Time-varying filtering of perturbed linear systems

The design of filters guaranteeing a bounded error covariance in the face of parameter uncertainties affecting the system has been investigated in previous works. These results hinge on the minimization of an upper bound of the steady-state filtering error covariance and are limited to the class of time-invariant and stable systems. In the present paper, the authors consider the more general problem of designing a time-varying robust filter for time-varying uncertain systems, starting from the knowledge of a bound on the covariance of the initial state. The analysis is not restricted to the steady-state behavior and does not require any stability assumption. The first step is the derivation of an upper bound for the state covariance of a time-varying perturbed linear system. Precisely, the bound is shown to be the solution of a time-varying differential Riccati equation. This result is applied to compute an upper bound for the filtering error covariance associated with any given linear time-varying filter. The minimization of such a bound leads to the optimal robust filter. The filter parameters depend on the solution of a further differential Riccati equation in which a scalar parameter function β(·) appears. Differently from previous contributions, the authors suggest a possible choice of β(·) for improving the filter performance. Contrary to the standard Kalman filter Riccati equation, the Riccati equations involved in the solution of the robust filtering problem can have finite escape times, so that the existence of the filter over arbitrarily long horizons is not guaranteed. When the system is time-invariant, the authors give a rather complete characterization of the escape times for the state covariance bound. This allows the authors to provide a sufficient condition for the extendibility over (t ₀,∞) of the (time varying) robust filter