動態系統最佳化設計之計算方案與其應用

Abstract

動態系統所引發的特性一直困擾著工程設計人員，而只在靜態系統模式下，採用最佳化設計方法所求得的設計，則往往在實際的應用上有所不足。本文主要依據最佳設計與最佳控制理論基礎，結合動態分析與數值分析求解技巧，發展一套通用之動態系統最佳設計方法與軟體。 一般動態系統之最佳化問題可以轉換成標準的最佳控制問題，再透過離散技術轉換成非線性規劃問題，如此便可利用現有之最佳化軟體進行求解。在本文中，首先將動態系統的解題方法與流程發展為最佳控制分析模組，再將該模組與最佳化分析軟體 (MOST) 整合得到整合最佳控制軟體，可以用來解決各種類型的最佳控制問題。為驗證軟體的效能與準確性，利用本文所發展之整合最佳控制軟體求解文獻資料中所提出之各類型最佳控制問題。藉由分析結果之數值與控制軌跡曲線的比對，整合最佳控制軟體所求出之數值解，在效能與準確性上都能與文獻資料所獲得的最佳解吻合，確認該整合最佳控制軟體的確可以用來解決我們工程應用上的最佳控制問題。 另外，針對工程設計中存在的離散（整數）最佳控制問題，本文依據混合整數非線性規劃法(mixed integer nonlinear programming) 做進一步的研究。猛撞型控制 (bang-bang type control) 是常見的離散最佳控制問題，其複雜與難解的特性更是吸引諸多文獻探討的主因。許多文獻針對此一問題所提出的方法多在控制函數的切換點數量為已知的假設條件下所推導，但這並不符合實際工程上的應用需求，因為控制函數的切換點數量大多在求解完成後才會得知。因此，本文針對此類型問題發展出兩階段求解的方法，第一階段先粗略求解該問題在連續空間下的解，並藉此求得控制函數可能的切換點資訊，第二階段再利用混合整數非線性規劃法求解該問題的真實解。發展過程中，加強型的分支界定演算法 (enhanced branch-and-bound method)被實際應用並且納入前一階段所開發的整合最佳控制軟體中，這也使得這個軟體可以同時處理實際動態系統中最常見的連續及離散最佳控制問題。 最後，本文將所發展的整合最佳控制軟體用來求解兩個實際的工程應用問題：飛航高度控制問題與車輛避震系統設計問題。兩個問題都屬於高階非線性控制問題，首先利用本文中所建議的解題步驟建構完成這兩個問題的數學模型，接著直接利用本研究所發展的軟體求解符合問題要求的最佳解。經由這些實際應用案例的驗證，顯示本文所發展的方法與軟體的確可以提供工程師、學者與學生一個便利可靠的動態系統設計工具。

The nonlinear behaviors of dynamic system have been of continual concern to both engineers and system designers. In most applications, the designs – based on a static model and obtained by traditional optimization methods – can never work perfectly in dynamic cases. Therefore, researchers have devoted themselves to find an optimal design that is able to meet dynamic requirements. This dissertation focuses on developing a general-purpose optimization method, based on optimization and optimal control theory, one that integrates dynamic system analysis with numerical technology to deal with dynamic system design problems. A dynamic system optimal design problem can be transformed into an optimal control problem (OCP). Many scholars have proposed methods to solve optimal control problems and have outlined discretization techniques to convert the optimal control problem into a nonlinear programming problem that can then be solved using extant optimization solvers. This dissertation applies this method to develop a direct optimal control analysis module that is then integrated into the optimization solver, MOST. The numerical results of the study indicate that the solver produces quite accurate results and performs even better than those reported in the earlier literatures. Therefore, the capability and accuracy of the optimal control problem solver is indisputable, as is its suitability for engineering applications. A second theme of this dissertation is the development of a novel method for solving discrete-valued optimal control problems arisen in many practical designs; for example, the bang-bang type control that is a common problem in time-optimal control problems. Mixed-integer nonlinear programming methods are applied to deal with those problems in this dissertation. When the controls are assumed to be of the bang-bang type, the time-optimal control problem becomes one of determining the switching times. Whereas several methods for determining the time-optimal control problem (TOCP) switching times have been studied extensively in the literature, these methods require that the number of switching times be known before their algorithms can be applied. Thus, they cannot meet practical demands because the number of switching times is usually unknown before the control problems are solved. To address this weakness, this dissertation focuses on developing a computational method to solve discrete-valued optimal control problems that consists of two computational phases: first, switching times are calculated using existing continuous optimal control methods; and second, the information obtained in the first phase is used to compute the discrete-valued control strategy. The proposed algorithm combines the proposed OCP solver with an enhanced branch-and-bound method and hence can deal with both continuous and discrete optimal control problems. Finally, two highly nonlinear engineering problems – the flight level control problem and the vehicle suspension design problem – are used to demonstrate the capability and accuracy of the proposed solver. The mathematical models for these two problems can be successfully established and solved by using the procedure suggested in this dissertation. The results show that the proposed solver allows engineers to solve their control problems in a systematic and efficient manner.