exercises for lectures 14 modern frequency response methods · 14 –modern frequency response...

Exercises for lectures14 – Modern Frequency Response

Methods

Michael ŠebekAutomatic control 2016

31-3-17

Automatické řízení - Kybernetika a robotika

• Feedback transfer functions

• E.g. Influence of disturbance will be small for small S and influence of noise for small T. However, small S and small T is not possible at the same time.Proof

• This is a serious limitation for the controller design. s=jω,and for each individual ω(It is complex number)

• Therefore, design is a compromise: We have to choose priorities for individual frequency ranges.

• That is called frequency response shaping (loop shaping)

Revision: Closed loop transfer functions

Michael Šebek 2ARI-14-2013

u( )K s ( )G s

( ) ( ) 1S s T s

( ) ( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

y s T s r s S s d s T s n s

u s K s S s r s K s S s d s K s S s n s

e s r s y s S s r s S s d s T s n s

1 ( ) 1 ( )( ) ( ) 1

1 ( ) 1 ( ) 1 ( )

L s L sS s T s

L s L s L s

( ) ( ) 1S j T j


• In the classic version of loop shaping It shapes OL characteristics, in spite the aim to shapes CL characteristics.

• In certain frequency ranges, it is sufficient to shape|L(jω)|, thereforefrom the |L(jω)| shape follows |S(jω)| a |T(jω)|,

• The precise relations

• The approximate relations• Typically for

low frequency:• Typically for

high frequencyIn the neighborhood of frequency ωc (where L is not low neither high)from the|L(jω)| shape It doesnt follow shapes of |S(jω)| , |T(jω)|,

• It also depends on the phase• E.g. |S(jω)| , |T(jω)| It can have a large peaks for L(jω) ~ -1

Revision - Loop Shaping – classic version

Michael Šebek 3

( )( )

1 ( )

L jT j

L j

1( )

1 ( )S j

L j

1( )L j

1( )L j

( ) 1T j 1

( )( )

S jL j

( ) ( )T j L j ( ) 1S j

ARI-14-2013


1 12arcsin [rad]

2 S S

PMM M

radPM

SMTM

1 12arcsin [rad]

2 T T

PMM M

E.g.

or

Therefoee it is easier to use MS or MT for specifications

Graphically:

PM , MS and MS Relations

Michael Šebek 4

2SM 2, 29GM PM

2TM 1.5, 29GM PM

1

S

S

MGM

M

SM

GM1

1T

GMM

TM

GM

ARI-14-2015


• unit FB should have a deviation less than 0.005 for all sinus signalswith amplitude 1 and frequency below 100 Hz

• We can formulate these requirements by using the frequency weighting function :

• Reference signals spectrumis = 1 for

• because eb = 0.005, the functionis rectangle with height1/0.005 = 200for a given frequency range

Example – input signal scale

Michael Šebek 5

1

10

100

1000

0.1

110210 310 410

510200

2000 200

ARI-14-2013


• The steady state error to step response

• Classical requirement for steady state error to step response

• It can be written as

• This requirement is extend to frequency range as:

• W1 is zero out of this range, the relation applies for all frequencies.

Example: Comparison with the classical req.

Michael Šebek 6

step,ss step,ss

11b

b

e eee

step,ss0

0

1lim ( ) (0)

1 lim ( ) s

s

e S s SL s

step,ss 1

1 11(0) (0)

b b

e WS Se e

1 1S W 11: ( ) ( ) 10, S j W

10,

1: ( ) ( ) 10, S j W

ARI-14-2013


• Requirement for control

• It can be also expressed with open loop transfer function• For low frequencies

• approximately

• Or in more detail

Requirement to function L

Michael Šebek 7

1 11 : ( ) ( ) 1S W S j W

1 1( )

1 ( ) ( )S j

L j L j

0 10, : ( ) ( )L j W

1 1S W 1 1W

L

1L W

Stea

dy

stat

eer

ror

limit

1c

( )L j

ARI-14-2013


Example: Uncertain system Nyquist graph

Michael Šebek 8

g0=2.5/((s+1)^3);

k=rdf(1);w=rdf(.5);

omega=0:.01:2;

ball(0,k,w,1,j*omega);

g0=2.5/(2.5*s+1);

w=(4*s+.2)/(10*s+1);a=2*pi;

for m=0:1/10:1,for alfa=0:a/10:a,

delta=m*exp(-j*alfa);

bode(tf(g0)*(1+tf(w)*delta),'b'),

hold on,

end,end

bode(tf(g0),'r')

bode(tf(w),'g')

0 2

0 2

2.5( )

( ) ( ) 1 ( ) ( )

2.

4 0.2( )

10 1

1,( )

5

,

1,

G j G j W

jW

jG

s

j j

s

dB

abs

ARI-14-2012


• system with multiplicative uncertainty

• nominal frequency response• The total frequency characteristic

Example: Uncertain system Nyquist and Bode graph

Michael Šebek 9

0 2( ) ( ) 1 ( ) ( ) , ( ) 1G j G j W j j

0 23

2.5( ) , ( ) 0.5

( 1)G s W s

s

g0=2.5/((s+1)^3);

k=rdf(1);w=rdf(.5);

omega=0:.01:2;

ball(g0,k,w,1,j*omega);

( ) ( )G jj G

0 ( )G j

ARI-14-2013


• Consider system with transfer function

• where G0(s) is given, • But f (s) is neglected and replace by multiplicative uncertainty.

• Intensity of relative uncertainty is

• We discuss two cases:• Time delay neglecting

• First order term neglecting

Uncertainty caused by dynamics neglecting

Michael Šebek 10

0( ) ( ) ( )G s G s f s

0

( ) ( )0

( ) ( )( ) max max ( ) 1

( )I

G s f s

G j G jl f j

G j

ARI-14-2015

max( ) 1 ( 1),0p pf s s

max( ) ,0sf s e


• Consider , where

• For maximum delay is variance depictedin the picture (for )

• reaches 1 for• maximum (=2) for• Then oscillates between 0 and 2• It is similar for other θ

• Rational function replacement1st order and 3rd order

Time delay neglecting

Michael Šebek 11

omega=.01:.01:100;plot(omega,abs(1-exp(-2*j.*omega))),

hold on

w1=(2*j.*omega)./(j*omega+1);plot(omega,abs(w1),'r--')

w3=((2/2.363)^2.*omega.^2+2*0.838*2/2.363*j.*omega+1)./...

((2/2.363)^2.*omega.^2+2*0.4*2/2.363*j.*omega+1);

w2=w1.*w3;plot(omega,abs(w2),'g--')

2

max1 max

0( ) ( ) ( )G s G s f smax( ) ,0sf s e

max1

max( ) | 1|j

Il e

max

max 2

max

max

max

| 1|,( )

2

j

I

el

ARI-14-2013


Time delay neglecting

Michael Šebek 12

0

1 1( )

, , {2,2.5,3}

ske

s

G s

k

2,1

4 0.2( )

10 1 s j

sW j

s

2

2,2 2,1 2

2.1 1( ) ( )

1.4 1 s j

s sW j W j

s s

ARI-14-2013


Consider , with

• variation

• From figure• for red• and for smaller τ blue

• it is represented by a rational weighting function

1st order time delay neglecting

Michael Šebek 13

omega=.01:.01:100;taumax=2;

for tau=.01:.01:taumax,plot(omega,abs(1-1./(1+tau.*j.*omega)))

hold on,end

plot(omega,abs(1-1./(1+taumax.*j.*omega)),'r--')

0( ) ( ) ( )G s G s f s max( ) 1 ( 1),0p pf s s

max( ) |1 1 ( 1) |Il s

max 2

| ( ) | ( )I Iw j l

max

max max

1( ) 1

1 1I

sw j

s s

ARI-14-2013


• Consider nominal CL system stability. Nyquist graph L0(s)=D(s)G0(s) It therefore meets the Nyquist stability criterion.

• Furthermore CL system doesn't have a pole on the imaginary axisdoesn't have a zero on the imaginary axis

• For robust stability it must also apply forfor each ω and ∆

nominalstability

Robust stability conditions - proof

Michael Šebek 14

1 ( )L s

0

0

0 0 2

0 0 2

0 0 2

0 2

0 2

0

1 ( )

1 ( )

1 ( ) 0 , ( ) 1

1 ( ) ( ) ( ) ( ) 0 , ( ) 1

1 ( ) ( ) ( ) ( ) 0 , ( ) 1

1 ( ) 1 ( ) ( ) ( ) 0 , ( ) 1

1 ( ) ( ) ( ) 0 , ( ) 1

( ) (

L j

L j

L j j

L j L j W j j

L j L j W j j

L j T j W j j

T j W j j

T j W

0 2

) ( ) 1 , ( ) 1

( ) ( ) 1

j j

T j W

01 ( )L s 01 ( ) 0,L j

ARI-14-2013


• European Southern Observatory: four 8 m telescopes (~16 m), Atacama in Chile

• Precise targeting and disturbance (wind) suppressionwith robust control methods

• Department project, Z. Hurák: AUTOMA 1/05• video

Example: VLT ESO



Example: VLT ESO

Michael Šebek 16

Model order: 60(the finite element

method)

Wind gusts

The first low damped structure mode

Model order reduction

Reduced model of order: 24

ARI-14-2013


Active and adaptive optics• plan 2015:• OWL Telescope, mirror 100m, • deformable segments• 500.000 actuators• design ?• Numerical methods?• VIDEO OWL.mpeg

Sci Fi



• For with one unstable pole

• is S stable

• however

• from Bode diagram

• It is understandable becausewe must pay something for stabilization

Example: Waterbed effect I

Michael Šebek 18

0ln ( )S j d

4( )

( 2)( 1)

L j

s s

2

2

2( )

2

s sS j

s s

2

2

2( )

2

s sS j

s s

( ) 1S j

ARI-14-2013


• Relative order condition can be omitted• Then general relationship applies

Waterbed effect I - more generally

Michael Šebek 19

unstable,00ln ( ) Re lim ( )

2

pn

is

S j d p sL s

ARI-14-2016


• let's compare the non-minimum phase transfer function

• With its minimum phase "counterpart"

• Phase lag (unstable zero) shift the graphto the red circle.

Example: Waterbed effect II.

Michael Šebek 20

1 1( )

1 1

sL j

s s

1( )

1mL j

s

L=(1-s)/(1+s)^2;Lm=1/(1+s);

t=0:2*pi/100:2*pi;fill(sin(t)-1,cos(t),'r')

nyquist(ss(L),'b',ss(Lm),'g')

L

mL

1S

ARI-14-2013


• Consider the non-minimal phase system and controlleri.e.

• Sensitivity S Bode diagram for k = 0.1, 0.5, 1.0 and 2.0

• With increasing gain also increaseinfluence of unstable zero(and sensitivity peak)

• System becomes unstable for k = 2

Example: Waterbed effect II.

Michael Šebek 21

G=(2-s)/(2+s);L01=(0.1/s)*G;L05=(0.5/s)*G;

L10=(1.0/s)*G;L20=(2.0/s)*G;

S01=1/(1+L01);S05=1/(1+L05);S10=1/(1+L10);S20=1/(1+L20);

bode(ss(S01),ss(S05),ss(S10),ss(S20),.01:.01:15)

2( )

2

sG s

s

k

s2

( )2

k sL s

s s

ARI-14-2013

exercises for lectures 14 modern frequency response methods · 14 –modern frequency response...

Documents