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Second Semester 
Teaching style
Lingua Insegnamento

Informazioni aggiuntive

Course Curriculum CFU Length(h)
[70/82]  ELECTRICAL ENGINEERING [82/00 - Ord. 2020]  PERCORSO COMUNE 6 60


The overall aim of the course is to develop the student skills related to the analysis and understanding of the behaviour of linear and nonlinear systems and to the ability of designing controllers and observers for some classes of dynamical systems.
In particular it will be developed:

*) Knowledge and understanding of the analogies among different dynamical systems and their representation by means of dynamical models, for linear, non linear systems. Knowledge and understanding of some obesrvation and control design approaches and techniques for linear and nonlinear systems by state feedback.

*) Applying knowledge and understanding in order to
be able to define and evaluate the structural properties of linear and nonlinear systems and their implications in the controller design.

*) Making judgements: as to be able to a critical and synergetic use of the methods and techniques for the analysis and control of dynamical systems.

*) Communication skills:
To be able to clearly express technical ideas and proposals.

*) learning skills:
To be able to get and integrate the information from several sources such that it is possible to achieve a deeper and conscious insight in the physical phenomena characterising dynamical systems.


The student has to know and to be able to critically apply the tools related to the following topics:
differential calculus in multi-dimensional spaces; ordinary differential equations; vectorial algebra; matrices; eigenvalues and eigenvectors; Fourier and Laplace transform; knowledge of the main physical phenomena; analysis and control of linear SISO systems. State-space representation of a dynamical system.


Introduction and fundamentals of dynamical systems (lectures: 4 hours)
Course presentation. Mathematical models for dynamical systems. Transfer functions and state space models. State variables and internal energy of the system. Realisation of a state-space dynamical system.

Linear systems analysis (lectures: 10 hours; exercises: 4 hours)
Canonical forms and invariant state-space transformations. Laplace transform of the free and forced response. The state transition matrix and its properties. Caley-Hamilton theorem (statement only) and the Sylvester formula of the transition matrix. Controllability and controllability matrices. Observability and observability test. Diagonal and Jordan forms. The modal matrix. Vandermonde matrix. Stability of linear systems. System modes, eigenvalues and eigenvectors.

Nonlinear systems analysis (lectures: 12 hours; exercises: 4 hours)
Local linearisation. Local stability. Equilibrium states and trajectories in nonlinear systems. Asymtotic, exponential and finite time stability of an equilibrium point. First and second Lyapunov methods for stability check. Lyapunov equation for time-invariant linear systems. Instability theorems. Invariant sets and La Salle theorem. Stability definitions and criteria for time-varying system. Absolute stability, Popov's and Circle criteria. Limit cycles. Describing funtion approximate analysis.

State-feedbcak control of linear systems (lectures: 8 hours; exercises: 2 hours)
Kalman canonical decomposition of linear systems. Canonical controllable form for SISO systems. Generalised canonical controllable form for MIMO systems. Pole placement via static state-feedback. Observability. Luenberger observer. Separation principle and output feedback. LQR controller.

Control of nonlinear systems (lectures: 12 hours; exercises: 4 hours)
Relative degree in nonlinear systems. Input-output and internal dynamics. State-feedback linearisation. Robust control via sliding modes. Basics on observers for nonlinear systems.

Teaching Methods

The course includes:
* lectures with the support of slides about specific topics (46 hours);
* guided exercises with individual and spontaneous collaborative activities with the support of the teacher and possible tutors (14 hours). Part of these exercises are carried on by means of the Matlab Control System Toolbox.
Specific laboratory tests on the implementation of linear and nonlinear controller will be presented.

During lectures students are actively involved by prompts from the teacher regarding the interpretation of the analytic results, the critical analysis of applicative aspects, and relationships with other topics.

Whenever needed, either by teacher's initiative or students' request, specific support and tutoring activities will be organized.

Lectures could be integrated by on-line strategies to improve their effectiveness and inclusivity

Verification of learning

Student assessment is achieved by means of an oral examination during which some problems about the analysis and control of linear and nonlinear dynamical systems in the state-space representation are proposed.
The student has to know and be able to appropriately apply the methods and techniques for the modelling and analysis of simple linear and nonlinear systems, beening able to identify their general framework.
The student has to know and be able to critically apply some control design techniques for simple linear and nonlinear systems at a at least sufficient degree.
Furthermore, the student has to show adequate skills in speaking and using a technical language as well as a sufficient synthesis and critical analysis ability.
The final mark is defined by means of a weighted combination of the marks in the specific topics.
In order to pass the exam the student must demonstrate a sufficient knowledge of the topics and to be able to both analyse the problems and find a possible solution.
Significant leaks about the required preliminary skills will imply the exam fail.


Alessandro GIUA, Carla SEATZU, Analisi dei sistemi
dinamici-2ª edizione, Springer-Verlag Italia, MIlano, 2009.

J.-J. E. Slotine, W. Li, Applied nonlinear control, Prentice Hall,
Englewood Cliffs, New Jersey, 1991

R.A. DeCarlo, S.H. Zak, G.P. Matthews, Variable Structure Control of Nonlinear Systems: a Tutorial, Proceedings of the IEEE, vol. 76, n. 3, pp. 212-232, 1988.

A. Pisano, E. Usai, Sliding mode control: A survey with applications in math, Mathematics and Computers in Simulation, vol. 81, pp. 954–979, 2011.

A. Tornambé, Use of asymtotic observers having
High-Gain in the state and parameter estimation,
Proc. of the 28th Conference on Decision and Control
(CDC 89), pp. 1791-1794, Tampa, Florida, December

More Information

Additional material in the teacher web site includes:
* exercise text with solution trace;
* slides on nonlinear systems analysis;
* slides on describing function (in English).
* slides on variable structures systems with sliding modes (in English)
* references texts under copyright (password requested to access)

Specific complementary activities could be programmed for part-time students in order to compensate for absences at lectures. Furthermore, the final exam could be completed by means of simulation or experimental tests on a problem agreed with the teacher.

Questionnaire and social

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