INTRODUCTION TO DYNAMIC SYSTEMS
Expected results according to the Dublin descriptors.
- Knowledge and ability to understand: the student must be able to analyze linear systems and to identify their main characteristics, in the domain of time and frequency.
- Ability to apply knowledge and understanding: Design of simple linear systems with assigned characteristics in terms of response to canonical signals.
- Making judgments: develop the ability to critically assess the behavior of a linear system in the presence of external inputs or assigned initial conditions.
- Communication skills: ability to clearly express procedures and results regarding system analysis.
- Ability to learn: know how to integrate knowledge from various sources in order to achieve a broad vision of the problems related to the behavior of dynamic systems.
It is necessary to have acquired the main contents provided by the courses of linear algebra, physics and mathematical analysis, and in particular:
- operations between matrices;
- differential equations with constant coefficients;
- equations of simple mechanical systems.
- Some topics of linear algebra (4 hours).
Linear Algebra: Calculation of the Inverse Matrix. Definition of the exponential of a matrix.
- Representation of dynamic systems (8 hours).
Dynamic systems and mathematical models. Input and output variables. Algebraic Systems and Dynamic Systems. State variables. Representations in the state space form. Linear and nonlinear systems, time variant and time invariant, SISO and MIMO.
- Analysis of discrete-time and continuous-time time-invariant linear systems (LTIs) in the time domain (14 hours).
Representation of LTI systems. Superposition principle. Free evolution and forced evolution. Dominant modes. Calculation of forced response to impulse and step imputs. Transient response and steady-state response. Stability of LTI systems. Routh's criterion.
- Laplace domain analysis of LTI continuous-time systems (12 hours).
Laplace's transformation: definitions and major properties. Inverse Laplace transform. Definition of transfer function. Zeros and poles. Calculation of the evolution of an LTI system in the Laplace domain. Block diagrams. Serial, parallel and feedback connections.
- Analysis in the frequency domain of LTI continuous-time systems (10 hours).
The harmonic response function and its interpretation. Steady-state sinusoidal response. Filtering action. Bode diagrams: asymptotic diagrams and corrections. Main parameters of the harmonic response.
Frontal lectures; classroom exercises.
Antsaklis, P., Michel, Anthony N., A Linear Systems Primer, Birkhauser, 2007
The exam is divided into a written test and an oral test. The written test consists of three numerical exercises with an assigned score. The written test is intended to be passed with a minimum score of 16. In the oral exam, which can be performed on the same day or after a few days, the ability to link and compare different aspects treated during the course is evaluated. The final score is calculated as an average of the score of the written exam and the score of the oral exam.