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DESCRIBING FUNCTION ANALYSIS:
The frequency response method is a powerful tool for the analysis and design of
linear control systems. It is based on describing a linear system by a complex-
valued function, the frequency response, instead of differential equation. Thepower of the method comes from a number of sources. First, graphical
representations can be used to facilitate analysis and design. Second, physical
insights can be used, because the frequency response functions have clear
physical meanings. Finally, the methods complexity only increases mildly with
system order. Frequency domain analysis, however, can not be directly applied
to nonlinear systems because frequency response functions !F"F# cannot bedefined for nonlinear systems.
)t(fx50x2x5 =++
50s2s5
1
)s(F
)s(X)s(F)s(X50)s(sX2)s(xs5
2
2
++=
=++ is =
( )
( ) i25501
)s(F
)s(X
50i2i5
1
)s(F
)s(X
2
2
+=
++=
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( )
( )
( ) i2550i2550
1
)s(F
)s(X
2
2
+=
( )( )
( )( ) ( )
i4550
2
4550
550
4550
i2550
)s(F
)s(X
222222
2
222
2
+
+
=
+
=
=== iAebia)s(F
)s(X)s(G
=+= a
btan,baA 122
)bia(angle,)bia(absA ==
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0 5 10 15
0
0.05
0.1
0.15
0.2Frequency Response Function
G(i)
0 5 10 15
-200
-150
-100
-50
0
PhaseAngle
(Degree)
R
clc;clear
w=0:0.01:15;
s=w!;
Gs=1."#5s.$%&%s&50';
s()*l+,#%11'
*l+,#w-a)s#Gs''
,!,le#Fre/(ec Res*+se F(c,!+'2la)el#3+4ea'
la)el#G#3+4ea!''
r!6 +
s()*l+,#%1%'
*l+,#w-ale#Gs'170"*!'
2la)el#3+4ea'
la)el#89ase Ale 3*9! #Deree''
r!6 +
0 5 10 150
0.05
0.1
0.15
0.2Frequency Response Function
G(i)
0 5 10 15-200
-150
-100
-50
0
Ph
aseAngle(Degree)
R
clc;clear
w=0:0.01:15;
r=#505w.$%'."##505w.$%'.$%&w.$%';
!4=#%w'."##505w.$%'.$%&w.$%';
Gs=r!4!;
s()*l+,#%11'
*l+,#w-a)s#Gs''
,!,le#Fre/(ec Res*+se F(c,!+'
2la)el#3+4ea'
la)el#G#3+4ea!''
r!6 +
s()*l+,#%1%'
*l+,#w-ale#Gs'170"*!'
2la)el#3+4ea'
la)el#89ase Ale 3*9! #Deree''
r!6 +
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For some nonlinear systems, an extended version of the frequency response
method, called the 6escr!)!
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s
#.'%
0 1
1ss2 + x
-
- x
( )2xxElementNonlinear 'inear %lement
w
Figure . Feedbac& interpretation of the +an der ol oscillator.
'inear element is unstable
0 1 2 3 4 5 6 7 8 9 100
0.2
0.4
0.6
0.8
1
1.2
1.4Frequency Response Function
G(i)
0 5 10 15 20 25 30-20
-15
-10
-5
0
5x 10
5Step Response
Time (sec)
Amplitude
'inear element has low pass behavior
2
2
Slotine and 'i, (pplied )onlinear *ontrol
s
#.'%
0 1
1ss2 + x
-
- x
( )2xxElementNonlinear 'inear %lement
w
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'et us assume that there is a limit cycle in the system and the oscillation
signal x is in the form of
Slotine and 'i, (pplied )onlinear *ontrol
( )tsinA)t(x =with ( being the limit cycle amplitude and 3 being the frequency. Thus,
( )tcosA)t(x =Therfore, the output of the nonlinear bloc& is
( ) ( )tcosAtsinAxxw 222 ==
( )[ ]
( ) ( )[ ]t3costcos4
Aw
tcost2cos1
2
Aw
3
3
=
=
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It is seen that w contains a third harmonic term. Since the linear bloc& has low-
pass propertiee, we can reasonably assume that this third harmonic term is
sufficiently attenuated by the linear bloc& and its effect is not presented in the
signal flow after the linear bloc&. This means that we can approximate w by
( ) ( )[ ]tsinAt
4
Atcos
4
Aw
23
=
so that the nonlinear bloc& in Figure can be approximated by the equivalent
4quasi-linear5 bloc& in Figure 6. the 4transfer function5 of the quasi-linear bloc&
depends on the signal amplitude (, unli&e a linear system transfer function!which is independent of the input magnitude#.
0 11ss2 + x-
- x
iona!!roximatlinear"#asi 'inear %lement
ws4
A2
Figure 6. 7uasi-linear approximation of the +an der ol oscillator.
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In the frequency domain, this corresponds to
( ) ( )
( ) ( )i4
As4
A,AN
x,ANw
22==
=
That is, the nonlinear bloc& can be approximated by the frequency response
function )!(,3#. Since the system is assumed to contain a sinusoidaloscillation, we have
( ) ( ) ( ) ( ) ( )x,ANiGwiGtsinAx ===
'inear part
where 8!3i# is the linear component transfer function. This implies that
( )
( ) ( ) 01ii4
iA1
2
2
=+
+
Slotine and 'i, (pplied )onlinear *ontrol
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( )
( ) ( ) 01ii4
iA1
2
2
=+
+
( ) 0i444
iA1
2
2
=
+
Solving this equation, we obtain (26 and 32
Slotine and 'i, (pplied )onlinear *ontrol
)ote that in terms of the 'aplace variable s23i, the closed loop characteristic
equation of this system is
01ss4
sA1 2
2
=+
+ whose roots are
( ) ( ) 14A$4
14A
%
1s
2222
2,1 =
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*orresponding to (26, we obtain the eigenvalues s,629i. This indicates the
existence of a limit cycle of amplitude 6 and frequency . It is interesting to note
neither the amplitude nor the frequency obtained above depends on the
parameter /.
/20. /2
/26 /2:
In the phase plane, the aboveapproximate analysis suggests
that the limit cycle is a circle of
radius 6, regardless of the value
of /. To verify the plausibility of
this result, the real limit cycles
corresponding to the different
values of / are plotted in Figure;.
It is seen that the above
approximation is reasonable for
small value of /, but that theinaccuracy grows as / increases.
This is understandable because
as / grows the nonlinearity
becomes more significant and the
quasi-linear approximation
becomes less accurate.
Figure ;. "eal limit cycle on the phase plane.
/20. /2
/26 /2:
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)ote that, in the approximate analysis, the critical step is to replace the nonlinear
bloc& by the quasi-linear bloc& which has the frequency response function !(6
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The first important class consists of 4almost5 linear systems. =y 4almost5 linear
systems, we refer to systems which contain hard nonlinearities in the control loop
but are otherwise linear. Such systems arise when a control system is designed
using linear control but its implementation involves hard nonlinearities, such as
motor saturation, actuator or sensor dead-$ones, *oulomb friction, or hysteresisin the plant. (n example is shown in Figure >, which involves hard nonlinearitie in
the actuator.
*onsider the control system shown in Figure >. the plant is linear and the
controller is also linear. ?owever, the actuator involves a hard nonlinearity. This
system can be rearranged into the for of Figure : by regarding 8p
8
86
as the
linear component 8, and the actuator nonlinearity as the nonlinear element.
Figure >. ( control system with hard nonlinearity.
r!t#20 18!s#
w!t# y!t#
-
x!t# u!t#8p!s#
86!s#
Dea6 +e a6 sa,(ra,!+
Slotine and 'i, (pplied )onlinear *ontrol
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4(lmost5 linear systems involving sensor or plant nonlinearities can be similarly
rearranged into the form of Figure :.
The second class of systems consists of genuinely nonlinear systems whose
dynamic equations can actually be rearranged into the form of Figure :. e saw
an example of such systems in the previous example !+an der ol equation#.
For systems such as the one in Figure >, limit cycles can often occur due to the
nonlinearity. ?owever, linear control cannot predict such problems. @escribing
functions, on the other hand, can be conveniently used to discover the existence
of limit cycles and determine their stability, regardless of whether the nonlinearity
is 4hard5 or 4soft5. The applicability to limit cycle analysis is due to the fact that the
form of the signals in a limit-cycling system is usually approximately sinusoidal.
Indeed, assume that the linear element in Figure : has low-pass properties
!which is the case of most physical systems#. If there is a limit cycle in the
system, then the system signals must all be periodic. Since, as a periodic signal,
the input to the linear element in Figure : can be expanded as the sum of many
harmonics, and since the linear element, because of its low-pass property, filters
out higher frequency signals, the output y!t# must be composed mostly of the
lowest harmonics.Slotine and 'i, (pplied )onlinear *ontrol
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rediction of limit cycles is very important, because limit cycles can occur frequently
in physical nonlinear system. Sometimes, a limit cycle can be desirable. This is the
case of limit cycles in the electronic oscillators used in laboratories. (nother
example is the so-called dither technique which can be used to minimi$e the
negative effects of *oulomb friction in mechanical systems. In most controlsystems, however, limit cycles are undesirable. This may be due to a number of
reasonsA
'imit cycle, as a way of instability, tends to cause poor control accuracy
The constant oscillation associated with the limit cycle can cause increasing
wear or even mechanical failure of the control system hardware
'imit cycling may also cause other undesirable effects, such as passenger
discomfort in an aircraft under autopilot.
In general, although a precise &nowledge of the waveform of a limit cycle is
usually not mondatory, the &nowledge of the limit cycles existence, as well as thatof its approximate amplitude and frequency, is critical. The describing function
method can be used for this purpose. It can also guide the design of
compensators so as to avoid limit cycles.
Slotine and 'i, (pplied )onlinear *ontrol
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Bas!c Ass(4*,!+s ! Descr!)! F(c,!+ Aals!s:
*onsider a nonlinear system in the general form of Figure :. In order todevelop the basic version of the describing function method, the system has
to satisfy the following four conditionsA
. Ther is only a single nonlinear components
6. The nonlinear component is time invariant !saturation, bac&lash, *oulomb friction, etc.#
;. *orresponding to a sinusoidal input x2sin!3t#, only the fundamental
component w!t# in the output w!t# has to be considered.
:. The nonlinearity is odd !symmetry about the origin#.
( ) ( ) ,3,2nforinGiG =>>
Sa,(ra,!+
x
w
Slotine and 'i, (pplied )onlinear *ontrol
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Bas!c De
'et us now discuss how to represent a nonlinear component by a describing
function. 'et us consider a sinusoidal input to the nonlinear element, of amplitude
( and frequency 3, i.e., x!t#2(sin!3t# as shown in Figure B.
N.L.( )tsinA )t(w
N#A->'( )tsinA ( )+tsin&
Figure B. ( nonlinear element and its describing function representation
The output of a nonlinear component w!t# is often a periodic, though generally
non-sinusoidal, function. )ote that this is always the case if the nonlinearity f!x#
is single-valued, because the output is fC(sin!w!t16/3##D2fC(sin!3t#D.
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Esing Fourier series, the periodic function w!t# can be expanded as
( ) ( )[ ]
= ++= 1n nn0
tnsinbtncosa2
a
)t(w
where the Fourier coefficients ais and bis are generally functions of ( and 3,
determined by
( )
( )
=
=
=
)t(tnsin)t(w1
b
)t(tncos)t(w1
a
)t()t(w1a
n
n
0
Slotine and 'i, (pplied )onlinear *ontrol
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@ue to the fourth assumtion above, one as a020. Furthermore, the third
assumption implies that we only need to consider the fundamental component
w!t#, namely
( ) ( ) ( )+=+== tsin&tsinbtcosa)t(w)t(w 111
where
( ) ( )
=+=
1
112
1
2
1b
atan,Aanba,A&
%xpression for w!t# indicates that the fundamental component corresponding to
a sinusoidal input is a sinusoid at the same frequency. In complexrepresentation, this sinusoid can be written as
( )+= ti1 &ew
Slotine and 'i, (pplied )onlinear *ontrol
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Similarly to the concept of frequency response function, which is the frequency-
domain ratio of the sinusoidal input and the sinusoidal output of a system, we
define the describing function of the nonlinear element to be the complex ratio of
the fundamental component of the nonlinear element by the input sinusoid, i.e.,
( )( )
+
== iti
ti
eA
&
Ae
&e,AN
ith a describing function representing the nonlinear component, the nonlinear
element, in the presence of sinusoidal input, can be treated as if it were a linear
element with a frequency response function )!(,3# as shown in Figure B.
The concept of a describing function can thus be regarded as an extention of the
notion of frequency response. For a linear dynamic system with frequency
response function ?!i3#, the describing function is independent of the input gain,
as can be easily shown. ?owever, the describing function of a nonlinear element
differs from the frequency response function of a linear element in that it
depends on the input amplitude (. Therefore, representing the nonlinear element
as in Figure B is also called quasi-lineari$ation.
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%xampleA @escribing function of a hardening spring
2
xxw3
+=
The characteristics of a hardening spring are given by
with x being the input and w being the output. 8iven an input x!t#2(sin!3t#, the
output
( ) ( )tsin2AtsinA)t(w 3
3+=
The output can be expanded as a Fourier series, with the fundamental being
( ) ( )tsinbtcosa)t(w 11 +==ecause w!t# is an odd function, one has a20 and the coefficient bis
-3 -2 -1 0 1 2 3-1.5
-1
-0.5
0
0.5
1
1.5
Time (seconds)
w(t)
( ) ( ) ( )
333
1 A%
3
Attsintsin2
A
tsinA
1
b +=
+=
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Therefore the fundamental is
( )tsinA%
3Aw
3
1
+=
and the describing function ofthis nonlinear component is
( ) 2A%
31)A(N,AN +==
( ) )t(x,ANw1 =
( )tsinAA%
31w 21
+=
)ote that due to the odd nature of this nonlinearity, the describing function is
real, being a function only of the amplitude of the sinusoidal input.
Sl ti d 'i ( li d ) li * t l