Operator Realization Theory

The realization problem for time-varying systems is to find a (minimal) state space description for the input-output operator of a time-varying system, solely based on the collection of time-varying impulse responses. An important role in its solution is played by the Hankel operator, which is a restriction or suboperator of the input-output operator. It maps input signals with support in the “past” to output signals restricted to the future. Its relevance to the realization theory of time-invariant systems has been known since the early 1960s and resulted in Ho and Kalman’s canonical realization algorithm in 1966 [HK66]. The fundamental properties that enable one to derive a realization are not the linearity or time invariance of the system (although these properties greatly simplify the problem), but rather its causality and the existence of a factorization of the Hankel operator into a surjective and an injective part [KFA70]. Thus, the problem of realization was brought into the algebraic context of the characterization of the Hankel operator. The algorithm derived by Ho and Kalman does not require knowledge of these invariant factors but uses the underlying structure to find the state representation of the system.