A First Course in FUZZY and NEURAL CONTROL

2.4. STABILITY 35

Figure 2.10. Disturbance in system

In general, a dynamical system can have several equilibrium states. Also,
the concept of stability about an equilibrium state can be formulated in many
different ways. Below is a popular concept of stability.

Definition 2.1The equilibrium state 0 of Equation 2.30 is said to be

1.stable(in the sense of Lyapunov) if for allε> 0 ,thereexistsδ> 0 such

that ifkx(t 0 )k<δthenkx(t)k<εfor allt≥t 0.

2.asymptotically stableif 0 is stable andlimt→∞x(t)= 0.

3.asymptotically stable in the largeif 0 is asymptotically stable and

limt→∞x(t)= 0 regardless of how large are the perturbations around 0.

In this definition we usekx(t)kto denote the Euclidean norm, noting that
the state space is someRn.Ofcourse 0 is unstable if there is anε> 0 such
that for allδ> 0 there existsx(t 0 )withkx(t 0 )k<δandkx(t 1 )k>εfor some
t 1 ≥t 0.

Figure 2.11. Equilibrium points

The notions ofstableandasymptotically stablerefer to two different proper-
ties of stability of 0. In other words, the nature of stability may vary from one
equilibrium point to another. The intuitive idea for stability is clear: for small