Design of P2D observers for second-order dynamic systems

A class of new type of observers for second-order dynamic systems named proportional plus second-order derivative (P2D) observers are proposed in this paper and their forms are given. Based on a complete general solution to a class of generalized Sylvester matrix equations, the design method of P2D observers in second-order dynamic systems is proposed. The method provides the complete parameterizations for all of the gain matrices and the parametric expression for the left eigenvector matrix of the observer system. The design method provides the system with all the degrees of freedom in design that can be utilized in order to optimize the system, achieving various desired specifications and performances such as LTR, disturbance decoupling and robustness. Besides, this method is relatively easy since it directly utilizes the original system data and only manipulates on n-dimensional matrices. An illustrative example is also given to show the simplicity and effectiveness of the proposed method.