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Constrained optimal control of an electronic throttleM. Vašak a , M. Baotić a b , M. Morari b , I. Petrović a & N. Perić aa Faculty of Electrical Engineering and Computing, University of Zagreb, Unska 3, HR-10000Zagreb, Croatiab Automatic Control Laboratory, Swiss Federal Institute of Technology, Physikstrasse 3,CH-8092 Zurich, Switzerland

Version of record first published: 20 Feb 2007.

To cite this article: M. Vašak, M. Baotić, M. Morari, I. Petrović & N. Perić (2006): Constrained optimal control of an electronicthrottle, International Journal of Control, 79:05, 465-478

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International Journal of ControlVol. 79, No. 5, May–June 2006, 465–478

Constrained optimal control of an electronic throttle

M. VASAK{, M. BAOTIC*{{, M. MORARI{,I. PETROVIC{ and N. PERIC{

{Faculty of Electrical Engineering and Computing, University of Zagreb,Unska 3, HR-10000 Zagreb, Croatia

{Automatic Control Laboratory, Swiss Federal Institute of Technology,Physikstrasse 3, CH-8092 Zurich, Switzerland

(Received 24 May 2005; in final form 16 November 2005)

The overall vehicle performance is strongly influenced by the quality of the control of theelectronic throttle – a DC motor driven valve that regulates the inflow of air to the vehicle’sengine. Designing a controller for the throttle system is a challenging task since one has tocope with two strong non-linearities: the gearbox friction and the so-called ‘‘limp-home’’

non-linearity. In this paper we address these issues by solving a constrained optimal controlproblem formulated for the discrete-time piecewise affine (PWA) model of the throttle.In an off-line, dynamic programming procedure we obtain the look-up table like solution to

the optimal control problem. Such a solution allows the real-time controller implementationthat would otherwise be impossible to achieve due to the small sampling time needed forthe application at hand.

We address the issue of the PWA friction modelling in more detail by considering bothstatic and dynamic friction models. Two different control strategies are studied: constrainedfinite time optimal control (CFTOC), used in the regulator case, and constrained time-optimalcontrol (CTOC), used in the reference tracking case. We report experimental results with

both control strategies. The reference tracking controller significantly outperformed a tunedPID controller with a feedforward compensation of non-linearities in terms of the responsespeed while preserving the response quality regarding the absence of an overshoot and the

static accuracy within the measurement resolution.

1. Introduction

In the past many vital systems and commands in

cars were directly mechanically controlled by drivers

(e.g., brakes, steering angle, gas command, etc.).

Today direct mechanical links between the driver and

the actuating objects are being replaced by control

loops comprising microprocessor-controlled actuators.

Modern cars incorporate several such control loops

usually called x-by-wire systems (e.g., brake-by-wire,

steer-by-wire, drive-by-wire, etc.) that improve driving

performance, passenger safety and comfort, and

decrease vehicle fuel consumption and air pollution.

Thanks to the x-by-wire systems one can implement

advanced control strategies that aim at achieving those

benefits.Many car related systems are fast and non-linear,

usually with very different and distinct modes of opera-

tion, and it is not uncommon for some of the process

variables to take values from a discrete domain

(Kiencke and Nielsen 2005). Furthermore, pushed by

the reduction of production costs, microprocessors

used in cars are rather simple, which makes the design

and implementation of control algorithms even more

challenging. Currently the most common choice of the

control algorithm in car systems is a PID controller,

mainly due to its simple realization. However, novel

research in the area of optimal control of hybrid systems

(systems with both discrete and continuous variables)*Corresponding author. Email: mato.baotic@fer.hr

International Journal of ControlISSN 0020–7179 print/ISSN 1366–5820 online � 2006 Taylor & Francis

http://www.tandf.co.uk/journalsDOI: 10.1080/00207170600587572

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opens the door to the design of implementablecontrollers that satisfy all system specifications,guarantee closed-loop system stability and outperformthe PID controller with respect to the overall carperformance.In this paper we discuss optimal control of the electro-

nic throttle system, a vital part of a drive-by-wire systemthat regulates air inflow to the car engine (Visteon 2000)and is used as an inner control loop in the engine torquecascade control. A desired inflow rate, calculated fromthe gas pedal position sensor information and thecurrent engine state, can be achieved by an appropriateelectronic throttle valve plate position (opening angle).The plate positioning control system must satisfy pre-scribed requirements: a fast transient response withoutovershoot, accuracy within the measurement resolution,and a control action that does not cause excessive wearon the system sensor and actuator. The fast transientresponse makes the engine torque control system fasterand directly improves drivability of the vehicle, whilethe static accuracy within the measurement resolutionbit rules out the need for an additional idle speed controlactuator thus reducing the car production costs.Furthermore, fulfilling the performance requirementsalso reduces fuel consumption and air pollution.The controller synthesis for the electronic throttle is

a hard task due to the physical limitations of the inputand state variables and the presence of two strongnon-linearities: friction in the gearbox and the ‘‘Limp-Home’’ (LH) non-linearity. Furthermore, to protectthe components of the throttle, the controller shouldensure that the process variables always stay withinthe values prescribed by the manufacturer (Visteon2000). Regarding the sensory information, there isonly one quantization-noise-corrupted measurementavailable, that of the throttle valve plate position givenby a dual potentiometer attached to an A/D converterof low resolution. No other sensors can be used sincethis would significantly increase the production costsof the throttle control system.Considering everything mentioned above, it is not a

surprise that this challenging problem from automotivecontrol has attracted significant attention of the auto-matic control research community, cf. Canudas deWitt et al. (2001), Deur et al. (2004), Baric et al.(2005) and Pavkovic et al. (2006). We use here a con-strained optimal control strategy, also known as amodel predictive control (MPC) strategy (Mayne et al.2000), designed for hybrid systems. Application of theMPC strategy in automotive control systems is a newbut active area as can be seen from the literature(Bemporad et al. 2001, Borrelli et al. 2001, Giorgettiet al. 2005a, b). The MPC strategy systematicallytakes into account all modelled process non-linearitiesby using a mathematical model of the system – we

use piecewise affine (PWA) approximations ofthe non-linearities resulting in a PWA processmodel – and provides the optimal control systemperformance with respect to the chosen criterion whilefulfilling all imposed constraints. Finding the optimalcontroller action can be formulated as a mixed-integerprogram (Borrelli 2003) but, since the process samplingtime is small, on-line computation of the optimalcontrol input using mixed-integer programming solversis prohibitive. To overcome this problem an easilyimplementable, optimal explicit state feedback controllaw is pre-computed in an off-line dynamic progra-mming procedure (Borrelli 2003, Baotic 2005).

Here we report the design procedure of an explicitmodel predictive controller for this fast-samplingmechanical servosystem with two interacting non-lineareffects. Two different friction models in the electronicthrottle PWA model were tried (a static and a dynamicone), and we consider two different optimal control strat-egies: constrained finite time optimal control (CFTOC)and constrained time-optimal control (CTOC).

The paper is organized in five sections. Section 2introduces two electronic throttle discrete-time PWAmodels, detailing the friction modelling issue. Section 3summarizes basic off-line and on-line computationsregarding the two optimal control strategies considered:CFTOC and CTOC. Experimental results for theelectronic throttle regulator using the CFTOC strategyand the reference tracking using the CTOC strategyare reported in x 4. The concluding x 5 summarizes theresults of the paper.

2. Discrete-time piecewise affine model of

an electronic throttle

A first step in the optimal controller design is to obtaina representative mathematical model of the system.We use a class of discrete-time PWA systems (Sontag1981, Heemels et al. 2001) for that purpose. Discrete-time PWA models entail several state-update equations,each defined over a polyhedron (polyhedron is definedas an intersection of a finite number of halfspaces)in the stateþ input space

xkþ1 ¼ Aixk þ Biuk þ fi

yk ¼ Cixk þDiuk

ifxkuk

� �2 Di, i ¼ 1, . . . , s,

9>>>=>>>;

ð1Þ

where x 2 Rn is the model state, y 2 R

p is the modeloutput, u 2 R

m is the model control input, fDigsi¼1 is

a polyhedral partition of the stateþ input space Rnþm,

and k denotes the sampling instant.

466 M. Vasak et al.

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We derive a discrete-time PWA model from thecontinuous-time non-linear model of the electronicthrottle.

2.1 Electronic throttle continuous-time model

The electronic throttle is a DC motor driven valve(see functional scheme in figure 1(a)) in which a bipolarchopper supplies the DC drive. The motor rotation istransmitted through the gearbox to the throttle plate,and the motor torque is counterbalanced by the dualreturn spring attached to the throttle plate’s shaft. Thecontrol system’s task is to achieve a desired air inflowto the engine combustion system by positioning thevalve plate at the required opening angle.The dynamical behaviour of the throttle can be

described with the following equations

Ladiadt

þ Raia ¼ Kchu� Kv!m, ð2Þ

mm ¼ Ktia, ð3Þ

mapp ¼ mm �mS �mL, ð4Þ

Jd!�

m

dt¼ mapp �mf, ð5Þ

!m ¼ !mð!�mÞ ð6Þ

ffdmf

dt,mf,!m,mapp

� �¼ 0, ð7Þ

d�

dt¼ Kl!m, ð8Þ

mS ¼ mSð�Þ, ð9Þ

where u½V� is the input control voltage, Kch is thechopper gain, ia ½A� is the armature current, mm ½Nm� isthe motor torque, mS ½Nm� is the return spring torque,mL ½Nm� is the load torque, mapp ½Nm� is the appliedtorque, mf ½Nm� is the friction torque, !�

m ½rad=s� is anauxiliary angular velocity variable (needed in somefrictionmodels),!m ½rad=s� is the motor angular velocity,� ½

�� is the position (opening angle) of the throttle plate,

Ra ½�� is the overall armature resistance, La ½H� is thearmature inductance, Kt ½Nm=A� is the motor torqueconstant,Kv ½Vs=rad� is the electromotive force constant,Kl is the gear ratio. The full non-linear continuous-timemodel of the process is shown in figure 1(b), where sdenotes the Laplace variable.

System sampling time is chosen with respect to thedominant time constant of the linearized electronicthrottle model presented in Deur et al. (2004) andis set to T ¼ 5ms. The armature current dynamics canbe neglected since the time constant Ta ¼ La=Ra � T.Therefore equation (2) is replaced with

ia ¼ KaðKchu� Kv!mÞ, ð10Þ

where Ka ¼ 1=Ra.To build a PWA model starting from the sketched

continuous one, process non-linearities need to beapproximated with PWA functions. This is straight-forward for the return spring stress-strain characteris-tics (9), which is characterized by the LH non-linearity(see figure 2), since it already has a PWA form

mS ¼

KlðKS,11� þ KS,01Þ if � � �LH,

KlðKS,12� þ KS,02Þ if �LH < � � �LH,

KlðKS,13� þ KS,03Þ if � > �LH,

8><>: ð11Þ

where KS,1j and KS,0j are coefficients of the jth affineexpression, j 2 f1, 2, 3g, and �LH, �LH are the angleswhere the affine law changes. The name ‘‘limp-home’’comes from an embedded security feature: in case of atotal power failure the valve plate is set to the position�LH so that there is enough air-inflow to the enginefor the car to ‘‘limp’’ to the nearest repairing facility.Note that the opening angle � we measure representsthe angle between the valve plate and the valve tubecross section (cf. Visteon 2000), and it spans from 13�

(closed valve – no air inflow) to 90� (totally openvalve). Therefore the limp-home position �LH of about

M

+

uia

wm

wm

q

Carbattery

Electronic throttle body

GearboxValveplate

Returnspring

Air inflowPositionsensor

Chopper

u ia

qLH

mm

mapp

mL

mS

Kch Kl++

−

−−Ka

1+Tas

Kv

Kt

K1

1s

Frictionmodel

(a)(b)

Figure 1. Electronic throttle: (a) Process scheme; (b) Non-linear continuous-time model.

Constrained optimal control of an electronic throttle 467

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20:0� reported in this paper actually corresponds to 7�

of the relative opening of the valve plate.The modelling of friction non-linearity (6)–(7) is

described in more detail in the following subsection.

2.2 Friction

The friction, unlike the LH non-linearity described inthe previous subsection, is not introduced in the throttlesystem on purpose. It is a consequence of the relativelyinexpensive construction of the throttle, where large-series mechanical components, such as gearboxes andbearings exhibit a substantial friction effect.Friction occurs between the contacting surfaces in the

gearbox that (try to) move relatively to each other. It ismanifested by the friction torque mf, that always actsto suppress the movement. Depending on the existenceof a relative movement we distinguish two modes offriction: stiction friction mode (no movement) andsliding (movement exists). The transition between thestiction and sliding is called presliding, and is basicallyintroduced to model the straining of the asperitycontacts on the contacting surfaces at the very beginningof sliding (Haessig and Friedland 1991). Friction inthe throttle’s gearbox is characterized by a significantpresliding effect (Deur et al. 2004) since a relativelylarge throttle valve displacement can be observedbefore the applied torque reaches the value of themaximum friction torque in stiction (i.e. static frictiontorque). To model this effect accurately one needs touse a dynamic friction model where the asperity contactsdisplacement, directly proportional to the resultingfriction torque, is used as an additional model state.However, since a static friction model is simpler thana dynamic one, we first describe the former.

2.2.1 Karnopp friction model. A straightforwardapproach to modelling the friction is to describe it as astatic function. In such a case the state space dimensionof the overall friction model is reduced (mf is not a statevariable of the model), which, eventually, results ina simpler feedback control law. Additionally, since mf

is not a state variable, there is no need to estimate thevalue of mf for the real-time implementation of thestate feedback control law.

The Karnopp friction model (Karnopp 1985) is anexample of a static model, where mf is a static functionof mapp and !m only (no derivative term in (7).Depending on the motor speed (see figure 3) theKarnopp model is either in the stiction or in the slidingfriction mode. An additional internal variable !�

m isintroduced so that !m can be easily set to 0 in stiction.Denoting with �! � 0 the angular velocity very closeto zero we say that the friction model is in stictionif j!�

mj � �!, otherwise it is in sliding. Equation (6)becomes

!m ¼0 if j!�

mj � �!,

!�m if j!�

mj > �!:

�ð12Þ

In stiction mf is equal to the applied torque mapp unlessit reaches the saturation point

mf ¼

�MS if j!mj � �! and mapp < �MS,

mapp if j!mj � �! and jmappj � MS,

MS if j!mj � �! and mapp > MS,

8><>: ð13Þ

where MS is the maximum friction torque in stiction. Insliding the friction torque is expressed as (Karnopp 1985)

mf ¼ MC þ ðMS �MCÞe� ðj!mj��!Þ=!sð Þ

�

þ bj!mj

h isgnð!mÞ

ð14Þ

×

mapp

mf

wmw*m

MS

MS MC

∆w

∆w

∆w

1Js

+

+

+

−

Figure 3. Block scheme of the Karnopp friction model.

19.5 19.6 19.7 19.8 19.9 20 20.1 20.2 20.3 20.4 20.5−0.015

−0.01

−0.005

0

0.005

0.01

qLH

qLH

qLH

q [°]

mS

[Nm

]

Figure 2. Return spring stress-strain characteristics (Visteon2000). Detail around �LH showing LH non-linearity.

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where MC is the Coulomb friction torque, !s is theStriebeck angular velocity, � is the Striebeck coefficient,and b is the viscous friction coefficient. In ourapplication, Striebeck and viscous friction effects canbe neglected (Deur et al. 2004), resulting in simple piece-wise constant friction model in sliding (j!�

mj > �!)

mf ¼MC if !�

m > �!,

�MC if !�m < ��!:

�ð15Þ

The equations (12), (13) and (15) give a PWA functionaldescription of the friction, which, together with the LHnon-linearity (11), is used to derive an overall discrete-time PWA model (1) of the electronic throttle.Discrete-time affine submodels in (1) are obtained byZOH discretization of continuous-time affine submodelswhich in turn were obtained by combining every affinepart of the friction non-linearity with the affine partsof the LH non-linearity. Regions Di in (1) follow fromthe localization of the affine approximations (if-termsin (11), (13) and (15)).The resulting discrete-time PWA model of the throttle

obtained from the Karnopp friction model will bereferred to as PWA-1 model in the rest of the paper.Since friction is modelled with 5 and LH non-linearity (11) with 2 affine regions PWA-1 modelcomprises s¼ 10 different affine dynamics. (We neglectthe narrow LH region in (11). This is justifiable whenwe are designing a regulator with a set-point angle farenough from �LH.)

2.2.2 Reset-integrator friction model. One of the maindemands on the electronic throttle control system isthat it should be accurate up to the measurementresolution (approximately 0:1� in our application).A serious obstacle in achieving such an accuracy withthe controller based on the static friction modelis the existence of a presliding effect. We denote with�ps the presliding displacement, i.e. the angle traversedby the valve plate from the beginning of its motionuntil the moment when the friction reaches the fullamount of the static friction torque MC. In our applica-tion the (estimated) presliding displacement is roughlythree times larger than the measurement resolution(Deur et al. 2004) and therefore the presliding effecthas to be taken into account when modelling friction.Note that the controller based on the static Karnopp

friction model would overcompensate the frictiontorque for the displacements in the presliding range.As a consequence, for a given set-point such a controllercould cause oscillations in the response around thedesired steady state (see experimental results reportedin x 4.1). To capture the presliding phenomenonand avoid friction overcompensation we therefore use

a dynamic friction model, in particular a reset-integratormodel (Haessig and Friedland 1991, Bozic et al. 2001)that is essentially modelling friction non-linearity (7)as a switched linear system of the following form(see model block scheme in figure 4)

dmf

dt¼

0 if!m � 0 and mf � MC

!m � 0 and mf � �MC

�,

Kf!m otherwise,

8<:

ð16Þ

where Kf ½Nm=rad� is a parameter, simply obtained bytime-integration of the presliding equation in (16)during the whole presliding displacement time

Kf ¼MC

ð�=180KlÞ�ps: ð17Þ

In the reset-integrator model there is no need for anauxiliary variable !�

m and we can write (6) as

!m ¼ !�m: ð18Þ

Note that the reset-integrator model (16) is well suitedfor the derivation of a discrete-time PWA processmodel, which is not necessarily the case with otherdynamic friction models. For example, the LuGrefriction model (Canudas de Witt et al. 1995) canaccurately describe the presliding effect but it isgoverned by a non-linear differential equation that isdifficult to linearize.

The choice of a continuous-time reset-integratorfriction model (Haessig and Friedland 1991) is just afirst step in deriving a PWA process model in the form(1). Similarly to the static friction modelling allcontinuous-time affine dynamics with different affineparts of friction model (16) are sampled with ZOH(with the same sampling time T ) to obtain their corre-

×

×

mapp

mf

OR| | +

MC

1Js

+

−

−

Kfs

w*m = wm

≤0

≤0•

Figure 4. Block scheme of the reset-integrator frictionmodel.

Constrained optimal control of an electronic throttle 469

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sponding discrete-time representations. Here, however,we need to perform additional corrections of themodel. Consider the discrete-time PWA system thatcomprises only ZOH generated affine dynamics andcompare it to the original continuous-time non-linearsystem (2)–(9). Even though ZOH is used for eachsub-system there is no guarantee that the outputs ofthe discrete-time and continuous-time system will notdiffer at sampling instances (unlike in the linear systemcase). This discrepancy is caused by the fixed samplingtime of a discrete-time model (1) which implies thatswitching between dynamics can happen only at multi-ples of the sampling time. On the other hand, in the con-tinuous-time model (16), the switching between twomodes can happen at any moment. Thus, if a samplingtime is too big, it could happen that the state updateequation xkþ1 ¼ Aixk þ Biuk þ fi ‘‘changes’’ the statetoo much. Unfortunately, in our application the abovementioned effect cannot be neglected since the samplingtime has to be large enough to predict the whole transi-ent response of the system in a small number of steps,otherwise the finally computed control law couldbecome too complex. We cope with this problem byintroducing additional affine dynamics in the PWAmodel (1) as explained below.It is critical to accurately model friction in the PWA

form around zero-velocity (!m ¼ 0) since at that pointaffine models obtained by ZOH discretization oflinear subsystems in (16) differ drastically. We predict!m-zero-crossing in the PWA model in order to usedifferent affine dynamics in the zero-crossing and thenon-zero-crossing case. This is achieved by introducingadditional partitioning in the stateþ input space,defined by the borders !m,i,kþ1 ¼ 0, where !m,i,kþ1 ¼ 0denotes the one-step ahead prediction of !m at discreteinstant k using affine dynamics indexed with i.Although in (16) both stiction and presliding are

modelled using _mf ¼ Kf!m, in the discrete-time modelwe differentiate between the two modes, see figure 5.In presliding, i.e. when mf and !m are of the same signwe use _mf ¼ Kf!m for discretization. In stiction,however, we do not permit valve plate displacement inthe model, thus preventing the unnatural situation inwhich mf supports the motion since this could be‘‘misused’’ by the controller. Border between stictionand presliding is introduced at mf ¼ sgnð!mÞ�MC,where �>0 is a small positive number and ��MC

denotes the applied torque at which the preslidingstarts. (In our application � ¼ 0:05. It is beneficial for� to be small since this makes the controller actionsmoother, but there is no need for its experimentalidentification.) The borders of the friction model in the!m �mf plane are depicted in figure 5, where " denotesa small positive number also. Note that the bordersintroduced by one-step-ahead prediction of !m are not

depicted, since they involve all the states and inputsand cannot be displayed in 2D. Note also that thisadditional partitioning happens only in the preslidingand sliding dynamics, i.e. when the valve moves.

The linearization of the reset-integrator friction modelin sliding and presliding regimes when zero-crossingof !m is not detected is straightforward. In case ofstiction or !m zero-crossing we use the following stateupdate equations

!m,kþ1 ¼ Kmmapp,k, ð19Þ

�kþ1 ¼ �k þ �180Kl

�

T

2!m,k, ð20Þ

mf,kþ1 ¼ mapp,k, ð21Þ

where Km ¼ 1 rad=Nms½ �, so that the torques around0:005Nm observed in practice produce a very smallspeed for stiction. We use equation (19) for the!m-update because resetting !m to the value 0 wouldcause problems at the next time step since such a statewould belong to more than one region Di in (1).Coefficient �, experimentally determined at value 0.7,is used to model the change of the valve position whenentering the stiction regime.

The PWA friction model consists of 10 affinesegments (six visible in figure 5, and additional fourintroduced by the splitting in two of each sliding andpresliding model due to the velocity zero-crossingdetection). The resulting discrete-time PWA model ofthe throttle obtained from the reset-integrator frictionmodel will be referred to as PWA-2 model in the restof the paper. It comprises 30 affine dynamicscorresponding to the combinations of 10 affine frictionsegments with 3 regions in the LH non-linearity (11).Clearly the PWA-2 throttle model is significantly more

wm

mf

Presliding

Presliding

Sliding

Sliding

Stiction

Stiction MC

−MC

−aMC

.mf = Kfwm

.mf = Kfwm

wm,k+1 = Kmmapp,k

wm,k+1 = Kmmapp,k

mf,k+1 = mapp,k

mf,k+1 = mapp,k

mf,k+1 = −MC−e

mf,k+1 = MC + e

0

aMC

Figure 5. Borders of affine characteristics regions for

the dynamic PWA friction model in the !m �mf plane(thick lines).

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complex than the PWA-1 model described in theprevious subsection (30 affine dynamics comparedto 10) but this is the price we have to pay if we wantto model the presliding effect.

3. Constrained optimal control strategies

Consider the PWA system (1) subject to input and stateconstraints

C xxk þ C uuk � C 0 ð22Þ

for k� 0. The restriction of the PWA system (1) overthe set of states and inputs defined by (22) is referredto as constrained PWA system (CPWA)

xkþ1 ¼ Aixk þ Biuk þ fi

yk ¼ Cixk þDiuk ifxk

uk

� �2 ~Di,

9=; ð23Þ

where f ~Digsi¼1 is the new polyhedral partition of the sets

of stateþinput space Rnþm obtained by intersecting the

polyhedrons Di in (1) with the polyhedron describedby (22). Let ~D :¼ [s

i¼1~Di. In the following we will

express the CPWA system equations (23) in the shorterform

xkþ1 ¼~f PWAðxk, ukÞ, ð24Þ

where ~f PWA: ~D ! Rn and ~f PWAðx, uÞ ¼ Aixþ Biuþ fi if

xu

� �2 ~Di, i ¼ 1, . . . , s:

3.1 CFTOC strategy

We define the following cost function

JðUN, x0Þ :¼ kPðxN � xeÞk2 þXN�1

i¼0

kQðxi � xeÞk2

þ kRðui � ueÞk2 ð25Þ

and consider the constrained finite-time optimal controlproblem (CFTOC)

J�ðx0Þ :¼ minUN

JðUN, x0Þ ð26Þ

subj. toxiþ1 ¼

~f PWAðxi, uiÞ

xN 2 X f

(ð27Þ

where the column vector UN :¼ ½uT0 uTN�1�T2 R

mN,is the optimization vector, xe is the equilibrium state,ue is the equilibrium control action, N is the time

horizon and X f is the terminal region. In (25),kQxk2 ¼ xTQx and R ¼ RT 0, Q ¼ QT � 0,P ¼

PT � 0. We denote by X 0 � Rn the set of initial states

x0 for which the optimal control problem (25)–(27) isfeasible. Similarly X i denotes the set of feasible statesxi, i ¼ 1, . . . ,N at instant i for the optimal controlproblem (25)–(27).

Let U�Nðx0Þ ¼ ½ðu�0Þ

T ðu�N�1Þ

T�T denote the

optimizer of the problem (25)–(27). We recall themain property enjoyed by U�

N. For more details see(Borrelli 2003, Baotic 2005).

Theorem 1: The solution to the optimal control problem(25)–(27) is a time-varying PWA state feedback controllaw of the form

u�i ðxiÞ ¼ Fi, jxi þ Gi, j if xi 2 Ri, j ð28Þ

where Ri, j, j ¼ 1, . . . ,Ni is a partition of the set X i offeasible states xi and the closure �Ri, j of the sets Ri, j hasthe following form

�Ri, j :¼ x j xTLi, j, qxþMi, j, qx � Ni, j, q, q ¼ 1, . . . , ni, j� �

,

i ¼ 0, . . . ,N� 1: ð29Þ

As shown in Borrelli (2003), the problem (25)–(27) canbe solved via dynamic programming

J�i ðxÞ :¼ minu

kQðx� xeÞk2 þ kRðu� ueÞk2

þ J�iþ1~f PWAðx, uÞ

� subj. to ~f PWAðx, uÞ 2 X iþ1

ð30Þ

for i ¼ N� 1, . . . , 0, with the boundary conditions

XN ¼ X f, ð31Þ

J�NðxÞ ¼ kPðx� xeÞk2, ð32Þ

where X i is the set of all initial states for which problem(30) is feasible

X i ¼ x 2 Rnj 9u, ~f PWAðx, uÞ 2 X iþ1

n o: ð33Þ

The dynamic program (30)–(32) can be solved off-line,backwards in time, by using a multi-parametricquadratic programming solver. The solution (28) isfurther simplified by using basic polyhedral manipula-tions and quadratic cost function comparisons overpolyhedral sets (Borrelli 2003, Baotic 2005).

In the on-line implementation receding horizon con-trol (RHC) is used, where at each sampling instant kthe solution to the optimal control problem (25)–(27)

Constrained optimal control of an electronic throttle 471

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with x0 xk is computed, and only the first controlmove u�0 is applied to the process while the rest ofthe control sequence U�

NðxkÞ is discarded. Since RHCis used, only the control law u�0ðxÞ consisting of gainsF0, j, G0, j and regions R0, j, j ¼ 1, . . . ,N0 needs to bestored. The control action computation is reduced toa simple look-up table evaluation that is implementableeven if the sampling time is in the range of milliseconds(as is typically the case in the automotive applications).Unfortunately, the complexity of such a control law

significantly rises as the PWA model becomes morecomplex (larger values of s and n that are neededfor the more accurate process modelling), and as theprediction horizon N increases.

3.2 CTOC strategy

The off-line and on-line computational complexity ofthe constrained optimal controller can be reduced ifa constrained time-optimal control (CTOC) strategy isadopted. In the CTOC formulation all constraintsremain fulfilled while a simple, constant cost-to-goterm is used – the number of time-instants needed forthe system state to reach a pre-defined invariant set X I

(for more details see Rakovic et al. (2004) and Griederet al. (2005)). The invariant set X I together with anasymptotically stabilizing control law uIðxÞ on it isdefined as follows

X I ¼ x0j xi 2 X I, xiþ1 ¼~f PWAðxi, u

IðxiÞÞ, 8i > 0n o

:

ð34Þ

The invariant set is computed in two steps. First a PWAcontrol law uIðxÞ is found, e.g., with the linear matrixinequalities (LMI) approach presented in Grieder et al.(2005), and then the set X I is constructed using theiterative procedure described in Rakovic et al. (2004).Once X I is found, the optimal control problem forminimizing the number Nt of time-instants needed fora state to enter the invariant set is solved

Jt�ðxNt Þ ¼ minUNt ,Nt

Nt

subj. to

xi�1 ¼~fPWAðxi, uiÞ

x0 2 X I

1 � i � Nt,

8><>:

9>>>>>=>>>>>;

ð35Þ

where UtNt ¼ ½ðutNtÞ

T ðut1Þ

T�T. The control problem

(35) is transformed into a sequence of reachabilityproblems

find all x 2 Rn, Jt�i ðÞ, u

tiðÞ such that

~f PWAðx, utiðxÞÞ 2 X ti�1 ð36Þ

for i ¼ 1, 2, . . . ,Nmax, with Jt�i ðxÞ :¼ 1þ Jt�i�1ðxÞ and theboundary conditions

X t0 ¼ X I, ð37Þ

X tNmaxþ1 ¼ ;, ð38Þ

Jt�0 ðxÞ ¼ 0, ð39Þ

where X ti is the so-called i-cost-to-go set

X ti :¼ x 2 R

nj 9u, ~f PWAðx, uÞ 2 X t

i�1

n on [i�1

j¼0 Xtj : ð40Þ

Reachability problems (36) are solved by using multi-parametric programming and polyhedral manipulations(there is no need for quadratic functions comparisons).In general, the optimal sequence Ut

Nt (and thus theoptimal control law utNt ) is not unique for a given x0.It seems that the simplest control law among the optimalones is the one that minimizes the switchings betweenaffine dynamics during the predicted transient outsideX I. In Vasak et al. (2005) a procedure has beenproposed to obtain the time-optimal control law withminimum switchings between dynamics. The resultingcontrol law is PWA over polytopes, drastically simplerthan the CFTOC law, and thus implementable fora broader range of PWA systems. Moreover, thepolytopes are within each X t

i sorted by affine dynamicsof ~f PWA that generated them. On-line computationof the control action is very simple. For a given x wedefine the set of active dynamics

AðxÞ ¼ ij1 � i � s, x 2 Dxi

� �, ð41Þ

where Dxi denotes the projection of Di on the X space.

Starting from an invariant (zero-cost-to-go) set towardsthe higher cost-to-go sets we search for a first polytopegenerated by some dynamics in AðxÞ that contains thestate x and compute the control input u by using theaffine control law corresponding to that polytope.

4. Experimental results

Here we report experimental results obtained on theelectronic throttle system with the optimal controlstrategies described in x 3.

4.1 Regulator problem

For the regulator problem, implemented mainly to testthe applicability of the RHC strategy to the electronicthrottle, we use the PWA-1 model. The state vectorfor this model is x ¼ ½!�

m ��T, the input is thecontrol voltage u, and the output y ¼ �. The sampling

472 M. Vasak et al.

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time is T ¼ 5 ms. Maximum possible absolute value ofthe control voltage u is 5 V. Unlike the classic controllerdesigns, where some ad-hoc strategies such asanti-windup are used to limit the controller output,the RHC strategy naturally incorporates the physicalconstraint u 2 �5, 5½ � in the control problem formu-lation. To guarantee the safety of the system and toprevent excessive throttle components wear, additionalconstraints are introduced along the prediction horizon:ia 2 ½�2, 2�, !m 2 ½�100, 100� – both to protect theDC motor, and � 2 ½13, 90� – to prevent the plate hittingthe mechanical stops (Visteon 2000). All theseconstraints are stacked into the matrices in (22) andthus included in the synthesis procedure.We design a receding horizon controller by solving

the CFTOC problem (25)–(27) with

N ¼ 5, Q ¼0:001 0

0 100

� �, R ¼ 0:01, P ¼ Q,

X f ¼ ½�0:2 24:95�T � ½0:2 25:05�T,

xe ¼ !em �e

�T¼ ½0 25�T, ue ¼ 1:21:

The corresponding optimal solution comprising 1190regions in !�

m-� space depicted in figure 6 was computedwith the MPT toolbox (Kvasnica et al. 2003).The optimal controller is verified in the experimental

throttle system setup. The control algorithm is runwithin Matlab Real-Time Workshop�. Since bothCFTOC and CTOC are full-state feedback controlstrategies, the system states that are not measuredmust be estimated on-line before the control actioncan be computed. In our application the only measured

state is �, while !m and mf (needed only if PWA-2 modelis used for controller synthesis) are not available.Because the system model is known we use a (PWA)model-based state estimation procedure. Between thetwo most commonly used non-linear-model-based stateestimators: extended kalman filter (EKF) and unscentedkalman filter (UKF) (see Wan and van der Merwe2001), after extensive testing we have chosen UKF dueto its superior performance (Vasak et al. 2003)compared to the EKF (UKF was able to cope wellwith the hard and almost discontinuous non-linearitiespresent in our system).

A typical control system response is presented infigure 7, where !m and � denote the estimated valuesof the states !m and �, respectively. Variable !diff

m iscalculated by time differentiation of � and is used toverify the performance of the UKF estimator. Thevalve plate is moved to the starting position (ca. 18:4�,below �LH) and then at t ¼ 4:5 s the regulator designedfor the set-point �e ¼ 25� is connected to the process.

Although in our experimental setup the armaturecurrent measurement is also available we do not use itin the control algorithm since such sensors are notpart of the standard electronic throttle setup in cars.However, we use this sensor in our experiments tocheck if the control system respects the current limit.One can observe in figure 7 a slight armature currentconstraint violation. This can happen since the currentconstraint is constructed using u and !m with fixedprocess parameters in (2) that may differ from theactual ones on the throttle due to the change ofcar-battery voltage and armature resistance. In orderto guarantee the constraint satisfaction in the presenceof such a model uncertainty a robust MPC techniqueshould be used.

In figure 7 we also see stick-slip oscillations in theangle response, which are caused by the overcompensa-tion of the friction. Namely, the controller based onthe static Karnopp friction model has no informationon the presliding effect and it tends to overcompensatethe friction torque in the presliding mode.

4.2 Reference tracking

To model friction well even for small valve openingangle changes, the dynamic friction model is used, asdetailed in x 2.2. We use the PWA-2 model withx ¼ ½!m � mf�

T. The sampling time T is again 5 ms.For numerical reasons the variable mf (jmfj � 0:007)was multiplied by 10000 to put it in a range similarto the other two state variables. After verifying thatthe electronic throttle regulator using CFTOC basedon the PWA-2 model does not exhibit the stick-slipbehaviour (Vasak et al. 2004), we have extended the

−150 −100 −50 0 50 100 15016

18

20

22

242526

28

30

32

34q

[°] q e

wem

[ rads ]w*

m

Figure 6. Regulation using CFTOC based on the Karnoppfriction model. Control law partition of the state space.

Constrained optimal control of an electronic throttle 473

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RHC approach to the reference tracking problem(Vasak et al. 2005).The optimal reference tracking problem for the

throttle is defined in the augmented stateþ input space

�xk ¼

xk

uk�1

rk

264

375, ð42Þ

�uk ¼ uk � uk�1, ð43Þ

where rk 2 R denotes the reference that the systemoutput yk should follow, and �uk is the change of acontroller action (which is, conveniently, equal to zeroat steady-state for any value of the reference). Theextended state-space is �X � R

5. The output �yk of suchan extended system is the tracking error itself

�yk ¼ rk � yk: ð44Þ

The extended PWA system can be constructed from (1)by using equations (42)–(44) and rk ¼ const:

�xkþ1 ¼ �Ai �xk þ �Bi �uk þ �fi,

�yk ¼ �Ci �xk

if ½ �xk �uk�T2 �Di, i ¼ 1, . . . , s:

ð45Þ

Constraints (22) are already included in the descriptionof the regions �Di. In the following we will denote the

state-update equation in (45) with the shorter form

�xkþ1 ¼�f PWAð �xk, �ukÞ: ð46Þ

Since the CFTOC algorithm used with such a complexPWA model in 5D results in far too complex off-linecomputations when a reasonable prediction horizon Nis used, that would result in a too complex optimalcontrol law, the CTOC problem was solved instead,resulting in an implementable time-optimal control law.

The constraints on the system variables are the sameas those for the regulator problem, except the constrainton !m, that is relaxed compared to the CFTOC caseto ensure faster tracking: !m 2 ½�150, 150�. Since theextended model input is �u, we use the constraint�u 2 ½�5, 5� to prevent excessive chattering of the controlsignal.

The time-optimal control law for the first three cost-to-go sets around the invariant set was computed withthe MPT toolbox (Kvasnica et al. 2003). It comprises2486 distinct affine control laws defined over 11639polyhedral regions (some regions have the same controllaw) in R

5. To represent polyhedral partition we need tostore 21256 distinct hyperplanes in R

5 and an index listof 131103 integers that matches constraints of everyregion in the polyhedral partition to the appropriatehyperplanes. Assuming 2 bytes are used for the signedinteger and 4 bytes for the float a straightforwardcomputation shows that to store the time-optimalcontrol law we need ð131103þ 11639Þ 2þ ð2486þ21256Þ 6 4 ¼ 855292 bytes of RAM. Since many

4.5 4.55 4.6 4.65 4.7 4.75 4.8 4.85

4.5 4.55 4.6 4.65 4.7 4.75 4.8 4.85

4.5 4.55 4.6 4.65 4.7 4.75 4.8 4.85

−2

0

2

4

−100

−50

0

50

100

150

18

20

22

242526

28

u[V

], i a

[A] RHC

ia limit

wm limit

ia

u

wm[ra

d s] wm

diff

wm

q [°

]

q e

qLH

qq

Figure 7. Experimental verification of the regulator based on the Karnopp friction model.

474 M. Vasak et al.

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regions have the same control law the memory require-ment for the storage of the controller could be furtherreduced by the optimal merging procedure describedin Geyer et al. (2004). Unfortunately, one should notexpect any merging procedure to reduce the memorydemand below 200 kB and requiring so much RAMfor the implementation of the control algorithm in acurrent car setup is a serious drawback.One should not confuse the high memory requirement

(i.e. the existence of 11639 polyhedral regions) withthe speed of the computation of the control action.Namely, as explained in subsection 3.2, in the on-lineimplementation we first search for the active dynamics(corresponding to 30 simple polytope checks), andthen search only through regions corresponding to theactive dynamics. As the controller regions are moreor less equally distributed per all dynamics, and sincein our case no more than two dynamics can be activesimultaneously, in the worst case we must check about11639=15 � 800 polytopes in R

5 to find the one thatcontains given �x. In our experimental setup the com-putation was carried out without problems within 5ms sampling time period with such a simple search.If a small computation time is crucial, one can usemore efficient computation algorithm from Tøndelet al. (2003) based on the binary search tree explorationstrategy. At the expense of additional off-line computa-tions and somewhat larger memory demand thealgorithm in Tøndel et al. (2003) finds the correspondingregion index in Oðlog2 NrÞ time, where Nr is the numberof regions (Nr¼ 800 in our case). Therefore, for a givenmeasurement of the state vector one can obtain optimalcontrol action very quickly.The time-optimal controller is tested on the same

experimental throttle setup as in the previous section.This time, however, due to the high RAM requirementswe use a computer system running Real-Time Linux�.The on-line control scheme is shown in figure 8.To eliminate an overshoot in the angle response and

to reduce the number of cost-to-go sets around theinvariant set needed for the on-line computation, aprefilter with 15ms time constant is introduced in thereference path. Namely, since only three cost-to-gosets around the invariant set are used in the on-linecontrol input computation for the abrupt referencechanges the extended state could fall outside the spacecovered with those sets. The step response settling timeof the applied prefilter is less than 50ms, and its useis thus justifiable since the car manufacturers demandthe throttle control system step response settling timeto not exceed 100ms (Deur et al. 2004).In the subsequent figures we report experimental

results obtained using CTOC strategy for referencetracking. Ramp reference following presented infigure 9 shows the capability of the controller to cope

both with the LH non-linearity (figure 9b) and thefriction (figure 9c). In figure 10 we report the systemresponse to a 2� square reference. In both step changedirections the settling time within the measurementresolution amounts approximately 30ms (figure 10band 10c), while the responses show no overshoot.

The obtained control system settling time is more thantwo times smaller compared to the one of the throttlecontrol system that comprises a tuned PID controllerand a feedforward compensation of the LH and frictionnon-linearities (Deur et al. 2004). From the applicationpoint of view considering idle-speed engine control,a very important reference is the 0:2� square reference,that should be precisely followed, although its changelies in the presliding displacement range and is only50% larger of the measurement resolution. The controlsystem response to such a reference is shown infigure 11. Two enlarged details of the response shownin figure 11(b) and (c) are pointed out to show thatthe settling time is again at about 30ms while theresponse again has no overshoot. In figure 12 we showthe control system response to the 1� stairs referencefor a longer time period.

5. Conclusion

Constrained optimal control of an electronic throttleprocess is considered. Two strong process non-linearities(friction and a so-called ‘‘limp-home’’ non-linearity)are approximated in a piecewise affine (PWA) formand a discrete-time PWA model of the throttle isderived. We address the issue of the PWA frictionmodelling in more detail by considering both staticand dynamic friction models. For the electronic throttleapplication where a high accuracy is needed a dynamicfriction model has to be used since it can capture thepresliding behaviour of the friction.

The PWA model of the throttle is used for themodel predictive controller synthesis based onhybrid systems theory. Two different control strategies

r rf

Look-uptable

UKF

(1−af)z−1

1−afz−1

Experimentalsetup

Real-Time Linuxcontrol computer

mf

wm q

q

z−1

++

uk uk

uk−1

¯Electronic

throttleAcquisition card

Figure 8. On-line control scheme.

Constrained optimal control of an electronic throttle 475

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are considered: constrained finite time optimalcontrol (CFTOC) used in the regulator case, andconstrained time-optimal control (CTOC) used inthe reference tracking case. All physical limitationsof process variables and constraints introduced bythe designer that prevent excessive wear of thethrottle components are naturally incorporated inthe control problem formulations. Since the samplingtime needed for the throttle process falls in the range

of milliseconds, an on-line control action comput-ation involving integer programming is prohibitive.In this paper the optimal control law is pre-computedin an off-line procedure for a range of processstates by using a dynamic programming strategy incombination with the multi-parametric programmingsolver and basic polyhedral manipulations. Theoptimal control law may require substantialstorage space but it has the form of a look-up table

7 8 9 10 11 12 13 14 15

7 8 9 10 11 12 13 14 15

7 8 9 10 11 12 13 14 15

7 8 9 10 11 12 13 14 15

−2

0

2

−20−10

01020

202224

−200

0

200

u [V

]w

m[ra

d s]

q [°

]

t [s]

qqrf

qLH

detail in 9(b)

detail in 9(c)

(a) (b)

(c)

8.1 8.6 9.1

8.1 8.6 9.1

−2

0

2

19

20

21

u [V

]u

[V]

q [°

]q

[°]

qLH

t [s]

t [s]

13 13.5 14

13 13.5 14

1

2

25

24.5

mf [1

0−4N

m]

ˆ

Figure 9. CTOC reference tracking system with the reset-integrator friction model: (a) Response to the ramp reference throughthe LH region; (b) A detail of the response around �LH; (c) A detail of the response in case of abrupt ramp slope change.

23.5 24 24.5 25 25.5 26 26.5 27 27.5

23.5 24 24.5 25 25.5 26 26.5 27 27.5

23.5 24 24.5 25 25.5 26 26.5 27 27.5

23.5 24 24.5 25 25.5 26 26.5 27 27.5

−10123

−50

0

50

2323.5

2424.5

25

−200

0

200

u [V

]w

m[ra

d s]

q [°

]

t [s]

qqrf

detail in 10(b) detail in 10(c)

(a) (b)

25.1 25.2 25.3

25.1 25.2 25.3

123

23

25

24

u [V

]

(c)

u [V

]q

[°]

q [°

]

t [s]

t [s]

26.1 26.2 26.3

26.1 26.2 26.3

0

1

2

23

25

24

mf [

10−4

Nm

]ˆ

Figure 10. CTOC reference tracking system with the reset-integrator friction model: (a) Response to the 2� step reference change;(b) A detail of the response to the positive step change; (c) A detail of the response to the negative step change.

476 M. Vasak et al.

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and is therefore easily implementable on cheap

hardware.We report experimental results of both the regulator

and the tracking controllers. The reference tracking

controller obtained by solving the CTOC problem

outperforms the classical PID controller with a

feedforward compensation of non-linearities with

respect to the speed of the response, while preserving

the other important performance measures, like the

absence of overshoot and static accuracy within the

measurement resolution. The controller synthesisdescribed in this paper shows prospects for applicationto other automotive control systems and their furtherimprovement.

Acknowledgments

We wish to thank to Danijel Pavkovic for sharing hisknowledge about the throttle, to Miroslav Baric for

13.5 14 14.5 15 15.5 16 16.5 17 17.5

13.5 14 14.5 15 15.5 16 16.5 17 17.5

13.5 14 14.5 15 15.5 16 16.5 17 17.5

13.5 14 14.5 15 15.5 16 16.5 17 17.5

−10−5

05

10

−100−50

050

100

22.923

23.123.223.3

t [s]

ˆ

detail in 11(b) detail in 11(c)

15.9 16.0 16.1

15.9 16.0 16.1

0.51

1.5

23.0

23.4

23.2

t [s]

16.9 17.0 17.1

16.9 17.0 17.1

1

2

23.0

23.4

23.2

t [s]

0

1

2u

[V](a)

wm[ra

d s]

q [°

]m

f [10

−4N

m]

ˆ

qqrf

(b)

u [V

]q

[°]

(c)

u [V

]q

[°]

Figure 11. CTOC reference tracking system with the reset-integrator friction model: (a) Response to the 0.2� step referencechange; (b) A detail of the response to the negative step change; (c) A detail of the response to the positive step change.

20 21 22 23 24 25 26 27 28 29 30

1

2

3

20 21 22 23 24 25 26 27 28 29 30

010

−10

203040

20 21 22 23 24 25 26 27 28 29 3036384042444648

20 21 22 23 24 25 26 27 28 29 30−100

0

50

100

u [V

]

−50

mf [1

0−4N

m]

q [°

]

qrf

t [s]

ˆw

m[ra

d s]

Figure 12. CTOC reference tracking for the reset-integrator friction model. Response to the 1� stairs reference.

Constrained optimal control of an electronic throttle 477

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help with the Real-Time Linux, and to Davor Hrovatand Josko Deur for helpful discussions regardingthe experimental results. This research was supportedby the Swiss National Science Foundation and theMinistry of Science, Education and Sports of theRepublic of Croatia.

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