1. Time domain response specifications
Dynamic Response
Unit step signal:
Step response: y(s)=H(s)/s, y(t)=L-1{H(s)/s}
Time domain response specifications
Defined based on unit step response
Defined for closed-loop system

3. 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

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

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

6. 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

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

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

9. 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

10. Unit ramp response:

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

13. Unit impulse response:

14. Prototype 2nd order system:

17. 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

18. 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

19. 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');

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

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

23. To find y(t) max:

26. 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

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

31. 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:

32. Useful
Range
td=( z)/wn

33. Useful
Range
tr=4.5(z-0.2)/wn
Or about 2/wn

34. Putting all things together:
Settling time: