# System theory (2019/2020)

Course code
4S02785
Credits
6
Coordinator
Paolo Fiorini
INF/01 - INFORMATICS
Language of instruction
Italian
Teaching is organised as follows:
Activity Credits Period Academic staff Timetable
Teoria 3 II semestre Paolo Fiorini

Laboratorio 3 II semestre Bogdan Mihai Maris

### Learning outcomes

The course objective is to give students the mathematical tools required for the analysis and modeling of linear, time-invariant (LTI) systems, and of the input/output signals to an LTI system. The model will allow students to study the main system properties and to address the general concepts of controller and filters to perform simple control actions on the dynamic system and filter operations on the input/output signals.

The mathematical tools will be based on analysis methods in the time domain, as well as of the complex variables s, z and the frequency of the input/output signals. Analysis and synthesis will be carried out both for continuous and discrete time systems and signals. The theoretical concepts acquired during the course will be consolidated with exercise sessions addressing the solution of basic problems with analytical approach and with numerical simulations.

### Syllabus

1. Review: complex numbers, functions of complex variables, series of complex powers, convergence, Euler's formula
2. Distributions: impulse, step, ramp. Sampling and reproducibility. Sinusoidal exponential functions, time translation. Discrete signals
3. Continuous time systems. Causal LTI systems. Stability.
4. Characteristic equation of a system, elementary modes, convergence.
5. Convolution, impulsive response, forced response, BIBO stability, asymptotic stability
6. Frequency response
7. The Laplace transform. Convergence region. Properties
8. Free response and forced response in the complex plane, transfer function, zeros and poles, stability.
9. The Laplace's antitrasform, poles, residues.
10. The Fourier series for periodic signals. Frequency, pulsation, linear combination of periodic signals, synthesis equation, analysis equation. The energy and power of a signal, the discrete spectrum.
11. The Fourier transform. Conditions of existence, discontinuities.
12. Convolution and modulation. Spectrum.
13. Fourier trasform of the ideal sampling train, replication and sampling, reconstruction filters, Nyquist frequency, Shannon forumula.
14. Bode diagrams
15. Block diagrams
16. Discrete-time LTI systems. ARMA model. Impulsive response, forced response.
16. z trasform. Properties
17. The antitransformed z. Frequency response. Discrete Fourier transform

The students can integrate the reference books ('Segnali e Sistemi' by M.E. Valcher and 'Regolazione Automatica' by Di Stefano et al. for the laboratory part) with the free text at the link:
https://ptolemy.berkeley.edu/books/leevaraiya/

Office hours: after class

### Assessment methods and criteria

The exam is written. There may be an oral exam. During the course two optional intermediate written tests will be offered, that can replace, in full or partially, the written exam.

 Activity Author Title Publisher Year ISBN Note Teoria P. Bolzern, R. Scattolini, N. Schiavoni Fondamenti di controlli automatici McGraw-Hill 1998 Teoria Di Stefano, Stubberud, Williams Regolazione Automatica Schaum -- Etas 1974 Teoria M.E. Valcher Segnali e Sistemi Ediitrice Progresso 2002 Laboratorio Di Stefano, Stubberud, Williams Regolazione Automatica Schaum -- Etas 1974