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This paper will study stochastic losslessness theory for nonlinear stochastic discrete-time systems, which are expressed by the Itô-type difference equations. A necessary and sufficient condition is developed for a nonlinear stochastic discrete-time system to be lossless. By the stochastic lossless theory, we show that a nonlinear stochastic discrete-time system can be lossless via state feedback if and only if it has relative degree

Since Willems [

According to the above results, we have been interested in the concepts of losslessness, relative degree, and zero dynamics for stochastic discrete-time systems. This paper will study losslessness theory for nonlinear stochastic discrete-time systems, which are expressed by the Itô-type difference equations. The main contributions of this paper can be summarized as follows. A necessary and sufficient condition is developed for a nonlinear stochastic discrete-time system to be lossless, which can be viewed as a stochastic generalized version of [

The rest of this paper is organized as follows. Section

Before concluding this section, let us introduce some notations.

Consider the following nonlinear stochastic discrete-time system governed by the Itô difference equation:

We denote by

A function

System (

In this paper, we mainly study the dissipative systems with supply rate

System (

For simplicity of our discussion, we give the following definitions.

System (

It is equivalent to the following inequality:

System (

It is easy to show that system (

In what follows, we give a fundamental property of lossless systems.

System (

If system (

Hence, we can obtain that

Conversely, since (

Together with (

Consider a discrete-time linear system

By Theorem

In the sequel, we give the following definitions about stochastic stability and zero-state observability, which are useful in treating the stabilization problem for lossless systems.

Consider the following stochastic system:

System (

By Definition

Below, we point out the zero-state observability condition for lossless system.

Assume that system (

Because system (

Based on the above, we study the problem of stabilization for system (

If system (

By Definition

For any

By Definition

The necessity is similar to the proof given in [

In this section, we solve the problem of feedback equivalence to a lossless system via state feedback. To this end, some preliminary definitions are needed, such as relative degree and zero dynamics. These concepts play crucial roles in this paper. It will be shown that the losslessness of system (

System (

By Definition

Let

Systems (

If system (

In the following, we give definition of system (

Assume that

System (

Now, we analyze a lossless system which has relative degree

Assume that system (

System (

(1) By Theorem

On the other hand, by

(2) From (1),

If linear system (

The following theorem can be viewed as the global version of Theorem

If system (

System (

The proof of Theorem

From the above, we give a necessary condition under which a lossless system has lossless zero dynamics at

If

By Theorem

Moreover, by the losslessness of system (

By (

Now, we attempt to consider a state feedback as follows:

If

System (

System (

If there exists a state feedback (

Since

In addition, we know that the zero dynamics of system (

Because system (

Then,

On the other hand, when system (

We apply (

Set

Since

Define

Moreover, by the fact that

We can show that

Taking expectation on both sides of (

By Definition

Finally, the losslessness of system (

Similar to the proof of Theorem

If

System (

System (

In the following, an example is presented to illustrate the effectiveness of our results.

Consider the following discrete-time nonlinear stochastic system

Let us choose the storage function as

We consider a static feedback of the form

System

This paper has investigated the problem of losslessness and feedback equivalence for nonlinear stochastic discrete-time systems. A necessary and sufficient condition is developed for a nonlinear stochastic discrete-time system to be lossless. Under some conditions, it has been shown that a nonlinear stochastic discrete-time system can be lossless via state feedback if and only if the system have relative degree

The authors declare that there is no conflict of interests regarding the publication of this paper.

This work is supported by China National Science Foundation (61170054 and 61402265).