#### 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.

#### 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 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: denotes the -dimensional real column vectors and is the real matrices with dimensions . A real-valued function represents that is a continuously twice differentiable function. represents the 2-norm. represents diagonal matrix with as its diagonal elements. We denote as the smallest eigenvalue operator, as the expectation operator, and as the covariance operator, respectively. For the probability space , denotes the sample space, denotes the -algebra of the observable random events, and is the probability measure on .

#### 2. Problem Formulation and Preliminaries

Consider the following two stochastic PCH systems:where , are the system state vectors, is the control input which satisfies , and are the outputs of systems. The signals and are both dimensional independent standard Wiener process defined on probability space . We assume , , , and . is the gradient of the Hamilton function , which is defined as , and , , for all . and are both skew-symmetric structure matrices; and are positive definite strict dissipation matrices; , , , and are known real constant gain matrices. In addition, by setting , . Suppose that there exist constants and such thathold for all , , , and .

For generalized PCH systems, it is shown in [13] that the two PCH systems can be simultaneously stabilized by a controller over constraint conditions, where 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 Figure 1. Suppose that the control signal suffers both constant transmission delay and signal attenuation in the closed-loop system. The transmission delay is caused by the message delivery from the controller to the actuator. The transmission of signal is accomplished in a form of components through independent parallel channels. Then, the control signal arriving at the actuator is modeled by the following multiple independent and memoryless forms:where and are the input and output of channels, respectively. represents the multiplicative noise with the following form:

, , , is known power spectral density. is an additive white Gaussian process noise with and known power spectral density , i.e., , denotes the Dirac delta function, . We make the following assumption for .

*Assumption 1. * (1) and are uncorrelated for , i.e., , , , and (2) is uncorrelated with and

*Remark 1. *We consider interference channels noise in the systems and input channels noise. The conditions of Assumption 1 avoid the possible occurrence of noise coupling phenomenon.

DenoteObviously, is nonsingular since . Without loss of generality, we assume , i.e., for simplicity hereinafter, .

Substituting (4) into (1) and (2), we get

*Remark 2. * is a white noise, which is formally regarded as the derivative of a Brownian motion (see [39]), i.e., , so we can further write that .

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 values , the solution of systems (7) and (8) exists and is unique, where (2)for every and , the first hitting time is finite almost surely; and are called the settling time and the compact set, respectively(3)for all , the solution of systems (7) and (8) satisfies

Lemma 1. *Consider the following Itô form stochastic system:*

Suppose and are locally Lipschitz continuous in and locally bounded, , and . If, for any , there exist class- functions and , real numbers , , and a positive definite, function such thatthen system (9) is SGFSSP. Furthermore, the compact set is expressed asand the settling time of system (9) with respect to satisfies

Lemma 2. *For any real number , , and any positive real numbers , which satisfy , it holds*

In Lemma 2, if and , thenfor any real number , .

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 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 functions and are given aswhere is a real number.

*Assumption 3. *There exist constants and such that

Lemma 3. *For any matrices , , it follows that*

#### 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 , , , and such that the following matrix inequalityholds, wherethen systems (7) and (8) are SGFSSP under the output feedback control law*

*Proof. *First of all, substituting (21) into (7) and (8), we obtainApplying Newton–Leibnitz formula, we haveThen, systems (22) and (23) can be rewritten asDefining the vectors , , the above equations can be further rewritten into an augmented Itô form stochastic PCH system described aswhere , ,Next, choosing the following Lyapunov function candidate:and according to Itô differential formula, we havewhereLetting and based on Lemma 3, we conclude thatwhere and are the components of the matrices and , respectively. Similarly, we haveThen, denotingwe obtain thatThus, taking expectations of both sides of (32), we haveDue to the fact thatwhereand the fact that , we getFurthermore, the following inequality holds:where ,Since Assumption 3 holds, the following inequalitiesare true. Then, (41) becomeswhere .

Assume that there exist matrices , , and such thatCombining (45) and (46), we deduce thatwhereSince , we further obtainAccording to Assumption 2 and Lemma 2, we haveThen inequality in (49) becomeswhere , and . Thus, we obtain that inequality (51) satisfies (11) in Lemma 1. In addition, there exist two class- functions and such that (10) in Lemma 1 holds.

Eventually, in view of Lemma 1, we arrive at a conclusion that systems (7) and (8) are SGFSSP under the output feedback controller (21). Furthermore, the settling time is obtained and satisfieswhere . In addition, the compact set is expressed asThe 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 Lemma 1, how to choose a suitable Lyapunov function is an essential difficulty during the research. Accordingly, we have overcome this difficulty by taking as a Lyapunov function, and has a concrete form which is given in (19) in Assumption 2.

*Remark 5. *Through the proof of Theorem 1, we can see that even if the dimensions of are not the same as that of , 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:where is the number of stochastic systems, is the plant state vector, is the outputs of the plant, and the signal is the independent scalar Wiener process with and . is the gradient of the Hamilton function , which is defined as , and , , for all . is a skew-symmetric structure matrix; is a positive definite strict dissipation matrix; and are known real constant gain matrices. In addition, , , , and satisfy locally Lipschitz condition.

*Assumption 4. *The Hamilton functions are given as

*Assumption 5. *There exist constants and such thatAssume that we can find out an arbitrary permutation from the positive integer set and that is a positive integer which satisfies . In addition, taking , , , and , we divide the stochastic PCH systems into two parts: and .

Defining the vectors , , , , then system (54) becomeswhereSubstituting (4) into systems (57) and (58), we haveFurthermore, we can obtain the following corollary.

Corollary 1. *Consider systems (60) and (61). Assumptions 4 and 5 are satisfied. If there exist matrices , , , , an arbitrary permutation of and a positive integer such thatholds, wherethen systems (60) and (61) are SGFSSP under the output feedback control law: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 . Next, choose the following Lyapunov function candidate:*