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P
=
0
.
0021 0
.
0021
0
.
0021 8
.
4511
leading to a Lyapunov function
V
(
v
−
v
0
,
s
)=0
.
0021
∗
s
2
+0
.
0042
∗
(
v
−
v
0
)
∗
s
+8
.
4511
∗
(
v
−
v
0
)
2
.
The contour lines of
V
are visualized in Fig. 16. These contour lines are only
passed “outside-in” by all trajectories, resulting in convergence to the center,
which represents
v
v
0
=0and
s
=0. Therefore, the velocity
v
will converge to
the desired velocity
v
0
and the integral value
s
of the PI-controller will converge
to 0.
The existence of this Lyapunov function is sucient to prove global asymp-
totic stability for the drive train system. Using the YALMIP [33] frontend under
Matlab, this computation took around 0.65 seconds. The problem consists of 17
scalar constraints and 6 three-by-three matrix inequality constraints, on a total
of 23 scalar variables. Therefore, the convex search space visualized in Fig. 15(b)
is 23-dimensional and bounded by 17 + 6 = 23 constraint surfaces.
−
100
80
60
40
20
0
−20
−40
−60
−80
−100
−5000
−4000
−3000
−2000
−1000
0
1000
2000
3000
4000
5000
s
Fig. 16. Lyapunov function contour lines
6.4
Stability of the Discretized Drive Train
For a time-discretized version of the drive train, stability can be shown in a very
similar manner. The discrete-time system is obtained by choosing an appropriate
sampling rate. Too slow sampling might destroy stability, while too fast sampling
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