Estimating regions of asymptotic stability of nonlinear systems with applications to power electronics systems
A region of asymptotic stability is a set of points surrounding a stable equilibrium point for which every system trajectory starting at a point in the set asymptotically returns to the equilibrium point. The objective of this research is to develop and validate computationally tractable methods of estimating regions of asymptotic stability of nonlinear systems and apply them to power electronics systems. Contributions are made in the areas of Lyapunov function generation and finding a Lyapunov function level set that bounds a region of asymptotic stability. The matrix positive image is defined and characterized, and a method of finding Lyapunov functions using the positive image is set forth. A more computationally efficient method of finding Lyapunov functions using linear matrix inequality techniques is proposed. A third Lyapunov function generation method that only requires diagonalization of the Jacobian matrix of the nonlinear system is developed and analyzed. A genetic algorithm optimization approach to finding the boundary of a region of asymptotic stability estimate is applied to power electronics system models of order 6, 8, 17, 33, 54, and 75. Simulation studies validate the region of asymptotic stability estimates. The genetic algorithm approach is compared to two other optimization techniques and is found to be the most effective of the three. Finally, a method of finding differently shaped region of asymptotic stability estimates to be combined in a union of estimates is set forth and applied to the 33rd order system.
Zak, Purdue University.
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