TY - JOUR

T1 - On the Sample Complexity of Stabilizing Linear Dynamical Systems from Data

AU - Werner, Steffen W.R.

AU - Peherstorfer, Benjamin

N1 - Funding Information:
The authors acknowledge support from the Air Force Office of Scientific Research (AFOSR) award FA9550-21-1-0222 (Dr. Fariba Fahroo). The second author additionally acknowledges support from the National Science Foundation under Grant No. 2012250.
Publisher Copyright:
© 2023, SFoCM.

PY - 2023

Y1 - 2023

N2 - Learning controllers from data for stabilizing dynamical systems typically follows a two-step process of first identifying a model and then constructing a controller based on the identified model. However, learning models means identifying generic descriptions of the dynamics of systems, which can require large amounts of data and extracting information that are unnecessary for the specific task of stabilization. The contribution of this work is to show that if a linear dynamical system has dimension (McMillan degree) n, then there always exist n states from which a stabilizing feedback controller can be constructed, independent of the dimension of the representation of the observed states and the number of inputs. By building on previous work, this finding implies that any linear dynamical system can be stabilized from fewer observed states than the minimal number of states required for learning a model of the dynamics. The theoretical findings are demonstrated with numerical experiments that show the stabilization of the flow behind a cylinder from less data than necessary for learning a model.

AB - Learning controllers from data for stabilizing dynamical systems typically follows a two-step process of first identifying a model and then constructing a controller based on the identified model. However, learning models means identifying generic descriptions of the dynamics of systems, which can require large amounts of data and extracting information that are unnecessary for the specific task of stabilization. The contribution of this work is to show that if a linear dynamical system has dimension (McMillan degree) n, then there always exist n states from which a stabilizing feedback controller can be constructed, independent of the dimension of the representation of the observed states and the number of inputs. By building on previous work, this finding implies that any linear dynamical system can be stabilized from fewer observed states than the minimal number of states required for learning a model of the dynamics. The theoretical findings are demonstrated with numerical experiments that show the stabilization of the flow behind a cylinder from less data than necessary for learning a model.

KW - Data-driven control

KW - Data-driven modeling

KW - Dynamical systems

KW - Model reduction

KW - Numerical linear algebra

KW - Scientific machine learning

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U2 - 10.1007/s10208-023-09605-y

DO - 10.1007/s10208-023-09605-y

M3 - Article

AN - SCOPUS:85146697385

SN - 1615-3375

JO - Foundations of Computational Mathematics

JF - Foundations of Computational Mathematics

ER -