Page 370 - 8th European Congress of Mathematics ∙ 20-26 June 2021 ∙ Portorož, Slovenia ∙ Book of Abstracts
P. 370
TIONAL APPROXIMATION FOR DATA-DRIVEN MODELING AND COMPLEXITY
REDUCTION OF LINEAR AND NONLINEAR DYNAMICAL SYSTEMS (MS-69)
complexity and the accuracy of the reduced model. These two approaches are compared first
for the case of fitting linear systems and then, for fitting systems with nonlinear dynamics, such
as bilinear systems. We discuss recent extensions for the latter case.
Rational interpolation and model order reduction for data-driven
controller design
Pauline Kergus, pauline.kergus@control.lth.se
Lund University, Sweden
In many control applications, a mathematical description of the system, derived from physical
laws, is not available. In this case, the controller has to be designed on the basis of experimental
measurements. This work presents a data-driven control strategy based on the Loewner frame-
work, where a reduced-order controller is directly obtained from the available experimental
data. Rational interpolation is also used to build achievable specifications and to ensure closed-
loop stability for the controlled system. No parametric model of the system is used, allowing
to handle applications in which the model of the system might be too complicated or too dif-
ficult to obtain for traditional model-based strategies. This technique is particularly appealing
to control infinite dimensional systems, such as the ones described by linear partial differential
equations. Such an example, a crystallizer (common in the chemical industry), is tackled in this
work.
Comparison of greedy-type approaches involving the Loewner matrix for
rational modeling
Sanda Lefteriu, sanda.lefteriu@imt-lille-douai.fr
IMT Lille Douai, France
The Loewner framework has established itself as a popular choice for building rational approx-
imations in barycentric form. The Loewner together with the shifted Loewner matrices are built
from measurements and, together with the data matrices, yield a high-order rational model for
a potentially non-rational function. To eliminate the redundancy in the data, an SVD step of the
Loewner matrix pencil is involved. However, as the size of these matrices is equal to half of the
number of measurements, this step is rather costly for large data sets when the traditional SVD
is employed.
This talk compares several greedy-type approaches which use the same principle: starting
from an order 1 approximant, points from the available data set are added in a greedy fashion by
minimizing the error measure of choice. These approaches considered in the comparison are:
AAA [1], the CUR decomposition [2], DEIM-CUR decomposition [3], as well as the adaptive
and recursive approaches [4].
References
[1] Y. Nakatsukasa, O. Sète, L. N. Trefethen. The AAA algorithm for rational approximation.
SIAM Journal on Scientific Computing, 40(3):A1494–A1522, Jan. 2018.
[2] D. S. Karachalios, I. V. Gosea, A. C. Antoulas. Data-driven approximation methods applied
to non-rational functions. Proc. Appl. Math. Mech., 18(1), 2018.
368
REDUCTION OF LINEAR AND NONLINEAR DYNAMICAL SYSTEMS (MS-69)
complexity and the accuracy of the reduced model. These two approaches are compared first
for the case of fitting linear systems and then, for fitting systems with nonlinear dynamics, such
as bilinear systems. We discuss recent extensions for the latter case.
Rational interpolation and model order reduction for data-driven
controller design
Pauline Kergus, pauline.kergus@control.lth.se
Lund University, Sweden
In many control applications, a mathematical description of the system, derived from physical
laws, is not available. In this case, the controller has to be designed on the basis of experimental
measurements. This work presents a data-driven control strategy based on the Loewner frame-
work, where a reduced-order controller is directly obtained from the available experimental
data. Rational interpolation is also used to build achievable specifications and to ensure closed-
loop stability for the controlled system. No parametric model of the system is used, allowing
to handle applications in which the model of the system might be too complicated or too dif-
ficult to obtain for traditional model-based strategies. This technique is particularly appealing
to control infinite dimensional systems, such as the ones described by linear partial differential
equations. Such an example, a crystallizer (common in the chemical industry), is tackled in this
work.
Comparison of greedy-type approaches involving the Loewner matrix for
rational modeling
Sanda Lefteriu, sanda.lefteriu@imt-lille-douai.fr
IMT Lille Douai, France
The Loewner framework has established itself as a popular choice for building rational approx-
imations in barycentric form. The Loewner together with the shifted Loewner matrices are built
from measurements and, together with the data matrices, yield a high-order rational model for
a potentially non-rational function. To eliminate the redundancy in the data, an SVD step of the
Loewner matrix pencil is involved. However, as the size of these matrices is equal to half of the
number of measurements, this step is rather costly for large data sets when the traditional SVD
is employed.
This talk compares several greedy-type approaches which use the same principle: starting
from an order 1 approximant, points from the available data set are added in a greedy fashion by
minimizing the error measure of choice. These approaches considered in the comparison are:
AAA [1], the CUR decomposition [2], DEIM-CUR decomposition [3], as well as the adaptive
and recursive approaches [4].
References
[1] Y. Nakatsukasa, O. Sète, L. N. Trefethen. The AAA algorithm for rational approximation.
SIAM Journal on Scientific Computing, 40(3):A1494–A1522, Jan. 2018.
[2] D. S. Karachalios, I. V. Gosea, A. C. Antoulas. Data-driven approximation methods applied
to non-rational functions. Proc. Appl. Math. Mech., 18(1), 2018.
368
