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

Stability from Nyquist plot G(s) The complete Nyquist plot: Plot G(jω) for ω = 0+ 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

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

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

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

Example: 4/(s-1)

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

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.

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 :

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

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

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

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? ________

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? ______

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 ∞?

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: _______

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?

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

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

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.

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.

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

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

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

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.

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

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

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

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: ______

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: ______

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;

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([-3 1 -2 2]); %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

nyquist(23. conv([1. 25],[1 0. 7 1]),conv([0 nyquist(23*conv([1 .25],[1 0.7 1]),conv([0.15 1],conv([1 10 0 0],[1 0.5 0.25])))

mynyquist(23. conv([1. 25],[1 0. 7 1]),conv([0 mynyquist(23*conv([1 .25],[1 0.7 1]),conv([0.15 1],conv([1 10 0 0],[1 0.5 0.25])))

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

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

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