Posted Aug 9, 1996

Some Preliminary Results on Information State System Stability

By J. W. Helton and M. R. James

The purpose of this paper is to present some preliminary results on the stability of the information state system. The information state system underlies the (infinite dimensional) dynamics of an H controller for a nonlinear system. Thus it is important to understand its stability and the structure of its equilibrium points. We analyse the important case corresponding to the mixed sensitivity problem. We prove the existence of an equilibrium information state, convergence under very general conditions to such an equilibrium state $p_e$ and uniqueness of this state (up to an irrelevant constant). In this case the equilibrium $p_e$ is usually singular in the sense that it takes on the value $-\infty$ except on a low dimensional subset of its domain. This meshes with the article \cite{HJ2} which analysed the effect of using $p_e$ to initialize the information state controller and gave explicit formulas which in many cases produce a dramatic reduction in the amount of computation required to implement the controller. What this article suggests is that indeed $p_e$ is the only equilibrium initialization possible.