A Design Framework for Predictive Engine Control · A Design Framework for Predictive Engine Control X. Wang, H. Waschl, ... Moreover, at high EGR flow (e.g.70% EGR-valve position)

A Design Framework for Predictive Engine ControlX. Wang, H. Waschl, D. Alberer* and L. del Re

Institute for Design and Control of Mechatronical Systems, Johannes Kepler University Linz, 4040 Linz - Austria e-mail: [email protected] - [email protected] - [email protected] - [email protected]

* Corresponding author

Résumé — Un cadre de conception pour la commande prédictive de moteurs — La commandeprédictive par modèle (Model Predictive Control ; MPC) a été proposée plusieurs fois dans l’automatiquepour l’automobile, avec des résultats prometteurs, principalement à partir d’une approche MPC linéaire.Toutefois, comme la plupart des systèmes automobiles sont non linéaires, la commande prédictive nonlinéaire (Nonlinear MPC ; NMPC) pourrait représenter une option intéressante. Malheureusement, laconception d’une commande optimale à partir d’un modèle non linéaire générique conduit généralementà un problème non convexe complexe. Dans ce contexte, cet article présente deux schémas différentspour prendre en compte la non linéarité du système en vue de la conception de la commande. En premierlieu, une méthode MPC multilinéaire est présentée sur la base d’une segmentation du système et, ensecond lieu, une conception de système de commande basée sur une identification de système nonlinéaire utilisant une structure quasi linéaire à paramètres variants (Linear Parameter Varying ; LPV) estproposée ; celle-ci est alors utilisée dans un cadre de conception de NMPC. Cet article présente cesapproches et leur application à un système bien connu, le module d’air (air path) d’un moteur Diesel.

Abstract — A Design Framework for Predictive Engine Control — Model Predictive Control (MPC)has been proposed several times for automotive control, with promising results, mostly based on a linearMPC approach. However, as most automotive systems are nonlinear, Nonlinear MPC (NMPC) would bean interesting option. Unfortunately, an optimal control design with a generic nonlinear model usuallyleads to a complex, non convex problem. Against this background, this paper presents two differentschemes to take into account the system nonlinearity in the control design. First, a multi-linear MPCmethod is shown based on the segmentation of the system and then a control system design based on anonlinear system identification using a quasi Linear Parameter Varying (LPV) structure is proposed,which is then used in a NMPC design framework. This paper presents the approaches and theapplication to a well studied system, the air path of a Diesel engine.

Oil & Gas Science and Technology – Rev. IFP Energies nouvelles, Vol. 66 (2011), No. 4, pp. 599-612Copyright © 2011, IFP Energies nouvellesDOI: 10.2516/ogst/2011107

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Oil & Gas Science and Technology – Rev. IFP Energies nouvelles, Vol. 66 (2011), No. 4600

INTRODUCTION

The continuously tightened emission limits oblige theautomotive industry to improve their combustion engines.This can either be done by changes in the hardware (e.g. byapplication of catalysts or advanced injection system etc.) orby modifications in the software, especially the control strat-egy for fuel injection and the air path system. Standard pro-duction engine controls are still developed in a very heuristicway and reach a huge and fault-prone complexity. Eventhough this is widely recognized and model based control isalmost unanimously advocated as the solution of choice, theenormous legacy of decades of heuristic solutions makes thetransition slow and uncertain. Against this background thispaper suggests a systematic approach for the identification ofcontrol models and their application in the framework ofModel Predictive Control (MPC) to the air path of a produc-tion Diesel engine, whereas the focus clearly lies on thedesign method.

From a control point of view the air path of a Dieselengine is a very challenging system, namely a highly couplednonlinear MIMO system with constraints and a limited feasi-ble working range. Figure 1, for instance, depicts stationarymeasurements of fresh air mass flow (MAF) and boost pres-sure (MAP) for changing air path actuator values. As it canbe seen, the DC-gains are not constant and even change theirsign1, which obviously cannot be captured by a linear model.

Many works present the application of model basedMIMO control to the Diesel engine air path. Stefanopoulouet al. (2000) for instance show a combination of a nonlinearfeedforward and a gain scheduled MIMO controller for thecontrol of the burnt mass fraction of the intake and the airfuel ratio or van Nieuwstadt et al. (2000) compared severalcontroller approaches. Robust nonlinear air path control wasapplied in Jankovic and Kolmanovsky (1998) and more indetail in Jankovic et al. (2000), where input-output lineariza-tion was used for the generation of a control Lyapunov func-tion and a controller with a guaranteed robustness propertywas determined. Flatness based polynomial control is shownin Ayadi et al. (2004), and Jung (2003) as well as Wei (2006)show approaches of robust Linear Parameter Varying (LPV)control for the air path.

Due to its capability of handling constraints on manipulatedvariables and system states and the explicit minimization of acost function, the use of Model Predictive Control for enginecontrol has recently attracted much interest. Rückert et al.(2004) implemented a gain scheduled Generalized PredictiveControl (GPC) strategy for a truck Diesel engine and

Dynamic Matrix Control (DMC) and GPC strategies weredeveloped in Garcia-Ortiz (2004) and Salcedo et al. (2004).To cope with the nonlinearity of the system, differentapproaches exist, e.g. Ortner and del Re (2007) and Langthaler(2007) introduce a segmentation of the working range anddesign separate linear MPCs for every segment. The logicalternative is to design a single Nonlinear MPC (NMPC) forthe whole engine operating range, e.g. Herceg et al. (2006)present a simulation study for the application of NMPC to aDiesel engine air path.

Unfortunately, very few attempts have been done in thisway in practice, because of several reasons, the two mostimportant ones being the lack of models required to computethe solution and the limited CPU power, in particular inthe case of a production Engine Control Unit (ECU).Furthermore, efficient solvers are required which are able tocompute the solution of the Quadratic Programs (QPs) whichare arising in Model Predictive Control. In Ferreau et al.(2006, 2008) an online active set strategy for QPs is pre-sented, which has already been successfully applied to airpath control at an engine test-bed (Ferreau et al., 2007).

The actual paper builds on this experience and comparestwo different methods of how to take into account the systemnonlinearities in the control design. First, the above men-tioned approach of the segmentation is discussed. To thisend, multiple linear models were identified for restricted

1 Due to the high exhaust back pressure at minimum turbine guide vaneopening (VGT → 100%) the EGR flow is increased and consequentlythe fresh air mass flow is reduced (Fig. 1a). Moreover, at high EGRflow (e.g. 70% EGR-valve position) the turbine flow and thus theturbine power is reduced, which causes a drop of boost pressure(Fig. 1b). All this strongly depends on the engine design.

0 4020 60 80 1000

100

50

150

200

250

VGT (%)a)

MA

F (

kg/h

)

EGR (%)

5%

10%15%

20%

25%

30%

40%50%

70%

b)0 4020 60 80 100

VGT (%)

EGR (%)

5%

10%15%

20%

25%

30%

40%50%

70%1000

1200

1400

1600

1800

2 000

MA

P (

hPa)

Figure 1

Stationary measurements of fresh air mass flow a) and boostpressure b) at different air path actuator positions.

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X Wang et al. / A Design Framework for Predictive Engine Control 601

operating regimes and several linear MPCs were applied tothe engine – according to the actual system state only onelinear MPC had the control authority. With the secondmethod a generalization is presented using a single nonlinearmodel, albeit one which is easy to linearize at each operatingpoint. This is done combining an LPV identification tech-nique with a simple formulation of NMPC based on therecursive use of the above mentioned QP solver with a guar-anteed computation time. The paper explains the methodsand presents their use for the very well known example of thefresh air mass flow (MAF) and boost pressure (MAP) controlof a Diesel engine air path.

1 THE DESIGN FRAMEWORK

As well known, MPC uses model information to predict thestates of the system over a finite horizon and to computeaccordingly the optimal trajectory (see Fig. 2). The problemtypically solved by MPC in the case of a linear system withconstraints on the inputs and the states can be resumed asfollows:

(1)

The solution of this optimization problem is the future controlsequence u* = [uk, uk+1, ..., uk+NCH–1], where only the firstvalue will be applied and the optimization is repeated at thenext sampling instant. NCH and NPH are called control andprediction horizon. The coefficients Q and R are positive def-inite weighting matrices to penalize the tracking error and thecontrol effort respectively. yref is the output reference trajec-tory, x denotes the vector of system states, u the vector ofmanipulated variables and ν forms the vector of measureddisturbances. x–,x– and u–,u– represent lower and upper boundson the states and the manipulated variables, which meansthat constraints can be taken into account explicitly in thecontroller design.

Inserting the linear model xk+1 = Akxk + Bkuk, + Zv,yk = Ckxk into the optimization task (1) yields a so calledQuadratic Program (QP), in which the sequence u* can beseparated from the prediction part and this convex problemcan be solved using a standard QP solver. Nonlinear MPC

minu u

k refT

k ref kT

kk

y y Q y y u R u∈

−( ) −( ) +⎡⎣⎢

⎤⎦

1

2Δ Δ ⎥⎥

= −= + +

=

−

+

∑k

NPH

k k k

k k k k k

k

s t

u u ux A x B u Zvy

0

1

1

. .

Δ

==≤ ≤ ∀ =≤ ≤ ∀ = −

= ∀

C xx x x k NPHu u u k NCH

u k

k k

k

k

k

00 1

0

……

Δ == NCH NPH...

(NMPC) is the logic extension of MPC in which the linearmodel is substituted by a nonlinear one. Unfortunately,inserting this nonlinear model into the problem (1) leads ingeneral to a non convex problem which may be difficult tosolve, e.g. by a sequential approximation with QuadraticPrograms at each time step, which is computationally expen-sive. Although efficient solvers for nonlinear programs exist(e.g. Bock et al., 2000), in this work we propose to apply aspecial nonlinear model class – the class of Linear ParameterVarying (LPV) systems – which allows to use an active setbased QP solver in order to keep the computational burdenlow. Hence, the nonlinear system of the Diesel engine airpath can be captured by a single model without a significantchange in complexity for the online numerical solver.

1.1 Determination of Multi-Linear Control Models

As mentioned above, due to the nonlinear behavior in thewide operating range of a Diesel engine (e.g. engine speedfrom 800 rpm to 4 500 rpm) it is not possible to obtain asingle linear model with a high precision. One evident possi-bility consists in working out several linear models to coverthe whole operating range. A very new approach to accom-plish this task is to use a clustering technique for the identifi-cation of piecewise affine systems (Ferrari-Trecate et al.,2003) or a more usual way is to divide the operating rangeempirically in small operating areas and identify severallinear models (see Fig. 3).

The air path dynamics depend on many factors, but for theconsidered case of a warm engine it proved sufficient to con-sider besides the two manipulated variables (EGR and VGT)engine speed and injected fuel amount as model inputs,which shall be treated as measured disturbances. Neglectingthe two measured disturbances would lead to unacceptablemodel quality.

Past

Future

Time

Control horizon

Prediction horizon

Predicted output

Optimized controlsequence

Figure 2

MPC strategy.

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As it will be shown in Section 2.3, for the determinationof the linear control models a separate identification of statespace models was done for the 12 regions, using a PredictionError Method (PEM) (see e.g. Ljung, 1999).

1.2 LPV Control Models

In Wei (2006) a quasi-LPV model from the simplification ofthe physical model of a production Diesel engine was consid-ered. The LPV model is defined by:

(2)

Note that A(z,ρ) and B(z,ρ) represent polynomials in the shiftoperator z and in the scheduling vector ρ, in contrast to equa-tion (1) where A and B represent matrices. m(k) is the error inthe actual sampling instant. In a standard LPV case, ρ isassumed to be an external variable, but in the quasi-LPVframework it may be a function of an internal state. The LPVsystem class has two strong advantages: it represents a con-tinuous gain scheduling model and thus can approximatewell many systems, and due to its structure and description ofthe input-output behavior, the parameters can be estimatedfrom measured data by linear algorithms extended from theclassical identification algorithms (Wei, 2006). Indeed, using:

(3)

ϕ

θ

k k k k n k k k m

i i i

y y y u u u

a a

= − − −[ ]

=

− − − − − −1 2 1 2

1 2

… …

…… …

…

a b b b

F f f

in i i imT

k N

1 2

1 1 2 1 11

[ ]

= ( ) −ρ ρ ρ, , ... ,ρρ2 , ...( )⎡⎣ ⎤⎦

A z y k B z u k m k, ( ) , ( ) ( )ρ ρ( ) = ( ) +

where the functions f1(ρ1, ρ2,...)... fN–1(ρ1, ρ2,...) are designparameters, we can rewrite (2) as:

(4)

(4) can be addressed by standard identification algorithms toestimate the parameter vector Θ̂:

(5)

where Nm is the number of measurements.

1.3 MPC Formulation

The formulation of MPC as Quadratic Program has alreadybeen shown in many works, for instance in Ferreau et al.(2006). In this work linear as well as LPV systems are con-sidered, whereas linear systems can be considered as a spe-cial case of LPV. Hence, in the following the MPC algorithmformulation is presented for the LPV systems and, obviously,the same holds for the linear case with the only differencebeing constant system parameters. The parameters of theLPV system may change every sample, thus the recursiveevaluation of the prediction model is: see Equation (6)This can be simplified to:

(7)

where V is the disturbance matrix:

(8)

and:

(9)

S

I

A

A A

S

B

A

NPH

N

00

1 0

1

0

0 0

0

=

⎛

⎝

⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟

=

−

��

��

� � �

PPH

NPH NCH NCH

NPH

A BA A B

B−

− −

−+ +

⎛

⎝

⎜⎜

1 1 0

1 1

1

� ��

...

⎜⎜⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟⎟⎟⎟

VZv

A A Zv ZvNPH

=

+ +

⎛

⎝

⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟

−

0

1 1

�... ...

X S Sx

UV S x S U V= ( )

⎛

⎝⎜

⎞

⎠⎟+ = + +0 1

00 0 1

ˆ ( )

[ , , ..., ]

[ , , ...,

Θ Γ Γ Γ

Γ Γ Γ Γ

=

=

=

−T T

N

Y

Y y y

m

1

1 2

1 2 yyNm]

y k F m mk k k k k

T TN

T

( ) = ⊗( ) + = +

= ( ) ( ) ( )⎡⎣

ϕ

θ θ θ

Θ Γ Θ

Θ 1 2 …⎢⎢⎤⎦⎥

= ⊗( )Γk k kF ϕ

Fuel

Speed

121110

9876

1 2 3 4 5

Figure 3

Segmentation of the engine working range.

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X Wang et al. / A Design Framework for Predictive Engine Control 603

Substituting (7) into (1) leads to:

(10)

which can be solved using standard QP solvers. As for theLPV case the matrices A and B depend on the schedulingvector, whereas they are updated according to the result ofthe previous optimization step and the problem (10) is set upagain each sampling instant. Note again, that in the specialcase of linear systems the matrices A and B are constant,which results in a single QP for all times and makes the QPreformulation at every time instant unnecessary.

1.4 State Estimation

For the application of state space MPC, information aboutthe system state is necessary to obtain the optimal controlsequence for the next time instant. For the presented results, aKalman filter (Kalman, 1960) was used to estimate thecurrent system states based on a system model and measuredin- and output signals. In the following, the basic ideas arebriefly summarized.Considering a linear discrete time system description:

(11)x A x B u G w

y C x D u H w vk k k k

k k k k k

+ = ⋅ + ⋅ + ⋅

= ⋅ + ⋅ + ⋅ +1

minU

kT T

diag diag k kT TU S Q S R U U S S x V

1

2 1 1 1 0 0+( ) + + − XXref

X I x

Q diag Q Q Q

R di

ref ref

diag

diag

( )

= ⋅

=

=

( , )�

aag R R R

U u u

X x x x

NCH

NPH

( , )

( ; , )

( ; ; , )

�

�

�

=

=

1

0 1

with the input u, the output y, a process noise input w and ameasurement noise v, it is assumed that both noise inputs arewhite and with normal probability distribution (E{wk} =E{νk} = 0). Although in practice the noise covariances:

(12)

might change, for the following it is assumed that they areconstant. Then, the Kalman filter constructs a state estimatex̂k that minimizes the steady state error covariance:

(13)

The implementation of the algorithm can be split into twoparts, a prediction and an update phase. During the first phasethe filter predicts the future system state, based on the currentstate and the system input signals. The expected errorcovariance matrix P is calculated using this information:

(14)

During the update phase the error between predicted systemoutputs and measured, but noisy, signals is used to correct thefilter gain and update the current state prediction:

(15)

Since in most cases the error covariances are not knownexactly, the matrices Q and R can be seen as tuning factors,which either can be found with adaptive methods, or need tobe determined by empirical tests. For the considered nonlin-ear models of the NMPC framework an Extended Kalman

ˆ ˆ| |

| ,

x A x B u

P A P A Q

k k k k k

k k k kT

+

+

= ⋅ + ⋅

= ⋅ +

1

1

P E x x x xk

k k k kT= − ⋅ −

→∞lim {( ˆ ) ( ˆ ) }

E w w Q E v v R E w v Nk kT

k kT

k kT{ } { } { }⋅ = ⋅ = ⋅ =

x

x

x

I

A B

ANPH NPH

0

1

0 0

1

0 0

0

�

��

� � � �

�

⎛

⎝

⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟

=

− AA A A BA A B

BNPH

NPH NCH NCH

NPH

0 1 1 0

1 1

1

−

− −

−+ +� �

�

...

⎛⎛

⎝

⎜⎜⎜⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟⎟⎟⎟

⎛

⎝

⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟

+

−

x

u

uNCH

0

0

1

�

00

1 1

Zv

A A Zv ZvNPH

�

− + +

⎛

⎝

⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟

... ...

(6)

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Oil & Gas Science and Technology – Rev. IFP Energies nouvelles, Vol. 66 (2011), No. 4604

Filter (EKF) was used for state estimation. Based on ageneric nonlinear system of the given structure:

(16)

and with the same assumptions on the noise characteristics asfor the linear plant model, the Kalman filter update equationsremain identical as above, only the system matrices A and Care now derived by a linearization in the Operating Point(OP) of the system at each time instant:

(17)

To take into account steady state offsets the control model isaugmented with an output disturbance model (Muske andBadgwell, 2002). In the linear case the augmented systemcan be described by:

(18)

where for each measured output the disturbance model isextended with an additional state.

x A

I

x B

e ek

k

plant k

k

+

+

⎡

⎣⎢

⎤

⎦⎥ =

⎡

⎣⎢

⎤

⎦⎥⋅

⎡

⎣⎢

⎤

⎦⎥+1

1

0

0pplant

k

k plantk

k

u

y C Ix

eD

0

⎡

⎣⎢

⎤

⎦⎥⋅

= ⎡⎣ ⎤⎦⋅⎡

⎣⎢

⎤

⎦⎥+ pplant ku⋅

A k x uf x u k

x

C k x ug x

k kk k

k

k kk

OP

, ,, ,

, ,,

( ) =∂ ( )

∂

( ) =∂ uu k

xk

k OP

,( )∂

x f x u k w

y g x u k v

k k k k

k k k k

+ = ( ) +

= ( ) +

1 , ,

, ,

1.5 Control Structure

Both, the multi-linear MPC and the LPV based NMPCclosed loop systems can be described by the control structurepresented in Figure 4. The MPC is provided with the estimatedstates, the measured disturbance and of course the referenceset points. The estimated states are obtained by a standardKalman filter in the linear case or an EKF in the nonlinearapplication.

2 MODEL PREDICTIVE ENGINE CONTROL

2.1 Diesel Engine Air Path

The air path of a modern Diesel engine is a complex thermo-dynamic and mechanic system. The schematic diagram of aturbocharged Diesel engine with high pressure Exhaust GasRecirculation (EGR) is illustrated in Figure 5. The inlet andexhaust manifolds of a Diesel engine are coupled twice –firstly through the Variable Geometry Turbocharger (VGT)and secondly through the Exhaust Gas Recirculation (EGR)path.

In this system key influent factors on MAF and MAP arepositions of EGR and VGT, which are the considered controlinput variables in this work. Since both actors affect both out-put variables, engine air path control is indeed a MIMO con-trol problem. The actual standard is to apply a heuristicallytuned feedforward part in combination with two SISO (PID)feedback controllers. Typically, most of the control action isgenerated by the with high effort determined feedforward,while the task of the feedback controllers is mainly to

Measureddisturbances

ReferencesMAFrefMAPref

Measured outputs

Controlled outputsState

estimatorEngineairpath

MPC

xegrxvgt

MAFMAP

Nmf

Figure 4

Closed-loop control structure.

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X Wang et al. / A Design Framework for Predictive Engine Control 605

compensate production tolerances, aging and wear as wellas to reject disturbances. In the following, as an alternativeto the standard method the application of the MPCapproaches is presented, using EGR valve position (con-strained from 0 to 100%), VGT position (constrained from 0to 100%) as control inputs and engine speed N and injectedfuel per stroke mf as measured disturbance v. The controltask is to track references for the two control outputs MAFin kg/h and MAP in hPa.

2.2 Real Time System Setup

Figure 6 depicts the real time implementation setup of theMPC approaches. The control design was done in Matlab/Simulink and finally the real time controller was imple-mented on a rapid prototyping system. In order to get controlauthority of the air path, a development Engine Control Unitwas used, which allowed to bypass the standard air pathcontrol loops while all the remaining engine control loopswere kept at their standard.

2.3 Implementation of Multi-Linear MPC

According to the segmentation of Figure 3 several linearmodels were determined independently. For identification thechoice of excitation signals and patterns for MIMO identifi-cation is known to be critical (Gevers et al., 2006). Figure 7presents the applied identification input signals as well as themeasured system outputs. Second order PEM models turnedout to be sufficient for the restricted operating ranges, e.g.Figure 8 shows the satisfactory validation results of themodel for region 5.

The engine speed and the injected fuel amount wereconsidered as measured disturbances ν for the control modeland furthermore an output error disturbance model e wasapplied in order to guarantee offset free tracking, leading to:

(19)

x

u

A B

I

e

k

k

k

MAF MAP MAF MAP+

+

+

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

=1

1

1

, , 0

0 0

00 0

0

0

0 0I

x

u

I

I

e

k

k

k

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥⋅

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥+

⎡

⎣

⎢⎢⎢

⎤⎤

⎦

⎥⎥⎥⋅

Δ

⎡

⎣⎢

⎤

⎦⎥

= ⎡⎣ ⎤⎦⋅

⎡

v

u

y C I

x

u

e

k

k

k MAF MAP

k

k

k

, 0

⎣⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

EGR

MAP

MAF

VGT

mj

N

Variable geometryturbine

Compressor

EGRcooler

Intake manifold

Swirl valve

Fuel injectors

EGRvalve

Intercooler Exhaust manifold

Real-time systems Monitoring and programming

10/100 MBit Ethernet

Bypass

ES910

CAN1

AVLPUMA

AutoBox

RT-Controller

Testbench

Matlab simulinkdSpace RTIControl desk

IntecrioINCA

CAN

CAN2 ETK

ETK

ECU

AVLPUMA

Figure 5

Structure of a Diesel engine air path.

Figure 6

Schematic representation of the combustion engine test bench.

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where:

Based on the identified models, in the next step, Kalman filtersas well as linear Model Predictive Controllers were designedfor each region. For the actual case explicit MPC wasselected, where affine control laws are computed a priori forevery possible initial state and consequently no QP solver isrequired in the real-time implementation (see Bemporad et al.(2002) for more details on explicit MPC). Note that the finalcontrol signal of this explicit version is the same as an onlineoptimizer would obtain2 (see Tab. 1 for the applied parametersettings). Finally the multiple controllers according to Figure 43

have been implemented on a real time system (Fig. 6),

x R u R R u R v Rek k k k k∈ ∈ ∈ Δ ∈ ∈2 2 2 2 2, , , and

50

EGR

0 100

20

30

40

50

10

50

Fuel

0 100

10

15

20

25

30

35

550

Mass air flow

0 100

500

600

700

800

900

1000

40050

Boost pressure

0 100

1100

1200

1300

1400

1500

1000

50

VGT

0 100

30

40

50

60

2050

Speed

0 100

2100

2200

2300

2400

2500

2000

Figure 7

Identification signals (%, %, rpm, mg/stroke, mg/stroke, hPa vs time).

10 6020 30 40 500

MA

F

300

-200

100

0

-100

200

Measured output and simulated model output

Measured

Simulated

Measured output and simulated model output

10 6020 30 40 500

MA

P

300

-200

100

0

-100

200Measured

Simulated

Figure 8

Validation of model 5 (mg/stroke and hPa vs time, mean value compensated).

2 Subject to applying the same linear system as well as identicalconstraints and an equal tuning.

3 The block “MPC” of Figure 4 contains the affine control laws.

TABLE 1

MPC tuning parameters

Property Value

QMPC [0.5 0; 0 0.5]

RMPC [1 0; 0 1]

ΔuMPC [–10 –5; 3.3 5]

NPH 120

NCH 1

Ts 0.05 s

whereas a controller output switching was realized.Therefore, only a single controller – the one belonging to theactually effective segment according to Figure 3 – was inclosed loop control authority. In order to provide a smoothtransition between the segments, a continuous update of all

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engine speed N, the injected fuel amount mf and the positionof the EGR and VGT actuators.

The sampling time of the identification data was Ts = 0.05 s.As already mentioned the model is a combination of twoMISO systems (one for MAF and one for MAP) which wereidentified separately. For the excitation of the inputs EGRand VGT and the external disturbances N and mf white noisesignals with different sample times were used around a speci-fied mean value which is shown in Figure 10. To avoid a

damage of the engine, caused by too high manifold airpressures, the maximum VGT position was constrained athigher engine speeds.

Figure 11 shows the satisfactory validation results of theidentified model for the two outputs MAP and MAF.

TABLE 2

LPV model quality

VAF

MAF 89.46%

MAP 77.88%

In Table 2 the VAF (Variance Accounted For) is used asmodel quality criterion. The VAF is defined as:

(22)

Transforming the two MISO models from the transfer func-tion model into state-space form yields to description (19)with non constant matrices AMAF,MAP, BMAF,MAP, CMAF,MAP,where:

The implementation of the closed-loop control structure inFigure 4 has been done using the setup shown schematicallyin Figure 6, whereas the NMPC was designed in C++ and

x R u R R u R v Rek k k k k∈ ∈ ∈ Δ ∈ ∈8 2 2 2 2, , , and

VAFy y

yk k

k

=−( )

( )⎛

⎝⎜⎜

⎞

⎠⎟⎟ ⋅max

var ˆ

var[%]1– 100

state estimators with the current system input was done atevery time instant.

2.4 Implementation of LPV-NMPC

As presented in Section 1.2, the proposed LPV identificationalgorithm has been used to model the air path system. Itturned out that using VGT and N as scheduling parameters ρ1and ρ2 is a valuable choice. The MAF model and the MAPmodel each consist of four parallel transfer functions (one foreach input) of the type:

(20)

with the following coefficients:

(21)

In Figure 9 as an example the described structure is depictedfor the MAP part of the LPV model. The model inputs are the

a a VGT N a a VGT a N a Vi i i i i i( , ) ( , )ρ ρ1 2 1 2 3 4= = + ⋅ + ⋅ + ⋅ GGT N

b b VGT N b b VGT b Ni i i i i

⋅

= = + ⋅ + ⋅ +( , ) ( , )ρ ρ1 2 1 2 3 bb VGT N

i

i4 ⋅ ⋅

∀ ∈ [ ]1,2,3,4

G z

b

z aii

i

( , )( , )

( , )ρ

ρ ρρ ρ

=+

1 2

1 2

Figure 10

Used input excitation signals for the LPV air path modelidentification.

VGT

EGR

mf

N

MAP

bMAP1(VGT,N)

+ aMAP(VGT,N)

bMAP2(VGT,N)

+ aMAP(VGT,N)

bMAP3(VGT,N)

+ aMAP(VGT,N)

+

bMAP4(VGT,N)

+ aMAP(VGT,N)

z

z

z

z

6000 400200

80

60

40

20

100

0

Time (s)6000 400200

80

60

40

20

100

0

Time (s)

EG

R p

ositi

on (

%)

VG

T p

ositi

on (

%)

6000 400200

3000

2000

1000

4000

0

Time (s)6000 400200

10

20

30

0

Time (s)

N (

rpm

)

mf (

mg/

st)

Figure 9

Structure of the LPV model for MAP.

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implemented on the rapid prototyping hardware system. Forsolving the resulting QP online, the software packageqpOASES was applied (Ferreau, et al., 2008), whereas in thegeneral nonlinear case a nonlinear program (NLP) instead ofa QP has to be treated. However as already mentioned above,for the considered case of the LPV model formulation theproblem can be stated as a QP at every time instant, whichagain can be treated by a standard QP solver. Further infor-mation about different solution variants for special QPs canbe found in Ferreau et al. (2006, 2008). The inputs (EGR,VGT, N, mf ) and outputs (MAF, MAP) of the engine weremeasured together with the model states by the EKF. Theinitial setup of tuning parameters is shown in Table 3.

TABLE 3

NMPC tuning parameters

Property Value

QNMPC [ 1 0; 0 1 ]

RNMPC [ 20 0; 0 20 ]

ΔuNMPC [– 5 – 5; 5 5]

NPH [60, 80, 100, 120, 140, 160]

NCH 3

REKF [1 0; 0 1]

QEKF *

Ts 0.05 s

* The last two terms of QEKF (errors of MAF and MAP) QEKF (9,9) and QEKF (10,10)are equal to 106, the other terms on diagonal are 10-2, for the purpose of offset freetracking.

Note that, due to the nonlinearity, it is difficult to choosethe exact horizons that ensure feasibility and stability of theoptimization problem. In order to enforce the closed-loopstability, a conservative choice is to set the prediction horizonmuch longer than the control horizon (see e.g. Chen andAllgöwer, 1998).

3 EXPERIMENTAL RESULTS

3.1 Multi-linear MPC Results

Figure 12a depicts a comparison of the multi-linear MPCagainst a standard control setup during reference changes ofMAF. As it can be seen, in particular during the transients theMPC shows a much faster reaction and consequently achievesa significant improvement of the dynamical response. InFigure 12b steps of engine speed during constant MAP/MAF-references were applied. The multi-linear MPC was able tosatisfactorily suppress the output impact of the disturbancesteps, also during region transitions.

3.2 NMPC Results

3.2.1 Choice of Control and Prediction Horizon

The computational burden is an important issue for the realtime implementation. In particular the choice of the controlhorizon has a high impact on the required computation time,whereas the feasible maximum was limited to NCH = 3. To

Figure 11

Validation result for MAP and MAF.

Measured

Simulated

Measured

Simulated

850 900800750700

MA

P (

hPa)

1600

900

1000

1100

1200

1300

1400

1500

Time (s)850 900800750700

MA

F (

kg/h

)

180

0

20

40

60

80

100

120

140

160

Time (s)

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X Wang et al. / A Design Framework for Predictive Engine Control 609

analyze the significance of the prediction horizon, severalMPCs with equal NCH and different NPH were tested (seeTab. 3). The comparison was done on a test sequence and thetracking errors for MAF and MAP were used for evaluation(relative sum of squared errors). Figure 13 depicts the changein performance with different prediction horizons, whereasNPH = 120 gave the best results for the considered case4.

3.2.2 Performance Comparison of NMPC to the StandardControl Setup

To show the feasibility of the Nonlinear MPC approach, itwas compared against the standard control setup during refer-ence changes of MAF and MAP. Figure 14 depicts the resultsand Table 4 presents a comparison of the cumulative trackingerror during the changes. Although a difference in the controlsignals can be noticed, the overall tracking performance issimilar – while NMPC had advantages in MAF tracking, the

Measured

a) b)

References

Measured References

EGR VGT

5 10 15 20 25 30 35 40380

400

420

440

460

480

MA

F (

mg/

st)

5 10 15 20 25 30 35 401030

1040

1050

1060

1070

1080

MA

P (

hPa)

5 10 15 20 25 30 35 400

50

100

EG

R V

GT

(%

)5 10 15 20 25 30 35 40

1000

1200

1400

1600

Sp

ee

d (

rpm

)

5 10 15 20 25 30 35 40

2

1

3

4R

egio

n

Time (s)

450

500

3500

400

MA

F (

mg/

st)

5 10 15 20 25

Time (s)

MPCSetpointSTD

30 35 40 45

1075

1090

10700

1080

1085

MA

P (

hPa)

5 10 15 20 25 30 35 40 45

100

00

50

EG

R p

ositi

on (

%)

5 10 15 20 25 30 35 40 45

50

200

30

40

VG

T p

ositi

on (

%)

5 10 15 20 25 30 35 40 45

MPCSetpointSTD

MPC

STD

MPC

STD

70 80 90 100 110 120 130 140 15060

Vls

(su

n of

qua

drat

ic e

rror

)

100

30

90

80

70

60

50

40

NPH160

Vls MAF

Vls MAP

Figure 12

Multi-linear MPC evaluation during reference changes a) and region transitions b).

Figure 13

Relative sum of squared errors Vls for the tracking of MAPand MAF as a function of the prediction horizon (NCH = 3).

4 Note that increasing the prediction horizon not necessarily improves thecontrol performance of MPC.

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50 10 15 20 25 30 35 40 45

120

110

100

90

80

70

100

80

60

40

20

0

1300

1250

1200

MA

F (

kg/h

)E

GR

/VG

T (

%)

MA

P (

hPa)

N/m

f

1150

1100

24

22

20

18

16

Time (s)

50 10 15 20 25 30 35 40 45Time (s)

NMPC

STD

REF.

N in 100 rpm

mf in mg/cyc

EGRNMPC

EGRSTD

VGTNMPC

VGTSTD

50 10 15 20 25 30 35 40 45Time (s)

50 10 15 20 25 30 35 40 45Time (s)

Figure 14

Comparison of NMPC with the standard setup for a reference change.

standard control had a slightly better MAP tracking result.Here two issues have to be mentioned:

– The focus of the LPV-NMPC approach was on providinga basic and straight forward method. Therefore, by intentionperformance was sacrificed in order to keep the complexitylow (e.g. by identifying very simple LPV models).

– The comparison of both controllers was done in a regionwhere the standard calibration already performed well.Obviously any controller cannot rule out the physics – ifthe standard setup is already close to the optimum, even thebest NMPC is not able to show significant performancegains.

TABLE 4

Tracking performance comparison (sum of relative quadratic error)

NMPC STD

eMAF 77% 100%

eMAP 107% 100%

3.2.3 Performance Comparison of NMPC to MPC

In a next step the NMPC is compared to a single linear MPCto provide insight in the difference between both approaches

and information whether the nonlinear extension leads to bet-ter results than a single linear MPC. To this end the samesetup was used for a linear framework, i.e. equal weights andhorizons in the formulation (1). Even the same identifiedmodel as for the NMPC was applied, however for the linearplant model the parameters (coefficients ai and bi of Eq. 20)were kept at constant values. Moreover, it should be notedthat during this comparison only one linear MPC was usedand not a multi-linear MPC as presented in Section 3.1.

TABLE 5

Tracking performance comparison (sum of relative quadratic error)

NMPC Linear MPC

eMAF 49% 100%

eMAP 14% 100%

In Table 5, the tracking performance for both target quan-tities is shown for the reference change depicted in Figure 15.Obviously, the single model of the linear MPC represents thenonlinear system well only in a restricted operating range.Therefore, the control performance of the NMPC is significantlybetter than the MPC during the considered transient. Here itshould be kept in mind that even the steady state gains

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0 8070605040302010 90

120

110

100

90

80

70

60

50

100

0

1500

1400

1300

1200

1100

80

60

40

20

22

20

18

16

1000

24

14

Time (s)0 8070605040302010 90

Time (s)

0 8070605040302010 90Time (s)

0 8070605040302010 90Time (s)

MA

F (

kg/h

)E

GR

/VG

T (

%)

MA

P (

hPa)

N/m

f

NMPC

STD

REF.

EGRNMPCEGRMPC

VGTNMPCVGTMPC

N in 100 rpm

mf in mg/cyc

Figure 15

Comparison of NMPC with MPC for a reference change.

change over the operating range (see. Fig. 1) and if such achange occurs the linear MPC will use an incorrect model forprediction, thus leading to a poor performance (e.g. betweent = 37 s and t = 53 s).

CONCLUSIONS

Two different frameworks for Model Predictive Control of aDiesel engine air path have been presented, namely a multi-linear MPC and a NMPC approach. The main idea was topresent practically relevant methods providing a simple andstraight forward setup and yielding a similar performancethan a with a huge effort heuristically determined standardcontrol setup.

Compared to the standard control in the evaluation scenarioan improvement of the multi-linear MPC tracking perfor-mance was achieved especially during fast transients. Adrawback of the multi linear MPC is that all controllers mighthave to be designed separately to reach the best performanceand therefore increase the tuning effort.

For the NMPC strategy simple Linear Parameter Varyingmodels proved sufficient to capture the essential nonlinear-ity of the system. The LPV identification as well as theNMPC were successfully applied to the engine and show a

similar performance of the MAF/MAP tracking than thewell working standard control. Although further improve-ments would be possible with the NMPC setup, in view ofthe lowest feasible complexity, performance was sacrificedto keep the tuning effort as low as possible. In this way,predictive control strategies (NMPC and multi linearMPC) might become attractive for a possible future use inproduction engine control.

The future work will focus on a further enhancement ofthe LPV identification scheme and a possible self tuningimplementation of the NMPC. With the continuouslyincreasing computational power of the available hardware,more complex model structures and also higher controlhorizons for the MPC can soon be tested, which lets usexpect an increase in the possible performance comparedto the current implementation. Another interesting and pos-sible valuable extension of the proposed framework will bethe combination with a moving horizon observer for stateestimation.

ACKNOWLEDGMENTS

The authors gratefully thank the support from RichardFuerhapter during the engine tests as well as Hans Joachim

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Ferreau for the QP solver issues. The sponsoring of this workby the COMET K2 Center “Austrian Center of Competencein Mechatronics (ACCM)” is gratefully acknowledged. TheCOMET Program is funded by the Austrian FederalGovernment, the Federal State Upper Austria and the ScientificPartners of ACCM.

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Final manuscript received in January 2011Published online in September 2011

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