Robust Positively Invariant Sets for Switched Systems under Smooth Control Transitions

This paper investigates the construction of robust positively invariant (RPI) sets for a class of discrete-time switched systems subject to smooth control and bounded exogenous disturbances. A tube-based framework is developed to characterize the evolution of system trajectories during controller switching, in which smooth control laws are incorporated to regulate transient behaviors. Within this framework, RPI sets for both system states and control inputs are computed in advance, guaranteeing boundedness under disturbances. By invariant set analysis, the proposed formulation provides a systematic tool to evaluate and constrain the variations of control inputs induced by smooth control transitions. The resulting RPI sets can be directly employed in controller design to ensure admissible switching behaviors and constraint satisfaction. In contrast to conventional tube-based designs with fixed controllers, the proposed method explicitly incorporates smooth controller transitions into invariant set analysis. Simulation studies are presented to demonstrate the effectiveness of the proposed method.