HOSVD-based canonical form of TP functions and qLPV models

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 ${\displaystyle n=1,\ldots ,N}$. 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.

HOSVD-based canonical form of TP function

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

• Singular functions of ${\displaystyle f(\mathbf {x} )}$: The weighting functions

${\displaystyle w_{n,i_{n}}(x_{n})}$, ${\displaystyle i_{n}=1,\ldots ,r_{n}}$ (termed as the ${\displaystyle i_{n}}$-th singular function on the ${\displaystyle n}$-th dimension, ${\displaystyle n=1,\ldots ,N}$) in vector ${\displaystyle \mathbf {w} _{n}(x_{n})}$ form an orthonormal set:

${\displaystyle \forall n:\int _{a_{n}}^{b_{n}}{\tilde {w}}_{n,i}(p_{n}){\tilde {w}}_{n,j}(p_{n})\,dp_{n}=\delta _{i,j},\quad 1\leq i,j\leq I_{n},}$

where ${\displaystyle \delta _{i,j}}$ is the Kronecker delta function (${\displaystyle \delta _{ij}=1}$, if ${\displaystyle i=j}$ and ${\displaystyle \delta _{ij}=0}$, if ${\displaystyle i\neq j}$).

• The subtensors ${\displaystyle {\mathcal {A}}_{i_{n}=i}}$ have the properties of

• all-orthogonality: two sub tensors ${\displaystyle {\mathcal {A}}_{i_{n}=i}}$ and ${\displaystyle {\mathcal {A}}_{i_{n}=j}}$ are orthogonal for all possible values of ${\displaystyle n,i}$ and ${\displaystyle j:\left\langle {\mathcal {A}}_{i_{n}=i},{\mathcal {A}}_{i_{n}=j}\right\rangle =0}$ when ${\displaystyle i\neq j}$,
• ordering: ${\displaystyle \left\|{\mathcal {A}}_{i_{n}=1}\right\|\geq \left\|{\mathcal {A}}_{i_{n}=2}\right\|\geq \cdots \geq \left\|{\mathcal {A}}_{i_{n}=r_{n}}\right\|>0}$ for all possible values of ${\displaystyle n=1,\ldots ,N+2}$.

• ${\displaystyle n}$-mode singular values of ${\displaystyle f(\mathbf {x} )}$: The Frobenius-norm ${\displaystyle \left\|{\mathcal {A}}_{i_{n}=i}\right\|}$, symbolized by ${\displaystyle \sigma _{i}^{(n)}}$, are ${\displaystyle n}$-mode singular values of ${\displaystyle {\mathcal {A}}}$ and, hence, the given TP function.
• ${\displaystyle {\mathcal {A}}}$ is termed core tensor.
• The ${\displaystyle n}$-mode rank of ${\displaystyle f(\mathbf {x} )}$: The rank in dimension ${\displaystyle n}$ denoted by ${\displaystyle rank_{n}(f(\mathbf {x} ))}$ is equals the number of non-zero singular values in dimension ${\displaystyle n}$.