【正文】 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) are arbitrary plex polynomials. In this section, we focus on the problem of determining those real values of, if any, for which (4) is Hurwitz stable. Note that by setting L(s) = sD(s) and M(s) = N(s), the problem stated above bees PID stabilization. We, therefore, refer to stabilization of (4) as the extended PID stabilization problem. Based on a plex version of the generalized Hermite–Biehler theorem, [14] provided a putational characterization of all admissiblevalues for this extended PID stabilization problem. For detailed derivation, the reader is referred to [14].To set the stage for presenting this characterization, we first introduce some notion and definitions. The standard signum function sgn : is defined byWe consider a polynomial of degree nDefine the signature of the polynomial denoted by, asLet≠0≠0Consider the following “realimaginary” depositions of L(s) and M(s):=whereDefineAlso, let n, m be the degrees of and M(s), respectively. Multiplying by and evaluating the resulting polynomial at , we obtainwhere Also, defineDefinition 1: Let be as already defined. Denote to be the leading coefficient of . For a given fixed , let be the real, distinct finite zeros of with odd multiplicities. Also, define and. Define a sequence of numbers as shown in the first equation at the bottom of the page, where the second equation shown at the bottom of the page holds. Ne。’s Theorem [16], we conclude thatis Hurwitz for all and[0,2).Sufficiency Proceeding by contradiction, we assume that conditionsa) and b) are true, butSince is a continuous function ofandthen there must exist at least one such thatTherefore, it implies that there exist and[0,2)such thatand this obviously contradicts condition b).Consider the stable weighting functions and , where, and are some real polynomials. Also, we denote the closedloop characteristic polynomial to beFor notational simplicity, we define the plex polynomialThe objective of this note is to determine stabilizing PID controllers such that the robust performance condition (2) holds. Based on Lemma 1, the problem of synthesizing PID controllers for robust performance can be converted into the problem of determining values of for which the following conditions hold:1) is Hurwitz。 robust performance.I. INTRODUCTIONThe proportional–integral–derivative (PID) controller is the most widely used controller structure in industrial applications. Its structural simplicity and sufficient ability of solving many practical control problems have greatly contributed to this wide acceptance. Over the past decades, many PID design techniques [1] have been proposed for industrial use. Most of these design techniques are based on simple characterizations of process dynamics, such as the characterization by a first order model with time delay. In spite of this, for plants having higher order, there are very few generally accepted design methods existing. Robust performance design is one 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. R