Ellipsoidal Approximations in Set-Membership State Estimation of Linear Discrete-Time Systems

There are mainly two approaches to the state estimation or the system identification problems, where the system contains additive unknown values. One is the well-known stochastic approach, and the other is the setmembership approach, where the unknown values are treated as unknown deterministic values in bounded sets. In this paper, we consider the state estimation of linear systems when the initial state vector, input vector and observation noise lie in ellipsoids. Then the state vector is guaranteed to exist in the membership set of the state, where all the points satisfy the system equations with unknown values in given ellipsoids. We first show ellipsoidal approximations that contain the smallest membership set at the prediction step and also at observation-update step, where each step has one free parameter. It is shown that the smallest membership set can be formally obtained by the intersection of infinitely many ellipsoids. But, for practical applications, we derive the optimal ellipsoids whose volumes are the smallest at the prediction step, and at the observation-update step based on the chosen prediction ellipsoid, respectively, for SISO systems. This scheme is called 'semioptimal' in this paper, and is free from using nonlinear optimization methods. However, there exist smaller ellipsoids if we do not divide the scheme into two steps. We show a direct method to minimize the volume of the ellipsoid using nonlinear optimization. This method is also applicable to MIMO system. The smoothing problem is also considered.