Hankel matrices for system identification

The coefficients of a linear system, even if it is a part of a block-oriented nonlinear system, normally satisfy some linear algebraic equations via Hankel matrices composed of impulse responses or correlation functions. In order to determine or to estimate the coefficients of a linear system it is important to require the associated Hankel matrix be of row-full-rank. The paper first discusses the equivalent conditions for identifiability of the system. Then, it is shown that the row-full-rank of the Hankel matrix composed of impulse responses is equivalent to identifiability of the system. Finally, for the row-full-rank of the Hankel matrix composed of correlation functions, the necessary and sufficient conditions are presented, which appear slightly stronger than the identifiability condition. In comparison with existing results, here the minimum phase condition is no longer required for the case where the dimension of the system input and output is the same, though the paper does not make such a dimensional restriction.