A First Course in FUZZY and NEURAL CONTROL

40 CHAPTER 2. MATHEMATICAL MODELS IN CONTROL

are simultaneously satisfied. Also, sinceV ̇(x)≤ 0 for allxin the neighborhood
kxk<kof 0 ,t 0 ≤t 1 ≤kimplies

V(x(t 1 ))≤V(x(t 0 ))<φ(ε)

Thus, for allxin the neighborhoodkxk<kof 0 ,t 0 ≤t 1 ≤kimplies

kx(t 1 )k<ε

since we know thatφ(kx(t 1 )k)≤V(x(t 1 ))<φ(ε),andkx(t 1 )k≥εwould
implyφ(kx(t 1 )k)≥φ(ε)by the property thatφis strictly increasing on[0,k].
Takingδ=ε,weseebyDefinition 2.1 that the origin is stable.

The proof of the following theorem is similar. A functionV ̇:Rn→Ris said
to benegative definiteifV ̇( 0 )=0and for allx 6 =0in some neighborhood
kxk<kof 0 ,V ̇(x)< 0.

Theorem 2.3For a nonlinear system of the form

x ̇=f(x),f(0) = 0 (2.34)

the origin is asymptotically stable if there is a Lyapunov functionV for the
system, withV ̇ negative definite.

Here is an example.

Example 2.3Consider the nonlinear system

x ̇=f(x)

where

x=

μ

x 1

x 2

∂

,x ̇=

μ

x ̇ 1

x ̇ 2

∂

=f(x)=

μ

f 1 (x)

f 2 (x)

∂

with

f 1 (x)=x 1

°

x^21 +x^22 − 1

¢

−x 2

f 2 (x)=x 1 +x 2

°

x^21 +x^22 − 1

¢

The origin(0,0)is an equilibrium position. The positive definite function

V(x)=x^21 +x^22

has its derivative along any system trajectory

V ̇(x)=∂V

∂x 1

x ̇ 1 +

∂V

∂x 2

x ̇ 2

=2x 1

£

x 1

°

x^21 +x^22 − 1

¢

−x 2

§

+2x 2

£

x 1 +x 2

°

x^21 +x^22 − 1

¢§

=2

°

x^21 +x^22 − 1

¢°

x^21 +x^22

¢

Whenx^21 +x^22 < 0 , we haveV ̇(x)< 0 , so that(0,0)is asymptotically stable.