Nonlinear Control Sy.. - Free

9.9. NONLINEAR L2 GAIN 245
(iii) IHy1I2
2
Combining these results, we have that
or
2IIy1I2 = 2(x2 +x2)2 = 2 IIxII4
= -IIxII4 + IIXI14 + 1 IIXI14 < 0
9.9.1 Linear Time-Invariant Systems
It is enlightening to consider the case of linear time-invariant systems as a special case.
Assuming that this is the case, we have that
( i=Ax+Bu
Sl y=Cx
that is, f (x) = Ax, g(x) = B, and h(x) = Cx. Taking 4(x) = a xT Px (P > p, p = PT )
and substituting in (9.36), we obtain
Taking the transpose of (9.37), we obtain
7{ = xT + 212 PBBT P + 2CTCl x < 0. (9.37)
7 J
7IT = xT 1ATp + PBBT P + 2 CT CJ
7
Thus, adding (9.38) and (9.38), we obtain
71 + NT = xT
x < 0. (9.38)
PA+ATP+ 212PBBTP+ 2CTC x < 0. (9.39)
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