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基于a粒子群算法的控制系統(tǒng)pid參數(shù)優(yōu)化設(shè)計(jì)分析畢業(yè)論文-資料下載頁(yè)

2025-06-27 17:38本頁(yè)面
  

【正文】 onal 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) 。b) is Hurwitz for all and[0,2).Proof: Necessity Suppose that (3) holds. Then, condition a) is obvious. The necessity of condition b) is established in the following way. Sinceit follows thatBecause B(s) and F(s) are Hurwitz, using Rouch233?!痵 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。2) is Hurwitz for all and [0,2)。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
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