Two-stage stochastic problems with correlated normal variables: computational experiences

Two-stage models are frequently used when making decisions under the influence of randomness. The case of normally distributed right hand side vector – with independent or correlated components – is treated here. The expected recourse function is computed by an enhanced Monte Carlo integration technique. Successive regression approximation technique is used for computing the optimal solution of the problem. Computational issues of the algorithm are discussed, improvements are proposed and numerical results are presented for random right hand side and a random matrix in the second stage problems.     

一种显式子步应力点积分算法及其在SMA数值模拟中的应用

形状记忆合金（shape memory alloys,简称SMA）具有复杂的热力本构关系,为了模拟SMA及其组合结构复杂的受力和变形行为,在数值模拟中需要采用可靠且高效的应力点积分算法．隐式应力点回映算法已经成功应用于形状记忆合金的数值模拟,但在复杂加载条件下,荷载增量较大时有可能导致整体非线性迭代求解不收敛．推广了局部误差控制的显式子步积分算法,首次将其应用于形状记忆合金及其组合结构这类热力相变问题的应力点积分,并通过数值算例对所提算法和隐式应力点回映算法进行了比较．数值结果表明:对于大规模数值模拟和计算,整体子步步数决定着总体计算时间;所提出的修正Euler自动子步方案可以有效减少整体子步步数,在保证相同计算精度的前提下能够大幅提高有限元计算效率,因而更适合大规模形状记忆合金智能结构的数值模拟.     

An applied cell mapping method for optimal control problems

From the application point of view, a series of modifications are proposed for the cell mapping method discussed in Ref. 1 for the optimal control analysis of dynamical systems. The cell order around the target set is rearranged. A set of common discriminate principles is used for the selection of the optimal one among competing control strategies of the same cost. Inequality constraints of the system are taken into account. The number of elements in the set of allowable time intervals is not prescribed, but left open. These modifications seem to make the cell mapping method more efficient for analyzing feedback systems and for obtaining their global optimal control information. The algorithms presented in this paper could broaden the application of the cell mapping approach of Ref. 1 to a wider class of engineering problems.     

Optimal synchronization of Rössler system with complete uncertain parameters

The paper discusses the optimal control for the chaos synchronization of Rössler systems with complete uncertain parameters during finite and infinite time intervals. Based on the Liapunov–Bellman technique, optimal control laws are derived from the conditions that ensure asymptotic stability of the error dynamical system and minimizes the cost transfer of this system from arbitrary state to its equilibrium state. The derived control laws make the states of two identical Rössler systems asymptotically synchronized. Some special cases are introduced. Important numerical simulation is included to show the effectiveness of the optimal synchronization technique.     

解并行处理机最优映射问题的遗传算法

该文给出了一种用于多处理机系统中实现并行计算的最优映射问题的遗传算法,它对于在固定结构的并行系统中充分利用计算资源,提高计算效率具有实用价值,实践表明,采用遗传算法是解决任务最优映射问题的有效的方法．     

A parallel approach to improvement and estimation of the approximate optimal control

In this paper the method for computing a priori estimates of the approximate optimal control is considered. These estimates provide us with information about the quality of the approximate optimal solution obtained by applying the improvement control procedure. The method is implemented in the form of a parallel algorithm. This algorithm is an essential part of the developed software package intended for optimization of controllable dynamical systems. We also consider the scalability of the parallel algorithm in the OpenTS parallel programming system for chemical and biochemical engineering problems.     

陀螺系统特征值问题的收缩Jacobi-Davidson方法

本文研究陀螺系统特征值问题的Jacobi-Davidson方法. 利用陀螺系统的结构性质,给出了求解Jacobi-Davidson方法中校正方程的有效方法. 基于非等价低秩收缩技术,给出了计算陀螺系统一些特征值的收缩Jacobi-Davidson方法. 数值结果表明本文所给算法是有效的.     

Real-time calculation of current optimal feedbacks for a delay system

A linear optimal control problem for a nonstationary system with a single delay state variable is examined. A fast implementation of the dual method is proposed in which a key role is played by a quasi-reduction of the fundamental matrices of solutions to the homogeneous part of the delay models under analysis. As a result, an iteration step of the dual method involves only the integration of auxiliary systems of ordinary differential equations over short time intervals. A real-time algorithm is described for calculating optimal feedback controls. The results are illustrated by the optimal control problem for a second-order stationary system with a fixed delay.     

Hamilton系统下基于相位误差的精细辛算法

Hamilton系统是一类重要的动力系统，辛算法（如生成函数法、SRK法、SPRK法、多步法等）是针对Hamilton系统所设计的具有保持相空间辛结构不变或保Hamilton函数不变的算法.但是，时域上，同阶的辛算法与Runge-Kutta法具有相同的数值精度，即辛算法在计算过程中也存在相位误差，导致时域上解的数值精度不高.经过长时间计算后，计算结果在时域上也会变得“面目全非”.为了提高辛算法在时域上解的精度，将精细算法引入到辛差分格式中，提出了基于相位误差的精细辛算法（HPD-symplectic method），这种算法满足辛格式的要求，因此在离散过程中具有保Hamilton系统辛结构的优良特性.同时，由于精细化时间步长，极大地减小了辛算法的相位误差，大幅度提高了时域上解的数值精度，几乎可以达到计算机的精度，误差为O（10-13）.对于高低混频系统和刚性系统，常规的辛算法很难在较大的步长下同时实现对高低频精确仿真，精细辛算法通过精细计算时间步长，在大步长情况下，没有额外增加计算量，实现了高低混频的精确仿真.数值结果验证了此方法的有效性和可靠性.     

Remarks on the algorithm of sakawa for optimal control problems

A general optimal control problem for ordinary differential equations is considered. For this problem, some improvements of the algorithm of Sakawa are discussed. We avoid any convexity assumption and show with an example that the algorithm is even applicable in cases in which no optimal control exists.     

Fractional set systems with few disjoint pairs

Ahlswede (1980) [1] and Frankl (1977) [5] independently found a result about the structure of set systems with few disjoint pairs. Bollobás and Leader (2003) [3] gave an alternate proof by generalizing to fractional set systems and noting that the optimal fractional set systems are {0,1}-valued. In this paper we show that this technique does not extend to t-intersecting families. We find optimal fractional set systems for some infinite classes of parameters, and we point out that they are strictly better than the corresponding {0,1}-valued fractional set systems. We prove some results about the structure of an optimal fractional set system, which we use to produce an algorithm for finding such systems. The run time of the algorithm is polynomial in the size of the ground set.     

An Effective Algorithm for Finding Multidimensional Conservation Laws for Integrable Systems of Hydrodynamic Type

We study a new property of integrable systems, the existence of infinitely many local three-dimensional conservation laws for pairs of integrable two-dimensional hydrodynamic chains. We describe an effective algorithm for successively computing an infinite set of three-dimensional conservation laws for the Benney pair of commuting flows.     

A penalty function-based random search algorithm for optimal control of switched systems with stochastic constraints and its application in automobile test-driving with gear shifts

Practical industrial process is usually a dynamic process including uncertainty. Stochastic constraints can be used for industrial process modeling, when system sate and/or control input constraints cannot be strictly satisfied. Thus, optimal control of switched systems with stochastic constraints can be available to address practical industrial process problems with different modes. In general, obtaining an analytical solution of the optimal control problem is usually very difficult due to the discrete nature of the switching law and the complexity of stochastic constraints. To obtain a numerical solution, this problem is formulated as a constrained nonlinear parameter selection problem (CNPSP) based on a relaxation transformation (RT) technique, an adaptive sample approximation (ASA) method, a smooth approximation (SA) technique, and a control parameterization (CP) method. Following that, a penalty function-based random search (PFRS) algorithm is designed for solving the CNPSP based on a novel search rule-based penalty function (NSRPF) method and a novel random search (NRS) algorithm. The convergence results show that the proposed method is globally convergent. Finally, an optimal control problem in automobile test-driving with gear shifts (ATGS) is further extended to illustrate the effectiveness of the proposed method by taking into account some stochastic constraints. Numerical results show that compared with other typical methods, the proposed method is less conservative and can obtain a stable and robust performance when considering the small perturbations in initial system state. In addition, to balance the computation amount and the numerical solution accuracy, a tolerance setting method is also provided by the numerical analysis technique.     

Determination of Inner and Outer Bounds of Reachable Sets Through Subpavings

The computation of the reachable set of states of a given dynamic system is an important step to verify its safety during operation. There are different methods of computing reachable sets, namely interval integration, capture basin, methods involving the minimum time to reach function, and level set methods. This work deals with interval integration to compute subpavings to over or under approximate reachable sets of low dimensional systems. The main advantage of this method is that, compared to guaranteed integration, it allows to control the amount of over-estimation at the cost of increased computational effort. An algorithm to over and under estimate sets through subpavings, which potentially reduces the computational load when the test function or the contractor is computationally heavy, is implemented and tested. This algorithm is used to compute inner and outer approximations of reachable sets. The test function and the contractors used in this work to obtain the subpavings involve guaranteed integration, provided either by the Euler method or by another guaranteed integration method. The methods developed were applied to compute inner and outer approximations of reachable sets for the double integrator example. From the results it was observed that using contractors instead of test functions yields much tighter results. It was also confirmed that for a given minimum box size there is an optimum time step such that with a greater or smaller time step worse results are obtained.     

The optimal control of just-in-time-based production and distribution systems and performance comparisons with optimized pull systems

In just-in-time (JIT) production systems, there is both input stock in the form of parts and output stock in the form of product at each stage. These activities are controlled by production-ordering and withdrawal kanbans. This paper discusses a discrete-time optimal control problem in a multistage JIT-based production and distribution system with stochastic demand and capacity, developed to minimize the expected total cost per unit of time. The problem can be formulated as an undiscounted Markov decision process (UMDP); however, the curse of dimensionality makes it very difficult to find an exact solution. The author proposes a new neuro-dynamic programming (NDP) algorithm, the simulation-based modified policy iteration method (SBMPIM), to solve the optimal control problem. The existing NDP algorithms and SBMPIM are numerically compared with a traditional UMDP algorithm for a single-stage JIT production system. It is shown that all NDP algorithms except the SBMPIM fail to converge to an optimal control.Additionally, a new algorithm for finding the optimal parameters of pull systems is proposed. Numerical comparisons between near-optimal controls computed using the SBMPIM and optimized pull systems are conducted for three-stage JIT-based production and distribution systems. UMDPs with 42 million states are solved using the SBMPIM. The pull systems discussed are the kanban, base stock, CONWIP, hybrid and extended kanban.     

A PROCESS FOR SOLVING A FEW EXTREME EIGENPAIRS OF LARGE SPARSE POSITIVE DEFINITE GENERALIZED EIGENVALUE PROBLEM

1. IntroductionWe are concerned in this work with finding a few extreme eigenvalues and theircorresponding eigenvectors of a generalized large scale eigenvalue problem in which thematrices are sparse and symmetric positive definite.Although finding a few extreme eigenpairs is of interest both in theory and practice,there are only few usable and efficient methods up to now. Reinsch and Baner ([12]),suggested a oR algorithm with Newton shift for the standard eigenproblem which included an ingen…     

Multi-objective optimal design of feedback controls for dynamical systems with hybrid simple cell mapping algorithm

This paper presents a study of multi-objective optimal design of full state feedback controls. The goal of the design is to minimize several conflicting performance objective functions at the same time. The simple cell mapping method with a hybrid algorithm is used to find the multi-objective optimal design solutions. The multi-objective optimal design comes in a set of gains representing various compromises of the control system. Examples of regulation and tracking controls are presented to validate the control design.     

Adaptive fuzzy sliding-mode control for multi-input multi-output chaotic systems

This paper describes an adaptive fuzzy sliding-mode control algorithm for controlling unknown or uncertain, multi-input multi-output (MIMO), possibly chaotic, dynamical systems. The control approach encompasses a fuzzy system and a robust controller. The fuzzy system is designed to mimic an ideal sliding-mode controller, and the robust controller compensates the difference between the fuzzy controller and the ideal one. The parameters of the fuzzy system, as well as the uncertainty bound of the robust controller, are tuned adaptively. The adaptive laws are derived in the Lyapunov sense to guarantee the asymptotic stability and tracking of the controlled system. The effectiveness of the proposed method is shown by applying it to some well-known chaotic systems.