【正文】 ne of fundamental problems in control. The problem of robust performance design is to synthesize a controller for which the closedloop system is internally stabilized and the desired performance specifications are satisfied despite plant uncertainty. Although H∞ [2] and μsynthesis [3], [4] techniques have been successfully applied to solve the problem of robust performance design, design of the optimal or robust PID controller is a putationally intractable task [5] using H∞ andμsynthesis design techniques. Instead of directly using H∞ or μsynthesis design techniques, there were several approaches proposed to synthesize PID controllers for robust performance. For a given plant, [6] parameterized the stabilizing pensators that consist of a PID controller with a free parameter and the H∞ design techniques were then used to select an appropriate free parameter to achieve robust performance. However, the order of the resulting PIDbased pensator is always greater than the plant order. A parameter optimization approach was proposed in [7] and an LMIbased iterative optimization method was given in [8]. None of these guarantees global convergence with a reasonable amount of putation. By searching in the prescribed controller parameter space, [9] proposed a procedure for constructing the space of the admissible PID controller gain values for multiple performance specifications. Unfortunately, this method suffered from putational intractability. Based on gain and phase margin design, several simple tuning formulas for robust performance were given in [10] and [11]. These tuning formulas are limited to simple characterizations of process dynamics such as the characterization by a firstorder model with time delay.The aim of this note is to effectively solve the problem of robust performance PID design for an arbitrary plant. In particular, we focus on the problem of synthesizing a stabilizing PID controller, if any, for which the disturbance rejection design specification is achieved for a plant with multiplicative uncertainty. In the earlier work [12], based on the generalized Hermite–Biehler theorem [12], [13], a putational characterization of all stabilizing PID controllers was given for an arbitrary plant. This solution of the PID stabilization problem is an essential first step to any rational design of PID controllers. Recently, an extension of PID stabilization to the case of plex polynomials was developed in [14] and it was shown that such an extension could be exploited to carry out many robust stability PID design problems. In this note, we show that the results from [14] can be also used to provide a putational characterization of all admissible PID controllers for robust performance. Such a characterization for all admissible PID controllers involves the solution of a linear programming problem. Accordingly, efficient algorithms are available for generating the parametric space of the entire admissible PID gain values. It will be clear from the exposition that PID design problems with other robust performance objectives can be treated in a similar fashion. The note is organized as follows. In Section II, we show that the robust performance design problem of interest to us in this note can be converted into simultaneous stabilization of plex polynomials. The results of [14] can be then used for solving the resulting simultaneous stabilization problem. In Section III, we state the results of [14] on determining admissible real values of, if any, for which a plex polynomial of the form is Hurwitz, where and are some given plex polynomials. A linear programming characterization of all admissible values is provided. These results immediately lead to a solution to the problem of synthesizing PID controllers for robust performance. In Section IV, a detailed synthesis procedure is presented and illustrated in a simple example. Finally, Section V contains some concluding remarks.II. ROBUST PERFORMANCE DESIGN VIA SIMULTANEOUSPOLYNOMIAL STABILIZATIONConsider the singleinput–singleoutput feedback control system shown in Fig. 1. Here, r is the mand signal, y is the output, and d is an energybounded disturbance. is the plant to be controlled, where and are coprimeFig. 1. Feedback control system with multiplicative uncertaintypolynomials. △s is any stable and proper transfer function with ||△||∞≤1. The weights W1(s) and W2(s) describe the the frequencydomain characteristics of the performance specifications and model uncertainty, respectively. C(s) is the controller used for making the closedloop system stable and achieving desired design specifications. In this note, the controller C(s) is chosen to be a PID controller, ., (1)Then, the plementary sensitivity function isand the sensitivity function isEspecially, we consider the problem of disturbance rejection for the plant with multiplicative uncertainty. This problem can be formulated as the following robust performance condition [15]: ||||+||||∞＜1 (2)Note that many other robust performance specifications can be formulated as (2). To convert the robust performance condition (2) into simultaneous polynomial stabilization, we first consider the following lemma.Lemma 1: Letandbe stable and proper rational functions with . Then (3)if and only ifa) 。3) .In view of these conditions, the problem of synthesizing PID controllers for robust performance has been expressed as simultaneous polynomial stabilization. With a fixed and a fixed , both andbee in the form of where L(s) and M(s) are some given plex polynomials. The result of stabilizing the plex polynomial L(s)+(kds2+kps+ki) M(s) is stated in Section III.III. CHARACTERIZATION OF ALL ADMISSIBLE GAIN VALUES FOR EXTENDED PID STABILIZATIONConsider the plex polynomial of the form (4)where L(s) and M(s) a