# Introduction to systems theory

### Introduzione al Corso

This course is aimed at anyone who wants to learn the basics of systems theory and who has a university background in mathematical analysis (including Laplace transform), linear algebra, and physics.

State models with one input and one output will be covered, highlighting their importance and main properties; it will also be shown how to construct such models from the equations of physics.

The topics covered in the course are:

- Definitions of state models, linear systems, construction of a state model, choice of state variables, equilibrium points and linearization, feedback linearization.
- Analysis of linear systems: evolution of state and output of a linear system, “superposition of effects” principle, free and forced evolution of state and output, transfer function and impulse response.
- Stability: stability in linear systems, stability conditions for equilibrium points of non-linear systems, BIBO stability. Examples: compartmental models, predator-prey model (Lotka-Volterra), electric motor model

Sciences

Ore di Formazione0

LivelloBase

Modalità CorsoTutoraggio

Italian

DurataTipologia

Online

Stato del CorsoTutoraggio Soft

### Agenda del Corso

Avvio Iscrizioni

Apertura Corso

Inizo Tutoraggio

Fine Tutoraggio

Tutoraggio Soft

Chiusura Corso

### Risultati Attesi

Upon completion of this course, students will be able to:

Understand the field of systems theory and its areas of application.

Understand and apply state models.

Build simple state models from physics equations.

Understand the concept of an equilibrium point in a state model.

Understand the concepts of simple and asymptotic stability of an equilibrium point.

Linearize a state model around an equilibrium point.

Describe the evolution of state and output of a linear system.

Understand the concept of transfer function and its relationship with the state model.

Understand the concept of feedback linearization.

Understand the concepts of simple and asymptotic stability of a linear system.

Understand the concept of BIBO stability of a linear system.

Understand the relationship between the stability of an equilibrium point and the stability of the linearized system.

Apply the learned concepts to simple physical systems.

### Pre-requisiti

In order to fully understand the material presented in this course, it is necessary to be familiar with the following concepts.

**Mathematical analysis:**scalar and vector-valued functions of one or more variables. Continuity, derivability, development in Taylor series. Integrals. Complex numbers and their use. Complex exponentials. Polynomials and their representations. Differential equations.**Linear algebra:**linear functions, matrices and vectors, eigenvectors, eigenvalues and their multiplicities. Diagonalization.**Physics:**kinematics and dynamics. Electro-magnetic field. Resistors, capacitors and inductances. Kirchhoff's laws.

**Signals and systems:**Laplace transform; inverse Laplace transform of rational functions.

### Libri di testo e letture consigliate

All the material covered in this course can be found in the book

Augusto Ferrante, "Appunti di Automatica per Ingegneria Biomedica con esercizi e temi d'esame risolti", Edizioni Progetto, Padova, 2023. (In Italian)

More advanced material for further study can be found in the books

Ettore Fornasini, "Appunti di teoria dei sistemi", Libreria Progetto, ISBN: 978-8896477328, 2011. (In Italian)

Thomas Kailath, “Linear systems”, Englewood Cliffs, NJ: Prentice-Hall, 1980.

Joao P. Hespanha, “Linear systems theory”, Princeton university press, 2018.