A Direct-Construction Approach to Multidimensional Realization and LFR Uncertainty Modeling

This paper proposes a direct-construction realization procedure which simultaneously treats all the involved variables and/or uncertain parameters and directly generates an overall Roesser model realization or LFR model for a given nD causal transfer matrix with rational function or polynomial entries. It is shown that the nD realization problem for an nD transfer matrixG(z1, . . . , zn), which is assumed without loss of generality to be strictly causal and given in the form of G(z1, . . . , zn) = N(z1, . . . , zn)D(z1, . . . , zn) −1 withD(0, ..., 0) = I and N(0, ..., 0) = 0, can be essentially reduced to the construction of an admissible nD polynomial matrix Ψ for which there exist matrices A,B,C such that N(z1, . . . , zn) = CZΨ and ΨD(z1, . . . , zn) −1 = (I−AZ)−1B with Z being the corresponding variable and/or uncertainty block structure, i.e., Z = diag{z1Ir1 , . . . , znIrn}. This important fact reveals a substantial difference between the 1D and nD (n ≥ 2) realization problems as in the 1D case Ψ can only be a monomial matrix and never a polynomial one, and implies the possibility to achieve a realization with further lower order than the constructive realization method given recently by the authors. Necessary and sufficient conditions that ensure Ψ to be admissible are given and, based on these conditions, algorithms are proposed for construction of an admissible Ψ with lower order and the corresponding realization. Illustrative examples are presented to illustrate the basic ideas and the effectiveness of the proposed method.