# H

Authors: Ming Gao; Weihai Zhang; Zhengmao Zhu
Journal: Mathematical Problems in Engineering
 (2015)

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## Abstract

This paper studies the infinite horizonH∞ control problem for a general class of nonlinear stochastic systems with time-delay and multiplicative noise. The exponential/asymptotic mean square H∞ control design of delayed nonlinear stochastic systems is presented by solving Hamilton-Jacobi inequalities. Two numerical examples are provided to show the effectiveness of the proposed design method.


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## Body

## 1. Introduction

It is well known thatH∞ control is one of the most effective approaches to eliminate the effect of the external disturbance [1]. For deterministic linear systems, H∞ norm is defined by a norm of the transfer function, which cannot be extended to stochastic or nonlinear systems directly. In 1989, Doyle et al. found that, from the view point of time-domain, the norm of a transfer function was the L2-induced norm of the input-output operator [2], which made it possible to develop the nonlinear or stochastic H∞ theory [3, 4]. Following along the lines of [4], Zhang and Chen developed infinite and finite horizon nonlinear stochastic H∞ control designs by means of Hamilton-Jacobi equations [5]. Moreover, the mixed H2/H∞ control has also received much attention due to its important significance in practical applications [6].The phenomena of time-delay are frequently encountered in many engineering applications owing to the finite speed of information processing [7]. Time-delay, nonlinearity, and stochasticity are arguably three of the main sources in reality which result in the complexity of a system. Over the past years, the stability of delayed nonlinear stochastic systems (DNSSs) has gained significant research interests [8–15]. In [8], Mao established the LaSalle-type theorems for the solutions of stochastic differential delay equations, which was applied to establish sufficient criteria for the stochastically asymptotic stability of the delay equations. In [10], the problem of exponential stability for a class of impulsive nonlinear stochastic differential equations with mixed time-delays was investigated, and some interesting results were derived. In [13], the delay-dependent stability conditions for DNSSs were derived based on the convergence theorem for semimartingale inequalities.Although many results for the stability analysis of DNSSs have been published, theH∞ control problem of DNSSs has received relatively little attention [16–18]. In [16], the H∞ analysis problem was studied for a general class of nonlinear stochastic systems with time-delay by using the Razumikhin-type method. In [17], the problem of robust H∞ output feedback control was studied for a class of uncertain discrete-time DNSSs with missing measurements. In [18], the quantized H∞ control problem was investigated for delayed nonlinear stochastic network-based systems with data missing. However, most of the above literatures only considered the stochastic systems with state-dependent noise. As pointed in [19], the control input and external disturbance may also be corrupted by noise. Therefore, it is necessary to study the stochastic systems with state, control, and disturbance-dependent noise [20, 21].Motivated by the preceding discussion, this paper will investigate the infinite horizonH∞ control for a class of nonlinear stochastic state-delayed systems with multiplicative noise. Compared with [16–18, 22], the considered system in this paper is more general since state, control, and disturbance enter into the diffusion term simultaneously. By means of Hamilton-Jacobi inequalities (HJIs), a sufficient condition is derived for the exponential and asymptotic mean square H∞ control of DNSSs, respectively. In contrast to the conditions for delay-free H∞ control [20, 21], the current HJIs depend on more variables owing to the appearance of time-delay. Finally, two numerical examples are given to demonstrate the effectiveness of the obtained results.Throughout this paper, the following notations will be used.Rn is n-dimensional Euclidean space. Rn×m is the set of all n×m real matrices. A′ is the transpose of a matrix A. A>0  (A≥0): A is a positive definite (positive semidefinite) symmetric matrix. E[·] is the mathematical expectation. x is the Euclidean norm of a vector x. LF2(R+;Rl) is the space of nonanticipative stochastic processes y(t)∈Rl with respect to an increasing σ-algebras Ft  (t≥0) satisfying y(t)LF2(R+;Rl)=(E∫0∞‍y(t)2dt)1/2<∞. C2,1(U,T) is the class of functions V(x,t) twice continuously differential with respect to x∈U and once continuously differential with respect to t∈T, except possibly at the point x=0. C([-τ,0],Rn) is a vector space of all continuous Rn-valued functions defined on [-τ,0]. symM: M+M′.


## 2. Definitions and Preliminaries

Consider the following delayed nonlinear stochastic system with multiplicative noise:(1)dxt=fxt,xt-τ,t+gxt,xt-τ,tutfxt,xt-τ,t+gxt,xt-τ,tut+hxt,xt-τ,tvtdt+lxt,xt-τ,t+qxt,xt-τ,tutlxt,xt-τ,t+qxt,xt-τ,tut+sxt,xt-τ,tvtdwt,zt=colmxt,xt-τ,t,ut∶=mxt,xt-τ,tut,t≥0,x(t)=ϕt∈CF0b-τ,0;Rn,-τ≤t≤0,
where x(t)∈Rn, u(t)∈Rnu, v(t)∈Rnv, and z(t)∈Rnz represent the system state, control input, exogenous disturbance, and regulated output, respectively. w(t) is the one-dimensional standard Wiener process defined on a complete filtered space (Ω,F,{Ft}t∈R+,P), a filtration {Ft}t∈R+ satisfying usual conditions. CF0b([-τ,0];Rn) denotes all F0-measurable bounded C([-τ,0];Rn)-valued random variable ξ={ξ(θ):-τ≤θ≤0}. Assume that f, g, h, l, q, s, and m satisfy the local Lipschitz condition and the linear growth condition, which guarantee system (1) has a unique strong solution [23]. Moreover, suppose that f(0,0,t)=l(0,0,t)=m(0,0,t)≡0; hence x≡0 is an equilibrium point of (1). For simplicity, we denote x∶=x(t) and xτ=x(t-τ).For eachV∈C2,1(Rn×R+;R+), an operator LV:Rn×Rn×R+→R associated with (1) is defined as follows [8]:
(2)LVx,y,t=Vtx,t+Vx′x,t×fx,y,t+gx,y,tut+hx,y,tvt+12lx,y,t+qx,y,tut+sx,y,tvt′Vxxx,t×lx,y,t+qx,y,tut+sx,y,tvt,
where Vt(x,t)=∂V(x,t)/∂t, Vx(x,t)=∂V(x,t)/∂x1,…,∂V(x,t)/∂xn′, and Vxx(x,t)=∂2V(x,t)/∂xi∂xjn×n.To deal with the infinite horizonH∞ control of system (1), the following internal stability is needed.Definition 1 (see [23]).
The delayed nonlinear stochastic system,(3)dxt=fxt,xt-τ,tdt+lxt,xt-τ,tdwt,t≥0,xt=ϕt∈CF0b-τ,0;Rn,-τ≤t≤0,
is exponentially mean square stable, if there exist positive constants ρ>0 and ϱ>0 such that every solution x(t) of (3) satisfies
(4)Ext2≤ρϕ2exp⁡-ϱt,
where ϕ2=Emax⁡-τ≤t≤0ϕt2.Lemma 2 (see [24]).
System (3) is exponentially mean square stable, if there exist a positive definite Lyapunov function V(x,t)∈C2,1(Rn,R+;R+) and c1,c2,c3,c4>0 with c1c3>c2c4 such that(i)
c1x2≤Vx,t≤c2x2, ∀x,t∈Rn×-τ,∞,(ii)
LV(x,y,t)v=0≤-c3x2+c4y2, ∀t>0.Definition 3.
For givenγ>0, u(t)=u*(t)∈LF2(R+;Rnu) is said to be an exponential mean square H∞ control of system (1), if(i)
for any nonzerov(t)∈LF2(R+;Rnv) and x(t)≡0, t∈[-τ,0], one always has
(5)ztLF2(R+;Rnz)≤γvtLF2R+;Rnv;(ii)
system (1) with v(t)=0 and u(t)=u*(t) is internally stable; that is, the system
(6)dxt=fxt,xt-τ,t+gxt,xt-τ,tu*tdt+lxt,xt-τ,tqxt,xt-τ,tu*t+qxt,xt-τ,tu*tdwtis exponentially mean square stable.Equation (5) is equivalent to Lzv∞≤γ, where the perturbation operator Lzv is defined by Lzv:LF2(R+;Rnv)↦LF2(R+;Rnz) as
(7)Lzvv=zxt,u*,v,xτ,t,t≥0,v∈LF2R+;Rnv,Lzv∞=sup⁡v∈LF2(R+;Rnv),v≠0,x(0)=0zLF2(R+;Rnz)vLF2(R+;Rnv).Definition 4.
In (ii) of Definition3, if the equilibrium point of system (6) is asymptotically mean square stable, that is,
(8)lim⁡t→∞Ext2=0,
and (5) holds, then u(t)=u*(t) is called an asymptotic mean square H∞ control.Lemma 5 (see [25]).
For a positive definite symmetric matrixP>0∈Rn×n and any matrices (or vectors) N1∈Rn×m and N2∈Rn×m, one has
(9)N1′PN2+N2′PN1≤N1′PN1+N2′PN2.Lemma 6 (see [21]).
For any vectorsx,b∈Rn and symmetric matrix A∈Rn×n, A-1 exists, and one has
(10)x′Ax+x′b+b′x=x+A-1b′Ax+A-1b-b′A-1b.


## 3. Infinite Horizon StochasticH∞ Control

In this section, several sufficient conditions are presented for the infinite horizonH∞ control of system (1) by using inequality technique.Theorem 7.
Assume that there exist a positive functionV(x,t)∈C2,1(Rn×R+;R+) and c1,c2,c3,c4>0 with c1c3>c2c4 such that(i)
c1x2≤V(x,t)≤c2x2, ∀x,t∈Rn×-τ,∞,(ii)
-m(x,y,t)2≤-c3x2+c4y2, ∀t>0.For given γ>0, if  V(x,t) solves the Hamilton-Jacobi inequalities (HJIs)
(11)Vtx,t+Vx′x,tfx,y,t+12l′x,y,tVxxx,tlx,y,t+m′x,y,tmx,y,t+14l′x,y,tVxxx,tsx,y,t+Vx′x,thx,y,t×γ2I-s′x,y,tVxxx,tsx,y,t-1·s′x,y,tVxxx,tlx,y,t+h′x,y,tVxx,t-14l′x,y,tVxxx,tqx,y,t+Vx′x,tgx,y,t×I+q′x,y,tVx,xx,tqx,y,t-1·q′x,y,tVx,xx,tlx,y,t+g′x,y,tVxx,t<0,(12)γ2I-s′x,y,tVx,xx,tsx,y,t>0,
then
(13)u*=-12I+q′x,y,tVx,xx,tqx,y,t-1×q′x,y,tVx,xx,tlx,y,t+g′x,y,tVxx,t
is an exponential mean square H∞ control of (1).Proof.
Applying Itô’s formula toV(x,t), we have
(14)VxT,T=Vx0,0+∫0T‍LVx,xτ,tdt+∫0T‍Vxx,t×lx,xτ,t+qx,xτ,tux2+sx,xτ,tvqx,xτ,tux2dwt.
Taking mathematical expectation on both sides of (14), we obtain
(15)EVxT,T-Vx0,0=E∫0T‍Vtx,t+Vx′x,t12×fx,xτ,t+gx,xτ,tu+hx,xτ,tv+12lx,xτ,t+qx,xτ,tu+sx,xτ,tv′×Vxxx,tlx,xτ,t+qx,xτ,tu+sx,xτ,tv+m(x,xτ,t)2+u2-12z2-γ2v2+γ2v2dt=E∫0T‍Ω1v,x,xτ,t+Ω2x,xτ,t12+Ω3u,x,xτ,t-z2+γ2v2+12symu′q′x,xτ,tVxxx,tsx,xτ,tvdt,
where
(16)Ω1v,x,xτ,t=v′-γ2I+12s′x,xτ,tVxxx,tsx,xτ,tv+12syml′x,xτ,tVxxx,tsx,xτ,t+Vx′x,thx,xτ,tv,Ω2x,xτ,t=Vt(x,t)+Vx′x,tfx,xτ,t+12l′x,xτ,tVxxx,tlx,xτ,t+m′x,xτ,tmx,xτ,t,Ω3u,x,xτ,t=u′I+12q′x,xτ,tVxxx,tqx,xτ,tu+12syml′x,xτ,tVxxx,tqx,xτ,t+Vx′x,tgx,xτ,tu.
Considering Vxx(x,t)>0 and Lemma 5, we have
(17)12symu′q′x,xτ,tVxxx,tsx,xτ,tv≤12u′q′x,xτ,tVxxx,tqx,xτ,tu+12v′s′x,xτ,tVxxx,tsx,xτ,tv.
Therefore,
(18)EVxT,T-Vx0,0≤E∫0T‍Ω1v,x,xτ,t+Ω2x,xτ,t12+Ω3(u,x,xτ,t)-z2+γ2v2+12u′q′x,xτ,tVxxx,tqx,xτ,tu+v′s′x,xτ,tVxxx,tsx,xτ,tv12dt=E∫0T‍Ω~1v,x,xτ,t+Ω2x,xτ,t+Ω~3(u,x,xτ,t)-z2+γ2v2dt,
where
(19)Ω~1v,x,xτ,t=v′-γ2I+s′x,xτ,tVxxx,tsx,xτ,tv+12syml′x,xτ,tVxxx,tsx,xτ,t+Vx′x,thx,xτ,tv,Ω~3u,x,xτ,t=u′I+q′x,xτ,tVxxx,tqx,xτ,tu+12syml′x,xτ,tVxxx,tqx,xτ,t+Vx′x,tgx,xτ,tu.
Set(20)A1=-γ2I+s′x,xτ,tVxxx,tsx,xτ,t,b1′=12l′x,xτ,tVxxx,tsx,xτ,t+Vx′x,thx,xτ,t,A3=I+q′x,xτ,tVxxx,tqx,xτ,t,b3′=12l′x,xτ,tVxxx,tqx,xτ,t+Vx′x,tgx,xτ,t.
According to Lemma 6, Ω~1(v,x,xτ,t) and Ω~3(u,x,xτ,t) can be rewritten as
(21)Ω~1v,x,xτ,t=v+A1-1b1′A1v+A1-1b1-b1′A1b1,Ω~3u,x,xτ,t=u+A3-1b3′A3u+A3-1b3-b3′A3b3.
Implementing (21) and Ω2(x,xτ,t) into (18) yields
(22)E[V(x(T),T)-V(x(0),0)]=E∫0T‍γ2v2-z2+v+A1-1b1′A1v+A1-1b112+u+A3-1b3′A3u+A3-1b3+Vtx,t+Vx′x,tfx,xτ,t+12l′x,xτ,tVxxx,tlx,xτ,t+m′x,xτ,tmx,xτ,t+12b1′(-A1)b1-b3′A3b3dt.
According to (11), we have
(23)EVxT,T-Vx0,0≤E∫0T‍γ2v2-z2+v+A1-1b1′A1v+A1-1b1+u+A3-1b3′A3u+A3-1b3dt.
Considering (12) and taking u=u*=-A3-1b3, (23) leads to
(24)E∫0T‍z2dt≤-EVxT,T+γ2E∫0T‍v2dt-E∫0T‍v+A1-1b1′-A1v+A1-1b1dt≤γ2E∫0T‍v2dt.
Let T→∞, and then (5) of Definition 3 is proved.
Next, we will prove system (6) to be exponentially mean square stable. Let Lu* be the infinitesimal generator of the system (6), and then
(25)Lu*Vx,y,t=Vtx,t+Vx′x,tfx,y,t+gx,y,tu*+12lx,y,t+qx,y,tu*′×Vxxx,tlx,y,t+qx,y,tu*=Vtx,t+Vx′x,tfx,y,t+12l′x,y,tVxxx,tlx,y,t+Vx′x,tgx,y,tu*+12syml′x,y,tVxxx,tqx,y,tu*+12u*′qx,y,t′Vxxx,tqx,y,tu*.
Setting
(26)Σ1=Vx′x,tgx,y,tu*+12syml′x,y,tVxxx,tqx,y,tu*,Σ2=12u*′qx,y,t′Vxxx,tqx,y,tu*
and implementing
(27)u*=-12I+q′x,y,tVx,xx,tqx,y,t-1×q′x,y,tVx,xx,tlx,y,t+g′x,y,tVxx,t
into Σ1 and Σ2, it yields
(28)Σ1=-12Vx′x,tgx,y,t×I+q′x,y,tVx,xx,tqx,y,t-1·q′x,y,tVx,xx,tlx,y,t+g′x,y,tVxx,t-14syml′x,y,tVxxx,tqx,y,tI+q′x,y,tVx,xx,tqx,y,t-1×I+q′x,y,tVx,xx,tqx,y,t-1·q′x,y,tVx,xx,tlx,y,t+g′x,y,tVxx,tI+q′x,y,tVx,xx,tqx,y,t-1,=-12l′x,y,tVxxx,tqx,y,t+Vx′x,tgx,y,t×I+q′x,y,tVx,xx,tqx,y,t-1·q′x,y,tVx,xx,tlx,y,t+g′x,y,tVxx,t,Σ2=18l′x,y,tVxxx,tqx,y,t+Vx′x,tgx,y,t×I+q′x,y,tVx,xx,tqx,y,t-1·q′x,y,tVxxx,tqx,y,t×I+q′x,y,tVx,xx,tqx,y,t-1·q′x,y,tVx,xx,tlx,y,t+g′x,y,tVxx,t≤18l′x,y,tVxxx,tqx,y,t+Vx′x,tgx,y,t×I+q′x,y,tVx,xx,tqx,y,t-1·q′x,y,tVx,xx,tlx,y,t+g′x,y,tVxx,t.
Substituting (28) into (25) and considering conditions (i), (ii), and (11) in Theorem 7, it follows that
(29)Lu*Vx,y,t≤Vtx,t+Vx′x,tfx,y,t+12l′x,y,tVxxx,tlx,y,t-38l′x,y,tVxxx,tqx,y,t+Vx′x,tgx,y,t×I+q′x,y,tVx,xx,tqx,y,t-1·q′x,y,tVx,xx,tlx,y,t+g′x,y,tVxx,t≤Vtx,t+Vx′x,tfx,y,t+12l′x,y,tVxxx,tlx,y,t-14l′x,y,tVxxx,tqx,y,t+Vx′x,tgx,y,t×I+q′x,y,tVx,xx,tqx,y,t-1·q′x,y,tVx,xx,tlx,y,t+g′x,y,tVxx,t<-14l′x,y,tVxxx,tsx,y,t+Vx′x,thx,y,t×γ2I-s′x,y,tVxxx,tsx,y,t-1·s′x,y,tVxxx,tlx,y,t+h′x,y,tVxx,t-m′x,y,tmx,y,t≤-mx,y,t2≤-c3x2+c4y2.
From Lemma 2, system (6) is exponentially mean square stable. This theorem is proved.The following theorem is derived for the asymptotic mean squareH∞ control, which is weaker than the exponential mean square H∞ control.Theorem 8.
Assume thatV(x,t)∈C2,1(Rn×R+;R+)  has an infinitesimal upper limit; that is, lim⁡x→∞inf⁡t>0V(x,t)=∞ and V(x,t)>cx2 for some c>0. If  V(x,t) solves HJIs (11)-(12), then (13) is an asymptotic mean square H∞ control of (1).Proof.
It only needs to prove that system (6) is asymptotically mean square stable when v=0. We know that Lu*V(x,y,t)<0 from (29), which implies that system (6) is globally asymptotically stable in probability 1 [26]. According to Itô formula and the property of stochastic integration, we obtain
(30)EVxt,t=EVx0,0+E∫0t‍Lu*Vxs,sv=0ds+E∫0t‍Vxxs,s×lx,xτ,t+qx,xτ,tu*v=0dws=EVx0,0+E∫0t‍Lu*Vxs,sv=0ds≤EVx0,0-E∫0t‍mxs,xs-τ,s2ds≤EVx0,0<∞.
Let F~t=Ft∪σ(y(s),0≤s≤t), and then (30) leads to
(31)EVxt,t∣F~s≤Vxs,sa.s.,
which means that {V(x(t),t),F~t,0≤s≤t} is a nonnegative supermartingale with respect to {F~t}t≥0. According to Doob’s convergence theorem [27] and lim⁡t→∞x(t)=0  a.s., we have V(x(∞),∞)=lim⁡t→∞V(x(t),t)=0  a.s. Furthermore, lim⁡t→∞EV(x(t),t)=EV(x(∞),∞)=EV(0,∞)=0. Since V(x,t)>cx2 for some c>0, it yields that lim⁡t→∞Ex(t)2=0. The proof is completed.Remark 9.
In [24], Zhang et al. studied the robust H∞ filtering problem of nonlinear stochastic systems with time delay. However, the H∞ control problem was not tackled in [24], mainly due to mathematical difficulties in dealing with the case that state, control, and disturbance enter into the diffusion term simultaneously. In this paper, Lemma 6 is applied to solve this problem, and two sufficient conditions for H∞ control of delayed nonlinear stochastic systems are obtained in Theorems 7 and 8.Remark 10.
A further development of the present issue is twofold. On the one hand, in order to avoid solving HJIs (11) and (12), the global linearization approach [25] or fuzzy approach based on Takagi-Sugeno model [28] can be used to design H∞ control for delayed nonlinear stochastic systems. On the other hand, Lévy noise is more versatile and interesting with a wider range of applications in comparison to the standard Gaussian noise [29, 30]. Therefore, the H∞ control of stochastic differential equations with Lévy noise is another valuable research topic.


## 4. Numerical Examples

In this section, two numerical examples are given to illustrate the proposedH∞ control design.Example 1.
Consider the following one-dimensional nonlinear stochastic state-delayed system:(32)dxt=-2xt+xtx2t-τ+xtx2t-τ4xt-τut+xt-τvtdt+xtxt-τ+ut+vtdwt,zt=2xtut,t≥0,xt=ϕt∈CF0b-τ,0;Rn,-τ≤t≤0.
SetV(x)=px2, p>0 to be determined, and then HJIs (11)-(12) become
(33)2px·-2x+xxτ2+12xxτ·2p·xxτ+2x·2x+14xxτ·2p+2px·xτ2γ2-2p-1-14xxτ·2p+2px·4xτ21+2p-1<0,γ2-2p>0.
Given γ=3, the above inequalities have a solution p=1. From Theorem 7, the H∞ control of system (32) is u*=-(5/3)xxτ.The initial condition is chosen asϕ(t)=1.2 for any t∈[-τ,0] with τ=0.2 and v(t)=e-t. Applying the Euler-Maruyama method [31], the state responses of the unforced system (u=0) and the controlled system (u=u*) and the control input are shown in Figures 1, 2, and 3, respectively. It is found that the controlled system can achieve stability and attenuation performance by using the proposed H∞ control.Figure 1
The statex(t) of the unforced system in Example 1.Figure 2
The statex(t) of the controlled system in Example 1.Figure 3
The control inputu(t) in Example 1.Example 2.
Consider a two-dimensional system (1) with the following parameters:
(34)fx=x2t-x23t-x2t-x1t,gx=02x2t-τ,mx=2x2t,hx=0x2t-τ,lx=0x2tx2t-τ,qx=0,sx=0.
Take Vx=x′Px with P=diag⁡{p1,p2}<0 to be determined. For a given disturbance attenuation level γ=1, P=diag⁡{1,1} is a solution to (11)-(12). According to Theorem 7, u-∞=-2x2x2τ is an H∞ control of system (1). The initial condition is chosen as ϕ(t)=0.20.5′ for any t∈[-0.2,0], and take v(t)=e-t. By using a similar method in Example 1, the states of the controlled system and the control input are shown in Figures 4-5, which show the effectiveness of the designed controller.Figure 4
The statesx1(t) and x2(t) of the controlled system in Example 2.Figure 5
The control inputu(t) in Example 2.


## 5. Conclusions

For general delayed nonlinear stochastic systems with state, control, and disturbance-dependent noise, this paper has presented a sufficient condition for exponential/asymptotic mean squareH∞ control problem in terms of HJIs. There still remain many interesting topics, for example, how to derive delay-dependent conditions or how to design H2/H∞ control for delayed nonlinear stochastic systems. These issues deserve further research.



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*Source: 382859-2015-03-02.xml*