Generalized Luenberger observer design for matrix second-order linear systems

In this paper, the issue of designing a type of generalized Luenberger observers for matrix second-order linear (MSOL) systems is addressed in the matrix second-order framework. By introducing the concept of stable matrix pair for MSOL systems, sufficient and necessary conditions for the design of the type of generalized Luenberger observers are given under the assumption of controllability and observability of the MSOL system. Based on the proposed conditions and the right coprime factorization of the system, a parametric approach to the design of such type of observers is presented, and a corresponding design algorithm is proposed. The proposed approach provides all the degrees of design freedom, which can be further utilized to achieve additional system specifications. A spring-mass system is utilized to show the effect of the proposed algorithm.