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研究生:林敏朗
研究生(外文):Min-Lang Lin
論文名稱:線性及非線性程序之建模與控制
論文名稱(外文):Modeling and Control of Linear and Nonlinear Processes
指導教授:黃世宏
指導教授(外文):Shyh-Hong 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:168
中文關鍵詞:建模與控制線性參數化模型時間加權轉換模型結構之決定簡單結構控制器極點配置設計非線性系統之模式簡化
外文關鍵詞:Pole Placement DesignModel Reduction of Nonlinear SystemsModeling and ControlLinear Parametric ModelsTime-Weighted TransformsModel Structure DeterminationSimple-Structure Controllers
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  • 下載下載:116
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摘要

化學工業程序多數包含複雜多變甚至非線性的動態特性,但線性模型仍然是較常被採用的動態描述方式,因此針對線性系統的建模與控制的研究一直深受重視。以線性模型來描述非線性動態的作法,可視為針對特定平衡點將程序線性化以描述在平衡點附近的動態,也因此線性模型往往只能代表該程序的局部特性。但在有限範圍操作時,線性模型仍能滿足大部份預測暨控制方面的需求。

本論文首先提出一鑑別法,利用時間加權積分轉換式,在開環或閉環操作下以一次實驗測試來識別連續系統的線性參數模型。所提的移動區間演算法可在受到雜訊干擾的情況下獲得模型參數之無偏估測,而據以發展出的模型結構辨識技術,更可由量測資料有效地辨識出系統階次與時延。此外,此鑑別法可輕易地應用於模型簡化,在任意指定的階次下辨識出包含適當時延的理想簡化模型。經由選擇轉換式内適當的時間加權函數,本法不需要程序的起始態與終止態的資訊,因此可利用任意時間區段和狀態的量測資料進行鑑別,使得鑑別試驗的操作相當容易。

本文亦提出一新型鑑別方法,以在開環路或閉環路操作下辨識出線性離散參數模型。待鑑別程序的階次與時延均可能未知,且量測訊號存在雜訊干擾。本法引入一時間加權離散濾波器,將不同時間區間內所得到的輸出入取樣訊號轉換成一實數,並進一步將差分方程式變換成代數方程組,最後再以遞迴移動區間最小平方法來估測模型參數。此外,本法亦提供用來辨別模型的階次以及時延的技術。相對於傳統的最小平方法,本鑑別法除了可以獲致無偏參數估測外,在模型結構不吻合或是取樣時間選擇不恰當的情況下,仍然具備良好的效能。所提出的估測器可迅速地收斂至真正參數值,大幅縮短鑑別試驗所需的時間。

程序的離散時間模型建立以後,本論文接著探討具備完整結構的離散時間控制器的設計方法,此種控制器的階次由程序模型來決定。該法利用修正內部模式控制與優勢極點配置演算法,透過調諧一至三個易理解的參數即可完成控制器設計。經由巧妙地近似完整結構控制器可得到相應之簡單結構控制器,例如比例積分微分控制器或更高階次的控制器。此外,配合發展出的新型強韌性指標,進一步提出強健控制器最佳設計演算法。模擬結果顯示,簡單結構的控制器效能常優於完整結構控制器。本設計法可處理包含時延、右半平面與左半平面零點、低階至高階的延遲、積分器、與不穩定極點等各種類型的動態。

許多真實程序是極為複雜的,須要花費相當努力利用理論推導方式來建立能夠正確描述程序動態的數學模型,且往往都需考慮到其非線性特性。一般而言,直接由原始非線性模型來瞭解程序所具有的各種非線性動態是相當困難的。另外,分歧理論指出在奇異點鄰域,數學模型之線性部分的影響式微,非線性效應可完全顯現出來。本文藉由分歧理論發展一模式簡化技術以有效處理該分析過程。此技術應用Center Manifold投映與Normal Form理論將非線性模型轉化為在奇異點附近有效之簡單正常型。最終的正常型只包含了必要的共振項,據此可描繪出完整的相軌跡,使得分析奇異點鄰域所具有的各種非線性動態更為簡易。本文針對在比例控制下串連迴流連續攪拌槽反應器系統,描繪其分歧圖與界定線性穩定區域,並進一步找出一般奇異點,以應用模式簡化技術來分析奇異點鄰域的非線性動態行為。
Abstract

Although a large variety of chemical processes contain rather complex or even nonlinear dynamics, linear models are often applied to describe their dynamics. Therefore, researches on the modeling and control of linear systems continually attract much attention. Using a linear model to describe nonlinear dynamics is justified if the system’s dynamic behavior in the vicinity of a specific equilibrium point is of prime interest. In the limited range of operating conditions, such linear models can still satisfy most requirements for prediction and control.

In this dissertation, a time-weighted integral transform is first presented to identify a linear continuous parametric model based on a single experimental test under open-loop or closed-loop operation. The resulting moving-horizon algorithm can arrive at unbiased estimates of the model parameters in the presence of noise. An effective technique is also developed to determine the model order and time delay from observed data in a simple manner. Furthermore, the proposed identification method can be easily applied as a model reduction technique that results in an ideal model with delay for any specified order. By selecting a suitable weighting function in the transform, the method does not require knowledge about initial and final states of the signals and hence allows the use of any interval and any state of data for identification. Consequently, the operation of a dynamic test is greatly simplified.

A novel method is then presented to identify a linear discrete parametric model under open-loop or closed-loop operation in a noisy environment. The order and time delay of the system are assumed unknown a priori. A time-weighted digital filter is introduced to convert the sampled data of input-output measurements over different sets of time horizons to a group of algebraic equations. The model parameters are then estimated by the moving horizon least-squares algorithm in a recursive fashion. An effective technique is also proposed to infer the model order and time delay from the observed data. In contrast with the conventional least-squares approach, the proposed method is able to yield unbiased parameter estimates despite the nature of noise. Furthermore, it is robust with respect to model structure mismatch and the selection of sampling period. The proposed estimator converges very rapidly to the true parameter values. This implies that the method does not require a long run to collect a large amount of data.

With a discrete parametric model, a method is developed to design a discrete-time general-structure controller the order of which is determined by the process model order. Through the modified internal model control and dominant pole placement algorithms, the direct design is achievable by specifying one to three readily understandable parameters. Simple-structure controllers with user-specified orders can then be derived based on ingenious approximations of the general-structure controller design. For example, a low-order proportional-integral- derivative and a high-order controller are obtainable in the same fashion. Furthermore, with a newly developed index for stability robustness, a procedure is presented to optimize the controller designs of various structures. It is concluded that the resultant simple-structure controllers are vastly superior to the general-structure controller in almost all respects. The method is demonstrated with a wide range of process dynamics including time delay, right-half-plane and left-half-plane zeros, low- to high-order lags, integrators, and unstable poles.

Many real processes are characterized by sophisticated or even nonlinear dynamics. The exact description demands a tedious procedure to establish a mathematical model theoretically by taking into account all nonlinear properties. In general, it is extremely difficult to understand various nonlinear dynamics by the direct use of the original nonlinear model. Furthermore, bifurcation theorem indicates that linear effects would vanish in the vicinity of singular points and nonlinear effects would dominant the dynamic behavior. In this work, a model reduction technique is developed based on bifurcation theorem to deal with the analysis of the nonlinear system effectively. The technique reduces the original nonlinear model to a simple normal form by applying center manifold projection and normal form theorem. The final normal form contains only the necessary resonance terms, which suffice to obtain the complete phase portrait and nonlinear dynamics in the vicinity of a singular point. The technique is applied to a proportional control system of two continuous stirred tank reactors with recycle connected in series. Linear stability boundaries are found and corresponding generic singular points are located for further investigation.
目 錄

表目錄 i
圖目錄 ii
符號說明 iv
第一章 緒論 ................................................ 1
1.1 線性連續時間系統鑑別之文獻回顧 ....................... 4

1.2 線性離散時間系統鑑別之文獻回顧 ....................... 5
1.3 強健離散時間控制器之最佳設計文獻回顧 ................. 7
1.4 分歧理論與CSTR非線性動態之文獻回顧 ................... 9
1.5 章節與組織 ........................................... 11
第二章 線性連續時間系統鑑別 ................................ 12
2.1時間加權積分轉換 ...................................... 13
2.2移動區間最小平方演算法 ................................ 15
2.3模型參數之無偏估測 .................................... 17
2.4開環與閉環鑑別實驗 .................................... 18
2.5模型階次與時延大小的決定 .............................. 21
2.6 MIMO系統上的應用 ..................................... 22
2.7模擬研究 .............................................. 23
第三章 線性離散時間系統鑑別 ................................ 38
3.1時間加權離散濾波器 .................................... 38
3.2 移動區間遞迴最小平方演算法 ........................... 41
3.3 開環路與閉環路鑑別實驗 ............................... 42
3.4 取樣週期及濾波區間的決定 ............................. 43
3.5 模型參數的幾近無偏估測 ............................... 44
3.6 系統階次及時延的決定 ................................. 46
3.7 分數時延 ............................................. 55
3.8 模擬研究 ............................................. 57
第四章 強健離散時間控制器之最佳設計 ........................ 63
4.1完整結構控制器 ........................................ 64
4.2優勢極點配置演算法 .................................... 65
4.3簡單結構控制器 ........................................ 68
4.4強韌性考量 ............................................ 73
4.5開環不穩定程序的增益邊距 .............................. 77
4.6控制器的設計步驟 ...................................... 78
4.7模擬研究 .............................................. 79
第五章 非線性系統之模式簡化 ................................ 88
5.1 Center Manifold 理論 ................................. 89
5.2 Normal Form 理論 ..................................... 93
5.3 Unfolding 分析 ....................................... 96
5.4 連續攪拌槽反應器之穩定性分析 ......................... 100
5.4.1 比例控制下單反應器之穩定性分析 ................. 102
5.4.2 比例控制下串連迴流連續攪拌槽反應器之穩定性分析 . 108
第六章 結論與展望 .......................................... 118
參考文獻 ..................................................... 122
附錄A、計算Center Manifold 投映函數之係數 ....................129
附錄B、Normal Form 係數計算 ..................................154
附錄C、近似特性轉移函數之係數計算 ............................ 165
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