Nonlinear Control Sy.. - Free

174 CHAPTER 6. INPUT-OUTPUT STABILITY
IG(7w')I. dIG
)I w w =0, and
d2I2w) I Iw- < 0.
Differentiating IC(.jw)I twice with respect to w we obtain w = v, and IG(j-w`)j = 1/ 12.
It follows that y(H1) = 1/ 12. The calculation of y(H2) is straightforward. We have
y(H2) = IKL. Applying the small gain condition y(Hl)y(H2) < 1, we obtain
y(Hl)y(H2) < 1 = IKI< 12.
Thus, if the absolute value of the slope of the nonlinearity N(.) is less than 12 the system
is closed loop stable.
Example 6.5 Let Hl be as in Example 6.4 and let H2 be a constant gain (i.e., H2 is linear
time-invariant and H2 = k). Since the gain of H2 is y(H2) = IkI, application of the small
gain theorem produces the same result obtained in Example 6.4, namely, if IkI < 12, the
system is closed loop stable. However, in this simple example we can find the closed loop
transfer function, denoted H(s), and check stability by obtaining the poles of H(s). We
have
G(s)
H(s) = 1 + kG(s)
s2 + 2s + (4 + k)
The system is closed-loop-stable if and only if the roots of the polynomial denominator of
H(s) lie in the open left half of the complex plane. For a second-order polynomial this is
satisfied if and only if all its coefficients have the same sign. It follows that the system is
closed-loop-stable if and only if (4 + k) > 0, or equivalently, k > -4. Comparing the results
obtained using this two methods, we have
Small gain theorem: - 12 < k < 12.
Pole analysis: -4 < k < oo.
We conclude that, in this case, the small gain theorem provides a poor estimate of the
stability region.
6.7 Loop Transformations
As we have seen, the small gain theorem provides sufficient conditions for the stability of
a feedback loop, and very often results in conservative estimates of the system stability.