The Numerical Solution of Continuous Time Optimal Control Problems with the Cutting Angle Method

Seyedalireza Seyedi
Università di Bologna, Bologna, Italy

Iraj Sadegh Amiri
Ho Chi Minh City, Vietnam

Sara Chaghervand
Islamic Azad University, Hamedan, Iran

Volker J. Sorger
The George Washington University, Washington, D.C., USA

Series: Mathematics Research Developments
BISAC: MAT027000

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This book consists of two parts. The first part is on the development of the proposition that “if there exists a type of function, then there exists a functional with the same type” based on the proposition of the inheritance and generalizability properties of a function in a functional. This study presents the abstract convex, increasing positively homogeneous and convex-along-rays functionals via this proposition. The second part concerns the investigation of the use of a global search optimization algorithm called the Cutting Angle Method (CAM) on Optimal Control Problems (OCP). Many algorithms are available for solving OCP, but they are basically local search algorithms. To overcome the problem associated with local searches, most OCP are modeled as Linear Quadratic Regulator (LQR) problems in the hope that the solution found estimates of the true global solution to the original problem. However, in doing so, a lot of information carried by the original problem might be lost in its translation into LQR models.

CAM being a global search algorithm is expected to overcome this problem. It can be used alone or in combination with a local search to find the global solution. CAM has been successfully used on functions, however, OCP are functionals. To do this, a model has been introduced based on inheritance and generalizability properties to demonstrate that the optimization algorithms that are used for functions can also be extended for use in functionals. Based on these properties, the study discovered that with the Unit Vectors Combinations Technique (UVCT) proposed in this research, CAM could successfully work on functionals in general and OCP particularly. To help speed up the convergence of CAM, the literature proposed the use of local searches for the determination of the initial solution. In a case study done in the research, CAM was successfully combined with a local search known as the Dynamic Integrated System Optimization and Parameter Estimation (DISOPE) algorithm. Moreover, the initial solution given by the DISOPE algorithm has been verified as a global influence by CAM.

Preface

Chapter 1. Introduction on Cutting Angle Method Inspired By Abstract Convexity for Solving Continuous Time Optimal Control Problems

Chapter 2. A Wide Literature Review on Building the Cutting Angle Method

Chapter 3. The Inheritance and Generalizability Properties Extended From Function Definitions into Functionals

Chapter 4. Study of Some New Type of Functionals Defined Based the Inheritance and Generalizability Properties of Functions

Chapter 5. Capability of Function Optimization Algorithms for Solving Optimal Control Problems With Respect to the Inheritance and Generalizability Properties

Chapter 6. A Generalized Version of Cutting Angle Method for Solving Continuous Time Optimal Control Problems

Chapter 7. A Combination of the Cutting Angle Method and a Local Search on Optimal Control Problems

Index

Chapter 1

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[2] Shang, Y. (1997). “Global search methods for solving nonlinear optimization problems,” University of Illinois at Urbana-Champaign.
[3] Beale, E. M. L. (1988). Introduction to optimization: Wiley-Interscience.
[4] Bundy, B. (1984). “Basic optimization methods”, Edward Arnold.
[5] Cesari, L. (1983). Optimization—theory and applications: Springer.
[6] Koo, D. (1977). Elements of optimization: Springer.
[7] Rao, S. “Optimization: theory and applications. 1984,” ed: Wiley, New York.
[8] Stoer, J. & Witzgall, C. (1970). Convexity and optimization in finite dimensions: Springer.
[9] Rubinov, A. M. (2000). Abstract convexity and global optimization, vol. 44, Springer.
[10] Kurdila, A. J. & Zabarankin, M. (2005). “Convex Functional Analysis, Systems and Control: Foundations and Applications,” ed: Birkhäuser, Basel.
[11] Nocedal, J. & Wright, S. J. (2006). “Numerical Optimization 2nd”.
[12] Rubinov, A. & Glover, B. (1999). “Increasing convex-along-rays functions with applications to global optimization”, Journal of Optimization Theory and Applications, 102(3), 615-642.
[13] Singh, M. G. & Titli, A. (1987). Systems: decomposition, optimisation, and control: Pergamon.
[14] Becerra, V. M. (1994). “Development and applications of novel optimal control algorithms,” City University.
[15] Roberts, P. (1993). “An algorithm for optimal control of nonlinear systems with model-reality differences”, in Proceedings of 12th IFAC World Congress on Automatic Control, 407-412.
[16] Bagirov, A. & Rubinov, A. (2000). “Global minimization of increasing positively homogeneous functions over the unit simplex”, Annals of Operations Research, 98(1-4), 171-187.

Chapter 2

[1] Martinez-Legaz, J. & Rubinov, A. (2001). “Increasing positively homogeneous functions defined on Rn”, Acta Math. Vietnam, 26(3), 313-331.
[2] Fletcher, R. (2013). Practical methods of optimization: John Wiley & Sons.
[3] Rao, S. “Optimization: theory and applications. 1984,” ed: Wiley, New York.
[4] Baldi, P. (1995). “Gradient descent learning algorithm overview: A general dynamical systems perspective”, Neural Networks, IEEE Transactions on, 6(1), 182-195.
[5] Beale, E. M. L. (1988). Introduction to optimization: Wiley-Interscience.
[6] Bundy, B. (1984). “Basic optimization methods”, Edward Arnold.
[7] Cesari, L. (1983). Optimization—theory and applications: Springer.
[8] Stoer, J. & Witzgall, C. (1970). Convexity and optimization in finite dimensions: Springer.
[9] Hansen, P. & Jaumard, B. (1995). Lipschitz optimization: Springer.
[10] Mladineo, R. H. (1986). “An algorithm for finding the global maximum of a multimodal, multivariate function”, Mathematical Programming, 34(2), 188-200.
[11] Bagirov, A. M. & Rubinov, A. M. (2003). “Cutting angle method and a local search”, Journal of Global Optimization, 27(2-3), 193-213.
[12] Siouris, G. M. (1996). An engineering approach to optimal control and estimation theory: John Wiley & Sons, Inc.
[13] Hocking, L. (2001). “Optimal control. An introduction to the theory with applications.” Clarendon, ed: Oxford.
[14] Brdyś, M. & Roberts, P. (1987). “Convergence and optimality of modified two-step algorithm for integrated system optimization and parameter estimation”, International journal of systems science, 18(7), 1305-1322.
[15] Ellis, J. & Roberts, P. (1981). “Simple models for integrated optimization and parameter estimation”, International Journal of Systems Science, 12(4), 455-472.
[16] Roberts, P. & Williams, T. (1981). “On an algorithm for combined system optimisation and parameter estimation”, Automatica, 17(1), 199-209.
[17] Stevenson, I., Brdys, M. & Roberts, P. (1985). “Integrated system optimization and parameter estimation for travelling load furnace control”, in Proceedings of 7th IFAC Symposium on Identification and Parameter Estimation, 1641-1646.
[18] Becerra, V. & Roberts, P. (1996). “Dynamic integrated system optimization and parameter estimation for discrete time optimal control of nonlinear systems”, International Journal of Control, 63(2), 257-281.
[19] Roberts, P. (1993). “An algorithm for optimal control of nonlinear systems with model-reality differences”, in Proceedings of 12th IFAC World Congress on Automatic Control, 407-412.
[20] Becerra, V. & Roberts, P. (1998). “Application of a novel optimal control algorithm to a benchmark fed-batch fermentation process”, Transactions of the Institute of Measurement and Control, 20(1), 11-18.

Chapter 3

[1] Munkres, J. R. (2000). “Topology Prentice Hall”, Upper Saddle River, NJ.
[2] Barbu, V. & Precupanu, T. (2012). Convexity and optimization in Banach spaces: Springer.
[3] Roekafellar, R. (1970). “Convex analysis”, Princeton.
[4] Kudryavtsev, L. (2002). “Homogeneous function”, Encyclopaedia of mathematics. Springer, Berlin.
[5] Kurdila, A. J. & Zabarankin, M. (2005). “Convex Functional Analysis, Systems and Control: Foundations and Applications,” ed: Birkhäuser, Basel.
[6] Sloughter, D. (2001). “The Calculus of Functions of Several Variables,” ed: Furman University, 260p.
[7] Boyd, S. P., El Ghaoui, L., Feron, E. & Balakrishnan, V. (1994). Linear matrix inequalities in system and control theory, vol. 15, SIAM.
[8] Bittner, L. (1974). “On moment equations and inequalities”, Lithuanian Mathematical Journal, 14(2), 178-185.
[9] Castagnoli, E. & Maccheroni, F. (2000). “Restricting independence to convex cones”, Journal of Mathematical Economics, 34(2), 215-223.

Chapter 4

[1] Bylinski, C. (1989). “Some basic properties of sets”, Journal of Formalized Mathematics, 1(198), 9.
[2] Hryniewiecki, K. (1990). “Basic properties of real numbers”, Formalized Mathematics, 1(1), 35-40.
[3] Kotowicz, J. & Sakai, Y. (1992). “Properties of partial functions from a domain to the set of real numbers”, Formalized Mathematics, 3(2), 279-288.
[4] Nowak, B. & Trybulec, A. (1993). “Hahn-Banach theorem”, Journal of Formalized Mathematics, 5(199), 3.
[5] Hamel, A. (2004). “From real to set-valued coherent risk measures”, Reports of the Institute of Optimization and Stochastics, Martin-Luther-University Halle-Wittenberg, 19, 10-22.
[6] Anger, B. (1977). “Representation of capacities”, Mathematische Annalen, 229(3), 245-258.
[7] Kurdila, A. J. & Zabarankin, M. (2005). “Convex Functional Analysis, Systems and Control: Foundations and Applications,” ed: Birkhäuser, Basel.
[8] Koliha, J. J. & Leung, A. (1975). “Ergodic families of affine operators”, Mathematische Annalen, 216(3), 273-284.
[9] Rubinov, A. M. (2000). Abstract convexity and global optimization vol. 44, Springer.

Chapter 5

[1] Becerra, V. M. (2008). “Optimal control”, Scholarpedia, 3(1), 5354.
[2] Seyedi, S., Ahmad, R. & Aziz, M. I. A. (2009). “Inheritance of Function Properties for Functionals”.
[3] Kazemi, M. & Miri, M. (1993). “Numerical solution of optimal control problems”, in Southeastcon’93, Proceedings, IEEE, 2 p.
[4] Fard, O. S. & Borzabadi, A. H. (2007). “Optimal control problem, quasi-assignment problem and genetic algorithm”, in Proceedings of World Academy of Science, Engineering and Technology, 70-43.

Chapter 6

[1] Singh, M. G. & Titli, A. (1978). Systems: decomposition, optimisation, and control: Pergamon.
[2] Becerra, V. M. (1994). “Development and applications of novel optimal control algorithms,” City University.
[3] Roberts, P. (1993), “An algorithm for optimal control of nonlinear systems with model-reality differences”, in Proceedings of 12th IFAC World Congress on Automatic Control, 407-412.
[4] Leithold, L. (1971). The Calculus Book.
[5] Fard, O. S. & Borzabadi, A. H. (2007). “Optimal control problem, quasi-assignment problem and genetic algorithm”, in Proceedings of World Academy of Science, Engineering and Technology, 70-43.

Chapter 7

[1] Bagirov, A. & Rubinov, A. (2000). “Global minimization of increasing positively homogeneous functions over the unit simplex”, Annals of Operations Research, 98(1-4), 171-187.
[2] Bagirov, A. M. & Rubinov, A. M. (2003). “Cutting angle method and a local search”, Journal of Global Optimization, 27(2-3), 193-213.
[3] Roberts, P. (1999). “Stability properties of an iterative optimal control algorithm”, in Proceedings of 14th IFAC World Congress on Automatic Control, 269-274.
[4] Bryson, Jr. A. E. (1996) “Optimal control-1950 to 1985”, Control Systems, IEEE, 16(3), 26-33.

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