Transcript Linearity

Spring 2008
Linear Systems and Signals
Lecture 6
Stability
Stability
• Many possible
definitions
• Two key issues for
practical systems
– System response to
zero input
– System response to
non-zero but finite
amplitude (bounded)
input
• For zero-input response
– If a system remains in a
particular state (or condition)
indefinitely, then state is an
equilibrium state of system
– System’s output due to
nonzero initial conditions
should approach 0 as t
– System’s output generated
by initial conditions is made
up of characteristic modes
6-2
Stability
• Three cases for zero-input response
– A system is stable if and only if all characteristic modes
 0 as t  
– A system is unstable if and only if at least one of the
characteristic modes grows without bound as t  
– A system is marginally stable if and only if the zeroinput response remains bounded (e.g. oscillates
between lower and upper bounds) as t  
6-3
Characteristic Modes
• Distinct characteristic roots l1, l2, …, ln
n
y0 t    c j e
l jt
j 1
0

lt
lim e  e j  t
t 


if Reλ  0
if Reλ  0
Im{l}
Right-hand
plane (RHP)
if Reλ  0
Stable
– Where l = s + j 
in Cartesian form
– Units of  are in
Left-hand
plane (LHP)
radians/second
Unstable
Re{l}
Marginally
Stable
6-4
Characteristic Modes
• Repeated roots
r
y0 t    ci t i 1 elt
i 1
– For r repeated roots of
value l.
0 if Rel   0

k lt
lim t e   if Rel   0
t 

 if Rel   0
– For positive k
• Decaying exponential
decays faster than
tk increases for any
value of k
– One can see this by
using the Taylor Series
approximation for elt
about t = 0:
1 2 2 1 33
1  lt  l t  l t  ...
2
6
6-5
Stability Conditions
• An LTIC system is asymptotically stable if and
only if all characteristic roots are in LHP. The
roots may be simple (not repeated) or repeated.
• An LTIC system is unstable if and only if either
one or both of the following conditions exist:
(i) at least one root is in the right-hand plane (RHP)
(ii) there are repeated roots on the imaginary axis.
• An LTIC system is marginally stable if and
only if there are no roots in the RHP, and there
are no repeated roots on imaginary axis.
6-6
Response to Bounded Inputs
• Stable system: a bounded input (in amplitude)
should give a bounded response (in amplitude)
• Linear-time-invariant (LTI) system
y t 
 ht   f t 
f(t)

 h  f t   d
y t    h  f t   d   h   f t    d
If f (t ) is bounded, i.e. f t   C   t , then

f t     C  , and yt   C  hτ  dτ







h(t)
y(t)
• Bounded-Input Bounded-Output (BIBO) stable
6-7
Impact of Characteristic Modes
• Zero-input response consists of the system’s
characteristic modes
• Stable system  characteristic modes decay
exponentially and eventually vanish
• If input has the form of a characteristic mode,
then the system will respond strongly
• If input is very different from the characteristic
modes, then the response will be weak
6-8
Impact of Characteristic Modes
• Example: First-order system with characteristic
mode e l t
y t   ht   f t 
 Ael t u t   e t u t 
A

e t  e l t u t 
 l


• Three cases
t elt u t 



  l resonance


yt   large amplitude   l strongresponse


small amplitude   l weak response
6-9
System Time Constant
• When an input is applied to a system, a certain
amount of time elapses before the system fully
responds to that input
– Time lag or response time is the system time constant
– No single mathematical definition for all cases
• Special case: RC filter
– Time constant is  = RC
t
1  RC
ht  
e
u t 
RC
h(t)
1/RC
e-1/RC

– Instant of time at which
h(t) decays to e-1  0.367 of its maximum value
t
6 - 10
System Time Constant
• General case:
h(t)
ĥ(t)
h(t0)
h(t)
t0
t
th
– Effective duration is th seconds where area under ĥ(t)

th
0

ˆ
h(t ) dt  th h(t0 )  C  h(t ) dt
0
– C is an arbitrary constant between 0 and 1
– Choose th to satisfy this inequality
• General case applied
to RC time constant:
t
 1

1
th

e RC dt
0 RC
RC
t h  RC

6 - 11
Step Response
• y(t) = h(t) * u(t)
u(t)
h(t)
y(t)
h(t)
u(t)
y(t)
A
1
t
A th
tr
t
tr
t
• Here, tr is the rise time of the system
• How does the rise time tr relate to the system
time constant of the impulse response?
• A system generally does not respond to an
input instantaneously
6 - 12
Filtering
• A system cannot effectively respond to periodic
signals with periods shorter than th
• This is equivalent to a filter that passes
frequencies from 0 to 1/th Hz and attenuates
frequencies greater than 1/th Hz (lowpass filter)
1/th is called the cutoff frequency
1/tr is called the system’s bandwidth (tr = th)
• Bandwidth is the width of the band of positive
frequencies that are passed “unchanged” from
input to output
6 - 13
Transmission of Pulses
• Transmission of pulses through a system (e.g.
communication channel) increases the pulse
duration (a.k.a. spreading or dispersion)
• If the impulse response of the system has
duration th and pulse had duration tp seconds,
then the output will have duration th + tp
• Refer to slides 5-2, 5-3 and 5-4
6 - 14