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Versione italiana
Academic year
Didactic period
Secondo Semestre

Training objectives

The aim of the course is to present the features of the mathematical models used to describe the behaviour of dynamical systems and to provide the basic tools for the design of feedback control devices.
Knowledge and understandings:
- basic methods for the analysis of linear time-invariant dynamic systems, with multiple-inputs/multiple-outputs (MIMO)
- basic tools for frequency-domain analysis of linear time-invariant systems (single-input/single-output, SISO)
- stability properties of feedback systems (Nyquist theorems, Routh-Hurwitz criterion)
- graphic tools (Bode diagrams, root loci) for the analysis of loop transfer functions
Such a knowledge can be applied by students to:
- analysis the response of engineering systems (mechanical systems, electrical networks, etc.), as a function of initial conditions or input signals
- design controllers for the stabilization of feedback systems
- solve simple control design problems according to given requirements on static (e.g. steady-state error) and dynamic (e.g settling time) performances of the closed-loop system


Matrix algebra, differential equations,complex variables.

Course programme

Mathematical models of dynamic systems.Continous-time and discrete-time models. Linear and nonlinear models.
Time-invariant and time-varying models. Linear time-invariant systems. State evaluation of dynamic systems. The response function. State transition matrix and its properties.
Eigenvalues and modes of oscillation. Impulse response.
Linear time-invariant systems with single input and single output.Transfer functions and block diagrams.
Relation between input-output and input-state-output representation. Typical test signals for the time response of control systems.
Steady-state errors. Frequency response. Frequency domain analysis: Nyquist and Bode plots, gain margin and phase margin.
General properties of feedback systems. Stability of linear control systems: Routh Hurwitz criterion, Nyquist criterion, stability margins.
The root-locus technique.

Didactic methods

The teaching activity is organized so that the required knowledge is acquired during lectures.
Some of these lectures will be dedicated to the solution of numerical exercises, similar to those that students are required to solve during the final exam.
Practical sessions will be organized to introduce the use of Matlab for linear systems analysis and control design.

Learning assessment procedures

The aim of the exam is to verify at which level the learning objectives previously described have been acquired.

The exam consists in a WRITTEN EXAM with numerical exercises and questions with multiple choice answers.
NOTE: Students repeating the exam will be evaluated on the last given one.

Passing the exam proves that the student has acquired the knowledge and abilities related to the analysis of linear dynamical systems and the design of simple control systems fulfilling both static and dynamic requirements, by means of the solution of numerical exercises and the selection of the correct answers to questions on theoretical aspects.

Reference texts

Lecture slides (in Italian).

Books suggested for optional in-depth analysis:
R. Dorf, R. Bishop: "Modern Control Systems", Prentice-Hall, 2010 (English vesion)
G.Marro: "Controlli automatici", Zanichelli, Bologna, 1992 (In Italian).
P.Bolzern, R.Scattolini, N.Schiavoni: "Fondamenti di controlli automatici", McGraw-Hill, 1998 (In Italian).
K.J.Astrom,R.Murray "Feedback Systems: An Introduction for Scientists and Engineers", online