Model Predictive Control of Engine Speed During Vehicle Deceleration

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IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY 1

Model Predictive Control of Engine SpeedDuring Vehicle Deceleration

Stefano Di Cairano, Member, IEEE, Jeff Doering, Ilya V. Kolmanovsky, Fellow, IEEE,and Davor Hrovat, Fellow, IEEE

Abstract— We consider the speed control of a spark ignitionengine during vehicle deceleration. When the torque converterbypass clutch is open, the engine speed needs to be keptclose to the turbine speed to guarantee responsiveness of thevehicle for subsequent accelerations. However, to maintain vehicledrivability, undesired crossing between engine speed and turbinespeed must not occur, despite the presence of significant torquedisturbances. Hence, the engine speed during vehicle decelera-tions needs to be precisely controlled by feedback control, whichhas to coordinate airflow and spark timing and enforce severalconstraints including engine stall avoidance, combustion stability,and actuator limits. We develop a model predictive controllerthat manipulates airflow and spark to track the reference signalfor engine speed while enforcing constraints, and synthesize itin the form of a feedback law. The controller is evaluatedin simulations and in a vehicle, and it is shown to achievea responsive and consistent deceleration and the potential forreducing fuel consumption.

Index Terms— Automotive control systems, constrained con-trol, engine control, model predictive control (MPC).

I. INTRODUCTION

IN today’s vehicles, the powertrain control system is beingcontinuously refined to improve fuel economy, emissions,

safety, and drivability, thus making the vehicles more eco-nomical, sustainable, safe, and fun to drive. Drivability is thecapability of the vehicle to be responsive and predictable to thedriver’s commands. While good drivability often goes unno-ticed by the driver, poor drivability may lead to low driver con-fidence in the vehicle. Examples of poor drivability include,among others, lack of responsiveness, undesired responses tothe driver commands, and/or excessive vehicle jerk. Thus, asignificant effort is devoted to refining the powertrain responseby appropriate control system design and calibration. Due tomultiple objectives and constraints, achieving good drivabilitycan be complex and time consuming.

For vehicles equipped with standard automatic transmis-sions, the drivability effects are noticeable during vehicle

Manuscript received June 16, 2013; revised November 20, 2013; acceptedFebruary 5, 2014. Manuscript received in final form March 2, 2014. Recom-mended by Associate Editor K. Butts.

S. Di Cairano was with Ford Research and Advanced Engineering, Dear-born, MI 48126 USA. He is now with the Mitsubishi Electric ResearchLaboratories, Cambridge, MA 02139 USA (e-mail: [email protected]).

J. Doering and D. Hrovat are with Ford Research and AdvancedEngineering, Dearborn, MI 48126 USA (e-mail: [email protected];[email protected]).

I. V. Kolmanovsky is with the Department of Aerospace Engineering, Uni-versity of Michigan, Ann Arbor, MI 48109 USA (e-mail: [email protected]).

Color versions of one or more of the figures in this paper are availableonline at http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TCST.2014.2309671

deceleration. When the vehicle is decelerating and the torqueconverter bypass clutch is open, the torque provided to thevehicle is related to the turbine–engine speed ratio [1]–[4],the ratio between the speeds of the engine and the torqueconverter turbine, which is connected to the gearbox inputshaft. To obtain a consistent behavior, to maintain vehicleresponsiveness, and to achieve high fuel economy, the enginespeed during deceleration has to be precisely controlled.In spark ignition (SI) engines, such control is achieved bymanipulating the airflow and the spark timing, which, however,are subject to several constraints, including limits imposed bycombustion stability and engine breathing.

Model predictive control (MPC) [5] systematicallyaddresses the problem of achieving multiple objectives whileenforcing constraints on vehicle inputs and states. Thus, it hasthe potential of significantly simplifying design and calibrationof multivariable control systems. Furthermore, the executionof MPC controllers at high rate and with low computingpower can be achieved using the explicit MPC feedbacklaw [6]. In this paper, we demonstrate the effectiveness ofMPC in engine speed control during vehicle decelerationfor vehicles equipped with standard automatic transmissions.MPC has been investigated for several automotive controlapplications, including engine control, driveline control,lateral vehicle dynamics control, and hybrid electric vehicleenergy management [7]–[14]. See also the survey in [15].Based on the authors’ previous experience with idle speedcontrol [16], MPC is a promising candidate for engine speedcontrol during deceleration (engine deceleration control, forshortness).

This paper is organized as follows.1 In Section II, wediscuss the behavior of the powertrain during deceleration,and we introduce the engine deceleration control problem. InSection III, we describe the physical model of the relevantdynamics during deceleration. Since the model is nonlinear, acontrol-oriented reparameterization is proposed. In Section IV,we design the MPC deceleration controller, which is synthe-sized as a feedback law for implementation in the enginecontrol unit (ECU). In Section V, the controller is evaluatedin simulations with disturbances and setpoint values recordedfrom real vehicle maneuvers. Then, in Section VI, experi-mental tests spanning nominal and nonnominal decelerationconditions are reported. The conclusion is given in Section VII.

Notation: Real, positive real, nonnegative real, and integer,positive integer, nonnegative integer numbers are denoted by

1Preliminary studies related to this paper appeared in [17].

1063-6536 © 2014 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission.See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.


2 IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY

Fig. 1. Schematic diagram of the powertrain components involved in enginedeceleration control.

R, R+, R0+, Z, Z+, Z0+, respectively. For a continuous signala(t) sampled with period Ts , a(k), k ∈ Z0+, denotes the valueat the kth sampling instant a(k) = a(kTs) and a(i |k), i ∈ Z0+denotes the value a(k + i) predicted based on data at kTs . Fora vector a, [a]i is the i th component, and inequalities betweenvectors are intended as componentwise. In all figures, the unitsare those introduced for the corresponding variables in thispaper, unless otherwise indicated.

II. ENGINE SPEED CONTROL DURING

VEHICLE DECELERATION

Fig. 1 shows the schematic diagram of the powertrain ofa vehicle with standard automatic transmission. The engineand the transmission are connected through a torque converter,which shapes the torque transfer between the two. Withsome simplifications, the torque converter is composed of animpeller, rigidly connected to the crankshaft and thus rotatingat the same speed as the engine, and of a turbine, rigidlyconnected to the transmission input shaft, that connects thetorque converter to the gearbox. Turbine and impeller areimmersed in oil, which realizes a hydraulic coupling betweenthe two (the impeller spins the oil, which spins the turbine, andthe other way around), hence allowing torque transfer betweenengine and transmission through the speed difference.

The engine speed dynamics are described by

Je N = (Mbrk − Mld) (1)

where N [rpm] is the engine speed, Je [Nm/(rpm/s)] is thelumped inertia of engine, flywheel, impeller, and the oilrotating with it, Mbrk[Nm] is the engine brake torque (the netengine output torque), and Mld[Nm] is the load torque on theengine transmitted from the torque converter.

Neglecting the driveline dynamics, when the torque con-verter bypass clutch that directly connects the crankshaft tothe transmission shaft is engaged and the transmission is in-gear (i.e., not shifting), the engine speed and turbine speed(NT [rpm]) are equal, N = NT , and the turbine speed isdetermined by the wheel speed, NT = �fdr�gi Nw , whereNw[rpm] is the vehicle wheel speed, �gi , gi = 1, . . . , n� is thegear ratio, and �fdr is the final drive ratio. In these conditionsand assuming a flat road, the load on the engine is a functionof the wheel speed Mld = �fdr�gi fw(Nw), where fw is aquadratic function [18].

When the torque converter bypass clutch is open, the enginespeed and the turbine speed are in general not equal. In this

Fig. 2. Torque converter load on the engine Mld as a function of the enginespeed N and speed ratio ρTQC = NT /N . Segment with ρTQC = 1 andMld = 0 shown by the red (dash) line.

case, the load on the engine, i.e., on the crankshaft, Mld is thetorque transferred from the torque converter [1], [2], [4]

Mld = sign(N − NT ) · max{N, NT }2

K(

NTN

)2 , (2)

where K is a positive nonlinear function defined by thetorque converter mechanical and hydrodynamical design. Thisfunction has maximum at one, is monotonically increasingfor arguments smaller than one, and monotonically decreasing(although not symmetrically) for arguments larger than one.For the vehicle used for experimental validation, a plot ofthe torque converter load on the engine Mld as function ofthe engine speed N and the torque converter speed ratioρTQC = NT /N is shown in Fig. 2. The curve for ρTQC > 1 issignificantly different from the one for ρTQC < 1. This is dueto the mechanical design of the torque converter [3], whichmaximizes the power transfer from the engine to the wheels,and minimizes the transfer from the wheels to the engine.In particular, for speed ratios larger than one, the torqueconverter load changes are relatively small.

By (2), in the case of null external torques, when the enginespeed is greater than the transmission speed, the turbine tendsto accelerate and the impeller tends to decelerate, while whenthe engine speed is lower than the transmission speed, theopposite occurs. The hydraulic coupling introduces damping,hence smoothing the torque transfer from the engine to thedriveline, and avoiding excessive jerk due to torque variations.The bypass clutch allows bypassing the hydraulic couplingfor improved fuel economy, when damping is not needed (forinstance, in high gears).

A typical deceleration profile with brake application isshown in Fig. 3, where different phases are highlighted.In the first phase (a), at high speed and in high gear, thetorque converter bypass clutch is locked, so that N = NT .In this phase, usually, there is no closed-loop control of theengine speed, and fuel injection may be shutoff. When thevehicle speed reduces and lower gears are engaged, the torqueconverter bypass clutch is opened (b) to avoid excessive jerk,and hence to maintain drivability. When the torque converteris open, the crankshaft and the transmission input shaft are


DI CAIRANO et al.: MPC OF ENGINE SPEED DURING VEHICLE DECELERATION 3

Fig. 3. Example of deceleration profile. Upper plot: engine speed(black line) and turbine speed (red line). Lower plot: gear (black line) andvehicle speed (red line) in miles per hour (mph).

coupled through the oil in the torque converter, and theengine speed and the turbine speed become different. Initially,N < NT , so that by (2), torque is transferred from the vehicleto the crankshaft, which increases the deceleration. At lowgears and low vehicle speed (c), N > NT and, if the brakesare not applied, eventually the vehicle creeps. If the brakes areapplied, the vehicle speed and the turbine speed reach zero (d)and the engine speed is stabilized at idle.

When the torque converter is open (b − d), to provide goodresponse to a driver’s application of the accelerator pedal (theso-called tip-in) while ensuring smooth and predictable vehicledeceleration without penalizing fuel consumption, the enginespeed needs to be tightly controlled. If the engine speed istoo low prior to a tip-in, it is difficult to provide smooth andresponsive behavior since the engine has to be acceleratedabove transmission input shaft speed to provide positive torqueto the wheels. If the engine speed is too high prior to a tip-in,the engine consumes more fuel than needed while decelerating.In addition, the driveline has lash [19], and if the enginespeed varies above and below transmission input shaft speeddue to poor control, unintentional driveline lash crossingscausing nonuniform deceleration and noise, vibration, andharshness (NVH) occur. Oscillations due to lash crossingmay be felt by the driver as undesired accelerations thatcompromise the vehicle drivability. Thus, during deceleration,to provide fuel economy, powertrain responsiveness, and drivesmoothness, precise feedback control of the engine speed isneeded. Since the modification of the engine torque throughthe airflow is subject to delays and gas dynamics, a sparktorque reserve is created for feedback control by advancingthe spark timing from the optimal ignition timing. In this way,when disturbances occur, torque modifications can be achievedrapidly (in both directions) by manipulating the spark timing.However, the authority of the spark reserve is limited and itdepends on the current engine airflow.

A well-known speed control problem in automotive is idlespeed control (ISC) [20]. In ISC, the engine speed is regulatedto a constant setpoint using throttle (or bypass valve, in olderengines) and spark timing, while rejecting disturbances caused

Fig. 4. Architecture for engine speed control during deceleration.

for instance by air-conditioning (AC) and power steering pumpload. Since the setpoint remains mostly constant, ISC operatesin very specific, and almost stationary, engine operating con-ditions. The challenges in ISC are the time delay in the airpathand the reduced and varying authority of the spark actuation.Several control strategies have been proposed for ISC ( [20]and the references therein). Recently, Di Cairano et al. [16]have proposed an idle speed controller based on MPC thatwas experimentally shown to improve closed-loop robustnessand reduce the need for spark reserve, thus resulting in a moreefficient engine operation.

While similar in terms of manipulated and controlled vari-ables, engine speed control during deceleration is significantlymore challenging than ISC, since the controller operates acrossseveral engine operating conditions, i.e., speeds and loads. Thecontroller has to track various reference profiles generated bythe vehicle control system according to the type of deceleration(braking, coasting, creeping, etc.). Some reference profiles canbe rapidly changing (e.g., during braking), while others may bealmost constant (e.g., during creeping). In all these conditions,the controller must track the reference profile with small error(e.g., less than 50 rpm), to provide a smooth and predictabledeceleration (i.e., consistent in repeated maneuvers), and tomaintain vehicle responsiveness to subsequent accelerationswhile avoiding undesired lash crossing events. The enginedeceleration controller needs to counteract the disturbancesthat affect the idle controller. In addition, it has to regulatethe engine speed after a gear shift, to engage from transientconditions, and to respond to reference profiles changing withrates that may not be achievable.

The architecture for engine speed control during decelera-tion is shown in Fig. 4. The deceleration controller receives thepowertrain state (engine speed, torque, etc.) from the vehiclesensors and estimators, and the desired engine speed andconstraints from the vehicle control system. The controllercommands the base torque, the maximum achievable indicatedtorque for the current airflow, and the torque ratio, the percent-age of the base torque that is actually produced through sparkmodulation. The commanded base torque and the torque ratioare achieved by electronic throttle control and spark timingcontrol implemented in lower level control functions that mayexploit additional powertrain state variables, such as maximumbrake torque (MBT) ignition angle [21], engine temperature,



manifold pressure, and so on. By this architecture, the enginedeceleration control is decoupled from the actuator control,and hence the complexity of control design is reduced, andthe resulting control system is modular.

Remark 1: In today’s SI engines, the air-to-fuel ratio istightly controlled to stoichiometry to maintain efficiency ofthe three-way catalyst, which reduces emissions [22], and, asa consequence, it is not a manipulated variable for the enginedeceleration control. In addition, since it is tightly controlledto a constant value, the impact of air-to-fuel ratio fluctuationson the engine speed dynamics can be neglected [16].

The deceleration controller needs also to enforce constraintson state and control variables. The airflow cannot decreasebelow a minimum value, otherwise the combustion becomesunstable [21]. Similarly, an upper bound on the airflow hasto be enforced, due to engine breathing conditions. Both thesebounds vary significantly during deceleration. The torque ratioachieved by spark timing is also limited. Finally, the enginespeed must be maintained above a given threshold designedconservatively to avoid engine stalls [16].

The design of classical controllers (e.g., PID-based) thatachieve all these requirements can be cumbersome due tothe need for selecting an appropriate logic to coordinatethe control channels and enforce saturation and antiwindupfor dealing with the constraints. This may result in longdevelopment and calibration time and suboptimal performance.The MPC approach proposed in this paper allows to obtain acontroller that (optimally) coordinates the commands, whileenforcing constraints by design.

III. FURTHER POWERTRAIN MODELING DETAILS

Next, we model in greater details the powertrain dynamicsthat are relevant for engine speed control during deceleration.The engine brake torque in (1) is

Mbrk = Mind − Mls (3)

where Mind[Nm] is the engine indicated torque and Mls[Nm]are the torque losses due to pumping, friction, and accessoryload. Mls can be modeled as a complicated function ofengine speed, ambient pressure and intake manifold pressure(pamb[Pa], pman[Pa]), engine temperature (Teng[K]), and acces-sory load (Macc[Nm])

Mls = fls(N, Teng, pman, pamb) + Macc.

Assuming stoichiometric operation according to Remark 1, theindicated torque is related to the torque ratio, κspk ∈ [0, 1], andto the base torque, Mair[Nm], by

Mind = κspk Mair . (4)

The torque ratio is controlled by modulating spark timingσ [deg] (expressed as degrees with respect to top deadcenter) for the current engine speed and mass cylinder airflowwair[kg/s], κspk = fσ (σ,wair, N), where κspk = 1 for MBTspark timing (σ = σMBT(N, wair)). A commonly used formof fσ is discussed in [16], [23], and the references therein.By (1)–(4), we obtain

N = 1

Je(κspk Mair − Mls − Mld). (5)

Several variables in (5) are subject to operating constraints.The base torque is limited to the range

Mminair (t) ≤ Mair(t) ≤ Mmax

air (t) (6)

where the lower bound Mminair (t) varies with time, and is

imposed to avoid combustion instability (e.g., misfires) thatoccurs when too little air is trapped in the cylinder. In (6),the time-varying upper bound Mmax

air (t) reflects the maximumachievable torque, due to current engine operating conditionsand operational requirements in terms of emissions, drivability,and NVH. Both Mmin

air and Mmaxair mainly depend on engine

speed, load, and temperature. Similarly, the torque ratio islower and upper bounded due to emissions, combustion sta-bility, and physical limitations

0 ≤ κminspk ≤ κspk ≤ κmax

spk ≤ 1 (7)

where κminspk and κmax

spk are constant during deceleration.In the torque domain, the control variables for engine speed

regulation are the commanded base torque Mair[Nm] and thecommanded torque ratio κspk. The action of Mair , κspk isnot instantaneous due to the presence of dynamics and timedelays. The base torque generation process is subject to thecombustion time delay δair[s], which is usually assumed tobe 360 crank angle degrees, and hence its time duration2

depends on the engine speed, δair = 60/N . Furthermore, dueto manifold filling dynamics, the base torque generation issubject to a first-order lag with a time constant τair[s]. Thus,the dynamics from commanded to generated base torque are

Mair(t) = 1

τair(−Mair(t) + Mair(t − δair(t))). (8)

Remark 2: From physics, the torque dynamics are themanifold filling dynamics, a first-order lag, followed by acombustion time delay. Here, by linearity, we use an equivalentmodel with input time delay.

The effect of the torque ratio actuation via spark is alsosubject to a delay due to the discrete firing of the cylinders,which, for a six-cylinder vehicle, is 120 crank angle degrees,i.e., δspk = 60/3N . Hence

κspk(t) = κspk(t − δspk). (9)

By collecting (2) and (5)–(9), the complete dynamicalmodel is

N(t) = 1

Je(κspk(t − δspk)Mair(t)

− Mls(t) − Mld(t)) (10a)

Mair(t) = 1

τair(−Mair(t) + Mair(t − δair(t))) (10b)

Mld(t) = sign(N − NT ) · max{N, NT }2

K(

NTN

)2 (10c)

Mminair (t) ≤ Mair(t) ≤ Mmax

air (t) (10d)

κminspk ≤ κspk(t) ≤ κmax

spk (10e)

2Alternatively, crank angle can be used as independent variable, resultingin a constant delay model. However, other aspects of the dynamics becomesignificantly more complicated [24].



where the state variables are the engine speed N , the basetorque Mair , and the past commands for base torque andspark torque ratio along the time delay windows, Mair(τ ),τ ∈ [t−δair, t), and κspk(τ ), τ ∈ [t−δspk, t); the control inputsare the commanded base torque Mair and the commandedspark torque ratio κspk; and the measured disturbances are theturbine speed NT and the torque losses Mls. In (10), the torquelosses dynamics have been neglected, since their changes aresmall and/or slow in the operating conditions of interest.

Remark 3: The powertrain dynamics in (10) are connectedto the longitudinal vehicle dynamics. A simple longitudinalvehicle dynamics model is

Jv Nw = �gi �fdr MT − Mrl (11)

where Jv [Nm/(rpm/s)] is the vehicle inertia at the wheels,MT [Nm] is the output torque of the torque converter, and Mrlis the road load. For zero road grade, the road load is a functionof the vehicle speed only, Mrl = fw(Nw). Based on (2), thetorque converter output torque is a function of the engine andturbine speed, MT = fis (N, NT ). When the transmission isin-gear, Nw = NT /(�gi � f dr ). By substituting into (11)

Jv Nw = �gi �fdr fis (N, �gi � f dr Nw) − fw(Nw) (12)

and hence the vehicle speed depends on the engine speed. Dueto the torque converter design [3], when the engine speed isbelow the turbine speed, the torque transmitted through theconverter is small and does not change significantly with thespeed ratio (Fig. 3). Hence, during deceleration, the enginespeed can be modulated with limited impact on the vehicledeceleration profile. However, precise engine speed control isneeded to maintain high fuel economy and vehicle respon-siveness while avoiding lash crossings, which cause NVH andreduced drivability.

A. Control-Oriented Parameterization

It is difficult to use (10) for MPC design since it is nonlinearand subject to (variable) time delays. A reparameterization canbe performed to remove the nonlinearity as described by thefollowing proposition.

Proposition 1: Consider

z(t) = x(t)u(t) (13a)

umin ≤ u(t) ≤ umax (13b)

where x, u, z ∈ R, and

ζ(t) = v(t) (14a)

uminx(t) ≤ v(t) ≤ umaxx(t) (14b)

where ζ, v ∈ R. Let x : [t0, t f ] → R+. Then, for all u :[t0, t f ] → R satisfying (13b) for all t ∈ [t0, t f ], there existsa unique v : [t0, t f ] → R satisfying (14b) for all t ∈ [t0, t f ]such that ζ(t) = z(t) for all t ∈ [t0, t f ].

The proof of Proposition 1 is immediate. One obtains (14)from (13) by introducing v ∈ R as v(t) = u(t)x(t). Theresult follows from the fact that x(t) > 0 for all t ∈ [t0, t f ],and hence the transformation is invertible. Note also that whenx(t) = 0, v(t) = 0, but still ζ(t) = z(t) independently of u(t).

In (14), we have reformulated a bilinear relation subjectto bounds, as a linear relation subject to linear constraints.For applying the result in Proposition 1 to (10), we noticethat Mair(t) > 0 for all t , and define a nominal torqueratio κspk

κspk(t) = κspk + κspk(t) (15a)

κminspk ≤ κspk(t) ≤ κmax

spk (15b)

where κminspk = κmin

spk − κspk and κmaxspk = κmax

spk − κspk. Bysubstituting (15) into (10a)

N (t) = 1

Je(κspk Mair(t) + Mspk(t) − Mls(t) − Mld(t))

where κspk is the indicated torque obtained from the currentbase torque and the nominal torque ratio and Mspk(t) =κspk(t)Mair(t) is the additional torque obtained by applyinga torque ratio different from nominal. The box constraintson torque ratio difference from nominal (15b) become linearconstraints between torque ratio and base torque

κminspk Mair(t) ≤ Mspk(t) ≤ κmax

spk Mair(t). (16)

In the new parameterization, the control variables arethe commanded engine indicated torque for spark at nomi-nal value, uair, and the commanded additional torque, uspk,from modulating spark timing around nominal value. Thus,from (10), (15), and (16), we obtain

N(t) = 1

Je(κspk Mair(t) + uspk(t − δspk)

− Mls(t) − Mld(t)) (17a)

Mair(t) = 1

τair(−Mair(t) + uair(t − δair(t))) (17b)

Mld(t) = max{N(t), NT (t)}2sign(N(t)−NT (t))

K(

NT (t)N(t)

)2 (17c)

Mminair (t) ≤ Mair(t)≤ Mmax

air (t) (17d)

κminMair(t) ≤ uspk(t−δspk)≤κmax Mair(t) (17e)

which is a constrained system subject to linear constraintswith controls uair, uspk, states N , Mair, and past commandsalong the delay periods, uair(τ ), τ ∈ [t − δair, t), uspk(τ ),τ ∈ [t − δspk, t), and measured disturbances Mls, NT .

Remark 4: For most of the control strategies, (14) doesnot simplify the design, because constraints involving multiplestates (or states and inputs) are difficult to enforce. However,MPC has the capability of enforcing such constraints bydesign, and hence it can control the bound constrained bilinearrelation (13) though controlling the linearly constrained linearrelation (14).

IV. MPC DESIGN

The advantage of (17) with respect to (10) is that, exceptfor (17c), the dynamics are linear, subject to linear constraintsand time delays. As a result, (17) can be used in a linearquadratic MPC algorithm.



A. Prediction Model Formulation

To simplify the computations, we remove (17c) and con-sider Mld, which is estimated by the vehicle control systemvia (17c), as a measured disturbance. The variable delaysin (17b) are approximated as constant multiples of the sam-pling period Ts , i.e., δair(t) = Ts δair, for all t ≥ 0, andδair ∈ Z0+. Similarly, the fixed spark-timing delay is approx-imated as a multiple of the sampling period, δspk = Ts δspk.Although better results can be achieved by gain schedulingmultiple controllers based on different delay durations, wehave verified that, during deceleration, this approximation doesnot significantly limit the performance. On the other hand,assuming that constant delays reduces the amount of storagememory needed in the ECU.

Under these simplifications, (17) sampled with period Ts

results in the discrete-time linear dynamics

x(k + 1) = Ax(k) + Bv(k) + w(k) (18)

where x = [N Mair]′, v(k) = [uair(k − δair)) uspk(k − δspk)]′,w = [Mld Mls]′. The model of the input time delay is

xδ(k + 1) = Aδxδ(k) + Bδu(k) (19a)

v(k) = Cδxδ(k) (19b)

where xδ is of the size of the number of delays in the system,in this case xδ ∈ R

δspk+δair , u = [uair uspk]′ ∈ R2, and Aδ, Bδ,

Cδ implement a delay buffer [16].The value of the measured disturbances is known/estimated

by the vehicle control system, but their dynamics are notknown. Hence, they are modeled as constant in prediction

w(k + 1) = w(k). (20)

This approximation is based on the assumption that themeasured disturbances changes are small and/or slow withrespect to the other variables. The receding horizon opera-tion of the MPC controller further compensates for such anapproximation. At any control cycle, the value of the measureddisturbances is acquired and used to initialize the constantprediction model (20) for prediction along the MPC horizon.At the subsequent step, the MPC controller acquires the newvalue of the measured disturbances and adjusts the controlsequence based on such an updated value.

Thus, the prediction model from (18)–(20) is

x(k + 1) = A f x f (k) + B f u f (k) (21)

where x f ∈ R4+δspk+δair , x f = [x ′ x ′

δ w′]′, and u f = u ∈ R2.

To obtain the full prediction model for MPC design, weneed to further extend (21) to enforce the control designspecifications for the engine deceleration controller.

For a constant reference and constant load, the spark timingshould return to the nominal value at steady state to maintaina torque reserve against future disturbances. Thus, the sparktorque Mspk should decrease to zero, while the base torqueMair should converge to an (unknown) steady state. As aconsequence, we formulate uair in incremental form, and add

an integrator on uspk, to be used in the cost function

ξu(k + 1) = ξu(k) + Tsuspk(k) (22a)

xu(k + 1) = xu(k) + uair(k) (22b)

uair(k) = xu(k) + uair(k) (22c)

where uair is the input variation from the previous step andξu is the integral of the spark torque command. Removinguair(k) in (22c) models an additional step of delay on thebase torque.

Since we want to obtain a controller that rejects constantdisturbances and achieves offset-free tracking of constantreferences, we add a model for the engine speed referencer [rpm] and an integrator on the tracking error

r(k + 1) = r(k) + Tsr(k) (23a)

r(k + 1) = r(k) (23b)

ξr (k + 1) = ξr (k) + Ts(N(k) − r(k)) (23c)

where r [rpm/s] is the gradient of the reference, which isknown in this application and assumed constant in prediction,and ξr [rpm s] is the integral of the tracking error.

Additional ancillary equations relates to the torque upperand lower bounds in (17d). These values change dependingon the engine operating conditions, and even if their currentvalues are known, their future values are difficult to predict.Thus, we treat them as constant in prediction

Mminair (k + 1) = Mmin

air (k) (24a)

Mmaxair r(k + 1) = Mmax

air (k). (24b)

As for the measured disturbances (20), the bounds in (24) areupdated at every control cycle with values computed by thevehicle control system, resulting in time-varying base torquebounds.

Equations (22)–(24) together with (21) define the full pre-diction model used in MPC

x p(k + 1) = A px(k) + Bpu p(k) (25)

where x p = [x ′f x ′

u r r ξu ξr Mminair Mmax

air ]′, x p ∈R

11+δspk+δair .

B. Cost Function and Constraints

The MPC engine deceleration controller is calibrated byappropriately shaping the cost function. The control objectiveis to track the engine speed reference with zero steady-state offset for constant references and constant disturbances.In addition, at steady state, the spark timing has to return toits setpoint. These objectives are encoded in the cost function

J =h−1∑i=0

(N(i) − r(i))2 + ρrξr (i)2 + ρuξu(i)2

+ ρs M2spk + ρau2

air (26)

where h ∈ Z+ is the MPC prediction horizon, i.e., the timeinterval along which the cost function is integrated based onthe prediction of the system dynamics, and ρr , ρu , ρs , and ρa

are positive weighting coefficients.



The remaining operating specifications, combustion stabil-ity, limits on base torque and spark timing, and minimumengine speed are enforced by the constraints

yp(k) = Cx p(k) (27a)

ymin ≤ yp(k) ≤ ymax (27b)

where [yp]1 = Mair − Mmaxair and [yp]2 = Mair − Mmin

airenforce (17d), [yp]3 = [Cδxδ]2 − κmax Mair and [yp]4 =[Cδxδ]2 − κminMair enforce (17e), and [yp]5 = N enforcesN ≥ N to avoid engine stalls. In (27b), upper and lowerbounds are assumed plus and minus infinity, respectively, forunbounded quantities.

C. Controller Design and Synthesis

The MPC finite-horizon optimal control problem is formu-lated based on (25)–(27)

minUp(k)

h−1∑i=0

x p(i |k)′Qx p(i |k) + u p(i |k)′Ru p(i |k) (28a)

s.t. x p(i + 1|k) = A px p(i |k) + Bpu p(i |k) (28b)

yp(i + 1|k) = Cx p(i |k) (28c)

ymini ≤ yp(i |k) ≤ ymax

i , i = δspk, . . . , hc + δair (28d)

u p(i |k) = 0, i = hu, . . . , h − 1 (28e)

x p(0|k) = x p(k) (28f)

where Up(k) = (u p(0|k), . . . , u p(h − 1|k)), and hu ∈ Z+is the control horizon, the number of free control movesavailable for the controller along the prediction horizon.In (28d), the bounds change depending on the time step togeneralize the constraint horizon for system with multipletime delays. It is not possible to enforce the constraints fromthe beginning of the horizon, since the controller cannotmodify the system response in an interval shorter than theshortest input time delay. Hence, after defining hc ∈ Z+,we enforce the constraints on minimum and maximum basetorque ([yp]1(i |k), [yp]2(i |k)) for i = δair, . . . , δair + hc, theconstraints on minimum and maximum torque from sparkmodulation ([yp]3(i |k), [yp]4(i |k)) for i = δspk, . . . , δspk +hc,and the constraint on engine speed [yp]5(i |k) fori = δspk, . . . , δair + hc, where δspk < δair, according toSection III. In (28), we set [ymax

i ] j = −[ymini ] j = ∞ for the

steps i when the constraints are not enforced.At every step k ∈ Z0+, the MPC algorithm executes

the following: 1) builds the current state vector x p(k) fromthe current estimated and measured signals; 2) solves finite-horizon optimal control problem (28) initialized from x p(k);and 3) computes the command for the base torque and thetorque ratio from the optimal input sequence U∗

p(k) and thecurrent state x p(k) by

κspk(k) = u∗spk(0|k)

Mair(k + δspk)(29a)

Mair(k) = xu(k) + u∗air(k). (29b)

In (29a), the correct computation of the spark ratio commandrequires the prediction of the base torque to compensate for

the delay in the spark control channel. Since δspk < δair, Mair(k + δspk) in (29a) can be computed from the current state andpast inputs by integrating the system dynamics.

To meet the stringent chronometric and memory require-ments of automotive ECUs, we use multiparametric program-ming [6] to synthesize the explicit MPC feedback law as thepiecewise affine function

U∗p(k) = Fj (k)x p(k) + G j (k) (30a)

j (k) ∈ {1, . . . , s} : H j (k)x p(k) ≤ K j (k) (30b)

where (30b) defines the polyhedral region partitions, s isthe total number of regions, and (30a) assigns an affineexpression to each region. The explicit synthesis of the MPCcontroller presents several advantages in terms of functionalityand integration process. Specifically, there is no need tosolve the optimization problem at every control cycle, butonly to evaluate (30). In addition, the worst case number ofoperations to execute per control cycle, as well as the requiredmemory, can be precisely computed, and hence feasibility ofthe implementation of the controller in a given computingplatform can be verified. Finally, a closed-loop system modelis obtained from (25) and (30)

x p(k + 1) = (A p + Bp�Fj )x p(k) + �G j (31a)

j : H j x p(k) ≤ K j (31b)

where � = [I 0], � ∈ R2×2h , selects the first command

of the optimal control sequence. System (31) is a piecewiseaffine system whose local and global stability and robustnessmargins can be analyzed using tools and methods available in[25] and [16].

V. CONTROLLER EVALUATION IN SIMULATIONS

The MPC controller designed in Section IV is implementedwith a sampling period Ts = 16 ms and horizons h p = 20,hc = 3, and hu = 3. Based on the intermediate speedN = 1200 rpm, the numbers of delay samples in torqueand spark actuation channels have been set to δair = 3and δspk = 1, i.e., 48 and 16 ms, respectively. We set thelower bound on the engine speed as N = 400 rpm. Thesampling period is the same as the one of the productioncontroller in the ECU, which is based on the tradeoff betweenperformance and chronometric requirements. The choice of theprediction horizon is based on the bandwidth of the dominantsystem dynamics, which, for engine speeds in the interval of1000–1500 rpm is in the range 3–5 Hz. Finally, the controland constraint horizons are chosen to tradeoff the explicitcontroller performance versus complexity. For the chosenvalues, the explicit MPC controller (30) results in s = 78regions for a total of 469 inequalities. Using the techniquesin [25], we have verified that the closed-loop system (31) islocally asymptotically stable around the equilibrium. We haveverified that the robustness margin guarantees stability for atleast 0.011 s of additional delay in the base torque dynamics,which is sufficient for the operating conditions of interest.Thus, the controller design is slightly conservative at highengine speed. Such conservativeness may be reduced if more



Fig. 5. Simulation of deceleration in neutral with stationary vehicle.(a) Upper plot: engine speed (solid line) and reference (dashed line). Lowerplot: engine speed tracking error. (b) Upper plot: base torque (solid line) andtorque constraints (dashed line). Lower plot: torque ratio by spark (solid line),constraints (dashed line), and setpoint (dotted line).

advanced stability analysis methods for systems with variabledelay are used [26].

For evaluating the proposed control design, a simulationmodel based on (10) has been implemented, where theparameters have been identified from the prototype vehicleused later for experimental validation. The reference andmeasured disturbances for the simulations, r , NT , and Mlsin (10), have been recorded in the vehicle while executing abaseline controller.

Remark 5: The turbine speed (and as a consequence theload on the engine) and the torque losses are affected by theengine speed. The engine speed profile obtained by MPC is(obviously) not exactly the same as the engine speed in thedata recording experiments. As a consequence, the measureddisturbances used for simulations are not exactly those that thecontroller will experience. However, during the data recording,the engine speed is maintained close to the reference by thebaseline controller, and if the same occurs in simulation, theengine speed profiles in simulation and in the experimentare similar. Hence, the approximations of Mld and Mls arereasonable, given also that the changes of Mld and Mls inthe operating conditions occurring during deceleration arerelatively small and slow.

Fig. 6. Simulation of rapid deceleration with downshifts with movingvehicle. (a) Upper plot: engine speed (solid line), reference (dashed line),and turbine speed (blue line). Lower plot: engine speed tracking error.(b) Upper plot: base torque (solid line) and torque constraints (dashed line).Lower plot: torque ratio by spark (solid line), constraints (dashed line), andsetpoint (dotted line).

To verify the capability of the controller of tracking a rapidlychanging reference, in Fig. 5, we show the deceleration fromthe maximum (limiter) engine speed to idle speed when thevehicle is in neutral and not moving. In this test, the enginespeed reference decreases by 3000 rpm in less than 2 s, and atthe same time the load on the crankshaft is very small sincethe powertrain is in neutral, i.e., NT ≈ N . The controllercannot track the moving reference with zero offset in the initialpart of the deceleration. This is due to the small load on thecrankshaft, and to the combustion stability constraint, whichimposes a lower bound on the commanded torque. However,the controller is able to enforce all the constraints and to softlyland on the reference without undershoot.

In Fig. 6, we show the simulation of a deceleration fromhigh gear (6th , in this case) while the vehicle is moving. Thevehicle is decelerated with the application of brakes causingdownshifts to first gear (the last downshift is 3rd –1st ), whichare easily noticeable by the sudden increase in the turbinespeed. The reference profile to be tracked is designed forachieving good drivability, and is generated by a separatefunction in the vehicle control system. All the constraints aresatisfied, and the reference trajectory is tracked with a smallerror. Compared with the simulated deceleration in neutral



in Fig. 5, the reference engine speed decelerates at a slowerrate, and there is a higher load on the crankshaft, which helpsthe deceleration. Since when the engine speed is larger thanthe turbine speed the engine is driving the wheels, whilein the other case the wheels are driving the engine, thecontroller has to maintain the engine speed on the desired siderelative to the turbine speed to avoid unintended lash crossingsin the driveline. In the simulation, no undesired lash crossingsoccur.

VI. EXPERIMENTAL RESULTS

The control strategy is synthesized in C-code for imple-mentation in a dSPACE DS1401 rapid prototyping unit.The test platform is a FWD production vehicle equipped witha 3.7L V6 engine and a 6-speeds standard automatic trans-mission. The controller in the rapid prototyping unit receivesthe current targets (reference and constraint bounds) and themeasurements/estimates for the plant variables from the ECU,constructs the prediction model state vector, evaluates theMPC feedback law, computes the actuators commands, andsends them to the ECU, which actuates them by the existingactuator drivers and lower level controllers. The communica-tion between rapid prototyping unit and ECU takes place viaCAN network, which introduces additional delays that will notbe present when the control system is entirely implemented inthe target platform, and hence not included in the predictionmodel. We have verified that the impact of the additionaldelays is appreciable only when the controller is initiallyengaged, causing a slightly larger initial tracking error.

While executed in the rapid prototyping unit for the simplic-ity of development, the controller computational requirementshave been evaluated for implementation in the automotivepowertrain control unit. The worst case number of atomicoperations (scalar sums, multiplications, and comparisons tozero) is less than 106 per second, which amounts to lessthan 1% of the CPU capabilities. The total ROM size (mostlycalibration memory) is about 10 kB, and only six additionalvariables have to be stored in RAM, the commands along thedelay windows in (19) and the integrators in (22) and (23).In extensive experimental tests, we have verified that theaverage CPU capabilities used are less than 10% of the worstcase, and that the storage memory usage can be reduced by afactor of four by simple data compression.

A. Experiments in Stationary and Driving Conditions

In Fig. 7, we show the experimental results for the decelera-tion in neutral. The behavior is qualitatively similar to the oneobserved in the simulation in Fig. 5, where it is not possibleto closely track the reference speed due to the low load andthe combustion stability constraint (6). We also note that dueto the way the engine maps are identified, in these operatingconditions, the engine torque actuation error (the differencebetween requested and delivered torque) may be significant (upto 30%). The proposed controller provides soft landing on thereference also in the experiments, with negligible undershoot.We have repeated the test with the transmission in drive andthe vehicle kept stationary by braking (Fig. 8). In this case,

Fig. 7. Experimental test of deceleration in neutral with stationary vehicle.(a) Upper plot: engine speed (solid line), reference (dashed line), and enginespeed constraint (dotted line). Lower plot: controller enabling signal (dashedline, 0 when inactive), and gear (solid line, 0 means neutral). (b) Upper plot:tracking error (0 when inactive). Middle plot: base torque (solid line) andtorque constraints (dashed line). Lower plot: torque ratio by spark (solid line),constraints (dashed line), and setpoint (dotted line).

there is a large difference between the engine speed and theturbine speed (NT = 0), and hence there is a larger loadon the crankshaft, so that the engine speed can track thereference also during the transient. The error transients whenthe controller is activated (t = 4, 12, 20 s) are in part causedby the CAN-induced time delay.

Next, we show the results of a test where the vehiclewas driven along an appropriately designed test course. Thecourse involves different types of decelerations, includingcoastdowns, rapid decelerations using brakes, and low-speedcreeping. As a further challenge, the test was executed with theAC compressor engaged, where any error in the estimate ofAC load introduces additional disturbances. The AC loadduring this test varied unpredictably between 5 and 15 Nm.Fig. 9 describes the scenario and highlights the differentoperating conditions that the test exercises. Some relevantmaneuvers are discussed next in detail.

In Fig. 10, we show a maneuver where the vehicle iscreeping (Fig. 3, phase c) in a roughly paved parking area,with a 2nd –1st downshift, where the 1st gear of the vehicle hasan overrunning one-way clutch. Due to the low speed and lowtorque, this is a critical maneuver since any small disturbancecan be felt. However, the controller precisely tracks the desired



Fig. 8. Experimental test of deceleration in drive with stationary vehicle.(a) Upper plot: engine speed (solid line), reference (dashed line), and enginespeed constraint (dotted line). Lower plot: controller enabling signal (dashedline, 0 when inactive) and gear (solid line, 0 means neutral). (b) Upper plot:tracking error (0 when inactive). Middle plot: base torque (solid line) andtorque constraints (dashed line). Lower plot: torque ratio by spark (solid line),constraints (dashed line), and setpoint (dotted line).

Fig. 9. Scenario of normal on road driving along a test course. Upper plot:engine speed (solid line), reference (dashed line), and turbine speed (blueline). Middle plot: controller enabling signal (dashed line, 0 when inactive)and gear (solid line, 0 means neutral). Lower plot: vehicle speed [mph].

deceleration profile also after the downshift, which involves alarge change in the gear ratio.

In Fig. 11, we show a coastdown without the applicationof brakes (until t = 162 s) ending with several consecutivedownshifts. Initially, the torque converter is locked (Fig. 3,phase a) and N = NT . After the 6th–5th downshift, the

Fig. 10. Experimental test of low-speed maneuver (creeping). (a) Upper plot:engine speed (solid line), reference (dashed line), and turbine speed (blue line).Middle plot: controller enabling signal (dashed line, 0 when inactive) and gear(solid line, 0 means neutral). Lower plot: vehicle speed [mph]. (b) Upperplot: tracking error (0 when inactive). Middle plot: base torque (solid line)and torque constraints (dashed line). Lower plot: torque ratio by spark (solidline), constraints (dashed line), and setpoint (dotted line).

torque converter is unlocked and closed-loop speed controlbegins (Fig. 3, phase b). The controller follows the targetspeed profile, rejecting load disturbances (e.g., aroundt = 143 s), which are due to, for instance, power steeringand AC compressor load changes. During the downshifts, thecontroller maintains the engine speed on the desired side ofthe turbine speed.

An aggressive deceleration while braking is shown inFig. 12, similar to the simulation in Fig. 6. The powertrain goesthrough multiple downshifts, and even in this case, the MPCis able to always maintain the engine speed on the correct sidewith respect to the turbine speed. The crossing between turbinespeed and engine speed (Fig. 3, b to c), which changes thesign of the load torque per (2), is well controlled (t = 50.5 s),which results in the driver not experiencing a significant jerkor NVH.

B. Experiments in Nonnominal and Faulty Conditions

The controller operation was verified also in the case ofnonnominal and faulty conditions.

One of such tests is shown in Fig. 13, where the vehicle wasstationary with the transmission in neutral and near idle speed(Fig. 3, phase d), which, as discussed before, is a particularly



Fig. 11. Experimental test of coastdown with downshifts without braking.(a) Upper plot: engine speed (solid line), reference (dashed line), and turbinespeed (blue line). Middle plot: controller enabling signal (dashed line, 0 wheninactive) and gear (solid line, 0 means neutral). Lower plot: vehicle speed[mph]. (b) Upper plot: tracking error (0 when inactive). Middle plot: basetorque (solid line) and torque constraints (dashed line). Lower plot: torqueratio by spark (solid line), constraints (dashed line), and setpoint (dotted line).

challenging condition due to the limited load on the crankshaft,larger actuation errors, and longer delays. In the test, adisturbance ϑd [deg] is added to the commanded electronicthrottle angle ϑcmd [deg], simulating faults in the throttle.These cause large and sudden torque disturbances, since thebase torque is different from what expected. During the test,the throttle angle output disturbance is not corrected by anyother control loop. In Fig. 13(a), we observe that the steady-state throttle angle with the vehicle stationary in neutral isonly slightly above 1 deg. The disturbances are steps between−1 deg and +2 deg, causing large relative torque disturbances,since in this conditions, the torque is approximately linearlyproportional to the throttle angle. The controller is robust tothese disturbances, rejecting them rapidly and enforcing theconstraints.

When the throttle angle disturbance increases from 0 deg to2 deg (t = 65 s), the controller cannot bring the spark backto the setpoint, since the steady-state value of the commandedthrottle angle would need to be negative, in the predictionmodel. However, the controller cannot command a negativebase torque (i.e., a negative throttle angle) because it hasto maintain the base torque above the combustion stabilitylimit, and the disturbance on the throttle angle is unknown

Fig. 12. Experimental test of rapid deceleration with downshifts whilebraking. (a) Upper plot: engine speed (solid line) and reference (dashed line).Middle plot: controller enabling signal (dashed line, 0 when inactive) and gear(solid line, 0 means neutral). Lower plot: vehicle speed [mph]. (b) Upper plot:tracking error (0 when inactive). Middle plot: base torque (solid line), torqueconstraints (dashed line), and turbine speed (blue line). Lower plot: torqueratio by spark (solid line), constraints (dashed line), and setpoint (dotted line).

even to the function that computes such limit. Thus, MPCuses the spark, including at steady state, to compensate forthe disturbance. Finally, when the throttle angle disturbancedecreases from 2 deg to 0 deg, the lower bound on theengine speed is briefly violated (t = 88 s). The maximumviolation is approximately 50 rpm and lasts less than 250 ms.However, since the limit on the engine speed is designedslightly conservatively and implemented as a soft constraint,the controller continues operating and rapidly recovers bycommanding a rapid increase of the base torque request,which, when delivered, is modulated by the spark (t = 89 s).

C. Performance Assessment

We have compared the proposed control strategy with abaseline controller and assessed the difference in performance.The baseline controller uses a gain-scheduled PID design withfeedforward and antiwindup, which outputs the desired braketorque. The feedforward is computed to compensate the torqueload (Mld) and losses (Mls) estimated from the same enginemaps used by MPC. The brake torque request computed bythe control strategy is allocated to engine torque and sparktorque ratio by an appropriate algorithm. This also guarantees



Fig. 13. Experimental test of robustness to throttle faults. (a) Upper plot:engine speed (solid line), reference (dashed line), and engine speed constraint(dotted line). Lower plot: commanded throttle angle and additive throttle angledisturbance (in degrees). (b) Upper plot: tracking error (0 when inactive).Middle plot: base torque (solid line) and torque constraints (dashed line).Lower plot: torque ratio by spark (solid line), constraints (dashed line), andsetpoint (dotted line).

that an appropriate spark torque reserve is maintained byimplementing a form of midranging [27] on the base torque.The baseline controller is a robust and effective strategy,yet, since it does not optimally account for the dynamicinteractions between the control inputs in the presence ofconstraints, its performance can be improved. In addition, thedesign of the saturation and antiwindup is rather cumbersomedue to the many constraints, resulting in a complex controllogic. An immediate benefit of MPC is that the constraintsare dealt with by design in a systematic way. In terms ofperformance, the MPC controller has shown improvementsin terms of tracking and robustness. Quantitatively, in thedeceleration tests, the settling time was reduced on averageby 11% and 62% in neutral (Fig. 7) and drive (Fig. 8),respectively. The performance improvement in neutral is lesspronounced because all the actuators are saturated for most ofthe test, and hence there are no degrees of freedom to optimize.Along the whole driving scenario (Fig. 9), the RMS trackingerror is reduced by 16%. In addition, we have observed thatthe torque ratio is on average 1.15% higher when using theMPC controller, due to the more precise tracking of the enginespeed reference, which allows to bring the torque ratio to itsnominal value more rapidly. As a consequence, the engine runs

closer to MBT, and hence more efficiently, reducing the fuelconsumption by 1.32%. While a decisive improvement of theengine efficiency was not the primary objective, we note thatthis can be accomplished by increasing the spark ratio nominalvalue and therefore tightening the constraints on torque fromspark modulation. Preliminary tests have shown that, in thisway, the fuel economy during deceleration can be improved bymore than 3% while keeping the same tracking performanceas the baseline controller.

VII. CONCLUSION

We have proposed a systematic design for engine speedcontrol during vehicle deceleration, when the torque converteris open. The multivariable constrained controller must keepthe engine speed close to the turbine speed to maintain vehicleresponsiveness, yet it should avoid crossing the lash to min-imize disturbances on the driveline and degraded drivability.By reparameterizing the powertrain dynamics to eliminate amultiplicative nonlinearity, we have implemented a feedbackcontroller based on linear quadratic MPC, which enforces thepowertrain constraints. In the experimental tests, the controllerhas been shown to achieve improved tracking performance andfuel economy as compared with a baseline strategy. In termsof its computational footprint, the controller is adequate forimplementation in a production ECU.

ACKNOWLEDGMENT

The authors would like to thank C. Cox and S. Szwabowskiof Ford Motor Co., for support with the test vehicle, andDr. D. Yanakiev of Ford Motor Co., for useful discussions.

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Stefano Di Cairano (M’08) received the master’s(Laurea) and Ph.D. degrees in information engineer-ing from the University of Siena, Siena, Italy, in2004 and 2008, respectively. He was also grantedthe Int. Curr. Opt. of Doctoral Studies in HybridControl Systems.

He was a Visiting Student with the Technical Uni-versity of Denmark, from 2002 to 2003, and with theCalifornia Institute of Technology, Pasadena, CA,USA, from 2006 to 2007. From 2008 to 2011, hewas with Powertrain Control R&A, Ford Research

and Advanced Engineering, Dearborn, MI, USA. Since 2011, he has beenwith the Mechatronics Group, Mitsubishi Electric Research Laboratories,Cambridge, MA, USA, where he is currently a Team Leader. His research is onadvanced control strategies for complex mechatronic systems in automotive,factory automation, and aerospace. His current research interests includemodel predictive control, constrained control, networked control systems,hybrid systems, optimization, automotive, aerospace, and factory automation.

Dr. Di Cairano is the Chair of the IEEE CSS Technical Committee onAutomotive Controls, a member of the IEEE CSS Conference EditorialBoard, and an Associate Editor of the IEEE TRANSACTIONS ON CONTROL

SYSTEMS TECHNOLOGY.

Jeff Doering received the B.S. degree in aerospaceengineering from the University of Minnesota, Min-neapolis, MN, USA, the M.S. degree in mechanicalengineering from Stanford University, Stanford, CA,USA, and the Ph.D. degree in mechanical engi-neering from Michigan Technological University,Houghton, MI, USA, in 1991, 1992, and 2000,respectively.

He has been with Ford Motor Company, Dearborn,MI, USA, since 1992, involved in the areas of engineand vehicle controls in both product development

and research, where he is currently a Technical Leader in research andadvanced engineering.

Ilya V. Kolmanovsky (F’08) received the M.S.and Ph.D. degrees in aerospace engineering and theM.A. degree in mathematics from the University ofMichigan, Ann Arbor, MI, USA, in 1993, 1995, and1995, respectively.

He is currently a Professor with the Departmentof Aerospace Engineering, University of Michigan,with research interests in control theory for sys-tems with state and control constraints, control ofautomotive and aerospace propulsion systems, andspacecraft control applications. He was with Ford

Research and Advanced Engineering, Dearborn, MI.Dr. Kolmanovsky is a past recipient of the Donald P. Eckman Award

of American Automatic Control Council, the IEEE TRANSACTIONS ON

CONTROL SYSTEMS TECHNOLOGY Outstanding Paper Award, and severalFord Research and Advanced Engineering Technical Achievement, Innovation,and Publication awards.

Davor Hrovat (F’07) received the B.Sc. (Dipl.Ing.)degree in mechanical engineering from the Univer-sity of Zagreb, Zagreb, Croatia, and the M.Sc. andPh.D. degrees in mechanical engineering from theUniversity of California, Davis, CA, USA, in 1972,1976, and 1979, respectively.

He has been with the Ford Research Laboratory,Dearborn, MI, USA, since 1981, where he is a HenryFord Technical Fellow, coordinating and leadingresearch efforts on various aspects of vehicle/powertrain control systems, where he holds numerous

patents.Dr. Hrovat is a fellow of the American Society of Mechanical Engineers

(ASME) and a member of the National Academy of Engineering. He is therecipient of the ASME/Dynamic Systems and Control Innovative PracticeAward in 1996 and the AACC Control Engineering Practice Award in 1999.He has served on editorial boards for a number of ASME and IEEE journals,and is currently an Editor for the Control Engineering Practice journal of theInternational Federation of Automatic Control and the International Journalof Vehicle Autonomous Systems.

Model Predictive Control of Engine Speed During Vehicle Deceleration

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