Control and Optimization
Mehrnoosh Salehi Chegeni; Majid Yarahmadi
Abstract
Optimal control of certain singularly perturbed systems, with slow and fast dynamics, presents notable challenges, including ill-conditioning, high dimensionality, and ill-posed algebraic Riccati equations. In this paper, we introduce a novel ...
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Optimal control of certain singularly perturbed systems, with slow and fast dynamics, presents notable challenges, including ill-conditioning, high dimensionality, and ill-posed algebraic Riccati equations. In this paper, we introduce a novel inverse optimal control method based on the eigenvalue assignment approach to address these issues. The proposed method optimizes the objective function while ensuring system stability through the strategic placement of eigenvalues in the singular perturbed closed-loop system. To facilitate analysis and support the implementation, a new theorem is proved, and a corresponding algorithm is developed. The proposed algorithm is free of ill-conditioned numerical problems, making it more robust in terms of numerical diffusion and perturbation measurement. Finally, two simulation examples are presented to illustrate the advantages of the proposed method, demonstrating improvement in controller robustness, substantial reductions in cost functions, and decreased control amplitudes.
Control and Optimization
Afrah Kadhim Saud Al-tameemi; Mahmoud Mahmoudi; Majid Darehmiraki
Abstract
This study introduces an innovative approach for addressing optimal control problems related to parabolic partial differential equations (PDEs) through the application of rational radial basis functions (RBFs). Parabolic PDEs, which are instrumental in modeling time-dependent processes such as heat transfer ...
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This study introduces an innovative approach for addressing optimal control problems related to parabolic partial differential equations (PDEs) through the application of rational radial basis functions (RBFs). Parabolic PDEs, which are instrumental in modeling time-dependent processes such as heat transfer and diffusion, pose significant computational challenges in optimal control due to the requirement for precise approximations of both state and adjoint equations. The proposed approach exploits the adaptability and spectral accuracy of rational RBFs within a meshless framework, effectively addressing the limitations of traditional discretization methods. By enhancing the accuracy and efficiency of control strategies, this method significantly contributes to advancing the theory and application of optimal control in dynamic systems. The tunable shape parameters of rational RBFs allow for accurate representation of solution characteristics, including steep gradients and localized behaviors. Additionally, their meshless framework adeptly accommodates complex geometries and boundary conditions, ensuring computational efficiency through the generation of sparse and well-conditioned system matrices. This paper also introduces a novel hybrid rational RBF, termed the Gaussian rational hybrid RBF. The efficacy of the proposed approach is validated through a series of benchmark tests and practical applications, highlighting its ability to achieve high accuracy with reduced computational effort. The findings illustrate the potential of rational RBFs as a robust and versatile tool for solving optimal control problems governed by parabolic PDEs, paving the way for further exploration of advanced rational RBF-based techniques in the field of computational optimal control.
Control and Optimization
Akbar Hashemi Borzabadi; Mohammad Gholami Baladezaei; Morteza Ghachpazan
Abstract
This paper explores the advantages of Sub-ODE strategy in deriving near-exact solutions for a class of linear and nonlinear optimal control problems (OCPs) that can be transformed into nonlinear partial differential equations (PDEs). Recognizing that converting an OCP into ...
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This paper explores the advantages of Sub-ODE strategy in deriving near-exact solutions for a class of linear and nonlinear optimal control problems (OCPs) that can be transformed into nonlinear partial differential equations (PDEs). Recognizing that converting an OCP into differential equations typically increases the complexity by adding constraints, we adopt the Sub-ODE method, as a direct method, thereby negating the need for such transformations to extract near exact solutions. A key advantage of this method is its ability to produce control and state functions that closely resemble the explicit forms of optimal control and state functions. We present results that demonstrate the efficacy of this method through several numerical examples, comparing its performance to various other approaches, thereby illustrating its capability to achieve near-exact solutions.
Control and Optimization
Masoomeh Ebrahimipour; Saeed Nezhadhosein; Seyed Mehdi Mirhosseini-Alizamini
Abstract
This paper presents an optimal robust adaptive technique for controlling a certain class of uncertain nonlinear affine systems. The proposed approach combines sliding mode control, a linear quadratic regulator for optimality, and gradient descent as an adaptive controller. ...
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This paper presents an optimal robust adaptive technique for controlling a certain class of uncertain nonlinear affine systems. The proposed approach combines sliding mode control, a linear quadratic regulator for optimality, and gradient descent as an adaptive controller. The convergence of the sliding mode control process is proven using two theorems based on the Lyapunov function. Simulation results for pendulum and inverted pendulum systems demonstrate that the proposed method outperforms both the linear quadratic regulator technique and the sliding mode control regarding reduced chattering and improved reaching time.
Control and Optimization
Zeinab Barary; AllahBakhsh Yazdani Cherati; Somayeh Nemati
Abstract
This paper proposes and analyzes an applicable approach for numerically computing the solution of fractional optimal control-affine problems. The fractional derivative in the problem is considered in the sense of Caputo. The approach is based on a fractional-order hybrid of block-pulse functions and ...
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This paper proposes and analyzes an applicable approach for numerically computing the solution of fractional optimal control-affine problems. The fractional derivative in the problem is considered in the sense of Caputo. The approach is based on a fractional-order hybrid of block-pulse functions and Jacobi polynomials. First, the corresponding Riemann-Liouville fractional integral operator of the introduced basis functions is calculated. Then, an approximation of the fractional derivative of the unknown state function is obtained by considering an approximation in terms of these basis functions. Next, using the dynamical system and applying the fractional integral operator, an approximation of the unknown control function is obtained based on the given approximations of the state function and its derivatives. Subsequently, all the given approximations are substituted into the performance index. Finally, the optimality conditions transform the problem into a system of algebraic equations. An error upper bound of the approximation of a function based on the fractional hybrid functions is provided. The method is applied to several numerical examples, and the experimental results confirm the efficiency and capability of the method. Furthermore, they demonstrate a good agreement between the approximate and exact solutions.
Control and Optimization
seyed mehdy shafiof; Javad Askari; Maryam Shams Solary
Abstract
In this paper, a modern method is presented to solve a class of fractional optimal control problems (FOCPs) indirectly. First, the necessary optimality conditions for the FOCP are obtained in the form of two fractional differential equations (FDEs). Then, ...
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In this paper, a modern method is presented to solve a class of fractional optimal control problems (FOCPs) indirectly. First, the necessary optimality conditions for the FOCP are obtained in the form of two fractional differential equations (FDEs). Then, the unknown functions are approximated by the hybrid functions, including Bernoulli polynomials and Block-pulse functions based on the spectral Ritz method. Also, two new methods are proposed for calculating the left Caputo fractional derivative and right Riemann-Liouville fractional derivative operators of the hybrid functions that are proportional to the Ritz method. The FOCP is converted into a system of the algebraic equations by applying the fractional derivative operators and collocation method, which determines the solution of the problem. Error estimates for the hybrid function approximation, fractional operators and, the proposed method are provided. Finally, the efficiency of the proposed method and its accuracy in obtaining optimal solutions are shown by some test problems.
Control and Optimization
Saeed Nezhadhosein
Volume 2, Issue 2 , December 2017, , Pages 1-14
Abstract
In this paper, Haar wavelets are performed for solving continuous time-variant linear-quadratic optimal control problems. Firstly, using necessary conditions for optimality, the problem is changed into a two-boundary value problem (TBVP). Next, ...
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In this paper, Haar wavelets are performed for solving continuous time-variant linear-quadratic optimal control problems. Firstly, using necessary conditions for optimality, the problem is changed into a two-boundary value problem (TBVP). Next, Haar wavelets are applied for converting the TBVP, as a system of differential equations, in to a system of matrix algebraic equations, as Haar matrix equations using Kronecker product. Then the error analysis of the proposed method is presented. Some numerical examples are given to demonstrate the efficiency of the method. The solutions converge as the number of approximate terms increase.
Control and Optimization
Zahra Rafiei; Behzad Kafash; Seyyed Mehdi Karbassi
Volume 2, Issue 1 , April 2017, , Pages 1-13
Abstract
In order to obtain a solution to an optimal control problem, a numerical technique based on state-control parameterization method is presented. This method can be facilitated by the computation of performance index and state equation via approximating the control and state variable ...
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In order to obtain a solution to an optimal control problem, a numerical technique based on state-control parameterization method is presented. This method can be facilitated by the computation of performance index and state equation via approximating the control and state variable as a function of time. Several numerical examples are presented to confirm the analytical findings and illustrate the efficiency of the proposed method.
Control and Optimization
Seyed Mehdi Mirhosseini-Alizamini
Volume 2, Issue 1 , April 2017, , Pages 77-91
Abstract
The controlled harmonic oscillator with retarded damping, is an important class of optimal control problems which has an important role in oscillating phenomena in nonlinear engineering systems. In this paper, to solve this problem, we presented an analytical ...
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The controlled harmonic oscillator with retarded damping, is an important class of optimal control problems which has an important role in oscillating phenomena in nonlinear engineering systems. In this paper, to solve this problem, we presented an analytical method. This approach is based on the homotopy perturbation method. The solution procedure becomes easier, simpler and more straightforward. In order to use the proposed method, a control design algorithm with low computational complexity is presented. Through the finite iterations of the proposed algorithm, a suboptimal control law is obtained for the problems. Finally, the obtained results have been compared with the exact solution of the controlled harmonic oscillator and variational iteration method, so that the high accuracy of the results is clear.
Control and Optimization
Akbar Hashemi Borzabadi; Manije Hasanabadi; Navid Sadjadi
Volume 1, Issue 1 , April 2016, , Pages 1-19
Abstract
In this paper an approach based on evolutionary algorithms to find Pareto optimal pair of state and control for multi-objective optimal control problems (MOOCP)'s is introduced. In this approach, first a discretized form of the time-control space is considered and then, a ...
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In this paper an approach based on evolutionary algorithms to find Pareto optimal pair of state and control for multi-objective optimal control problems (MOOCP)'s is introduced. In this approach, first a discretized form of the time-control space is considered and then, a piecewise linear control and a piecewise linear trajectory are obtained from the discretized time-control space using a numerical method. To do that, a modified version of two famous evolutionary genetic algorithm (GA) and particle swarm optimization (PSO) to obtain Pareto optimal solutions of the problem is employed. Numerical examples are presented to show the efficiency of the given approach.
