Stabilization and regulation of class of non-linear singularly perturbed discrete-time systems (Q5942207)

A class of discrete-time nonlinear systems that are two-time-scale of the form NEWLINE\[NEWLINEh(k, x(k),\mu) x(k+1)- f(k,x(k), \mu)x(k)+\mu x(k- 1)= 0,\quad x(0)= x_0,NEWLINE\]NEWLINE where \(x\in \mathbb{R}^p\) and \(\mu\) is a small parameter, is treated.NEWLINENEWLINENEWLINEUsing the singular perturbation theory in a systematic way, a mode-decoupling approach that yields two separate subsystems containing the slow and fast parts is presented.NEWLINENEWLINENEWLINEFurthermore, a two-time-scale analysis and design procedure for stabilization and regulation is provided. The controllability and stabilizability invariance of the fast subsystem is shown, and an asymptotic stabilizing composite feedback control is proposed. Finally, it is shown that the composite control produces a finite cost that tends to the optimal cost of a slow problem as the singular perturbation parameter \(\mu\) tends to zero.