On robust filter design for uncertain neural systems: LMI optimization approach
Introduction
State estimation has been one of the fundamental issues in the control area. Recently, there has been a lot of interest on the problem of robust filtering for dynamic systems with parametric uncertainties. In the filtering, the exogenous noise signal is assumed to be energy bounded rather than Gaussian, and the problem is to design a filter such that the norm of the system, which reflects the worst-case gain of the system, is minimized [11].
On the other hand, the stability analysis and stabilization problem for delay differential systems has been received considerable attention during the last few decades [1], [2]. It is well-known that the delay is often main cause of instability and poor performance of systems. More recent years, the stability analysis for neutral delay-differential equations/systems of various types has been extensively studied [3], [4], [5], [6], [7], [8], [9]. Various methods has been introduced to derive less conservative and concise stability criteria for neutral systems [3], [4], [5], [6], [7], [8]. A control scheme to guarantee the adequate level of performance has been presented by Park [9], and the design problem on observer-based controller of a class of neutral system is investigated in Park [10]. However, so far, the robust filtering problem for neutral differential systems has not been fully investigated and remains to be important and challenging.
In this paper, the problem of robust filtering for neutral delay-differential systems subjected to parameter uncertainties is investigated. The uncertainties are assumed to be bounded. Using the Lyapunov functional technique combined with LMI technique, we develop a robust filter for this system, which makes the closed-loop system asymptotically stable and the bound of norm be minimized. A stability criterion for the existence of the filter is derived in terms of LMIs, and theirs solutions provide a parameterized representation of the filter. The LMIs can be easily solved by various efficient convex optimization algorithms [13].
Section snippets
Problem formulation
Consider a class of neutral delay-differential system of the form:where is the state vector, A0, A1, A2, B, C, and D are known constant real matrices of appropriate dimensions, is the noise signal vector (including process and measurement noises) in , is the measured output, h and τ are the positive constant time delays, ΔA0(t), ΔA1(t) and ΔC(t) are time-varying
Robust stability analysis and filter design
Before proceeding further, we will give two lemmas and a fact. Lemma 1 For any constant symmetric positive-definite matrix , a scalar σ>0, and the vector function such that the integrations in the following are well defined, then Lemma 2 For given positive scalars h and τ and any , the operator defined byis stable if there exist a positive definite matrix Γ and positive scalars α1 and α2 such that[14]
[12]
Conclusions
This paper investigated the design of robust filter for neutral delay-differential system with parametric uncertainties in the system state-space model. A methodology for the design of a stable linear filter that assures asymptotic stability and minimization of performance for the filtering error system is studied. The filter is obtained from a convex optimization problem described in terms of LMIs. The validity of proposed filter design algorithm has been checked through a numerical
Acknowledgements
J.H. Park is grateful to H.Y. Jung, H.J. Baek, W.C. Shim, and S.G. Kim for their good advice about my research, and to my students, I.G. Jung, Y.I. Ban, and S.D. Lee for their devoted contribution in my laboratory. This work was supported by grant No. R05-2003-000-10173-0 from Korea Science & Engineering Foundation.
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