Set-valued estimation approach to recursive robust H∞ filtering

A recursive robust H∞ filtering algorithm is proposed for a discrete-time uncertain linear system subject to the sum quadratic energy constraint. This type of uncertainty description can accommodate a large class of uncertainties. A set-valued estimation approach is used to tackle the problem. To this end, an augmented energy constraint is produced by combining an energy constraint on the H∞-norm condition of the error dynamics and an inequality relationship between the uncertainty input and output. The robust H∞ filtering problem is formulated as finding the set of estimates that satisfy the augmented constraint. The solutions are given in terms of ellipsoids whose centres are the minimums of the indefinite quadratic function defined by the augmented constraint. Krein space estimation theory is utilised to efficiently deal with the minimisation problem of the indefinite quadratic function it is shown that the robust H∞ filter is simply a special form of, the Krein space Kalman filter. The proposed robust filter has basically the same structure as the information form of a Kalman filter and therefore needs only a small computational effort is required in its implementation. In addition, it can be reduced into versions of robust and nominal filters by tuning the relevant parameters. Numerical examples are presented that verify that; (i) the proposed filter guarantees robustness in the presence of parametric uncertainties; and (ii) its bounding ellipsoidal sets of filtered estimates always contain true states.