A phase-space method for the synthesis of nonlinear servomechanisms

This paper introduces a method for the synthesis of nonlinear servomechanisms the behavior of which, when stimulated by step inputs of position, approaches as closely as desired the behavior of optimum contactor servomechanisms of the same order. The synthesis procedure begins with the selection of a phase trajectory in a phase space, the co-ordinates of which are system error and derivatives of system error. Theorems are proposed which place sufficient conditions on the given phase trajectory so that the synthesis procedure will yield a nonlinear differential equation of the desired type. The synthesis process is carried out in detail for nonlinear systems of order two, three, and four, and the results are verified by an analogue-computer study. The general conclusions are: 1. A synthesis procedure can be evolved which begins with a phase trajectory and terminates with a nonlinear differential equation describing the behavior of a stable, physically realizable, nonlinear servomechanism; 2. the synthesized nonlinear differential equation is closely related to the linear differential equation of the same order and has coefficients expressed in terms of design-specified quantities; 3. the synthesized nonlinear system´s response is superior to the equivalent linear system´s response; and 4. time solutions of the nonlinear differential equations can be obtained in a number of important cases.