Relative stability for control systems with adjustable parameters

Abstract

This paper presents an approach for finding regions Of specified gain margins (GM) and phase margins (PM) in the parameter space of a single-input single-output (SISO) control system with adjustable parameters. The method uses the Nyquist stability criterion of encirclement of a point in the polar plot. The necessary of the specified GM or PM regions in the parameter space are fully treated by mapping a point related to specified GM or PM in the polar plane into the parameter space. A general problem with previous mapping techniques is identified where points in the parameter space do not necessarily satisfy the required GM or PM. A searchbased technique is proposed to address this problem. Three examples illustrate the theoretical development. I. Introduction G AIN margin (GM) and phase margin (PM) are perhaps the two most important and widespread accepted criteria in analysis and design of practical control systems. These criteria are related to the open-loop transfer function of systems. In the robust stabilization problem of control systems, however, parameter space methods generally consider the closed-loop characteristic equation of the system.13 Parameter space techniques were initially studied by Vishnegradesky for a third-order system using a graphical technique that was popularized by Nimark and was referred to as D-partition or D-decomposition46 in its early development. This method actually establishes a direct correlation between the variable parameters of the closed-loop characteristic equation and the various important stability regions, e.g., open left half of the complex plane (OLHP). The method is directly suitable for graphical representation, in which case it is convenient to consider two parameters at a time. The idea of correlating the free parameters of the open-loop transfer function of a system and the relative stability regions appears to have been initially given in Ref. 7. Unfortunately , the method was not developed in detail, and the idea of isolating the GM region in the parameter plane was not correct. A recent improvement to the idea was presented by introducing the gain-phase margin tester into the system,8 which attempted to isolate specified GM and PM regions in the parameter space to allow a choice of the adjustable parameters. However, it is argued here that in this latter approach all the possible of the relative stability regions are not in fact considered, and some and possibly all points in the relative stability region do not necessarily satisfy the actual specified relative stability margin. In particular, the so called singular boundaries of the D-partition method6 need to be considered. Finally, arising through frequency loci with re-entrant or convoluted characteristics also need to be considered with care. In this paper, a method for accounting for the