Computing Balanced Realizations for Nonlinear Systems

Abstract

This paper addresses the problem of computability pertaining to the Scherpen(1994) theory and procedure for balancing of nonlinear systems. In contrastto Moore's (1981) balancing method for linear systems, the Scherpen procedurefor nonlinear balancing is not immediately amenable to computationalimplementation. For example, the controllability energy function correspondsto the value function for a nonlinear optimal control problem. Also, theMorse-Palais lemma guarantees the existence of a local coordinatetransformation under which the controllability energy function takes acanonical quadratic form, but provides no constructive procedure for obtainingit. Thus, tools have not yet appeared for computing balanced realizations fornonlinear systems, and the procedure has not yet been applied as a tool formodel reduction.First, we consider the problem of computing the controllability energyfunction without numerically solving the family of optimal control problems,or the associated Hamilton-Jacobi-Bellman equation, implied in its definition.Stochastically excited systems play a major role in our methodology. Wepresent a stochastic method for computing an estimate of the controllabilityfunction, and show that in certain situations the method provides an exactsolution. The procedure is tested on applications via Monte-Carlo experiments.Then, we address the problem of numerically determining a Morse transformationfor a function with non-degenerate critical point at 0. We develop analgorithm for computing the desired nonlinear transformation and estimatingthe neighborhood on which the transformed controllability function isquadratic. In the literature, examples of applied nonlinear balancing have been limited topseudo-balancing of 2-dimensional gradient systems and noting that in the caseof linear systems the energy functions approach reduces to the usual setting ofgramians. We apply our approach to numerically derive, for the first time,balanced representations of nonlinear state-space models. In particular, wepresent applications to a forced damped pendulum system and a forced dampeddouble pendulum system.The research and scientific content in this material has been published in theProceedings of the 14th International Symposium on Mathematical Theory of Networks and Systems, Perpignan, France, June 19-23, 2000.