Nonlinear Control Sy.. - Free

244 CHAPTER 9. DISSIPATWITY
This result is important. According to (9.35), if the system V) given by (9.31) is finite-gainstable
with gain ry, then it must satisfy the so-called Hamilton-Jacobi inequality:
(ax )T 1
H def
a-f(x) + 2,y2 a--gg
+ 2 IyI 2< 0 (9.36)
Finding a function that maximizes f and satisfies inequality (9.36) is, at best, very
difficult. Often, we will be content with "estimating" an upper bound for y. This can be
done by "guessing" a function 0 and then finding an approximate value for -y; a process that
resembles that of finding a suitable Lyapunov function. The true L2 gain of 0, denoted 'y',
is bounded above by -y, i.e.,
0