On the Application of Liapunov’s Second Method to the Stability Analysis of Time-invariant Control Systems

The investigation of the stability properties of the equilibrium point of a control system poses various problems -which, even if conceptually very similar, vary greatly in difficulty and in the methods appropriate for their solutions. The first and easier problem is what we may call the stability analysis of a completely defined system: given a particular control system to decide what stability properties its equilibrium point has. The second problem deals with a system having a fixed configuration but with parameters whose numerical values are to be determined. The problem is to find the boundaries in the parameter space at which the stability properties of the system undergo a change. The third problem is that of synthesizing stable systems and its solution implies knowledge of necessary and sufficient conditions for the equilibrium point of the system to be stable. This problem is far from being solved and it is also doubtful if its practical solution will emerge from the classical theory of stability. In fact stability problems of this generality arise only in very general formulations like the synthesis of systems which are optimal in some sense. It should he easier to include stability among the other constraints that the system must satisfy, then investigate this property separately. Obviously then many problems about structural stability will arise, but the logical procedure will still be to find at least a fixed structure for the system, by various variational methods, and then to investigate its stability properties; in other words, to reduce the general problem to the second problem of our classification. In the present report, this problem will be investigated by means of the Second Method of Liapunov. The Second Method of Liapunov is essentially based upon the now classical Grande Memorie [1] that the Russian scientist published in I893. This method can, however, be thought of in different ways. The first way is as a general procedure for tackling the problem of stability of systems, a way of thinking, a policy, more than a well defined stability criterion: in this fashion, it has mostly been used in mathematical works. In the Soviet Union especially, this state of mind prevails also in the area of control theory, in other words, the Liapunov Second Method has been almost exclusively applied in order to develop some algebraic condition of stability. In other words, the Liapunov method has been applied to stability problems in order to find stability criteria applicable to certain classes of systems. On the contrary, we regard the Liapunov Second Method as a stability criterion, based upon certain theorems ( 3). The main aim of this work is to develop a method ( 5) for the construction of Liapunov functions for autonomous systems. This method in contrast to others will always yield a solution of the stability problem. The price we have to pay for assuring that the method always works is the restriction to a particular class of Liapunov functions, solutions of a partial differential equation (19) \u27which turns out to be the generalization of an analogous equation proposed by Zubov [2]. Since the solution of this partial differential equation is by no means elementary and since there exist very easy methods [3, 4, 5] which give the answer in many cases (but fail in others.’) the investigation of the stability by applying this new method is advisable only in the case in which these simpler methods have failed. In order to make this paper reasonably self-contained, a short outline of these three methods are given. [10] and a comparison between them is made. On the basis of this critical study of the available methods for constructing Liapunov functions, we shall suggest a scheme for attacking the stability problem