Describing Functions

7/26/2019 Describing Functions

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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

+=

++=


2/21

( )

( )

( ) 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 ==


3/21

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


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 +


4/21

For some nonlinear systems, an extended version of the frequency response

method, called the 6escr!)!


5/21

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


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


6/21

'et us assume that there is a limit cycle in the system and the oscillation

signal x is in the form of


( )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

=

=


7/21

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.


8/21

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

=+

+



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( )

( ) ( ) 01ii4

iA1

2

2

=+

+

( ) 0i444

iA1

2

2

=

+

Solving this equation, we obtain (26 and 32


)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 =


10/21

*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:


11/21

)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


12/21

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,!+



13/21

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


14/21

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.



15/21

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



16/21

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.


17/21

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



18/21

@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



19/21

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.


20/21

%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

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Describing Functions