Radau Collocation Method Research Articles

Method for solving state‐path constrained optimal control problems using adaptive Radau collocation

AbstractA new method is developed for accurately approximating the solution to state‐variable inequality path constrained optimal control problems using a multiple‐domain adaptive Legendre–Gauss–Radau collocation method. The method consists of the following parts. First, a structure detection method is developed to estimate switch times in the activation and deactivation of state‐variable inequality path constraints. Second, using the detected structure, the domain is partitioned into multiple‐domains where each domain corresponds to either a constrained or an unconstrained segment. Furthermore, additional decision variables are introduced in the multiple‐domain formulation, where these additional decision variables represent the switch times of the detected active state‐variable inequality path constraints. Within a constrained domain, the path constraint is differentiated with respect to the independent variable until the control appears explicitly, and this derivative is set to zero along the constrained arc while all preceding derivatives are set to zero at the start of the constrained arc. The time derivatives of the active state‐variable inequality path constraints are computed using automatic differentiation and the properties of the chain rule. The method is demonstrated on two problems, the first being a benchmark optimal control problem which has a known analytical solution and the second being a challenging problem from the field of aerospace engineering in which there is no known analytical solution. When compared against previously developed adaptive Legendre–Gauss–Radau methods, the results show that the method developed in this paper is capable of computing accurate solutions to problems whose solution contain active state‐variable inequality path constraints.

On the discontinuous solutions of explicit neutral delay differential equations

This paper considers explicit neutral delay differential equations (NDDE) with piecewise continuous initial functions. We explain how the discontinuities in the solutions arise and present a perturbing scheme, in combination with an adaptive Legendre–Gauss–Radau collocation method, to deal with this type of problems computationally. The pointwise and mean convergence of the continuous solution of the perturbed NDDE to the discontinuous solution of the original NDDE are proved. Our new method for discontinuous NDDEs and the rigorous theoretical analysis provided are particularly important since explicit NDDEs have received little attention in the literature. Numerical results are given to show that the proposed method can be implemented in an efficient and accurate manner.

Accurate Laguerre collocation solutions to a class of Emden–Fowler type BVP

We solve numerically the nonlinear and double singular boundary value problem formed by the well known Emden–Fowler equation , s > 1 along with the boundary conditions and In order to capture the exponential decrease of its solution we use the Laguerre–Gauss–Radau collocation method and infer its convergence. We show that the value of u ʹ at origin, which plays a fundamental role in these problems, definitely satisfies some rigorous accepted bounds. A particular attention is paid to the Thomas–Fermi case, i.e. We treat the problems as boundary values ones without any involvement of ones with initial values. The method is robust with respect to scaling and order of approximation.

Shifted Legendre spectral collocation technique for solving stochastic Volterra–Fredholm integral equations

Abstract This paper addresses a spectral collocation technique to treat the stochastic Volterra–Fredholm integral equations (SVF-IEs). The shifted Legendre–Gauss–Radau collocation (SL-GR-C) method is developed for approximating the FSV-IDEs. The principal target in our technique is to transform the SVF-IEs to a system of algebraic equations. For computational purposes, the Brownian motion W(x) is discretized by Lagrange interpolation. While the integral terms are interpolated by Legendre–Gauss–Lobatto quadrature. Some numerical examples are given to test the accuracy and applicability of our technique. Also, an error analysis is introduced for the proposed method.

A Successive Pseudospectral-Based Approximation of the Solution of Regulator Equations

Through the combination of contraction mapping and pseudospectral method, we propose a successive approximation technique to approximate the solution of a class of regulator equations with periodic exosystems and hyperbolic zero dynamics. In this scheme, the initial points of flows on the zero-error constrained manifolds are approximated successively as the fixed point of a contractive integral mapping. Accordingly, flows are obtained by utilizing the scaled Fourier–Gauss–Radau collocation method. Appropriate error analysis, in association with both regulation error and the error resulting from the approximate solution of the center manifold equation, is provided. Our analysis shows that the regulation error becomes negligible as we start the approximation process at an adequately large order and maintain it for a proper number of iterations.

Numerical solutions for fractional initial value problems of distributed-order

This paper addresses spectral collocation techniques to treat with the fractional initial value problem of distributed-order. We introduce three algorithms based on shifted fractional order Jacobi orthogonal functions outputted by Jacobi polynomials. The shifted fractional order Jacobi–Gauss–Radau collocation method is developed for approximating the fractional initial value problem of distributed-order. The principal target in our techniques is to transform the fractional initial value problem of distributed-order to a system of algebraic equations. Some numerical examples are given to test the accuracy and applicability of our technique. It is known that the accuracy of numerical approaches for nonsmooth solution is deteriorated. Employing fractional order Jacobi functions instead of the classical Jacobi stopped this deterioration.

Modified Legendre–Gauss–Radau Collocation Method for Optimal Control Problems with Nonsmooth Solutions

A new method is developed for solving optimal control problems whose solutions are nonsmooth. The method developed in this paper employs a modified form of the Legendre–Gauss–Radau orthogonal direct collocation method. This modified Legendre–Gauss–Radau method adds two variables and two constraints at the end of a mesh interval when compared with a previously developed standard Legendre–Gauss–Radau collocation method. The two additional variables are the time at the interface between two mesh intervals and the control at the end of each mesh interval. The two additional constraints are a collocation condition for those differential equations that depend upon the control and an inequality constraint on the control at the endpoint of each mesh interval. The additional constraints modify the search space of the nonlinear programming problem such that an accurate approximation to the location of the nonsmoothness is obtained. The transformed adjoint system of the modified Legendre–Gauss–Radau method is then developed. Using this transformed adjoint system, a method is developed to transform the Lagrange multipliers of the nonlinear programming problem to the costate of the optimal control problem. Furthermore, it is shown that the costate estimate satisfies one of the Weierstrass–Erdmann optimality conditions. Finally, the method developed in this paper is demonstrated on an example whose solution is nonsmooth.

Shifted fractional Legendre spectral collocation technique for solving fractional stochastic Volterra integro-differential equations

This paper presents a spectral collocation technique to solve fractional stochastic Volterra integro-differential equations (FSV-IDEs). The algorithm is based on shifted fractional order Legendre orthogonal functions generated by Legendre polynomials. The shifted fractional order Legendre–Gauss–Radau collocation (SFL-GR-C) method is developed for approximating the FSV-IDEs, with the objective of obtaining a system of algebraic equations. For computational purposes, the Brownian motion function W(x) is discretized by Lagrange interpolation, while the integral terms are interpolated by Legendre–Gauss–Lobatto quadrature. Numerical examples demonstrate the accuracy and applicability of the proposed technique, even when dealing with non-smooth solutions.

Accurate spectral algorithm for two‐dimensional variable‐order fractional percolation equations

A highly accurate spectral algorithm for (2 + 1) fractional percolation equations with variable order (VO‐FPEs) is considered. We propose a shifted Legendre–Gauss–Lobatto collocation (SL‐GL‐C) method in conjunction with shifted Chebyshev–Gauss–Radau collocation (SC‐GR‐C) method to solve the two‐dimensional VO‐FPEs. A complete theoretical formulation is presented, and numerical results are given to illustrate the performance and efficiency of the algorithm. The superiority of the scheme to tackle VO‐FPEs is revealed, even when dealing with non‐smooth time solutions.

Generalized Lagrange–Jacobi–Gauss–Radau collocation method for solving a nonlinear optimal control problem with the classical diffusion equation

In this paper, a well-known nonlinear optimal control problem (OCP) arisen in a variety of biological, chemical, and physical applications is considered. The quadratic form of the nonlinear cost function is endowed with the state and control functions. In this problem, the dynamic constraint of the system is given by a classical diffusion equation. This article is concerned with a generalization of Lagrange functions. Besides, a generalized Lagrange–Jacobi–Gauss–Radau (GLJGR) collocation method is introduced and applied to solve the aforementioned OCP. Based on initial and boundary conditions, the time and space variables, t and x, are clustered with Jacobi–Gauss–Radau points. Then, to solve the OCP, Lagrange multipliers are used and the optimal control problem is reduced to a parameter optimization problem. Numerical results demonstrate its accuracy, efficiency, and versatility of the presented method.

Coordinated Optimization of Generation and Compensation to Enhance Short-Term Voltage Security of Power Systems Using Accelerated Multi-Objective Reinforcement Learning

High proportions of asynchronous motors in demand-side have pressured heavily on short-term voltage security of receiving-end power systems. To enhance short-term voltage security, this paper coordinates the optimal outputs of generation and compensation in a multi-objective dynamic optimization model. With equipment dynamics, network load flows, lower and upper limitations, and security constraints considered, this model simultaneously minimizes two objectives: the expense of control decision and the voltage deviation. The Radau collocation method is employed to handle dynamics, by transforming all differential algebraic equations into algebraic ones. Most importantly, Pareto solutions are obtained through an accelerated multi-objective reinforcement learning (AMORL) method by filtering the dominated solutions. The entire feasible region is partitioned into small independent regions, to eliminate the scope for Pareto solutions. Besides, the AMORL method redefines the state functions and introduces creative state sensitivities, which accelerate the switch from learning to applying, once the agent accumulates sufficient knowledge. Furthermore, Pareto solutions are diversified via introducing some potential solutions. Lastly, the Fuzzy decision-making methodology picks up the tradeoff solution. Case studies are implemented on a practical 748-node power grid, which validate the acceleration and efficiency of the AMORL method. The AMORL method is overall superior to conventional reinforcement learning (RL) method with more optimal non-dominated objective values, much shorter CPU time, and better convergence to accurate values. Moreover, compared with another three state-of-the-art RL methods, the AMORL method takes almost the same CPU time of several seconds, but is slightly superior to the state-of-the-art methods in terms of optimal objective values. Additionally, the calculated values of the AMORL method fit the best with the accurate values during each iteration, resulting in a good convergence.

Optimal Scheduling of Active Distribution Networks With Limited Switching Operations Using Mixed-Integer Dynamic Optimization

The optimal scheduling of active distribution networks (ADNs) significantly enhances voltage security and reduces costs, particularly as the numbers of distributed generation sources and energy storage devices increase. Therefore, this paper proposes a mixed-integer dynamic optimization (MIDO) model for the optimal scheduling of ADNs. This model incorporates loads and distributed generation outputs with continuous trajectories and aims to provide optimal continuous-trajectory schemes for ADNs. The optimization is conducted with the objective of minimizing the daily costs of electricity purchased from distribution substations. However, in practice, discrete control devices are required to adopt a limited number of switching operations, which increase the computational complexity of the MIDO model. Hence, a reduced convex relaxation method is utilized to achieve reduced convex transformation and tight relaxation of the MIDO model with respect to integer variables. This converts the MIDO model into a continuous dynamic optimization model, which is then further approximated as a nonlinear programming model using the Radau collocation method. Meanwhile, the absolute-value constraints limiting the number of switching operations are eliminated by an equivalent conversion to a series of linear inequalities. Numerical simulations on IEEE 33-bus, PG&E 69-bus, and real-world 110-bus ADNs demonstrate the effectiveness and efficiency of the proposed methodology.

Highly accurate numerical schemes for multi-dimensional space variable-order fractional Schrödinger equations

As a natural generalization of the fractional Schrödinger equation, the variable-order fractional Schrödinger equation has been exploited to study fractional quantum phenomena. In this paper, we develop an exponentially accurate Jacobi–Gauss–Lobatto collocation (J–GL-C) method to solve the variable-order fractional Schrödinger equations in one dimension (1D) and two dimensions (2D). In this method, the aforementioned problem is reduced to a system of ordinary differential equations (ODEs) in the time variable. As a result, we propose two efficient schemes for dealing with the numerical solutions of initial value problems for nonlinear system of ordinary differential equations, one based on the implicit Runge–Kutta (IRK) method of fourth order and the other based on Jacobi–Gauss–Radau collocation (J–GR-C) method. The validity and effectiveness of the two methods are demonstrated by solving three numerical examples in 1D and 2D. The convergence of the methods is graphically analyzed. The results demonstrate that the proposed methods are powerful algorithms with high accuracy for solving the variable-order nonlinear partial differential equations.

Solution of time-varying delay systems using an adaptive collocation method

An adaptive Legendre–Gauss–Radau collocation method is presented for solving time-varying delay systems. By dividing the domain of the problem into a uniform mesh, the delay system is replaced with a sequence of initial value problems. The method developed in this paper adjusts both the mesh spacing and the degree of the polynomial on each mesh interval in order a specified error tolerance is satisfied. We investigate some error estimates on the whole domain of interest. Some numerical examples are given and comparisons with other methods are made to demonstrate the efficiency and accuracy of the proposed method.

A Nonclassical Radau Collocation Method for Nonlinear Initial‐Value Problems with Applications to Lane‐Emden Type Equations

We propose a numerical method for solving nonlinear initial‐value problems of Lane‐Emden type. The method is based upon nonclassical Gauss‐Radau collocation points, and weighted interpolation. Nonclassical orthogonal polynomials, nonclassical Radau points and weighted interpolation are introduced on arbitrary intervals. Then they are utilized to reduce the computation of nonlinear initial‐value problems to a system of nonlinear algebraic equations. We also present the comparison of this work with some well‐known results and show that the present solution is very accurate.