In this chapter we consider the rst method the other two are discussed in the next
chapter.
9.1
Bode’s Log Sensitivities
9.1.1
First Order Fractional Sensitivity
H. Bode was the rst to systematically study the e ect of small changes in closed-
loop transfer functions due to small changes in the plant. He considered the I/O
transfer function of the SASS 1-DOF control system (see section 2.3.2),
T
=
0
P
K
T
1 + 0 :
P
K
He noted that for any frequency ,s
( )
( )
@
T
s
T
s
( )
(9.1)
0( )
0( ) =
1
1 + 0( ) ( ) = S s
@
P
s
P
s
P
s
K
s
which gave the name and symbol to the classical sensitivity transfer function.
S
We thus have a basic rule-of-thumb for the 1-DOF control system,
( )
0( )
T
s
s
( )
( ) P
'
S
s
0( )
(9.2)
T
s
P
s
( means equal to rst order), which we interpret as follows: the fractional or
'
relative change in the I/O transfer function is, to rst order, the sensitivity transfer
function times the fractional change in 0. For example, and roughly speaking, at
P
a frequency with ( ) = 0 1, a ten percent change in the complex number
!
jS
j
!
j
:
0( ) yields a change in ( ) of only (and approximately) one percent.
P
j
!
T
j
!
An important consequence of (9.2) is that a design speci cation that limits the
rst order fractional change in the I/O transfer function with respect to fractional
changes in 0 can be expressed as an equivalent closed-loop convex speci cation that
P
limits the size of the sensitivity transfer function. For example, the speci cation
( )
0( )
T
j
!
j
!
( )
0 05 P
bw
(9.3)
<
:
0( )
for ! !
T
j
!
P
j
!
( means holds to rst order), which limits the rst order fractional change in
<
T
to no more than 5% of the fractional change in 0 for frequencies less than bw, is
P
!
equivalent to the design speci cation
( ) 0 05 for
bw
(9.4)
jS
j
!
j
:
!
!
which is closed-loop convex.
9.1 BODE’S LOG SENSITIVITIES
197
The precise meaning of (9.3) is
lim
( )
0( )
T
j
!
P
j
!
( ) 0 05 for
(9.5)
( ) 0 ( )
0( ) =
bw
jS
j
!
j
:
!
!
:
T
j
!
P
j
!
P
j
!
!
0
In many cases, the limit in (9.5) is rapidly approached, i.e., the rst order approxi-
mation to the fractional change in accurately predicts the actual fractional change
T
in due to a (non-di erential) change in 0. We will see two examples of this in
T
P
sections 9.1.3 and 9.1.4.
9.1.2
Logarithmic Sensitivity
We can express (9.1) as
( ) = log ( )
@
T
s
(9.6)
S
s
log 0( ):
@
P
s
For this reason ( ) is called the logarithmic sensitivity of with respect to 0.
S
s
T
P
We must be careful about what (9.6) means. By log ( ) we mean
T
s
log ( ) = log ( ) +
( )
(9.7)
6
T
s
jT
s
j
j
T
s
where ( ) is a phase angle, in radians, of ( ). Whereas log ( ) is unambiguous
6
T
s
T
s
jT
s
j
and well-de ned for all for which ( ) = 0, the phase angle ( ) is ambiguous:
6
s
T
s
6
T
s
it is only de ned to within an integer multiple of 2 . On any simply-connected
region in the complex plane on which ( ) = 0, it is possible to make a selection
T
s
6
of particular phase angles in (9.7) at each in such a way that ( ) is continuous
6
s
T
s
on the region, and in fact log ( ) is analytic there. When this process of phase
T
s
selection is applied along the imaginary axis, it is called phase unwrapping.
In particular, if ( 0) = 0, then in some small disk around ( 0) in the complex
T
s
6
T
s
plane, we can de ne log ( ) (i.e., choose the phase angles) so that it is analytic
T
s
there. Moreover any two such de nitions of log ( ) will di er by a constant multiple
T
s
of 2 , and therefore yield the same result in the partial derivative in (9.6), evaluated
j
at 0. A similar discussion applies to the expression log 0( ): while it need not
s
P
s
make sense as an unambiguous function of the complex variable over the whole
s
complex plane, the result in (9.6) will nevertheless be unambiguous.
The real part of (9.7), which is the natural log of the magnitude, has an uncom-
mon unit in engineering, called nepers. One neper is the gain that corresponds to a
phase angle of one radian, and is approximately 8.7dB. In more familiar units, one
decibel corresponds to about 6 6 degrees of phase.
:
(9.2) can be expressed more completely as
log ( )
( ) log 0( )
( ) 0( )
(9.8)
6
jT
s
j
'
<S
s
jP
s
j
;
=S
s
P
s
( )
( ) log 0( ) + ( ) 0( )
(9.9)
6
6
T
s
'
=S
s
jP
s
j
<S
s
P
s
:
These formulas show the rst order change in the magnitude and phase of the I/O
transfer function caused by magnitude and phase variations in 0.
P
198