【正文】 ecently, 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) 。 附錄B（外文文獻）PID Controller Design for Robust PerformanceMingTzu Ho and ChiaYi LinAbstractThis note is devoted to the problem of synthesizing proportional–integral– derivative(PID)controllers for robust performance for a given singleinput–single output plant in the presence of uncertainty. First, the problem of robust performance design is converted into simultaneous stabilization of a plex polynomial family. An extension of the results on PID stabilization is then used to devise a linear programming design procedure for determining all admissible PID gain settings. The most important feature of the proposed approach is that it putationally characterizes the entire set of the admissible PID gain values for an arbitrary plant.Keywords linear programming。for i=1:D sum=sum+x(i)^2。)%算法結束DreamSun GL amp。)Result=fitness(pg,D)disp(39。disp(39。函數(shù)的全局最優(yōu)位置為：39。*************************************************************39。 end end Pbest(t)=fitness(pg,D)。 y(i,:)=x(i,:)。 x(i,:)=x(i,:)+v(i,:)。 %Pg為全局最優(yōu)for i=2:N if fitness(x(i,:),D)fitness(pg,D) pg=x(i,:)。 y(i,:)=x(i,:)。 %隨機初始化位置 v(i,j)=randn。 %初始化群體個體數(shù)目eps=10^(6)。 %最大迭代次數(shù)D=10。 %學習因子2w=。%給定初始化條件c1=。clc。還有大學里所有在生活和學習上幫助我、和我一起度過平淡而快樂的大學生活的同學們!感謝其他所有關心、幫助和支持我的朋友們!我還要特別感謝我的家人，是他們支持和理解是我前進的動力，并使我具有勇氣和信心去不斷克服前進中遇到的困難和障礙。老師淵博的知識、嚴謹?shù)目茖W態(tài)度和勤奮的工作作風，以及對問題清晰敏銳的洞察力、勤奮的工作作風都給我留下了深刻的印象，她對我學業(yè)上的指導和幫助將使我終生受益。首先，我要衷心地感謝帶我完成這次畢業(yè)設計的導師!本文的工作是在導師的悉心指導下完成的。致 謝 在我大學學習及這次論文的撰寫過程中，有許多人給我提供了熱情的幫助，正是他們的支持和鼓勵，才使我順利地完成了課題的研究和論文的撰寫。(3)增大會降低系統(tǒng)的超調量；減小會相對地增大系統(tǒng)超調量。除此之外，本設計還研究了P、I、D各參數(shù)對系統(tǒng)的影響以及Smith預估補償器對該系統(tǒng)的作用，敘述如下：(1)增大會增大系統(tǒng)的超調量，但降低了峰值時間和調節(jié)時間；而減少則相對地降低了超調量，但增加了峰值時間和調節(jié)時間。曲線的各個指標也有了明顯的提高，尤其是超調量有了明顯的減少，上升時間也有了明顯的縮短。結論 本設計采用多變量尋優(yōu)的粒子群算法對控制系統(tǒng)的PID參數(shù)進行優(yōu)化設計，通過采用工程上的整定方法（臨界比例度法）粗略的確定其初始的三個參數(shù)，并采用粒子群算法用SIMULINK的仿真工具對PID參數(shù)進行優(yōu)化，得出系統(tǒng)的響應曲線。 Smith預估補償器 Smith預估補償器所觀察到的圖形，其中Wc(s)中的參數(shù)應用經粒子群算法整定后的那組參數(shù)，仿真后與粒子群算法整定相比較。 改變不同的值所觀察到的圖形 改變不同的值所觀察到的圖形，增大增加了系統(tǒng)了超調量；而減少則相對地降低了系統(tǒng)超調量；無則系統(tǒng)存在余差。(3)分別增大和減小，保持、不變。具體步驟如下：(1)分別增大和減小，保持、不變。這正是我們所期待的。 用工程的方法整定后的曲線 粒子群算法整定后的波形 結果比較、（臨界 兩種不同方法的仿真曲線比例度法）粗略的確定其初始的三個參數(shù)，并通過仿真得到響應曲線，曲線的上升時間雖然比較快，但是過度時間比較長，超調量也過大，這對工程實踐是不利的。3. ，求出調節(jié)器各參數(shù)K，的數(shù)值。2. 待系統(tǒng)運行穩(wěn)定后，逐漸減小比例帶，直到出現(xiàn)等幅振蕩為止，即所謂的臨界振蕩過程。設在本控制系統(tǒng)中采用傳遞函數(shù)：第 4 章 系統(tǒng)仿真研究 工程上的參數(shù)整定對于本文選中的加熱爐模型。資料顯示，在大多數(shù)情況下，自衡對象的動態(tài)特性都可以用一階、一階滯后、二階、二階滯后4種模型來描述。當爐溫有反應時，燃料流量已超過了所需的設定值，從而引起燃料流量的浪費，造成爐溫大幅度波動。從上面的分析我們得知，造成爐溫滯后的原因是爐溫有了偏差后，控制爐溫的燃料流量變化迅速，而溫度要滯后一段時間才會改變。這樣控制系統(tǒng)認為爐溫偏差信號仍沒有得到校正而進一步對燃料流量實施作用，其結果導致了較長時間的超調。但由于爐溫信號的滯后，雖然燃料流量產生了變化，爐溫并不立即變化。這是因為控制爐溫是通過控制燃料流量間接控制的。使得最佳設定值變的不佳。在加熱爐最佳溫度控制系統(tǒng)中，配備了先進的計算機控制系統(tǒng)，建立了復雜的數(shù)學模型。(2)加熱爐爐溫滯后的特點及其克服