Pseudospectral Chebyshev Optimal Control of Constrained Nonlinear Dynamical Systems

A pseudospectral method for generating optimal trajectories of linear and nonlinear constrained dynamic systems is proposed. The method consists of representing the solution of the optimal control problem by an mth degree interpolating polynomial, using Chebyshev nodes, and then discretizing the problem using a cell-averaging technique. The optimal control problem is thereby transformed into an algebraic nonlinear programming problem. Due to its dynamic nature, the proposed method avoids many of the numerical difficulties typically encountered in solving standard optimal control problems. Furthermore, for discontinuous optimal control problems, we develop and implement a Chebyshev smoothing procedure which extracts the piecewise smooth solution from the oscillatory solution near the points of discontinuities. Numerical examples are provided, which confirm the convergence of the proposed method. Moreover, a comparison is made with optimal solutions obtained by closed-form analysis and/or other numerical methods in the literature.     

On certain optimization methods with finite-step inner algorithms for convex finite-dimensional problems with inequality constraints

Numerical methods are proposed for solving finite-dimensional convex problems with inequality constraints satisfying the Slater condition. A method based on solving the dual to the original regularized problem is proposed and justified for problems having a strictly uniformly convex sum of the objective function and the constraint functions. Conditions for the convergence of this method are derived, and convergence rate estimates are obtained for convergence with respect to the functional, convergence with respect to the argument to the set of optimizers, and convergence to the g-normal solution. For more general convex finite-dimensional minimization problems with inequality constraints, two methods with finite-step inner algorithms are proposed. The methods are based on the projected gradient and conditional gradient algorithms. The paper is focused on finite-dimensional problems obtained by approximating infinite-dimensional problems, in particular, optimal control problems for systems with lumped or distributed parameters.     

Numerical methods for solving terminal optimal control problems

Numerical methods for solving optimal control problems with equality constraints at the right end of the trajectory are discussed. Algorithms for optimal control search are proposed that are based on the multimethod technique for finding an approximate solution of prescribed accuracy that satisfies terminal conditions. High accuracy is achieved by applying a second-order method analogous to Newton’s method or Bellman’s quasilinearization method. In the solution of problems with direct control constraints, the variation of the control is computed using a finite-dimensional approximation of an auxiliary problem, which is solved by applying linear programming methods.     

Fast automatic differentiation as applied to the computation of second derivatives of composite functions

A technique for deriving formulas for the second derivatives of a composite function with constrained variables is proposed. The original system of constraint equations is associated with a linear system of equations, whose solution is used to determine the Hessian of the function. The resulting formulas are applied to discrete problems obtained by approximating optimal control problems with the use of Runge-Kutta methods of various orders. For a particular optimal control problem, the numerical results obtained by the gradient method and Newton’s method with the resulting formulas are described and analyzed in detail.     

Finite-time control of discrete-time systems with time-varying exogenous disturbance

This paper deals with some finite-time control problems for uncertain discrete-time linear systems subject to exogenous disturbance. Sufficient conditions are presented for finite-time stabilization via state feedback. These conditions can be reduced to feasibility problems involving linear matrix inequality (LMI). A detailed solving method is proposed for the restricted linear matrix inequalities. Finally, an example illustrates the proposed methodology.     

Parametric continuation method with correction and its applications

The article discusses the parametric continuation method for nonlinear equations. A continuation algorithm with correction is proposed, an approximation accuracy theorem is proved, and issues of efficient numerical implementation are considered. An approach is described to the application of the continuation method for seeking the Pontryagin extremal solution in the optimal control problem. Algorithms developed by the author are applied to optimal control problems nonlinear in control, to problems with a nonsmooth control region, and to affine problems with mixed constraints. Translated from Prikladnaya Matematika i Informatika, No. 30, 2008, pp. 55–94.     

Control parameterization enhancing transform for optimal control of switched systems

In this paper, a class of optimal switching control problems with prespecified order of the sequence of subsystems is considered, where the switching instants are included in the cost functional. Both the switching instants and the control function are to be chosen such that the cost functional is minimized. Through the discretization of the control space, each control component is approximated by a piecewise constant function. The partition points and the heights of each of these piecewise constant functions are taken as decision varibles. Using the control parameterization enhancing transform, we map both types of switching instants into preassigned knot points via the introduction of an additional control, known as the enhancing control. In this way, we construct a sequence of approximate optimal parameter selection problems with fixed switching time points. We then show that these approximate optimal parameter selection problems are solvable as mathematical programming problems. The convergence analysis of this approximation is investigated. Two examples are solved using the proposed method so as to demonstrate the effectiveness of the method proposed.     

Convergence of approximations vs. regularity of solutions for convex,control-constrained optimal-control problems

A method of estimating the rate of convergence of approximation to convex, control-constrained optimal-control problems is proposed. In the method, conditions of optimality involving projections on the set of admissible control are exploited. General results are illustrated by examples of Galerkin-type approximations to optimal-control problems for parabolic systems.     

基于非均匀参数化的自由终端时间最优控制问题求解

针对自由终端时间最优控制问题,提出了一种基于非均匀控制向量参数化的数值解法.将控制时域离散化为不同长度的时间段,各时间段长度作为新的控制变量.通过引入标准化的时间变量,原问题转化为均匀参数化的固定终端时间最优控制问题.建立目标和约束函数的Hamilton函数,通过求解伴随方程获得目标和约束函数的梯度,采用序列二次规划(SQP)获得数值解.针对两个经典的化工过程自由终端时间最优控制问题进行仿真研究,验证了所提出算法的可行性和有效性.     

A Globally Convergent and Efficient Method for Unconstrained Discrete-Time Optimal Control

Shift schemes are commonly used in non-convex situations when solving unconstrained discrete-time optimal control problems by the differential dynamic programming (DDP) method. However, the existing shift schemes are inefficient when the shift becomes too large. In this paper, a new method of combining the DDP method with a shift scheme and the steepest descent method is proposed to cope with non-convex situations. Under certain assumptions, the proposed method is globally convergent and has q-quadratic local conve rgence. Extensive numerical experiments on many test problems in the literature are reported. These numerical results illustrate the robustness and efficiency of the proposed method.     

Numerical optimization methods for controlled systems with parameters

First- and second-order numerical methods for optimizing controlled dynamical systems with parameters are discussed. In unconstrained-parameter problems, the control parameters are optimized by applying the conjugate gradient method. A more accurate numerical solution in these problems is produced by Newton’s method based on a second-order functional increment formula. Next, a general optimal control problem with state constraints and parameters involved on the righthand sides of the controlled system and in the initial conditions is considered. This complicated problem is reduced to a mathematical programming one, followed by the search for optimal parameter values and control functions by applying a multimethod algorithm. The performance of the proposed technique is demonstrated by solving application problems.     

Numerical solution of minimax optimal control problems by multiple shooting technique

This paper presents the application of the multiple shooting technique to minimax optimal control problems (optimal control problems with Chebyshev performance index). A standard transformation is used to convert the minimax problem into an equivalent optimal control problem with state variable inequality constraints. Using this technique, the highly developed theory on the necessary conditions for state-restricted optimal control problems can be applied advantageously. It is shown that, in general, these necessary conditions lead to a boundary-value problem with switching conditions, which can be treated numerically by a special version of the multiple shooting algorithm. The method is tested on the problem of the optimal heating and cooling of a house. This application shows some typical difficulties arising with minimax optimal control problems, i.e., the estimation of the switching structure which is dependent on the parameters of the problem. This difficulty can be overcome by a careful application of a continuity method. Numerical solutions for the example are presented which demonstrate the efficiency of the method proposed.     

Asymptotically suboptimal control of weakly interconnected dynamical systems

Optimal control problems for a group of systems with weak dynamical interconnections between its constituent subsystems are considered. A method for decentralized control is proposed which distributes the control actions between several controllers calculating in real time control inputs only for theirs subsystems based on the solution of the local optimal control problem. The local problem is solved by asymptotic methods that employ the representation of the weak interconnection by a small parameter. Combination of decentralized control and asymptotic methods allows to significantly reduce the dimension of the problems that have to be solved in the course of the control process.     

A general view to relaxation methods in control theory1

A general method of construction of relaxed optimal control problems is proposed. Two standard relaxed problem in optimal control of ordinary differential equation and relation between them are thus obtained as special cases, and besides these relaxed problems are justified in accord with their extremal properties.     

Non-recursive Haar Connection Coefficients Based Approach for Linear Optimal Control

In the present paper, two-fold contributions are made. First, non-recursive formulations of various Haar operational matrices, such as Haar connection coefficients matrix, backward integral matrix, and product matrix are developed. These non-recursive formulations result in computationally efficient algorithms, with respect to execution time and stack-and-memory overflows in computer implementations, as compared to corresponding recursive formulations. This is demonstrated with the help of MATLAB PROFILER. Later, a unified method is proposed, based on these non-recursive connection coefficients, for solving linear optimal control problems of all types, irrespective of order and nature of the system. This means that the single method is capable of optimizing both time-invariant and time-varying linear systems of any order efficiently; it has not been reported in the literature so far. The proposed method is applied to solve finite horizon LQR problems with final state control. Computational efficiency of the proposed method is established with the help of comparison on computation-time at different resolutions by taking several illustrative examples.     

Numerical method for solving a nonlinear time-optimal control problem with additive control

Nonlinear systems whose right-hand sides are divided by the state and control and are linear in control are considered. An iterative method is proposed for solving time-optimal control problems for such systems. The method is based on constructing finite sequences of adjacent simplexes with their vertices lying on the boundaries of reachability sets. For a controllable system, it is proved that the minimizing sequence converges to an ?-optimal solution in a finite number of iterations.     

A semismooth Newton method for topology optimization

This paper deals with elliptic optimal control problems for which the control function is constrained to assume values in {0, 1}. Based on an appropriate formulation of the optimality system, a semismooth Newton method is proposed for the solution. Convergence results are proved, and some numerical tests illustrate the efficiency of the method.     

Numerical solution of singular control problems using multiple shooting techniques

Algorithms for calculating the junction points between optimal nonsingular and singular subarcs of singular control problems are developed. The algorithms consist in formulating appropriate initialvalue and boundary-value problems; the boundary-value problems are solved with the method of multiple shooting. Two examples are detailed to illustrate the proposed numerical methods.The author would like to thank Professor Dr. R. Bulirsch, who stimulated and encouraged this work, which is part of the author's dissertation.     

On a Fourier Method of Embedding Domains Using an Optimal Distributed Control

We propose a domain embedding method to solve second order elliptic problems in arbitrary two-dimensional domains. This method can be easily extended to three-dimensional problems. The method is based on formulating the problem as an optimal distributed control problem inside a rectangle in which the arbitrary domain is embedded. A periodic solution of the equation under consideration is constructed easily by making use of Fourier series. Numerical results obtained for Dirichlet problems are presented. The numerical tests show a high accuracy of the proposed algorithm and the computed solutions are in very good agreement with the exact solutions.     

求解分布控制问题的预处理最小残量方法(英文)

通过分析Bai(Bai Z Z.Block preconditioners for elliptic PDE-constrained optimization problems.Computing,2011,91:379-395)给出的离散分布控制问题的块反对角预处理线性系统,提出了该问题的一个等价线性系统,并且运用带有预处理子的最小残量方法对该系统进行求解.理论分析和数值实验结果表明,所提出的预处理最小残量方法对于求解该类椭圆型偏微分方程约束最优分布控制问题非常有效,尤其当正则参数适当小的时候.