CPN1
Canonical polyadic norm-1 form
Contents
Description
Use CPN1 to create a canonical polyadic (CP) norm-1 tensor, i.e., a CPN1 tensor.
A tensor can be decomposed into a CP tensor by the sum of
outer products, where
is the so called tensor rank. All factors are represented in the so called factor matrices
for each dimension
of size
of the original tensor
up to
, which can be abreviated as
An element of is then given by
A CPN1 tensor is a CP tensor where
for all
from
to
and for each dimension
.
Particularly for mti systems, the parameter tensor factor matrices have dimensions for the first
dimensions corresponding to the monomial dimensions and the last
dimension called
is saved for parameterization, i.e.,
, such that
For further reference see [1].
Creation
Syntax
CPN1tens = CPN1(Umat,Phimat)
CPN1tens = CPN1(tens)
Description
CPN1tens = CPN1(Umat,Phimat) creates a CP norm-1 tensor Ttens of the form
CPN1tens = CPN1(tens) takes a CP decomposed ktensor of the form from the Tensor Toolbox and creates a CPN1 tensor object
Input Arguments
Umat - Structure matrix
Structure matrix, which in the context of the MTI, vertically stacks the second row of each CPN1 factor matrix corresponding to each dimension of the monomial. For example for explicit MTI systems:
leading to an by
matrix. Note that due to the norm-1, specifying both columns of the monomial factor matrices is redundant.
Phimat - Parameter matrix
Parameter matrix, which in the context of the MTI, contains the parameters for each rank element of the tensor. For example for the state transition tensor parameter for explicit MTI systems, it would lead to a
by
factor matrix
.
Output Arguments
CPN1tens - Output tensor
Output tensor returned as:
- A CP norm-1 (CPN1) tensor object.
Properties
U - Structure matrix
Structure matrix, which for MTI parameter tensors correspond to the monomial dimension by the rank .
phi - Parameter matrix
Parameter matrix, which for MTI parameter tensors correspond to the parameters of each equation by the rank .
Examples
SISO Multilinear Time-Invariant State-Space Model
Create the state-transition tensor of a SISO MTI state-space model defined by the following system equations:
The model has the following parameter matrix
and monomial vector
The same parameter matrix can be presented in CPN1 format as:
Initializing the tensor in CPN1 format
F_U = [1 0 1 0; 1 0 0 0; 0 1 1 0]; F_phi = [1 0.5 0 0; 0 0 2 7]; F=CPN1(F_U,F_phi);
Normalizing CP tensor to CPN1 tensor object
Create a ktensor for the mti system above with factor matrices
Fx1=[0 1 0 1;1 0 1 0]; Fx2=[0 1 1 1;1 0 0 0]; Fu=[1 0 0 1;0 1 1 0]; FPhi=[1 0.5 0 0;0 0 2 7]; lambda=[1;1;1;1]; CPtens=ktensor(lambda,Fx1,Fx2,Fu,FPhi);
Create a CPN1 tensor object by normalization
CPN1tens=CPN1(CPtens);
References
[1] Lichtenberg, Gerwald; Pangalos, Georg; Cateriano Yáñez, Carlos; Luxa, Aline; Jöres, Niklas; Schnelle, Leona; Kaufmann, Christoph (2022): Implicit multilinear modeling. In at - Automatisierungstechnik 70 (1), pp. 13–30. DOI: 10.1515/auto-2021-0133.