A direct method based on the Chebyshev polynomials for a new class of nonlinear variable-order fractional 2D optimal control problems

The main goal of this study is to develop an efficient matrix approach for a new class of nonlinear 2D optimal control problems (OCPs) affected by variable-order fractional dynamical systems. The offered approach is established upon the shifted Chebyshev polynomials (SCPs) and their operational matrices. Through the way, a new operational matrix (OM) of variable-order fractional derivative is derived for the mentioned polynomials.The necessary optimality conditions are reduced to algebraic systems of equations by using the SCPs expansions of the state and control variables, and applying the method of constrained extrema. More precisely, the state and control variables are expanded in components of the SCPs with undetermined coefficients. Then these expansions are substituted in the cost functional and the 2D Gauss-Legendre quadrature rule is utilized to compute the double integral and consequently achieve a nonlinear algebraic equation.After that, the generated OM is employed to extract some algebraic equations from the approximated fractional dynamical system. Finally, the procedure of the constrained extremum is used by coupling the algebraic constraints yielded from the dynamical system and the initial and boundary conditions with the algebraic equation extracted from the cost functional by a set of unknown Lagrange multipliers. The method is established for three various types of boundary conditions.The precision of the proposed approach is examined through various types of test examples.Numerical simulations confirm the suggested approach is very accurate to provide satisfactory results.     

Free final time fractional optimal control problems

A formulation and solution scheme of free final time fractional optimal control problems is presented in this paper. The dynamic constraint is described by a fractional differential equation. Performance index considered is a function of both the state and control variables. The necessary conditions of optimality and the transversality condition are obtained using Lagrange multiplier technique. A numerical technique similar to Shooting method is used for solving the optimal conditions. Numerical example is provided to show the effectiveness of the formulation and numerical solution scheme. It is interesting to note that the final time changes with the interchange of the boundary conditions, which does not occur in classical optimal control problems.     

A composite pseudospectral method for solving multi-delay optimal control problems involving piecewise constant delay functions

An adaptive numerical method for solving multi-delay optimal control problems with piecewise constant delay functions is introduced. The proposed method is based on composite pseudospectral method using the well-known Legendre–Gauss–Lobatto points. In this approach, the main problem converts to a mathematical optimization problem whose solution is much more easier than the original one. The necessary conditions of optimality associated to nonlinear piecewise constant delay systems are derived. The method is easy to implement and provides very accurate results.     

Direct solution of nonlinear optimal control problems using quasilinearization and Chebyshev polynomials

In this paper, a numerical method to solve nonlinear optimal control problems with terminal state constraints, control inequality constraints and simple bounds on the state variables, is presented. The method converts the optimal control problem into a sequence of quadratic programming problems. To this end, the quasilinearization method is used to replace the nonlinear optimal control problem with a sequence of constrained linear-quadratic optimal control problems, then each of the state variables is approximated by a finite length Chebyshev series with unknown parameters. The method gives the information of the quadratic programming problem explicitly (The Hessian, the gradient of the cost function and the Jacobian of the constraints). To show the effectiveness of the proposed method, the simulation results of two constrained nonlinear optimal control problems are presented.     

Solution of delay fractional optimal control problems using a hybrid of block-pulse functions and orthonormal Taylor polynomials

This paper introduces an efficient direct approach for solving delay fractional optimal control problems. The concepts of the fractional integral and the fractional derivative are considered in the Riemann–Liouville sense and the Caputo sense, respectively. The suggested framework is based on a hybrid of block-pulse functions and orthonormal Taylor polynomials. The convergence of the proposed hybrid functions with respect to the L²-norm is demonstrated. The operational matrix of fractional integration associated with the hybrid functions is constructed by using the Laplace transform method. The problem under consideration is transformed into a mathematical programming one. The method of Lagrange multipliers is then implemented for solving the resulting optimization problem. The performance and computational efficiency of the developed numerical scheme are assessed through various types of delay fractional optimal control problems. Our numerical findings are compared with either exact solutions or the existing results in the literature.     

A novel direct torque control method for brushless DC motors based on duty ratio control

Conventional direct torque control (DTC) suffers from large torque ripple and nonconstant switching frequency, which are caused by the hysteresis band amplitude and the motor speed. Many methods have been proposed to tackle these problems. However, these methods are usually complicated and parameter dependent. A novel DTC method for brushless DC motors based on duty ratio control is proposed to reduce torque ripple and maintain a constant switching frequency. During each switching period, an active voltage vector and a zero voltage vector are applied. A simple and effective method implemented to calculate the duty ratio relies only on the torque error, reducing the parameter dependence. The proposed method has the advantages of conventional DTC and effectively reduces torque ripple, which improves the performance of conventional DTC. Simulation and experimental results are given to confirm the method’s validity.     

A walsh series direct method for solving variational problems

This paper establishes a clear procedure for the variational problem solution via the Walsh functions.technique. First the Walsh functions are introduced and their properties briefly summarized. Then an operational matrix is derived for integration use. The variational problems are solved by means of the direct method using the Walsh series. An illustrative example and a practical application to a heat conduction problem are included.     

Necessary conditions for Chebyshev-Bolza optimal control problems

A performance index consisting of a Chebyshev absolute maximum functional plus terminal and integral cost is applied to the optimal control of dynamical systems. First-order necessary conditions are derived for a large class of systems. Utilizing the necessary conditions, analytic examples are worked in demonstrating many of the properties of this class of systems.     

A hybrid approximation scheme for discretizing constrained quadratic optimal control problems

In this paper, a composite Chebyshev finite difference method for solving linear quadratic optimal control problems with inequality constraints on state and control variables is introduced. This method is an extension of Chebyshev finite difference scheme and is based on a hybrid of block-pulse functions and Chebyshev polynomials using the well known Chebyshev–Gauss–Lobatto nodes. The excellent properties of hybrid functions are used to convert optimal control problem into a mathematical programming problem whose solution is much more easier than the original one. Various types of optimal control problems are investigated to demonstrate the effectiveness of the proposed approximation scheme. The method is simple, easy to implement and provides very accurate results.     

A direct approach using the finite element method for the solution of the linear optimal control problem with a quadratic criterion

The present paper proposes a numerical approach to a linear optimal control problem with a quadratic performance index. In this technique, the time interval is divided into a number of time segments and all of the unknown functions which appear in the performance index are either interpolated linearly with respect to time or assumed to be constant in each time segment. The augmented performance index is discretized within each time element through the ordinary finite element technique.The main advantage of the present method is as follows: all of the necessary conditions for the performance index to be stationary can be expressed in the form of algebraic equations and the performance sequence of the state variables can be eliminated. As a result, the optimal control problem is reduced to the simple one of finding the sequence of control variables alone, which minimizes the quadratic performance index.A general formulation of the method is given and simple numerical examples are shown to demonstrate the effectiveness of the technique.     

New approach to the optimal control of delay systems via Chebyshev series

Based on the Chebyshev series, a directly computational formulation in matrix form is established for evaluating the optimal control and trajectory of time-delay systems. In a comparison with the previous work (3, Int. J. Control, Vol. 41, pp. 1221-1234, 1985), the formulation is shown to be more straightforward and convenient for digital computation. Thus the difficulty in obtaining a solution of the two-point boundary-value problem with both delayed and advanced arguments is circumvented. An example compares the actual solution with the one obtained using the technique of this paper.     

基于电压跟踪的直接转矩控制系统

对感应电动机直接转矩控制（DTC,direct torque control）的磁链观测方法进行改进,减小观测误差,同时采用电压空间矢量调制技术（SVPWM）对DTC加以完善,提出了一种基于电压跟踪来实现直接转矩控制的方案。通过matlab7.4/simulink对改进前后的感应电机直接转矩控制系统进行仿真,从仿真结果来看,转矩、磁链脉动明显减少,本文提出的方案能够取得较好的控制效果。     

A novel mesh discretization strategy for numerical solution of optimal control problems in aerospace engineering

An effective approach is proposed for optimal control problems in aerospace engineering. First, several interval lengths are treated as optimization variables directly to localize the switching points accurately. Second, the variable intervals are usually refined into more subintervals homogeneously to obtain the trajectories with high accuracy. To reduce the number of optimization variables and improve the efficiency, the control and the state vectors are parameterized using different meshes in this paper such that the control can be approximated asynchronously by fewer parameters where the trajectories change slowly. Then, the variables are departed as independent variables and dependent variables, the gradient formulae, based on the partial derivatives of dependent parameters with respect to independent parameters, are computed to solve nonlinear programming problems. Finally, the proposed approach is applied to the classic moon lander and hang glider problems. For the moon lander problem, the proposed approach is compared with CVP, Fast-CVP and GPM methods, respectively. For the hang glider problem, the proposed approach is compared with trapezoidal discretization and stopping criteria methods, respectively. The numerical results validate the effectiveness of the proposed approach.     

A new singular system approach to output feedback sliding mode control for fractional order nonlinear systems

This paper studies the problem of output feedback sliding mode control (OFSMC) for fractional order nonlinear systems. A necessary and sufficient condition for the existence of a sliding surface is obtained by a new singular system approach and a linear matrix equality (LMI), which reduces the conservativeness of the system. Then an OFSMC law is designed based on a fractional order Lyapunov method, which ensures that the resulting fractional closed-loop system is asymptotically stable and the states of the fractional closed-loop system converge to the sliding surface in finite time. A fractional electrical circuit is discussed to illustrate the effectiveness of the proposed approach.     

Numerical solutions of optimal control for linear Volterra integrodifferential systems via hybrid functions

By applying hybrid functions of general block-pulse functions and Legendre polynomials, linear Volterra integrodifferential systems are converted into a system of algebraic equations. The approximate solutions of linear Volterra integrodifferential systems are derived. Using the results we obtain the optimal control and state as well as the optimal value of the objective functional. The numerical examples illustrate that the algorithms are valid.     

基于费用-效能分析的电气设备购置管理新方法

电力系统的快速发展带来大量的电气设备采购.目前,电气设备购置的决策方式尚停留在传统的招标方式上,缺乏更加科学客观的参考依据.该文提出采用费效分析方式对参与投标的设备进行综合评估排队,为设备的购置决策提供参考依据.论文建立了模糊费效分析模型,给出了费效分析方法的具体步骤和流程.文中给出的实例验证了所提方法的有效性.     

Operational matrix based set-membership method for fractional order systems parameter identification

In this paper, a new method is proposed to identify the coefficients and differentiation orders of fractional order systems with measurement noise. The proposed method combines the operational matrix method and the set-membership method. First, the block pulse functions operational matrix of the fractional differentiation is used to convert the fractional order system to an algebraic system. Then, the coefficients and differentiation orders are simultaneously estimated through a nest loop optimization process, where the optimal bounding ellipsoid set-membership algorithm is utilized to estimate the system’s coefficients and the orders are estimated with the interior-point method. The proposed method can accurately estimate the coefficients and differentiation orders of fractional order systems under any bounded measurement noise with less computational effort. Experimental results demonstrate the effectiveness of the proposed method.     

Adaptive pseudospectral methods for solving constrained linear and nonlinear time-delay optimal control problems

In this paper, we first develop an adaptive shifted Legendre–Gauss (ShLG) pseudospectral method for solving constrained linear time-delay optimal control problems. The delays in the problems are on the state and/or on the control input. By dividing the domain of the problem into a uniform mesh based on the delay terms, the constrained linear time-delay optimal control problem is reduced to a quadratic programming problem. Next, we extend the application of the adaptive ShLG pseudospectral method to nonlinear problems through quasilinearization. Using this scheme, the constrained nonlinear time-delay optimal control problem is replaced with a sequence of constrained linear-quadratic sub-problems whose solutions converge to the solution of the original nonlinear problem. The method is called the iterative-adaptive ShLG pseudospectral method. One of the most important advantages of the proposed method lies in the case with which nonsmooth optimal controls can be computed when inequality constraints and terminal constraints on the state vector are imposed. Moreover, a comparison is made with optimal solutions obtained analytically and/or other numerical methods in the literature to demonstrate the applicability and accuracy of the proposed methods.     

Numerical solutions of optimal control for linear time-varying systems with delays via hybrid functions

By applying hybrid functions of general block-pulse functions and Legendre polynomials, the linear-quadratic problem of linear time-varying systems with delays are transformed into the optimization problem of multivariate functions. The approximate solutions of the optimal control and state as well as the optimal value of the objective functional are derived. The numerical examples illustrate that the algorithms are valid.