H2 and H∞ low-gain theory

For stabilization of linear systems subject to input saturation, there exist four different approaches of low-gain design all of which are independently proposed in the literature, namely direct eigenstructure assignment, H
₂
and H
_∞
algebraic Riccati equation (ARE) based methods, and parametric Lyapunov equation based method. We show here all these methods are rooted in and can be unified under two fundamental control theories, H
₂
and H
_∞
theory. Moreover, both the H
₂
and H
_∞
ARE based methods are generalized to consider systems where all input channels are not subject to saturation, and explicit design methods are developed.