H∞ Multi-objective and Multi-Model MIMOcontrol design for Broadband noise

attenuation in a 3D enclosure

Paul LOISEAU, Philippe CHEVREL, Mohamed YAGOUBI, Jean-Marc DUFFAL

Mines Nantes, IRCCyN & Renault SAS

March 2016

Content

1 IntroductionGeneral contextPhD objectiveState of ArtScope of the presentation

2 System to control

3 Control Strategy

4 Results

5 Conclusions and Perspectives

2

General ContextBrief ANC overview

Duct

I Propagative wavesI Feedforward + feedback

Headphone

I SISO controlI Co-located actuator and sensor

Headrest

I SISO controlI Co-located actuator and sensor

3

General ContextActive Noise Control (ANC) in a cavity

Cavity

Feedback

Feedforward

Sensor

Feedback

Feedforward

Cavity

Sensor

Characteristics of ANC in a cavityI Stationary wavesI Actuators and sensors co-located or notI feedback or feedback + feedforwardI d narrow or broadband noiseI SISO or MIMO control

4

PhD objectiveActive control of broadband low frequency noise in car cabin

Engine noise

Aeroacoustic noise

ROAD noise

(Line spectrum)

(Low frequency, Broadband spectrum)

(Mainly in high frequency)

I Passive treatments for low frequency noise ⇒ Addition of weightI Active Noise Control (ANC) is a great opportunity to simultaneously:

I Reduce road noiseI Achieve car weight reduction

5

PhD objectiveActive Noise Control of broadband noise

Cavity

Feedback Feedback

Cavity

ANC problem characteristicsI 3D enclosureI Actuators and sensors not co-locatedI No measure of w is availableI d broadband low frequency noise

Limitations involvedI Waterbed effect (Bode integral)I Non minimum phase zeros

6

State of ArtAdaptive feedforward control (FxLMS)

1

1T. Sutton, S. J. Elliott, M. McDonald, et al., “Active control of road noiseinside vehicles”, Noise Control Engineering Journal, vol. 42, no. 4,pp. 137–147, 1994.

7

State of ArtInternal Model Control (feedback)

2

2J. Cheer, “Active control of the acoustic environment in an automobilecabin”, PhD thesis, University of Southampton, Southampton, 2012, p. 346.

8

Scope of the presentation

Feedback

Cavity ProblemI Attenuate broadband low frequency noise;I In a closed cavity;I by feedback.

Goal of the presentationCompare SISO and MIMO achievable performances.

9

Content

1 IntroductionGeneral contextPhD objectiveState of ArtScope of the presentation

2 System to controlExperimental Set upIdentification

3 Control StrategyControl problem formulationMulti-objective optimizationController StructureInitialization

4 Results

5 Conclusions and Perspectives

10

Content

1 Introduction

2 System to controlExperimental Set upIdentification

3 Control Strategy

4 Results

5 Conclusions and Perspectives

11

Experimental set up

Top view of the cavity

RC filter

Preamplifier

ADC DAC

Amplifier

Acquisition CardNI PCIe 6259 Cavity characteristics

I One predominant dimension: 1D acoustic field in low frequency;I One biased side: Attenuation of the first longitudinal mode;I Frequency complexity: Similar to vehicle one.

12

MIMO IdentificationFrequency Domain, Continuous time model

IdentificationI Algorithm: Subspace;I Model structure: Modal;I Frequency range: [20-1000]Hz;I Order: 80.

Fit indicatorLS1 LS2 LS3

M1 86.2326 84.1038 91.1196M2 84.6231 88.8484 91.1542

Remark: SISO transfers contain RHP zeros.

−20

0

20

40

60

From: LS2 To: M

1

Magnitude (

dB

)

200 400 600 800 1000 1200 1400 1600 1800 2000−180

−90

0

90

180

Phase (

deg)

Bode Diagram N = 80 (FIT : 84.1038)

Frequency (Hz)

Measure

Model

13

Content

1 Introduction

2 System to control

3 Control StrategyControl problem formulationMulti-objective optimizationController StructureInitialization

4 Results

5 Conclusions and Perspectives

14

Control problem formulation

|W1|

fmin fmax

1|W2|

fmaxG

Optimization problem

minK

∥∥W1Tw→e1

∥∥∞

subject to

∥∥W2Tw→ui

∥∥∞

< 1∥∥∥W3Td′j →ei

∥∥∥∞

< 1

|piK | < fe/N

Re(piK ) < 0

i = 1, 2 and j = 1, 2

15

Control problem formulationAdditional robustness needed

Environment conditions modify acoustic transfers

−10

0

10

20

30

40

50

Magnitude (

dB

)

100 200 300 400 500 600 700 800 900 1000−180

−90

0

90

180

Phase (

deg)

Measured frequency responses from LS2 to M1

Frequency (Hz)

FRF1

FRF2

FRF3 (nominal plant)

A multi-model approach was used to tackle system variations16

Control problem formulation

|W1|

fmin fmax

1|W2|

fmaxG

Optimization problem

minK

max1,...,N

∥∥W1Tw→e1

∥∥∞

subject to

max1,...,N

∥∥W2Tw→ui

∥∥∞

< 1

max1,...,N

∥∥∥W3Td′j →ei

∥∥∥∞

< 1

|piK | < fe/N

Re(piK ) < 0

i = 1, 2 and j = 1, 2

17

Multi-objective and Multi-model optimization

MotivationsI Be able to consider various constraints without pessimism;I Clearly distinguish objective and constraints;I Have the possibility to mix H2 and H∞ objectives, if needed;I Be able to structure the controller;I Be able to consider reduce order controller.

Optimization tool: systuneI Specialized in tuning fixed-structure control systems;I Based on non smooth optimization;I P. Apkarian, “Tuning controllers against multiple design

requirements”, in American Control Conference (ACC),Washington, 2013, pp. 3888–3893

DrawbackI May lead to local optima;I Necessity of ”good” initialization and controller structure.

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Controller StructureState feedback observer

Model of the systemI No real time measure of wI Gp is known{

x = Ax + Buu + Bw we = Cx + Duu + Dw w

Model of the controller{˙x = Ax + Buu + Kf (e − e)u = −Kcx

RemarksI Kf : observation gainI Kc : state feedback gainI full order controller

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InitializationLQG

LQ criteria

JLQ = minKc‖WLQe‖2

2 + ρ‖u‖22

I WLQ is a bandpass filter (attenuation frequency range)I ρ manages trade-off between performances and control energy

Kalman filter {xa = Aaxa + Bua u + Bwa we = Caxa + Dua u + Dwa w + v

I Tuning parameters are the covariances of noises v and w

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Content

1 Introduction

2 System to control

3 Control Strategy

4 Results

5 Conclusions and Perspectives

21

ResultsNarrow attenuation: [190-220] Hz

150 160 170 180 190 200 210 220 230 240 250

−10

0

10

20

30

40

50M

ag

nitu

de

(d

B)

Transfer e1w [190-220] Hz (SIMULATION)

Frequency (Hz)

Open loop

SISO (LS1)

SISO (LS2)

MISO

MIMO

22

ResultsNarrow attenuation: [190-300] Hz

150 200 250 300 350−5

0

5

10

15

20

25

30

35

40

45M

ag

nitu

de

(d

B)

Transfer e1w [190-300] Hz (SIMULATION)

Frequency (Hz)

Open loop

SISO (LS1)

SISO (LS2)

MISO

MIMO

23

ResultsExperimentation: 190-300 Hz (MIMO)

50 100 150 200 250 300 350 400 450 500−20

−10

0

10

20

30

40

50From: w To: e1

Ma

gn

itu

de

(d

B)

Transfer e1w [190-300] Hz (MIMO)

Frequency (Hz)

Simulation (nominal Plant)

Experimentation

24

Content

1 Introduction

2 System to control

3 Control Strategy

4 Results

5 Conclusions and Perspectives

25

Conclusions and Perspectives

ConclusionsI A general framework (for identification and control) was presented;I It allows to quantify and compare SISO and MIMO achievable performances

according to :I Frequency range of attenuation ;I Actuators and sensors position ;I Cavity geometryI . . .

Ongoing workI Compare feedback and feedforward controlI Apply methodology to the industrial problem where:

I Gp is unknownI System order and dimensions are higher

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