Recursive robust H_∞ filtering within the framework of set-valued estimation

A recursive robust H_∞ filtering algorithm is newly proposed for the discrete time uncertain linear system subject to the energy constraint called sum quadratic constraint (SQC). A set valued estimation approach will be used to tackle the given problem. To this end, by combining an SQC on the H_∞ norm condition of the error dynamics and an inequality relationship between the uncertainty input and output, we obtain an augmented SQC and then formulate the robust H_∞, filtering problem as the one of finding the set of estimates satisfying this constraint. The solutions will be given in terms of ellipsoids whose centers are the minimums of the indefinite quadratic function defined by the augmented SQC. The Krein space estimation theory will be utilized to efficiently deal with the minimization problem of the indefinite quadratic function and it is shown that the robust H_∞ filter turns out to be just the special form of Krein space Kalman filter. The proposed robust filter has basically the same recursive structure as the information form of Kalman filter and therefore demands less computational burdens for the implementation. Numerical examples will be given to verify that the proposed filter guarantees the robustness in the presence of parametric uncertainties and its bounding ellipsoidal sets of filtered estimates always contain true states.