Module 3
First – Order System
(1 hours)
First order systems
Time response of first order systems
Proportional feedback control of first
order systems
First order systems – systems
described by a single first order
differential equation
First order systems contain one
energy storage element:
lump mass
spring
capacitor
inductor
3.1 The Model of First-Order system
3.2 Impulse Response
1)0(?
C
2
1)0(
C
2
1

)(?
3.3 Step Response
0.632
3.4 Ramp Response
The steady state error in the
ramp response of first-order system is:ss
e
)()()( tctrte
)(
1 t
ett


t
e
1
t
sse?
3.5 Acceleration
Response
3
2 1)(;
2
1)(
ssRttr

1
11
1
1)( 22
233 sssssssC
0)1(
2
1)( 122 tetttc t
)()1()()()( 2 tettctrte
t

)(tr
)(tc
)(te
t
The conclusion for first – order
system:
A first – order system can follow a
impulse signal,a unit signal,and a
ramp signal (steady state error exists
in the ramp response),but can not
follow an acceleration signal !
3.6 The Relationship between input signals,
or output signals of a linear control system
ona c c e l e r a t ir a m ps t e pi m p u l s e trdt
dtr
dt
dtr
dt
dtr )()()()(
3
3
2
2

ona c c e l e r a t ir a m ps t e pi m p u l s e tcdt
dtc
dt
dtc
dt
dtc )()()()(
3
3
2
2

And we can find
It suits every linear time-invariant system.
3.7 Sine Signal Response
_ _
_
_
_ _ _
_
3.8 First-Order Feedback System
Closed loop,K=4
3.9 Complex-Plane Representation:
Poles and Zeros in the Complex Plane
Pole
Zero
Im
Re
0
3.10 Dominant Pole(s)
The Conclusion on Dominant Pole(s)
The closed-loop poles close to the
imaginary axis dominate those farther
away,and in many cases the response
may be approximated by the response
of the closed-loop poles closest to the
imaginary axis,(P48)
Instructional objectives
At the end of this lecture students should
be able to
write down the differential equations
governing simple first order mechanical
and electrical systems;
obtain the transfer function;
find the time response to impulse,step,
ramp and harmonic input.