1. MESB374 System Modeling and Analysis Forced Response
ME375 Handouts - Spring 2002
MESB374 System Modeling and Analysis Forced Response

2. Forced Responses of LTI Systems
ME375 Handouts - Spring 2002
Forced Responses of LTI Systems
Forced Responses of LTI Systems
Superposition Principle
Forced Responses to Specific Inputs
Forced Response of 1st Order Systems
Transfer Function and Poles/Zeros
Forced Response of Stable 1st Order Systems
Forced Response of 2nd Order Systems
Forced Response of Stable 2nd Order Systems

3. Forced Responses of LTI Systems
Superposition Principle
Input
Output
Linear System
u1 (t)
u2 (t)
u(t)=k1 u1 (t) + k2 u2 (t)
y1 (t)
y2 (t)
y(t)=k1 y1 (t) + k2 y2 (t)
The forced response of a linear system to a complicated input can be obtained by studying how the system responds to simple inputs, such as unit impulse input, unit step input, and sinusoidal inputs with different input frequencies.

4. Typical Forced Responses
Unit Impulse Response
Forced response to unit impulse input
Unit Step Response
Forced response to unit step input (u (t) = 1)
Sinusoidal Response
Forced response to sinusoidal inputs at different input frequencies
The steady state response of sinusoidal response is call the Frequency Response.
Time t
u(t)
If system is stable, SS is zero.
Time t
u(t) 1
u(t) y
Time t

5. Forced Response of 1st Order Systems
Standard Form of Stable 1st Order System
where
t : Time Constant
K : Static (Steady State, DC) Gain
TF and Poles/Zeros
Unit Step Response
( u=1 and zero ICs )
Stable system
y(t)
yss(t) = K K
yT(t) = - K e -t/t
Time t

6. Normalized Unit Step Response
Normalized Unit Step Response (u = 1 & zero ICs) 1t 2t 3t 4t 5t 6t
Time [ t ]
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9 1
Normalized Response
Time t t 2 t 3 t 4 t 5 t
( 1 - e - t/ t )
0.6321
0.8647
0.9502
0.9817
0.9933

7. Unit Step Response of Stable 1st Order System
Smallest
Effect of Time Constant t :
Normalized:
Initial Slope:
Q: What is your conclusion ?
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9 1
Time [sec]
Normalized Response
increases
Largest 2 4 6 8 10
The smaller
is,
the steeper the initial slope is, and
the faster the response approaches the steady state.

8. Forced Responses of Stable 1st Order System
Q: How would you calculate the forced response of a 1st order system to a unit pulse (not unit impulse)?
Q: How would you calculate the unit impulse response of a 1st order system?
Q: How would you calculate the sinusoidal response of a 1st order system?
Time t
u(t) 1
Q (Hint: superposition principle ?!) -1
Time t
u(t) 1

9. Standard Form of 2nd Order Systems
I/O Model
TF and Pole/Zeros
Stability Condition
Standard Form of Stable 2nd Order Systems without Zeros
where
wn : Natural Frequency [rad/s]
z : Damping Ratio
K : Static (Steady State, DC) Gain

10. Poles of Stable 2nd Order Systems
Stable 2nd Order Systems without Zeros
Pole Locations
Real
Img. wn
-wn
Over-damped ( )
Critically damped ( )
Under damped ( )
Two distinct real poles
Two identical real poles at
Two complex poles at

11. Under-damped 2nd Order System
Unit Step Response ( u=1 and zero ICs )

12. Unit Step Response of 2nd Order SystemsOS
yMAX tP 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Time [sec]
0.2K
0.4K
0.6K
0.8K K
1.2K
1.4K
1.6K
Unit Step Response Td tS

13. Unit Step Response of 2nd Order System
Peak Time (tP)
Time when output y(t) reaches its maximum value yMAX.
Percent Overshoot (%OS)
At peak time tP the maximum output
The overshoot (OS) is:
The percent overshoot is:

14. Unit Step Response of 2nd Order System
Settling Time (ts)
Time required for the response to be within a specific percent of the final (steady-state) value.
Some typical specifications for settling time are: 5%, 2% and 1%.
Look at the envelope of the response:
Q: Which parameters of a 2nd order system affect the peak time?
Damping ration and natural frequency
Q: Which parameters of a 2nd order system affect the % OS?
Damping ratio
x% band settling time:
Q: Which parameters of a 2nd order system affect the settling time?
Damping ratio and natural frequency % 1% 2% 5% t
Q: Can you obtain the formula for a 3% settling time? S

15. In Class Exercise Mass-Spring-Damper System
I/O Model:
Q: What is the static gain of the system ?
Q: How would the physical parameters (M, B, K) affect the step response of the system ?
(This is equivalent to asking you for the relationship between the physical parameters and the damping ratio, natural frequency and the static gain.) x Ks M B
f(t)