A Combined Adaptive Control Parametrization and Homotopy Continuation Technique for the Numerical Solution of Bang–bang Optimal Control Problems

Mehrpouya, Mohammad Ali; Shamsi, M.; Razzaghi, Mohsen

doi:10.1017/s1446181114000261

Cited by 4 publications

(3 citation statements)

References 43 publications

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“…In 2013, Tarzi et al [21] presented a numerical orbital averaging method to provide a simple and fast method for obtaining preliminary solutions for a wide range of minimum-fuel low-thrust orbit transfers, and they used the solution of the energy-optimal problem as an initial guess for the minimum-fuel problem. Later, Mehrpouya et al [22] combined the adaptive control parametrization method and the homotopy method to solve the bang-bang optimal control problems, which do not require any assumptions on the control structure and the number of switching points. Several works also incorporated homotopy methods to transfer between ephemeris models.…”

Section: Introductionmentioning

confidence: 99%

Double-Homotopy Method for Solving Optimal Control Problems

Pan

et al. 2016

Journal of Guidance, Control, and Dynamics

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The homotopy method has long served as a useful tool in solving optimal control problems, particularly highly nonlinear and sensitive ones for which good initial guesses are difficult to obtain, such as some of the well-known problems in aerospace trajectory optimization. However, the traditional homotopy method often fails midway: a fact that occasional practitioners are not aware of, and a topic which is rarely investigated in aerospace engineering. This paper first reviews the main reasons why traditional homotopy fails. A new double-homotopy method is developed to address the common failures of the traditional homotopy method. In this approach, the traditional homotopy is employed until it encounters a difficulty and stops moving forward. Another homotopy originally designed for finding multiple roots of nonlinear equations takes over at this point, and it finds a different solution to allow the traditional homotopy to continue on. This process is repeated whenever necessary. The proposed method overcomes some of the frequent difficulties of the traditional homotopy method. Numerical demonstrations in a nonlinear optimal control problem and a three-dimensional low-thrust orbital transfer problem are presented to illustrate the applications of the method.

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Section: Introductionmentioning

confidence: 99%

Double-Homotopy Method for Solving Optimal Control Problems

Pan

et al. 2016

Journal of Guidance, Control, and Dynamics

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“…By these parameters, we can adjust the relative and absolute error tolerances. 59 In our simulations, the error tolerances are set to RelTol = 10 −6 and AbsTol = 10 −8 . These considerations guarantee that the relative and absolute errors in satisfying IVPs are less than 10 −6 and 10 −8 , respectively.…”

Section: Numerical Resultsmentioning

confidence: 99%

An indirect shooting method based on the POD/DEIM technique for distributed optimal control of the wave equation

Sabeh

Shamsi

Navon

2017

Numerical Methods in Fluids

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Summary This paper presents a fast numerical method, based on the indirect shooting method and Proper Orthogonal Decomposition (POD) technique, for solving distributed optimal control of the wave equation. To solve this problem, we consider the first‐order optimality conditions and then by using finite element spatial discretization and shooting strategy, the solution of the optimality conditions is reduced to the solution of a series of initial value problems (IVPs). Generally, these IVPs are high‐order and thus their solution is time‐consuming. To overcome this drawback, we present a POD indirect shooting method, which uses the POD technique to approximate IVPs with smaller ones and faster run times. Moreover, in the presence of the nonlinear term, to reduce the order of the nonlinear calculations, a discrete empirical interpolation method (DEIM) is applied and a POD/DEIM indirect shooting method is developed. We investigate the performance and accuracy of the proposed methods by means of 4 numerical experiments. We show that the presented POD and POD/DEIM indirect shooting methods dramatically reduce the CPU time compared to the full indirect shooting method, whereas there is no significant difference between the accuracy of the reduced and full indirect shooting methods.

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“…The adaptation is built on a multiscale setting involving wavelets. Schlegel et al 41 presented an extension and modification of the work by Binder et al 40 An adaptive CVP method was also introduced in the study of Mehrpouya et al 42 for determining the control structure and the number of switching points using the homotopy continuation technique.…”

Section: Introductionmentioning

confidence: 99%

An Approach to Determining the Number of Time Intervals for Solving Dynamic Optimization Problems

Lazutkin

Geletu

2018

Ind. Eng. Chem. Res.

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To numerically solve a dynamic optimization problem, the model equations need to be discretized over a time horizon. The very first step therefore is to decide the number of time intervals. In principle, the decision is made to achieve a compromise between the numerical accuracy of the discretization and the computation load for solving the discretized optimization problem. However, there have been no comprehensive rules for this purpose. In the context of collocation on finite elements, we propose a novel bilevel approach to evaluate an upper limit of the approximation error by formulating and solving an error maximization problem. In this way, a minimum number of time intervals can be determined a priori, which guarantees a userdefined error tolerance. In addition, the impact of the initial conditions on the maximum approximation error is taken into account so that the determined number of intervals is valid for varying initial conditions and thus can be applied to nonlinear model predictive control. Several case studies are used to demonstrate the efficacy of the proposed approach.

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A Combined Adaptive Control Parametrization and Homotopy Continuation Technique for the Numerical Solution of Bang–bang Optimal Control Problems

Cited by 4 publications

References 43 publications

Double-Homotopy Method for Solving Optimal Control Problems

Double-Homotopy Method for Solving Optimal Control Problems

An indirect shooting method based on the POD/DEIM technique for distributed optimal control of the wave equation

An Approach to Determining the Number of Time Intervals for Solving Dynamic Optimization Problems

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