饱和约束控制系统的吸引域估计

饱和约束控制系统的吸引域估计是非线性系统领域的一个基本问题,主要涉及到饱和约束的处理和Lyapunov函数的选取.围绕这两个关键问题,梳理了解决饱和约束控制系统吸引域估计问题的一系列方法,总结了文献及最近的一些研究结果.     

基于系统输出的嵌套饱和系统抗饱和补偿器设计

针对嵌套输入饱和系统的吸引域扩大问题,本文提出了一种基于系统输出的抗饱和补偿器激发策略,将被控系统输出信号经性能补偿器馈入到抗饱和补偿器激发环节中,形成蕴含系统实时性能信息的抗饱和激发新机制,克服了传统抗饱和激发机制无法直接反映系统性能的缺点.基于上述抗饱和控制新框架,本文建立了抗饱和补偿器及性能补偿器存在的充分条件,并依此构建了优化问题求解最优补偿器增益以实现扩大闭环系统吸引域的目的.仿真结果表明本文方法的有效性.     

线性开关系统的饱和控制

利用开关系统切换的特点研究了线性开关系统的饱和控制问题,在较弱的条件下,得到了开关系统饱和镇定的结果,即构成开关系统的每个子系统可以是非稳定的,但假定由开关系统构成的某个凸组合系统是可稳的.借助于Lyapunov函数设计饱和控制器,基于凸组合系统的稳定性设计切换规律,引入设计参数满足控制器实现的条件,使开关系统局部渐近稳定.     

运用线性反馈分析设计饱和线性系统

给出了状态反馈控制饱和单输入系统以及动态输出反馈单输出饱和线性系统是全局渐近稳定还是区域渐近稳定的充分性条件,并在区域渐近稳定的情况下计算其不变吸引椭球.对于控制饱和系统,运用Ricatti方程迭代法设计控制器,以使所得椭球尽量大.仿真算例说明了所提出方法的有效性.     

输入饱和系统的离散增益调度控制及其在在轨交会中的应用

基于参量Lyapunov方法和不变集理论,针对具有输入饱和非线性约束的线性系统,提出了一种离散增益调度控制方法. 通过逐渐 增大代表闭环系统收敛速率参数的值,所提出的离散增益调度控制方法逐步加快闭环系统的收敛速度,达到改善闭环系统 动态性能的目的. 如果开环系统是非指数不稳定的,则所提出的离散增益调度控制器可实现半全局镇定;反之可实现局部镇定,并均可保证闭环系统的指数稳定性. 最后,将 所提出的方法应用于空间合作目标在轨交会控制系统的控制器设计,并直接在原始非线 性系统模型上进行仿真,结果验证了所提方法的有效性.     

基于LMI的小时滞饱和系统稳定域估计方法

基于Pade近似变换,将小时滞饱和系统的稳定域估计转化为估计奇异摄动饱和系统的稳定域问题.证明了此奇异摄动饱和系统的稳定域具有可解耦性,并在此基础上建立LMI优化模型并提出小时滞饱和系统稳定域估计的降阶方法.算例仿真验证了方法的正确性和有效性.     

跳变双线性随机系统饱和执行器的镇定

对于一类具有Markov跳变参数的双线性离散随机系统,研究其饱和执行器问题.分别采用一般二次型Lyapunov函数、饱和关联Lyapunov函数进行系统随机稳定性分析,以椭圆不变集构造随机稳定域,提出两种依赖于模态跳变率的饱和状态控制器设计方法,两种方法均以线性矩阵不等式的形式给出.     

一种估计奇异摄动饱和系统稳定域的方法

针对奇异摄动饱和系统, 提出了一种估计其稳定域的降阶方法. 结合饱和函数的特殊性质, 证明了此类系统的稳定域可分解为伴随系统的不变集与一个足够大球体的笛卡尔积. 将原系统稳定域估计问题转化为低阶伴随系统稳定域的估计问题, 利用线性矩阵不等式(Linear matrix inequality, LMI)优化方法估计伴随系统的稳定域以减少保守性. 本方法不仅可以克服奇异摄动饱和系统的奇异性, 还可以一定程度克服系统的``维数灾'等问题.     

基于N步不变集的时滞切换系统的饱和控制

构造离散时滞切换系统的不变集,提出基于N步不变集的切换控制器设计方法,估计执行器饱和非线性的吸引域范围。首先,考虑时滞的影响,选取依赖于时滞的Lyapunov函数,构造时滞切换系统的不变集,并将其表达为若干个椭球集的凸组合,椭球集的个数与时滞常数相关。其次,在系统的前N个采样时刻,分别施加不同的饱和约束,求解得到一组椭球集,椭球集的个数与常数N相关,而每一步计算得到的椭球集均为时滞切换系统的不变集。再将N个不变集用一组凸包系数拟合,即可获取较大的吸引域估计。最后,在满足平均驻留时间约束的条件下设计切换律,并设计状态反馈控制器,保证闭环系统渐近稳定。控制器的求解转化为线性矩阵不等式的可行性问题。仿真结果验证了所提方法的可行性和有效性。     

基于嵌套饱和方法的吊车系统非线性控制

针对吊车系统定位和防摆的控制要求,提出了一种基于嵌套饱和方法的非线性控制策略。对吊车系统动力学方程进行部分反馈线性化,并通过坐标变换将其转化为便于控制器设计的严格前馈级联规范型;在此基础上利用嵌套饱和非线性控制方法设计了吊车定位防摆控制器。仿真结果表明,该方法在较小的控制力作用下实现了吊车系统的定位和防摆,并且对于吊车系统参数的变化具有很强的鲁棒性。     

Stability and stabilization of discrete-time periodic linear systems with actuator saturation

This paper is concerned with the problems of stability and stabilization for discrete-time periodic linear systems subject to input saturation. Both local results and global results are obtained. For local stability and stabilization, the so-called periodic invariant set is used to estimate the domain of attraction. The conditions for periodic invariance of an ellipsoid can be expressed as linear matrix inequalities (LMIs) which can be used for both enlarging the domain of attraction with a given controller and synthesizing controllers. The periodic enhancement technique is introduced to reduce the conservatism in the methods. As a by-product, less conservative results for controller analysis and design for discrete-time time-invariant systems with input saturation are obtained. For global stability, by utilizing the special properties of the saturation function, a saturation dependent periodic Lyapunov function is constructed to derive sufficient conditions for guaranteeing the global stability of the system. The corresponding conditions are expressed in the form of LMIs and can be efficiently solved. Several numerical and practical examples are given to illustrate the theoretical results proposed in the paper.     

Stabilization of linear discrete-time systems with actuator saturation

This paper investigates the stabilization problem of linear discrete-time systems with actuator saturation. A simple method recently developed for analyzing the elemental perturbation bound of robust pole-assignment is extended to obtain the stabilization condition. The state feedback controller is designed to satisfy the requirement of stability under the nonlinear saturation. An example is given to demonstrate the use of the proposed approach.     

Analysis and design of delta operator systems with nested actuator saturation

In this paper, the problem of estimating the domain of attraction is considered for delta operator systems subject to nested actuator saturation. A set invariance condition is established for the delta operator system with nested actuator saturation in terms of auxiliary feedback matrices. Based on the set invariance condition, an optimisation approach is proposed to estimate the domain of attraction for the delta operator system. Thereby, the partial results of nested actuator saturation for both continuous-time systems and discrete-time systems are extended to delta operator system framework. A numerical example is provided to illustrate the effectiveness of the proposed design techniques.     

An analysis and design method for linear systems under nested saturation

This paper considers a linear system under nested saturation. Nested saturation arises, for example, when the actuator is subject to magnitude and rate saturation simultaneously. A condition is derived in terms of a set of auxiliary feedback gains for determining if a given ellipsoid is contractively invariant. Moreover, this condition is shown to be equivalent to linear matrix inequalities (LMIs) in the actual and auxiliary feedback gains. As a result, the estimation of the domain of attraction for a given set of feedback gains can be formulated as an optimization problem with LMI constraints. By viewing the feedback gains as extra free parameters, the optimization problem can be used for controller design.     

Analysis and design of discrete-time linear systems with nested actuator saturations

This paper is concerned with the analysis and design of discrete-time linear systems subject to nested saturation functions. By utilizing a new compact convex hull representation of the saturation nonlinearity, a linear matrix inequalities (LMIs) based condition is obtained for testing the local and global stability of the considered nonlinear system. The estimation of the domain of attraction and the design of feedback gains such that the estimation of the domain of attraction for the resulting closed-loop system is maximized are then converted into some LMIs based optimization problems. Compared with the existing results on the same problems, the proposed solutions are less conservative as more slack variables are introduced into the conditions. A couple of numerical examples are worked out to validate the effectiveness of the proposed approach.     

状态饱和离散线性系统的稳定性分析

讨论一类具有状态饱和非线性的离散线性系统稳定性分析问题. 通过引入无穷范数小于等于1 的自由矩阵与对角元素非正的对角矩阵, 将状态饱和离散线性系统的状态变量约束在一个凸多面体内, 进而以矩阵不等式形式给出状态饱和离散线性系统的稳定性判据, 并给出该矩阵不等式的迭代线性矩阵不等式算法. 基于这一稳定性判据, 给出了基于迭代线性矩阵不等式的状态反馈控制律设计算法. 通过状态饱和离散线性系统的状态空间分割方法, 给 出了保守性更小的稳定性判据, 并给出了相应的迭代线性矩阵不等式算法. 数值例子验证了所给出方法的正确性与有效性.

     

The Ellipsoidal Invariant Set of Fractional Order Systems Subject to Actuator Saturation: The Convex Combination Form

The domain of attraction of a class of fractional order systems subject to saturating actuators is investigated in this paper. We show the domain of attraction is the convex hull of a set of ellipsoids. In this paper, the Lyapunov direct approach and fractional order inequality are applied to estimating the domain of attraction for fractional order systems subject to actuator saturation. We demonstrate that the convex hull of ellipsoids can be made invariant for saturating actuators if each ellipsoid with a bounded control of the saturating actuators is invariant. The estimation on the contractively invariant ellipsoid and construction of the continuous feedback law are derived in terms of linear matrix inequalities (LMIs). Two numerical examples illustrate the effectiveness of the developed method.      

Saturation-based switching anti-windup design for linear systems with nested input saturation

This paper proposes a saturation-based switching anti-windup design for the enlargement of the domain of attraction of a linear system subject to nested saturation. A nestedly saturated linear feedback is expressed as a linear combination of a set of auxiliary linear feedbacks, which form a convex hull where the nestedly saturated linear feedback resides. This set of auxiliary linear feedbacks is then partitioned into several subsets. The auxiliary linear feedbacks in each of these subsets form a convex sub-hull of the original convex hull. When the value of the nestedly saturated linear feedback falls into a convex sub-hull, it can be expressed as a linear combination of the subset of all the auxiliary feedbacks that form the convex sub-hull. A separate anti-windup gain is designed for each convex sub-hull by using a common quadratic Lyapunov function and is implemented when the value of the nestedly saturated linear feedback falls into this convex sub-hull. Simulation results indicate that such a saturation-based switching anti-windup design has the ability to significantly enlarge the domain of attraction of the closed-loop system.