Simultaneous global external and internal stabilization of linear time-invariant discrete-time systems subject to actuator saturation

Simultaneous external and internal stabilization in a global framework for linear time-invariant discrete-time systems subject to actuator saturation is considered. Internal stabilization is in the sense of Lyapunov while external stabilization is in the sense of $\ell_p$ stability with different variations, e.g. with or without finite gain, with fixed or arbitrary initial conditions, with or without bias. Several simultaneous external and internal stabilization problems, all in the global framework, are studied in depth. Moreover, we present a design for appropriate adaptive low-and-high gain feedback controllers that achieve the intended simultaneous external and internal stabilization whenever such problems are solvable.