基于线性矩阵不等式的一类广义系统观测器设计

Design of Observer for a Class of Generalized System Based on Linear Matrix Inequality

状态反馈控制能有效进行系统控制,并使其正常工作。由于系统状态的测量具有一定的难度,因此需要设计状态观测器,利用重构的状态进行反馈。针对一类非线性部分满足Lipschitz条件的广义系统,讨论了系统全维观测器和降维观测器的设计问题。首先,根据系统的可观性,提出对广义系统进行两次变换,并将系统变换成易于设计观测器的形式。其次,利用系统的变换式对系统的两种观测器进行设计,通过Lyapunov稳定性理论给出观测器的状态估计误差系统稳定且误差收敛于零的充分条件。考虑到观测器增益矩阵求解的盲目性,利用Schur补引理给出了两种观测器存在的充分条件,并简化了增益矩阵的求解过程。最后,给出数值算例,求解出两种观测器的增益矩阵,证明了观测器设计的可行性。该研究为类似系统观测器的构建提供了设计方法。

State feedback control can effectively control the system and make it work normally.Because the measurement of system state is difficult,it is necessary to design a state observer and use the reconstructed state to feedback.For some of the generalized systems whose nonlinear part satisfies Lipschitz condition,the design of full dimensional observer and reduced dimensional observer is discussed.Firstly,according to the observability of the system,two transformations of the generalized system are proposed and the system is transformed into a form that is easy to design observers.Secondly,two kinds of observers of the system are designed by using the transformation of the system;through the Lyapunov stability theory,the sufficient condition that the state estimation error system of the observer is stable and the error converges to zero is given.Considering the blindness of solving the gain matrix of the observer,the Schur complement lemma is used to give the sufficient conditions for the existence of the two observers and simplify the solution process of the gain matrix.Finally,a numerical example is given to solve the gain matrix of the two observers,which proves the feasibility of the observer design.This study provides a design method for the construction of observers for similar systems.