Linear quadratic approach in designing robust controller of uncertain systems

• Main Authors: CAI, SHU-CHUAN, 蔡樹川
• Other Authors: GUO, DE-SHENG
• Format: Others
• Language: zh-TW
• Published: 1990
• Online Access: http://ndltd.ncl.edu.tw/handle/24091883197601032491

博士 === 國立臺灣大學 === 電機工程研究所 === 79 === In recent decades, linear quadratic (LQ) state feedback controller has been widely applied on a number of control problens. A munber of researchers (Kalman 1964, Anderson 1971, Atheans and Safonov 1977, Safonov 1981) have also found fany elegant features of LQ controller such as high gain margin, significant phase margin and large gain tolerance. In this dissertation, we shall follow this line of efforts to exploit the merits of LQ design and apply it to stabilize a class of uncertain systems. Both linear nucertainty and nonlinear uncertainty are tackled in this dissertation. In linear robust problem, where the system uncertainty is a linear function of the state and input, and satisfies the so-called matching conditions, a criterion is developed for choosing and the weighting matrices of the linear quadratic cost function. By utilizing the Lyapunov stability criterion and algebraic Riccati equation (ARE), it is shown that the optimal control law for nominal systems also makes the uncertain system achieve the desired system behavior, which is quadratic stability if there is no imput disturbance and output measurement error. For the case in which there is input disturbance or output measurement error, we show that practical stability can be achieved. This controller is also termed as robust optimal controller. In order to systematically obtain the weighting matrices, three efficient but simple algorithms are proposed in this dissertation. To handle a more general class of systems, in which the uncertainty does not exactly satisfy the matching conditions, a sufficient condition is derived to ensure that the linear system can be stabilized by the robust optimal controller. A method of adjusting the state weighting matrix is suggested to facilitate the computation of the mismatching threshold used to chedk the sufficient condition. To consider the cases where state variables cna not be accessed, the output control problem is considered. The robust full order and reduced-order Luenberger observers are studied to generate the estimated state variable. The purpose is to construct a robust output feedback controller by combining the estimated state vector and the robust optimal control law to guarantee the desired system behavior. Finally, a robust output tracking controller design method based on the robust optimal controller is also proposed. In the nonlinear robust control problem, a robust decetralized controller is designed to stabillize a group of interconnected systems with nonlinear uncertainties. It is assumed that the coupled elements and the nonlinear uncertain elements containing the local and coupled terms in the subsystem satisfy the matching conditions and all the nonlinear uncertainties satisfy the Lipschitz condition. We show in this dissertation that a set of weighting matrices can be selected to obtain the robust decetralized controller which ensure the interconnected system can achieve the desired system behavior. We use multi-machine power system control as a practical application of the robust decentralized controller. Finally, a test matrix is derived to ensure whether the robust decentralized controller can stabilize the nonlinear interconnected system with the uncertain nonlinearity and coupled element not exactly satisfying the matching conditions. In summary, in this disseration LQ controller has been showed to be able to stabilize the linear uncertain and nonlinear uncertain system. Comparing LQ robust control with the other robust control methods in the literatures, we find LQ robust control has the following advantages:(1) it has simple design procedure; (2) it generates linear controller; (3) no precompensator is needed; (4) it has wider application ranges. 傳統線性平方最佳狀態迴授控制已廣泛應用於控制領域。本論文應用此優良的設計方 法在系統具有不定性的控制上。主要探討的內容可分成二大部分：線性系統具有不定 性的控制與非線性系統具有不定性的控制。在線性系統具有不定性的控制方面，首先 假設不定量符合所匹配條件，然後應用李亞普諾穩定法則及李卡提方程式推演出對於 給定之系統不定性，二次成本函數加權矩陣之可用公式。如此所得對常態系狀的最佳 控制器亦能使得系統在不定參數的所有可能變化內，皆可達成所謂的二次穩定性或者 是實用穩定性。另外為了系統化地求得成本函數的加權矩陣，本論文提出三種有效而 且簡便的演算法。對於不定性必須符合匹配條件的限制，本論文亦推導放寬限制的必 要條件。對於輸出控制方面，本論文探討強健的全狀態與部分狀態Luenberger觀察器 設計，俾用以估測系統狀態變數向量。如此與強健二次控制律結合的迴授控制器，亦 可達成糸統的期望響應。在線性控制的應用上，本論文將探討有關強健輸出追蹤控制 的問題。 在非線性不定性的強健控制方面，本論文探討非線性不定連結系統的分散式控制問題 。首先假設各個子系統的交連項及本身的不定性元件都滿足匹配條件，而且所有的非 線性元都假設符合李匹奇條件，如此所有交連量及非線性不定量與個別子系統的加權 矩陣關係都可推得，如此經由個別子系統的所得的強健控制，將使得此連結系統達成 二次穩定性要求。多電力廠系統的分散LQ控制器設計是分散式控制的一實際應用的例 子。另外，當有不完全符合匹配條件的情形發生時，本論文另推導一測試矩陣，並推 得此測試矩陣為正定時，可保證前述的強健分散控制器亦可使得系統在不完全符合匹 配條件時為二次穩定。 綜觀本論文可發現，由LQ設計的強健控制不論在線性不定性或在非線性不定性控制上 ，都有很好的效果。與其他相關文獻比較，本論文具⑴容易設計，⑵線性控制，⑶不 需要前置控制器設計與⑷保有LQ設計特點等優點，所以應用範圍更為廣泛。最後對於 本論文所探討的每一問題，本論文都給予一些應用實例，以說明本論文所開發設計方 法之可行性及優點。