A ph mesh refinement method for optimal control

A mesh refinement method is described for solving a continuous‐time optimal control problem using collocation at Legendre–Gauss–Radau points. The method allows for changes in both the number of mesh intervals and the degree of the approximating polynomial within a mesh interval. First, a relative error estimate is derived based on the difference between the Lagrange polynomial approximation of the state and a Legendre–Gauss–Radau quadrature integration of the dynamics within a mesh interval. The derived relative error estimate is then used to decide if the degree of the approximating polynomial within a mesh should be increased or if the mesh interval should be divided into subintervals. The degree of the approximating polynomial within a mesh interval is increased if the polynomial degree estimated by the method remains below a maximum allowable degree. Otherwise, the mesh interval is divided into subintervals. The process of refining the mesh is repeated until a specified relative error tolerance is met. Three examples highlight various features of the method and show that the approach is more computationally efficient and produces significantly smaller mesh sizes for a given accuracy tolerance when compared with fixed‐order methods. Copyright © 2014 John Wiley & Sons, Ltd.     

A modified pseudospectral scheme for accurate solution of Bang‐Bang optimal control problems

In the present contribution, a modified Legendre pseudospectral scheme for accurate and efficient solution of bang‐bang optimal control problems is investigated. In this scheme control and state functions are considered as piecewise constant and piecewise continuous polynomials, respectively, and the switching points are also taken as decision variables. Furthermore, for simplicity in discretization, the integral formulation of the dynamical equations is considered. Thereby, the problem is converted into a mathematical programming problem which can be solved by the well‐developed parameter optimization algorithms. The main advantages of the present method are that: (i) it obtains good results even by using a small number of collocation points and the rate of convergence is high; (ii) the switching times can be captured accurately; and (iii) the wrongly chosen number of switching points can be detected by the results of the method. These are illustrated through a numerical implementation of the method on three examples and the efficiency of the method is reported. Copyright © 2010 John Wiley & Sons, Ltd.     

Multi‐player nonzero‐sum Nash differential game: variation and pseudo‐spectral method

A novel method is presented to solve the nonzero‐sum multi‐player Nash differential game. It combines the use of the variation and Legendre pseudo‐spectral methods. By the variation method, the original game is converted into a regular optimal control problem, avoiding the need to solve the associated Hamilton–Jacobi equation. Then the latter problem is converted into a common nonlinear programming problem via the Legendre pseudo‐spectral method, by which the saddle‐point for the original game can be achieved accurately. As an illustration, the air combat between two pursuers and an evader is formulated as a nonzero‐sum differential game. The simulation results show that numerical solutions can converge to the saddle‐points from different initial conditions, which demonstrates the feasibility and validity of the proposed method. Because the solution process requires little computational time, this method will allow for the development of a real time air combat control strategy. In addition, the simulations show that if the initial states of the two pursuers are fixed, there is an optimal initial heading angle for the evader to delay the interception time most effectively. Copyright © 2016 John Wiley & Sons, Ltd.     

Finding initial costates in finite‐horizon nonlinear‐quadratic optimal control problems

A procedure for obtaining the initial value of the costate in a regular, finite‐horizon, nonlinear‐quadratic problem is devised in dimension one. The optimal control can then be constructed from the solution to the Hamiltonian equations, integrated on‐line. The initial costate is found by successively solving two first‐order, quasi‐linear, partial differential equations (PDEs), whose independent variables are the time‐horizon duration T and the final‐penalty coefficient S. These PDEs need to be integrated off‐line, the solution rendering not only the initial condition for the costate sought in the particular (T, S)‐situation but also additional information on the boundary values of the whole two‐parameter family of control problems, that can be used for design purposes. Results are tested against exact solutions of the PDEs for linear systems and also compared with numerical solutions of the bilinear‐quadratic problem obtained through a power‐series' expansion approach. Bilinear systems are specially treated in their character of universal approximations of nonlinear systems with bounded controls during finite time‐periods. Copyright © 2007 John Wiley & Sons, Ltd.     

Structure‐preserving local optimal control of mechanical systems

While system dynamics are usually derived in continuous time, respective model‐based optimal control problems can only be solved numerically, ie, as discrete‐time approximations. Thus, the performance of control methods depends on the choice of numerical integration scheme. In this paper, we present a first‐order discretization of linear quadratic optimal control problems for mechanical systems that is structure preserving and hence preferable to standard methods. Our approach is based on symplectic integration schemes and thereby inherits structure from the original continuous‐time problem. Starting from a symplectic discretization of the system dynamics, modified discrete‐time Riccati equations are derived, which preserve the Hamiltonian structure of optimal control problems in addition to the mechanical structure of the control system. The method is extended to optimal tracking problems for nonlinear mechanical systems and evaluated in several numerical examples. Compared to standard discretization, it improves the approximation quality by orders of magnitude. This enables low‐bandwidth control and sensing in real‐time autonomous control applications.     

Parametric uncertainty and disturbance attenuation in the suboptimal control of a non‐linear electrochemical process

The optimal control of the hydrogen evolution reactions is attempted for the regulation and change of set‐point problems, taking into account that model parameters are uncertain and I/O signals are corrupted by noise. Bilinear approximations are constructed, and their dimension eventually increased to meet accuracy requirements with respect to the trajectories of the original plant. The current approximate model is used to evaluate the optimal feedback through integration of the Hamiltonian equations. The initial value for the costate is found by solving a state‐dependent algebraic Riccati equation, and the resulting control is then suboptimal for the electrochemical process. The bilinear model allows for an optimal Kalman–Bucy filter application to reduce external noise. The filtered output is reprocessed through a non‐linear observer in order to obtain a state‐estimation as independent as possible from the bilinear model. Uncertainties on parameters are attenuated through an adaptive control strategy that exploits sensitivity functions in a novel fashion. The whole approach to this control problem can be applied to a fairly general class of non‐linear continuous systems subject to analogous stochastic perturbations. All calculations can be handled on‐line by standard ordinary differential equations integration software. Copyright © 2007 John Wiley & Sons, Ltd.     

Distributed optimal control of viscous Burgers' equation via a high-order,linearization, integral,nodal discontinuous Gegenbauer-Galerkin method

We developed a novel direct optimization method to solve distributed optimal control of viscous Burgers' equation over a finite-time horizon by minimizing the distance between the state function and a desired target state profile along with the energy of the control. Through a novel linearization strategy, well-conditioned integral reformulations, optimal Gegenbauer barycentric quadratures, and nodal discontinuous Galerkin discretizations, the method reduces such optimal control problems into finite-dimensional, nonlinear programming problems subject to linear algebraic system of equations and discrete mixed path inequality constraints that can be solved easily using standard optimization software. The proposed method produces “an auxiliary control function” that provides a useful model to explicitly define the optimal controller of the state variable. We present an error analysis of the semidiscretization and full discretization of the weak form of the reduced equality constraint system equations to demonstrate the exponential convergence of the method. The accuracy of the proposed method is examined using two numerical examples for various target state functions in the existence/absence of control bounds. The proposed method is exponentially convergent in both space and time, thus producing highly accurate approximations using a significantly small number of collocation points.     

Optimal control of non‐linear chemical reactors via an initial‐value Hamiltonian problem

The problem of designing strategies for optimal feedback control of non‐linear processes, specially for regulation and set‐point changing, is attacked in this paper. A novel procedure based on the Hamiltonian equations associated to a bilinear approximation of the dynamics and a quadratic cost is presented. The usual boundary‐value situation for the coupled state–costate system is transformed into an initial‐value problem through the solution of a generalized algebraic Riccati equation. This allows to integrate the Hamiltonian equations on‐line, and to construct the feedback law by using the costate solution trajectory. Results are shown applied to a classical non‐linear chemical reactor model, and compared against suboptimal bilinear‐quadratic strategies based on power series expansions. Since state variables calculated from Hamiltonian equations may differ from the values of physical states, the proposed control strategy is suboptimal with respect to the original plant. Copyright © 2005 John Wiley & Sons, Ltd.     

Fractional Chebyshev pseudospectral method for fractional optimal control problems

In this paper, we introduce and apply a fractional pseudospectral method for indirectly solving a generic form of fractional optimal control problems. By employing the fractional Lagrange interpolating functions and discretizing the necessary optimality conditions at Chebyshev‐Gauss‐Lobatto points, the problem is converted into an algebraic system. By solving this system, the optimal solution of the main fractional optimal control problem is approximated. Finally, in some numerical examples, we show the applicability, efficiency, and accuracy of the proposed method comparing with some other methods.     

A variable time transformation method for mixed‐integer optimal control problems

The article discusses a variable time transformation method for the approximate solution of mixed‐integer non‐linear optimal control problems (MIOCP). Such optimal control problems enclose real‐valued and discrete‐valued controls. The method transforms MIOCP using a discretization into an optimal control problem with only real‐valued controls. The latter can be solved efficiently by direct shooting methods. Numerical results are obtained for a problem from automobile test‐driving that involves a discrete‐valued control for the gear shift of the car. The results are compared to those obtained by Branch&Bound and show a drastic reduction of computation time. This very good performance makes the suggested method applicable even for many discretization points. Copyright © 2006 John Wiley & Sons, Ltd.     

Numerical investigation of distributed‐order fractional optimal control problems via Bernstein wavelets

The aim of this article is to investigate an efficient computational method for solving distributed‐order fractional optimal control problems. In the proposed method, a new Riemann‐Liouville fractional integral operator for the Bernstein wavelet is given. This approach is based on a combination of the Bernstein wavelets basis, fractional integral operator, Gauss‐Legendre numerical integration, and Newton's method for solving obtained system. Easy implementation, simple operations, and accurate solutions are the essential features of the proposed method. The error analysis of the proposed method is carried out. Examples reveal the applicability of the proposed technique.     

Short communication: A collocation-type method for linear quadratic optimal control problems

This communication presents a spectral method for solving time-varying linear quadratic optimal control problems. Legendre–Gauss–Lobatto nodes are used to construct the mth-degree polynomial approximation of the state and control variables. The derivative x ·(t) of the state vector x (t) is approximaed by the analytic derivative of the corresponding interpolating polynomial. The performance index approximation is based on Gauss–Lobatto integration. The optimal control problem is then transformed into a linear programming problem. The proposed technique is easy to implement, efficient and yields accurate results. Numerical examples are included and a comparison is made with an existing result.     

Energy‐optimal trajectory planning for the Pendubot and the Acrobot

In this paper, we study the trajectory‐planning problem for planar underactuated robot manipulators with two revolute joints in the presence of gravity. This problem is studied as an optimal control problem. We consider both possible models of planar underactuated robot manipulators, namely with the elbow joint not actuated, called Pendubot, and with the shoulder joint not actuated, called Acrobot. Our method consists of a numerical resolution of a reformulation of the optimal control problem, as an unconstrained calculus of variations problem, in which the dynamic equations of the mechanical system are regarded as constraints and treated by means of special derivative multipliers. We solve the resulting calculus of variations problem by using a numerical approach based on the Euler–Lagrange necessary condition in integral form. In which, time is discretized, and admissible variations for each variable are approximated using a linear combination of the piecewise continuous basis functions of time. In this way, a general method for the solution of optimal control problems with constraints is obtained, which, in particular, can deal with nonintegrable differential constraints arising from the dynamic models of underactuated planar robot manipulators with two revolute joints in the presence of gravity. Copyright © 2012 John Wiley & Sons, Ltd.     

Optimal control for discrete‐time singular stochastic systems with input delay

The finite‐horizon linear quadratic regulation problem is considered in this paper for the discrete‐time singular systems with multiplicative noises and time delay in the input. Firstly, the extremum principle is discussed, and a stationary condition is derived for the singular stochastic system. Then, based on the relationships established between the state and the costate variables, the stationary condition is also shown to be a sufficient criterion assuring the existence of the solution for the stochastic control problem. The optimal controller is designed as a linear function of the current state and the past inputs information, which can be recursively calculated by effective algorithms. With the designed optimal controllers, the explicit expression is also derived for the minimal value of the performance index. One numerical example is provided in the end of the paper to illustrate the effectiveness of the obtained results. Copyright © 2016 John Wiley & Sons, Ltd.     

Linear quadratic regulation for discrete‐time systems with multiplicative noise and multiple input delays

This paper is concerned with the optimal linear quadratic regulation problem for discrete‐time systems with state and control dependent noises and multiple delays in the input. We show that the problem admits a unique solution if and only if a sequence of matrices, which are determined by coupled difference equations developed in this paper, are positive definite. Under this condition, the optimal feedback controller and the optimal cost are presented via some coupled difference equations. Our approach is based on the stochastic maximum principle. The key technique is to establish relations between the costate and the state. Copyright © 2016 John Wiley & Sons, Ltd.     

Solution of a specific class of nonlinear fractional optimal control problems including multiple delays

This research provides a new framework based on a hybrid of block‐pulse functions and Legendre polynomials for the numerical examination of a special class of scalar nonlinear fractional optimal control problems involving delay. The concepts of the fractional derivative and the fractional integral are employed in the Caputo sense and the Riemann‐Liouville sense, respectively. In accordance with the notion of the Riemann‐Liouville integral, we derive a new integral operator related to the proposed basis called the operational matrix of fractional integration. By employing two significant operators, namely, the delay operator and the integral operator connected to the hybrid basis, the system dynamics of the primal optimal control problem converts to another system involving algebraic equations. Consequently, the optimal control problem under study is reduced to a static optimization one that is solved by existing well‐established optimization procedures. Some new theoretical results regarding the new basis are obtained. Various kinds of fractional optimal control problems containing delay are examined to measure the accuracy of the new method. The simulation results justify the merits and superiority of the devised procedure over the existing optimization methods in the literature.     

Space-time a posteriori error analysis of finite element approximation for parabolic optimal control problems: A reconstruction approach

We derive space-time a posteriori error estimates of finite element method for linear parabolic optimal control problems in a bounded convex polygonal domain. To discretize the control problem, we use piecewise linear and continuous finite elements for the approximations of the state and costate variables, whereas piecewise constant functions are employed for the control variable. The temporal discretization is based on the backward Euler implicit scheme. An elliptic reconstruction technique in conjunction with energy argument is used to derive a posteriori error estimates for the state, costate, and control variables in the L^∞(0,T;L²(Ω))-norm. Moreover, numerical experiments are performed to illustrate the performance of the derived estimators.     

Numerical solution of a class of two‐dimensional quadratic optimal control problems by using Ritz method

In this paper, we focus on a class of a two‐dimensional optimal control problem with quadratic performance index (cost function). We are going to solve the problem via the Ritz method. The method is based upon the Legendre polynomial basis. The key point of the Ritz method is that it provides greater flexibility in the initial and non‐local boundary conditions. By using this method, the given two‐dimensional continuous‐time quadratic optimal control problem is reduced to the problem of solving a system of algebraic equations. We extensively discuss the convergence of the method and finally present our numerical findings and demonstrate the efficiency and applicability of the numerical scheme by considering three examples. Copyright © 2015 John Wiley & Sons, Ltd.     

Quadratic and linear controls developing an optimal treatment for the use of BCG immunotherapy in superficial bladder cancer

Our prime objective of the study is to exhibit the advantage to introduce a quadratic control in place of linear control in a cost function to be minimized, and that is associated to an optimal control problem that we formulate for a pre‐validated model of bacillus Calmette‐Guérin (BCG) immunotherapy in superficial bladder cancer. The compartmental model of interest is in the form of a nonlinear system of four ordinary differential equations that describe interactions between the used BCG strain, tumor cells, and immune responses. Previous studies reported that the optimal dose of BCG for treating bladder cancer is yet unknown. Hence, we aim to establish the optimization approach that can be applied for determining the values of the optimal BCG concentration along the therapy period to stimulate immune‐system cells and reduce cancer cells growth during BCG intravesical therapy. Pontryagin's maximum principle and the generalized Legendre–Clebsch condition are employed to provide the explicit formulations of the sought optimal controls. The optimality system is resolved numerically based on a fourth‐order iterative Runge–Kutta progressive‐regressive scheme, which is used to solve a two‐point boundary value problem. Copyright © 2015 John Wiley & Sons, Ltd.     

A computational method for a general class of optimal control problems involving integrodifferential equations

In this paper we consider a general class of optimal control problems involving integrodifferential equations. The integral equation component is a Volterra integral equation with convolution kernel. A method is proposed to approximate the kernel which gives rise to a system of ordinary differential equations approximating the Volterra integral equation. Thus the optimal control problem is approximated by a sequence of standard optimal control problems involving only differential equations. Each of the approximate problems is solvable by any of the existing methods for standard optimal control problems, such as the gradient restoration algorithm and the control parametrization algorithm. Convergence analysis is also carried out to justify the proposed approximation. As an application we consider an optimal control problem involving production and marketing systems with distributed time lags. This problem is then solved numerically using the proposed method.