HOSVD-based canonical form of TP functions and qLPV models

Based on the key idea of higher-order singular value decomposition (HOSVD) in tensor algebra, Baranyi and Yam proposed the concept of HOSVD-based canonical form of TP functions and quasi-LPV system models. Szeidl et al. proved that the TP model transformation is capable of numerically reconstructing this canonical form.
Related definitions (on TP functions, finite element TP functions, and TP models) can be found here. Details on the control theoretical background (i.e., the TP type polytopic Linear Parameter-Varying state-space model) can be found here.
A free MATLAB implementation of the TP model transformation can be downloaded at or at MATLAB Central .
Existence of the HOSVD-based canonical form.
Assume a given finite element TP function:
where formula_2. Assume that, the weighting functions in formula_3 are othonormal (or we transform to) for formula_4. Then, the execution of the HOSVD on the core tensor formula_5 leads to:
Then,
that is:
where weighting functions of formula_9 are orthonormed (as both the formula_3 and formula_11 where orthonormed) and core tensor formula_12 contains the higher-order singular values.
Definition.
formula_15, formula_16 (termed as the formula_17-th singular function on the formula_18-th dimension, formula_19) in vector
formula_3 form an orthonormal set:
where formula_22 is the Kronecker delta function
(formula_23, if formula_24 and formula_25, if formula_26).