Observer-Type H_inf Filter Design for Structural Systems

The present paper addresses the problem of Luenberger-type H1 filter design for structural systems af- fected by external disturbance and sensor noise signals. An explicit bound on the H1 norm performance of the filtering error system is derived, and a bound on the filter gain matrix is proposed to guarantee a prescribed level of H1 performance for the augmented system. We will prove that the structure of the proposed filter is explicit, and the filter design does not require any burdensome computations. The numerical examples validate our proposed bounds on the filter gain matrix, as well as, the norm of the error system. The estimation of the state vector x(t) of a linear system from input and output information is an important problem in the systems and control area. The state vector x(t) can be estimated independently of the control problem, and the resulting estimate ^ x(t) can be used for control or fault detection purposes. Recently, several researchers have addressed the computational advantages of designing state estimators directly using the second-order structural models. 4 An optimal Kalman filter has been proposed by Hashemipour and Laub 10 for discretized second-order structural models. In the work published by Juang and Maghami 12 the eigenvalues and eigenvectors of the state estimator are assigned by solving the second-order eigenvalue problem of the structural system. A variable structure observer is designed by Cao et. al. 7 based on the natural matrix second order model for linear and nonlinear mechanical vibrating structures. The H1 filtering technique has been widely studied for the benefit of different time and frequency domain properties. In the H1 setting, the exogenous input signal is assumed to be energy bounded rather than Gaussian. An H1 filter is designed such that the H1 norm of the augmented system of the plant and the filter, which reflects the worst-case L2-gain of the system, is minimized. The advantage of using an H1 filter in comparison with an H2 filter is twofold. First, no statistical assumption on the input is needed. Second, the filter tends to be more robust when there exists additional uncertainty in the system, such as quantization errors, delays, and unmodeled dynamics. These features make H1 filtering technique useful in certain applications. The problem of H1 filtering contains two aspects: H1 filter analysis and H1 filter synthesis. The analysis aspect is to determine the worst-case H1 performance when a filter is given, whereas the synthesis aspect is to design a filter such that the worst-case H1 performance is satisfactory. The H1 filter design has been employed to identify civil engineering structural systems in the literature. 14, 16 In this paper first we address the problem of H1 filter design for the structural systems affected by external disturbance inputs. The proposed method provides an upper bound on H1 performance of the augmented system of the filter and the structural system and provides an explicit formula for the H1 filter matrix gain which guarantees a prescribed level of the H1 performance for the error system. In the next part of the paper we add measurement noise into the structural system equations and propose a solution for the unknown filter gain to satisfy a desired level of the H1 performance for the error system. Numerical examples validate the effectiveness of the results. The notation used in this paper is standard. Given a symmetric matrix X = X T 2 R n£n , X > 0 (X ‚ 0) denotes matrix positive definiteness (semi-definiteness). (:) T denotes the transpose of a real matrix, (:) y represents the Moore-Penrose generalized inverse of a matrix. Given a real n£m matrix Y with rank r, the orthogonal complement Y ? is defined as the (nir)£n matrix that satisfies Y ? Y = 0 and Y ? Y ?> > 0. The maximum eigenvalue of a matrix M will be denoted by ‚max(M).