On finite-horizon ℓ₂-induced norms of discrete-time switched linear systems

The finite-horizon ℓ₂-induced norm evaluation problem of discrete-time switched linear systems is considered in this paper. The set of worst-case switching laws is characterized, which is a subset of the whole set of switching laws under consideration. Based on the properties of the worst-case switching law set, a low-complexity complete solution to the ℓ₂-induced norm evaluation problem is provided for general switched linear systems. It is shown that under the assumption of exponential stability, the finite-horizon ℓ₂-induced norm monotonically converges to the infinite-horizon norm, and the volume of the worst-case switching law set gradually becomes constant as the time horizon approaches infinity. Some numerical properties of the worst-case switching law set are also exploited. A numerical example is presented to illustrate the proposed results.