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研究生:呂政芳
研究生(外文):Jeng-Fan Leu
論文名稱:SISO回饋控制系統之時域模擬與設計
論文名稱(外文):Time-Domain Simulation and Design of SISO Feedback Control Systems
指導教授:黃奇黃奇引用關係蔡三元
指導教授(外文):Chyi HwangSun-Tuan Tsay
學位類別:博士
校院名稱:國立成功大學
系所名稱:化學工程學系碩博士班
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:2002
畢業學年度:90
語文別:中文
論文頁數:139
中文關鍵詞:控制器調諧分數階次系統時域模擬
外文關鍵詞:Fractional-order systemTime-domain simulationController tuning
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本篇論文的主要工作重點在於開發出能夠精確且快速地求得SISO(Single-Input Single-Output)回饋控制系統的時間應答的計算方法,並藉此完成分數階次系統的時域模擬與最適分數階次控制器的設計。此外,對於SISO回饋系統的控制性能改進方面,有別於傳統型PID控制器,我們各別探討了分數階次PID控制器、含Smtih預測器(Smith predictor)之I-P控制器(Integral-Proportional controllers)以及PID時延控制器(PID-deadtime controller)的設計概念以及應用情形。其中,後兩者主要是用於含較長時延之程序控制。

在時域模擬計算方面,我們利用B-spline所組成的級數,表示回饋控制系統(含單純或分散時延)之暫態(transient)時間應答。更明確地說,即是將系統之Laplace轉移函數之逆轉換問題,以B-spline級數展開方式求得數值解。系統時間應答$f(t), t ge 0$對應之B-spline級數,將包括前幾項次的邊界(boundary)B-splines,而其餘則為內部(interior)B-splines所構成。至於各項次的係數值,則在滿足系統應答之起始條件下,以高效率之快速傅立葉轉換(Fast Fourier Transform; FFT)為基礎之演算法所獲得。

分數階次回饋系統(feedback fractional-order system)的相關研究一直受到相當程度的重視,其主要原因是由於許多物理系統可以分數階次模式充分地描述其特性,而另一方面,為突破傳統整數階次控制器的性能極限,分數階次控制器的使用也獲得高度地關切。對於線性分數回饋系統而言,尚缺乏有效的解析方式以進行時域分析及模擬研究,吾人在此介紹二種可靠、準確的數值方法,用以求取分數階次Laplace轉移函數的逆轉換。其中之一,主要是利用可控制精確度的數值積分法,進行Bromwich's積分的計算。另一種方法,則是應用前所提及之B-spline級數展開法,求得系統之時間應答。在本研究中,對於分數階次程序,我們分別以PD$^mu$控制器和分數階次頻帶限制之領先補償器(fractional-order band-limited lead compensator)進行回饋控制之時域模擬,從而說明出此二種方法較Gr"unwald--Letniknov近似法及Podlubny's解析式(由雙層無窮多項級數所構成)更能獲得精確的解答,並能減少運算的複雜度。由模擬結果亦可得知,Podlubny's無窮多項級數表示法確實存在有無法收斂的可能性,且此缺點並不能藉由級數加速收斂法(series acceleration scheme)予以克服。

本文中,我們也將考慮分數階次PID控制器之設計問題,其中,積分器(integrator)與(differentiator)皆為非整數階次。分數階次PID控制器的三個增益值與二個實數階次,將會被決定並使得系統的平方誤差積分(integral square error;ISE)之性能指數可為最小,且符合設定的增益邊限與相位邊限(gain and phase margins)。整個設計問題將可推導為一含條件之參數最適化形式,並應用微分演化演算法(differential evolution algorithm)搜尋出全域之最佳控制器參數。由於尚缺乏對於分數階次系統時域分析之解析方法,我們採用了一種高效率計算ISE性能指標,同時能夠從頻域驗證其穩定性的數值方法。所進行的設計範例包括了整數和分數階次程序的控制問題,而且亦將比較出最適分數階次PID控制器與最適整數階次PID控制器的性能差異。

對於含長時延程序的控制方面,使用含Smith預測器之I-P控制器之主要目的,除了是利用Smith預測器消弭系統的時延效應外,也考慮以I-P控制器改善PI回饋控制系統時間應答遲滯與超越量過大的缺點。配合閉環路因配對(matched)Smith預測器而形成不含時延項的模式,其系統與控制器輸出的時間應答亦能解析表示,故回饋控制系統的性能規範條件,包括時域與頻域規範,皆能以解析聯立方程組作數學描述。藉由此優勢,我們引用球形演算法(spherical solution tracing method)精確且有效率地求出各個性能規範的$K_p$-$tau_i$流形(manifold)。於是,由流形曲線所包圍的交集即為I-P控制器的可行解設計區域。而所得之I-P控制器將使對應的回饋控制系統能夠符合所望的性能規範,也避免傳統Smith預測器所導致的過大增益值缺點。

基於PID時延控制器對存在較長或分散時延之程序控制方面的發展潛力,在本研究中,我們將探討針對含時延之積分和不穩定程序的最適ISE之PID時延控制器設計。當PID時延控制器之補償時延與程序時延(即$theta_p$)設定相同時,回饋控制系統之二次性能函數,如ISE等,以及其偏微分都可透過殘值的計算方法,以控制器參數作解析表示。如此,再以球形演算法在滿足最小ISE的條件下,同時追蹤出PID時延控制器的三條最適參數對應程序時延的流形曲線:$K_p$-$theta_p$、$tau_i$-$theta_p$及$tau_d$-$theta_p$,並且能夠獲得PID時延控制器對於設定點與負載點之參數調諧規則。所設計的PID時延控制器,將與最適ISE之傳統PID控制器作比較,以了解其對於積分與不穩定程序控制的性能提升情形。
This dissertation is concerned with the time-domain simulation and design of SISO (Single-Input Single-Output) feedback control systems. The major effort is devoted to developing fast and accurate numerical methods for computing time responses of closed-loop systems. These methods greatly facilate the design of fractional-order feedback control systems. Effort is also devoted to enhancing the performance of feedback SISO control systems with the use of non-conventional controllers such as fractional-order PID controllers, I-P (Integral-Proportional) controllers with Smith predictors, and PID-deadtime controllers. The latter two types of control schemes are constructed mainly for the lag-dominant processes containing longer delay.

For achieving fast and accurate time-domain simulations for systems with fractional-order or having pure and/or distributed time delays, we represent the transient time response of a closed-loop control system in terms of B-spline series. More precisely, it regards to the inversion of Laplace transforms of irrational type with the B-spline series expansion approach. The B-spline series representation for a time function $f(t)$ with $t ge 0$ contains first several terms in boundary splines and the remaining terms in interior B-splines. It is shown that by matching the initial conditions of the response exactly, the coefficients associated with the interior B-splines can be accurately obtained by a computationally efficient FFT-based algorithm.

The study of feedback fractional-order systems has been receiving considerable attention due to the facts that many physical systems are well characterized by fractional-order models, and that fractional-order controllers are used in feedback systems with the intention of breaking through the performance limitation of integer-order controllers. Owing to the lack of effective analytic methods for the time-domain analysis and simulation of linear feedback fractional-order systems, we suggest two reliable and accurate numerical methods for inverting fractional-order Laplace transforms. One is based on computing Bromwich's integral with a numerical integration scheme capable of accuracy control, and the other is based on expanding the time response function in a B-spline series. In order to demonstrate the superiority in solution accuracy and computational complexity of these two numerical methods over the Gr"unwald--Letniknov approximation method and Podlubny's analytic formulas, which are in a form of double infinite series, the time-domain simulations of the feedback control of a fractional-order process with a PD$^mu$-controller and a fractional-order band-limited lead compensator are worked out. The simulation results indicate that a convergence problem indeed occurs in using Podlubny's infinite series expressions, and that the problem could not be overcome by a series acceleration scheme.

In this work, we are also concerned with the design of a fractional-order PID controller which involves noninteger-order integrator and differentiator. The three controller gains and two real orders of the fractional-order PID controller are determined to minimize an integral square error (ISE) performance index while satisfying the specified gain and phase margins. We formulate the design problem as a constrained parametric optimization and apply a differential evolution algorithm to search globally the optimal controller parameters. Due to the problem lack of analytic time-domain analysis methods for fractional-order systems, an effective numerical method is utilized to compute the ISE performance index as well as to check the stability in the frequency domain. Design examples are given to compare the performance of the optimal fractional-order PID controller with the optimal integer-order PID controller in controlling integer-order as well as fractional-order processes.

For lag-dominant process, the features gained by using a I-P controller with a Smith predictor are: 1) the dead time is compensated by the Smith predictor and 2) sluggish and overshoot can be improved owing to rearranging the controller scheme from P-I to I-P structure. A delay-free model is obtained for inducing a matched Smith predictor so that the time responses of system and controller output can be solved exactly, and the specifications in time and frequency domains of the feedback system can also be formulated in sets of analytical equations. Due to this benefit, a spherical solution tracing method is applied to computing the $K_p$-$tau_i$ manifold corresponding to each performance specification with great accuracy and efficiency. Thus, the feasible domain in which the I-P controllers synthesized always meet the desired constrains is enclosed by these mainfolds, and the high gain problem in controller design caused by classic Smith predictors is avoided in turn.

Since the potential of PID-deadtime controllers in control of process existing long or distributed delay, in our work, the optimal-ISE PID-deadtime controllers are designed for integral and first-order unstable plus time delay plants. Once the dead time of the compensator is designated to be the delay induced by process, $theta_p$, the quadratic performance index, such as ISE, and its derivatives of the feedback control system can be represented in terms of controller parameters analytically via a residue computation method. As a spherical method is invoked to meet the constraints of minimum ISE, we can trace the three $K_p$-$theta_p$, $tau_i$-$theta_p$ and $tau_d$-$theta_p$ manifolds simultaneously. These manifolds represent the optimal settings of the PID deadtime controller with respect to the process deadtime. The tuning rules for servo and regulatory control are also correlated and presented. As compared with the optimal-ISE PID controllers, the designed PID-deadtime controllers exhibit a significant performance promotion for integral and unstable process control.
1 緒論
1.1研究動機 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2文獻回顧 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2.1 Laplace逆轉換與時間應答計算. . . . . . . . . . . . . . . . . . . . 2
1.2.2 分數階次系統 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2.3整數與分數階次PID控制器. . . . . . . . . . . . . . . . . . . . . . 3
1.2.4 I-P控制器與Smith預測器 . . . . . . . . . . . . . . . . . . . . . . . 4
1.2.5 PID時延控制器. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.3 論文組織. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2 利用B-splines精確求取含時延閉環路控制系統之暫態應答 7
2.1簡介 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.2 區間B-splines之基本性質 . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.3 以B-spline級數表示暫態應答 . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.4 範例. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.4.1 例一. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.4.2 例二 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.4.3 例三 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.5 結論. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
3 分數階次回饋系統之時域模擬 31
3.1 簡介. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.2 文獻方法回顧 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.2.1 Podlubny's解析方法. . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.2.2 Grunwald-Letniknov近似法. . . . . . . . . . . . . . . . . . . . . . 32
3.3 級數加速收斂法. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.4 Bromwich's積分法與B-spline級數表示法. . . . . . . . . . . . . . . . . . . 34
3.5實例比較探討. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.6結論. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
4 最適分數階次PID控制器之設計 43
4.1 簡介. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
4.2 問題陳述與公式表示. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
4.3 ISE計算與穩定性測試之數值方法. . . . . . . . . . . . . . . . . . . . . . 46
4.4 應用微分演算法搜尋最適PI^lambdaD^mu控制器參數. . . . . . . . . . . . . . 47
4.5範例. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
4.5.1 範例一. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
4.5.2 範例二. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
4.5.3 範例三. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
4.6 結論. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
5 含Smith預測器之I-P控制器設計 63
5.1 簡介. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
5.2 含配對Smith預測器之I-P回饋控制器系統. . . . . . . . . . . . . . . . . . . 63
5.2.1 一階程序. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
5.2.2 二階程序. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
5.3 控制器規範之流形計算. . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
5.3.1 控制器輸出限制. . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
5.3.2 系統時間應答輸出限制. . . . . . . . . . . . . . . . . . . . . . . . 69
5.3.3 上升時間限制. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
5.3.4 增益邊限和相位邊限限制. . . . . . . . . . . . . . . . . . . . . . . 70
5.3.5 穩定時間限制. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
5.4 範例. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
5.4.1 對於一階系統的I-P控制器設計. . . . . . . . . . . . . . . . . . . . 71
5.4.2 對於二階系統的I-P控制器設計. . . . . . . . . . . . . . . . . . . . 72
5.4.3 對於非最小相系統的I-P控制器設計. . . . . . . . . . . . . . . . . 72
5.5 結論. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
6 對於積分與不穩定程序之最適PID時延控制器設計 97
6.1 簡介. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
6.2 PID時延回饋系統. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
6.3 PID時延回饋控制系統之ISE參數計算法. . . . . . . . . . . . . . . . . . 98
6.4 ISE之偏微分解析式. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
6.5 流形求取與PID時延控制器之最適ISE參數調諧規則. . . . . . . . . . . . 110
6.6實例. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
6.6.1 積分程序. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
6.6.2 不穩定程序. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
6.7 結論. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
7 總結與未來展望 123
7.1 總結. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
7.2 未來展望. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
A 以FFT為基礎之冪次級數展開演算法 135
B ISE參數表示法 137
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