CFU: 9
Prerequisites
Basic knowledge on the Laplace, Zeta and Fourier transforms.
Preliminary Courses
Analisi Matematica II, Geometria e Algebra, Fisica Generale II.
Learning Goals
Provide the students with: the basics of mathematical modelling of natural and/or artificial systems in continuous and discrete time, the techniques for the analysis of systems described by mathematical models in state space and input-output forms, with particular reference to linear time-invariant systems, the main techniques for the analysis of feedback systems. Introduce the student to the use of the main software tools for the analysis and simulation of systems.
Expected Learning Outcomes
Knowledge and understanding
The course aims to provide students with the methodological tools needed to describe simple engineering systems through an adequate mathematical model, derive small signal models of nonlinear systems, and characterize the time response and the main structural properties of linear systems. To this end, the student will be introduced to the main techniques for the analysis of dynamical systems, both in the time domain and in the complex domain. In addition, the analysis of systems in the frequency domain will also be treated by presenting the main parameters that, in this context, characterize linear systems.
Applying knowledge and understanding
At the end of the course, students will be able to analyze block diagrams, obtaining an overall model, and to evaluate the response of this model to assigned external signals. In addition, the student will be able to analyze the structural properties of this model with particular reference to the stability. Students will also be able to use the Matlab/Simulink software for system analysis and simulation.
Course Content - Syllabus
- Preliminary elements of matrix algebra: basic operations on matrices and vectors. Eigenvalues and eigenvectors of a matrix. Vector spaces. Banach spaces and Hilbert spaces. P-norms of matrices and vectors.
- Dynamical systems: input, state and output variables, state space and input-output representations, classification of dynamical systems.
- Modeling of dynamical systems and examples.
- Nonlinear systems: equilibrium points of a nonlinear system, linearization around trajectories and equilibrium points.
- Analysis of linear time-invariant systems in continuous and discrete time: the superposition principle, free and forced responses. Transition matrix calculation through diagonalization. Natural modes.
- Analysis of continuous-time linear time-invariant systems with the Laplace transform: transfer function, impulse and step response, characteristic parameters of step response, response to polynomial and sinusoidal signals, steady state and transient response.
- Analysis of discrete-time linear time-invariant systems with the Zeta transform, transfer function, impulse and step response, steady state and transient response.
- Stability of equilibrium points: simple and asymptotic stability, instability. Examples of analysis of the stability of equilibrium points for nonlinear systems (pendulum, etc.). Notes on Lyapunov's stability theory. Stability of linear systems, Routh criterion, application of the Routh criterion to discrete time systems. Input-output stability of linear systems.
- Interconnected systems and block diagrams: series, parallel, and feedback interconnections. Representations of interconnected systems. Stability of interconnected systems.
- Realization theory for single-input-single-output systems, observability and controllability canonical forms.
- Discretization of continuous time systems. Sampler and ZOH filters. Sampled data representation of a finite-dimensional linear system.
- Fourier series and Fourier transform. Frequency response of a linear time-invariant system.
- Bode diagrams. Algorithms to draw the Bode diagrams.
- Linear systems as filters: low pass, high pass, band pass, and notch filters.
- Stability analysis of feedback systems: Nyquist diagrams and the Nyquist criterion. Stability margins
- Structural properties of linear systems: reachability, controllability, and observability. Kalman canonical forms.
- Analysis and simulation of dynamical systems using the Matlab and Simulink software tool.
Readings/Bibliography
G. Celentano, L. Celentano – “Modellistica, Simulazione, Analisi, Controllo e Tecnologie dei Sistemi Dinamici - Fondamenti di Dinamica dei Sistemi”, Vol. II, EdiSES, 2010.
Other Books and Notes provided by the lectures.
Teaching Methods
The teaching activities will be organized as follows: a) lectures for 80% of the total hours, b) classroom exercises also involving the use of the MATLAB/SIMULINK software tool (https://www.mathworks.com/), for about 20% of the total hours.
Examination/Evaluation criteria
Exam type
Written and oral. Questions of the written exam refer to: open answers and numerical exercises.
The written exam is aimed at verifying the student's ability to calculate the response of a linear system to assigned signals, to draw the Bode diagrams, and to analyze the stability properties of interconnected systems.
The oral exam, which follows the written one, consists of a discussion on the theoretical topics covered in the course, and of a simple project Matlab/Simulink project developed by the student.
Evaluation pattern
The written exam performance is binding to access to the oral exam. Passing the written exam is only a necessary condition to pass the overall exam.