Nonlinear analysis of generalized tracking systems (1969)

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 nonlinearity, 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|y <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</inf> , t <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</inf> ) is the solution to an (N+1)-dimensional Fokker-Planck equation. The vector (y, t)=(φ, y <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</inf> ,..., y <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</inf> , t) is Markov and φ represents the system phase error. According to the theory the transition pdf's {p(φ, t|φ <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</inf> , t <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</inf> ), P(y <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</inf> , t <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</inf> |y <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k0</inf> , t <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</inf> ); 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 U <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</inf> (y <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</inf> , t); which is related to the nonlinear restoring force h <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</inf> (y <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</inf> , t)=-∇U <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</inf> (y <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</inf> , t); k=0, 1,..., N. In turn the potential functions are completely determined by the set of conditional expectations {E(y <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</inf> , t|φ), E(g(φ), t|y); 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.