A Convex Optimization Approach to Synthesizing Bounded Complexity $\ell ^{\infty }$ Filters

We consider the worst-case estimation problem in the presence of unknown but bounded noise. Contrary to stochastic approaches, the goal here is to confine the estimation error within a bounded set. Previous work dealing with the problem has shown that the complexity of estimators based upon the idea of constructing the state consistency set (e.g., the set of all states consistent with the a priori information and experimental data) cannot be bounded a priori, and can, in principle, continuously increase with time. To avoid this difficulty we propose a class of bounded complexity filters, based upon the idea of confining r-length error sequences (rather than states) to hyperrectangles. The main result of the technical note shows that this can be accomplished by using linear time invariant filters of order no larger than r. Further, synthesizing these filters reduces to a combination of convex optimization and line search.