# Losslessness of Nonlinear Stochastic Discrete-Time Systems

Authors: Xikui Liu; Yan Li; Ning Gao
Journal: Discrete Dynamics in Nature and Society
 (2015)

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## 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 degree0,…,0 and lossless zero dynamics. The effectiveness of the proposed results is illustrated by a numerical example.


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## Body

## 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 H∞ control problem for nonlinear stochastic Itô systems was discussed in [12, 17]. Moreover, Zhang et al. [18] gave two sufficient conditions for H∞ control of nonlinear stochastic Itô systems by Hamilton-Jacobi inequality. Reference [19] studies the H∞ 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 H2/H∞ 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. Section2 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.MT represents the transpose of a matrix M; M>0(M≥0) means that M is positive definite (positive semidefinite) symmetric matrix; E[x] represents the mathematical expectation of a random variable x; Rk is the k-dimensional Euclidean space with the usual 2-norm ·; Rm×n is the vector space of all m×n matrices with entries in R; I is the identity matrix with appropriate dimension; C2 is the class of functions V(x) twice continuously differential with respect to x; N={0,1,2,…}; NK={0,1,2,…,K}.


## 2. Lossless Systems

Consider the following nonlinear stochastic discrete-time system governed by the Itô difference equation:(1)xt+1=f1xt+g1xtut+f2xtωtzt=Jxt+hxtut,where f1, g1, f2, h, and J are smooth mappings with appropriate dimensions and f1(0)=0 and f2(0)=0. x∈Rn is called the system state, u∈Rm is the control input, and z∈Rm is the regulated output. Let Ω be a nonempty set, F a σ-field consisting of subsets of Ω, and P a probability measure; that is, P is a map from F to [0,1]. We call the triple (Ω,F,P) a probability space. ω(t) is a sequence of second-order stationary random variables defined on the complete probability space (Ω,F,P), such that E[ω(t)]=0 and E[ω(t)ω(s)]=δst, where δst is the Kronecker delta. x, u, f1, g1, and f2 are supposed to be independent of ω.We denote byFt the σ-algebra generated by ω(t), t∈NK; that is, Ft=σ(ω(t):t∈NK). Let L2(Ω,Rk) represent the space of Rk-valued, square integrable random vectors and lω2(NK,Rk) consists of all finite sequences y={y(t):y(t)∈Rk}t∈NK, such that y(t)∈L2(Ω,Rk) is Ft-1 measurable for t∈NK, where F-1={ϕ,Ω}; that is, y(0) is constant. The l2-norm of lω2(NK,Rk) is defined by y(·)lω2(NK,Rk)=(∑t=0KEy(t)2)1/2.A functionW(·,·):Rm×Rm is called the supply rate on NK if for any u∈lω2(NK,Rk), x(t)∈Rn, z(t) of (1) is satisfied ∑t=0KEW(u(t),z(t))<∞. A nonnegative function V:Rn→R+ with V(0)=0 is called the storage function.Definition 1.
System (1) with supply rate W is said to be dissipative on NK if there exists a storage function V, such that for all u(t)∈Rm and t∈NK,(2)EVxt+1-EVxt≤EWzt,ut.
In this paper, we mainly study the dissipative systems with supply rateW(z(t),u(t))=z(t)Tu(t).Definition 2.
System (1) is said to be passive if there is a storage function V such that ∀t∈NK and ∀u(t)∈Rm(3)EVxt+1-EVxt≤EztTut.
For simplicity of our discussion, we give the following definitions.Definition 3.
System (1) with storage function V is said to be strictly passive if for all t∈NK and u(t)∈Rm(4)EVxt+1-EVxt<EztTut.It is equivalent to the following inequality:(5)EVxt+1-EVxt≤EztTut-ESxt,where S:Rn→R+.Definition 4.
System (1) with storage function V is said to be lossless if for all t∈NK and u(t)∈Rm(6)EVxt+1-EVxt=EztTut.
It is easy to show that system (1) with storage function V is lossless if and only if(7)EVxK-Vx0=∑t=0K-1EztTut,∀t∈NK,∀ut,∀x0.
In what follows, we give a fundamental property of lossless systems.Theorem 5.
System (1) with a C2-storage function V is lossless if and only if V satisfies(8)EVf1xt+f2xtωt=EVxt,(9)E∂Vα∂αα=f1x+f2xωg1x=EJxT,(10)Eg1xT∂V2α∂α2α=f1x+f2xωg1x=Ehx+hxT,(11)Vxt+1=Vf1x+g1xu+f2xωwhich is quadratic in u.Proof.
If system (1) is lossless, by Definition 4, there exists a storage function V such that(12)EVxt+1=EVxt+EztTut=EVxt+EJxTu+EuThx+hxT2u.It is clear that V(x(t+1))=V(f1(x)+g1(x)u+f2(x)ω) is quadratic in u. Let u=0, then E[V(f1(x(t))+f2(x(t))ω(t))]=E[V(x(t))].
Hence, we can obtain that(13)E∂Vxt+1∂u=E∂Vα∂αα=f1x+g1xu+f2xωg1x=E∂Vα∂αα=f1x+g1xu+f2xωg1x=EJxT+EuThx+hxT,E∂2Vxt+1∂u2=Eg1xT∂2Vα∂α2α=f1x+g1xu+f2xωg1x=Ehx+hxT.Taking u=0, (9) and (10) are given.
Conversely, since (11) is quadratic in u, we use Taylor’s expansion formula and have(14)Vxt+1=Vf1x+g1xu+f2xω=Ax+Bxu+12uTCxu.Obviously,(15)Ax=Vf1xt+f2xtωt,Bx=∂Vf1x+g1xu+f2xω∂uu=0g1x=∂Vα∂αα=f1x+f2xωg1x,Cx=g1xT∂2Vf1x+g1xu+f2xω∂u2u=0·g1x=g1xT∂2Vα∂α2α=f1x+f2xωg1x.
Together with (8)–(10) and (1), we conclude that(16)EVxt+1-EVxt=EztTut,∀t∈NK,∀ut∈Rm.This implies that system (1) is lossless with V∈C2.Corollary 6.
Consider a discrete-time linear system(17)xt+1=A1xt+B1ut+A2xtωt,zt=Cxt+Dut.
By Theorem5, system (17) with V(x)=xTPx is lossless if and only if there exists a matrix P≥0 such that(18)ExTA1TPA1+A2TPA2x=ExTPx,EB1TPA1+PA2x=ECx,B1TPB1=D+DT.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:(19)xt+1=fxt+gxtωt,x0=x0∈Rn,f0=0,g0=0.(a)
x=0 of (19) is called stable in probability if for any ϵ>0,(20)limx0→0⁡Psup⁡xt>ϵ,t≥0=0.(b)
x=0 of (19) is called locally asymptotically stable in probability if (20) holds and(21)limx0→0⁡Plimt→∞⁡xt=0=1.(c)
x=0 of (19) is called globally asymptotically stable in probability if (20) holds and(22)Plimt→∞⁡xt=0=1.Definition 8.
System (1) is called locally (resp., globally) zero-state observable if there is a neighborhood U of x=0 such that, for all x(0)=x0∈U (resp., Rn),(23)Eztut=0=EJxt=0impliesx0=0.Remark 9.
By Definition7, it is clear that system (1) is zero-state observable iff(24)x0∈Rn∣EJxt=0,∀t∈NK=0.Below, we point out the zero-state observability condition for lossless system.Theorem 10.
Assume that system (1) with V∈C2 is lossless. Let(25)S=x0∈Rn∣E∂Vα∂αα=f1x+f2xωg1x=0,∀t∈NK;then, system (1) is zero-state observable iff S=0.Proof.
Because system (1) is lossless, by Theorem 5 it follows that(26)EJxtT=E∂Vα∂αα=f1x+f2xωg1x,∀t∈NK.We can show that S=0 iff(27)x0∈Rn∣EJxt=0,∀t∈NK=0.Together 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, ψ is a smooth mapping with ψ(0)=0, and zTψ(z)>0 for any z≠0, then the close-loop system(28)xt+1=f1xt-g1xtψzt+f2xtωtis locally asymptotically stable in probability if and only if system (1) is locally zero-state observable. If V is proper (i.e., for any a>0, V-1([0,a]) is compact), then it is globally asymptotically stable in probability if and only if system (1) is globally zero-state observable.Proof (sufficiency).
By Definition4 and (28), we have(29)EVxt+1-EVxt=-EztTψzt≤0.According to Theorem  2.1 in [26], system (1) is stable in probability. Define(30)γ=x∈Rn:EΔVxt=EVxt+1-EVxt=0.
For anyx0=x∈γ, it follows that(31)0=EVxt+1-EVxt=-EztTψzt≤0.It means that z(t)Tψ(z(t))=0; otherwise, it contradicts zTψ(z)>0, ∀z≠0. Then,(32)zt=0,ut=-ψzt=0,∀t∈N.
By Definition8, it yields x=0. By Theorem  2.3 in [26], x=0 is locally asymptotically stable in probability. If V is proper, then x=0 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 (0,…,0) and lossless zero dynamics.Definition 12.
System (1) is said to have relative degree (resp., uniform relative degree) (0,…,0) at x=0 if h(0) (resp., h(x), ∀x∈Rn) is nonsingular.By Definition12, if system (1) has relative degree (0,…,0), there is a neighborhood U⊂Rn of x=0 such that h-1(x) exists. Set u∗(t)=-h-1(x(t))J(x(t)), then E[z(t)]≡0 for any x(t)∈U and(33)xt+1=f1xt+g1xtu∗t+f2xtωt.LetZ∗={x∈Rn∣E[z(t)]=0}, it is obvious that Z∗=U, given(34)u∗t=-h-1xtJxt,∀x∈U=Z∗.Definition 13.
Systems (33)-(34) are called the zero dynamics of system (1).
If system (1) has uniform relative degree (0,…,0), then Z∗=U≡Rn and the global zero dynamics (33)-(34) exist.In the following, we give definition of system (1) having lossless zero dynamics.Definition 14.
Assume thath(0) is nonsingular. System (1) is said to have locally lossless zero dynamics if there exists a function V∈C2, which is positive definite and is locally defined on the neighborhood U of x=0 such that the following conditions are satisfied:(35)EVf1x+g1xu∗+f2xω=EVx,Vf1x+g1xu∗+f2xω+g1xuwhich is quadratic in u.
System (1) has globally lossless zero dynamics if there exists a positive definite function V∈C2 satisfying (35) for all x∈Rn.
Now, we analyze a lossless system which has relative degree(0,…,0) at x=0 and present the following theorem.Theorem 15.
Assume that system (1) is lossless with V∈C2 and V is nondegenerate at x=0, then(1)
Rank{g1(0)}=m if and only if E[h(0)T+h(0)]>0;(2)
System (1) has relative degree (0,…,0) at x=0.Proof.
(1) By Theorem5 and (10), we have(36)Eg10T∂V2α∂α2α=f1x+f2xωg10=Eh0+h0T.It is obvious that for all x∈Rm(37)Eg10xT∂V2α∂α2α=f1x+f2xωg10x=ExTh0+h0x.It is easy to show that Rank{g1(0)}=m implies E[h(0)T+h(0)]>0.
On the other hand, byE[h(0)T+h(0)]>0, it yields that(38)Eg10xT∂V2α∂α2α=f1x+f2xωg10x>0and g1(0)x≠0, ∀x≠0. This means that g1(0)x=0 has a unique solution x=0. Then, Rank{g1(0)}=m.
(2) From (1),J(0) is nonsingular. By Definition 12, system (1) has relative degree (0,…,0) at x=0.Remark 16.
If linear system (17) is lossless with V(x)=xTPx/2>0, then ∂V2(x)/∂x2=P>0, so V is nondegenerate at x=0.The following theorem can be viewed as the global version of Theorem15.Theorem 17.
If system (1) with V∈C2 is lossless and satisfies one of the following conditions:(39)∂V2α∂α2α=f1x+f2xω>0,orVisastrictlyconvex.Then, for all x∈Rn,(1)
Rankg1(x)=m if and only if E[h(x)T+h(x)]>0;(2)
System (1) has uniform relative degree (0,…,0).Proof.
The proof of Theorem17 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 atx=0.Theorem 18.
IfRank{g1(0)}=m and the conditions of Theorem 15 hold, then the zero dynamics of system (1) at x=0 are lossless.Proof.
By Theorem15, the zero dynamics (33)-(34) of system (1) locally exist at x=0.
Moreover, by the losslessness of system (1), we can obtain that(40)EVf1x+g1xu+f2xω-EVxt=EztTut.Take u(t)=u∗(t), we have(41)EVf1x+g1xu∗t+f2xω=EVx,∀x∈Z∗.
By (40), we can show that V(f1(x)+g1(x)u∗(t)+f2(x)ω+g1(x)u~) is quadratic in u~ if u=u∗+u~. So, we conclude that the zero dynamics of system (1) are lossless.
Now, we attempt to consider a state feedback as follows:(42)ut=αxtυt+βxt,β0=0,where α(x) and β(x) are smooth functions defined near x=0 locally or globally and α(x) is nonsingular. We solve the problem of feedback equivalence to a lossless system.Theorem 19.
IfRank{g1(0)}=m and V is nondegenerate at x=0, then the following conditions are equivalent.(1)
System (1) is locally feedback equivalent to a lossless system with V∈C2;(2)
System (1) has relative degree (0,…,0) at x=0 and has lossless zero dynamics.Proof.
If there exists a state feedback (42) such that system (1) is lossless. It can be known that system(43)xt+1=f1xt+g1xtβxt+f2xtωt+g1xtαxtυt,zt=Jxt+hxtβxt+hxtαxtυtis lossless with V∈C2.
Sinceα(0) is invertible, it yields Rank{g~(0)}=m. According to Theorem 15, it is clear that h(0)α(0) is nonsingular and so is h(0). Therefore, system (1) has relative degree (0,…,0) at x=0.
In addition, we know that the zero dynamics of system (43) are(44)xt+1=f~∗xt=f1xt-g1xth-1xtJxt+f2xtωtxt,∀x∈Z~∗.
Because system (43) is lossless, proceed along the same lines of Theorem 18, we have E[V(f~∗(x))]=E[V(x)], and V(f~∗(x)+g~(x)υ) is quadratic in υ.
Then,E[V(f∗(x))]=E[V(x)] and V(f∗(x)+g1(x)u) is quadratic in u. This means that zero dynamics of system (1) are lossless.
On the other hand, when system (1) has relative degree (0,…,0) at x=0, we define(45)ut=u∗t+h-1xtξt.
We apply (45) to system (1), system (1) can be rewritten as(46)xt+1=f1xt+g1xtu∗xt+f2xtωt+g1xtξt,zt=Jxt+hxtu∗t+ξt=ξt.
Setz(t)=ξ(t)=J~(x(t))+h~(x(t))ν(t), then system (46) is replaced by(47)xt+1=f1xt+g1xtu∗xt+f2xtωt+g1xtJ~xt+h~xtνt,zt=J~xt+h~xtνt.
SinceRank{g1(0)}=m and V is nondegenerate at x=0, we can see that there exists a neighborhood U of x=0 satisfying g1(x)T(∂V2(α)/∂α2)α=f∗(x)g1(x)>0.
Defineh~(x) and J~(x) as(48)h~x=12g1xT∂V2α∂α2α=f∗xg1x-1,J~x=-h~x∂Vα∂αα=f∗xg1xT.
Moreover, by the fact thatV(f∗(x)+g1(x)u) is quadratic in u, then(49)Vf∗x+g1xJ~x=Vf∗x+∂Vα∂αα=f∗xg1xJ~x+12J~xTg1Tx∂2Vα∂α2α=f∗xg1xJ~x.
We can show that(50)E∂Vα∂αα=f∗x+g1xJ~xg1xh~x=E∂Vα∂αα=f∗xg1xh~x+EJ~xTg1xT∂2Vα∂α2α=f∗xg1xh~x=EJ~xT,Eg1xT∂V2α∂α2α=f∗x+g1xJ~xg1xh~x=Eg1xT∂2Vα∂α2α=f∗xg1xh~x=Eh~x+Eh~xT.
Taking expectation on both sides of (49) and using (50), it is easy to show that(51)EVf∗x+g1xJ~x=EVf∗x.
By Definition14, we have(52)EVf∗x+g1xJ~x=EVxand V(f∗(x)+g1(x)J~(x)+g1(x)h~(x)ν) is quadratic in ν.
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 Theorem19, we present the global version of Theorem 19.Theorem 20.
IfRank{g1(x)}=m and V satisfies one of the conditions, (∂V2(α)/∂α2)α=f1(x)+f2(x)ω>0 or V is a strictly convex Lyapunov function. Then, the following are equivalent:(1)
System (1) is globally feedback equivalent to a lossless system with V∈C2;(2)
System (1) has uniform relative degree (0,…,0) at x=0 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∑‍:(53)xt+1=1212122122x1x2+1-x1x11+x1ut+12-12122-122x1x2ωt,zt=x1+x21+2x1-x12+1-x121+x1+2x11+x1ut.
Let us choose the storage function asV(x)=xTPx, P=1002.
We consider a static feedback of the form(54)ut=1+x1υt,by Theorem 5, system ∑‍~(55)xt+1=1212122122x1x2+12-12122-122x1x2ωt+1-x12x1υt,zt=x1+x21+2x1-x12+1-x122+2x1utis lossless.
System∑‍ is locally feedback equivalent to system ∑‍~; by Theorem 19, system ∑‍ have relative degree (0,…,0) at x=0 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(0,…,0) and lossless zero dynamics.



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