$L_{\infty}$ and $L_{2}$ Low-Gain Feedback: Their Properties, Characterizations and Applications in Constrained Control

Low-gain feedback has found several applications in constrained control, robust control, and nonlinear control. In this paper, we first generalize the existing low-gain design methods by introducing the notion of
L
_∞
-vanishment and by providing a full characterization of feedback gains that achieve such a property. We observe that
L
_∞
low-gain feedback can lead to energy peaking, namely, the control energy required by
L
_∞
low-gain feedback increases toward infinity as the low-gain parameter decreases to zero. Motivated by this observation, we consider the notion of
L
₂
-vanishment and establish several of its characterizations, based on which a new design approach referred to as the
L
₂
low-gain feedback approach for linear systems is developed. Different from the
L
_∞
low-gain feedback, the
L
₂
low-gain feedback is instrumental in the control of systems with control energy constraints. As an application of
L
₂
low-gain feedback, the problem of semiglobal stabilization of linear systems with control energy constraints is solved in this paper. The notions of
L
_∞
and
L
₂
-vanishment also allow us to establish a systematic approach to the design of
L
_∞
and
L
₂
low-gain feedback. The advantage of this new design approach is that it results in a family of control laws, including those resulting from the existing design methods.