Nonlinear Control Sy.. - Free

228 CHAPTER 9. DISSIPATIVITY
Thus, defining ,0 f - O(xo), (9.11) is identical to Definition (8.2). This formulation
is also important in that it gives 3 a precise interpretation: 3 is the stored energy at
time t = 0, given by the initial conditions xo.
2- Strictly passive systems: The system Vi is strictly passive if and only if it is
dissipative with respect to Q = 0, R = -b, and S =
these values in (9.10) and obtain
2I. To see this, we substitute
or
(y, u)T + (U, -bU)T >- 0(x1) - O(x0) >- -O(xO)
(u, y)T b(u, U) T + Q = 5IIuIIT + 3.
3- Finite-gain-stable: The system 7/i is finite-gain-stable if and only if it is dissipative
with respect to Q = - i I, R = 2 I, and S = 0. To see this, we substitute these values
in (9.10) and obtain
f Q
1
2 (y, y)T + 2 ('a, u)T > O(xl) - O(x0) > -qS(xo)
(y,y)T >_ -72(u,u)T - 20(x0)
or (y, y)T < 72 (n,u)T + 20(xo)
IIyTIIGZ 0, a -+b2 < (a + b), then defining 3 = 2 (xo), we conclude
that
IIyTIIGZ - 0(x1) - O(xo) ? -O(xo)
fT
J UT y dt = (u, y)T ? E(y, y)T +
0