# 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 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.

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 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

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. [7 hours]

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. The transfer function. Relations between the initial conditions of a differential equation. First and second order linear systems. The concept of dominant poles. [14 hours]

3) Stability of dynamical systems: stability to perturbations, BIBO (bounded-input bounded-output) stability and related theorems. Routh’s Criterion. [5 hours]

4) 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 minimumphase systems. [7 hours]

5) Properties of feedback systems. The Nyquist criterion. Phase and magnitude margins: traditional definitions and their extensions. The Padé approximants of the time delay. [6 hours]

6) 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. [5 hours]

7) Control system design: the approach with fixed-structure controllers. Specification requirements and their compatibility. Phase-lead and phase-lag Compensation. The pole-zero cancellation technique and the internal stability of a feedback connection. Frequency synthesis with the inversion formulas. The Diophantine equation for the direct synthesis. Regulation of dynamic systems. The PID regulators: frequency design, tuning and implementation. Control of systems with time delay. A glimpse on feedforward-feedback schemes [13 hours]

8) 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. [13 hours]

9) A design example: position regulation of a DC servo electric motor. Modeling and design of a PD controller by means of the root locus and simulations. Digital implementation with the Arduino board. Experimental results and final considerations. [2 hous]

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. [7 hours]

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. The transfer function. Relations between the initial conditions of a differential equation. First and second order linear systems. The concept of dominant poles. [14 hours]

3) Stability of dynamical systems: stability to perturbations, BIBO (bounded-input bounded-output) stability and related theorems. Routh’s Criterion. [5 hours]

4) 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 minimumphase systems. [7 hours]

5) Properties of feedback systems. The Nyquist criterion. Phase and magnitude margins: traditional definitions and their extensions. The Padé approximants of the time delay. [6 hours]

6) 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. [5 hours]

7) Control system design: the approach with fixed-structure controllers. Specification requirements and their compatibility. Phase-lead and phase-lag Compensation. The pole-zero cancellation technique and the internal stability of a feedback connection. Frequency synthesis with the inversion formulas. The Diophantine equation for the direct synthesis. Regulation of dynamic systems. The PID regulators: frequency design, tuning and implementation. Control of systems with time delay. A glimpse on feedforward-feedback schemes [13 hours]

8) 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. [13 hours]

9) A design example: position regulation of a DC servo electric motor. Modeling and design of a PD controller by means of the root locus and simulations. Digital implementation with the Arduino board. Experimental results and final considerations. [2 hous]

## 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”, quarta edizione, McGraw-Hill Education, 2015.

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.

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”, quarta edizione, McGraw-Hill Education, 2015.

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.

## Teaching methods

Lessons with theory illustrated by examples. Exercitations with problem solving on all teaching topics. A glimpse of computer aided control systems design using MATLAB and Control Systems Toolbox.

The slides used to support the lessons and exercitations are available on the online teaching site (Elly) and constitute the main teaching material of the course.

The modality of all teaching (lectures and exercitations) is defined by the official university provisions that depend on the pandemic situation in progress. A mixed mode is expected, i.e. in presence or remotely online with Microsoft Teams.

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

The slides used to support lessons are available on the online teaching site. They are the main didactical material for the lessons.

## Assessment methods and criteria

The assessment of learning is carried out with a Written Test which includes a part with questions (multiple-choice or open) on all course topics and one or more parts with analysis or synthesis exercises to be solved.

To participate in a Written Test, registration on the University's ESSE3 online site is mandatory. During the Written Test, it is not allowed to consult notes, handouts, books, etc. The use of a basic scientific calculator is recommended and allowed.

The final grade is expressed out of thirty (0-30) and is obtained as the sum of the scores achieved in the various parts of the Written Test.

The modality of the Written Tests is defined by the official university provisions that depend on the pandemic situation in progress. It is expected an online modality with Microsoft Teams.

Assessment of learning is carried out in one of the following forms to be chosen by the student:

1) a written test in the middle of course lessons followed by a final written test at the end of the course.

2) Full written test (at least one for each exam session)

3) Simplified written test followed by an oral exam (at least one for each exam session).

To enroll in the written tests, it is mandatory the registration on the ESSE3 website of the University. During the assessment tests, it is not permitted to read notes, manuals, books, etc. Some parts of the written tests require the use of a basic scientific calculator.

The final vote is expressed in 0-30 scale and is obtained as a weighted average in the assessment forms that have two distinct parts.