Verriest, Erik I. (Georgia Inst. of Tech.), Gray, W. Steven (Old Dominion Univ.)

Geometry and Topology of the State Space Via Balancing

Scheduled for presentation during the Mini-Symposium "Model order reduction for nonlinear systems" (TuA12), Tuesday, July 25, 2006, 11:15−11:40, Room 104b

17th International Symposium on Mathematical Theory of Networks and Systems, July 24-28, 2006, Kyoto, Japan

This information is tentative and subject to change. Compiled on April 26, 2024

Abstract

In earlier work we extended the linear balancing method to the class of nonlinear systems. While sharing many similarities with the method first researched by Scherpen, there are some fundamental differences. Sliding-Interval-Balancing (SIB), which forms the basis of our method, is briefly reviewed. Our original approach towards the generalization of balancing from linear to nonlinear systems was based upon three principles: 1) Balancing should be defined with respect to a nominal flow; 2) Only Gramians defined over small time intervals should be used in order to preserve the accuracy of the linear perturbation model and; 3) Linearization should commute with balancing, in the sense that the linearization of a globally balanced model should correspond to the balanced linearized model in the original coordinates. However, it is known that an integrability condition generically provides an obstruction towards such a notion of a globally balanced realization in the strict sense. Whereas an interpolation method (Mayer-Lie interpolation) has been proposed to get around this, in this paper, it will be shown that the information obtained by local balancing already provides a lot of useful information about the dominant dynamics of the system and the topology of the state space. The state manifold M is endowed with two Riemannian metrics: One modeling the local reachability properties and one modeling the local observability properties. These metrics give an (in general different) intrinsic curvature to these manifolds. Local balancing at a point P corresponds to bending and reshaping the manifolds without tearing so that near P there is a snug fit (osculating contact). In general, however, there will be a misfit further away from P if the two manifolds have a different Euler characteristic. This explains why global balancing and pseudo-balancing may not be possible except in particular cases.