OPTEC - K.U.Leuven Center of Excellence: Optimization in Engineering

"Nonlinear Balancing and Model Reduction"
Erik Verriest,

Abstract:
A method for generalizing balancing to nonlinear systems is sketched. It differs from nonlinear balancing as introduced by Scherpen in 1994, and is 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 linearizedmodel in the original coordinates.
The first two principles lead to local balancing, but it is shown that an integrability condition generically provides an obstruction towards a notion of a  globally balanced realization in the strict sense. The information obtained by local balancing of a nonlinear system already provides a lot of useful information about the dominant dynamics of the system and the topology of the state space. To accomplish local balancing, two Riemannian metrics are specified: One models the local reachability properties and one models the local observability properties. In general these are incompatible, inducing a different global topology, and thus explaining the aforementioned obstruction. Locally, it still may be possible to match these up, and 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) between the induced manifolds. Unlike the linear case, sensitivity and reduced modeling must be local concepts, and lead at best to a hybrid reduced model with modes of different dimension.   
Finally, the use of stochastic reduced models will be mentioned, introducing a notion of uncertainty equivalence.

Biography:
Erik I. Verriest received the degree of ‘Burgerlijk Electrotechnisch Ingenieur’ from the State University of Ghent, Ghent, Belgium and the M.Sc. and Ph.D. degrees from Stanford University.  He was with  the Control Systems Laboratory and the Hybrid Computation Centre, Ghent, Belgium in 1973-74.  He joined the faculty of Electrical and Computer Engineering at Georgia Tech in 1980, and spent three years at Georgia Tech Lorraine in Metz, France.  His  interests are in mathematical system theory, with focus on periodic and hybrid systems, delay – differential systems, model reduction for nonlinear systems, and control with communication constraints. He served on several IPC’s and is a member of the IFAC Committee on Linear Systems.  He is presently with ESAT, KULeuven, Belgium.