In this project an optimal control problem solver, the OCP solver, based on the Sequential Quadratic Programming (SQP) method and integrated with many well-developed numerical routines is implemented. A systematic procedure for solving optimal control problems is also offered in this project.
This project also presents a novel method for solving discrete-valued optimal control problems. Most traditional methods focus on the continuous optimal control problems and fail when applied to a discrete-valued optimal control problem. One common type of such problems is the bang-bang type control problem arising from time-optimal control problems.
When the controls are assumed to be of the bang-bang type, the time-optimal control problem becomes one of determining the TOCP switching times. Several methods for such determination have been studied extensively in the literature; however, these methods require that the number of switching times be known before their algorithms can be applied. As a result, they cannot meet practical situations in which the number of switching times is usually unknown before the control problem is solved. Therefore, to solve discrete-valued optimal control problems, this dissertation has focused on developing a computational method consisting of two phases: (a) the calculation of switching times using existing optimal control methods and (b) the use of the information obtained in the first phase to compute the discrete-valued control strategy.
The proposed algorithm combines the proposed OCP solver with an enhanced branch-and-bound method. To demonstrate the proposed computational scheme, the study applied third-order systems and an F-8 fighter aircraft control problem considered in several pioneering studies. Comparing the results of this study with the results from the literature indicates that the proposed method provides a better solution and the accuracy of the terminal constraints is acceptable.
REFERENCES
Banks, S.P., and Mhana, K.J., “Optimal Control and Stabilization of Nonlinear systems,”
IMA Journal of Mathematical Control and Information, Vol. 9, pp. 179-196, 1992.
Bellman, R., 1957, Dynamic Programming, Princeton University Press, Princeton, NJ, USA.
Betts, J. T., 2000, “Very Low-Thrust Trajectory Optimization Using a Direct SQP Method,”
Journal of Computational and Applied Mathematics, Vol. 120, pp. 27-40.
C.H. Huang, and C.H. Tseng, 2003, “Computational Algorithm for Solving A Class of Optimal Control Problems,” IASTED International Conference on Modelling, Identification, and Control (MIC’2003), Innsbruck, Austria, pp. 118-123.
C.H. Huang, and C.H. Tseng, 2004, “Numerical Approaches for Solving Dynamic System Design Problems: An Application to Flight Level Control Problem,” Proceedings of the Fourth IASTED International Conference on Modelling, Simulation, and Optimization (MSO’2004), Kauai, Hawaii, USA, pp. 49-54.
C.H. Tseng, W.C. Liao, and T.C. Yang, 1996, MOST 1.1 User's Manual, Technical Report No.
AODL-93-01, Department of Mechanical Engineering, National Chiao Tung Univ., Taiwan, R.O.C.
Gill, P. E., Murray, W., and Saunders, M. A., 2002, “SNOPT: An SQP Algorithm for Large-Scale Constrained Optimization,” SIAM Journal on Optimization, Vol. 12, No. 4, pp. 979-1006.
G.S. Hu, C.J. ONG, and C.L. Teo, 2002, “An Enhanced Transcribing Scheme for The Numerical Solution of A Class of Optimal Control Problems,” Engineering Optimization, Vol. 34, No. 2, pp. 155-173.
Jaddu, H., and Shimemura, E., 1999, “Computational Method Based on State Parameterization for Solving Constrained Nonlinear Optimal Control Problems,”
International Journal of Systems Science, Vol. 30, No. 3, pp. 275-282.
Kaya, C.Y., and Noakes, J.L., “Computations and Time-Optimal Controls,” Optimal Control Applications and Methods, Vol. 17, pp. 171-185, 1996.
Lee, H.W., Jennings, L.S., Teo, K.L., and Rehbock, V., “Control Parameterization Enhancing Technique for Time Optimal Control Problems,” Dynamic Systems and Applications, Vol.
6, pp. 243-262, 1997.
Lucas, S.K., and Kaya, C.Y., “Switching-Time Computation for Bang-Bang Control Laws,”
Proceedings of the 2001 American Control Conference, pp. 176-181, 2001.
Simakov, S.T., Kaya, C.Y., and Lucas, S.K., “Computations for Time-Optimal Bang-Bang Control Using A Lagrangian Formulation,” 15th Triennial World Congress, Barcelona, Spain, 2002.
Pontryagin, L. S., Boltyanskii, V. G., Gamkrelidze, R. V., and Mischenko, E. F., 1962, The Mathematical Theory of Optimal Processes, Wiley.
Tseng, C.H., Wang, L.W., and Ling, S.F., “Enhancing Branch-And-Bound Method for Structural Optimization,” Journal of Structural Engineering, Vol. 121, No. 5, pp. 831-837, 1995.
Wu, S.T., “Time-Optimal Control and High-Gain Linear State Feedback,” International Journal of Control, Vol. 72, No.9, pp. 764-772, 1999.
Table 1 Results of various methods for the F-8 fight aircraft problem.
Method
t
f Switching TimesAccuracy of Terminal Constraints STC
(Kaya and Noakes, 1996)
6.3867 0.0761, 5.4672, 5.8241, 6.3867 ≤ 10-5
CPET
(Lee et al., 1997)
6.0350 2.188, 2.352, 5.233, 5.563 ≤ 10-10
Two-phase scheme 5.7422 0.098, 2.027, 2.199, 4.944, 5.265 ≤ 10-10
Figure 1 flow chart of the NLP method for solving OCP
ūi ūi+1
J0
Pj Continuous optimum point
Figure 2 Conceptual layout of the branching process.
Initialization
1. Relax all discrete-valued restrictions 2. Place the resulting continuous NLP
problem on the branching tree.
3. Set the cost bound Jmax=
Is the branching tree empty?
A
Select an unexplored node from branching tree
Solve subproblem by applying the OCP Solver (AOCP)
(NLP is optimal) &&
(J0<Jmax)
B
B
Yes
NO NO
Is feasible ?
Drop this node
NO
Yes
Yes
Evaluate the values of criteria for selecting branch object
Decide the branch of design variable
Branching process
Two new nodes are added into branching tree Subdomain that
does not contain any allowable value
Add a new upper and lower bounds Bounding process
Add the new node to feasible
node matrix Set new cost bound
Jmax= J0
Show results
Figure 3 Flow chart of the algorithm for solving discrete-valued optimal control problems.
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Time (sec.)
-12 -10 -8 -6 -4 -2 0 2 4 6 8 10 12 14
Control (u)
Mixed Discrete NLP method (tf = 0.7813) AOCP (tf = 0.7826)
AOCP vs. a mixed-integer NLP method.
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
Time (sec.) -12
-10 -8 -6 -4 -2 0 2 4 6 8 10 12 14
Control (u)
Phase I (tf = 0.7864) Phase II (tf = 0.7813)
Phase I vs. Phase II.
Figure 4 Control trajectories for the third-order system.
0.0 1.0 2.0 3.0 4.0 5.0 6.0 Time(sec.)
-0.06 -0.05 -0.04 -0.03 -0.02 -0.01 0.00 0.01 0.02 0.03 0.04 0.05 0.06
Tail deflection angle (rad.)
Phase I (tf = 5.74173) Phase II (tf = 5.74216)
Figure 5 Control trajectories for the F-8 fighter aircraft.
0.0 1.0 2.0 3.0 4.0 5.0 6.0 Time(sec.)
-0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 0.4 0.5 0.6
State and control varialbes
Tail deflection angle (rad.) Angle of attack (rad.) Pitch angle (rad.) Pitch rate (rad./s)
(a) Phase I.
0 1 2 3 4 5 6
Time(sec.) -0.6
-0.5 -0.4 -0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 0.4 0.5 0.6
State and control variables
Tail deflection angle (rad.) Angle of attack (rad.) Pitch angle (rad.) Pitch rate (rad./s)
(b) Phase II.
Figure 6 Trajectories of the states and control input for the F-8 fighter aircraft.
計畫成果自評
傳統最佳化設計方法單靠靜態分析所得到產品在實際動態工作環境下往往表現不 佳，甚至有時會面臨無法正常動作的窘境。因此，本研究計畫主要發展目標是依據動態 系統分析與最佳化理論，整合數值分析技巧，發展一套有效率的方法與軟體來協助設計 者處理動態系統最佳設計問題，其中必須將非線性與離散參數設計問題皆需納入考量，
以因應實際工程需求。
本研究計畫分兩年實施，第一年研究重點在於發展出一套整合動態分析與最佳化技 巧的方法，並將之實作成一套泛用型之動態最佳化軟體，此軟體的發展將有助於動態系 統設計者縮短分析及設計的時程，並對產品的更新與市場佔有率的提升提供顯著的幫 助。而要發展一套整合動態分析與最佳化技巧的方法，首先必須解決工程系統非線性的 問題，系統動態特性分析時所解得的系統方程式常常是高度非線性的微分方程組，此時 通常無法求得解析解，而藉助電腦數值分析技巧來求得收斂解是必要的。 第二個困難 點是在於許多動態分析必須藉助專業的分析軟體來進行，此時如何整合最佳化與分析軟 體便成為另一項挑戰。
本計畫利用數值分析方法與程式設計技巧，順利完成預期目標中所要發展的泛用型 之動態最佳化軟體，並利用許多文獻上著名的動態設計與控制問題來進行驗證，從軟體 求得的數值解與文獻的結果作比對，發現本計畫所發展之軟體所得的結果與文獻的結果 是吻合的，甚至有些問題利用本計畫所發展出來的軟體求得的解優於文獻上的結果，由 這些結果我們得到的初步的驗證，也順利完成本計畫中第一年所預期想要達到的目標。
本研究計畫第二年除了延續第一年的主題外，更將工程設計常會遭遇到某些設計尺
寸是有限離散數值（或整數）的情況納入考量。這個看似簡單的問題，其實是讓原本單 純的連續變數最佳化設計問題，轉換成複雜難解的混合型整數最佳化設計問題，但這類 型的問題卻是實際工程上所常常會遭遇的，如果能進一步提供解決這類型的動態最佳化 問題，將可大幅提升研發設計的能力。本研究計畫，採納第一年所發展的解題軟體做為 核心，並將加強型的分支界定演算法 (enhanced branch-and-bound method)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。
本研究計畫的成果除了實作成功能強大的泛用型動態最佳化分析軟體外，其方法與 應用也發表於國際期刊與會議中。由整個計畫執行過程中，於每個計畫執行階段，計畫 執行所預期之目標均已完成，而整個研究過程中所執行之目標與成果敘述如下：
1. 發展一系統化解決動態最佳化問題的方法與流程，使用者只要依循所建議的方法將 問題定義成標準形式，便可以利用本研究計畫所發展的軟體求得其最佳解。
2. 發展一新穎的方法來求解離散數值最佳控制問題(混合整數之離散數值動態設計問 題)，此方法可以讓使用者在對於原來連續最佳控制問題上對於某些設計變數作些微 的設定修改，就可以求解複雜的離散數值最佳控制問題。
3. 發展一新穎的方法來求解猛撞型的最佳控制問題(Bang-bang Control problems)使用 者無須事先知道控制變數的切換時間點數量，即可求解出最佳的Bang-bang control law，對於非線性的問題以往要求得其Bang-bang control law是相當困難的，但利用本 研究計畫所發展的方法，搭配數值計算技巧，可以順利求得符合拘束條件的控制法 則。
本研究目前已發表兩篇國際期期刊及兩篇研討會論文
期刊論文
1. C.H. Huang and C.H. Tseng, “An Integrated Two-Phase Scheme for Solving Bang-Bang Control Problems,” Accepted for publication in Optimization and Engineering (SCI Expended/ISI).
2. C.H. Huang and C.H. Tseng, “A Convenient Solver for Solving Optimal Control Problems,” Journal of the Chinese Institute of Engineers, Vol. 28, pp. 727-733, 2005 (Ei/SCI).
研討會論文
1. C.H. Huang, and Tseng, C.H., “Numerical Approaches for Solving Dynamic System Design Problems: An Application to Flight Level Control Problem,” Proceedings of the Fourth IASTED International Conference on Modelling, Simulation, and Optimization (MSO2004), Kauai, Hawaii, USA, 2004, pp. 49-54.
2. C.H. Huang, and Tseng, C.H., “Computational Algorithm for Solving A Class of Optimal Control Problems,” IASTED International Conference on Modelling, Identification, and Control (MIC2003), Innsbruck, Austria, 2003, pp. 118-123.
附錄
1. (期刊論文) C.H. Huang and C.H. Tseng, “A Convenient Solver for Solving Optimal Control Problems,” Journal of the Chinese Institute of Engineers, Vol. 28, pp. 727-733, 2005 (Ei/SCI).
2. (會議論文) C.H. Huang, and Tseng, C.H., “Numerical Approaches for Solving Dynamic System Design Problems: An Application to Flight Level Control Problem,” Proceedings of the Fourth IASTED International Conference on Modelling, Simulation, and
Optimization (MSO2004), Kauai, Hawaii, USA, 2004, pp. 49-54.
3. (會議論文) C.H. Huang, and Tseng, C.H., “Computational Algorithm for Solving A Class of Optimal Control Problems,” IASTED International Conference on Modelling,
Identification, and Control (MIC2003), Innsbruck, Austria, 2003, pp. 118-123.
Numerical Approaches for Solving Dynamic System Design Problems:
An Application to Flight Level Control Problem
C.H. Huang* and C.H. Tseng**
Department of Mechanical Engineering, National Chiao Tung University Hsinchu 30056, Taiwan, R. O. C. E-mail: chtseng@mail.nctu.edu.tw
TEL: 886-3-5726111 Ext. 55155 FAX: 886-3-5717243 (*Research Assistant, **Professor)
ABSTRACT
The optimal control theory can be applied to solve the optimization problems of dynamic system. Two major approaches which are used commonly to solve optimal control problems (OCP) are discussed in this paper. A numerical method based on discretization and nonlinear programming techniques is proposed and implemented an OCP solver. In addition, a systematic procedure for solving optimal control problems by using the OCP solver is suggested. Two various types of OCP, A flight level tracking problem and minimum time problem, are modeled according the proposed NLP formulation and solved by applying the OCP solver. The results reveal that the proposed method constitutes a viable method for solving optimal control problems.
KEY WORDS
optimal control problem, nonlinear programming, flight level tracking problem, minimum time problem, SQP, AOCP.
1. INTRODUCTION
Over the past decade, applications in dynamic system have increased significantly in the engineering.
Most of the engineering applications are modeled dynamically using differential-algebraic equations (DAEs). The DAE formulation consists of differential equations that describe the dynamic behavior of the system, such as mass and energy balances, and algebraic equations that ensure physical and dynamic relations. By applying modeling and optimization technologies, a dynamic system optimization problem can be re-formulated as an optimal control problem (OCP). There are many approaches can be used to deal with these OCPs.
In particular, OCPs can be solved by a variational method [1, 2] or by Nonlinear Programming (NLP) approaches [3-5].
The indirect or variational method is based on the solution of the first order necessary conditions for optimality that are obtained from Pontryagin’s Maximum Principle (PMP) [1]. For problems without inequality constraints, the optimality conditions can be formulated as
a set of differential-algebraic equations which is often in the form of two-point boundary value problem (TPBVP).
The TPBVP can be addressed with many approaches, including single shooting, multiple shooting, invariant embedding, or some discretization method such as collocation on finite elements. On the other hand, if the problem requires the handling of active inequality constraints, finding the correct switching structure as well as suitable initial guesses for state and co-state variable is often very difficult.
Much attention has been paid to the development of numerical methods for solving optimal control problems [6, 7]. Most popular approach in this field turned to be reduction of the original problem to a NLP. A NLP consists of a multivariable function subject to multiple inequality and equality constraints. The solution of the nonlinear programming problem is to find the Kuhn-Tucker points of equalities by the first-order necessary conditions. This is the conceptual analogy in solving the optimal control problem by the PMP. NLP approaches for OCPs can be classified into two groups: the sequential and the simultaneous strategies. In simultaneous strategy the state and control variable are fully discretized, but in the sequential strategy only discretizes the control variables. The simultaneous strategy often leads the optimization problems to large-scale NLP problems which usually require special strategies to solve them efficiently. On the other hand, instability questions will arise if the discretizations of control and state profiles are applied inappropriately. Comparing to the simultaneous NLP, the sequential NLP is more efficient and robust when the system contains stable modes. Therefore, the admissible optimal control problems which bases on the sequential NLP strategy is propose to solve the dynamic optimization problems in this paper. To facilitate engineers to solve their optimal control problems, a general optimal control problem solver which integrates proposed method with SQP algorithm is developed.
In spite of extensive use of nonlinear programming methods to solve optimal control problems, engineers still spend much effort reformulating nonlinear programming problems for different control problems. Moreover, implementing the corresponding programs of the
nonlinear programming problem is tedious and time-consuming. Therefore, a systematic computational procedure for various optimal control problems has become an imperative for engineers, particularly for those who are inexperienced in optimal control theory or numerical techniques. Hence, the other purpose of this paper is to apply nonlinear mathematical programming techniques to implement a general optimal control problem solver that facilitates engineers in solving optimal control problems with a systematic and efficient procedure.
Flight level tracking plays an important role in autopilot systems receives considerable attentions in many researches [8-12]. For a commercial aircraft, its cruising altitude is typically assigned a flight level by Air Traffic Control (ATC). To ensure aircraft separation, each aircraft has its own flight level and the flight level is
Flight level tracking plays an important role in autopilot systems receives considerable attentions in many researches [8-12]. For a commercial aircraft, its cruising altitude is typically assigned a flight level by Air Traffic Control (ATC). To ensure aircraft separation, each aircraft has its own flight level and the flight level is