Solution Of Optimal Control Problem Research Articles

Boundary optimal control of time-space SIR model with nonlinear Robin boundary condition.

A boundary optimal control problem arising in time-space SIR epidemic models is treated. In this work we aim with the control of the flux of infected individuals crossing part of boundary. On the other side of the domain, we suppose a nonlinear boundary condition of third kind: nonlinear Robin boundary condition, this condition models immersing individual crossing this part of the boundary of the domain of study. We give the existence and uniqueness of the solution of both state and optimal control problem ending some numerical tests throughout a simple example.

Numerical solution of time-delay optimal control problems by the operational matrix based on Hartley series

In this paper, a numerical technique based on the Hartley series for solving a class of time-delayed optimal control problems (TDOCPs) is introduced. The main idea is converting such TDOCPs into a system of algebraic equations. Thus, we first expand the state and control variables in terms of the Hartley series with undetermined coefficients. The delay terms in the problem under consideration are expanded in terms of the Hartley series. Applying the operational matrices of the Hartley series including integration, differentiation, dual, product, delay, and substituting the estimated functions into the cost function, the given TDOCP is reduced to a system of algebraic equations to be solved. The convergence of the proposed method is extensively investigated. At last, the precision and applicability of the proposed method is studied through different types of numerical examples.

SENTIMENT ANALYSIS OF KASPI PRODUCT REVIEWS

This article discusses the study of problems of optimal control for electric power systems. The numerical solution of optimal control problems for complex electric power systems using an iterative algorithm is shown. Also considered are issues of solving the optimal control of a nonlinear system of ordinary differential equations in two different cases. The proposed solution methods follow the principle of continuation of extremal problems based on sufficient conditions for optimality of V. F. Krotov. A special case of optimal control problems is considered. Numerical experiments showed sufficient efficiency of the implemented algorithms. The problem of optimal motion control of a two-system electric power system is graphically illustrated in the proposed numerical example.

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

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, we consider the construction of a feedback according to the Kalman algorithm. For this, the solution of an auxiliary optimal control problem with a quadratic functional is used by analogy with the SDRE approach.Since 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, then on a finite time interval it becomes necessary to solve a complicated matrix differential Riccati equations, with state-dependent coefficient matrices. This circumstance, due to the nonlinearity of the system, in comparison with the Kalman algorithm for linear-quadratic problems, 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. The proposed synthesis construction algorithm is constructed using the extension principle proposed by V. F. Krotov and developed by V. I. Gurman and allows 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. The article establishes the correctness of the application of the extension principle by introducing analogs of the Lagrange multipliers, depending 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.

Discussion on Optimality Conditions of Fuzzy Optimal Control Problems

In this study, a novel methodology for detecting fuzzy optimal solutions of fuzzy optimal control problems governed by fuzzy differential equations is presented. For this purpose, first the concept of optimal solution is proposed by considering a partial ordering on fuzzy numbers space which is based on a new parametric representation for α-level set of fuzzy numbers. Using this partial ordering and parametric representation, the discussed fuzzy optimal control problems are reduced to general optimal control problems in parametric form. Then by Pontryagin’s principle, a candidate for the solution of the original problem is derived. The procedure of extracting candidate solution is illustrated for two experimental problems, which improve the recent results of Refs. 10 and 17.

Application of periodic Bernoulli polynomials in approximating the solution of linear time-delay optimal control problems

In this paper, a numerical method based on application of periodic Bernoulli polynomials for solving optimal control of linear time-delay systems with quadratic performance index is presented. Some important properties of Bernoulli polynomials are reviewed and a new operational matrix of delay based on these functions is introduced. Also an error bound for approximating a function with periodic Bernoulli polynomials is presented. Using operational matrices of differentiation, delay and the integration of the cross product of two Bernoulli polynomials, the linear time delay optimal control problem with quadratic objective function is converted to a nonlinear optimization problem. Finally, numerous illustrative examples are studied to show the efficiency and accuracy of the proposed method compared with some other methods.

New class of hybrid explicit methods for numerical solution of optimal control problems

Forward-backward sweep method (FBSM) is an indirect numerical method used for solving optimal control problems, in which the differential equation arising from this method is solved by the Pontryagin’s maximum principle. In this paper, a set of hybrid methods based on explicit 6th-order RungeKutta method is presented for the FBSM solution of optimal control problems. Order of truncation error, stability region, and numerical results of the new hybrid methods were compared with those of the 6th-order Runge Kutta method. Numerical results show that new hybrid methods are more accurate than the 6th-order Runge–Kutta method and that their stability regions are also wider than that of the 6th-order Runge–Kutta method.

Solving quantum optimal control problems by wavelets method

We present the quantum equation and synthesize an optimal control proce dure for this equation. We develop a theoretical method for the analysis of quantum optimal control system given by the time depending Schrodinger equation. The Legendre wavelet method is proposed for solving this problem. This can be used as an efficient and accurate computational method for obtaining numerical solutions of different quantum optimal control problems. The distinguishing feature of this paper is that it makes the method, previously used to solve non-quantum control equations based on Legendre wavelets, usable by using a change of variables for quantum control equations.

Optimization problem of attitude control of a spacecraft with bounded rotary energy using quaternions

The original control problem of optimal reorientation from a state of rest to a state of rest is considered and solved. The control function is torque vector. Problem of optimal control is investigated in detail for statement when control is restricted and the used functional of optimality includes kinematical rotary energy and time of maneuver. For solving and synthesis of the optimal control program, the quaternion method and the Pontryagin’s maximum principle are applied. Analytic solution of the proposed problem is presented basing on the differential equation connecting the angular velocity vector and quaternion of spacecraft attitude. It is shown that a chosen criterion of optimality provides a turn of a spacecraft with rotation energy which does not exceed the required value. This property of proposed control increases safety of flight. The time-optimal problem was solved also. The control law is formulated in the form of an explicit dependence of the control variables on the phase coordinates. All key expressions and equations are written in quaternion form which is convenient for onboard realization and implementation. The analysis of the special control regime of the spacecraft was made. Analytical formulas were written for duration of acceleration and braking. For specific cases of spacecraft configurations (dynamically symmetric spacecraft and spheric-symmetrical spacecraft as particular cases), complete solution of optimal control problem in closed form is given. Numerical example and the results of mathematical simulation for spacecraft motion under optimal control are demonstrated. This data supplements the made theoretical descriptions, and illustrates the practical feasibility of the designed algorithm for controlling the spatial orientation of the spacecraft showing reorientation process in visual form.

On the estimation of optimal state-of-charge trajectory for plug-in hybrid electric buses using trip information

Plug-in hybrid electric buses are a viable solution to increase the fuel economy. In this framework, precise estimation of optimal state-of-charge trajectory along the upcoming driving cycle appears to play a pivotal role in the way to approach the globally optimal fuel economy. This paper aims to conduct a parametric study on the key factors affecting the estimation of optimal state-of-charge trajectory, including trip information availability and trip segment distance, and to provide a guideline for the design and implementation of predictive energy management systems. To accomplish this, the dynamic programming algorithm is employed to obtain the solution of optimal control problem for the sampled driving cycles in a particular bus route. A large database comprising of driving features of the cycles and the optimal solution is developed which then is used to construct a neural network based estimator for obtaining the optimal state-of-charge trajectory. The main results show promising performance of the proposed method with about 76% reduction in the root mean square error of the estimated trajectory comparing to the linear state-of-charge trajectory assumption. Moreover, the robustness of the estimator is verified through simulation and it is observed that appropriate choice of trip segment distance is vital to improve the estimation accuracy, especially in case of uncertain prediction of trip information.

Radial Basis Function Interpolation and Galerkin Projection for Direct Trajectory Optimization and Costate Estimation

This work presents a novel approach combining radial basis function (RBF) interpolation with Galerkin projection to efficiently solve general optimal control problems. The goal is to develop a highly flexible solution to optimal control problems, especially nonsmooth problems involving discontinuities, while accounting for trajectory accuracy and computational efficiency simultaneously. The proposed solution, called the RBF-Galerkin method, offers a highly flexible framework for direct transcription by using any interpolant functions from the broad class of global RBFs and any arbitrary discretization points that do not necessarily need to be on a mesh of points. The RBF-Galerkin costate mapping theorem is developed that describes an exact equivalency between the Karush-Kuhn-Tucker (KKT) conditions of the nonlinear programming problem resulted from the RBF-Galerkin method and the discretized form of the first-order necessary conditions of the optimal control problem, if a set of discrete conditions holds. The efficacy of the proposed method along with the accuracy of the RBF-Galerkin costate mapping theorem is confirmed against an analytical solution for a bang-bang optimal control problem. In addition, the proposed approach is compared against both local and global polynomial methods for a robot motion planning problem to verify its accuracy and computational efficiency.

A pseudospectral method for budget-constrained infinite horizon optimal control problems

In this paper a generalization of the indirect pseudo-spectral method, presented in [17], for the numerical solution of budget-constrained infinite horizon optimal control problems is presented. Consideration of the problem statement in the framework of weighted functional spaces allows to arrive at a good approximation for the initial value of the adjoint variable, which is inevitable for obtaining good numerical solutions. The presented method is illustrated by applying it to the budget-constrained linear-quadratic regulator model. The quality of approximate solutions is demonstrated by an example.

Approaches to Numerical Solution of Optimal Control Problem Using Evolutionary Computations

Two approaches to the numerical solution of the optimal control problem are studied. The direct approach is based on the reduction of the optimal control problem to a nonlinear programming problem. Another approach is so-called synthesized optimal control, and it includes the solution of the control synthesis problem and stabilization at some point in the state space, followed by the search of stabilization points and movement of the control object along these points. The comparison of these two approaches was carried out as the solution of the optimal control problem as a time function cannot be directly used in the control system, although the obtained discretized control can be embedded. The control object was a group of interacting mobile robots. Dynamic and static constraints were included in the quality criterion. Implemented methods were evolutionary algorithms and a random parameter search of piecewise linear approximation and coordinates of stabilization points, along with a multilayer network operator for control synthesis.

Three-Stage Numerical Solution for Optimal Control of COVID-19

In this paper, we present a three-stage algorithm for finding numerical solutions for optimal control problems. The algorithm first performs an exhaustive search through a discrete set of widely dispersed solutions which are representative of large subregions of the search space; then, it uses the search results to initialize a Monte Carlo process that searches quasi-randomly for a best solution; then, it finally uses a Newton-type iteration to converge to a solution that satisfies mathematical conditions of local optimality. We demonstrate our methodology on an epidemiological model of the coronavirus disease with testing and distancing controls applied over a period of 180 days to two different subpopulations (low-risk and high-risk), where model parameters are chosen to fit the city of Houston, Texas, USA. In order to enable the user to select his/her preferred trade-off between (number of deaths) and (herd immunity) outcomes, the objective function includes costs for deaths and non-immunity. Optimal strategies are estimated for a grid of (death cost) × (non-immunity cost) combinations, in order to obtain a Pareto curve that represents optimum trade-offs. The levels of the four controls for the different Pareto-optimal solutions over the 180-day period are visually represented and their characteristics discussed. Three different variants of the algorithm are run in order to determine the relative importance of the three stages in the optimization. Results from the three algorithm variants are fairly consistent, indicating that solutions are robust. Results also show that the Monte Carlo stage plays an especially prominent role in the optimization, but that all three stages of the process make significant contributions towards finding lower-cost, more effective control strategies.

Numerical solution of optimal control problem

Numerical solution of optimal control problem

Nonlinear optimal control for multi‐DOF robotic manipulators with flexible joints

AbstractA main problem in the control of flexible‐joint robots and in the precise positioning of their end‐effector, is synchronization between the turn angles of the robots' joints and the turn angles of the motors that provide actuation to these joints. In this article, a nonlinear optimal (H‐infinity) control method is proposed for multi‐DOF robotic manipulators with flexible joints. The dynamic model of these robotic manipulators is shown to be underactuated and to satisfy differential flatness properties. The state‐space description of these robots undergoes approximate linearization around a temporary operating point which is recomputed at each iteration of the control method. The linearization procedure makes use of first‐order Taylor‐series expansion and relies on the computation of Jacobian matrices. For the approximately linearized model of these multi‐DOF robotic systems an H‐infinity feedback controller is designed. This controller stands for the solution of the robots' optimal control problem under model uncertainty and external disturbances. The computation of the controller's feedback gain requires the solution of an algebraic Riccati equation, which is performed again at each time‐step of the control algorithm. The stability properties of the control scheme are proven trough Lyapunov analysis. First, it is confirmed that the controller satisfies the H‐infinity tracking performance criterion which ascertains its robustness. Moreover, it is proven that the control loop is globally asymptotically stable. Finally, to implement sensorless control for such robotic systems the H‐infinity Kalman Filter is used as a robust state estimator. The proposed control method for multi‐DOF robotic manipulators with flexible joints retains the advantages of linear optimal control, that is fast and accurate tracking of its reference setpoints under moderate variations of the control inputs.

Nonlinear optimal control for the dynamics of the underactuated slosh-container system

ABSTRACT Using Euler-Lagrange analysis the state-space model of the slosh-container system is obtained. This state-space description undergoes approximate linearization with the use of Taylor series expansion and the computation of Jacobian matrices around a temporary operating point which is recomputed at each iteration of the control method. For the approximately linearized model, an H-infinity feedback controller is designed. The proposed control method stands for the solution of the nonlinear optimal control problem for the slosh-container system, under model uncertainties and external perturbations. To find the controller's feedback gains an algebraic Riccati equation is solved at each time-step of the control method. The new nonlinear optimal control approach achieves fast and accurate tracking of setpoints for all state variables of the slosh-container system, under moderate variations of the control inputs. The stability properties of the control scheme are proven through Lyapunov analysis.

Solving fractional optimal control problems with inequality constraints by a new kind of Chebyshev wavelets method

In this paper, a new orthonormal wavelets based on the sixth-kind Chebyshev polynomials is constructed to obtain the solution of fractional optimal control problems (FOCPs) with inequality constraints. The convergence analysis and error estimate of the sixth-kind Chebyshev wavelets function expansion are investigated. By using the relationship between the second-kind and sixth-kind Chebyshev polynomials, the exact formula of Riemann-Liouville fractional integral operator of Chebyshev wavelet is derived. For solving FOCPs, positive slack functions are added to inequality conditions, and then the problem is simplified to the problem of solving algebraic equations by fractional integral operational matrix and collocation method. The applicability and validity of the proposed method are verified by several examples and the results are compared with those reported in literature.

Controlled Process of Crystallization in Weld Pool

In this report the problem of control of solidification crack formation in welded plates is considered. In this problem the welding source is determined in dependence on the preset configuration and curvature of the rear part weld pool. A double ellipsoid model of weld pool with preset semi-axes may be used for the first approximation of preset weld pool configuration. It is an inverse problem which may be more efficiently solved as optimal control problem. The Function of welding source as a controlling function obtained in the result of solution is determined in a class of piecewise continuous functions which is more common class and the continuous-smooth functions are special partial case of common class. Recent methods of optimal control which use for solution of optimal control problems require to present the controlling functions in class of piecewise constant functions. Laser influence, electron beam, plasma, arc and submerged arc are the welding sources with high concentrated energy. A mathematical models of these welding sources may be introduced in class of piecewise continuous function with an efficient accuracy.

Optimal control and approximation for elliptic bilateral obstacle problems

The aim of this paper is to study an elliptic bilateral obstacle system (EBOS, for short) involving a nonlinear and nonhomogeneous partial differential operator and a multivalued term which is described by Clarke’s generalized gradient. First, we obtain the weak formulation of (EBOS) which is a variational-hemivariational inequality, and prove the unique solvability of the bilateral obstacle problem. Second, we consider a nonlinear optimal control problem governed by (EBOS) in which the control variable is the bilateral obstacle, and establish the existence of an optimal solution to the obstacle control problem under mild conditions. Then, employing the regularization technique and penalty approach, we introduce a family of approximating problems corresponding to the optimal control problem under consideration. Finally, a convergence result that any sequence of solutions for the approximating problems converges to an optimal solution of the original optimal control problem is delivered.