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An explicit model predictivecontrol framework for

turbocharged diesel engines

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Citation: ZHAO, D. ... et al., 2014. An explicit model predictive controlframework for turbocharged diesel engines. IEEE Transactions on IndustrialElectronics, 61 (7), pp.3540-3552.

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1

An Explicit Model Predictive Control Frameworkfor Turbocharged Diesel Engines

Abstract—The turbocharged diesel engine is a typical multi-input multi-output (MIMO) system with strong couplings, ac-tuator constraints, and fast dynamics. This paper addresses theexhaust emissions regulation in turbocharged diesel engines usingan explicit model predictive control (EMPC) approach, whichallows tracking of the time-varying setpoint values generatedby the supervisory level controller while satisfying the actu-ator constraints. The proposed EMPC framework consists ofcalibration, engine model identification, controller formulation,and state observer design. The proposed EMPC approach has alow computation requirement and is suitable for implementationin the engine control unit (ECU) on board. The experimentalresults on a turbocharged Cat® C6.6 diesel engine demonstratethat the EMPC controller significantly improves the trackingperformance of the exhaust emission variables in comparisonwith the decoupled single-input single-output (SISO) controlmethods.

Index Terms—Explicit model predictive control, turbochargeddiesel engines, exhaust emissions regulation

NOMENCLATURE

N Engine speed.Wf Engine fueling rate.Wc Compressor air mass flow rate.Wegr EGR mass flow rate.We Engine total mass flow rate.Wt Turbine gas mass flow rate.Pc Compressor power.Pt Turbine power.pin Intake manifold pressure.pexh Exhaust manifold pressure.pa Ambient pressure.Vin Intake manifold volume.Vexh Exhaust manifold volume.Tin Intake manifold temperature.Texh Exhaust manifold temperature.Ta Ambient temperature.F1 Burnt gas fraction in the intake manifold.Ntc Turbocharger shaft rotating speed.λa In-cylinder air-fuel ratio.Qpilot Amount of fuel in the pilot injection.Qmain Amount of fuel in the main injection.Qtotal Qpilot +Qmain.θsoi Crank angle of start-of-injection.prail Fuel rail pressure.Rfuel Fuel ratio, defined as Qpilot/Qtotal.ρnox Density of NOx.θca50 Crank angle of 50% heat release.χegr EGR valve position.χvgt VGT vane position.ηm Turbocharger mechanical efficiency.Rg Specific gas constant.

τ Turbocharger time constant.nc Compressor isentropic efficiency.nt Turbine isentropic efficiency.cp Specific heat at constant pressure.γ Specific heat ratio, 1.4 for air.µ γ−1

γ .

I. INTRODUCTION

D IESEL engines are widely used in heavy duty vehi-cles and off-road applications, due to their merits of

high thermal efficiency. However, there is a tradeoff betweenefficiency and further reduction of exhaust emissions suchas nitrogen oxides (NOx) and particulate matter (PM). Asthe increasingly tighter pollution standards are required forvehicles with internal combustion engines (ICE), more exhaustgas regulation systems are fitted to the diesel engines, such asthe diesel particulate filter (DPF) and the selective catalyticreduction (SCR) [1], [2]. On a higher level, developing moreeffective control strategies on the fuel injection and air pathsubsystem are now essential for improving exhaust emissions[3].

Modern diesel engines are normally equipped with a vari-able geometry turbocharger (VGT) and exhaust gas recircu-lation (EGR) valves [4], [5]. Turbocharging the diesel enginereduces fuel consumption, and together with EGR enables areduction in exhaust emissions, in particular NOx [6]. VGTand EGR actuators are strongly coupled because they are bothexposed to the exhaust gas. They should be well tuned for reg-ulating the intake mass flow for combustion with the desiredburnt gas fraction F1 to minimize NOx, without violating theair-fuel ratio λa associated with PM generation. Unfortunately,the performance variables, F1 and λa, are unmeasurable usingnormal sensors. As a consequence, two intermediate variables,Wc and pin are introduced as the new controlled variables,which are closely related with the previous ones [7]–[9]. Theregulation of F1 and λa is correspondingly transformed tothe control of Wc and pin, which are measurable but exhibitnonlinear dynamics due to the coupling between VGT andEGR actuators.

Besides the air path, the exhaust emissions of diesel enginesare also massively affected by the inputs on the fuel path,such as the crank angle of start-of-injection θsoi, fuel railpressure prail, fuel ratio Rfuel, and dwell time between thepilot fuel injection and main fuel injection [10], [11]. Themanipulation of these variables is a challenging task, dueto the strong coupling that affects fuel injection. Moreover,a suitable mathematical model of the engine fuel path islacking. Nowadays, highly sophisticated measurement deviceshave been developed and implemented on diesel engines, such

2

Supervisory Controller

N

fW Air Path Controller

State Observer

Fuel Path Controller

State Observer

Automotive Engine

Lower Level Controller

Fig. 1. Hierarchical control structure of the automotive engines

as the air mass flow sensor, boost pressure sensor, and railpressure sensor. However, on the system control level, mostof the fuel path subsystems are feedforward control systems,which implies that the effects of the engine dynamics onexhaust emissions cannot be compensated simply by adjustingthe fuel injection [12].

A typical automotive engine controller can be presentedin a hierarchical structure, as shown in Fig. 1. The super-visory controller determines the time-varying setpoint valuesof the lower level controller according to the engine states, interms of fuel economy [13], exhaust emissions [14], or theirweighted cost function [15]. Normally the engine speed N andthe fueling rate Wf are selected to determine operation points.The lower level controller consists of the fuel path controllerand the air path controller, whose main functions are regulatingthe fuel injection and the fresh air boosting respectively. Thelower level controllers regulate the performance variables totheir desired setpoint values distributed by the supervisorycontroller, where the unmeasurable engine states are estimatedusing state observers.

Focusing on the lower level controllers, most of thecommercial ECUs use SISO proportional-integral-differential(PID) controllers in diesel engine air path control, where oneregulates the Wc by tuning χvgt, while another regulates thepin by tuning χegr. However, with the increasingly stricteremission standards, it is more difficult for the decoupledSISO methods to meet the setpoint values tracking withoutconsideration of coupling [16]. Therefore, developing controlalgorithms that can deal with both sets of nonlinear dynam-ics is required. For the ability to handle the constraints onmanipulated variables in MIMO systems, model predictivecontrol (MPC) is one of the most promising control strategiesin industrial applications [17]–[23]. Compared with linearquadratic regulators (LQR) that typically optimize the systemperformance around a given initial state, MPC optimizes ateach time step, and results in higher flexibility in dealing withthe constraints on inputs, outputs, and states. Several typesof MPC methods including generalized predictive control(GPC) [24], nonlinear MPC (NMPC) [7], [25], [26], andadaptive predictive control [27] have been applied in theengine control field. However, the real-time implementation ofMPC brings a high computation burden, due to a finite horizonoptimal control problem that is solved in each sampling period[28]. Higher computation burden brings higher requirementson the processing power of ECUs, and furthermore, higher

VGT

Intake Manifold

Exhaust Manifold

Compressor

EGR Valve

Fuel Tank

fW

Cylinders

tcN

N

cWegrW

tW

Turbine Shaft

Crank Shaft

1,,,, FTpW ainine

exhexh Tp ,

Fig. 2. Turbocharged diesel engine

requirements on the cost. Therefore, high computational costcontrol methods are unsuited to production diesel engines.Recently, the explicit MPC (EMPC) has attracted interestin engineering with the potential ability to reduce hardwarecost and online computation time [29]–[32]. In the EMPCapproach, the optimal control laws are pre-computed andread from a look-up table, resulting in reduced computationalresource requirements [33]–[35].

In this paper, several aspects of establishing an EMPCframework on diesel engines are proposed. The contributionsof this paper mainly focus on the complete procedure informulating an EMPC controller for exhaust emissions regu-lation according to the development experience, particularlyon the general method of obtaining the multi-linear dieselengine model and building an augmented EMPC controller.Experimental results support the proposed method.

The paper is organized as following. After the introductionin section I, the diesel engine model is described in sectionII. The EMPC control framework is formulated in section III.The experiment results are stated in section IV. Finally, theconclusions are summarized in section V.

II. SYSTEM DESCRIPTION

The schematic of a turbocharged diesel engine is shown inFig. 2. The turbocharger consists of the VGT and compressor,where the VGT takes energy from the exhaust gas to powerthe compressor which is mounted on the same shaft and inturns compresses more fresh air, resulting in higher pressurein the intake manifold. The EGR loop feeds back part ofthe burnt exhaust gas to the intake manifold to dilute thefresh air, causing lower combustion peak temperature andlowering NOx concentration. The air and burnt gas are mixedand pumped into the cylinders from the intake manifold. Asthe piston reaches the top of its compression stroke, fuel isinjected into the cylinders and burnt in the now compressedair, producing torque on the crank shaft. The hot burnt exhaustgas is pumped into the exhaust manifold from the cylinders,where part of the exhaust gas flows out of the engine through

3

Engine ModelEmission

Model

N

cW

1FNOx

PM

Measurements

Engine Performance

Variables

Exhaust Performance

Variables

LT

Control Signals

External Signals

egr

vgt a

inp

Fig. 3. Signals employed in the air path [5]

the VGT, and the other part is recirculated back to the intakemanifold through the EGR valves.

Since the EGR valves and VGT vanes are both in theexhaust gas flow, there is a strong coupling between the EGRflow and VGT flow. The reduction of the pumped fresh airin the intake manifold leads to an increase of PM emissionswhile a low value of EGR flow fraction results in higher NOx

emissions. The dilemma is known as the NOx-PM tradeoff.Strong coupling also exists in the fuel path control variables.

Generally speaking, the dynamics of the fuel path are oftenpresented by empirical calibration models [36], [37]. Compar-ing with the air path with relatively slow dynamics due tothe turbocharger dynamics and inertia, the fuel path has fasterdynamics owing to its more direct influence on the cylinderprocesses [38], [39].

A. Diesel Engine Air Path

The signals employed in the air path are shown in Fig. 3.The exhaust performance variables of the diesel engine aredefined as NOx and PM, while their reduction is achieved bykeeping a sufficient large value of F1 and λa in the intakemanifold, respectively. Therefore, F1 and λa are employed asthe engine performance variables, defined by

F1 =Wegr

Wc +Wegrλa =

Wc

Wf. (1)

Precise tracking of F1 and λa to their optimal setpoint valuesF ∗1 and λ∗a is desired, where F ∗1 and λ∗a can be obtained using asupervisory controller. In conventional environments, Wc andpin are measured by the compressor air flow sensor and theboost pressure sensor, respectively, to provide the informationabout the intake gas process [5]. Wc and pin have strongcoupling and are regulated by tuning both of the VGT vaneand EGR valves.

Ignoring the slow deviation of Tin and Texh, a third-ordernonlinear control-oriented air path model is formulated withrespect to pin, pexh, and Pc:

pin =RgTinVin

(Wc +Wegr −We), (2a)

pexh =RgTexhVexh

(We −Wegr −Wt +Wf ), (2b)

Pc =1

τ(ηmPt − Pc). (2c)

Wc is related to Pc with

Wc =ηccpTa

Pcpµin − 1

, (3)

while Pt can be expressed by Wt:

Pt = ηtcpTexh(1− p−µexh)Wt. (4)

The mass flow rate through the EGR valves can be obtainedby the actuator map given by:

Wegr =

Aegr(χegr)pexh√RgTexh

Ψ

(pinpexh

),

if pin < pexh;

0, if pin = pexh;

Aegr(χegr)pin√RgTin

Ψ

(pexhpin

),

if pexh < pin;

(5)

where

Ψ(pipj

) =

γ0.5

(2

γ + 1

)(γ+1)/(2(γ−1))

,

ifpipj≤(

2

γ + 1

)γ/(γ−1)

;√√√√ 2γ

γ − 1

((pipj

)2/γ

−(pipj

)(γ+1)/γ),

ifpipj

>

(2

γ + 1

)γ/(γ−1)

;

(6)

and Aegr is the EGR effective flow with a quadratic functionwith respect to χegr. The turbine mass flow rate is representedby a modified version of the orifice equation:

Wt = Avgt(χvgt)pexh√RgTexh

Φ

(papexh

, χvgt

), (7)

where Avgt is a quadratic function with respect to χvgt andΦ(

papexh

, χvgt

)is obtained from a VGT mass flow rate map.

The mass flow maps on Wt and Wegr together with theefficiency maps on ηt and ηc are all generated from calibrationtests. The reader can refer to [4] for further details on the airpath dynamics.

B. Diesel Engine Fuel Path

The predominant function of the fuel path is to satisfythe power requirement of the diesel engine under varyingloads. The input/output signals employed in the diesel enginefuel path emissions regulation are illustrated in Fig. 4, wherethe exhaust performance variables are Texh, ρnox, θca50, thecontrol signals are θsoi, prail, and Rfuel, and the measurabledisturbances are N and TL. The unit of θsoi is the degrees aftertop dead center (◦CA ATDC). The exhaust emissions PM iseffected by Texh and θca50 [40]. For ease of understanding.graphical illustrations of the definition of θca50 and the fuelinjection sequence in a sampling period are given in Fig. 5(a)Fig. 5(b), respectively.

4

Diesel Engine

N

Exhaust Performance

Variables

LT

Control Signals

External Signals

railpexhTsoi

fuelR

nox

50ca

Fig. 4. Signals employed in the fuel path

Crank Shaft Position (°CA ATDC)

Accu

mul

ated

Hea

t Rel

ease

50%

100%

0-10-20-30 10 20 30 40 50

50ca

(a) definition of θca50

mainpilot

pilotQ mainQ+ totalQ

pilotduration

Start of injection

mainduration

pilotdwelling time

(b) fuel injection sequence

Fig. 5. Demonstration of variables used in the diesel engine fuel path

C. Linearized Model

Generally, Wf reveals the influence of the load torque TLon the engine. At an engine determined operation point withfixed N and TL, the diesel engine air path and fuel path canbe modeled as linear systems in form of discrete state spaceequations: {

x(k + 1) = Ax(k) +Bu(k)y(k) = Cx(k)

, (8)

where x ∈ Rn is the state vector, u ∈ Rm is the input vector,y ∈ Rp is the output vector, and (A,B) is a controllable pair.Both inputs and outputs of the system are constrained withrespect to their maximum and minimum bounds:

umin ≤ u ≤ umax, ymin ≤ y ≤ ymax.

The coefficient matrices A, B, and C are obtained usingsystem identification at the selected operation point. The inputsand outputs of the linearized model for the air path are selectedas

u = [ χegr χvgt ]T

y = [ Wc pin ]T, (9)

while the inputs and outputs for the fuel path are chosen as

u = [ θsoi prail Rfuel ]T

y = [ Texh ρnox θca50 ]T. (10)

According to different engine speed and load torque, theengine operation region is segregated into several subzones[29]. The number of subzones depends on the precision ofthe required identified model. Within each subzone, a linearmodel is identified at the geometrical central point, as shownin Fig. 6.

Engine speed

Load

torq

ue

0

Maximum torque

Fig. 6. Segmentation of the engine operation region

III. EMPC FRAMEWORK DESIGN

As shown in Fig. 7, the implementation of EMPC can bedivided into two stages: offline and online. In the offline stage,the diesel engine model is identified using the calibration data.Based on the identified model, the EMPC control laws are cal-culated via multi-parametric quadratic programming methodand stored in the look-up table within the ECU. In the onlinestage, the diesel engine is controlled by the pre-computedcontrol law, and the feedback states are estimated via the stateobserver, fulfilling the closed-loop control function.

A. Calibration and System Identification

Considering the nonlinear behavior of the diesel engine, itis infeasible to obtain a unified linear model in the engineoperation range. A more practical approach is to identify thepiecewise affine models in smaller operation ranges.

The calibration data set should cover the operation rangeof the diesel engine to be tested. System identification isimplemented based on the calibration data which are separatedinto two parts: training data and validation data. A group ofcandidate models with different orders should be generatedfrom the training data. The one with the highest fitting scorein the validation data is selected as the proper linear model in

Multi-Parametric Programming

System IdentificationCalibration

Diesel Engine

Offline

Online

EMPC control Law Look-up Table

State Observer

Real-Time Engine Control

by ECU

Fig. 7. Implementation procedure of EMPC on the diesel engine

5

0 200 400 600 800 1000 1200−100

−50

0

50

100

Tex

h

time(s)

0 200 400 600 800 1000 1200−400

−200

0

200

ρ nox

time(s)

0 200 400 600 800 1000 1200−20

−10

0

10

θ ca50

time(s)

1200 1400 1600 1800 2000 2200 2400−100

−50

0

50

100

time(s)

Tex

h

RealPredicted

1200 1400 1600 1800 2000 2200 2400−400

−200

0

200

time(s)

ρ nox

RealPredicted

1200 1400 1600 1800 2000 2200 2400−20

−10

0

10

time(s)θ ca

50

RealPredicted

Training data Validation data

Fig. 8. System identification of the diesel engine fuel path

the assigned zone. Generally speaking, the air path model isranging from 2nd to 4th order, while the fuel path model isranging from 3rd to 6th order. As an example, the trainingdata and validation data employed in the fuel path modelidentification, together with the prediction data using theidentified model are illustrated in Fig. 8. The calibration hascontinued for 40 minutes at the operation point of 1550 rpmand 375 Nm, where the first half 20 minutes data are usedfor training, and the latter half 20 minutes data are used forvalidation. By comparing the fitting values, a 5th-order linearmodel with the highest score is selected as the fuel path modelat the specified engine operation point. The red curves showthe predicted data using the identified linear model.

B. EMPC Controller Design

The EMPC controller described in this subsection adopts thelinear MPC technique to achieve tracking of the given outputvariables.

1) Problem Formulation: The setpoints of output variablesare incorporated into the standard EMPC formulation fortracking of time varying reference values. An augmentedformulation of (8) including the input dynamics u(k) =u (k − 1) + ∆u(k) and the setpoints of outputs is representedas:

x(k + 1)u(k)

r(k + 1)

︸ ︷︷ ︸

xk+1

=

A B 00 I 00 0 I

︸ ︷︷ ︸

A

x(k)u(k − 1)r(k)

︸ ︷︷ ︸

xk

+

BI0

︸ ︷︷ ︸B

∆u(k)︸ ︷︷ ︸uk

,

yk = Cxk,

(11)

where xk ∈ Rn+m+p, uk ∈ Rm, yk ∈ Rp, with C =[C, 0, − I]; r(k) is the desired setpoint of y(k).

For a generalized system, the system performance indexwith the initial state xk at time instant k can be specified bya quadratic cost function to be minimized:

minU

J(xk, U) =∥∥xk+Hp

∥∥2

P︸ ︷︷ ︸J(xk+Hp )

+

Hp−1∑i=0

(‖xk+i‖2Q + ‖uk+i‖2R

)︸ ︷︷ ︸

J(xk+i, uk+i)

,

(12a)

s.t. xk+i+1 = Axk+i + Buk+i, (12b)umin ≤ uk+i ≤ umax, i = 0, 1, . . . ,Hc − 1, (12c)uk+i = uk+Hc−1, i = Hc, . . . ,Hp, (12d)ymin ≤ yk+i ≤ ymax, i = 1, . . . ,Hp, (12e)

where ‖x‖2S = xTSx; Hp and Hc are the prediction horizonand control horizon, respectively; xk+i and yk+i denote thepredictions of x and y at time k + i, made at time k,respectively; uk+i is the value of control input u at timek + i; U = [uT

k , . . . , uk+Hp−1]T ∈ RmHp is the controlsequence within which the optimal control inputs be deter-mined; X = [xT

k+1, · · · , xTHp

]T ∈ R(n+m+p)Hp is the vectorof predicted states; J(xk+Hp

) is the terminal penalty function,while J(xk+i, uk+i) is the stage cost at time k + i; Fromphysical point of view, Hc ≤ Hp should be guaranteed.P = PT > 0, Q = QT > 0 and R = RT > 0 are assumed.

2) Multi-Parametric Quadratic Programming: The opti-mization problem (12) can be converted into a quadraticprogramming (QP) formation for which fast and numericallyreliable algorithms are available. The cost function (12a) is

6

re-written in the following quadratic form:

J(xk, U) = XQX + U RU , (13)

where the augmented matrices Q and R are Q =diag(Q, · · · , Q, P ) and R = diag(R, · · · , R), respectively.

For each element xk+i, the evolution of the system (11),i.e. the equality constraint (12b), can be represented by

xk+i = Aixk +

i−1∑j=0

AjBuk+i−1−j , (14)

for i = 1, 2, . . . ,Hp, which means the system states at anytime can be expressed in terms of the initial state xk and theinputs vector U . Therefore, a matrix expression of the systemstates evolution can be derived from (14):

X = Axk + BU , (15)

with

A =

A

A2

...AHc

...AHp ,

B =

B 0 · · · 0

AB B · · · 0...

......

...AHc−1B AHc−2B · · · 0

......

......

AHp−1B AHp−2B · · · B

.

(16)Substituting (15) into (13), the optimization problem can betranslated into solving the following QP problem :

J(xk, U) = xTkY xk + min

U{UTHU + 2xT

kFU}, (17a)

s.t. GU ≤W + Exk, (17b)

where Y = ATQA, H = BTQB + R, and F = ATQB. Thematrices G, W , and E can be obtained from the constraints(12c) and (12e). The reader can refer to [32] for more details.

Introducing z ∆= U +H−1FTxk ∈ RmHp , the optimization

problem (17) can be rewritten as

Jz(xk, U) = minzzTHz, (18a)

s.t. Gz ≤W + Sxk, (18b)

where S ∆= E + GH−1FT, and Jz(xk, U) = J(xk, U) −

12 x

Tk (Y − FH−1FT)xk. The QP problem (18) can be solved

by applying the Karush-Kuhn-Tucker (KKT) optimality con-ditions [41]:

Hz +GT = 0, (19a)λi(Giz −Wi − Sixk) = 0, i = 1, . . . , q, (19b)

λ ≥ 0, (19c)Gz −W − Sxk ≤ 0, (19d)

where λ ∈ Rq denotes the Lagrange multipliers, and q is thenumber of inequalities in (18b). The subscript i denotes thei-th row of the corresponding matrix.

Let z∗(xk) be the optimal solution to (18) for a given xk, theconstraint (19b) is called active if Giz∗(xk)−Wi−Sixk = 0is held. Accordingly, the matrices on the corresponding rows

are denoted as G, W , and S, respectively. Substituting (19b)into (19a), the active Lagrange multiplier λ is solved as:

λ = −(GH−1GT)−1(W + Sxk). (20)

Substituting (20) into (19a) yields

z = H−1GT(GH−1GT)−1(W + Sxk). (21)

It is clear from (20) and (21) that λ and z are affine functionsof xk. Substituting (20) and (21) into (19c) and (19d), theregion satisfying the constraints can be determined by

−(GH−1GT)−1(W + Sxk) ≥ 0, (22a)

GH−1(GH−1GT)−1(W + Sxk) ≤W + Sxk. (22b)

After removing redundant constraints, (22) describes a poly-hedron in the xk-space, denoted as the critical region CR0,which is represented by

CR0 = {xk | Dxk ≤ d,D ∈ RNc×(n+m+p), d ∈ RNc},(23)

where Nc = dim(d) is the total number of inequalities inDxk ≤ d, which is translated from the constraint (17b). Asimilar method is also used on dividing the rest of the region:

Rrest = {xk | Dxk ≥ d}. (24)

As shown in [33], the optimal EMPC control law is a contin-uous piecewise affine function of xk on each divided region:

U∗(xk) = fj xk + gj , j = 1, . . . , Nj , (25)

where Nj is the number of polyhedral sets defined by (23)and (24).

3) Implementation: In the offline stage, the optimal EMPCcontrol laws are computed explicitly. The multi-parametric QPsolver in polyhedral sets partition is briefly summarized asAlgorithm 1.

Algorithm 1 Polyhedral Sets Partition of X

1) Select an initial state x0 ∈ X;2) Compute the optimal solution (z0, λ0) for x0 by solving

(18) according to (19);3) Determine the set of active constraints when z = z0 and

λ = λ0, and build G, W , S;4) Characterize the solution λ(x) and z(x) from (20) and

(21);5) Compute the critical region CR0 (23) from (22);6) Compute the rest of the region Ri = {Dix ≥ di}

⋂X ,

i = 1, . . . , Nc;7) For each new sub-region Ri, partition Ri from step 1).

Once the multi-parametric QP problem is solved offline, theEMPC control law (25) is available explicitly. Only the firstcomponent of the vector U∗(xk) be applied:

uk = [I, 0, . . . , 0] U∗(xk). (26)

The open-source MPT toolbox 1 based on MATLAB is usedto calculate the linear MPC law. On the next step, U∗(xk+1)

1The MPT toolbox is available at http://control.ee.ethz.ch/∼mpt/.

7

Prediction(28)

Correction(29)

Diesel Engine(27)

Measurement

Kalman Filter

Fig. 9. Workflow of the Kalman filter

is found in the pre-calculated look-up table again, and uk+1 isupdated accordingly. There are several quick search methods,such as the sequential search, binary search tree, and traversesearch tree, that can be used in searching EMPC control lawsfrom the look-up tables [42]. It shares the same function asthe normal MPC method, but results in lower hardware costand reduced online computation resources.

A switched EMPC control scheme is adopted in the real-time control. The identified model of each subzone and thepre-computed control laws are stored in look-up tables. Eachsubzone may have an identified model with a different order.Accordingly, the designed EMPC in this subzone would havethe matched dimension to deal with the plant model. Byonline identification of the engine speed and the load torque,the EMPC controller in the corresponding subzone would beenabled.

The main advantage of using the EMPC as the enginecontroller is the ability to handle the control and outputconstraints, so that the exhaust emissions of the diesel enginecan be retained in a reasonable range without violating thepractical limits of actuators.

C. State Estimation

In this subsection, a Kalman filter is selected for stateestimation. The Kalman filter is considered to be the theoptimal recursive data processing algorithm, with the highestefficiency across a range of engineering applications [43]–[45]. Kalman filter uses the state space equation and recursivemethod to observe the states, and has no requirements on thesmooth or time invariant characteristics of the signal. Withoutloss of generality, considering the diesel engine disturbed byGaussian white noises, which are represented by the processerror v(k) and the measurement error w(k), the augmentedsystem model (11) is transformed to{

xk+1 = Axk + Buk + v(k)

yk = Cxk + w(k), (27)

where v(k) and w(k) are independent, and hold the covarianceof Qv and Rw, respectively.

The Kalman filter is composed of two sequential steps:prediction and correction. In the prediction step, the prediction

of the states are

ˆxk+1|k = Aˆxk|k + Buk, (28)

where ˆxk+1|k and ˆxk|k are the estimates of xk+1 and xk bythe given output sequence of [yk, yk−1, · · · ], respectively. Inthe correction step, the updated estimate of xk+1 is obtainedby

ˆxk+1|k+1 = ˆxk+1|k + ∆xk+1, (29)

where

∆xk+1 = Kk+1∆yk+1, (30a)

Kk+1 = Pk+1|kC(CPk+1|kC

T)−1

, (30b)

∆yk+1 = yk+1 − C ˆxk+1|k, (30c)

Pk+1|k = APk|kAT +Qv, (30d)

Pk|k = E

((xk − ˆxk|k

)(xk − ˆxk|k

)T). (30e)

Finally, Pk+1|k+1 is updated:

Pk+1|k+1 = Pk+1|k −Kk+1CPk+1|kCTKT

k+1. (31)

In the diesel engine model, the covariance matrices of theprocess noise and measurement noise are specified as Qv =diag[Qa1 , Q

a2 ] and Rw = diag[Ra1 , R

a2 ] in the air path control

system, respectively; or Qv = diag[Qf1 , Qf2 , Q

f3 ] and Rw =

diag[Rf1 , Rf2 , R

f3 ] in the fuel path control system, respectively.

The workflow of the Kalman filter is illustrated in Fig. 9.

IV. EXPERIMENTAL RESULTS

The performance of the proposed control method is evalu-ated in test-cell experiments. Air path tests and fuel path testsare implemented separately. In the air path test, a comparisonbetween an EMPC scheme and a valves-fixed control schemeis made. In the fuel path test, a comparison of an EMPCscheme and a embedded ECU control scheme is demonstrated.

A. Experiment Setup

The investigated engine is a Cat® C6.6 ACERT™ heavy-duty off-highway engine. The engine is a 6-cylinder, 6.6-literengine equipped with a common rail fuel system. The enginecalibration used for this work produces up to 159 kW at ratedspeed of 2200 rpm with peak torque of 920 Nm at 1400 rpm.The engine has been modified with a high pressure loopEGR and a Honeywell servo-actuated twin-stage VGT. Theengine is fully instrumented to measure air, fuel and coolingsystem pressure, temperatures, and flow rates. Emissions datais gathered principally from AVL 415 smoke meter, AVL439 opacity meter, and a Horiba 9100 exhaust gas analyzermeasuring NOx, CO2, hydrocarbons, and oxygen.

The engine is equipped with a CP Engineering Cadet V14dynamometer control system coupled to a Froude AG400-HS eddy current dynamometer, which are used to managethe engine speed and load torque. The air path control lawis implemented by xPC, which communicates with the ECUvia CAN protocol. The fuel path control system has beendeveloped using FPGA-centric hardware components. The

8

(a) turbocharged diesel engine

PXI real time control

XPC air path control

Opacity meter Test cellmonitor

Fuel meter

Cylinders pressuremeasurement

Dynamometercontrol

(b) operation platform

Fig. 10. The turbocharged diesel engine and its operation platform

control code of the air path and fuel path are both developedbased on Labview 8.5. The key devices in the fuel path controlhardware include a PXI-1042Q chassis with a PXI-8106embedded controller, and three Drivven Diesel Injector DriverModules which slot into an NI-9151 C-series I/O expansionchassis. Fig. 10 shows the turbocharged diesel engine and itsoperation platform. The updated performance of PXI real-timecontrol, xPC air path control, and dynamometer control canbe observed on the screens. PM and fueling rate are measuredby an opacity meter and a fuel meter, respectively, and areboth displayed on the screens.

B. Air Path Control Performance Evaluation

In air path control tests, the commanded engine speed is1550 rpm, while the load torque changes from 375 Nm to475 Nm during a ramp time, tramp. The two selected operationpoints are both located in a subzone, which means the sameidentified model is used. To verify the EMPC controllerperformance under different dynamics, tramp values are setas 10 s and 2 s, respectively. The sampling period of the ECUis 0.1 s. The setpoint value of Wc changes from 8 kg/min to9.5 kg/min with the ramping change of the load torque, whilethe setpoint value of pin changes from 162 kPa to 190 kPa.The constraints on the inputs u = [χegr χvgt]

T are definedas

umin =

[5%45%

], umax =

[15%65%

]. (32)

The prediction horizon and control horizon are set as Hp = 8,Hc = 2, respectively. The fuel path input variables are fixedat θsoi

prailRfuel

=

3 ◦CA ATDC75 kPa

80%

. (33)

The air path control performance evaluation with tramp =10 s is given in Fig. 11, where the engine operation conditionis illustrated in Fig. 11(a). Both of the valves-fixed controlmode and the proposed EMPC control method are tested. In

the valves-fixed control mode, the air path dynamics behaveas an open-loop control system.

Fig. 11(b) and Fig. 11(c) shows a clear improvement ofthe transient dynamics of the EMPC controller over the fixedcontrol mode. This is because the EMPC is a MIMO controlmethod which considers both the internal plant coupling andactuator constraints. In the fixed control mode, χvgt is kept at55% by means of a single loop PID controller, and the χegr isheld at 10%. As a result of the open-loop control, the transientperformance of the air path is slow and the steady state error ishigh. It is clear that the tracking performance on the air pathis faster and more accurate using the EMPC controller, forthe optimal control laws can be selected from pre-calculatedsolutions quickly. The oscillation is due to the noise producedin the VGT and EGR actuators.

Fig. 11(d) shows the tuning process of χvgt and χegraccording to the variations of engine operation points. Withthe increasing setting values of Wc and pin, χvgt increases andχegr decreases, which means more fresh air is added and lessexhaust gas is recirculated, in order to meet a higher λa. Withthe decreasing setting values of Wc and pin, χvgt decreasesand χegr increases, since less fresh air and more recirculatedexhaust gas are required.

In the fast dynamics test, the engine operation condition isillustrated in Fig. 12(a). In the results given in Fig. 12(b) andFig. 12(c), the EMPC method also shows superior performancecompared with the fixed control mode. The correspondinginputs are illustrated in Fig. 12(d), where χegr touched bothof the upper limit and lower limit, which reveals the inputconstraints are active in this test environment. In practice,umax and umin are set according to industrial requirements andsafety risk assessments. These considerations imply that theinput constraints would be activated only under some specialtest conditions.

C. Fuel Path Control Performance Evaluation

In fuel path control tests, the torque changes with the ramp-ing time of 5 s, while the other test environment parameters

9

0 10 20 30 40 50 601500

1550

1600

time (s)

Eng

ine

spee

d (r

pm)

0 10 20 30 40 50 60350

400

450

500

time (s)

Load

torq

ue (

Nm

)

(a) Engine operation condition

0 10 20 30 40 50 607.8

8

8.2

8.4

8.6

8.8

9

9.2

9.4

9.6

9.8

time (s)

Mas

s A

ir F

low

Rat

e (k

g/m

in)

in fixed control mode

setpoint

using EMPC

(b) Wc

0 10 20 30 40 50 60155

160

165

170

175

180

185

190

195

time (s)

Boo

st P

ress

ure

(kP

a)

setpoint

using EMPC

in fixed control mode

(c) pin

0 10 20 30 40 50 6052

54

56

58

60

time (s)

VG

T p

ositi

on (

%)

χvgt

in EMPC method

χvgt

in fixed control mode

0 10 20 30 40 50 606

8

10

12

14

16

time (s)

EG

R p

ositi

on (

%)

χegr

in EMPC method

χegr

in fixed control mode

(d) Valves position

Fig. 11. Diesel engine air path control performance evaluation with tramp =10 s

0 10 20 30 40 50 601500

1550

1600

time (s)

Eng

ine

spee

d (r

pm)

0 10 20 30 40 50 60350

400

450

500

time (s)

Load

torq

ue (

Nm

)

(a) Engine operation condition

0 10 20 30 40 50 607.5

8

8.5

9

9.5

10

10.5

time (s)

Mas

s A

ir F

low

Rat

e (k

g/m

in)

using EMPC

in fixed control mode

setpoint

(b) Wc

0 10 20 30 40 50 60155

160

165

170

175

180

185

190

195

200

time (s)

Boo

st P

ress

ure

(kP

a)

setpointusing EMPC

in fixed control mode

(c) pin

0 10 20 30 40 50 6050

55

60

65

time (s)

VG

T p

ositi

on (

%)

χvgt

in EMPC method

χvgt

in fixed control mode

0 10 20 30 40 50 605

10

15

time (s)

EG

R p

ositi

on (

%)

χegr

in EMPC method

χegr

in fixed control mode

(d) Valves position

Fig. 12. Diesel engine air path control performance evaluation with tramp =2 s

10

0 10 20 30 40 50 60400

410

420

430

440

450

460

470

time (s)

Exh

aust

Tem

pera

ture

(de

gree

C)

setpoint

using EMPC

using ECU control

(a) Texh

0 10 20 30 40 50 60280

290

300

310

320

330

340

350

360

time (s)

NO

x (p

pm)

using ECU control

setpoint

using EMPC

(b) ρnox

0 10 20 30 40 50 6017

18

19

20

21

22

23

time (s)

CA

50 (

degr

ee)

using EMPC

using ECU control

setpoint

(c) θca50

Fig. 13. Diesel engine fuel path control performance evaluation

maintain the same values as those in air path control tests. Theconstraints on the input variables u = [θsoi, prail, Rfuel]

T

are

umin =

−4 ◦CA ATDC55 kPa

65%

, umax =

10 ◦CA ATDC95 kPa

95%

.(34)

0 10 20 30 40 50 600

5

time (s)θ soi (

° CA

AT

DC

)

0 10 20 30 40 50 6060

80

100

time (s)

p rail (

kPa)

0 10 20 30 40 50 6085

90

95

time (s)

Rfu

el (

%)

Fig. 14. Fuel path input variables tuning process in the ECU control scheme

0 10 20 30 40 50 602.8

3

3.2

time (s)θ soi (

° CA

AT

DC

)

0 10 20 30 40 50 6070

75

80

time (s)

p rail (

kPa)

0 10 20 30 40 50 6079.5

80

80.5

time (s)

Rfu

el (

%)

Fig. 15. Fuel path input variables tuning process in the EMPC scheme

The parameters in (12a) are set as Hy = 10, Hc = 2, Q =diag[10, 2, 5], and R = I3. The air path input variables arefixed at [

χegrχvgt

]=

[10%55%

]. (35)

The fuel path outputs from the embedded ECU controllerand the EMPC controller are demonstrated in Fig. 13. Thesetpoint value of Texh changes from 410 ◦C to 460 ◦C withthe ramping change of the load torque. Fig. 13(a) shows thatthe tracking of Texh in the ECU scheme is not precise, and theovershoot is rather large. In comparison, the performance ofEMPC is improved due to the ability to compute the optimalcontrol law online.

The contrast between the respective tracking performancesof NOx is shown in Fig. 13(b), which is more impressive, withits setpoint value changes from 285 ppm to 325 ppm. Becauseof the NOx-PM tradeoff, it is a challenging task to guaranteeprecise tracking of both NOx and PM, simultaneously. TheNOx tracking performance in ECU scheme is very weak sincethe PID controller just uses the feedback output to computethe control law for the corresponding variable, ignoring thecoupling between the multi-output variables. When the loadtorque is changed to 475 Nm, the tracking performance of

11

NOx is slightly improved, but the tracking performance inother stages is far from acceptable. The tracking performanceof NOx using the EMPC controller is much better, becausethe receding horizon is employed in the MPC technique. Theoptimal control action can be implemented on the diesel engineat every sampling instant, leading to a smaller tracking error.

The biggest advantage of the EMPC over the ECU control,in comparing the θca50 tracking performance, is in the reducedsteady state error, while the setpoint value changes from 19 ◦

to 21.25 ◦. This is shown in Fig. 13(c). The integrator in thePID controller was set as a reasonable value, to avoid theresponse lagging due to a too strong integrator according to theexperimental experience. In the formulated model (11) for theEMPC controller design, integration on the step changes of thecontrol law is included, which plays a key role in diminishingthe steady state error.

Fig. 14 and Fig. 15 show the tuning process of input vari-ables using the SISO PID controller and the EMPC controller,respectively. In Fig. 14, there is no variation in θsoi, while prailand Rfuel are regulated in opposite directions, respondingto the change of the setpoint values. Furthermore, Prail andRfuel are very close to the physical constraints defined above.The unexpected phenomenon indicates SISO controllers arenot good choices for the high-order turbocharged diesel en-gines, because there is no explicit relation between manipu-lated variables and exhaust emission performance variables.

In Fig. 15, θsoi, prail, and Rfuel change in a similar wayaround the mean values of 3 ◦CA ATDC, 75 kPa, and 80%,respectively. Only minor variations are introduced in responseto the change on the setpoint values of the exhaust performancevariables with better tracking performance. The variation inthe input variables shows that EMPC is feasible in the wideoperation range of the diesel engine.

V. CONCLUSIONS

An EMPC scheme has been proposed for the diesel en-gine exhaust emissions regulation. Diesel engines demonstratestrong nonlinearities so that the emission variables cannot bewell controlled independently by regulating actuators inde-pendently. An EMPC controller is designed to cope with thehighly nonlinear dynamics in a straight-forward way, whoserobustness is guaranteed by the Kalman filter against the dis-turbances resulting from process and sensor noises. Comparingwith the traditional MPC, the EMPC uses far fewer resources,while maintaining the identical performance as MPC. Thestate-space models of the diesel engine including the airpath and fuel path are identified at several steady operationpoints, under a reasonable hypothesis. An augmented EMPCcontroller is formulated with the increment of control actions,such that the tracking is improved and the steady state errorsare significantly diminished. Experimental results show thatthe proposed EMPC strategy demonstrates high precision intrajectory tracking, as well as the high robustness against thestep change in load torque.

This paper has a potential reference value in the designand implementation of real-time control algorithms for tur-bocharged diesel engines, and in a wider range, for more

industrial application fields. As a topic for future work, thedisturbance compensator on the forward channel should bedesigned to reduce the influence of the noise on VGT andEGR actuators.

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