The Stability of Linear Feedback Systems

5.1 The Concept of Stability
209
Stable
Neutral
Unstable
Figure 5.2. Stability in the s-plane.
How strong this particular signal is depends upon the distance between
speaker and the microphone. Because of the attenuating properties of
Iaqer this distance is, the weaker the sign~1 is that reaches the micro­
ID addition, due to the finite propagation s*ed of sound waves, there is
y between the signal produced by the lou~speaker and that sensed by
one. In this case, the output from the feedback path is added to the
input. This is an example of positive feedback.
the distance between the loud speaker and the microphone decreases, we
ifthe microphone is placed too close to the speaker then the system will
. The result ofthis instability is an excessive amplification and distoraf'audio
signals and an oscillatory squeal.
",otI>erexamp!e ofan unstable system is shown in Fig. 5.3. The flrst bridge
the Tacoma Narrows at Puget Sound, Washington, was opened to traffic
1.1940. The bridge was found to oscillate whenever the wind blew. After
!hI. on November 7,1940, a wind produced an oscillation that grew in
until the bridge broke apart. Figure 5.3(a) shows the condition of
oscillation; Fig. 5.3(b) shows the catastrophic failure [II].
1enns of linear systems, we recognize that the stability requirement may
in terms of the location of the poles of the closed-loop transfer funcclosed-loop
system transfer function is written as
KIt: I (s + Zi)
p(s)
T( .). - - (5.1)
q(s) s"'rr~_1 (s + O"k) II:_I [~ + 2u m s + (u;n + w;,,)] ,
1(&) • 4(s) is the characteristic equation whose roots are the poles of the
system. The output response for an impulse function input is then
c(t) = t. A k e- Ok1 + t B m (-!...) e-"",1 sin w",l (5.2)
k_1 m_1 W m
O. Clearly, in order to obtain a bounded response, the poles of the
GIld syst~m must ~ in the left-hand portion of the s-plane. Thus, a necnifficlent
conditIOn that afeedback system be stable is that a/1 the poles
em transfer function ha~'e negative real parts.