Off-line Robustification of Multivariable Model ... - cpnl.esigelec.frcpnl.esigelec.fr/archives/Presentation_Stoica_310108.pdf · 31/01/2008 3 Introduction Off-line state-space methodology

OffOff--line line RobustificationRobustification of Multivariableof MultivariableModel Predictive ControlModel Predictive Control

Cristina StoicaPedro Rodríguez-Ayerbe

Didier Dumur

Department of Automatic Control, SUPELEC Gif-sur-Yvette, France

GT CPNL, 31GT CPNL, 31--0101--20082008

31/01/2008 2

Introduction

Multivariable MPC

Robustness using the Youla parameter

Robustified MIMO MPCRobust stability (RS)Nominal performance

Application to a stirred tank reactor

Conclusions

ContentContent

Introduction

31/01/2008 3

IntroductionIntroduction

Off-line state-space methodology for enhancing the robustness of multivariable MPC

Initial stabilizing MIMO Model Predictive Controller

Robustification of this initial controller by convex optimization of a multivariable Youla parameter

Off-line methodology to improve robust stability under unstructured uncertainties, while respecting nominal performance specifications LMI tools

Starting point

Means

Goal

31/01/2008 4

ContentContent

IntroductionMultivariable MPC

Model descriptionCost functionPrediction equationControl law synthesis

Robustness using the Youla parameterRobustified MIMO MPCApplication to a stirred tank reactorConclusions

31/01/2008 5

Model description (MIMO system )with

Steady-state errors cancellation

Predicted output vector

Observer

∑ ∑−

=

+

=

−−

⎥⎥⎦

⎤

⎢⎢⎣

⎡+Δ+−+=+

1

0

)(

0

1 )()1()(ˆ)(ˆi

j

jk

j

l

jii lkkkik

4444 34444 21u

uuBACxACy

MultivariableMultivariable MPC MPC

)](ˆ)([)()(ˆ)1(ˆ kkkkk eeeeee xCyKuBxAx −+Δ+=+

)()1()( kkk uuu Δ+−=

⎩⎨⎧

=+=+

)()()()()1(

kkkkk

xCyuBxAx

pm×npmnnn ××× ∈∈∈ RCRBRA ,,

⎩⎨⎧

=Δ+=+

)()()()()1(

kkkkk

ee

eeee

xCyuBxAx

⎥⎦

⎤⎢⎣

⎡−

=)1(

)()(

kk

ke ux

xExtended state-space

31/01/2008 6

MultivariableMultivariable MPCMPC

Quadratic objective function minimization

withObjective function in the matrix formalism

with

∑∑−

==

+Δ++−+=1

0

2)(~

2)(~ )()()(ˆ

2

1

u

JJ

N

ii

N

Niir ikikikJ RQ uyy

2222 )()()()()()(JJJJ

kkkkkkJ r RQRQ UΘUΦUYY Δ+−Δ=Δ+−= Δ

)1()(ˆ)()()()1()(ˆ)(

−−−=Δ+−+= Δ

kkkkkkkk

r uΦxΨYΘUΦuΦxΨY

uNiik ≥∀=+Δ ,0)(u

Output prediction horizons Control horizonWeightings

)(kUΔ

0)(=

Δ∂∂

kJU

Future control actions Extended state-space and future setpoints

31/01/2008 7


Future control laws sequence

with

where

The first component of each future control sequence

with

)()()( T1T kk JJJ ΘQΦΦQΦRU Δ−

ΔΔ+=Δ

∑=

−=i

j

jii

0

BACΣ

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=

−−−−−

−

Δ

−

−

uNNNNNNN

N

N

N

N

N

212122

1

2

1

2

1

11

01

1

1 00,,

ΣΣΣΣ

ΣΣΦ

Σ

ΣΦ

CA

CAΨ

LL

MOMMLM

LL

MM

)()( kk Θμu =Δ

[ ] JJJNmmm uQΦΦQΦR0Iμ T1T

)1(, )( Δ−

ΔΔ− +=

Receding horizon principle

31/01/2008 8


Block diagram of MIMO MPC)(ˆ)()( 2 kNkk err xLyFu −+=Δ

[ ] [ ]μΦμΨLLL == 21with

)(ky⎥⎦

⎤⎢⎣

⎡0CBA)( 2Nkr +y

)(kuΔ

rF)(ku∫

)(ˆ ky

)(ˆ kexObserver

+−

[ ]⎥⎦

⎤⎢⎣

⎡ −0IKBKCA eee

eC

L

31/01/2008 9

ContentContent

IntroductionMultivariable MPCRobustness using the Youla parameter

Initial stabilizing control lawRobust stability under frequency constraintsNominal performance specifications

Robustified MIMO MPCApplication to a stirred tank reactorConclusions

31/01/2008 10

RobustnessRobustness usingusing the the YoulaYoula parameterparameter

Initial stabilizing controller

Set of stabilizing controllers

Affine dependence in

0K

Youlaparametrization

sK

Q

y′u′

sK

MIMO System

0K

w z

yu

w z

Q

⎟⎟⎠

⎞⎜⎜⎝

⎛0TTT

21

1211

y′u′

211211 QTTTTzw +=

Q

?=Q

∞ℜ∈ HQ

Convex specifications in closed-loop

31/01/2008 11

),,,( clclclcl DCBA∞H γ

0/0TT

T1

T

11

T11 pf

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

−−

−−

=∃

−

IDC0DI0BC0XA0BAX

XX

γγclcl

clcl

clcl

clcl

Robustification under unstructured uncertainties

TheoremA discrete-time system is stable and admits a norm lower than if and only if

New optimization problem


w z

uΔ

zwT

uΔ

∞ℜ∈ ∞zwQT

Hmin Small gain theorem

Transformation

into LMI1γ1

minLMI

31/01/2008 12

Nominal performance specifications as output time-domain templates for disturbances rejection


tNkkkkk ≤≤∀≤≤ 0/),()()( maxmin yyy

Time

Time-domain template for disturbance rejection

Tim

e re

spon

se

Transformation

into LMI2

γ21,

minLMILMI

31/01/2008 13

ContentContent

IntroductionMultivariable MPCRobustness using the Youla parameterRobustified MIMO MPC

Initial stabilizing control lawRobust stability under frequency constraintsNominal performance specifications

Application to a stirred tank reactorConclusions

31/01/2008 14

RobustifiedRobustified MIMO MPCMIMO MPC

⎥⎦

⎤⎢⎣

⎡

QQ

QQ

DCBA

)( 2Nkr +y )(ky)(kuΔ

rF )(ku∫

)(kd

)(kuz

uΔ

⎥⎦

⎤⎢⎣

⎡0CBA

)(kb

[ ]⎥⎦

⎤⎢⎣

⎡0CIBA

e

ee )(ˆ ky

K

)(ˆ kxeL

)(ky′)(ku′

uW

Observer

+−

+

−

+++

+−

Initial stabilizing controlMIMO MPC

Robust stabilityAdditive uncertainties

Nominal performanceTime-domain templates for disturbances rejection

∞∞= uubbz WTT

uminmin

uΔ

Block diagram of MIMO MPC with the parameterQ

)()( kk yd →

31/01/2008 15

)]()(ˆ)([)()(ˆ)1(ˆ kkkkkk eeeeee bxCyKuBxAx +−+Δ+=+


GoalMultivariable Model Predictive Control law

with the observer

Error

Weighting

Youla parameter

)(ˆ)()( kkk ee xxε −=

⎥⎦

⎤⎢⎣

⎡

ww

ww

DCBA

Wu :Extended state-space

of MIMO system

)()( kk yQu ′=′

)()( kk uzb →

)()(ˆ)()( 2 kkNkk err uxLyFu ′−−+=Δ

31/01/2008 16

Aim: find the Youla parameter which minimizes the norm convex optimization problemSub-optimal solution

for each pair a polynomial or a FIR filter

State-space representation


⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

=⎥⎦

⎤⎢⎣

⎡

mpQ

mQ

mpQ

mQ

pQQ

pQQ

QQ

QQ

QQ

QQ

dd

dd

LL

MOMMOM

LL

OO

11

111111

cc

ccb0a0

0b0a

DCBA

[ ] ijijQ

ijn

ijijQ

nQ

nn

nQ qdqq

QQQQ

Q01

1,11,11

1,1 ,,1

,0

==⎥⎥⎦

⎤

⎢⎢⎣

⎡=

⎥⎥⎦

⎤

⎢⎢⎣

⎡=

−−−

−Lc0b0I

0a

pjmiqqQQn

l

lijl

ij ,1,,1,0

===∑=

−

variable

fixed

∞bzuT∞ℜ∈ HQ

),( ji ijQ

order of Youla parameter

unknown parameters

31/01/2008 17

Closed-loop system in state-space formulation

Maximization of robust stability under :


⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

−−−

−−−−

=⎥⎦

⎤⎢⎣

⎡′′′

QwQweQw

QQeQ

QuQueQu

clcl

clcl

DDCDCDDCCBACB0K0A0DBCBCDBAA

DCBA

21

2

31 111

)()( kk uzb →

∞bzuTminuΔγ

1minLMI

0

*************

*********

*****

*

TT

TTT1

T1222

TT2

T1

T111211

T111

22T1222222

T1222

121112122111211

121112121312211111111 111

p

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

−−−

−−

−+−−−

+−+−

+−+−−−+−−−−+−−− ′′′

IDDIDCCS0TDDCCCS0TT

CR000R0BTKTATCBTAT0T0BTKTATCBTAT0TT0BSKSDBCBASSACDBACBSASSARA00R

γγ wQ

wQ

wTQ

Te

QQeQ

QQeQ

QQuQuQeQueQ

1LMI

31/01/2008 18


Nominal performance for disturbances rejectionTransfer for

Time domain templates

⎪⎩

⎪⎨

⎧

′=′

⎥⎦

⎤⎢⎣

⎡′⎥

⎥⎦

⎤

⎢⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡′

yQu

ud

0TTT

yy

yd

ydyd

21

1211

)()()( 211211 kkk dQTTdTyydydyd

+=

Affine in Q)()( kk yd → tNk ,0=

)()()( maxmin kkk yyy ≤≤

0)()()( max211211 ≤−+ kkk ydQTTdTydydyd

0)()()( min211211 ≤++ kkk ydQTTdTydydyd

2LMIManipulations

γ21,

minLMILMI

31/01/2008 19

ContentContent

IntroductionMultivariable MPCRobustness using the Youla parameterRobustified MIMO MPCApplication to a stirred tank reactor

System description & MPC parametersRobust stability under frequency constraintsNominal performance specifications

Decoupled system(2 templates)Coupled system(4 templates)

Conclusions

31/01/2008 20

Simplified MIMO model of reactor

Discretized for

MPC parameters

Weightings

Application to a stirred tank reactorApplication to a stirred tank reactor

⎥⎦

⎤⎢⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

++

++=⎥⎦

⎤⎢⎣

⎡)()(

4.012

5.011

3.015

7.011

)()(

2

1

2

1

sUsU

ss

sssYsY

Feed flow rate

Coolant flow

Reactor temperature

Effluent concentration1y

1u

2u ∞

2y

min03.0=eT

⎥⎦

⎤⎢⎣

⎡0CBA

2,2,4 === pmn

State-space representation

1

21

12

~,05.0~2,3,1

+−==

===

NNJNJ

u

u

NNN

IQIR

3.0/)7.01( 12

−−= qIWu ⎥⎦

⎤⎢⎣

⎡

ww

ww

DCBA

31/01/2008 21


MPC0 : initial MPC controller (before robustification)

MPC1 : robustnessunder frequencyconstraints only

MIMO System

1Q

y′u′0MPC

yu

uΔ

buz

MPC2 : robustness underfrequency and time-domainconstraints – 2 templates

Système MIMO

MIMO

System

2Q

y′u′0MPC

1d 1y

yu

uΔ

2d 2y

MPC3 : robustness underfrequency and time-domainconstraints – 4 templates

3Q

y′u′

MIMO

System

0MPC

yu

uΔ

1d 1y2d 2y

31/01/2008 22


Robustification using a Youla parameter of orderDecreasing norm

Increasing stability robustness

40=Qn

110-1 100 10 102

-40

-30

-20

-10

0

10

20

Sing

ular

Valu

es(d

B)

MPC0MPC1MPC2MPC3

Frequency (rad/min)

∞H

MPC0 = initial MPC

MPC1 = MPC0+LMI1

MPC2 = MPC0+LMI1+LMI2 (2 templates)

MPC3 = MPC0+LMI1+LMI2 (4 templates)

Singular values of ubT

31/01/2008 23


Time-domain responsesDisturbances rejection 3.0,5.0

3.0,5.0

21

21

==

==

dd

yy rr

0 1 2 3 4 5 60

0.5

1

1.5

Effl

uent

con

cent

ratio

n

MPC0MPC1MPC2MPC3Setpoint

Time (min)

1d 2d1d 2d

0 1 2 3 4 5 60

0.1

0.2

0.3

0.4

0.5

Rea

ctor

tem

pera

ture

MPC0MPC1MPC2MPC3Setpoint

Time (min)

Coupling influence

31/01/2008 24


Control signals

0 1 2 3 4 5 6

-1.5

-1

-0.5

0

0.5

1

Feed

flow

rate

MPC0MPC1MPC2MPC3

Time (min)

1d 2d

1d 2d

0 1 2 3 4 5 6-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

Time (min)

Coo

lant

flow

MPC0MPC1MPC2MPC3

Coupling influence

31/01/2008 25


Time-domain templates for disturbances rejection

11 yd →

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

-0.02

0

0.02

0.04

0.06

0.08

0.1

Effl

uent

con

cent

ratio

n

Time (min)

MPC0MPC1MPC2MPC3

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-0.25-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

0.25

Rea

ctor

tem

pera

ture

Time (min)

MPC0MPC1MPC2MPC3

22 yd →

31/01/2008 26


Time-domain templates for disturbances rejection

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.91

Effl

uent

con

cent

ratio

n

Time (min)

MPC0MPC1MPC2MPC3

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

-0.02

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

Rea

ctor

tem

pera

ture

Time (min)

MPC0MPC1MPC2MPC3

-0.04

12 yd → 21 yd →

31/01/2008 27

Validation of robust stability under a neglected dynamics of

corresponding to the transfer

Without robustification – unstable After robustification – stable


0 0.5 1 1.5 2 2.50

0.1

0.2

0.3

0.4

0.5

Time (min)

Tim

e re

spon

ses

MPC0MPC3Setpoint

)07.01)(7.01(1

ss ++

Coupling influence

1u

11 yu →

31/01/2008 28

ContentContent

IntroductionMultivariable MPCRobustness using the Youla parameterRobustified MIMO MPCApplication to a stirred tank reactorConclusions

31/01/2008 29

Conclusions & Conclusions & currentcurrent workwork

Complete off-line methodology which enables robustification of an initial MIMO MPCInput/output behaviour remains unchangedImproved robustness towards unstructured uncertainties through a convex optimization problem (LMI techniques)Use of four time-domain templates to manage the coupling effect in an efficient wayCompromise between robust stability and nominal performance specificationsReduced computational effort due to state-space representation

31/01/2008 30


Developing a new MATLAB toolbox (MIMOptMPC) based on the theoretical feature of this off-line robustification procedure

An user friendly and easily extensible toolbox Choice of tuning parameters and robustification options

Visualization tools enabling performances evaluation

A helpful solution for non-specialist users as well as researchers working in the field of robust MPC

31/01/2008 31


31/01/2008 32

Conclusions & current workConclusions & current work

Time

y

maxupL

minupL

mindownL

maxdownL

stTN

stTn

Documents

Off-line Robustification of Multivariable Model ... - cpnl.esigelec.frcpnl.esigelec.fr/archives/Presentation_Stoica_310108.pdf · 31/01/2008 3 Introduction Off-line state-space methodology