Summary: APPROXIMATE REDUCTION OF DYNAMICAL SYSTEMS
PAULO TABUADA, AARON D. AMES, AGUNG JULIUS, AND GEORGE J. PAPPAS
Abstract. The reduction of dynamical systems has a rich history, with many important applications related
to stability, control and verification. Reduction of nonlinear systems is typically performed in an "exact"
manner--as is the case with mechanical systems with symmetry--which, unfortunately, limits the type of
systems to which it can be applied. The goal of this paper is to consider a more general form of reduction,
termed approximate reduction, in order to extend the class of systems that can be reduced. Using notions
related to incremental stability, we give conditions on when a dynamical system can be projected to a lower
dimensional space while providing hard bounds on the induced errors, i.e., when it is behaviorally similar to
a dynamical system on a lower dimensional space. These concepts are illustrated on a series of examples.
Modeling is an essential part of many engineering disciplines and often a key ingredient for successful designs.
Although it is widely recognized that models are only approximate descriptions of reality, their value lies
precisely on the ability to describe, within certain bounds, the modeled phenomena. In this paper we consider
modeling of closed-loop nonlinear control systems, i.e., differential equations, with the purpose of simplifying
the analysis of these systems. The goal of this paper is to reduce the dimensionality of the differential
equations being analyzed while providing hard bounds on the introduced errors. One promising application
of these techniques is to the verification of hybrid systems, which is currently constrained by the complexity
of high dimensional differential equations.
Reducing differential equations--and in particular mechanical systems--is a subject with a long and rich