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were t o b e d isturbed slightly, i t w ould eventually r eturn t o i ts original equilibrium
s tate if left t o itself. I n t his case, t he cone is s aid t o b e in s table e quilibrium.
I n c ontrast, i f t he cone s tands o n i ts a pex, t hen t he s lightest d isturbance will cause
t he cone t o move f arther a nd f arther away from i ts e quilibrium s tate. T he cone in
t his case is said t o b e i n a n u nstable e quilibrium. T he cone lying o n i ts side, if 148 2 TimeDomain Analysis of ContinuousTime Systems disturbed, will neither go back to the original s tate nor continue t o move farther
away from t he original state. The cone in this case is said t o be in a n eutral
e quilibrium.
Let us a pply these observations t o systems in general. I f, in t he absence of a n
e xternal input, a system remains in a particular s tate (or condition) indefinitely,
then t hat s tate is said t o be an e quilibrium s tate o f t he s ystem. For an LTI
system this equilibrium s tate is t he zero state, in which all initial conditions are
zero. Now s uppose an LTI system is in equilibrium (zero state) and we change this
s tate by creating some nonzero initial conditions. By analogy with the cone, if the
system is s table i t should eventually r eturn t o zero state. In other words, when left
to itself, t he s ystem's o utput due t o t he nonzero initial conditions should approach
o as t ~ 00. B ut t he system o utput generated by initial conditions (zeroinput
response) is m ade up of its characteristic modes. For this reason we define stability
as follows: a s ystem is ( asymptotically) s table if, and only if, all its characteristic
modes ~ 0 as t ~ 0 0. I f any of t he modes grows without bound as t ~ 0 0, t he
system is u nstable. T here is also a borderline situation in which the zeroinput
response remains bounded (approaches neither zero nor infinity), approaching a
constant or oscillating with a constant amplitude as t + 0 0. For this borderline
situation, t he s ystem is said t o b e m arginally s table or just stable.
I f a n LTIC system has n d istinct characteristic roots AI, A2, . .. , An, t he zeroinput response is given by
n Yo(t) = 2:~>jeAJt (2.62) j =1 t Jmag R eA<O stable marginally stable .... R eA=O F ig. 2 .15 Characteristic roots location and system stability. t ~ 0 0, t ke At + 0, if ReA < 0 (A in LHP). Therefore, repeated roots in LH~
do not cause instability. B ut when the repeated roots are on the imaginary aXIs
(A = jw), t he corresponding modes t ke iwt ~ppro~~h infi?ity as t + 0 0. Therefore,
repeated roots on t he imaginary axis cause instabIlity. ~Igure 2.1.6 sh.ows characteristic modes corresponding t o characteristic roots a t various locatIOn I n t he com?le.x
plane. Observe the central role played by t he c haracteristic roots or characteristIc
modes in determining the system's stability. We have shown elsewhere [see Eq. (B.14)]
lim
t +oo eM = {O 00 149 2.6 System Stability To summarize:
Re A < 0...
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This note was uploaded on 04/14/2013 for the course ENG 350 taught by Professor Bayliss during the Spring '13 term at Northwestern.
 Spring '13
 Bayliss
 Signal Processing, The Land

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