Solving Optimal Control Problems on the Class of Step Functions

Linear optimal control problems with multipoint non-separated conditions with a quadratic performance criterion of control are analyzed. An approach with a violation of intermediate conditions is proposed. The original problem is reduced to the classical quadratic programming problem with the dimension determined by the number of intermediate points. __________ Translated from Kibernetika i Sistemnyi Analiz, No. 4, pp. 153–162, July–August 2005.     

带不等式路径约束最优控制问题的惩罚函数法

控制变量参数化（Control variable parameterization,CVP）方法是目前求解流程工业中最优操作问题的主流数值方法,但如果问题中包含路径约束,特别是不等式路径约束时,CVP方法则需要考虑专门的处理手段.为了克服该缺点,本文提出一种基于L1精确惩罚函数的方法,能够有效处理关于控制变量、状态变量、甚至控制变量/状态变量复杂耦合形式下的不等式路径约束.此外,为了能使用基于梯度的成熟优化算法,本文还引进了最新出现的光滑化技巧对非光滑的惩罚项进行磨光.最终得到了能高效处理不等式路径约束的改进型CVP架构,并给出相应数值算法.经典的带不等式路径约束最优控制问题上的测试结果及与国外文献报道的比较研究表明：本文所提出的改进型CVP 架构及相应算法在精度和效率上兼有良好表现.     

基于拟线性化和Haar 函数的最优控制问题的直接求解方法

针对有约束条件的非线性最优控制问题,提出一种基于拟线性化和Haar函数的数值求解方法.首先将最优控制问题转化为一系列的二次规划问题,并使用系数未知的Haar函数对问题中的状态变量进行近似;然后应用拟线性化法将原非线性最优控制问题转化为相应的一系列受限的二次最优控制问题进行求解;最后基于所提出的方法对2个受限非线性最优控制问题进行求解,并通过仿真结果表明了采用所提出的算法求解最优控制问题的有效性.     

求解含复杂约束非线性最优控制问题的改进Gauss伪谱法

针对含有复杂约束条件的非线性最优控制问题,提出了一种改进的Gauss伪谱法 (Improved Gauss pseudospectral method, IGPM). 这类问题难以得到解析解,特别是有些问题不存在解析的模型, 一些参数只能通过查表得到,使得传统方法难以求解. 在传统的Gauss伪谱法的基础上,将非线性的终端状态积分约束等价地转化为线性形式,提出了IGPM, 通过协态映射定理可以计算出协态变量,检验最优性,使得IGPM具有间接法一样的精度. 并且给出了初始时刻协态变量和端点时刻控制变量的计算方法. 为了提高解的精度,基于IGPM提出了迭代算法, 最后将该算法应用于求解高超声速飞行器上升段轨迹优化问题,结果表明最优轨迹基本满足路径约束条件和最优性条件.     

Nonlinear Adaptive Model Predictive Control of Constrained Systems with Offset‐Free Tracking Behavior

In this paper, a nonlinear model‐based predictive control strategy for constrained systems based on an adaptive neural network (NN) predictor is proposed. The proposed controller is robust against the model uncertainties and external bounded disturbances. Moreover, it provides offset‐free tracking behavior using the adaptive structure in the model. Based on the uncertainties bounds, the restriction of the system constraints causes robust feasibility and stability of the closed‐loop system. It is shown that the output of the NN predictor converges to the system output. Moreover, offset‐free behavior of the closed‐loop system is investigated using the Lyapunov theorem. Simulation results show the effectiveness of the proposed method as compared to the recently proposed model predictive control methods in the literature.     

Uncertain Method for Optimal Control Problems With Uncertainties Using Chebyshev Inclusion Functions

In this paper, a new uncertain analysis method is developed for optimal control problems, including interval variables (uncertainties) based on truncated Chebyshev polynomials. The interval arithmetic in this research is employed for analyzing the uncertainties in optimal control problems comprising uncertain‐but‐bounded parameters with only lower and upper bounds of uncertain parameters. In this research, the Chebyshev method is utilized because it generates sharper bounds for meaningful solutions of interval functions, rather than the Taylor inclusion function, which is efficient in handling the overestimation derived from the wrapping effect due to interval computations. For utilizing the proposed interval method on the optimal control problems with uncertainties, the Lagrange multiplier method is first applied to achieve the necessary conditions and then, by using some algebraic manipulations, they are converted into the ordinary differential equation. Afterwards, the Chebyshev inclusion method is employed to achieve the solution of the system. The final results of the Chebyshev inclusion method are compared with the interval Taylor method. The results show that the proposed Chebyshev inclusion function based method better handle the wrapping effect than the interval Taylor method.     

A Comparative Approach for Time‐Delay Fractional Optimal Control Problems: Discrete Versus Continuous Chebyshev Polynomials

This paper aims to demonstrate the superiority of the discrete Chebyshev polynomials over the classical Chebyshev polynomials for solving time‐delay fractional optimal control problems (TDFOCPs). The discrete Chebyshev polynomials have been introduced and their properties are investigated thoroughly. Then, the fractional derivative of the state function in the dynamic constraint of TDFOCPs is approximated by these polynomials with unknown coefficients. The operational matrix of fractional integration together with the dynamical constraints is used to approximate the control function directly as a function of the state function. Finally, these approximations were put in the performance index and necessary conditions for optimality transform the under consideration TDFOCPs into an algabric system. A comparison has been made between the required CPU time and accuracy of the discrete and continuous Chebyshev polynomials methods. The obtained numerical results reveal that utilizing discrete Chebyshev polynomials is more efficient and less time‐consuming in comparison to the continuous Chebyshev polynomials.     

Optimal Control of Nonlinear Inverted Pendulum System Using PID Controller and LQR: Performance Analysis Without and With Disturbance Input

Linear quadratic regulator(LQR) and proportional-integral-derivative(PID) control methods, which are generally used for control of linear dynamical systems, are used in this paper to control the nonlinear dynamical system. LQR is one of the optimal control techniques, which takes into account the states of the dynamical system and control input to make the optimal control decisions.The nonlinear system states are fed to LQR which is designed using a linear state-space model. This is simple as well as robust. The inverted pendulum, a highly nonlinear unstable system, is used as a benchmark for implementing the control methods. Here the control objective is to control the system such that the cart reaches a desired position and the inverted pendulum stabilizes in the upright position. In this paper, the modeling and simulation for optimal control design of nonlinear inverted pendulum-cart dynamic system using PID controller and LQR have been presented for both cases of without and with disturbance input. The Matlab-Simulink models have been developed for simulation and performance analysis of the control schemes. The simulation results justify the comparative advantage of LQR control method.     

The Use of the Ritz Method and Laplace Transform for Solving 2D Fractional‐Order Optimal Control Problems Described by the Roesser Model

In this article a numerical solution is presented for a class of two‐dimensional fractional‐order optimal control problems (2D‐FOOCPs) with one input and two outputs. To implement the numerical method, the Legendre polynomial basis is used with the aid of the Ritz method and the Laplace transform. By taking the Ritz method as a basic scheme into account and applying a new constructed fractional operational matrix to estimate the fractional and integer order derivatives of the basis, the given 2D‐FOOCP is reduced to a system of algebraic equations. One of the advantages of the proposed method is that it provides greater flexibility in which the given initial and boundary conditions of the problem are imposed. Moreover, satisfactory results are obtained in just a small number of polynomials order. The convergence of the method is extensively investigated and finally two illustrative examples are included to show the validity and applicability of the novel proposed technique in the current work.