A new two-point approximation approach for structural optimization

The objective of this work is to build up a high-quality approximation scheme to realize computational savings for the solution of structural optimization problems. To this end, a newly developed two-point approximation scheme is proposed. This scheme is constructed by the linear combination of Taylor expansions in terms of both original and reciprocal variables. The coefficients of the combination are determined by utilizing both the function and gradient information of two different design points obtained during the process of optimization. Based on this approach, the accuracy of the existing constraint approximation methods can be improved. The effectiveness of the proposed approach is demonstrated on a number of numerical examples. The numerical results are also compared with those of previously published work. Received April 2, 1999     

Automatic evaluation of move-limits in structural optimization

The two-point exponential approximation method was introduced by Fadelet al. (1990) and tested on structural optimization problems with stress and displacement constraints. It was subsequently tested on problems with frequency constraints (Sareenet al. 1991). The results reported in earlier papers showed reductions in the number of iterations needed to reach an optimum, and a smoother convergence towards that optimum. The method, which consists in correcting Taylor series approximations using previous design history, is used in the present paper to automatically determine move-limits. Move-limits are the allowable changes in design variables during the optimization of the approximate problem. The exponents, computed in the two-point exponential approximation by matching slopes between two design iterations, are used as a measure of non-linearity of the objective function and constraints with respect to each of the design variables. The relationships between the move-limits and the exponents are established and individual move-limits are computed and applied to each design variable. The method is applied to two classical structural optimization examples. It provides the engineer with more flexibility when choosing the move-limits and typically converges in less iterations.     

Solution of linear dynamic systems with initial or boundary value conditions by shifted Chebyshev approximations

The shifted Chebyshev polynomial approximation is employed to solve the linear, constant parameter, ordinary differential equations of initial or two-point boundary value problems. An effective recursive algorithm is developed to calculate the expansion coefficients of the shifted Chebyshev series. An effective transformation is proposed to transform the two-point boundary value problem into an initial value problem. An illustrative example is included to show that the computational results are accurate.     

Observer-based approximate optimal tracking control for time-delay systems with external disturbances

This paper proposes a successive approximation design approach of observer-based optimal tracking controllers for time-delay systems with external disturbances. To solve a two-point boundary value problem with time-delay and time-advance terms and obtain the optimal tracking control law, two sequences of vector differential equations are constructed first. Second, the convergence of the sequences of the vector differential equations is proved to guarantee the existence and uniqueness of the control law. Third, a design algorithm of the optimal tracking control law is presented and the physically realisable problem is addressed by designing a disturbance state observer and a reference input state observer. An example of an industrial electric heater is given to demonstrate the efficiency of the proposed approach.     

非线性时滞系统次优控制的逐次逼近法

对状态变量含有时滞的非线性系统的次优控制问题进行了研究，提出了一种次优控制的逐次逼近设计方法．针对由最优控制理论导出的既含有时滞项又含有超前项的非线性两点边值问题，构造了其解序列一致收敛于原问题最优解的非齐次线性两点边值问题序列．从而将两点边值问题解序列的有限次迭代结果作为系统的次优控制律．仿真结果表明了所提出方法的有效性．     

一种非线性离散动态投入产出系统的最优控制

阐述了基于动态投入产出模型的最优控制理论,并对当前国内外的研究成果进行了对比研究。在分析其优缺点的同时,针对非线性离散动态投入产出系统的特点,提出了一种动态投入产出系统最优控制的逐次逼近方法。此方法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,然后通过构造线性两点边值问题族,将非线性两点边值问题转化为非齐次线性两点边值问题族;得到的最优控制律由精确控制项和非线性补偿项两部分组成,精确控制项可以通过求解Riccati方程求出其精确解,非线性补偿项由逐次逼近法求解一族线性伴随向量方程的解序列求得;最优控制律的最终目标是在规划期内使实际产出尽可能地与理想产出接近。实验仿真测试表明,采用逐次逼近法获得了非线性离散动态投入产出的最优系统控制,从而为最优控制问题的有效解决提供了参考和借鉴。     

关于一般图形Voronoi图的近似构造法的研究

（?）1.引 言 随着计算机处理图形图象能力的增强,对计算几何理论与应用的研究,越来越为人们所重视.计算几何研究的是,如何高效处理通过视觉器官等途径得到的几何图形信息,开发高速解决几何问题的方法,从理论上探寻几何计算的复杂性与可行性,并对其性能做出评价.     

A sixth order method for the integration of two point boundary value problems

A finite difference method for obtaining sixth order accurate approximation to the solution of the two-point linear boundary value problem is given. The convergence of the method is proved. Numerical results for a typical problem are tabulated and in each case the observed error is compared with its theoretical estimate. The numerical results are compared with those obtained from an earlier method of the author and the method of Noumerov.     

Computation of the eigenelements of singular sturm-liouville problems

We investigate a Galerkin method for the approximation of the eigenvalues and eigenfunctions of singular two-point boundary value problems. The method can handle limit circle and a class of limit point cases. Optimal orders of convergence are derived.     

Monotonic bicompact schemes for linear transport equations

The biocompact difference scheme earlier proposed by these authors for a linear transport equation, which has the fourth-order approximation in spatial coordinate on the two-point stencil and the first-order approximation in time, is monotonic. This implicit scheme is absolutely stable and can be solved by explicit formulas of a running calculation. On the basis of this scheme a monotone non-linear homogeneous difference scheme of high (third for smooth solutions) order accuracy in time is constructed. Calculations of test problems with discontinuous solutions have demonstrated that the proposed scheme has a significant advantage in accuracy over the known nonoscillatory schemes of high-order approximation.     

An analogue technique for solving trajectory optimization problems

Analogue or hybrid computer methods for solving trajectory optimization problems usually require the solution of a two-point boundary value problem. A method of solving this problem is presented which does not require a good initial approximation to ensure convergence of the iteration. Analogue computer results are given with the method applied to a double integrator plant. Two different forms of performance index are considered.     

Family of quadratic spline difference schemes for a convection–diffusion problem

We consider the finite difference approximation of a singularly perturbed one-dimensional convection–diffusion two-point boundary value problem. It is discretized using quadratic splines as approximation functions, equations with various piecewise constant coefficients as collocation equations and a piecewise uniform mesh of Shishkin type. The family of schemes is derived using the collocation method. The numerical methods developed here are non-monotone and therefore apart from the consistency error we use Green's grid function analysis to prove uniform convergence. We prove the almost first order of convergence and furthermore show that some of the schemes have almost second-order convergence. Numerical experiments presented in the paper confirm our theoretical results.     

非线性相似组合大系统最优控制的逐次逼近过程

研究一类仿射非线性相似组合大系统关于二次型性能指标的最优控制问题．首先通过模型简化,将非线性相似组合大系统化为若干个准解耦的子系统;然后利用非线性系统最优控制的逐次逼近设计方法,将求解高阶强耦合的非线性两点边值问题简化为求解一族解耦的线性两点边值问题序列．该线性两点边值问题序列的解一致收敛于非线性相似组合大系统的最优控制,得到的最优控制律由线性最优控制的解析项与非线性补偿序列的极限项组成．通过截取最优控制非线性补偿序列的有限次逼近值．得到了非线性组合大系统的次优控制律．     

A globally convergent sequential convex programming using an enhanced two-point diagonal quadratic approximation for structural optimization

In this study, we propose a sequential convex programming (SCP) method that uses an enhanced two-point diagonal quadratic approximation (eTDQA) to generate diagonal Hessian terms of approximate functions. In addition, we use nonlinear programming (NLP) filtering, conservatism, and trust region reduction to enforce global convergence. By using the diagonal Hessian terms of a highly accurate two-point approximation, eTDQA, the efficiency of SCP can be improved. Moreover, by using an appropriate procedure using NLP filtering, conservatism, and trust region reduction, the convergence can be improved without worsening the efficiency. To investigate the performance of the proposed algorithm, several benchmark numerical examples and a structural topology optimization problem are solved. Numerical tests show that the proposed algorithm is generally more efficient than competing algorithms. In particular, in the case of the topology optimization problem of minimizing compliance subject to a volume constraint with a penalization parameter of three, the proposed algorithm is found to converge well to the optimum solution while the other algorithms tested do not converge in the maximum number of iterations specified.     

Stochastic optimal open-loop feedback control: Approximate solution of the Hamiltonian system

Considering a dynamic control system with random model parameters and using the stochastic Hamilton approach stochastic open-loop feedback controls can be determined by solving a two-point boundary value problem (BVP) that describes the optimal state and costate trajectory. In general an analytical solution of the BVP cannot be found. This paper presents two approaches for approximate solutions, each consisting of two independent approximation stages. One stage consists of an iteration process with linearized BVPs that will terminate when the optimal trajectories are represented. These linearized BVPs are then solved by either approximation fixed-point equations (first approach) or Taylor-Expansions in the underlying stochastic model parameters (second approach). This approximation results in a deterministic linear BVP, which can be handled by solving a matrix Riccati differential equation.     

离散线性时滞系统的次优控制:逐次逼近法

A successive approximation approach for designing optimal controllers is presented for discrete linear time-delay systems with a quadratic performance index. By using the successive approximation approach, the original optimal control problem is transformed into a sequence of nonhomogeneous linear two-point boundary value (TPBV) problems without time-delay and time-advance terms. The optimal control law obtained consists of an accurate feedback terms and a time-delay compensation term which is the limit of the solution sequence of the adjoint equations. By using a finite-step iteration of the time-delay compensation term of the optimal solution sequence, a suboptimal control law is obtained. Simulation examples are employed to test the validity of the proposed approach.     

Suboptimal Control for Discrete Linear Systems with Time-delay:A Successive Approximation Approach

A successive approximation approach for designing optimal controllers is presented for discrete linear time-delay systems with a quadratic performance index.By using the successive approximation approach,the original optimal,control problem is transformed into a sequence of nonhomogeneous linear two-point boundary value (TPBV) problems without time-delay and time- advance terms.The optimal control law obtained consists of an accurate feedback terms and a time-delay compensation term which is the limit of the solution sequence of the adjoint equations. By using a finite-step iteration of the time-delay compensation term of the optimal solution sequence, a suboptimal control law is obtained.Simulation examples are employed to test the validity of the proposed approach.     

Optimal control of linear time-delay systems via general orthogonal polynomials

A Padé approximation is introduced to transform the delay differential equation into an ordinary equation. The optimal control of a linear time delay system with quadratic performance index is then derived via general orthogonal polynomials. The adjoint equation is also extracted using a maximum principle. The state and adjoint equations can be solved by using the general orthogonal polynomials and the difficulty encountered when using a two-point boundary condition is circumvented. Examples are given to demonstrate the usefulness of the proposed method.     

离散时滞系统的最优滑模控制

针对离散线性时滞系统,设计了最优滑模面和最优离散变结构控制律.在设计最优滑模面的过程中,利用逐次逼近算法,将既含有时滞项又含有超前项的两点边值问题转化为不含时滞项和超前项的线性两点边值问题,并证明了其解序列一致收敛于原系统的最优滑模.进一步给出了原系统的变结构控制律.     

Solution of Polynomial Systems Derived from Differential Equations

Nonlinear two-point boundary value problems arise in numerous areas of application. The existence and number of solutions for various cases has been studied from a theoretical standpoint. These results generally rely upon growth conditions of the nonlinearity. However, in general, one cannot forecast how many solutions a boundary value problem may possess or even determine the existence of a solution. In recent years numerical continuation methods have been developed which permit the numerical approximation of all complex solutions of systems of polynomial equations. In this paper, numerical continuation methods are adapted to numerically calculate the solutions of finite difference discretizations of nonlinear two-point boundary value problems. The approach taken here is to perform a homotopy deformation to successively refine discretizations. In this way additional new solutions on finer meshes are obtained from solutions on coarser meshes. The complicating issue which the complex polynomial system setting introduces is that the number of solutions grows with the number of mesh points of the discretization. To counter this, the use of filters to limit the number of paths to be followed at each stage is considered.