## INTRODUCTION TO CONTROL SYSTEMS

For an effective approach to the study of system and control theory, students should have the following skills.

Knowledge:

Basics of linear algebra. Trigonometirc, esponential and logarithmic functions and their properties. Ordinary linear differential equations. Integrals. Complex numbers. Polynomials. Fourier transforms. Phisic principia of the solid body and of electrical systems.

Capacity:

Algebraic, differential and integral calculus. Analysis and representation of functions depending on one or more variables. Variable transformations.

Competence:

To be able to apply the methods of algebra, differential calculus and function analysis to represent physical systems.

The final exam consists on a written test to verify the ability of solving simple analysis problems and of designing the controller of simple single-loop feedback linear systems.

Students that achieve a sufficient mark in the written test are required to discuss and explain the achieved results in the written test, showing their understanding of the used methods and the knoledge of possible alternatives.

The final mark will take into account both the results in the written test and in the oral discussion.

In order to pass the exam, students are required to show a good knonwledge of the mathematical tools and methods for the analysis and design of feedback control systems both in the continuous and discrete-time domain.

Full marks are given to students that show an excellent knowledge, a good comprehension of the methods and a sufficient critical mind of the topic.

Students who are attending the lectures are admitted to a mid-term evaluation, which is scheduled after completing the lectures on system analysis. The evaluation includes a written test on the analysis of dynamical systems described by both a state-space representation and a transfer function. The mid-term results can be considered in the final evaluation under student's request; in this case the analysis section of the final written test will be skipped.

According to the scope of the undergraduate course in Computer Engineering, the aim of the course is to give the basic knowledge, capacity and competence on automatic control.

Knowledge:

mathematical models to represent dynamical systems and to identify their properties; the meaning of the functional blocks in a control system; basic tools to design the controllers in a single-loop feedback system; basic language and terminology of automatic control systems.

Ability:

to compute the performance of linear dynamical systems on the basis of their mathematical and graphical representations; to compute the performance of a feedback system on the basis of the transfer functions of the component blocks.

Competence:

to design the transfer function of the controller in single-loop feedback systems in order to satisfy prescribed requirements; to be able to chose the proper design technique; to critically evaluate the results of the analysis and of the design by means of formal and approximate methods; to be able to clearly express technical and scientific concepts on control systems.

1. Introduction and mathematical recalls (lecture: 4 h)

Automatic control and systems. Differential equations. Complex numbers. Eulero's relationships. Vectors. Matrices. Fourier transform.

2. Dynamical systems (lecture: 2 h, ex: 3h)

Input, state and output variables. Input-output (IO) and state variables (SV) models. Causality principle. SV and internal energy. Classification of dynamical systems. Local linearization of non linear systems.

3. Laplace transform (lecture: 4 h)

Laplace transform of: impulse, step, exponential and sinusoidal function. Properties: time an frequency shift; derivation, integration and convolution theorems. Laplace anti-transform.

4. Linear SV models (lecture: 11 h, ex: 8 h)

State transition matrix. Characteristic polynomial, eigenvalues and poles. Lagrange formula: free and forced response. System stability, modes and eigenvalues. Equivalence transformations. Diagonalization. Jordan form. Generalised eigenvectors. Controllability and observability matrices. State-feedback eigenvalues assignment. Luenberger observer. Separation principle.

5. IO models (lecture: 8 h, ex: 4 h)

Transfer matrix. Transfer function. Step response. Harmonic response. Bode diagram. Filtering parameters of the harmonic response. Nyquist diagram.

6. Feedback systems (lecture: 5 h, ex: 3 h)

Control systems structure. Systems with dominant modes. Block algebra. Load effects. Transfer function of connected systems. Stability of feedback systems: Nyquist and Bode criteria. Stability margins. Roots locus: modal analysis of feedback systems.

7. Requirements in control systems (lectures: 7h, exercise: 3h)

Sensitivity function with respect to external disturbances. Tracking error. Steady state tracking error with respect to canonical inputs and disturbances. Transient behaviour and closed loop characteristics. Relationships between step and frequency response. Relationships between open-loop and closed-loop characteristics.

8. Controller design (lectures: 9 h, exercise: 7 h)

Controller design by loop-shaping. Lead, lag, lead-lag compensators. Normalized diagrams for the lead compensator. Control design by means of zero-pole compensation using the roots locus. PID tuning via open-loop and closed-loop Ziegler and Nichols methods. Relay feedback test.

9. Z transform (lectures: 4 h)

Z transform: Kroneker impulse, constant, power, harmonic sequences. Properties:forward and backward shift; initial value, final value and convolution theorems. Z anti-transform.

10. Discrete time systems (lez: 6 ore, es: 2 ore)

SV models. Free evolution, modes and stability. Forced response. Transfer function. Digital control systems. Sampling. ZOH D/A converter. Sampled systems. Aliasing. Filtering characteristics of the ZOH. Correspondence between s-plane and z-plane. Spectrum of sampled signals. Feedback system stability: roots locus analysis. Digital controller design by roots locus.Discretization of continuous-time controllers: zero-pole correspondence, forward and backward difference methods, Tustin method.

P. Bolzern, R. Scattolini, N. Schiavoni, Fondamenti di controlli automatici – ed. 4a, McGraw Hill, 2015

Per ulteriori esercizi

A.V. Papadopulos, M Prandini, Fondamenti di automatica – esercizi, Pearson, 2016

The course includes:

* 60 hours of lectur with the support of the blackboard and slide projections as a hint to discussion.;

* 30 hours of exercise in a the software laboratory with the use of Scilab and Xcos.

Students are always solicited by questions, requests of interpretation of the analitical results and thir critical interpretation with respect to applications and connections with other matters.