1. With respect to reference input:
Name:________________
With respect to reference input:
A critical condition for using these is that:
closed-loop system must __________

2. Prototype 2nd order system:

3. Unit step response:

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

5. 2) When ζ = 1, ωd = 0

6. The tracking error:

7. 3) Over damped: ζ > 1

9. General Transient Response
Recall 1st order system step response:
2nd order:

11. Pole location determines transient

15. 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

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

17. 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.

18. Find conditions on σ and ωd.

19. In the complex plane :

20. Constant σ : vertical lines
σ > # is half plane

21. Constant ωd : horizontal line
ωd < · · · is a band
ωd > · · · is the plane excluding band

22. Constant ωn : circles
ωn < · · · inside of a circle
ωn > · · · outside of a circle

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

24. 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.

26. Try to remember these:

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

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

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

33. 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.

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

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