Luenberger-like interval observer design in the physical framework of uncertain vector second-order dynamic systems

This paper is the first to investigate the design problem of Luenberger-like dynamic interval observers (LD-IOs) for a class of uncertain vector second-order (VS) systems. First, a coordinate transformation-based classical LD-IO structure is re-built within the original physical framework of VS systems, and an improved generalized proportional–differential LD-IO structure with more design degrees of freedom is further developed through introducing the differential elements and re-representing its uncertainty boundaries and outputs. Meanwhile, the existence conditions of such two LD-IOs are formulated as a class of constrained VS matrix equations on the original physical coefficients of VS systems, which better retains certain advantages on computational amount and physical background than the order-reduction ones. We prove the solvability of the existence conditions under the common observable condition, and the parametric expressions of two LD-IOs are obtained by solving the constrained VS matrix equations, more clearly showing the available design parameters and thus providing greater operability and less conservatism than the LMIs-based one. Besides, the above results are discussed and extended in different coupling modes for the special decentralized outputs. Finally, a non-linear robotic manipulator system and a spacecraft relative motion system are separately simulated to verify the superiority and correctness of the proposed methods.