Lecture on Optimization-based High Order Sliding Mode ...bipopsummerschool2015.inria.fr/wp-content/uploads/sites/...Lecture on Optimization-based High Order Sliding Mode Control and

Lecture on Optimization-based High OrderSliding Mode Control and Applications.

Part 4: Automotive Applications

Spring School 2015Slding-Mode Control _ Theory and Applications

Aussois, FranceJune 8-12, 2015

Dipartimento di Ingegneria Industriale e dell’InformazioneUniversity of Pavia, Italy

Antonella Ferrara

Summary• Traction Control with Fastest Acceleration/Decelerationvia Second Order Sliding Mode Control

• Vehicles Platooning via Second Order Sliding Mode Control

• Braking Control of Two-Wheeled Vehicles via Switched Second Order Sliding Mode Control

• Torque Vectoring Control in Fully Electric Vehicles via SMC

2Spring School on SMC, Aussois, June 2015

Antonella Ferrara

Traction Control with FastestAcceleration/Deceleration Via Second Order

Sliding Mode Control

3Spring School on SMC, Aussois,

June 2015 Antonella Ferrara

Traction Force Control• The traction force control is an automatic driver assistancesystem which increases vehicle drivability expecially in difficult wheater conditions, allowing antianti--skidskid brakingbraking and antianti--spinspin accelerationacceleration.

•The traction force produced by a vehicle is stronglyinfluenced by road road conditionsconditions.

• It is necessary to design a robust traction force controllertaking into account the time-varying tire/road interaction.

• As a further requirement, the designed control law has toprevent the generation of vibrationsvibrations which could increase the discomfort index.


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xiii

x

ii

xi F

Jv

rvf 2ωω

−−=&

( ) rollxairloss FvFF +=

The Vehicle Model

},{ rfi∈

( ) ( )[ ]( )

)(2

)(2

)(

hf

xhfzr

hf

xhrzf

ixiii

ixiiiii

lossrxrfxfx

llvmlmgl

F

llvmlmglF

Tvhf

FrTJ

FFFvm

+

+=

+−

=

+=

−=

−+=

&

&

&

&

&

λ

λω

λλ

• Bicycle model (“single-track model”)

• Roll and yaw dynamics, lateral and vertical motions, and actuators dynamics are neglected

• To design the controller, the normal forces are assumed to be constant (thisassumption is removed in simulations)

( )2)(

iii

ixxi

rJrv

vhω

=


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• The traction force depends on the tire/road interactiontire/road interaction, (which depends on the road conditions), on the the normalforce , and on the ““slip slip ratioratio””

The Traction Force Control ProblemxF

zF λ

( )x

x

vrvr,max

:ω

ωλ

−=

• The ““tractiontraction controllercontroller”” can be realized as a ““slip slip controller"controller": to control so that the desired traction force isproduced, by using the torques acting on the front and rearwheel, , , as control variables.

λ

fT rT

• To design the slip controller it is necessary to model thetire/road interactiontire/road interaction.


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Tire/Road Interaction Models

Table from: Li Li, Fei-Yue Wang, and Qunzhi Zhou, “Integrated Longitudinal and Lateral Tire/Road Friction Modeling and Monitoring for Vehicle Motion Control”, IEEE Trans. On Intelligent Transportation Systems, Vol. 7, No. 1, March 2006.


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The well-known ““MagicMagic FormulaFormula”” by Bakker-Pacejkamodels the traction force as a function of the slip ratio

parametrized by the tiretire--roadroad adhesionadhesion coefficientcoefficient

),( ztpx FfF λμ ⋅=

pμ

The Tire/Road Interaction

pμ

xF

λ

depends on roadconditions and needsto be estimated


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The control objective for the FADC FADC problemproblem is to maximizethe generated traction force

The Fastest Acceleration/DecelerationControl (FADC) Problem

λ

xF

The desireddesired slip slip ratioratiois the abscissa of

the extremal value of the curve corresponding

to the actual road conditionxF−λ

pμ̂

dλ

dλ

9Spring School on SMC, Aussois,


1.1. ToTo estimate onestimate on--line theline the adhesionadhesion coefficientcoefficient toidentify the actual curve

2.2. ToTo calcuatecalcuate the the desireddesired slip slip ratioratio as the abscissaof the extremal value of the actual curve

3.3. ToTo design a design a controlcontrol lawlaw that makes the actual slip ratiotrack the desired value

xF−λpμ

dλ

Solution of the FADC Problem

To solve the FADC FADC problemproblem the control scheme mustaccomplish the following tasks

xF−λ

dλλ


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The Adhesion Coefficient Estimate 1/2The adhesion coefficient can estimated using a recursiverecursiveleastleast squaresquare (RLS)(RLS) technique with “forgetting factor”.

The objective is to find the value that minimizes the following cost

( ) ( ) ( ) ( )∫ −= ∫−t

pdrr dtyeJ

t

0

2ˆ ττφμττ ρ

( ) ( ) ( )( ) ( ) ( )tttFtvmty plossx φμ=−= &21

( ) 0≥tρ

( ) ( ) ( )tftftrf tt +=φ

forgetting factormeasurablemeasurablequantitiesquantities

pμ̂


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The parameterparameter update update lawlaw is given by

where and is the update update gaingainof parameter

The update update gaingain can for instance be adjusted as

( ) ( ) )(ˆ tettPp φμ −=&

)()(ˆ)( tytte p −= φμ )(tPp

The Adhesion Coefficient Estimate 2/2

μ̂

( ) ( ) ( ) ( ) ( )22 tPttPttP φρ −=&


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The Slip Controller 1/2

1iiii ds σλλ =−= { }rfi ,∈

21 )(0 iii t Γ≤≤Γ< γii t Φ<)(ϕ

{ }}

⎪⎩

⎪⎨

⎧

+−+==

∈==

i

t

i

t

iiiiii

iii

ThThfs

rfis

ii

d&

4484476&&&&&&&

&&

)()(

2

21 ,γϕ

λσ

σσAuxiliaryAuxiliary

systemsystem““auxiliaryauxiliarycontrolcontrol””

unmeasurableunmeasurable

only the knowledge of the upperboundsupperbounds is necessary to design the control law

The selected slidingsliding variablesvariables are the ““slip slip trackingtracking errorerror””at the front and rear wheel, respectively


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0=−=diiis λλ

As a consequence, the slip slip errorerror relevant to the frontfront and and rearrear wheelwheel vanishes in a finite time

The auxiliaryauxiliary systemssystems are in a form suitable to apply a secondsecond orderorder slidingsliding mode mode techinquetechinque


so that, the sliding variables and their first time derivatives,

iiii ss &== 21 σσ , are steered to zero in a finite time.


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ControlControl lawlaw

( ) { }rfistsVtT MiiiMii ,,21sgn)( ∈⎟

⎠⎞

⎜⎝⎛ −−= α&

⎟⎟⎠

⎞⎜⎜⎝

⎛Γ−Γ

ΦΓΦ

>2

*11

* 34;max

iii

i

ii

iiMV

αα

⎪⎩

⎪⎨⎧

=1

*i

iαα ( )[ ] ( )[ ] 05.0 11 >−⋅− txxxtx iMiMii

( ] ⎟⎟⎠

⎞⎜⎜⎝

⎛ΓΓ

∩∈2

1* 3,01,0

i

iiα

The Sub-Optimal Controller

otherwise15Spring School on SMC, Aussois,


( ) ( ) ( ) },{,sgn 1 rfitusstT iiiiii ∈+−= ρη

( ) ( )iii sWtu sgn1 −=&

1i

iiW

ΓΦ

>( )

( )iii

iiiii W

WΦ−ΓΦ+ΓΦ

> 31

22 4η

The Super Twisting Controller

ControlControl lawlaw

5.00 ≤< iρ


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Simulation Test

The total total availableavailable brakingbraking torquetorque and the engineengine torquetorque exerted on the driving shaft can be determined from

For instance, for a front-wheel-driven(FWD) vehicle:

braker

brakeshaftf

TT

TTT

2.0

3.05.0

−=

−=

{ }rfjT j ,, ∈

In the simulated scenario, the vehicle performs anacceleration manoeuvre for 6 s starting from the velocity of20 m/s. The road condition changes at 3 s from dry asphaltto wet asphalt

17

Antonella Ferrara, Spring School on SMC, Aussois, June 2015

Slip error at the front wheelSlip at the front wheel

Slip error at the rear wheelSlip at the rear wheel

Sub-Optimal: Simulation Results 1/2


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Sub-Optimal: Simulation Results 2/2

Engine and Braking Torque

Real road condition

Estimatedroad condition

VehicleVelocity

Spring School on SMC, Aussois, June 2015

Antonella Ferrara

Super Twisting: Simulation Results 1/2Slip error at the front wheelSlip at the front wheel

Slip error at the rear wheelSlip at the rear wheel


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Engine and Braking TorqueReal road condition

Estimatedroad condition

VehicleVelocity

Super Twisting: Simulation Results 2/2


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Simulation Results: A Comparison

Braking torque and enginetorque with a conventionalfirst order sliding mode controller

Braking torque and engine torque with the considered second ordersliding mode controllers


Antonella Ferrara

Vehicles Platooning via Second OrderSliding Mode Control

Spring School 2015 Slding-Mode Control _ Theory and Applications,Aussois, France, June 8-12, 2015


Antonella Ferrara

Control of a Platoon of Vehicles• The longitudinal control of platoons of vehicles is effective toimprove traffictraffic capacitycapacity and road road safetysafety by reducing trafficdishomogeneities.

• As for traction control, the performances of a ““platooningplatooningcontrolcontrol systemsystem”” are strongly influenced by road road conditionsconditions.

• The ““platooningplatooning control problemcontrol problem”” has to be solved in a robust way.

• As a further requirement, also in this case, the designedcontrol law has to prevent the generation of vibrationsvibrationsinduced by the controller.


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• Control objective: make the distance between two subsequentvehicles of the platoon be equal to a pre-specified ““safetysafety distancedistance”” .

( )ixspacingiii vxxxd =−= −1

The Platooning Control Problem 1/2

• Solving the platooningplatooning problemproblem even in presence of possible changes of the road conditions requires to re-formulate the problem asa tractiontraction controlcontrol problemproblem and take again into account the tire/road tire/road interactioninteraction.


Antonella Ferrara

A possible way to determine the ““safetysafety distancedistance”” is torely on the following ““variablevariable spacingspacing policypolicy””

( )ii xdxspacing hvSvx +=

0

headwayheadway timetime

followerfollower’’s s velocityvelocity

distancedistance betweenbetweenstoppedstopped vehiclesvehicles

To solve the platooning control problem as a traction controlproblem means to design a controller capable of generatingthe traction force suitable to steer to zero the ““spacingspacingerrorerror””

xF

( ) ixspacingi dvxei−=

The Platooning Control Problem 2/2

26Spring School on SMC, Aussois, June 2015 Antonella Ferrara

• The traction force depends on the tire/road interactiontire/road interaction, (which depends on the road conditions), on the the normalforce , and on the ““slip slip ratioratio””

xF

zF

The Traction Control Problem

λ

( )x

x

vrvr,max

:ω

ωλ

−=

• The ““tractiontraction controlcontrol”” can be realized as a ““slip slip controlcontrol"": to control so that the desired traction force is produced, byusing the torques acting on the front and rear wheel, , .

•To design the slip control it is necessary to model the tire/road interactiontire/road interaction. Also in this case the Bakker-Pacejkamodel can be used.

λfT rT


The Control Scheme

pμ̂

( )xspacing vx xv

i

Outer loop:Outer loop: on the basis of the spacing errorspacing error the controller determines the desired traction forcedesired traction force. Relying on this latter and on the current tire/road adhesion coefficient the desired slip ratiosdesired slip ratios are calculated. These are references for the inner loop.

Inner loop:Inner loop: has the objective of attaining the desired slip ratios by actingon the torques at the front and rear wheelstorques at the front and rear wheels.


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SlidingSliding variablevariable: the ““spacingspacing errorerror”” associated with thei-th vehicle

iiixdii xxhvSeS +−+== −10

The Longitudinal Controller Design 1/4

( )xspacing vx xvpμ̂

i

To design a ““secondsecond orderorder slidingsliding mode mode controlcontrol lawlaw””, determine the firstand second timederivatives

ixixixii vhvveS &&& +−== −1

ixixixii vhvveS &&&&&&&& +−== −1 29Antonella Ferrara

only the knowledge of the upperboundupperbound is necessaryto design the control law

iiyy 21 =&

ii wyi

+= ε2&

the previous differential equations can be rewritten as

By introducing the auxiliaryauxiliary variablesvariables

ii SySyii

&== 21

where iixixi vv Γ≤−= − || 1&&ε and ixi vhw &&=

{ ““auxiliaryauxiliary controlcontrol””


unmeasurableunmeasurable


Antonella Ferrara

It can be proved that, if the auxiliaryauxiliary controlcontrol signalsignal is chosen(for instance) as

011112

,21)(

==

⎭⎬⎫

⎩⎨⎧ −−=

iiiiii yMAXMAXMi yyyysignWtw

iMiW Γ> 2where

then, the sliding variable and its first time derivative, ii SySy

ii&== 21 , are steered to zero in a finite time.

010=+−+== − iiixdii xxhvSeS

As a consequence, the spacing error between twosubsequent vehicles vanishes in a finite time



Antonella Ferrara

⎟⎟⎠

⎞⎜⎜⎝

⎛+−

⎟⎟⎠

⎞⎜⎜⎝

⎛++

=

ii

ilosshf

ii

ilosshr

xr

xf

gmF

ll

gmF

ll

F

F

ii

ii

desi

desi

μ

μ

Note that, as usual in this context, it is assumed that the force distribution is described by

( )ilossxixrxf FvmFFdesidesidesi

+=+ &21

On the basis of the auxiliary signal , the desired tractionforce (the actualactual controlcontrol signalsignal !!) can be determined as

( )∫=t

tix dw

hv

desi

0

1 ττ&where

)(twi



The slip reference for the inner loopof the i-th vehicle control system

ixjF

λ

pμ̂

desijλ

slip ratio reference for the inner loopof the i-th vehicle control system

traction force determined by thelongitudinal controller as the forcenecessary to steer the spacing errorbetween vehicles to zero

desixjF

{ }rfj ,∈

33Antonella Ferrara

SlidingSliding variablevariable: the ““slip slip errorerror””for each wheel j ofthe i-th vehicle


1jjjj dess σλλ =−=

( )xspacing vx xvpμ̂

i

(the subscript i is omitted !){ }rfj ,∈

{ }( )}

⎪⎪⎩

⎪⎪⎨

⎧

+−+==

∈==

t

j

t

xj

t

jjxjjjj

jjj

jjj

dTvhTvhfs

rfjs

ν

λσ

σσ

&876444 8444 76

&&&&&&&

&&

)()(

2

21

)()(

,

gF

21)(0 jjj GtG ≤≤< gjj Ft <)(F

AuxiliaryAuxiliary systemsystem““auxiliaryauxiliary controlcontrol””

34Antonella Ferrara

( ) ( )0

11112

,21sgn)(

==⎟

⎠⎞

⎜⎝⎛ −Ν−==

jjjjj MAXMAXjjjj ttTtσ

σσσσαν &

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

−>Ν

2*

11* 3

4;max

jjj

j

jj

jj GG

F

G

F

αα

It can be proved that, if the auxiliaryauxiliary controlcontrol signalsignal is chosenas


and { }rfj ,∈with

then, the sliding variable and its first time derivative, jj ss

jj&== 21 σσ

0=−=desjjjs λλ

As a consequence, the slip slip errorerror relevant to eachwheel of the ii--thth vehiclevehicle vanishes in a finite time

, are steered to zero in a finite time.


Simulation TestTwo vehicles, a leaderand a follower.

Antonella Ferrara 36

Simulation ResultsSpacing error

Slip error at the front wheel Slip error at the rear wheel

Inter-vehicular distance


Antonella Ferrara

Simulation Results

The total available braking torque and the engine torque exerted on the drivingshaft can be determined fromFor instance, for front-wheel-driven (FWD) vehicles:

braker

brakeshaftf

TT

TTT

2.0

3.05.0

−=

−=

{ }rfjT j ,, ∈

Braking torque and engine torque


Simulation Results

Braking torque and engine torquewith the proposed second ordersliding mode controller

Braking torque and engine torquewith a conventional first ordersliding mode controller


Simulation Results

Inter-vehicular distancewith the designed second ordersliding mode controller(ideal sliding mode!)

Inter-vehicular distancewith a conventional first ordersliding mode controller withcontinuous approximationof the sign function(quasi sliding mode)

40Antonella FerraraSpring School on SMC, Aussois, June 2015

Braking Control of Two-Wheeled Vehicles via Switched Second Order Sliding Mode Control

Antonella Ferrara41Spring School on SMC, Aussois,


Braking Control• Braking control is based on the regulation of the wheel slip (relative speed between wheel and center of mass). In motorbikes the slip dynamics is dependent on the vehicle speed.

• As an alternative to conventional adaptive control, we have proposed to apply the switched formulation of second order sliding mode controllers (S-SOSM), since it can be useful in all contexts where different uncertainties and/or performance specs are associated with different regions of the state space.

MAIN IDEAMAIN IDEA:partition the state-space into different regionstune a dedicated S-SOSM controller for each of them


Antonella Ferrara

Switched SOSM: state-space partitioning

The state space Z is partitioned in k regions Ri, i=1,…, k, allcontaining the origin, such that ∪iRi =Z and with Ri+1⊂ Ri. Let us define as switching surfaces Si=∂Ri+1,i=1,…,k-1.


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Switched SOSM: uncertainty description

• Case 1: Outermost Region Z1

• Case 2: Inner Regions Zi,i=2,…,k

• As in the inner regions the state norm can be bounded, one can write:


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• Two different algorithms:

– Full switched S-SOSM (FS-SOSM) both the gain V and α*are adapted in the different regions (different uncertainties and different performance specifications)

– Gain switched S-SOSM (GS-SOSM) only the gain V is varied (uniform uncertainties in the whole state space but different performance specifications)

Switched SOSM: control algoritms


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Switched SOSM: FS S-SOSM Control Algorithm

•Outermost Region Z1

1. Initialization: for 0≤ t≤ tM1

2. For tMj≤t≤tMj+1 such that z(t)∈ Z1


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Switched SOSM: FS S-SOSM Control (2)

If

•Inner Regions:If


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Hydraulic or electro-

hydraulicbrake

BrakeControl

vehicledynamicsSlip

Control

wheel slip set point

throttle position

front and rearwheel slip

Main components:

• Servo-control loop for the actuator: brake pressure control system

• Wheel slip estimation (from indirect measurements): needed if front and rearbrake is used. If front brake only (common on motorcycles), the rear wheel speedgives an estimate of the vehicle speed

• Slip control: the goal is to track a wheel slip target

We have considered Braking Control on a straight line(target slip definition “easier”)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.2

0.4

0.6

0.8

1

1.2

1.4

λ (longitudinal tire slip)

Fric

tion

coef

ficie

nt μ

( λ)

Asphalt, dry

Cobblestone, dry

Asphalt, wet

Snow

wheel slip estimation

wheel speed, acceleration,…

Braking control of two wheeled vehicles


Antonella Ferrara

Load transfer between front and rear wheels –> proportionalto longitudinal acceleration

Braking control of two wheeled vehicles: vehicle model

• The wheel slip in braking is defined as

h

rf

lr lf

l

vM, J

rrr f

FxfFxr wheel slipwheel sideslip (neglectedas we consider straightline maneuvers)

tire/road modelparametrization


Antonella Ferrara

Tire/Road Interaction

Table from: Li Li, Fei-Yue Wang, and Qunzhi Zhou, “Integrated Longitudinal and Lateral Tire/Road Friction Modeling and Monitoring for Vehicle Motion Control”, IEEE Trans. On Intelligent Transportation Systems, Vol. 7, No. 1, March 2006.


Antonella Ferrara

The “Burkhardt model” describes the tire-road friction as a function ofthe wheel slip

By changing the values of the parameters different road conditionscan be modeled

The Tire/Road Interaction

Note: the friction curve model, together with the fact that

implies that longitudinal forces are bounded

Due to tire relaxation dynamics, also the forces first time derivativesare bounded

51Antonella Ferrara

Braking Dynamics: slip dynamics

h

rf

lr lf

l

vM, J

rrr f

FxfFxr

Wheel speed and wheel slip are linked by analgebraic relationship can use the wheel slip as

state variable for wheel dynamics

The dependence of Ψron λf can be easily

managed within a SM framework Ψr

bounded for all λf and this is all one needs toknow for designing a

SM controller

Two SISO controllerscan be designed, and the coupling betweenfront and rear wheel

only affects the boundson the uncertainties

52Antonella Ferrara

AuxiliaryAuxiliary

systemsystem

The selected slidingsliding variablevariable are the slip slip trackingtracking errorserrors

““auxiliaryauxiliarycontrolcontrol

signalsignal””: : itit isis the the controlcontrol lawlaw v(t)v(t) ofof

the Sthe S--SOSM SOSM algorithmalgorithm

S-SOSM Wheel Slip Controller

BothBoth ϕϕii and and hhii are are fnsfns ofof vv!!53Spring School on SMC, Aussois,


AuxiliaryAuxiliary

systemsystem

““auxiliaryauxiliarycontrolcontrolsignalsignal””

Only the knowledge of the upperupper--boundsbounds is necessary to design the control law

S-SOSM Wheel Slip Controller (2)


Antonella Ferrara

Simulation Results

Results on a Simulink model including brakes dynamics, tireelasticity and tire relaxation (definitely more complex than thatused for the design!)

Braking maneuver with set-point step-changes of width0.05 from λ*=0 to λ*=0.2

Three algorithms compared: Suboptimal SOSM, FS-SOSM, GS-SOSM


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Simulation Results (2)

The switched algorithms allow for a 30% improvement with respect to the standard SOSM one

The small difference between FS-SOSM and GS-SOSM probably due tothe fact that Js captures performance and both algorithms behave

similarly in this respect


Antonella Ferrara

Torque Vectoring Control in FullyElectric Vehicles via SMC


Antonella Ferrara

Antonella Ferrara Spring School on SMC, Aussois, June 2015

58

Torque Vectoring via SMC

• The torque-vectoring control of a four-wheel-drive fully electric vehicle with in-wheel drivetrains has been studied in recent years, and at ACC14 an ISM control based solution has be presented in collaboration with:

• The ISM controller is easy to tune and robust with respect to a significant set of uncertainties and disturbances.

• Its low complexity and other positive features make it appropriate for practical implementations.

10/06/2015Antonella Ferrara


59

European Project E-VECTOORC

Vehicle Conceptand Layout

Powertrain Design and Safety

Brake Design and EM-compatibility

Vehicle Dynamics and Control

60

Electronic Stability Program (ESP) VS Torque-Vectoring (TV)

• Only when |rref – r|>th.• ON-OFF control

• By means of:

- Friction brakes

- Wheel torques (-)

ESP

• In any condition (v, ax , ay , δ)• At any instant of time

• By means of:

- Friction brakes

- Wheel torques (+/-)

TV

61

Why Sliding Mode for Yaw Rate Control?

• Conventional yaw rate control systems: PID + FF

– Good tracking performance and large bandwidth of the closed-loop system– Non robust and smooth enough in critical conditions (high lateral acceleration)

• Many other approaches have been investigated (some examples):

– Internal Model Control;– MPC/Linear Quadratic Control;– Optimal Control based on LMI.

• Major motivations for using Integral Sliding Mode Control:

– Ease of implementation (few parameters only);– High level of robustness against unmodeled dynamics and external disturbances,

even when implemented with sampling times typical of real automotive controlapplications (experimental tests support this claim).

62

The Vehicle Model

( )( )( ) ( )

LRzRRzLFzRFz

rLRxRRxLRyRRy

LFLFxRFRFxLFLFyRFRFy

fLFLFxRFRFxLFLFyRFRFyz

MMMM

TFFbFF

aFFFF

TFFFFJr

,,,,

,,,,

,,,,

,,,,

2

sinsincoscos2

coscossinsin

−−−

−⋅−+⋅+

+⋅+++

+⋅−++−=•

δδδδ

δδδδ),()()( txhuxBxfx ++=•

The proposed controller has been designed considering the vehicle yaw dynamics


63

Design of the proposed ISM controller

( )( )( ) LRzRRzLFzRFzLRyRRy

LFLFyRFRFy

fLFLFyRFRFy

z

MMMMbFF

aFF

TFF

Jrk

,,,,,,

,,

,,

coscos2

sinsin1),,(

−−−−⋅+

−⋅+

+⋅+−=

δδ

δδδβYaw Acceleration contribution due to lateral forces and self-

aligning moments

distzz

MJ

rkh ,1),,( += δβUnknown part: refr

•

−Known part:

distzz

ISMzz

refreferr MJ

MJ

rrkrrr ,,11),,( ++−=−=−

••••

δβYaw RateError Model

In the previous model, the overall yaw moment is the control variable

10/06/2015Antonella Ferrara


64

The Yaw Rate Control Scheme

Demanded Wheel Motor Torque

Pedal brake pressure

Wheel torques due to driver’s intention

Torque and Yaw Moment

65

Integral Sliding Mode Control

filSMzPIDzISMz MMM ,,,, +=

SMzfilSMzfilSMzSM MMM ,,,,, =+•

τ

( )ssignKJM SMzSMz −=,

( ) ( )⎥⎦

⎤⎢⎣

⎡−+−

−∂∂

−=••

SMzISMzz

referr

MMJ

rrsz ,,

0 1Where with the initial condition

zss += 0

( )( )0)0( 0 errrsz −=

The sliding mode starts from the first sampling instant, without any reaching-phase transient


66

• η-reaching condition fulfilled for any value

• a sequence of step steer maneuvers at v=90 km/h (high critical situation) has provided the upper

bound

• a conservative value has been chosen:

• The value has been selected in order to avoid the chattering effect, so that the yaw moment generated by the high-level controller does not induce vibrations perceivable by the passengers.

η+⋅> maxhJK zSMsss η−<&

NmhJ z 000,9max ≅⋅

NmKSM 000,15=

sSM 05.0=τ

Controller Parameters Design

Only two parameters needs to be tuned: and SMK SMτ


67

The Show Case

68

• The performance assessment has been carried out using an accurate IPG CarMaker model of the vehicle(validated on the basis of experimental tests at the Lommel Proving Ground in Belgium)

• Two conventional test maneuvers have beenconsidered:

- Ramp Steer Meneuver

- Step Steer Maneuver

• The controlled vehicle (“Sport Mode”) has beencompared with the passive vehicle (“Baseline Mode”)

The Show Case

69

Ramp Steer Maneuver (I)

• lower value of the understeer gradient;• extension of the linear region of the vehicle understeer

characteristic;• higher values of lateral acceleration.

Benefits of the adoption of the ISM yaw rate controller:

70

Reference yaw rate tracking performance at different speeds.

The reference (dashed dark line) and the yaw rate of the controlled vehicle (blue line) are practically overlapped. The red line is the yaw rate of the passive vehicle.

Ramp Steer Maneuver (II)

71

Step Steer Maneuver

• Evident degradation of the passive vehicle tracking performance at increasing speeds

• Good tracking of the yaw rate reference exhibited by the controlled vehicle, with very short rise-time and negligibleovershoot.

• Obtained by means of a control signal perfectly acceptable in automotive applications.


72

Summary Results

( )∫−= end

in

t

t errinend

dtrtt

RMSE 21 BLUE: Controlled veh. RED: Passive veh.

Some Final Remarks••SlidingSliding mode mode controlcontrol is an interesting methodology for its simplicity and robustness features, but it may be non appropriate to be applied, in itsconventional formulation, to the automotive context due to the discontinuous control variable.

• The secondsecond orderorder slidingsliding mode mode controller controller generates continuous control actions, thus limiting the vibrations it can induce and propagate throughout the vehicle subsystems because of chattering effects.

• The switched versionswitched version of Suboptimal SOSM control law allows to tune the controller parameters taking into account the vehicle speed. For this reason it is particularly appropriate for motorbikes, the slip dynamics of which strictly depends on speed.

• Integral SMCIntegral SMC can also be an effective solution.73Spring School on SMC, Aussois,


• Ferrara A., Vecchio C., Collision avoidance strategies and coordinated control of passenger vehicles, Nonlinear Dynamics. Vol. 49, Nr. 4, Sept 2007, pp. 443-577.

•• Ferrara A., Vecchio C., Second order sliding mode control of a platoon of vehicles, Int. Journal of Modelling, Identification and Control, Vol. 3, Nr. 3, 2008.

• Canale M., Fagiano L., Ferrara A., Vecchio C., Vehicle Yaw Control via Second Order Sliding Mode Technique, IEEE Transactions on Industrial Electronics, Vol. 55, Nr. 11, pp. 3908-3916, 2008.

• Canale M., Fagiano L., Ferrara A., Vecchio C., Comparing Internal Model Control and Sliding Mode Approaches for Vehicle Yaw Control, IEEE Transactions on Intelligent Transportation Systems Vol. 10, Nr. 1, pp. 31 - 41, 2009.

• Ferrara A., Vecchio C., Second order sliding mode control of vehicles with distributed collision avoidance capabilities, Mechatronics, Vol. 19, Nr. 4, pp. 471-477, June 2009.

• M. Tanelli, C. Vecchio, M. Corno, A. Ferrara, S.M. Savaresi. "Traction Control for Ride-by-Wire Sport Motorcycles: a Second Order Sliding Mode Approach". IEEE Transactions on Industrial Electronics. Vol. 56, Nr. 9, pp. 3347-3356, Sept. 2009.

• Amodeo M., Ferrara A., Terzaghi R., Vecchio C., Wheel Slip Control via Second Order Sliding Modes Generation, IEEE Transactions on Intelligent Transportation Systems, Vol. 11 , Nr. 1, pp. 122 – 131, 2010 .

References


• Tanelli M., Ferrara A., Wheel Slip Control of Road Vehicles via Switched Second Order Sliding Modes, International Journal of Vehicle Design, to appear.

Other topics in the automotive field:• Ferrara A., Pisu P., Minimum sensor second order sliding longitudinal control of passenger vehicles, IEEE Transactions on Intelligent Transportation Systems, Vol. 5 , Nr. 1, pp. 20-32, March 2004.

• Ferrara A., Paderno J., Application of switching control for automatic pre-crash collision avoidance in cars, Nonlinear Dynamics, Vol. 46, pp. 307-321, June 2006

• De Nicolao G., Ferrara A., Giacomini L., On board sensor-based collision risk assessment to improve pedestrians’ safety, IEEE Transaction on Vehicular Technology, Vol. 56, Nr. 5, Part 1, pp. 2405 – 2413, Sept. 2007.

• Goggia, T., Sorniotti, A., De Novellis, L., Ferrara, A., Gruber, P., Theunissen, J., Steenbeke, D., Knauder, B., Zehetner, J., Integral Sliding Mode for the Torque-Vectoring Control of Fully Electric Vehicles: Theoretical Design and Experimental Assessment, IEEE Transactions on Vehicular Technology, Vol. 64 , Nr.5, pp. 1701 – 1715, 2015.

• Goggia, T., Sorniotti, A., Ferrara, A., De Novellis, L., Pennycott, A., Gruber, P. 'Integral Sliding Mode for the Yaw Moment Control of Four-Wheel-Drive Fully Electric Vehicles with In-Wheel Motors', International Journal of Powertrains, 2015 (to appear) .

References


Last Part of the Lecture

MARIE SKŁODOWSKA-CURIE ACTIONS Innovative Training Networks (ITN) Call: H2020-MSCA-ITN-2015

Interdisciplinary Training Network in Multi-Actuated Ground Vehicles

“ITEAM”

1 Ph.D. Position possibly available at University of Pavia since June 1st, 2016!Please contact Prof. Antonella Ferrara since October 1st, 2015 for application informations

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