1. What damping ratio range do we typically want?___0.4~1____
For 5% Mp, what damping ratio do we need:___0.7___
For 10% Mp, what damping ratio do we need:__0.6____
For 15% Mp, what damping ratio do we need:___0.5___
Overshoot formula in terms of damping ratio is: Mp =___e^(-pi*zeta/rt(1-zeta^2))
For +-1% settling the approximate settling time is: ts = _5/sigma__
For +-5% settling the approximate settling time is: ts = _3/sigma_
A slightly more accurate formula for settling time is ts = _-ln(tol*rt(1-zeta^2))/sigma_
A very crude formula for rise time is tr = __1.8/wn, or 1.5 to 2.2 / wn____

2. 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:

3. Effects of additional pole
One additional R.L. branch shoots out
It increases # asymp. by one
More asymptotes go towards +Re-axis
More likely to be unstable
Poles tend to push R.L. away from them
Don’t introduce poles unless required by other concerns

4. Examples:

5. Effects of additional zero
It sinks one branch of R.L.
It reduces the # asymp. by one
Asymptotes move more towards –Re-axis
More likely to be stable
Zeros attract R.L.
Each zero attracts one branch
If > 1 branches nearby, they go to Re-axis & split, the one branch goes to zero
Never have >= 2 branches go to a zero

6. Examples:

7. If we put that additional zero near (0,0):
The dominant pole pair are more negative
But there is one pole (real) close to s = 0, which will settle very slowly (sluggish settling)

8. Controller design by R.L.
Typical setup:
C(s)
G(s)

9. 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.

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

11. When to use: What is that region: If R.L. of G(s) goes through
the desired region for 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 / plot these in s-plane

12. 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:
C(s)
G(s)

13. 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

15. 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.

17. 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

18. 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

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

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

23. 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?

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

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

26. 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

27. 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

28. 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

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

31. 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:

32. Pick –z to the left of pd