Automatics M - Z

Module SYSTEM THEORY

• SYSTEMS THEORY

The module aims to achieve the following objectives, in line with the Dublin descriptors:

1. Knowledge and understanding

Students will learn to:

• Analyze a time-invariant system, obtaining the model in the form of a state and subsequently solving the equations of dynamics also with the aid of the Laplace transform;
• Determine the properties of stability, controllability, observability;
• Formulate the transfer function of a linear time-invariant system and determine the frequency response.

2. Applying knowledge and understanding

Apply the above knowledge to the design of the linear state regulator for a linear dynamic system and its observer.

3. Making judgments

Students will be able to indicate the potential and limits of Linear and Time-Invariant Theory (LTI), in particular, both to modeling aspects and in relation to stability.

4. Communication skills

Students will be able to illustrate the basic aspects of LTI Systems Theory, interact and collaborate in groups with other colleagues and external experts.

5. Learning skills

Students will be able to autonomously extend their knowledge on LTI Dynamic Systems Theory, drawing on the vast literature available in the sector

The teaching methods used during the course essentially consist of lectures both performed on the blackboard, and with the aid of personal computers through which slides on theoretical topics and examples of application and computer simulations can be projected.

There are also exercises in which some students are invited to actively participate in the exercise, in order to stimulate collective attention and also to obtain a 'sample' evaluation of the learning outcomes.

Should teaching be carried out in mixed mode or remotely, it may be necessary to introduce changes with respect to previous statements, in line with the program planned and outlined in the syllabus.

SYSTEM THEORY

• Algebra of complex numbers.
• Linear differential equations
• Matrix algebra.

The course does not require mandatory attendance.

However, assiduous attendance at lessons and exercises is strongly recommended to achieve the expected training objectives within the expected time frame.

• SYSTEMS THEORY

Module 1:

Concept of a dynamic system - MIMO, SISO, MISO, SIMO systems, state variables; Block diagram algebra.

(Hours of teaching: 5)

Laplace transform, Dirac impulse, finite duration impulse. Theorems of: translation in frequency, delay, derivative and integral, initial and final value. Antitransform of Laplace - poles and zeros - simple fractions - the concept of transfer function; antitransform of complex and conjugated poles, simple and with multiplicity; Transfer function as a derivative of the impulse response; invariance of the f.d.t;

(Hours of teaching: 9)

Module 2:

Lagrange formula for continuous and discrete systems; Transition matrix: Properties; Definition and calculation by means of inv [sI-A]; minimal form; poles and eigenvalues; demonstration of the Lagrange formula; Cayley-Hamilton theorem; Use of the C-H theorem for the computation of exp(At);

(Hours of teaching: 5)

Module 3.

Movement; trajectory; equilibrium; definition of a stable equilibrium state according to Lyapunov; Stability in non-linear systems; application of the equilibrium state definition for a simple first-order non-linear system with a cubic generating function; basin of attraction; stability in linear time continuous and time discrete systems through eigenvalues; BIBO stability; construction in diagonal form through blocks and robustness characteristics: minimal form and role of residues in the diagonal form; Routh criterion; Lyapunov stability criteria for nonlinear systems - Lyapunov equation for continuous and discrete linear systems; linearization; Diagonalization and Jordan form, geometric stability-multiplicity algebraic multiplicity; linearization;

(Hours of teaching: 9)

Module 4:

Reachability; reachability matrix; zero controllability, controllability, and reachability, A-invariance, controllability matrix, Kalman canonical form for controllability, canonical control form; linear state regulator: arbitrary allocation of eigenvalues; Ackermann's formula; stabilizability; observability; Kalman canonical form, minimal form, the canonical form of observability, observer; compensator - separation theorem;

(Hours of teaching: 9)

Module 5:

First and second-order systems - harmonic response function; Bode diagrams; transformed zeta; anti-deformation zeta; Bilinear transformation.

(Didactic hours: 7)

Module 6:

Exercises through the Matlab environment. In particular, the aspects relating to the frequency response, the determination of properties, and the calculation of characteristic parameters of linear dynamic systems are studied in depth.

(Hours of teaching: 6)

1 - Giua, Seatzu. Analisi dei sistemi dinamici, Springer; II Edizione.

2 - Norman Nise, Controlli Automatici, CittàStudi.

3 - Dorf, Bishop, Controlli Automatici, Pearson.