Dynamic output feedback regulation for a class of nonlinear systems

In this paper results are presented on the problem of regulating nonlinear systems by output feedback, using Lyapunov-based techniques. In all the cases considered here, we assume that the part of the state which is not measured enters linearly in the equations. Sufficient conditions for the global stabilization of the observed states via dynamic output feedback are obtained, assuming that such stabilization is possible using state feedback. Systems satisfying these conditions include a natural class of bilinear systems and systems which reduce to linear observable systems when the nonlinear terms in the measured states are removed. Some simple examples are included to illustrate our approach.This work was supported in part by the Natural Sciences and Engineering Research Council of Canada. J.-B. Pomet is now with Laboratoire d'Automatique de Nantes (URA C.N.R.S. 823), E.C.N., 44072 NANTES cedex 03, France; most of this work was done when he was with Queen's University.     

Stabilization via homogeneous feedback controls

In this paper, we provide an explicit homogeneous feedback control with the requirement that a control Lyapunov function exists for an affine control system and satisfies an homogeneous condition. We use a modified version of the Sontag formula to achieve our main goal. Moreover, we prove that the existence of an homogeneous control Lyapunov function for an homogeneous affine system leads to an homogeneous closed-loop system by using the previous feedback control.     

Adding an integrator for the stabilization problem

We study the relationship between the following two properties: P1: The system is locally asymptotically stabilizable; and P2: The system is locally asymptotically stabilizable; where . Dayawansa, Martin and Knowles have proved that these properties are equivalent if the dimension n = 1. Here, using the so called Control Lyapunov function approach, (a) we propose another more constructive and somewhat simpler proof of Dayawansa, Martin and Knowles's result; (b) we show that, in general, P1 does not imply P2 for dimensions n larger than 1; (c) we prove that P2 implies P1 if some extra assumptions are added like homogeneity of the system. By using the latter result recursively, we obtain a sufficient condition for the local asymptotic stabilizability of systems in a triangular form.     

Nonsmooth feedback stabilizer for strict-feedback nonlinear systems that may not be linearizable at the origin

We present a continuous feedback stabilizer for nonlinear systems in the strict-feedback form, whose chained integrator part has the power of positive odd rational numbers. Since the power is not restricted to be larger than or equal to one, the linearization of the system at the origin may fail. Nevertheless, we show that the closed loop system is globally asymptotically stable (GAS) with the proposed continuous (but, possibly not differentiable) feedback. We formulate a condition that enables our design by characterizing the powers of the given system. The condition also shows that our result is an extension of Qian and Lin [Non-lipschitz continuous stabilizers for nonlinear systems with uncontrollable unstable linearization, Systems Control Lett. 42 (2001) 185–200] where the power of odd positive integers has been considered. New result on the global finite time stabilization problem is also presented.     

线性时滞不确定系统的鲁棒性研究

本文研究了具有一般摄动表达形式的时滞不确定系统,将不确定项分解为“秩1”形式和相乘形式,利用Lyapunov方法和二次镇定的概念,求得了较为完善的充分条件,并举例验证。     

欠驱动两杆机器人的统一控制策略和全局稳定性分析

针对包括 Acrobot 和 Pendubot 在内的欠驱动两杆机器人, 提出了一种统一的运动控制策略. 欠驱动两杆机器人的整个运动空间分为两个区域: 摇起区和平衡区, 并对这两个区域分别设计控制律. 首先, 在摇起区, 应用一种基于弱控制 Lyapunov 函数 (Weak-control Lyapunov function, WCLF) 的控制方法, 来增加系统能量和控制驱动杆的姿势. 其次, 为了避免奇异值的出现, 选择弱控制 Lyapunov 函数中的一个参数为系统状态空间的非线性函数. 然后, 通过系统状态调节基于弱控制 Lyapunov 函数的控制律中的另一个设计参数, 来改进系统控制效果. 使用弱控制 Lyapunov 设计的摇起区控制律, 可基于最大不变集原理保证其稳定性; 而机器人离开摇起区后, 利用非光滑 Lyapunov 函数 (Non-smooth Lyapunov function, NSLF) 来保证其稳定. 最后, 结合 WCLF 和 NSLF 保证了控制系统的全局稳定.     

Constructive stabilization for quadratic input nonlinear systems

In this paper stabilization of nonlinear systems with quadratic multi-input is considered. With the help of control Lyapunov function (CLF), a constructive parameterization of controls that globally asymptotically stabilize the system is proposed. Two different cases are considered. Firstly, under certain regularity assumptions, the feasible control set is parameterized, and continuous feedback stabilizing controls are designed. Then for the general case, piecewise continuous stabilizing controls are proposed. The design procedure can also be used to verify whether a candidate CLF is indeed a CLF. Several illustrative examples are presented as well.     

关于Sontag-Type控制的注记

基于控制李雅普诺夫函数的Sontag-Type控制是仿射系统鲁棒镇定中的重要控制律.首先揭示该控制律本质上是一种变结构控制且闭环的切换面总可达,受此启发并为了相对容易地构造控制李雅普诺夫函数,运用零状态可检测概念定义弱控制李雅普诺夫函数,并证明了基于弱控制李雅普诺夫函数的Sontag-Type控制的优化镇定性.文中还证明,在温和条件下,基于弱控制李雅普诺夫函数的Sontag＿Type控制为仿射系统的输入到状态镇定控制.     

具有对称初始数据的二维反应扩散方程的边界镇定

研究了二维圆盘上具有对称初始数据的反应扩散方程的边界控制. 由于初始条件和边界条件关于圆心旋转对称, 系统可以转化为等价的极坐标系下的一维抛物方程. 此时, 极点的奇异性成为了控制器设计中的难点. 本文设计了一系列方程变换, 消除了核函数方程中极点奇异性的影响, 将其转化为修正的Bessel方程, 求出了显式的核函数表达式和精确的边界反馈控制律, 扩展了偏微分方程的backstepping方法. 系统的收敛速度可通过改变控制器中的一个参数来调节. 然后用Lyapunov函数法证明了闭环系统在H1范数下指数稳定, 表明了系统对初值的连续依赖. 最后用数值仿真验证了方法的有效性.     

二维切换系统指数镇定问题研究

对于由两个不稳定的线性时不变子系统构成的二维切换系统,利用逐段光滑Lyapunov函数方法研究其指数镇定问题．通过切换面及相应Lyapunov函数的适当参数化,导出了关于可指数镇定性的代数判据,由此得到切换面及Lyapunov函数的求解方法．算例验证了该理论的灵活性与有效性．     

带有未知参数的惯性轮摆系统的自适应控制

针对带有未知参数的惯性轮摆系统,提出了一种自适应控制律设计方法。首先利用坐标变换将惯性轮摆系统的动力学模型转化为级联系统的形式。然后,针对系统参数未知的问题,在已有的惯性轮摆系统反馈控制律的基础上,利用控制器迭代设计思想,设计了惯性轮摆系统的自适应控制律,并利用李雅普诺夫稳定性理论证明了所得自适应控制律可以使得带有未知参数的惯性轮摆系统保持在摆杆垂直向上的平衡状态。最后以一个实际的惯性轮摆系统为例,采用该系统的物理参数进行仿真,分析了不同自适应参数下惯性轮摆系统各状态的收敛速度及摆起和稳定时间。仿真结果验证了所设计自适应控制器能够使惯性轮摆系统从垂直向下的平衡位置摆起并稳定在垂直向上的平衡位置。     

基于混合观测器的混合反馈控制

对于一类混合动态系统,研究基于混合观测器的混合反馈控制问题．通过系统线性部分和离散事件部分的Lyapunov函数构造了整个混合系统的Lyapunov函数．据此设计了使整个系统稳定的混合反馈控制且证明了闭环系统的稳定性．仿真实例说明该方法的有效性．     

Backstepping design for global stabilization of switched nonlinear systems in lower triangular form under arbitrary switchings

This paper is concerned with the global stabilization problem for switched nonlinear systems in lower triangular form under arbitrary switchings. Two classes of state feedback controllers and a common Lyapunov function (CLF) are simultaneously constructed by backstepping. The first class uses the common state feedback controller which is independent of switching signals; the other class utilizes individual state feedback controllers for the subsystems. As an extension of the designed method, the global stabilization problem under arbitrary switchings for switched nonlinear systems in nested lower triangular form is also studied. An example is given to show the effectiveness of the proposed method.     

Controller design for nonlinear affine systems by control Lyapunov functions

This paper addresses the stabilization problems for nonlinear affine systems. First of all, the explicit feedback controller is developed for a nonlinear multiple-input affine system by assuming that there exists a control Lyapunov function. Next, based upon the homogeneous property, sufficient conditions for the continuity of the derived controller are developed. And then the developed control design methodology is applied to stabilize a class of nonlinear affine cascaded systems. It is shown that under some homogeneous assumptions on control Lyapunov functions and the interconnection term, the cascaded system can be globally stabilized. Finally, some interesting results of finite-time stabilization for nonlinear affine systems are also obtained.     

FEEDBACK STABILIZATION OF NONHOLONOMIC CONTROL SYSTEMS WITH DRIFT

This paper presents a differential geometric approach for feedback stabilization of nonholonomic control systems with drift and its applicability is tested on two different systems possessing different algebraic structures: a system with six state variables and three controls, and a knife edge example. The approach is universal in the sense that it is independent of the vector fields determining the motion of the system, or of the choice of a Lyapunov function. The proposed feedback law is as a composition of a standard stabilizing feedback control for a Lie bracket extension of the original system and a periodic continuation of a specific solution to an open loop control problem stated for an abstract equation on a Lie group, an equation which describes the evolution of flows of both the original and extended systems. The open loop problem is solved as a trajectory interception problem in logarithmic coordinates of flows.