MASSACHUSETTS INSTITUTE OF TECHNOLOGY
DEPARTMENT OF MECHANICAL ENGINEERING
2.151 Advanced System Dynamics and Control
Review of First and SecondOrder System Response
1
1
FirstOrder Linear System Transient Response
The dynamics of many systems of interest to engineers may be represented by a simple model
containing one independent energy storage element. For example, the braking of an automobile,
the discharge of an electronic camera flash, the flow of fluid from a tank, and the cooling of a cup
of coffee may all be approximated by a firstorder differential equation, which may be written in a
standard form as
τ
dy
dt
+
y
(
t
) =
f
(
t
)
(1)
where the system is defined by the single parameter
τ
, the system time constant, and
f
(
t
) is a
forcing function. For example, if the system is described by a linear firstorder state equation and
an associated output equation:
˙
x
=
ax
+
bu
(2)
y
=
cx
+
du.
(3)
and the selected output variable is the statevariable, that is
y
(
t
) =
x
(
t
), Eq. (3) may be rearranged
dy
dt

ay
=
bu,
(4)
and rewritten in the standard form (in terms of a time constant
τ
=

1
/a
), by dividing through
by

a
:

1
a
dy
dt
+
y
(
t
) =

b
a
u
(
t
)
(5)
where the forcing function is
f
(
t
) = (

b/a
)
u
(
t
).
If the chosen output variable
y
(
t
) is not the state variable, Eqs. (2) and (3) may be combined
to form an input/output differential equation in the variable
y
(
t
):
dy
dt

ay
=
d
du
dt
+ (
bc

ad
)
u.
(6)
To obtain the standard form we again divide through by

a
:

1
a
dy
dt
+
y
(
t
) =

d
a
du
dt
+
ad

bc
a
u
(
t
)
.
(7)
Comparison with Eq. (1) shows the time constant is again
τ
=

1
/a
, but in this case the forcing
function is a combination of the input and its derivative
f
(
t
) =

d
a
du
dt
+
ad

bc
a
u
(
t
)
.
(8)
In both Eqs. (5) and (7) the lefthand side is a function of the time constant
τ
=

1
/a
only, and
is independent of the particular output variable chosen.
1
D. Rowell 10/22/04
1
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Example 1
A sample of fluid, modeled as a thermal capacitance
C
t
, is contained within an insulating
vacuum flask. Find a pair of differential equations that describe 1) the temperature of
the fluid, and 2) the heat flow through the walls of the flask as a function of the external
ambient temperature. Identify the system time constant.
T
C
T
a m
b
q
w
a l l s
R
t
f l u i d
C
t
C
t
R
t
T
C
T
a m
b
T
r e f
h e a t
f l o w
Figure 1: A firstorder thermal model representing the heat exchange between a laboratory vacuum
flask and the environment.
Solution:
The walls of the flask may be modeled as a single lumped thermal resistance
R
t
and a linear graph for the system drawn as in Fig. 1. The environment is assumed
to act as a temperature source
T
amb
(
t
). The state equation for the system, in terms of
the temperature
T
C
of the fluid, is
dT
C
dt
=

1
R
t
C
t
T
C
+
1
R
t
C
t
T
amb
(
t
)
.
(i)
The output equation for the flow
q
R
through the walls of the flask is
q
R
=
1
R
t
T
R
=

1
R
t
T
C
+
1
R
t
T
amb
(
t
)
.
(ii)
The differential equation describing the dynamics of the fluid temperature
T
C
is found
directly by rearranging Eq. (i):
R
t
C
t
dT
C
dt
+
T
C
=
T
amb
(
t
)
.
(iii)
from which the system time constant
τ
may be seen to be
τ
=
R
t
C
t
.
This is the end of the preview.
Sign up
to
access the rest of the document.
 Spring '11
 Klaous
 Mechanical Engineering, dt dt, Eqs.

Click to edit the document details