On the Estimation of Asymptotic Stability Region of Nonlinear Polynomial Systems: Geometrical Approaches

In recent years, the problem of determining the asymptotic stability region of autonomous nonlinear dynamic systems has been developed in several researches. Many methods, usually based on approaches using Lyapunov’s candidate functions (Davidson & Kurak, 1971) and (Tesi et al., 1996) which altogether allow for a sufficient stability region around an equilibrium point. Particularly, the method of Zubov (Zubov, 1962) is a vital contribution. In fact, it provides necessary and sufficient conditions characterizing areas which are deemed as a region of asymptotic stability around stable equilibrium points. Such a technique has been applied for the first time by Margolis (Margolis & Vogt, 1963) on second order systems. Moreover, a numerical approach of the method was also handled by Rodden (Rodden, 1964) who suggested a numerical solution for the determination of optimum Lyapunov function. Some applications on nonlinear models of electrical machines, using the last method, were also presented in the Literature (Willems, 1971), (Abu Hassan & Storey, 1981), (Chiang, 1991) and (Chiang et al., 1995). In the same direction, the work presented in (Vanelli & Vidyasagar, 1985) deals with the problem of maximizing Lyapunov’s candidate functions to obtain the widest domain of attraction around equilibrium points of autonomous nonlinear systems. Burnand and Sarlos (Burnand & Sarlos, 1968) have presented a method of construction of the attraction area using the Zubov method. All these methods of estimating or widening the area of stability of dynamic nonlinear systems, called Lyapunov Methods, are based either on the Characterization of necessary and sufficient conditions for the optimization of Lyapunov’s candidate functions, or on some approaches using Zubov’s digital theorem. Equally important, however, they also have some constraints that prevented obtaining an exact asymptotic stability domain of the considered systems. Nevertheless, several other approaches nether use Lyapunov’s functions nor Zubov’s which have been dealt with in recent researches. Among these works cited are those based on topological considerations of the Stability Regions (Benhadj Braiek et al., 1995), (Genesio et al., 1985) and (Loccufier & Noldus, 2000). Indeed, the first method based on optimization approaches and methods using the consideration of Lasalle have been developed to ensure a practical continuous stability