Induced Norm from L2 to L∞ in SISO Sampled-Data Systems

This paper investigates the maximum amplitude (i. e., the L-infinity norm) of the output for the worst input with a unit energy (i. e., a unit L-2 norm) in single-input/single-output (SISO) linear time-invariant (LTI) sampled-data systems, by which we mean the generalized plant and the controller are both LTI. It is known that the induced norm from L-2 to L-infinity coincides with the H-2 norm in SISO LTI systems. To highlight the arguments tailored to (SISO) sampled-data systems in this paper, we first see how this induced norm reduces to H-2 norms in the continuous-time and discrete-time cases. Through the lifting-based arguments, we next give a closed-form representation of the induced norm from L-2 to L-infinity in SISO LTI sampled-data systems. We further exploit the associated arguments to compare this induced norm with two existing definitions of the H-2 norm for sampled-data systems, and show that the induced norm coincides with neither of them in SISO LTI sampled-data systems. We further develop a more sophisticated closed-form representation for the induced norm and give an approximate but asymptotically exact method for its computation.