Dissipativity preserving balancing for nonlinear systems - A Hankel operator approach

In this paper we present a version of the balancing technique for nonlinear systems which are dissipative with respect to a general quadratic supply rate that depends on the input and the output of the system. We discuss an input output approach that allows us to apply the theory of balancing based upon Hankel singular value analysis. In order to do that we prove that the available storage and the required supply of the original system, solutions of a (general) Hamilton-Jacobi-Bellman equation, are the controllability and the observability functions of asymptotically stable realizations. Then the Hankel singular value theory can be applied and a relation between the axis singular value functions obtained for these realizations and the axis singular value functions with respect to the available storage and the required supply of the original system, is provided. Thus balanced realizations are provided and truncation is performed yielding dissipative reduced order models.