1. Controller design by R.L.
Typical setup:

2. This is the R.L. eq.
With no zi, pi, controller design means to pick good K for R.L.
Those zi, pi means to pick additional poles /zeros to R.L.

3. Proportional control design
Draw R.L. with given
Pick a point on R.L. to be desired c.l. pole: PD
Compute

4. When to use: What is that region: If R.L. of G(s) goes through
the region for desired c.l. poles
What is that region:
From design specs, get desired Mp, ts, tr, etc.
Use formulae for 2nd order system to get desired ωn , ζ, σ, ωd
Identify these in s-plane

5. Example:
When C(s) = 1, things are okay
But we want initial response speed as fast as possible; yet we can only tolerate 10% overshoot.
Sol: From the above, we need that means:

6. This is a cone around –Re axis with ±60° area
We also want tr to be as small as possible.
i.e. : want ωn as large as possible
i.e. : want pd to be as far away from s = 0 as possible
Draw R.L.
Pick pd on R.L. in cone & | pd | max

8. Example:
Want: , as fast as possible
Sol:
Draw R.L. for
Draw cone ±45° about –Re axis
Pick pd as the cross point of the ζ = 0.7 line & R.L.

10. Controller tuning:
First design typically may not work
Identify trends of specs changes as K is increased. e.g.: as KP , pole
Perform closed-loop step response
Adjust K to improve specs e.g. If MP too much, the 2. says reduce KP

11. 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

12. Design steps:
From specs, draw desired region for pole. Pick from region.
Compute
Select
Select:

13. Example:
Want:
Sol:
(pd not on R.L.)
(Need a zero to attract R.L. to pd)

16. Closed-loop step response simulation:
» ng = [1] ;
» dg = [ ] ;
» nc = [Kp, Kd]
» n = conv(nc, ng) ;
» d = dg + n ;
» stepspec(n, d) ;
Tuning: for fixed z:
Q: What’s the effect of tuning z?

17. Drawbacks of PD Not proper : deg of num > deg of den
High frequency gain → ∞:
High gain for noise
Saturated circuits
cannot be implemented physically

18. Lead Controller Approximation to PD Same usefulness as PD
It contributes a lead angle:

19. Lead Design:
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

20. There are many choices of z, p
More neg. (–z) & (–p) → more close to PD & more sensitive to noise, and worse steady-state error
But if –z is > Re(pd), pd may not dominate

21. 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

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

24. 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:

25. Pick –z to the left of pd

27. Suppose we select –z = -1 instead.

28. Which one is better? The new R.L. closed-loop has pole pair
at pd & pd* but
the 3rd pole is
at somewhere
(-1, 0)
Which one is better?

34. Particular choice of z :