Nonlinear Control Sy.. - Free

178 CHAPTER 6. INPUT-OUTPUT STABILITY
Figure 6.12: The nonlinearity 0(t*, x) in the sector [a, 3].
6.8 The Circle Criterion
Historically, one of the first applications of the small gain theorem was in the derivation of
the celebrated circle criterion for the G2 stability of a class of nonlinear systems. We first
define the nonlinearities to be considered.
Definition 6.17 A function 0 : R+ x 1l -a IR is said to belong to the sector [a, Q] where
a 0, Vx E R. (6.38)
According to this definition, if 0 satisfies a sector condition then, in general, it is timevarying
and for each fixed t = t', t(t', x) is confined to a graph on the plane, as shown in
Figure 6.12.
We assume that the reader is familiar with the Nyquist stability criterion. The Nyquist
criterion provides necessary and sufficient conditions for closed-loop stability of lumped
linear time-invariant systems. Given a proper transfer function
G(s) = p(s)
q(s)
where p(s) and q(s) are polynomials in s with no common zeros, expanding d(s) in partial
fractions it is possible to express this transfer function in the following form:
where
G(s) = 9(s) + d(s)