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
Maha Mohsin Mohammed Ali
Abstract
A core challenge in numerically solving partial differential equations (PDEs) is achieving accuracy without sacrificing computational tractability. Globally supported methods provide high-order precision but at a computational cost that grows prohibitively with problem size, while locally supported schemes ...
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A core challenge in numerically solving partial differential equations (PDEs) is achieving accuracy without sacrificing computational tractability. Globally supported methods provide high-order precision but at a computational cost that grows prohibitively with problem size, while locally supported schemes scale more efficiently, though often with a reduction in solution accuracy. The proposed localized meshless framework resolves this tension through the integration of Radial Basis Function (RBF) approximation with the Partition of Unity (PU) decomposition, with applications to elliptic, convection-diffusion, and Helmholtz boundary value problems and their corresponding PDE-constrained optimal control problems. The domain is partitioned into overlapping subdomains, each accommodating an independent RBF expansion that is subsequently blended by compactly supported PU weight functions. This localization strategy replaces the single ill-conditioned dense system of global RBF collocation with a collection of smaller, well-conditioned local systems, reducing the theoretical computational complexity from O(N3) to O(N log N). Three canonical benchmark problems validate the framework: Poisson’s equation, a singularly perturbed convection-diffusion equation, and the Helmholtz equation on a circular domain, each embedded in a distributed optimal control formulation. The proposed scheme achieves L2-norm errors of order 10-4 at N = 1,000 nodes, outperforming both standard finite difference methods and global RBF collocation by factors of 28 to 49 in accuracy. Optimal control problems are handled through a Lagrange multiplier formulation yielding coupled state and adjoint systems, both discretized within the same RBF-PU architecture. The framework is illustrated through a detailed engineering case study — optimal heat-source regulation in a thermally loaded plate — in which the complete optimality system is derived, physically interpreted, and numerically solved. Numerical stability, current limitations, and directions for future investigation are discussed.
Control Theory & Systems
Mahnaz Abedini; Gorbanali Haghighatdoost; Hossein Kheiri
Abstract
This study develops and analyzes a four-compartment predator–prey model incorporating disease dynamics and optimal control. The first prey population follows logistic growth and is susceptible to infection, whereas the second prey population remains disease-free and exhibits greater competitive ...
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This study develops and analyzes a four-compartment predator–prey model incorporating disease dynamics and optimal control. The first prey population follows logistic growth and is susceptible to infection, whereas the second prey population remains disease-free and exhibits greater competitive ability. The model incorporates an epidemic process within the first prey population and considers two control measures: reducing disease transmission and treating or removing infected individuals. We investigate the resulting system theoretically to assess the effects of these interventions on disease prevalence, prey persistence, predator dynamics, and ecosystem stability. In particular, stability analysis, sensitivity analysis, and optimal control theory are employed to characterize the model dynamics and identify the most influential parameters. The results indicate that the carrying capacity (k), infection rate (γ), and disease-induced mortality rate (µ) play major roles in determining the persistence of infection. Moreover, the optimal isolation (u1) and treatment (u2) strategies substantially reduce disease prevalence, increase the healthy prey population, and promote predator stability compared with the uncontrolled scenario. Numerical simulations further demonstrate the effectiveness of the proposed control strategies in mitigating infection and maintaining ecological balance. These findings demonstrate that the model provides a unified framework for examining the interactions between heterogeneous prey populations, disease transmission, predation, and targeted control interventions.
Control Theory & Systems
Asiyeh Ebrahimzadeh
Abstract
This paper presents a chickenpox-inspired mathematical modeling framework for analyzing varicella-like disease transmission dynamics, using a modified compartmental model that incorporates vaccination, isolation-induced contact reduction, treatment efficacy, and public awareness interventions. The objective ...
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This paper presents a chickenpox-inspired mathematical modeling framework for analyzing varicella-like disease transmission dynamics, using a modified compartmental model that incorporates vaccination, isolation-induced contact reduction, treatment efficacy, and public awareness interventions. The objective is to develop an optimal-control framework and compare scenario-based intervention policies aimed at decreasing infection rates while considering economic and public-health constraints. For scenario comparison, the numerical simulations use fixed representative control levels; they do not numerically solve for time-dependent optimal control trajectories. The research applies Pontryagin’s Maximum Principle to formulate a theoretical optimal control framework, including the most critical intervention measures such as vaccination, isolation-induced contact reduction, treatment efficacy, and public awareness. Sensitivity analysis is utilized to examine the impact of model parameters on disease transmission, focusing on the basic reproduction number ℜ0. Numerical computations under various control regimes reveal that an appropriate combination of vaccination, isolation-induced contact reduction, treatment improvement, and public awareness campaigns can effectively reduce infection rates and provide theoretical insights into community-level intervention planning. The findings are highly beneficial for policymakers by supporting the evaluation of epidemic intervention scenarios and resource allocation decisions in public health administration.
Control Theory & Systems
Zargham Heydari; Hamed Gorginpour; Nariman Heidari
Abstract
High-power multiphase induction motors are attractive for safety-critical and power-intensive applications because of their inherent advantages, including superior fault tolerance, lower per-phase power requirements, and reduced torque pulsation. Under open-phase faults, however, the loss of phase symmetry ...
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High-power multiphase induction motors are attractive for safety-critical and power-intensive applications because of their inherent advantages, including superior fault tolerance, lower per-phase power requirements, and reduced torque pulsation. Under open-phase faults, however, the loss of phase symmetry couples the fundamental and harmonic subspaces, increasing torque ripple and current distortion. This paper develops a hierarchical control framework for a high-power seven-phase induction motor that combines online harmonic-reference optimization with enhanced nonlinear model predictive control (ENMPC). The upper layer estimates the machine state and slowly varying electrical parameters using an Extended Kalman Filter (EKF) and weighted recursive least squares (WRLS), and determines reference components in the third- and fifth-harmonic subspaces. The lower layer directly regulates the fundamental and harmonic current components subject to voltage, current, and terminal-state constraints. An explicit fault-constrained state-space representation is derived for one open phase and two adjacent open phases, and a terminal-set Lyapunov argument is given for nominal recursive feasibility and asymptotic closed-loop stability. MATLAB/Simulink simulations of the 1-MW plant consider a healthy interval, a single-phase open fault at 4 s, and an adjacent two-phase open fault at 8 s. For the reported cases, the compensated hierarchical controller maintains 5.76 kN·m and 4.68 kN·m of effective torque under the single- and two-phase fault conditions, respectively, while the corresponding torque-ripple factors are 4.8% and 8.5%. These results support the proposed architecture as a fault-tolerant control strategy while also defining the present study as simulation-based and motivating hardware-in-the-loop and experimental validation.
Control Theory & Systems
Fidelis Nofertinus Zai; Rian Kurnia; Juan Prihanda Nainggolan
Abstract
In this study, we examine solutions to Optimal Tracking Control (OTC) problems for both Linear Quadratic (LQ) and nonlinear systems. Classical approaches to OTC rely on formulating and solving the Hamilton-Jacobi-Bellman (HJB) equation, which typically requires numerical solutions of the state, co-state, ...
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In this study, we examine solutions to Optimal Tracking Control (OTC) problems for both Linear Quadratic (LQ) and nonlinear systems. Classical approaches to OTC rely on formulating and solving the Hamilton-Jacobi-Bellman (HJB) equation, which typically requires numerical solutions of the state, co-state, and stationary equations using the forward-backward method. Such methods often involve intricate mathematical analysis and substantial computational effort. To address these challenges, we explored the use of Physics Informed Neural Networks (PINN) as an alternative framework for solving OTC problems. The PINN approach is implemented by constructing a problem-specific loss function that directly incorporates the governing dynamics and control objectives. This method is comparatively simpler and more flexible to implement. The performance of PINNs is evaluated through quantitative error analysis and benchmarked against the classical Runge-Kutta (RK) method. A detailed comparison is presented using tabulated error metrics and time-domain plots of absolute errors. Numerical results demonstrate that PINNs achieve lower approximation errors than Runge-Kutta method for both LQ and nonlinear tracking problems, indicating their effectiveness as a viable alternative solution strategy for OTC problems.
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
Maha Mohsin Mohammed Ali; Mahmoud Mahmoudi; Majid Darehmiraki
Abstract
This study addresses the numerical solution of an optimal control problem governed by a fractional convection–reaction–diffusion partial differential equation. The approach utilizes Radial Basis Function–Partition of Unity (RBF-PU) methods combined with the Grünwald-Letnikov approximation ...
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This study addresses the numerical solution of an optimal control problem governed by a fractional convection–reaction–diffusion partial differential equation. The approach utilizes Radial Basis Function–Partition of Unity (RBF-PU) methods combined with the Grünwald-Letnikov approximation for fractional derivatives, which provides a fundamental extension of classical derivatives in fractional calculus. To enhance sparsity in the control strategy, an $L_2$ norm is integrated into the objective function, along with quadratic penalties to reduce deviations from the desired state. This hybrid formulation facilitates the effective management of spatially sparse controllers, relevant in many practical applications. The RBF-PU technique offers a flexible and efficient framework by partitioning the domain into overlapping subregions, applying local RBF approximations, and synthesizing the global solution with compactly supported weight functions. Numerical experiments demonstrate the accuracy and effectiveness of this method.
Control Theory & Systems
Maria Afsharirad
Abstract
This paper presents a hybrid scheme for solving optimal control problems. Discretizing the time interval and assuming a constant control value on each sub-interval transforms the optimal control problem into an assignment problem. To cluster feasible solutions, a novel method is proposed ...
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This paper presents a hybrid scheme for solving optimal control problems. Discretizing the time interval and assuming a constant control value on each sub-interval transforms the optimal control problem into an assignment problem. To cluster feasible solutions, a novel method is proposed in this paper, which applies metaheuristic algorithms—specifically, genetic algorithms and particle swarm optimization—to generate a large number of solutions. Subsequently, the K-means clustering method is employed to classify these solutions into clusters. Enhancing the median of each cluster, using metaheuristic techniques, ultimately results in improved medians. The best median from the final iteration of the algorithm serves as an acceptable solution for the optimal control problem. In some cases, it even succeeds in discovering a new best solution.
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.
Control Theory & Systems
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 Theory & Systems
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 Theory & Systems
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 Theory & Systems
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 Theory & Systems
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 Theory & Systems
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 Theory & Systems
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 Theory & Systems
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 Theory & Systems
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.
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