# Digital Control

## Subject description

Basic concepts of digital control, schematic representation of a digital control system, quantisation of time, quantisation of the signal value.

Mathematical bases of discrete systems, sampled signals, z-transform, inverse z-transform, Parseval’s theorem, relations among different forms of Fourier transform, relation between z- and s-planes, transfer function, discrete convolution.

States of discrete systems, state-space representation and transfer function, relation between system response and system eigen-values and eigen-vectors, system response as a function of the system matrix, fundamental matrix, methods for determination of a state transition matrix, the response of non-homogenous linear systems, equilibrium states of the systems.

Frequency response of discrete systems.

Discrete equivalent of continuous systems, discrete equivalent of continuous transfer functions, discrete equivalent of continuous systems given by state-space representations, the relation between continuous and discrete representations, transformation of continuous PID controllers into discrete ones.

Controllability and observability of discrete systems, canonical forms.

Stability of discrete systems Stability criteria, stability of nonlinear systems, direct Lyapunov method.

State controller with a state observer. Basic state controller, optimal state controller, state observer, Kalman filter, duality principle.

## Objectives and competences

• To present the area of discrete control systems, i.e. the systems, given in a form suitable for digital control
• To present complex methods of discrete systems analysis and design.
• To show the methods of conversion of continuous systems into discrete form.
• To present some modern control algorithms to be implemented in digital control.
• To introduce the problems of digital control robustness.

## Teaching and learning methods

Lectures and laboratory work

## Expected study results

After successful completion of the course, students should be able to:

– understand the basic structure of digital control systems,

– analyse the behaviour of discrete systems in time domain and frequency domain,

– describe a discrete system in state space,

– design digital control to achieve a prescribed goal,

– design observers for estimation of internal process states,

– implement the algorithms of digital control on a micro-controller platform.

## Basic sources and literature

1. Sašo Blažič, Digitalno vodenje, Založba FE in FRI, Ljubljana, 2013.
2. Drago Matko, Diskretni regulacijski sistemi,  Univerza v Ljubljani, Fakulteta za elektrotehniko, 1991.
3. Sašo Blažič, Diskretni regulacijski sistemi, Zbirka vaj, Univerza v Ljubljani, Fakulteta za elektrotehniko, 2007.
4. Drago Matko, Računalniško vodenje procesov,  Univerza v Ljubljani, Fakulteta za elektrotehniko, 1995.
5. Gene F. Franklin, J. David Powell, Michael L. Workman, Digital Control of Dynamic Systems, Third Edition, Addison-Wesley, 1997.
6. Karl Johan Astrom , Bjorn Wittenmark, Computer-Controlled Systems: Theory and Design Third Edition, Prentice Hall 1997.
7. Gurvinder Singh Virk, Digital Computer Control Systems, Macmillan, 1991.
8. Rajko Svečko,  Diskretni regulacijski sistemi, Univerza v Mariboru, Fakulteta za elektrotehniko, računalništvo in informatiko, Maribor 2003.

## Stay up to date

University of Ljubljana, Faculty of Electrical Engineering Tržaška cesta 25, 1000 Ljubljana