Small-Gain Theorem. Consider the closed-loop system shown in Figure 10–42. In
the figure and M(s)are stable and proper transfer functions.
The small-gain theorem states that if
then this closed-loop system is stable. That is, if the norm of M(s)is smaller
than 1, this closed-loop system is stable. This theorem is an extension of the Nyquist
stability criterion.
It is important to note that the small-gain theorem gives a sufficient condition for sta-
bility. That is, a system may be stable even if it does not satisfy this theorem. However,
if a system satisfies the small-gain theorem, it is always stable.
System with Unstructured Uncertainty. In some cases an unstructured uncer-
tainty error may be considered multiplicative such that
where is the true plant dynamics and Gis the model plant dynamics. In other cases
an unstructured uncertainty error may be considered additive such that
In either case we assume that the norm of or is bounded such that
where and are positive constants.
EXAMPLE 10–14 Consider a control system with unstructured multiplicative uncertainty. We shall consider robust
stability and robust performance of the system. (A system with unstructured additive uncertain-
ty will be discussed in Problem A–10–18.)Robust Stability. Let us define
true plant dynamics
G=model of plant dynamics
unstructured multiplicative uncertaintyWe assume that is stable and its upper bound is known.We also assume that and Gare
related byG=G(I+¢m)G
¢m¢m=G
=
gm ga
7 ¢m 7 6 gm, 7 ¢a 7 6 ga
¢m ¢a
G
=G+¢a
G
G
=G( 1 +¢m)
Hq ¢(s)
7 ¢(s)M(s) (^7) q 61
¢(s)
Section 10–9 / Robust Control Systems 809(s)M(s)Figure 10–42
Closed-loop system.