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
1. Knowledge and understanding: at the end of the course the sthudents will :
- understand the basics of retroactive control of a linear dynamic system, continuous time and discrete time;
- analyze the stability of closed-loop systems produced by external disturbances or parametric variations;
- know the specifications of a control system, both in the time and frequency domain;
2. Applying knowledge and understanding: at the end of the course the students will be able to:
- perform the design of a feedback control systems for a linear time-invariant continuous-time system and performing its discrete-time realization;
- carry out the project using standard PID type controllers
3. Making judgements: students will be able to judge the potential and limits of the control of Linear and Time-Invariant Systems (LTI).
4. Communication skills: students will be able to illustrate the basic aspects of LTI Systems, interact and collaborate as a team with other experts in the field of control.
5. Learning skills: students will be able to autonomously extend their knowledge on the Theory of Control of LTI Systems, drawing on the vast literature available in the field.
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.
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)
Module 1 Introduction to control systems; response performance of linear systems of the first and second order in the time domain: time constants, response time, time of
climb, settling time. Dependence of the characteristics of the response on the position of the system poles in the plane s. Characteristics of the frequency response of systems of the first and of the
second order, crossing pulsation, pass band, resonance module. Non-minimum phase systems. Polar diagrams. (Teaching hours: 9)
Module 2 Open and closed chain control. Effect of feedback on sensitivity to parametric variations, on chain and feedback chain disturbances and on the band
pass of a linear system. Accuracy at steady state of a feedback system for step, ramp, parabolic entrances, classification of feedback control systems in types. Analysis
of the stability of linear systems fed back by the Nyquist criterion. Phase and earning margin. Root site method - Tracking rules and examples. (Teaching hours: 12)
Module 3 Specifications of a control system: static and dynamic specifications. Transformation of time-specific specifications into harmonic response specifications. Nichols Charter. Synthesis for
attempts. Elementary compensating networks: anticipatory networks and attenuating networks. Synthesis by trial and error for compensation of frequency response. Synthesis with the help of the place of the roots. (Hours of
Module 4 Realization of compensating networks through both passive electrical networks and operational amplifiers. Standard PID type controllers: empirical calibration methods, analytical methods of calibration. (Teaching hours: 6)
Module 5 Relationship between the z-plane and the s-plane. Discretization and reconstruction. Shannon theorem. Specifications of a discrete control system. Design of a control system
discreet. Synthesis of the discrete controller for translation. (Teaching hours: 5)
Module 6. Exercises with the help of the Matlab code (Teaching hours: 6)
Giua, Seatzu. Analisi dei sistemi dinamici, Springer; II Edizione.
1. Norman Nise, Controlli Automatici, CittàStudi;
2. Dorf, Bishop, Controlli Automatici, Pearson