Nonlinear Control Sy.. - Free

318 APPENDIX A. PROOFS
We have
x
Figure A.3: Characteristics of H2 and H2.
Hi
= H [1+
s q
Hi
]-s =
G(s)
[1 + qd (s)]
(A . 14)
H2' = H2-q. (A.15)
By Theorem 6.4, the stability of the original system S can be analyzed using the
modified system SK. The significance is that the type I loop transformation has a considerable
effect on the nonlinearity 0. Namely, if H2 = ¢ E [a, I3], it is easy to see that
H2 = 0' E [-r, r], where
f (Q a)
r 2 (see Figure A.3.)
Thus, the gain of H2 is
-y(H2) = r (A.16)
According to the small gain theorem, if the following two conditions are satisfied, then the
system is stables.
(i) Hl and H2 are G2 stable.
(ii) 7(Hi)7(Hz) < 1.
Condition (i): H2 is bounded by assumption, and moreover, 7(H2) = r. By lemma 6.9.1,
Hi is G2 stable if and only if the Nyquist plot of d(s) encircles the point (-q-s + 30) v
times in counterclockwise direction.
'Here condition (ii) actually implies condition (i). Separation into two conditions will help clarify the
rest of the proof.