MODEL ORDER REDUCTION

Need for model order reduction (MOR)

Methodology for MOR

Classical approaches

Supercapacitor

Data driven approaches

Design of experiment

Linear approach: Batteries

Nonlinear SISO / MISO approach: Power converter

Nonlinear MIMO approach: Fuel cell

Summary of findings

OUTLINE

• MOR is required to reduce computational complexity of model yet retain

sufficient accuracy for a specific purpose, i.e. control, diagnosis, prognosis

Model order reduction procedure to obtain banks of linear models

MODEL ORDER REDUCTION OVERVIEW

Models for purpose:

Control

Diagnosis

Prognosis

Approaches for MOR:

Classical

Based on mathematical

manipulation of system equations

Data-driven

Models derived from data collected from complex models or hardware

Include System Identification and Machine Learning methods

All approaches retain dominant modes whilst discarding modes with

low contribution to system dynamics

METHODOLOGY

classical

data-driven

METHODOLOGY

Single input single output (SISO) nonlinear model

60th order model used as baseline for model order reduction

Truncation without model order reduction methods

100

101

102

103

-55

-54.5

-54

-53.5

-53

-52.5

-52

-51.5

-51

-50.5

-50

Frequency [rad/s]

Ma

gn

itu

de

[d

B]

analytical

2nd order

3rd order

5th order

60th order

CASE STUDY: SUPERCAPACITOR

Comparison of 3 rd order model variants

obtained via selected reduced order

modelling techniques

100

101

102

103

-55

-54.5

-54

-53.5

-53

-52.5

-52

-51.5

-51

-50.5

-50

Frequency [rad/s]

Ma

gn

itu

de

[d

B]

system

SPA

TBR

Arnoldi

3rd order

100

101

102

103

-55

-54.5

-54

-53.5

-53

-52.5

-52

-51.5

-51

-50.5

-50

Frequency [rad/s]

Ma

gn

itu

de

[d

B]

system

SPA

TBR

Arnoldi

3rd order

RIV

CASE STUDY: SUPERCAPACITOR

0 2 4 6 8 100

1

2

3

4

5x 10

-4Hankel Singular Values (State Contributions)

State

Sta

te E

ne

rgy

Stable modes

SPA – Singular perturbation

approximation

TBR – Truncated balanced

residualisation

Arnoldi – Krylov subspace

method

RIV – refined instrumental

variable

DATA DRIVEN APPROACHES

An experimental process, whereby

making use of available input -output

data and a priori knowledge, one aims

to mathematically describe causalities

that govern behaviour of system

Different approaches to modelling

based on a priori knowledge

White box

Black box

Grey box

1. Data acquisition

Voltage and SOC (outputs) responses to current input

36 short (80 seconds) data sets starting at different SOC

(positive current input – charge mode)

Experiment repeated for negative current (discharging)

2. Obtained set of 144 LTI models using simplified refined

instrumental variable (SRIVC) method

Current to SOC models

Current to voltage models

3. Assumption: low order model can have linear structure,

where parameters depend on SOC and sign of current

CASE STUDY: BATTERY

CASE STUDY: BATTERY

0 100 200 300 400 500 600 700 80027

28

29

30

31

SO

C [

%]

0 100 200 300 400 500 600 700 800-4

-2

0

2

4

Curr

ent

[A]

0 100 200 300 400 500 600 700 8003.4

3.5

3.6

Voltage [

V]

Time [s]

0 100 200 300 400 500 600 700 800-5

0

5

Curr

ent

[A]

High fidelity

Low order

Current

Model structure : 10 single input - s ingle output (SISO)

Hammerstein models with 4 th order polynomial

static nonlinearity

Inputs: Input voltage and duty cycle

Output: Output current

Model order: 1

RT2= 99.83 %

CASE STUDY: BUCK-BOOST CONVERTER

0.5 1 1.5 2 2.5 3 3.5 4

-2

-1.5

-1

-0.5

outp

ut

Actual

Identified

0.5 1 1.5 2 2.5 3 3.5 4

-2

-1.5

-1

-0.5

outp

ut

Actual

Identified

0.5 1 1.5 2 2.5 3 3.5 4

8

10

12

14

time

inputs

duty ratio

Vin

CASE STUDY: BUCK-BOOST CONVERTER

Model structure : Bil inear multiple inputs – s ingle output (MISO)

Inputs: Input voltage and duty cycle

Output: Output current

Model order: 5

RT2= 93.7 %

a – 5

b – 2

Bilinear term – 1

0 0.5 1 1.5 2 2.5 3 3.5 4-2.5

-2

-1.5

-1

-0.5

0

Time [Sec]

True output

Identification data

Validation data

CASE STUDY: FUEL CELL STACK

J. Pukrushpan, A. Stefanopoulou, and H. Peng, “Control of fuel cell breathing,” IEEE

Contr. Syst. Mag., vol. 24, no. 2, pp. 30–46, Apr. 2004.

Model s tructure : MIMO NARX

Inputs: stack current, compressor

voltage

Outputs: stack voltage, oxygen

excess, net power

Logarithmic type nonlinearity on

the input

Model order: 15

Oxygen excess RT2= 96.8 %

Stack voltage RT2 =94.2 %

Net power RT2 =99.3 %

10 15 20 25

1.4

1.6

1.8

2

2.2

Time [Sec]

Oxyg

en

exce

ss

True output

Identification data

Validation data

10 15 20 25

3

4

5

6

x 104

Time [Sec]

Ne

t p

ow

er

True output

Identification data

Validation data

10 15 20 25 30

200

220

240

260

Time [Sec]

Sta

ck v

olta

ge

True output

Identification data

Validation data

MOR effectively retains fidelity of high order model whilst

reducing the model order

Data driven approaches are effective for reduced order

modelling

Purpose of model and a priori information determines the

modelling method

Outline of methodology for model order reduction

Control

Diagnosis

Prognosis

Guidelines for MOR

Methodology is robust for multiple case studies

SUMMARY OF FINDINGS