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Nonlinear pressure control for BBW systems viadead zone and anti-windup compensation

Fabio Todeschini, Simone Formentin, Giulio Panzani, Matteo Corno, Sergio Savaresi and Luca Zaccarian

Abstract

In the automotive field, brake-by-wire (BBW) systems are electronically regulated actuators, which are capable of applying adesired braking torque to the vehicle’s wheel. Specifically, the electro-hydraulic technology is the most widely used in commercialvehicles, as it offers a good trade-off in terms of size, weight and cost. However, control of BBW actuators in such a configurationis a challenging problem for many reasons, among which the most critical are the dead zone due to the fluid reservoir and theinput saturation limits of the electric motor that moves the pump. In this paper, a complete control architecture accounting forthis nonlinear behavior is presented, where the main components are a linear controller, a dead zone compensator and an anti-windup block, designed in a cascade fashion. With such a configuration, the achieved equilibrium point is guaranteed to beglobally asymptotically stable and the overall system shows to be robust with respect to variations of the position-pressure curve.Simulation and experiments on a production prototype are proposed to show the effectiveness of the proposed strategy.

Index Terms

Brake-by-wire, Anti-windup compensator, Actuator control, Non linear systems.

I. INTRODUCTION

During recent years, in the automotive field - both from the industrial and the academic side - a significant effort has beendevoted to developing active safety and/or performance vehicle dynamic control strategies, as well as autonomous vehicles,see, e.g., [2], [6], [11], [12], [16], [18], [29], [32], [33]. The practical application of these control systems is naturally subduedto the availability of on-board actuators that allow one to regulate the desired control variable independently - or within certaindegrees of freedom - from the driver’s commands. This need has led to the so called drive-by-wire paradigm, where the standardmechanical connection between the driver and actuators is replaced by an electronic system, devoted to the regulation of theactuator according to driver or vehicle control logic requests.

Among others, the brake-by-wire (BBW) technology focuses on the design of an actuator capable of applying the desiredbraking torque to the vehicle’s wheel. During the years, different technical solutions have been explored. The most successfulones are the electro-mechanical (EMB) and the electro-hydraulic (EHB) architectures. In the former, an electric motor directlyexerts the requested torque on the wheel [21], [22]. A promising variant of such architecture is the Electric Wedge Brake(EWB), see e.g. [4], which exploits the DC motor torque to move a wedge where the braking pads are installed, ratherthan exerting a clamping force directly on the braking disk. The EHB - directly derived from the most spread vehicle brakearchitecture - employs an hydraulic system, activated by an electronically commanded motor/pump, to generate the requestedbraking force [10], [26], [30], [31].

In this work, a modified EHB solution, firstly presented in [7], is considered: a traditional hydraulic brake is employed and,using an electric motor mechanically connected to the master cylinder, the desired pressure on the braking pads is generated.With respect to the well known EMB and EHB solutions, the proposed one has the advantage of keeping the usual vehiclehydraulic brake layout, adding just the electric actuator, thus saving space, weight and cost. With such an architecture, theactuator control problem consists in the tracking of a reference pressure that comes from the driver-vehicle interface (i.e. thebraking pedal) or from other dynamic control strategies. For such a BBW technology, the control problem, however, turnsout to be quite challenging: firstly because of the typical non-linearities of the traditional hydraulic layout - e.g. presence ofbrake fluid reservoir, oil compressibility - along with those related to friction and temperature variations. Moreover, thanks toits appealing characteristics, this BBW architecture is particularly employed in racing applications where control performance(static and dynamic precision, closed loop-bandwidth, robustness) are highly demanding.

To solve some of the above mentioned control problems, in [7] a standard pressure-based feedback strategy is flanked with afeed-forward action that mainly compensates for the presence of the braking reservoir. However, this approach shows long-termrobustness issues since the feed-forward action causes a non proper fluid compensation that is instead necessary to guaranteethe correct mechanical operation of the device, also in the presence of pad wear and temperature changes. For this reason, in[28] a hybrid position-pressure switching control strategy has been proposed, which yields regular fluid compensation, whilestill achieving the desired closed-loop control performance. Essentially, during the first phase of the braking maneuver - i.e.

F. Todeschini, S. Formentin, G. Panzani, M. Corno and S. Savaresi are with Dipartimento di Elettronica, Informazione e Bioingegneria, Politecnico diMilano, Piazza Leonardo da Vinci 32, 20133 Milano, Italy.

L. Zaccarian is with CNRS, LAAS, 7 avenue du colonel Roche, F-31400 Toulouse, France, Univ. de Toulouse, LAAS, F-31400 Toulouse, France, andDipartimento di Ingegneria Industriale, University of Trento, Italy. Email to: simone.formentin@polimi.it

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in the so called dead zone, where, due to the fluid reservoir, the brake pressure cannot increase - a master cylinder positioncontrol is active; when the system is eventually able to generate a pressure in the hydraulic plant, the pressure-based feedbackcontrol is activated. This solution, although attractive, has the disadvantage that, if the static position-pressure relationship isnot precisely identified, the dynamic closed-loop performance can significantly worsen, showing large overshoots (even thoughthis problem could be overcome using adaptive estimation solutions like the one presented in [35]). Moreover, none of theproposed techniques takes into account the limits of the control action, i.e., the saturation of the motor current, which maylead to windup behavior in case of aggressive requirements.

The goal of this paper is to devise a control architecture for the brake-by-wire system that directly addresses the main nonlinearcomponents, i.e. the dead zone due to the presence of the fluid reservoir, and the saturation of the control input. The controldesign is carried out in three steps. In the first step we assume that a linear controller is designed, which specifies a prescribedlocal behavior of the controlled system in a small neighborhood of the desired pressure value. This linear controller should bedesigned to meet a certain level of performance vs robustness in the proximity of the equilibrium and would typically involvean integral action to induce a zero steady-state error in the presence of a constant reference. As a second step, we recognizethat the presence of the fluid reservoir induces a peculiar nonlinear dead zone effect in the pressure/position nonlinearityand we propose a specific nonlinear dead zone compensation scheme that account for this undesired effect. In particular, thelinear controller augmented with this nonlinear compensation becomes a nonlinear (so-called unconstrained) controller forwhich we guarantee suitable asymptotic regulation properties when neglecting actuator saturation. As a final and third step,we address the issue of input saturation, which becomes relevant in our scheme because we are pushing the performance ofthe closed-loop system. To address input saturation we pursue a modular architecture and add a further anti-windup (AW)augmentation to the controller, which consists of a dynamic filter suitably designed based on the nonlinear BBW modeland interconnected with the nonlinear unconstrained regulator to ensure global asymptotic stability and asymptotic recoveryof the unconstrained performance. The underlying anti-windup construction is not trivial because most existing anti-windupconstructions require linearity of both controller and plant (see, e.g., [13], [34]). Here, instead, we address a windup problemwith a nonlinear plant and a nonlinear controller. A possible solution is presented in [23], where a gain-scheduling feedbacklinearization control scheme is used in combination with an MPC approach that explicitly deals with actuator saturation (seealso the early MPC-based works [1], [3] that may be more computationally demanding than a direct solution such as the oneadopted here). Alternative nonlinear anti-windup solutions can be found in the early works [9], [20] and references therein,based on feedback linearization approaches. A similar philosophy was followed more recently in [15] even though the proposedarchitecture is different. The few remaining existing approaches [17], [27], [36] are all special cases of the so-called ModelRecovery Anti-Windup (MRAW) paradigm [38, §6.3.1], which addresses the nonlinear anti-windup problem. In particular, thenonlinear IMC anti-windup scheme of [17], provides global asymptotic stability guarantees just as the design proposed here.We implement that scheme here for comparison purposes.

The paper is organized as follows. In Section II we describe the BBW system setup. The simple dynamical model typicallyemployed in the scientific literature for control design is presented in Section III, where the pressure regulation problem isalso formally defined. The dead zone compensation scheme not accounting for actuator saturation is presented in Section IV.Then, in Section V the proposed nonlinear anti-windup augmentation is described. The effectiveness of the presented strategyis first assessed in a simulation environment of Section VII and then experimental results on a BBW prototype system aregiven in Section VIII.

II. SYSTEM DESCRIPTION

The BBW system is composed of two parts:• an electro-mechanical actuator consisting of a DC motor that, through a mechanical transmission, moves a piston forward

and backward (this is the component that provides the required braking force);• a traditional hydraulic brake consisting of a hydraulic pump, a pipeline, a brake caliper, pads and discs.

The actuator is mechanically connected to the hydraulic pump, thus, moving the piston forward and backward, the pressure inthe hydraulic part can be modulated. An important part of the system is the brake reservoir: at the end of each braking event,the master cylinder must retract before the brake reservoir holes to allow for fluid volume compensation due to temperaturechanging and brake pad wear. This component is the main cause of the non-linearity in the position-pressure map that is widelydescribed in the sequel. A pictorial representation of the system can be seen in Figure 1, while Figure 2 shows a picture ofthe considered electro-mechanical actuator.

The pressure y in the master cylinder, the linear position x of the master cylinder and the current i flowing in the motor areavailable measurements. The final goal of the control algorithm is to control the pressure in the master cylinder, minimizingthe tracking delays and the overshoots.

From physical equations governing each component of the system, a complete model can be derived following [7], [8].Although the complete model describes accurately all dynamics in the system, it is too complex to be used for controlalgorithm design. In this section a control oriented model is derived. In doing so, we make the following simplifications.

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Figure 1. Pictorial representation of the BBW system.

Figure 2. Picture of the electro-mechanical actuator.

• There is a current loop that regulates the current flowing in the DC motor which is much faster than the pressure dynamics.This allows to neglect the electrical transients and to consider the DC motor current as the control variable.

• The static Coulomb friction is discarded. This can be done if a dithering signal is added to the control input [25], or if afriction compensator is properly designed [8], [28], [35].

• The pressure in the master cylinder and the pressure in the brake caliper are the same. Since the first resonant mode ofthe pressure wave propagation is usually beyond 100 Hz, this is a reasonable hypothesis.

• The amount of fluid volume in the hydraulic part is not influenced by the master cylinder movements. Also this is areasonable hypothesis since the fluid volume in the master cylinder is negligible compared to the one into the pipe and

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into the brake caliper.It follows that, by applying a force balance at the master cylinder of the hydraulic pump, a simple control-oriented model

that captures the most relevant system dynamics can be derived, similarly to [28], as:

m∗x =−Kdampx−Kspringx−Amc p(x)+Qeqi, (1)

where m∗, Kdampx and Kspringx represent the inertia, the damping force and the spring force, respectively. Amc p(x) is thereturn force due to the pressure in the pipeline. The coefficient Qeq lumps all the parameters (electric motor torque coefficient,mechanical transmission ratio) required to convert the motor current into the linear force exerted on the master cylinder. Thefunction p(x) is the position/pressure static relationship and it is a nonlinear function of x.

III. SYSTEM MODEL AND PROBLEM STATEMENT

Model (1) can be written in the compact form:

S :

x = vv = −k2v− k1x− kp p(x)+ kuuy = p(x),

(2)

where x is the position and v is the velocity of the master cylinder, k2 =Kdamp/m∗, k1 =Kspring/m∗, kp = Amc/m∗, ku =Qeq/m∗,the scalar input u is the current requested from the DC motor and the scalar output y is the pressure to be regulated.

By applying the Laplace transform to the linear components, the model can be seen as the block diagram depicted inFigure 3. Note that the system, except for the position/pressure map, is linear and has an intrinsic feedback structure.

1s2+k2s+k1ku

u yp(x)x

kp

Figure 3. Block diagram of the brake-by-wire plant model S in (1), (4).

Due to the presence of the dead zone, for every x smaller than xdz, defined as the position where the brake reservoir holesare reachable by the fluid, the pressure remains zero because the fluid can flow to the brake reservoir. Furthermore, once thesystem is beyond the dead zone, i.e. for all x≥ xdz, the position pressure relationship is nonlinear, due to fluid compressibilityvariations.

Figure 4 shows the experimental position/pressure pairs (blue dots) in the considered brake, in addition to the followingabstractions:• p`(x) (red dashed line): a linear static approximation of the position/pressure relation in the operating range x ≥ xdz :=−h0/h1, defined as:

p`(x) = h0 +h1x, (3)

where h0 and h1 > 0 are suitable real scalars;• p(x) (green solid line): a piecewise linear static approximation, also taking into account the dead zone effect for small

values of x≤ xdz :=−h0/h1, defined as:p(x) = max{0,h0 +h1x}. (4)

In this paper, a suitable control strategy is proposed to deal with the dead zone both without and with current saturationlimits on the input u. In both cases, we will address the following problem.

Problem 1 (Pressure regulation). Given two scalars pmax > pmin > 0, determine a control system such that, for each constantpressure reference value r ∈ [pmin, pmax], the equilibrium point (xc,x,v) = (x∗c ,x

∗,0), satisfying r = p(x∗), is GAS (globallyasymptotically stable) for the closed loop dynamics.

Remark 1. Note that in Problem 1 we are only focusing on the braking phase, disregarding the converse phase when thebrake is released. This is due to the fact that at the end of the braking, i.e. when r = 0, it is enough to switch the control actionoff. This creates a reasonable release of the brake because, when no force is exerted by the actuator, the contrast spring inthe master cylinder pulls the master cylinder back to the rest position with a satisfactory transient.

5

0 1 2 3 4 5 6−20

−10

0

10

20

x [mm]

y [bar]

Experimental Data

p(x)

pl(x)

Figure 4. Position - pressure maps: experimental points collected (isolated dots), p(x) (solid line) and p`(x) (dashed line). xdz is the position where p(x)changes slope. In this paper, xdz = 4 mm.

IV. LINEAR CONTROL WITH DEAD ZONE COMPENSATION

In this section we propose a control strategy to obtain regulation of the nonlinear BBW system (1), (4) of Figure 3. Theproposed scheme follows a two-step design. In the first step we focus on the target linear system that follows. By consideringthe linear approximation of the position/pressure map after the dead zone, the system model (2) can be expressed in itsstate-space form as:

S` :

x` = v`v` =−k1x`− k2v`− kp p`(x`)+ kuu`y` = p`(x`),

(5)

which comes from the linear approximation in (3) (corresponding to the red dashed line in Figure 4). In equation (5) we usesubscript “`” to denote quantities related to the linear plant dynamics. Note that model (5) only corresponds to the BBW plantdynamics for x ≥ xdz but has the advantage of corresponding to a linear system under the action of a constant disturbanceinput (equal to h0). Indeed, (5) can be rewritten as

S` :

x` = v`v` =−(k1 + kph1)x`− k2v`− kph0 + kuu`y` = h1x`+h0.

(6)

Therefore, focusing on (5), we consider the design of a linear error feedback controller whose equations are:

xc = Acxc +Bc(r− yl)ul = Ccxc +Dc(r− yl),

(7)

where we use subscript “l” to denote input and output of the linear controller (LC) described by (7). This controller is designedin such a way as to induce desirable properties of the ideal linear closed loop via interconnection

u` = ul , y` = yl . (8)

In particular, (5), (7), (8) are required to induce suitable asymptotic regulation of the output y according to the specificationsof Problem 1. This regulation property will be achieved typically by including an integral action in the linear controller (7) toinduce zero steady-state regulation error and reject the constant disturbance h0 in (6).

Assumption 1. The linear closed loop (5), (7), (8) is internally stable. Moreover, for each constant value r ∈ [pmin, pmax], theequilibrium point (x∗c ,x

∗,0), satisfying r = p(x∗), is GAS for (5), (7), (8).

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LCy

BBWul

DC

+

--

u

udc

ydc

yl

rknc

Figure 5. Linear controller and dead zone compensator seen as an unconstrained controller.

kpku

p(x)x udc

ydc

DC

Figure 6. Dead zone compensator: block diagram representation.

Starting from the linear controller in (7), we propose in this section the dead zone compensation (DC) scheme of Figure 5.In the scheme, the “DC” block injects suitable nonlinear correction signals (udc, ydc) at the input and output of the plant:

u = ul−udcy = yl− ydc.

(9)

In particular, udc, ydc are selected in such a way that the dynamic relation between u` and y` is linear and coincident withthat of the target linear system S` in (5). To this aim, and as clearly represented in Figure 6, it is enough to use the positionmeasurement x to select the DC block outputs as:

ydc = p(x) = p`(x)− p(x) (10a)

udc =kp

kuydc. (10b)

The following theorem establishes desirable properties of the overall control scheme (2), (7), (9), (10).

Theorem 1 (Dead zone compensation). If the linear controller (7) satisfies Assumption 1, then the control scheme (2), (7),(9), (10) shown in Figure 5 solves Problem 1.

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Proof: Substituting (9) in (2) we verify that the closed-loop dynamics corresponds to (7) interconnected to:

x = v

v = −k1x− k2v− kp p(x)+ ku(ul−udc)

= −k1x− k2v− kp p`(x)+ kuul

yl = y+ ydc = p(x)+ p(x)

= p`(x),

which coincides with the linear closed loop (5), (7), (8). Then, from Assumption 1, the control scheme induces the desiredregulation property on the output yl = y+ydc. Moreover, since the set-point satisfies r≥ pmin > 0, then the stabilized equilibrium(x∗c ,x

∗,0), satisfies p(x∗) = p`(x∗). Therefore from (10a), limt→∞ ydc(t) = 0 and the asymptotic regulation property also holdsfor the actual output y.

V. MODEL RECOVERY ANTI-WINDUP DESIGN

In this section we deal with the more realistic case where the input u is limited. The system to be controlled is the onepresented in (2), (4) with the addition of control input saturation:

Sσ :

x = vv =−k1x− k2v− kp p(x)+ kuσ(u)y = p(x),

(11)

where σ(u) represents the control variable saturation:

σ(u) =

u, i f u≥ uu, i f u < u < uu, i f u≤ u.

(12)

For Problem 1 to be solvable it is necessary that we make the following compatibility assumption between the saturation limits(u,u) and the pressure range [pmin, pmax], essentially requiring that the control input has enough authority to induce the desiredsteady state (note that this assumption is indeed necessary and not restrictive).

Assumption 2. The saturation limits (u,u) are such that

kuu > k1h−11 (pmax−h0)+ kp pmax

kuu < k1h−11 (pmin−h0)+ kp pmin.

(13)

The linear controller with dead zone compensator (7), (9), (10) presented in Section IV can be seen as a generic nonlinearcontroller (Knc), as represented in Figures 5, 6, which, under Assumption 1, is guaranteed by Theorem 1 to induce regulation forthe unsaturated plant (2). As customary in the anti-windup literature, we may refer to this controller (Knc) as the “unconstrainedcontroller”. To implement the control scheme of Section IV also in the presence of input saturation, we modify interconnection(9), (10) as:

u = ul−udc +uaw = unc +uawy = yl− ydc + yaw = ync + yaw

ydc = p(x− xaw), udc =kpku

ydc,(14)

where the selection of uaw is specified below, and (xaw,yaw) arise from the following anti-windup filter having states ξaw =[xaw vaw]:

xaw = vawvaw = −k1xaw− k2vaw− kpyaw

+ku(σ(σ(unc)+uaw)−unc)yaw = y− ync = p(x)− p(x− xaw).

(15)

The structure of this solution corresponds to the one suggested in [38, §6.3.1], which requires the synthesis of a suitablestabilizer uaw. As compared to typical model recovery anti-windup solutions (see, e.g., [38, Part II]), the structure proposedhere presents a “nested saturations” feature arising from the pre-saturation of signal unc before adding it to uaw. This operationis not necessary for stabilization purposes, but yields improved transient responses when transient evolutions of unc exhibitlarge peaks that may negatively affect the action of the stabilizer uaw. The proposed pre-saturation allows removing thosepossible peaks.

For the specific plant under consideration, due to its exponential stability properties (essentially coming from the dissipativeterm caused by friction) the stabilizer uaw may be selected to be zero, which actually leads to (see, [38, §6.5.1]) a nonlinearversion of the well-known IMC-based anti-windup solution [17]. In Theorem 2 below, we propose a nonzero selection of uaw.

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Indeed, as well known, and as it will be confirmed by our simulations and experiments, IMC-based anti-windup solutionstypically provide slow transients and their performance may be unacceptable in practice.

When no saturation occurs, as long as uaw remains at zero, the anti-windup filter above remains at zero, so that, in accordancewith typical anti-windup features, no modification to the unconstrained law of Section IV is performed until saturation isactivated (indeed (uaw,yaw,xaw) = (0,0,0)). Moreover, defining the “unconstrained” coordinates:

ξnc =

[xncvnc

]:=[

x− xawv− vaw

], (16)

one obtains from (11), (14), and (15): {xnc = vncvnc =−k1xnc− k2vnc− kp p(xnc)+ kuunc

ync = p(xnc)

ydc = p(xnc), udc =kpku

ydc

unc = ul−udc, ync = yl− ydc,

(17)

which coincides with the unconstrained interconnection (2), (9), (10), thus revealing that the unconstrained controller Knc isvirtually connected to the (nonlinear) plant without saturation, coinciding with (2), and that the feedback interconnection withthe anti-windup compensator (15) has an intrinsic cascaded structure.

The following statement proposes a selection for uaw exploiting this cascaded structure and shows the effectiveness of thearising scheme in solving Problem 1 when dealing with the saturated plant (11).

yBBW

unc

AW

u

r

Knc

++

-

--

uawxawyaw

yncxnc

σ(u)

x

Figure 7. Anti-windup compensator block interconnected to the unconstrained controller.

Theorem 2 (Anti-windup compensation). Under Assumptions 1 and 2, consider the following selection:

uaw = Kaw

[xawvaw

], (18)

where Kaw is selected as Kaw = XQ−1 for any solution to the following LMI, in the variables Q = QT > 0, X ∈ℜ1×2, W ∈ℜ,

He

[[ 0 1−k1 −k2

]Q+

[ 0ku

]X −

[ 0ku

]W

X −W

]< 0,

He

[[0 1

−k1−kph1 −k2

]Q+

[ 0ku

]X −

[ 0ku

]W

X −W

]< 0,

(19)

where He(Ξ) = Ξ+ΞT . Then the control scheme (7), (14), (15) shown in Figure 7 solves Problem 1.

Proof: According to the coordinates ξnc introduced in (16) and the derivations in (17), the overall closed loop writtenin the coordinates (xc,ξnc,ξaw) corresponds to the cascaded interconnection between the unconstrained closed loop (7), (17)driving, by way of signals unc and xnc, the following anti-windup dynamics derived from (15):

xaw = vawvaw = −k1xaw− k2vaw− kpyaw

+ku(σ(σ(unc)+uaw)−unc

)yaw = p(xnc + xaw)− p(xnc).

(20)

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From Theorem 1 and Assumption 1, we know that the upper subsystem of the cascade converges to (xc,ξnc) = (x∗c ,x∗nc,0),

where p(x∗nc) = r > pmin. Therefore, there exists a finite time t1 after which one has p(xnc(t)) ≥ pmin/2, for all t ≥ t1. As aconsequence, one has

p(xnc(t)) = max{0,h0 +h1xnc(t)}= h0 +h1xnc(t), (21)

for all t ≥ t1. In this regime, one can prove: 1

yaw = k1(t)xaw, (22a)

wherek1(t) ∈ [0,h1], ∀t ≥ t1. (22b)

Indeed, from (21), for h0 +h1(xnc + xaw)≥ 0 we have

yaw = max{0,h0 +h1(xnc + xaw)}− (h0 +h1xnc)= h1xaw,

where we used the fact that the maximizer is the second term. Whereas for h0 + h1(xnc + xaw) ≤ 0, where the maximizer iszero, we observe that necessarily xaw < 0 (from positivity of h1), and then inclusion (22) must be established with reversedsigns (indeed, yaw < 0 too). In particular, we get on one side (using (21),

yaw = −(h0 +h1xnc)≤ 0 = 0xaw,

and on the other sideyaw = −(h0 +h1xnc)

≥ −(h0 +h1xnc)+h1(xnc + xaw)+h0= h1xaw,

thereby establishing (22).Consider now inequalities (13) in Assumption 2. Due to the fact that they are strict, there exists a small enough ε > 0 such

that the following strengthened versions of (13) hold:

ku(u−2ε)> k1h−11 (pmax−h0)+ kp pmax

ku(u+2ε)< k1h−11 (pmin−h0)+ kp pmin.

(23)

Moreover, due to the asymptotic convergence of unc to the steady-state value u∗ satisfying kuu∗ = k1h−11 (r−h0)+kpr (which

follows from Assumption 1 and the results of Theorem 1), from (23), there exists t2 such that for all t ≥ t2, the unconstrainedinput unc(t) is in the interior of the set [u,u] with a distance of at least ε from its boundary. As a consequence, for all t ≥ t2we may write

σ(σ(unc(t))+uaw(t))−unc(t) = σε(t,uaw(t)), (24a)

where σε(t, ·) is a time-varying saturation function having, for all t ≥ t2, strictly positive and strictly negative time-varyingupper and lower saturation levels. To better illustrate this function, Figure 8 shows a possible scenario for this function (notethat σ(unc(t)) = unc(t) for all t ≥ t2). In the figure it is clear that the red thin function can never become smaller that the ε

margin corresponding to the vertical arrows. Also, the figure corresponds to a small negative value of σ(unc(t)) and one shouldkeep in mind that this nonlinearity is time-varying. Due to (24a) and from the sector properties of the saturation function, wehave that

q(t) := uaw(t)−σε(t,uaw(t)) (24b)

satisfies:q(t)(uaw(t)−q(t))≥ 0, ∀t ≥ t2. (24c)

Based on (22) and (24), we may rewrite dynamics (20) as the following uncertain linear system subject to a time-varyingbounded uncertainty and subject to input saturation:

ξaw :=[

xawvaw

]=

[vaw

−(k1 + kpk1(t))xaw− k2vaw + ku(uaw−q)

], (25)

which should be considered, together with selection (18) for stability analysis.In particular, for all t ≥max{t1, t2}, denoting ξaw = [ xaw

vaw ], dynamics (25), (18) satisfies:

ξaw ∈ co{A0ξaw,A1ξaw}+[

0ku

](Kawξaw−q), (26)

1An alternative approach for establishing (22) is to apply the generalized mean value theorem in [5, page 41], exploiting the fact that the generalizedgradient of p(·) always belongs to [0,h1]. Nevertheless, to keep the discussion at a simpler level, we provide here a self-contained proof of (22).

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uaw

σ(uaw)

ε

σ(uaw + σ(unc))

σ(uaw + σ(unc))− σ(unc)

Figure 8. A typical shape for the time-varying saturation function σε (t,uaw(t)) defined in (24).

whereA0 =

[0 1−k1 −k2

], A1 =

[0 1

−k1− kph1 −k2

].

Consider now the candidate common quadratic Lyapunov function V (ξaw) = ξ TawPξaw, where P = Q−1 comes from the

solution to (19). Then, using (24c), and (26) multiplied by the positive scalar U =W−1, (where U , coming from the solutionto (19), is necessarily positive because it is the opposite of a diagonal element of a negative definite matrix), we have:

〈∇V (ξaw), ξaw〉 ≤ maxζ∈co{PA0ξaw,

PA1ξaw}

ξTaw

(2ζ +2P

[0ku

](Kawξaw−q)

)

≤ maxζ∈co{PA0ξaw,

PA1ξaw}

ξTaw

(2ζ +2P

[0ku

](Kawξaw−q)

)+2qTU(Kawξaw−q)

=

[ξawq

]T

M[

ξawq

],

where for some λ ∈ [0,1],

M = He[

P(λA0 +(1−λ )A1)+P[ 0

ku

]Kaw −P

[ 0ku

]UKaw −U

].

Multiplying matrix M on both sides by[

P−1 00 U−1

]=[Q 0

0 W

], and keeping in mind that KawQ=X , we get that V = 〈∇V (ξaw), ξaw〉

is uniformly negative definite if M < 0, which holds if and only if

He[(λA0 +(1−λ )A1)Q+

[ 0ku

]X −

[ 0ku

]W

X −W

]< 0.

Finally, the negativity of the above matrix follows in a straightforward way by convex combination of inequalities (19).

Remark 2. Due to the possible uncertainty and time variation of the curve p(x) (see the blue curves represented in Figure 4),one may wonder if some robustness holds for the proposed solution. A partial answer to this arises from the intrinsic robustnessthat we may provide from the regularity properties of our compensation scheme, whose equations are globally Lipschitz functions(this is easily seen when considering the right hand sides of (7), (14), (15)). In particular, robust global asymptotic stability

11

holds from the derivations in [14, Thm. 7.21], which addresses small parameter variations. For larger parameter variations,we can conclude semiglobal practical asymptotic stability from [14, Thm. 7.20]. Note that even though the results in [14]address general nonlinear hybrid systems, they do apply to our continuous-time system, which is a special case of the broaderscenario considered in [14]. Finally, notice that in the braking application at hand, since xdz is fixed, the only significantvariations are those related to h1.

Note that LMIs (19) essentially treat the uncertain anti-windup dynamics using a common quadratic Lyapunov function.While for the parameters of our experiments the corresponding LMIs are feasible, for the general case there may be situationsrequiring more sophisticated conditions involving nonquadratic Lyapunov functions.

Remark 3. While Theorem 2 gives feasibility conditions for a stabilizing anti-windup gain Kaw, we may be interested inoptimized synthesis for Kaw. As explained in detail in [38, Ch. 6], the state xaw memorizes the mismatch between the actualplant state response and ideal plant state response without saturation. Therefore it is desirable to drive xaw to zero aseffectively as possible. To this end, we may take two alternative routes. On one hand we may try to maximize the decay rateα of the exponential convergence in the linear regime while preserving the stability constraints (19). Namely, we may selectKaw = XQ−1 where X and Q come from the solution of the following (quasi-convex) generalized eigenvalue problem in thevariables Q = QT > 0, Z = ZT , X, W and α: 2

maxX ,Q,Z,W,α

α, subject to:

He

[[ 0 1−k1 −k2

]Q−Z +

[ 0ku

]X −

[ 0ku

]W

X −W

]< 0,

He

[[0 1

−k1−kph1 −k2

]Q−Z +

[ 0ku

]X −

[ 0ku

]W

X −W

]< 0,

Z <−αQ. (27)

A second alternative is to use the quasi-LQR technique in [37, Remark 1] aiming at optimizing locally (namely in the regionwhere p(x) has slope h1) the quadratic performance index

J =∫

∞

0ξ

TawQLQRξaw +uT

awRLQRuawdt, (28)

under the global asymptotic stability constraints (19). This is accomplished by solving the following LMI optimization in thevariables Q = QT > 0, Z = ZT , X and γ:

minX ,Q,γ

γ, subject to:[γI II Q

]> 0, (19),

He

[

0 1−k1−kph1 −k2

]Q+

[ 0ku

]X Q 0

0 − 12 Q−1

LQR 0X 0 − 1

2 R−1LQR

< 0,

(29)

where we need to explicity require (19) so that local optimality comes with global stability guarantees. As typically experiencedwith optimal control synthesis, it is unclear what is the best optimality criterion and alternative ones could also be formulated.For our application, we used the design paradigm in (29) with QLQR essentially penalizing the position (therefore the pressure)error, and with RLQR selected to obtain a reasonable size of Kaw, in terms of implementability in the sampled-data digitalcontrol system of the BBW application. This choice induces desirable results, as illustrated in Sections VII and VIII.

Note that while solving the optimization problems (27) or (29) requires the use of sophisticated convex optimizationtools, once the gain Kaw is determined the implementation of the proposed algorithm in a real ECU does not require morecomputational effort than a typical control scheme. This fact is clearly illustrated in Section VIII, where a typical ECU hardwareis used together with a standard CAN bus communication line.

Remark 4. Notice that, in case of slow variation of the pressure curve (due, e.g. to deterioration of calipers, pads or disks),such a change could be accurately estimated online as illustrated in [35]. Then, the linear control system could be keptinvariant with respect to the nominal closed-loop behavior, by simply adjusting, in a gain-scheduling fashion, the controllergain.

2(27) can be solved, e.g., using the Matlab command gevp.

12

uPRBS

Rx(S) BBWxid

yu

xur

Figure 9. Closed-loop architecture for model identification.

u

y

xb1

1z2+a1z+a0

b2

Figure 10. BBW model seen as an ARX model with inputs u and y and output x.

VI. IDENTIFICATION

Consider the brake-by-wire actuator introduced and described in Section II. Let the DC-motor be controlled by an innercurrent loop, such that the control variable to be considered is directly the current set-point (from now on, it will be assumed thatthe desired and the actual values of the current coincide). In the following tests, the clock of the microprocessor is configuredat 1 kHz, whereas the interface with the device is obtained via CANbus, which is the industrial standard for automotiveapplications. The sampling time for the signal acquisition is 1ms.

To identify the control-oriented model of the system, the current control-loop is excited by means of a Pseudo RandomBinary Sequence (PRBS) [24]. The excitation is provided according to the scheme depicted in Figure 9. The position controllerRx(s) is a low performance controller which has been designed to maintain the system around a specific position in the operativezone, specifically around xid = 4.5 mm, such that the pressure is about 10 bar (see Figure 4). A dither signal at 100 Hz is alsoapplied in order to compensate for the static friction [35].

To obtain a model in the correct form, an ARX structure, as shown in Figure 10, has been selected:

x(k) =−a1 x(k−1)−a0 x(k−2)++b1 u(k−2)+b2 y(k−2),

(30)

where the output y is inserted in the regressor (it represents the pressure, which affects the position dynamics in feedback).In this way, by rewriting the model as

x(k) = θT

φ(k)+ e(k), (31)

with

θ =[a1 a0 b1 b2

]Tφ(k) =

[−x(k−1) −x(k−2) u(k−2) y(k−2)

]T,

(32)

it is guaranteed that, by solvingmin

θ‖x(k)− x(k)‖2, (33)

13

parameter value unitsk1 56.68 s−1

k2 24.85 s−2

kp 27.04 mms−2/barku 169.3 mms−2/barpmin 0.1 barpmax 40 barh1 10.68 bar/mmh0 −43.9 baru 15 Au −15 A

Table IIDENTIFIED MODEL PARAMETERS.

the identification problem becomes a convex optimization problem and the identification algorithm will find the global optimalθ o. Notice that it is also possible to enforce the asymptotic stability of the identified model by applying the Jury criterion [19]as a constraint of the optimization problem (33): 1+a1 +a0 > 0

1−a1 +a0 > 0|a0|< 1.

(34)

The continuous time counterpart of the identified model is in the form expressed in (2) and in (3), and yields the parametersshown in Tab. I. The matching between the discrete time and the continuous time model parameters is ensured by the Matlabcommand d2c using the Backward Euler method. In Figure 11, the quality of the obtained model is assessed on a validationdataset.

VII. SIMULATION RESULTS

In this section, the dead zone compensation scheme introduced in Section IV is first tested on a simulator of the BBW systemwithout saturation of the motor current. This preliminary test will highlight the performance of the compensator, with respectto the use of a linear controller alone. Then, a bound on the input is added and the anti-windup compensation is included inthe scheme.

The simulator is based on model (2) with the parameters in Table I. Moreover, in order to better validate the controlarchitecture for a real-world system, a nonlinear function representing the position-pressure experimental points in Figure 4is used instead of p(x). More specifically, for each value of x, we compute the corresponding p by averaging out the valuesobtained in different quasi-stationary experiments (to compensate for the effect of noise) and then we linearly interpolate themto obtain a piece-wise linear map.

The brake by wire model is developed without considering the international system unit of measurements, due to numericalconsistency. In fact, the usual operative range of this brake by wire system has very small positions expressed in meters andvery large pressures expressed in Pascals. Instead, using millimeters and bars, the order of magnitude of the signals in thesystem are comparable.

In all the simulations, the linear controller has been designed starting from the linear target system (5), i.e., starting fromthe transfer function:

GS`(s) =

kuh1

s2 + k2s+h1kp + k1. (35)

Specifically, a PID controller is tuned. Its parameters have been tuned following a model-based approach, using the identifiedsystem in its continuous time form, in order to guarantee a closed-loop bandwidth of 16Hz and a phase margin of approximately72o. The resulting controller parameters are: Kp = 1.39, Ti = 0.0578, Td = 0.0401 and N = 12.5, when the PID is written asfollows:

U(s) = Kp

(1+

1sTi

+Tds

TdN s+1

)E(s). (36)

where U(s) and E(s) are, respectively, the Laplace transform of the control action and of the tracking error. Figure 12 showsthe simulation results with the PID controller and the dead zone compensator presented in Sec. IV simulated on S . Thereference is a 0→ 20 bar step input. The aim of this test is to simulate the most critical part of a braking action, namely thebeginning, when the master cylinder needs to overcome the dead zone and reach the desired pressure.

The performance is as expected, and the settling time is about 25 ms. Notice that the dead zone compensator works onlywhen the system is within the dead zone. In other words, when x≥ xdz, udc and ydc become identically zero

To better appreciate the benefits of the dead zone compensation, in Figure 13 we compare two closed-loop responses usingthe same step reference (black dotted line in the upper plot). The blue solid curve shows the closed-loop response without dead

14

30 30.5 31 31.5 32−0.5

0

0.5

1P

ositio

n [m

m]

30 30.5 31 31.5 32−6

−4

−2

0

2

4

Time [s]

u [A

]

x

x

Figure 11. Measured position x and simulated position x using the identified model on a validation dataset.

zone compensation that reveals an overshoot and an undesirable delay of the response. The green dashed curve, corresponding tothe response of Figure 12, shows the desired response where the dead zone compensator enhances the closed-loop performanceboth in terms of settling time and overshoot.

A crucial point arising from the simulations is the following one: to perfectly compensate the dead zone, the compensatorneeds to reach values of the motor current that would be unattainable on a real device (see the trajectory of the current inFigure 13). This is the main reason why we introduced an anti-windup compensator. Figure 14 shows how the bounds on uaffect the closed-loop performance when the dead zone compensator is applied to the saturated system but no anti-windup ispresent. Note that, due to saturation, the system needs some time to overcome the dead zone (the effect is similar to a timedelay). In fact, the response considerably deteriorates both in terms of settling time and overshoot.

In order to mitigate the saturation effect, the anti-windup compensator is then designed according to the method proposedin Section V. The quasi-LQR technique is applied, i.e. the LMI problem (29) is solved with

RLQR = 1 QLQR =

[40 00 0.0001

],

which yield Kaw = [−3.96−0.21]. Notice that, as usually done in the LQR control approach, QLQR and RLQR can be seen asdesign parameters. In particular, in the application at hand, the tuning is based on the physics of the system as follows. Sincethe position is the variable of interest, the element QLQR(1,1) needs to be weighted much more than QLQR(2,2), which needsto be just slightly greater than zero, so that QLQR > 0. At the same time, RLQR is set to 1 to obtain u of a physically reasonableorder of magnitude for Kaw. Note that the norm of Kaw can be adjusted by either changing RLQR or QLQR(1,1).

Figure 15 depicts the simulation results with both the dead zone and the anti-windup compensator (on the the model (2), reddash-dotted line). The same control architecture is tested also on the model with the experimental position-pressure points usedinstead of p(x) (green dashed line). It can be seen that the reference tracking slightly worsens. However, by comparing this

15

0 0.02 0.04 0.06 0.08 0.1−40

−20

0

20

40P

ressure

[bar]

0 0.02 0.04 0.06 0.08 0.1

−100

−50

0

50

100

Curr

ent [A

]

Time [s]

ry

ydc

uudc

Figure 12. Simulated closed-loop trajectories of pressure and current with the linear controller and the dead zone compensator.

case to the one without anti-windup compensator (blue solid line), the improvement of the closed-loop performance is evidentboth in terms of response time and overshoot. Notice also that the scheme with only the dead zone compensator achieves worseperformance than the simple PID control (magenta dots). This is not surprising because the dead zone compensator ideallywould require a too large control variable that the actuator is not able to provide. In Figure 15 we also report (black circles)the simulation results obtained when using the dead zone compensation and the nonlinear IMC-based anti-windup scheme of[17], which corresponds to selecting Kaw = 0 in our scheme. One clearly sees that this response is excessively slow, which isa typical drawback of IMC-based anti-windup solutions, where the convergence rate coincides with the slowest mode of theplant.

In Figure 16 the dead zone and the anti-windup compensator contributions to the control action can be appreciated.

Remark 5. The small difference between the red dash-dotted and green dashed responses of Figure 15 is caused by the factthat the system in the operating zone is not completely linear (recall that we used a nonlinear position-pressure curve). Clearly,the control architecture with dead zone compensation simulated using p(x) instead of the experimental points would provideexactly the same response.

VIII. EXPERIMENTAL RESULTS

The different scenarios illustrated by simulations in Section VII have been also tested on the experimental setup describedin Section II and whose model has been identified in Section VI. In the experiments, the same PID and Kaw gains tuned forthe previous simulations have been used.

For testing the control strategy proposed in this paper, a dither signal needs to be added to the current set-point to avoidfriction induced stick-and-slip effects, analogously to what we did for the identification experiment. Notice that such an open-

16

0 0.05 0.1 0.15 0.2 0.25 0.30

5

10

15

20

25P

ressure

[bar]

0 0.05 0.1 0.15 0.2 0.25 0.3−500

0

500

1000

Curr

ent [A

]

Time [s]

r

PID on p(x)

PID + DC on p(x)

Figure 13. Dead zone compensator simulations results: reference (black dotted line), linear controller with the dead zone compensator simulated on p(x)(green dashed line) and linear controller simulated on p(x) without dead zone compensator (blue solid line).

loop action causes oscillations at the same dither frequency on the pressure. However, by filtering the measured pressure usinga notch filter at the dither frequency the pressure oscillations are hidden to the controller.

The current saturation is imposed by the power electronics and the DC motor characteristics, and it is set to ±15 A.Figure 17 shows a 0→ 20 bar step response on the real plant, starting from a zero position. This experiment shows how

the system would behave in a realistic braking maneuver. As a matter of fact, when the driver starts to pull the brake lever,the master cylinder piston position is before the brake reservoir holes, i.e. at the beginning of the dead zone.

In this experiment, the performance of the simple PID controller (magenta dots) is unacceptable due to the large overshoot,which would be very critical for the safety of the driver. Ideally, by adding the dead zone compensator this problem couldbe solved without any additional position controller (like, e.g. in [28], [35]). The performance of the PID pressure controllerwith the dead zone compensator is shown by the blue solid line. Unfortunately, compared to the simple PID controller alone,the overshoot increases. As a matter of fact, by adding the dead zone compensation, the plant input at the beginning of theresponse is larger than the one with the PID only and the saturation effects are worse (this behavior matches the one seen inour simulations, see again Figure 15). Therefore, in this scenario, the employment of an anti-windup compensator becomeseven more important, to make the dead zone compensation effective. The experiments using the complete control architecture(green dashed line) confirm this expectation. Notice that the settling time decreases and now there is no overshoot. Notice alsothat the experimental response is somewhat faster than the simulated one of Figure 15, which indicates a non-negligible levelof uncertainty in the dynamic model. In particular, we believe that this mismatch is caused by the fact that inspecting Figure 4one clearly sees that the actual experimental curves already exhibit an increase of the pressure for positions smaller than xdz,anticipating for a more prompt response. The curve in black circles in the figure represents the experimental response whenusing the IMC-based anti-windup solution of [17] in combination with the dead zone compensator. The resulting response,similar to the simulated case, leads to an unacceptably large settling time. Finally, Figure 18 shows the contributions of the

17

0 0.05 0.1 0.15 0.2 0.25 0.30

10

20

30

40

50

60

Time [s]

Pre

ssure

[bar]

PID + DC on p(x)

PID + DC + SAT on p(x)

Figure 14. Effect of the input saturation on the closed loop performance of the system without anti-windup compensation: with (solid) and without (dashed)input saturation.

nonlinear compensators during the step response discussed above. From this figure, it can be observed that these contributionsare comparable to the ones obtained in simulation (see again Figure 16).

IX. CONCLUSIONS

In this paper, a control architecture for dead zone and anti-windup compensation in electro-hydraulic brake-by-wire systemshas been proposed. Specifically, the pressure regulation problem has been dealt with and it has been shown that the providedscheme guarantees global asymptotic stability of the achieved equilibrium point. The design procedure is based on simplealgebraic calculations and on the solution of an LMI problem. The effectiveness of the proposed approach has been illustratedboth in simulations and experiments carried out on a prototype device. In the light of the satisfactory experimental results,follow-up work includes verifying the effectiveness of the proposed control algorithm during road test drives on a real vehicle.

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0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.40

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PID + DC + SAT + AW on p(x)

PID + DC + SAT + AW on real points

PID + DC + SAT + IMC on real points

Figure 15. Simulation analysis of anti-windup and dead zone compensators: reference (black dotted line), dead zone and anti-windup compensators applied to(2) (red dash-dotted line), dead zone and anti-windup compensators applied to (2) with an interpolation of the position-pressure experimental points instead ofp(x) (green dashed line). The blue solid line shows the system response with the dead zone compensator and input saturation without anti-windup compensator(same curve as the blue trace in Figure 14); the magenta dots show the response of the PID controller without compensators and with input saturation. Finally,the black circles illustrate the behavior of the system with the nonlinear IMC-based anti-windup scheme.

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0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4−40

−20

0

20

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ressure

[bar]

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−10

0

10

20

30

40

Curr

ent [A

]

Time [s]

ryydcyaw

uudc

uaw

Figure 16. Signals involved for the nonlinear compensation in the simulation test (green dashed curve) of Figure 15.

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2011.

20

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

10

20

30

Pre

ssure

[bar]

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4−20

−10

0

10

20

Time [s]

Curr

ent [A

]

r

PID + SAT

PID + DC + SAT

PID + DC + SAT + AW

PID + DC + SAT + IMC

Figure 17. Experimental step response of the brake-by-wire system with different control architectures.

21

0.05 0.1 0.15 0.2 0.25 0.3−40

−20

0

20

40

Pre

ssure

[bar]

0.05 0.1 0.15 0.2 0.25 0.3

−10

0

10

20

30

40

Curr

ent [A

]

Time [s]

ryydcyaw

uudcuaw

Figure 18. Signals involved in the nonlinear compensation for the experimental test (green dashed curve) of Figure 17.