New Slip Control System Considering Actuator Dynamics-2015 … · 2020-02-17 · A new control law...

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2015-01-0656

New Slip Control System Considering Actuator Dynamics

Amir Soltani*, Francis Assadian***Cranfield University, Cranfield, UK **University of California, Davis, USA

Abstract

A new control strategy for wheel slip control, considering thecomplete dynamics of the electro-hydraulic brake (EHB) system, isdeveloped and experimentally validated in Cranfield University’sHiL system. The control system is based on closed loop shapingYoula-parameterization method. The plant model is linearized aboutthe nominal operating point, a Youla parameter is defined for allstabilizing feedback controller and control performance is achievedby employing closed loop shaping technique. The stability andperformance of the controller are investigated in frequency and timedomain, and verified by experiments using real EHB smart actuatorfitted into the HiL system with driver in the loop.

Introduction

Wheel slip dynamics is characterized by highly nonlinear anduncertain behaviour of tire forces and fast changing dynamics of thewheel during braking. Because of the complex and variable dynamicsof the system, design of a slip control system is a challenging task.Several linear and nonlinear control design approaches have beenproposed in the literatures, ranging from linear and nonlinear PIDcontrol system [1], fuzzy logic [2], gain scheduling [3] to nonlinearmethodologies such as sliding mode [4] and Lyapunov-based [5]control design approaches. However, it is worth noting that most ofthe proposed brake controllers, which can be found in the publishedliteratures, consider wheel slip (and/or angular velocity) differentialequations as the plant model, without taking the complete dynamicsof the brake system (such as calliper and value dynamics) and theirconstraints into consideration. Interestingly, it is shown in [5] that itis impossible to employ a single linear PID controller for completebrake dynamics to provide stability and performance at alloperational conditions.

In this paper, a new wheel slip control system based on Youlaparameterization approach is proposed [6]. The controller providesstability and good control performance over the whole range ofoperating conditions of the system, considers all the existingdynamics and constraints of the brake system. In the next sections,the mathematical model of the system is introduced, then theproposed closed loop control system based on Youlaparameterization is presented, and finally the stability andperformance of the control system is validated by simulation and HiLtesting at different driving conditions.

System Modelling

A new control law for continues control of wheel slip is presented inthis paper. It is assumed that the smart brake actuation (i.e. brakepressure control) is already provided by an off-the-shelf electrohydraulic brake (EHB) system, also called Sensotronic Brake Control(SBC), which has been developed by Daimler and Bosch [7].Schematic diagram of Bosch EHB SBC system is shown in Fig 1 [8].

Figure 1: Schematic diagram of Bosch SBC system [8]

Employing electro hydraulic brake system as a smart brake actuatorprovides the possibility to control individual wheel brake linepressure to follow the desired target pressure continuously [9]. Therelevant reference brake pressure for each wheel (to be generated byEHB system) is provided by the proposed slip control system. Havingcontinuous control over brake pressure provides several advantagessuch as driver comfort as well as the possibility of fast and precisecontrol over the tires longitudinal force (and slip) [10].

As the brake pressure is regulated for each wheel individually(through EHB valve modulation unit, as shown in Figure 1), there are4 independent inputs and 4 similar plants, and four similar SISOclosed loop brake control systems exist in this architecture. Design ofthe closed loop brake control system based on wheel slip feedbackfor one wheel ( wheel), as shown in Figure 2, is presented inthis section.

SharonTextboxSAE SAE International Journal of Passenger Cars - Mechanical Systems. 8(2):512-520, 2015, DOI:10.4271/2015-01-0656.

SharonTextboxPublished by SAE. This is the Author Accepted Manuscript issued with: Creative Commons Attribution Non-Commercial License (CC:BY:NC 3.0).The final published version (version of record) is available online at DOI:10.4271/2015-01-0656. Please refer to any applicable publisher terms of use.

Figure

The starting point in designing the control system is to derive theplant dynamics model. Here the plant, so calledmodela single independent rotating wheel subject to braking torque

the wheel hub (centre) and longitudinal forceground contact patch, as shown inthe quarter car model relies on some simplificationsa simple and effective model which considers the major brakingdynamics and is widely applied in active braking control systemdesigns [4,

The equation of wheel dynamics is:

where

radius anddynamics in the longitudinal direction can be wr

where

vehicle longitudinal acceleration.

nonlinear function of slip

The tire

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Figure 2: Closed Loop Brake Control system

The starting point in designing the control system is to derive thedynamics model. Here the plant, so called

model, consists of the quarter of car massa single independent rotating wheel subject to braking torque

the wheel hub (centre) and longitudinal forceground contact patch, as shown inthe quarter car model relies on some simplificationsa simple and effective model which considers the major brakingdynamics and is widely applied in active braking control systemdesigns [4, 5, 11].

Figure


௬,

where ௬ǡ is the (

radius and ோǡ is the wheel angular velocitydynamics in the longitudinal direction can be wr

where ௦ is the quarter car mass (as described above) and

vehicle longitudinal acceleration.

nonlinear function of slip

௫,

tire slip ratio during braking

ǡௗ,

௫

Brake Control system

The starting point in designing the control system is to derive thedynamics model. Here the plant, so called

of the quarter of car massa single independent rotating wheel subject to braking torque

the wheel hub (centre) and longitudinal forceground contact patch, as shown in Figurethe quarter car model relies on some simplificationsa simple and effective model which considers the major brakingdynamics and is widely applied in active braking control system

Figure 3: Wheel Dynamics


ோ, , ௗ

) wheel moment of inertia

is the wheel angular velocitydynamics in the longitudinal direction can be wr

ݔ ݏ ݔ

is the quarter car mass (as described above) and

vehicle longitudinal acceleration. The

nonlinear function of slip ǡ݅ܤ , i.e

, ௫,

slip ratio during braking ǡ݅ܤ was defined as

ோ,

௫,

Brake Control system

The starting point in designing the control system is to derive thedynamics model. Here the plant, so called single corner wheel

of the quarter of car mass moving ina single independent rotating wheel subject to braking torque

the wheel hub (centre) and longitudinal force ௫ǡat the centre ofFigure 3. It should be noted that

the quarter car model relies on some simplifications; however, ta simple and effective model which considers the major brakingdynamics and is widely applied in active braking control system

Wheel Dynamics Model

ௗ, ௫,

) wheel moment of inertia, ௗis the wheel angular velocity. The quarter car

dynamics in the longitudinal direction can be written as

݅,ݔ

is the quarter car mass (as described above) and

he tire longitudinal force

, ௭,

was defined as [

ǡ

The starting point in designing the control system is to derive thesingle corner wheel

direction witha single independent rotating wheel subject to braking torque ǡat

at the centre of tire-It should be noted that

; however, this isa simple and effective model which considers the major brakingdynamics and is widely applied in active braking control system

(1)

ǡ is the wheel

. The quarter caritten as:

(2)

is the quarter car mass (as described above) and ݔ is the

longitudinal force ǡ݅ݔ is a

(3)

[7]:

(4)

with

-

his is

whereassumed that is equal to vehicle longitudinal speed

Combining Eq.torque input and

Substitutingcan be formulated as a first order model of the wheel slip dynamicsas:

Eq. (6)

To be able to design a linear control system, one(6) arounthis nominal operating point in the following paragraphs).Considering

௪ǡ

brake torque input tofirst order Tylor series expansion as

where

around a nomof ௫,could be obtained either by differentiation of aBurckhardtexample), or by implementation of an online algorithm to detect thesign of the friction curveapplications

The linearizedform of:

with a gain of:

and a single pole

where ௫ǡ is the forward speed of theassumed that is equal to vehicle longitudinal speed

Combining Eq. (1) with Eq.torque input and tire slip output can be derived as

ǡ௫

௫

Substituting ݔ from Eq.can be formulated as a first order model of the wheel slip dynamics

ǡ௫

(6) is a nonlinear

To be able to design a linear control system, onearound nominal operating points

this nominal operating point in the following paragraphs).onsidering ǡ

௪ǡǡ; the linear transfer function for plant dynamics (frombrake torque input tofirst order Tylor series expansion as

ఒಳǡ

,

௪,

ௗǡ

௬ǡ ௫ǡ

where ௫ǡǡ ǡǡ

around a nominal operating point. It should be noted that calculation

,ǡ in Eq.(7) is not dependent on any specificcould be obtained either by differentiation of aBurckhardt tire model as cited in the literatures (sexample), or by implementation of an online algorithm to detect thesign of the friction curveapplications [5,13].

linearized plant (7)form of:

with a gain of:

and a single pole of:

is the forward speed of theassumed that is equal to vehicle longitudinal speed

with Eq. (4), the relationship between brakingslip output can be derived as

ǡௗ,ଶ

௬,

from Eq. (2) into Eq.can be formulated as a first order model of the wheel slip dynamics

,

௦

ௗ,ଶ

௬,

function of two state variables

To be able to design a linear control system, oned nominal operating points

this nominal operating point in the following paragraphs).

ǡ ǡǡ and

; the linear transfer function for plant dynamics (frombrake torque input to tire slip output) can be derived by application offirst order Tylor series expansion as [5

,

,ǡ

௫ǡǡ ௭ǡ

௦ ௫ǡ

డఓೣǡ

డఒಳǡ ఒಳǡǡ

, represents the

inal operating point. It should be noted that calculationis not dependent on any specific

could be obtained either by differentiation of amodel as cited in the literatures (s

example), or by implementation of an online algorithm to detect thesign of the friction curve slope, which is suitable for practical

(7) (herein after

ఒಳǡ

ௗ,

௬, ௫,

is the forward speed of the tire centre whichassumed that is equal to vehicle longitudinal speed ௫

, the relationship between brakingslip output can be derived as:

௫ǡ

ௗ,

௬, ௫

into Eq. (5), the dynamics of the systemcan be formulated as a first order model of the wheel slip dynamics

௫ǡௗ,

௬, ௫௪

function of two state variables

To be able to design a linear control system, one should

௫ǡ and ǡǡ (we will discussthis nominal operating point in the following paragraphs).

and ௫ ௫ ௫ǡ; the linear transfer function for plant dynamics (from

slip output) can be derived by application of[5]

ǡǡ௦ ௗ,

ଶ

௬ǡ

represents the slope

inal operating point. It should be noted that calculationis not dependent on any specific tire

could be obtained either by differentiation of a tire model, such asmodel as cited in the literatures (see [12]

example), or by implementation of an online algorithm to detect the, which is suitable for practical

called plant) is of first order in

centre which is here

௫.

, the relationship between braking

௪ǡ(5)

dynamics of the systemcan be formulated as a first order model of the wheel slip dynamics

௪ǡ (6)

ݔ and ǡ݅ܤ .

should linearize Eq.(we will discuss

this nominal operating point in the following paragraphs).

and ௪ǡ; the linear transfer function for plant dynamics (from

slip output) can be derived by application of

,

(7)

slope of the ௫ǡcurve

inal operating point. It should be noted that calculationtire model. Itmodel, such as

[12] forexample), or by implementation of an online algorithm to detect the

, which is suitable for practical

called plant) is of first order in

(8)

(9)

dynamics of the system

Eq.

slip output) can be derived by application of

curve

inal operating point. It should be noted that calculation

To derive a controlan appropriate operating point, where the plant is beingaround it. By studying the slip dynamics transfer function, Eqone can conclude that the

velocitythe main parameters that are considerably changing during braking

actuation time perioddynamics, it is requireof longitudinal speedfunction, Eq.

As ௫cannot make the plant unstabl( ). On the other hand, the change oflocation of pole onof plant. More specifically, the effect ofdynamics is significant as it can make the plant unstable, as discussedbelow

Figure 4:

Considering a typical longitudinalFigure 4function

point (operating points are corresponding to high longitudinal speednormal loadequivalent to the smallest distancein planof the curve, i.e.

1By other word, it is assumed that the vehicle mass, wheel inertia,

and wheel radius are consta

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௫ǡǡ

௦ ௫

To derive a control oriented linear plant model, it is essential to selectan appropriate operating point, where the plant is beingaround it. By studying the slip dynamics transfer function, Eqone can conclude that the

velocity, ௫ǡ, and the road condition (which is reflected onthe main parameters that are considerably changing during braking

actuation time period1

dynamics, it is requireof longitudinal speedfunction, Eq. (7).

is always positive (forward driving assumption), its variationcannot make the plant unstabl

). On the other hand, the change oflocation of pole on plan, which will have an effect on the stabilityof plant. More specifically, the effect ofdynamics is significant as it can make the plant unstable, as discussedbelow.

Figure 4: Slope of μ(x, i)

Considering a typical longitudinalFigure 4, one can conclude from Eq.function ǡ is stable before slip peak (as

, and become unstable when the slip goes beyondpoint (as ௫ǡǡ , thereforeoperating points are corresponding to high longitudinal speednormal load ௭ǡ and smallest positiveequivalent to the smallest distance

plan. Moreover, when the linearization point is close to the peakof the curve, i.e. ௫ǡǡ

By other word, it is assumed that the vehicle mass, wheel inertia,and wheel radius are consta

௭ǡ

௫ǡǡǡ

oriented linear plant model, it is essential to selectan appropriate operating point, where the plant is beingaround it. By studying the slip dynamics transfer function, Eqone can conclude that the tire normal load,

, and the road condition (which is reflected onthe main parameters that are considerably changing during braking

1. Therefore, to investigate the variation of plant

dynamics, it is required to derive the maximum and minimum values

௫ , ௭ǡand ௫ǡ

is always positive (forward driving assumption), its variationcannot make the plant unstable, but, it will impact the value of gain

). On the other hand, the change ofplan, which will have an effect on the stability

of plant. More specifically, the effect ofdynamics is significant as it can make the plant unstable, as discussed

) at different operating points

Considering a typical longitudinal tireonclude from Eq. (

is stable before slip peak (as

, and become unstable when the slip goes beyond, therefore ). Therefore, the worst case

operating points are corresponding to high longitudinal speedand smallest positive

equivalent to the smallest distance of stable pole from imaginary axisMoreover, when the linearization point is close to the peak

, the plant becomes a pure integrator

ఒಳǡ

By other word, it is assumed that the vehicle mass, wheel inertia,and wheel radius are constant.

௦ ௗǡଶ

௬ǡ

oriented linear plant model, it is essential to selectan appropriate operating point, where the plant is beingaround it. By studying the slip dynamics transfer function, Eq

normal load, ௭ǡ, the veh

, and the road condition (which is reflected onthe main parameters that are considerably changing during braking

. Therefore, to investigate the variation of plantd to derive the maximum and minimum values

in the plant dynamics transfer

is always positive (forward driving assumption), its variatione, but, it will impact the value of gain

). On the other hand, the change of ௭ǡand ௫ǡǡ will move theplan, which will have an effect on the stability

of plant. More specifically, the effect of ௫ǡǡ variation on the plantdynamics is significant as it can make the plant unstable, as discussed

at different operating points

tire force/slip curve, as shown in(10) that the open loop transfer

is stable before slip peak (as ௫ǡǡ, and become unstable when the slip goes beyond

). Therefore, the worst caseoperating points are corresponding to high longitudinal speed

and smallest positive slope of ௫ǡwhich isof stable pole from imaginary axis

Moreover, when the linearization point is close to the peak, the plant becomes a pure integrator

By other word, it is assumed that the vehicle mass, wheel inertia,

௫,

(10)

oriented linear plant model, it is essential to selectan appropriate operating point, where the plant is being linearizedaround it. By studying the slip dynamics transfer function, Eq. (7),

he vehicle

, and the road condition (which is reflected on ௫ǡǡ) arethe main parameters that are considerably changing during braking


in the plant dynamics transfer

is always positive (forward driving assumption), its variatione, but, it will impact the value of gain

will move theplan, which will have an effect on the stability

variation on the plantdynamics is significant as it can make the plant unstable, as discussed

force/slip curve, as shown inthat the open loop transfer

, therefore

, and become unstable when the slip goes beyond the peak). Therefore, the worst case

operating points are corresponding to high longitudinal speed ௫ , lowwhich is

of stable pole from imaginary axisMoreover, when the linearization point is close to the peak

, the plant becomes a pure integrator:

(11)

By other word, it is assumed that the vehicle mass, wheel inertia,

oriented linear plant model, it is essential to select

) are


dynamics is significant as it can make the plant unstable, as discussed

, low

of stable pole from imaginary axis

To obtain the value ofthe Burckhardtrelationship betweendefined as

wheresurfaces

Table

By investigating different road surfaces at different slips, one canconclude that the maximum

dry asphalt at

at slipthe slip

The tire

weight distribution (static load) and land lateral accelerations (dynamic loads),from the following equations

௭ǡଵ

௭ǡଶ

௭ǡଷ

௭ǡସ

where

front and rearis the trackindicesleft wheels, respectively

To obtain the value ofthe Burckhardt tire modelrelationship betweendefined as:

௫ǡ ǡ௫,

where ଵ, ଶ and ଷ are constants,surfaces in Table 1. Therefore

௫ǡ ǡ

Table 1: Values of the Burckhardt

Road Surface

Asphalt, dryAsphalt, wetConcrete, dryCobblestone, drySnowIce

By investigating different road surfaces at different slips, one canconclude that the maximum

dry asphalt at ǡat slip ǡ with the value ofthe slip slope bound is

tire normal load on each wheel

weight distribution (static load) and land lateral accelerations (dynamic loads),from the following equations

here is the total sprung and unsprung vehicle mass,

front and rear tire distance to Centre of Gravityhe track width, is the height of CG from the ground

ndicesleft wheels, respectively

To obtain the value of tire friction slopemodel [12]. Based on Burckhardt

relationship between tire slip and normalised longitudinal

,

௭,ଵ

are constants, asTherefore

௫, ,

,ଵ

: Values of the Burckhardt tire model coefficients

Road Surface

1.28010.857

Concrete, dry 1.1973Cobblestone, dry 1.3713

0.19460.05

By investigating different road surfaces at different slips, one canconclude that the maximum and minimum

with the value of

with the value ofbound is:

௫,

normal load on each wheel

weight distribution (static load) and land lateral accelerations (dynamic loads),from the following equations [14]:

௫

௫

௫

௫

is the total sprung and unsprung vehicle mass,

distance to Centre of Gravityis the height of CG from the groundrefer to front left, front right, front left, and rear

left wheels, respectively.

slope, ௫ǡ , one can employ. Based on Burckhardt

slip and normalised longitudinal

ିమఒಳǡଷ

as defined for different road

ଵ ଶିమఒಳǡ

ଷ

model coefficients [12]

2801 23.99857 33.822

1973 25.1683713 6.45651946 94.129

05 306.39

By investigating different road surfaces at different slips, one canand minimum values of

with the value of and on dr

, respectively

ǡ݅ݖ is a function of the vehicle

weight distribution (static load) and load transfer due to longitudinaland lateral accelerations (dynamic loads), which can be estimated

is the total sprung and unsprung vehicle mass,

distance to Centre of Gravity (CG)is the height of CG from the groundrefer to front left, front right, front left, and rear

, one can employ. Based on Burckhardt tire model, the

slip and normalised longitudinal tire force is

ǡ (12)

defined for different road

(13)

[12]

0.520.347

0.53730.66910.0646

0

By investigating different road surfaces at different slips, one can

௫ǡhappen on

on dry cobblestone

, respectively. Therefore

(14)

is a function of the vehicle

oad transfer due to longitudinalcan be estimated

௬௪

(15)

௬௪

௬௪

௬௪

is the total sprung and unsprung vehicle mass, and are

(CG), respectively, ௪is the height of CG from the ground andrefer to front left, front right, front left, and rear

force is

cobblestone

oad transfer due to longitudinal

(15)

are

௪

refer to front left, front right, front left, and rear

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Table 2: Vehicle parameters

Considering the vehicle parameters, as indicated in Table 2, andassuming the extreme magnitudes of ௫ and ௬ , the

maximum normal force will be applied on the front right tire is:

௭ ೌೣ ௭,ଶ

The same conclusion can be made when ௬ , however, the

maximum normal force will be applied to front left tire ( ௭,ଵ),instead. Therefore the normal force range limit is:

௭, (16)

Finally, it is assumed that the longitudinal velocity range (in whichthe safety brake actuation will be activated) is between 10 to 50 m/s,i.e.

௫ (17)

Employing the above operational limits of ௫, , ௭,and ௫, , themaximum and minimum values of gain and pole of the plant (for thevehicle with the values indicated in Table 2) can be obtained fromEqs. (9) and (10), respectively. By investigating the possiblemagnitudes of the plant’s gain and pole locations, it is concluded thatthe plant dynamics is highly sensitive to its operating conditions.Therefore, selection of appropriate nominal operating point plays animportant role in design of the proposed control system.

Finally, it should be noted that the complete brake plant model (i.e.from pressure input to wheel slip output, as shown in Figure 1),includes the slip dynamics, the calliper dynamics and the EHP smartactuator dynamics. The required barking torque about the wheel spinaxis and the subsequence braking force is produced by application ofhydraulic brake pressure at the brake callipers. The relationshipbetween brake line pressure ,and wheel (bake) torque ௪, isdefined as [11]:

௪, , ,

(18)

where ,and are the calliper gain and time constant (time lag),respectively. The calliper dynamics , is

,௪,

,

,

(19)

Similarly, the EHB smart actuator dynamics ாு, can beconsidered as a stable first order transfer functions as [21]:

ாு ,ாு

(20)

where ܤܪܧ is the EHB time constant (time lag). Therefore, thecomplete plant dynamics takes the form of:

, ఒಳ , ாு, ,

,

,

ாு

,

ாு (21)

Bode diagram of the plant dynamics at different longitudinal slips isplotted in Figure 5, assuming the vehicle and brake parameters asindicated in Table 2, ாு [11] , dry asphalt,

௭, and ௫ . By investigating the phase angle ofthe plant, it is clear that the plant is stable at the slip values lowerthan ௫, and become unstable at higher slips. As discussed

before, the tire slip of ௫, corresponds to the peak point of

the tire friction curve on dry asphalt, where ௫, (see also Figures4).

Figure 5: Plant dynamics G,୧for different slips (dry asphalt, F,୧= 10KN andV୶ = 50 m/s)

Control System Design

Due to the fact that the dynamics of the system is changingconsiderably during its operational envelope, and there are severaluncertainties that exist in the system (such as brake pad coefficient offriction and so on); it is necessary to employ a feedback controlsystem, as shown in Figure 2, to provide stability as well as goodperformance for the system at all operating conditions. In this paper,

Parameters Abbreviation value Unit

Vehicle Mass ݉ 1226 ݇݃

Front Tire distance to Centre of Gravity ݈ 0.863 ݉

Rear Tire distance to Centre of Gravity ݈ 1.567 ݉Half track ௪݈ 0.71 ݉Height of CG from the ground ℎ 0.519 ݉Vehicle Inertia ௭ܫ 1458.76 ݇݃ ݉

ଶ

Gravitational Acceleration ݃ 9.8 ݉ ଶݏ/

Wheel inertia ௬,ܫ 1.17 ݇݃ ݉ଶ

Wheel dynamic radius ܴௗ௬ 0.266 ݉

Brake gain factor (front) ,ܭ ,݅= 1,2 10 ܰ݉ /ܾܽ ݎ

Brake gain factor (rear) ,ܭ ,݅= 3,4 5 ܰ݉ /ܾܽ ݎ

line pressure build up time lag ߬ 0.1 ݁ݏ ܿ

EHB actuator build up time lag ா߬ு 0.1 ݁ݏ ܿ

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a new closed loop wheel slip control system based on Youlaparameterization approach is proposed [15, 16, and 17].

Investigating the complete (linearized) plant dynamics, as describedby Eq. (21), one can conclude that the plant dynamics consists ofthree first order transfer functions. We take the Youla parameter asthe inverse of the plant transfer function at a nominal operatingpoint, ,,, multiply to three stable first order filters with adjustablepoles corresponding to the three dynamics exist in the system, suchas:

,,, ଵ ଶ ଷ

ଵ ଶ ଷ (22)

By selecting a stable nominal plant ,,, the proposed Youlatransfer function is stable (and minimum phase), therefore, internalstability of the feedback system is guaranteed [17]. The tuneableparameters ଵ, ଶ and ଷ can be employed to shape of the closed looptransfer functions ,and ,and control system bandwidth suchthat it could provide robust performance considering plant dynamicsuncertainties at low frequencies and attenuate sensor noise at highfrequencies.

The complementary sensitivity ,and sensitivity , transferfunctions are:

, , ,,ଵ ଶ ଷ

(23)

, ,,ଵ ଶ ଷ

ଵ ଶ ଷ(24)

The controller transfer function can݅,ܤ be derived as:

,,

, ,, ଵ ଶ ଷ(25)

and the open loop transfer function :is݅,ܤ

, , ,,ଵ ଶ ଷ

(26)

Recall, ,,, is the linearized transfer function of plant dynamics ata nominal operating point where the nonlinear differential equation ofthe slip dynamics was linearized around its nominal point. Asexplained before, the dynamics and stability of the plant is highlydependent on these parameters; therefore, selecting differentoperating points results in Youla parameters (and controllers) withdifferent behaviours. To obtain an appropriate plant dynamics, ,,,(for our proposed control design approach), the nominal operatingpoint for ௫,, should be selected at a slip value, ௫,,, where the tirefriction curve ௫, is near to its peak value but is in stable region (i.e

௫, ). Note that the wheel slip value corresponding to the

abscissa of the maximum of ௫,is different at various road surfaces,as illustrated in Figure 6.

Figure 6: Longitudinal friction curve

Taking the nominal operating points as: dry asphalt friction curve(from Burckhardt tire model), ௫,, , ௭,, and

௫, , , ாு and ଵ ଶ ଷ ; thecontrol system transfer functions (for the vehicle parameters asindicated in Table 2) can be derived as:

, ଶ (27)

,

ଷ (28)

,ଷ ଶ

ଶ

ଷ

ଷ (29)

,

ଶ

ଷ (30)

,

ଶ

ଷ ଶ

ଶ

(31)

,ଷ ଶ

ଶ

(32)

To shape the close loop response of the system, tuning parameter can be employed. The magnitude Bode plot of ,and , transferfunctions for two different values of time constant are shown inFigure 7. The system bandwidth increase by decreasing the timeconstant, however, the peak values of and transfer functions ( ௌand ்) are less than 2db so the 6db gain margin is guaranteed whichmeans that good control performance is met [19]. Moreover, thecrossover gain is less than zero which means the minimum of °

phase margin is also guaranteed [20].

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Figure 7: Brake Control System S & T

By selecting , the brake controller takes the form of:

,

ଶ

ଶ (33)

All the transfer functions of the brake control system, including plant,Youla parameter, controller, sensitivity, open loop and closed looptransfer functions, are shown in Figure 8, confirm our previousconclusion for the control system performance.

Figure 8: Frequency response of the brake control transfer functions

To investigate the behaviour of the control system in time domain,the response of the closed loop control system subject to unit stepinput at nominal operating point is shown in Figure 9. The resultconfirms a good dynamic response of the controlled linear systemwith no overshoot.

Figure 9: Unit step response of the brake control at nominal operating point

It is worth mentioning that the proposed controller was designedbased on the linearized plant transfer function at a nominal operatingcondition (dry asphalt friction curve (Burckhardt tire model, ௫,,

, ௭,, and ௫, ). However, the dynamicsof the plant is highly sensitive to variation of the parameters such asroad surfaces, tire slip, vehicle velocity and tire normal forces, asdiscussed before. More importantly, increasing the tire slip (abovethe peak point of tire friction cure) make the plant unstable. Toinvestigate the stability and robustness of the control system at theentire range of operational envelope, the response of closed loopbrake control system subject to slip step input at different operationalconditions and surfaces are plotted in Figure 10. Interestingly, thecontroller can stabilize the closed loop system on different surfacesand at all conditions, even in the worse-case conditions in which theplant is unstable (i.e. at tire high slip and high normal load, andvehicle low longitudinal velocity). Meanwhile, the performance ofthe control system exhibits a sizable variation (from underdamped tounderdamped (oscillatory) behaviour). Moreover, the settling time ofthe system range from few millisecond to one second, depending alsoon the vehicle speed and normal load. This is due to utilizing a fixedstructure controller for the whole ranges of the system operations, atwhich the dynamics of the plant is changing considerably. However,considering the fact that the controller can stabilise the plant at allconditions and track the reference value within few milliseconds inmost cases (except in worse case scenarios, which cannot happen inreality), the utilisation of one fixed structure controller could bejustified.

௫ܸ (݉ (ݏ/ ௭,ܨ (ܰܭ) ௫,ߣ

1 10 1 0.12 10 1 0.53 30 5 0.14 30 5 0.55 50 10 0.16 50 10 0.5

Figure 10: Brake control step response at various operational conditions

Control System Validation

Simulation Results

To examine the stability and performances of the proposed closedloop slip control system in simulation, a single wheel model, asdescribed by Eqs. (1) and (2), in conjunction with the EHB andcalliper dynamics is constructed in Simulink® environment. Thebrake actuator constraint is also included in the model, by limiting thepressure command to EHB within the range of [0,200] bar, as shownin Figure 11.

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Figure 11: Closed loop brake control with brake dynamics and constraint, andquarter car vehicle dynamics

The results of simulations subject to two different driving conditionson dry asphalt surfaces are presented in Figure 12. The first drivingcondition ( ௫ ௭, ௫, )corresponds to low vehicle speed and nominal tire load where the slipis below its threshold limit, therefore the plant is stable. In thisscenario, the slip reach to its target ( ௫, ) within 1 sec with anoverdamped response and the vehicle stops within 2.3 sec. Thecommanded brake pressure and the tire longitudinal force are withintheir limits. The second driving condition ( ௫ ௭,

௫, ) corresponds to a severe driving conditionwhere the tire is operating beyond its saturation limit, therefore theplant is highly unstable. It could be observed that the control systemcan stabilise the plant even at slips greater than 0.16, however, due toactuator saturation, the system exhibits an overshoot, but it finallycould track the reference value within 2 sec. The slip overshoots,which is clearly reflected from the difference between vehicle speedand tire longitudinal speed ( ௗ, ோ,), is generated because of thebrake pressure has reached its limit of 200 bar. In spite of the fact thatthe tire slip is in unstable region and also brake pressure is saturated;the control system is stable and can reduce the vehicle speed from 50m/s to 0 in less than 4 sec (without locking the wheel).

Figure 12: Brake control step response, dry asphalt

To investigate the performance of the control system on low musurfaces, a similar simulation is performed with the same operationalconditions, but on snow. The simulation results, as shown in Figure13, confirm the stability and performance of the control system at lowspeeds. Moreover, the control system can stabilize the plant and

provide acceptable tracking (but with high overshoot) even in severedriving conditions.

Figure 13: Brake control step response, snow

HiL Testing Results

To validate the proposed control system in real time environmentwith the existence of real dynamics of the brake system, the controlsystem is implemented in dSPACE MicroAutoBox rapid controlprototyping (RCP) platform in the Cranfield’s integrated Brake &Steering HiL rig system [21]. The HiL rig consists of a complete realbrake system (including brake pedal, master cylinder, disk, calliperand the EHB unit), integrated with IPG CarMaker/HiL® as an off-the-shelf high fidelity real time vehicle model [22] running in adSPACE ds1006 Simulator and a driver in the loop facility to form acomprehensive vehicle dynamic rapid control development platform.

The vehicle is driving at speed of around 33 m/s (120 Kph) wassubjected to step slip input (slip target was set to 0.03). The testresults, as shown in Figure 14, confirm the good stability andperformance of the slip control system. The brake pressure increaseto 40 bar and the wheel slip reach to its target within 0.5 sec, and as aresult, the vehicle longitudinal speed reduces (from 33 m/s) to 10 m/swithin 1 sec without wheel locking.

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Summary/Conclusions

A new control strategy for wheel slip control, considering thecomplete dynamics and constraints of the electro-hydraulic brake(EHB) system, is proposed. The control system is based on closedloop shaping Youla-parameterization method. The stability andperformance of the controller are experimentally validated utilizingCranfield University’s HiL system. The real time experimentsinclude the proposed control system implemented in a MicroAutoBoxrapid control prototyping platform, with the existence of real EHBsmart actuator integrated with CarMaker/HiL® vehicle model anddriver in the loop (driving simulator) facility. The experimentalresults, verified the stability and good performances of the slipcontrol system, even in hazardous driving conditions. The proposedslip control system can be employed in standalone brake based safetysystems such as ABS or ESP as well as integrated vehicle dynamicscontrol systems [21].

References

1. Jiang, F., and Z. Gao. "An Application of Nonlinear PID Controlto a Class of Truck ABS Problems." Decision and Control,2001. Proceedings of the 40th IEEE Conference on. IEEE, 2001.516-521.

2. Mauer, G. F. "A fuzzy logic controller for an ABS brakingsystem." Fuzzy Systems, IEEE Transactions on 3, no. 4 (1995):381-388.

3. Johansen, T. A., I. Petersen, J. Kalkkuhl, and J. Ludemann."Gain-scheduled wheel slip control in automotive brakesystems." Control Systems Technology, IEEE Transactions on11, no. 6 (2003): 799-811.

4. Drakunov, S., U. Ozguner, P. Dix, and B. Ashrafi. "ABS controlusing optimum search via sliding modes." Control SystemsTechnology, IEEE Transactions on 3, no. 1 (1995): 79-85.

5. Savaresi, S. M., and M. Tanelli. Active braking control systemsdesign for vehicles. Springer, 2010.

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Automatic Control, IEEE Transactions on 21, no. 3 (1976): 319-338.

7. Robert Bosch GmbH. Automotive Handbook. 8th. Edited by K.Reif. Cambridge, USA: Bentley Publishers, 2011.

8. Gunther Plapp. Electronic Brake Control Systems. Nov. 20,2001.http://www.autospeed.com/cms/article.html?&title=Electronic-Brake-Control-Systems&A=1202 (accessed April 22,2014).

9. Van Zanten, A. T. "Evolution of electronic control systems forimproving the vehicle dynamic behavior." In Proceedings of the6th International Symposium on Advanced Vehicle Control.2002. 1-9.

10. Schöner, H. P. "Automotive mechatronics." Control engineeringpractice 12, no. 11 (2004): 1343-1351.

11. Limpert, R. Brake Design and Safety. 3rd. USA: SAEInternational, 2011.

12. Kiencke , U., and L. Nielsen . Automotive Control Systems: ForEngine, Driveline, and Vehicle, 2nd ed. Berlin: Springer-Verlag,2005.

13. Van Zanten, A. T., R. Erhardt, G. Pfaff, F. Kost, U. Hartmann,and T. Ehret. "Control aspects of the Bosch-VDC." AVEC’ 96.Achen, 1996.

14. Milliken, W. F., and D. L. Milliken. Race Car VehicleDynamics. USA: SAE International, 1995.

15. Assadian, F. “Neo-Classic Control Approach.” UnpublishedLecture Notes. Cranfield University, 2011.

16. Assadian, F., "Mixed H∞ and Fuzzy Logic Controllers for the Automobile ABS," SAE Technical Paper 2001-01-0594, 2001,doi:10.4271/2001-01-0594.

17. Vidyasagar, M. Control system synthesis: a factorizationapproach. Morgan & Claypool Publishers, 2011.

18. Doyle, J. C., B. A. Francis, and A. Tannenbaum. FeedbackControl Theory. New York: Macmillan Publishing Co., 1992.

19. Skogestad, S., and I. Postlethwaite. Multivariable feedbackcontrol: analysis and design. 2. New York: Wiley, 2007.

20. Ogata, K. Modern Control Engineering. Prentice-Hall, 2010.21. Soltani, A. M. Low Cost Integration of Enhanced Stability

Control (ESC) with Electric Power Steering (EPS) (PhDThesis). Cranfield University, 2014.

22. IPG Automotive GmbH. “CarMaker® Reference ManualVersion 4.0.6.” 2013.

Contact Information

Dr. Amir Soltani, Cranfield University, Cranfield, MK43 0AL, UK,[email protected]

Prof. Francis Assaidan, University of California, Davis, USA,[email protected]

Acknowledgments

This work is undertaken within the Evoque_e project, co-funded bythe UK's innovation agency, Innovate UK.

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New Slip Control System Considering Actuator Dynamics-2015 … · 2020-02-17 · A new control law...

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