Discrete-time l∞l∞ and l2l2 norm vanishment and low gain feedback with their applications in constrained control

The existing low gain feedback, which is a parameterized family of stabilizing state feedback gains whose magnitudes approach zero as the parameter decreases to zero, has been designed in very specific ways. In this paper, by recognizing the l∞l∞ and l2l2 slow peaking phenomenon that exists in discrete-time systems under low gain feedback, more general notions of l∞l∞ and l2l2 norm vanishment are considered so as to provide a full characterization of the nonexistence of slow peaking phenomenon in some measured signals. Low gain feedback that does not lead to l∞l∞ and l2l2 slow peaking in the control input are respectively referred to as l∞l∞ and l2l2 low gain feedback. Based on the notions of l∞l∞ and l2l2 vanishment, not only can the existing low gain feedback be recognized as an l∞l∞ low gain feedback, but also a new design approach referred to as the l2l2 low gain feedback approach is developed for discrete-time linear systems. Parallel to the effectiveness of l∞l∞ low gain feedback in magnitude constrained control, the l2l2 low gain feedback is instrumental in the control of discrete-time systems with control energy constraints. The notions of l∞l∞ and l2l2-vanishment also result in a systematic approach to the design of l∞l∞ and l2l2 low gain feedback by providing a family of solutions including those resulting from the existing design methods.