Norm vanishment and its applications in constrained control - Part I: the L∞ case

Low gain feedback has found several applications in constrained control, robust control and nonlinear control. Low gain feedback refers to a family of stabilizing state feedback gains that are parameterized in a scalar and go to zero as the scalar decreases to zero. Slow peaking, the peak value of the state of the closed-loop system increasing towards infinity as the time goes to infinity and the value of the low gain parameter decreases to zero, is a prominent phenomenon resulting from low gain feedback. The success of low gain feedback design lies in the fact that such slow peaking gets canceled when the state is multiplied by the feedback gain to obtain the control signal. In this paper, we generalize the existing low gain design methods by introducing the notion of L_∞-vanishment and providing a full characterization of low gain feedback that achieves such an L_∞-vanishment property. A time domain signal parameterized by a scalar ε possesses the L_∞-vanishment property if its L_∞ norm approaches zero as ε does. This notion of L_∞-vanishment not only facilitates the description of low gain feedback and the resulting slow peaking phenomenon, but also allows us to establish a systematic approach to the design of low gain feedback. A salient feature of this new design approach is that it results in all possible low gain feedback laws, including those resulting from the existing design methods.