An Efficient Numerical Solution of Fractional Optimal Control Problems by using the Ritz Method and Bernstein Operational Matrix

This paper deals with the Ritz spectral method to solve a class of fractional optimal control problems (FOCPs). The developed numerical procedure is based on the function approximation by the Bernstein polynomials along with fractional operational matrix usage. The approximation method is computationally consistent and moreover, has a good flexibility in the sense of satisfying the initial and boundary conditions of the optimal control problems. We construct a new fractional operational matrix applicable in the Ritz method to estimate the fractional and integer order derivatives of the basis. As a result, we achieve an unconstrained optimization problem. Next, by applying the necessary conditions of optimality, a system of algebraic equations is obtained. The resultant problem is solved via Newton's iterative method. Finally, the convergence of the proposed method is investigated and several illustrative examples are added to demonstrate the effectiveness of the new methodology.     

Fractional optimal control problems with both integer-order and Atangana–Baleanu Caputo derivatives

This article considers fractional optimal control problems (FOCPs) including both integer-order and Atangana–Baleanu Caputo derivatives. First, the existence and uniqueness of the solution of a fractional Cauchy problem is given. Then, applying calculus of variations and Lagrange multiplier method, we present necessary optimality conditions of FOCPs and sufficient optimality conditions are also given under some assumptions. Next, a collection method is developed to derive numerical solutions by using shifted Legendre polynomials. Finally, error estimate of numerical solutions is also provided, and numerical examples further show the accuracy and feasibility of our method.     

A numerical technique for solving fractional optimal control problems

This paper presents a numerical method for solving a class of fractional optimal control problems (FOCPs). The fractional derivative in these problems is in the Caputo sense. The method is based upon the Legendre orthonormal polynomial basis. The operational matrices of fractional Riemann-Liouville integration and multiplication, along with the Lagrange multiplier method for the constrained extremum are considered. By this method, the given optimization problem reduces to the problem of solving a system of algebraic equations. By solving this system, we achieve the solution of the FOCP. Illustrative examples are included to demonstrate the validity and applicability of the new technique.     

Solving Fixed Final Time Fractional Optimal Control Problems Using the Parametric Optimization Method

In this paper, the parametric optimization method is used to find optimal control laws for fractional systems. The proposed approach is based on the use for the fractional variational iteration method to convert the original optimal control problem into a nonlinear optimization one. The control variable is parameterized by unknown parameters to be determined, then its expression is substituted into the system state‐space model. The resulting fractional ordinary differential equations are solved by the fractional variational iteration method, which provides an approximate analytical expression of the closed‐form solution of the state equations. This solution is a function of time and the unknown parameters of the control law. By substituting this solution into the performance index, the original fractional optimal control problem reduces to a nonlinear optimization problem where the unknown parameters, introduced in the parameterization procedure, are the optimization variables. To solve the nonlinear optimization problem and find the optimal values of the control parameters, the Alienor global optimization method is used to achieve the global optimal values of the control law parameters. The proposed approach is illustrated by two application examples taken from the literature.     

A new Wavelet Method for Variable‐Order Fractional Optimal Control Problems

In this paper, a new computational method based on the Legendre wavelets (LWs) is proposed for solving a class of variable‐order fractional optimal control problems (V‐FOCPs). To do this, a new operational matrix of variable‐order fractional integration (OMV‐FI) in the Riemann‐Liouville sense for the LWs is derived and used to obtain an approximate solution for the problem under study. Along the way the hat functions (HFs) are introduced and employed to derive a general procedure to compute this matrix. In the proposed method, the variable‐order fractional dynamical system is transformed to an equivalent variable‐order fractional integro‐differential dynamical system, at first. Then, the highest integer order of the derivative of the state variable and the control variable are expanded by the LWs with unknown coefficients. Next, the OMV‐FI in the the Riemann‐Liouville sense together with some properties of the LWs are employed to achieve a nonlinear algebraic equation in place of the performance index and a nonlinear system of algebraic equations in place of the dynamical system in terms of the unknown coefficients. Finally, the method of constrained extremum is applied which consists of adjoining the constraint equations derived from the given dynamical system to the performance index by a set of undetermined Lagrange multipliers. As a result, the necessary conditions of optimality are derived as a system of algebraic equations in the unknown coefficients of the state variable, control variable and Lagrange multipliers. Furthermore, the efficiency and accuracy of the proposed method are demonstrated for some concrete examples. The obtained results show that the proposed method is very efficient and accurate.     

最优控制问题的Legendre 伪谱法求解及其应用

伪谱法通过全局插值多项式参数化状态和控制变量,将最优控制问题(OCP)转化为非线性规划问题(NLP)进行求解,是一类具有更高求解效率的直接法。总结Legendre伪谱法转化Bolza型最优控制问题的基本框架,推导OCP伴随变量与NLP问题KKT乘子的映射关系,建立基于拟牛顿法的LGL配点数值计算方法,并针对非光滑系统,进一步研究分段伪谱逼近策略。基于上述理论开发通用OCP求解器,并对3个典型最优控制问题进行求解,结果表明了所提出方法和求解器的有效性。     

挡位离散型车辆经济性加速策略的伪谱法优化

加速过程中, 车辆的油耗与驾驶员的操作策略密切相关. 本文通过最优控制方法定量化地研究了挡位离散型车辆的经济性加速策略. 将加速策略的辨识构建为一个Bolza型最优控制问题(Optimal control problem, OCP), 设计了考虑加速距离影响的经济性定量评价指标. 该问题含有离散型控制变量, 隶属于混合整型最优控制问题, 且性能函数和状态方程呈现强非线性. 为高效地求解该问题, 结合变速器挡位切换规律, 将该整型问题转化为多段光滑问题的协同优化, 采用Legendre伪谱拼接法实现变速器挡位、换挡时机、发动机力矩的数值求解. 解析分析了经济性加速策略的形成机理, 总结了实用化的经济性加速度选择策略和挡位切换规律. 仿真验证了所求策略的节油潜力.     

Global Optimal Feedback‐Linearizing Control of Robot Manipulators

The present paper offers a new optimal feedback‐linearizing control scheme for robot manipulators. The method presented aims at solving a special form of the unconstrained optimal control problem (OCP) of robot manipulators globally using the results of the Lyaponov method and feedback‐linearizing strategy and without using the calculus of variations (indirect method), direct methods, or the dynamic programming approach. Most of these methods and their sub‐branches yield a local optimal solution for the considered OCP by satisfying some necessary conditions to find the stationary point of the considered cost functional. In addition, the proposed method can be used for both set‐point regulating (point‐to‐point) tasks (e.g. pick‐and‐place operation or spot welding tasks) and trajectory tracking tasks such as painting or welding tasks. However, the proposed method can not support the physical constraints on robot manipulators and requires precise dynamics of the robot, as well. Instead, it can be used as an on‐line optimal control algorithm which produces the optimal solution without performing any kind of optimization algorithms which require time to find the optimal solution.     

Optimal control of viscous Burgers equation via an adaptive nonmonotone Barzilai–Borwein gradient method

An adaptive nonmonotone spectral gradient method for the solution of distributed optimal control problem (OCP) for the viscous Burgers equation is presented in a black-box framework. Regarding the implicit function theorem, the OCP is transformed into an unconstrained nonlinear optimization problem (UNOP). For solving UNOP, an adaptive nonmonotone Barzilai–Borwein gradient method is proposed in which to make a globalization strategy, first an adaptive nonmonotone strategy which properly controls the degree of nonmonotonicity is presented and then is incorporated into an inexact line search approach to construct a more relaxed line search procedure. Also an adjoint technique is used to effectively evaluate the gradient. The low memory requirement and the guaranteed convergence property make the proposed method quite useful for large-scale OCPs. The efficiency of the presented method is supported by numerical experiments.     

Optimal combustion control with application to engine design and development

The need to reduce development time whilst simultaneously improving engine performance has motivated this application of optimal control to product development processes for engines and powertrains. The optimisation of the fuel consumption is formulated as a constrained Optimal Control Problem (OCP) and solved using pseudospectral methods, giving the optimum heat release and injection profiles in the presence of cylinder pressure rate and cylinder pressure constraints. The technique is applied to an engine design problem and used to reduce fuel consumption by optimising compression ratio within a cylinder pressure limit, also providing new insights into the combustion processes.     

Optimal control of polymer flooding based on mixed-integer iterative dynamic programming

Polymer flooding is one of the most important technologies for enhanced oil recovery. In this article, a mixed-integer optimal control model of distributed parameter systems (DPS) for the injection strategies is established, which involves the performance index as maximum of the profit, the governing equations as the fluid flow equations of polymer flooding and some inequalities constraints, such as polymer concentration and injection amount limitation. The control variables are the volume size, the injection concentration of each slug and the terminal flooding time. For the constant injection rate, the slug size is determined by the integer time stage length, and thus the integer variables are introduced in the DPS. To cope with the optimal control problem (OCP) of this DPS, a mixed-integer iterative dynamic programming incorporating a special truncation procedure to handle integer restrictions on stage lengths is proposed. First, the OCP with variable time stage lengths is transformed into a fixed time stage problem by introducing a normalised time variable. Then, the optimisation procedure is carried out at each stage and preceded backwards in a systematic way. Finally, the numerical results of an example illustrate the effectiveness of the proposed method.     

Control-oriented model validation and errors quantification in the /spl lscr//sub 1/ setup

A priori information required for robust synthesis includes a nominal model and a model of uncertainty. The latter is typically in the form of additive exogenous disturbance and plant perturbations with assumed bounds. If these bounds are unknown or too conservative, they have to be estimated from measurement data. In this paper, the problem of errors quantification is considered in the framework of the /spl lscr//sub 1/ optimal robust control theory associated with the /spl lscr//sub /spl infin// signal space. The optimal errors quantification is to find errors bounds that are not falsified by measurement data and provide the minimum value of a given control criterion. For model with unstructured uncertainty entering the system in a linear fractional manner, the optimal errors quantification is reduced to quadratic fractional programming. For system under coprime factor perturbations, the optimal errors quantification is reduced to linear fractional programming.     

Impulsive Synchronization of Stochastic Neural Networks via Controlling Partial States

In this paper the perceptron neural networks are applied to approximate the solution of fractional optimal control problems. The necessary (and also sufficient in most cases) optimality conditions are stated in a form of fractional two-point boundary value problem. Then this problem is converted to a Volterra integral equation. By using perceptron neural network’s ability in approximating a nonlinear function, first we propose approximating functions to estimate control, state and co-state functions which they satisfy the initial or boundary conditions. The approximating functions contain neural network with unknown weights. Using an optimization approach, the weights are adjusted such that the approximating functions satisfy the optimality conditions of fractional optimal control problem. Numerical results illustrate the advantages of the method.     

Existence of χ‐efficient solution of multiobjective fractional programming problem with bounded parameters

In this paper, a methodology is developed to solve a multiobjective fractional programming problem in which the coefficients of the objective functions and constraints are intervals. This model is transformed into an interval‐free equivalent optimization problem. A new partial ordering is introduced and the relation between the original problem and the transformed problem is established using this partial ordering. The proposed methodology is illustrated through a numerical example.     

On the two-degree-of-freedom Wiener–Hopf optimal design with tracking and disturbance rejection constraints

In this paper, the design of the two-degree-of-freedom optimal Wiener–Hopf controller with asymptotic tracking and disturbance rejection constraints is studied. Based on the solvability conditions and the solution parameterization of Diophantine equations over the stable rational fractional ring M(s), the disturbance rejection problem in the general case is solved. A simple cost function is defined to reflect the transient performance of control signal caused by deterministic signals. Finally, the design of the two-degree-of-freedom optimal Wiener–Hopf controller is presented.     

A Proximal Point Based Approach to Optimal Control of Affine Switched Systems

This paper focuses on the proximal point regularization technique for a class of optimal control processes governed by affine switched systems. We consider switched control systems described by nonlinear ordinary differential equations which are affine in the input. The affine structure of the dynamical models under consideration makes it possible to establish some continuity/approximability properties and to specify these models as convex control systems. We show that, for some classes of cost functionals, the associated optimal control problem (OCP) corresponds to a conventional convex optimization problem in a suitable Hilbert space. The latter can be reliably solved using standard first-order optimization algorithms and consistent regularization schemes. In particular, we propose a conceptual numerical approach based on the gradient-type method and classic proximal point techniques.     

Time-Optimal Boundary Control for Systems Defined by a Fractional Order Diffusion Equation

We consider the optimal control problem for a system defined by a one-dimensional diffusion equation with a fractional time derivative. We consider the case when the controls occur only in the boundary conditions. The optimal control problem is posed as the problem of transferring an object from the initial state to a given final state in minimal possible time with a restriction on the norm of the controls. We assume that admissible controls belong to the class of functions L_∞[0, T ]. The optimal control problem is reduced to an infinite-dimensional problem of moments. We also consider the approximation of the problem constructed on the basis of approximating the exact solution of the diffusion equation and leading to a finitedimensional problem of moments. We study an example of boundary control computation and dependencies of the control time and the form of how temporal dependencies in the control dependent on the fractional derivative index.     

Shared control for lane keeping assistance system based on multiple-phase handling inverse dynamics

To achieve user-friendly design, a novel shared control scheme for Lane Keeping Assistance (LKA) system is proposed in this paper. Instead of adjusting the control authority allocation, the proposed scheme achieves shared control by adjusting the displacement interval between each phase. Through this scheme, we aim to design a LKA system which can realize continuously shared control between assistant controller and human driver with a fixed driving authority allocation strategy. Firstly, the shared control problem is formulated as a multiple-phase inverse dynamics problem; then we covert the inverse dynamics problem into an Optimal Control Problem (OCP). Secondly, by applying Radau Pseudospectral Method (RPM), the OCP is converted into a Nonlinear Programming Problem (NLP), which is then solved by Sequential Quadratic Programming (SQP). Finally, simulations under two typical conditions are implemented to validate the effectiveness of the proposed shared control approach.     

Piecewise fractional Legendre functions for nonlinear fractional optimal control problems with ABC fractional derivative and non-smooth solutions

In this study, to solve fractional problems with non-smooth solutions (which include some terms in the form of piecewise or fractional powers), a new category of basis functions called the orthonormal piecewise fractional Legendre functions is introduced. The upper bound of the error of the series expansion of these functions is obtained. Two explicit formulas for computing the Riemann–Liouville and Atangana–Baleanu fractional integrals of these functions are derived. A direct method based on these functions and their fractional integral is proposed to solve a family of optimal control problems involving the ABC fractional differentiation whose solutions are non-smooth in the above expressed forms. By the proposed technique, solving the original fractional problem turns into solving an equivalent system of algebraic equations. The established method accuracy is studied by solving some examples.