Course roadmap
ME451: Control Systems
Modeling
Analysis
Laplace transform
Lecture 14
Time response of 1st-order systems
Transfer function
Models for systems
• electrical
• mechanical
• electromechanical
Block diagrams
Linearization
Dr. Jongeun Choi
Department of Mechanical Engineering
Michigan State University
Time response
• Transient
• Steady state
Frequency response
• Bode plot
Stability
• RouthRouth-Hurwitz
• Nyquist
Design
Design specs
Root locus
Frequency domain
PID & LeadLead-lag
Design examples
(Matlab simulations &) laboratories
1
Performance measures (review)
ƒ Transient response
2
First-order system
ƒ A standard form of the first-order system:
(Today’
(Today’s lecture)
ƒ Peak value
ƒ Peak time
ƒ Percent overshoot
ƒ Delay time
ƒ Rise time
Next, we will connect
these measures
with ss-domain.
ƒ DC motor example
ƒ Settling time
ƒ Steady state response
ƒ Steady state error
(Done)
3
4
DC motor example (cont’d)
Step response for 1st-order system
ƒ If La<<Ra, we can obtain a 1st-order system
ƒ Input a unit step function to a first-order system.
Then, what is the output?
u(t)
u(t)
y(t)
y(t)
1
0
0
ƒ TF from motor input voltage to
ƒ motor speed is 1st1st-oder
ƒ motor position is 2nd2nd-order
(Partial fraction expansion)
5
How to eliminate steady-state error
ƒ Make a feedback system with a controller having
an integrator (copy of Laplace transform of a unit
step function):
Meaning of K and T
ƒ K : Gain
ƒ Final (steady(steady-state) value
K=1,T=1
1
0.8
Controller
ƒ T : Time constant
1
ƒ Time when response
0
rises 63% of final value
ƒ Indication of speed of
response (convergence)
ƒ Response is faster as T
becomes smaller.
One has to select controller parameters
to stabilize the feedback system.
Suppose K=T=1, and obtain such parameters!
7
Amplitude
u(t)
u(t)
6
0.6
0.4
0.2
0
0
1
2
3
Time
4
5
6
8
DC gain for a general system
Settling time of 1st-order systems
ƒ DC gain : Final value of a unit step response
ƒ For firstfirst-order systems, DC gain is K.
ƒ Relation between time and exponential decay
ƒ For a general stable system G,
G, DC gain is G(0).
Final value theorem
ƒ Examples
5% settling time is about 3T!
2% settling time is about 4T!
9
Step response for some K & T
K=1,T=1
Amplitude
Amplitude
5
Time
K=2,T=1
5
Time
K=2,T=2
10
5
Time
10
ƒ Obtain step response
2
Amplitude
Amplitude
Unknown
1
0
0
10
2
1
0
0
ƒ Suppose that we have a “black-box” system
2
1
0
0
System identification
K=1,T=2
2
10
5
Time
10
1
0
0
ƒ Can you obtain a transfer function? How?
11
12
Ramp response for 1st-order system
Ramp response for 1st-order system
K =1 ,T =1
ƒ Input a unit ramp function to a 1st-order system.
Then, what is the output?
y(t)
y(t)
0
0
K=1,T=1
4
Amplitude
Amplitude
u(t)=t
u(t)=t
5
y(t)
y(t)
u(t)=t
u(t)=t
3
slope
2
1
0
0
1
2
T im e
Time
3
4
5
ƒ Steady state response
ƒ We may want to modify the system s.t.
(Partial fraction expansion)
13
How to eliminate steady-state error
ƒ Make a feedback system with a controller having
a double integrator (copy of Laplace transform of
ramp function):
u(t)=t
u(t)=t
14
Summary and exercises
ƒ Time response for 1st-order systems
ƒ Step and ramp responses
ƒ Time constant and DC gain
ƒ System identification
Controller
ƒ Next, time response for 2nd-order systems
ƒ Exercises
0
ƒ Review examples in this lecture.
One has to select controller parameters
to stabilize the feedback system.
Suppose K=T=1, and obtain such parameters!
15
16