Quasi-Stability Regions: Analysis and Characterization
The stability boundary structure of general nonlinear dynamical systems can be very complex. A simple three-dimensional example given in Zaborszky et al. (1988b) shows that the closure of stability regions may contain subsets of the stability boundary. The stability boundary of a simple swing equation may have a truncated fractal structure (Varghese and Thorp, 1988). There are several factors that contribute to the complexity of the stability boundary. One of them is the presence of critical elements (i.e., the equilibrium point and limit cycle) in the interior of the closure of the stability region. This motivates a study of the notions of the quasi-stability boundary and region. Indeed, from an engineering viewpoint, the quasi-stability region is a “practical” stability boundary while the quasi-stability boundary is less complex than the stability boundary.
We present in this chapter the quasi-stability boundary and develop some characterizations of the quasi-stability region that are useful for the analysis of direct methods. The closest unstable equilibrium point (UEP) and controlling UEP will then be characterized by using the characterization of equilibrium points on the quasi-stability region developed in this chapter. The topic of optimal estimation of the quasi-stability region is also presented. Most of the proofs presented in this chapter are taken from Chiang and Fekih-Ahmed (1996a,b) ...