Robustness margins and robustification of nominal …folk.ntnu.no/torarnj/Rodriguez-Ayerbe ECC 2016.pdf · XXNN ... Robustness margins and robustification of nominal explicit MPC

P. Rodríguez-Ayerbe, S. Olaru, A. Nguyen, R. Koduri

L2S, CentraleSupélec – CNRS – UPSUniversité Paris-Saclay

29 June 2016

European Control Conference

Robustness margins and robustification of nominal explicit MPC

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Outline

Problem statement

Robustness margins• Polytopic uncertainty

• Gain margin

• Neglected dynamics

Robustification of explicit solutions

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Problem statement

A nominal discrete time linear systems( 1) ( ) ( )( ) ( )

x k Ax k Bu ky k Cx k

A function defined over a polyhedral partition

: ppwau X

N

ni I iX X

1

( )pwa i i i

p n pi i

u x G x g for x X

G g

Is a piecewise affine control law over the partition X

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Polyhedral state partition

2

2

1) ,

2) is polyhedral

3) int( ) int( ) with , ( , )

4) ( , ) are neighbours if ( , ) ,

and dim( ) 1

Ni I i

i N

i j N

i j N

i j

X X N N

X i I

X X i j i j I

X X i j I

i j X X n

A polyhedral partition of a compact set is defined as follows: nX

Problem statement

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Problem statement• Several popular techniques lead to such a problem formulation,

among which we can mention for example the Model Predictive Control in its explicit form (Bemporad et al, Seronet al, Tondel et al., Olaru and Dumur).

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Robustness margins and robustification of nominal explicit MPCPr

ésen

tatio

n

Goals:

• To analyse the inherent robustness of PWA control towards model uncertainties → robustness margin problems.

• To further tune the original controller to improve robustness margin.

• Synthesize explicit solution with good robustness margin.

Problem statement

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Outline

Problem statement


• Gain margin



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ésen

tatio

n

Robustness margins• Nominal closed loop

The nominal closed loop dynamics represent a PWA system:

0 0 0( 1) ( ) ( ) for ( )pwa i i ix k f x k A B G x k B g x k X

• Positive invariance

0

A set is posive invariant with respect to the system ( 1) ( ), if for any ( ) ,

( )

n

pwa

Xx k f x k w x k X

x k X k

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ésen

tatio

n

Robustness margins• Assumptions

• The set X is a polytope

• The set X is positive invariant with respect to the nominal model

• The control is continuous

• The origin is the only fixed point of the nominal dynamics

• The origin is asymptotically stable with X as basin of attraction

: n mpwau

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ésen

tatio

n

Robustness margins• Robust invariance for polytopic uncertainty

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

x k A k x k B k u ky k Cx k

1 1[ ( ) ( )] ([A , B ], [A ,B ])L LA k B k conv

[ ( ) ( )] robA k B k

[ ( ) ( )] robA k B k

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ésen

tatio

n

Robustness margins• Robust invariance for polytopic uncertainty

rob is polytopic

can be computed using vertex or half-space representation ofrob,

Ni I iX X N N

rob

1, , 1

LL T

j j j i i j j i i Nj

A B G X B g X I

1

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ésen

tatio

n

Robustness margins• For half-space representation:

Lrob =Proj P

11

1

,, ,

,

L

j j j i i ij

N NL

i i j j ij

F A B G M FP M M i I

M h h F B g

nX x Fx h ni i iX x F x h

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i iF x h

2P x Fx h

1 2

1 2

1 2

. .( ) 0,

P P M s tvec MMF FMh h

1 ( ) ( )( )i i i iP x F A k B k G x g h

Positive Invariance if : 1 2P P

Robustness margins

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ojet

de

rech

erch

e

Robustness margins• Numerical example

1 2 3

1 2 3

1 1 2 1 1.5 10.9 0.5 1.5 1.5 3.8 1

11

A A A

B B B

In presence of constraints on the control variable and the output variable

0.2 0.20.5 0.5

k

k

uy

With the nominal model chosen to synthesize a PWA controller law

0 1 2 3

0 1

0.5 0.3 0.2A A A AB B

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e


The partition of the obtained PWA control law

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e


Ψrob projected on the plane [α1,α2]. Note that α3 = 1-α1-α2

1

2

Nominal model

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ojet

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e

Robustness margins• Gain margin

The gain margin is represented by the set , such that:

( 1) ( ) ( ) ( ) , ( )m K pwa Kx k Ax k B I diag u x k X x k X and K

mK

1

,

p

qq

mq q

K K

K z u U Mz u

Proj

( ), ,

( )

q U q

m rq

pwa

U H

x kH u and A B B u W

u x k

W stores the vertices of X : 1, [ , ]n rrW W w w

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ojet

de

rech

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Robustness margins• Gain margin: numerical example

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ojet

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Robustness margins• Gain margin: numerical example

2

1

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Robustness margins


The augmented model is:(1 )

( 1) ( ) ( )0e e

A B Bx k x k u k

( ) ( ) 0 ( ) for ( )e i e i iu k x k G x k g x k X

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ojet

de

rech

erch

e

Robustness margins


1 1 2 2

Proj

, (1 )r pe e e e

T

T AV BU A V B U W

W stores the vertices of X, Ve stores de vertices of Xi :

(1 )( 1) ( ) ( )

0e e

A B Bx k x k u k

Polytopic description of extended model

1 1 2 1

1 1 2 2

0 00 0 1 0 1 0

[ ] (1 )[ ] [ ]e e

A B A BA B A B

A B A B A B

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ojet

de

rech

erch

e

Robustness margins

• Neglected dynamics : numerical example

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Outline

Problem statement


• Gain margin



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Robustification of linear control law

Youla-Kučera parameterisation

Para

mét

risa

tion

de Y

oula

-Kuč

era


Stable transferts

Stabilizing controllers

Youla-Kučera parametrisation

Linear time invariant transfers set

Linear time invariant transfers set

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Q2(z-1) modifies tracking performance

Q1(z-1) modifies regulation dynamics

Processu

G

Observer

x̂

y

xCyy ˆ~

Q1

yref

u~

d

Fr

Q2

Para

mét

risa

tion

de Y

oula

-Kuč

era

Youla-Kučera parameterisationRobustification of explicit solutions

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• Youla-Kučera parametrisation can not be directly used to robustify a PWA controller because the continuity of the control law is lost

• The following approach is follow:

1. Robustification of unconstrained controller2. Find the perturbance/noise model associated to the obtained

controller3. Use the new model to modify or the initial PWA controller

- This modification conserves the initial tracking behaviour- The number of partitions is kept, as depends on the constraints

Loi

s exp

licite

s


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With white noise

Initial model

Cost function minimization without constraints

Q Parameter

ttt

tttt

eCxyKeBuAxx

1

1

1

* minargN

k ptktptktptNtu uRQxxPuk

k

tt

ttt

xLuKyxBLKCAx

ˆˆ)(ˆ 1

Initial unconstrained

controller

tQtQ

tQQt

tQ

Q

tQ

t

QQ

QQ

tQ

t

yDXx

CCDLu

yB

BDKXx

ACBBCCBDBLKCA

Xx

ˆ

ˆˆ

1

1

Robustified controller

1

1

* minargN

k ptktptktptNtu uRQxxPuk

k

ttv

t

C

vt

t

K

vt

B

tv

t

A

v

v

x

tv

t

exx

CCy

eBK

uB

xx

AKCA

xx

e

eeete

1

11

1

001

teet

teteeeeeete

xLuyKxLBCKAx

ˆˆ)(ˆ 1

Find Av, Bv, Cv to obtain

With white noise

teAugmented model

te

(C1)

(C2)

)()( 21 CC

Loi

s exp

licite

s


Cost function minimization without constraints

Unconstrained controller

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),(

0

0

0

1

2

vvvv

vQvQ

vvvQ

eQ

CAFHL

CDLC

CBAA

KLD

Non-linear equations :• depends on cost function• Optimisation techniques can be used to solve the non-linear equation system• The existence of the solution haven't been proved yet

Two controllers are equivalents if the following equations are verified :

),( vvv CAF

Loi

s exp

licite

s


ECC – 29 June 2016 29


Buck converter example

102

103

104

105

-50

-45

-40

-35

-30

-25

-20

-15M

agni

tude

(dB

)

Pulsation (rad/s)

InitialRobustified

Unconstrained controller robustification towards additive disturbances

Loi

s exp

licite

s


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-40-20

020

40

-40-20

020

40-10

-5

0

5

10

x3x4

x 6

Active region

Initial controller Robustified controller

iL il iL il 1 76,0014,0093,006100 , 0 0139,0074,076,0014,0093,006100 , 0

2 00001 4,0 0000001 4,0

3 839,1007,0123,00890005,1 , 4,0 024,0093,0839,1007,0123,00890005,1 , 4,0

4 00001 4,0 0000001 4,0

5 839,1007,0123,00890005,1 , 4,0 024,0093,0839,1007,0123,00890005,1 , 4,0

6 00001 4,0 0000001 4,0

7 00001 4,0 0000001 4,0

8 00001 4,0 0000001 4,0

9 00001 4,0 0000001 4,0

-25 -20 -15 -10 -5 0 5 10 15 20 25-25

-20

-15

-10

-5

0

5

10

15

20

25Regions du correcteur pwa

courant I

tens

ion

V c

Robustified PWA controller, x1=0, x2=0, x5=0, x7=0Initial PWA controller,

x1=0, x2=0, x5=0

Loi

s exp

licite

s

Buck converter example


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4 5 6 7 8 9 10

x 10-3

5

10

15

20Tension de sortie sans filtre de mesure

InitialRobustifié

4 5 6 7 8 9 10

x 10-3

-0.2

0

0.2

0.4

0.6Rapport cyclique.

temps (s)

4 5 6 7 8 9 10

x 10-3

5

10

15

20Tension de sortie avec filtre de mesure.

InitialRobustifié

4 5 6 7 8 9 10

x 10-3

-0.2

0

0.2

0.4

0.6Rapport cyclique.

temps (s)

Output voltage step response

Perturbed processNominal process

Loi

s exp

licite

s

Buck converter exampleRobustification of explicit solutions

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References

Loi

s exp

licite

s

N. A. Nguyen, S. Olaru, P. Rodriguez-Ayerbe, “Explicit robustness and fragilitymargins for linear discrete systems with piecewise affine control law”,Automatica, 2016, Vol. 68, pp. 334–343.

P. Rodriguez-Ayerbe and S. Olaru, “On the disturbance model in therobustification of explicit predictive control”, International Journal ofSystems Science, 2013, Vol. 44(5), pp. 853–864.

Documents

Robustness margins and robustification of nominal …folk.ntnu.no/torarnj/Rodriguez-Ayerbe ECC 2016.pdf · XXNN ... Robustness margins and robustification of nominal explicit MPC