热传导方程的时间最优脉冲控制问题的有限元逼近

时间最优控制问题是一类典型的最优控制问题, 受到研究者的广泛关注. 脉冲控制是一种在工程控制中被广泛应用的控制方式. 偏微分方程描述系统的最优控制问题的数值逼近的收敛性为数值求解方法的可行性提供了定性依据. 本文研究热传导方程的时间最优脉冲控制问题的有限元逼近的收敛性. 通过利用投影算子的特性和系统状态的误差估计, 证明了逼近问题的最优时间收敛到原问题的最优时间. 由此进一步利用原问题最优控制的bangbang性证明了最优控制的收敛性.     

A numerical algorithm for optimal control of a class of hybrid systems: differential transformation based approach

A novel numerical algorithm based on differential transformation is proposed for optimal control of a class of hybrid systems with a predefined mode sequence. From the necessary conditions for optimality of hybrid systems, the hybrid optimal control problem is first converted into a two-point boundary value problem (TPBVP) with additional transverse conditions at the switching times. Then we propose a differential transformation algorithm for solving the TPBVP which may have discontinuities in the state and/or control input at the switching times. Using differential transformation, the hybrid optimal control problem reduces to a problem of solving a system of algebraic equations. The numerical solution is obtained in the form of a truncated Taylor series. By taking advantage of the special properties of the linear subsystems and a quadratic cost functional, the differential transformation algorithm can be further simplified for the switched linear quadratic optimal control problem. We analyse the error of the numerical solution computed by the differential transformation algorithm and some computational aspects are also discussed. The performance of the differential transformation algorithm is demonstrated through illustrative examples. The differential transformation algorithm has been shown to be simple to be implemented and computationally efficient.     

Optimal design for 2D wave equations

In this paper We consider a problem of optimal design in 2D for the wave equation with Dirichlet boundary conditions. We introduce a finite element discrete version of this problem in which the domains under consideration are polygons defined on the numerical mesh. We prove that, as the mesh size tends to zero, any limit, in the sense of the complementary-Hausdorff convergence, of discrete optimal shapes is an optimal domain for the continuous optimal design problem. We work in the functional and geometric setting introduced by V. ?veràk in which the domains under consideration are assumed to have an a priori limited number of holes. We present in detail a numerical algorithm and show the efficiency of the method through various numerical experiments.     

Optimization of systems with pulse-width-modulated control

The problem of finding pulse-width-modulated signals which provide optimal control for a specified plant is considered. A set of necessary conditions which must be satisfied by the optimal control is derived using the calculus of variations. A steepest-descent procedure is developed to obtain a numerical solution to the resulting nonlinear two-point boundary value problem. Special features of the numerical procedure are automatic step size adjustment and the generation of a sequence of performance indices which is monotonic. An example which illustrates the range of application is given.     

A solution of the optimal turn problem of an axially symmetric spacecraft with bounded and pulse control under arbitrary boundary conditions

Given a constraint on the modulus of the control action in the quaternion statement, the optimal turn problem of a spacecraft regarded as a rigid body with one symmetry axis is considered under arbitrary boundary conditions with respect to its attitude and angular velocity. The functional combining the time and the integral value of the control vector spent to the spacecraft turn is used as the optimality criterion. Changes of variables allow reducing (in terms of dynamic Euler equations) the initial optimal turn problem of an axially symmetric spacecraft to the optimal turn problem of a rigid body with spherical mass distribution. Two ways of solving the optimal control problem are proposed. In the first case, the Pontryagin maximum principle is used to obtain the expressions of the optimal control and the adjoint system of equations. In the second case, the passage to the limit with the upper value of the control action increasing indefinitely is performed to construct an analytical solution to the pulse optimal turn problem of a spacecraft that implements the double-pulse control scheme. The original procedure of finding the numerical solution to the continuous optimal turn problem of a spacecraft with bounded control is described, and the examples of calculations are given. The numerical approbation of the proposed analytical algorithm of solving the pulse optimal turn problem of a spacecraft is given     

Remarks on the application of dynamic programming to the optimal path timing of robot manipulators

In this paper, we investigate the use of the dynamic programming approach in the solution of the optimal path timing problem in robotics. This problem is computationally feasible because the path constraint reduces the dimension of the state in the problem to two. The Hamilton–Jacobi–Bellman equation of dynamic programming, a nonlinear first order partial differential equation, is presented and is solved approximately using finite difference methods. Numerical solution of this results in the optimal policy which can then be used to define the optimal path timing by numerical integration. Issues relating to the convergence of the numerical schemes are discussed, and the results are applied to an experimental SCARA manipulator. © 1998 John Wiley & Sons, Ltd.     

主子阵约束下矩阵方程AX=B的对称最小二乘解

本文主要讨论主子阵约束下矩阵方程AX=B的对称最小二乘解．基于投影定理,巧妙的把最小二乘问题转化为等式问题求解,并利用奇异值分解的方法,给出了该对称最小二乘解的一般表达式．此外,文章还考虑了此对称最小二乘解集合对任一给定矩阵的最佳逼近问题,得到了最佳逼近解,并给出了相应的算法步骤和数值例子．     

动态规划-粒子群算法在水库优化调度中的应用

水库优化调度是一个典型的具有多约束条件的、动态的、非线性的优化问题。针对这些问题,利用动态规划-粒子群(DP-PSO)算法加以求解。利用动态规划中的多阶段最优策略原理,将水库优化调度问题转化为多阶段决策子问题,各个子问题采用粒子群算法优化求解。数值实验表明,在计算时段较多时,DP-PSO算法计算的可靠性明显优于一般的动态规划(DP)算法,在计算时间上,DP-PSO算法用时较动态规划-遗传算法(DP-GA)少。     

An optimal control model for a system of degenerate parabolic integro-differential equations

An initial-boundary-value problem for a system of degenerate parabolic integro-differential equations is considered. The sufficient conditions for the existence and uniqueness of its generalized solution and for the existence of at least one optimal control for a given performance functional are obtained. A stable numerical solution to the initial-boundary-value problem is derived for a locally one-dimensional case and conditions are formulated for constructing a stable numerical algorithm of the optimal control problem on a class of piecewise-smooth control functions. __________ Translated from Kibernetika i Sistemnyi Analiz, No. 6, pp. 90–102, November–December 2007.     

Mathematical programs with complementarity constraints in the context of inverse optimal control for locomotion

In this paper an inverse optimal control problem in the form of a mathematical program with complementarity constraints (MPCC) is considered and numerical experiences are discussed. The inverse optimal control problem arises in the context of human navigation where the body is modelled as a dynamical system and it is assumed that the motions are optimally controlled with respect to an unknown cost function. The goal of the inversion is now to find a cost function within a given parametrized family of candidate cost functions such that the corresponding optimal motion minimizes the deviation from given data. MPCCs are known to be a challenging class of optimization problems typically violating all standard constraint qualifications (CQs). We show that under certain assumptions the resulting MPCC fulfills CQs for MPCCs being the basis for theory on MPCC optimality conditions and consequently for numerical solution techniques. Finally, numerical results are presented for the discretized inverse optimal control problem of locomotion using different solution techniques based on relaxation and lifting.     

A global approach for the optimal path generation of redundant robot manipulators

In this article the optimal path generation of redundant robot manipulators is considered as an optimization problem, with given kinematics and subject to the robot requirements and a singularities avoidance constraint. This problem is formulated as a constrained continuous optimal control problem, which allows to consider joints and velocities constraints and/or manipulator dynamics. This approach is exemplified for a planar redundant manipulator and the resultant state constrained problem is solved by an efficient iterative numerical technique.     

线性混杂系统优化控制的Monte Carlo统计预测方法

针对含扩散项的线性混杂切换系统优化控制问题, 为降低优化求解的计算复杂性, 提出了Monte Carlo统计预测方法. 首先通过数值求解技术把连续时间优化控制问题转化为离散时间的Markov决策过程问题; 然后在若干有限状态子空间内, 利用反射边界技术来求解相应子空间的最优控制策略; 最后根据最优控制策略的结构特性, 采用统计预测方法来预测出整个状态空间的最优控制策略. 该方法能有效降低求解涉及大状态空间及多维变量的线性混杂切换系统优化控制的计算复杂性, 文末的仿真结果验证了方法的有效性.     

Numerical simulation of wave processes in inhomogeneous media with impedance boundaries

The paper is concerned with numerical simulation and properties of wave processes in infinite inhomogeneous media with impedance boundaries. A numerical method for solving a boundary-value problem and optimal control problem for a parabolic Schrödinger-type wave equation with a complex nonself-adjoint operator is proposed and analyzed.     

Optimal activation policies for continuous scanning observations in parameter estimation of distributed systems

The problem of determining an optimal measurement scheduling for identification of unknown parameters in distributed systems described by partial differential equations is discussed. The discrete-scanning observations are performed by an optimal selection of measurement data from spatially fixed sensors. In the adopted approach, the sensor scheduling problem is converted to a constrained optimal control problem. In this framework, the control value represents the selected sensor configuration. Thus the control variable is constrained to take values in a discrete set and switchings between sensors may occur in continuous time. By applying the control parameterization enhancing transform technique, a computational procedure for solving the optimal scanning measurement problem is obtained. The numerical scheme is then tested on a computer example regarding an advection-diffusion problem.     

Optimal stopping and control with two kinds of boundary conditions: application to dynamic routeing in networks

A stochastic optimal stopping and optimal control problem concerning the dynamic routeing of a randomly perturbed flow is considered. Equations for the solution of the optimal stopping and control problem are given. These equations lead to the solution of an elliptic problem with a given condition on a ‘ free boundary’ within a given domain D and a condition given on the boundary ? D of D. An example is given and a numerical study is conducted on it     

Impulse control of piecewise-deterministic processes

In a recent paper we presented a numerical technique for solving the optimal stopping problem for a piecewise-deterministic process (P.D.P.) by discretization of the state space. In this paper we apply these results to the impulse control problem. In the first part of the paper we study the impulse control of P.D.P.s. under general conditions. We show that iteration of the single-jump-or-intervention operator generates a sequence of functions converging to the value function of the problem. In the second part of the paper we present a numerical technique for computing optimal impulse controls for P.D.P.s. This technique reduces the problem to a sequence of one-dimensional minimizations. We conclude by presenting some numerical examples.     

Near optimal control for a class of stochastic hybrid systems

To address a computationally intractable optimal control problem for a class of stochastic hybrid systems, this paper proposes a near optimal state feedback control scheme, which is constructed by using a statistical prediction method based on approximate numerical solution that samples over the entire state space. A numerical example illustrates the potential of the approach.     

Constrained optimal control of switched systems based on modified BFGS algorithm and filled function method

In this paper, we consider an optimal control problem of switched systems with a continuous-time inequality constraint. Because of the complexity of this constraint, it is difficult to solve this problem by standard optimization techniques. To overcome this difficulty, the problem is divided into a bi-level optimization problem involving a combination of a continuous-time optimal control problem and a discrete optimization problem. Then, a modified Broyden-Fletcher-Goldfarb-Shanno algorithm and a discrete filled function method is first proposed to solve this bi-level optimization problem. Finally, a numerical example is presented to illustrate the efficiency of our method.     

Optimal control of LQR for discrete time-varying systems with input delays

In this work, we consider the optimal control problem of linear quadratic regulation for discrete time-variant systems with single input and multiple input delays. An innovative and simple method to derive the optimal controller is given. The studied problem is first equivalently converted into a problem subject to a constraint condition. Last, with the established duality, the problem is transformed into a static mathematical optimisation problem without input delays. The optimal control input solution to minimise performance index function is derived by solving this optimisation problem with two methods. A numerical simulation example is carried out and its results show that our two approaches are both feasible and very effective.