Nonlinear Control Sy.. - Free

6.8. THE CIRCLE CRITERION 179
(i) g(s) has no poles in the open left-half plane (i.e., g(s) is the transfer function of an
exponentially stable system).
(ii) n(s) and d(s) are polynomials, and
is a proper transfer function.
n(s)
d(s)
(iii) All zeros of d(s) are in the closed right half plane. Thus,
n(s)
d(s)
contains the unstable part of G. The number of open right half plane zeros of d(s)
will be denoted by v.
With this notation, the Nyquist criteria can be stated as in the following lemma.
Lemma 6.1 (Nyquist) Consider the feedback interconnection of the systems Hl and H2.
Let Hl be linear time-invariant with a proper transfer function d(s) satisfying (i)-(iii)
above, and let H2 be a constant gain K. Under these conditions the feedback system is
closed-loop-stable in LP , 1 < p < oo, if and only if the Nyquist plot of G(s) [i.e., the polar
plot of G(3w) with the standard indentations at each 3w-axis pole of G(3w) if required] is
bounded away from the critical point (-1/K + 30), Vw E R and encircles it exactly v times
in the counterclockwise direction as w increases from -oo to oo.
The circle criterion of Theorem 6.6 analyzes the L2 stability of a feedback system
formed by the interconnection of a linear time-invariant system in the forward path and a
nonlinearity in the sector [a, ,Q] in the feedback path. This is sometimes referred to as the
absolute stability problem, because it encompasses not a particular system but an entire
class of systems.
In the following theorem, whenever we refer to the gain -y(H) of a system H, it will
be understood in the L2 sense.
Theorem 6.6 Consider the feedback interconnection of the subsystems Hl and H2 : Gee -+
Gee. Assume H2 is a nonlinearity 0 in the sector [a, Q], and let Hl be a linear timeinvariant
system with a proper transfer function G(s) that satisfies assumptions (i)-(iii)
above. Under these assumptions, if one of the following conditions is satisfied, then the
system is £2-stable: