Solving optimal control problems of the time-delayed systems by Haar wavelet

Nazemi, Alireza; Mansoori, Masoomeh

doi:10.1177/1077546314550698

Cited by 19 publications

(6 citation statements)

References 42 publications

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“…Various computational procedures have been devised for solving this problem, including the hybrid of block-pulse functions and Bernoulli polynomials, 26 the adaptive shifted Legendre-Gauss PS method, 39 the semigroup approximation method, 41 the control parametrization approach, 42 the combined parameter and function optimization algorithm, 43 the linear Legendre multiwavelets, 44 the composite Chebyshev finite-difference scheme, 45 the hybrid of block-pulse functions and Bernoulli polynomials, 46 and the Haar wavelet. 47 In Table 4, the approximate values of the performance index J obtained by the current approach for N = 2 and different values of M are compared with the existing results in the literature. This Table verifies that there is a good agreement between the results obtained by the proposed approximation scheme and those reported in the works of Maleki and Hashim 39 and Marzban and Hoseini.…”

Section: Examplementioning

confidence: 99%

A novel approach for the numerical investigation of optimal control problems containing multiple delays

Marzban

Pirmoradian

2017

Optim Control Appl Methods

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SummaryThis paper deals with the numerical solution of optimal control problems with multiple delays in both state and control variables. A direct approach based on a hybrid of block-pulse functions and Lagrange interpolating polynomials is used to convert the original problem into a mathematical programming one. The resulting optimization problem is then solved numerically by the Lagrange multipliers method. The operational matrix of delay for the presented framework is derived. This matrix plays an imperative role to transfer information between 2 consecutive switching points.

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Section: Examplementioning

confidence: 99%

A novel approach for the numerical investigation of optimal control problems containing multiple delays

Marzban

Pirmoradian

2017

Optim Control Appl Methods

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“…The control of systems with time delay and obtaining their approximate solutions are very important issues in control theory. Optimal control problems with delay arise in many important applications in science and engineering . Some researchers have extended this model to the scope of fractional calculus.…”

Section: Introductionmentioning

confidence: 99%

Fractional Chebyshev functional link neural network‐optimization method for solving delay fractional optimal control problems with Atangana‐Baleanu derivative

Kheyrinataj

Nazemi

2020

Optim Control Appl Methods

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In this article, we propose a higher order neural network, namely the functional link neural network (FLNN), for the model of linear and nonlinear delay fractional optimal control problems (DFOCPs) with mixed control-state constraints. We consider DFOCPs using a new fractional derivative with nonlocal and nonsingular kernel that was recently proposed by Atangana and Baleanu.The derivative possesses more important characteristics that are very useful in modelling. In the proposed method, a fractional Chebyshev FLNN is developed. At the first step, the delay problem is transformed to a nondelay problem, using a Padé approximation. The necessary optimality condition is stated in a form of fractional two-point boundary value problem. By applying the fractional integration by parts and by constructing an error function, we then define an unconstrained minimization problem. In the optimization problem, trial solutions for state, co-state and control functions are utilized where these trial solutions are constructed by using single-layer fractional Chebyshev neural network model. We then minimize the error function using an unconstrained optimization scheme based on the gradient descent algorithm for updating the network parameters (weights and bias) associated with all neurons. To show the effectiveness of the proposed neural network, some numerical results are provided. K E Y W O R D SAtangana-Baleanu derivative, delay fractional optimal control problems, fractional Chebyshev polynomials, functional link neural network, optimization, Padé approximation 1 808 /journal/oca Optim Control Appl Meth. 2020;41:808-832.KHEYRINATAJ and NAZEMI 809 cannot be used in modeling of the complex control systems. To bypass these difficulties, Caputo and Fabrizio proposed a so-called Caputo-Fabrizio derivative based on the exponential kernel 10 instead of the power principle. After that, Atangana and Baleanu (AB) 2 generalized the Caputo-Fabrizio derivative using the Mittag-Leffler function as one of the best candidates among existing kernels which is both nonsingular and nonlocal. The control of systems with time delay and obtaining their approximate solutions are very important issues in control theory. Optimal control problems with delay arise in many important applications in science and engineering. [11][12][13] Some researchers have extended this model to the scope of fractional calculus. From these works, we can mention, Chelyshkov wavelets, 14 Legendre operational method, 15 Müntz-Legendre spectral method 16 , block-pulse function, 17 Bernoulli wavelet 18 , Bernstein polynomials 19 , discretization-based methods, 20,21 and embedding approach. 22 In the above-mentioned works, the functions are approximated locally, using the discretization of domain into the number of finite elements. Despite providing good approximations to the solution of problem, these methods have difficulties. Due to discretization of domain via meshing, these methods need a high computational time when the control problems have high-order dimension. Also, d...

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“…Among direct methods, parametrization technique [17][18][19] is known to minimize decision variables compared to the discretization of the problem [7]. It is worth noting that parametrization relies basically on orthogonal functions or wavelets [20][21][22]; however that tool have been used to solve various other problems of dynamic systems like identification (see [8]), tracking control (see [23]), observer based control (see [24]), or minimum time control (see [25]). The main characteristic of this pseudo-spectral technique is that it allows transforming complex dynamic optimization problems to solving a set of algebraic equations in the least square sense in the linear systems case [26,27] or permits formulating an equivalent nonlinear static programming problem for problems related to nonlinear systems [13,28,29].…”

Section: Introductionmentioning

confidence: 99%

Hybrid Functions Direct Approach and State Feedback Optimal Solutions for a Class of Nonlinear Polynomial Time Delay Systems

Bouafoura

Braïek

2019

Complexity

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The aim of this paper is to determine the optimal open loop solution and a nonlinear delay-dependent state feedback suboptimal control for a class of nonlinear polynomial time delay systems. The proposed method uses a hybrid of block pulse functions and Legendre polynomials as an orthogonal base for system’s states and input expansion. Hence, the complex dynamic optimization problem is then reduced, with the help of operational properties of the hybrid basis and Kronecker tensor product lemmas, to a nonlinear programming problem that could be solved with available NLP solvers. A practical nonlinear feedback controller gains are deduced with respect to a least square formalism based on the optimal open loop control results. Simulation results show efficiency of the proposed numerical optimal approach.

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Solving optimal control problems of the time-delayed systems by Haar wavelet

Cited by 19 publications

References 42 publications

A novel approach for the numerical investigation of optimal control problems containing multiple delays

A novel approach for the numerical investigation of optimal control problems containing multiple delays

Fractional Chebyshev functional link neural network‐optimization method for solving delay fractional optimal control problems with Atangana‐Baleanu derivative

Hybrid Functions Direct Approach and State Feedback Optimal Solutions for a Class of Nonlinear Polynomial Time Delay Systems

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