The goal of this course is to acquaint the students with the basic principles of system identification and model reduction of systems in state-space representation This course will cover the theoretical foundations of modern system identification (learning from data) and model reduction of control systems in state-space form. In this course, the well-established system identification and model reduction algorithms will be covered. In addition, we will also cover the most important theoretical results on these algorithms: their statistical consistency, error bounds, etc. The goal of this course is to teach students not only the algorithms, but also why they work.
To this end, the course will cover the necessary elements of mathematical systems theory, especially realization theory. Realization theory is a classical, but not widely taught subject. The goal of realization theory is to establish a correspondence between input-output behaviors and (minimal dimensional) state-space representations. As such, it can be viewed as an abstract version of the system identification and model reduction problems. Realization theory provides the theoretical foundations for system identification and model reduction, and it will be used in the course to explain the theoretical properties of the presented system identification and model reduction methods. In this course we will cover the following topics:
Theoretical foundations for deterministic systems: realization theory:
Linear time-invariant state-space representations: motivation, control (stability, stabilization), relationship with non-linear systems.
Observability, controllability, minimality
Ho-Kalman realization algorithm, Hankel-matrices
Input-output equations, elements of behavioral theory
Existence of controllers and observers
Dissipativity, storage functions, positive real lemma
Ressources:
Book by E. Sontag: Mathematical Control Theory. Chapter 1, Chapter 2 (all sections except 2.9, 2.11), Chapter 3 (Section 3.1-3.3), Chapter 5 (Section 5.1, 5.5, 5.7 (especially pages 230 - 246), Section 5.8 (important)), Chapter 6 (Sectioin 6.1, 6.2 (Theorem 23)), Section 6.5)
Jan C. Willems Behavioral approach to open and interconnected systems, IEEE CONTROL SYSTEMS MAGAZINE, 2007 (untill page 78, but the whole paper is nice to read).
Stephen Boyd, Laurent El Ghaoui, Eric Feron, and Venkataramanan Balakrishnan Linear Matrix Inequalities in System and Control Theory (Chapter 6, Section 6.3.2, 6.3.3).
Model reduction and system identification of deterministic linear systems
Model reduction: balanced truncation, moment matching
System identification: deterministic subspace methods
Ressources:
Book by A.C. Antoulas. Approximation of Large Scale Dynamical Systems. SIAM 2005. (Chapter 5, Section 5.1,5.3, 5.5, 5.8, Chapter 7, Section 7.1, Chapter 11, Section 11.1, 11.2)
K. Glover and J. Willems, "Parametrizations of linear dynamical systems: Canonical forms and identifiability," in IEEE Transactions on Automatic Control, vol. 19, no. 6, pp. 640-646, December 1974.
Thesis B. Hanzon. Identifiability, Recursive Identification and Spaces of Linear Systems 1986. Link to download Section 4.4 .
Tomas McKelvey, Anders Helmersson, Thomas Ribarits, Data driven local coordinates for multivariable linear systems and their application to system identification, Automatica, 40(9), 2004
Peter Van Overschee, Bart L.R. De Moor Subspace identification for linear systems. Theory, implementation, applications, Kluwer, 1996 (Chapter 2), Link
Tohru Katayama. Subspace Methods for System Identification, Springer, 2005, (Chapter 6, Sectio 6.1-6.7) Link
Theoretical foundations for stochastic systems: realization theory
Existence and minimality of a realization, spectral factorization
Systems in innovation form, relationship with Kalman filters
Covariance realization algorithm
Stochastic system identification algorithm