| Abstract: | 強穩定問題即是設計穩定的控制器使閉迴路系統穩定。歷年來的研究成果豐碩，然而仍有不足處，例如有解條件的保守性、解法的保守性，受控體接近違反pip特性時如何求解的問題等尚待改良。因此，本論文針對強穩定控制器設計問題與H∞強穩定控制器設計問題，提出新型的更寬鬆的有解條件，並提供具體的狀態空間設計方法。質言之，我們將這兩個問題分別轉換為多目標、正實控制器的設計問題、正實結構化控制器的設計問題，以及混合正實及H∞性能要求的設計問題。除了部份可採用Scherer等人的解法外，我們也發展出這些問題的新解法，特別是在正實結構化控制器的設計問題(Scherer等人的解法無法應用於此種問題)上。而當受控體接近違反pip性質時，面對頻域內插法繁複的設計程序(特別是受控體為多輸入多輸出時)，而現存狀態空間設計法皆可能失效的情形，我們特別提出一套設計過程簡單許多的交替式狀態空間設計法(每一步驟皆為解決一凸最佳化問題)，以彌補現有時域方法之不足。
本論文所提出的條件可表為線性矩陣不等式，故可運用許多現存的相關軟體有效求解。相較於若干內插法或黎卡迪方程式的方法，我們提出的方法條件較不保守、設計過程簡單、適用範圍較廣。數值模擬結果證實我們所提出的方法確實有效。
The strong stabilization problem is to design a stable controller that achieves close-loop stability. For years profound results relevant to this subject have been addressed. However, certain in-sufficiency of the current results can be improved further, for example, the conservatism incurred from the solvability conditions and the technical treatment exploited. Moreover, there still lacks a state-space approach for dealing with the case that a plant nearly violates the parity interlacing property (pip), the necessary and sufficient condition for the existence of stable stabilizing controllers. Therefore, this thesis presents new less conservative solvability conditions for the strongly stabilization problem and H-infinity strongly stabilization problem, Explicit state-space design procedures are provided. Specifically, we recast these two problems as the solvability conditions of multi-objective positive real controller synthesis problem, positive real structured controller synthesis problem and multi-objective (a mixture of positive realness and H-infinity performance requirements) control problem, respectively. New methods are derived for solving the cases, particularly on the positive real structured controller synthesis problem, where Scherer et al’s method is not applicable. In addition, this thesis also presents a new method to deal with the case that a plant nearly violates pip. Considering that the design procedure of the (frequency-domain) interpolation approach is quite involved, especially for multi-input multi-output systems, furthermore, all the currently existing state-space approaches may fail to yield a solution in this situation, we propose an iterative design with much simpler synthesis procedure (each step of which can be efficiently solved by performing convex optimization), so as to alleviate the in-sufficiency of the current time-domain designs.
The conditions proposed in this thesis can be converted to be LMIs, which can be efficiently solved with several LMI solvers. Compared with the interpolation approach and Riccati equation-based approach, our methods are less conservative and applicable to a broader class of plants. Numerical examples are presented to demonstrate the effectiveness of the proposed methods. |
