1. The root locus technique
Obtain closed-loop TF and char eq d(s) = 0
Re-arrange to get
Mark zeros with “o” and poles with “x”
High light segments of x-axis and put arrows
Decide #asymptotes, their angles, and x-axis meeting place:
Determine jw-axis crossing using Routh table
Compute breakaway:
Departure/arrival angle:

2. Effects of additional pole
Examples:

3. Effects of additional zero
Examples:

4. Controller design by R.L.
Typical setup:
C(s)
G(s)
Controller Design Goal:
Select poles and zero of C(s) so that R.L. pass through desired region
Select K corresponding to a good choice of dominant pole pair

5. Types of classical controllers
Proportional control
Needed to make a specific point on RL to be closed-loop system dominant pole
Proportional plus derivative control (PD control)
Needed to “bend” R.L. into the desired region
Lead control
Similar to PD, but without the high frequency noise problem; max angle contribution limited to < 75 deg
Proportional plus Integral Control (PI control)
Needed to “eliminate” a non-zero steady state tracking error
Lag control
Needed to reduce a non-zero steady state error, no type increase
PID control
When both PD and PI are needed, PID = PD * PI
Lead-Lag control
When both lead and lag are needed, lead-lag = lead * lag

6. Proportional control design
Draw R.L. for given plant
Draw desired region for poles from specs
Pick a point on R.L. and in desired region
Use ginput to get point and convert to complex #
Compute
Obtain closed-loop TF
Obtain step response and compute specs
Decide if modification is needed
nump=…; denp= …; sysp=tf(nump, denp); rlocus(sysp);
[x,y]=ginput(1); pd=x+j*y;
Gpd=evalfr(sysp,pd);
K=1/Gpd;
sysc = K;
syscl = feedback(sysc*sysp,1);
use your program from several weeks ago to do all these

7. Example code for Matlab
rlocus([9], [ ]);
grid;
yl=ylim;
omega_n=5; %this may be from tr or td requirement
rectangle('Position',[-omega_n,-omega_n,2*omega_n,2*omega_n],'Curvature',[1,1]);
sigma = 4; %this may be from ts requirement
line([-sigma -sigma],yl);
zeta=0.7; %this may be from Mp requiremet
line([0 yl(1)*zeta/sqrt(1-zeta^2)],[0 yl(1)]);
line([0 -yl(2)*zeta/sqrt(1-zeta^2)],[0 yl(2)]);
xl=xlim;
xl(2)=0;
xlim(xl);
Draws circle with r=5
Draws line at -4
Draws 2 rays for zeta=0.7

9. xl=xlim;
xl(2)=0;
xlim(xl);

10. axis equal;

11. Can use mouse to zoom in
Or use: xlim(-2*wn 0]);
ylim([-wn wn]);

12. PD controller design
This is introducing an additional zero to the R.L. for G(s)
Use this if the dominant pole pair branches of G(s) do not pass through the desired region
Place additional zero to “bend” the RL into the desired region

13. PD controller design Design steps:
From specs, draw desired region for pole. Pick from region, not on RL
Compute
Select
Select:
[x,y]=ginput(1); pd=x+j*y;
Gpd=evalfr(sysp,pd)
phi=pi - angle(Gpd)
z=abs(real(pd))+abs(imag(pd)/tan(phi));
Kd=1/abs(pd+z)/abs(Gpd);
Kp=z*Kd;

14. Drawbacks of PD Not proper : deg of num > deg of den
High frequency gain → ∞:
High gain for noise
Saturates circuits
Cannot be implemented physically

15. Lead Controller
Transfer function of Lead
It contributes a lead angle:

16. From specs draw region for desired c.l. poles Select pd from region
Approximation to PD
Same usefulness as PD
Lead Control:
Draw R.L. for G
From specs draw region for desired c.l. poles
Select pd from region
Let Pick –z somewhere below pd on –Re axis Let Select
[x,y]=ginput(1); pd=x+j*y;
Gpd=evalfr(sysp,pd)
phi=pi - angle(Gpd)
[x,y]=ginput(1); z=abs(x);
phi1=angle(pd+z); phi2=phi1-phi;
p=abs(real(pd))+abs(imag(pd)/tan(phi2));
K=abs((pd+p)/(pd+z)/Gpd);
sysc=tf(K*[1 z],[1 p]);
Hold on;
rlocus(sysc*sysp);

17. Want: Don’t increase MP but double the resp. speed
Example: Lead Design
MP is fine,
but too slow.
Want: Don’t increase MP
but double the resp. speed
Sol: Original system: C(s) = 1
Since MP is a function of ζ, speed is proportional to ωn
C(s)

19. Draw R.L. & desired region
Pick pd right at the
vertex:
(Could pick pd a little
inside the region
to allow “flex”)

20. Clearly, R. L. does not pass through pd, nor the desired region
Clearly, R.L. does not pass through pd, nor the desired region. need PD or Lead to “bend” the R.L. into region. (Note our choice may be the easiest to achieve)
Let’s do Lead:

21. Pick –z to the left of pd

22. Speed is doubled, but over shoot is too much.

23. Change controller from to
To reduce the gain a bit, and make it a little closer to PD

24. Particular choice of z : angle bisector

28. Alternative Lead Control
Draw R.L. for G
From specs draw region for desired c.l. poles
Select pd from region
Let
Select
phipd=angle(pd);
phi1=(phipd+phi)/2; phi2=phi1-phi;

29. Lag controller design It has “destabilizing” effect (lag)
Not used for improving MP, tr, …
Use it to improve ess
Use it when R.L. of G(s) go through the desired region but ess is too large.

30. Lag Design steps Draw R.L. for G(s).
From specs, draw desired pole region
Select pd on R.L. & in region
Get
With that K, compute error constant (Kpa, Kva, Kaa) from KG(s)
From specs, compute Kpd, Kvd, Kad
sysol = sysc*sysp;
[nol, dol]=tfdata(sysol,'v');
dn0=dol(dol~=0); Kact=nol(end)/dn0(end);
Kdes = 1/ess;

31. If K#a > K#d , done else: pick
Re-compute
Closed-loop simulation & tuning as necessary
z=-real(pd)/…;
p=z*Kact/Kdes/(1+…);
0.05 or 0.1

32. Example:
Want:
Solution:
C(s)

34. Draw R.L.
Pick pd on R.L. & in Region pick pd = – j0.5
Since there is one in G(s)

37. Lead-lag design example
Too much overshoot, too slow & ess to ramp is too large.
R(s)
E(s)
C(s)
U(s)
Gp(s)
Y(s)

40. Draw R.L. for G(s) & the desired region

41. Clearly R.L. does not pass through desired region. need PD or lead.
Let’s do lead. Pick pd in region

42. Now choose zlead & plead.
Could use bisection.
Let’s pick zlead to cancel plant pole s + 0.5

43. Use our formula to get plead
Now compute K :
Now evaluate error constant Kva

45. Could re-compute K, but let’s skip:
do step response.