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Decoupling multivariate polynomials
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MATLAB package for approximate polynomial decoupling
The starting point for this package are two coupled multivariate polynomials with real coefficients. The MATLAB function DecouplePolynomial.m (click here to download the source code) can be used to find an approximate decoupled representation of this function, using different kinds of covariance matrices as weighting matrices, as described in this article. It attempts to find the best possible decoupled representation, according to a cost function defined on the output of the system. In this decoupled representation, the output is written as a linear combination of parallel univariate polynomials of linear forms of the input (see Figure):
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In this figure, f represents the coupled multivariate polynomial before the decoupling step. After the decoupling, V and W represent two transformation matrices, and g represent r different univariate polynomials.
Usage
In this section, the usage of the MATLAB function DecouplePolynomial.m is discussed in detail.
Input structure
The function takes the MATLAB structure coupledPolynomial as argument and outputs the structure decoupledPolynomial. The argument structure should contain the following fields
coupledCoeffs, the coefficients of the coupled multivariate polynomial,
covarianceMatrix, the covariance matrix of coupledCoeffs, used in the decoupling method,
CPDtype ('no’, 'diag’, 'blockdiag’ or 'full’), type of weighted decoupling,
r, the number of branches of the decoupling.
lambda, a metaparameter to be used in the case the covarance matrix has low rank. If not given, the default value is 1.
In the case no covariance matrix or an empty covariance matrix is given, then the implementation performs an unweighted decoupling. This implementation performs similarly to other implementations like Tensorlab or the Nway toolbox.
Output structure
The output of the function is the MATLAB structure decoupledPolynomial containing all the information about the decoupled polynomial. It contains the following fields:
type, the type of the decoupling,
We, the transformation matrix W ('e’ stands for 'estimated’),
Ve, the transformation matrix V ('e’ stands for 'estimated’),
Ge, the coefficients of the decoupled polynomials,
iteration_count, the number of iterations before the end of the algorithm,
relerr, a measure for the relative error of the decoupling.
Example code and output
This part shows a small example using this implementation. We define the coefficients of the coupled polynomial function with the statement
coupledPolynomial.approximatedCoeffs = …
These coefficients are organized in columns, one for every output of the function, and are ordered using the list of monomials (for 2 inputs and degree 3)
1, u1, u2, u1^2, u1u2, u2^2, u1^3, u1^2u2, u1*u2^2, u2^3.
Next, the covariance matrix is defined by the statement
coupledPolynomial.covarianceMatrix = …
Also the number of branches of the decoupled representation is defined by
coupledPolynomial.r = …
and the type of weighted decoupled is given by coupledPolynomial.CPDtype = …;
In the case the covariance matrix is rank-deficient, the hyperparameter lambda can be defined (it is set to 1 as a default value) by
coupledPolynomial.lambda = …;
Finally, the decoupling process is performed with the command
decoupledPolynomial = DecouplePolynomial(coupledPolynomial);
The output of this function is the structure decoupledPolynomial with the following fields:
type: decoupling with full weight
We: 2x2 double
Ve: 2x2 double
Ge: 2x4 double
iteration_count: 20
relerr: 0.0096
The matrix Ge contains the coefficients of the decoupled polynomials, ordered from high to low degree, one row per polynomial.
Click here to download the source code with the minimal example.
Authors
This work has been developed by Gabriel Hollander under the supervision of the following people: