IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 48, NO. 8, AUGUST 2003
Performance Limitations for Linear Feedback Systems in the Presence of Plant Uncertainty
Graham C. Goodwin, Mario E. Salgado, and Juan I. Yuz
Abstract—The goal of this paper is to contribute to the understanding of fundamental performance limits for feedback control systems. In the literature to date on this topic,all available results assume that the designer has an exact model of the plant. Heuristically, however, one would expect that plant uncertainty should play a significant role in determining the best achievable performance. The goal of this paper is to investigate performance limitations for linear feedback control systems in the presence of plant uncertainty. We formulate the problem by utilizingstochastic embedding of the uncertainty. The results allow one to evaluate the best average performance in the presence of uncertainty. They also allow one to judge whether uncertainty or other properties, e.g., nonminimum phase behavior, are dominant limiting factors. Index Terms—Performance limitations, stochastic embedding, uncertainty.
Feedback control loop.
I. INTRODUCTIONUNDAMENTAL limitations on the performance of feedback control loops have been a topic of interest since the seminal work of Bode during the 1940s related to feedback amplifier design . There are several well-known examples where one can readily appreciate the link between structure and the associated limits on control-loop performance; see, for example, the discussion of the inverted pendulum or the flight controller for the X-29 aircraft . The tools for analyzing limits of performance for systems without uncertainty include logarithmic sensitivity integrals, limiting quadratic optimal control and entropy measures. Early work focused on linear feedback systems; see, for example, –. There has also been growing interest in performance limitations for nonlinear feedback systems;see, for example, –. To give a flavor of the results achieved to date, we note that one can distinguish two types of performance constraints, namely: i) those that hold for all designs, independent of the criterion used to design the controller; ii) those that hold for a best design, based upon some given optimality criterion. Examples of fundamental limitations of type i) are theBode–Poisson integral formulas for a control loop as in Fig. 1,
Manuscript received March 30, 2002; revised November 19, 2002. Recommended by Guest Editor R. H. Middleton. G. C. Goodwin and J. I. Yuz are with the School of Electrical Engineering and Computer Science, The University of Newcastle, Newcastle, NSW 2308, Australia (e-mail: firstname.lastname@example.org; email@example.com). M. E. Salgadois with Departamento de Electrónica, Universidad Técnica Federico Santa María, Casilla 110-V, Valparaíso, Chile (e-mail: firstname.lastname@example.org). Digital Object Identifier 10.1109/TAC.2003.815011
where the plant and the controller are linear. For example, in the single-input–single-output (SISO) case, the achieved complementary sensitivity is known to satisfy an integral constraintdepending on open right-half plane (ORHP) zeros and time delays. Similar, the achieved sensitivity function is known to satisfy an integral constraint depending on ORHP poles. Well known examples of the type ii) limitations are best cheap control. These give the minimum achievable integral square output error due to unit step output disturbances and/or impulsive measurement noise. These bounds can beestablished via frequency domain arguments , , , and can be extended to plants which are both nonminimum phase or unstable . There also exist interesting connections between the cheap control results and the Bode–Poisson integral equations . A key point here is that results of type i) hold for all stabilizing controllers, whereas bounds of type ii) require that a very...