Abstract

We consider the problem of robust filtering of an observable signal to estimate another signal. Both signals are assumed to be generated by a finite dimensional linear discrete time invariant system driven by a disturbance signal. We characterize the performance of an estimator by the maximum energy-to-energy gain from the disturbance to the estimation error signals over all stationary Gaussian disturbances whose mean anisotropy is bounded from above by a given nonnegative parameter a. The mean anisotropy is an information theoretical measure of colouredness and spatial nonroundness of a signal. We construct an optimal estimator which minimizes the performance index, the so called a-anisotropic norm, of the operator relating the estimation error and disturbance signals. The resulting a-anisotropic estimator retains the structure of the steady-state Kalman filter. However, computing the gain matrices of the estimator reduces to a system of two cross-coupled algebraic matrix Riccati equations and one more equation involving the logarithm of the determinant of a matrix. A homotopy method is considered for numerical solution of the set of equations. In two limiting cases, a = 0 and a tending to infinity, the a-anisotropic estimator yields the classical Kalman filter and H-optimal estimator, respectively. However, for intermediate values of the anisotropy level a, the a-anisotropic estimator provides for more robustness than the Kalman filter, and yet is less conservative than the H-optimal estimator.