ΨΗΦΙΑΚΟΣ ΕΛΕΓΧΟΣ (22Δ802) Β΄ ΕΞΑΜΗΝΟ Καθηγητής Πέτρος Π. Γρουμπός  Ώρες Γραφείου: Τετάρτη Πέμπτη Παρασκευή 11:00-12:00 Γραφείο: 1 ος όροφος Τομέας Συστημάτων & Αυτομάτου Ελέγχου Τμήμα ΗΜ&ΤΥ ΚΕΦΑΛΑΙΟ 7

2 Digitization Analog Control System ctrl. filter D(s) plant G(s) sensor 1 r(t)u(t)y(t)e(t) + - continuous controller For example, PID control

3 Digitization Digital Control System T is the sample time (s) Sampled signal : x(kT) = x(k) control: difference equations D/A and hold sensor 1 r(t)u(kT)u(t)e(kT) + - r(kT) plant G(s) y(t) clock A/D T T y(kT) digital controller voltage → bit bit → voltage

4 Digitization Continuous control vs. digital control  Basically, we want to simulate the cont. filter D(s)  D(s) contains differential equations (time domain) – must be translated into difference equations. Derivatives are approximated (Euler’s method)

5 Digitization Example (3.1) Using Euler’s method, find the difference equations. Differential equation Using Euler’s method

6 Digitization Compare – investigate using Matlab 1)Closed loop step response with continuous controller. 2)Closed loop step response with discrete controller. Sample rate = 20 Hz 3)Closed loop step response with discrete controller. Sample rate = 40 Hz Significance of sampling time T Example controller D(s) and plant G(s)

7 Digitization Matlab - continuous controller numD = 70*[1 2]; denD = [1 10]; numG = 1; denG = [1 1 0]; sysOL = tf(numD,denD) * tf(numG,denG); sysCL = feedback(sysOL,1); step(sysCL); Matlab - discrete controller numD = 70*[1 2]; denD = [1 10]; sysDd = c2d(tf(numD,denD),T); numG = 1; denG = [1 1 0]; sysOL = sysDd * tf(numG,denG); sysCL = feedback(sysOL,1); step(sysCL); Controller D(s) and plant G(s)

8 Digitization Notice, high sample frequency (small sample time T ) gives a good approximation to the continuous controller

9 Effect of sampling D/A in output from controller The single most important impact of implementing a control digitally is the delay associated with the hold.

10 Effect of sampling Analysis  Approximately 1/2 sample time delay  Can be approx. by Padè (and cont. analysis as usual) ctrl. filter D(s) Padé P(s) sensor 1 r(t)u(t)y(t)e(t) + - plant G(s)

11 Effect of sampling Example of phase lag by sampling Example from before with sample rate = 10 Hz Notice PM reduction

12 Spectrum of a Sampled Signal Spectrum  Consider a cont. signal r(t)  with sampled signal r*(t)  Laplace transform R*(s) can be calculated r(t)r*(t) T

13 Spectrum of a Sampled Signal

14 Spectrum of a Sampled Signal High frequency signal and low frequency signal – same digital representation.

15 Spectrum of a Sampled Signal Removing (unnecessary) high frequencies – anti-aliasing filter control: difference equations D/A and hold sensor 1 r(t)u(kT)u(t)e(kT) + - r(kT) plant G(s) y(t) clock A/D T T y(kT) digital controller anti- aliasing filter

16 Spectrum of a Sampled Signal

17 Sampling Theorem Nyquist sampling theorem  One can recover a signal from its samples if the sampling frequency f s =1/T (  s =2  /T) is at least twice the highest frequency in the signal, i.e.   s > 2  b (closed loop band-width) In practice, we need  20  b <  s < 40  b