DESCRIBING FUNCTION

DESCRIBING FUNCTION

DESCRIBING FUNCTION

• Is there a way to analytically analyze oscillations in nonlinear control loops?

• Simulation is OK, but if control design is also required, then simulation approach is not easy

• Describing function or harmonic linearization analysis provides analytical approach for both analysis and design of controllers

DESCRIBING FUNCTION

N(e) G(j)+

-

r(t) = 0 e(t)

A

Nonlinearity Linear part

c(t)n(t)

B

DESCRIBING FUNCTION

• Assumptions– No input, r(t) = 0– Linear part acts as a low-pass filter, that is,

higher order harmonic components are damped– Nonlinearity does not generate subharmonics– Nonlinearity is symmetric– Nonlinearity does not depend on frequency– Assume that at point A, e(t) = E sin(t)

DESCRIBING FUNCTION

• Consider the nonlinearity– At the output, point B

n(t) = N(e(t)) = N(E sin(t)) = n(t)

– Fourier series

0

1 1

( ) cos( ) sin( )2 k k

k k

An t A k t B k t

DESCRIBING FUNCTION

• From assumptions

– where

1 1( ) cos( ) sin( )n t A t B t

2

2

1

2 1( )cos( ) ( )cos( )

T

T

A n t t dt n dT

2

2

1

2 1( )sin( ) ( )sin( )

T

T

B n t t dt n dT

DESCRIBING FUNCTION

1 1( ) cos( ) sin( )n t A t B t

2 21 1 1 1

1

( , ) tanA B A

N j EE B

sin( )E tN(j,E)

( )n t

Only fundamentalfrequency

sin( )E t ( )n tM

M

Tehonsäätö langattomassa tietoliikenteessä

MS MS

PSTN

M S C

RNCRNC RNC

BS BS

RNC = Radio Network Controller

MSC = Mobile Services Switching Center

PSTN = Public Switched Telephone Network

Closed-Loop Power Control for CDMA Systems

SIRtarget

Outagearea

SIR at receiver

Probability

SIR target(dB)

Return channelerror ±1

Controller

Loopdelay

Stepsize

Tp

kTp

Integrator

Interference(dB)

Channelvariation(dB)

ReceivedSIR (dB)

TPC command±1 dB

-+

Basestation

Radiochannel

Mobileunit

Outer loop

power control

Closed-loop

power control

Tehonsäätö langattomassa tietoliikenteessä

controller

loop delaystepsize

BASE STATION

MOBILE STATION

RADIO CHANNEL

SIR target

measuredSIR

channel fading

interference

pc error

u(k)e(k)

-

+

transmitted power

RETURN RADIOCHANNEL

+1 dB or –1 dB

DESCRIBING FUNCTION - Relay

Input signal

Output signalNonlinearity

This is developed into Fourier series

sinE t

e(t)E

t

0-E

e

n=N(e)

M

-M

M

-M

( )n t

t

1t

1t

DESCRIBING FUNCTION - Relay

• Odd nonlinearity, therefore A1 = 0.

1

1( )sin( )B n d

0

0

1sin( ) sin( )M d M d

4M

04 4( , ) ( ) 0

M MN j E N j

E E

DESCRIBING FUNCTION - Saturation

1 1 1 1sin ; sin SK E K S E

11 1 1

2( ) cos( ) , sin( )

K S SN j

E E

1

1

1

( ) sin( ) -

K S e S

n t K E t S e S

K S e S

DESCRIBING FUNCTION - Saturation

11 1 1

2( ) cos( ) , sin( )

K S SN j

E E

DESCRIBING FUNCTION - Backlash

DESCRIBING FUNCTION - Criterion

DESCRIBING FUNCTION - Criterion

• Criterion for predicting oscillations• Analogous to linear case

• Can also be written as

( ) ( ) 1G j N j

1G

N

DESCRIBING FUNCTION – Stable limit cycle

G

-1/NStabiili rajajakso

E

Im

Re

G

-1/N

Stabiili rajajakso

E

Vaihekulma

Vahvistus(dB)

Nyquist diagram Nichols diagram

DESCRIBING FUNCTION – Unstable limit cycle

G

-1/N

Epästabiili rajajaksoE

Im

Re

G

-1/N

EpästabiilirajajaksoE

Vaihekulma

Vahvistus(dB)


DESCRIBING FUNCTION – Unstable limit cycle

G

-1/N

Stabiili rajajakso

E

Vaihekulma

Vahvistus(dB)

Epästabiili rajajakso

G

-1/N Stabiili rajajakso

E

Im

Re

Epästabiili rajajakso


DESCRIBING FUNCTION – Example of dead zone

Use describing function method to investigate oscillations in the feedback system below.

D=1(1 0.5 )(1 0.1 )

K

j j j

c(t)r(t)=0 +

-

k=1

DESCRIBING FUNCTION – Example of dead zone, describing functions

D/E N/K1 (1/N)dB

0 1 0

0.2 0.75 2.5

0.4 0.5 6.0

0.6 0.3 10.4

0.27 0.63 4

DESCRIBING FUNCTION – Dead-zone example

g=zpk([],[0 -2 -10],20) ;

Zero/pole/gain: 20-------------s (s+2) (s+10)

Nichols(g);grid

Multiply gain by 17g=zpk([],[0 -2 -10],20*17) %Zero/pole/gain: 340--------------s (s+2) (s+10)

Describing function

E=1.01:0.01:100; C=1;N=(2/pi)*acos(C./E)-(C./E).*sqrt(1-(C./E).^2);

hold on;Nichols(g);gridplot (-1./N, zeros(size(N)));

DESCRIBING FUNCTION – Type of limit cycle


Oscillationoccurs.

DESCRIBING FUNCTION – Backlash example


D/E N/K1 (1/N) (1/N)dB arg(N) 180-arg(N)

0 1 1.0 0 0 -180

0.2 0.87 1.15 1.1 -12 -168

0.4 0.69 1.45 3.2 -25 -155

0.6 0.45 2.2 6.9 -39 -141

0.8 0.23 4.4 12.8 -55 -125

0.9 0.1 10 -70 -110

1.0 0 20 -90 -90

DESCRIBING FUNCTION – Relay with Hysteresis

M = relay amplitude

= relay width

2 21

( ) 4 4E j

N E d d

-1/N(E)

Re

jIm

Constant

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DESCRIBING FUNCTION