Closed-Loop System Analysis Using Lyapunov Stability Theory

A special class of closed-loop systems composed of a controller and observer in cascade are analyzed. The plant dynamics are assumed to be linear and time-varying but the system parameters are uncertain. The class of observation functions is restricted to those that can be transformed into a linear structure in the state called pseudo-linear measurements where the coefficient may be an explicit function of the original measurements. If along a given path the state vector is observable, then the estimation error of a linear observer structure can be shown to be asymptotically stable. The emphasis is on deriving and analyzing general Lyapunov functions which indicate system stability or a measure of system performance under parameter variations. The first Lyapunov function is developed by combining the separate controller and observer Lyapunov functions, both of which are quadratic. This combined Lyapunov function is not valid for all linear, time-varying, closed-loop systems. A second Lyapunov function is derived to account for the system where the controller is a function of the estimated states. This Lyapunov function is valid for linear, time-varying, closed-loop systems. A third Lyapunov function is derived to directly account for parameter uncertainties in the system model. Keywords Homing missile guidance Theses.