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

Abstract: 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 and slow peaking phenomenon that exists in discrete-time systems under low gain feedback, more general notions of and 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 and slow peaking in the control input are respectively referred to as and low gain feedback. Based on the notions of and vanishment, not only can the existing low gain feedback be recognized as an low gain feedback, but also a new design approach referred to as the low gain feedback approach is developed for discrete-time linear systems. Parallel to the effectiveness of low gain feedback in magnitude constrained control, the low gain feedback is instrumental in the control of discrete-time systems with control energy constraints. The notions of and -vanishment also result in a systematic approach to the design of and low gain feedback by providing a family of solutions including those resulting from the existing design methods. [Copyright &y& Elsevier]