Novel mechanical classification and energy function construction for fourth-order linear systems and its applications

This study presents a systematic classification of fourth-order linear systems from a mechanical vibration perspective. For each category, the corresponding vibration model and a physically meaningful energy function are established. According to the distribution of eigenvalues and the structure of the Jordan blocks, fourth-order linear systems can be divided into four distinct types: decoupled vibration models, coupled vibration models, vibration models with nonholonomic constraints, and models fundamentally irreducible to any vibration form. Furthermore, based on the proposed vibration models and the derived energy functions, suitable state-dependent switching rules are developed for fourth-order linear switching systems with positive real eigenvalues in all subsystems. These rules ensure that the system states converge to the origin with the fastest possible convergence rate. Finally, simulation results and comparative analyses demonstrate that the energy functions derived from the proposed theoretical framework effectively resolve the switching stabilization problem for fourth-order switching systems composed entirely of unstable subsystem. This offers a new, practical approach to designing switching rules for complex dynamical systems.