The l∞ l_\infty -induced norm of multivariable discrete-time linear systems: Upper and lower bounds with convergence rate analysis

This paper develops a method for computing the $l_{\infty}$-induced norm of a multivariable discrete-time linear system, for which an infinite-dimensional matrix should be intrinsically concerned with. To make such a computation feasible, we treat the infinite-dimensional matrix in a truncated fashion, and an upper bound and a lower bound on the $l_\infty$-induced norm of the original multivariable discrete-time linear system are derived. More precisely, the matrix $\infty$-norm of the (infinite-dimensional) tail part can be approximately computed by deriving its upper and lower bounds, while that of the (finite-dimensional) truncated part can be exactly obtained. With these values, an upper bound and a lower bound on the original $l_\infty$-induced norm can be computed. Furthermore, these bounds are shown to converge to each other within an exponential order of $N$, where $N$ is the corresponding truncation parameter. Finally, some numerical examples are provided to demonstrate the theoretical validity and practical effectiveness of the developed computation method