EL3340 Introduction to Model Order Reduction

Ph.D. course, 7 credits, Spring 2019

In engineering and science it is often desirable to use the simplest possible mathematical model that “does the job”. First-principles modeling or system identification often result in dynamical models with an unnecessarily large state space. Model (order) reduction concerns the systematic approximation of such models. There are many advantages of working with models with a low-dimensional state space. For example, low-dimensional models are easier to rigorously analyze and faster to simulate. Model-reduction methods have successfully been used to solve large-scale problems in areas such as control engineering, signal processing, image compression, fluid mechanics, and power systems. In this Ph.D. course, we give an introduction to some available reduction techniques, as well as to the required underlying mathematics.

Instructor:

Meeting times: See schedule below. First lecture on Tuesday March 19th, 2019.

Prerequisites: Basic linear systems theory (state-space realizations, Laplace transforms etc.), matrix theory, and basic MATLAB-programming.

Intended learning outcomes

After the course, the student should:

·         be able to distinguish between difficult and simple model-reduction problems;

·         have a thorough understanding of Principle Component Analysis (PCA) and Singular Value Decomposition (SVD);

·         understand the interplay between linear operators on Hilbert spaces, controllability, observability, and model reduction;

·         know the theory behind balanced truncation and Hankel-norm approximation;

·         be able to reduce systems while preserving certain system structures, such as interconnection topology;

·         be able to reduce linear feedback controllers while taking the overall system performance into account; and

·         to understand, and be able to contribute to, current research in model order reduction.

### Schedule

Tentative Program

 Date Time Event Room Topic 2019-03-19 10:15-12 Lecture 1 MV10* Introduction. The model-reduction problem. 2019-03-21 10:15-12 Lecture 2 MV10* Model truncation, singular perturbation, and Padé approximation, model structures (port-Hamiltonian) 2019-03-26 10:15-12 Lecture 3 MV10* Model structures (Laplacian matrices), heuristics (Padé, half rule, pole/zero truncation) 2019-03-28 10:15-12 Lecture 4 MV10* Review of linear systems and Hilbert spaces. H2 and H-infinity norms. 2019-04-02 10:15-12 Lecture 5 MV10* POD/PCA/SVD-based reduction. 2019-04-04 10:15-12 Lecture 6 MV10* Gramians and balanced realizations. 2019-04-09 13:15-15 Lecture 7 MV10* Balanced truncation. 2019-04-11 10:15-12 Lecture 8 MV10* Weighted balanced truncation and controller reduction (part 1). 2019-05-02 10:15-12 Lecture 9 MV10* Optimal model-order reduction in the Hankel norm. 2019-05-07 10:15-12 Lecture 10 MV10* Optimal model-order reduction in the H2/H-infinity norms. 2019-05-14 13:15-15 Lecture 11 MV10* Controller reduction (part 2): Coprime factorization and gap metrics