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ME 131 Vehicle Dynamics Homework 4 Solutions
1
Cruise Control Design and Simulation
1.1
Upper Level Controller Design
We would like to simulate the engine dynamics of our vehicle for testing and analysis. For designing purposes
it has been suggested that we use a ﬁrstorder ﬁlter model for our vehicle:
τ
d
dt
¨
x
+ ¨
x
= ¨
x
des
(1)
Let’s choose a PI control law of the form: ¨
x
des
=

k
p
(
v

v
des
)

k
i
R
(
v

v
des
)
dt
(a) Find the ordinary diﬀerential equation relating
v
des
to
v
. Show that if
v
des
=
constant
then
v
ss
=
v
des
.
By assuming the solution to be
v
(
t
) =
e
λt
and setting
v
des
= ˙
v
des
= 0, ﬁnd the characteristic equation.
The ode may be found by plugging the control law into the ﬁrstorder ﬁlter model and diﬀerentiating
to get rid of the integral.
τ
d
3
v
dt
3
+
d
2
v
dt
2
+
k
p
dv
dt
+
k
i
v
=
k
p
˙
v
des
+
k
i
v
des
(2)
Since
v
ss
=
constant
all derivatives are zero and
v
ss
=
v
des
. Plugging
v
(
t
) =
e
λt
into the ode will yield
e
λt
(
τλ
3
+
λ
2
+
k
p
λ
+
k
i
) = 0. The characteristic equation is then Δ(
λ
) =
τλ
3
+
λ
2
+
k
p
λ
+
k
i
.
It is often easier in control system design to work in the Laplace domain. In this domain the plant model
and the PI controller may be represented as:
P
(
s
) =
v
¨
x
des
=
1
s
(
τs
+ 1)
(3)
C
(
s
) =
k
p
+
k
i
s
(4)
The closedloop system representing the transfer function from
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 Spring '09

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