1. Modern Control Theory (Digital Control)
Lecture 1

2. Course Overview Analog and digital control systems
MM 1 – introduction, discrete systems, sampling.
MM 2 – discrete systems, specifications, frequency response methods.
MM 3 – discrete equivalents, design by emulation.
MM 4 – root locus design.
MM 5 – root locus design.

3. Outline Short repetition of analog control methods
Introduction to digital control
Digitization
Effect of sampling
Sampling
Spectrum of a sampled signals
Sampling theorem
Discrete Systems
Z-transform
Transfer function
Pulse response
Stability

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

5. System caracteristics
Transfer function
Characteristic equation
1+D(s)G(s)H(s) = 0
Poles are the roots of the characteristic equation

6. Time functions associated with poles

7. Second-order system Transfer function is the damping ratio
is the undamped natural frequency

8. Rise time, overshoot and settling time

9. Response og second-order system versus

10. Bode-plot design Determin the open loop gain end phase as function of
Evaluate the phase margin and gain margin
Adjust the margins by use of poles, zeros and gain scheduling.

11. Bode plot

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

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

15. 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)

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

17. Digitization Significance of sampling time T
Example controller D(s) and plant G(s)
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

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

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

20. 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.

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

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

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

24. Spectrum of a Sampled Signal

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

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

27. Spectrum of a Sampled Signal

28. Sampling Theorem Nyquist sampling theorem In practice, we need
One can recover a signal from its samples if the sampling frequency fs=1/T (ws=2p /T) is at least twice the highest frequency in the signal, i.e.
ws > 2 wb (closed loop band-width)
In practice, we need
20 wb < ws < 40 wb

29. Discrete Systems Discrete Systems Z-transform Transfer function
Pulse response
Stability