以根軌跡法探討具有全實數極點的線性輸入延遲系統之PID控制器設計

Abstract

本論文研究具實數極點的輸入非時變延遲系統，所有極點中至多包含一個不穩定極點。由於此種無限維時間延遲系統的特徵多項式為一超越函數，欲求出所有的特徵值十分困難，然而藉由Lambert函數，一階時間延遲系統經輸出回授後的所有特徵值軌跡可以作圖，且此作圖方法可延伸至二階具重根極點的時間延遲系統。透過這些成果，本論文亦討論包含相異實根極點的二階時間延遲系統特徵值軌跡，以及包含PID控制器的多階時間延遲系統之特徵值軌跡。 除建構一階、二階及多階時間延遲系統的特徵值軌跡，本論文還定義與判斷所謂最右邊或領導軌跡，並用以保證所有的閉迴路特徵值皆落在複數平面之左半平面，且此軌跡最左邊的點可用來分析評估輸出響應的效能。因此最右邊或領導軌跡不僅可作為穩定系統的準則，亦可用於分析輸出響應效能。除以上所述之外，本論文尚提出一種新類型之PID控制器，將傳統PD與PID控制器的優勢結合，其設計方法與參數調整皆有完整討論。最後，本論文所有提出的方法與理論將藉由例子與模擬進行說明，以驗證其應用流程與效能。

This dissertation studies linear time-invariant systems with an input delay and only real open-loop poles, in which at most one pole may be unstable. The retarded-type time-delay systems considered are infinite dimensional and have transcendental characteristic equations with an infinite spectrum. It is generally hard to find the entire spectrum of such systems. However, via Lambert function, all eigenvalues of first-order time-delay systems with an output feedback are found, and the method is extended to second-order time-delay systems with repeating poles. Based on these achievements, the eigenvalue-loci of second-order time-delay systems with distinct real poles are constructed, and nth-order time-delay systems with PID-control are discussed. When eigenvalue-loci of the first, second and nth-order time-delay systems are investigated, the concepts of the rightmost or leading locus is defined and identified. The leading locus can be used not only to derive stabilization criteria but also to analyze the performance of output response. Next, a new type PID-controller, which combines the advantages of conventional PD and PID-control, is proposed. Finally, all proposed methods are tested with examples to illustrate the application process and demonstrate the effectiveness.