Identification of noncontrollable systems from impulse response measurements

We analyze the problem of modeling an observed impulse response by means of a finite-dimensional, linear, time-invariant system. Our approach differs from classical realization theory in the following respects. The modeling problem is split in two steps, namely, identification for determining a model for the observations, and realization for determining parameters which describe the model. Systems are considered as sets of time series, not as input-output maps. In particular, the partitioning of variables into inputs and outputs need not be known, and it is not required that there exist a causal relationship between inputs and outputs. Further, we make no assumptions concerning initial conditions, which in particular may be nonzero. Determination of initial conditions is part of the modeling problem. A final significant distinction from classical realization theory is that the systems need not be controllable.We characterize the class of systems which can be identified from impulse response measurements. Necessary and sufficient conditions are formulated in terms of state-space realizations. It turns out that noncontrollable systems are also identifiable. For causal systems, the condition is that the state transition matrix, restricted to the noncontrollable states, has sufficiently small cyclic index. For noncausal systems, the condition is expressed in terms of the rank of the (singular) state evolution equation.