Generalization of the parameter plane method

Abstract

Absimcf-This paper presents a generalization of the parameter plane method in that it considers the case when the characteristic equation coefficients are nonlinear functions of the system adjustable parameters. The generalized method is then applied to the system analysis in which the coefficients are linear functions of two parameters and their product. As a simple and rapid procedure for factoring polynomials in the parameter plane, the method is used in the design of linear continuous multivariable control systems. A nonlinear system with two nonlinearities is also considered whereby the stability and existence of limit cycles are investigated. 1 N PREVIOUS papers [1]-[3]1 a parameter plane method has been applied to the analysis and synthesis of linear continuous, sampled-data, and nonlinear control s);stems. In the linear analysis [I], the method relates tn-o adjustable system parameters, which appear linearly in the coefficients of the relevant characteristic equation, to all the root locations. Asimple and rapid procedure has been developed for factoring characteristic polynomials in the plane of these two parameters. This enables the designer to maintain control over salient system characteristics of both transient and frequency responses [4]. The procedure extends in a straightforn-ard manner to sampled-data control systems [2], [SI. Recently, the parameter plane method has been applied to the simultaneous consideration of steady-state and transient responses of linear control systems [6].In nonlinear system analysis [SI, the stability of selfexcited oscillations has been investigated in the parameter plane with respect to both the system parameters and the initial conditions. Control systems with two nonlinearities and with amplitude-and frequencydependent describing functions have been designed. The proposed method has been extended to relative stability and sensitivity problems in nonlinear controls [7], [8].Limitations of the parameter plane method may arise when the adjustable system parameters enter nonlinearly into the coefficients of the characteristic equation. This paper considers a general case in v,-hich coefficients depend nonlinearly on two system parameters and applies the obtained results to the specific case when the coefficients are functions of the linear combination of two parameters and their product. The presented procedure is then applied to the design of a linear multivariable control system by giving in evidence the polezero locations of the closed-loop transfer functions as a function of adjustable system parameters. -4 nonlinear multivariable control system with tn-o nonlinearities is used to illustrate the stability anall-sis of nonlinear systems.In addition, certain rules for the mapping of the contours from the complex variable plane into the parameter plane, 11-hich have been given intuitively in the reference papers [1]- [3], are now proved. Some additional theorems are introduced to facilitate the interpretations of parameter plane diagrams. I t is also to be noted that ...

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