Nonlinear Control Sy.. - Free

6.4. L GAINS FOR LTI SYSTEMS
where
It follows that lullc_ = 1, and
and the result follows.
6.4.2 G2 Gain
sgn[h(t)]
1 if h(T) > 0
l 0 if h(T) < 0
y(t) = (h * u)(t) = bou(t) + t ha,(T)U(t - r)dr
J0 t
Ihol + f Iha(T)ldr = IhIIA
0
This space consists of all the functions of t that are square integrable or, to state this in
different words, functions that have finite energy. Although from the input-output point of
view this class of functions is not as important as the previous case (e.g. sinusoids and step
functions are not in this class), the space G2 is the most widely used in control theory because
of its connection with the frequency domain that we study next. We consider a linear timeinvariant
system H, and let E A be the kernel of H., i.e., (Hu)(t) = h(t) *u(t), du E G2.
We have that
-Y2(H) = sup III
(6.18)
x xIIIC2
1/2
IIxIIc2 = { f Ix(t)I2 dt }
We will show that in this case, the gain of the system H is given by
00
'Y2(H) = sup I H(.)I
J
f IIHIIoo
167
(6.19)
(6.20)
where H(tw) = .F[h(t)], the Fourier transform of h(t). The norm (6.20) is the so-called
H-infinity norm of the system H. To see this, consider the output y of the system to an
input u
Ilylicz = IIHuIIcz =f [h(t) * u(t)]dt = 21r
ii:
where the last identity follows from Parseval's equality. From here we conclude that
IIyIIcz < {supIH(Jw)I}2 I
2r1 00
r i-
IU(JW)12dw}