Stability of Second-Order Discrete Linear Switching Systems: The Case of All Subsystems With Eigenvalue Modulus > 1 $$ >1 $$ Under State-Dependent Switching

This article investigates the asymptotic stability of discrete-time switching systems in which the modulus of all subsystem eigenvalues are greater than one. The study begins by constructing an energy function with practical physical meaning for each discrete-time subsystem. Specifically, a standard linear discrete system is derived by discretizing a single-degree-of-freedom vibratory model, with its energy function defined as the sum of the model's kinetic and potential energies. An invertible transform is then applied to convert a general linear discrete system into standard form, enabling the construction of a general energy function based on that of a standard one. Since the construction of subsystem energy functions depends on the choice of invertible transform, which is not unique, the resulting energy functions for different subsystems under distinct transforms cannot be directly compared. To address this issue, an intermediate subsystem is introduced to define a relative energy ratio function between two subsystems. Using this relative energy ratio function, a state-dependent switching rule is developed to maximize energy loss within a switch circuit, thereby achieving rapid asymptotic stability of the discrete switching system. Finally, numerical examples are provided to validate the effectiveness of the proposed method.