From an analog system K, digital systems K-d and K-bt are often obtained via the step-invariant and bilinear transformations, respectively. For the case when K is stable, it is shown that under certain conditions, K-bt converges to K-d in l(p)-induced norms as the sampling period tends to zero for l less than or equal to p less than or equal to infinity. Furthermore, K-bt converges to K-d in the graph metric as the sampling period tends to zero no matter whether K is stable or not. (C) 2003 Elsevier Ltd. All rights reserved.