HOSVD-based canonical form of TP functions and qLPV models

Based on the key idea of higher-order singular value decomposition{{cite journal

|author = Lieven De Lathauwer and Bart De Moor and Joos Vandewalle

|title = A Multilinear Singular Value Decomposition

|journal = SIAM Journal on Matrix Analysis and Applications

|year = 2000

|volume = 21

|number = 4

|pages = 1253–1278

|doi=10.1137/s0895479896305696

|citeseerx =10.1.1.3.4043

}} (HOSVD) in tensor algebra, Baranyi and Yam proposed the concept of HOSVD-based canonical form of TP functions and quasi-LPV system models.{{cite conference

|author = P. Baranyi and L. Szeidl and P. Várlaki and Y. Yam

|title = Definition of the HOSVD-based canonical form of polytopic dynamic models

|book-title = 3rd International Conference on Mechatronics (ICM 2006)

|date = July 3–5, 2006

|pages = 660–665

|place = Budapest, Hungary

}}{{cite book

|author = P. Baranyi, Y. Yam and P. Várlaki

|title = Tensor Product model transformation in polytopic model-based control

|publisher= Taylor & Francis |location=Boca Raton FL

|year = 2013

|pages = 240

|isbn = 978-1-43-981816-9

}} Szeidl et al.{{cite journal

|author = L. Szeidl and P. Várlaki

|title = HOSVD Based Canonical Form for Polytopic Models of Dynamic Systems

|journal = Journal of Advanced Computational Intelligence and Intelligent Informatics

|year = 2009

|volume = 13

|number = 1

|pages = 52–60

|doi = 10.20965/jaciii.2009.p0052

|doi-access =free

}} proved that the TP model transformation{{cite journal

|author = P. Baranyi

|title = TP model transformation as a way to LMI based controller design

|journal = IEEE Transactions on Industrial Electronics

|date=April 2004

|volume = 51

|number = 2

|pages = 387–400

|doi=10.1109/tie.2003.822037

|s2cid = 7957799

}}{{cite journal

|author = P. Baranyi and D. Tikk and Y. Yam and R. J. Patton

|title = From Differential Equations to PDC Controller Design via Numerical Transformation

|journal = Computers in Industry

|year = 2003

|volume = 51

|issue = 3

|pages = 281–297

|doi=10.1016/s0166-3615(03)00058-7

}} 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 [https://web.archive.org/web/20120229061018/http://tptool.sztaki.hu/] or at MATLAB Central [http://www.mathworks.com/matlabcentral/fileexchange/25514-tp-tool].

Existence of the HOSVD-based canonical form

Assume a given finite element TP function:

: $f(\mathbf{x})=\mathcal{S}\boxtimes_{n=1}^N \mathbf{w}_n(x_n),$

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

: $\mathcal{S}=\mathcal{A}\boxtimes_{n=1}^N \mathbf{U}_n.$

Then,

: $f(\mathbf{x})=\mathcal{S}\boxtimes_{n=1}^N \mathbf{w}_n(x_n) = \left(\mathcal{A}\boxtimes_{n=1}^N \mathbf{U}_n\right) \boxtimes_{n=1}^N \mathbf{w}_n(x_n),$

that is:

: $f(\mathbf{x})=\mathcal{A}\boxtimes_{n=1}^N \left( \mathbf{w}_n(x_n) \mathbf{U}_n\right) = \mathcal{A}\boxtimes_{n=1}^N \mathbf{w'}_n(x_n),$

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

Definition

;HOSVD-based canonical form of TP function

:: $f(\mathbf{x})=\mathcal{A}\boxtimes_{n=1}^N \mathbf{w}_n(x_n),$

Singular functions of $f(\mathbf{x})$ : The weighting functions $w_{n,i_n}(x_n)$ , $i_n=1,\ldots,r_n$ (termed as the $i_n$ -th singular function on the $n$ -th dimension, $n=1,\ldots,N$ ) in vector $\mathbf{w}_n(x_n)$ form an orthonormal set:

:: $\forall n:\int_{a_{n}}^{b_{n}}\tilde{w}_{n,i}(p_{n})\tilde{w}_{n,j}(p_{n}) \, dp_n=\delta_{i,j},\quad1\leq i,j\leq I_n,$

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

The subtensors ${\mathcal{A}}_{i_n = i}$ have the properties of
all-orthogonality: two sub tensors ${\mathcal{A}}_{i_{n}=i}$ and ${\mathcal{A}}_{i_{n}=j}$ are orthogonal for all possible values of $n,i$ and $j:\left\langle$

{\mathcal{A}}_{i_{n}=i},{\mathcal{A}}_{i_{n}=j}\right\rangle =0 when

i\neq j

&* ordering: $\left\|
{\mathcal{A}}_{i_n=1}\right\| \geq\left\|
{\mathcal{A}}_{i_n=2}\right\| \geq\cdots\geq\left\|
{\mathcal{A}}_{i_n=r_n}\right\| >0$ for all possible values of $n=1,\ldots,N+2$ .

$n$ -mode singular values of $f(\mathbf{x})$ : The Frobenius-norm $\left\|$

{\mathcal{A}}_{i_n=i}\right\| , symbolized by

\sigma_i^{(n)}

, are

n

-mode singular values of

\mathcal{A}

and, hence, the given TP function.

${\mathcal{A}}$ is termed core tensor.
The $n$ -mode rank of $f(\mathbf{x})$ : The rank in dimension $n$ denoted by $rank_n(f(\mathbf{x}))$ equals the number of non-zero singular values in dimension $n$ .

References

Category:Multilinear algebra