Balancing via Gradient Flows

Abstract

In the previous chapter we investigate balanced realizations from a variational viewpoint, i.e. as the critical points of objective functions defined on the manifold of realizations of a given transfer function. This leads us naturally to the computation of balanced realizations using steepest descent methods. In this chapter, gradient flows for the balancing cost functions are constructed which evolve on the class of positive definite matrices and converge exponentially fast to the class of balanced realizations. Also gradient flows are considered which evolve on the Lie group of invertible coordinate transformations. Again there is exponential convergence to the class of balancing transformations. Of course, explicit algebraic methods are available to compute balanced realizations which are reliable and comparatively easy to implement on a digital computer, see Laub, Heath, Paige and Ward (1987) and Safonov and Chiang (1989).