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An approximate method for solving a class of nonlinear optimal control problems

optimal control APPLICATIONS & METHODS 2014年第3期35卷 324-339页

作者： Nik, H. Saberi Effati, S. Ferdowsi Univ Mashhad Sch Math Sci Dept Appl Math Mashhad Iran Ferdowsi Univ Mashhad Ctr Excellence Soft Comp & Intelligent Informat P Mashhad Iran

This paper presents a novel computational approach to generate the suboptimal solutions for a class of nonlinear optimal control problems (OCP's) with a quadratic performance index. Our method is based on the one-dimensional differential transform method (DTM) and new polynomials that are called DT's polynomials. This method simplifies the difficulties and massive computational work for calculating the differential transform of nonlinear function. The convergence of proposed method are discussed in detail. This method consists of a new modified version of the DTM together with a shooting method such as procedure, for solving the extreme conditions obtained from the Pontryagin's maximum principle. The results reveal that the proposed methods are very effective and simple. Comparisons are made between new DTM generated results, results from literature, and MATLAB bvp4c generated results, and good agreement is observed. Copyright (c) 2013 John Wiley & Sons, Ltd.

关键词： Pontryagin's maximum principle nonlinear optimal control problems DT's polynomials differential transform method

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The Group-Theoretical Analysis of nonlinear optimal control problems with Hamiltonian Formalism

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JOURNAL OF nonlinear MATHEMATICAL PHYSICS 2020年第1期27卷 106-129页

作者： Polat, Gulden Gun Ozer, Teoman Istanbul Kultur Univ Dept Math & Comp Sci TR-34156 Istanbul Turkey Istanbul Tech Univ Dept Civil Engn TR-34469 Istanbul Turkey

In this study, we pay attention to novel explicit closed-form solutions of optimal control problems in economic growth models described by Hamiltonian formalism by utilizing mathematical approaches based on the theory of Lie groups. For this analysis, the Hamiltonian functions, which are used to define an optimal control problem, are considered in two different types, namely, the current and present value Hamiltonians. Furthermore, the first-order conditions (FOCs) that deal with Pontrygain maximum principle satisfying both Hamiltonian functions are considered. FOCs for optimal control in the problem are studied here to deal with the first-order coupled systems. This study mainly focuses on the analysis of these systems concerning for to the theory of symmetry groups and related analytical approaches. First, Lie point symmetries of the first-order coupled systems are derived, and then by using the relationships between symmetries and Jacobi last multiplier method, the first integrals and corresponding invariant solutions for two different economic models are investigated. Additionally, the solutions of initial-value problems based on the transversality conditions in the optimal control theory of economic growth models are analyzed.

关键词： nonlinear optimal control problems theory of Lie groups dynamic optimization Hamiltonian formalism Pontryagin?s maximum principle economic growth models

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Fitness Dependent Optimizer Based Computational Technique for Solving optimal control problems of nonlinear Dynamical Systems

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IEEE ACCESS 2023年 11卷 38485-38501页

作者： Laghari, Ghulam Fareed Malik, Suheel Abdullah Khan, Irfan Ahmed Daraz, Amil AlQahtani, Salman A. Ullah, Hayat Int Islamic Univ Islamabad IIUI Fac Engn & Technol Dept Elect & Comp Engn Islamabad Pakistan Univ Malaya Fac Engn Dept Elect Engn Kuala Lumpur Malaysia NingboTech Univ Sch Informat Sci & Engn Ningbo Peoples R China King Saud Univ Coll Comp & Informat Sci Dept Comp Engn Riyadh Saudi Arabia NYU Tandon Sch Engn Dept Elect & Comp Engn New York NY USA

This paper presents a pragmatic approach established on the hybridization of nature-inspired optimization algorithms and Bernstein Polynomials (BPs), achieving the optimum numeric solution for nonlinear optimal control problems (NOCPs) of dynamical systems. The approximated solution for NOCPs is obtained by the linear combination of BPs with unknown parameters. The unknown parameters are evaluated by the conversion of NOCP to an error minimization problem and the formulation of an objective function. The Fitness Dependent Optimizer (FDO) and Genetic Algorithm (GA) are used to solve the objective function, and subsequently the optimal values of unknown parameters and the optimum solution of NOCP are attained. The efficacy of the proposed technique is assessed on three real-world NOCPs, including Van der Pol (VDP) oscillator problem, Chemical Reactor Problem (CRP), and Continuous Stirred-Tank Chemical Reactor Problem (CSTCRP). The final results and statistical outcomes suggest that the proposed technique generates a better solution and surpasses the recently represented methods in the literature, which eventually verifies the efficiency and credibility of the recommended approach.

关键词： Optimization Convergence Approximation algorithms optimal control Linear programming Artificial neural networks Search problems Bernstein polynomials dynamical systems fitness dependent optimizer genetic algorithm nonlinear optimal control problems optimization problem optimization techniques

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Iterative Chebyshev approximation method for optimal control problems

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ISA TRANSACTIONS 2024年 152卷 277-289页

作者： Wu, Di Yu, Changjun Wang, Hailing Bai, Yanqin Teo, Kok-Lay Toh, Kim-Chuan Shanghai Univ Dept Math Shanghai 200444 Peoples R China Sunway Univ Sch Math Sci Selangor Malaysia Natl Univ Singapore Dept Math 10 Lower Kent Ridge Rd Singapore 119076 Singapore Natl Univ Singapore Inst Operat Res & Analyt 10 Lower Kent Ridge Rd Singapore 119076 Singapore

We present a novel numerical approach for solving nonlinear constrained optimal control problems (NCOCPs). Instead of directly solving the NCOCPs, we start by linearizing the constraints and dynamic system, which results in a sequence of sub-problems. For each sub-problem, we use finite number of Chebyshev polynomials to estimate the control and state vectors. To eliminate the errors at non-collocation points caused by conventional collocation methods, we additionally estimate the coefficient functions involved in the linear constraints and dynamic system by Chebyshev polynomials. By leveraging the characteristics of Chebyshev polynomials, the approximate sub-problem is changed into an equivalent nonlinear optimization problem with linear equality constraints. Consequently, any feasible point of the approximate sub-problem will satisfy the constraints and dynamic system throughout the entire time scale. To validate the efficacy of the new method, we solve three examples and assess the accuracy of the method through the computation of its approximation error. Numerical results obtained show that our approach achieves lower approximation error when compared to the Chebyshev pseudo-spectral method. The proposed method is particularly suitable for scenarios that require high-precision approximation, such as aerospace and precision instrument production.

关键词： nonlinear optimal control problems Linearization Chebyshev polynomials Global convergence

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Numerical solution of some initial optimal control problems using the reproducing kernel Hilbert space technique

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INTERNATIONAL JOURNAL OF control 2020年第6期93卷 1345-1352页

作者： Isfahani, F. Toutian Mokhtari, R. Loghmani, G. B. Mohammadi, M. Yazd Univ Dept Math Yazd 89195741 Iran Isfahan Univ Technol Dept Math Sci Esfahan Iran Kharazmi Univ Dept Math Sci & Comp Tehran Iran

In this paper, we present a general technique for solving a class of linear/nonlinear optimal control problems. In fact, an analytical solution of the state variable is represented in the form of a series in a reproducing kernel Hilbert space. Sometimes with the aid of this series form, we can also present the optimal control variable in a series form. An iterative method is given to obtain the approximate optimal control and state variables and the cost functional is numerically obtained. Convergence analysis of the method is also provided. Several numerical examples are tested to demonstrate the applicability and efficiency of the method.

关键词： Linear nonlinear optimal control problems reproducing kernel Hilbert space

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A new operational matrix based on Boubaker wavelet for solving optimal control problems of arbitrary order

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TRANSACTIONS OF THE INSTITUTE OF MEASUREMENT AND control 2020年第10期42卷 1858-1870页

作者： Rabiei, Kobra Ordokhani, Yadollah Alzahra Univ Fac Math Sci Dept Math Vanak St Tehran *** Iran

This paper presents numerical solution for solving the nonlinear one and two-dimensional optimal control problems of arbitrary order. First, we have constructed Boubaker wavelet for the first time and defined a general formulation for its fractional derivative operational matrix. To solve the one-dimensional problem, we have transformed the problems into an optimization one. The similar process together with the Ritz method are applied to find a solution for two-dimensional problems as well. Then, the necessary conditions of optimality result in a system of algebraic equations with unknown coefficients and then control parameters can be simply solved. The error vector is considered to show the convergence of the used approximation in this method. Finally, some illustrative examples are given to demonstrate accuracy and efficiency of the proposed method.

关键词： Boubaker wavelet nonlinear optimal control problems two-dimensional optimal control problems Caputo derivative operational matrix convergence analysis

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A Numerical Algorithm for optimal control of Systems with Parameter Uncertainty

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IFAC-PapersOnLine 2016年第18期49卷 468-475页

作者： Walton, Claire Phelps, Chris Gong, Qi Kaminer, Isaac Naval Postgraduate School MontereyCA93943 United States University of California Santa Cruz Santa CruzCA95060 United States

This paper describes a numerical scheme for computing optimal solutions to a class of nonlinear optimal control problems in which parameter uncertainty may be a feature of the state dynamics or objective function. Consistency results are provided for states and controls generated by the algorithm as well as for the adjoint variables. © 2016

关键词： Parameter estimation Numerical methods optimal control systems Consistency result Non linear control nonlinear optimal control problems Numerical algorithms Objective functions optimal controls optimal solutions Parameter uncertainty

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Energy-efficient communication in mobile aerial relay-assisted networks using predictive control⁎

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IFAC-PapersOnLine 2018年第20期51卷 197-202页

作者： Faqir, Omar J. Nie, Yuanbo Kerrigan, Eric C. Gündüz, Deniz Department of Electrical & Electronic Engineering Imperial College London LondonSW7 2AZ United Kingdom Department of Aeronautics Imperial College London LondonSW7 2AZ United Kingdom

Energy-efficient communication in wireless networks of mobile autonomous agents mandates joint optimization of both transmission and propulsion energy. In Faqir et al. (2017) we developed communication-theoretic data transmission and Newtonian flight mechanics models to formulate a nonlinear optimal control problem. Here we extend the previous work by generalizing the communication model to include UAV-appropriate slow fading channels and specifically investigate the potential from joint optimization of mobility and communication over a multiple access channel. Numerical results exemplify the potential energy savings available to all nodes through this joint optimization. Finally, using the slow fading channel problem formulation, we generate a chance-constrained nonlinear model predictive control scheme for control of a terrestrial network served by a single UAV relay. Closed-loop simulations are performed subject to uncertainties in both transmission and mobility models. © 2018

关键词： Energy efficiency Autonomous agents Channel capacity Energy conservation Fading channels Mobile agents Mobile antennas Model predictive control optimal control systems Potential energy Predictive control systems Telecommunication networks Unmanned aerial vehicles (UAV) Autonomous Mobile Robot Energy efficient communications Mobile autonomous agents Model based controls Multiple access channels Non linear control nonlinear optimal control problems optimal controls

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PID2018 Benchmark Challenge: Model Predictive control With Conditional Integral control Using A General Purpose optimal control Problem Solver – RIOTS.

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IFAC-PapersOnLine 2018年第4期51卷 882-887页

作者： Dehghan, Sina T., Zhao Y., Zhao J., Yuan A., Ates Y., Chen University of California Merced MercedCA95340 United States School of Control Science and Engineering Shandong University JinanShandong250061 China School of Control Science and Engineering Southeast University Nanjing China Engineering Faculty Computer Engineering Department Inonu University Malatya44280 Turkey

This paper presents a multi-variable Model Predictive control (MPC) based controller for the one-staged refrigeration cycle model described in the PID2018 Benchmark Challenge. This model represents a two-input, two-output system with strong nonlinearities and high coupling between its variables. A general purpose optimal control problem (OCP) solver Matlab toolbox called RIOTS is used as the OCP solver for the proposed MPC scheme which allows for straightforward implementation of the method and for solving a wide range of constrained linear and nonlinear optimal control problems. A conditional integral (CI) compensator is embedded in the controller to compensate for the small steady state errors. This method shows significant improvements in performance compared to both discrete decentralized control (C1) and multi-variable PID controller (C2) originally given in PID2018 Benchmark Challenge as a baseline. Our solution is introduced in detail in this paper and our final results using the overall relative index, J, are 0.2 over C1 and 0.3 over C2, respectively. In other words, we achieved 80% improvement over C1 and 70% improvement over C2. We expect to achieve further improvements when some optimized searching efforts are used for MPC and CI parameter tuning. © 2018

关键词： Model predictive control Benchmarking controllers Decentralized control Multivariable systems optimal control systems Predictive control systems Proportional control systems Refrigeration Three term control systems Multi variable models nonlinear optimal control problems optimal control problem performance improvement PID2018 Benchmark Challenge Refrigeration cycles RIOTS Steady state errors

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Autonomous Driving of a Mobile Robot Using a Combined Multiple-Shooting and Collocation Method

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IFAC-PapersOnLine 2016年第15期49卷 193-198页

作者： Drozdova, Evgeniya Hopfgarten, Siegbert Lazutkin, Evgeny Li, Pu Technische Universität Ilmenau Group Simulation and Optimal Processes P.O. Box 10 05 65 Ilmenau98693 Germany Krasnokazarmennaya 14 Moscow111250 Russia

A nonlinear model predictive control (NMPC) approach for steering an autonomous mobile robot is presented. The vehicle dynamics with a counter steering system is described by a nonlinear bicycle model. The NMPC problem is formulated taking into account the obstacles description as inequality constraints which will be updated at each sampling time based on a laser scanner detection. The nonlinear optimal control problem (NOCP) is efficiently solved by a combined multiple-shooting and collocation method. Experimentation results illustrate the viability of our approach for active autonomous steering in avoiding spontaneous obstacles. © 2016

关键词： Navigation Automobile steering equipment Constraint theory Mobile robots Model predictive control nonlinear systems Obstacle detectors optimal control systems Predictive control systems Autonomous Mobile Robot Autonomous steering Bicycle model Collocation method Inequality constraint nonlinear model predictive control nonlinear optimal control problems Obstacle detection

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