Stability

Instructor: Mohammed Farrag

Dept. of Electrical Engineering
Assiut University

mohammed.farrag@eng.au.edu.eg
Stability

• Many possible definitions • For zero-input response

• If a system remains in a
particular state (or
• Two key issues for practical
condition) indefinitely, then

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systems state is an equilibrium state
• System response to zero of system

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input: internal stability • System’s output due to
• System response to non- nonzero initial conditions
zero but finite amplitude should approach 0 as t
(bounded) input: bounded • System’s output generated
input bounded output by initial conditions is made
(BIBO) stability up of characteristic modes

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Stability
• Three cases for zero-input response
• A system is stable if and only if all characteristic modes go to 0 as t  

• A system is unstable if and only if at least one of the characteristic

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modes grows without bound as t  

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• A system is marginally stable if and only if the zero-input response
remains bounded (e.g. oscillates between lower and upper bounds) as t


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Characteristic Modes

• Distinct characteristic roots l1, l2, …, ln

n
y0 t    c j e
l jt

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j 1

0 if Reλ  0

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
Right-hand
lim elt  e j  t if Reλ  0 plane (RHP)
t  Im{l}
 if Reλ  0


• Where λ    j Stable Unstable Re{l}

in Cartesian form
• Units of w are in
radians/second Left-hand
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plane (LHP) Marginally 13
Stable
Characteristic Modes
• Repeated roots • Decaying exponential decays
r faster than tk increases for
y0 t    ci t i 1 e lt any value of k
i 1

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For r repeated roots of value l. • One can see this by using the

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Taylor Series approximation
0 if Rel   0 for elt about t = 0:

lim t e   if Rel   0
k lt
1 2 2 1 33
t 
 1  lt  l t  l t  ...
 if Rel   0 2 6
For positive k

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Stability Conditions
• An LTI system is asymptotically stable if and only if all characteristic
roots are in LHP. The roots may be simple or repeated.

• An LTI system is unstable if and only if either one or both of the

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following conditions exist:

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i. at least one root is in the right-hand plane (RHP)
ii. there are repeated roots on the imaginary axis.

• An LTI system is marginally stable if and only if there are no roots in

the RHP, and there are no repeated roots on imaginary axis

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Response to Bounded Inputs

• Stable system: a bounded input (in amplitude) should give a

bounded response (in amplitude)
• Test for linear-time-invariant (LTI) systems

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y t   h t   f t 

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
  h  f t   d

f(t) h(t) y(t)
 
y t    h  f t   d   h   f t    d
 

If f (t ) is bounded, i.e. f t   C   t , then


 hτ  dτ 

f t     C  , and y t   C  hτ  dτ 


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• Bounded-Input Bounded-Output (BIBO) stable
Fourier Analysis
Fourier Series
Course Outline

• Time domain analysis

• Signals and systems in continuous and discrete time

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• Convolution: finding system response in time domain
• Frequency domain analysis
• Fourier series

Signal Processing
• Fourier transforms
• Frequency responses of systems
• Generalized frequency domain analysis
• Laplace and z transforms of signals
• Tests for system stability
• Transfer functions of linear time-invariant systems

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Periodic Signals
• For some positive constant T0
• f(t) is periodic if f(t) = f(t + T0) for all values of t  (-, )
• Smallest value of T0 is the period of f(t)

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  2     1 
sin( 2f 0t )  sin( 2f 0t  2 )  sin  2f 0  t    sin  2f 0  t   
 
  2f 0     f 0  

Signal Processing
• A periodic signal f(t)
• Unchanged when time-shifted by one period
• May be generated by periodically extending one period
• Area under f(t) over any interval of duration equal to the period is same;
e.g., integrating from 0 to T0 would give the same value as integrating
from –T0/2 to T0 /2

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 Periodic Signals
• Complex Exponential Fourier Series
T
 m m
j 2   t 2  j 2   t
xt    xt  e
1
C e T 
 Cm  T 
dt

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m
m   T T

2

Signal Processing
• For all t, x(t + T) = x(t)
• x(t) is a periodic signal
• Smallest value of T is the fundamental period
• Fundamental frequency 1/T

• Periodic signals have a Fourier series representation

• Fourier series coefficient Cm quantifies the strength of the component
of x(t) at frequency m/T
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• Fourier transforms are for both periodic and aperiodic signals
Trigonometric Fourier Series

• General representation f t   a0   an cosn 0t   bn sin n 0t 
of a periodic signal n 1

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 f t dt
1 T0
a0 
T0 0

 f t cosn t dt
2 T0

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• Fourier series coefficients an  0
T0 0

 f t sin n t dt

2 T0
bn  0
T0 0


f t   C0   Cn cosn0t   n 
• Harmonic Form Fourier series n 1

where C0  a0 , Cn  an2  bn2 , and

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 bn  12
 n  tan  1

 an 
Existence of the Fourier Series
f t  dt  
T0
• Existence

0

• Convergence for all t f t    t

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• Finite number of maxima and minima in one period of f(t)

Signal Processing
• What about periodic extensions of

g t  
1 1
for - 1  t  1 st   sin   for 0  t  1
t t 
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Power content of a periodic signal

• The average power of a periodic signal x ( t ) over any period is

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• If x ( t) is represented by the complex exponential Fourier series,

Signal Processing
then it can be

• This equation is called Parseval's identity (or Parseval's theorem)

for the Fourier series.
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Example #1

f t   a0   an cos( π n t )  bn sin π n t 
f(t) n 1

a0  0 (by inspection of the plot)

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A an  0 (because it is odd symmetric)
1/ 2

1 1 bn   2 A t sin( π n t ) dt 

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0
1 / 2
-A
3/ 2

 (2 A  2 A t ) sin( π n t ) dt
1/ 2
• Fundamental period
• T0 = 2 
 0
• Fundamental frequency  8 A
n is even
• f0 = 1/T0 = 1/2 Hz bn   2 2 n  1,5,9,13, 
 n 11 -
• 0 = 2/T0 =  rad/s  8 A n  3,7,11,15,  15
 n 2 2
 Thanks

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