# HOSVD based canonical form of TP functions and qLPV models

Updated on
Covid-19

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.

## Contents

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:

where x Ω R N . Assume that, the weighting functions in w n ( x n ) are othonormal (or we transform to) for n = 1 , , N . Then, the execution of the HOSVD on the core tensor S leads to:

Then,

that is:

where weighting functions of w n ( x n ) , are orthonormed (as both the w n ( x n ) and U n where orthonormed) and core tensor A contains the higher-order singular values.

## Definition

HOSVD-based canonical form of TP function
• Singular functions of f ( x ) : The weighting functions
• w n , i n ( x n ) , i n = 1 , , r n (termed as the i n -th singular function on the n -th dimension, n = 1 , , N ) in vector w n ( x n ) form an orthonormal set:

where δ i , j is the Kronecker delta function ( δ i j = 1 , if i = j and δ i j = 0 , if i j ).

• The subtensors A i n = i have the properties of
• all-orthogonality: two sub tensors A i n = i and A i n = j are orthogonal for all possible values of n , i and j : A i n = i , A i n = j = 0 when i j ,
• ordering: A i n = 1 A i n = 2 A i n = r n > 0 for all possible values of n = 1 , , N + 2 .
• n -mode singular values of f ( x ) : The Frobenius-norm A i n = i , symbolized by σ i ( n ) , are n -mode singular values of A and, hence, the given TP function.
• A is termed core tensor.
• The n -mode rank of f ( x ) : The rank in dimension n denoted by r a n k n ( f ( x ) ) is equals the number of non-zero singular values in dimension n .
• ## References

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

Similar Topics
Giorgi Seturidze
Kathy Tomlinson
David Wilder (baseball)
Topics