CDS Lecture Series 1999

Friday, February 12, 1999, 2:00 p.m.

W. Steven Gray
Old Dominion University
Department of Electrical and Computer Engineering

General Input Balancing and Model Reduction for Linear and Nonlinear Systems

Model reduction for linear state space systems has been investigated by many researchers over the past twenty years. Perhaps the most widely used methods in control applications are those related to the method of balance realizations introduced by Moore. In this context, a sufficient condition for a balanced realization to exist is minimality, i.e., joint controllability and observability, which can be determined independently from the class of admissible inputs. The input class does, however, play a role in the model reduction process. For example, a state may be strongly influenced by an input stimulus at a certain frequency and unaffected otherwise. Hence it could be deleted from the state variable model if this resonant input is never generated by the applied controller. In both the linear and nonlinear cases, various notions of closed-loop balancing exist where the admissible inputs are assumed to be generated by specific controller types (LQG-optimal, H_inf-optimal, etc.). In the nonlinear setting, the input class plays two roles. As in the linear case, it still plays a direct role in the state deletion decision, but it also plays a role in the observability property. In short, the choice of input class is linked to the existence of a balanced nonlinear realization. In the well known method due to Scherpen, the standing assumption is that the nonlinear system is zero-state observable, i.e., the zero-input is a universal input. In this talk, we present a less restrictive notion of nonlinear balancing where observability is required only over the set of finite energy inputs. This idea then leads naturally to a generalized notion of closed-loop balancing.

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