1. Prototype 2nd order system:
target

2. Type 1:
Settling time:
= (3 or 4 or 5)/s
For 5%, 2%, 1% tol

3. Pole location determines transient

4. All closed-loop poles must be strictly in the left half planes
Transient dies away
Dominant poles: those which contribute the most to the transient
Typically have dominant pole pair
(complex conjugate)
Closest to jω-axis (i.e. the least negative)
Slowest to die away

5. Typical design specifications
Steady-state:
ess to step ≤ # %
ts ≤ · · ·
Speed (responsiveness)
tr ≤ · · ·
td ≤ · · ·
Relative stability
Mp ≤ · · · %

6. These specs translate into requirements
on ζ, ωn or on closed-loop pole location :
Find ranges for ζ and ωn so that all 3 are satisfied.

7. Find conditions on σ and ωd.

8. In the complex plane : Any pole on the ray have the same z p=-s + jwd
Mp < … or
z > … corresponds to a conic region about the neg real axis j -s

9. Constant σ : vertical lines σ > # is half plane
Any poles on the same vertical line have the same s, and the same settling time ts
s=1 or
ts=5
s>2 or
ts=2.5
ts < … corresponds to a half plane to the left of a vertical line

10. Constant ωd : horizontal line ωd < · · · is a band
ωd > · · · is the plane excluding band
Any poles on the same horizontal line have the same wd, and the same oscillation frequency
A centered horizontal band corresponds to oscillation frequency < …
The plane excluding a centered horizontal band corresponds to oscillation frequency > …

11. ωn < · · · inside of a circle ωn > · · · outside of a circle
Constant ωn : circles
ωn < · · · inside of a circle
ωn > · · · outside of a circle
Any poles on the same circle have the same wn, and similar rise time/delay time
Inside circle corresponds to tr> …
Outside a circle corresponds to tr < …

12. Constant ζ : φ = cos-1ζ constant
Constant ζ = ray from the origin
ζ > · · · is the cone
ζ < · · · is the other part

13. If more than one requirement, get the common (overlapped) area
e.g. ζ > 0.5, σ > 2, ωn > 3 gives
Sometimes
meeting two
will also meet
the third, but
not always.

15. Try to remember these:

16. When given unit step input, the output looks like:
Example: + -
When given unit step input, the output looks like:
Q: estimate k and τ.

19. When given unit step input, the output looks like:
Example: + -
When given unit step input, the output looks like:
Q: estimate k and τ.
Yss = ____; is it prototype?
Mp = ____; tp = ____; 5% ts = ____

21. Solve for J and t

22. Effects of additional zeros
Suppose we originally have:
i.e. step response
Now introduce a zero at s = -z
The new step response:

24. Effects:
Increased speed,
Larger overshoot,
Might increase ts

25. When z < 0, the zero s = -z is > 0,
is in the right half plane.
Such a zero is called a nonminimum phase zero.
A system with nonminimum phase zeros is called a nonminimum phase system.
Nonminimum phase zero should be
avoided in design.
i.e. Do not introduce such a zero in your controller.

26. Effects of additional pole
Suppose, instead of a zero, we introduce
a pole at s = -p, i.e.

27. L.P.F. has smoothing effect, or
averaging effect
Effects:
Slower,
Reduced overshoot,
May increase or decrease ts