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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.

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 [

On the other hand, due to the great many applications of stochastic Itô systems in real world [

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 [

The remainder of the paper is organized as follows: Section

For convenience, we adopt the following notations:

First of all, let

Consider the following uncertain nonlinear stochastic control system governed by Itô's differential equation:

An uncertain system of the form (

Here, (

In view of Definition

System (

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.

System (

Sufficiency. According to the Cauchy inequality and considering the fact of

Denote

Indeed, Theorem

From the proof of Theorem

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

System (

System (

In this section, we discuss the feedback equivalence and global stabilization problems for the general nonlinear stochastic system (

The nonlinear stochastic system (

In what follows, we only consider the simple case of

Motivated by the works of [

Moreover, we set the control input as

Let zero-output system describe the internal dynamic of a system which is consistent with the constraint

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

System (

(i) Robust weakly minimum phase if the zero-output system (

(ii) Robust minimum phase if for the zero-output system (

(iii) Robust strongly minimum phase, if for the zero-output system (

The following theorem discusses the relationship between the passivity of system (

If system (

We construct the storage function

On the other hand, suppose that there is a feedback control law

Note that

If the zero-output system (

From Remark

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

Suppose that system (

We construct the storage function

If the zero-output system (

The proof is similar to that of Theorem

Obviously, if the conditions of Theorem

In this work, we only consider the relative degree

Consider the following nonlinear stochastic system:

Firstly, we construct the function

Then, we change system (

Taking

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.

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.