Solution Of Optimal Control Problem Research Articles

Fitness Dependent Optimizer Based Computational Technique for Solving Optimal Control Problems of Nonlinear Dynamical Systems

This paper presents a pragmatic approach established on the hybridization of nature-inspired optimization algorithms and Bernstein Polynomials (BPs), achieving the optimum numeric solution for Nonlinear Optimal Control Problems (NOCPs) of dynamical systems. The approximated solution for NOCPs is obtained by the linear combination of BPs with unknown parameters. The unknown parameters are evaluated by the conversion of NOCP to an error minimization problem and the formulation of an objective function. The Fitness Dependent Optimizer (FDO) and Genetic Algorithm (GA) are used to solve the objective function, and subsequently the optimal values of unknown parameters and the optimum solution of NOCP are attained. The efficacy of the proposed technique is assessed on three real-world NOCPs, including Van der Pol (VDP) oscillator problem, Chemical Reactor Problem (CRP), and Continuous Stirred-Tank Chemical Reactor Problem (CSTCRP). The final results and statistical outcomes suggest that the proposed technique generates a better solution and surpasses the recently represented methods in the literature, which eventually verifies the efficiency and credibility of the recommended approach.

A Neural Network Approach for High-Dimensional Optimal Control Applied to Multiagent Path Finding

We propose a neural network approach that yields approximate solutions for high-dimensional optimal control problems and demonstrate its effectiveness using examples from multi-agent path finding. Our approach yields controls in a feedback form, where the policy function is given by a neural network (NN). Specifically, we fuse the Hamilton-Jacobi-Bellman (HJB) and Pontryagin Maximum Principle (PMP) approaches by parameterizing the value function with an NN. Our approach enables us to obtain approximately optimal controls in real-time without having to solve an optimization problem. Once the policy function is trained, generating a control at a given space-time location takes milliseconds; in contrast, efficient nonlinear programming methods typically perform the same task in seconds. We train the NN offline using the objective function of the control problem and penalty terms that enforce the HJB equations. Therefore, our training algorithm does not involve data generated by another algorithm. By training on a distribution of initial states, we ensure the controls' optimality on a large portion of the state-space. Our grid-free approach scales efficiently to dimensions where grids become impractical or infeasible. We apply our approach to several multi-agent collision-avoidance problems in up to 150 dimensions. Furthermore, we empirically observe that the number of parameters in our approach scales linearly with the dimension of the control problem, thereby mitigating the curse of dimensionality.

Personalized Constrained MPC for glucose regulation

The technological advances reached in the last years overcome the relevant hardware limitations of the past artificial pancreas releasing portable devices with high computational capabilities. In view of this, the use of a Constrained MPC (CMPC) that needs the solution of a finite horizon optimal control problem can be considered. The real advantage of the explicit inclusion of the state constraints in the optimization problem is mainly related to the quality of the model used for the predictions. In a recent work, a CMPC based on an average model has been tested on the 100 in silico patients of the new UVA/Padova simulator, showing satisfactory results in terms of average glucose, time spent in tight range and time above 180 mg/dl. A percentage of time below 70 mg/dl grater than 3% is present for more than 25% of the patients. In this work, a new personalized CMPC is proposed to mitigate the hypoglycemia phenomena for these patients, using personalized impulse-response models for glucose prediction.

Variational Principles for Mirror Descent and Mirror Langevin Dynamics

Mirror descent, introduced by Nemirovski and Yudin in the 1970s, is a primal-dual convex optimization method that can be tailored to the geometry of the optimization problem at hand through the choice of a strongly convex potential function. It arises as a basic primitive in a variety of applications, including large-scale optimization, machine learning, and control. This paper proposes a variational formulation of mirror descent and of its stochastic variant, mirror Langevin dynamics. The main idea, inspired by the classic work of Brezis and Ekeland on variational principles for gradient flows, is to show that mirror descent emerges as a closed-loop solution for a certain optimal control problem, and the Bellman value function is given by the Bregman divergence between the initial condition and the global minimizer of the objective function.

The Numerical Solution for Quadratic Optimal Control Problems by Using Chebyshev and Legendre Polynomials

الغرض من هذا البحث هو حل مشاكل التحكم الأمثل من الدرجة الثانية (QOCP) عدديًا بمساعدة متعددة حدود Chebyshev و Legendre كوظائف أساسية لإيجاد حل للتحكم الأمثل (QOC) تقريبًا. سنشرح خوارزميات الحل بالأمثلة ونستخدم برنامج Mathcad للوصول إلى النتيجة الدقيقة.مقدمةتتمثل مشكلة التحكم المثلى في العثور على عنصر تحكم u ^ * (t) يقلل من مؤشر أداء معين مع تلبية معادلات حالة النظام والقيود. [1]نستخدم طرق التقريب لحل مشكلة التحكم الأمثل اعتمادًا على متعدد حدود Chebyshev في المرة الأولى ومتعدد حدود Legendre ، وبعد ذلك سنقترب من هذه الحلول ذات الوقت الخطي المستمر. للوصول إلى الحلول التقريبية ، نستخدم المعادلات التفاضلية الخطية متعددة المدى لـ u (t) و x (t) لكل من متعدد حدود Chebyshev و Legendre ونجعل شروط هذه المعادلات كمصفوفة مربعة للعثور على هذه القيم بواسطة نظام المصفوفات.عندما نستخدم هذه كثيرات الحدود في حلول تقريبية ، تم تقييم النتائج باستخدام الفهرس مع n = 5.سنشرح هذه الخوارزميات من خلال أخذ بعض الأمثلة لمشاكل التحكم التربيعي.تم تحديد المشكلة التربيعية الخطية على النحو التالي ؛قلل من الوقت المستمر التربيعيدالة التكلفة J = ∫_ (t_0) ^ (t_f) ▒ 〖(x ^ T Qx + u ^ T Ru) □ (24 & dt)〗 ... (1)مع مراعاة معادلات حالة النظام الخطي ؛س ̇ (t) = Dx (t) + Eu (t) ، ... (2)حيث الشرط الأولي x (0) = x_0 والمصفوفات (D و E و Q و R) هي

An Algorithm for Finding Feedback in a Problem with Constraints for One Class of Nonlinear Control Systems

We consider the construction of a feedback according to the Kalman algorithm for a continuous nonlinear control system on a finite time interval with control constraints where the right-hand side of the dynamics equations is linear in control and linearizable in the vicinity of the zero equilibrium position. The solution of an auxiliary optimal control problem with a quadratic functional is used for this task by analogy with the SDRE approach. Because this approach is used in the literature to find suboptimal synthesis in optimal control problems with a quadratic functional with formally linear systems where all coefficient matrices in differential equations and criteria can contain state variables, on a finite time interval it becomes necessary to solve a complicated matrix differential Riccati equations with state-dependent coefficient matrices. Due to the nonlinearity of the system this issue significantly increases the number of calculations for obtaining the coefficients of the gain matrix in the feedback and for obtaining synthesis with a given accuracy in comparison with the Kalman algorithm for linear-quadratic problems. The proposed synthesis construction algorithm is constructed using the extension principle proposed by V.F. Krotov and developed by V.I. Gurman and allows one not only to expand the scope of the SDRE approach to nonlinear control problems with control constraints in the form of closed inequalities, but also to propose a more efficient computational algorithm for finding the matrix of feedback gains in control problems on a finite interval. This article establishes the correctness of the application of the extension principle by introducing analogs of the Lagrange multipliers, which depend on the state and time, and also derives a formula for the suboptimal value of the quality criterion. The presented theoretical results are illustrated by calculating suboptimal feedbacks in the problems of managing three-sector economic systems.

A numerical study on fractional optimal control problems described by Caputo‐Fabrizio fractional integro‐differential equation

AbstractThis paper provides a numerical technique for evaluating the approximate solution of fractional optimal control problems with the Caputo‐Fabrizio (CF) fractional integro‐differential equation. First, due to the Gegenbauer polynomials definition and CF‐fractional derivative, we present the modified operational matrix and complement vector of integration and CF‐fractional derivative. Then, the corresponding discretization of the problem is obtained with the help of the optimization method and the Legendre‐Gauss‐Lobatto (LGL) quadrature rule. The technique of obtaining the proposed matrices and LGL‐quadrature method leads to obtaining the approximate solution with high precision. Moreover, the error of the performance index obtained by the computational method is investigated. Finally, we implement the methodology in several examples. So that, the effect of various parameters defined in the method is illustrated in tables and plots.

Influence of the control strategy on the performance of hybrid polygeneration energy system using a prescient model predictive control

Hybrid Polygeneration Energy Systems (HPES) can be effective solutions to reach COP26 goals. In particular, Combined Heat and Power (CHP) systems can increase the total system efficiency when both electrical and thermal power are required. The integration of a Battery Energy Storage System (BESS) can further improve the plant efficiency and economy, assuring higher operational flexibility. Configuration, sizing and control strategy definition are of primary concern for these systems when the best possible performances are sought. This work aims to quantitatively assess the importance of the adopted control strategy in the operation and performance of a possible sub-system of a grid-connected Hybrid Polygeneration Energy Systems (HPES), consisting in this study of a CHP plant assisted by a BESS. A simulation code of the plant from an energetic point of view was used, and the main economic indicator was also calculated according to the legislative reference scenario. Then a Prescient Model Predictive Control (MPC) was coded to achieve near-optimal plant operation, using a novel system state description to reduce the computational burden linked to the Optimal Control Problem (OCP) solution. The BESS system was modelled, including cycle battery ageing. Subsequently, the performances of the adopted prescient MPC have been compared to the previous results given by a multi-objective plant sizing with a rule-based control. The results show that the control strategy can enhance the performance of the CHP system, achieving remarkable overall better performances, with up to 12% higher Primary Energy Savings. Moreover, the research findings highlight how the proposed control variable ensure a reduction of the computational time by more than 70%, also improving the quality of the found solutions to the OCP. The results also suggest that proper control strategies should be adopted even in a preliminary optimal sizing phase of the CHP plant, since there is a large room of improvement in predicting the achievable plant performance when more basic rule-based control strategies are overcome and replaced by MPC.

A switched system approach for the direct solution of singular optimal control problems

Optimal control problems where the control variable appears linearly in the formulation lead to singular control arcs that introduce analytical and numerical complications during the solution process. In this work, the switching behavior (switching between singular and nonsingular control arcs) is exploited to solve singular optimal control problems using a direct method. The problem is reformulated as a switched system optimal control problem where each control arc defines a different subsystem, i.e., lower and upper limits and a singular arc. The switching between control arcs is handled using binary functions that depend on time. The switched system is embedded into a larger family by relaxing the binary functions to avoid integer variables. The variable switching times at which the control switches between arcs are mapped to fixed points in a new time domain using a time-scaling transformation. A direct transcription method along with control parametrization is applied to transform the embedded problem into a nonlinear programming formulation that contains few variables and constraints. A penalty term is introduced to obtain binary solutions for the optimal selection of control arcs. The proposed reformulation avoids the use of mesh refinement techniques and continuation methods, and allows to solve challenging problems in very short CPU times with high accuracy.

An Efficient Algorithm for the Multi-Scale Solution of Nonlinear Fractional Optimal Control Problems

An efficient algorithm based on the wavelet collocation method is introduced in order to solve nonlinear fractional optimal control problems (FOCPs) with inequality constraints. By using the interpolation properties of Hermite cubic spline functions, we construct an operational matrix of the Caputo fractional derivative for the first time. Using this matrix, we reduce the nonlinear fractional optimal control problem to a nonlinear programming problem that can be solved with some suitable optimization algorithms. Illustrative examples are examined to demonstrate the important features of the new method.

ON RITZ APPROXIMATION FOR A CLASS OF FRACTIONAL OPTIMAL CONTROL PROBLEMS

We apply the Ritz method to approximate the solution of optimal control problems through the use of polynomials. The constraints of the problem take the form of differential equations of fractional order accompanied by the boundary and initial conditions. The ultimate goal of the algorithm is to set up a system of equations whose number matches the unknowns. Computing the unknowns enables us to approximate the solution of the objective function in the form of polynomials.

Error Estimates for Approximate Solutions to Seminlinear Elliptic Optimal Control Problems with Nonlinear and Mixed Constraints

This paper gives some sufficient conditions for convergence of approximate solutions to seminlinear elliptic optimal control problems with mixed pointwise constraints. We build discrete optimal control problems by the finite element method in type of the full control discretization. We show that if the strictly second-order sufficient condition is valid, then some error estimates between approximate solutions of discrete optimal control problems and optimal solutions of the original problem are obtained.

Numerical solution of delay fractional optimal control problems with free terminal time

Numerical solution of delay fractional optimal control problems with free terminal time

On Approximate Solution of Optimal Control Problems by Parabolic Equations

On Approximate Solution of Optimal Control Problems by Parabolic Equations

Linear programming based optimality conditions and approximate solution of a deterministic periodic optimal control problem in discrete time

This paper is focused on a discrete time periodic optimal control problem. We establish linear programming based optimality conditions and present a method for the construction of near optimal controls. A numerical example is included to demonstrate the results.

Numerical solutions of two-dimensional PDE-constrained optimal control problems via bilinear pseudo-spectral method

Numerical solutions of two-dimensional PDE-constrained optimal control problems via bilinear pseudo-spectral method

Use of mathematical modeling for solving tasks of optimal control of an electric drive

The object of study in this article is the control processes of electromechanical systems with an electric drive. The subject matter is a mathematical model in the problem of optimal control of the following electric drive. This article is devoted to the mathematical modeling of the control processes of electromechanical systems with an electric drive. In this paper, we use the approximation of vector-valued functions of one variable by splines of the first degree to solve the problem of rotation of the motor shaft in electromechanical system. As you know, it is not always possible to find exact solutions in optimal control problems. In this regard, there is a need to obtain approximate solutions for optimal control problems. Therefore, an urgent task is the development of new methods for the approximate solution to the problems of optimal control of the electric drive, which provide higher approximation accuracy. The following results are presented: Graphs of phase coordinates and optimal control of approximation by splines of the first order are obtained. The results obtained were compared with the exact solution. The given examples illustrate the high accuracy of approximate solutions to the control problem obtained using the proposed method. Conclusions. The scientific novelty of the results obtained lies in the fact that a method is proposed for an approximate solution to the problem of optimal control of a servo drive using spline functions, in which unknown control parameters are found simultaneously with unknown parameters of phase coordinates, which gives the best approximation to the exact solution.

A numerical method for approximating the solution of fuzzy fractional optimal control problems in caputo sense using legendre functions

Fractional order differential equations accurately model dynamic systems and processes. In some of the fractional optimal control problems (FOCPs), due to the ambiguity in the initial conditions and the transfer of ambiguity to the solution, it is necessary to use fuzzy mathematics. In this paper, a numerical method is presented to approximate the solution for a class of Fuzzy Fractional Optimal Control Problems (FFOCPs) using the Legendre basis functions. The fuzzy fractional derivative is described in the Caputo sense. The performance index of an FFOCP is considered as a function of both the state and the control variables, and the dynamic constraints are expressed by a set of Fuzzy Fractional Differential Equations (FFDEs). After obtaining Euler–Lagrange equations for FFOCPs and the necessary and sufficient conditions for the existence of solutions, using the definition of generalized Hukuhara differentiability (types I, II), the problem is considered in two cases. Then the distance function and an approach similar to the variational type along with the Lagrange multiplier method are used to formulate and solve the equations in a system. Time-invariant and time-varying examples are provided to assess the presented method. Numerical results show a similar trend for the state and control variables for various numbers of Legendre polynomials. Also, the convergence of state and control variables for the time-invariant system can be seen, and the same is true for control variables for the time-varying system.

Parallel solution of optimal control problems using the graphics processing unit

AbstractThis paper presents an indirect method for solving optimal control problems (OCPs) using graphics processing unit (GPU). The OCPs considered here include control variable inequality constraints (CVICs), state variable inequality constraints (SVICs), and parameters. The necessary conditions of the minimum for the OCPs are written as a boundary value problem with index‐1 differential algebraic equations (BVP‐DAEs). The complementarity conditions associated with those inequality constraints are approximated using Kanzow's smoothed Fisher‐Burmeister formula. The numerical solution for solving the BVP‐DAEs is based on the multiple shooting technique and the DAEs are solved using a single step linearly implicit Runge–Kutta (Rosenbrock–Wanner, ROW) method. A Newton's continuation method is performed to solve the BVP‐DAEs system and the descent direction is found by solving a sparse bordered almost block diagonal (BABD) linear system with a structured orthogonal factorization algorithm. Parallel computing techniques are used to accelerate the code which is implemented using Python and CUDA on GPU. Numerical examples are presented to illustrate the efficiency of the code. The GPU based parallel implementation is shown to be significantly faster than the implementation using central processing unit (CPU) alone.

Ritz approximate method for solving delay fractional optimal control problems

In this paper, a numerical method based on the Müntz–Legendre polynomials for solving a class of time-delay fractional optimal control problems (TDFOCPs) is presented. The concept of fractional derivatives is considered in the Caputo sense. The key point of the applied method is that the initial and boundary conditions are imposed upon the approximations of state and control functions. First, the unknown state or control and their delayed functions are approximated by the Müntz–Legendre polynomials basis using the Ritz method then the next one is calculated through the given fractional differential equation. By substituting the estimated values in the cost function, the TDFOCP is converted to an unconstrained linear or nonlinear optimization problem. The convergence of the suggested method is extensively argued and several illustrative examples are presented to show the applicability and effectiveness of the method. It is shown that only a few terms of Müntz–Legendre polynomials are needed for obtaining the high accuracy and satisfactory results. The obtained results are compared with the available results in the literature to demonstrate the superiority of the applied approach. Furthermore, the approximated solutions agree with exact solutions and as α approaches an integer value, the method provides solution for the integer-order optimal control problem.