Design and experimental validation of a nonlinear wheel slip control algorithm

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Automatica 48 (2012) 1852–1859

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Automatica

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Brief paper

Design and experimental validation of a nonlinear wheel slip control algorithm✩

William Pasillas-Lépine a,1, Antonio Loría a, Mathieu Gerard b

a CNRS (Centre National de la Recherche Scientifique), L2S, SUPELEC – Univ Paris-Sud, 3 rue Joliot-Curie, 91192, Gif-sur-Yvette cedex, Franceb DCSC (Delft Center for Systems and Control), Delft University of Technology, Mekelweg 2, 2628CD, Delft, The Netherlands

a r t i c l e i n f o

Article history:

Received 21 April 2011

Received in revised form21 November 2011

Accepted 24 February 2012

Available online 26 June 2012

Keywords:

Anti-lock braking systems

Wheel slip control

Lyapunov designCascaded design

a b s t r a c t

We present a newwheel slip controller and validateit both experimentally andin simulation. Thecontrolstrategy is based on both wheel slip and wheel acceleration regulation, and ensures global asymptotic

stabilityin closed loop.The stabilityanalysisis established using toolsfor cascaded systems.This approachhas the advantage of setting conditions on the control gains that are considerably milder, if comparedwith those obtained from Lyapunov’s direct method. Simulations are provided to illustrate the robustnessagainst vertical loadvariations and uncertaintiesin tyre–roadfriction. Furthermore, testson a tyre-in-the-loop facility show that with our control law the wheel slip converges precisely to the assigned reference.

© 2012 Elsevier Ltd. All rights reserved.

1. Introduction

There are several applications where wheel slip control is

important. The most famousis probably theanti-lock brake system(ABS) in passenger cars (Robert Bosch GmbH, 2003). Nevertheless,in addition to the ABS, wheel slip control is equally important forother applications. Also present on passenger cars, one can findthe electronic stability program (ESP), in which the slip of one of the wheels is controlled in order to maintain the vehicle insideits lateral stability domain (see, e.g., van Zanten, 2002). Anotherapplication is found in racing-motorcycle braking strategies (see,e.g., Corno, Savaresi, Tanelli, & Fabbri, 2008). Finally, themeasure of tyre characteristics is another area where wheel slip control mightbe useful (see, e.g., Pacejka, 2006).

Among these applications, one must distinguish those in whichthe vehicle’s velocity is available (either as a measurement oras an estimation) from those for which it is not. For example,in the case of ABS, one typically avoids the use of the vehicle’sspeed while in the other three applications described above

✩ This work was financially supported by the European Commission through the

FP7 NoE HYCON2. The Ph.D. thesis of M. Gerard was supported by TNO Automotive

and SKF Automotive Division. The research of W. Pasillas-Lépine was supported

by the Region Ile-de-France through the Regeneo project (RTRA Digiteo and DIM

LSC). The material in this paper was partially presented at the IFAC Nonlinear

Control Systems Symposium (NOLCOS 2010), September 1–3 2010, Bologna, Italy.This paper was recommended for publication in revised form by Associate Editor

Yoshikazu Hayakawa under the direction of Editor Toshiharu Sugie.

E-mail addresses: [email protected] (W. Pasillas-Lépine),

[email protected] (A. Loría), [email protected] (M. Gerard).1 Tel.: +33 1 69 85 17 42; fax: +33 1 69 85 17 65.

(ESP, motorcycles, and tyre measures), velocity measurements arecommonly available.

In this paper we present a continuous slip control algorithm

designed for the case in which the velocity of the vehicle is knownprecisely. In a different context, hybrid algorithms would probablybe bettersuited. A description of these methods canbe found in theworks of Gerard, Pasillas-Lépine, de Vries, and Verhaegen (2010),Kiencke and Nielsen (2000), Pasillas-Lépine (2006), Robert BoschGmbH (2003), Tanelli, Osorio, di Bernardo, Savaresi, and Astolfi(2009), and in the references given therein. They are based onlogic switchings triggered by wheel acceleration thresholds. For anexperimental comparison of hybrid and continuous methods, werefer the reader to the recent work of Shida, Sakurai, Watanabe,Kano, and Abe (2010).

In the last ten years, several algorithms have been proposedfor wheel slip regulation — see, e.g., Chapters 5–8 of Johanssonand Rantzer (2003). These methods are however confronted tothree delicate points. First, they are mainly based on linearisationarguments; the nonlinear system is linearised around the desiredequilibrium point hence, the stability analysis is valid onlylocally — see, e.g., Choi (2008), Petersen, Johansen, Kalkkuhl,and Lüdemann (2001) and Savaresi, Tanelli, and Cantoni (2007).Second, when nonlinear terms are considered, some methodsgenerate a limit cycle if the setpoint is in the unstable region of thetyre model (see, e.g., Tanelli, Savaresi, & Astolfi, 2008). Third, theavailableapproaches are mainly based on pure wheel slip feedback(there are no feedforward terms), which considerably limits thebandwidth of the closed-loop system.

In these works, the main variable used for control is wheelslip (wheel acceleration is, in general, not used). Nevertheless,there exists an approach that combines both wheel slip and wheel

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W. Pasillas-Lépine et al. / Automatica 48 (2012) 1852–1859 1853

acceleration measurements (see, e.g., Savaresi et al., 2007). Ouralgorithm also follows this idea but, in addition, it presents severalnew interesting points. First, it provides global asymptotic stabilityto the desired setpoint by taking into account the nonlinear natureof the system. Second, it gives precise bounds on the gains of thecontrol law for which stability is proved mathematically. Third,it contains a feedforward term that improves the bandwidth of the closed-loop system. Fourth, the theoretical algorithm has been

implemented and tested on TU Delft’s experimental setup. Theexperimental results agreecompletely with the theory and confirmthe interest of our approach.

Last but not least we remark that global asymptotic stabilityimplies, via converse theorems, local input to state stability; arobustness feature that cannot be overestimated and which wecorroborate in a simulation scenario (under relaxed assumptionsthat cannot be tested with the current test rig).

2. System modelling

The design of our control law is based on a simplified single-wheel model. Only the longitudinal dynamics of a single loadedwheel braking on a straight line is considered. Even though weighttransfer and combined slip are ignored, all the basic phenomenarelated to ABS already appear in this very simple model.

2.1. Wheel speed dynamics

The angular velocity ω of thewheel hasthe followingdynamics:

I dω

dt = −RF x + T , (1)

where I denotes the inertia of the wheel, R its radius, F x thelongitudinal tyre force, and T the torque applied to the wheel.

The torque T = T e − T b is composed of the engine torque T eand the brake torque T b. We assume that during ABS braking theclutch is open andthus neglect theenginetorque. The brake torqueis given by

T b = γ bP b, (2)where P b ≥ 0 denotes the brake pressure and γ b > 0 the brakeefficiency.

2.2. Tyre force modelling

The longitudinal tyre force F x is often modelled by the relation

F x(λ, F z ) = µ(λ)F z . (3)

That is, by a function that depends linearly on the vertical load F z

and nonlinearly on the wheel slip

λ =Rω − v x

v x

, (4)

where v x

is the longitudinal speed of the vehicle. We assume thatv x > 0. In a braking manoeuvre, we have λ < 0 and F x < 0.

The nondimensional tyre characteristic µ(·) is an odd functionsuch that µ(0) = 0and µ′(0) > 0 (see Fig.1), where µ′(λ) denotesthe derivative of µ with respect to λ. This function µ(·) has a max-imum at λ = λ∗, hence µ′(λ∗) = 0. The part of the curve |λ| < λ∗

with a positive slope µ′ > 0 is called the ‘‘stable zone’’ of the tyrewhile the other side of the peak with |λ| > λ∗ and µ′ < 0 is calledthe ‘‘unstable zone’’. This denomination is motivated by the sta-bility analysis of the single-wheel model (see, e.g., Olson, Shaw,& Stépán, 2003). This static tyre force model is only valid at highspeeds (typically v x > 45 km/h). For lower velocities, the tyre re-laxation length should be taken into account. See, e.g., Canudas-deWit, Tsiotras, Velenis, Basset, and Gissinger (2003), Pacejka (2006)and Zegelaar (1998).

Several mathematical formulas have been used to describe tyreforces. Trigonometric functions are used by Pacejka (2006)

Fig. 1. Measurement and fitting of the nondimensional tyre characteristic µ(·)for the tyre-in-the-loop facility. The maximum braking force is achieved with a

longitudinal slip of 10%. The parameters associated to the tyre mounted on the

test rig are: a1 = 36, a2 = 217, a3 = 13, and a4 = 271. Moreover, we have

F z = 2500 N, R = 0.3 m, and I = 1.2 kg m2.

µ(λ) = D sin (C arctan (Bλ)) , (5)

exponentials by Burckhardt (1993), and second order rationalfractions by both Kiencke and Nielsen (2000) and Pasillas-Lépine(2006)

µ(λ) =a1λ − a2λ2

1 − a3λ + a4λ2, for λ ≤ 0. (6)

Formulas on how to relatethe coefficients ai of therational fractionto more common features as the slope at the origin, or the positionof the peak, can be found in Pasillas-Lépine (2006). The model iscompared to real tyre measurements in Fig. 1.

These mathematical models use coefficients that depend on

tyre characteristics, road conditions, tyre pressure, temperature,and several other factors. Therefore, for the sake of robustness,wheel slip control algorithms should be able to handle theuncertainty associated to these coefficients.

2.3. Wheel slip and acceleration dynamics

In this subsection, we derive the equations that describe theevolution of wheel slip and wheel acceleration. Other wheeldeceleration models proposed in the literature include thosereported by Olson et al. (2003); Ono et al. (2003) and Tanelli et al.(2008).

We define the longitudinal acceleration a x(t ) by the relation

dv x

dt = a x(t ) (7)and introduce the state variables

x1 = λ (8)

x2 = Rdω

dt − a x(t ). (9)

The state x1 is the wheel slip, the variable that we want to control.The state x2 is the acceleration offset. That is, the differencebetween the acceleration of the wheel and that of the vehicle.

If the tyre load F z is assumed to be constant, then we have thefollowing dynamics

dx1

dt =

1

v x(t )(−a x(t ) x1 + x2)

dx2

dt = − c µ

′

( x1)v x(t )

(−a x(t ) x1 + x2) + RI

dT dt

− da x(t )dt

,



1854 W. Pasillas-Lépine et al. / Automatica 48 (2012) 1852–1859

Fig. 2. Simulation of a hard braking ABS scenario, with a change of road conditions. A first moderate braking manoeuvre (first 4 s of simulation) exhibits the effects of

braking on load transfer and the behavior of the algorithm in the tyre’s stable domain. A second hard braking manoeuvre (last 8 s of simulation) shows the behavior of the

algorithm in the tyre’s unstable domain. Between the time instants 5.5 and 7.5, the road’s friction coefficient is drastically modified. The algorithm does not know the value

of this coefficient; it assumes that the road is dry. The robustness of the algorithm is a consequence of Corollary 1. The speed of the vehicle is 180 km/h at the beginning of

the simulation and 10 km/h at the end.

where

c =R2

I F z . (10)

This assumption on F z is made in order to simplify the control de-sign and analysis. Nevertheless, in our simulations (see Section 4.1and Fig. 2), we took a non-constant vertical load that depends onthe suspension dynamics.

3. Control design

The objective is to define a control law u that drives the wheelslip x1 towards a given time-dependent reference λ∗(t ). Inordertosimplify the control design, we consider as the control input u thederivative of the torque applied to the wheel, instead of the torqueitself. That is,

u = v x(t )R

I

dT

dt . (11)

Depending on the technology used in the brake actuator (EMB orEHB), it might be necessary to integrate the control in order toobtain a brake torque reference

T (t ) =I

R

t

0

u(τ )dτ

v x(τ ). (12)

Several steps are taken to determine u. Firstly, the time-scaleis normalised in order to obtain equations that are independent of

thevehicle speed.This is enabled by applying a filterto thesetpointλ∗. Secondly, the problem of steering x1 towards its setpoint λ∗

is translated into the problem of driving x2 towards a dynamicreference x∗

2. Finally, the control law u is derived in a way that theclosed-loop system has a cascaded structure.

3.1. An homogeneous setpoint filter

For any piecewise continuous wheel slip reference λ∗(t ), wedefine a filtered setpoint

dλ1

dt =

λ2

v x(t )

dλ2

dt =

−γ 1(λ1 − λ∗) − γ 2λ2

v x(t ),

where γ 1 > 0 and γ 2 > 0 are two constant gains.

The aim of this setpoint filter is twofold. On the one hand,it allows to have a smooth setpoint (that one can differentiatetwice) even if the original setpoint is discontinuous (for example,piecewise constant). On the other hand, it allows to have a systemfor which all equations are divided by the vehicle’s velocity. Thishomogeneity allows us to analyse the system in a new (nonlinear)time-scale in which the dependence on speed disappears.

3.2. A new time-scale

For further development, we assume that a x is constant. Thisassumption is made in order to simplify the control design and

analysis. Nevertheless, in our simulations (see Section 4.1 andFig. 2), we took a non-constant acceleration that depends on tyreforces.

This assumption allows us to apply a change of time-scale as inPasillas-Lépine (2006). Let

s(t ) :=

t

0

dτ

v x(τ )(13)

hence, dt = v x(t )ds and consequently, for any function ϕ : R →R

n we have

dϕ

ds=

dϕ

dt

dt

ds. (14)

Therefore, defining ϕ(s) = dϕ(s)

dswe obtain

˙ x1 = −a x x1 + x2 (15)

˙ x2 = −c µ′( x1)[−a x x1 + x2] + u (16)

λ1 = λ2 (17)

λ2 = −γ 1(λ1 − λ∗) − γ 2λ2. (18)

In the sequel, all derivatives are considered with respect to thescaled time and, with an abuse of notation we use the variable t to denote it.

The fact that the previous equations do not depend on thevehicle speed might be misleading. Indeed, one might think thatthe velocity of the vehicle is not used by the control law. But this isnot the case. First, the velocity appears in the computation of the

wheel slip. Second, the expression of the control law in the originaltime-scale (12) also uses this variable.




Fig. 3. Experimental comparison of the performances with and without feedforward. The speed of the drum is 65 km/h. The control law is changed at t = 10 s.

3.3. Reference setpoints and error variables

For wheel slip, the natural setpoint x∗1 is clearly x∗1 = λ1. Thetracking error is

z 1 = x1 − x∗1, (19)

which leads to the error dynamics

˙ z 1 = −a x x1 + x2 − λ2. (20)

To drive the solution z 1(t ) of the previous equation to zero, weconsider x2 as a virtual control input. A direct computation showsthat x2 = x∗

2 , with

x∗2 = a x x1 + λ2 − α z 1, (21)

yields ˙ z 1 = −α z 1, whose origin is exponentially stable. Now,define z 2 = x2 − x∗

2. Then, when x2 = x∗2 , we obtain the error

dynamics

˙ z 1 = −α z 1 + z 2, (22)

which is input-to-state stable. Therefore, the natural objectiveis to define a control u(t ) such that z 2(t ) converges to zeroexponentially.

Observe that while x∗1 is only based on λ1(t ), the setpoint x∗

2 isdynamic. The steady state part is a x x1, while the two other termsare used to decrease the error on z 1, both using cascaded feedback−α z 1 and feedforward λ2. Thanks to this dynamic setpoint, thesystem converges towards the desired wheel slip irrespectivelyfrom thetyre characteristic.This is notthe case in other mixed slip-acceleration control approaches (see, e.g., Savaresi et al., 2007).

3.4. The control law

With the purpose of steering z 2 to zero, we define

u = −γ 1(λ1 − λ∗) +

−γ 2 + a x + aµ′( x1)

λ2 feedforward

−k1 z 1 − k2 z 2 feedback

, (23)

where we have dropped the argument t from λ∗(t ) and x1(t ). Notethatthe control u is composed of a feedforwardand a feedback part.

On the one hand, if a x and µ′( x1) are preciselyknown,the originof the controlled system is proven to be globally exponentiallystable for any arbitrary piecewise continuous reference setpointλ∗(t ). This fact is formalised in Theorem 1. On the other hand, if the system is uncertain, the feedforward cannot be implemented

precisely. This leads to a decrease of performance (see Fig. 3).Nevertheless, in this degraded case, the feedback term is still able

to maintain global exponential stability for a constant reference λ∗.This robustness issue is addressed in Corollary 1 below.

In other words, the feedforward term increases bandwidthand the feedback term robustness. To the best of our knowledge,similar feedforward or feedback terms have never been proposedfor wheel slip control in the literature.

3.5. Main results

Consider a tyre model given by Pacejka’s (2006) magic formula

µ( x1) = D sin (C arctan (Bx1)) , (24)

where the constants B, C , and D are strictly positive. Define η(t ) =c µ′( x1(t )) and denote by −ηm the infimum of c µ′( x), over x ∈ R.

Theorem 1. Consider an arbitrary piecewise-continuous wheel slip

reference λ∗

(t ). Let such λ∗

(t ) be injected into the filtered setpoint Eqs. (17) and (18), and consider the control law (23) in closed-loopwith the system defined by Eqs. (15) and (16). Then, the closed-loopsystem has the form

˙ z = A(t ) z , (25)

where

A(t ) =

−α 1

−k1 + a xα − α2 + αη(t ) −k2 + α − a x − η(t )

(26)

exists for all t and is uniformly bounded.Furthermore, if the control gains k1 and k2 satisfy

k1 > a xα − α2 and k2 > α − a x + ηm, (27)

then the origin of the closed loop system (25) is globally exponentiallystable.

Define w1 = λ1 − λ∗ and w2 = λ2. In order to exhibit therobustness of our control strategy, we will consider an approxima-tion of the tyre curve

µ( x1) = D sin

C arctan

Bx1

, (28)

where the constants B, C , and D are strictly positive. It is impor-tant to observe that, independently of the precision of the chosenapproximation, the function

φ(t ) = c µ′( x1(t )) − c µ′( x1(t )) (29)

is bounded uniformly for all t such that x1(

t )

exists. This observa-tion is the key point that leads to the following robustness result.




Corollary 1. Consider a constant wheel slip reference λ∗ . Let such λ∗

be injected into the filtered setpoint Eqs. (17) and (18), and consider the control law

u = −γ 1(λ1 − λ∗) +

−γ 2 + a x + c µ′( x1)

λ2 − k1 z 1 − k2 z 2 (30)

in closed-loop with the system defined by Eqs. (15) and (16). Then theclosed-loop system system has the form

˙ z = A(t ) z + B(t )w (31)

w = C (t )w, (32)

with the same matrix A(t ) as in Theorem 1, and with

B(t ) =

0 00 φ(t )

and C (t ) =

0 1

−γ 1 −γ 2

. (33)

Furthermore, if the control gains k1 and k2 satisfy the bounds (27) of Theorem 1, then the origin of the closed loop system (25) is globallyexponentially stable.

Even though this result seems purely technical, it has animportant practical consequence: for constant references, our control law is able to stabilise the wheel slip around its desired

value even when the exact tyre characteristic is unknown . In oursimulations and experiments, this result is illustrated in twodifferentcontexts. First,in thecase of a changeof road conditions. If the controller uses the tyre characteristics of a dryroad in differentroad conditions, like in Fig. 2, the convergence to the right setpointis ensuredby Corollary 1. Second, in thecase when thefeedforwardterms are omitted, like in Fig. 3, it is again this result that providesthe desired robustness.

4. Simulation and experiments

4.1. Simulation results

The first aim of our simulations was to test the effects of perturbations that were not taken into account in the simplequarter-car model of Section 2. We took a two-axles model, witha non-constant vertical load on each axle that depends on the loadtransfer dynamics (Genta, 1997), see Fig. 2. We also took a non-constant acceleration, given by the sum of the longitudinal tyreforces on each axle divided by the total mass of the vehicle.

We injected in this vehicle model the control law of Section 3.4.The value of F z in the control was taken constant and equal to theequilibrium value at full braking (with an optimal value of λ). Forthe gains k1, k2, and α, we took the following values,

k1 = 0, k2 = 2000, and α = 1000. (34)

These gains satisfy the conditions imposed by Theorem 1. A typicalsimulation result is shown on Fig. 2. The details of the brakingscenario are given in the figure’s caption. Clearly, our control law

seems to be robust to the unmodeled dynamics that were omittedfor control design.

The second aim of the simulations is to observe the effects of the feedback and feedforward terms, and to quantify the effects of perturbations, in order to evaluate the robustness of the controllaw (when both feedback and feedforward terms are used). Thesesimulations are reported in a preliminary version of our work(Pasillas-Lépine & Loría, 2010). Since the results were satisfactory,we were able to proceed to an experimentalimplementation on TUDelft’s setup.

4.2. Experimental results

The tyre-in-the-loop experimental facility of Delft University of

Technology on which the control law is tested consist of a largesteel drum of 2.5 m diameter on top of which the tyre is rolling.

The setup has been used for many years for tyre modelling andidentification using open-loop excitation (see Pacejka, 2006, andthe references therein). Recently, the electronics was upgraded inorder to allow closed-loop tests to be performed and, in particular,rapid prototyping and testing of ABS controllers. The drum isdriven by a large electromotor and can run up to 300 km/h.The speed of the drum can be accurately measured thanks toan encoder. The weight of the drum makes it more suitable forkeeping a constant speed.

The wheel (together with the tyre) is attached to an axle witha rigidly constrained height. The axle is supported by two bearingson both side of the wheel. The bearing housings are connectedto the fixed frame by means of piezo-electric force transducers.An hydraulic disk brake is mounted on one side of the axle. Thepressure in thecalliperis locally controlled byan analogelectronicsmodule, which is connected to a servo-valve in order to match thereference pressure given by the dSpace computer. The encoder ismounted on the other side of the axle to provide highly accuratewheel speed measurements.

For our experiments, the gains k1, k2, and α, were set to

k1 = 250, k2 = 1000, and α = 250. (35)

The gain values are lower than in Section 4.1. Indeed, the tyre testrig has a high delay at the level of the brake pressure actuator.Our gains were chosen in a way that increases the delay marginof the system. It is important to stress that these gains satisfy theconditions imposed by Theorem 1.

A typical experimental result is shown on Fig. 3. The details of the braking scenario are given in the figure’s caption. During thistest, performances with and without feedforward are compared.During the first 10 s, only the feedback is enabled. Then thefeedforward is turned on. On Fig. 3, it can be observed thatthe convergence to a new reference is much faster when thefeedforward term is switched on. This is particularly noticeable atlow slip, where the pure feedback controller is particularly slow.

5. Proofs of Theorem 1 and Corollary 1

5.1. Proof of Theorem 1

First step. In order to derive the closed-loop error dynamics (25),the following relation is used several times:

− a x x1 + x2 = −α z 1 + z 2 + λ2. (36)

Indeed,

−a x x1 + x2 = −a x( z 1 + x∗1) + ( z 2 + x∗

2)

= −a x( z 1 + x∗1) + ( z 2 + a x x1 + λ2 − α z 1)

= a x( x1 − x∗1) − a x z 1 + z 2 + λ2 − α z 1

= −α z 1 + z 2 + λ2.

Now, on the one hand, differentiating z 1 = x1 − x∗1 we obtain

˙ z 1 = −a x x1 + x2 − λ2

and using (36) we get

˙ z 1 = −α z 1 + z 2. (37)

On the other hand, differentiating z 2 and using relations (36) and(37) leads to

˙ z 2 = ˙ x2 − ˙ x∗2

=

−c µ′( x1)(−a x x1 + x2) + u

−

a x ˙ x1 + λ2 − α ˙ z 1

= −c µ′( x1)(−a x z 1 + z 2 + λ2) + u +

−

a x(−α z 1 + z 2 + λ2) + λ2 − α(−α z 1 + z 2)

.




Grouping the factors of z 1 and z 2 together, we obtain

˙ z 2 =

αc µ′( x1) + αa x − α2

z 1 +

−c µ′( x1) − a x + α

z 2

− c µ′( x1)λ2 − a xλ2 − λ2 + u.

Substituting in the previous expression the control law u definedby (23) yields

˙ z 2 = −k1 + αc µ′( x1) + αa x − α2 z 1

+

−k2 − c µ′( x1) − a x + α

z 2. (38)

Reintroduce the argument t of x1, which wasomitted forclarity,in Eqs. (37) and (38). Recalling that η(t ) = c µ′( x1(t )) and

A(t ) =

−α 1

−k1 + a xα − α2 + αη(t ) −k2 + α − a x − η(t )

,

we see that Eqs. (37) and (38) may be written in matrix form

˙ z = A(t ) z .

This ends the proof of the first statement of Theorem 1.At this point, it is important to observe that forward complete-

ness of (25) can be proved using a quadratic Lyapunov function.

Indeed, since µ

′

( x1) is continuous and bounded from above andbelow, the existence of x1(t ) is granted for all t > 0.

Second step. The constraint imposed by the theorem on the gain k2

is

k2 > α − a x + ηm.

Therefore, there exists an ϵ > 0 such that

k2 > α − a x + ηm + ϵ.

Denote β = ηm + ϵ. By definition, we have η(t ) ≥ −ηm for all t .Therefore, we have both

k2 − α + a x − β > 0 and η(t ) + β ≥ ϵ > 0, (39)

for all t . The constraint imposed by the theorem on the gain k1

implies that we additionally havek1 + α2 − a xα > 0. (40)

Now, by decomposing

−k1 + a xα − α2 + αη(t ) =

−k1 + a xα − α2

+ (αη(t ))

and

−k2 + α − a x − η(t ) = − (k2 − α + a x − β) + (−η(t ) − β) ,

the matrix A(t ) can be written as

A =

0 1

−(k1 + α2 − a xα) −(k2 − α + a x − β)

A1

+−α 0

αη −(η + β)

A2

.

Observe that relations (39) and (40) imply that the matrix A1 isHurwitz.

Third step. The key step of the proof is to deduce the exponentialstability of (25) from that of the two systems

ξ 1 = A1ξ 1 (41a)

ξ 2 = A2ξ 2. (41b)

To this end, define y = ξ 1 + ξ 2, B = [− A1 − A2], ξ = [ξ 1 ξ 2]⊤, and

A = A1 00 A2

.

Then, ξ = Aξ and

˙ y = A1ξ 1 + A2ξ 2

= A1( y − ξ 2) + A2( y − ξ 1)

= ( A1 + A2) y − ( A1ξ 2 + A2ξ 1)

= Ay − ( A1ξ 2 + A2ξ 1)

= Ay + Bξ .

Therefore, the dynamics of the variables y and ξ are interconnectedand described by

˙ y = Ay + Bξ (42a)

ξ = Aξ , (42b)

which is a cascaded system.

Claim 1. Assume that: (a) The equilibrium {ξ = 0} of (42b) is globally exponentially stable. (b) The function ∥ y(t )∥, where y(t ) isany solution of (42a), is upper bounded by an estimate of the formc ∥[ y0 ξ 0]∥e−λ(t −t 0) with c and λ independent of the initial conditions.Then, necessarily, the equilibrium { y = 0} is a globally exponentiallystable equilibrium of ˙ y = Ay.

Proof of Claim 1. The claim is a direct consequence of standardconverse results on Lyapunov functions. Firstly, items (a) and (b)imply that the origin is a globally exponentially stable equilibriumof (42). Hence, converse theorems (see, e.g., Khalil, 2000, Theorem4.14) can be used in order to build a Lyapunov function for thesystem (42). Secondly, this converse Lyapunov function can berestricted to the set {ξ = 0}, which yields a Lyapunov functionfor ˙ y = Ay. The global exponential stability of { y = 0} is then adirect consequence of the existence of this Lyapunov function (see,e.g., Khalil, 2000, Theorem 4.10).

Fourth step. Since Theorem 1 is a direct consequence of Claim 1,it is only left to show that the claim’s assumptions (a) and (b)hold.

Itis clear that(b) holds if ξ 1(t ) and ξ 2(t ) also satisfy exponentialestimates. Hence if {ξ 1 = 0} and {ξ 2 = 0} are globally exponen-tially stable equilibria of (41a) and (41b), respectively. Indeed, wehave by definition y(t ) = ξ 1(t ) + ξ 2(t ). That is, (a) implies (b).

To prove (a), memustshowthat ξ = 0 is globally exponentiallystable for (42b). Firstly, the origin of (41a) is GES since A1 isHurwitz, under conditions (39) and (40). Indeed, in this case bothcoefficients of the characteristic polynomial are negative. For theorigin of (41b) we proceed as follows. Let x1(t ) be defined for all t and recall that η(t ) = c µ′( x1(t )). Under forward completeness wehave η(t ) ≥ −ηm. The trajectories of system (41b) satisfy

ξ 22 = −[β + η(t )]ξ 22 + αη(t )ξ 21 (43a)

ξ 21 = −αξ 21 (43b)

which is a cascaded system. We have: the origin of (43b) is GES;the interconnection term αη(t ) is uniformly bounded; the originof

ξ 22 = −[β + η(t )]ξ 22

is also GES, since β + η ≥ ϵ > 0. The latter follows by invokingthe comparison theorem and from relation (39). Therefore, theequilibrium {ξ 2 = 0} of (41b) is globally exponentially stable andso is the origin of (41).

5.2. Proof of Corollary 1

We have a time-varying system

˙ z = A(t ) z + B(t )w

w = C (t )w.




On the one hand, the subsystem

w = C (t )w

is globally exponentially stable, since the gains γ 1 and γ 2 arestrictly positive. On the other hand, the subsystem

˙ z = A(t ) z

is globally exponentially stable, as it has been already shown in the

Proof of Theorem 1. Finally, the matrix B(t ) is obviously bounded,since both µ′( x) and µ′( x) are bounded. Therefore, by a standardcascade argument (Loria & Panteley, 2005), the system defined byEqs. (31) and (32) is globally exponentially stable.

6. Conclusions

We presented in this paper a cascaded wheel slip control algo-rithm, based on wheel slip and wheel acceleration measurements.We proved that our control law stabilises globally and asymptot-ically the wheel slip around any prescribed setpoint, both in thestable and unstable regions of the tyre. A stability proof, based ona quadratic Lyapunov function, was already available in a prelim-inary version of our work (Pasillas-Lépine & Loría, 2010). Never-theless, the constraints imposed by this approach on the controller

gains were toorestrictive, andwere notrespected bythe gains usedin our experiments.

In order to release these constraints, three different approachescan be followed. The first approach, based on LMI’s (Gerard, 2011),has the inconvenient of being purely numerical (without analyticconditions for the solvability of the obtained LMI’s) but has theadvantage of proving stability in cases that were not covered byour original quadratic Lyapunov function and, in particular, in thecase of the controller gains used in our experiments. The secondapproach, is to find analytic conditions for the solvability of theseLMI’s, using a recent classification result for second-order time-varying linear systems (Balde, Boscain, & Mason, 2009). The thirdapproach is the one presented in Section 5. Its main advantage isthat it gives the weakest possible conditions for stability. Namely,

if the conditions of Theorem 1 are not satisfied then there existwheel slip reference setpoints for which the closed-loop system isnot asymptotically stable. Moreover, with this third approach, theconditions imposed on the controller gains are simpler than withthe other two approaches.

Finally, this paper supports the proposed theory by showinga practical implementation and validation of the controller on atyre-in-the-loop test facility. The obtained results are good butcould still be improved by reducing the gain between the wheelacceleration measurement noise and the wheel slip tracking error,or by proposing a proof that takes into account actuator delays ortyre relaxation length. This is a point that we plan to address in ourfuture works on wheel slip control.

Of course, our algorithm has also some limitations. First, itis primarily oriented to situations where the impact of the tyre

relaxation length is low. The speed range where our results can beapplied depends on both the tyre characteristics and the type of vehicle. In the case of an automobile, the velocity should be higherthan 45 km/h. But this value will be different, for example, in thecase of an airplane or a motorcycle. Second, we only considered thecase where the vehicle is braking in a straight line. When brakingin a curve, the effects of combined slip should also be considered.Third, we assumed that the vehicle’s speed was available. In thecontext of ABS, it might be possible to use this kind of algorithm inthe front axle of the vehicle, provided that an alternating hybridalgorithm is used on the rear axle (Ait-Hammouda & Pasillas-Lépine, 2008), in order to maintain the vehicle in a configurationfor which the velocity can be observed easily. Finally, anotherdifficulty is related to the choice of an optimal target for wheel slip.This problem is not easy to solve, but some works in the literaturehave been devoted to its study (Tanelli, Piroddi, & Savaresi, 2009).

References

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Pasillas-Lépine, W., & Loría, A. (2010). A new mixed wheel slip and accelerationcontrol based on a cascaded design. In Proceedings of the IFAC symposium onnonlinear control systems. Bologna, Italy.

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William Pasillas-Lépine was born in Amiens, France, in1971. He received hisB.Sc. degreein AppliedMathematicsfrom the National Institute for Applied Sciences (INSA deRouen), in 1995. And both his M.Sc. and Ph.D., in AppliedMathematics and Control Theory, from the University of Rouen (France), in 1995 and 2000, respectively. Since2002, he has been serving as a CNRS researcher atLaboratoire des Signaux et Systèmes (Supélec, Gif-sur-Yvette). From 2005 to 2007, he was on leave for theFrench company PSA Peugeot Citroën. His main interest is

focused on applied control theory problems, in the areasof automotive systems and neurosciences.




Antonio Loria was born in Mexico City in 1969. Heobtained the B.Sc. degree in Electronic Engineering fromthe ITESM, Monterrey, Mexico in 1991. He obtained theM.Sc. and Ph.D. degrees in Control Engineering from theUTC, France in 1993 and Nov. 1996 respectively. Hehas been an IEEE (student) member since 1990. FromDecember1996 through Dec. 1998, hewas successivelyanassociate researcher at Univ. of Twente, The Netherlands;NTNU, Norway and the CCEC of the Univ. of Californiaat Sta Barbara, USA. A. Loria has the honour of holding

a research position as ‘‘Directeur de Recherche’’ (seniorresearcher), at the French National Centre of Scientific Research (CNRS). Hisresearch interests include control systems theory and practice, electrical systems,analysis and control of chaos. He serves as an associate editor for Systemsand Control Letters, Automatica, IEEE Transactions on Automatic Control, IEEETransactions on Control Systems Technology and is a member of the IEEE CSSConference Editorial Board. See also http://www.l2s.supelec.fr/perso/loria.

Mathieu Gerard was born in Gosselies, Belgium, in 1983.He studied Electrical Engineering at the University of Liege, Belgium, and received his Masters degree SummaCum Laude in 2006. During his studies, he spent oneyear in the department of Automatic Control of LundUniversity, Sweden. Between 2006 and 2010, he workedon his Ph.D. project at the Delft Center for Systems andControl, Delft University of Technology, The Netherlands.His Ph.D. research was about global chassis controland braking control using tyre forces measurement,

and was performed under the supervision of Prof.Dr.ir.Michel Verhaegen. In 2009, he started developing mobile applications to bringconference programs into mobile phones, and co-founded Conference Compasstogether with Dr. Jelmer van Ast. He is now a Managing Director at ConferenceCompass B.V.

http://www.l2s.supelec.fr/perso/loria



Design and experimental validation of a nonlinear wheel slip control algorithm

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