A game theory approach to mixed H2/H∞ control for a class of stochastic time-varying systems with randomly occurring nonlinearities

doi:10.1016/j.sysconle.2011.08.009

Systems & Control Letters

Volume 60, Issue 12, December 2011, Pages 1009-1015

https://doi.org/10.1016/j.sysconle.2011.08.009 Get rights and content

Abstract

This paper is concerned with the mixed $H_{2} / H_{\infty}$ control problem for a class of stochastic time-varying systems with nonlinearities. The nonlinearities are described by statistical means and could cover several kinds of well-studied nonlinearities as special cases. The occurrence of the addressed nonlinearities is governed by two sequences of Bernoulli distributed white sequences with known probabilities. Such nonlinearities are named as randomly occurring nonlinearities (RONs) as they appear in a probabilistic way. The purpose of the problem under investigation is to design a controller such that the closed-loop system achieves the expected $H_{2}$ performance requirements with a guaranteed $H_{\infty}$ disturbance attenuation level. A sufficient condition is given for the existence of the desired controller by means of solvability of certain coupled matrix equations. By resorting to the game theory approach, an algorithm is developed to obtain the controller gain at each sampling instant. A numerical example is presented to show the effectiveness and applicability of the proposed method.

Section snippets

Problem formulation

Consider the following discrete-time nonlinear stochastic system defined on $k \in [0, N - 1]$ : ${\begin{cases} x_{k + 1} = A_{k} x_{k} + B_{k} u_{k} + D_{k} ω_{k} + α_{k} f_{k} (x_{k}, u_{k}, k) \\ y_{k} = C_{k} x_{k} + β_{k} g_{k} (x_{k}, u_{k}, k), x_{0} = x_{0} \end{cases}$ where $x_{k} \in R^{n}$ is the system state, $u_{k} \in R^{m}$ is the control input, $y_{k} \in R^{r}$ is the system output, $ω_{k} \in R^{p}$ is the external disturbance belonging to $l_{2} ([0, N - 1], R^{p})$ . $A_{k}$ , $B_{k}$ , $C_{k}$ and $D_{k}$ are time-varying matrices with appropriate dimensions.

For notation simplicity, let $h_{k} ≜ {[\begin{matrix} f_{k}^{T} (x_{k}, u_{k}, k) & g_{k}^{T} (x_{k}, u_{k}, k) \end{matrix}]}^{T}$ . For all $x_{k}$ and $u_{k}$ , $h_{k}$ is assumed to satisfy: $E {h_{k} | x_{k}} = 0,$ $E {h_{k} h_{j}^{T} | x_{k}$

$H_{\infty}$ performance

In this section, we first analyze the $H_{\infty}$ performance for the time-varying stochastic system with randomly occurring nonlinearities. Then, a sufficient condition is given for satisfying the pre-specified $H_{\infty}$ requirement. To this end, setting $u_{k} = 0$ , we obtain the corresponding unforced system for system (1) as follows: ${\begin{cases} x_{k + 1} = A_{k} x_{k} + D_{k} ω_{k} + α_{k} f_{k} (x_{k}, k) \\ y_{k} = C_{k} x_{k} + β_{k} g_{k} (x_{k}, k), x_{0} = x_{0} . \end{cases}$

We now consider the following recursion: $Q_{k} = A_{k}^{T} Q_{k + 1} A_{k} + \sum_{i = 1}^{q} Γ_{i k} (tr [μ Q_{k + 1} Π_{i k}^{11}] + tr [ν R_{k} Π_{i k}^{22}]) + C_{k}^{T} R_{k} C_{k} + {(D_{k}^{T} Q_{k + 1} A_{k})}^{T} Θ_{k}^{- 1} (D_{k}^{T} Q_{k + 1} A_{k}),$

Mixed $H_{2} / H_{\infty}$ controller design

In this section, we shall design a state feedback controller for system (1) such that the design objectives $(R 1)$ and $(R 2)$ are satisfied simultaneously. It turns out that the solvability of the addressed mixed $H_{2} / H_{\infty}$ control problem can be determined by the solvability of certain coupled backwards Riccati-type matrix equations. A computational algorithm is proposed in the sequel to solve such set of matrix equations.

Numerical example

Let us use the networked control systems as the background to illustrate the numerical example. There are different ways to define Quality-of-Service (QoS) for NCS [24]. In this paper, we take into account two of the most popular QoS measures: (1) the point-to-point network allowable data dropout rate that is used to indicate the probability of data packet dropout in data transmission leading to the randomly occurring nonlinearities; and (2) the point-to-point network throughput that is used to

Conclusion

In this paper, the mixed $H_{2} / H_{\infty}$ controller design problem has been dealt with for a class of nonlinear stochastic systems with randomly occurring nonlinearities that are characterized by two Bernoulli distributed white sequences with known probabilities. The stochastic nonlinearities addressed cover several well-studied nonlinearities in the literature. For the multiobjective controller design problem, the sufficient condition of the solvability of the mixed $H_{2} / H_{\infty}$ control problem has been

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A game theory approach to mixed H2/H∞ control for a class of stochastic time-varying systems with randomly occurring nonlinearities

Abstract

Section snippets

Problem formulation

H∞ performance

Mixed H2/H∞ controller design

Numerical example

Conclusion

Automatica

Automatica

Control Engineering Practice

Systems & Control Letters

Automatica

Automatica

Automatica

Induced l2 and generalized H∞ filtering for systems with repeated scalar nonlinearities

IEEE Transactions on Signal Processing

Robust H∞ filtering for stochastic time-delay systems with missing measurements

IEEE Transactions on Signal Processing

New approach to mixed H2/H∞ filtering for polytopic discrete-time systems

IEEE Transactions on Signal Processing

Fault detection in a mixed H2/H∞ seetting

Proceedings of IEEE Conference on Decision and Control

Mixed H2/H∞ filtering theory and an aerospace application

International Journal of Robust and Nonlinear Control

A game theory approach to mixed $H_{2} / H_{\infty}$ control for a class of stochastic time-varying systems with randomly occurring nonlinearities

$H_{\infty}$ performance

Mixed $H_{2} / H_{\infty}$ controller design

Induced $l_{2}$ and generalized $H_{\infty}$ filtering for systems with repeated scalar nonlinearities

Robust $H_{\infty}$ filtering for stochastic time-delay systems with missing measurements

New approach to mixed $H_{2} / H_{\infty}$ filtering for polytopic discrete-time systems

Fault detection in a mixed $H_{2} / H_{\infty}$ seetting

Mixed $H_{2} / H_{\infty}$ filtering theory and an aerospace application