Measure-controlled dynamic systems: Polyhedral approximation of their reachable set boundary

An algorithm for polyhedral approximation of the reachable set of impulsive dynamic control systems is designed. The boundary points of the reachable set are determined by recursively generating and solving a family of auxiliary optimal impulsive control problems with state-linear objective functional. The impulsive control problem is solved with an algorithm that implicitly reduces the problem an ordinary optimal control problem. The reduced problem thus obtained is solved with an algorithm based on local approximations of the reachable set.     

On the asymptotic stability of boundary trajectories

For a class of dynamical systems, it is shown that trajectories which remain on the boundary of a non-empty, connected subset of a reachable set must be asymptotically stable. Using this result and a reachability maximum principle, we obtain a control law which will drive the system from the origin to the boundary of the reachable set for a class of two-dimensional linear control systems. Using this control law, the reachable set is easily constructed.     

Construction of switching sequences for reachability realization of switched impulsive control systems

This paper studies the construction of switching sequences for reachability realization of switched impulsive control systems. An approach is proposed to design switching sequences so that the reachable subspace of switched impulsive control systems is expressed in terms of the reachable state sets of the designed switching sequences. For a class of switched impulsive control systems, it is proved that a single switching sequence can be designed with its reachable state set coinciding with the reachable subspace. Periodic switching sequences are also constructed in this paper for the reachability realization problem. The results present a new way to exploit the switching mechanism to achieve the reachability and controllability of switched impulsive control systems. Copyright © 2007 John Wiley & Sons, Ltd.     

Time-optimal control strategy for saturating linear discrete systems

An easily implemented strategy for obtaining the required minimal-time and time-optimal control sequence for a given initial state of an nth-order linear discrete system with constraints on the input is described. This strategy uses the boundary faces of the reachable set for the determination of the minimal time. The time-optimal control sequence is achieved by dividing the reachable set into linear and saturated parts.     

Asymptotics of reachable sets of linear dynamical systems with impulsive control

Explicit asymptotic formulas for the reachable sets of linear dynamical systems with constraints on the total impulse of control action under different assumptions concerning the spectrum of the system’s matrix are obtained. It is shown that for large time, the reachable sets can be approximately represented in the form of the product of the scaling matrix and the normalized reachable set, where the matrix is an elementary function of time, and the normalized reachable set depends on time quasi-periodically. Analysis of asymptotic formulas has shown that, generally speaking, there exists a continuum of limit shapes, the aggregate of which produces a multidimensional attractor.     

Properties of the reachable set of control systems

We investigate some properties of the reachable set of a control system. Representing the system as a differential inclusion and using proximal Hamilton–Jacobi equation we describe its graph. We work in infinitely dimensional Hilbert space and use one sided Lipschitz approach. The funnel equation is considered in the last section. That equation describes the reachable set in arbitrary Banach space. We consider also the autonomous case and prove the existence of a limit of the reachable set.     

带有运行成本函数的Hamilton-Jacobi可达性分析

水平集方法将可达集表示为Hamilton-Jacobi方程解的零水平集,保存多个不同时间范围的可达集则需要保存Hamilton-Jacobi方程在多个时刻的解,这不仅需要消耗大量的存储空间还为控制律的设计造成了困难.针对这些局限性,提出了一种改进的基于Hamilton-Jacobi方程的可达集表示方法.该方法在Hamilton-Jacobi方程中加入了一项运行成本函数,可以用同一个时刻的解的多个非零水平集表示多个不同时间范围的可达集,极大地节省了存储空间并为控制律的设计提供了便利.为了求解所构造的带有运行成本函数的Hamilton-Jacobi方程,采用了一种基于递归和插值的方法.最后,通过一些数值算例验证了所提出的方法的精确性、在存储空间方面的优越性以及设计的控制律的有效性.     

车辆跟随控制策略的状态可达集建模及验证方法

车辆动态行为的不确定性会造成跟随车辆控制的不确定性.传统的车辆跟随控制方法只是针对车辆的单一行为动态的不确定性进行建模,无法遍历车辆所有可能的控制输入,因而,既无法一次性提供跟随策略下完整的可行控制方案,也不足以在理论上保证对策略安全检测的可信性.为此,提出车辆跟随控制策略的状态可达集建模及验证方法.该方法将控制策略转换为能用可达集计算和表征的多级安全判定事件,利用随机可达集的状态遍历特征描述车辆控制输入的不确定性,通过对可达集交集的判断,辨识所有初始条件对应的危险与安全控制行为,为驾驶员提供完整的可选择控制方案;然后利用马尔科夫链逼近可达集,近似表达车辆行为的不确定性,依据驾驶员行为习惯统计验证策略的安全性,实现对控制策略的有效建模分析.实验结果表明,所提出的建模及验证方法不仅可以完备地表征车辆不确定行为,提供交通情形中跟随策略相应的完整控制方案,也可实现对策略安全性的精确验证.     

基于Hamilton-Jacobi方程的飞行器机动动作可达集分析

为了给驾驶员完成标准机动动作提供决策支持, 提出一种使用哈密尔顿-雅克比(Hamilton-Jacobi)方程求解机动动作可行状态空间的研究方法.使用关键点将机动动作划分为不同阶段, 将各关键点的标准状态约束作为目标集, 逆时间求解目标集对应的可达集得到各阶段的边界状态范围, 目标集和可达集均由零水平集表示.使用该方法得到斤斗动作三维度运动模型下各阶段的可达集及斤斗动作的可行状态空间, 为了使运动模型的控制量与驾驶员实际操纵更为接近, 构建了以迎角变化率为控制量的四维度运动模型, 在此基础上对斤斗动作各阶段的可达集进行了分析.     

一种求解线性控制系统可达集的数值方法

针对线性控制系统,研究应用常微分方程数值方法和优化技术相结合的近似可达集的方法.首先,用常微分方程数值方法对系统进行离散化.然后,提出基于优化技术的外部投影法来近似离散系统的可达集.外部投影法构造有限多个投影问题,每个都对应一个凸优化问题,通过求解这些凸优化问题最终可以得到可达集的近似描述.最后,通过数值仿真结果验证了所提出方法的有效性.与文献中已有的方法相比,在求解相同数量凸优化问题的情况下,外部投影法的近似精度更高.     

基于可达集的鲁棒模型预测控制

设计了一种基于可达集的鲁棒模型预测控制算法.首先确定了一个鲁棒不变集,并将此不变集用作模型预测控制的终端约束集;接着采用终端约束集对可达集的包含度作为优化指标;最后,采用预测时域逐渐减小的控制策略以保证在线优化存在可行解.从理论上证明了吸引域内的任意点在有限时域内都会被引导至终端约束集并始终停留在此集之内,并由仿真算例验证了本文所设计鲁棒模型预测控制算法的可行性.     

Overapproximating Reachable Sets by Hamilton-Jacobi Projections

In earlier work, we showed that the set of states which can reach a target set of a continuous dynamic game is the zero sublevel set of the viscosity solution of a time dependent Hamilton-Jacobi-Isaacs (HJI) partial differential equation (PDE). We have developed a numerical tool—based on the level set methods of Osher and Sethian—for computing these sets, and we can accurately calculate them for a range of continuous and hybrid systems in which control inputs are pitted against disturbance inputs. The cost of our algorithm, like that of all convergent numerical schemes, increases exponentially with the dimension of the state space. In this paper, we devise and implement a method that projects the true reachable set of a high dimensional system into a collection of lower dimensional subspaces where computation is less expensive. We formulate a method to evolve the lower dimensional reachable sets such that they are each an overapproximation of the full reachable set, and thus their intersection will also be an overapproximation of the reachable set. The method uses a lower dimensional HJI PDE for each projection with a set of disturbance inputs augmented with the unmodeled dimensions of that projection's subspace. We illustrate our method on two examples in three dimensions using two dimensional projections, and we discuss issues related to the selection of appropriate projection subspaces.     

Zonotopic reachable set estimation for linear discrete‐time systems with time delay

This article studies reachable set estimation for linear discrete‐time systems with time delay, which are influenced by unknown but bounded disturbances. We propose a novel reachable set estimation method based on zonotopes for the considered systems. The proposed method can estimate real‐time reachable set under nonzero initial conditions. In order to increase estimation accuracy, we propose an iterative method to reduce the conservatism caused by the couplings between reachable sets at different instants. The effectiveness of the proposed method is illustrated by three numerical simulations.     

Optimal technique for estimating the reachable set of a controlled n-dimensional linear system

The reachable set from a given point of a controlled dynamical system is the set of all states to which the system can be driven from that point in a finite time by the allowed controls. A technique is presented Tor estimating the reachable set from the asymptotically-stable origin of a class of n-dimensional linear systems under bounded control. The technique is an optimal version of a Lyapunov method, and provides an (over-)estimate of the full reachable set; it involves the minimization of a quadratic constraint, followed by the maximization of a quadratic form subject to this constraint. The non-linear optimization problems can be routinely solved by means of computer algebra and commonly available computer software. In general, the technique produces a much-improved estimate of the reachable set compared to that given by the standard Lyapunov method. Another advantage of the technique is that it is truly applicable to higher-dimensional systems (n ≥ 3). Since the estimate produced is in the convenient form of an n-dimensional ellipsoid, projections of the estimate onto any space of dimension ≤ n — 1 can be readily found. Problems of two, three, and four dimensions are solved to illustrate the technique.     

Determination of reachable set for a linear discrete system

An easily implemented method for obtaining the reachable setR_{N}, N > n, of annth-order discrete system with constraints on the input is described. This method uses elementary matrix calculations at each step to construct the boundary hyperplanes of the reachable set. The result can be applied to determine the minimum control time required for a given terminal state. IfN leq n, the method of echelon matrix analysis is employed to solve the same problem.     

Under-estimates of reachable sets for linear control systems

A new algorithm is presented for providing under-estimates of the reachable set from the origin for a class ofn-dimensional linear systems with bounded controls. This algorithm is based on the novel approach of choosing a feedback control which makes all the eigenvalues of the closed loop system unstable. Results from feedback control and Liapunov stability theory are then used to formulate the problem as the minimization of a nonlinear function subject to constraints on certain matrices. The solution of this optimization problem provides an under-estimate of the reachable set in the form of ann-dimensional ellipsoid. Examples of both continuous and discrete-time systems are presented to illustrate the method. Comparison with existing exact results for some 2-dimensional systems shows that the method provides good approximations in these cases.     

Reachable set of bilinear control systems with time varying drift

In this article, we give a complete characterization of the reachable set at all times for a class of bilinear control systems with time varying drift and unbounded control amplitude. These results are of fundamental interest in geometric control theory and have important applications to control of coupled spins in solid state NMR spectroscopy.     

不确定线性系统的可达集估计

本文针对线性不确定系统,研究了系统状态可达集估计问题.在已知参考输入,未知扰动有界的条件下,提出了一种新的可达集估计方法,并把所提方法推广到系统不稳定的情况,分别估计出不稳定系统的闭环可达集和开环可达集.通过分析动态系统的李雅普诺夫函数,将求取系统可达集的问题转化为线性矩阵不等式优化问题,并将可达集范围用椭球集形式表述.最后分别通过数值仿真分析,验证了所提出方法对线性系统可达集估计的有效性.     

On the reachability of quantized control systems

In this paper, we study control systems whose input sets are quantized, i.e., finite or regularly distributed on a mesh. We specifically focus on problems relating to the structure of the reachable set of such systems, which may turn out to be either dense or discrete. We report results on the reachable set of linear quantized systems, and on a particular but interesting class of nonlinear systems, i.e., nonholonomic chained-form systems. For such systems, we provide a complete characterization of the reachable set, and, in case the set is discrete, a computable method to completely and succinctly describe its structure. Implications and open problems in the analysis and synthesis of quantized control systems are addressed