Study of certain aspects of a class of nonlinear multiplicative systems.

Abstract

In this thesis we present a study of certain aspects of a class of nonlinear multiplicative systems. This study is carried out in three directions: (i) linear representation of nonlinear multiplicative systems; (ii) improvement in the performance of systems by introducing intentional multiplicative nonlinearities; (iii) stability study of systems in (ii). In Chapter II, it is shown that an open-loop nonlinear multiplicative system, if excited by a step function input, can be represented exactly by a linear system multiplied by a "gain factor". The method presented here is straightforward, and the representation for the nth order nonlinear multiplicative systems can be written after two complex convolution operations. In Chapter III, it is shown that the performance of linear systems up to third order, and some second order nonlinear differential systems, can be improved by introducing intentional multiplicative nonlinearities in the forward path of the feedback-control system. The resulting control systems are more sensitive to parameter variations than direct feedback control systems. The transient response is extremely insensitive to parameter variations due to high controller gain and presence of multipliers. However, the steady state output of the system changes with parameter variations, but can be corrected by changing the gains of the linear systems H2(s) and H3(s) as discussed in Chapter III. The improvement in the response of the given system, the relation between the number of multipliers and the order of the given linear system, and the relation of the poles of the given linear system to the poles of H2(s) and H3(s), are consistent in every case considered. In Chapter IV, the stability of the systems discussed in Chapter III is studied and shows that, according to Zubov's formulation, a closed form Liapunov function is not possible. However, these systems always have an equilibrium point, which is asymptotically stable and a Liapunov function of the quadratic form can be constructed to give a conservative estimate of the region of asymptotic stability around the equilibrium point.