A time-optimal control algorithm for digital computer control

A time-optimal control algorithm for digital computer control allowing bounds on control variables and state variables is presented. Through linear programming techniques a time-optimal sequence is computed by using a linear discrete model of the process. For real-time applications feedback control is achieved by recalculating the control sequence each sampling period. In addition, an adaptive control strategy based on on-line estimation of the state variables and the parameters of the system is introduced. In order to be able to apply the algorithm in a real-time environment, computational efficiency is emphasized. An application of the algorithm to a sixth-order multivariable system is given.     

Shortest correlation control strategy

A new stochastic control strategy is developed for discrete linear constant systems. The strategy is optimal in the sense of yielding shortest correlation in the error sequence, i.e., the resulting error sequence is as white as possible. This is a stochastic counterpart of the deterministic discrete-time time-optimal control problem. Algebraic manipulations are used to reduce the synthesis procedure to solving a linear polynomial equation.     

Reachability and optimal control for linear hybrid automata: a quantifier elimination approach

This paper considers an optimal control problem for linear hybrid automata (LHA). First, we present a controller synthesis algorithm based on reachability analysis. The algorithm computes the maximal initial set from which the controller drives the system to a given target set. It is shown that, using quantifier elimination (QE), an under-approximation of the maximal reachable set can be derived. Next, a weighted time-optimal control problem is solved by transforming it into a constrained optimization problem whose constraints are a set of inequalities with quantifiers. Quantifier elimination (QE) techniques are employed in order to derive the quantifier free inequalities that are shown to be linear. Thus, the optimal cost is obtained using linear programming. For any state belonging to the maximal initial set the optimal switching times and the optimal continuous control inputs are computed. These are used in order to derive a hybrid controller which is optimal with respect to the cost function. Our results are applied to an air traffic management example which is of practical interest.     

On time-optimal control of a sequence of projects of activitiesunder time-variable resource

A sequence of k projects of independent activities, each project composed of activities available for realization at the same time, is considered. It is assumed that the activities are continuous dynamical systems with dynamics that depend continuously on the alloted amounts of the resource and that the initial and terminal states are fixed. The problem is to allocate the time-variable renewable, continuously divisible resource (e.g. power, fuel flow, approximate manpower) to the activities in order to minimize the performance time of the sequence of projects. A solution is presented that is based on the notion of the set of reachable states under the assumption that the allowable level of the total usage of the resource is piecewise constant. Necessary and sufficient optimality conditions are stated in terms of the performance time, and the existence of the optimal control is proved. An algorithm for the time optimal control is proved. An algorithm for the time-optimal resource allocation is derived and an example is given to illustrate the approach     

Bilinear Control Systems. Special Types

Several classes of bilinear control systems in n-space are identified. Convexity of reachable and attainable sets is treated, with reference to time-optimal regime synthesis.     

From Dubins’ car to Reeds and Shepp’s mobile robot

In the paper, a control system with intermediate dynamics between Dubins’ car and Reeds and Shepp’s mobile robot is investigated. Time-limited reachable sets and reachable sets at given time are computed. Families of semipermeable curves that are useful for the detection of jumps of the value function of time-optimal control problem are constructed. Research is partly supported by the Russian Foundation for Fundamental Research under Grants 06-01-00414 and 07-01-96085.     

Sequential synthesis of the time-optimal control in real time

The paper considers the sequential synthesis method for time-optimal control of linear systems. The method is based on piecewise constant finite controls that ensure approximate solutions for time-optimal control problems. A sequence of finite controls is thereafter transformed into the optimal control. The appropriate computations are reduced to a sequence of linear algebraic equation problems and the integration of a matrix differential equation over the intervals of control switching points and final point change. It is proven that the sequence of finite controls converge to the optimal control. The sliding mode conditions are obtained, as well as the control structure modifications for the motions on switching manifolds. The initial approximations reducing considerably computational complexity are considered. The computational algorithm, together with the modelling and numerical results, are presented.     

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.     

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.     

Generalized bang-bang control for feedforward constrained regulation

In the behavioral framework for continuous-time linear scalar systems, simple sufficient conditions for the solution of the minimum-time rest-to-rest feedforward constrained control problem are provided. The investigation of the time-optimal input-output pair reveals that the input or the output saturates on the assigned constraints at all times except for a set of zero measure. The resulting optimal input is composed of sequences of bang-bang functions and linear combinations of the modes associated to the zero dynamics. This signal behavior constitutes a generalized bang-bang control that can be fruitfully exploited for feedforward constrained regulation. Using discretization, an arbitrarily good approximation of the optimal generalized bang-bang control is found by solving a sequence of linear programming problems. Numerical examples are included.     

Time-scale decomposition of the reachable set of constrained linear systems

We consider a linear control system with a multiparameter singular perturbation representing multiple time scales and with constraints for the control and the slow state. The Hausdorff limit of the reachable set when the small parameters tend to zero is found. The result provides a basis for a time-scale approximation of the reachable set.on leave from the Institute of Mathematics, Bulgarian Academy of Sciences, Sofia, Bulgaria.     

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.     

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.     

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.     

近似时间最优的舵机多模位置控制策略

高超声速飞行器要求其伺服作动系统具有高动态高精度的特性,针对传统控制方法难以兼顾系统动态和精度的难题,本文设计了一种近似时间最优的舵机多模位置控制策略.首先建立了以一阶惯性环节串联积分器表征舵机输入输出特性的特征模型.然后以相平面为分析工具,给出了其近似时间最优控制的切换区;在切换区外以快速性为目标而采用bang-bang最优控制,在切换区内以避免振荡和超调为目标而采用bang-bang次优控制;为提高稳态性能,在小误差时时采用线性控制.实验表明,该方法响应快速,避免了振荡和超调,很好地满足了对伺服作动系统高动态高精度的要求.     

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.     

交流位置伺服系统的时间次优滑模控制器设计

为了提高伺服系统定位过程中的动态特性和鲁棒性,提出一种新的时间次优滑模控制器的设计方法。首先,借鉴时间最优控制对伺服定位过程中的系统状态进行规划,得到时间最优的位置和速度响应轨迹,从而保证定位的快速性。其次,针对传统时间最优控制鲁棒性差和工程实现较难等问题,根据期望的时间最优状态轨迹设计非线性滑模面,并采用鲁棒趋近律方法设计滑模控制律,实现了伺服定位的鲁棒时间次优控制,仿真结果验证了该方法的有效性。最后,在永磁同步电机伺服控制器开发平台中进行了实验分析,结果表明所提出的控制方法具有很好的动态响应性能和鲁棒     

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.     

A complete characterization of reachable sets for constrained linear time-varying systems

We consider linear time-varying systems with linear constraints on both control and state variables. The reachable set is a convex polyhedron that we completely characterize in terms of its boundary hyperplanes.