A First Course in FUZZY and NEURAL CONTROL

2.4. STABILITY 39

2.4.3 Stability of nonlinear systems................

Stability analysis for nonlinear systems is more complicated than for linear sys-
tems. There is, nevertheless, an extensive theory for control of nonlinear sys-
tems, based on Lyapunov functions. This theory depends on the mathematical
models of the systems, and when considering fuzzy and neural control as an
alternative to standard control of nonlinear systems, we will need to consider
other approaches to stability analysis.
Foranonlinearsystemoftheform

x ̇=f(x),f( 0 )=0 (2.33)

withx(t 0 )=x 0 , it is possible to determine the nature of stability of the origin
without solving the equation to obtainx(t).Sufficient conditions for stability
of an equilibrium state are given in terms of theLyapunov function.These
conditions generalize the well-known property that an equilibrium point is stable
if the energy is a minimum.

Definition 2.2ALyapunov functionfor a systemx ̇=f(x)is a function
V:Rn→Rsuch that

1.Vand all its partial derivatives∂x∂Vi,i=1, 2 ,...,nare continuous.

2.V( 0 )=0andV(x)> 0 for allx 6 = 0 in some neighborhoodkxk<kof

0 .Thatis,V ispositive definite.

3. Forx(t)=(x 1 (t),...,xn(t))satisfyingx ̇=f(x)withf( 0 )=0,

V ̇(x)=∂V

∂x 1

x ̇ 1 +∑∑∑+

∂V

∂xn

x ̇n

is such thatV ̇( 0 )=0andV ̇(x)≤ 0 for allxin some neighborhood of 0.

In other words,V ̇ isnegative semidefinite.

Theorem 2.2For a nonlinear system of the form

x ̇=f(x),f( 0 )=0

the origin is stable if there is a Lyapunov functionVfor the systemx ̇=f(x),
f( 0 )=0.

Proof.Take a numberk> 0 satisfying bothV(x)> 0 andV ̇(x)≤ 0 for all
x 6 = 0 in the neighborhoodkxk<kof 0. Then there exists a continuous scalar
functionφ:R→Rwithφ(0) = 0that is strictly increasing on the interval
[0,k]such that
φ(kxk)≤V(x)

for allxin the neighborhoodkxk<kof 0 .Givenε> 0 ,thensinceφ(ε)> 0 ,
V(0) = 0andV(x)is continuous, andx 0 =x(t 0 )canbechosensufficiently
close to the origin so that the inequalities

kx 0 k<ε,V(x 0 )<φ(ε)