Appunti di Automation and control in electric and hybrid vehicles

Politecnico di Milano

Notes of Automation and Control in

Electric and Hybrid Vehicles

Alberto Giacalone

Course held by

Prof. Sergio Savaresi

July 19, 2022

Contents

0 Preface 2

1 Vertical dynamics control 3

1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.2 Introduction - Suspensions main elements . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.3 Mathematical model - Suspensions system . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.4 Introduction - Classification and side-concepts . . . . . . . . . . . . . . . . . . . . . . . . 6

1.5 Control - Adaptive suspensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.6 Control - Semi-active suspensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.6.1 Damping control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.6.2 Control - Stiffness control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.7 Control - Load-leveling suspensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

1.7.1 Pneumatic suspensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

1.7.2 Hydro-Pneumatic suspensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

1.8 Control - Full-active suspensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

1.9 Sensors configurations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

1.10 Appendix - Towards the autonomous car . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2 Longitudinal dynamics control 14

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.1.1 Tire-road contact forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.1.2 Aerodynamic forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.1.3 Braking and acceleration forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.2 Architecture and actuators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.3 Mathematical model - Slip dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.4 Control - Proportional actuators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.5 Control - 3-states actuators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.6 Appendix - Longitudinal dynamics L1 autonomy level systems . . . . . . . . . . . . . . . 22

2.7 Appendix - Traction control for motorcycles . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.8 Appendix - Slip estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3 Lateral dynamics control 24

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

3.2 Mathematical model - Bicycle model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

3.3 Mathematical model - Under-steering gain . . . . . . . . . . . . . . . . . . . . . . . . . . 24

3.4 Architecture and actuators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

4 Electric and Hybrid vehicles 27

4.1 Powertrain analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

4.2 Powertrain modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

4.2.1 Energy utilizer - Vehicle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

4.2.2 Energy links - Inverter or Gearbox . . . . . . . . . . . . . . . . . . . . . . . . . . 28

4.2.3 Energy converters - Internal Combustion Engines and Electric Motors . . . . . . . 29

4.2.4 Primary energy sources - Fuel and Battery . . . . . . . . . . . . . . . . . . . . . . 30

4.3 Energy management . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

4.4 Energy management problem solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

4.4.1 Dynamic Programming (DP) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

4.4.2 Equivalent Fuel Consumption Minimization (ECMS) . . . . . . . . . . . . . . . . 31

5 Future mobility 33

5.1 Prologue: issues and trends . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

5.2 CO issue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

2

5.3 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

5.4 Main technology development streams . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

5.4.1 Energy accumulation and powertrains . . . . . . . . . . . . . . . . . . . . . . . . . 34

5.4.2 Vehicles architecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

5.4.3 The autonomous car . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

5.4.4 The shared/service car . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

5.5 “Digital convergence” of mobility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

1

0 Preface

This document is an approximation of Course slides, Lectures and Books for the course Automation and

(ACHEV), AY 2021/2022 held by Prof. Sergio Savaresi.

Control for Electric and Hybrid Vehicles

The content is categorized in the same way as the course slides but it has been reorganized in some

sections in order to follow a simpler information flow. For the sake of simplicity some information

is sometimes omitted or simplified, thus I strongly suggest using this document after following the

professor’s lectures.

To download the latest version of this document, please check this link.

For any suggestions or corrections please contact me at alberto.giacalone@hotmail.com.

Alberto. 2

1 Vertical dynamics control

1.1 Introduction

General goal: filter the road-to-vehicle interaction.

Related movements: vertical movement, roll, pitch.

Objectives:

• Comfort: we would like small body acceleration (low pass filter).

Handling-Performance-Safety: we would like small vertical load variations: must be maximized

• F Z

to maximize and . Note that = (M + + + and

F F F m)g DynamicLoad AerodynamicLoad

X Y Z

suspensions can act on the Dynamic Load.

Stroke limitation: usually to End-stops or hard-stop brushes made of rubber are

• ±5cm ±50cm.

used.

1.2 Introduction - Suspensions main elements

Suspensions are composed of a spring and a damper.

basic behavior in linear region is = We usually put an air spring in series to the

−c ·

Damper F v.

damper for various motivations:

• Filtering action: in case of step inputs so the damper would break, so a low-pass filtering

→ ∞

v .

action is needed. The transfer function (Fig. 1a) is k

→ g

x x

1 2 sc+k

g

• Volume and temperature compensation: remember the stem volume varies and oil is in-compressible

(Fig. 1b).

• Cavitation (Fig. 1c): at 0bar oil bubbles get bigger and cause problems to the damper, so a

pre-loaded gas chamber is needed.

a Filtering action b Volume and Temperature com- c Cavitation

pensation Figure 1

Three types of dampeners will be presented:

Electro-Hydraulic controlled with mechanical variations of the orifices size.

Magneto-Rheologic controlled with magnetic fields that change the fluid viscosity.

Electro-Rheologic controlled with electric fields that change the fluid viscosity.

can be coiled spring or air spring.

Spring

Air springs can have linear (F = progressive or regressive behavior. Progressive is the most

−k · x),

secure because it helps prevent hitting the end of stroke.

Figure 2: Deflection-Force relation for a spring

two main configurations are used:

Complete suspensions 3

a Pneumatic b Hydro-Pneumatic

Figure 3

A simple comparison between them is here shown:

Size/Space Hydro-Pneumatic (remotization property)

Static friction (stick-slip) Pneumatic (less oil leakage in piston-rod contact)

Fluid/Gas management Pneumatic (no oil required)

Damping management Pneumatic (can achieve much more sophisticated control)

1.3 Mathematical model - Suspensions system

Quarter-Car model Figure 4: Quarter-Car Model

 = ) ∆ )

−c( − − − − −

M z̈ ż ż k(z z M g

 t t s

= +c( ) + ∆ ) (z ∆ )

− − − − − − −

mz̈ ż ż k(z z k z mg

t t t s t t r t



The control variable is = , the states = [z, ] and the output = [z, ]. This is a 4th order

u z x ż, z , ż y z

r t t t

LTI system. Note that the major limit of validity is when ∆ 0: the road can’t pull the wheel

− −

z z <

t r t

and the force must be set to 0.

Quarter-Car model - controllable damping

 = ) ∆ )

−c( − − − − −

M z̈ ż ż k(z z M g

 t t s

= +c( ) + ∆ ) (z ∆ )

− − − − − − −

mz̈ ż ż k(z z k z mg

t t t s t t r t



The control variable becomes = [z This is a 4th order non-linear system.

u , c].

r

Quarter-Car model - controllable damping and actuator dynamics

= ) ∆ )

 −c( − − − − −

M z̈ ż ż k(z z M g

t t s





 = +c( ) + ∆ ) (z ∆ )

− − − − − − −

mz̈ ż ż k(z z k z mg

t t t s t t r t

 = +

 −βc

ċ βc

 in

A classic low pass filter is introduced (solenoid valve), so the control variable becomes = [z ]. This

u , c

r in

is a 5th order non-linear system.

Equilibrium points  = ∆ + ∆ M g M g+mg

− −

z̄

 s t

= 0 k k

−→

ẋ t

= ∆ M g+mg

−

z̄

t t

 k t

in reality the model is already linear, but we want to focus on variations around the

Linearization

equilibrium 4

 = ) )

−c(δ − − −

M δz̈ ż δ ż k(δz δz

 t t

= +c(δ ) + ) (δz )

− − − −

mδz̈ ż δ ż k(δz δz k δz

t t t t t r



The model transfer functions (Fig. 5a) and (Fig. 5b) can then be computed: both have the same

F F

z z t

denominator, 4 poles (so full observability and controllability is guaranteed) and relative order 3 and 2.

→ →

z z z z

a b

r r t

Figure 5

We also use three transfer functions:

performance

= (s) or transfer function (objective).

2

→ Comfort acceleration

δz δz̈ s F

r z

= (s) 1 transfer function (objective).

→ − − Road-contact

δz δz δz F

r t r t

= (s) (s) transfer function (limitation).

→ − − Elongation

δz δz δz F F

r t z t

Quarter-Car model - Single-Mass simplification

Figure 6: Single-Mass Model

= ) ∆ )

−c( − − − − −

M z̈ ż ż k(z z M g

r r s

= ) )

−c(δ − − −

M δz̈ ż δ ż k(δz δz

r r +

+ c k

s

sc k =

(s) = M M

F

z + + + +

c k

2 2

M s cs k s s

M M

q q

The DC gain is 1, = and = . This simplified model has order 2 and relative order 1, so

k c 1

ω ξ

n M kM

2

lower filtering at high frequencies. Of course it doesn’t have the wheel resonance nor filtering effect of

wheel mass. This model is often used for off-highway vehicles seat suspensions, where 3 or 4 levels of

suspensions are used: in these cases the seat has no influence on the cabin and the cabin height itself can

be taken as an input. on the simplified model:

Sensitivity analysis Comment Drawbacks

ω ξ

n

increases Lower is better Risk of hitting stops, low bandwidth

↗ ↘

k q

increases = Tuning is difficult point at = 2k

↗

c c-invariant ω

i M

increases Higher is better Low acceleration

↘ ↘

M

RIVEDERE AGGIUNGERE SENSITIVITY AL FULL MODEL? SLIDE 58 + REMARK 61

k c

a Sensitivity to b Sensitivity to

Figure 7

5

Considering a specific road profile (t), a trade-off map is drawn by setting the experiment with nominal

z

r

parameters in (1, 1) and calculating every other point changing one parameter at a time (Fig. 8).

Figure 8: Trade-off comfort-contact map

1.4 Introduction - Classification and side-concepts

Suspensions can be classified according to their bandwidth and power request (Fig. 9):

Figure 9: Suspensions systems classification

Note:

• Development is usually carried out in simulation, single component test-rig, 4 poster test-rig and

finally in the full vehicle.

• Road profiles are modeled as sums of sinusoids with decreasing amplitude and can be approximated

as white noises filtered with a very low-frequency low pass filter.

• In theory road preview with stereo cameras or LIDARs could be used for adaptive control, in

practice it’s very difficult.

• Comfort is evaluated numerically through weighting of 3 variances: vertical acceleration, pitch

angular acceleration and longitudinal acceleration.

1.5 Control - Adaptive suspensions

Basic idea: slowly the damping ratio according to the characteristics of the road or to the driving

adapt

style through a manual switch. This isn’t a feedback approach, so no sensors are used and slow stepper

motors are used.

Note that the damping trade-off is not solved, but makes the system more flexible.

adaptation

6

1.6 Control - Semi-active suspensions

1.6.1 Damping control

Semi-active damping control is largely used because of easy modulation, fast actuation and low cost.

Assuming a controllable damping it is possible to remove the trade-off of a passive or adaptive

c(t)

suspension, however intrinsic performance limits hold in semi-active suspensions (Fig. 10).

Figure 10: Theoretical optimal bound

Skyhook control

The principle of this approach is to design an active suspension control so that the chassis is to

linked

the sky. Thus a fictitious damper is considered.

Figure 11: Sky-hook model

The desired behaviour is then modeled as: = )

−c(δ − −

M δz̈ ż) k(δz δz t

k

(s) =

F z + +

2

M s cs k

We basically gain 1 order of filtering with respect to the conventional model, but the wheel resonance is

undampened.

To design the controller, the idea is to mimic the ideal skyhook, so we compare the traditional model

with the skyhook model. Two approximations are then presented. c ż

SH

) = =

−c(t)( − −c −→

ż ż ż c(t)

t SH −

ż ż

t

Skyhook control - Two-States switching algorithm

 if ) 0

− ≥

c ż(

ż ż

c ż

 max t

SH

= =

−→

c(t) sat c(t) if ) 0

−

ż ż −

c ż(

ż ż <

t min t



Sensors requirements: body speed and stroke speed.

Actuator requirements: two states only.

Skyhook control - Linear approximation algorithm ( ) + (1

( )

− −

c ż αc ż ż α)c ż

SH SH t SH

= =

−→

c(t) sat c(t) sat

− −

ż ż ż ż

t t

Sensors requirements: body speed and stroke speed.

7

Actuator requirements: continuous modulation in [c ].

, c

min max

Note that 0 1 is a tuning parameter that modifies the closed-loop performances: if = 0 we go

≤ ≤

α α

back to the linear skyhook control, if = 1 we have the damper in the traditional configuration.

α

Groundhook control - Two-States and Linear approximation algorithm

 if ( ) 0

− − ≥

c ż ż ż

 max t t

) = =

− −c −→

c(t)(

ż ż ż c(t)

t GH t if ( ) 0

− −

c ż ż ż <

min t t

 −c ż

GH t

=

−→ c(t) sat −

ż ż

t

Sensors requirements: wheel speed and stroke speed.

Actuator requirements: respectively two states and continuous modulation.

Stroke-Threshold control  if )

− ≥

c δ(z z T

 max t e

=

c(t) if )

−

c δ(z z < T

min t e



Sensors requirements: stroke.

Actuator requirements: two states only.

Acceleration-Drive-Damping (ADD) control

 if ) 0

− ≥

c z̈(

ż ż

 max t

=

c(t) if ) 0

−

c z̈(

ż ż <

min t



Sensors requirements: body acceleration and stroke speed.

Actuator requirements: two states only.

This algorithm is optimal in the sense that it minimizes vertical body acceleration when no road

information is available. Also mixed SH-ADD algorithms can be applied: they are based on the idea

that if speed is dominant (low frequency) then SH is applied, if acceleration is dominant (high frequency)

then ADD is applied. To this aim, a time domain frequency selector is used . An intuitive

2 2 2

−

z̈ α ż

explanation of why this switching choice is chosen can be seen in Fig. 12.

Figure 12: Algorithms comparison

Let’s go back to the comfort-contact map (Fig. 13): notice all modern suspensions algorithms stay in the

optimal comfort zone and not in the optimal road-holding zone (groundhook), not even very sporty cars.

This is because in this zone the pilot feels like floating and cannot feel what’s going on.

Figure 13: Comfort-contact map

Note: 8

• All-mechanical solutions are possible. They require no sensors but are not flexible.

• The design of semi-active control algorithms looks simple but is very complex because of system and

controller non-smooth nonlinearities. Moreover, local linearization around equilibrium is usually

not a viable option. For this r