Automatics M - Z

Module SYSTEM 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. Ability to apply knowledge and understanding, apply the above knowledge to the design of the linear state regulator for a linear dynamic system and its observer.

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

4. Communication skills Students will be able to illustrate the basic aspects of LTI Systems Theory, and 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.

Module 1: Concept of a dynamic system - MIMO, SISO, MISO, SIMO systems, state variables; Block diagram algebra; Models in the form of a state. (Didactic hours: 5) Laplace transform the Dirac impulse, the impulse of finite duration. Theorems of: translation in frequency, delay, derivative and integral, initial and final value. Anti Laplace transform - 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 using inv [sI-A]; minimal form; poles and eigenvalues; proof of Lagrange's formula; Cayley-Hamilton theorem; Use of the C-H theorem for the computation of exp (At); (Teaching hours: 5)

Module 3. Movement; trajectory; equilibrium; Lyapunov definition of a stable equilibrium state; Stability in non-linear systems; application of the equilibrium state definition for a simple first-order non-linear system with a cubic generating function; stability in linear time-continuous and time-discrete systems using 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; Diagonalization and Jordan form, linearization; (Hours of teaching: 9)

Module 4. reachability; reachability matrix; 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 (Hours of teaching: 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 d

The exams at the end of the course refer to the entire "Automatic" subject. These include a written and an oral test. The written test will consist of a task divided into two parts: one related to Systems Theory (max. 15 points), the other to Automatic Controls (max 15 points), each consisting of exercises and questions.

Should the organizational didactic conditions exist and agreed with the teachers of the Automatica A-L course, two ongoing tests may be provided, which would allow, in the event of a positive outcome, not to carry out the System Theory part within the written task of Automatica, limited to the summer session exams.

Dynamic systems models: determination of equations of state and/or transfer function

Calculation of free and forced evolution. Analysis of controllability and observability stability

Design of a linear regulator for a SISO system

Relevant observer project

Plotting of the frequency response of a linear dynamic system.