Induced l2 and generalized H2 filtering for systems with repeated scalar nonlinearities

This paper provides complete results on the filtering problem for a class of nonlinear systems described by a discrete-time state equation containing a repeated scalar nonlinearity as in recurrent neural networks. Both induced l₂ and generalized H₂ indexes are introduced to evaluate the filtering performance. For a given stable discrete-time systems with repeated scalar nonlinearities, our purpose is to design a stable full-order or reduced-order filter with the same repeated scalar nonlinearities such that the filtering error system is asymptotically stable and has a guaranteed induced l₂ or generalized H₂ performance. Sufficient conditions are obtained for the existence of admissible filters. Since these conditions involve matrix equalities, the cone complementarity linearization procedure is employed to cast the nonconvex feasibility problem into a sequential minimization problem subject to linear matrix inequalities, which can be readily solved by using standard numerical software. If these conditions are feasible, a desired filter can be easily constructed. These filtering results are further extended to discrete-time systems with both state delay and repeated scalar nonlinearities. The techniques used in this paper are very different from those used for previous controller synthesis problems, which enable us to circumvent the difficulty of dilating a positive diagonally dominant matrix. A numerical example is provided to show the applicability of the proposed theories.