PERFORMANCE FUNCTIONS AND SUBLEVEL SETS FOR FREQUENCY DOMAIN DESIGN OF SINGLE LOOP FEEDBACK CONTROL SYSTEMS WITH PLANT UNCERTAINTY

The frequency domain design of control systems involves the fitting of a designable complex function of frequency (e.g., loop transfer function, compensator transfer function) to specification derived constraints in the design space ℂ×ℝ. In the classical Bode approach and its recent modifications the designable function is the loop transfer function, the specifications lead to constraints on the magnitude of the loop transfer function and the constraints have circular cross-sections in ℂ. In more recent design procedures (e.g., H _∞ optimization or QFT) the specifications lead to more complex constraint surfaces in ℂ×ℝ and the design procedure must fit the designable function to these constraints. In this paper we develop explicit representations of some important ℂ×ℝ constraint surfaces encountered in the design of SISO control systems with both non-parametric (unstructured) and parametric plant uncertainty and study the characteristics of their frequency axis cross-sections or level sets: important information use in the fitting process. The results are presented in two systems of co-ordinates (i.e., designable functions): nominal closed loop transfer function T_o(jω) and and nominal open loop transfer function L_o(jω). While the results in T_o co-ordinates are more useful when using mathematical optimization (e.g., H _∞ optimization techniques), the results in L_o co-ordinates have significant advantages of insight and ease of graphical manipulation as demonstrated in QFT. The inclusion of results in both sets of co-ordinates increases their utility for workers in both areas and also reveals some links between these two different approaches. © 1997 by John Wiley & Sons, Ltd.