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

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.


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][4][5]).
anks 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][7][8][9][10]). In recent years, stabilization as well as simultaneous stabilization problem has been extensively studied for PCH systems ( [11][12][13][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][18][19][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. Finitetime 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.
us, many investigations about finite-time stabilization controller design have been carried out ( [21][22][23][24][25][26][27][28][29]). For stochastic nonlinear systems which are written as Itô differential form, [23] has proposed a method to solve the finite-time stabilization problem. e 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][31][32][33][34][35][36][37][38]). Under memoryless fading channels environment, the study in [33] has illustrated state feedback stabilization problem for linear continuous systems. is 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. e 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 finitetime 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.
rough the Lyapunov function method and Itô differential formula, the closedloop 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. e 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: R n denotes the n-dimensional real column vectors and R n×m is the real matrices with dimensions n × m. A real-valued function f(x) ∈ C 2 represents that f(x) is a continuously twice differentiable function. ‖ · ‖ represents the 2-norm. diag a 1 , a 2 , . . . , a m represents diagonal matrix with a 1 , a 2 , . . . , a m as its diagonal elements. We denote λ min (·) 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.

Problem Formulation and Preliminaries
Consider the following two stochastic PCH systems: where x(t), ξ(t) ∈ R n are the system state vectors, u(t) ∈ R m is the control input which satisfies E t 0 ‖u(s)‖ 2 ds < ∞, and y(t), η(t) ∈ R m are the outputs of systems. e signals ω(t) and ρ(t) are both κ−dimensional independent standard Wiener process defined on probability space (Ω, F, P). We assume E dω(t) is the gradient of the Hamilton function H i (x): R n ↦R, which is defined as ∇H i (x) � (zH i (x)/zx), and H i (x) ≥ 0, H i (0) � 0, for all t ≥ 0. J(x) ∈ R n×n and J(ξ) ∈ R n×n are both skew-symmetric structure matrices; R(x) ∈ R n×n and R(ξ) ∈ R n×n are positive definite strict dissipation matrices; g 1 , g 2 , h 1 , and h 2 are known real constant gain matrices. In addition, by setting Suppose that there exist constants k F > 0 and k G > 0 such that hold for all θ 1 , θ 2 ∈ R n , u(t) ∈ R m , 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 � −K(y(t) − η(t)) 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.

Complexity
Let us focus on the NCS as depicted in Figure 1. Suppose that the control signal u(t) suffers both constant transmission delay d > 0 and signal attenuation in the closed-loop system. e transmission delay d is caused by the message delivery from the controller to the actuator. e transmission of signal u(t) is accomplished in a form of components through independent parallel channels. en, the control signal u(t) arriving at the actuator is modeled by the following multiple independent and memoryless forms: where u(t) ∈ R m and (ε(t)u(t − d) + q(t)) ∈ R m are the input and output of channels, respectively. ε(t) ∈ R m×m represents the multiplicative noise with the following form: We make the following assumption for ε(t).
Remark 1. We consider interference channels noise in the systems and input channels noise. e conditions of Assumption 1 avoid the possible occurrence of noise coupling phenomenon. Denote Obviously, M is nonsingular since μ i ≠ 0. Without loss of generality, we assume Substituting (4) into (1) and (2), we get is a white noise, which is formally regarded as the derivative of a Brownian motion ϖ(t) (see [39]), i.e., q(t) � (dϖ(t)/dt), so we can further write that Before proceeding further, we need to put forward a definition as follows.

Definition 1.
e stochastic PCH systems (7) and (8) are said to be semiglobally finite-time simultaneous stable in probability (SGFSSP) if (1) for any initial values X 0 ∈ R 2n , the solution X(t) of systems (7) and (8) exists and is unique, where and Ω are called the settling time and the compact set, respectively (3) for all t ≥ T(X 0 , ε), the solution X(t) of systems (7) and (8) satisfies E(‖X(t; X 0 )‖) < ϵ Lemma 1. Consider the following Itô form stochastic system: Suppose f(x) and g(x) are locally Lipschitz continuous in x and locally bounded, f(0) � 0, and g(0) � 0. If, for any x 0 ∈ R n , there exist class-K ∞ functions c 1 and c 2 , real numbers c > 0, 0 < Z < 1, a > 0 and a positive definite, function V(x) ∈ C 2 such that then system (9) is SGFSSP. Furthermore, the compact set Ω is expressed as and the settling time of system (9) with respect to x 0 satisfies x (t) Figure 1: NCS over delayed and fading noisy channels. Complexity Lemma 2. For any real number z i , i � 1, . . . , n, and any positive real numbers ϵ 1 , In Lemma 2, if ϵ 1 � β ≥ 1 and ϵ 2 � 1, then for any real number z i , 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 u(t − 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.
e Hamilton functions H 1 (x) and H 2 (ξ) are given as where α > 1 is a real number.

Assumption 3.
ere exist constants c 1 > 0 and c 2 > 0 such that Lemma 3. For any matrices P 1 , P 2 ∈ R m×n , it follows that

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. (7) and (8). Assumptions 2 and 3 are satisfied. If there exist matrices K � K T , L 1 � L T 1 , L 2 � L T 2 , and L 3 � L T 3 such that the following matrix inequality

Theorem 1. Consider systems
then systems (7) and (8) are SGFSSP under the output feedback control law Proof. First of all, substituting (21) into (7) and (8), we obtain Applying Newton-Leibnitz formula, we have 4 Complexity en, systems (22) and (23) can be rewritten as Defining the vectors T , the above equations can be further rewritten into an augmented Itô form stochastic PCH system described as
Letting λ � sup t≥0 ‖Hess(H 1 (x) + H 2 (ξ))‖ 2 and based on Lemma 3, we conclude that tr g T Hess(H(X))g ≤ 1 2 tr g T g + 1 2 tr g T Hess(H(X))Hess T (H(X))g where g ij and g ij are the components of the matrices g 1 � (g ij ) n×m and g 2 � (g ij ) n×m , respectively. Similarly, we have tr g T Hess(H(X))h en, denoting we obtain that Complexity tr g T Hess(H(X))g + h T Hess(H(X))h us, taking expectations of both sides of (32), we have
□ 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. us, 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 Lemma 1, how to choose a suitable
Lyapunov function is an essential difficulty during the research. Accordingly, we have overcome this difficulty by taking H(x) as a Lyapunov function, and H(x) has a concrete form which is given in (19) in Assumption 2.

Remark 5.
rough the proof of eorem 1, we can see that even if the dimensions of x(t) are not the same as that of ξ(t), the result of eorem 1 still holds. us, the design strategy of controllers in eorem 1 can be extended to multiple systems. us, we have the following analysis about SGFSSP of multiple stochastic PCH systems.
Next, consider the following multiple stochastic PCH systems: where V is the number of stochastic systems, x j (t) ∈ R n j is the plant state vector, y j (t) ∈ R m is the outputs of the plant, and the signal ω j (t) ∈ R r j is the independent scalar Wiener process with is a skew-symmetric structure matrix; R j (x j ) ∈ R n j ×n j is a positive definite strict dissipation matrix; g j and h j are known real constant gain matrices. In addition, J j (x j ), R j (x j ), g j , and h j satisfy locally Lipschitz condition.
From Figures 2 and 3 in the simulation, we can see that a single controller (76) can simultaneously stabilize systems (69) and (70) in finite time. e settling time T * 1 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 eorem 1 performs well in the SGFSSP of systems (69) and (70).

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. e 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.

Data Availability
No data were used to support this study.

Conflicts of Interest
e authors declare that they have no conflicts of interest. 10 Complexity