近線性系統的強韌控制

Robust Control for Pseudo-Linear Systems

莊斌鎰(Bin-Yih Juang)

指導教授 ： 顏家鈺

國立臺灣大學/工學院/機械工程學研究所/博士(2015年)

https://doi.org/10.6342/NTU.2015.02772

摘要

本論文研究近線性系統(pseudo-linear systems)的強韌控制技術。近線性系統是一種完全可回饋線性化(fully feedback linearizable)的非線性系統。這個強韌控制技術的主要觀念是以線性強韌控制器結合非線性系統的回饋線性化控制器(考慮輸出變數)再引入受干擾非線性系統使得這個受干擾非線性系統的閉迴路具有良好的強韌穩定(robust stability, RS)的控制特性。研究成果主要分成線性強韌控制器、回饋線性化控制器及近線性系統的強韌控制技術三個部份。這些研究成果分別呈現在第3章、第4章、第5章及第6章。最後一章是結論與建議。
第3章發展一個區間多項式的強韌極點指定(robust pole-assignment, RPA)技術基於正的區間羅斯表(positive interval Routh-table, PIRT)利用二項式係數型回饋增益(binomial coefficients feedback gain, BCFG)。這個強韌技術將區間特性多項式的強韌極點指定問題轉成解一個最大下界值(greatest lower bound value, GLBV)確保一組穩定條件多項式(stability condition polynomials, SCP)所有的函數值恆正的問題。解存在的條件是二項式係數羅斯表(Routh-table of binomial coefficients, RTBC)所有的項為正。重要的，本章提供一個先進的二項式係數羅斯表所有產生項(all generated entries of RTBC)的通式，證明二項式係數羅斯表所有項恆為正。一個例題的模擬結果顯示這個線性強韌極點指定技術是可行的。
第4章展示一個區間多項式的強韌極點指定技術基於Kharitonov定理使用二項式係數回饋增益。這個強韌技術將區間特性多項式的強韌極點指定問題轉成解存在一個最大下界值確保四組穩定條件多項式所有的函數值恆正的問題。這個強韌技術成立的條件是二項式係數羅斯表所有的項為正。兩個正定的二項式係數羅斯表所有產生項的通式被證明是等效的。此外，這個強韌技術不僅確保Kharitonov’s 四個多項式強韌穩定同時也確保Kharitonov’s 四個多項式係數線性相依的干擾系統及區間多項式強韌穩定。一個例題的模擬結果顯示這個線性強韌極點指定技術是可行的。
第5章表現一個受干擾近線性系統的強韌控制技術。這個強韌技術結合近似回饋線性化控制器(approximate feedback linearization control, AFL)及第3章或第4章的強韌極點指定控制器使得這個受干擾非線性系統強韌穩定。首先，基於恰當回饋線性化(exact feedback linearization, EFL)的結果，我們得到一個非線性系統的近似回饋線性化控制技術的解。接著，建立這個近似回饋線性化技術與恰當回饋線性化技術的類相似轉換關係式(quasi-similarity transformations)。這個類相似轉換關係式包括座標變換、回饋線性化控制器及強韌控制器轉換三個部份。本文的類相似轉換關係是強調強韌控制器轉換的部份。最後，一個區間線性系統的強韌極點指定控制器透過這個類相似轉換公式結合近似回饋線性化控制器被應用到受干擾的非線性系統使得這個受干擾非線性系統的閉迴路具有良好的強韌極點指定控制特性的響應。一個例題的模擬結果顯示這個近線性系統的強韌控制技術是可行的且不受有限度參數變化及平衡點變化的影響。相對的，這些模擬結果顯示傳統的線性系統的強韌控制效應受參數變化及平衡點變化的影響。此外，模擬結果亦顯示近線性系統的強韌控制比傳統的線性系統的強韌控制有較大的操作區間。
第6章呈現一個受干擾近線性系統的強韌控制技術。這個強韌控制技術結合恰當回饋線性化控制器及H_∞強韌控制器。這個強韌控制技術是延續第5章的研究成果。一個從近似回饋線性化控制器轉到恰當回饋線性化控制器的類相似轉換公式被建立。一個狀態回饋線性 H_∞強韌控制器透過這個類相似轉換公式結合恰當回饋線性化控制器被應用到受干擾的非線性系統使得這個受干擾非線性系統的閉迴路具有H_∞ 強韌控制的響應特性。一個例題的模擬結果顯示這個近線性系統的強韌控制技術有較大的操作區間且不受有限度參數變化及平衡點變化的影響。相對的，這些模擬結果顯示傳統的線性系統的強韌控制受參數變化及平衡點變化的影響，不僅操作區間小，一旦操作區間有微量變化則系統的特性呈現不可預測的行為。

關鍵字

強韌極點指定 ； 區間多項式 ； 二項式係數羅斯表 ； 正定區間羅斯表 ； Kharitonov定理 ； 非線性系統 ； 近似回饋線性化 ； 類相似轉換 ； 恰當回饋線性化 ； H_∞強韌控制器.

並列摘要

This dissertation proposes a robust control scheme for pseudo-linear systems with perturbed terms. The pseudo-linear systems are nonlinear systems, which are also fully feedback linearizable systems. The concept of the scheme utilizes a linear robust controller combine with a feedback linearization controller such that the closed-loop perturbed nonlinear system possess an excellent robust stability (RS) properities.
In chapter 3, we elucidate a robust pole-assignment (RPA) scheme for an)interval polynomial based on positive interval Routh-table (PIRT) using a binomial coefficients feedback gain (BCFG). The robust scheme transforms the problem of RPA into the problem of solving a family of stability condition polynomials (SCP) that have positive function values. The condition for the existence of a solution, the greatest lower bound value, GLBV, is that all entries of the Routh-table of binomial coefficients (RTBC) are positive. Importantly, in this chapter, we provide an advanced general formula for the generated entries of the RTBC, confirming that all entries of RTBC are positive. An illustrative example reveals the effectiveness of the scheme.
In chapter 4, we elucidate a RPA scheme for an interval polynomial that is based on the Kharitonov theorem, and uses a BCFG. The robust scheme transforms the problem of RPA into the problem of solving four families of SCP that have positive function values. The existence of a solution, GLBV, requires that all entries of the RTBC are positive. Two general formulas, which both exhibit positive property, have been proven to be mutually equivalent. This robust scheme ensures not only that the four Kharitonov polynomials are all stable, but also that their linear combinations are stable; it also guarantees the interval polynomial robust stability. An illustrative example demonstrates the effectiveness of the scheme.
In chapter 5, we elucidate a robust(control for nonlinear systems that is subject to some uncertainties, using approximate feedback linearization (AFL) and a RPA controller. A close form solution of AFL is obtained; it transforms the nonlinear system into an equivalent Taylor-linearized form. A formula for transformation, quasi-similarity transformations, between the exact feedback linearization (EFL) form and the AFL is presented. The perturbed Brunovsky-linearized form is utilized to design a RPA controller. The robust controller, which combines the RPA controller, the transformation formula and the AFL controller, is applied to a perturbed nonlinear system such that the closed-loop perturbed nonlinear system exhibits RPA properties. An illustrative example reveals the robust scheme is not influenced under the limited parameter uncertainties and the equilibrium point variants with great effectiveness. Moreover, these simulation results also indicate that the robust scheme for pseudo-linear systems has larger operation region than that of the traditional robust scheme for linear systems.
In chapter 6, we elucidate a robust control for nonlinear systems, which is subject to some uncertainties, using exact feedback linearization with H_∞ robust controller. A transfer formula, quasi-similarity transformations, from the AFL to the EFL is presented. Through the transfer formula, the H_∞ robust controller combining the EFL controller is applied to the perturbed nonlinear system such that the closed-loop perturbed nonlinear system has the H_∞ robust properties. An illustrative example reveals the robust scheme is not affected under the limited parameter uncertainties and the equilibrium point variants with great validity. Moreover, these simulation results also imply that the robust scheme for pseudo-linear systems has larger operation region than that of the traditional robust scheme for linear systems.

並列關鍵字

robust pole assignment ； interval polynomial ； Routh-table of binomial coefficients ； positive interval Routh-table ； the Kharitonov theorem ； Nonlinear systems ； approximate feedback linearization ； quasi-similarity transformations ； exact feedback linearization ； H_∞ robust controller.

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