${\mathcal{H}}_{\infty }$ control for Poisson-driven stochastic systems

Highlights

• This paper uses the Doob-Meyer decomposition and measure theory to give a model transfor-mation method, and Poisson-driven stochastic systems are transformed into stochastic systems driven by compensated Poisson process. Since compensated Poisson process is a martingale, we can use effective properties and tools in martingale theory to investigate the ${\mathcal{H}}_{\infty }$ control problem.

• The martingale theory is utilized to deal with the jump item and the sum of stochastic integrals with respect to the continuous part of states in Itô formula, and then we gives an equivalent Itô formula for stochastic differential equations driven by compensated Poisson process.

• Based on these, this paper designs an ${\mathcal{H}}_{\infty }$ controller by means of a linear matrix inequality, which is fairly straightforward to be solved.

• The designed result contains information about the average number of jump random events in a unit time, and one can use convex optimization algorithm easily to estimate the maximum average number of jump random events in a unit time that the system can undergo to achieve the stability and control performance.

Abstract

The paper aims to investigate the ${\mathcal{H}}_{\infty }$ control problem for systems perturbed by jump random noise, i.e., Poisson-driven stochastic systems (SSs). Firstly, this paper uses the Doob-Meyer decomposition and measure theory to give a model transformation method, and Poisson-driven SSs, which are SSs driven by semi-martingale, are transformed into SSs driven by compensated Poisson process. Since SSs driven by compensated Poisson process are SSs driven by martingale, we can use effective properties and tools in martingale theory to investigate the ${\mathcal{H}}_{\infty }$ control problem. Secondly, this paper utilizes martingale theory to deal with the jump item and the sum of stochastic integrals (SIs) with respect to (w.r.t.) the continuous part of states in Itô formula, and then gives an equivalent Itô formula for SDEs driven by compensated Poisson process. Thirdly, on the basis of these, this paper presents a simple ${\mathcal{H}}_{\infty }$ controller design method. Furthermore, the design criterion contains information about the average number of jump random events in a unit time, and one can utilize convex optimization algorithm to estimate the maximum average number of jump random events in a unit time, which the closed-loop system can tolerate to achieve the stability and ${\mathcal{H}}_{\infty }$ performance. Finally, the usefulness of the design result is verified by two numerical examples.

Introduction

To model systems in the real world, two kinds of equations are always used. One is differential equation (DE), and another is stochastic differential equation (SDE) [1], [2], [3], [4], [5]. For example, in a simple population growth model

$$dv\left(t\right)=\vartheta \left(t\right)v\left(t\right)dt,$$

Mao [2] considered that the relative rate of the growth ϑ(t) is perturbed by random environmental effects, i.e., $\vartheta \left(t\right)=q\left(t\right)+p\left(t\right)\beta \left(t\right),$ where β(t) is the random effect. Consequently, above equation becomes

$$dv\left(t\right)=q\left(t\right)v\left(t\right)dt+p\left(t\right)v\left(t\right)\beta \left(t\right)dt.$$

If the “random effect” β(t) is continuous, it turns out that a reasonable interpretation is the white noise formally defined as the derivative of Wiener process, i.e. $\dot{w}\left(t\right)=\frac{dw\left(t\right)}{dt}$. Then, Eq. (1) would be

$$dv\left(t\right)=q\left(t\right)v\left(t\right)dt+p\left(t\right)v\left(t\right)dw\left(t\right),$$

which is a Wiener-driven SDE.

During past decades, Wiener-driven SDEs have been employed widely to model systems perturbed by random noises [1], [2], [6], [7], [8], [9]. As an efficacious control method, ${\mathcal{H}}_{\infty }$ control has been a hot topic [10], [11], [12], [13], and discussions on ${\mathcal{H}}_{\infty }$ control for Wiener-driven SSs can be found in [14], [15], etc.

Although used widely to model random phenomena in many disciplines including sciences and engineering, Wiener process, which is a continuous stochastic process, can not describe jump random phenomena [16], [17], [18], [19]. Jump random phenomena exist widely in physical, biological, chemical, engineering, and finance (e.g., [16], [20], [21], [22], [23]), etc. To describe jump random phenomena, Poisson process is a natural model [16], and some results on Poisson-driven SSs in control field have been presented in [24], [25], [26], [27], [28], [29].

Here we briefly comment on related results for Poisson-driven SSs in control field. Westman and Hanson [24] presented an SDE to model the multistage manufacturing system, where Poisson process was employed to depict the machine failure and repair. Then Westman and Hanson improved this result and proposed a computational method to solve the optimal control problem in [25]. Dong et al. [26] introduced Poisson process to model faults in power systems, and the Milstein and Euler numerical methods were adopted to investigate the transient stability of power systems. Moreover, for Poisson-driven SSs, Kolmanovsky and Maizenberg [27] adopted the successive approximations algorithm to solve the optimal control problem. Basically, the methods, which were used in above results to solve control problems, were based on numerical methods, and it is relatively complicated to obtain the solutions of the derived criteria. More significantly, the amplitudes of jump random noise in stochastic models of [24], [25], [27] were functions of state x(t). But these jump amplitudes should be functions of $x(t-)$ rather than x(t), and the specific reasons were given in Remark 1 and 2 of our paper [28]. The work in [28] presented a stability criterion for neural networks subject to Poisson noise, and the result was obtained by the semimartingale theory such as predictable projection operator. However, it should be noted that there is a flaw in the Itô formula used in [28] (See Remark 6). And when using predictable project operator, one needs to verify some strict mathematical conditions, which are hard to be verified sometimes. Lin and Zhang [29] addressed the ${\mathcal{H}}_{\infty }$ control problem for SSs disturbed by random jumps, where random jumps were depicted by Poisson random measure. Nevertheless, it is difficult to understand the physical meaning of Poisson random measure, and the system model in [29] does not have the explicit application background. By contrast, the physical meaning of Poisson-driven SSs is obvious (See Remark 1). More importantly, there is no effective algorithm to solve the design criterion of [29], which is given by means of a generalized algebraic Riccati equality. Thus it is rather difficult to solve the equality.

A careful analysis of above results on Poisson-driven SSs indicates that there are unsolved problems listed below, which motivate us to conduct the research.

• Problem 1: Can we design a controller by using effective properties and tools in martingale theory?

Wiener-driven SSs, e.g.,

$$dy\left(t\right)=\ell \left(y\right(t\left)\right)dt+\hslash \left(y\right(t\left)\right)dw\left(t\right),$$

are SSs driven by martingale, and one can use effective properties and tools in martingale theory. For instance, when using Lyapunov theory to study control problems for above system, one would utilize the Itô formula:

$$\begin{array}{ccc}& & f(t,y(t\left)\right)-f(0,y(0\left)\right)\hfill \\ & & ={\int }_0^t(f_{\alpha }(\alpha ,y\left(\alpha \right))+f_y(\alpha ,y\left(\alpha \right))\ell \left(y\left(\alpha \right)\right)+\frac{1}{2}trace\left(\hslash {\left(y\left(\alpha \right)\right)}^T{f}_{yy}(\alpha ,y\left(\alpha \right))\hslash \left(y\left(\alpha \right)\right)\right))d\alpha \hfill \\ & & +{\int }_0^t{f}_y(\alpha ,y\left(\alpha \right))\hslash \left(y\left(\alpha \right)\right)dw\left(\alpha \right),\hfill \end{array}$$

where the first integral is a Lebesgue-Stieltjes (L-S) integral, which is very common in Lyapunov theory. The second one, ${\int }_0^t{f}_y(\alpha ,y\left(\alpha \right))\hslash \left(y\left(\alpha \right)\right)dw\left(\alpha \right),$ is an SI w.r.t. martingale, and it would satisfy many properties such as martingale property, continuity property and isometry property. By these properties, it is straightforward to utilize Itô formula (2) and Lyapunov theory to investigate control problems of Wiener-driven SSs. However, if investigating control synthesis of SSs driven by Poisson process χ(t):

$$dy\left(t\right)=\ell \left(y\right(t\left)\right)dt+\hslash \left(y\right(t-\left)\right)d\chi \left(t\right),$$

one can not use effective properties and tools in martingale theory to study since Poisson process is a semimartingale. In contrast with semimartingale theory, martingale theory has more effective tools and properties. For example, SIs w.r.t. Poisson process do not satisfy the properties such as martingale property, continuity property and isometry property. And when using semimartingale theory, one needs to verify stricter mathematical conditions. Therefore, a natural idea is to present a new method, which can transform Poisson-driven SSs into SSs driven by martingale. Then one can investigate the SSs more easily.

• Problem 2: Can we bridge over the difficulties induced by the jump term and the sum of SIs w.r.t. the continuous part of states in Itô formula to develop a straightforward controller design method?

Since the Poisson-driven SS (3) is an SDE driven by discontinuous semimartingale, we should adopt the Itô formula (4) to investigate stability or control problems:

$$\begin{array}{ccc}& & f(t,y(t\left)\right)-f(0,y(0\left)\right)\hfill \\ & =& \sum _{\alpha \le t}\left(f\right(\alpha ,y\left(\alpha \right))-f(\alpha -,y(\alpha -)\left)\right)\hfill \\ & & +{\int }_0^t\frac{\partial f(\alpha -,y(\alpha -\left)\right)}{\partial \alpha }d\alpha +\sum _{i=1}^n{\int }_0^t\left[\frac{\partial f(\alpha -,y(\alpha -\left)\right)}{\partial y_i}\right]d{y}_i^c\left(\alpha \right),\hfill \end{array}$$

where t > 0, $y\left(t\right)={\left(y_i\left(t\right)\right)}_{n\times 1}$ and $y_i^c\left(t\right)$ is the continuous part of y_i(t) [34].

Unlike Itô formula (2) for Wiener-driven SDE, Eq. (4) includes the jump term

$$J_t:=\sum _{\alpha \le t}\left(f\right(\alpha ,y\left(\alpha \right))-f(\alpha -,y(\alpha -)\left)\right)$$

and the sum of SIs w.r.t. the continuous part of states

$$S_t:=\sum _{i=1}^n{\int }_0^t\left[\frac{\partial f(\alpha -,y(\alpha -\left)\right)}{\partial y_i}\right]d{y}_i^c\left(\alpha \right).$$

When one uses Lyapunov theory, the existence of J_t, S_t and the left limit of state $y(\alpha -)$ in (4) result in failure to obtain a controller design criterion by matrix inequalities, and the design criterion is not easy to be verified in the practice. Thus it is valuable to deal with J_t, S_t and the left limit of state $y(\alpha -)$ to present a simple controller design method for Poisson-driven SSs.

• Problem 3: For control synthesis of Poisson-driven SSs, can we estimate the maximum average number of random jump events in a unit time that the closed-loop system can undergo to achieve the stability and ${\mathcal{H}}_{\infty }$ performance?

Since this paper concentrates on SSs perturbed by discrete jump random events, the average number of such random events in a unit time obviously would influence the dynamical behavior of the SSs. Generally speaking, the more the average number of jump random events in a unit time is, the less likely the system can achieve stability and control performance. So, when considering jump random effects on a system, we will wonder if an approach can be presented to estimate the maximum average number of jump random events in a unit time that the system can undergo to achieve the stability and control performance. Due to difficulties mentioned in problem 1 and 2, it is hard to estimate the above maximum average number when investigating ${\mathcal{H}}_{\infty }$ control problem of Poisson driven SSs.

Given the above discussion, this paper aims to address the ${\mathcal{H}}_{\infty }$ control problem of Poisson-driven SSs.

The main contributions lie in the following aspects:

(1) This paper uses the Doob-Meyer decomposition and measure theory to give a model transformation method, and Poisson-driven SSs are transformed into SSs driven by compensated Poisson process. Since compensated Poisson process is a martingale, we can use effective properties and tools in martingale theory to investigate the ${\mathcal{H}}_{\infty }$ control problem.

(2) The martingale theory is utilized to deal with the jump item and the sum of SIs w.r.t. the continuous part of states in Itô formula, and then we gives an equivalent Itô formula for SDEs driven by compensated Poisson process.

(3) Based on these, this paper designs an ${\mathcal{H}}_{\infty }$ controller by means of a linear matrix inequality (LMI), which is fairly straightforward to be solved.

(4) The designed result contains information about the average number of jump random events in a unit time, and one can use convex optimization algorithm easily to estimate the maximum average number of jump random events in a unit time that the system can undergo to achieve the stability and control performance.