# CPN1

Canonical polyadic norm-1 form

## Contents

- Description
- Creation
- Properties
- Examples
- References

## 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.

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