Abstract: Design of a high-order proportional plus integral (PI) observer in second-order dynamic systems is investigated.Based on the solutions to Sylvester matrix equations,a parametric method for designing this observer is proposed.The method presents the parametric expression for the gain matrix of the high-order PI observer.The contained parameters satisfy the needs of two constraints and are completely free as well.These parameters offer all the design degrees of freedom,which can be further selected to satisfy certain performances by means of optimization,such as disturbance decoupling,fault detection and robustness.In addition,the parametric method of designing the PI observers directly utilizes the original parameter matrices of the second-order dynamic systems and involves manipu-lations only on n-dimensional matrices of the second-order dynamic systems.A numerical example is presented to show the simplicity and effectiveness of the proposed method.