A direct-construction approach to multidimensional realization and LFR uncertainty modeling

Abstract

This article proposes a direct-construction realization procedure that simultaneously treats all the involved variables and/or uncertain parameters and directly generates an overall multidimensional (n-D) Roesser model realization or linear fractional representation (LFR) model for a given n-D polynomial or causal rational transfer matrix. It is shown for the first time that the realization problem for an n-D transfer matrix G(z ₁, . . . , z _n ), which is assumed without loss of generality to be strictly causal and given in the form of G(z ₁, . . . , z _n )=N _r (z ₁, . . . , z _n )D ⁻¹ _r (z ₁,..., z _n ) with D _r (0, . . . , 0)=I and N _r (0, . . . , 0)  = 0, can be essentially reduced to the construction of an admissible n-D polynomial matrix Ψ for which there exist real matrices A, B, C such that N _r (z ₁, . . . , z _n ) = CZΨ and Ψ D ⁻¹ _r (z ₁, . . . , z _n ) = (I − AZ)⁻¹ B with Z being the corresponding variable and/or uncertainty block structure, i.e., ${Z={\rm diag} \{z_1I_{r_1},\ldots,z_nI_{r_n} \}}$ . This important fact reveals a substantial difference between the 1-D and n-D (n ≥  2) realization problems as in the 1-D case Ψ can only be a monomial matrix and never a polynomial one. Necessary and sufficient conditions for Ψ to satisfy the above restrictions are given and algorithms are proposed for the construction of such an admissible n-D polynomial matrix Ψ with low order (for an arbitrary but fixed field of coefficients) and the corresponding realization. Symbolic and numerical examples are presented to illustrate the basic ideas as well as the effectiveness of the proposed procedure.