Analýza stability diskrétních dynamických systémů

Abstract

This thesis presents a comprehensive study of the stability properties of discretetime dynamical systems, with a primary focus on linear difference equations (LDEs) with constant coefficients. After introducing firstorder stability concepts, we extend the analysis to higherorder LDEs via the Schur–Cohn criterion, deriving regions of asymptotic stability in the complex parameter plane. Two illustrative higherorder examples are examined: MATLAB codes perform a bruteforce parameter sweep to map stability regions, while the root locus technique delineates their exact boundaries. Crossvalidation of these methods confirms the accuracy of the computed stability domains. A brief exploration of nonlinear difference equations and nonautonomous linear systems, characterizing their stability under timevarying coefficients through Lyapunov theory. Future work may extend these methods to stochastic or realtime control settings, and to nonlinear or distributedparameter systems.

This thesis presents a comprehensive study of the stability properties of discretetime dynamical systems, with a primary focus on linear difference equations (LDEs) with constant coefficients. After introducing firstorder stability concepts, we extend the analysis to higherorder LDEs via the Schur–Cohn criterion, deriving regions of asymptotic stability in the complex parameter plane. Two illustrative higherorder examples are examined: MATLAB codes perform a bruteforce parameter sweep to map stability regions, while the root locus technique delineates their exact boundaries. Crossvalidation of these methods confirms the accuracy of the computed stability domains. A brief exploration of nonlinear difference equations and nonautonomous linear systems, characterizing their stability under timevarying coefficients through Lyapunov theory. Future work may extend these methods to stochastic or realtime control settings, and to nonlinear or distributedparameter systems.