Robust Passivity and Feedback Design for Nonlinear Stochastic Systems with Structural Uncertainty

Abstract

This paper discusses the robust passivity and global stabilization problems for a class of uncertain nonlinear stochastic systems with structural uncertainties. A robust version of stochastic Kalman-Yakubovitch-Popov (KYP) lemma is established, which sustains the robust passivity of the system. Moreover, a robust strongly minimum phase system is defined, based on which the uncertain nonlinear stochastic system can be feedback equivalent to a robust passive system. Following with the robust passivity theory, a global stabilizing control is designed, which guarantees that the closed-loop system is globally asymptotically stable in probability (GASP). A numerical example is presented to illustrate the effectiveness of our results.

1. Introduction

It is well known that passivity theory plays an important role in many engineering problems, which is a powerful technique in handling stability issue. Many problems on related topics have been investigated; see [1–4]. In 1990s, much more attention has been focused on the development of synthesis technique that combines the passivity theory with geometric nonlinear control theory; see [5–7] and the references therein. The work in [5] developed a framework for studying the stabilizability of minimum phase deterministic nonlinear systems. Especially, [8] has proposed a robust enhancement of the result in [5], which discussed the passivity and global stabilization for a class of uncertain minimum phase nonlinear systems, which considers the nonlinear systems with structural uncertainties, that is, gain-bounded uncertainties. The work in [9] extended the corresponding approach to solve the robust almost disturbance decoupling problem for such a class uncertain nonlinear systems.

On the other hand, due to the great many applications of stochastic Itô systems in real world [10], the study on feedback controller design for such a class of systems has received a great deal of attention; see [11–19] and the references therein. Especially, many researchers have extended the existing passivity theory from deterministic systems to stochastic systems. The work in [11] obtained necessary and sufficient conditions for the existence of diffeomorphisms that transform stochastic nonlinear systems to various canonical forms. The work in [12] studied the problem of feedback equivalent to a passive system for a particular class of interconnected stochastic systems. The work in [13] developed a systematic method for global asymptotic stabilization in probability of nonlinear control stochastic differential systems based on the passivity theory. However, compared with the deterministic case, up to date, there still requires much work of investigating the passivity theory of uncertain nonlinear stochastic systems with structural uncertainties.

This paper considers a class of uncertain nonlinear stochastic systems which are expressed by the Itô-type stochastic differential equations with structural uncertainty. We shall investigate the problem of feedback equivalent to a robust passive system and global stabilization for uncertain nonlinear stochastic system via robust passivity theory. A robust stochastic KYP lemma is proposed, which can be regarded as a robust stochastic extended results in deterministic case [2, 5]. Based on the above, we investigate the relationship between a robust passive system and the corresponding zero-output dynamics, where a robust strongly minimum phase property is proposed. Then, sufficient conditions for global asymptotic stabilization in probability are provided.

The remainder of the paper is organized as follows: Section 2 develops the robust passivity theory for a class of uncertain nonlinear stochastic systems, which presents a robust stochastic version of KYP lemma. In Section 3, through adopting the appropriate diffeomorphism, we discuss the relationship between the robust passivity of the system and the stability of the zero-output system, that is, the robust minimum phase system. Then, the global stabilizing control of the system can be determined through discussing the robust strongly minimum phase property. Based on the robust stochastic passivity theory, sufficient conditions are given for global stabilization of such a class of uncertain nonlinear stochastic systems. In Section 4, an example is given to illustrate the usefulness of our results. Section 5 concludes this paper.

For convenience, we adopt the following notations: 𝒮_𝑛: the set of all real 𝑛×𝑛 symmetric matrices; 𝐴^′: the transpose of a matrix or vector 𝐴; 𝐴≥0(𝐴>0): 𝐴 is a positive semidefinite (positive definite) matrix; 𝒞²(𝑈): the class of functions 𝑉(𝑥) twice continuously differentiable with respect to 𝑥∈𝑈; 𝐿_𝑔𝜆: the Lie derivative of a smooth function 𝜆 along the vector field 𝑔, that is, 𝐿_𝑔𝜆∶=(𝜕𝜆/𝜕𝑥)𝑔; 𝑅^𝑛: 𝑛-dimensional Euclidean space; ‖𝑥‖: 2-norm of a vector 𝑥∈𝑅^𝑛.

2. Robust Stochastic Passivity Theory

First of all, let (Ω,ℱ,{ℱ_𝑡}_𝑡≥0,𝑃) be a given filtered probability space where there lives a standard one-dimensional Brownian motion 𝑤(𝑡) on [0,+∞) with 𝑤(0)=0 and ℱ_𝑡=𝜎{𝑤(𝑠)∣0≤𝑠≤𝑡}. The Brownian motion is assumed to be one dimensional only for simplicity, because there is no essential difference from the multidimensional case.

Consider the following uncertain nonlinear stochastic control system governed by Itô's differential equation: 𝑓(𝑥)(𝑥)(𝑥)𝑢(𝑥)(0)(0)(𝑥)(0)𝑑𝑥=+Δ𝑓+𝑔𝑑𝑡+𝑙𝑑𝑤,𝑓=0,𝑙=0,𝑦=ℎ,ℎ=0, (1) where 𝑥(𝑡)∈𝑅^𝑛 is the state vector, 𝑥₀∈𝑅^𝑛 is the initial state, and 𝑦(𝑡)∈𝑅 is the controlled output. 𝑤(𝑡) is an one-dimensional Brownian motion. 𝑢(𝑡)∈𝑅 is the control input, which is an adaptive process with respect to {ℱ_𝑡}_𝑡≥0. 𝑓, 𝑔, 𝑙, and ℎ are assumed to be smooth functions of appropriate dimensions. Δ𝑓(𝑥)∶𝑅^𝑛→𝑅^𝑛 represents structural uncertainty or uncertain perturbation characterized by Δ𝑓(𝑥)=𝑒(𝑥)𝛿(𝑥),Δ𝑓(0)=0, (2) where 𝑒(𝑥)∶𝑅^𝑛→𝑅^𝑛×𝑚, with 𝑒(0)=0 being a known matrix whose entries are given smooth functions, and 𝛿(𝑥)∶𝑅^𝑛→𝑅^𝑚 is an unknown vector-valued function. It is assumed that 𝛿(𝑥) is constrained to a given smooth function 𝑛(𝑥)∶𝑅^𝑛→𝑅^𝑚, that is, Γ={𝛿(𝑥)∶‖𝛿(𝑥)‖≤‖𝑛(𝑥)‖}. (3) If 𝛿(𝑥)∈Γ, then Δ𝑓(𝑥) is said to be admissible.

Definition 1. An uncertain system of the form (1) is said to be robust passive on [𝑠,∞), 𝑠≥0 if there exists a nonnegative continuous function 𝑉(𝑥)∶𝑅^𝑛→𝑅⁺, called the storage function, such that for all 𝑡≥𝑠≥0, 𝑥(𝑠)∈𝑅^𝑛, and for all admissible Δ𝑓(𝑥), 𝐸𝑉(𝑥(𝑡))−𝑉(𝑥(𝑠))≤𝐸^𝑡_𝑠𝑦^′(𝜏)𝑢(𝜏)𝑑𝜏. (4)
Here, (4) can be regarded as the robust version of the passive inequality for stochastic systems. For simplicity of our following discussion, we assume that the storage function of (4), if it exists, belongs to 𝒞²(𝑅^𝑛).

In view of Definition 1 and [8], it is quite natural to introduce the following concept for uncertain nonlinear stochastic systems.

Definition 2. System (1) is said to have the robust KYP property if there exists a nonnegative function 𝑉∈𝒞²(𝑅^𝑛)∶𝑅^𝑛→𝑅⁺, with 𝑉(0)=0, such that 𝐿_𝑓+Δ𝑓𝑉(𝑥)+12𝑙^′𝜕²𝑉(𝑥)𝜕𝑥²𝐿𝑙≤0,_𝑔𝑉(𝑥)=ℎ^′. (5) If the above inequality becomes a strict inequality that holds for a positive definite function 𝑉(𝑥), then system (1) is said to be robust strictly passive.

Now, we derive conditions under which an uncertain nonlinear stochastic system is robust passive, which can be viewed as a robust stochastic version of the nonlinear KYP lemma, which plays an important role in studying global robust stabilization for uncertain nonlinear stochastic systems.

Theorem 3. System (1) is robust passive if and only if there exists a nonnegative function 𝑉∈𝒞²(𝑅^𝑛)∶𝑅^𝑛→𝑅⁺, with 𝑉(0)=0, such that 𝐿_𝑓𝑉(𝑥)+𝐿^′_𝑒𝑉(𝑥𝑛(𝑥+1)‖)‖2𝑙^′𝜕²𝑉(𝑥)𝜕𝑥²𝐿𝑙≤0,_𝑔𝑉(𝑥)=ℎ^′, (6) where 𝐿^′_𝑒𝑉(𝑥)=(𝐿_𝑒𝑉(𝑥))^′.

Proof. Sufficiency. According to the Cauchy inequality and considering the fact of ‖𝛿(𝑥)‖≤‖𝑛(𝑥)‖, ∀𝛿(𝑥)∈Γ, we have 𝐿_𝑒𝑉(𝑥)𝛿(𝑥)≤𝐿^′_𝑒𝑉(𝑥𝛿(𝑥≤𝐿)‖)‖_′𝑒𝑉(𝑥)‖𝑛(𝑥)‖,∀𝛿(𝑥)∈Γ, (7) which reduces to 𝐿_𝑓+Δ𝑓𝑉(𝑥)+12𝑙^′𝜕²𝑉(𝑥)𝜕𝑥²𝑙=𝐿_𝑓𝑉(𝑥)+𝐿_𝑒𝑉(𝑥)𝛿(𝑥)+12𝑙^′𝜕²𝑉(𝑥)𝜕𝑥²𝑙≤0. (8) The robust KYP property of system (1) is guaranteed through Definition 2. Then, by Itô's formula, we have ℒ𝑉(𝑥)=𝐿_𝑓𝑉(𝑥)+𝐿_Δ𝑓𝑉(𝑥)+𝐿_𝑔𝑉(𝑥)𝑢+12𝑙^′𝜕²𝑉(𝑥)𝜕𝑥²𝑙≤𝐿_𝑔𝑉(𝑥)𝑢, (9) where ℒ is the infinitesimal generator of system (1). From the second equation in (6), it follows that ℒ𝑉(𝑥)≤𝑦^′𝑢. (10) Integrating the above inequality from 𝑠 to 𝑡 for any 𝑡≥𝑠≥0, 𝑥(𝑠)∈𝑅^𝑛, and for all admissible Δ𝑓(𝑥), 𝐸𝑉(𝑥(𝑡))−𝑉(𝑥(𝑠))≤𝐸^𝑡_𝑠𝑦^′(𝜏)𝑢(𝜏)𝑑𝜏, (11) which implies that system (1) is robust passive by Definition 1.
Necessity. Assume that system (1) is robust passive which sustains a storage function 𝑉(𝑥); that is, (11) always holds for ∀𝛿(𝑥)∈Γ. By (11) with any 𝑥(𝑠)∈𝑅^𝑛, we have 𝐸𝑉(𝑥(𝑡))−𝑉(𝑥(𝑠))−𝑡−𝑠𝐸∫^𝑡_𝑠𝑦^′(𝜏)𝑢(𝜏)𝑑𝜏𝑡−𝑠≤0,∀𝑡>𝑠. (12) By Itô's formula, 𝐸𝑉(𝑥(𝑡))−𝑉(𝑥(𝑠))=𝐸^𝑡_𝑠𝐿_𝑓𝑉(𝑥)+𝐿_𝑒𝑉(𝑥)𝛿(𝑥)+𝐿_𝑔𝑉(𝑥)1𝑢+2𝑙^′𝜕²𝑉(𝑥)𝜕𝑥²𝑙𝑑𝜏. (13) Let 𝑡↓𝑠, it follows that 𝐿_𝑓𝑉(𝑥)+𝐿_𝑒𝑉(𝑥)𝛿(𝑥)+12𝑙^′𝜕²𝑉(𝑥)𝜕𝑥²𝑙+𝐿^′_𝑔𝑉(𝑥)−𝑦^′𝑢≤0 (14) always holds for any control input 𝑢, which implies that (5) holds; that is, the robust passive system (1) has the robust KYP property.
Denote 𝛼(𝑥)=𝐿_𝑓𝑉(𝑥)+12𝑙^′𝜕²𝑉(𝑥)𝜕𝑥²𝑙,𝛽^′(𝑥)=𝐿_𝑒𝑉(𝑥). (15) So, 𝐿_𝑓+Δ𝑓𝑉(𝑥)+12𝑙^′𝜕²𝑉(𝑥)𝜕𝑥²𝑙=𝛼(𝑥)+𝛽^′(𝑥)𝛿(𝑥)≤0,∀𝛿(𝑥)∈Γ. (16) The rest is similar to the proof of Lemma 2 in [8], and we only note that through defining 𝛿₀=‖𝑛(𝑥)‖𝛽(𝑥𝛽(𝑥)(𝑥)),𝛽≠0, (17) it can be found that 𝛿₀∈Γ, which also deduces that 𝛽^′(𝑥)𝛿₀(𝑥)=𝛽^′(𝑥)𝛽(𝑥)‖𝑛(𝑥)‖𝛽(𝑥=𝛽(𝑥𝑛(𝑥.))‖)‖ (18) Obviously, the above equality also holds when 𝛽(𝑥)=0. Considering the fact that sup_𝛿∈Γ𝛽^′(𝑥)𝛿(𝑥)≤𝛽(𝑥)‖𝑛(𝑥)‖, (19) we get to the result that sup_𝛿∈Γ𝛽^′(𝑥)𝛿(𝑥)=𝛽(𝑥)‖𝑛(𝑥)‖. (20) Hence, from (16) and (20), we have 𝛼(𝑥)≤−sup_𝛿∈Γ𝛽^′(𝑥)𝛿(𝑥)=−𝛽(𝑥)‖𝑛(𝑥)‖, (21) which also implies that 𝛼(𝑥)+𝛽(𝑥)‖𝑛(𝑥)‖=𝐿_𝑓𝑉(𝑥)+𝐿^′_𝑒𝑉(𝑥𝑛‖+1)‖2𝑙^′𝜕²𝑉(𝑥)𝜕𝑥²𝑙≤0. (22) The proof of Theorem 3 is completed.

Remark 4. Indeed, Theorem 3 has established the equivalent relations between (4), (5), and (6). Obviously, the condition (6) is more convenient than (4) or (5), which has taken the bounding function 𝑛(𝑥) instead of the unknown function 𝛿(𝑥).

Remark 5. From the proof of Theorem 3, we can also discuss the robust strict passivity for system (1). There is no difference except that the inequality in (6) becomes a strict inequality which holds for a positive definite function 𝑉(𝑥). Moreover, it is easy to deduce from Theorem 3 that a sufficient condition for the robust passivity of system (1) is that there exists a real-valued function 𝜆(𝑥)>0, such that 𝐿_𝑓𝑉(𝑥)+𝜆2𝐿_𝑒𝑉(𝑥)²+1‖‖2𝜆𝑛(𝑥)²+12𝑙^′𝜕²𝑉(𝑥)𝜕𝑥²𝐿𝑙≤0,_𝑔𝑉(𝑥)=ℎ^′. (23) Indeed, if (23) is also a strict inequality, it is equivalent to the strict inequalities (5) or (6). The proof can follow the line of Theorem 3 and [8] and is omitted.
In what follows, we recall some facts in the theory of stochastic stability, where only global stability is considered. Obviously, local stability results may also be achieved in a similar way.
Consider the uncertain stochastic unforced system with 𝑓(𝑥)(𝑥)(𝑥)𝑥(0)𝑑𝑥=+Δ𝑓𝑑𝑡+𝑙𝑑𝑤,=𝑥₀∈𝑅^𝑛(0)(0)(0),𝑓=0,Δ𝑓=0,𝑙=0. (24)

Definition 6 (see [20]). System (24) with the structural uncertainty is said to be stable in probability if for any 𝜀>0, lim_𝑥→0𝑃sup_𝑡≥0‖𝑥(𝑡)‖>𝜀=0. (25) Additionally, if we also have 𝑃(lim_𝑡→∞𝑥(𝑡)=0)=1 for ∀𝑥₀∈𝑅^𝑛, system (24) is said to be GASP.

Definition 7 (20). System (24) with the structural uncertainty is called zero-state detectable if for all 𝑥(0)=𝑥₀∈𝑅^𝑛, ℎ(𝑥(𝑡))=0,a.s.∀𝑡≥0⇒𝑃lim_𝑡→∞𝑥(𝑡)=0,𝑥(0)=𝑥₀=1. (26) System (24) is globally zero-state observable if for all 𝑥₀∈𝑅^𝑛, ℎ(𝑥(𝑡))≡0 implies 𝑥₀≡0.

3. Feedback Equivalence and Global Stabilization

In this section, we discuss the feedback equivalence and global stabilization problems for the general nonlinear stochastic system (1) based on the above robust passivity theory. Similarly as the deterministic system case, we first present the following relative degree definition for uncertain nonlinear stochastic systems.

Definition 8 (relative degree). The nonlinear stochastic system (1) is said to have the relative degree 𝜌 if 𝐿_𝑔𝐿_𝑓^𝑗−1ℎ(𝑥)𝐿=0,_𝑔𝐿_𝑓^𝜌−1ℎ(𝑥)≠0, (27) where 𝑗=1,2,…,𝜌−1.
In what follows, we only consider the simple case of 𝜌=1. For system (1), by Itô's formula, we know that 𝑑𝑦=𝜕ℎ^′()+1𝜕𝑥𝑑𝑥2(𝑑𝑥)^′𝜕²ℎ𝜕𝑥²()=𝐿𝑑𝑥_𝑓ℎ+𝐿_Δ𝑓ℎ+𝐿_𝑔1ℎ𝑢+2𝑙^′𝜕²ℎ𝜕𝑥²𝑙𝑑𝑡+𝐿_𝑙ℎ𝑑𝑤. (28) From the above assumption, we have 𝐿_𝑔ℎ(𝑥)≠0. If we take 𝑢 as 𝑢=𝐿_𝑔ℎ⁻¹𝑣−𝐿_𝑓1ℎ−2𝑙^′𝜕²ℎ𝜕𝑥²𝑙, (29) where 𝑣 can be regarded as a new control input instead of 𝑢. Then, we obtain that 𝑑𝑦=𝑣𝑑𝑡+𝐿_Δ𝑓ℎ𝑑𝑡+𝐿_𝑙ℎ𝑑𝑤. (30)
Motivated by the works of [5, 8], we assume that the vector fields 𝑔₁(𝑥),𝑔₂(𝑥),…,𝑔_𝑝(𝑥) are complete, where [𝑔₁(𝑥),…,𝑔_𝑝(𝑥)]=𝑔(𝑥)[𝐿_𝑔ℎ(𝑥)]⁻¹ and the distribution spanned by 𝑔₁(𝑥),𝑔₂(𝑥),…,𝑔_𝑝(𝑥) is involutive. Under these hypotheses, it is possible to find a global diffeomorphism 𝜙(𝑥), such that (𝑥)=𝜃(𝑥)ℎ(𝑥)=𝜂𝑦𝜉=𝜙, (31) where 𝜂=𝜃(𝑥)∈𝑅^𝑛−1. Then, system (1) can be transformed into an interconnected system of the form 𝑑𝜂=𝐿_𝑓𝜃+𝐿_Δ𝑓1𝜃+2𝑙^′𝜕²𝜃𝜕𝑥²𝑙𝑑𝑡+𝐿_𝑙𝜃𝑑𝑤,𝑑𝑦=𝑣𝑑𝑡+𝐿_Δ𝑓ℎ𝑑𝑡+𝐿_𝑙ℎ𝑑𝑤. (32) For simplicity, we adopt that ℎ(𝜉)=ℎ(𝜙⁻¹(𝜉)). Then, by taking the transformation 𝑥=𝜙⁻¹(𝜉), system (1), in the new coordinate, is formulated as 𝑓(𝜉)𝑒̃(𝜉)𝛿(𝜉)𝑙̃(𝜉)𝑒̂(𝜉)𝑙̂(𝜉)𝑑𝜂=𝑑𝑡+𝑑𝑡+𝑑𝑤,𝑑𝑦=𝑣𝑑𝑡+𝛿𝑑𝑡+𝑑𝑤, (33) where 𝑓(𝜉)=𝐿_𝑓1𝜃+2𝑙^′𝜕²𝜃𝜕𝑥²𝑙_{𝑥=𝜙−1(𝜉)},𝑒̃(𝜉)=𝐿_𝑒𝜃|_𝑥=𝜙⁻¹_(𝜉),𝛿(𝜉)=𝛿𝜙⁻¹(𝜉),𝑙̃(𝜉)=𝐿_𝑙𝜃|_𝑥=𝜙⁻¹_(𝜉),𝑒_𝑖̂(𝜉)=𝐿_𝑒ℎ|_𝑥=𝜙⁻¹_(𝜉),𝑙_𝑖̂(𝜉)=𝐿_𝑙ℎ|_𝑥=𝜙⁻¹_(𝜉). (34) Here, we define 𝑓₁(𝜉)=¹₀𝜕𝑓𝜂,𝜁𝜕𝜁_{𝜁=𝜏𝑦}𝑒̃𝑑𝜏,₁(𝜉)=¹₀𝜕𝑒̃𝜂,𝜁𝜕𝜁_{𝜁=𝜏𝑦}𝑙̃𝑑𝜏,₁(𝜉)=¹₀𝜕𝑙̃𝜂,𝜁𝜕𝜁_{𝜁=𝜏𝑦}𝑒̂𝑑𝜏,₁(𝜉)=¹₀𝜕𝑒̂𝜂,𝜁𝜕𝜁_{𝜁=𝜏𝑦}𝑙̂𝑑𝜏,₁(𝜉)=¹₀𝜕𝑙̂𝜂,𝜁𝜕𝜁_{𝜁=𝜏𝑦}𝑑𝜏. (35) Then, the following decomposition holds 𝑓(𝜉)=𝑓_∗𝜂+𝑓₁(𝜉)𝑦,where𝑓_∗𝜂=𝑓𝜂,0. (36) Similarly, we can also find 𝑙̃, 𝑙̂, 𝑒̃, and 𝑛 with the following structures, respectively 𝑙̃(𝜉)=𝑙_∗̃𝜂+𝑙₁̃(𝜉)𝑦,𝑒̃(𝜉)=𝑒_∗̃𝜂+𝑒₁̃(𝜉)𝑙̂(𝜉)=𝑙̂𝑦,_∗𝜂+𝑙₁̂(𝜉)𝑦,𝑛(𝜉)=𝑛𝜂,0+𝑛₁(𝜉)𝑦. (37)
Moreover, we set the control input as 𝑣=𝑐(𝜉)+𝜔, (38) where 𝑐 is a smooth function defined on 𝑅^𝑛 with 𝑐(0)=0, and 𝜔 is the new control input. Equivalently, system (33) can be expressed with the form of (𝜉)(𝜉)(𝜉)(𝜉)ℎ(𝜉),𝑑𝜉=𝐹𝑑𝑡+𝐸𝛿𝑑𝑡+𝐺𝜔𝑑𝑡+𝐿𝑑𝑤,𝑦= (39) where 𝐹(𝜉)=𝑓(𝜉)𝑐(𝜉)(𝜉)=𝑒̃(𝜉)𝑒̂(𝜉),𝐺(𝜉)=0𝐼(𝜉)=𝑙̃(𝜉)𝑙̂(𝜉).,𝐸,𝐿 (40)
Let zero-output system describe the internal dynamic of a system which is consistent with the constraint 𝑦≡0. Obviously, the zero-output dynamic of system (1) comes down to 𝑑𝜂=𝑓_∗𝜂𝑑𝑡+𝑒_∗̃𝜂𝛿𝜂,0+𝑙_∗̃𝜂𝑑𝑤. (41)

As follows, we define the minimum phase system for nonlinear stochastic system (1).

Definition 9. System (1) is said to be
(i) Robust weakly minimum phase if the zero-output system (41) is stable in probability; that is, there exists a nonnegative function 𝑊(𝜂)∈𝒞²∶𝑅^𝑛−1→𝑅⁺ with 𝑊(0)=0, satisfying the following Lyapunov inequality: 𝐿𝑓_∗𝑊𝜂+𝐿𝑒^′_∗̃𝑊𝜂𝑛+1𝜂,02𝑙^′_∗̃𝜕²𝑊𝜂𝜕𝜂²𝑙_∗̃≤0. (42)
(ii) Robust minimum phase if for the zero-output system (41) is GASP; that is, there exists a positive definite function 𝑊(𝜂)∈𝒞²∶𝑅^𝑛−1→𝑅⁺ with 𝑊(0)=0, satisfying the following Lyapunov inequality: 𝐿𝑓_∗𝑊𝜂+𝐿𝑒^′_∗̃𝑊𝜂𝑛+1𝜂,02𝑙^′_∗̃𝜕²𝑊𝜂𝜕𝜂²𝑙_∗̃<0. (43)
(iii) Robust strongly minimum phase, if for the zero-output system (41) there exists positive definite functions 𝑊(𝜂)∈𝒞²∶𝑅^𝑛−1→𝑅⁺ with 𝑊(0)=0, and 𝜆(𝜉)∶𝑅^𝑛→𝑅⁺, satisfying the following Lyapunov-like inequality constraint: 𝐿𝑓_∗𝑊𝜂+𝜆2𝐿𝑒_∗̃𝑊(𝜂)²+12𝜆𝑛(𝜂,0)²+12𝑙^′_∗̃𝜕²𝑊𝜂𝜕𝜂²𝑙_∗̃≤−12𝑙^′_∗̂𝑙_∗̂. (44)

The following theorem discusses the relationship between the passivity of system (39) and the stability of the zero-output system (41).

Theorem 10. If system (1) is robust strongly minimum phase, then the system is feedback equivalent to a robust passive system (39). Conversely, the robust passivity of system (39) with a positive storage function implies that system (1) is robust weakly minimum phase.

Proof. We construct the storage function 𝑉(𝜉)=𝑊(𝜂)+(1/2)𝑦^′𝑦 for system (39), where 𝑊(𝜂) satisfies the inequality (44). Obviously, we have 𝐿_𝐹𝑉(𝜉)+𝐿^′_𝐸𝑉(𝜉𝑛(𝜉+1))2𝐿^′𝜕²𝑉(𝜉)𝜕𝜉²𝐿≤𝐿_𝐹𝑉(𝜉)+𝜆2𝐿_𝐸𝑉(𝜉)²+1𝑛(𝜉2𝜆)²+12𝐿^′𝜕²𝑉(𝜉)𝜕𝜉²𝐿=𝐿_𝑓𝑊𝜂+𝑦^′𝜆𝑐+2𝐿_̃𝑒𝑊𝜂+𝑦^′𝑒̂²+1𝑛(𝜉2𝜆)²+12𝑙^′̃𝜕²𝑊𝜂𝜕𝜂²𝑙̃+12𝑙^′̂𝑙̂𝑓=𝐿_∗+𝑓_1𝑦𝑊𝜂+𝑦^′𝑐+𝜆2𝐿𝑒_∗̃𝑊𝜂+𝑒₁̃𝑦+𝑦^′𝑒̂²+12𝜆𝑛(𝜂,0)+𝑛₁(𝜉)𝑦²+12𝑙_∗̃+𝑙₁̃𝑦^′𝜕²𝑊𝜂𝜕𝜂²𝑙_∗̃+𝑙₁̃𝑦+12𝑙_∗̂+𝑙₁̂𝑦^′𝑙_∗̂+𝑙₁̂𝑦𝑓=𝐿_∗𝑊𝜂𝑓+𝐿₁𝑊𝜂𝑦+𝑦^′𝑐+𝐿𝑒_∗̃𝑊𝜂²+𝜆2𝑦^′𝑐₁+1𝑛2𝜆𝜂,0²+1𝑦2𝜆^′𝑐₂+12𝑙^′_∗̃𝜕²𝑊𝜂𝜕𝜂²𝑙_∗̃+𝑦^′𝑙^′₁̃𝜕²𝑊𝜂𝜕𝜂²𝑙_∗̃+12𝑙^′₁̃𝜕²𝑊𝜂𝜕𝜂²𝑙₁̃𝑦+12𝑙^′_∗̂𝑙_∗̂+𝑦^′𝑙^′₁̂𝑙_∗̂+12𝑙^′₁̂𝑙₁̂𝑦𝑓=𝐿_∗𝑊𝜂+𝐿𝑒_∗̃𝑊𝜂²+12𝜆𝑛(𝜂,0)²+12𝑙^′_∗̃𝜕²𝑊𝜂𝜕𝜂²𝑙_∗̃+12𝑙^′_∗̂𝑙_∗̂+𝑦^′𝐿^′𝑓₁𝑊𝜂𝜆+𝑐+2𝑐₁+1𝑐2𝜆²+𝑙^′₁̃𝜕²𝑊𝜂𝜕𝜂²𝑙_∗̃+12𝑙^′₁̃𝜕²𝑊𝜂𝜕𝜂²𝑙₁̃𝑦+𝑙^′₁̂𝑙_∗̂+12𝑙^′₁̂𝑙₁̂, (45) where 𝑐₁=2(𝐿𝑒₁̃𝑊+𝑒)̂^′𝐿𝑒^′_∗̃𝑊+𝐿𝑒₁̃𝑊+𝑒̂(𝐿𝑒₁̃𝑊+𝑒)̂^′𝑐𝑦,₂=(𝑛₁)^′2𝑛𝜂,0+𝑛₁𝑦. (46) For (45), taking 𝑐=−𝐿^′𝑓₁𝜆𝑊−2𝑐₁−1𝑐2𝜆²−𝑙^′₁̃𝜕²𝑊𝜕𝜂²𝑙_∗̃−12𝑙^′₁̃𝜕²𝑊𝜕𝜂²𝑙₁̃𝑦−𝑙^′₁̂𝑙_∗̂−12𝑙^′₁̂𝑙₁̂, (47) and considering the robust strongly minimum phase property, we have 𝐿_𝐹𝑉(𝜉)+𝐿^′_𝐸𝑉(𝜉𝑛(𝜉+1))2𝐿^′𝜕²𝑉(𝜉)𝜕𝜉²𝐿≤0. (48) Besides, it is obvious that 𝐿_𝐺𝑉(𝜉)=𝜕𝑊^′𝜕𝜂,𝑦^′0𝐼=ℎ^′. (49) Then, by Theorem 3, we know that system (39) is robust passive.
On the other hand, suppose that there is a feedback control law 𝑣_∗ that renders the closed-loop system (39) robust passive with a storage function 𝑉(𝜉), which is positive definite and satisfies 𝐿_𝐹𝑉(𝜉)+𝐿^′_𝐸𝑉(𝜉𝑛(𝜉+1))2𝐿^′𝜕²𝑉(𝜉)𝜕𝜉²𝐿𝐿≤0,_𝐺𝑉(𝜉)=ℎ^′. (50)
Note that 𝜕𝑉^′(𝜉)=𝜕𝜉𝜕𝑉^′(𝜉),𝜕𝜂𝜕𝑉^′(𝜉),𝜕𝜕𝑦₂𝑉(𝜉)𝜕𝜉²=𝜕²𝑉(𝜉)𝜕𝜂²𝜕²𝑉(𝜉)𝜕𝜕𝜂𝑦₂𝑉(𝜉)𝜕𝜕𝑦𝜂²𝑉(𝜉)𝜕𝑦². (51) Then, (50) deduces that 𝜕𝑉^′(𝜉)𝑓(𝜉)+𝜕𝜂𝜕𝑉^′(𝜉)𝑐(𝜉)+𝜕𝑦𝜕𝑉^′(𝜉)̃𝜕𝜂𝑒(𝜉)+𝜕𝑉^′(𝜉)̂𝜕𝑦𝑒(𝜉)^′𝑛(𝜉+1)2𝑙̃𝑙̂^′𝜕²𝑉(𝜉)𝜕𝜂²𝜕²𝑉(𝜉)𝜕𝜕𝜂𝑦₂𝑉(𝜉)𝜕𝜕𝑦𝜂²𝑉(𝜉)𝜕𝑦²𝑙̃𝑙̂≤0 (52) holds for all 𝜉∈𝑅^𝑛. Obviously, 𝜕𝑉𝜂,0𝜕𝜕𝑦=0,²𝑉𝜂,0𝜕𝑦²=0,∀𝜂∈𝑅^𝑛−1. (53) Setting 𝑦=0 in (52), it reduces to 𝐿𝑓_∗𝑉𝜂,0+𝐿𝑒^′_∗̃𝑉𝑛+1𝜂,0𝜂,02𝑙^′_∗̃𝜕²𝑉𝜂,0𝜕𝜂²𝑙_∗̃≤0. (54) Then, by [20] and Definition 9(i), we know that the zero-output system (41) is stable in probability, and that system (1) is robust weakly minimum phase. The proof of Theorem 10 is completed.

Corollary 11. If the zero-output system (41) satisfies the following Lyapunov-like equality constraint, that is, there exists a positive definite and proper function 𝑊(𝜂)∈𝒞²∶𝑅^𝑛−1→𝑅⁺ with 𝑊(0)=0 and 𝜆(𝜉)∶𝑅^𝑛→𝑅⁺, such that 𝐿𝑓_∗𝑊𝜂+𝜆2𝐿𝑒_∗̃𝑊𝜂²+12𝜆𝑛(𝜂,0)²+12𝑙^′_∗̃𝜕²𝑊𝜂𝜕𝜂²𝑙_∗̃<−12𝑙^′_∗̂𝑙_∗̂, (55) then system (1) is feedback equivalent to a robust strictly passive system (39). Conversely, the robust strict passivity of system (39) with a positive definite and proper storage function implies system (1) to be robust minimum phase.

Proof. From Remark 5 and the proof of Theorem 10, the above conclusion is obvious.

As follows, we use the above results to study the problem of global stabilization for system (39).

Theorem 12. Suppose that system (1) is robust strongly minimum phase which sustains a positive definite and proper function 𝑊(𝜂)∈𝒞²∶𝑅^𝑛−1→𝑅⁺ with 𝑊(0)=0 and a positive definition function 𝜆(𝜉)∶𝑅^𝑛→𝑅⁺ satisfying (44), and that system (39) is zero-state detectable, then system (39) is globally asymptotically stabilizable in probability.

Proof. We construct the storage function 𝑉(𝜂,𝑦)=𝑊(𝜂)+(1/2)𝑦^′𝑦 for system (39), where 𝑊(𝜂) satisfies (44). Obviously, we have ℒ_𝜔𝑉=𝐿_𝐹𝑉(𝜉)+𝐿_𝐸𝑉(𝜉)𝛿+𝐿_𝐺𝑉(𝜉)1𝜔+2𝐿^′𝜕²𝑉(𝜉)𝜕𝜉²𝐿≤𝐿_𝐹𝑉(𝜉)+𝐿_𝐺𝑉(𝜉)𝜔+𝐿^′_𝐸𝑉(𝜉𝑛(𝜉+1))2𝐿^′𝜕²𝑉(𝜉)𝜕𝜉²𝐿≤𝐿_𝐹𝑉(𝜉)+𝐿_𝐺𝑉(𝜉)1𝜔+𝐿2𝜆^𝐸𝑉(𝜉)²+12𝜆𝑛(𝜉)²+12𝐿^′𝜕²𝑉(𝜉)𝜕𝜉²𝐿, (56) where ℒ_𝜔 is the infinitesimal generator of system (39). Similar as the proof of Theorem 10, we have ℒ_𝜔𝑓𝑉≤𝐿_∗𝑊𝜂+𝐿𝑒_∗̃𝑊(𝜂)²+12𝜆𝑛(𝜂,0)²+12𝑙^′_∗̃𝜕²𝑊𝜂𝜕𝜂²𝑙_∗̃+12𝑙^′_∗̂𝑙_∗̂+𝑦^′𝜆𝜔+𝑐+2𝑐₁+1𝑐2𝜆²+𝐿^′𝑓₁𝑊𝜂+𝑙^′₁̃𝜕²𝑊𝜂𝜕𝜂²𝑙_∗̃+12𝑙^′₁̃𝜕²𝑊𝜂𝜕𝜂²𝑙₁̃𝑦+𝑙^′₁̂𝑙_∗̂+12𝑙^′₁̂𝑙₁̂. (57) Taking the same 𝑐₁, 𝑐₂ and the corresponding 𝑐 in (46)-(47) as in Theorem 10, we have ℒ_𝜔𝑉≤𝑦^′𝜔. (58) Let 𝜓(𝑦)∶𝑅→𝑅 be any smooth function satisfying 𝜓(0)=0 and 𝑦^′𝜓(𝑦)>0 for each nonzero 𝑦. Then, we can construct the control law as 𝜔=−𝜓(𝑦), which follows that ℒ_{𝜔=−𝜓(𝑦)}𝑉≤−𝑦^′𝜓𝑦≤0. (59) The rest is proved by using the version of LaSalle's invariance principle [21], which is similar to Theorem 4.6 of [12] and is omitted. The proof of Theorem 12 is completed.

Corollary 13. If the zero-output system (41) satisfies the Lyapunov-like equality constraint (55) with a positive definite and proper function 𝑊(𝜂)∈𝒞²∶𝑅^𝑛−1→𝑅⁺ with 𝑊(0)=0, then system (39) is globally asymptotically stabilizable in probability.

Proof. The proof is similar to that of Theorem 12, and we only note that ℒ_{𝜔=−𝜓(𝑦)}𝑉≤−𝑦^′𝜓𝑦−𝑆<0. (60)

Remark 14. Obviously, if the conditions of Theorem 12 or Corollary 13 are satisfied, then system (39) is GASP under the control law 𝜔=−𝑦. Furthermore, system (1) is GASP under the corresponding control 𝑢=𝐿_𝑔ℎ⁻¹−𝑦+𝑐(𝜉)|_{𝜉=𝜙(𝑥)}−𝐿_𝑓1ℎ−2𝑙^′𝜕²ℎ𝜕𝑥²𝑙. (61)

Remark 15. In this work, we only consider the relative degree 𝜌=1 case. Further efforts should be concentrated on the robust passive control design for uncertain nonlinear stochastic system with any arbitrary relative degree case. In that case, we need to discuss the robust passivity through applying the backstepping technique for nonlinear stochastic systems.

4. Numerical Example

Consider the following nonlinear stochastic system: 𝑥𝑑𝑥=¹𝑥₂−2𝑥₁8𝑥³₁+𝑥₁𝑥₂𝑥𝑑𝑡+¹𝑥₂𝛿(𝑥)+11𝑥𝑑𝑡𝑢𝑑𝑡+¹𝑥₂𝑑𝑤,𝑦=𝑥₁+𝑥₂. (62) The uncertainty 𝛿(𝑥) satisfies ‖𝛿(𝑥)‖≤‖𝑛(𝑥)‖, (63) where 𝑛(𝑥)=𝑘𝑥²₁, (64) and 𝑘 is a given constant satisfying 0<𝑘≤1.

Firstly, we construct the function 𝜙(𝑥) as (𝑥)=𝜃(𝑥)ℎ(𝑥)=𝑥𝜉=𝜙¹−𝑥₂𝑥₁+𝑥₂=𝜂𝑦, (65) then it is obvious that the following holds 𝜙⁻¹=𝑥𝜂,𝑦¹𝑥₂=121𝜂+𝑦2−𝜂+𝑦. (66) Also, it can be found that (𝜕ℎ^′/𝜕𝑥)𝑔=2, (𝜕𝜃^′/𝜕𝑥)𝑔=0 hold for the above system. Thus, all the assumptions are satisfied. According to the discussion in Section 3, we design the controller 𝑢 as 𝑢=𝑣−8𝑥³₁−2𝑥₁𝑥₂+2𝑥₁. (67)

Then, we change system (62) into the following form 𝑑𝜂=−𝜂+𝑦³𝛿𝛿−𝜂−𝑦+𝜂𝑑𝑡+𝜂𝑑𝑤,𝑑𝑦=𝑣+𝑦𝑑𝑡+𝑦𝑑𝑤, (68) with 𝑓_∗=−𝜂³−𝜂,𝑓₁𝑒̃=−1,_∗=𝜂,𝑒₁̃𝑙̃=0,_∗=𝜂,𝑙₁̃𝑒̂=0,_∗=0,𝑒₁̂=12,𝑙_∗̂=0,𝑙₁̂=12,𝑛=𝑘𝜂,04𝜂²,𝑛₁=𝑘42𝜂+𝑦. (69)

Taking 𝑊=𝜂², then from (44), we obtain that 𝐿𝑓_∗𝑊𝜂+𝜆2𝐿𝑒_∗̃𝑊𝜂²+1𝑛2𝜆𝜂,0²+12𝑙^′_∗̃𝜕²𝑊𝜂𝜕𝜂²𝑙_∗̃+12𝑙^′_∗̂𝑙_∗̂=2𝜂−𝜂³−𝜂+2𝜆𝜂⁴+𝑘²𝜂32𝜆⁴+𝜂². (70) Leting 𝑘=1 and  𝜆=1/2, we have −15𝜂16⁴−𝜂²≤0. (71) Hence, system (62) is robust strongly minimum phase. By Theorem 10, we know that system (62) is feedback equivalent to the robust passive system (68), and we have 𝑐₁=2𝜂²1𝑦+4𝑦³,𝑐₂=14𝜂³+38𝜂²1𝑦+4𝜂𝑦²+1𝑦16³. (72) Then, from (47), we have 1𝑐=−4𝜂³−18𝑦³−78𝜂²1𝑦−4𝜂𝑦²1+2𝜂−8. (73) Besides, according to Theorem 12 and Remark 14, we can construct the control law as 𝜔=−𝑦, and it follows that 1𝑣=𝑐𝜂,𝑦−𝑦=−4𝜂³−18𝑦³−78𝜂²1𝑦−4𝜂𝑦²1+2𝜂−𝑦−8, (74) which guarantees that the closed-loop system (68) is GASP.

5. Conclusion

In this paper, we have discussed the robust passivity, feedback equivalence, and global stabilization problems for a class of uncertain nonlinear stochastic systems, which contain the structural uncertainty. Through establishing the robust passivity theory, a robust stochastic version of KYP lemma has been presented for such a class of systems. Then, the feedback equivalence and global stabilization problems have been discussed through the robust strongly minimum phase property. However, more efforts should be concentrated on the robust passive control of uncertain nonlinear stochastic systems with any arbitrary relative degree.

Acknowledgments

This work is partially supported by the National Basic Research Program of China (973 Program) (Grant no. 2012CB215203), the National Natural Science Foundation of China (no. 61203043 and no. 51036002), and the Fundamental Research Funds for the Central Universities.