The performance of many engineering systems, subsystems and components often critically depends on what the designers like to address as “second order effects”. These are typically unwanted or parasitic phenomena that can be captured accurately only by computationally demanding partial differential equation solvers (e.g., Maxwell or heat diffusion field solvers). Designers, however, would greatly benefit from the availability of compact small-size models capturing the dominant dynamical input-output behavior of such complex systems, but still characterized by a controlled accuracy with respect to first-principle descriptions. In this multidisciplinary course we will survey several techniques to automatically “compress” dynamical system models, while preserving accuracy and important physical properties such as stability and passivity. Two main approaches will be discussed under a system-theoretical standpoint: the so-called "model-driven" or "intrusive" methods and the complementary "data-driven" or "non-intrusive" methods. The starting point of the presented model-driven approaches are large-scale systems of Ordinary Differential Equations (ODE) or Differential Algebraic Equations (DAE), for which various Model Order Reduction (MOR) schemes will be discussed based on Moment Matching and Balanced Truncation, within a classical projection framework. Conversely, the presented data-driven approaches build compact state-space equations starting from samples of the system response, either in frequency or time domain in a completely black-box framework. Multivariate model-driven and data-driven approaches will also be discussed, where dependence on additional parameters (for design or uncertainty quantification) is embedded in the compact models. The above techniques will be applied to a number of examples from various application fields. It is expected that the know-how provided by this course will be helpful in research projects involving analysis, design and optimization problems in a variety of different engineering and science disciplines.

Basic linear algebra, signal and system theory, Laplace and Fourier transforms. Most of the required background will be reviewed during the first lectures.

Most of the course material is found in the textbook: S. Grivet-Talocia and B. Gustavsen, "Passive Macromodeling: Theory and Applications". New York: John Wiley and Sons, 2016 PART I: Introduction [4 hours] 1. Motivations 2. Sample problems from various engineering disciplines 3. Math background: 4. Background on finite-dimensional LTI dynamical models (state-space and descriptor systems): time-domain and frequency-domain characterization 5. Fundamental properties LTI systems (e.g. stability, passivity, dissipativity) PART II: Model-driven MOR [8 hours] 1. Compressing Linear Time Invariant (LTI) Systems: early approaches a. Modal analysis b. Rational interpolation c. Padé approximation and Asymptotic Waveform Evaluation 2. Compressing LTI Systems a. The Projection Framework b. Krylov subspaces c. Moment matching (Arnoldi, Lanczos, PRIMA algorithms) d. Single and multi-point moment matching e. Truncated Balanced Realizations (TBR) 3. Preserving or enforcing stability and passivity (dissipativity) Part III: Data-Driven MOR [8 hours] 1. Representation of rational functions and barycentric forms 2. Rational approximation of frequency responses a. Classical approaches (Levy, Sanathanan-Koerner) b. The Vector Fitting algorithm for single-input single-output systems c. Extension to multi-input multi-output systems 3. The Loewner framework a. Rational interpolation via Loewner matrices b. The AAA algorithm c. Direct realization from frequency samples d. Stability enforcement 4. Passivity characterization and enforcement a. Kalman Yakubovich Popov Lemma and Linear Matrix Inequalities b. Spectral characterization through associated Hamiltonian matrices c. Frequency-domain inequalities d. Model perturbation under passivity constraints 5. Synthesis of state-space equations and equivalent circuits from matrix rational functions 6. Application examples Part IV: Data-driven MOR of parameterized systems [5 hours] 1. The Parameterized Sanathanan-Koerner iteration 2. Multivariate state-space realizations 3. Stability and Passivity enforcement a. via structural constraints b. via constrained (convex) optimization 4. Extension to Linear Parameter Varying (LPV) systems

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P.D.1-1 - Febbraio