| Abstract: | Both initial state stability and structural stability of a class of nonlinear systems are studied in this dissertation. These are two fundamental issues underlying many engineering problems such as the monitoring of electric power system stability for initial state stability and system parameter specifications for robustness design in power or general electronic circuits. Yet to be able to decide stability for both cases, one has to know the structure of the stability region and its boundary in the state space and in the system parameter space. Such knowledge serves as the foundation of any numerical implementation and rigorous stability test. For years, such knowledge was lacking either in systems theory or practical application. This dissertation presents results on both stability issues of a class of large nonlinear dynamic systems which includes the power system. The bounds of the number of unstable equilibria on the stability boundary, the parameter space variation bound to preserve dynamical behavior are obtained. To derive these results, some mathematical tools need to be developed. Specifically, Morse theory is generalized to noncompact manifolds and general position concept is developed for phase portraits of nonlinear dynamical systems. Since qualitative and conceptual results are emphasized in this dissertation, algebraic topology becomes a natural tool to characterize the dynamical behavior. Equilibrium Equivalence theorem, which is both mathematically and engineering-wise important is derived to characterize the dynamical behavior of different systems, which is an important step toward classifying systems with similar dynamical behaviors. The applications of these results to power systems are also given in details. |
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