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研究生:程石良
研究生(外文):Cheng, Shyr-Lian
論文名稱:基因與微分演化法在控制系統參數最適化問題之應用
論文名稱(外文):The Application of Genetic and Differential Evolution Algorithms in Parametric Optimization Problems of Control Systems
指導教授:黃奇黃奇引用關係
指導教授(外文):Chyi Hwang
學位類別:博士
校院名稱:國立成功大學
系所名稱:化學工程學系
學門:工程學門
學類:化學工程學類
論文種類:學術論文
論文出版年:1999
畢業學年度:87
語文別:中文
論文頁數:164
中文關鍵詞:時延系統D分割基因演化法微分演化法絕對誤差積分最佳簡化模式PID控制器數值最適化
外文關鍵詞:time-delay systemsD-partitiongenetic algorithmdifferential evolution algorithmintegral of absolute erroroptimal reduced modelPID controllernumerical optimization
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本論文主要在應用D分割(D partition)原理分析時延系統回饋控制的穩
定性,以及利用基因與微分演化法來求得最佳PID控制器參數和最佳簡化模式參數。所謂的$D$分割是將參數空間分割成穩定區域與非穩定區域,直接求出時延相對於增益值之穩定區域圖,這種方式會比根軌跡法更有效率,特別是要利用參數最佳化來設計最佳控制器。在本論文中我們針對一個一階及二階不穩定時延系統,串接P控制器、PI控制器、PD控制器與PID控制器,分析其回饋控制系統的穩定性,分析的方式是針對回饋系統的特徵方程式,求出其超越平面(hyperplane),以及超越界面ypersurface),透過端點分析包括極點與零點,就可以求出各條分支線,將分支線連接起來就將控制器參數空間分割成穩定區域與非穩定區域。
在控制的方面,我們不僅要求系統要能穩定控制,另外我們還要求在時間領域實際操作的性能要好。在本論文我們是以絕對誤差積分(IAE)為性能指標,理論上是無法以傳統的梯度法求得最佳PID控制器參數,因為絕對誤差積分很難求出微分項,因此我們利用微分演化法搜尋最佳的PID控制器參數。基本上,微分演化法是屬於隨機搜尋法。起始時,先在搜尋空間隨機產生一群待選值,之後再利用一系列的演化方式,如突變、交配、評估適應值與選擇等機制來搜尋最佳PID控制器參數。由於價值函數是IAE的型式,計算上會花費很長的時間。為了節省時間,在應用DEA之前,我們先應用$D$分割原理,分析出PID控制器參數的穩定空間, DEA就在這個穩定空間搜尋最佳化的參數值。
其次,我們也利用微分演化法求得最佳簡化模式的參數,能夠足以描述原始系統模式的主要特性。所選用的目標函數為頻域的L^2-norm,但在最佳簡化模式的搜尋上會有一個先天上的問題,那就是無法事先知道最佳簡化模式的參數其範圍是多大,雖然DEA可以將參數的上下限取消,本身可以克服這個問題,但是如果參數範圍定得太小,疊代的次數會激增。反之如果定得太大,DEA族群內就會充斥很多不理想的待選值,執行上會很沒效率。因此我們在DEA的機制加上擴大搜尋空間的功能,可以使DEA的收斂速度加快。因此改良型的DEA比一般的DEA更具有強韌性與實用性,也由於可以避開不知如何設定參數上下限的窘境,在使用上更加簡便。
最後,針對非時延的區間系統,我們改用基因演化法(GAs)搜尋最佳PID控制器參數,區間系統我們是選用ISE。對於這種最大值的最小化問題必須使用兩層GAs,上一層是搜尋最佳的PID控制器參數,下一層GAs是在固定的PID控制器參數之情形下,搜尋控制性能最差的系統參數,使得閉環控制系統的ISE為最大。由於上一層GAs的價值函數值是由下層的GAs所計算而來的,如果下層的GAs找到的不是ISE的全域最大值,將會誤導上層GAs搜尋的方向。因此要解決最大值的最小化問題,最重要的關鍵是在於最大值必須是全域最大值,這也是我們之所以選用GAs的原因之一,因為GAs是比較能找到全域值(global optimum)。為了使GAs更有效率的搜尋,我們利用邊線理論(Edge Theorem)與邊界穿越理論(boundary crossing
theorem)分析系統32條分節多項式(segment polynomial)的強韌穩定性。事先將PID參數的穩定區間找出來,使得GAs能在這些穩定區域中迅速有效的找到最佳的PID控制器參數。
The purposes of this dissertation are to apply the method of D-partition to determine feedback stabilizability of time-delay and unstable processes, design the optimal PID controller and solve the optimal approximation of linear systems by genetic algorithms (GAs) and a differential evolution algorithm (DEA), respectively. In this dissertation, the method of $D$-partition is proposed for determining feedback stabilizability of first- and second-order time-delay and unstable processes. using traditional P-, PI-, PD-, and PID-type controllers. We show that it is more convenient to determine stabilizability
conditions than the root-locus method because the former directly gives rise to stability regions in the delay-gain plane. Moreover, construction of stability region in the controller parameter space, which is particularly necessary when a parametric optimization method is adopted to determine optimal PID-stabilizers. The method of partitioning the parameter space into sets $D(k)$ is called the $D$-partition method and for a system to be asymptotically stable, the parameter vector $\bx$ must be inside the set $D(0)$.
Computating the conformal mapping of the Nyquist contour $C$ into the $G$-plane, it is easy to find the boundary of the D-partition of the $x$-parameter line, which are formed by the hyperplanes($H_0$ and $H_{\infty}$) and the hypersurface($H_{\omega}$). Connecting those $D$-partition boundaries with the analysis of end-point in the ($\theta,k$) plane, we can deterimine the stable regions for feedback stabilization of $G_p(s)$.
A excellent controller ont only ensures a required level of stability and speed of transient response of the system but also improves the tracking behavior. In this dissertation,
the integral of absolute error (IAE) is used as a performance measure for controller evaluation. Unfortunately, the IAE objective function is not differentiable everywhere in the parameter space and hence the gradient-based optimization techniques cannot be applied to such an optimization problem. Moreover, there are no general closed-form expressions
or efficient algorithms for evaluating the value of an IAE performance index. To avoid the difficulty associated with the non-differentiability of the objective functions, we apply in this work a differential evolution algorithm to search for the optimal controller parameters in the feasible gain domain $\bK(\sigma,\theta)$ obtained by the method of $D$-partition to minimize the IAE performance index. Basically, DEA are stochastic search algorithms that were originally motivated by the mechanisms of natural selection and evolutionary genetics. They manipulate a population of candidate solutions which are sampled at random from the search space initially. The population of candidate solutions is then successively
improved by applying genetic operators such as selection, creossover, and mutation to solve a parametric optimization problem.
The optimal approximate rational model with/without a time delay for a system described by its rational or irrational transfer function is sought such that a frequency-domain $L^2$-error criterion is minimized. In order to overcome the inherent difficulty of specifying a priori proper bounds of parameters within which the optimal solution locates, we incorporate in the algorithm a search-space expansion scheme.
The distinct feature of the proposed model approximation approach is that the search-space expansion scheme can enhance the possibility of converging to a global optimum in the DE search. This feature and the chosen frequency-domain error criterion make the proposed approach quite efficacious for
optimally approximating unstable and/or nonmimimum-phase linear systems.
An interval plant family is described by a rational transfer function with its coefficients taking values independently from their respective intervals. This dissertation presents the use of genetic algorithms (GAs) to design the optimal PID controller that not only stabilizes simultaneously all members
of the interval plant family but also minimizes the integral of squared error (ISE) of the worst-case plant. The controller design problem is essentially a min-max optimization problem and the design procedure involves two loops of GA searches. The inner-loop GA searches for, under a fixed set of controller
parameters, the plant''s coefficients such that the orresponding feedback control system has maximum ISE value, while the outer-loop GA searches for the controller parameters such that the worst-case ISE of the plant is minimized. If the worst-case ISE obtained by the inner-loop GA is local maximum, this optimal paramerets of the robust PID controllers will be not the global optimum. Hence we shoose GAs which is to solove this min-max problem. In order to eliminate the need of performing robust stability test in the process of searching for the optimal
controller parameters via genetic algorithms, it is helpful to find the stabilizing controller parameter domain $\bK_s$ prior to the search for optimal controller parameters. We contructed $\bK_s$ previously according to the Edge Theorem and the Boundary crossing Theorem as shown in this dissertation, which the task of testing if a controller stabilizes an interval plant family can be completed through testing the robust stability of 32 edge polynomials.
中文摘要....................................i
英文摘要..................................iii
圖目錄....................................vii
表目錄......................................x
第一章 緒論.................................1
1-1 研究動機...............................1
1-2 文獻回顧...............................7
1-3 論文章節組織..........................12
第二章 時延不穩定系統D分割.................14
2-1 前言..................................14
2-2 D分割.................................14
2-3 一階不穩定系統的穩定性分析............17
2-4 二階不穩定系統的穩定性分析............23
2-5 結論..................................32
第三章 微分演化法.........................50
3-1 簡介..................................50
3-2 演化機制..............................52
3-3 演化流程..............................55
3-4 結論..................................56
第四章 時延與非時延系統最佳控制器設計......58
4-1 前言..................................58
4-2 二度自由PID控制器.....................58
4-3 可行控制器參數空間....................60
4-4 價函數之估算..........................64
4-5 範例..................................66
4-5 結論..................................70
第五章 最佳簡化模式........................86
5-1 前言..................................86
5-2 時域L^2-誤差之簡化模式................86
5-3 搜尋空間擴大機制......................87
5-4 計算流程..............................88
5-5 範例..................................89
5-6 結論..................................94
第六章 基因演化法.........................113
6-1 簡介.................................113
6-2 演化機制.............................114
6-3 演化流程.............................119
6-4 結論.................................120
第七章 區間系統最佳控制器設計.............122
7-1 前言.................................122
7-2 區間系統.............................122
7-3 控制器穩定區域Ks之估算...............124
7-4 串級基因演化法.......................127
7-5 計算流程.............................131
7-6 範例.................................133
7-7 結論.................................138
第八章 總結與未來展望.....................151
8-1 總結.................................151
8-2 未來展望.............................152
參考文獻.................................153
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