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