Journal of Convex Analysis 28 (2021), No. 2, 311--328
Copyright Heldermann Verlag 2021
Identification of Linear Dynamical Systems and Machine Learning
Alain Bensoussan
International Center for Decision and Risk Analysis, Jindal School of Management, University of Texas, Dallas, U.S.A.
and: School of Data Science, City University, Hong Kong
alain.bensoussan@utdallas.edu
Fatih Gelir
Department of Mathematics, University of Texas, Dallas, U.S.A.
fxg150330@utdallas.edu
Viswanath Ramakrishna
Department of Mathematics, University of Texas, Dallas, U.S.A.
vish@utdallas.edu
Minh-Binh Tran
Department of Mathematics, Southern Methodist University, University Park, U.S.A.
minhbinht@mail.smu.edu
The identification of dynamical systems is core to control theory. Driven by the advances in machine learning, data driven approaches are becoming important. In this paper, we study such an approach to the identification of a linear dynamical system under observation. The problem is formulated as an optimization problem to which gradient descent is applied. Surprisingly the fact that the state is available only through observations renders this a non-convex optimization problem. We study this problem in detail, including performing an asymptotic analysis and showing that the cost function is guaranteed to decrease along successive iterates.
Keywords: Control theory, machine learning, gradient descent, system identification.
MSC: 37N35, 93B30.
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