線性多變數系統之穩定條件、參數化、以及計算機輔助解耦控制器設計

Abstract

針對單一回授的線性多變數系統，本論文提出系統閉迴路穩定的簡化條件 ，以及系統在感測器或致動器毀損時仍然能保持閉迴路穩定之充份必要條 件。藉著這些穩定條件，本論文將系統能達到的所有解耦輸入輸出轉移矩 陣以及相關的解耦控制器參數化，並進而提出簡便的法則來計算解耦控制 器、開迴路穩定的解耦控制器、以及使系統在感測器毀損時仍然能保持閉 迴路穩定的解耦控制器。利用前面所提的解耦控制器參數化，本論文以最 佳化理論為基礎，建立了一套計算機輔助的解耦控制器設計法則，而這項 法則所處理的待控體可以同時是不穩定及非最小相位，並且也可以只是單 輸入單輸出。這項設計法則的主要優點在於參數的調整相當俱有系統化， 而且許多實際的工程規格像爬升時間、最大超越量、制動器最大輸入限制 、以及穩定強韌性等，都可以很容易地列入為設計的考量。目前這項設計 方法已經發展成 MATLAB上的一套交談式輔助設計軟體。本論文最後並以 兩個實際的應用設計範例，來驗證這個方法的效率以及可行性。 We establish in this dissertation some simplified conditions for the closed-loop stability of the linear multivariable unity- feedback system and for the system to remain stable under sensor or actuator failures. We also propose the parametrizations of all stabilizing controllers and the descriptions of all achievable I/O maps. Such controller parametrizations and I/O map descriptions are applied to characterize the set of all decoupling controllers, and lead to simple computational algorithms for the construction ofdecoupling controller, stable decoupling controller, and decoupling controllers that retain the closed-loop stability under sensor failures. By the algebraic property of our analysis, most results in this dissertation can apply to continuous-time systems as well as discrete-time systems. Based on our characterization of decoupling controllers and the corresponding computational algorithm, we develop an optimization-based decoupling control design procedure which can also apply to SISO system design without any modifications. This design procedure is systematic in that the design is improved in each iteration based on a well-defined performance index. It is also practical in that many engineering-level design specifications such as rise time, maximum overshoot, plant input limit, and robust stability can be easily incorporated into the optimization program. By our formulation, there is no equality constraints which are in general hard to achieve in optimization problems. This design procedure has been implemented as an interactive CAD package for use under MATLAB. Two illustrative design examples are also proposed to verify the effectiveness of this design approach.