State Estimation for Second-order Linear Systems: A Sructure-improved First-order Observer

In this paper, a structure-improved first-order observer is proposed to more accurately estimate the state and its derivative in the second-order system. Firstly, under the time-invariant change of coordinates, the novel structure of an improved first-order observer is given to deal with the problem where the derivative of the position estimation is not equal to the estimation of state vector for all time, and further, the sufficient conditions for the existence of such an observer is obtained as a type of the full-actuated second-order Sylvester equations, directly related to the second-order system coefficient matrices. Secondly, on the basis of the proposed existence conditions, the parametric method of the improved first-order observer is presented by adopting the solutions of the Sylvester equation. Finally, the correctness of the proposed method is illustrated through numerical simulation about a simple flexible-joint mechanism.