Optimal Control Formulation for Complementarity Dynamical Systems

In this paper, we show that the complementarity dynamical systems can be reformulated as optimal control problems. By using this reformulation, we present a pseudospectral scheme to discretize the complementarity dynamical systems. Applying this discretization, the complementarity dynamical system is reduced to a sequence of nonlinear programming problems. Numerical examples and comparison with two other methods are included to demonstrate the capability of the proposed method.     

Optimal Mathematical Models for Nonlinear Dynamical Systems

We propose a new method for the optimal causal representation of nonlinear systems. The proposed approach is based on the best constrained approximation of mappings in probability spaces by operators constructed from matrices of special form so that the approximant preserves the causality property. It is supposed that the observable input is contaminated with noise. The approximant minimises the mean-square difference between a desired output signal and the output signal of the approximating model. The method provides a numerically realisable mathematical model of the system. An analysis is given of the error associated with this representation.     

Optimal Rate for Nonparametric Estimation in Deterministic Dynamical Systems

Statistical Inference for Stochastic Processes - We propose a minimax statistical framework adapted to nonparametric estimation in deterministic dynamical system, which highlights the importance of...     

Discontinuous Dynamical Systems in Optimal Control Theory

We develop foundations of the theory of discontinuous Hamiltonian systems appearing in the problems of optimal control. We consider analogs of the classical Poisson and Liouville theorems for discontinuous Hamiltonian systems. We study the local geometry of discontinuous dynamical systems and describe singularities in general position and the behavior of integral trajectories near an elliptical submanifold (sliding mode).     

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.     

Solving Optimal Control Problems by Exploiting Inherent Dynamical Systems Structures

Computing globally efficient solutions is a major challenge in optimal control of nonlinear dynamical systems. This work proposes a method combining local optimization and motion planning techniques based on exploiting inherent dynamical systems structures, such as symmetries and invariant manifolds. Prior to the optimal control, the dynamical system is analyzed for structural properties that can be used to compute pieces of trajectories that are stored in a motion planning library. In the context of mechanical systems, these motion planning candidates, termed primitives, are given by relative equilibria induced by symmetries and motions on stable or unstable manifolds of e.g. fixed points in the natural dynamics. The existence of controlled relative equilibria is studied through Lagrangian mechanics and symmetry reduction techniques. The proposed framework can be used to solve boundary value problems by performing a search in the space of sequences of motion primitives connected using optimized maneuvers. The optimal sequence can be used as an admissible initial guess for a post-optimization. The approach is illustrated by two numerical examples, the single and the double spherical pendula, which demonstrates its benefit compared to standard local optimization techniques.     

Optimal Point Placement for Mesh Smoothing

We study the problem of moving a vertex in an unstructured mesh of triangular, quadrilateral, or tetrahedral elements to optimize the shapes of adjacent elements. We show that many such problems can be solved in linear time using generalized linear programming. We also give efficient algorithms for some mesh smooth ing problems that do not fit into the generalized linear programming paradigm.     

不可压缩Navier-Stokes方程最优动力系统建模和分析

研究了同时满足任意速度边界条件和速度不可压条件的Navier-Stokes方程最优动力系统的建模方法.通过对方柱绕流问题的最优动力系统的建模与分析,发现该最优动力系统的动力学特性为极限环.同时,该最优动力系统仅使用了三个最优基函数就很好地描述了所有主要的流场特征和该问题的动力学特性,故满足任意速度边界条件和速度不可压条件Navier-Stokes方程最优动力系统的建模方法,能够用最少的基函数最大限度地描述复杂流体问题及其动力学特性.     

Gradient Descent Approach to Optimal Mode Scheduling in Hybrid Dynamical Systems

This paper concerns the problem of optimally scheduling the sequence of dynamic response functions in nonlinear switched-mode hybrid dynamical systems. The control parameter has a discrete component and a continuous component, namely the sequence of modes and the duration of each mode, while the performance criterion consists of a cost functional on the state trajectory. The problem is naturally cast in the framework of optimal control. This framework has established techniques sufficient to address the continuous part of the parameter, but lacks adequate tools to consider the discrete element. To get around this difficulty, the paper proposes a bilevel hierarchical algorithm. At the lower level, the algorithm considers a fixed mode sequence and minimizes the cost functional with respect to the mode durations; at the upper level, it updates the mode sequence by using a gradient technique that is tailored to the special structure of the discrete variable (mode sequencing). The resulting algorithm is not defined on a single parameter space, but rather on a sequence of Euclidean spaces of increasing dimensions, an unusual setting for which there is no established notion of convergence. The paper suggests first a suitable definition of convergence based on the concepts of optimality functions; then, it proves that the proposed algorithm converges in that sense. Research supported in part by the National Science Foundation under Grant #0509064.     

离散系统迭代学习型瞬时最优控制及作动器位置优化研究

以线性离散系统为研究对象,以瞬时最优化控制和智能算法中的迭代学习控制为基础,以系统响应期望值与实际值之差为反馈信号,以离散系统的二次型性能泛函为目标函数,提出了迭代学习型瞬时最优控制算法.该方法以瞬时最优化控制算法初始化控制信号,并采用迭代学习控制在线实时修正控制信号以提高主动控制的效果.针对迭代学习型瞬时最优化控制算法迭代的特性,采用范数方法给出了该算法收敛的充分条件.数值算例表明,迭代学习型瞬时最优控制算法较离散瞬时最优控制算法有较明显的优势.同时,基于改进遗传算法,对主动控制器位置优化进行了讨论.数值分析结果表明:部分楼层设置主动控制器且安装位置经过优化后,其控制效果可接近甚至优于全楼层设置主动控制器时的控制效果.     

Optimal Robotic Component Placement

An increasingly common feature of manufacturing companies is the use of robotic component placement machines in the assembly of electrical goods. These machines place electronic components at predefined positions on printed circuit boards. A robotic arm, which carries one component at a time, collects components from feeders and places them, under computer control, onto the printed circuit board.Using examples provided by a major New Zealand manufacturer, we investigated the problem of assigning component reels to feeder positions and then constructing efficient placement sequences for such a robotic placement machine. An integer programming formulation is developed for this problem that, unlike previous solution techniques in the literature, can produce solutions which provably minimize total placement time for all company supplied sample problems. This solution method generates solutions approximately 10% better than those generated manually by the manufacturer's engineers.     

Invariant Measures for Set-Valued Dynamical Systems

A continuous map on a compact metric space, regarded as a dynamical system by iteration, admits invariant measures. For a closed relation on such a space, or, equivalently, an upper semicontinuous set-valued map, there are several concepts which extend this idea of invariance for a measure. We prove that four such are equivalent. In particular, such relation invariant measures arise as projections from shift invariant measures on the space of sample paths. There is a similarly close relationship between the ideas of chain recurrence for the set-valued system and for the shift on the sample path space.

     

广义控制系统的动态阶配置

本文考虑广义控制系统。通过使用输出导数反馈配置系统的动态阶。     

树形多刚体系统动力学方程

本文提出描述联结成树形多刚体系统位置与排列的“位形图”.在不需要引入“增广体”和“子系统”等概念的情况下,本文利用“位形矩阵”研究了树形多刚体系统的运动,并推导出其动力学方程.树形多刚体系统的动态参数与其结构的密切关系在这样的动力学方程中明显地表示出来.     

Navier-Stokes方程的脉动速度方程的最优动力系统建模和动力学分析

研究了基于Navier-Stokes方程的脉动速度方程的最优低维动力系统建模理论.最优目标泛函为脉动速度基函数的不可压缩性和正交性.数值计算了充分发展的并排双方柱绕流问题,并基于双尺度全局最优化方法,建立了它的脉动速度的最优动力系统模型.对其相空间轨道、Poincaré截面、分岔特性、功率谱和Lyapunov指数集等动力学特性进行了分析.随着Reynolds数的增加,双方柱绕流的脉动速度方程最优动力系统具有复杂的类倍周期分岔行为.     

Controllability Results for Nonlinear Fractional-Order Dynamical Systems

This paper establishes a set of sufficient conditions for the controllability of nonlinear fractional dynamical system of order 1<α<2 in finite dimensional spaces. The main tools are the Mittag–Leffler matrix function and the Schaefer’s fixed-point theorem. An example is provided to illustrate the theory.