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Unformatted text preview: : roots of n(s)
n(s) system is BIBO stable Pole : roots of d(s)
d(s) All the poles of G(s) are in the open left
half of the complex plane. Characteristic polynomial : d(s)
system is asymptotically stable Characteristic equation : d(s)=0
2008/09 MECH466 : Automatic Control 7 2008/09 MECH466 : Automatic Control 8 2 Idea of stability condition Remarks on stability Example For a general system (nonlinear etc.), BIBO
stability condition and asymptotic stability
condition are different.
For linear time-invariant (LTI) systems (to which
timewe can use Laplace transform and we can
obtain a transfer function), the conditions
happen to be the same.
In this course, we are interested in only LTI
systems, we use simply “stable” to mean both
BIBO and asymptotic stability. Asym. Stability:
(y(0)=0) Bounded if Re(α)>0
2008/09 MECH466 : Automatic Control 9 2008/09 Remarks on stability (cont’d)
G(s) has no pole in the open RHP (Right Half Plane), &
G(s) has at least one simple pole on jw-axis, &
jwG(s) has no multiple pole on jw-axis.
jw- NOT marginally stable Unstable if a system is neither stable nor
2008/09 MECH466 : Automatic Control 10 Stability summary Marginally stable if
Marginally Marginally stable MECH466 : Automatic Control Let si be poles of G.
Then, G is …
(BIBO, asymptotically) stable if
Re(si)<0 for all i.
marginally stable if
Re(si)<=0 for all i, and
simple root for Re(si)=0
it is neither stable nor
marginally stable. 11 2008/09 MECH466 : Automatic Control 12 3 Mechanical examples: revisited
f(t) M Examples
x(t) ? x(t)
stable? ? K B M f(t)
x(t) 2008/09 M
stable? ? f(t)
f(t) ? x(t)
stable? MECH466 : Automatic Contr...
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- Winter '09