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controllers using Lyapunov function (Andújar et al. 2006 ; Andújar and Bravo 2005 ;
Andújar et al. 2004 ; Tanaka 1995 ; Wang 1997 ). However, probably the main problem
to be solved is to find a suitable Lyapunov function. Moreover, since for nonlinear
systems, the compliance of the theorem is only sufficient, its non-compliance for a
candidate function throws no knowledge on the stability or otherwise of the system
under analysis, the only conclusion one can draw is that the elected function may
not be a suitable candidate.
The generalized Krasovskii theorem (Krasovskii 1959 ; Slotine and Li 1991 )
provides a Lyapunov function for the system, and imposes sufficient conditions of
asymptotic stability more flexible that in its simplified version. This theorem can be
applied as a tool for synthesis of fuzzy controllers stable by design, i.e. where the
stability is ensured during the design process of the controller (Andújar and Barragán
2004 , 2005a , b ). Although the idea may seem very intuitive, its implementation is
complex for several reasons: (1) it should to obtain the necessary terms and prop-
erly parameterized for verification; (2) there should be an adjustment algorithm for
the adaptable parameters of the fuzzy controller so that the condition is satisfied to
ensure asymptotic stability of the system; and (3) must take into account that, since
Krasovskii theorem must be satisfied in a neighborhood of the equilibrium state, is
not enough to rate it only at that point, but rather it should ensure compliance in a
sufficiently large region around the equilibrium state.
In relatively complex problems, the space of solutions that enable compliance
Krasovskii theorem in a sufficiently large region around the equilibrium state, is too
restrictive.Therefore, the use of Krasovskii theorem, even in its generalized version,
as overall design methodology of fuzzy controllers is only feasible in relatively simple
cases (Andújar and Barragán 2004 , 2005a , b ).
The following theorem is proposed in order to use Lyapunov theory to solve more
complex problems than can be addressed by Krasovskii theorem:
Consider the general autonomous system defined by 1 :
Theorem 4.1
˙
x
(
t
) =
f
(
x
(
t
)),
(4.27)
where the equilibrium state of interest is located, without loss of generality, in the
origin, 2
be s
∈ N
regions
q , with q
=
1
,...,
s, defined in the state space by s
n
1 , such that
1 , and
continuously differentiable functions
ϕ q (
x
) : R
→ R
ϕ q (
x
) C
be J
the jacobian matrix of the system ( 4.27 ) ; a sufficient condition for the origin
to be asymptotically stable is that there exist s symmetric positive-definite matrices,
P 1 ,...,
(
x
)
P s , such that
x
=
0 ,
1 The dependence of the state variables with respect to time will be omitted in all the development
of the theorem to simplify the notation and to improve clarity.
2 In the case that the steady state was not located at the origin, it is always possible to establish a
base change of the coordinate system to place it in that position.
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