# Finite-Time Simultaneous Stabilization for Stochastic Port-Controlled Hamiltonian Systems over Delayed and Fading Channels

Authors: Yaping Tang; Weiwei Sun; Dongqing Liu
Journal: Complexity
 (2020)

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## Abstract

In this paper, a finite-time simultaneous stabilization problem is investigated for a set of stochastic port-controlled Hamiltonian (PCH) systems over delayed and fading noisy channels. The feedback control signals transmitted via a communication network suffer from both constant transmission delay and fading channels which are modeled as a time-varying stochastic model. First, on the basis of dissipative Hamiltonian structural properties, two stochastic PCH systems are combined to form an augmented system by a single output feedback controller and then sufficient conditions are developed for the semiglobally finite-time simultaneous stability in probability (SGFSSP) of the resulting closed-loop systems. The case of multiple stochastic PCH systems is also considered and a new control scheme is proposed for the systems to save costs and achieve computational simplification. Finally, an example is provided to verify the feasibility of the proposed simultaneous stabilization method.


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## Body

## 1. Introduction

Port-controlled Hamiltonian (PCH) systems are known as an important class of nonlinear systems ([1, 2]). Compared to the general nonlinear systems, an excellent benefit of PCH systems is that the Hamiltonian function in the systems can be used as a Lyapunov function candidate in stability analysis (see, for instance, [3–5]). Thanks to the special system structure and clear physical meaning, applications of PCH systems can be found in a variety of engineering systems including power systems, robotic systems, and irreversible thermodynamic systems ([6–10]). In recent years, stabilization as well as simultaneous stabilization problem has been extensively studied for PCH systems ([11–14]). In terms of PCH systems with disturbances, the above stabilization problem has been resolved in [11, 13]. Taking actuator saturation into account, the study in [14] has proposed an adaptive control strategy to simultaneously stabilize PCH systems with parameter uncertainties.On the other hand, there usually exist stochastic components and random disturbances in practical control plants, which often result in performance degradation, as well as destabilization of the systems. In the last few decades, many researchers have made efforts to deal with the stabilization problem of stochastic systems ([15, 16]). For example, in [15], output feedback stabilization has been studied using the backstepping approach for Itô-type stochastic systems. As for stochastic PCH systems, the control problem has also captured public attentions ([17–20]). Exploiting an energy-based feedback control scheme, the authors of [17] have raised stochastic feedback stabilization results. In regard to time-varying stochastic PCH systems, the study in [18] has come up with a kind of stochastic generalized canonical transformations approach to stabilize stochastic PCH systems. In addition, the adaptive control topic for nonlinear stochastic Hamiltonian systems has been introduced in [19, 20]. Parameter uncertainty, randomness, and time delay are all considered in above references.In many practical problems, the fast convergence within a fixed finite time interval plays an important role. Finite-time stabilization makes closed-loop systems enjoy fast convergence. In addition, disturbance rejection properties and better robustness both can be reflected in the finite-time stabilization. Thus, many investigations about finite-time stabilization controller design have been carried out ([21–29]). For stochastic nonlinear systems which are written as Itô differential form, [23] has proposed a method to solve the finite-time stabilization problem. The finite-time stabilization of the Hamiltonian systems has been studied in [21, 25, 27, 28]. For instance, the finite-time feedback control manner is developed in [21] to deal with finite-time stabilization problem for PCH systems with nonvanishing disturbances.Generally speaking, the phenomenon of fading channels as well as network-induced delay is very likely to occur in the networked control system, which can lead to various distortions and information constraints. By now, a considerable number of researches have been done for continuous and discrete systems over network-induced phenomenon ([30–38]). Under memoryless fading channels environment, the study in [33] has illustrated state feedback stabilization problem for linear continuous systems. This problem has been solved by realizing the balance between the demand of communication resource and the supply of that. Different from [33], an output feedback control scheme has been introduced in [35] to achieve mean-square stabilization over multiplicative fading channels for discrete systems. For continuous linear network control systems over delayed and fading channels, a necessary and sufficient condition has been established by algebraic Riccati equation method in [37], and mean-square stabilization problem has also been resolved. Recently, the problem of H∞ filtering design has been solved in [36] for a class of nonlinear Hamiltonian systems considering fading channel and saturation.Summarizing the above discussion, in this paper, we try to solve the finite-time simultaneous stabilization problem of stochastic PCH systems over delayed and fading channels and propose some new results that serve for the design of feedback controllers. The fading noisy channels modeled as multiple independent and memoryless forms exist between the controller and the plant. We try to design feedback controller to render closed-loop systems semiglobally finite-time simultaneous stable in probability (SGFSSP). To begin with, two stochastic PCH systems are considered. We will design a single output feedback controller which contributes to SGFSSP for the systems. Utilizing the structural properties of dissipative Hamiltonian systems, the two stochastic PCH systems form an augmented stochastic PCH system, which makes the problem solved easily. Through the Lyapunov function method and Itô differential formula, the closed-loop systems will be SGFSSP. Besides, we will extend our approach to the case of multiple stochastic PCH systems over delayed and fading channels. A feedback control strategy is proposed. At last, the feasibility of the above method is illustrated by the simulation.The contributions of this paper mainly lie in the following two aspects: (1) taking network-induced delay and fading noisy channels environment into consideration, a new single output feedback controller design method is raised to deal with the SGFSSP problem for stochastic PCH systems. In this way, the controller implementation costs can be greatly reduced, and the computational simplification of control can be achieved. (2) We make an in-depth study of the proposed method by extending the approach to the case of multiple PCH systems. SGFSSP result for multiple PCH systems over delayed and fading channels is given.Notation:ℝn denotes the n-dimensional real column vectors and ℝn×m is the real matrices with dimensions n×m. A real-valued function fx∈C2 represents that fx is a continuously twice differentiable function. · represents the 2-norm. diaga1,a2,…,am represents diagonal matrix with a1,a2,…,am as its diagonal elements. We denote λmin· as the smallest eigenvalue operator, E· as the expectation operator, and Cov· as the covariance operator, respectively. For the probability space Ω,ℱ,P, Ω denotes the sample space, ℱ denotes the σ-algebra of the observable random events, and P is the probability measure on Ω.


## 2. Problem Formulation and Preliminaries

Consider the following two stochastic PCH systems:(1)dxt=Jxt−Rxt∇H1xtdt+g1utdt+h1dωt,yt=g1T∇H1xt,(2)dξt=J^ξt−R^ξt∇H2ξtdt+g2utdt+h2dρt,ηt=g2T∇H2ξt,where xt, ξt∈ℝn are the system state vectors, ut∈ℝm is the control input which satisfies E∫0tus2ds<∞, and yt,ηt∈ℝm are the outputs of systems. The signals ωt and ρt are both κ−dimensional independent standard Wiener process defined on probability space Ω,ℱ,P. We assume Edωt=0, Edρt=0, Edωt2=dt, and Edρt2=dt. ∇Hix∈ℝn×1 is the gradient of the Hamilton function Hix:ℝn↦ℝ, which is defined as ∇Hix=∂Hix/∂x, and Hix≥0, Hi0=0, for all t≥0. Jx∈ℝn×n and J^ξ∈ℝn×n are both skew-symmetric structure matrices; Rx∈ℝn×n and R^ξ∈ℝn×n are positive definite strict dissipation matrices; g1, g2, h1, and h2 are known real constant gain matrices. In addition, by setting f1x,u=Jx−Rx∇H1x+g1u, f2ξ,u=J^ξ−R^ξ∇H2ξ+g2u. Suppose that there exist constants kF>0 and kG>0 such that(3)fiθ1,u−fiθ2,u≤kFθ1−θ2,fiθ1,u+hi≤kG1+θ1+u,hold for all θ1,θ2∈ℝn, ut∈ℝm, t≥0, and i=1,2.For generalized PCH systems, it is shown in [13] that the two PCH systems can be simultaneously stabilized by a controller u=−Kyt−ηt over constraint conditions, where K is a gain matrix with appropriate dimension. Unfortunately, when it comes to the stochastic networked control system (NCS), the feedback control signals transmitted via a communication network may suffer from delayed and fading noisy channels.Let us focus on the NCS as depicted in Figure1. Suppose that the control signal ut suffers both constant transmission delay d>0 and signal attenuation in the closed-loop system. The transmission delay d is caused by the message delivery from the controller to the actuator. The transmission of signal ut is accomplished in a form of components through independent parallel channels. Then, the control signal ut arriving at the actuator is modeled by the following multiple independent and memoryless forms:(4)εtut−d+qt,where ut∈ℝm and εtut−d+qt∈ℝm are the input and output of channels, respectively. εt∈ℝm×m represents the multiplicative noise with the following form:(5)εt=diagε1t,ε2t,…,εmt.Figure 1
NCS over delayed and fading noisy channels.Eεit=μi, Covεit,εis=σi2δt−s, μi≠0, σi2 is known power spectral density. qt=q1t,q2t,…,qmtT∈ℝm is an additive white Gaussian process noise with Eqit=0 and known power spectral density ϱi2, i.e., Covqit,qis=ϱi2δt−s, δ denotes the Dirac delta function, i=1,…,m. We make the following assumption for εt.Assumption 1.
(1)
εit and εjt are uncorrelated for i≠j, i.e., Eεit1εjt2=0, ∀t1, t2>0, and i≠j(2)
εt is uncorrelated with ωt and ρtRemark 1.
We consider interference channels noise in the systems and input channels noise. The conditions of Assumption1 avoid the possible occurrence of noise coupling phenomenon.
Denote(6)M≔diagμ1,μ2,…,μm,Q≔diagϱ12,ϱ22,…,ϱm2.
Obviously,M is nonsingular since μi≠0. Without loss of generality, we assume ϱi2=1, i.e., Q=Im∈ℝm×m for simplicity hereinafter, i=1,2,…,m.
Substituting (4) into (1) and (2), we get(7)dxt=Jxt−Rxt∇H1xtdt+g1εtut−ddt+g1dϖt+h1dωt,yt=g1T∇H1xt,(8)dξt=J^ξt−R^ξt∇H2ξtdt+g2εtut−ddt+g2dϖt+h2dρt,ηt=g2T∇H2ξt.Remark 2.
qt is a white noise, which is formally regarded as the derivative of a Brownian motion ϖt (see [39]), i.e., qt=dϖt/dt, so we can further write that qtdt=dϖt.
Before proceeding further, we need to put forward a definition as follows.Definition 1.
The stochastic PCH systems (7) and (8) are said to be semiglobally finite-time simultaneous stable in probability (SGFSSP) if(1)
for any initial valuesX0∈ℝ2n, the solution Xt of systems (7) and (8) exists and is unique, where Xt=xTtξTtT(2)
for everyX0∈ℝ2n\Ω and ϵ>0, the first hitting time TX0,ϵ=inft:Xt;X0∈Ω is finite almost surely; TX0,ε and Ω are called the settling time and the compact set, respectively(3)
for allt≥TX0,ε, the solution Xt of systems (7) and (8) satisfies EXt;X0<ϵLemma 1.
Consider the following Itô form stochastic system:(9)dxt=fxdt+gxdω.Supposefx and gx are locally Lipschitz continuous in x and locally bounded, f0=0, and g0=0. If, for any x0∈ℝn, there exist class-K∞ functions γ1 and γ2, real numbers c>0, 0<ℏ<1, a>0 and a positive definite, function Vx∈C2 such that(10)γ1x≤Vx≤γ2x,(11)EℒVx≤−cVℏx+a,then system (9) is SGFSSP. Furthermore, the compact set Ω is expressed as(12)Ω=xVℏx≤ac1−b,∀0<b<1,and the settling time of system (9) with respect to x0 satisfies(13)T∗=1bc1−ℏV1−ℏx0−ac1−b1−ℏ/ℏ.Lemma 2.
For any real numberzi, i=1,…,n, and any positive real numbers ϵ1, ϵ2 which satisfy 0<ϵ1≤ϵ2, it holds(14)∑i=1nziϵ21/ϵ2≤∑i=1nziϵ11/ϵ1.In Lemma2, if ϵ1=β≥1 and ϵ2=1, then(15)∑i=1nzi1/β≤∑i=1nzi1/β,for any real number zi, i=1,…,n.In this paper, our main goal is to make the two systems (1) and (2) with the delayed and fading noisy channels SGFSSP. More specifically, based on Lemma 1, we have an interest in designing a suitable output feedback controller ut−d such that systems (7) and (8) satisfy (10) and (11). Besides, we extend our results to multiple stochastic PCH systems.For the above purpose, the following assumptions and lemmas are essential in the sequel.Assumption 2.
The Hamilton functionsH1x and H2ξ are given as(16)H1x=∑i=1nxi2α/2α−1,H2ξ=∑i=1nξi2α/2α−1,where α>1 is a real number.Assumption 3.
There exist constantsc1>0 and c2>0 such that(17)c1=inft≥0λminRxt,c2=inft≥0λminR^ξt.Lemma 3.
For any matricesP1, P2∈ℝm×n, it follows that(18)P1TP2+P2TP1≤P1TP1+P2TP2.


## 3. SGFSSP of Two Stochastic PCH Systems and That of Multiple Stochastic PCH Systems

In this section, we will give the analysis result that serves for the SGFSSP of two stochastic PCH systems.Theorem 1.
Consider systems (7) and (8). Assumptions 2 and 3 are satisfied. If there exist matrices K=KT, L1=L1T, L2=L2T, and L3=L3T such that the following matrix inequality(19)Π1=A11A1200A15−g1Kg2T∗−L200A250∗∗A33A34g2Kg1TA36∗∗∗−12L20A46∗∗∗∗−L30∗∗∗∗−L3<0holds, where(20)A11=−2c1In−2g1Kg1T−L1,A12=12L1+L2,A15=g1Kg1T+12L1+L3,A25=−12L2+L3,A33=−2c2In+2g2Kg2T−L1,A34=12L1+L2,A36=−g2Kg2T+12L1+L3,A46=−12L2+L3,then systems (7) and (8) are SGFSSP under the output feedback control law(21)ut−d=−M−1Kyt−d−ηt−d.Proof.
First of all, substituting (21) into (7) and (8), we obtain(22)dxt=Jxt−Rxt∇H1xtdt−g1εtM−1Kg1T∇H1xt−ddt+g1εtM−1Kg2T∇H2ξt−ddt+g1dϖt+h1dωt,(23)dξt=J^ξt−R^ξt∇H2ξtdt−g2εtM−1Kg1T∇H1xt−ddt+g2εtM−1Kg2T∇H2ξt−ddt+g2dϖt+h2dρt.
Applying Newton–Leibnitz formula, we have(24)∇H1xt−∇H1xt−d=∫t−dt∇H1xs′ds,(25)∇H2ξt−∇H2ξt−d=∫t−dt∇H2ξs′ds.
Then, systems (22) and (23) can be rewritten as(26)dxt=Jxt−Rxt∇H1xtdt−g1εtM−1Kg1T∇H1xdt+g1εtM−1Kg2T∇H2ξdt+g1εtM−1Kg1T∫t−dt∇H1xs′dsdt−g1εtM−1Kg2T∫t−dt∇H2ξs′dsdt+g1dϖt+h1dωt,(27)dξt=−g2εtM−1Kg1T∇H1xdt+J^ξt−R^ξt∇H2ξtdt+g2εtM−1Kg2T∇H2ξdt+g2εtM−1Kg1T∫t−dt∇H1xs′dsdt−g2εtM−1Kg2T∫t−dt∇H2ξs′dsdt+g2dϖt+h2dρt.
Defining the vectorsXt=xTtξTtT, ζt=ωTtρTtT, the above equations can be further rewritten into an augmented Itô form stochastic PCH system described as(28)dXt=JXt−RXt∇HXtdt+Gt∫t−dt∇HXs′dsdt+gdϖt+hdζt,where HXt=H1xt+H2ξt, RXt=diagRxt+g1εtM−1Kg1T,R^ξt−g2εtM−1Kg2T,(29)JXt=Jxtg1εtM−1Kg2T−g2εtM−1Kg1TJ^ξt,Gt=g1εtM−1Kg1T−g1εtM−1Kg2Tg2εtM−1Kg1T−g2εtM−1Kg2T,∫t−dt∇HXs′ds=∫t−dt∇H1xs′ds∫t−dt∇H2ξs′ds,g=g1g2,h=h100h2.
Next, choosing the following Lyapunov function candidate:(30)VXt=2HXt,and according to Itô differential formula, we have(31)dVXt=ℒVXtdt+∂VXt∂Xtgdϖt+∂VXt∂Xthdζt,where(32)ℒVXt=2∇THXtJXt−RXt∇HXt+2∇THXtGt∫t−dt∇HXs′ds+trgTHessHXg+trhTHessHXh.
Lettingλ=supt≥0HessH1x+H2ξ2 and based on Lemma 3, we conclude that(33)trgTHessHXg≤12trgTg+12trgTHessHXHessTHXg≤12λ+1trgTg=12λ+1∑j=1m∑i=1ng¯ij2+g^ij2,where g¯ij and g^ij are the components of the matrices g1=g¯ijn×m and g2=g^ijn×m, respectively. Similarly, we have(34)trgTHessHXh≤12trhTh+12trhTHessHXHessTHXh≤12λ+1trhTh=12λ+1∑k=1κ∑i=1nh¯ik2+h^ik2.
Then, denoting(35)τ≔∑j=1m∑i=1ng¯ij2+g^ij2+∑k=1κ∑i=1nh¯ik2+h^ik2,
we obtain that(36)trgTHessHXg+hTHessHXh≤12λ+1∑j=1m∑i=1ng¯ij2+g^ij2+∑k=1κ∑i=1nh¯ik2+h^ik2=12λ+1τ.
Thus, taking expectations of both sides of (32), we have(37)EℒVXt≤2E∇THXtJXt−RXt∇HXt+2E∇THXtGt∫t−dt∇HXs′ds+12λ+1τ.
Due to the fact that(38)E∇THXtJXt∇HXt=∇THXtJ¯Xt∇HXt,where(39)J¯Xt=Jxtg1Kg2T−g2Kg1TJ^ξt=Jxtg1Kg2T−g1Kg2TTJ^ξt,and the fact that J¯Xt=−J¯TXt, we get(40)∇THXtJ¯Xt∇HXt=0.
Furthermore, the following inequality holds:(41)EℒVXt≤−∇THXtR¯Xt+R¯TXt∇HXt+2∇THXtG¯∫t−dt∇HXs′ds+12λ+1τ,where R¯Xt=diagRxt+g1Kg1T,R^ξt−g2Kg2T,(42)G¯=g1Kg1T−g1Kg2Tg2Kg1T−g2Kg2T.
Since Assumption3 holds, the following inequalities(43)−∇TH1xtRxt∇H1xt≤−λminRxt∇TH1xt∇H1xt≤−c1∇TH1xt∇H1xt=−∇TH1xtc1In∇H1xt,(44)−∇TH2ξtR^ξt∇H2ξt≤−λminR^ξt∇TH2ξt∇H2ξt≤−c2∇TH2ξt∇H2ξt=−∇TH2ξtc2In∇H2ξt,are true. Then, (41) becomes(45)EℒVXt≤−∇THXtR¯+R¯T∇HXt+2∇THXtG¯∫t−dt∇HXs′ds+12λ+1τ,where R¯=diagc1In+g1Kg1T,c2In−g2Kg2T.
Assume that there exist matricesL1=L1T, L2=L2T, and L3=L3T such that(46)∇HXt−∇HXt−d−∫t−dt∇HXs′dsT·−diagL1,L1∇HXt+diagL2,L2∇HXt−d+diagL3,L3∫t−dt∇HXs′ds≡0.
Combining (45) and (46), we deduce that(47)EℒVXt≤−∇THXtR¯+R¯T∇HXt+2∇THXtG¯∫t−dt∇HXs′ds+∇HXt−∇HXt−d−∫t−dt∇HXs′dsT·−diagL1,L1∇HXt+diagL2,L2∇HXt−d+diagL3,L3∫t−dt∇HXs′ds+12λ+1τ=υTtΠ1υt+12λ+1τ,where(48)υt=∇H1xt∇H1xt−d∇H2ξt∇H2ξt−d∫t−dt∇H1xs′ds∫t−dt∇H2ξs′ds.
SinceΠ1<0, we further obtain(49)EℒVX≤−λmin−Π1∇TH1xt∇H1xt+∇TH2ξt∇H2ξt+12λ+1τ=−λmin−Π1∇THX∇HX+12λ+1τ.
According to Assumption2 and Lemma 2, we have(50)∇THX∇HX=2α2α−1x121−α/2α−1x1,…,2α2α−1xn21−α/2α−1xn,2α2α−1ξ121−α/2α−1ξ1,…,2α2α−1ξn21−α/2α−1ξn·2α2α−1x121−α/2α−1x1,…,2α2α−1xn21−α/2α−1xn,2α2α−1ξ121−α/2α−1ξ1,…,2α2α−1ξn21−α/2α−1ξnT=2α2α−12∑i=1nxi21/2α−1+∑i=1nξi21/2α−1=2α2α−12∑i=12nXi21/2α−1=2α2α−12∑i=12nXi2α/2α−11/α≥2α2α−12∑i=12nXi2α/2α−11/α=2α2α−12V1/αX.
Then inequality in (49) becomes(51)EℒVX≤−λmin−Π12α2α−12V1/αX+12λ+1τ=−c¯Vℏ1X+a1,where c¯=λmin−Π12α/2α−12>0, 0<ℏ1=1/α<1 and a1=1/2λ+1τ>0. Thus, we obtain that inequality (51) satisfies (11) in Lemma 1. In addition, there exist two class-K∞ functions γ1X=2X and γ2X=2X2 such that (10) in Lemma 1 holds.
Eventually, in view of Lemma1, we arrive at a conclusion that systems (7) and (8) are SGFSSP under the output feedback controller (21). Furthermore, the settling time T1∗ is obtained and satisfies(52)T1∗=1b1c¯1−ℏ1V1−ℏ1X0−1b1c¯1−ℏ1a1c¯1−b11−ℏ1/ℏ1,where 0<b1<1. In addition, the compact set Ω1 is expressed as(53)Ω1=xVℏ1X≤a1c¯1−b1.
The proof of this theorem is now completed.Remark 3.
In [33], the channel is modeled as a cascade of a multiplicative noise and an additive white Gaussian noise. Based on this channel, we take the constant transmission delay into consideration. Thus, the channel model in this paper is more general. In addition, [33] proposes a state feedback controller design strategy to stabilize linear systems. Meanwhile, this paper deals with the output feedback simultaneous stabilization problem for stochastic PCH systems in finite time.Remark 4.
Under Lemma1, how to choose a suitable Lyapunov function is an essential difficulty during the research. Accordingly, we have overcome this difficulty by taking Hx as a Lyapunov function, and Hx has a concrete form which is given in (19) in Assumption 2.Remark 5.
Through the proof of Theorem1, we can see that even if the dimensions of xt are not the same as that of ξt, the result of Theorem 1 still holds. Thus, the design strategy of controllers in Theorem 1 can be extended to multiple systems. Thus, we have the following analysis about SGFSSP of multiple stochastic PCH systems.
Next, consider the following multiple stochastic PCH systems:(54)dxjt=Jjxjt−Rjxjt∇Hjxjtdt+gjutdt+hjdωjt,yjt=gjTxj∇Hjxj,j=1,2,…,V,where V is the number of stochastic systems, xjt∈ℝnj is the plant state vector, yjt∈ℝm is the outputs of the plant, and the signal ωjt∈ℝrj is the independent scalar Wiener process with Edωjt=0 and Edωjt2=dt. ∇Hjxj∈ℝnj×1 is the gradient of the Hamilton function Hjxj:ℝnj↦ℝ, which is defined as ∇Hjxj=∂Hjxj/∂xj, and Hjxj≥0, Hj0=0, for all t≥0. Jjxj is a skew-symmetric structure matrix; Rjxj∈ℝnj×nj is a positive definite strict dissipation matrix; gj and hj are known real constant gain matrices. In addition, Jjxj, Rjxj, gj, and hj satisfy locally Lipschitz condition.Assumption 4.
The Hamilton functionsHjxj are given as(55)Hjxj=∑i=1njxji2α/2α−1.Assumption 5.
There exist constantsc3>0 and c4>0 such that(56)c3=inft≥0λminRj1xj1t,…,RjSxjSt,c4=inft≥0λminRjS+1xjS+1t,…,RjVxjVt.
Assume that we can find out an arbitrary permutationj1,j2,…,jV from the positive integer set 1,2,…,V and that S is a positive integer which satisfies 1≤S≤V−1. In addition, taking V1=nj1+⋯+njS, V2=njS+1+⋯+njV, r¯1=rj1+⋯+rjS, and r¯2=rjS+1+⋯+rjV, we divide the V stochastic PCH systems into two parts: j1,…,jS and jS+1,…,jV.
Defining the vectorsX1t=xj1Tt,…,xjSTtT∈ℝV1, W1t=ωj1Tt,…,ωjSTtT∈ℝr¯1, X2t=xjS+1Tt…,xjVTtT∈ℝV2, W2t=ωjS+1Tt,…,ωjVTtT∈ℝr¯2, then system (54) becomes(57)dX1t=J˜1X1t−R˜1X1t∇H˜1X1tdt+g˜1utdt+h˜1dW1t,Y1t=g˜1T∇H˜1X1t,(58)dX2t=J˜2X2t−R˜2X2t∇H˜2X2tdt+g˜2utdt+h˜2dW2t,Y2t=g˜2T∇H˜2X2t,where(59)J˜1X1=−J˜1TX1=diagJj1xj1,…,JjSxjS∈ℝV1×V1,R˜1X1=diagRj1xj1,…,RjSxjS>0,g˜1=gj1,…,gjST∈ℝV1×m,h˜1=diaghj1,…,hjS∈ℝV1×r¯1,J˜2X2=−J˜2TX2=diagJjS+1xjS+1,…,JjVxjV∈ℝV2×V2,R˜2X2=diagRjS+1xjS+1,…,RjVxjV>0,g˜2=gjS+1,…,gjVT∈ℝV2×m,h˜2=diaghjS+1,…,hjV∈ℝV2×r¯2,H˜1X1=∑k=1SHjkxjk,H˜2X2=∑k=S+1VHjkxjk,Y1=yj1+⋯+yjS,Y2=yjS+1+⋯+yjV.
Substituting (4) into systems (57) and (58), we have(60)dX1t=J˜1X1t−R˜1X1t∇H˜1X1tdt+g˜1εtut−ddt+g˜1dϖt+h˜1dW1t,Y1t=g˜1T∇H˜1X1t,(61)dX2t=J˜2X2t−R˜2X2t∇H˜2X2tdt+g˜2εtut−ddt+g˜2dϖt+h˜2dW2t,Y2t=g˜2T∇H˜2X2t.
Furthermore, we can obtain the following corollary.Corollary 1.
Consider systems (60) and (61). Assumptions 4 and 5 are satisfied. If there exist matrices K=KT, N1=N1T, N2=N2T, N3=N3T, an arbitrary permutation j1,j2,…,jV of 1,2,…,V and a positive integer S1≤S≤V such that(62)Π2=B11B1200B15−g˜1Kg˜2T∗−N200B250∗∗B33B34g˜2Kg˜1TB36∗∗∗−12N20B46∗∗∗∗−N30∗∗∗∗∗−N3<0holds, where(63)B11=−2c3IV1−2g˜1Kg˜1T−N1,B12=12N1+N2,B15=g˜1Kg˜1T+12N1+N3,B25=−12N2+N3,B33=−2c4IV2+2g˜2Kg˜2T−N1,B34=12N1+N2,B36=−g˜2Kg˜2T+12N1+N3,B46=−12N2+N3,then systems (60) and (61) are SGFSSP under the output feedback control law:(64)ut−d=−M−1KY1t−d−Y2t−d.Proof: the proof of this corollary is similar to the proof of Theorem 1, and we give some main analysis. First, similar to the proof of Theorem 1, we can obtain an augmented Itô form stochastic PCH system with state X¯t=X1tX2tT. Next, choose the following Lyapunov function candidate:(65)VX¯t=2H˜1X1t+H˜2X2t.
Letλ1=supt≥0Hess∑j=1VHjxj2 and τ1=tr∑j=1VgjTgj+∑j=1VhjThj. Then, we have(66)EℒVX¯≤−λmin−Π22α2α−12V1/αX¯+12λ1+1τ1=−c˜Vℏ1X¯+a˜,where c˜=λmin−Π22α/2α−12>0 and a˜=1/2λ1+1τ1>0. In the end, under Lemma 1, we can see that systems (60) and (61) are SGFSSP under the output feedback controller (64). Furthermore, the settling time T2∗ satisfies(67)T2∗=1b˜c˜1−ℏ1V1−ℏ1X¯0−1b˜c˜1−ℏ1a˜c˜1−b˜1−ℏ1/ℏ1,where 0<b˜<1. In addition, we have(68)Ω2=xVℏ1X¯≤a˜c˜1−b˜.
The proof of this corollary is now completed.


## 4. Illustrative Example

In this section, a numerical example is performed to illustrate the stabilization scheme for stochastic PCH systems subject to delayed and fading channels.The considered systems are two stochastic PCH systems:(69)dxt=Jx−Rx∇H1xdt+g1utdt+h1dωt,yt=g1T∇H1xt,(70)dξt=J^ξ−R^ξ∇H2ξd+g2utdt+H2dρt,ηt=g2T∇H2ξt,where x=x1x2T∈ℝ2, ξ=ξ1ξ2T∈ℝ2, H1x=x14/3+x24/3α=2, H2ξ=ξ14/3+ξ24/3α=2,(71)Jx=01−10,Rx=1002,J^ξ=02−20,R^ξ=2003,g1=0.0250.05T,h1=0.050.05T,g2=0.050.05T,h2=0.050.025T.We take the constant transmission delayd=0.25. The control input signal ut−d is sent through the delayed and fading noisy channels to the actuator, so a single controller for systems (69) and (70) is modeled as (4). Through statistical experiments, the probability density function of multiplicative noise εt with mean μ=0.9 and variance σ2=0.065 is listed as(72)pts¯=0.05,s¯=0,0.10,s¯=0.5,0.85,s¯=1.Obviously, M=μ=0.9.Substituting (4) into (69) and (70) leads to(73)dxt=Jx−Rx∇H1xdt+g1εtut−0.25dt+g1dϖt+h1dωt,(74)dξt=J^ξ−R^2ξ∇H2ξdt+g2εtut−0.25dt+g2dϖt+h2dρt.ChooseL1=211−1, L2=1116, and L3=3111. Then we can use the LMI toolbox of MATLAB to obtain(75)K=4.447.To sum up, all the conditions of Theorem1 hold. Consequently, from Theorem 1, systems (73) and (74) are SGFSSP under the output feedback control law:(76)ut−d=−0.16x11/3t−d−0.33x21/3t−d+0.33ξ11/3t−d+0.33ξ21/3t−d.In the simulation, we choose the initial states of the systems asx0=−0.30.3T, ξ0=0.60.4T, and set the parameter b1=0.6. Then, considering (52) in Theorem 1, it is easy to obtain that the settling time T1∗=0.32s. The state trajectories of x and ξ are shown in Figures 2 and 3, respectively.Figure 2
State trajectory ofx1 and x2.Figure 3
State trajectory ofξ1 and ξ2.From Figures2 and 3 in the simulation, we can see that a single controller (76) can simultaneously stabilize systems (69) and (70) in finite time. The settling time T1∗ is consistent with that of (52), and the states converge to the origin in 0.32 s. In summary, the output feedback controller proposed in Theorem 1 performs well in the SGFSSP of systems (69) and (70).


## 5. Conclusion

In this paper, the finite-time simultaneous stabilization in probability of stochastic PCH systems over delayed and fading channels has been investigated. On the basis of the dissipative Hamiltonian structural properties and Lyapunov functional technique, a single output feedback controller has been designed for two stochastic PCH systems, which guarantees the SGFSSP of the closed-loop Hamiltonian systems. The case of multiple stochastic PCH systems also has been studied. Sufficient conditions for the existence of the stabilization controllers have been derived in consideration of the phenomenon of delayed and fading channels. At last, a numerical example has highlighted the effectiveness of the stabilization technology proposed in this paper. As for network-based control, sometimes the states of systems are not fully measured, so a possible future research will be involved in observer-based simultaneous stabilization of a set of stochastic PCH systems over delayed and fading channels.



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