1. What is the system type? What are Kp, Kv, Ka?
What are wgc, wpc, GM, PM?

2. Stability from Nyquist plot
G(s)
The complete Nyquist plot:
Plot G(jω) for ω = to +∞
Get complex conjugate of plot, that’s G(jω) for ω = 0– to –∞
If G(s) has pole on jω-axis, treat separately
Mark direction of ω increasing
Locate point: –1

3. As you follow along the G(jω) curve for one complete cycle, you may “encircle” the –1 point
Going around in c.w. once is +1 encirclement
c.c.w. once is –1 encirclement

5. Nyquist criterion: # (unstable poles of closed-loop)
Nyquist criterion: # (unstable poles of closed-loop) Z = # (unstable poles of open-loop) P + # encirclement N or: Z = P + N To have closed-loop stable: need Z = 0, i.e. N = –P

6. That is: G(jω) needs to encircle the “–1” point c.c.w. P times.
If open loop is stable to begin with, G(jω) cannot encircle the “–1” point for closed-loop stability
In previous example:
No encirclement, N = 0.
Open-loop stable, P = 0
Z = P + N = 0, no unstable poles in closed-loop, stable

7. Example:
4/(s-1)

8. As you move around
from ω = –∞ to 0–,
to 0+, to +∞, you go
around “–1” c.c.w.
once.
# encirclement N = – 1.
# unstable pole P = 1

9. Check: c.l. pole at s = –3, stable.
i.e. # unstable poles of closed-loop = 0
closed-loop system is stable.
Check:
c.l. pole at s = –3, stable.

10. Example:
Get G(jω) for ω = 0+ to +∞
Use conjugate to get G(jω) for ω = –∞ to 0–
How to go from ω = 0– to ω = 0+? At ω ≈ 0 :

11. Tiny radius
CCW
Huge radius CW
Around a simple jw-axis pole, G(jw) sweeps CW 180 deg
Around a double pole on jw-axis, G(jw) sweeps CW 360 deg

13. # encirclement N = _____
# open-loop unstable poles P = _____
Z = P + N = ________
= # closed-loop unstable poles.
closed-loop stability: _______

14. Q: 1. Find stability margins
Example:
Given:
G(s) is stable
With K = 1, performed open-loop sinusoidal tests, and G(jω) is on next page
Q: 1. Find stability margins
2. Use Nyquist criterion to determine closed-loop stability
G(s) K

16. Solution:
Where does G(jω) cross the unit circle? ________ Phase margin ≈ ________ Where does G(jω) cross the negative real axis? ________ Gain margin ≈ ________ Is closed-loop system stable with K = 1? ________

17. Note that the total loop T.F. is KG(s).
If K is not = 1, Nyquist plot of KG(s) is a scaling of G(jω).
e.g. If K = 2, scale G(jω) by a factor of 2 in all directions.
Q: How much can K increase before GM becomes lost? ________
How much can K decrease? ______

18. Some people say the gain margin is 0 to 5 in this example
Q: As K is increased from 1 to 5, GM is lost, what happens to PM?
What’s the max PM as K is reduced to 0 and GM becomes ∞?

19. To use Nyquist criterion, need complete Nyquist plot.
Get complex conjugate
Connect ω = 0– to ω = 0+ through an infinite circle
Count # encirclement N
Apply: Z = P + N
o.l. stable, P = _______
Z = _______
c.l. stability: _______

21. G(jω) for ω > 0 as given. Get G(jω) for ω < 0 by conjugate
G(s)
Example:
G(s) stable, P = 0
G(jω) for ω > 0 as given.
Get G(jω) for ω < 0 by conjugate
Connect ω = 0– to ω = 0+. But how?

22. Choice a) :
Where’s “–1” ?
# encirclement N = _______
Z = P + N = _______
Make sense? _______

23. Choice b) :
Where is “–1” ?
# encir. N = _____
Z = P + N = _______
closed-loop stability _______

24. Note: If G(jω) is along –Re axis to ∞ as ω→0+, it means G(s) has in it.
when s makes a half circle near ω = 0, G(s) makes a full circle near ∞.
choice a) is impossible, but choice b) is possible.

26. Example: G(s) stable, P = 0
Get conjugate for ω < 0
Connect ω = 0– to ω = 0+. Needs to go one full circle with radius ∞. Two choices.

27. Choice a) :
N = 0
Z = P + N = 0
closed-loop stable

28. Choice b) :
N = 2
Z = P + N = 2
Closed loop has two unstable poles

29. Which way is correct?
For stable & non-minimum phase systems,

30. Example: G(s) has one unstable pole
P = 1, no unstable zeros
Get conjugate
Connect ω = 0– to ω = 0+. How? One unstable pole/zero If connect in c.c.w.

31. # encirclement N = ?
If “–1” is to the left of A
i.e. A > –1
then N = 0
Z = P + N = = 1
but if a gain is increased, “–1” could be inside, N = –2
Z = P + N = –1
c.c.w. is impossible

32. If connect c.w.:
For A > –1 N = ______
Z = P + N
= ______
For A < –1 N = ______
Z = ______
No contradiction This is correct way.

33. Example: G(s) stable, minimum phase
G(jω) as given:
get conjugate.
Connect ω = 0–
to ω = 0+,

34. If A < –1 < 0 :. N = ______. Z = P + N = ______. stability of c
If A < –1 < 0 : N = ______ Z = P + N = ______ stability of c.l. : ______
If B < –1 < A : N = ______ Z = P + N = ______ closed-loop stability: ______

35. If C < –1 < B :. N = ______. Z = P + N = ______
If C < –1 < B : N = ______ Z = P + N = ______ closed-loop stability: ______
If –1 < C : N = ______ Z = P + N = ______ closed-loop stability: ______

36. function mynyquist(n,d,w)
%mynyquist-- plots a nonlinearly compressed nyquist plot.
% Near the origin (distance less than 2), it preserves the
% original Nyquist plot. Far away from the origin (distance
% more than 2), the distance is compressed by taking the log. %
%Syntax: mynyquist(n,d), or mynyquist(n,d,w)
% where n and d are the numerator and denominator of the transfer
% function, w is the desired frequency points.
%The program does not handle jw-axis poles.
if nargin <3
[m,p,w]=bode(n,d);
d1=log10(min(w));
d2=log10(max(w));
w=logspace(d1, d2, 200);
end;

37. G=polyval(n, j*w)./polyval(d,j*w); %or G=evalfr(tf(n,d),j*w);
GA=abs(G);
GP=angle(G);
GAA=(GA<2).*GA+(GA>=2).*(2+2*log(GA/2)); %if GA<2, do nothing; if>2, do log compression
plot(GAA.*cos(GP),GAA.*sin(GP),'b');
hold;
plot(GAA.*cos(GP),-GAA.*sin(GP),'r'); %this plots the negative freq part
axis equal;
%axis([ ]); %this sets x, y limits for the graph window
grid;
a=0:2*pi/200:2*pi;
plot(cos(a),sin(a),':'); %this plots the unit circle for phase margin

38. nyquist(23. conv([1. 25],[1 0. 7 1]),conv([0
nyquist(23*conv([1 .25],[ ]),conv([0.15 1],conv([ ],[ ])))

39. mynyquist(23. conv([1. 25],[1 0. 7 1]),conv([0
mynyquist(23*conv([1 .25],[ ]),conv([0.15 1],conv([ ],[ ])))

44. For small zeta,
resonance freq
is about wn
BW ranges from
0.5wn to 1.5wn
For good z range,
BW is 0.8 to 1.1 wn
So take BW = wn
z=0.1
0.2
0.3
No resonance
for z <= 0.7
Mr=1dB for z=0.6
Mr=3dB for z=0.5
Mr=7dB for z=0.4

46. 0.2
z=0.1
0.3
0.4
wgc
In the range of good zeta,
wgc is about 0.65 times to 0.8 times wn
w/wn

48. In the range of good zeta,
PM is about 100*z
z=0.1
0.2
0.3
0.4
w/wn