Control Theory & Systems
Ghania Idiri
Abstract
In this paper, a systematic design approach for the parametric optimization of uncertain dynamical systems with bounded parameter uncertainties is proposed. The methodology proceeds in two stages. In the first stage, the control input is expressed as a finite linear combination of polynomial basis functions, ...
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In this paper, a systematic design approach for the parametric optimization of uncertain dynamical systems with bounded parameter uncertainties is proposed. The methodology proceeds in two stages. In the first stage, the control input is expressed as a finite linear combination of polynomial basis functions, and an approximate analytical solution of the state equation is derived using the Variational Iteration Method, which is a semi-analytical iterative technique that avoids discretization and linearization. Substituting this approximate trajectory into the performance index yields a robust min–max optimization problem parameterized by the control coefficients. In the second stage, the robust optimization problem is converted into a tractable scenario optimization problem by drawing a finite number of independent and identically distributed samples from the uncertainty set. The resulting problem is solved using a genetic algorithm. The effectiveness of the proposed approach is demonstrated through two application examples. The first example concerns an uncertain linear-quadratic regulator, and the second addresses an uncertain nonlinear optimal control problem. Optimal control and state trajectories are provided for a set of samples. In addition, the optimal value of the performance index is reported, showing that this value does not exceed the threshold imposed by the proposed approach. The paper concludes by discussing limitations, including dependence on the accuracy of the Variational Iteration Method approximation and the assumption of a known probability distribution over the uncertainty set, and identifies directions for future research.
Control Theory & Systems
Muhammed Hassanein Al-Hakeem; Mahmoud Mahmoudi; Ahmed Sabah Ahmed Al-Jilawi
Abstract
This paper presents a novel hybrid orthogonal polynomial method for solving optimal control problems governed by fractional parabolic PDEs. By strategically weighting and combining these polynomial bases, the method adaptively leverages their respective strengths to achieve superior approximation properties. ...
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This paper presents a novel hybrid orthogonal polynomial method for solving optimal control problems governed by fractional parabolic PDEs. By strategically weighting and combining these polynomial bases, the method adaptively leverages their respective strengths to achieve superior approximation properties. The proposed approach combines the spectral accuracy of Legendre polynomials, the minimax properties of Chebyshev polynomials, and the flexibility of Jacobi polynomials to create a robust numerical framework. The hybrid orthogonal polynomial method is applied to discretize the fractional parabolic PDEs, and an efficient numerical scheme is developed to solve the resulting optimal control problem. Numerical experiments demonstrate the accuracy, efficiency, and applicability of the proposed approach, showing significant improvements over traditional radial basis function methods. The results highlight the potential of the hybrid orthogonal polynomial method for solving complex optimal control problems in science and engineering.
Control Theory & Systems
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.
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