The Important State Coordinates of a Nonlinear System

We offer an alternative way of evalating the relative importance of the state coordinates of a nonlinear control system. Our approach is based on making changes of state coordinates to bring the controllability and observability functions into input normal form. These changes of coordinates are done degree by degree and the resulting normal form is unique through terms of degree seven. 1 The Problem The theory of model reduction for linear control systems was initiated by B. C. Moore [6]. His method is applicable to controllable, observable and exponentially stable linear systems. The reduction is accomplished by making a linear change of state coordinates to simultaneously diagonalize the controllability and observability gramians and make them equal. The diagonal entries of the gramians are the singular values of the Hankel map from past inputs to future outputs. The reduction is accomplished by Galerkin projection onto the states associated to large singular values. The method is intrinsic, the reduced order model depends only on the dimension of the reduced state space. Jonckheere and Silverman [4] extended Moore's methodology to control-lable, observable but not necessarily stable linear system. Their method is based on simultaneously diagonalizing the positive definite solutions of the control and filtering Riccati equations and making them equal. The diagonal entries are called the characteristic values of the system and reduction is achieved by Galerkin projection ontothe states associated to large characteristic values. The method is sometimes called LQG balancing and reduction. Two nice features of their approach is that it is applicable to unstable systems and LQG controller of the reduced order model is the Galerkin projec