## Weihai Zhang on the Study of General Nonlinear Stochastic

#### Fast Moving Front Commentary, September 2010

Article: State feedback H-infinity control for a class of nonlinear stochastic systems
Authors: |

Weihai Zhang talks with ScienceWatch.com and answers a few questions about this month's Fast Moving Fronts paper in the field of Mathematics.

**Why do you think your paper is highly
cited?**

Before this paper was published in 2006, I believed that it would attract many researchers' attention after its publication. My self-confidence was based on the following facts:

Firstly, this paper initiated the study of general nonlinear stochastic H-infinity control for the systems described by Ito stochastic differential equations, which brings about many subsequent works in the direction of stochastic H-infinity theory.

Secondly, in order to develop stochastic H-infinity theory, this paper generalized dissipative theory of deterministic systems to stochastic case. More importantly, this paper showed that stochastic dissipativity is equivalent to the solvability of a nonlinear Lure equation, which plays an essential role in further developing stochastic dissipative theory.

Beyond my imagination, recently, some research groups, such as a group at National Tsing Hua University of Taiwan, have applied nonlinear stochastic H-infinity theory developed in the paper to biochemical network design and gene regulator network reconstruction, which makes the paper more attractive and valuable in the study of systems biology.

**Does it describe a new discovery, methodology, or
synthesis of knowledge?**

This paper first showed that, different from H-infinity control of deterministic systems, the existence of nonlinear stochastic H-infinity control is associated with the solvability of a class of Hamilton-Jacobi-Bellman equations (HJBEs), which are nonlinear constrained second-order partial differential equations (PDEs) not first-order PDEs as in nonlinear deterministic systems.

A gap between the sufficient condition and necessary condition for nonlinear stochastic H-infinity control is revealed. How to solve such constrained HJBEs is in itself a challenging and interesting problem and becomes a key in nonlinear stochastic H-infinity design.

From the methodological point of view, this paper combines an essential technique of squares completion with stochastic dynamic programming developed in 1990s. The above two skills have been shown to be very powerful in dealing with nonlinear stochastic quadratic regulator and mixed H2/H-infinity controller (filtering) design.

**Would you summarize the significance of your paper
in layman's terms?**

"Based on our study in this paper, it is easy to solve H8 filtering design of affine nonlinear stochastic systems, including a class of stochastic time-delay systems."

In many applied fields, such as systems biology, mathematical finance, signal processing, network systems, etc., stochastic Ito equations are ideal models in describing natural phenomena. Meanwhile, in the actual modeling process, the system state is not only affected by intrinsic noise but also by external random disturbance.

Popularly speaking, the objective of stochastic H-infinity control is to search for a desired controller to efficiently attenuate the external random disturbance below a given level. In the last 10 years, stochastic H-infinity has become one of the most important robust control methodologies and has attracted a great deal of attention.

This paper theoretically proved that a desired H-infinity controller can be obtained by solving a corresponding HJBE. Although, at the present stage, it is difficult to solve general HJBEs, the paper, after all, presented a clue to stochastic H-infinity design.

**How did you become involved in this research, and
how would you describe the particular challenges, setbacks, and
successes that you've encountered along the way?**

In 1998, linear stochastic H-infinity control was completely solved; see
the excellent paper *SIAM J. Control and Optimization*, 36(5):
1504-1538, 1998. After that, we began to consider the nonlinear stochastic
H-infinity control problem.

As is well known, there is big difference between linear and nonlinear stochastic H-infinity issues. For example, for linear stochastic H-infinity design, it is easy to present a bounded real lemma (BRL) by squares completion technique and linear matrix inequality (LMI) representation, which is a key lemma in H-infinity design.

Unfortunately, LMI-based approach is not suitable for deriving a BRL of nonlinear stochastic systems. In order to obtain a nonlinear BRL, we have introduced stochastic dissipativity and a nonlinear Lure equation, and have proven the equivalence between stochastic dissipativity and the solvability of nonlinear Lure equations.

Finally, I would like to point out that, although the squares completion technique is still useful in studying nonlinear stochastic H-infinity issue, more skill is required in practice.

**Where do you see your research leading in the
future?**

From a mathematical point of view, the finding of this paper will lead the following studies:

Stochastic H-infinity control has becomes an attractive area in recent years, however, up to now, most work is confined to stochastic Ito systems with deterministic coefficients or parameters. The methodology used in this paper together with backward stochastic equation theory makes it possible to solve H-infinity design of stochastic systems with random parameters, which is very valuable in network control and mathematical finance.

The paper transformed the H-infinity design into solving a new class of constrained HJBEs, how to solve such constrained HJBEs will attract many researchers' attention. In addition, I am sure that what we have obtained a new nonlinear Lure equation will be an essential equation in nonlinear stochastic frequency theory.

Due to my knowledge limitation, I am not able to foresee the applied scope of nonlinear stochastic H-infinity theory, which is beyond my capability.

**Do you foresee any social or political
implications for your research?**

Based on our study in this paper, it is easy to solve H-infinity filtering design of affine nonlinear stochastic systems, including a class of stochastic time-delay systems. In addition, a stochastic dissipativity theorem obtained in this paper will play an important role in the study of nonlinear stochastic stability and frequency theory. I am sure that our research is helpful to systems biology, mathematical finance and other applied sciences.

**Weihai Zhang, Professor**

**College** **of Information** **and
Electrical Engineering**

**Shandong** **University** **of Science
and Technology**

**Qingdao****, Shandong** **Province, P. R.
China**

KEYWORDS: TIME-VARYING SYSTEMS; JUMP LINEAR-SYSTEMS; H-2/H-INFINITY CONTROL; OUTPUT-FEEDBACK; DEPENDENT NOISE; CONTROL DESIGN; STABILIZATION; UNCERTAINTY; ATTENUATION; SPACECRAFT.