1. Transient Response First order system transient response
Step response specs and relationship to pole location
Second order system transient response
Effects of additional poles and zeros

2. Prototype first order system1 τs
Y(s)
U(s) + -

3. First order system step resp
Normalized time t/t

4. Prototype first order system
No overshoot, tp=inf, Mp = 0
Yss=1, ess=0
Settling time ts = [-ln(tol)]/p
Delay time td = [-ln(0.5)]/p
Rise time tr = [ln(0.9) – ln(0.1)]/p
All times proportional to 1/p= t
Larger p means faster response

5. The error signal: e(t) = 1-y(t)=e-ptus(t)
Normalized time t/t

6. In every τ seconds, the error is reduced by 63.2%

7. We know how this responds to input
General First-order system
We know how this responds to input
Step response starts at y(0+)=k, final value kz/p
1/p = t is still time constant; in every t, y(t) moves 63.2% closer to final value

8. Step response by MATLAB:
>> n = [ b1 b0 ]
>> d = [ 1 p ]
>> step ( n , d )
Other MATLAB commands to explore:
plot, hold, axis, xlabel, ylabel, title, text, gtext, semilogx, semilogy, loglog, subplot

9. Unit ramp response:

11. Note: In step response, the steady-state tracking error = zero.

12. Unit impulse response:

13. Prototype 2nd order system:

16. Can use \omega in stead of w
xi=[ ];
x=['\zeta=0.7'; '\zeta=1 '; '\zeta=2 '; '\zeta=5 '; '\zeta=10 '; '\zeta=0.1'; '\zeta=0.2'; '\zeta=0.3'; '\zeta=0.4'; '\zeta=0.5'; '\zeta=0.6'];
T=0:0.01:16;
figure;
hold;
for k=1:length(xi)
n=[1];
d=[1 2*xi(k) 1];
y=step(n,d,T);
plot(T,y);
if xi(k)>=0.7
text(T(290),y(310),x(k,:));
else
text(T(290),max(y)+0.02,x(k,:));
end
grid;
text(9,1.65,'G(s)=w_n^2/(s^2+2\zetaw_ns+w_n^2)')
title('Unit step responses for various \zeta')
xlabel('w_nt (radians)')
Can use \omega in stead of w

17. annotation
Create annotations including lines, arrows, text arrows, double arrows, text boxes, rectangles, and ellipses
xlabel, ylabel, zlabel
Add a text label to the respective axis
title
Add a title to a graph
colorbar
Add a colorbar to a graph
legend
Add a legend to a graph

18. For example:
“help annotation” explains how to use the annotation command to add text, lines, arrows, and so on at desired positions in the graph
ANNOTATION('textbox',POSITION) creates a textbox annotation at the
position specified in normalized figure units by the vector POSITION
ANNOTATION('line',X,Y) creates a line annotation with endpoints
specified in normalized figure coordinates by the vectors X and Y
ANNOTATION('arrow',X,Y) creates an arrow annotation with endpoints
specified
Example:
ah=annotation('arrow',[.9 .5],[.9,.5],'Color','r');
th=annotation('textarrow',[.3,.6],[.7,.4],'String','ABC');

19. Unit step response:
1) Under damped, 0 < ζ < 1

20. cosq = z =-Re/|root|
= cos-1(Re/|root|)
= tan-1(-Re/Im)
=Im d s
=-Re

22. To find y(t) max:

25. z=0.3:0.1:0.8;
Mp=exp(-pi*z./sqrt(1-z.*z))*100
plot(z,Mp)
grid;
Then preference -> figure…
->powerpoint -> apply to figure
Then copy figure

29. For 5% tolerance
Ts ~= 3/zwn

30. Delay time is not used very much
For delay time, solve y(t)=0.5 and solve for t
For rise time, set y(t) = 0.1 & 0.9, solve for t
This is very difficult
Based on numerical simulation:

31. Useful
Range
Td=( z)/wn

32. Useful
Range
Tr=4.5(z-0.2)/wn
Or about 2/wn

33. Putting all things together:
Settling time:

34. 2) When ζ = 1, ωd = 0

35. The tracking error:

36. 3) Over damped: ζ > 1

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

40. Pole location determines transient

43. All closed-loop poles must be strictly in the left half planes
Transient dies away
Dominant poles: the single real pole or the complex pole pair 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

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

45. 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.

46. Find conditions on σ and ωd.

47. In the complex plane :

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

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

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

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

52. 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.

54. Try to remember these:

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

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

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

61. 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.

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

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