Stability of Nonlinear Control Systems by the Second Method of Liapunov

This report investigates the stability of autonomous closed-loop control systems containing nonlinear elements. An n-th order nonlinear autonomous system is described by a set of n first order differential equations of the type dxi/dt=xi(x1, x2, …xn) i=1,2,…n. Liapunov\u27s second (direct) method is used in the stability analysis of such systems. This method enables one to prove that a system is stable (or unstable) if a function V=V (x1, x2, … xn) can be found which, together with its time derivative, satisfies the requirements of Liapunov\u27s stability (or instability) theorems. At the present time, there are no generally applicable straight forward procedures available for constructing these Liapunov\u27s functions. Several Liapunov\u27s functions, applicable to systems described in the canonic form of differential equations, have been reported in the literature. In this report, it is shown that any autonomous closed-loop system containing a single nonlinear element can be described by canonic differential equations. The stability criteria derived from the Liapunov\u27s functions for canonic systems give sufficient and not necessary conditions for stability. It is known that these criteria reject many systems which are actually stable. The reasons why stable systems are sometimes rejected by these simplified stability criteria are investigated in the report. It is found that a closed-loop system will always be rejected by these simplified stabi1ity criteria if the root locus of the transfer function G(s), representing the linear portion of the system, is not confined to the left-half of the s-plane for all positive values of the loop gain. A pole-shifting technique and a zero-shifting technique, extending -the applicability of the simplified stability criteria to systems that are stable for sufficiently high and/or sufficiently low values of the loop gain, are proposed in this report. New simplified stability criteria have been developed which incorporate the changes in the canonic form of differential equations caused by the application of the zero-shifting technique. Other methods of constructing Liapunov\u27s functions for nonlinear control systems are presented in Chapter III, These include the work of Pliss, Aizerman and Krasovski. Numerous other procedures, which have been reported in literature, apply to only very special cases of automatic control systems. No attempt has been made to account for all of these special cases and the presentation of methods of constructing Liapunov’s functions is limited to only those which are more generally applicable. A pseudo-canonic transformation has been developed which enables one to find stability criteria of canonic systems without the use of complex variables. The results of this research indicate that the second method of Liapunov is a very powerfuI tool of exact stability analysis of nonlinear systems. Additional research, especially in the direction of the methods of construction of Liapunov’s functions, will not only yield new analysis and synthesis procedures, but also will aid in arriving at a set of meaningful performance specifications for nonlinear control systems