二阶Lipschitz非线性系统自然观测器设计

Design of natural observers for second-order Lipschitz nonlinear systems

机械系统的动力学方程是用状态速度和状态加速度描述的二阶系统,而传统的基于一阶系统表示的观测器理论并不完全适用于二阶系统,为此,文中针对一类二阶Lipschitz非线性系统提出了自然观测器的设计问题。所提出的自然观测器与被观测的二阶系统具有相同的代数结构,能保证位置估计量的导函数恰好是速度估计量。为降低结果的保守性,利用非线性函数的Lipschitz性质,将估计误差系统表示为LPV(Linear Parameter Varying)系统,在此基础上,引入参数依赖的Lyapunov函数分析估计误差系统的稳定性,并建立了自然观测器存在的充分条件。该条件表示为一组带有可调参数的线性矩阵不等式,通过求解该组线性矩阵不等式,便可获得自然观测器的增益矩阵和估计误差的收敛速率。最后通过数值例子验证了本文结果的有效性和实用性。

The dynamics of mechanical systems are described by second-order systems in terms of both state velocities and state accelerations. The traditional observer theory based on the representation of one-order form is not fully suitable to second-order systems. In this paper,the design problem of natural observers for a class of second-order Lipschitz nonlinear systems is addressed. The proposed natural observer has the same structure as the observed second-order system,which ensures that the derivative of the estimated position is indeed the estimated velocity. In order to reduce conservatism,the estimation error system is reformulated as a linear parameter varying system（ LPV） by taking the properties of the Lipschitz nonlinear function into account. Based on the representation,a parameter-dependent Lyapunov function is introduced to analyze the stability of the error dynamics. A sufficient condition for the existence of natural observers is established. The sufficientcondition is expressed as a set of linear matrix inequalities with a tuning parameter. Finally,a numerical example is presented to validate the effectiveness and practicability of the proposed method.