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 analog filters based on the desired frequency characteristics; use of software (MATLAB, OCTAVE ...) in order to evaluate the behavior of linear and nonlinear systems.
- 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;
- eigenvalues and eigenvectors;
- differential equations with constant coefficients;
- equations of simple mechanical systems.
- Linear algebra review (6 hours).
Linear Algebra: Calculation of the Inverse Matrix. Definition of the exponential of a matrix.
- Representation of dynamic systems (12 hours).
Dynamic systems and mathematical models. Input and output variables. Algebraic systems and dynamic systems. State variables. Representation in the state-space form. Linear and non-linear systems, time-variant and time-invariant, SISO and MIMO. Equilibrium state and output. Stability of an equilibrium state. Linearization of non-linear systems around a state of equilibrium.
- Time domain analysis of linear time-invariant systems (LTI), discrete time and continuous time (22 hours).
Representation of LTI systems. Equivalent representations. Superposition principle. Free evolution and forced evolution. Free evolution modes. Dominant modes. Calculation of forced response to impulse and step. Parameters of the step response. Steady-state response and transient response. Stability of LTI systems. Routh's criterion.
- Analysis in the Laplace domain of LTI systems with continuous time (16 hours).
The Laplace transform: definition and main properties. Inverse Laplace transform. Definition of transfer function. Zeros and poles. Calculation of the evolution of an LTI system in the Laplace domain. From the transfer function representation to an i-s-u representation. Block diagrams. Series connections, parallel and feedback.
- Analysis in the frequency domain of LTI systems with continuous time (16 hours).
The harmonic response function and its interpretation. Steady-state response to sinusoidal signals. Filtering action of dynamic systems. Bode diagrams: asymptotic diagrams and corrections. Main parameters of the harmonic response.
- Use of scientific software such as Matlab or Octave for the analysis of dynamic systems (24 hours).
Brief introduction to Matlab / Octave. Main commands for the manipulation of vectors and matrices. Main graphic commands. Analysis of LTI systems with the aid of Matlab / Octave.
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 each 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 a few days later, the ability to link and compare different aspects treated during the course and the ability to use Matlab (or Octave) to analyze dynamic systems are evaluated.