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Stabilization of Nonholonomic Systems 197

To discuss generalized sampled-data control laws, consider the perturbed
model

˙ x = g(x)u(x, t) + d(x, t) (5.8)
where d ∈ R m is a disturbance. Assume the system is between a sampler
9
and zero-order hold. Then it is possible to define a parameterized family of
discrete-time models of (5.8) described by
x(k + 1) = F T (k, x(k), u(k), d(k)) (5.9)

where the free parameter T > 0 is the sampling period, and x(k) =
x(kT), u(k) = u(kT), and d(k) = d(kT). If we use the approximate model
(5.9) to design a discrete-time controller we obtain a discrete-time controller
u T (x(k), k) that is also parameterized by T. Consider now the resulting closed
loop system, namely

x(k + 1) = F T (k, x(k), u T (x(k), k), d(k)) (5.10)

Definition 5.4 [39] The family of systems (5.10) is semiglobally practically
input-to-state stable (SP-ISS) if there exist β ∈ KL and γ ∈ K, such that for
any strictly positive real numbers x , d , δ there exists T > 0 such that the
∗
solutions of the closed loop system satisfy
|x(k, k o , x o , d)|≤ β(|x o |, (k − k o )) + γ( d ∞ ) + δ (5.11)

∗
for all k ≥ k o , T ∈ (0, T ), |x o |≤ x , and d ∞ ≤ d . Moreover, if d = 0,
andtheaboveholds, thesystemissemigloballypracticallyasymptoticallystable
(SP-AS) and u T is called a SP-AS controller.

We stress that, in practice, when designing a discrete-time controller for a
continuous-time plant the final goal is to achieve stabilization for the sampled-
data system. It is therefore important to note that, as discussed in References 40
and 42, SP-ISS (SP-AS) of the discrete time closed-loop systems implies, under
the considered assumptions, SP-ISS (SP-AS) of the sampled-data controlled
systems.

5.3 DISCONTINUOUS STABILIZATION

Discontinuous, time invariant, control laws have been dealt with in several
research papers, see for example, References 17, 19, and 43; however, our

9
The approximate model, to be useful for control design, has to satisfy the so-called one-step
consistency property [40,41].

© 2006 by Taylor & Francis Group, LLC

FRANKL: “dk6033_c005” — 2006/3/31 — 16:42 — page 197 — #11