Page 235 - Autonomous Mobile Robots

220 Autonomous Mobile Robots

n
c
∗
a i
W(x) = i=2 i |x i | , with c i > 0 a i ∈{2, 3, ...}. Then there exists T > 0
∗
such that for all T ∈ (0, T ) the controller u T := (u 1T , u 2T ) where 20

u 1T =− g 1 x 1 − ρ(W)(cos((k + 1)T) − sin((k + 1)T))
2

+ ρ sin((k + 1)T)
2
α
W| (2ρ(W) + 2(g 1 x 1 + ρ(W) cos((k + 1)T))
u 2T =− g 2 sign(L f 2 W)|L f 2
× cos((k + 1)T) − g 1 x 1 sin((k + 1)T)) (5.37)
with g 1 > 0, g 2 > 0, a > 0 and a sufficiently small > 0, is a SP-AS controller
for the system (5.36) and the function
2
2
V T (k, x) = (g 1 x 1 + ρ(W) cos(kT)) + ρ(W) − g 1 x 1 ρ(W) sin(kT)
(5.38)
is a (strict) SP-AS Lyapunov function for the closed loop system (5.36), (5.37).

The control law is similar to the one proposed in Reference 50 and the
Lyapunov function is a modification of the one proposed in Reference 10.
The proposed result provides a discrete-time counterpart and to some extent a
generalization of Theorem 2 of Reference 10. Theorem 5.8 states that V T is a
strict SP-AS Lyapunov function for the closed loop system. It is well known that
the existence of a strict SP-AS negative Lyapunov function allows to address
the stabilization problem in the presence of disturbances.

∗
∗
Proposition 5.5 [42] There exist T > 0 such that for all T ∈ (0, T ) the
controller (5.37) is a SP-ISS controller for system (5.36) and the function (5.38)
is a SP-ISS Lyapunov function for the closed loop system (5.36), (5.37.)

5.8.2 An Example: A Car-Like Vehicle Revisited
In this section we apply the proposed result to the model of a car-like vehicle
introduced in Section 5.5.1. Consider the model (5.30) and the coordinate
transformation [50,52]

x 1 = x
3
x 2 = sec (θ) tan(φ)
(5.39)
3
x 3 = x sec (θ) tan(φ) − l tan(θ)
1 2
3
x 4 = ly + x sec (θ) tan(φ) − lx tan(θ)
2
20 f 2 denotes the vector [0, 1, x 1 , ... , 1 x n−2 ∂W f
(n−2)! 1 ] and L f 2 W = ∂x 2 .

© 2006 by Taylor & Francis Group, LLC

FRANKL: “dk6033_c005” — 2006/3/31 — 16:42 — page 220 — #34