#### Abstract

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 and lossless zero dynamics. The effectiveness of the proposed results is illustrated by a numerical example.

#### 1. Introduction

Since Willems [1] first founded the concepts of dissipativity and passivity of nonlinear deterministic systems [1], dissipativity and passivity have been studied by many authors; see [2–9] and the references therein. The central result in [2] was a sufficient and necessary condition for an affine nonlinear system to be passive. Reference [6] dealt with the problem of a nonlinear deterministic system feedback equivalence to a passive system. Some results in [7] were derived for a nonlinear deterministic discrete-time system, which are parallel to analogous ones in [6]. For the past decade, many researchers have extended the existing methodology from deterministic systems to stochastic systems; see [10–26] and the references therein. Based on the dissipative point of view, the control problem for nonlinear stochastic Itô systems was discussed in [12, 17]. Moreover, Zhang et al. [18] gave two sufficient conditions for control of nonlinear stochastic Itô systems by Hamilton-Jacobi inequality. Reference [19] studies the control problem for nonlinear Markovian jump by means of geometric control method. Li [20–22] discussed the problems of state-feedback stabilization for high-order stochastic nonlinear systems. Lin et al. [23] addressed the issues of stochastic passivity, feedback equivalence, and global stabilization for a general nonlinear stochastic system. Liu et al. [24] was devoted to the indefinite stochastic discrete-time systems with a linear equality constraint on the terminal state. Sheng et al. [25] have investigated the relationship between Nash equilibrium strategies and the finite horizon control of stochastic Markov jump systems with multiplicative noise.

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 [7]. Likewise, we show that a stochastic discrete-time lossless system can be asymptotically stabilized in probability by output feedback. Then, a necessary and sufficient condition is yielded for a nonlinear stochastic discrete-time system to be feedback equivalent to a lossless system.

The rest of this paper is organized as follows. Section 2 considers the lossless theory for a nonlinear stochastic discrete-time system paralleling that of [7]. Theorem 5 is a necessary and sufficient condition for system to be lossless, which extends Theorem 2.6 of [7] and is used in Section 3. Under some given conditions, Section 3 is concerned with the problem of feedback equivalence to a lossless system. An numerical example is presented to illustrate the effectiveness of our results. In Section 4, conclusions are drawn.

Before concluding this section, let us introduce some notations. represents the transpose of a matrix ; means that is positive definite (positive semidefinite) symmetric matrix; represents the mathematical expectation of a random variable ; is the -dimensional Euclidean space with the usual -norm ; is the vector space of all matrices with entries in ; is the identity matrix with appropriate dimension; is the class of functions twice continuously differential with respect to ; ; .

#### 2. Lossless Systems

Consider the following nonlinear stochastic discrete-time system governed by the Itô difference equation:where , , , , and are smooth mappings with appropriate dimensions and and . is called the system state, is the control input, and is the regulated output. Let be a nonempty set, a -field consisting of subsets of , and a probability measure; that is, is a map from to . We call the triple a probability space. is a sequence of second-order stationary random variables defined on the complete probability space , such that and , where is the Kronecker delta. , , , , and are supposed to be independent of .

We denote by the -algebra generated by , ; that is, . Let represent the space of -valued, square integrable random vectors and consists of all finite sequences , such that is measurable for , where ; that is, is constant. The -norm of is defined by .

A function is called the supply rate on if for any , , of (1) is satisfied . A nonnegative function with is called the storage function.

*Definition 1. *System (1) with supply rate is said to be dissipative on if there exists a storage function , such that for all and ,In this paper, we mainly study the dissipative systems with supply rate .

*Definition 2. *System (1) is said to be passive if there is a storage function such that and For simplicity of our discussion, we give the following definitions.

*Definition 3. *System (1) with storage function is said to be strictly passive if for all and

It is equivalent to the following inequality:where .

*Definition 4. *System (1) with storage function is said to be lossless if for all and It is easy to show that system (1) with storage function is lossless if and only ifIn what follows, we give a fundamental property of lossless systems.

Theorem 5. *System (1) with a -storage function is lossless if and only if satisfies**which is quadratic in .*

*Proof. *If system (1) is lossless, by Definition 4, there exists a storage function such thatIt is clear that is quadratic in . Let , then .

Hence, we can obtain thatTaking , (9) and (10) are given.

Conversely, since (11) is quadratic in , we use Taylor’s expansion formula and haveObviously,Together with (8)–(10) and (1), we conclude thatThis implies that system (1) is lossless with .

Corollary 6. *Consider a discrete-time linear system**By Theorem 5, system (17) with is lossless if and only if there exists a matrix such that*

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.

*Definition 7 (see [26]). *Consider the following stochastic system:(a) of (19) is called stable in probability if for any ,(b) of (19) is called locally asymptotically stable in probability if (20) holds and(c) of (19) is called globally asymptotically stable in probability if (20) holds and

*Definition 8. *System (1) is called locally (resp., globally) zero-state observable if there is a neighborhood of such that, for all (resp., ),

*Remark 9. *By Definition 7, it is clear that system (1) is zero-state observable iff

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

Theorem 10. *Assume that system (1) with is lossless. Let**then, system (1) is zero-state observable iff .*

*Proof. *Because system (1) is lossless, by Theorem 5 it follows thatWe can show that iffTogether with Definition 7 and Remark 9, Theorem 10 is easily obtained.

Based on the above, we study the problem of stabilization for system (1).

Theorem 11. *If system (1) is lossless, is a smooth mapping with , and for any , then the close-loop system**is locally asymptotically stable in probability if and only if system (1) is locally zero-state observable. If is proper (i.e., for any , is compact), then it is globally asymptotically stable in probability if and only if system (1) is globally zero-state observable.*

*Proof (sufficiency). *By Definition 4 and (28), we haveAccording to Theorem 2.1 in [26], system (1) is stable in probability. DefineFor any , it follows thatIt means that ; otherwise, it contradicts , . Then,By Definition 8, it yields . By Theorem in [26], is locally asymptotically stable in probability. If is proper, then is globally asymptotically stable in probability.

The necessity is similar to the proof given in [27] and is omitted.

#### 3. Feedback Equivalence to a Lossless System

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 (1) can be achieved by means of state feedback if and only if system (1) have relative degree and lossless zero dynamics.

*Definition 12. *System (1) is said to have relative degree (resp., uniform relative degree) at if (resp., , ) is nonsingular.

By Definition 12, if system (1) has relative degree , there is a neighborhood of such that exists. Set , then for any and

Let , it is obvious that , given

*Definition 13. *Systems (33)-(34) are called the zero dynamics of system (1).

If system (1) has uniform relative degree , then and the global zero dynamics (33)-(34) exist.

In the following, we give definition of system (1) having lossless zero dynamics.

*Definition 14. *Assume that is nonsingular. System (1) is said to have locally lossless zero dynamics if there exists a function , which is positive definite and is locally defined on the neighborhood of such that the following conditions are satisfied:which is quadratic in .

System (1) has globally lossless zero dynamics if there exists a positive definite function satisfying (35) for all .

Now, we analyze a lossless system which has relative degree at and present the following theorem.

Theorem 15. *Assume that system (1) is lossless with and is nondegenerate at , then* (1)* if and only if ;* (2)*System (1) has relative degree at .*

*Proof. *(1) By Theorem 5 and (10), we haveIt is obvious that for all It is easy to show that implies .

On the other hand, by , it yields thatand , . This means that has a unique solution . Then, .

(2) From (1), is nonsingular. By Definition 12, system (1) has relative degree at .

*Remark 16. *If linear system (17) is lossless with , then , so is nondegenerate at .

The following theorem can be viewed as the global version of Theorem 15.

Theorem 17. *If system (1) with is lossless and satisfies one of the following conditions:**Then, for all ,* (1)* if and only if ;* (2)*System (1) has uniform relative degree .*

*Proof. *The proof of Theorem 17 is the same as that of Theorem 15.

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

Theorem 18. *If and the conditions of Theorem 15 hold, then the zero dynamics of system (1) at are lossless.*

*Proof. *By Theorem 15, the zero dynamics (33)-(34) of system (1) locally exist at .

Moreover, by the losslessness of system (1), we can obtain thatTake , we haveBy (40), we can show that is quadratic in if . So, we conclude that the zero dynamics of system (1) are lossless.

Now, we attempt to consider a state feedback as follows:where and are smooth functions defined near locally or globally and is nonsingular. We solve the problem of feedback equivalence to a lossless system.

Theorem 19. *If and is nondegenerate at , then the following conditions are equivalent.* (1)*System (1) is locally feedback equivalent to a lossless system with ;* (2)*System (1) has relative degree at and has lossless zero dynamics.*

*Proof. *If there exists a state feedback (42) such that system (1) is lossless. It can be known that systemis lossless with .

Since is invertible, it yields . According to Theorem 15, it is clear that is nonsingular and so is . Therefore, system (1) has relative degree at .

In addition, we know that the zero dynamics of system (43) areBecause system (43) is lossless, proceed along the same lines of Theorem 18, we have , and is quadratic in .

Then, and is quadratic in . This means that zero dynamics of system (1) are lossless.

On the other hand, when system (1) has relative degree at , we defineWe apply (45) to system (1), system (1) can be rewritten asSet , then system (46) is replaced bySince and is nondegenerate at , we can see that there exists a neighborhood of satisfying .

Define and asMoreover, by the fact that is quadratic in , thenWe can show thatTaking expectation on both sides of (49) and using (50), it is easy to show thatBy Definition 14, we haveand is quadratic in .

Finally, the losslessness of system (47) is achieved by Theorem 5. This means system (1) is locally feedback equivalent to a lossless system (47).

Similar to the proof of Theorem 19, we present the global version of Theorem 19.

Theorem 20. *If and satisfies one of the conditions, or is a strictly convex Lyapunov function. Then, the following are equivalent:* (1)*System (1) is globally feedback equivalent to a lossless system with ;* (2)*System (1) has uniform relative degree at and has globally lossless zero dynamics.*

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

*Example 21. *Consider the following discrete-time nonlinear stochastic system :Let us choose the storage function as , .

We consider a static feedback of the formby Theorem 5, system is lossless.

System is locally feedback equivalent to system ; by Theorem 19, system have relative degree at and lossless zero dynamics.

#### 4. Conclusions

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 and lossless zero dynamics.

#### Conflict of Interests

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

#### Acknowledgment

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