EML5311 Lyapunov Stability & Robust Control Design

In Robust control design of nonlinear uncertain systems, stability theory plays an important role in engineering systems. For any given control system, it is crucial to have a stable system since an unstable control system is useless. Lyapunov . stability criterion is a general and useful procedure for studying the stability of nonlinear systems. The Lyapunov stability theory include two methods, Lyapunov’s first method and Lyapunov’s direct method. Lyapunov’s first method is a technique which simply uses the idea of system linearization(lowest order approximation) around a given point and one can only achieve local stability results with small stability regions. Lyapunov’s direct method is the most important tool for design and analysis of nonlinear systems. Lyapunov’s direct method is directly applied to nonlinear systems without the need to linearization and thus achieves global stability. The basic concept behind Lyapunov’s direct method is that if the total energy of a system, electrical/mechanical; linear/nonlinear, is continuously dissipating, then the system will eventually reach an equilibrium point and remain at that point. Hence, Lyapunov’s direct method include two steps, first find a appropriate scalar function, referred to as Lyapunov function, second evaluate its first-order time derivative along the trajectory of the system. If the Lyapunov function derivative is decreasing along the system trajectory as time increases, then the system energy is dissipating and thus the system will eventually settle down. The definitions below give a more formal statement of admissible choices of Lyapunov function candidate.