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Adaptive Control – Landau, Lozano, M’Saad, Karimi1

Adaptive Control

Chapter 8: Robust digital control design


Chapter 8:Robust digital control design


r(t)

m

m

AB

TS1

ABq d−

R

u(t) y(t)

Controller

PlantModel

+

-

The R-S-T Digital Controller

Plant Model:)(

)(*)(

)()()(1

11

1

111

−

−−−

−

−−−− ===

qAqBq

qAqBqqHqG

dd

A

A

nn qaqaqA −−− +++= ...1)( 1

11 )(*...)( 111

11 −−−−− =++= qBqqbqbqB B

B

nn

R-S-T Controller: )()()1(*)()()( 111 tyqRdtyqTtuqS −−− −++=

Characteristic polynomial (closed loop poles):

)()()()()( 11111 −−−−−− += qRqBqqSqAqP d


Digital control in the presence of disturbances and noise

Output sensitivity function(p y) )()()()(

)()()( 1111

111

−−−−

−−−

+=

zRzBzSzAzSzAzS yp

)()()()()()()( 1111

111

−−−−

−−−

+−

=zRzBzSzA

zRzAzSup

)()()()()()()( 1111

111

−−−−

−−−

+−

=zRzBzSzA

zRzBzS yb

)()()()()()()( 1111

111

−−−−

−−−

+=

zRzBzSzAzSzBzS yv

Input sensitivity function(p u)

Noise-output sensitivity function(b y)

Input disturbance-output sensitivity function(v y)

All four sensitivity functions should be stable !

T

R

1/S B/A

PLANT

r(t)u(t) y(t)+

-

p(t)

b(t)

++

++

(disturbance)

(measurement noise)

v(t)++


Stability of closed loop discrete time systems

The Nyquist is used like in continuous time(can be displayed with WinReg ou Nyquist_OL.sci(.m))

)()()()()(

ωω

ωωω

jj

jjj

OL eSeAeReBeH

−−

−−− =

)()()()()()()(1)(

11

1111111

−−

−−−−−−− +

=+=zSzA

zRzBzSzAzHzS OLyp

Nyquist criterion (discrete time –O.L. is stable)

The Nyquist plot of the open loop transfer fct. HOL(e-jω) traversed in the sense of growingfrequencies (from 0 to 0.5fS) leaves the critical point[-1, j0] on the left

ω =0

HBO

(e )S = 1 +H

BO(e )

yp-1

Critical point

-1

Im H

Re Hω = π

-jω-jω


The Nyquist plot of the open loop transfer fct. HOL(e-jω) traversed in the sense of growingfrequencies (from 0 et fS) leaves the critical point[-1, j0] on the left and the number of encirclements of the critical pointcounter clockwise should be equal to the number ofunstable poles in open loop.

Remarks:-The controller poles may becomeunstable if high performances arerequired without using an appropriatedesign method

-The Nyquist plot from 0.5fS to fS is the symmetric with respect to the real axisof the Nyquist plot from 0 to 0.5fS

Stability of closed loop discrete time systems

Nyquist criterion (discrete time –O.L. is unstable)

1 unstable pole in Open Loop

ω =0

ω = π

ω = 2π

-1

Im H

Re H

Stable Closed Loop (a)

ω = π

Stable Open Loop Unstable Closed Loop(b)

ω =0


Marges de robustesse

The minimal distance with respect to the critical pointcharacterizes the robustness of the CL with respect touncertainties on the plant model parameters( or their variations)

-Gain margin ΔG-Phase margin Δφ-Delay margin Δτ-Modulus margin ΔM

-1

ΔΦ

Δ1

ΔΜ

1

Crossoverfrequency

Re H

Im H

G

|HOL|=1

ωCR


( )fj

ypypOL

ezpourzRzBzSzA

zSzA

zSzSzHM

π21

1

max

1111

11

1

max

1

min

11

min

1

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

)()()(1

−−

−

−−−−

−−

−−−−−

=⎟⎟⎠

⎞⎜⎜⎝

⎛+

===+=Δ

dBMdBMdBeS jyp )( 1

maxΔ−=Δ= −− ω

Modulus margin and sensitivity function

ω

dB

Syp

Syp

Syp

-1

min

-1

= - MΔ

= MΔ

Sypmax

0

= - Sypmax


Robust stabilityTo assure stability in the presence of uncertainties (or variations)on the dynamic chatacteristics of the plant model

ωj

ypOLOLOL

ezzSzA

zPzSzA

zRzBzSzA

zSzHzHzH

−−

−−

−

−−

−−−−

−−−−−

==+

==+<−′

1

11

1

11

1111

11111

;)()(

)()()(

)()()()(

)()(1)()(Robust stabilitycondition

(sufficient cond.):

HOL – nominal F.T.; H’OL –Different from HOL (perturbed)

ω

dB

Syp

Syp

Syp

-1

min

-1

= - MΔ

= MΔ

Sypmax

0

= - Sypmax

Size of the tolerated uncertainity on HOL at each frequency (radius)

(*)

H

HOL

OL'

1 +

-1

Im H

Re H

HOL


Frequency templates on the sur sensitivity functionsThe robust stability conditions allow to define frequency templateson the sensitivity functions which guarantee the delay margin andthe modulus margin;The templates are essential for designing a good controller

Frequency template on the noise-outputsensitivity function Syb for Δτ = TS

Frequency template on the output sensitivityfunction Syp for Δτ = TS and ΔΜ = 0.5

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5-8

-7

-6

-5

-4

-3

-2

-1

0Noise-output Sensitivity Function Template

Mag

nitu

de (d

B)

Frequency (f/fs)

Template for Delay margin Δτ = Ts

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5-25

-20

-15

-10

-5

0

5

10Output Sensitivity Function Template

Mag

nitu

de (d

B)

Frequency (f/fs)

Modulus margin = 0.5Delay margin = Ts

Delay margin = Ts

Modulus margin templateDelay margin = Ts template

Output Sensitivity function


)(1

180ωjHG

OL

=Δ o180)( 180 −=∠ ωφ

1)( )(1800 =∠−=Δ crBOcr jHpour ωωφφ

ω

φτ

icr

ii

Δ=Δ min

( ) 1

maxmin

1

min)()()(1

−− ==+=Δ ωωω jSjSjHM ypypOL

Modulus margin

Gain margin

Phase margin

Delay margin

ωφτcr

Δ=Δ Several intersections points:

φφ iiΔ=Δ min If there are several intersections with the unit circle

pour

Robustness margins


Robustness margins – typical values

Gain margin : ΔG ≥ 2 (6 dB) [min : 1,6 (4 dB)]

Phase margin : 30° ≤ Δφ ≤ 60°

Delay margin : fraction of system delay (10%) or of time response (10%) (often 1.TS)

Modulus margin : Δ M ≥ 0.5 (- 6 dB) [min : 0,4 (-8 dB)]

A modulus margin Δ M ≥ 0.5 implies ΔG ≥ 2 et Δφ > 29°Attention ! The converse is not generally true

The modulus margin defines also the tolerance with respectto nonlinearities


Good gain and phase marginBad delay margin

Robustness margins

Good gain and phase marginBad modulus margin

.


ω

dB

Syp

Syp

Syp

-1

min

-1

= - MΔ

= MΔ

Sypmax

0

= - Sypmax

( )fj

ypypOL

ezpourzRzBzSzA

zSzA

zSzSzHM

π21

1

max

1111

11

1

max

1

min

11

min

1

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

)()()(1

−−

−

−−−−

−−

−−−−−

=⎟⎟⎠

⎞⎜⎜⎝

⎛+

===+=Δ

dBMdBMdBeS jyp )( 1

maxΔ−=Δ= −− ω

Modulus margin and sensitivity function

Minimum distance with respect to thecritical point

Critical regionfor design


Correspondance Output Sensitivity Nyquist Plot

-1

ΔΦΔΜ

1

Crossoverfrequency

Re H

Im H

|HOL|=1

ωCR

ω

dB

Syp

Syp

Syp

-1

min

-1

= - MΔ

= MΔ

Sypmax

0

= - Sypmax


– The open loop being stable, one has the property:

∫ =−SS

f.πf/fj

yp df)(eS50

0

2 0log

The sum of the areas between the curve of Syp and the axis 0dB taken withtheir sign is null

Disturbance attenuation in a frequency region implies amplificationof the disturbances in other frequency regions!

Properties of the output sensitivity function


Augmenting the attenuation or widening the attenuation zone

Reduction of the robustness(reduction of the modulus margin)

Higher amplification of disturbancesouside the attenuation zone



Robust stabilityTo assure stability in the presence of uncertainties (or variations)on the dynamic chatacteristics of the plant model

ωj

ypOLOLOL

ezzSzA

zPzSzA

zRzBzSzA

zSzHzHzH

−−

−−

−

−−

−−−−

−−−−−

==+

==+<−′

1

11

1

11

1111

11111

;)()(

)()()(

)()()()(

)()(1)()(Robust stabilitycondition

(sufficient cond.):

HOL – nominal F.T.; H’OL –Different from HOL (perturbed)

ω

dB

Syp

Syp

Syp

-1

min

-1

= - MΔ

= MΔ

Sypmax

0

= - Sypmax

Size of the tolerated uncertainity on HOL at each frequency (radius)

(*)

H

HOL

OL'

1 +

-1

Im H

Re H

HOL

1111 )()()( −−−− −′< zHzHzS OLOLyp


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

)()(

)()(

)()(

)()()()(

)()()()(

11

1111

1

1

1

1

1

1

11

11

11

11

−−

−−−−

−

−

−

−

−

−

−−

−−

−−

−− +<−

′′

⋅=−′′

zSzAzRzBzSzA

zAzB

zAzB

zSzR

zSzAzRzB

zSzAzRzB

)()()(

)()()(

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

)()( 11

11

1

11

1111

1

1

1

1−−

−−

−

−−

−−−−

−

−

−

−

==+

<−′′

zSzRzA

zPzRzA

zRzBzSzAzAzB

zAzB

up

Sup dB

Limitation of the actuator stress

00.5f e

Sup

-1

(Size of tolerated uncertainties)

Tolerance to plant additive uncertainty

H’OL HOL

(*)

From previous slide :

G’ G

G’ G

(**)

1111 )()(')(−−−− −< zGzGzSup


Tolerance to plant normalized uncertainty(multiplicative uncertainty)

)()()(

)()()(

)()()()(

)()(

)()(

)()(

11

11

1

11

1111

1

1

1

1

1

1

−−

−−

−

−−

−−−−

−

−

−

−

−

−

==+

<−

′′

zSzRzB

zPzRzB

zRzBzSzA

zAzB

zAzB

zAzB

yb

From (**), previous slide:

The inverse of the modulus of the “complementary sensitivity function”gives at each frequency the tolerance with respect to “normalized(multiplicative) uncertainty”

Relation between additive and multiplicative uncertainty:

)'1()'('G

GGGGGGG −+=−+=


Important message

Large values of the modulus of the sensitivity functions in a certain frequency region

Low tolerance to model uncertainty

Critical regions for control designNeed for a good model in these regions


Small gain theorem

S1

S2

-u1

u2

y1

y2

11 <∞

S

12 ≤∞

S

S1: linear time invariant (state x)11 <

∞S

S2: 12 ≤∞

S

Then:

0)(lim;0)(lim;0)(lim 11 ===∞→∞→∞→

tytutxttt

It will be used to characterize “robust stability”


Description of uncertainties in the frequency domain

Re H

Im H

Uncertainty disk(at a certain frequency)

1) It needs a description by a transfer function which may have any phase but a modulus < 12) The size of the radius will vary with the frequency and is characterized by a transfer function


Additive uncertainty

)()()()(' 1111 −−−− += zWzzGzG aδ

)( 1−zδ any stable transfer function with 1)( 1 ≤∞

−zδ)( 1−zWa a stable transfer function

∞

−

∞

−−−− =−=− )()()(')()(' 111

max

11 zWzGzGzGzG a

ABzHSRK d /;/ −==

δ aW

K G-

+

+

aWupS−

δ-

1)()( 11 <∞

−− zWzS aupRobust stability condition:

Apply small gain theorem


Multiplicative uncertainties

[ ])()(1)()(' 1111 −−−− += zWzzGzG mδ

)( 1−zδ any stable transfer function with 1)( 1 ≤∞

−zδ)( 1−zWm a stable transfer function

)()()( 111 −−− = zWzHzW ma

mWybS−

δ-

δ

K- +

+

G

mW

1)()( 11 <∞

−− zWzS mybRobust stability condition:


Feedback uncertainties on the input

[ ])()(1)()('

11

11

−−

−−

+=

zWzzGzG

rδ)( 1−zδ any stable transfer function with 1)( 1 ≤

∞

−zδ

)( 1−zWr a stable transfer function

rWypS−

δ-

1)()( 11 <∞

−− zWzS rypRobust stability condition:

δ

K- +

-

G

mW


Robust stability conditions

),(', δWHH Ρ∈ Family (set) of plant modelsRobust stability :The feedback system is asymptotically stable for all the plant models belonging to the family ),( δWΡ

• Additive uncertainties

1)()( 11 <∞

−− zWzS aup πωωω ≤≤< −−− 0)()( 1j

a

j

up eWeS

• Multiplicative uncertainties1)()( 11 <

∞

−− zWzS myb πωωω ≤≤< −−− 0)()( 1j

m

j

yb eWeS

• Feedback uncertainties on the input (or output)1)()( 11 <

∞

−− zWzS ryp πωωω ≤≤< −−− 0)()( 1j

r

j

yp eWeS


Im G

Re Gω = π

G (e )-j ω

uncertaintydisk

δW

ω = 0

Robust Stability

Family of plant models:),,(' xyWGFG δ∈

G – nominal model; 1)( 1 ≤∞

−zδ

)( 1−zWxy - size of uncertainty

Robust stability condition:a related sensitivity

functiona type of uncertainty

1<∞xyxyWS

⇓1−

< xyxy WSdefines the size of thetolerated uncertainty

defines an upper templatefor the modulus of the

sensitivity function

There also lower templates (because of the relationship between various sensitivity fct.)


Robust stability and templates for the sensitivity functions

Robust stability condition:

•The functions (the inverse of the size of the uncertainties) define an “upper” template for the sensitivity functions

• Conversely the frequency profile of can be interpreted interms of tolerated uncertainties

11 )( −−zW

πωωω ≤≤< −−− 0)()( 1j

z

j

xy eWeS

)( ωj

xy eS −

ω

dB

Syp

Syp

Syp

-1

min

-1

= - MΔ

= MΔ

Sypmax

0

= - Sypmax

Tolerated feedback uncertainty on the input

Sup dB

00.5f e

Sup

-1

Tolerated additive uncertainty


( G ’ = G + δWa )

Sup dB

actuator effort

size of the tolerated additive uncertainty Wa

00.5f s

Sup-1

Templates for the Sensitivity Functions

Output SensitivityFunction

Input SensitivityFunction

Syp max= - MΔSyp dB

0.5fs0

delaymarginnominal

perform.

Dangerous zones.Need for good models inthese regions


Templates for the output sensitivity functions Syp

Syp dB

0,5fe

Syp max= - MΔ

0

Performances

Robustness

Syp dB

0,5fe

Syp max= - MΔ

0

Attenuation zone

openingthe loop


Shaping the sensitivity functions

1. Choice of the dominants et auxiliary poles of the closed loop2. Choice of the fixed part of the controller (HS and HR )3. Simultaneous choice of the fixed parts and the auxiliary poles

Procedure:

Basic shaping : use 1 and 2Fine shaping: use 3

There exist also tools for automatic sensitivity function shapingbased on convex optimization (Optreg from Adaptech)

Tools for sensitivity shaping: WinReg (Adaptech) and ppmaster.m


Pole placement with sensitivity functions shaping

Performance specification for pole placement :• Desired dominant poles for the closed loop• The reference trajectory (tracking reference model)

Questions:• How to take into account the specifications in certain frequencyregions?

• How to guarantee the robustness of the controllers ?• How to take advantage from the degree of freedom forthe maximum number of poles which can be assigned ?

Answer:Shaping the sensitivity functions by:

- introducing auxiliary poles- introducing filters in the controllers


Sensitivity functions - review

)()()()()()()( 1111

111

−−−−−

−−−

+=

qRqBqqSqAqSqAqS dyp

)()()()()()()( 1111

111

−−−−−

−−−

+−=

qRqBqqSqAqRqAqS dup

)()(')( 111 −−− = qHqRqR R )()(')( 111 −−− = qHqSqS S

)()()()()()()( 1111111 −−−−−−−− ==+ qPqPqPqRqBqqSqA FDd

Pre specified parts (filters)

Output sensitivity function:

Input sensitivity function:

Controller structure :

Dominant and auxiliary filters:

Study of the properties of the sensitivity functions in the frequency domain: q=z=ejω



P.1- The modulus of the output sensitivity function at a certainfrequency gives the amplification or attenuation factor of thedisturbance on the output

Syp(ω) < 1(0 dB) attenuation Syp(ω) > 1 amplification

Syp(ω) = 1 operation in open loop

P.2 ( ) 1max

)(−

=Δ ωjSM ypModulus margin


P.3 – The open loop (KG) being stable one has the property:

∫ =−SS

f.πf/fj

yp df)(eS50

0

2 0log

The sum of the areas between the curve of Syp and the axis 0dB taken withtheir sign is null

Disturbance attenuation in a frequency region implies amplificationof the disturbances in other frequency regions!



The asymptotically stable auxiliary poles (PF) lead in generalto the reduction of in the frequency regionscorresponding to the attenuation regions for 1/PF


)( ωjS yp

FPnF qpqP )1()( 11 −− ′+= 05.05.0 −≤′≤− p

DF PPP nnn −≤

In many applications the introduction of damped high frequency auxiliarypoles is enough for assuring the required robustness margins


22

11

22

11

1

1

11

)(

)(−−

−−

−

−

++

++=

qqqq

qP

qH

i

i

F

S

ααββ

Obtained by the discretization of :

200

2

200

2

22

)(ωωζωωζ

++

++=

ssss

sFden

num1

1

112

−

−

+−

=zz

Ts

e

with:

produce and attenuation (hole) at the normalized discretized frequency:

⎟⎠

⎞⎜⎝

⎛=

2arctan2 0 e

discTω

ω with attenuation: ⎟⎟⎠

⎞⎜⎜⎝

⎛=

den

numtM

ζζ

log20 dennum ζζ <( )

and has negligible effects at f << fdisc and at f >> fdisc


Simultaneous introduction of a fixed part HSi and of a pair of auxiliary poles PFi of the form:


For details see Landau: Commande des Systèmes, HermesEfective computation using: filter22.sci (.m)



0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5-25

-20

-15

-10

-5

0

5

Syp Magnitude Frequency Responses

Frequency (f/fs)

Mag

nitu

de (d

B)

ω = 0.4 rad/secω = 0.6 rad/secω = 1 rad/secTemplate for Modulus marginTemplate for Delay margin = Ts

Augmenting the attenuation or widening the attenuation zone

Higher amplification of disturbancesoutside the attenuation zone

Reduction of the robustness(reduction of the modulus margin)



P.4 – Cancellation of the disturbance effect at a certain frequency:

sjj

Sjjj ffeSeHeAeSeA /20)()()()()( ; πωωωωωω ==′= −−−−−

{Zeros of Syp Allows introduction of zeros at desired frequencies


0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5-40

-35

-30

-25

-20

-15

-10

-5

0

5


Frequency (f/fs)

Mag

nitu

de (d

B)

HS = 1 - q-1

HS = (1 + q-2)(1 - q-1)


0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5-25

-20

-15

-10

-5

0

5


Frequency (f/fs)

Mag

nitu

de (d

B)

HR = 1HR = 1 + q-2

P.5 - at frequencies where:)0(1)( dBjS yp =ω

sjj

Rjjj ffeReHeBeReB /20)()()()()( ** ; πωωωωωω ==′= −−−−−


Allows introduction of zeros at desired frequencies


0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5-25

-20

-15

-10

-5

0

5


Frequency (f/fs)

Mag

nitu

de (d

B)

PF = 1PF = (1 - 0.375q-1)2

Template for Modulus marginTemplate for Delay margin = Ts

P.6 – Asymptotically stable auxiliary poles (PF) lead (in general) to the reduction of in the attenuationband of 1/PF

)( ωjS yp

FPnF qpqP )1()( 11 −− ′+= 05.05.0 −≤′≤− p

DF PPP nnn −≤

In many applications, introduction of high frequency auxiliary polesis enough for assuring the required robustness margins



P.7 – Simultaneous introduction of a fixed part HSi and of a pairof auxiliary poles PFi having the form:

22

11

22

11

1

1

11

)(

)(−−

−−

−

−

++

++=

qqqq

qP

qH

i

i

F

S

ααββ

resulting from the dicretization of :

200

2

200

2

22

)(ωωζωωζ

++

++=

ssss

sFden

num1

1

112

−

−

+−

=zz

Ts

e

with:

introduces an attenuation at the normalized discretized frequency:

⎟⎠

⎞⎜⎝

⎛=

2arctan2 0 e

discTω

ω with the attenuation: ⎟⎟⎠

⎞⎜⎜⎝

⎛=

den

numtM

ζζ


and with negligible effect at f << fdisc and at f >> fdisc



Effective computation with the function: filter22.sci (.m)


0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5-25

-20

-15

-10

-5

0

5


Frequency (f/fs)

Mag

nitu

de (d

B)

HS = 1, PF = 1HS = ( ω = 1.005, ζ = 0.21), PF = ( ω = 1.025, ζ = 0.34)


0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5-40

-30

-20

-10

0

10

20Sup Magnitude Frequency Responses

Frequency (f/fs)

Mag

nitu

de (d

B)

HR = 1HR = 1 + 0.5q-1

HR = 1 + q-1

P.1 – Cancellation of the disturbance effect on the input at a certain frequency (Sup = 0):

sjj

Rj ffeReHeA /20)()()( ; πωωωω ==′ −−−

101)( 11 ≤<+= −− ββqqH R( active at 0.5fS)

Properties of the input sensitivity function

Allows introduction of zeros at desired frequencies

Rem: The system operate in open loop at this frequency


P.2 – At frequencies where:

sjj

Sj ffeSeHeA /20)()()( ; πωωωω ==′ −−−

)()()( ω

ωω

j

jj

up eBeAeS

−

−− =0)( =ωjS yp

One has:

Consequence : strong attenuation of the disturbances should bedone only in the frequency regions where the system gainis enough large ( in order to preserve robustness and avoidtoo much stress on the actuator)

Inverse ofthe systemgain

Remember: gives the tolerance with respect to additive uncertainties on the model (high = weak robustness)

1)(

−ωjSup

)( ωjSup



P.3 – Simultaneous introduction of a fixed part HRi and of a pairof auxiliary poles PFi having the form:

resulting from the dicretization of :

200

2

200

2

22

)(ωωζωωζ

++

++=

ssss

sFden

num1

1

112

−

−

+−

=zz

Ts

s

with:

introduces an attenuation at the normalized discretized frequency:

⎟⎠

⎞⎜⎝

⎛=

2arctan2 0 e

discTω

ω with the attenuation: ⎟⎟⎠

⎞⎜⎜⎝

⎛=

den

numtM

ζζ


and with negligible effect at f << fdisc and at f >> fdisc

22

11

22

11

1

1

11

)(

)(−−

−−

−

−

++

++=

qqqq

qP

qH

i

i

F

R

ααββ



Shaping the sensitivity functions - Example I

sTdqBqA e 1;2;3.0;7.01 11 ===−= −−Plant:

• Integrator• Dominant poles: discretization of a cont. time 2nd order system : ω0 = 1 rad/s, ζ = 0.9

Controller A :Attenuation band: 0 up to 0.058 Hz but ΔM < -6 dB and Δτ < Ts

Objective: same attenuation band but with ΔM > -6 dB and Δτ > Ts- insertion of auxiliary poles:

Specifications:

( )214.01 −−= qPF

Controller B : good margins but reduction of the attenuation band-insertion of pole-aero filter HS/PF centered at ω0 = 0.4 rad/s (0.064 Hz)

Controller C : good attenuation band but Syp > 6 dB - larger (slower) auxiliary poles (0.4 0.44)

Controller D : Correct


Shaping the sensitivity functions - Example I

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5-25

-20

-15

-10

-5

0

5

10Syp Magnitude Frequency Responses

Frequency (f/fs)

Mag

nitu

de (d

B)

ABCDTemplate for Modulus marginTemplate for Delay margin = Ts


Shaping the sensitivity functions - Example II

Plant (integrator): sTdqBqA s 1;2;5.0;1 11 ===−= −−

q-dBA

u(t) y(t)

Sinusoidal disturbance (0.25 Hz)

Low frequencies disturbances

+

+

+

Specifications:1. No attenuation of the sinusoidal disturbance at (0.25 Hz)2. Attenuation band at low frequencies : 0 à 0.03 Hz3. Disturbances amplification at 0.07 Hz: < 3dB 4. Modulus margin > -6 dB and Delay margin > T5. No integrator in the controller



- Fixed parts design : 1;1 2 =+= −SR HqH

Opening the loop at 0.25 Hz

-Dominant poles: discretization of a cont. time 2nd order system:ω0 = 0.628 rad/s, ζ = 0.9

Controller A : the specs. at 0.07 Hz are not fulfilled- insertion of a pole-zero filter HS/PF centered at ω0 = 0.44 rad/s

Controller B : Attenuation band smaller than that specified- dominant poles acceleration: ω0 = 0.9 rad/s

Controller C : Correct



0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5-15

-10

-5

0

5

10Syp Magnitude Frequency Responses

Frequency (f/fs)

Mag

nitu

de (d

B)

ABCTemplate for Modulus marginTemplate for Delay margin = Ts


Robust Controller Design

Pole placement with sensitivity functions shaping

FD PPP =

RHRR '=

SHSS '=

Nominal performance: SRD HandHofpartandP

Allow to shape the sensitivity functions

-IterativeChoosing and using band stop filters(matlab toolbox « ppmaster » )

FjSjFiRi PHPH /,/FP

Several approaches to design :

-Convex optimization(see Langer, Landau, Automatica, June99, Optreg (Adaptech) )


Position Control by means of a Flexible Transmission

Φm

LOAD

Φref

MOTORAXIS

AXISPOSITION

DAC

ADCCONTROLLER

DCMOTOR

R-S-TCONTROLLER

POSITIONTRANSDUCER

u(t) y(t)

For details see next slide and book


Position Control by means of a Flexible Transmission

12 13 14 15 16 17 18-1.5

-1

-0.5

0

0.5

1

1.5Flexible Transmission: Output

Time (s) (Ts = 50 ms)A

mpl

itude

(Vol

t)

12 13 14 15 16 17 18

-0.2

0

0.2

0.4

Flexible Transmission: Control Signal

Time (s) (Ts = 50 ms)

Am

plitu

de (V

olt)

5 10 15 20 25-0.2

0

0.2

0.4

0.6

0.8

1

Flexible Transmission: Output


Am

plitu

de (V

olt)

PositionReference

5 10 15 20 25

0

0.5

1

Flexible Transmission: Control Signal


Am

plitu

de (V

olt)

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5-30

-25

-20

-15

-10

-5

0

5

10

15

20Flexible Transmission: Sup Magnitude Frequency Responses

Frequency (f/fs)

Mag

nitu

de (d

B)

HR = 1HR = 1, 4 aux. poles in α = 0.2HR = 1 + q-1, 4 aux. poles in α = 0.2

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5-20

-15

-10

-5

0

5

10Flexible Transmission: Syp Magnitude Frequency Responses

Frequency (f/fs)

Mag

nitu

de (d

B)

HR = 1HR = 1, 4 aux. poles in α = 0.2HR = 1 + q-1, 4 aux. poles in α = 0.2

Template for Modulus marginTemplate for Delay margin = Ts

RegulationTracking


MIRROR

DETECTOR

RIGID FRAMES

LIGHTSOURCE

POT.

ENCODER

SERVO.

LOCALPOSITION

COMPUTERTACH.

ALUMINIUM

360° Flexible Arm


Frequency characteristics Poles-Zeros

360° Flexible Arm

(Identified Model)


1 2 3 4 5 6 7 8 9-30

-25

-20

-15

-10

-5

0

5

10

15

20

25

Frequence [Hz]

Mod

ule

[dB]

Syp - Sensibilité perturbation-sortie

A B

D C

gabarit

1 2 3 4 5 6 7 8 9-30

-20

-10

0

10

20

30

40

50

60

Frequence [Hz]M

odul

e [d

B]

Sup - Sensibilité perturbation-entrée

A

D

B

C

gabarit

A- without auxiliary polesB- with auxiliary polesC- with stop band filterD- with stop band filter

11 / FS PH22 / FR PH

Shaping the Sensitivity Functions

Output Sensitivity Function - Syp Input Sensitivity Function - Sup

template

template

adaptive control - gipsa-labioandore.landau/adaptivecontrol... · adaptive control – landau,...

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