A unified framework of convex stability conditions for 2-D switched systems with stable or unstable modes

The discrete-time two-dimensional (2-D) switched system is a powerful tool to model many practical physical processes with abrupt changes. However, the stability analysis problem for 2-D switched systems struggles to provide a unified framework for stable or unstable modes. Users have to delicately tune lots of parameters to shape the Lyapunov function for each mode. Besides, the resulting stability conditions are non-convex and thus the feasibility highly depends on the tuning parameters. In this paper, we propose a unified stability framework for 2-D switched systems, formulated by the well-known Fornasini–Marchesini local state-space model and the Roesser model. The concepts called unit switching sequence (USS) and sequence generator are presented. A novel Lyapunov function is also accordingly established which views the switching sequence as a concatenation of USSs, different from the perspective that regards the switching sequence as a combination of subsystems in previous literatures. Upon this framework, a convex stability criterion is developed, and the conservativeness can be lowered by a lifting technique at the expense of a higher off-line computational complexity. The derived theories are further extended for a standard l₂-gain which is achieved under restricted switching signals for the first time. Numerical examples are provided for validity verification, conservativeness comparison, and l₂-gain computation.