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

作者：Xu Li^*; Fan Huijin; Lin Zhiping; Bose N K

来源：Multidimensional Systems and Signal Processing, 2008, 19(3-4): 323-359.

DOI：10.1007/s11045-008-0057-0

• multidimensional systems
• realization
• LFR
• LFT
• uncertainty modeling
• state-space realization
• minimal realization
• Roesser model

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 transfermatrix. It is shown for the first time that the realization problem for an n-D transfer matrix G(z(1),..., zn), which is assumed without loss of generality to be strictly causal and given in the form of G( z(1),...,zn) = N(r)(z(1),..., z(n)) D(r)(-1)(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 Psi for which there exist real matrices A, B, C such that N(r)( z(1),...,z(n)) = CZ Psi and Psi D(r)(-1) r(z(1),...,z(n)) = (I - AZ)(-1) B with Z being the corresponding variable and/ or uncertainty block structure, i. e., Z = diag{z(1) I(r1),..., z(n) I(rn)}. This important fact reveals a substantial difference between the 1-D and n-D (n >= 2) realization problems as in the 1-D case Psi can only be a monomial matrix and never a polynomial one. Necessary and sufficient conditions for Psi to satisfy the above restrictions are given and algorithms are proposed for the construction of such an admissible n-D polynomial matrix Psi 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.

• 出版日期2008-12
• 单位华中科技大学; 南阳理工学院