Discrete-time $H_2$ and $H_\infty$ 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, H2 and Hoo algebraic Riccati equation (ARE) based methods, and parametric Lyapunov equation based method. It is shown in Wang et al. [2010b] that for continuous-time linear systems, all these methods are rooted in and can be unified under two fundamental control theories, H2 and Hoo theory. In this paper, we extend such a result to a discrete-time setting. Both the H2 and Hoo ARE based methods are generalized to consider systems where all input channels are not necessarily subject to saturation, and explicit design methods are developed.