Norm vanishment and its applications in constrained control - Part II: the ℒ2 case

In Part I of this paper, we introduced the notion of L_¿-vanishment and established several kinds of characterizations of this notion. Based on this notion, L_¿ low gain feedback was reconsidered and a systematic approach to L_¿ low gain feedback design was proposed. In this second part of the paper, we consider the parallel notion of L₂-vanishment. We establish several characterizations of the L₂-vanishment property, based on which a new design approach referred to as the L₂ low gain feedback approach is developed. Just as the L_¿ low gain design is useful in the control of linear systems subject to actuator magnitude saturation, the L₂ low gain feedback is instrumental in the control of systems with control energy constraints. As applications of L₂ low gain feedback, the problem of semi-global stabilization of linear systems with energy constraints and the problem of linear time-delay system with energy constraints are solved in this paper. The notion of L₂-vanishment and the resulting L₂ low gain feedback are also extended to nonlinear systems. As in the L_¿ low gain feedback considered in Part I, a systematic approach is also developed for L₂ low gain feedback design in this second part of the paper. Finally, an example involving a linearized model of the relative motion with respect to another in a circular orbit around the Earth is used to illustrate the effectiveness of the results developed in this paper.