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WHEEL SLIP CONTROL IN ABS BRAKES USING GAINSCHEDULED CONSTRAINED LQR

Idar Petersen , Tor A. Johansen , Jens Kalkkuhl and Jens Ldemann

SINTEF Electronics and Cybernetics, N-7465 Trondheim, Norway. e-mail: Idar.Petersen@ecy.sintef.noDepartment of Engineering Cybernetics, Norwegian University of Science and Technology, N-7491 Trondheim, Norway.

DaimlerChrysler, Research and Technology, Alt-Moabit 96a, D-10559 Berlin, Germany.

Keywords: Automotive control, gain scheduling, nonlinearsystems, robustness, stability

Abstract

A wheel slip controller for ABS brakes is formulated using anexplicit constrained LQR design. The controller gain matricesare designed and scheduled on the vehicle speed based on locallinearizations. A Lyapunov function for the nonlinear control

system is dervied using the Riccati equation solution in orderto prove stability and robustness with respect to uncertaintyin the road/tyre friction characteristic. Experimental results

from a test vehicle with electromechanical brake actuators andbrake-by-wire show that high performance and robustness areachieved.

1 Introduction

An anti-lock brake system (ABS) controls the slip of eachwheel to prevent it from locking such that a high frictionis achieved and steerability is maintained. ABS controllersare characterized by robust adaptive behaviour with respect tohighly uncertain tyre characteristics and fast changing road sur-face properties [1].

The introduction of advanced functionality such as ESP (elec-tronic stability program), drive-by-wire and more sophisticatedactuators offers both new opportunities and requirements forhigher performance in ABS brakes. The contribution of thepresent work is a study of a model-based design of ABS con-trollers, see also [2], [3]. We consider electromechanical ac-tuators [4] rather than hydraulic actuators, which allow con-tinuous adjustment of the clamping force. Here, we designthe wheel slip controller based on a recently developed explicitLQR design method that takes into account input and state con-straints [5], see also [6]. The wheel slip dynamics are highlynonlinear. Despite this, our control design relies on local lin-earization and gain-scheduling. In order to analyse the effectsof this simplication, we develop a Lyapunov based stabilityand robustness analysis. Results from experiments using a testvehicle are also included. Other studies of model-based wheelslip control in ABS can be found in [7], [8], [9], [10].

2 Modelling

In this section, we review a mathematical model of the wheelslip dynamics, see also [1], [8] and [7]. The problem of wheelslip control is best explained by looking at a quarter car modelas shown in Figure 1. The model consists of a single wheelattached to a mass p. As the wheel rotates, driven by inertiaof the mass p in the direction of the velocity y> a tyre reactionforce I is generated by the friction between the tyre surfaceand the road surface. The tyre reaction force will generate atorque that initiates a rolling motion of the wheel causing anangular velocity $. A brake torque applied to the wheel willact against the spinning of the wheel causing a negative angular

Figure 1: Quarter car forces and torques.

acceleration. The equations of motion of the quarter car are

p by @ I (1)

Mb$ @ u I W vljq+$, (2)

where

y horizontal speed at which the car travels

$ angular speed of the wheelI vertical forceI tyre friction forceW brake torqueu wheel radiusM wheel inertia

The tyre friction force I is given by

I @ I +> > , (3)

where the friction coefcient is a nonlinear function of

tyre slip friction coefcient between tyre and road slip angle of the wheel

and the longitudial slip dened by

@y $u

y(4)

describes the normalised difference between horizontalspeed y and the speed of the wheel perimeter $u. The slip valueof @ 3 characterises the free motion of the wheel where nofriction force I is exerted. If the slip attains the value @ 4>then the wheel is locked which means that it has come to astandstill.

The friction coefcient can span over a very wide range,but is differentiable with the property +3> > , @ 3 and

+> > , A 3 for A 3. Its qualitative dependence onslip is shown in Figure 2. The friction coefcient increaseswith slip up to a value where it attains its maximum .

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0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

= 0 o

= 10 o

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

x

()

H

=0.95

H

=0.8

H

=0.5

H

=0.3

H

=0.1

x

(longitudinal slip)

Figure 2: Tyre slip/friction curves +> > ,

For higher slip values, the friction coefcient will decrease toa smaller value where the wheel is locked and only the slid-ing friction will act on the wheel. The dependence of frictionon the road condition is shown in the bottom part of Figure 2.For wet or icy roads, the maximum friction is small and theright part of the curve is atter. The tyre friction curve will alsodepend on the brand of the tyre. In particular, for winter tyres,the curve will cease to have a pronounced maximum.If the motion of the wheel is extended to two dimensions, thenthe lateral slip of the tyre must also be considered. The slip an-gle is where the wheel moves with velocity y in the longi-tudinal direction and with a velocity y in the lateral direction.

In this case, the longitudinal slip @ and the lateralslip @ vlq are distinguished as well as the correspond-ing friction coefcients and . The top part of Figure 2shows the dependence of the friction coefcient on the sideslip angel = The side force I depends greatly on the side slipangle . For large side slip angles, the force gets smaller. Inthe sequel, for simplication purposes unless otherwise stated,the side slip angle will be considered to be zero with @

and y @ y=Using (1)-(4) we get for y A 3 and $ 3

b @ 4

y

4

p+4 , .

u

M

I +> > , .

4

y

u

MW (5)

by @ 4

pI +> > , (6)

Note that when y $ 3, the dynamics of the open loop system(from W to ) becomes innitely fast with innite gain. Thisleads to a loss of controllability and the slip controller must beswitched off for small y.

Proposition 1 Consider the system (5)-(6) with W +w, 3 forall w 3. Ify+3, A 3 and+3, 5 ^3> 4`> then +w, 5 ^3> 4` andby+w, 3 for all w 3 where y+w, A 3=

Proof. Note that +w, is a continuous trajectory. Hence, thepossibile escape points are @ 3 and @ 4. Consider rst @ 3. Since +3, @ 3, it follows from (5) that b @ W 3due to W 3. Hence, +3, 3 implies +w, 3 for all w 3.Consider next @ 4. Then, $ @ 3 and from (2) it follows thatb$ 3 due to the discontinuity vljq+$, in (2). From (4), weconclude that b 3 and +3, 4, which implies +w, 4 forall w 3. Finally, note that by 3 from (1) because I 3 for 5 ^3> 4`.

3 Control problem

The control problem is essentially to control the value of thelongitudinal slip to a given setpoint that is either con-stant or commanded from a higher-level control system such asESP (electronic stability program). The controller must be ro-bust with respect to uncertainties in the tyre characteristic andvariations in the road surface conditions.

The control input is the clamping force I that is related tothe brake torque as W @ n I . A maximum or a minimumforce can be applied to the brake pads by the actuator duringbraking. The (small) minimum is to ensure that the pads arepositioned close to the brake disk (no air-gap). Maximum iswhat the actuator is capable of. There is also a rate limit at howfast the torque can be changed by the actuator.

Integral action or adaptationmust be incorporated to remove er-rors due to model inaccuracies, in particular the road/tyre fric-tion coefcient . It is essential that the controller maintainsa high performance and is robust w.r.t. to any road/tyre frictioncurve that can be encountered cf. Figure 2.

4 Gain-scheduled LQRC controller design andanalysis

The dynamics of the wheel and car body are given by (5) and(6), respectively. Due to large differences in inertia, the wheeldynamics and car dynamics will evolve on signicantly differ-ent time scales. The speed y will change much more slowlythan the slip and is therefore a natural candidate for gainscheduling. Thus, for the control design, we consider only(5) and regard y as a slow time-varying parameter. A con-strained LQR controller [5] is applied. This requires a nominallinearized model for design.

4.1 Linearized dynamics

Let ( > aW ) be an equilibrium point for (5) dened by the con-stants > I and , where is the desired slip

aW @

M

pu+4 , . u

I + > > ,

The linearized slip dynamics are given by

b @

y+ , .

y+W aW , . h.o.t. (7)

where and are linearization constants given by

@ I

4

p+4 , .

u

M

C

C+ > > ,

. I4

p+ > > , (8)

@ uM

A 3 (9)

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The wheel slip dynamics (5) can be written in the form

b{ @!+{ ,

y.

y+W W , (10)

where

!+{ , @

4

p+4 { , .

u

M

I +{ . > > ,

.u

MW (11)

and { @ and

W @

M

pu+4 , . u

I + > > , (12)

It can be seen that this system has an equilibrium point givenby { @ 3, W @ W and !+3, @ 3, and the linearized slipmodel (7) with a perturbation term written as follows

b{ @

y{ .

y+W W , .

+{ ,

y(13)

where +{ , @ !+{ , { .

4.2 Wheel slip control with integral action

Let the system dynamics be augmented with a slip error inte-grator b{ @ @ {

b{b{

@ D+y,

{{

. E+y, +x W , . Z+y, +{ ,

(14)

where

D+y, @

3 43

> E+y, @

3

> Z+y, @

3

Since W is assumed unknown, we have dened x @ W , andthe equilibrium point to be { @ +{ > 3, and x @ W where{ is still unspecied and we get

b{ @ D+y, +{ { , . E+y, +x x , . Z+y, +{ , (15)

Next, dene the quadratic cost function for the purpose of localLQ design based on the nominal part of (15):

M+{+w,> x^w>4,, @]

++{+, { , T+y, +{+, { ,

. +x+, x , U +x+, x ,,g (16)

Assuming constant y, the optimal control law is given by

ax @ N+y,{ (17)

where the gain matrix N+y, @ U E +y,S+y, and wechoose to neglect the unknown { and x which will be ac-counted for due to the integral action. The symmetric matrixS+y, A 3 is dened by the solution to the algebraic Riccatiequation for the design:

S+y,D+y, . D +y,S+y, S+y,E+y,U E +y,S+y, @ T+y, (18)

namely

S +y, @

. U

T +y, . y

+T +y,U,

S +y, @ y

+T +y,U,

S +y, @

Uy

.

. U

T +y, . y

U

y

and the controller gains are

N +y,@

T +y,U

N +y,@

. . U T +y, . y

We introduce the saturated control

x @

+W > ax A WW > ax ? Wax> otherwise

(19)

for some W A W 3. Furthermore, we dene the error(note that both x and ax depend on y and {)

+{> y, @ +x ax, (20)

With the denition +{> y, @ +{> y, . +{ , and assuming{ @ W @N +y,, the closed loop dynamics can be written as

b{ @ +D+y, . E+y,N+y,, { . Z+y,+{ > y, (21)

with { @ { { .

Proposition 2 Consider the system (5) with controller (19).Assume U A 3 and the smooth matrix-valued function T sat-ises T +y, @ T +y, @ 3> T +y, A 3> T +y, A 3>

3> 3 for all y A 3. Moreover, suppose

T +y,AS +y,F

y+4 , (22)

T +y,{ A

#5 +{ ,S +y,{

y.

S +y, +{ ,

yF

$+4 ,

(23)

G+y,@S +y,S +y, S +y,S +y, A 3 (24)

are satised for all y A 3, { 5 ^ > 4 ` and some F A 3and5 +3> 4,. IfW A W A W , the equilibrium { @ 3is uniformly exponentially stable.

Proof. Consider a neighbourhood [ of the origin in U suchthat +{> y, @ 3 for all { 5 [ and y A 3. Let a Lyapunov

function candidate beY+{, @ { S+y,{

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Its time-derivative along trajectories of (21)

bY @ {

CS+y,

Cyby

{ . b{ S{ . { Sb{ (25)

is found by substituting for (15), (17) and (18) in (25):

bY @ {CS+y,

Cyby{ { T+y,{

. +{ ,+Z +y,S+y,{ . { S+y,Z+y,, (26)

From Proposition 1, it is clear that by 3. Thus, the nega-

tivity of {

by

{ requires S +y, @ A 3 for all

y A 3= For S +y, to be positive denite, it is sufcient thatS +y, A 3 and G+y, @A 3 which follows due to (24). Notethat since T +y,> T +y,> A 3, it follows immediatelythat S +y, A 3. Then, (26) becomes:

bY +{ ,+Z S+y,{ . { S+y,Z+y,, { T+y,{

@ T +y,{ T +y,{

.5

y +{ ,+S +y,{ . S +y,{ , (27)

To obtain all { -terms in (27) in a quadratic form, we applyYoungs inequality 5de d @F. Fe . Hence,

bY T +y,{ .S +y,

y{ FT +y,{

.5

y +{ ,S +y,{ .

S +y,

y

+{ ,

F(28)

Due to (22) and (23) it follows that

bY T +y,{ T +y,{

and we conclude that the equilibrium is uniformly exponen-tially stable using Corollary 4.2, [11].

Inequality (22) states that the error weight T +y, must be suf-ciently large, leading to a sufeciently large controller gain.Note that S +y, depends on T +y,> but it is evident thatS +y, increases with

sT +y, such that (22) will indeed

hold for a sufciently large T +y,, except when y $ 3=

Inequality (23) states that the error weight T +y, must also besufciently large, leading to a sufciently high gain to stabilizethe system. Note that S +y, increases with

sT +y, such

that this is also possible, except for y $ 3.

Note that T +y, m{ m is essentially chosen in (23) to dominatethe perturbation +{ ,. Inequality (23) should be checked withrespect to the perturbations that are generated by all possiblefriction curves +, to ensure robust stability.

Inequality (24) can be seen to be non-restrictive since it willalways be satised for close to zero [3]. This correspondsto generating the nominal model by linearizing near the peak ofthe friction curve. Experience shows that high performance isindeed achieved this way. Note that for @ 3, no informationon the friction curves is actually utilized in the control design.

The constant F A 3 should be chosen to minimize conserva-tiveness. However, the choice T +y, @ T +y, @ 3 and

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-1

-0.5

0

0.5

1

1.5

2

2.5

3x 10

7

x

(longitudinal slip)

Left hand side

Right hand sides

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-1

-0.5

0

0.5

1

1.5

2

2.5

3x 10

7

x

(longitudinal slip)

Left hand side

Right hand sides

Figure 3: Illustration of the robust stability requirement, i.e.the left and right hand sides of eq. (23). The upper part is fory @ 4 m/s and the bottom part is for y @ 65 m/s.

taking S+y, from the solutions of the Riccati equation are pos-sibly conservative.

The controller gain N+y, depends on the speed (gain schedul-ing). From a practical point of view, a useful gain schedule is

achieved by letting A 3 and A 3, as this re-duces the gain as y $ 3. As mentioned earlier, this is necessaryto avoid instability due to the unmodelled (actuator) dynamicssince these tend to dominate as y $ 3.

An important aspect is the initialization of the integrator state{ . Note that { is a controller state that can be initialized ar-bitrarily, while { is unknown since it depends on the road/tyrefriction coefcient . Hence, an intelligent initialization of{ +3, based on any a priori information on and possibly onthe known { +3, may signicantly improve the transient per-formance [2].

4.3 An idealized design example

We consider a design example with the following parametersp @ 783 nj, I @ 7747 Q, u @ 3=65 p, M @ 4=3 nj pand the friction model in the right part of Figure 2. Assuming @ 3=47 and the nominal design @ 3=; and @ 3,we get @ 43=5 and @ 3=65. We choose U @ 4 andT+y, @ Ty with T @ 9 43 and T @ 73 43 . Notethe scaling due to the different magnitudes of W and . Thechoice for T+y, leads to a gain schedule with reduced gain asy $ 3, that is useful to avoid instability due to unmodelledacatuator dynamics as y $ 3.

Figure 3 shows that the robust stability requirement (23) is sat-ised for all { 5 ^ > 4 ` for all the friction curves inthe right part of Figure 2. Although curves are shown only fory @ 4 m/s and y @ 65 m/s, we have veried that (24) is fullled

for intermediate values ofy. The control design is also satis-ed the stability requirements (22) and (24), and we concluderobust uniform exponential stability of the equilibrium.

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5 Implementation

The experimental vehicle is a Mercedes E220 equipped withelectromechanical disc brake actuators and a brake-by-wiresystem. A previous version of this system is described in [4].The system consists of four independent servo controllers, one

for each brake, and a central electronic control unit (ECU)where the wheel slip controllers run. With reference to thetheoretical study presented in previous sections, we note thefollowing implementation details:

The actual control implementation is based on the dis-cretization of the models and a discrete-time control de-sign. The discrete-time model also takes into accountcommunication delays on the brake-by-wire electronics.The sampling rate for the wheel slip control loop is 7 ms.

The electromechanical brake actuator is assumed to be lin-ear,

bI @ I . I (29)

where is the time-constant and I is the commandedclamping force. This is reasonable since it contains a lo-cal feedback loop. The model used for control design isaugmented with a discretized version of (29) and the mea-sured clamping force I is used for feedback.

The controller actually computes the change of the clamp-ing force, rather than the clamping force itself. This is im-plemented in order to enjoy the benets of velocity-basedgain scheduling [12], as well as it easily account for theactuator constraint (see [5])

bI bI bI (30)

Anti-windup is implemented on the integrator state { . Gain scheduling is implemented by computing gain ma-

trices for a nite number of operating points and thenswitching gain matrices. To achieve bumpless transfer theintegrator is reset at the switching instants. The gain ma-trices are scheduled on both speed and slip. The schedul-ing on slip only has an effect for very small slip val-ues, and will improve the transient performance when thewheel slip controller is activated.

The slip and speed y are estimated online using an ex-tended Kalman lter based on wheel speed ($) and accel-eration measurements.

The ABS system monitors the commands given by thedriver using the brake pedal. Essentially, we set W @n I where I is the clamping force commanded by thedriver using the brake pedal. The slip controller is deacti-vated when the speed is below 1 m/s.

The tuning of the implemented controller is similar to thetuning in the idealized set-up in section 4.3. More speci-cally, the dominant pole of the nominal linear closed loopis almost the same in both cases (about 12.0 for a typicaly).

6 Experimental results

The rst test, Figure 4, is braking on dry asphalt, starting aty+3, @ 54=8 m/s and without any steering manoeuvers. The

slip setpoint is @ 3=3< and we note that the regulation ishighly accurate and satisfactory. When the speed approacheszero, signicant variability in the slip emerges. Since the

0 0.5 1 1.5 2 2.5 30

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

t, [s]

slip (lambda)

0 0.5 1 1.5 2 2.5 30

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2x 10

4

t, [s]

clamping force (Fb, N) from driver (dashed) and ABS (solid)

0 0.5 1 1.5 2 2.5 30

5

10

15

20

25

t, [s]

speed (v, m/s) and wheel speed (w r, m/s)

Figure 4: Experimental results with braking on dry asphalt.

clamping force does not oscillate, we conclude that this is ac-tually sensor noise that is known to increase at the speed goesto zero. We also note that the initial transient is not satisfactoryas the claming force does not increase fast enough so that theslip is too low and the resulting friction force is too low in theinterval w 5 +3=5> 3=:, leading to increased braking distance.This is due to the signicant model inaccuracy in the low-slipregion, and redesign of the slip controller is necessary for thisregion.

The second test, Figure 5, is braking on snow, starting aty+3, @ 55=3 m/s and without any steering manoeuvers. Theslip setpoint is @ 3=3: and we note that the regulation issatisfactory. There are some small oscillations that we believeare due to unmodeled actuator phenomenon (non-linearities)

that are expected to be more pronounced when operating at lowclamping force levels due to low friction on snow.

The third test, Figure 6, is braking on a wet inhomogeneoussurface (asphalt that is partially covered by a plastic coatingand water) without any steering manoeuvers. The intial speedis y+3, @ 55=3 m/s and the slip setpoint is @ 3=3

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0 1 2 3 4 5 6 7 8 90

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

t, [s]

slip (lambda)

0 1 2 3 4 5 6 7 8 90

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2x 10

4

t, [s]

clamping force (Fb, N) from driver (dashed) and ABS (solid)

0 1 2 3 4 5 6 7 8 90

5

10

15

20

25

t, [s]

speed (v, m/s) and wheel speed (w r, m/s)

Figure 5: Experimental results with braking on snow.

achieve the robustness, the approach does not rely on knowl-edge of the tyre/road friction curve. Static uncertainty (dueto unknown ) is eliminated using integral action, while dy-namic uncertainty (due to unknown shape of +,) is handledby a robust design with sufcient stability margin.

The results are somewhat prelimenary, and can be possibly beextended and improved. In particular, the analysis might takeinto consideration some of the details that are present in theimplementation, but omitted from the current analysis. Also,the use of the Riccati-solution to dene the Lyapunov functionis not always the best choice. Finally, means for improving thetransient performance of the controller (especially for low slip)are under investigation.

Although a detailed comparison with existing off-the-shelf

ABS has not been conducted, the present results are encour-aging, in particular when taking into account the modest timetaken to design, tune and comission this model-based approach.

Acknowledgements

The work was sponsored by the European Commission underthe ESPRIT LTR-project 28104 H C.

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0 1 2 3 4 5 6 7 8 90

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t, [s]

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speed (v, m/s) and wheel speed (w r, m/s)

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