A local I/O structure theory for multivariable systems and its application to minimal cascade realization

Abstract

In this paper a systematic study of the local behavior of a multivarlable transfer function <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">T_{(p))</tex> is undertaken. Starting from a Laurent expansion of the transfer function in a pole or zero, spaces generated by block-Toeplitz matrices are defined and a systematic calculus for these spaces is developed. The relationship between these objects, classical Smith-McMillan theory and coprime factorization techniques is discussed and a number of Interesting results are deduced, e.g., an algorithm to determine characteristics of the inverse system <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">T^{-1}(p)</tex> without actually computing the inverse. Finally, the main result Is deduced: necessary and sufficient conditions for a given <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">T_{1}(p)</tex> to be a minimal factor of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">T(p)</tex> . The theorem provides the mathematical conditions needed for cascade synthesis of a multivarlable system. This result shows how classical Smith-McMillan theory or coprime factorization techniques do not provide enough information on T(p) to allow a cascade synthesis. The Toeplitz calculus developed in the paper does provide the correct information needed, and appears to be the natural vehicle for multivariable cascade synthesis.