On semi-global stabilization of linear periodic systems with control magnitude and energy saturations

This paper establishes a systematic approach to solve the L_∞(l_∞) and L₂(l₂) semi-global stabilization problem of linear periodic systems with controls having bounded magnitude and energy, respectively. The developed approach will be referred to as L_∞(l_∞) and L₂(l₂) low gain feedback. Definitions, properties, and characterizations of this new concept are also provided, and particularly, the characterizations are based upon differential (difference) Lyapunov inequalities. Design of L_∞(l_∞) and L₂(l₂) low gain feedback by solving differential (difference) Riccati equations is proposed. Both continuous-time and discrete-time linear periodic systems are studied and both state feedback and observer based output feedback are considered. In the discrete-time setting, a linear matrix inequalities (LMIs) based solution to the l_∞ and l₂ semi-global stabilization problem is also established and the LMIs conditions are shown to be always solvable. Applications of the proposed approach to the elliptical spacecraft rendezvous system show the effectiveness of the established theory.