1. Modern Control Theory (Digital Control)
Lecture 2

2. Outline Signal analysis and dynamic response s-Plane specifications
Discrete signals
Discrete time – discrete signal plot
z-Transform – poles and zeros in the z-plane
Correspondence with continuous signals
Step response
Effect of additional zeros
Effect of additional poles
s-Plane specifications
z-Plane specifications
Frequency response

3. Signal analysis – discrete signals
look at different characteristic signals
z-transform, poles and zeros
signals
unit pulse
unit step
exponential
general sinusoid

4. Signal analysis – discrete signals
The z transform

5. Signal analysis – discrete signals
The Unit Pulse

6. Signal analysis – discrete signals
The unit Step
Zeros : z=0
Poles : z=1

7. Signal analysis – discrete signals
Exponential
Zeros : z=0
Poles : z=r

8. Signal analysis – discrete signals
General Sinusoid
(let us look at the terms, one by one, and use linearity)

9. Signal analysis – discrete signals
Plots shown for
Zeros : z=0, z=r cos(q)
Poles : z=r exp(jq) , z=r exp(-jq)

10. Signal analysis – discrete signals
Conclusions
General sinusoid
Transients
r > 1, growing signal (unstable)
r = 1, constant amplitude signal
r < 1, decreasing signal (the closer r is to 0 the shorter the settling time. In fact, we can compute settling time in terms of samples N.)

11. Signal analysis – discrete signals
Samples per oscillation (cycle)
number of samples in a cycle is determined by q
or, N = samples/cycle depends on q
pole placements depend on q 4 5 3
We have 2 1
k=0
dependence of q

12. Signal analysis – discrete signals
Samples per oscillation (cycle), cont.

13. Signal analysis – discrete signals

14. Signal analysis – discrete signals
Pole
placements

15. Correspondence with cont. signals
Continuous signal
Poles:
s = -a + jb,
s = -a - jb
Pole map
Discrete signal
Poles:
z = exp(-aT - jbT)
z = exp(-aT + jbT)

16. Correspondence with cont. signals
Pole map

17. Correspondence with cont. signals
Recall, poles in the s-plane

18. Correspondence with cont. signals
Fixed z,
varying wn
Pole map
Fixed z,
varying wn

19. Correspondence with cont. signals
Fixed wn,
varying z
Fixed z,
varying wn

20. Correspondence with cont. signals
Notice, in the vicinity of z = 1, the map of z and wn looks like the s-plane in the vicinity of s = 0.

21. Signal analysis – step response
Investigate effect of zeros
fix z1 = p1, and explore effect of z2
a (delayed) second order sys is obtained
z = {0.5, 0.707} (by adjusting a1 and a2)
q = {18°,45°,72°} (by adj. a1 and a2)
a unit step U(z) = z/(z-1) is applied to the system (pole, z=1, and zero, z=0)

22. Signal analysis – step response
Discrete step
responses
for q = 18°
Overshoot
increases with
the zero Z2

23. Signal analysis – step response
The zero has little infuence on the negative axis, large influence near +1

24. Signal analysis – step response

25. Signal analysis – step response
Investigate effect of extra pole
fix z1 = z2 = -1, and explore effect of moving singularity p1 (from -1 to 1)
z = 0.5
q = {18°,45°,72°}
a unit step is applied to the system

26. Signal analysis – step response
Mainly effect on rise time
Rise time expressed
as number of samples.
The rise time increases
with the pole

27. Signal analysis – step response
Conclusions
Addition of a pole or a zero between -1 and 0
Only small effect
Addition of a zero between 0 and +1
Increasing overshoot when the zero is moving towards +1
Addition of a pole between 0 and +1
Increasing rise time when the pole is moving towards +1 (the pole dominates)

28. s-Plane specifications
Spec. on transients of dominant modes
dominant first order
time constant t (related to 3 dB bandwidth)
dominant second order
rise time tr (related to natural frequency wn ≈ 1.8/tr )
settling time ts (related to real part s = 4.6 ts )
overshoot Mp, or damping ratio z.
Spec. on reference tracking
typically step or ramp input specification
i.e. specifications on Kp and Kv , ess = r0 /Kv
ess is the steady state error for a ramp input of slope r0

29. s-Plane specifications
Example
We have system with dominant 2. order mode
Specifications:
Notice, spec. on wn not shown

30. z-Plane specifications
Discrete system
similar specifications
in addition, sample time T
Example (continued)
Notice, sample time T must be chosen.
If fixed wn

31. z-Plane specifications
Specifications are
1) Overshoot Mp less than 16%
2) Settling time ts (1%) less than 10 sec.
3) Chose sample time T such that
Example (7.2 and 7.5)
A system is given by

32. z-Plane specifications
1) Overshoot Mp less than 16%
2) Settling time ts (1%) less than 10 sec.

33. z-Plane specifications
Damping, radius r z wn
Possible
region
Also, we might have an additional specification on rise time tr

34. z-Plane specifications
Steady-state errors
ZOH of plant transfer function, i.e. G(s) to G(z)
Transfer function from R(z) to E(z), for investigating the error.
R(z)
E(z)
controller
D(z)
U(z)
plant
G(z)
Y(z) + -

35. z-Plane specifications
Now, if r(kT) is a step, then

36. Frequency response Frequency response methods
Gain and phase can easily be plotted.
Freq. response can be measured directly on a physical plant.
Nyquist's stability criterion can be applied.
Error constants can be seen on gain plot.
Corrections to gain a phase by additional poles and zeros. Effect can easily be observed – in terms of cross over frequency, gain margin, phase margin.
Frequency response methods can also be applied for discrete systems (example).

37. Frequency response Discrete Bode Plot, Example (7.8)
Plot the discrete frequency response corresponding to
Transform to z-domain by ZOH, with sample time T = 0.2, 1 and 2.
Solution. Use Matlab c2d(sys,T).
Matlab
sysc = tf([1],[ ]);
sysd1 = c2d(sysc,0.2);
sysd2 = c2d(sysc,1);
sysd3 = c2d(sysc,2);
bode(sysc,'-',sysd1,'-.', sysd2,':', sysd3,'-',)

38. Frequency response Half sample frequency Primary effect,
Additional lag
Approx.
phase lag
Df = wT/2

39. Frequency response
Approx.
phase lag
Df = wT/2
Accurate up
to wT = p/2

40. Discrete Equivalents - Overview
r(t)
e(t)
controller
D(s)
u(t)
plant
G(s)
y(t) + -
Translation to
discrete plant
Zero order hold (ZOH)
Translation to discrete controller (emulation)
Numerical Integration
• Forward rectangular rule
• Trapeziod rule (Tustin’s method, bilinear transformation)
• Bilinear with prewarping
Zero-Pole Matching
Hold Equivalents
• Zero order hold (ZOH)
• Triangle hold
Lecture 3