An iterative method for suboptimal control of a class of nonlinear time-delayed systems

This work presents an approximate solution method for the infinite-horizon nonlinear time-delay optimal control problem. A variational iteration method (VIM) is applied to design feedforward and feedback optimal controllers. By using the VIM, the original optimal control is transformed into a sequence of nonhomogeneous linear two-point boundary value problems (TPBVPs). The existence and uniqueness of the optimal control law are proved. The optimal control law obtained consists of an accurate linear feedback term and a nonlinear compensation term which is the limit of an adjoint vector sequence. The feedback term is determined by solving Riccati matrix differential equation. By using the finite-step iteration of a nonlinear compensation sequence, we can obtain a suboptimal control law. Simulation results demonstrate the validity and applicability of the VIM.     

状态定义粒子群算法在非线性方程组中的应用

提出了一种新颖的状态定义粒子群优化算法。该算法针对粒子群算法容易陷入局部最优和搜索精度不高的缺点,结合爬山算法和粒子群算法的特点,根据粒子状态的实时更新采用不同的搜索方法,在迭代过程中搜索到尽可能多的局部最优解,从而使算法可以更容易地跳出局部最优,更高效地搜索到全局最优解。对测试函数和非线性方程组求解问题进行实例仿真,仿真结果验证了算法的有效性,具有一定的实际应用价值。     

Nonlinear model predictive control using deterministic global optimization

This paper presents a Nonlinear Model Predictive Control (NMPC) algorithm utilizing a deterministic global optimization method. Utilizing local techniques on nonlinear nonconvex problems leaves one susceptible to suboptimal solutions at each iteration. In complex problems, local solver reliability is difficult to predict and dependent upon the choice of initial guess. This paper demonstrates the application of a deterministic global solution technique to an example NMPC problem. A terminal state constraint is used in the example case study. In some cases the local solution method becomes infeasible, while the global solution correctly finds the feasible global solution. Increased computational burden is the most significant limitation for global optimization based online control techniques. This paper provides methods for improving the global optimization rates of convergence. This paper also shows that globally optimal NMPC methods can provide benefits over local techniques and can successfully be used for online control.     

Global optimization of an optical chaotic system by Chaotic Multi Swarm Particle Swarm Optimization

The control and estimation of unknown parameters of chaotic systems are a daunting task till date from the perspective of nonlinear science. Inspired from ecological co-habitation, we propose a variant of Particle Swarm Optimization (PSO), known as Chaotic Multi Swarm Particle Swarm Optimization (CMS-PSO), by modifying the generic PSO with the help of the chaotic sequence for multi-dimension unknown parameter estimation and optimization by forming multiple cooperating swarms. This achieves load balancing by delegating the global optimizing task to concurrently operating swarms. We apply it successfully in estimating the unknown parameters of an autonomous chaotic laser system derived from Maxwell-Bloch equations. Numerical results and comparison demonstrate that for the given system parameters, CMS-PSO can identify the optimized parameters effectively evolving at each iteration to attain the pareto optimal solution with small population size and enhanced convergence speedup.     

A symplectic local pseudospectral method for solving nonlinear state‐delayed optimal control problems with inequality constraints

In this paper, a symplectic local pseudospectral (PS) method for solving nonlinear state‐delayed optimal control problems with inequality constraints is proposed. We first convert the original nonlinear problem into a sequence of linear quadratic optimal control problems using quasi‐linearization techniques. Then, based on local Legendre‐Gauss‐Lobatto PS methods and the dual variational principle, a PS method to solve these converted linear quadratic constrained optimal control problems is developed. The developed method transforms the converted problems into a coupling of a system of linear algebraic equations and a linear complementarity problem. The coefficient matrix involved is sparse and symmetric due to the benefit of the dual variational principle. Converged solutions can be obtained with few iterations because of the local PS method and quasi‐linearization techniques are used. The proposed method can be applied to problems with fixed terminal states or free terminal states, and the boundary conditions and constraints are strictly satisfied. Numerical simulations show that the developed method is highly efficient and accurate.     

结合权余量和粒子群方法求解变分优化问题

鉴于对变分问题的研究,发现了可以将微分方程与变分问题进行转化的原理,将求解微分方程的权余量方法与标准粒子群算法进行结合,旨在找到一种求解变分优化问题的较好算法。在权余量方法的计算中,把方程组转化为求近似目标函数的极小值的变分问题,接着用标准粒子群算法对其求解。对于因变量的幂为高次时的变分问题,可以直接用改进的算法对其求解。对于因变量的幂为一次时的变分问题,需要对等式左右分别平方后,再转化为求极小值的变分问题并求解。用一个变分问题的实例进行检验,发现改进后的算法容易,计算过程简单,结果较好。对近似目标函数的基的个数取三四个进行比较,对于具体问题,应该选取较好的适应实际问题的次数,得到较好的结果。从理论上分析,该算法具有可行性与一定的实际意义。     

一类分数阶种群扩散模型解的研究

从推广的Fick扩散定律出发研究了一类时间分数阶Fisher单种群扩散模型。利用变分迭代法求解了三种不同情况下的近似解,对结果进行了讨论和数值模拟。     

A mixed PI/VI design method for nonlinear H∞ Control

This paper proposes a new mixed policy iteration and value iteration (PI/VI) design method for nonlinear H_∞ control based on the theories of polynomial optimization and Lasserre's hierarchy. The design of a mixed PI/VI controller can be carried out in four steps: firstly, initialize design parameters and expand nonlinear system matrices; secondly, obtain a polynomial matrix inequality for policy improvement; thirdly, obtain the Lasserre's hierarchy of a global polynomial optimization problem for value improvement; fourthly, perform the mixed PI/VI algorithm to approximate the optimal nonlinear H_∞ control law. The novelty of this work lies in that the problem of designing a nonlinear H_∞ controller is translated into a polynomial global optimization problem, which can be solved by Lasserre's hierarchy directly, and then, the mixed PI/VI algorithm is presented to approximate the optimal nonlinear H_∞ control law by updating global optimizers iteratively. The main results of this paper consist of the mixed PI/VI algorithm and the related three theorems, which guarantee robust stability and performance of the closed‐loop nonlinear system. Numerical simulations show that the mixed PI/VI algorithm converges very fast and achieves good robust stability and performance in transient behavior, disturbance rejection, and enlarging the domain of attraction of the close‐loop system.     

State waypoint approach to continuous‐time nonlinear optimal control problems

In this paper, we propose an optimal control technique for a class of continuous‐time nonlinear systems. The key idea of the proposed approach is to parametrize continuous state trajectories by sequences of a finite number of intermediate target states; namely, waypoint sequences. It is shown that the optimal control problem for transferring the state from one waypoint to the next is given an explicit‐form suboptimal solution, by means of linear approximation. Thus the original continuous‐time nonlinear control problem reduces to a finite‐dimensional optimization problem of waypoint sequences. Any efficient numerical optimization method, such as the interior‐reflection Newton method, can be applied to solve this optimization problem. Finally, we solve the optimal control problem for a simple nonlinear system example to illustrate the effectiveness of this approach. Copyright © 2009 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society     

A hybrid splitting method for variational inequality problems with separable structure

Alternating direction method (ADM), which decomposes a large-scale original variational inequality (VI) problem into a series of smaller scale subproblems, is very attractive for solving a class of VI problems with a separable structure. This type of method can greatly improve the efficiency, but cannot avoid solving VI subproblems. In this paper, we propose a hybrid splitting method with variable parameters for separable VI problems. Specifically, the proposed method solves only one strongly monotone VI subproblem and a well-posed system of nonlinear equations in each iteration. The global convergence of the new method is established under some standard assumptions as those in classical ADMs. Finally, some preliminary numerical results show that the proposed method performs favourably in practice.     

Synthesis of four-bar mechanism in a hydraulic support using a global optimization algorithm

The paper discusses optimal synthesis of a four-bar mechanism. The general optimization problem is addressed in the form of a nonlinear programming problem. The objective of this approach is to determine the optimal values of the mechanism link lengths, to minimize the difference between the trajectory T of the arbitrary point C on the mechanism coupler link and the prescribed trajectory P, while hinge forces have to remain within the prescribed values. The global optimization method is used in order to find the global optimal solution. The procedure uses the adaptive grid refinement algorithm. The algorithm is based on the identification of feasible nodes at each iteration, which define the solution set. Nodes far from the current optimum are pruned from the solution. The algorithm identifies optimal regions that satisfy predefined conditions, rather than only a single optimal point.     

一种基于改进模拟退火算法的程序性能优化参数搜索算法

高级程序变换是提高程序性能的重要手段,很多涉及到优化参数的选择问题,如为循环分块选择适当的分块因子。由于优化参数搜索问题本身是NP难问题,目前尚没有确定性的算法可以有效解决该问题,针对于此,将该问题转化为一个非线性全局最优化问题,提出一种基于改进模拟退火算法的程序性能优化参数搜索算法,实验结果验证了算法的有效性。     

SENSITIVITY APPROACH TO OPTIMAL CONTROL FOR AFFINE NONLINEAR DISCRETE‐TIME SYSTEMS

This paper deals with the optimal control problem for a class of affine nonlinear discrete‐time systems. By introducing a sensitivity parameter and expanding the system variables into a Maclaurin series around it, we transform the original optimal control problem for affine nonlinear discrete‐time systems into the optimal control problem for a sequence of linear discrete‐time systems. The optimal control law consists of an accurate linear term and a nonlinear compensating term, which is an infinite sequence of adjoint vectors. In the present approach, iteration is required only for the nonlinear compensation series. By intercepting a finite sum of the series, we obtain a suboptimal control law that reduces the complexity of the calculations. A numerical simulation shows that the algorithm can be easily implemented and has a fast convergence rate.     

A Computational Method for Stochastic Optimal Control Problems in Financial Mathematics

Principle of optimality or dynamic programming leads to derivation of a partial differential equation (PDE) for solving optimal control problems, namely the Hamilton‐Jacobi‐Bellman (HJB) equation. In general, this equation cannot be solved analytically; thus many computing strategies have been developed for optimal control problems. Many problems in financial mathematics involve the solution of stochastic optimal control (SOC) problems. In this work, the variational iteration method (VIM) is applied for solving SOC problems. In fact, solutions for the value function and the corresponding optimal strategies are obtained numerically. We solve a stochastic linear regulator problem to investigate the applicability and simplicity of the presented method and prove its convergence. In particular, for Merton's portfolio selection model as a problem of portfolio optimization, the proposed numerical method is applied for the first time and its usefulness is demonstrated. For the nonlinear case, we investigate its convergence using Banach's fixed point theorem. The numerical results confirm the simplicity and efficiency of our method.     

受扰非线性离散系统的前馈反馈最优控制

利用逐次逼近法研究含外部扰动的非线性离散系统的线性二次型前馈反馈最优控制问题.首先将系统的最优控制问题转化为非线性两点边值问题族.其次,构造了该问题族的由精确线性项和非线性补偿项组成的解序列,并证明了解序列一致收敛到系统的最优解.最后,通过截取最优控制序列解中非线性补偿项的有限项,得到系统的前馈反馈次优控制(FFSOC)律及设计算法.仿真算例表明,该算法容易实现,且对抑制外部扰动的鲁棒性优于经典的反馈次优控制(FSOC).     

Semi‐Global Robust Output Regulation for A Class of Singular Nonlinear Systems with Unknown Algebraic Equations

This paper considers semi‐global robust output regulation problem for a class of singular nonlinear systems whose algebraic equations are not precisely known. Since the algebraic equations are not known, the output regulation problem of singular nonlinear systems cannot be solved by directly reducing the singular nonlinear system into a normal nonlinear system. Based on internal model principle, we convert the robust output regulation problem of singular nonlinear systems into a robust stabilization problem of an augmented singular nonlinear system. The augmented singular nonlinear system is also with unknown algebraic equations. However, without transforming the singular nonlinear system into a normal nonlinear system, it is shown that the augmented singular nonlinear system can be semi‐globally stabilized by a high gain output feedback control law under some reasonable assumptions. Moreover, the semi‐global stabilization control law of the augmented singular nonlinear systems also solves the semi‐global robust output regulation problem of the original singular nonlinear system.     

具有未知参数的非线性系统动态优化

针对具有未知参数和不等式路径约束的非线性系统动态优化问题,提出一种新颖有效的数值求解方法.首先,将未知参数视为一个动态优化问题的决策变量;其次,利用多重打靶法将无限维的含未知参数动态优化问题转化为有限维的非线性规划问题,进而在不等式路径约束违反的时间段内,用有限多个内点约束替代原不等式路径约束;然后,用内点法求解转化后的非线性规划问题,在路径约束违反的一定容许度下,经过有限多次步数迭代后得到未知参数值的同时得到控制策略,并在理论上对所提出算法的收敛性进行相应证明;最后,对两个经典的含未知参数非线性系统的动态优化问题进行数值仿真以验证所提出算法的有效性.     

A new Wavelet Method for Variable‐Order Fractional Optimal Control Problems

In this paper, a new computational method based on the Legendre wavelets (LWs) is proposed for solving a class of variable‐order fractional optimal control problems (V‐FOCPs). To do this, a new operational matrix of variable‐order fractional integration (OMV‐FI) in the Riemann‐Liouville sense for the LWs is derived and used to obtain an approximate solution for the problem under study. Along the way the hat functions (HFs) are introduced and employed to derive a general procedure to compute this matrix. In the proposed method, the variable‐order fractional dynamical system is transformed to an equivalent variable‐order fractional integro‐differential dynamical system, at first. Then, the highest integer order of the derivative of the state variable and the control variable are expanded by the LWs with unknown coefficients. Next, the OMV‐FI in the the Riemann‐Liouville sense together with some properties of the LWs are employed to achieve a nonlinear algebraic equation in place of the performance index and a nonlinear system of algebraic equations in place of the dynamical system in terms of the unknown coefficients. Finally, the method of constrained extremum is applied which consists of adjoining the constraint equations derived from the given dynamical system to the performance index by a set of undetermined Lagrange multipliers. As a result, the necessary conditions of optimality are derived as a system of algebraic equations in the unknown coefficients of the state variable, control variable and Lagrange multipliers. Furthermore, the efficiency and accuracy of the proposed method are demonstrated for some concrete examples. The obtained results show that the proposed method is very efficient and accurate.     

带有复形法局部搜索的粒子群算法求解最优控制问题*

对最优控制问题的求解是实现计算机控制的前提条件,但有效地求出控制系统的全局最优解是困难的。现今各种求解此问题的算法需要增强算法的优化性能与简洁性。遗传算法求此类问题需要进行离散化;复形法、粒子群算法求解此类问题容易陷入局部极值。为此提出用带有复形法局部搜索的粒子群算法求解此类问题。此算法充分考虑粒子群算法与复形法的特性,将复形法的局部搜索与粒子群算法的全局搜索结合起来,以提高算法搜索能力,克服粒子群算法与复形法易陷局部极值的不足。通过性能测试效果良好,同时算法简便、可行、高效。最后将所提算法用于求解Park-Ramirez生物反应器补料流率的动态优化,取得了满意的效果。     

一种求解非线性约束优化全局最优的新方法

本文提出了一种求解非线性约束优化的全局最优的新方法—它是基于利用非线性互补函数和不断增加新的约束来重复解库恩-塔克条件的非线性方程组的新方法。因为库恩-塔克条件是非线性约束优化的必要条件,得到的解未必是非线性约束优化的全局最优解,为此,本文首次给出了通过利用该优化问题的先验知识,不断地增加约束来限制全局最优解范围的方法,一些仿真例子表明提出的方法和理论有效的,并且可行的。