Induced norm from L₂ to L∞in SISO sampled-data systems

This paper tackled the problem of characterizing the induced norm from L₂ to L∞ in single-input/single-output (SISO) LTI sampled-data systems. Behind the interest in this problem lied the two facts that (i) this induced norm coincides with the H₂ norm when we confine ourselves to SISO continuous-time LTI systems as a special class of systems under consideration, while (ii) there exist two conceptually different definitions for the H₂ norm of sampled-data systems. We first gave a closed-form expression for the induced norm and argued that it coincides with neither of the two existing definitions for the H₂ norm. In particular, we showed that it is at least as large as (a more commonly used) one of the two definitions. We then gave a more sophisticated closed-form expression, by which we established an approximate but asymptotically exact method for computing the induced norm. These results are believed to shed a new light on the consequences of the hybrid and periodically time-varying nature of sampled-data systems. Finally, we would like to remark that the induced norm studied in this paper can be regarded as a new definition of the H₂ norm of sampled-data systems, and the optimal controller synthesis problem of minimizing the induced norm may be an interesting future topic