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

This paper investigates the maximum amplitude (i.e., the L_∞ norm) of the output for the worst input with a unit energy (i.e., a unit L₂ 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 L₂ to L_∞ coincides with the H₂ 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 H₂ 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 L₂ to L_∞ in SISO LTI sampled-data systems. We further exploit the associated arguments to compare this induced norm with two existing definitions of the H₂ 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.