stability and stabilization are obtained by Lyapunov stability theory and linear matrix inequalities method. An example is presented to illustrate the effectiveness of the approach. Although only the time-delay issue for NCS is addressed in this paper, the method can be extended to the NCS with the random packet dropouts, time delays, and packet dropouts and to the MJLS with the uncertain Markovian transition probabilities.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
This work was supported partly by the National Natural Science Foundation of China (Grant nos. 61174070 and 61104107), the Natural Science Foundation of Guangdong Province (Grant no. S2013040016183), the Foundation of Key Laboratory of Autonomous Systems and Networked Control, Ministry of Education, China, and the Natural Science Foundation of Shenzhen University (Grant no. 201207).
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