Mathematical Problems in Engineering

Volume 2014, Article ID 635758, 7 pages

http://dx.doi.org/10.1155/2014/635758

## The Control for Bilinear Systems with Poisson Jumps

^{1}College of Electronic Communication and Physics, Shandong University of Science and Technology, Qingdao 266590, China^{2}College of Electrical Engineering and Automation, Shandong University of Science and Technology, Qingdao 266590, China^{3}College of Mathematics and System Science, Shandong University of Science and Technology, Qingdao 266590, China

Received 3 April 2014; Accepted 5 May 2014; Published 21 May 2014

Academic Editor: Xuejun Xie

Copyright © 2014 Rui Zhang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

This paper discusses the state feedback control problem for a class of bilinear stochastic systems driven by both Brownian motion and Poisson jumps. By completing square method, we obtain the 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 control is represented with the form of tensor power series.

#### 1. Introduction

The main purpose of control design is to find the law to efficiently eliminate the effect of the disturbance [1, 2]. Theoretically, study of control first starts from the deterministic linear systems, and the derivation of the state-space formulation of the standard control leads to a breakthrough, which can be found in the paper [3]. In recent years, stochastic control systems, such as Markovian jump systems [4–6], 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 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 -type theory over infinite time horizon for the disturbance attenuation of stochastic bilinear systems with Poisson jumps by dynamic state feedback.

Generally, the key of 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 under control. This paper will follow along the lines of [22] to study the stochastic 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 Section 2, we review Itô’s theories about the system driven by Brownian motion and Poisson jumps. In Section 3, we obtain the by solving the HJE which is proved by the completing square method. In Section 4, we discuss the problem of finite horizon control with jumps, and using the tensor power series approach, we discuss the approximating control given in the paper. For convenience, we adopt the following notation.

denotes the set of all real symmetric matrices; is the transpose of the corresponding matrix ; is the positive definite (semidefinite) matrix ; is the identity matrix; is the expectation of random variable ; is the Euclidean norm of vector and is the dimension of ; is the set of -dimensional stochastic process defined on interval ( can take ), taking values in , with norm is the class of function twice continuously differential with respect to and once continuously differential with respect to ; is the inner product of two vectors .

#### 2. Preliminaries

For a given complete probability space , let and be the Brownian motion and the Poisson random measure, respectively, which are mutually independent:(i)a 1-dimensional standard Brownian motion ;(ii)a Poisson random measure on , where is a nonempty open set equipped with its Borel field , with the compensator , such that is a martingale for all satisfying . Here is an arbitrarily given finite Lévy measure on , that is, a measure on with the property that . We also let where denotes the totality of 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. *Let be a square integral continuous martingale; is a continuous adapted process with finite variance. is locally square integral due to and ; satisfies the following Itô type stochastic process:
**
Then for , we have (see [23] Chapter I, 3, Theorem 11)
**
where denotes the predictable compensator of martingale .**In the paper, for convenience, is shorten as . Furthermore, for , if using Itô formula to and integrating from to , then taking expectation with both sides
**
we can see that is continuous with respect to time . Since we mainly use the results of expectations of some well functions on and those expectations are continuous with respect to time , so, for briefness, in the rest of this paper the sign under integration is also shortened as .*

#### 3. The Control for Bilinear Systems with Jumps

We consider the following bilinear system driven by Poisson jumps: where represents the exogenous disturbance, , , and are constant matrices, , and only depends on . If there exists an such that for any given and all , , the closed-loop system satisfies we call the control of (6).

Theorem 2. *Suppose there exists a nonnegative solution to the HJE
**
Then is an control for system (6).*

*Proof . *Applying Itô’s formula to , we have
Taking expectation with both sides and applying and , we obtain
Completing square for and , respectively, we have
where
By HJE (8) and let , we have
So, the following inequality is true:
This proves that is an control for system (6).

*Remark 3. *From the proof of Theorems 2 and (13), we can see that given by (12) is a saddle point for the following stochastic game problem:

#### 4. The Tensor Power Series Representation of 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 satisfying (8) has the following form: where , , are symmetrically and continuously differential matrix-valued functions on , is the Kronecker product of matrix (or vectors), and is times Kronecker product of .

Theorem 4. *For given , suppose satisfy the following Riccati equations:
**
Then the control for system (6) can be given by
**
where and is given by following Lemma 6.**In order to prove Theorem 4, we need the following lemmas, and Lemmas 5–8 are given without proofs.*

Lemma 5. *For any , , , , , and we have
*

Lemma 6. *For any matrix , , and integer , we have
**
where , , and .*

Lemma 7. *Let exist. We have
*

Lemma 8. *For any matrix , , and integer , we have
**
where , and is the th row vector of matrix .*

Lemma 9. *For any matrix and integer , we have
**
where , , is the th row vector of matrix C, and is determined as in Lemma 8.*

*Proof. *Let ; then we have
By Lemma 5,
So we can obtain (23).

* Proof of Theorem 4. *Applying Lemmas 6–9, we have
Substituting (26)–(31) and (21) into (8) with terminal conditions , we can prove that satisfies HJE (8). By Theorem 2, the control for system (6) can be given as
Similar to (29), we prove that the control can be represented with the form of (18).

By Theorem 4, we can obtain the approximation of control for system (6).

Now we apply the result of tensor power approach to an example.

*Example 10. *Consider the system (6) with coefficients
is Poisson measure with parameter ; is 1-dimensional Brownian motion and . Here the approximation of is given by
and Figure 1 is the simulation of and control , where is the approximation of of system (6). For the theories of simulation, we will discuss them in another paper. Here we only give the results of simulation.

#### 5. Concluding Remarks

We have discussed the state feedback 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.

#### Conflict of Interests

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

#### Acknowledgment

This work was supported by the NSF of China, Grant nos. 61174078, 11271007, and 61170054.

#### References

- C. S. Wu and B. S. Chen, “Unified design for ${H}_{2}$, ${H}_{\infty}$, and mixed control of spacecraft,”
*Journal of Guidance, Control, and Dynamics*, vol. 22, pp. 884–896, 1999. View at Publisher · View at Google Scholar - C. S. Wu and B. S. Chen, “Adaptive attitude control of spacecraft: Mixed ${H}_{2}$/ ${H}_{\infty}$ approach,”
*Journal of Guidance, Control, and Dynamics*, vol. 24, pp. 755–766, 2001. View at Google Scholar - J. C. Doyle, K. Glover, P. P. Khargonekar, and B. A. Francis, “State-space solutions to standard ${H}_{2}$ and ${H}_{\infty}$ control problems,”
*IEEE Transactions on Automatic Control*, vol. 34, no. 8, pp. 831–847, 1989. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - M. D. S. Aliyu and E. K. Boukas, “${H}_{\mathrm{\infty}}$ control for Markovian jump nonlinear systems,” in
*Proceedings of the 37th IEEE Conference on Decision and Control*, pp. 766–771, Tampa, Fla, USA, 1998. - E. K. Boukas and Z. K. Liu, “Robust ${H}_{\infty}$ control of discrete-time Markovian jump linear systems with mode-dependent time-delays,”
*IEEE Transactions on Automatic Control*, vol. 46, no. 12, pp. 1918–1924, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - D. P. de Farias, J. C. Geromel, J. B. R. do Val, and O. L. V. Costa, “Output feedback control of Markov jump linear systems in continuous-time,”
*IEEE Transactions on Automatic Control*, vol. 45, no. 5, pp. 944–949, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - X. Chen and K. Zhou, “Multiobjective ${H}_{2}/{H}_{\infty}$ control design,”
*SIAM Journal on Control and Optimization*, vol. 40, no. 2, pp. 628–660, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - D. Hinrichsen and A. J. Pritchard, “Stochastic ${H}_{\mathrm{\infty}}$,”
*SIAM Journal on Control and Optimization*, vol. 36, no. 5, pp. 1504–1538, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - T. Damm, “State-feedback ${H}_{\mathrm{\infty}}$-type control of linear systems with time-varying parameter uncertainty,”
*Linear Algebra and Its Applications*, vol. 351-352, pp. 185–210, 2002. View at Publisher · View at Google Scholar · View at MathSciNet - B.-S. Chen and W. Zhang, “Stochastic ${H}_{2}/{H}_{\infty}$ control with state-dependent noise,”
*IEEE Transactions on Automatic Control*, vol. 49, no. 1, pp. 45–57, 2004. View at Publisher · View at Google Scholar · View at MathSciNet - V. A. Ugrinovskii, “Robust ${H}_{\infty}$ control in the presence of stochastic uncertainty,”
*International Journal of Control*, vol. 71, no. 2, pp. 219–237, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - V. Dragan, A. Halanay, and A. Stoica, “The $\gamma $-attenuation problem for systems with state dependent noise,”
*Stochastic Analysis and Applications*, vol. 17, no. 3, pp. 395–404, 1999. View at Publisher · View at Google Scholar · View at MathSciNet - V. Dragan and T. Morozan, “Global solutions to a game-theoretic Riccati equation of stochastic control,”
*Journal of Differential Equations*, vol. 138, no. 2, pp. 328–350, 1997. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - R. Boel and P. Varaiya, “Optimal control of jump processes,”
*SIAM Journal on Control and Optimization*, vol. 15, no. 1, pp. 92–119, 1977. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - R. Rishel, “A minimum principle for controlled jump processes,” in
*Control Theory, Numerical Methods and Computer Systems Modelling*, vol. 107 of*Lecture Notes in Economics and Mathematical Systems*, pp. 493–508, Springer, 1975. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - S. J. Tang and X. J. Li, “Necessary conditions for optimal control of stochastic systems with random jumps,”
*SIAM Journal on Control and Optimization*, vol. 32, no. 5, pp. 1447–1475, 1994. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - Z. Wu, “Fully coupled FBSDE with Brownian motion and Poisson process in stopping time duration,”
*Journal of the Australian Mathematical Society*, vol. 74, no. 2, pp. 249–266, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - I. I. Gihman and A. V. Skorohod,
*Stochastic Differential Equations*, Springer, 1972. View at MathSciNet - C. Graham, “McKean-Vlasov Itô-Skorohod equations, and nonlinear diffusions with discrete jump sets,”
*Stochastic Processes and Their Applications*, vol. 40, no. 1, pp. 69–82, 1992. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - Z. Wu, “A general maximum principle for optimal control of forward-backward stochastic systems,”
*Automatica*, vol. 49, no. 5, pp. 1473–1480, 2013. View at Publisher · View at Google Scholar · View at MathSciNet - W. Zhang and B.-S. Chen, “Stochastic affine quadratic regulator with applications to tracking control of quantum systems,”
*Automatica*, vol. 44, no. 11, pp. 2869–2875, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - W. Zhang and B.-S. Chen, “State feedback ${H}_{\infty}$ control for a class of nonlinear stochastic systems,”
*SIAM Journal on Control and Optimization*, vol. 44, no. 6, pp. 1973–1991, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - I. I. Gihman and A. V. Skorohod,
*The Theory of Stochastic Processes III*, Springer, 1979. View at MathSciNet