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Stabilization of Nonholonomic Systems 221

yielding

˙ x 1 = u 1
˙ x 2 = u 2
(5.40)
˙ x 3 = x 1 u 2
1 2
˙ x 4 = x u 2
2 1
d 3
where u 1 = v 1 cos θ and u 2 = (sec (θ) tan(φ)). Applying Theorem 5.8 we
dt
construct the controller

u 1T =−3x 1 + ρ(W)(cos((k + 1)T) − sin((k + 1)T))
2
u 2T = u 2 (2ρ(W) − 3 x 1 sin((k + 1)T) + 2(3x 1 + ρ(W) (5.41)
× cos((k + 1)T)) cos((k + 1)T))
√
4 6 3
5
with k = 1, ρ(W) = W(x), and u 2 =− sign(L f 2 W(x)) |L f 2 W(x)|,
10 100
which is a SP-AS controller for the Euler model (5.40). Figure 5.5 shows
simulation results when the controller (5.41) is applied to control the plant
(5.40). We have used x o = (0, 0, 0, 1) , T = 0.2, and = 0.35.

5.8.3 Discussion
The problem of robust stabilization of nonholonomic systems in power form
has been addressed and solved in the framework of nonlinear sampled-data
control theory. It has been shown that, by modifying the periodic controller in
Reference 10, SP-AS and SP-ISS can be achieved. The main drawback of the
proposed controllers is the slow convergence rate, which is, however, intrinsic
to smooth time-varying controllers [12].

5.9 CONCLUSIONS
The problem of (discontinuous) stabilization and robust stabilization for non-
holonomic systems has been discussed from various perspectives. It has been
shown that, in ideal situations, a class of discontinuous controllers allow to
obtain fast convergence and efficient trajectories. This approach is, however,
inadequate in the presence of disturbances and measurement noise, hence it is
necessary to modify the proposed control by introducing a second controller,
a hybrid variable, and a switching strategy, which together guarantee robust
stability. Both these controllers have been designed in continuous time. It is
therefore difficult to quantify the loss of performance arising from a sampled-
data implementation. As a result, we have discussed the robust stabilization

© 2006 by Taylor & Francis Group, LLC

FRANKL: “dk6033_c005” — 2006/3/31 — 16:42 — page 221 — #35