Comments on the H2 Norm of Discrete-Time Stochastic Jump Parameter Linear Systems

In this note, we consider a general class of stochastic, jump parameter systems and show that the so-called H-2 norm is the Frobenius norm of the impulse response matrix of the system. We present a necessary and sufficient condition for the H-2 norm to be the induced norm from the input to the output for a particular type of deterministic input signal. As expected in the context of time-varying systems, we cannot extend this result to more general deterministic input signals. Moreover, finite H-2 norm does not prevent infinite norm output in response to a finite norm input, unless additional conditions are imposed on the system. Here, we give relatively mild conditions - mean square stability and jumps driven by a finite-dimensional Markov chain starting in equilibrium - and we show that finite H-2 norm implies a bounded ratio between the norms of the output and the input, for certain suitable norms.