Nonlinear Control Sy.. - Free

A.3. CHAPTER 6 319
Condition (ii): H1 and Hi are linear time-invariant systems; thus, the ,C2 gain of H1 is
Thus
if and only if
or
'Y(H1) = j1G1j. = sup IG(7w)I-
w
ry(Hl)ry(H2) < 1 (A.17)
G(7w)
r [sup l[1+gG(.7w)J < 1 (A.18)
rlG(yw)l < l1+gG(.7w)l dwER. (A.19)
Let G(3w) = x + jy. Then equation (A.19) can be written in the form
r2(x2 + y2)
x2(82 - r2) + y2(g2 - r2) + 2qx + 1
a,3(x2 + y2) + (a + 3)x + 1
We now analyze conditions (a)-(c) separately:
< (1 + qx)2 + g2y2
> 0
> 0.
(a) 0 < a 0
which can be expressed in the following form
where
a
(x + a)2 + y2 > R2
_ a+Q _ (/3-a)
R
2af
2a,3
(A.20)
(A.21)
(A.22)
(A.23)
Inequality (A.22) divides the complex plane into two regions separated by the circle C
of center (-a + 30) and radius R. Notice that if y = 0, then C satisfies (x + a)2 = R2,
which has the following solutions: x1 = (-/3-1 + 30) and x2 = (-a-1 + 30). Thus
C = C', i.e. C is equal to the critical circle C' defined in Theorem 6.6. It is easy to see
that inequality (A.22) is satisfied by points outside the critical circle. Therefore, the
gain condition y(Hi)y(H2) < 1 is satisfied if the Nyquist plot of G(s) lies outside the
circle C*. To satisfy the stability condition (i), the Nyquist plot of d(s) must encircle
the point (-q-1 +30), v times in counterclockwise direction. Since the critical point
(-q-1 +, j0) is located inside the circle C*, it follows that the Nyquist plot of d(s)
must encircle the entire critical circle C' v times in the counterclockwise direction,
and part (a) is proved.