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Stabilization of Nonholonomic Systems 213

To have almost exponential stability of the closed loop system it is necessary
−
to have k > 0 and to set p 2 , p 3 , and p 4 such that 16 σ(A 4 ) ∈ /C , where
 
p 2 p 3 p 4
A 4 =  −k k 0 
0 −k 2k

It must be noticed that the matrix A 4 is a submatrix of the matrix A + b 2 p
considered in Proposition 5.2. This is without lack of generality as only n − 1
eigenvalues of the matrix A + b 2 p can be set with the vector p, whereas one
eigenvalue is always equal to −k. Figure 5.3 and Figure 5.4 show the results of
simulations carried out with the proposed controller.

5.5.2 Discussion
Discontinuous, state feedback, control laws to almost exponentially stabilize
chained systems, have been presented. In contrast to other results, the given
controllawsareextremelysimpleandpossessanintuitiveinterpretationinterms
of linear feedback with state dependent gain scheduling. It is worth stressing
that the design of the stabilizing control law involves mainly linear control tools,
that is, stability of the closed loop system depends on the stability of some linear
systems. A drawback of the proposed approach is the possibility for numerical
problems to appear in real time implementations. In fact, most of the features
of the closed loop system derive from the simplification in the product 1 u 1 .
x 1
If such a simplification takes place only approximately, for example, for the
1 ∗ ∗
presence of measurement noise, the limit lim x 1 →0 u 1 (x ), where x is the
1
1
x 1
available measure on x 1 , may be unbounded.
5.6 ROBUST STABILIZATION —PART I
The results in Section 5.4 can be interpreted as follows. For nominal and ideal
conditions (e.g., exact integration, noise free measurements) and as long as
x 1 (0) = 0, the discontinuous controllers proposed therein are well defined
and yield bounded control action, along the trajectories of the closed loop
system. Moreover, as detailed in Reference 53, the analysis carried out in
References 19–21, 43, and 54–56 is correct and yields an adequate picture
of the ideal properties of this class of discontinuous controllers. However, a
substantial difference is to be expected in a nonideal situation, as the control
law blows up, that is, provides unbounded control action, whenever the discon-
tinuity surface x 1 = 0 is intersected, for example, in the presence of external

16 σ(A) denotes the spectrum of the square matrix A and /C − denotes the open left-half of the
complex plane.

© 2006 by Taylor & Francis Group, LLC

FRANKL: “dk6033_c005” — 2006/3/31 — 16:42 — page 213 — #27