# HOSVD-based canonical form of TP functions and qLPV models

Based on the key idea of higher-order singular value decomposition [1] (HOSVD) in tensor algebra Baranyi and Yam proposed the concept of HOSVD-based canonical form of TP functions and quasi-LPV system models.[2] [3] Szeidl et al.[4] proved that the TP model transformation [5] [6] 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 [1] or at MATLAB Central [2].

## Existence of the HOSVD based canonical form

Assume a given finite element TP function:

${\displaystyle f({\mathbf {x} })={\mathcal {S}}\boxtimes _{n=1}^{N}{\mathbf {w} }_{n}(x_{n}),}$

where ${\displaystyle {\mathbf {x} }\in \Omega \subset R^{N}}$. Assume that, the weighting functions in ${\displaystyle \mathbf {w} _{n}(x_{n})}$ are othonormal (or we transform to) for all ${\displaystyle n=1\ldots N}$. Then, the execution of the HOSVD on the core tensor ${\displaystyle {\mathcal {S}}}$ leads to:

${\displaystyle {\mathcal {S}}={\mathcal {A}}\boxtimes _{n=1}^{N}\mathbf {U} _{n}}$.

Then,

${\displaystyle f({\mathbf {x} })={\mathcal {S}}\boxtimes _{n=1}^{N}{\mathbf {w} }_{n}(x_{n})=\left({\mathcal {A}}\boxtimes _{n=1}^{N}{\mathbf {U} }_{n}\right)\boxtimes _{n=1}^{N}{\mathbf {w} }_{n}(x_{n}),}$

that is:

${\displaystyle f({\mathbf {x} })={\mathcal {A}}\boxtimes _{n=1}^{N}\left({\mathbf {w} }_{n}(x_{n}){\mathbf {U} }_{n}\right)={\mathcal {A}}\boxtimes _{n=1}^{N}{\mathbf {w'} }_{n}(x_{n}),}$

where weighting functions of ${\displaystyle \mathbf {w'} _{n}(x_{n}),}$ are orthonormed (as both the ${\displaystyle \mathbf {w} _{n}(x_{n})}$ and ${\displaystyle {\mathbf {U} }_{n}}$ where orthonormed) and core tensor ${\displaystyle {\mathcal {A}}}$ contains the higher order singular values.

## Definition

HOSVD based canonical from of TP function
${\displaystyle f({\mathbf {x} })={\mathcal {A}}\boxtimes _{n=1}^{N}{\mathbf {w} }_{n}(x_{n}),}$
${\displaystyle \forall n:\int _{a_{n}}^{b_{n}}{\tilde {w}}_{n,i}(p_{n}){\tilde {w}}_{n,j}(p_{n})\,dp_{n}=\delta _{i,j},\quad 1\leq i,j\leq I_{n},}$

## References

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