Parametrization of Hankel-norm approximants of time-varying systems

The classical time-invariant Hankel-norm approximation problem is generalized to the time-varying context. The input-output operator of a time-varying bounded causal linear system acting in discrete time may be specified as a bounded upper-triangular operator T with block matrix entries Tij. For such an operator T, we will define the Hankel norm as a generalization of the time-invariant Hankel norm. Subsequently, we describe all operators T which are closer to T in (operator) norm than some prespecified error tolerance Γ, and whose upper triangular part admits a state realization of minimal dimensions. The upper triangular part of T can be regarded as the input-output operator of a causal time-varying system that approximates T in Hankel norm. 1