1

Synthesis of State Estimation and H∞

Predictive Control for Networked Control Systems in automotive

applications

Yiming ZHANG, Vincent SIRCOULOMB, Nicolas LANGLOIS

EA 4353

Brighton, the 6th of March, 2015

Introduction

Robust estimation for networked control systems: • Application to battery state of charge estimation through a CAN bus

H∞ predictive control for networked control systems: • Application to electromagnetic valve control through a CAN bus

Conclusions and prospects

Outline

2

Outline Introduction Battery SOC estimation Electromagnetic valve control Con.&Pros. ● ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○

Industrial background

• Connect cyberspace to physical space

• Economical investments (unnecessary wiring)

• Data sharing, exchange

• Battery Management Unit （Battery state-of-charge estimation）

3

Network

• Engine Management Unit （electromagnetic valve control）

Outline Introduction Battery SOC estimation Electromagnetic valve control Con.&Pros. ○ ● ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○

State estimator

Controller

Estimated state

Overview of main issues

Process

(State) Transmitted Measurements

Sensors

Measurement noise

Dynamic model of the process

Desired output

Network

Actuators

Control signal

Received measurements

State constraints

4

Process noise Parameter

uncertainties

Network-induced

issues

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skT

…3ky 1ky 2ky

1 sk T

…3ky 1ky

2 sk T

…3ky

3 sk T

…3ky

…

Buffer

Network-induced issues

5

𝑦𝑘

𝑦𝑘−3

𝑦𝑘−2

𝑦𝑘−1

Network Process Sensors Estimator 𝑦𝑘 𝑦𝑘−𝜏

(𝑘 − 3)𝑇𝑠 :

(𝑘 − 2)𝑇𝑠 :

(𝑘 − 1)𝑇𝑠 :

𝑘𝑇𝑠 : • Packet dropouts

𝑦𝑘−3

𝑦𝑘−2

𝑦𝑘−1

𝑦𝑘

• Network-induced delay

• Packet disordering

Model of a networked system

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Modeling of network characteristics

6

N

E

T

W

O

R

K

Actuator System

Sensor

System

output

Observer

Controller

Estimated

state

Desired

input

Uncertain link

(network uncertainties)

Uncertain link

Measurement sequence over networks:

1, , kkY y y

1 1, , k krY y y

Incomplete and intermittent

Measurements :

r kY Y

1kE

Packet arrival probability [Sinopoli, 2004]

1, , , 1 or 0k i Bernoulli process :

Outline Introduction Battery SOC estimation Electromagnetic valve control Con.&Pros. ○ ○ ○ ○ ● ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○

Full measurement sequence

Robust state estimation for LTI system

controlled over a network

Application to battery state of charge

estimation

7

the a priori error-covariance

process noise covariance

measurement noise

covariance

the filtering gain

The estimator design

1k k k

k k

k

k

x x u

y cx

1 11ˆ

kk k kxx u

11ˆ ˆ ˆk k k kk k k k kx x K y Cx Du

1

11

T Tk kk k k

K P C CP C R

11 1ˆ

k

T TkTx F Fx F l

Kalman filter :

11 kk k c

TP P Q

Prediction :

Update :

Projection :

8

• State constraints • Varying process noise covariance • Packet arrival probability • Measurement noise covariance

Dynamic model of the process :

The estimation error affected by :

1k kP

kK

cQ

R

T = I when , it is the state-unconstrained system

Outline Introduction Battery SOC estimation Electromagnetic valve control Con.&Pros. ○ ○ ○ ○ ○ ● ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○

A critical value [Zhang et al., 2013a, 2013b]

c c we can find a critical value so that for

1 : , , , , , , , ,

0 : , , , , , , , ,

c cc

c cc

p Q p Q p p Q p Q p

p Q p Q p p Q p Q p

For , 0cQ Q Q 0, , , , , ,cp p pQ QpQ

0, , , , , ,cp p pQ QpQ

1 2 If , 1 2, , , ,c cp Q p Q 1 2, , , ,c cp Q p Q

• The existence of a critical packet arrival probability

• The existence of upper bound of admitted process noise covariance

9

1

, ,T T T T TTp Tp T Tp C Cp C R CpQ TQ

1

, ,T T T T TT T T T T T Tp T p T TQT T Tp T C C Tp T C R C TpQ Q Q QT T

• Stability : Riccati equations with packet arrival probability

Stability

Unstability

Outline Introduction Battery SOC estimation Electromagnetic valve control Con.&Pros. ○ ○ ○ ○ ○ ○ ● ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○

HEV motorization management system

• HEV motorization management. • What is the battery state-of-charge? (0%~100%) • Internal variable in battery pack : SOC

Internal Combustion Engine

Electric Motor

10

Supervision unit

Driver intention (torque)

Outline Introduction Battery SOC estimation Electromagnetic valve control Con.&Pros. ○ ○ ○ ○ ○ ○ ○ ● ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○

Battery model and contraints

The battery electrical equivalent model [Banghu et al., 2005]:

i tLead-acid

Battery

tR( )i t

( )tV t

( )CsV t

sRenR

( )CbV t

bC sC

( )bi t ( )si t

( )tV t

1 1

( ) ( ) ( ) 

1 1

( ) ( ) ( )

 

s

b en s b en s b en s

Cs enCs

s en s s en s s en s

s en en st

Csen s en

Cb

Cb

Cb

s en s

R

C R R C R R C R Ri t

V RdV

C R R C R R C R Rdt

R R R RV R

VR R R R R

dV t

V tdt

V t

t R

tt

t

i t

Cb SOC bV a n

Known parameters Estimating the state of the battery model

White noise

Uncertainties

11

The relationship between the bulk voltage and the SOC:

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Input current and output voltage: experimental data

• Low and limited input current

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Algorithm results for the unconstrained-state system

The SOC estimation problem:

13

• the nominal value of process noise covariance Q0 ( ) 0 0.005*Q I

,

2.1662 6,

0.0018

[0.5 0.5], 0

0.999

.

7 3.19 4

0.4126 0.58

41

74

00

e

C

e

D

Outline Introduction Battery SOC estimation Electromagnetic valve control Con.&Pros. ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ● ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○

Algorithm results for the unconstrained-state system

A priori form

(Point A)

A posteriori form

(Point B)

0.0117pritrace Q

0.7533post

• the upper value of uncertainty covariance Qu ( ) when 0.01*uQ I 1

14

0.5954pri

0.0119posttrace Q

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Estimation performance for the unconstrained-state system

0.0934 0.9761

0.0049 0.8876

1 0.6016 2 0.1851

priAve

postAve

Comparison of mean absolute estimation Errors

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The augmented model with state constraints

Coulomb counting equation [Ng et al., 2009]:

13600

sk k k

Ah

Tu

CSOC SOC

Augmented system model:

, 1

1 , 1

1

0

03600

0

cb k

k

k cs k k ks s

k

Ahk

k k k k

V

x V x uTI

C

y C x D

SOC

u

Cb a C bV SO

Linear hard equality constraint equation:

16

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The robust results for the constrained-state system

The SOC estimation problem:

• the nominal value of process noise covariance Q0 ( ) 0 30.005*Q I

0.0682 0.0749 0.2042

0.0472 0.0682 0.1969

1 0.55 3 0.1851

priAve

postAve

2 0.5

Comparison of mean absolute estimation errors

17

A priori form

(Point A)

A posteriori form

(Point B)

0.0213pritrace Q 0.0212posttrace Q

0.505pri 0.6561post

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Comparison of constrained and unconstrained approaches

The state unconstrained approach The state constrained approach

0.0934 0.013

0.0049 0.0045

0.6016

priAve

postAve

Comparison of mean absolute estimation errors

The a priori model form: The a posteriori model form :

18

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Controller

Estimated state

Process

(State) Sensors

Dynamic model of the process

Desired output

Network

Actuators

Control signal

19

State estimator

From robust estimation to robust control

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Control for LTI system controlled over a

network

Application to electromagnetic valve

control

20

Design of a predictive controller

• The cost function:

1 2, , ...,T

T T Tk m k m k NkR

1 0 0

0 0

0 0uN

2 2

11 1

,uNN

k k j k k j j k j kj m j

J k U Exp y u

• The future reference sequence:

ˆ ˆ,T T

k k k k k k kJ k U Y R Y R U U

• Vector form of the cost function:

1 1, ,...,

u

TT T T

k k k k k k N kU u u u

1 2ˆ ˆ ˆ ˆ, , ...,

TT T T

k k m k k m k k N kY y y y

• The predicted output sequence: • The predicted control sequence:

N m uN

21

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Considered state-space model

.

The state-space model:

1k k k

k

m

k k k k

kx x u E

y z Cx

1 1k k k k m k

k k k k k

x I x x u E

y z Cx

1 1 1

1

1

k k k m k

k k k

k m k m k m

x x u E

u u u

Incremental state-space model:

22

• Proposal of a deterministic approach to compute incremental control signal in the case of parameter and process noise uncertainties

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H∞ design for uncertainties in parameter and process noise

• The state-space model after introducing incremental control signal:

r

2 2

1k kCx r

2[0, )k

1 1 1 21 1T T

b b b bk k k kx I I G I x I G I x E

2

00T T T

k k k kk

x C Cx r

• We define H∞ control performance index with a positive scalar so that the system under the incremental control law should satisfy: if (1) is satisfied for , the system is H∞ norm stabilizable.

min ( )kU

J N

1 2 1ˆ ˆT

b a k b k m k b k m kkU H I G I x I x

• Control law computation from :

23

1

H T

b bI G I G I

1 1* 21 0

uN

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Proposed theorem [Zhang et al., 2013]

.

0r P QFor a given , if there are symmetric positive definite matrices and so that the LMI is solvable, then the system is H∞ norm stable.

where ,

2

1 2

0 0

0 00

0 0

T

T Tb b b b

Q P C C

Q

r I

P P X I A G I A X I A G I A P PE P

1X P 1K P X

• Computation of the control-weighting parameter:

11

1 T T T T Tb bT T

K K I G I GKKK

……

……

1 2 n

1 2 n

• Parameter tuning :

24

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Electromagnetic valve

• Electromagnetic valve : the mechanical and the electrical parts [Zhao, 2010]. • What does the electromagnetic valve do? • The displacement of the valve A valve lift (from -4mm to 4mm) A valve drop (from 4mm to -4mm)

Opening Spring

positionMass

Closing Spring

The kinetic model of the mechanical part of an solenoid valve :

eL

eI

u

R

d

d t

+-

cI eR

The equivalent circuit of the electrical part of an solenoid valve :

min max, 4 , 4mm mm

the coil current

the eddy current

25

plunger

M: the mass weight

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Modelling and contraints

The linearized state-space valve model:

• State : • Uncertainties: process noise parameter

26

max max max

maxmax

max m

max

axmax

0 1 0 00

3 20 0

202 2

1

0 0

f

c

e

ee

e

c

e

e

k k d kf

R RkI

IR R R LL

I

L

I

0

1

0

0

u

the displacement of the mass the velocity of the mass

the coil current

the eddy current

0P

Line B

Line AP

ress

ure

Position max

0P

min

max

0P

the relation between the combustion pressure and the valve position

parameter

Friction factor Displacement factor Spring factor Coil current factor

Outline Introduction Battery SOC estimation Electromagnetic valve control Con.&Pros. ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ● ○ ○ ○ ○ ○ ○ ○ ○

Model Parameters of the EM-driven Valve

27

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Electromagnetic valve control system

Engine

Management

Unit

m size buffer

Desired

Torque from

the Driver

CAN Bus

Electronic

Control Unit

Valve

Position

Reference

Control signal

Valve

Valve

Position

Valve Position

Feedback

k m ku

k ku

Control signal

r

k m

Engine ECU

Measurements

(pressure, temperature...)

References for other

local control units

Sensor

28

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Reference setting

29

Valve lift Valve drop

Time interval between two complete valve motions

• A complete valve motion Discretization : 10 microseconds • Time interval determines the engine

speed (from 18000rpm to 12000rpm) 18000rpm : 0.00667s 12000rpm : 0.01s

Driver-desired reference of valve position

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Predictive reference strategy and controller setting Zoom-in figure

30

5 1 4 9

10 1 4 14

20 1 4 24

m uN N

Parameters setting for GPC

5.5e-20 2e-20 0.78e-20

8.7e-20 3e-20 1e-20

10e-20 5.45e-20 2.45e-20

0.2 0.5 0.7

with 5m

with 10m

with 20m

Setting for parameter tuning smT

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Valve position error-covariance for the whole simulation.

System output, error and valve velocity Zoom-in between 103.5ms and 106ms

31

Tracking error

Zero valve velocity

0.014026e-4 0.017219e-4 0.018528e-4

2.3731e-4 3.1546e-4 3.8832e-4

563e-4 572e-4 577e-4

cov

5m

10m

20m

0.2 0.5 0.7

Outline Introduction Battery SOC estimation Electromagnetic valve control Con.&Pros. ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ● ○ ○ ○

sec landing opening landing opening landing opening

2.1e-4 2.1e-4 2.11e-4 2.11e-4 2.13e-4 2.13e-4

2.1e-4 2.1e-4 2.11e-4 2.11e-4 2.13e-4 2.13e-4

2.1e-4 2.1e-4 2.11e-4 2.11e-4 2.13e-4 2.13e-4

Features : valve motion and control response

32

The valve opening/landing velocity (18000rpm)

10m

20m

5m

0.2 0.5 0.7

• The expected zero overshoot

• A short rise time (t2–t1 and t4–t3)

2 1t t s 4 3t t s 2 1t t s 2 1t t s 4 3t t s 4 3t t s

mm landing opening landing opening landing opening

0.0205 -0.0205 0.0171 -0.0171 0.017 -0.017

0.5125% 0.5125% 0.4275% 0.4275% 0.425% 0.425%

0.02 -0.02 0.0193 -0.0193 0.0166 -0.0166

0.5% 0.5% 0.4825% 0.4825% 0.415% 0.415%

0.0201 -0.0201 0.0184 -0.0184 0.0177 -0.0177

0.5025% 0.5025% 0.46% 0.46% 0.4425% 0.4425%

0.2 0.5 0.7

10m

20m

5m

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Features : valve motion and control response

33

The valve opening/landing velocity (18000rpm)

landing opening

7.4674 4.1488 -7.6841 -4.1463

6.7163 4.7286 -6.7853 -4.7363

6.5229 4.8937 -6.5421 -4.9001

m saV m sbV m scV m sdV

5m

10m

20m

0.2

landing opening

7.4674 4.1488 -7.6841 -4.1463

8.4944 3.426 -8.6696 -3.4305

8.8851 3.1824 -8.8915 -3.1918

m saV m sbV m scV m sdV5m

0.2

0.5

0.7

• A large starting valve velocity (Va and Vc)

• A small arrival valve velocity (Vb and Vd)

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Conclusion and prospects

1) Design of a network-faced state-constrained estimator robust to varying uncertainties in both parameters and process noise

• Application to battery state-of-charge estimation problem

2) Proposal of a deterministic control law from a H∞ predictive controller

• Application to electromagnetic valve control problem

1) Relax the strong assumption for noise

2) Consider nonlinear systems

3) Consider a variable sampling period method

Conclusion :

Prospects :

34

Outline Introduction Battery SOC estimation Electromagnetic valve control Con.&Pros. ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ●

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