# FUNDAMENTALS OF AUTOMATIC CONTROL

## Learning outcomes of the course unit

The aims of the course in relation to understanding and knowledge are:

- Understanding of the two principles of active control, feedforward and feedback, and of the broad applications to automation.

- Understanding of the methods, based on Laplace and Zeta transforms, to determine the time-evolution of linear scalar dynamic systems.

- Knowledge of the harmonic analysis and of the stability theory for linear systems.

- Knowledge of the main methods of analysis and synthesis for feedback control systems.

In relation to the ability to apply knowledge and understanding, the aims are:

- Skill to analyze feedback control systems.

- Skill to set up and solve simple problems of regulation and control with a single controlled variable.

## Prerequisites

Mathematical Analysis 1, General Physics 1.

## Course contents summary

1) Fundamental concepts: systems and mathematical models. Block diagrams. Feedforward and feedback. Robustness of feedback with respect to feedforward. Mathematical modelling of physical systems: examples from electric networks, mechanical systems, and thermal systems.

2) Analysis methods of LTI (linear time-invariant) SISO (single-input single-output) systems. Ordinary differential equations and Laplace transform. Inverse Laplace transform of rational functions. Generalized derivatives and elements of impulse function theory. Relations between the initial conditions of a differential equation. First and second order linear systems.The concept of dominant poles.

3) Frequency-domain analysis: the frequency response function. Relation between the impulse response and the frequency response. Bode’s diagrams. Nyquist’s or polar diagrams. Asymptote of the polar diagrams. Bode’s formula and minimum-phase systems.

4) Stability to perturbations and BIBO (bounded-input bounded-output) stability of LTI systems: definitions and theorems. The Routh criterion. Properties of feedback systems. The Nyquist criterion. Phase and magnitude margins: traditional definitions and their extensions. The Padé approximants of the time delay.

5) The root locus of a feedback systems: properties for the plotting. Generalization of the root locus: the “root contour”. Examples. Stability degree on the complex plane of a stable systems.

6) Control system design: the approach with fixed-structure controllers. Specification requirements and their compatibility. Phase-lead and phase-lag compensation. Pole-zero cancellations and the internal stability of a feedback connection. The PID regulator. Frequency synthesis with the inversion formulas. The Diophantine equation for the direct synthesis.

7) Digital control systems: The z-transform. Conversion from continuous-time to discrete-time. Sampling frequency and anti-aliasing filtering. SISO discrete-time linear systems: free and forced response, stability and Jury’s Criterion. Glimpse on the synthesis of discrete-time controllers.

## Recommended readings

Pdf slides of the lessons on the web site of the course.

FURTHER READINGS

1) G. Marro, ``Controlli Automatici'', quinta edizione, Zanichelli, Bologna, 2004.

2) P. Bolzern, R. Scattolini, N. Schiavoni, “Fondamenti di Controlli Automatici”, terza edizione, McGraw-Hill, 2008.

3) M. Basso, L. Chisci, P. Falugi, “Fondamenti di Automatica”, CittàStudi, 2007.

4) A. Ferrante, A. Lepschy, U. Viaro, “Introduzione ai Controlli Automatici”, UTET, 2000.

5) J.C. Doyle, A. Tannembaum, B. Francis, “Feedback Control Theory”, MacMillan, 1992.

6) M.P. Fanti, M. Dotoli, “MATLAB: Guida al laboratorio di automatica”, CittàStudi, 2008.

7) A. Cavallo, R. Setola, F. Vasca, “La nuova Guida a MATLAB: Simulink e Control Toolbox, Liguori, 2002.

## Teaching methods

Classroom sessions with alternate use of slides and explanations at the blackboard. Discussion and resolution of exercises at the blackboard on all topics of the course. A glimpse on computer aided control systems design using MATLAB and Control Systems Toolbox.

## Assessment methods and criteria

The exam consists of a written test. It is also possible to take the exam with a written test in the middle of the course lessons and a final written test at the end of the lessons.