Nonlinear Analysis of Generalized Tracking Systems

Abstract

This paper sets forth a rather general analysis pertaining to the performance and synthesis of generalized tracking systems. The analysis is based upon the theory of continuous Markov processes. in particular. the Fokker-Planck equation. We point out the interconnection between the theory of continuous Markov processes and Maxwell's wave equations by interpreting the charge density as a transition probability density function (pdf). These topics presently go under the name of probabilistic potential theory. Although the theory is valid for (N+1)-order tracking systems with an arbitrary, memoryless periodic nonlineerity we study in detail the case of greatest practical interest, viz., a second-order tracking system with sinusoidal nonlinearity In general we show that the transition pdf p(y, t/y0, to) is the solution to an (N+1)-dimensional Fokker-Planck equation. The vector (y t)=Φ, Y1, YN, t) is Markov and., Φ the system phase error. According to the theory the transition pdf's (y t)=Φ, Y1, YN, t) k=1 •. • •, N} of the state variables satisfy a set of second-order partial differential equations which represent equations of flow taking place in each direction of (N+1)-space. Each equation, and solution is characterized by a potential function Uk(yk, t) which is related to the nonlinear restoring force [formula omitted]. In turn the potential functions are completeely determined by the set of conditional expectations {E(yk, t|¢, E(g(¢, tly); k=1 2…, N}. It is conjectured that the potential functions represent the projections of the system Lyapunov function which characterizes system stability. This paper explores these relationships in detail. Copyright © 1969 by The Institute of Electrical and Electronics Engineers, Inc.