cpn2LinSparse

Computes the Jacobian of a sparse CPN1 MTI object around an operating point

Contents

Syntax

  cpn2LinSparse(obj,x)

Description

cpn2LinSparse(obj,x) Computes the Jacobian of an explicit multilinear model stored in the CP-decomposed norm-1 format obj around an operating point x. This implementation makes use of the sparsity of the model for an efficient computation of the Jacobian. For further information on the implementation of the algorithm see [1].

Input Arguments

obj: CPN1 object

x : operating point, scalar or vector

Output Arguments

Jsparse: sparse Jacobian matrix

Example:

Assume we have the following thrird-order explicit MTI model $\dot{\mathbf{x}}=\left(\begin{array}{cc} x_2 + x_1x_2\\ 2x_2 + 3x_3 + 2x_1x_2 - 3x_1x_3 \\ 4x_3 - 4x_1x_3 \end{array}\right)$. We write the explicit MTI model as a CPN1 object by defining the structural matrix S to form the monomial

    S=[0.5 -0.5; 1 0; 0 1];

and the parameter matrix with corresponding coefficients of the summands:

    phi=[2 0; 4 6; 0 8];

Then we can create explicit MTI model as a CPN1 object

    obj=CPN1(S,phi);

and assume the operating point

    x_op=[-1;1;3];

Then we can calcute the Jacobian

    Jsparse = cpn2LinSparse(obj,x_op)
Jsparse =

     1     0     0
    -7     0     6
   -12     0     8

References

[1] C. Kaufmann, D. Crespí, G. Lichtenberg, G. Pangalos, and C. Cateriano Yáñez, "Efficient Linearization of Explicit Multilinear Systems using Normalized Decomposed Tensors," IFAC-PapersOnLine, vol. 56, no. 2, pp. 7312–7317, Jan. 2023, doi: https://doi.org/10.1016/j.ifacol.2023.10.344.

See also

jacobian, cpn2Lin, linearize