# TheH∞ Control for Bilinear Systems with Poisson Jumps

Authors: Rui Zhang; Weihai Zhang; Xiangyun Lin
Journal: Mathematical Problems in Engineering
 (2014)

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## Abstract

This paper discusses the state feedbackH∞ control problem for a class of bilinear stochastic systems driven by both Brownian motion and Poisson jumps. By completing square method, we obtain the H∞ control by solutions of the corresponding Hamilton-Jacobi equations (HJE). By the tensor power series method, we also shift such HJEs into a kind of Riccati equations, and the H∞ control is represented with the form of tensor power series.


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## Body

## 1. Introduction

The main purpose ofH∞ control design is to find the law to efficiently eliminate the effect of the disturbance [1, 2]. Theoretically, study of H∞ control first starts from the deterministic linear systems, and the derivation of the state-space formulation of the standard H∞ control leads to a breakthrough, which can be found in the paper [3]. In recent years, stochastic H∞ control systems, such as Markovian jump systems [4–6], H∞ Gaussian control design [7], and Itô differential systems [8–13], have received a great deal of attention. However, up to now, most of the work on stochastic H∞ control is confined to Itô type or Markovian jump systems. Yet, there are still many systems which contain Poisson jumps in economics and natural science. In 1970s, Boel and Varaiya [14] and Rishel [15] considered the optimal control problem with random Poisson jumps, and many basic results have been made. From then on, many scholars and economists also study the system and its applications; for further reference, we refer to [16–20] and their references. But those results mostly concentrate on optimal control and its application in financial market or corresponding theories. Of course, such model still can be disturbed by exogenous disturbance and its robustness is also an important problem. The objective of this paper is to develop an H∞-type theory over infinite time horizon for the disturbance attenuation of stochastic bilinear systems with Poisson jumps by dynamic state feedback.Generally, the key ofH∞ control design is to solve a general Hamilton-Jacobi equation (HJE). However, up to now, there is still no effective algorithm to solve such a general HJE. In order to solve the HJE given in this paper, we extend a tensor power series approach which is used in [21] and also give the simulation of the trajectory of output z under H∞ control. This paper will follow along the lines of [22] to study the stochastic H∞ control with infinite horizons and finite horizon for a class of nonlinear stochastic differential systems with Poisson jumps. The paper is organized as follows.In Section2, we review Itô’s theories about the system driven by Brownian motion and Poisson jumps. In Section 3, we obtain the H∞ by solving the HJE which is proved by the completing square method. In Section 4, we discuss the problem of finite horizon H∞ control with jumps, and using the tensor power series approach, we discuss the approximating H∞ control given in the paper. For convenience, we adopt the following notation.S
n
(
R
) denotes the set of all real n×n symmetric matrices; A′ is the transpose of the corresponding matrix A; A>0(A≥0) is the positive definite (semidefinite) matrix A; I is the identity matrix; Eξ is the expectation of random variable ξ; ∥x∥ is the Euclidean norm of vector x∈Rnx and nx is the dimension of x; L2([0,T],Rny) is the set of ny-dimensional stochastic process y defined on interval [0,T] (T can take ∞), taking values in Rny, with norm
(1)∥y∥L2([0,T],Rny)=(E∫0T‍∥y(t)∥2dt)1/2<∞;C1,2(R+,Rnx) is the class of function V(t,x) twice continuously differential with respect to x∈Rnx and once continuously differential with respect to t; 〈x,y〉 is the inner product of two vectors x,y∈Rn.


## 2. Preliminaries

For a given complete probability space(Ω,F,P), let Wt and μ be the Brownian motion and the Poisson random measure, respectively, which are mutually independent:(i)
a 1-dimensional standard Brownian motion{Wt}t≥0;(ii)
a Poisson random measureμ on R+×E, where E⊂Rl is a nonempty open set equipped with its Borel field B(E), with the compensator μ^(de,dt)=λ(de)dt, such that {μ~((0,t]×A)=(μ-μ^)((0,t]×A)}t≥0 is a martingale for all A∈B(E) satisfying λ(A)<∞. Here λ is an arbitrarily given σ-finite Lévy measure on (E,B(E)), that is, a measure on (E,B(E)) with the property that ∫E‍(1∧|e|2)λ(de)<∞. We also let
(2)Ft=σ[∫‍∫(0,s]×A‍μ~(de,ds):s≤t,A∈B(E)]∨σ×[Ws:s≤t]∨N,
where N denotes the totality of P-null sets.In order to discuss the systems driven by Brownian motion and Poisson jumps, we first review the theorem about Itô’s formula for such stochastic processes.Theorem 1.
LetMt be a square integral continuous martingale; At is a continuous adapted process with finite variance. γ(s,e) is locally square integral due to μ and P; x(t) satisfies the following Itô type stochastic process:
(3)x(t)=x(0)+Mt+At+∫0t‍∫E‍γ(s,e)μ~(de,ds).
Then for F(t,x)∈C1,2(R+,Rnx), we have (see [23] Chapter I, §3, Theorem 11)
(4)dF(t,xt)=Ft(t,xt)dt+Fx(t,xt)d(Mt+At)+12Fxx(t,xt)d〈M〉t+∫E‍[F(t,xt+γ(t,e))-F(t,xt)sssssssssssssssssssssssssssssssss-Fx(t,xt)γ(t,e)(t,xt+γ(t,e))]λ(de)dt+∫E‍[F(t,xt+γ(t,e))-F(t,xt)]μ~(de,dt),
where 〈M〉 denotes the predictable compensator of martingale M.
In the paper, for convenience,xt is shorten as x. Furthermore, for F∈C1,2(R+,Rnx), if using Itô formula to F(t,xt) and integrating from s to t(0≤s<t), then taking expectation with both sides
(5)EF(t,xt)-EF(s,xs)=∫st‍EFt(r,xr)dr+12∫st‍E[σ(r,xr)′Fxx(r,xr)σ(r,xr)]dr+∫st‍E{∫E‍[F(r,xr+γ(r,xr-,e))ssssssssssssssssF(r,xr+γ(r,xr-,e))-F(r,xr)-〈Fx(r,xr),γ(r,e)〉]ssssssssssssssss×λ(de)∫E‍}dr,
we can see that EF(t,xt) is continuous with respect to time t. Since we mainly use the results of expectations of some well functions on xt and those expectations are continuous with respect to time t, so, for briefness, in the rest of this paper the sign xt- under integration ∫E‍ is also shortened as x.


## 3. TheH∞ Control for Bilinear Systems with Jumps

We consider the following bilinear system driven by Poisson jumps:(6)dx=(Ax+Bxu+Kv)dt+CxdW+∫E‍G(e)xμ~(de,dt),z=[Mxu],
where v∈L2([0,T],Rnd) represents the exogenous disturbance, A, B, and C are constant nx×nx matrices, K∈RK∈Rnx×nd, and G(e)∈Rnx×nx only depends on e. If there exists an uT*∈L2([0,T],Rnu) such that for any given γ>0 and all v∈L2([0,T],Rnd), x(0)=0, the closed-loop system satisfies
(7)∥z∥L2([0,T],Rnz)≤γ∥v∥L2([0,T],Rnd),
we call uT* the H∞ control of (6).Theorem 2.
Suppose there exists a nonnegative solutionV∈C1,2([0,T],Rnx) to the HJE
(8)HT1(VT(t,x)):=∂VT∂t+∂VT∂x′Ax+12x′M′Mx+12γ2∂VT∂x′KK′∂VT∂x+12x′C′∂2VT∂x2Cx-12∂VT∂x′Bxx′B′∂VT∂x+∫E[∂V∂x′VT(t,x+G(e)x)-VT(t,x)sssssssss-∂V∂x′G(e)x]λ(de)=0,VT(T,x)=0,VT(t,0)=0,∀(t,x)∈[0,T]×Rnx.
Then uT*=-x′B′(∂VT/∂x) is an H∞ control for system (6).Proof .
Applying Itô’s formula toV(t,x), we have
(9)V(T,xT)-V(0,0)=∫0T‍{∫E‍V(t,x)+Vx′(t,x)(Ax+Bxu+Kv)sssssssssss+12x′C′VxxCxsssssssssssss+∫E‍[V(t,x+G(e)x)-V(t,x)ssssssssssssssss-Vx(t,x)G(e)x]λ(de)∫E‍}dtvvvvv.+∫0T‍Vx′(t,x)CxdWtvvvvv.+∫0T‍[V(t,x+G(e)x)-V(t,x)]μ~(de,dt).Taking expectation with both sides and applying V(T,x)=0 and V(t,0)=0, we obtain
(10)0=∫0T‍E{∫E‍V(t,x)+Vx′(t,x)(Ax+Bxu+Kv)ssssssssss+12x′C′VxxCxsssssssssss+∫E‍[V(t,x+G(e)x)-V(t,x)ssssssssssssss-Vx(t,x)G(e)x]λ(de)sssssssssss+xt′M′Mx+ut′u-γ2vt′v∫E‍12}dt-∥z∥L2([0,T],Rnz)2+γ2∥v∥L2([0,T],Rnd)2.
Completing square for u and v, respectively, we have
(11)0=∫0T‍E{HT1(V(t,x))+∥u-uT*∥2-1γ2∥v-vT*∥2}dt-∥z∥L2([0,T],Rnz)2+γ2∥v∥L2([0,T],Rnd)2,
where
(12)uT*=-x′B′∂VT∂x,vT*=1γ2(K′∂VT∂x+K′∂2VT∂x2Cx).
By HJE (8) and let u=uT*, we have
(13)∥z∥L2([0,T],Rnz)2-γ2∥v∥L2([0,T],Rnd)2=-1γ2∫0T‍E{∥v-vT*∥2}dt.
So, the following inequality is true:
(14)∥z∥L2([0,T],Rnz)2≤γ2∥v∥L2([0,T],Rnd)2.
This proves that uT* is an H∞ control for system (6).Remark 3.
From the proof of Theorems2 and (13), we can see that (uT*,vT*) given by (12) is a saddle point for the following stochastic game problem:
(15)min⁡u∈L2([0,T],Rnu)max⁡d∈L2([0,T],Rnd)E∫0T‍(∥z∥2-γ2∥d∥2)dt.


## 4. The Tensor Power Series Representation ofH∞ Control

Generally speaking, it is very hard to solve HJE (8). Now we use an approximation algorithm which is called tensor power series approach to solve a special case of HJE (8). In what follows, suppose VT(t,x) satisfying (8) has the following form:
(16)VT(t,x)=∑i=1∞‍〈⊗ix,Pi(t)⊗ix〉,
where Pi(t), i≥1, are symmetrically and continuously differential matrix-valued functions on [0,T], ⊗ is the Kronecker product of matrix (or vectors), and ⊗ix=x⊗⋯⊗x is i times Kronecker product of x.Theorem 4.
For givenγ>0, suppose Pi(t)(i=1,2,…) satisfy the following Riccati equations:
(4)P˙1+A(1)P1+P1A(1)′+12M′M+2γ2(Q(1)P1)⊗(P1Q(1)′)+R(1)(P1)+∫E‍[(Inx+G(e))′P1(Inx+G(e))-P1ssssssssssssssssssss-G(1)(e)P1-P1G(1)′(e)]λ(de)=0,P˙i+A(i)Pi+PiA(i)′+2γ2∑r+j=i+1‍(K(r)Pr)⊗(PjK(j)′)+R(i)(Pi)-2∑r+j=i‍(B(r)Pr)⊗(PjB(j)′)+∫E‍[(⊗i(Inx+G(e)))′Pi(⊗i(Inx+G(e)))sssssssss-Pi-G(i)(e)Pi-PiG(i)′(e)(⊗i(Inx+G(e)))′]λ(de)=0,Pi(T)=0,i=1,2,….
Then the H∞ control uT* for system (6) can be given by
(18)uT*=-2∑i=1∞‍〈⊗ix,(B(i)Pi)⊗ix〉,
where B(i)=∑j=1nxBj′⊗D(i,j) and D(i,j) is given by following Lemma 6.
In order to prove Theorem4, we need the following lemmas, and Lemmas 5–8 are given without proofs.Lemma 5.
For anyx∈Rnx, y∈Rny, u∈Rnu, v∈Rnv, P∈Rnx×ny, and Q∈Rnu×nv we have
(19)〈x,Py〉〈u,Qv〉=〈x⊗u,(P⊗Q)(y⊗v)〉.Lemma 6.
For any matrixP∈Snxi(R), K∈Rnk, and integer i, we have
(20)∂〈⊗ix,P⊗ix〉′∂xK=2〈⊗i-1x,(K(i)P)⊗ix〉,
where K(i)=∑j=1nxkjD(i,j), D(i,j)=∑l=1iDl(i,j), and Dl(i,j)=Inxl-1⊗ej′⊗Inxi-l.Lemma 7.
LetVT(t,x)=∑i=1∞〈⊗ix,Pi(t)⊗ix〉 exist. We have
(21)∂VT∂x′KK′∂VT∂x=4∑m=1∞‍〈⊗mx,∑i+j=m+1‍(K(i)Pi)⊗(PjK(j)′)⊗mx〉.Lemma 8.
For any matrixP∈Snxi(R), x∈Rnx, and integer i, we have
(22)∂〈⊗ix,P⊗ix〉∂x′Ax=2〈⊗ix,(A(i)P)⊗ix〉,
where A(i)=∑j=1nxAj′⊗D(i,j), and Aj is the jth row vector of matrix A.Lemma 9.
For any matrixP∈Snxi(R) and integer i, we have
(23)x′C′∂2〈⊗ix,P⊗ix〉∂x2Cx=2〈⊗ix,R(i)(P)⊗ix〉,
where R(i)(P)=C(i)PC(i)′+Q(i)P, Q(i)=∑s=1nx∑t=1nxCs′⊗Ct′⊗(D(i-1,s)D(i,t)), Cs is the sth row vector of matrix C, and C(i) is determined as A(i) in Lemma 8.Proof.
LetK=Cx=(k1,k2,…,knx)′; then we have
(24)x′C′∂2〈⊗ix,P⊗ix〉∂x2Cx=∑s=1nx‍∑t=1nx‍∂2〈⊗ix,P⊗ix〉∂xs∂xtkskt=2∑s=1nx‍∑t=1nx‍kskt∂∂xs〈⊗i-1x,∑l=1i‍Dl(i,t)P〉=2∑s=1nx‍∑t=1nx‍kskt[∑m=1i-1〈∑l=1i‍Dl(i,t)P⊗ix⊗m-1x⊗es⊗⊗i-m-1x,ssssssssssssssssssssssssss∑l=1i‍Dl(i,t)P⊗ix〉+∑m=1i‍〈⊗i-1x,∑l=1i‍Dl(i,t)P⊗m-1x⊗es⊗⊗i-mx〉]=2∑s=1nx‍∑t=1nx‍kskt[〈⊗i-2x,∑m=1i-1‍Dmi-1,s∑l=1i‍Dl(i,t)P⊗ix〉sssssssssssssssssssss+〈⊗i-1x,∑m=1i‍∑l=1i‍Dl(i,t)PDm(i,t)′⊗i-1〉]=2∑s=1nx‍∑t=1nx‍[∑l=1i‍〈x,Cs′〉〈x,Ct′〉ssssssssssssssss×〈⊗i-2x,∑m=1i-1‍Dmi-1,s∑l=1i‍Dl(i,t)P⊗ix〉ssssssssssssssss+〈x,Cs′〉〈⊗i-1x,∑m=1i‍∑l=1i‍Dl(i,t)PDm(i,t)′⊗i-1〉ssssssssssssssss×∑l=1i‍〈Ct′,x〉].
By Lemma 5,
(25)x′C′∂2〈⊗ix,P⊗ix〉∂x2Cx=2∑s=1nx‍∑t=1nx‍[〈⊗ix,Cs′⊗Ct′⊗∑m=1i-1‍Dmi-1,sssssssssssssssssss×∑l=1i‍Dl(i,t)P⊗ix〉sssssssssssssssss+〈⊗ix,(Cs′⊗∑m=1i‍∑l=1i‍Dl(i,t)PDm(i,t)′sssssssssssssssssssssssssss∑m=1i‍⊗Ct)⊗ix〉].
So we can obtain (23).Proof of Theorem4.
Applying Lemmas6–9, we have
(26)VT(t,x+G(e)x)=∑i=1∞‍〈⊗ix,(⊗i(Inx+G(e)))′Pi(t)ssssssss×(⊗i(Inx+G(e)))⊗ix⊗ix,(⊗i(Inx+G(e)))′Pi(t)〉,(27)∂VT∂t=∑i=1∞‍〈⊗ix,P˙i⊗ix〉,(28)∂VT∂x′Ax=∑i=1∞‍〈⊗ix,(A(i)Pi)⊗ix〉+∑i=1∞‍〈⊗ix,(PiA(i)′)⊗ix〉,(29)∂V∂x′G(e)x=∑i=1∞‍〈⊗ix,(D(i)(e)Pi)⊗ix〉+∑i=1∞‍〈⊗ix,(PiD(i)′)⊗ix〉,(30)∂VT∂x′Bxx′B′∂VT∂x=4∑m=2∞‍〈⊗mx,∑i+j=m‍(B(i)Pi)⊗(PjB(j)′)⊗mx∑i+j=m‍〉,(31)x′C′∂2VT∂x2Cx=2∑i=1∞‍〈⊗ix,R(i)(Pi)⊗ix〉.
Substituting (26)–(31) and (21) into (8) with terminal conditions Pi(T)=0(i=1,2,…), we can prove that VT(t,x) satisfies HJE (8). By Theorem 2, the H∞ control for system (6) can be given as
(32)uT*=-x′B′∂VT∂x.
Similar to (29), we prove that the H∞ control uT* can be represented with the form of (18).By Theorem4, we can obtain the approximation of H∞ control for system (6).Now we apply the result of tensor power approach to an example.Example 10.
Consider the system (6) with coefficients
(33)A=[0.040.020.020.04],B=[0.020.020.020.02],K=[0.020.02],C=[-0.020.040.020.02],G=[0.02-0.020.020.02],M=[-0.040.02].N(t) is Poisson measure with parameter λ=2; W(t) is 1-dimensional Brownian motion and γ=1. Here the approximation of uT* is given by
(34)u*=-2∑i=16‍〈⊗ix,(B(i)Pi)⊗ix〉,
and Figure 1 is the simulation of m(x)=Mx and H∞ control u*, where u* is the approximation of uT* of system (6). For the theories of simulation, we will discuss them in another paper. Here we only give the results of simulation.Figure 1
Tracking performance of Example10.


## 5. Concluding Remarks

We have discussed the state feedbackH∞ control for a class of bilinear stochastic system where both Brownian motion and Poisson process are present. In order to solve the HJE given in the paper, we also discuss the method of tensor power series approach.



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