Research Article Optimal Control of Diesel Engines ...downloads.hindawi.com/journals/mpe/2014/286538.pdf · Research Article Optimal Control of Diesel Engines: Numerical Methods,

Research ArticleOptimal Control of Diesel Engines Numerical MethodsApplications and Experimental Validation

Jonas Asprion1 Oscar Chinellato2 and Lino Guzzella1

1 Institute for Dynamic Systems and Control ETH Zurich Sonneggstraszlige 3 8092 Zurich Switzerland2 FPT Motorenforschung AG Schlossgasse 2 9320 Arbon Switzerland

Correspondence should be addressed to Jonas Asprion jonasasprionalumniethzch

Received 6 October 2013 Revised 20 November 2013 Accepted 20 November 2013 Published 5 February 2014

Academic Editor Hui Zhang

Copyright copy 2014 Jonas Asprion et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

In response to the increasingly stringent emission regulations and a demand for ever lower fuel consumption diesel engineshave become complex systems The exploitation of any leftover potential during transient operation is crucial However evenan experienced calibration engineer cannot conceive all the dynamic cross couplings between the many actuators Therefore ahighly iterative procedure is required to obtain a single engine calibration which in turn causes a high demand for test-bench timePhysics-based mathematical models and a dynamic optimisation are the tools to alleviate this dilemma This paper presents themethods required to implement such an approach The optimisation-oriented modelling of diesel engines is summarised and thenumerical methods required to solve the corresponding large-scale optimal control problems are presented The resulting optimalcontrol input trajectories over long driving profiles are shown to provide enough information to allow conclusions to be drawn forcausal control strategies Ways of utilising this data are illustrated which indicate that a fully automated dynamic calibration of theengine control unit is conceivable An experimental validation demonstrates the meaningfulness of these resultsThemeasurementresults show that the optimisation predicts the reduction of the fuel consumption and the cumulative pollutant emissions with arelative error of around 10 on highly transient driving cycles

1 Introduction

Optimal control of diesel engines becomes increasingly im-portant More stringent emission regulations [1 2] requirethe exploitation of the remaining potential of reducing theemissions not only in stationary operation but also especiallyduring transients [3 4] Simultaneously the fuel consump-tion has to be minimised for economic and environmentalreasons

The classical approach to parameterise an engine controlunit (ECU) is to first derive stationary lookup maps Subse-quently transient corrections are added to these static mapsand heuristic feedforward parts are included This approachwhich relies mainly on engineering experience leads to ahighly iterative procedure when transient driving cycles needto be considered With the increasing complexity of mod-ern engine systems it becomes difficult for the calibrationengineer to conceive all the cross couplings between themany actuators A high demand for test-bench time results

Furthermore each new calibration of an engine requiresthis manual procedure to be executed again Particularly forheavy-duty engines which are usually employed in a varietyof applications a fast and partially automated calibrationprocess is desirable

Model-based approaches and mathematical optimisationtools provide themeans for such an automation For exampleduring the calibration of the ECU the static lookup mapsneed to be optimised This optimisation can be performeddirectly on the engine [5] or mathematical models may beused in place of the physical engine [6ndash8] An overviewof modelling principles suitable for this approach can befound in [9] The information about the driving cycle to beconsidered may be included by a weighting of the operatingrange of the engine This weighting matrix represents thetime during which the engine is operated at a certain enginespeed and load torque Alternatively a few relevant operatingpoints can be identified and the optimisation is restrictedto those points [10 Chapter 7] Identifying the parameters

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2014 Article ID 286538 21 pageshttpdxdoiorg1011552014286538

2 Mathematical Problems in Engineering

of a feedback controller can also be cast as an optimisationproblem Again the solution can be obtained directly onthe engine [11] or by a model-based approach [12] Adaptivemethods aim to automatically improve the engine calibrationduring normal operation [13]

Dynamics may be incorporated either online by model-predictive control [14ndash16] or offline by an optimisation of thecontrol-trajectories over a representative driving profile Thesolution of such optimal control problems (OCPs) whichin the past were also obtained directly on the engine [17]provides implications for well-suited control structures [18ndash21] The optimal control profiles with the correspondingfuel consumption and pollutant emissions can be used as abenchmark to disclose scenarios in which the performanceof the actual control system is suboptimal An analysis ofthese cases provides guidelines for the improvement of thecontrol structure the feedforward control and the reference-trajectory generation Another way of utilising the resultsfrom optimal control is to use this data to train functionapproximators such as artificial neural networks [22 23]

In the literature referenced it is stated that due to thecomplexity of the system OCPs for diesel engines can only besolved over short time horizons of around 3 s to 10 s Alterna-tively model simplifications such as a quasistationary treat-ment are performed However to derive implications for asuitable control structure or to train function approximatorsoptimal solutions over long time horizons are required basedon a detailed fully dynamic engine model Transient drivingcycles which are used during the homologation procedureand thus emulate a representative operation scenario of theengine are typically 20 to 30 minutes long

This paper presents the means to calculate such optimalsolutions The optimisation-oriented modelling of dieselengines is summarised and extended to engineswith exhaust-gas recirculation (EGR) in Section 22 Subsequently theproblem formulation is detailed and the numerical opti-misation methods are described in Sections 23ndash26 Thepresentation of the results is subdivided into three partsFirst the performance of the numerical methods is analysedin Section 31 and subsequently two case studies illustratehow the results from the optimisation can be utilised inSection 32 Finally the experimental validation providedin Section 33 demonstrates the meaningfulness and thesoundness of the approach

A reader interested in the engineering aspects of the prob-lem may focus on Sections 2ndash22 26 and 32 Conversely areader interested in the details of the optimisation methodand its performancemay concentrate on Sections 23ndash25 and31

Throughout the text the derivative with respect to timeis denoted by = 119889119909119889119905 whereas

lowast

119909 designates a flow Lowerand upper bounds are denoted by 119909 and 119909 and 119909 indicates areference value Bold-face symbols such as x or X representvectors and matrices respectively The abbreviations andsymbols are summarised in the nomenclature section at theend of the text except for the specific notation used inSections 24 and 25

Table 1 Main data of the engines used

Engine A BDispl volume 119881

119889(l) 87 30

Cylinders 119899cyl (mdash) 6 4Borestroke (mm) 117135 96104Compression ratio (mdash) 158 176EGR mdash High pressureRated power (kW) 300 (1650 rpm) 130 (2900 rpm)Max torque (Nm) 1720 (1200 rpm) 420 (1400 rpm)

2 Materials and Methods

This section introduces the engines used and presents allmethods required for the optimal control of diesel enginesThe numerical framework for optimal control as well as theengine model is implemented in MATLAB 2013b (Math-Works Natick MA USA) running under 64bit Windows 8All computations are performed on a laptop computer withan Intel Core i7-2760QM CPU running at a clock speed of240GHz

21 Engines The relevant data of the two engines consideredhere are provided in Table 1 Engine A is a heavy-duty enginewhich is used in on-road and off-road applications It doesnot have an EGR system but relies on a selective catalytic-reduction system (SCR) to reduce the engine-out NO

119909

emissions to the level imposed by the legislation Engine Bis a light-duty engine that is used mainly in light commercialvehicles It has an EGR system but no SCR Both engines areequipped with a common-rail injection system and a dieselparticulate filter (DPF) The soot emissions are low enoughsuch that no active regeneration of the DPF is necessaryThe optimisation thus has to maintain a similar level of sootemissions and should not produce large instantaneous sootpeaks

22 Engine Modelling During an optimisation a vast num-ber of model evaluations are performed Either the modelfunction itself is evaluated for example during static calibra-tion or when employing simultaneous methods for optimalcontrol [24] or a forward simulation of the model is used forexample for parametric studies or in the context of shootingmethods for optimal control [25 Sections 32ndash34]Thereforethe model has to be simple to enable a fast execution Inboth cases described partial derivatives need to be calculatedas well If only standard mathematical operations are usedautomatic differentiation is applicable which enables a fastand accurate evaluation of partial derivatives [26 27]

Besides being computationally efficient models apt foroptimisation have to be smooth and quantitatively accuratecapture all relevant qualitative trends and provide plausibleextrapolation These properties qualify the model itself or itssimulation as an ldquoaccurate and consistent function generatorrdquo[25 Section 38] As a fundamental requirement the modeloutputs have to be predicted using the control signalsknown parameters and the ambient conditions only Ideally

Mathematical Problems in Engineering 3

mfcc

prail120593SOI

Air-pathmodel

Cylinder andcombustion

model Tload

lowastmNO119909lowastmsoot

Neng

uVGT

uEF

uEGRx

Figure 1 Structure of the engine model The model for the enginewith EGR does include the grey signals but not the dashed ones

themodel can be identified using a small set ofmeasurementsin order not to cancel the reduction of test-bench time gainedby the application of model-based approaches If the modelis able to predict the effects induced by influences that canhardly be excited on the test bench its value for parametricstudies is further enhanced

Optimisation-orientedmodels are thus classified betweencontrol-oriented and phenomenological models [28 29]Theformer only capture the trends relevant to control while beingas simple as to be implementable on the ECU Due to thepresence of feedback control the quantitative accuracy is ofminor importance and the models have to be valid only ina region around the setpoints of the controller In contrastphenomenological models are used to perform predictiveparametric studies or to analyse specific effects Since a highexecution speed is not critical a single model evaluationtypically takes from a few seconds up to several hours [30]Due to their complexity such models often suffer from errorpropagation [31 32]

221 TheModel Used An optimisation-oriented model for adiesel engine without EGR is presented in [29 33] A classicalmean-value model for the air path is extended by thermalmodels for the intake and the exhaust manifolds Physics-based setpoint-relative models are used for the combustionefficiency the in-cylinder processes and the NO

119909emissions

However since all setpointmaps are replaced by polynomialsthe model is smooth and apt for algorithmic differentiationThe models for the fuel consumption and the NO

119909emissions

capture the influence of the air-path state the injectionpressure and the injection timing that is the start ofinjection (SOI)Themodelling errors are in the range of 06and 5 respectively For the soot emissions the simplemodeldescribed in [34] is used It captures the effects of the injectionpressure and the air-to-fuel ratio (AFR)

Figure 1 illustrates the general structure of the modelOnly the air path is modelled as a dynamic system whereasall phenomena occurring in the cylinders are assumed to beinstantaneous processes The state variables x of the air pathare inputs to this static part of the model The exhaust-gasaftertreatment system (ATS) is included in the model only asa flow restriction in the air pathThe engine-out emissions arelimited directly in the optimisation problem

222 Extension to EGR Several submodels need to be addedor adapted for an engine with EGR The dynamics in theintake manifold comprise balances for the mass the energy

and the burnt-gas fraction The corresponding differentialequations read

119889119901IM119889119905

=

119877120581

119881IM(

lowast

119898CP120599IC +lowast

119898EGR120599EGR minuslowast

119898cyl120599IM) (1a)

119889120599IM119889119905

=

119877120599IM119901IM119881IM119888V

[119888

119901(

lowast



119898cyl120599IM)

minus 119888V120599IM (

lowast

119898CP +lowast

119898EGR minuslowast

119898cyl)]

(1b)

119889119909BGIM

119889119905

=

119877120599IM119901IM119881IM

(

lowast

119898EGR (119909BGEM minus 119909BGIM)

minus 119909BGIMlowast

119898CP)

(1c)

The burnt-gas fraction in the exhaust gas is

119909BGEM =

119909BGIM sdot

lowast

119898cyl + (1 + 120590

0) sdot

lowast

119898fuellowast

119898cyl +lowast

119898fuel

(2)

where 1205900asymp 145 is the stoichiometric AFR

The mass-flow through the EGR valve is modelled by asimplified flow function for compressible fluids [1 Section235]

lowast

119898EGR = 119860EGR sdot119901EM

radic119877 sdot 120599EMsdot Ψ (

119901EM119901IM

) (3a)

Ψ (Π) =radic

2

119896

Πsdot Π

sdot (1 minus

1

119896

Πsdot Π

)(3b)

119860EGR = 119896EGR1 sdot 119906EGR + 119896EGR2 sdot 119906119896EGR119890EGR

(3c)

The factor 119896Π

asymp 104 accounts for flow phenomena thatchange the effective pressure ratio over the EGR valve Themodel for the EGR cooler has the same structure as theintercooler model presented in [33]

An exhaust flap (EF) is installed directly after the turbineIts purpose is to choke the flow of exhaust gas such that thepressure in the exhaust manifold increasesThis higher back-pressure enables higher EGRmass-flowsThe EF is modelledby a factor that reduces the effective opening area of the ATS

119909EF = 1 minus 119896EF1 sdot 119906119896EF2EF

(4)

Due to the lower oxygen concentration when EGR isapplied the combustion efficiency is slightly reduced Thefactor

120578

120585O2

= 1 minus 119886

120585O2

(119873eng 119898fcc) sdot1003817

1003817

1003817

1003817

1003817

120585O2

minus

120585O2

1003817

1003817

1003817

1003817

1003817

119887120585O2 (5a)

is appended as an additional multiplicative efficiency reduc-tion The reference oxygen mass-fraction before the com-bustion 120585O

2

as well as the exponential factor 119887120585O2

are scalarparameters The efficiency reduction is a function of theengine operating point

119886

120585O2

= 119886

0+ 119886

1sdot 119873eng + 119886

2sdot 119898fcc (5b)


The ignition-delay model also has a multiplicative struc-ture The additional correction factor

120591

120585O2

= 1 minus 119896lin sdot (120585O2

minus

120585O2

)

minus 119896quad (119873eng 119898fcc) sdot (120585O2

minus

120585O2

)

2

(6a)

is introduced where the coefficient of the second-order termis again a function of the engine operating point

119896quad = 119896quad0 + 119896quad1 sdot 119873eng + 119896quad2 sdot 119898fcc

+ 119896quad3 sdot 1198732

eng + 119896quad4 sdot 1198982

fcc + 119896quad5 sdot 119873eng sdot 119898fcc

(6b)

TheNO119909model inherently accounts for the change of the

temperature and of the composition of the intake air by theEGR However since the combustion speed is changed by thereduced oxygen availability a smaller region in the cylinderreaches a temperature that is sufficiently high for thermalNO119909formation This effect is found to depend on the engine

speed and therefore the factor

119909O2

= 120585

119896O2

sdot119873engO2

(7)

is applied to the formation volume in the original NO119909model

[29]The effect of the EGR which reduces the NO119909emissions

by a factor of up to 18 is predicted by the model with anaverage magnitude of the relative error of 8

The simple soot model that is suitable for engine A doesnot yield plausible results for engine B with EGR Thereforeno model for the soot emissions is used In the OCP thecorresponding limit is replaced by a lower bound on the AFR

23 Numerical Optimal Control A general formulation of anOCP reads

minx(sdot)u(sdot)

int

119879

0

119871 (x (119905) u (119905) 120587 (119905)) 119889119905 (8a)

st x (119905) minus f (x (119905) u (119905) 120587 (119905)) = 0 119905 isin [0 119879]

(8b)

int

119879

0

g (x (119905) u (119905) 120587 (119905)) 119889119905 minus g le 0(8c)

c (x (119905) u (119905) 120587 (119905)) le 0 119905 isin [0 119879] (8d)

x (119905) le x (119905) le x (119905) 119905 isin [0 119879] (8e)

u (119905) le u (119905) le u (119905) 119905 isin [0 119879] (8f)

The goal is to find trajectories for the state variables x(119905) andthe control inputs u(119905) that minimise the integral cost (8a)while satisfying the dynamics of the system (8b) the integralinequality constraints (8c) and the time-variable path con-straints (8d) (The integral cost 119871 is called a Lagrange termEvery differentiable end cost also called a Mayer term canbe replaced by an equivalent Lagrange termThe latter is to bepreferred from a numerical point of view [25 Section 49])

The simple bounds (8e) and (8f) represent the actuatorranges mechanical limits and fixed initial or end conditionsThe system has 119899

119909state variables that is x f x x isin R119899119909 and

there are 119899119906control inputs to the systemuu u isin R119899119906 Several

time-variable parameters 120587 isin R119899120587 may be present Finallythere are 119899

119892integral constraints and 119899

119888path constraints that

is g g isin R119899119892 and c isin R119899119888 respectivelyThemost common approaches to tackle continuous-time

OCPs are outlined in Figure 2 The top level is inspired by[35] Partial overviews are provided in [36ndash38] and [39]presents a brief historical outline Solving the Hamilton-Jacobi-Bellman equation which is a partial differential equa-tion subject to the model equations is practicable only for asystem with few state variables and control inputs Similar toits discrete-time equivalent dynamic programming it suffersfrom the curse of dimensionality for larger systems Likewisethe indirect approach can hardly be applied to large andcomplex problems If the first-order optimality conditionscan be derived analytically still a two-point boundary-value problem (BVP) needs to be solved The dynamics ofthe costate variables however are ill-conditioned and it isimpossible to derive a good initial guess for the general case

Directmethods start by discretising the full OCPAfinite-dimensional constrained nonlinear optimisation problemresults This type of problem is termed nonlinear program(NLP) Due to the performance and the robustness ofcurrent NLP solvers this approach is a means to an efficientnumerical solution of large-scale complex OCPs

Within direct methods two main approaches exist Thesequential approach also called single shooting discretisesonly the control inputs and a forward simulation is usedto evaluate the model Although this method is simpleto implement the resulting NLP is highly nonlinear andsensitive [25 Sections 33 and 34] which limits the lengthof the time horizon Furthermore consistent derivatives haveto be available for the solution of the NLP [25 Section38] which requires the use of custom ODE solvers Finallyinstabilities of the model can lead to a failure of the ODEsolver and thus a premature termination of the optimisation

The alternative to the sequential approach is to discretisethe state trajectories along with the control inputs The entirecontinous-time problem is ldquotranscribedrdquo into a large butextremely sparse NLP which fully includes the discretisedstate trajectories No forward simulation is performed andthe model ODEs are only satisfied after the solution of theNLPThis simultaneous optimisation and simulation resolvesthe problems encountered by the sequential approach Onedrawback is that the accuracy of the solution of the ODEsis coupled to the discretisation of the control inputs Aniterative mesh refinement may be necessary to obtain a suffi-ciently accurate solution A recent overview of simultaneousapproaches is provided in [24 40] while [41] unveils thebeginnings of these methods

231 Direct Collocation Collocation methods are a family ofintegration schemes often used for the direct transcriptionof OCPs [42ndash47] In contrast to other Runge-Kutta schemesthey represent the state trajectories on each integration inter-val as polynomials and thus provide a continuous solution


Numerical optimal control

Hamilton-Jacobi-Bellman equationldquoTabulation in state spacerdquorarr Dynamic programming (discrete systems)minus Curse of dimensionality

Indirect methodsldquoOptimise then discretiserdquorarr Pontryagin solve two-point BVPminus Hamiltonian dynamics ill-conditionedminus Initialisation of the BVP difficultminus First-order optim cond ldquoby handrdquo

Direct methodsldquoDiscretise then optimiserdquorarr Transform into NLP

+ Flexible+ Efficient (sparsity)+ Robust (with respect to

Control parameterisationldquoSequential approachrdquo rarr ODE always feasible+ Simple to implementminus Sensitivity (ldquotail wags the dogrdquo)minus ODE solver has to provide consistent sensitivities

Single shootingOnly discretised

control inputs in NLP

Multiple shootingSmall intervalsrarr Initial state of each interval in NLP

Direct transcriptionApply discretisation scheme to ODErarr All discretised control inputs and state vars in NLP

LocalLow order fine grid

+ Arbitrary mesh and order (local refinement)+ Sparsity

Direct collocationFamilies of implicit Runge-Kutta schemes that represent state trajectory by polynomial rarr continuous solution

Global pseudospectral methodsOne single interval extremely high order

+ Accuracy of integrationminus Grid prescribed poor approximation of nonsmoothness

Pseudospectral patchingFew intervals medium to high order

Hermite-SimpsonLobatto 3rd order

Crank-NicholsonLobatto 2nd order

Euler backwardRadau 1st order

Control and state parameterisationldquoSimultaneous approachrdquo rarr ODE satisfied only after optim+ Many degrees of freedom during optimisation+ Sparsity | + Can handle unstable systemsminus Same discretisation of control inputs and state variables

initialisation)

Figure 2 Overview of the most prominent methods for the numerical solution of optimal control problems

The method chosen here is the family of Radau col-location schemes [48] It provides stiff accuracy and stiffdecay (or L-stability) [48] [49 Section 35] L-stable schemesapproximate the true solution of a stiff system when thediscretisation is refined Furthermore the integral and thedifferential formulations of Radau collocation are equivalent[47] This property is necessary to construct a consistenttranscription of the OCP at hand

Historically two different branches of direct collocationmethods evolved which is indicated in Figure 2 On theone hand low-order methods originated from the forwardsimulation of ODEs A step-size adaptation is used tocompensate for the low order On the other hand pseu-dospectral methods originally evolved in the context of partialdifferential equations within fluid dynamics In their purestform these methods do not subdivide the time horizon intointervals but represent the state variables as single high-order polynomials The order of these polynomials maybe increased to achieve a higher accuracy In the contextof direct collocation these two branches can be cast in aunified framework that allows for an arbitrary order on eachcollocation interval Nowadays approaches that selectivelyrefine the step size and the collocation order represent thestate of the art [50 51]

232 Nonlinear Programming Before the equations fordirect collocation and the structure of the resulting NLP arederived the fundamentals of solving NLPs are summarised

This outline enables the interpretation of the performanceof different solvers on the problem at hand Neither com-pleteness is claimed nor a proper mathematical foundationis followed Several textbooks on the subject are available forexample [52 53]

An NLP is formulated as

min120596

119865 (120596) (9a)

st g (120596) = 0 g isin R119899119892

(9b)

h (120596) le 0 h isin R119899ℎ

(9c)

By the introduction of the Lagrangian function

L (120596120582119892120582ℎ) = 119865 (120596) + 120582

119879

119892g (120596) + 120582119879

ℎh (120596) (10)

the first-order necessary conditions for the NLP (9a)ndash(9c)also known as the Karush-Kuhn-Tucker (KKT) conditionsbecome

nabla

120596L (120596lowast

120582lowast

119892120582lowast

ℎ) = 0 (11a)

nabla

120582119892

L (120596lowast

120582lowast

119892120582lowast

ℎ) = g (120596lowast) = 0 (11b)

nabla

120582ℎ

L (120596lowast

120582lowast

119892120582lowast

ℎ) = h (120596lowast) le 0 (11c)


120582lowast

ℎge 0 (11d)

120582

lowast

ℎ119894sdot ℎ

119894(120596lowast

) = 0 119894 = 1 119899

ℎ (11e)

The matrix of the first partial derivatives of all constraintsis called the Jacobian The second-order sufficient condi-tions require the Hessian of the Lagrangian nabla2

120596L(120596120582

119892120582ℎ)

projected on the null-space of the Jacobian to be positivedefinite This projection is termed the ldquoreduced Hessianrdquo

Equality-constrained problems can be solved by applyingNewtonrsquos method to the nonlinear KKT conditions Aftersome rearrangements one obtains the KKT system

(

nabla119865 (120596119896)

g (120596119896)

) + [

nabla

2

120596L (120596119896120582119892119896) nablag (120596

119896)

nablag(120596119896)

119879

0

] sdot (

120596 minus 120596119896

120582119892

) = 0

(12)

which is solved during each Newton iteration 119896 The solutionof the quadratic program (QP)

minp

1

2

p119879nabla2120596L (120596119896120582119892119896) p + nabla119865(120596

119896)

119879p (13a)

st nablag(120596119896)

119879p + g (120596119896) = 0

(13b)

with p = 120596 minus 120596119896 is equivalent to the solution of (12)

Applying Newtonrsquos method to the KKT conditions thus canbe interpreted as sequential quadratic programming (SQP)

To ensure progress from remote starting points twodifferent globalisation strategies are used The line searchapproach first defines a direction and searches along thisdirection until an acceptable step length is found The mostpopular search direction is the Newton direction defined by(12) or (13a) and (13b) In contrast the trust-region approachdefines a region for example a ball around the currentpoint in which a model for 119865mdashusually also a quadraticapproximationmdashis assumed to be reliable The trust-regionradius is adjusted by assessing the model accuracy observedover the last step In proximity of the solution bothapproaches reduce to the standard Newton iteration on theKKT conditions which exhibits quadratic convergence

Quasi-Newton Approximations In the KKT system (12) orthe QP (13a) and (13b) the exact Hessian of the Lagrangiancan be replaced by a quasi-Newton (QN) approximationThissubstitution is possible since the current Lagrangemultipliers120582119892119896

only occur implicitly in the Hessian itself The basic ideais to utilise the curvature information obtained along theNLPiterations to construct an approximation of the exact HessianThe most prominent method is the limited-memory BFGSupdate [54] named after its discoverers C G Broyden RFletcher D Goldfarb andD F ShannoThis update preservesthe positive definiteness of a usually diagonal initialisation

Inequality Constraints To solve inequality-constrained (IC)problems two fundamentally different approaches are usednowadays On the one hand the idea of SQP can be extendedto the IC case These methods model (9a)ndash(9c) as an IC QP

at each iteration The search direction p119896for a line search is

thus the solution of the QP

minp

1

2

p119879nabla2120596L (120596119896120582119892119896120582ℎ119896) p + nabla119865(120596

119896)

119879p (14a)

st nablag(120596119896)

119879p + g (120596119896) = 0

(14b)

nablah(120596119896)

119879p + h (120596119896) le 0

(14c)

On the other hand interior-point (IP) methods penalise theviolation of the ICs by a logarithmic barrier function Theproblem

min120596

119865 (120596) minus 120591

119899ℎ

sum

119894=1

log (minusℎ119894) (15a)

st g (120596) = 0 (15b)

is solved while the barrier parameter 120591 is decreased itera-tively The solution for one value is used to initialise the nextiteration

Both approaches to handle IC problems exhibit someadvantages but also suffer from specific drawbacksThemaindifference is that within an SQP method the structure ofthe QP approximation changes from iteration to iterationwhereas for IP methods this structure is invariant As aconsequence an IP method can afford to derive a good fac-torisation of the KKT system once to ensure a fast calculationof the Newton steps Direct solvers for large sparse linearsystems are readily available In contrast an SQP methodrelies on a QP solver that detects the set of active ICs for thecurrent QP approximation by itself This solver thus has todeal with a constantly changing problem structure which isusually handled by updates of the initial factorisation

This difference is closely related to the two basicapproaches to solve the KKT system On the one hand thefull-matrix approach relies on a direct symmetric indefinitefactorisation of the KKT matrix In this context varioussoftware packages are available for example MA275797[55] MUMPS [56] or PARDISO [57] On the other handthe decomposition approach calculates and updates the null-space basis matrix However for large sparse problems thereduced Hessian is much denser than the Hessian itselfSince dense linear algebra has to be applied to the reducedsystem this approach is only computationally efficient whenthe number of the degrees of freedom is small

The most important points concerning the advantagesand the drawbacks of the SQP and IPmethods are stated next[58]

(i) It is difficult to implement an exact-Newton SQPmethod The main pitfalls are the nonconvexity ofthe QP subproblems when the exact Hessian is usedand that SQP methods often rely on custom-tailoredlinear algebra The latter limits the flexibility of thosealgorithms to adopt the latest developments in soft-ware and hardware technology

(ii) IP methods are most efficient when relying on theexact second derivatives Furthermore they usually


converge in fewer inner iterations even for very largeproblems and may utilize the latest ldquooff-the-shelf rdquolinear algebra software

(iii) Applying IP solvers to the QPs in an SQP frameworkhad limited success because they are hard to warm-start [25 Section 413] In short an IP solver whichfollows a central path and approaches the constraintsurface orthogonally faces sensitivity problems whenforced to step onto the solution path perpendicularly

NLP Solvers Used The following list characterises the NLPsolvers used here

SNOPT 72 [59] a proprietary solver implementing anSQP algorithm based on a decomposition approach(LU factorisation) and a line-search globalisation Itcannot use exact second derivatives but relies on aBFGS updateIPOPT 3110 [60] an open-source solver implement-ing a primal-dual IP method using a line-searchglobalisation As linear solver MUMPS [56] withMETIS preordering [61] is used IPOPT provides aBFGS update but can also exploit the exact secondderivativesWORHP 12-2533 [62] This solver tackles the QPswithin an SQP framework by an IP solver that can beefficiently warm-started A line-search globalisationand various partitioned BFGS updates are imple-mented The default linear solver MA97 is usedWORHP is free for academic useKNITRO 80 [63] a proprietary solver that pro-vides three algorithms namely an SQP trust-regionmethod and two IP methods The latter rely eitheron a direct solver and a line search or on a pro-jected conjugate-gradient (CG)method to solve trust-region subproblems Exact and QN methods areimplemented

233 Direct Collocation of Optimal Control Problems Radaucollocation represents the state 119909(119905) of the scalar ODE =

119891(119909) on the interval [1199050 119905

119891] as a polynomial say of degree

119904 The time derivative of this polynomial is then equated tothe values of 119891 at 119904 collocation points 119905

0lt 120591

1lt 120591

2sdot sdot sdot lt

120591

119873= 119905

119891 The left boundary 120591

0= 119905

0is a noncollocated point

The notation 119909

119895= 119909(120591

119895) is adopted The resulting system of

equations reads

(

1

2

119904

) asymp [d0

D]⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

=Dsdot(

119909

0

119909

1

119909

2

119909

119904

)

= (

119891(119909

1)

119891 (119909

2)

119891 (119909

119904)

) (16)

Each of these 119904 equations (index 119894) is a sum of linear termsand one nonlinear term

119889

0119894sdot 119909

0+

119904

sum

119895=1

119863

119894119895sdot 119909

119895minus 119891 (119909

119894) = 0 (17)

The collocation nodes are defined as the roots of Legendrepolynomials and have to be computed numerically [64Section 23] Lagrange interpolation by barycentric weightsis used to calculate the differentiation matrix D along withthe vector of quadrature weightsw [65] Here the step lengthℎ = 119905

119891minus 119905

0is assumed to be included in D and w The latter

is used to approximate the definite integral of a function 119892(119905)as int1199051198911199050

119892(119905)119889119905 asymp sum

119904

119895=1119908

119895119892(120591

119895)

The collocation naturally extends to a system of ODEsthat is x = f(x) To simplify the notation the state variablesare assumed to be stacked in a row vector that is x f isin R1times119899119909 Equation (16) becomes a matrix equation in R119904times119899119909

D sdot [

x0

X] = F (X) (18)

where the rows of X and F correspond to one collocationpoint each In turn the columns of X and F represent onestate variable and its corresponding right-hand side at allcollocation points

When the OCP (8a)ndash(8f) is transcribed all functionsand integrals have to be discretised consistently Here 119896 =

1 119898 integration intervals [119905119896minus1

119905

119896] are used with 0 =

119905

0lt 119905

1sdot sdot sdot lt 119905

119898= 119879 The collocation order 119904

119896can be

different for each interval Summing up the collocation pointsthroughout all integration intervals results in a total of 119872 =

119897(119898 119904

119898) discretisation points The ldquolinear indexrdquo 119897 thereby

corresponds to collocation node 119894 in interval 119896

119897 = 119897 (119896 119894) = 119894 +

119896minus1

sum

120572=1

119904

120572 (19)

The following NLP results

minxsdotusdot

119872

sum

119897=1

119882

119897sdot 119871 (x119897 u119897) (20a)

st D(119896) sdot [[

X(119896minus1)119904119896minus1sdot

X(119896)]

]

minus F (X(119896)U(119896)) = 0 119896 = 1 119898

(20b)

119872

sum

119897=1

119882

119897sdot g (x119897 u119897) minus g le 0 (20c)

c (x119897 u119897) le 0 119897 = 1 119872 (20d)

The simple bounds (8e) and (8f) are imposed at each discre-tisation point and are not restated here The vector of theldquoglobalrdquo quadrature weights W results from stacking thevectors of the quadrature weights w(119896) of each interval 119896 afterremoving the first element which is zero

The Lagrangian of the NLP (20a)ndash(20d) is the sum of theobjective (20a) and all constraints (20b) (20c) and (20d)which are weighted by the Lagrangemultipliers120582 To simplifythe notation the Lagrange multipliers are grouped accordingto the problem structure The 119899

119909sdot 119904

119896multipliers for the

discretised dynamic constraints on each integration interval


1 12 13 13 23 33 4

Dynamiccon-straints(k i)

Pathconstraint

4 1

Obj and integconstraints

0 1 2 3 4 5 6 7Linear index l

Figure 3 Sparsity structure of the Jacobian of the NLP resultingfrom the transcription of a continuous-time OCP by Radau colloca-tion Four collocation intervals are used all with collocation order1 except for the third interval which is of order 4 Characteristicallyfor Radau collocation the zeroth point only contributes linearly tothe first dynamic constraint of the first intervalThe variables at eachdiscretisation point are ordered as (u119879

119897 x119879119897)

119879

119896 are denoted by 120582(119896)119889 the 119899

119892multipliers for the integral

inequalities are stacked in 120582119892 and the 119899


path constraints at each discretisation point 119897 are gathered inthe vector 120582

119888119897

The Lagrangian can be separated in the primal variables120596119897= 120596(119896)

119894= (x(119896)

119894 u(119896)119894) The element Lagrangian L

(119896)

119894at

collocation node 119897 reads

L(119896)

119894= 119882

119897sdot 119871 (120596119897)

+ x(119896)119894

sdot (

119873119896

sum

119895=1

119863

(119896)

119895119894120582(119896)

119889119895+ 120575

(119896)

119894

119873119896+1

sum

119895=1

119889

(119896+1)

0119895120582(119896+1)

119889119895)

minus f (120596(119896)119894)120582(119896)

119889119894+119882

119897sdot 120582119879

119892g (120596119897) + 120582119879

119888119897c (120596119897)

(21)

The Lagrangian of the full NLP is obtained by summing theseelement Lagrangians The Hessian of the Lagrangian thus isa perfect block-diagonal matrix with uniformly sized squareblocks of size (119899

119906+ 119899

119909)

Similarly the Jacobian of the objective function and theconstraints exhibits a sparse structure Figure 3 provides anexample Exploiting the fact that only the derivatives of themodel functions at each discretisation point are required toconstruct all derivative information of the NLP is crucial foran efficient solution [66] In fact the least number of modelevaluations possible and thus a perfect sparsity exploitationare achieved by this procedure Shared-memory parallelcomputing can be applied to further speed up the modelevaluations and a sparse matrix format has to be used inorder not to be restricted by memory limitations for largeproblems

The KKT system (12) is constructed from the Hessian ofthe Lagrangian and the Jacobian of the constraintsThereforeit is very sparse in the case of direct collocation and directsolvers are able to efficiently solve this linear system ofequations

24 Mesh Refinement The idea of an iterative mesh refine-ment is to first solve a coarse approximation of thecontinuous-time OCP This solution is used to identifyregions where the discretisation needs refinement Step-sizerefinement for relatively low-order collocation is consideredhere and consistently the error is estimated by the localtruncation error For Radau collocation a more detailedanalysis of the convergence of the transcribed problemtowards the continuous one is provided in [46]

The truncation error 120591 is of order 119904 + 1

x = x(1) + 120591(1) (22a)

120591(1)

= c sdot ℎ119904+1 (22b)Here x denotes the state at the end of the interval thatis x = x(119905

119896+ ℎ) and x(119901) is its approximation using 119901

integration stepsThe factors c depend on themodel functionf and therefore are not constant However over the timehorizon of one integration step assuming a constant valueof c is reasonable (Another common assumption is that thesolution has a ldquomemoryrdquo and thus 119888

119894119896+1119888

119894119896asymp 119888

119894119896119888

119894119896minus1 In

step-size control for ODE solvers the two models may becombined [48])

By subdivision of the interval into 119901 equal steps a moreaccurate approximation of the exact solution is obtained

x = x(119901) + 120591(119901) (23a)

120591(119901)

= 119901 sdot c sdot ( ℎ119901

)

119904+1

=

c sdot ℎ119904+1

119901

119904

=

120591(1)

119901

119904

(23b)

By equating the exact solution x in (22a) and (23a) with119901 = 2 an estimate for the absolute and the relative truncationerror of the original approximation is obtained

120591(1)

asymp

x(2) minus x(1)

1 minus 2

minus119904

120591

(1)

119894=

120591

(1)

119894

119909

(1)

119894

for 119894 = 1 119899

119909 (24)

For the refined approximation the estimate

120591

(119901)

119894asymp

120591

(119901)

119894

119909

(1)

119894

asymp

1

119901

119904

sdot

120591

(1)

119894

119909

(1)

119894

=

120591

(1)

119894

119901

119904

(25)

results The magnitude of this relative error has to be smallerthan the desired relative tolerance 120576rel Solving for 119901 yields thestate-variable individual step refinement

119901

119894=

[

[

[

[

119896 sdot (

1003816

1003816

1003816

1003816

1003816

120591

(1)

119894

1003816

1003816

1003816

1003816

1003816

120576rel)

1119904

]

]

]

]

(26)

The ldquosafetymarginrdquo 119896 ensures that the desired tolerance ismeton the new grid The final step refinement is chosen as themaximum among all 119901

119894

In the context of optimal control a value of 119896 lt 1

may be convenient Since again an OCP is solved on thenew grid the control inputs may change Thus the exactsolution of theODEs as well as the error of the approximationchanges Due to this changing nature of the problem itmay be advantageous to refine the grid in multiple cautiousiterations instead of trying to enforce the desired accuracy bya single refinement step


25 Regularisation In continuous-time OCPs singular arcsare defined as time intervals over which the Hamiltonianbecomes affine in at least one control input (TheHamiltonianis the continuous-time equivalent of the Lagrangian It can beshown that the KKT conditions of the transcribed problemare equivalent to the continuous first-order necessary condi-tions for optimality [45 67]) On singular arcs the optimalsolution thus is not defined by the first-order optimalityconditions and the second-order sufficient conditions arenot strictly satisfied since the second derivatives are zeroIn indirect methods a workaround is to consider timederivatives of the Hamiltonian of increasing order until thesingular control input appears explicitly [68]

The presence of singular arcs introduces problems when adirectmethod is chosen to numerically solve theOCPA goodoverview is provided in [69 Section 45] Loosely spokenthe better the continuous problem is approximated the moreexactly the singular nature of the problem is revealed Thusa fine discretisation of the problem may lead to unwantedeffects Namely spurious oscillations can be observed alongsingular arcs This behaviour is intensified whenever thesolution follows trajectory constraints

In [69 Section 45] a method based on the ldquopiecewisederivative variation of the controlrdquo is proposed to regularisethe transcribed singular OCPsThereby the square of the dif-ference in the slope of each control input in two neighbouringintervals that is the local curvature is added to the objectiveas a penalty termThe regularisation term for a scalar controlinput 119906 is

119871 reg (119906sdot) = 119888

119872sdot

Var2119905119873

(119906

sdot) = 119888

119872sdot

1

2

119872minus1

sum

119897=3

1003816

1003816

1003816

1003816

119904

119897+1minus 119904

119897

1003816

1003816

1003816

1003816

2

(27)

where 119904119897= (119906

119897minus 119906

119897minus1)(119905

119897minus 119905

119897minus1) is its slope in interval 119897

The summation starts at 119897 = 3 since the first point is nota collocation point in Radau methods and thus the controlinput at this point does not have a meaning and is usuallyexcluded from the NLP The factor 119888

119872is chosen as

119888

119872=

119888reg

(119872 minus 3) (119872 minus 1)

2 (28)

The term (119872 minus 3) accounts for the number of summationterms whereas (119872 minus 1)

2 is an approximation of the averagestep size of the discretisation grid This formulation scalesthe regularisation term according to the resolution of thetranscription such that the influence of the user-specifiedparameter 119888reg is invariant with respect to a mesh refinement

The solution of the regularised problem converges tothe solution of the original problem for an increasingly fineresolution It is further shown in the original literature that119888

119872goes to zero fast enough as 119872 rarr infin such as to ensure

that the regularisation term stays bounded

26 Optimal Control of Diesel Engines The OCP of dieselengines can be cast in the form of the general OCP (8a)ndash(8f)The objective is to minimise the cumulative fuel consump-tion that is 119871 =

lowast

119898fuel in (8a) The dynamic constraints (8b)represent the air-path model The control inputs comprise

the air-path actuators the fuel injected per cylinder andcombustion cycle as well as the signals that control thecombustion namely the start of injection (SOI) and theinjection pressure delivered by the common-rail system Theengine speed its time derivative and the desired load torqueare introduced as time-variable parameters Therefore inthe most general case the vectors of the state variables thecontrol inputs and the time-variable parameters read

x =

(

(

(

(

(

(

(

(

(

(

119901IM119901EM120596TC119901

1

119901

4

120599IM119909BGIM120599EMC120599IMC

)

)

)

)

)

)

)

)

)

)

u =

(

(

(

119906VGT119906EGR119906EF119898fcc120593SOI119901rail

)

)

)

120587 = (

119873eng

119873eng

119879load

)

(29)

The fuel mass-flow is simplylowast

119898fuel = 119898fcc sdot 119873eng120 sdot 119899cylThe cumulative pollutant emissions are limited by the integralinequality constraints (8c) that is g = (

lowast

119898NO119909

lowast

119898soot)119879

The absolute limits g are calculated from the desired brake-specific values by multiplication with the integral of thenonnegative segments of the engine power119875eng = 119873eng sdot12058730sdot

119879loadThe desired load torque is imposed as a lower bound bya path constraint that is 119879load(119905) minus 119879load(119905) le 0 for all 119905 Therationale for this formulation and a detailed description areprovided in [70]

The simple bounds (8e) and (8f) represent variousconstraints First the physical actuator ranges need to berespected The fuel mass is limited from below by zero andfrom above by the maximum injection quantity allowed forthe engine at hand Second mechanical limits are imposedon several state variables such as the maximum turbochargerspeed and maximum pressures and temperatures in theintake and exhaust manifolds Finally some of the controlinputs are limited to a region in which the model is knownto deliver plausible results However these limits are foundnot to be active at the optimal solution

To initialise the OCP a feedforward simulation of themodel is performed The control signals recorded during atest-bench run using a preseries ECU calibration are usedThis initialisation is crucial in order not to obtain an infeasibleQP in the first NLP iteration

261 Engines with and without EGR For engine A whichdoes not have an EGR system the four state variables from119901

1to 119909BGIM as well as the two control inputs 119906EGR and 119906EF

are not present in the model Conversely the full state andcontrol vectors presented in (29) are required to representthe air path of engine B However since no satisfactory soot


0 200 400 600 800 1000 1200 1400 1600 18000

1000

2000

Time (s)

583671

0

1000

2000

Time (s)49 55

1 6

654 670 690 708

3

900 920 940 958

5

1629 1650 1670 1690 1710 1730 1746

Tload (Nm)Neng (rpm)

353 375 400 425 450 475 500 525 550 575 600 625 6480

1000

2000

Time (s)

24

Figure 4 The full WHTC (top) with the individual test cases used in this paper indicated by the shaded areas The test cycles are shown inthe middle and bottom plots scaled for engine A Here the shaded areas indicate drag phases Test cycles 2 and 4 are contained within cycle7 The detailed plot of test cycle 8 scaled for engine B is provided in the left-hand plot in Figure 11

model could be derived for this engine the integral constraintfor this emission species is not included in the problemformulation To account for themost prominent influence onthe soot emissions the AFR is limited by the additional pathconstraint 120582AFRmin minus 120582AFR(119905) le 0 for all 119905

The rail pressure defines a tradeoff between the soot emis-sions and the NO

119909emissions If the rail pressure is too low

the combustion is slow and incomplete which causes highsoot emissions Conversely this slow and cool combustionleads to less thermal NO

119909formation The negative effect of

a low rail pressure on the combustion efficiency is almostoutweighed by the reduced power consumption of the high-pressure pump of the common-rail system Thus if the sootemissions are not modelled and limited there is no tradeofffor the rail pressure and the optimisation would always setit to its lower bound For this reason the rail pressure isexcluded as a control input to themodel for engine B Insteadthe values defined by the calibration map are used

Dynamic Loops For the engine with EGR the optimisationterminates prematurely for all problem instances testedEither theQP subproblemof some outer iteration is infeasibleor the algorithm just diverges The reason is the additionaldynamic loop introduced by the EGR system It interacts withthe dynamic loop of the turbocharger and thus introduces ahigh degree of nonlinearity to the dynamic system A similarobservation is reported in [71] and the authors propose tobreak the loops and to apply a homotopy to close them again

For the problem at hand a more straightforward solutionis implemented The control inputs and the state variablesare restricted to a small ldquostability regionrdquo around the initialtrajectories After solving this problem the resulting trajec-tories are used as the new initial guess and the stabilityregion is relocated around these trajectories This procedureis repeated until the solution lies entirely inside the currentstability region An SQPmethod is able to efficiently solve thissequence of related problems see Section 312

262 Driving Cycle and Test Cases Various segments ofthe World-Harmonized Transient Cycle (WHTC) [72] areused as test cases This driving profile prescribes the enginespeed and the load torque over time (For passenger carsand light commercial vehicles a trajectory for the vehiclespeed is usually prescribed From this speed profile anoperating-point trajectory for the engine can be calculated bymeans of a vehicle emulation [73]) Figure 4 displays the fullWHTC as well as the segments used here Since the profiledemands values for the engine speed and the load torqueat each second only shape-preserving cubic splines are usedfor the interpolation Compared to a linear interpolationthis method smoothes the operating-point trajectory whichimproves the numerical properties of the resulting OCP

The shaded areas in the middle and bottom plots ofFigure 4 indicate drag phases During these intervals theinjection is cut off and the engine ismotored at the prescribed


Table 2 Performance of the NLP solvers on test cycle 1 number of outer iterations and overall time required for the solution (in brackets)For the discretisation with order 5lowast a piecewise-constant control was imposed by additional linear constraintsThis variant imitates multipleshooting by resolving the state variables finer than the control inputs The last row is the pure pseudospectral method Symbols used are ℎ(length of collocation intervals) 119904 (collocation order) 119899NLP (number of NLP variables) 119899DOF (degrees of freedom in the NLP) and QNEN(quasiexact Newton method) For WORHP only the exact Newton method is shown For the IP method using a direct solver in KNITRO(KN-IPDIR) only the QN method could solve the problem which is shown here The IP-CG method always performed worse and thus isnot shown An accuracy of 10minus6 is requested with respect to optimality and feasibility If this accuracy was not achieved within 200 iterationsthe optimisation was terminated which is indicated by italic script Bold script indicates the fastest solution for each case

ℎ 119904 119888reg 119899NLP 119899DOF SNOPT IPOPT QN IPOPT EN WORHP KN-SQP QN KN-SQP EN KN-IPDIR025 1 0 225 61 20 (35) 39 (64) 13 (45) 18 (67) 66 (113) 58 (296) 54 (78)025 3 10 657 190 69 (152) 40 (99) 15 (98) 18 (119) 151 (490) 65 (649) 70 (193)025 5lowast 0 1089 60 23 (71) 30 (107) 15 (143) 21 (232) 136 (822) 112 (1675) 137 (868)300 15 1 279 78 42 (68) 62 (102) 12 (48) 21 (85) 200 (438) 200 (1071) 144 (323)600 30 0 279 80 154 (270) 82 (219) 28 (107) 27 (137) 200 (459) 200 (1182) 200 (459)

speed The individual test cases are characterised in thefollowing list

(1) Short simple cycle used for first tests of the algorithmsand for illustration purposes

(2) Short segment with a singular arc The effect of theregularisation is illustrated on this cycle

(3) Longer realistic driving profile but without dragphases

(4) Longer realistic driving profile with many dragphases The presence of motored phases renders theOCPmore inhomogeneous and thus more difficult tobe solved Together with cycle 3 this test cycle is usedto assess the convergence properties of the variousalgorithms

(5) Long test cycle which is easy to be solved since almostno transients occur The cycle is periodic that is itends at the same operating point as it starts withBy a repetition of the cycle large-scale OCPs can beconstructed that retain the complexity of the singlecycle

(6) Longer realistic cycle that is used for the experimentalvalidation

(7) Long realistic cycle that is used for the experimentalvalidation as well as for the identification of a causalcontrol structure in Section 321

(8) A single load step at almost constant engine speedused for the parametric study in Section 322

3 Results and Discussion

The results are subdivided into three sections The first oneanalyses crucial numerical aspects and states the conclusionsthat can be drawn from these testsThe second part covers theengineering aspect by presenting various ways of utilising theoptimal solutionsThe last section describes the experimentalvalidation of the methodology which indicates that allfindings presented in this paper accurately transfer to the realengine

31 Numerical Aspects First the results from all tests per-formed are presented Based on these data conclusions aredrawn on what discretisation approach is suitable for theproblem at hand andwhich numericalmethods are preferableto solve the resulting NLPs

311 Test Results All NLP solvers introduced in Section 232are tested for their convergence behaviour and their compu-tational performance when applied to problems of varyingcomplexity and size Moreover the basic effects of meshrefinement and regularisation are analysed

Convergence Behaviour Table 2 summarises the performanceof all NLP solvers on the short simple test cycle 1 Dif-ferent discretisation approaches are tested from first-ordercollocation to the pseudospectral methodThe regularisationparameter 119888reg is chosen for each case individually such thata smooth solution results For the solver WORHP only theexact Newton method is applied None of the partitionedBFGS updates yields satisfactory results for the problem athand

Table 3 provides the same data for the more realistic testcycles 3 and 4 All methods implemented in KNITRO werenot able to solve any of the problem instances to the requiredtolerance within 200 outer iterations Therefore this data isnot shown Similarly IPOPT is most efficient with respectto the number of iterations and the solution time when theexact second derivatives are providedTheQNvariant did notconverge within 200 iterations for any of the tests on cycle 4

The first and second partial derivatives of the modelfunctions from which the KKT matrix of the NLP isconstructed are calculated by forward finite differences(FFD) The convergence behaviour of the NLP solvers isnot improved when more accurate derivatives are calculatedby algorithmic differentiation (AD) Even the exact Newtonmethods require the same number of iterations when usingAD instead of FFD Therefore AD is advantageous only ifan implementation is available that is faster than FFD Allimplementations of AD for Matlab tested namely ADiMat[74] ADMAT20 (Cayuga ResearchWaterloo ON Canada)INTLAB V6 [75] and the open-source implementation [76]are found to be at least a factor of 16 slower than FFD when


Table 3 Performance of the NLP solvers on test cycles 3 (top) and 4 (bottom) Only the exact Newton methods are shown for IPOPT andWORHP The regularisation is chosen individually for all discretisation variants such that smooth control-input trajectories are obtained

ℎ 119904 119888reg 119899NLP 119899DOF SNOPT IPOPT WORHP025 1 0 2097 666 35 (299) 16 (264) 24 (545)040 3 20 3924 1256 37 (1148) 19 (611) 24 (1081)025 5lowast 10 10449 669 24 (1145) 17 (1536) 33 (3389)300 15 10 2574 822 51 (1208) 18 (390) 28 (697)960 47 30 2547 810 179 (5546) 200 (5444) 26 (927)580 282 30 2547 791 200 (54955) 26 (4482) 24 (4337)025 1 0 2205 450 45 (297) 28 (467) 37 (1174)040 3 500 4113 1199 114 (3115) 29 (890) 49 (1965)025 5lowast 30 10989 617 66 (3861) 76 (6802) 33 (4058)300 15 200 2709 790 197 (3318) 25a (652) 41 (1045)101 50 200 2709 768 200 (6599) 36 (923) 58 (2042)610 300 300 2709 797 200 (62193) 30a (24937) 30 (7816)aA segmentation error crashed the optimisation at that iteration

the first and second partial derivatives of themodel functionsare calculated

Large-Scale Performance Test cycle 5 is repeated multipletimes to construct a problem of increasing size that retainsthe same complexity Therefore the performance of the NLPsolvers when applied to large problems can be assessedindependently of their general convergence behaviour First-order collocation on a uniform grid with a step size of 025 s isused and the cycle is repeated 1 to 6 times NLPs with around4000 to 25000 variables result

The main observations are summarised next All solversrequire a similar number of iterations to solve the differentlysized problems Even the QN methods perform well for thelarge-scale problems Therefore their poor performance onthe more difficult test cases is not induced by the size ofthe problem to be solved but rather by its complexity andnonlinearity

The number of model evaluations per iteration is pro-portional to the number of discretisation points and thus tothe size of the NLP Therefore the time required to solvethe KKT system or to perform updates of a decomposi-tion directly defines the computational performance of thesolvers for large-scale problems The time required for the(re)factorisations and for the dense algebra on the reducedproblem in SNOPT grows with a power of about 25 withrespect to the problem size All other solvers work on thefull but sparse KKT system and exhibit an approximatelylinear runtime The proportionality factor between runtimeand problem size differs by a factor of 3 from the QNmethodof IPOPT (fastest) to the exact Newton method in WORHP(slowest) In WORHP the linear solver has to recalculate orupdate the indefinite factorisation of the KKT system at thebeginning of each outer iteration

For the full-matrix approaches just described the fractionof the overall solution time required for the model eval-uations which are performed in parallel on four cores isbetween 40 (KNITRO) and 70 (IPOPT) For SNOPT thisfraction is below 1 for large problems

20

40

60

80

uVG

T(

)Re

lativ

e tru

nca-

tion

erro

r (mdash

)

10minus4

10minus3

10minus2

10minus1

49 50 51 52 53 54 55012

New

poi

nts

Time (s)

After refinement 1 After refinement 2

Desired accuracyInitial grid h = 025 s Fine grid h = 01 s

Figure 5 Mesh refinement on test cycle 1 First-order collocationis used and the initial step size is 025 s Two refinements areperformed with 119896 = 05 and 119896 = 1 respectively

Mesh Refinement Figure 5 illustrates the effect of the meshrefinement on test cycle 1 First-order collocation is usedand a relative tolerance of 5 sdot 10

minus3 is requested for theODE solution This accuracy is achieved by a uniformdiscretisation with a step size of 01 s WORHP requires 21iterations and 138 s for the solution of the resulting NLP

When mesh refinement is applied an initial step sizeof 025 s is used and two refinements are performed Thethree solution runs of the coarse initial problem and the tworefined problems require 17 6 and 5 iterations and a total


20

40

60

80

100

uVG

T(

)

0

05

1

(nor

mal

ised)

120593SO

I

05

354 355 356 357 358 359 3600

1

Time (s)

pra

il(n

orm

alise

d)

creg = 0

creg = 10

creg = 100

Bounds

Figure 6 Regularisation of the control inputs test cycle 2 The fuelconsumption is reduced by 2676 by the nonregularised solutionand by 2567 when a regularisation with 119888reg = 100 is appliedNormalised signals are shown for reasons of confidentiality Ahigher value of 120593SOI indicates an earlier injection

of 110 s After the second refinement the desired accuracy isachieved and the problem is discretised into 45 intervals onlyas compared to the 60 intervals of the uniform discretisation

When this procedure is applied to test cycle 7 the solutiontime is reduced from 2609 to 1015 s (or from around 43 to17 minutes) IPOPT requires 3234 s to solve the uniformlydiscretised test cycle 7 However in contrast to the SQPmethod implemented in WORHP IPOPT is not able toexploit the good initialisation of the refined problems Infact the solution of the initial and the first refined problemsrequires the same time as the one-off solution of the uni-formly discretised problem

Regularisation Figure 6 shows the optimal trajectories of thecontrol inputs when a second-order collocation on a uniformgrid with 02 s step size is applied to test cycle 2 As to beexpected oscillations occur mainly in the region where thetrajectory constraint for the rail pressure is active The fuelconsumption and the pollutant emissions predicted by theoptimisation are checked by an accurate forward simulationwhich yields the same results For a finer discretisation fasteroscillations result Also when only the state variables areresolved more accurately the oscillations persist

Consequently an oscillatory solution actually is optimalPossibly the gas-exchange losses are slightly reduced by theoscillations of the pressure in the exhaust manifold induced

by the oscillations of the VGT At the same time the intakepressure is hardly affected due to the slow dynamics of theturbocharger Therefore the air mass-flow remains the sameand the soot emissions do not increase

Since a fast oscillating actuation of the mechanical actu-ators is not sustainable regularisation has to be applied AsFigure 6 shows the regularisation does not change the gen-eral shape of the solution In particular the solution is identi-cal in the nonsingular regions Furthermore the loss in fuelefficiency when requiring a smooth solution is negligible

312 Discussion On the simple test case 1 all discretisa-tions and all methods to solve the resulting NLPs seem towork reasonably well However especially the QN methodsconverge faster for a local discretisation than for the pseu-dospectral approach The trust-region method implementedin KNITRO always gets stuck at a small trust radius and thusconverges only slowly towards the optimum The line-searchglobalisation thus seems to be preferable for the problem athand

When more meaningful driving profiles such as testcycles 3 and 4 are considered the QN methods become lessefficient also for local discretisation schemes SurprisinglySNOPT still manages to solve most problem instances in lessthan 200 iterations As the complexity and the problem sizeincrease the SQPmethod implemented inWORHP becomesmore reliable and consistent than the IP method of IPOPT

The pseudospectral approach yields a denser NLP asindicated in Figure 3 Although theHessian of the Lagrangianis still block diagonal the Jacobian of the constraints does notretain a near-diagonal shape The decomposition performedby SNOPT as well as the direct linear solvers of the full-matrix approaches become disproportionally slow when thecollocation order is increased but the problem size is notchanged

Combined with the effectiveness of a step-size refine-ment local discretisation schemes seem to be preferable forthe problem at hand A local discretisation enables a finerresolution of the problem only where necessary and an SQPsolver is able to exploit the good initialisation of the refinedproblem Conversely the pseudospectral method can onlyincrease the order of the collocation polynomial and thusalways refines the approximation of the problem over the fulltime horizon

Summarising these findings a full-matrix approach forthe solution of the KKT system utilising a direct linear solvershould be combinedwith an exactNewton SQPmethod and aline-search globalisation (The effect of the choice of themeritfunction or a filter was analysed too No unique trend couldbe observed that favours one approach The problem at handthus seems not to be susceptible to the Maratos effect [53Section 155]) WORHP implements such a method and infact this solver is found to perform well on all test cases aswell as to adapt to large problems best A relatively low-ordercollocation scheme and an iterative step-size refinementcombine well with this type of NLP solver Only for smallproblems arising for example in receding horizon controla decomposition approach and a QN method such as thoseimplemented in SNOPT prove to be more efficient


12502000

0 110

220

0

05

1

500m

fcc (mg) Neng (rpm)

120593SOI (normalised)

125020000 110 220

0

05

1

500

prail (normalised)

mfcc (mg) Neng (rpm)

+25 NOx

minus35 NOx

Reference NOx level

Figure 7 Maps obtained from the optimal control-trajectoriesand different NO

119909levels relative to the one resulting from the

current ECU calibration Normalised signals are shown for reasonsof confidentiality

32 Engineering Aspects Two ways of utilising the resultsfrom optimal control are proposed On the one hand theoptimal solution may be used to identify all maps andeven the control parameters of an entire feedback-controlstructure On the other hand control strategies for transientoperation can be derived from parametric studies The latteris illustrated by a case study in which the NO

119909emissions of

the engine with EGR have to be reduced over a load increaseAnother option is the repeated solution of a receding-

horizon OCP online on the ECU However in contrast tomodel-predictive feedback control where linear models maybe used that are valid only around the referencemaps [16] thefull nonlinear model would have to be considered Despitethe application of custom-tailored algorithms already thesimpler optimisation problems encountered within model-predictive control are difficult to be solved in realtime dueto the limited computational power and memory providedby the ECU [77] Therefore an online optimisation is not afeasible option at this time

321 Model-Based Engine Calibration Throughout this sec-tion engine A is considered From the solution of the OCPover a sufficiently long time horizon such as test case 7implications for the control structure can be derived [34]The optimal trajectories of the control inputs defining thecombustion that is the SOI and the rail pressure can berepresented accurately by static maps over the engine oper-ating range The same finding applies to the boost pressureAlthough these quantitiesmight be chosen freely over time bythe optimisation values that can be scheduled over the enginespeed and the injected fuel mass result If a quasistationaryrepresentation of the air path is used a static feedforwardmap for the VGT position is obtained from the solution ofthe corresponding OCP

Two applications of these findings are presented hereFirst the maps for the combustion-related control inputs canbe derived for different emission levels The maps for theSOI and the rail pressure for three different NO

119909levels are

shown in Figure 7 These maps can be parameterised by the

mfcc

mfcc lt 1mgNeng

uVGTdrag

uVGTFF

uVGTFB

pIMAirpathuVGT

+

+

+

minusPI

pIMref

dNeng

dt

Figure 8 Structure of the boost-pressure controller

requested emission level On the ECU the maps could beshifted adaptively depending on the current performance ofthe ATS or according to the current driving situation

The second application is the implementation of a feed-back controller for the boost pressure based on the maps forthe boost pressure and the static optimal VGT position Theformer serves as reference to be tracked by the controller andthe latter is used as feedforward control signal Figure 8 showsthe structure of the control system During drag phases apure feedforward control is applied The map for the VGTposition during motored operation is spanned by the enginespeed and its derivative and is identified using the optimaldata from the last 2 seconds of each drag phase [70] Alongwith this controller defining the VGT position the mapspresented in Figure 7 are used for the SOI and the railpressure

The PI controller for the VGT position is implemented as

119906VGTFB = 119896

119901sdot 119890

119901IM(119905) +

119896

119894

10

sdot int

119905

0

119890

119901IM(120591) 119889120591

(30)

with 119890119901IM

= 10

minus4

sdot(119901IMminus119901IMref)The scaling factors are used toprovide a similarmagnitude of the two controller parametersA classical anti-reset windup scheme [78] is applied to handleactuator saturation and the purely feedforward operationduring drag phases

The optimal values for the two PI parameters whichare not scheduled over any quantity may be obtained auto-matically Here a brute-force approach is proposed Theparameters are varied on a reasonable grid and a forwardsimulation of the model is performed for all combinationsFigure 9 displays the results The NO

119909emissions are rather

insensitive with respect to the choice of the PI parametersSimilarly the soot emissions increase rapidly only if a tooslow feedback controller is used In this case the boost-pressure buildup lags behind during load increases resultingin low AFRs An aggressive controller yields the lowest fuelconsumption However the control signal overshoots andoscillates for this parameter set Therefore the parameter setwhich yields the closest representation of the optimal control-input trajectory is chosen

The performances of the reference controller currentlyimplemented the optimal solution and the fully causal con-trol system derived from the optimal solution are illustratedin Figure 10The causal controller is able to closely reproducethe optimal solution Note that this causal control systemis identified by a fully automated procedure requiring onlythe stationary measurement data for the identification of theengine model as input


minus3 minus

2

minus1

minus0

5

minus0

5

1

10

20

30

40

50Soot with respect to optimal control ()

2

0

minus1

minus2

minus102

10

1

10

20

30

40

50

Fuel with respect to optimal control ()

07 0

4

03

025

025 0

3 04

02

18

1412

11

11

12

NOx with respect to optimal control ()

ki(mdash

)

ki(mdash

)

1

10

20

30

40

50

1 5 10 15

ki(mdash

)

kp (mdash)

1

10

20

30

40

50

1 5 10 15

ki(mdash

)

kp (mdash)

Normalised int(uopt minus uPI )2dt (mdash)

Figure 9 Selection of the parameter values for the PI controllerThe circle indicates the fuel-optimal pair while the asterisk denotesthe values that result in the closest possible approximation of theoptimal control-input trajectory

322 Optimal Transient 119873119874

119909-Reduction Strategy Several

control inputs that define a tradeoff between the fuel con-sumption and the NO

119909emissions are present on engine

B The EGR valve controls the flow of exhaust gas intothe intake manifold The burnt gas in the intake mixtureis inert and reduces the temperature of the combustionzones in the cylinders which in turn reduces the thermalNO119909formation Conversely the combustion becomes less

efficient The exhaust flap (EF) can be closed to increase thepressure difference over the EGR valve A higher EGR mass-flow results and also higher gas-exchange losses have to beovercome Finally a retardation of the injection yields a lessefficient combustion and less NO

119909emissions

Particularly during transient operation it is difficult toderive a control strategy that reduces the NO

119909emissions to

a desired level while maintaining the lowest possible fuelconsumption For a load increase the additional constraintof a lower limit on the AFR has to be honoured in ordernot to produce large peaks of soot emissions It is shownhow optimal control can be used to derive a general controlstrategy The single load step of test case 8 is considered seethe left-hand plot of Figure 11

The following procedure is applied First optimal trajec-tories for the VGT and the SOI are derived while keepingthe EGR valve closed and the EF fully open This solutionis used as reference for a successive reduction of the NO

119909

emissions For each desired level of the NO119909emissions

the minimum fuel consumption is calculated by solvingthe corresponding OCP with all four control inputs as freevariables Each resulting pair of cumulative NO

119909emissions

and fuel consumption is plotted in the right-hand plot of

Figure 11 The resulting line defines the optimal tradeoff thatis the Pareto front between the NO

119909emissions and the fuel

consumption for the load step under considerationFigure 12 shows the corresponding control-input trajec-

tories The following control strategy can be derived Theoptimisation does not close the EF at any point Thereforethis is the least efficient way to reduce the NO

119909emissions

and should be avoided During the load increase the VGTas well as the EGR valve needs to be closed such that theAFR stays above the lower limit which is chosen at 14here A feedforward part based on the gradient of the torquedemand could be derived from the solution of the OCPFinally the higher NO

119909emissions during the load step can

be compensated by a transient shift of the SOI and a higherEGR rate during the stationary operation before and after thestep

By extending this case study to a more representativelonger time horizon sufficient information could be collectedto derive an overall controller calibration as presented forengine A in Section 321

33 Experimental Validation In order to validate the resultsfrom the dynamic optimisation on the engine test benchthree critical points have to be resolved First the controlsignals have to be transmitted to the ECU and the test-bench brake synchronously and sufficiently fast A masterprocess writes all signals to a shared memory section ofthe automation system at a frequency of 100Hz Using theiLinkRT protocol developed by AVL (AVL List GmbH GrazAustria) and ETAS (ETAS GmbH Stuttgart Germany) thecontrol signals are transferred to an ETAS ES9103 proto-typing and interface module at the same frequency Thismodule immediately transfers the updated values to theECU using the ETK interface (ETAS) Simultaneously anadditional process transfers the desired engine speed from thesharedmemory section to the controller (SPARCbyHORIBALtd Kyoto Japan) of the test-bench dynamometer (HORIBAHD 700 LC) To this end the proprietary OpenSIM CANmessage protocol by HORIBA is used which ensures a time-synchronous transfer at 100Hz These CAN messages aswell as all relevant signals of the ECU are recorded at theirnatural sampling rate by INCA (ETAS) which runs on a hostcomputer

The secondproblemconsists in obtainingmeaningful andcomparable results When the engine is operated using theECU limits such as an operating-point dependent AFR limitare respected Furthermore a feedback controller is used tofollow the desired load torque resulting in deviations of up to3Therefore the engine does not exactly produce the torquedesired by the driving cycle In addition the optimal solutionhas to provide the same braking torque during drag phasesthat results from the current control strategy implementedon the ECU For these reasons the effective torque deliveredby the engine during a normal run is prescribed during theoptimisation Note that this different torque demand causesthe change in the predicted fuel savings as compared tothe values provided in the description of Figure 10 During


0

1000

2000


0

05

1

ReferenceOptimal

Causal

ReferenceOptimal

Causal

Dragphases

pIM

(nor

mal

ised)

Time (s)388 395 400 405 410 415 420 425 430 435 440 447

30

60

90

uVG

T(

)

Figure 10 Comparison of the reference (the current ECU calibration) the optimal and the causal control derived from the optimal solutionResults from a simulation of the model over a segment of test cycle 7 are shown All three cases produce the same cumulative pollutantemissions The optimal solution saves 221 fuel and the causal one 191 Normalised signals are shown for reasons of confidentiality

1470 1472 1474 1476

500

1000

1500

2000

2500

Time (s)

Neng (rpm)Tload middot 5 (Nm)

025 05

(a) (b)

075 1

1

102

104

106

108

Reference no EGR

Sweep with EGR

NOx with respect to reference (mdash)

Fuel

with

resp

ect t

o re

fere

nce (

mdash)

Figure 11 The driving profile of test case 8 scaled for engine B (a)and the optimal fuel-NO

119909tradeoff for engine B over this cycle (b)

the validation run the optimised control inputs are pre-scribed directly and no bounds are applied

Finally the ECU does not inject exactly the amount offuel demanded by the corresponding control signal In factthe injector maps are identified for a narrow operating region

and the ECU estimation becomes increasingly inaccurate forlarger deviations of the combustion-related control inputsfrom the current calibration Contrarily the model is iden-tified using data recorded by an accurate external fuel scaleTherefore the torque produced by the engine deviates fromthe desired one by up to 7 To resolve this mismatch acorrection is applied after the first validation run The fuelinjection is updated according to a Willans approximation[1 Section 251] The net torque is modelled as a time-variable affine function of the fuel mass

119879load (119905) = 120578

119882(119905) sdot 119898fcc (119905)

minus 119879

0(119873eng (119905) 119901EM (119905) minus 119901IM (119905))

(31)

The loss torque 119879

0is the sum of the friction the inertia

and the gas-exchange work The former two contributionsare estimated using the corresponding submodels of theengine model whereas the latter is estimated from thepressure difference over the engine measured during the firstvalidation run


100

200

300

400

Tlo

ad(N

m)

70

80

90

100

uVG

T(

)

0

05

1

(nor

mal

ised)

120593SO

I

1470 1471 1472 1473 1474 1475 14760

25

50

75

100

Time (s)

100 NOx (no EGR)100 NOx (with EGR)76 NOx (with EGR)

54 NOx (with EGR)44 NOx (with EGR)33 NOx (with EGR)

uEG

R(

)

Figure 12 Control-input trajectories for the fuel-NO119909tradeoff

depicted in the right-hand plot of Figure 11 The EGR valve is closedat 0The exhaust flap remains fully open for all cases and thus is notshown Normalised signals are shown for reasons of confidentialityA higher value of 120593SOI indicates an earlier injection

Themodel is identified using the fuel mass prescribed forthe first run 119898fcc and the resulting torque 119879load Thereforethe updated fuel injection is calculated by

119898fccNEW (119905) = 119898fcc (119905) sdot

119879load (119905) minus 119879

0(119905)

119879load (119905) minus 119879

0(119905)

(32)

After the correction the deviation of the torque produced bythe engine from the desired load torque hardly ever exceeds1

331 Results For each of the two test cycles 6 and 7 twoOCPs were solved The first one required the emissions toremain at the level of the current engine calibration (I)The second one allowed a prescribed increase of the NO

119909

emissions (II) This increase is larger for test cycle 6 since forcycle 7 already high brake-specific (BS) NO

119909emissions result

from the current calibration By performing this variationnot only the quantitative accuracy of the model is assessed

Table 4 Results of the experimental validationThe changes relativeto the values measured for the current engine calibration are shown

Test cycleOCP 6I 6II 7I 7IIFuel

Predicted minus128 minus285 minus152 minus249Measured minus085 minus239 minus142 minus250

NO119909

Predicted 0 25 0 10Measured minus105 2127 minus364 831

but also its ability to reproduce the tradeoff between fuelsavings and NO

119909emissions is evaluated

For test cycle 6 the measurement is repeated 5 timesfor each set of control inputs to assess the reproducibilityof the measurement The maximum deviation of any of the5 measurements from the average of all of them is used asa measure For the BS fuel consumption this deviation is009 and for the cumulative BS NO

119909emissions the figure

is 064 Based on this high reproducibility the longer testcycle 7 was only measured once for each set of control-inputtrajectories

Table 4 summarises the results from the experimentalvalidation On test cycle 6 the prediction overestimates thefuel savings but manages to predict the NO

119909emissions

accurately An interesting quantity is the factor that describesby how much the NO

119909emissions increase for a desired

reduction of the fuel consumption

119896ntf = minus

Δ119898NO119909

119898NO119909ref

sdot

119898fuelref

Δ119898fuel (33)

As can be extracted from the data in Table 4 the modelpredicts a factor of 159 when comparing cases I and II Themeasured factor is 145

On test cycle 7 the fuel savings are predicted accuratelyby the model However the increase of the NO

119909emissions is

underestimatedThis shift of themodel accuracy is due to thefact that the engine is operated in different operating regionson the two cycles Cycle 6 comprises high-power operatingpoints to the largest extent whereas cycle 7 prescribes a low-power profile For cycle 7 the predicted and the measuredvalues of the NO

119909-to-fuel factor 119896ntf are 103 and 111 respec-

tivelyFigure 13 shows the opacity of the exhaust gas measured

for the reference and the two optimal solutions As predictedby the model the overall level remains the same and thus noactive regeneration of the DPF becomes necessary Further-more most instantaneous peaks are even slightly reduced

4 Conclusion

A self-contained set of tools and numerical methods forthe efficient solution of optimal control problems for dieselengines are presented This framework enables the calcula-tion of optimal trajectories of the control inputs over longdriving profiles These solutions provide sufficient informa-tion to derive complete dynamic engine calibrations This


Time (s)

Opa

city

(1m

)

388 400 410 420 430 440 4470

005

01

015

02

ReferenceOptimal IOptimal II

Figure 13 Opacity of the exhaust gas measured for the referenceand the two optimal solutions The same segment of test cycle 7 asin Figure 10 is shown

fully automated model-based approach is illustrated for anengine without EGR For an engine with EGR it is shownhow optimal control can be utilised to develop a controlstrategy that provides an optimal transient fuel-NO

119909tradeoff

An experimental validation indicates that these findingsaccurately transfer to the real engines

Further work should focus on a more detailed model forthe soot emissions and on the analysis of the singular arcsoccurring in the optimal solutionsThe time-resolved controlinputs could be replaced by the parameters of a prescribedcontrol structure in the OCP The simultaneous nature ofthe solution process would be preserved but an optimalcontroller calibration could be obtained directly From anengineering point of view the exhaust-gas aftertreatmentsystem should be included in the model to optimise theinteraction between this system and the engine

Nomenclature

Abbreviations and Indices

AD Algorithmic differentiationAFR Air-to-fuel ratioATS Aftertreatment systemBG Burnt gasBS Brake-specificBVP Boundary-value problemCG Conjugate gradientCP Compressorcyl CylinderDPF Diesel particulate filterECU Engine control unitEF Exhaust flapEGR Exhaust-gas recirculationEM(C) Exhaust manifold (casing)eng EngineFB FF Feedback feedforwardfcc Fuel per cylinder and cycleFFD Forward finite differencesIC Intercooler inequality-constrainedIM(C) Intake manifold (casing)

IP Interior pointKKT Karush-Kuhn-Tuckerlin linearNLP Nonlinear program (or programming)ntf NO

119909-to-fuel

OCP Optimal control problemODE Ordinary differential equationopt OptimalQN Quasi-NewtonQP Quadratic program (or programming)quad Quadraticref ReferenceSCR Selective catalytic-reduction systemSOI Start of injectionSQP Sequential quadratic programmingTC TurbochargerVGT Variable-geometry turbineWHTC World-Harmonized Transient Cycle

Latin Symbols

119860 Area119886 119887 Generic model parameters119888 Specific heat path constraintD Differentiation matrix119890 Control error119891 Dynamic model function119865 Objective of a generic NLP119892 Integrand of a cumulative constraintℎ Step size inequality constraint of a generic NLP119896 Generic model parameter119871L Integrand of the objective Lagrangian119872 Total number of collocation points119898 Mass number of collocation intervals119899 Number119873 Rotational speed (rpm)119901 Pressure QP step119877 Gas constant119904 Order of the collocation polynomial119879 Torque time interval119905 Time119906 Control input119881 Volume119908119882 Quadrature weights119909 State variable fraction

Greek Symbols

120578 Efficiency120582 Lagrange multiplier AFR120585 Mass fractionΠ Pressure ratio120587 Time-varying parameter120591 Barrier parameter collocation points120593 Crank angleΨ Flow function120596 Rotational speed (rads) generic NLP variable


Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This work was partially funded by the Swiss InnovationPromotion Agency CTI under Grant no 108081 PFIW-IW

References

[1] L Guzzella and C H Onder Introduction to Modeling andControl of Internal Combustion Engine Systems Springer BerlinGermany 2nd edition 2010

[2] C Rakopoulos and E Giakoumis Diesel Engine TransientOperation Springer London UK 2009

[3] M Lindgren and P-A Hansson ldquoEffects of transient conditionson exhaust emissions from two non-road diesel enginesrdquoBiosystems Engineering vol 87 no 1 pp 57ndash66 2004

[4] J R Hagena Z S Filipi and D N Assanis ldquoTransient dieselemissions analysis of engine operation during a tip-inrdquo SAETechnical Paper 2006-01-1151 2006

[5] J F Cassidy ldquoA computerized on-line approach to calculatingoptimum engine calibrationsrdquo SAE Technical Paper 7700781977

[6] M Hafner and R Isermann ldquoMultiobjective optimizationof feedforward control maps in engine management systemstowards low consumption and low emissionsrdquo Transactions ofthe Institute of Measurement and Control vol 25 no 1 pp 57ndash74 2003

[7] C Atkinson M Allain and H Zhang ldquoUsing model-basedrapid transient calibration to reduce fuel consumption andemissions in diesel enginesrdquo SAETechnical Paper 2008-01-13652008

[8] M Guerrier and P Cawsey ldquoThe development of modelbased methodologies for gasoline IC engine calibrationrdquo SAETechnical Paper 2004-01-1466 2004

[9] I Brahma M C Sharp and T R Frazier ldquoEmpirical modelingof transient emissions and transient response for transientoptimizationrdquo SAE Technical Paper 2009-01-1508 2009

[10] E Alfieri Emissions-controlled diesel engine [PhD thesis] ETHZurich 2009

[11] A de Risi P Carlucci T Donateo andA Ficarella ldquoA combinedoptimization method for common rail diesel enginesrdquo inAmerican Society of Mechanical Engineers Internal CombustionEngine Division (Publication) ICE vol 38 pp 243ndash250 2002

[12] J Wahlstrom L Eriksson and L Nielsen ldquoController tuningbased on transient selection and optimization for a diesel enginewith EGR and VGTrdquo SAE Technical Paper 2008-01-0985 2008

[13] A A Malikopoulos D N Assanis and P Y PapalambrosldquoReal-time self-learning optimization of diesel engine calibra-tionrdquo Journal of Engineering for Gas Turbines and Power vol131 no 2 Article ID 022803 2009

[14] H J Ferreau G Lorini and M Diehl ldquoFast nonlinear modelpredictive control of gasoline enginesrdquo in Proceedings of theIEEE International Conference on Control Applications (CCArsquo06) pp 2754ndash2759 Munich Germany October 2006

[15] H J Ferreau P Ortner P Langthaler L del Re and M DiehlldquoPredictive control of a real-world Diesel engine using an

extended online active set strategyrdquo Annual Reviews in Controlvol 31 no 2 pp 293ndash301 2007

[16] L del Re P Ortner and D Alberer ldquoChances and challenges inautomotive predictive controlrdquo in Automotive Model PredictiveControl chapter 1 pp 1ndash22 Springer Berlin Germany 2010

[17] A R Dohner ldquoOptimal control solution of the automotiveemission-constrained minimum fuel problemrdquo Automaticavol 17 no 3 pp 441ndash458 1981

[18] Y Urano Y Nakano H Takada and M Sugita ldquoOptimizationtechnique for transient emission reduction of heavy duty dieselenginerdquo SAE Technical Paper 2005-01-1099 2005

[19] A Westlund and H E Angstrom ldquoFast physical emissionpredictions for offline calibration of transient control strategiesrdquoSAE Technical Paper 2009-01-1778 2009

[20] C Ericson B Westerberg I Odenbrand and R Egnell ldquoChar-acterisation and model based optimization of a complete dieselengineSCR systemrdquo SAE Technical Paper 2009-01-0896 2009

[21] M Benz M Hehn C H Onder and L Guzzella ldquoModel-basedactuator trajectories optimization for a diesel engine using adirect methodrdquo Journal of Engineering for Gas Turbines andPower vol 133 no 3 Article ID 032806 2011

[22] R Omran R Younes J C Champoussin D Fedeli F Massonand N Guerrassi ldquoGenetic algorithm for dynamic calibrationof enginersquos actuatorsrdquo SAE Technical Paper 2007-01-1079 2007

[23] R Omran R Younes and J-C Champoussin ldquoOptimal controlof a variable geometry turbocharged diesel engine using neuralnetworks applications on the ETC test cyclerdquo IEEETransactionson Control Systems Technology vol 17 no 2 pp 380ndash393 2009

[24] S Kameswaran and L T Biegler ldquoSimultaneous dynamic opti-mization strategies recent advances and challengesrdquoComputersand Chemical Engineering vol 30 no 10ndash12 pp 1560ndash15752006

[25] J T Betts Practical Methods for Optimal Control and EstimationUsing Nonlinear Programming vol 19 Society for Industrialand Applied Mathematics (SIAM) Philadelphia Pa USA 2ndedition 2010

[26] L B Rall Automatic Differentiation Techniques and Applica-tions vol 120 of Lecture Notes in Computer Science SpringerBerlin Germany 1981

[27] A Griewank and A Walter Evaluating Derivatives Principlesand Techniques of Algorithmic Differentiation Society for Indus-trial and Applied Mathematics (SIAM) Philadelphia Pa USA2nd edition 2008

[28] C Querel O Grondin and C Letellier ldquoState of the art andanalysis of control oriented NO

119909modelsrdquo SAE Technical Paper

2012-01-0723 2012[29] J Asprion O Chinellato and L Guzzella ldquoA fast and accurate

physics-based model for the NO119909emissions of Diesel enginesrdquo

Applied Energy vol 103 pp 221ndash233 2013[30] E G Pariotis G M Kosmadakis and C D Rakopoulos

ldquoComparative analysis of three simulation models applied ona motored internal combustion enginerdquo Energy Conversion andManagement vol 60 pp 45ndash55 2012

[31] G A Weisser Modelling of combustion and nitric oxide for-mation for medium-speed DI diesel engines a comparativeevaluation of zero- and three-dimensional approaches [PhDthesis] ETH Zurich 2001

[32] A Schilling A Amstutz C H Onder and L Guzzella ldquoA real-timemodel for the prediction of the NO

119909emissions in DI diesel

enginesrdquo in Proceedings of the IEEE International Conferenceon Control Applications (CCA rsquo06) pp 2042ndash2047 MunichGermany October 2006


[33] J Asprion O Chinellato and L Guzzella ldquoOptimisation-oriented modelling of the NO

119909emissions of a Diesel enginerdquo

Energy Conversion and Management vol 75 pp 61ndash73 2013[34] J Asprion O Chinellato C H Onder and L Guzzella ldquoFrom

static to dynamic optimisation of Diesel-engine controlrdquo inProceedings of the 52nd IEEE Conference on Decision andControl Florence Italy December 2013

[35] M Diehl H G Bock H Diedam and P-B Wieber ldquoFastdirect multiple shooting algorithms for optimal robot controlrdquoin Fast Motions in Biomechanics and Robotics M Diehl andK Mombaur Eds vol 340 of Lecture Notes in Control andInformation Sciences pp 65ndash93 Springer Berlin Germany2006

[36] J T Betts ldquoSurvey of numerical methods for trajectory opti-mizationrdquo Journal of Guidance Control and Dynamics vol 21no 2 pp 193ndash207 1998

[37] A V Rao ldquoA survey of numerical methods for optimal controlrdquoAdvances in the Astronautical Sciences vol 135 pp 497ndash5282009

[38] B A Conway ldquoA survey of methods available for the numericaloptimization of continuous dynamic systemsrdquo Journal of Opti-mization Theory and Applications vol 152 no 2 pp 271ndash3062012

[39] R W H Sargent ldquoOptimal controlrdquo Journal of Computationaland Applied Mathematics vol 124 no 1-2 pp 361ndash371 2000

[40] L T Biegler ldquoAn overview of simultaneous strategies fordynamic optimizationrdquo Chemical Engineering and Processingvol 46 no 11 pp 1043ndash1053 2007

[41] G Bashein and M Enns ldquoComputation of optimal controlsby a method combining quasi-linearization and quadraticprogrammingrdquo International Journal of Control vol 16 pp 177ndash187 1972

[42] L T Biegler ldquoSolution of dynamic optimization problems bysuccessive quadratic programming andorthogonal collocationrdquoComputers and Chemical Engineering vol 8 no 3-4 pp 243ndash247 1984

[43] D Tieu W R Cluett and A Penlidis ldquoA comparison of col-location methods for solving dynamic optimization problemsrdquoComputers and Chemical Engineering vol 19 no 4 pp 375ndash3811995

[44] A L Herman and B A Conway ldquoDirect optimization usingcollocation based on high-order Gauss-Lobatto quadraturerulesrdquo Journal of Guidance Control and Dynamics vol 19 no3 pp 592ndash599 1996

[45] D A Benson G T Huntington T P Thorvaldsen and AV Rao ldquoDirect trajectory optimization and costate estimationvia an orthogonal collocation methodrdquo Journal of GuidanceControl and Dynamics vol 29 no 6 pp 1435ndash1440 2006

[46] S Kameswaran and L T Biegler ldquoConvergence rates for directtranscription of optimal control problems using collocation atRadau pointsrdquo Computational Optimization and Applicationsvol 41 no 1 pp 81ndash126 2008

[47] D Garg M Patterson W W Hager A V Rao D A Bensonand G T Huntington ldquoA unified framework for the numericalsolution of optimal control problems using pseudospectralmethodsrdquo Automatica vol 46 no 11 pp 1843ndash1851 2010

[48] E Hairer and G Wanner ldquoStiff differential equations solvedby Radau methodsrdquo Journal of Computational and AppliedMathematics vol 111 no 1-2 pp 93ndash111 1999

[49] J C Butcher Numerical Methods for Ordinary DifferentialEquations John Wiley amp Sons Chichester UK 2003

[50] C L Darby W W Hager and A V Rao ldquoDirect trajectoryoptimization using a variable low-order adaptive pseudospec-tral methodrdquo Journal of Spacecraft and Rockets vol 48 no 3pp 433ndash445 2011

[51] C L Darby W W Hager and A V Rao ldquoAn ℎ119901-adaptivepseudospectral method for solving optimal control problemsrdquoOptimal Control Applications ampMethods vol 32 no 4 pp 476ndash502 2011

[52] J F Bonnans J C Gilbert C Lemarechal and C A Sagastiza-bal Numerical Optimization Theoretical and Practical AspectsSpringer Berlin Germany 2nd edition 2006

[53] J Nocedal and S J Wright Numerical Optimization SpringerSeries in Operations Research and Financial EngineeringSpringer New York Second edition 2006

[54] DC Liu and JNocedal ldquoOn the limitedmemoryBFGSmethodfor large scale optimizationrdquo Mathematical Programming vol45 no 3 pp 503ndash528 1989

[55] ldquoThe HSL mathematical software libraryrdquo httpwwwhslrlacuk

[56] ldquoMUMPS homepagerdquo httpgraalens-lyonfrMUMPS[57] ldquoPARDISO homepagerdquo httpwwwpardiso-projectorg [58] P E Gill and E Wong ldquoSequential quadratic programming

methodsrdquo inMixed IntegerNonlinear Programming J Lee and SLeyffer Eds vol 154 ofTheIMAVolumes inMathematics and ItsApplications pp 147ndash224 Springer New York NY USA 2012

[59] P E Gill W Murray and M A Saunders ldquoSNOPT anSQP algorithm for large-scale constrained optimizationrdquo SIAMReview vol 47 no 1 pp 99ndash131 2005

[60] A Wachter and L T Biegler ldquoOn the implementation ofan interior-point filter line-search algorithm for large-scalenonlinear programmingrdquoMathematical Programming vol 106no 1 pp 25ndash57 2006

[61] ldquoMETISmdashserial graph partitioning and fill-reducing matrix or-deringrdquo httpglarosdtcumnedugkhomemetismetisoverview

[62] C Buskens and D Wassel ldquoThe ESA NLP solver WORHPrdquo inModeling and Optimization in Space Engineering G Fasano andJ D Pinter Eds vol 73 ofOptimization and Its Applications pp85ndash110 Springer New York NY USA 2013

[63] RH Byrd J Nocedal andRAWaltz ldquoKNITRO an integratedpackage for nonlinear optimizationrdquo in Large-Scale NonlinearOptimization vol 83 of Nonconvex Optimization and Its Appli-cations pp 35ndash59 Springer New York NY USA 2006

[64] C Canuto M Y Hussaini A Quarteroni and T A ZangSpectral Methods Fundamentals in Single Domains SpringerBerlin Germany 2006

[65] J-P Berrut and L N Trefethen ldquoBarycentric Lagrange interpo-lationrdquo SIAM Review vol 46 no 3 pp 501ndash517 2004

[66] M A Patterson and A V Rao ldquoExploiting sparsity in directcollocation pseudospectral methods for solving optimal controlproblemsrdquo Journal of Spacecraft and Rockets vol 49 pp 364ndash377 2012

[67] J R Cloutier and C N DrsquoSouza ldquoUnification of the direct andindirect approaches to optimal controlrdquo AIAA Meeting PapersArchive 481 AIAA-92-4529-CP 1992

[68] H P Geering Optimal Control with Engineering ApplicationsSpringer Berlin Germany 2007

[69] A L Schwartz Theory and implementation of numerical meth-ods based on Runge-Kutta integration for solving optimal controlproblems [PhD thesis] University of California Berkeley CalifUSA 1996


[70] J Asprion C H Onder and L Guzzella ldquoIncluding drag phasesin numerical optimal control of Diesel enginesrdquo in Proceedingsof the 7th IFAC Symposium on Advances in Automotive ControlTokyo Japan 2013

[71] S Gros M Zanon and M Diehl ldquoA relaxation strategy forthe optimization of airborne wind energy systemsrdquo in EuropeanControl Conference (ECC rsquo13) Zurich Switzerland 2013

[72] E G Giakoumis and A I Alafouzos ldquoComparative study ofturbocharged diesel engine emissions during three differenttransient cyclesrdquo International Journal of Energy Research vol34 no 11 pp 1002ndash1015 2010

[73] T Ott C Onder and L Guzzella ldquoHybrid-electric vehicle withnatural gas-diesel enginerdquo Energies vol 6 pp 3571ndash3592 2013

[74] C H Bischof H M Bucker A Vehreschild and J WillkommAutomatic Differentiation for Matlab (ADiMat) MATLAB-DayAachen Germany 2012

[75] S M Rump ldquoVerification methods rigorous results usingfloating-point arithmeticrdquo Acta Numerica vol 19 pp 287ndash4492010

[76] M Fink ldquoAutomatic differentiation for Matlabrdquo 2007 httpwwwmathworkscommatlabcentralfileexchange15235

[77] G Stewart F Borrelli J Pekar D Germann D Pachner andD Kihas ldquoToward a systematic design for turbocharged enginecontrolrdquo inAutomotive Model Predictive Control chapter 14 pp211ndash230 Springer Berlin Germany 2010

[78] MV Kothare P J CampoMMorari andCNNett ldquoA unifiedframework for the study of anti-windup designsrdquo Automaticavol 30 no 12 pp 1869ndash1883 1994

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


of a feedback controller can also be cast as an optimisationproblem Again the solution can be obtained directly onthe engine [11] or by a model-based approach [12] Adaptivemethods aim to automatically improve the engine calibrationduring normal operation [13]

Dynamics may be incorporated either online by model-predictive control [14ndash16] or offline by an optimisation of thecontrol-trajectories over a representative driving profile Thesolution of such optimal control problems (OCPs) whichin the past were also obtained directly on the engine [17]provides implications for well-suited control structures [18ndash21] The optimal control profiles with the correspondingfuel consumption and pollutant emissions can be used as abenchmark to disclose scenarios in which the performanceof the actual control system is suboptimal An analysis ofthese cases provides guidelines for the improvement of thecontrol structure the feedforward control and the reference-trajectory generation Another way of utilising the resultsfrom optimal control is to use this data to train functionapproximators such as artificial neural networks [22 23]

In the literature referenced it is stated that due to thecomplexity of the system OCPs for diesel engines can only besolved over short time horizons of around 3 s to 10 s Alterna-tively model simplifications such as a quasistationary treat-ment are performed However to derive implications for asuitable control structure or to train function approximatorsoptimal solutions over long time horizons are required basedon a detailed fully dynamic engine model Transient drivingcycles which are used during the homologation procedureand thus emulate a representative operation scenario of theengine are typically 20 to 30 minutes long

This paper presents the means to calculate such optimalsolutions The optimisation-oriented modelling of dieselengines is summarised and extended to engineswith exhaust-gas recirculation (EGR) in Section 22 Subsequently theproblem formulation is detailed and the numerical opti-misation methods are described in Sections 23ndash26 Thepresentation of the results is subdivided into three partsFirst the performance of the numerical methods is analysedin Section 31 and subsequently two case studies illustratehow the results from the optimisation can be utilised inSection 32 Finally the experimental validation providedin Section 33 demonstrates the meaningfulness and thesoundness of the approach

A reader interested in the engineering aspects of the prob-lem may focus on Sections 2ndash22 26 and 32 Conversely areader interested in the details of the optimisation methodand its performancemay concentrate on Sections 23ndash25 and31

Throughout the text the derivative with respect to timeis denoted by = 119889119909119889119905 whereas

lowast

119909 designates a flow Lowerand upper bounds are denoted by 119909 and 119909 and 119909 indicates areference value Bold-face symbols such as x or X representvectors and matrices respectively The abbreviations andsymbols are summarised in the nomenclature section at theend of the text except for the specific notation used inSections 24 and 25

Table 1 Main data of the engines used

Engine A BDispl volume 119881

119889(l) 87 30

Cylinders 119899cyl (mdash) 6 4Borestroke (mm) 117135 96104Compression ratio (mdash) 158 176EGR mdash High pressureRated power (kW) 300 (1650 rpm) 130 (2900 rpm)Max torque (Nm) 1720 (1200 rpm) 420 (1400 rpm)

2 Materials and Methods

This section introduces the engines used and presents allmethods required for the optimal control of diesel enginesThe numerical framework for optimal control as well as theengine model is implemented in MATLAB 2013b (Math-Works Natick MA USA) running under 64bit Windows 8All computations are performed on a laptop computer withan Intel Core i7-2760QM CPU running at a clock speed of240GHz

21 Engines The relevant data of the two engines consideredhere are provided in Table 1 Engine A is a heavy-duty enginewhich is used in on-road and off-road applications It doesnot have an EGR system but relies on a selective catalytic-reduction system (SCR) to reduce the engine-out NO

119909

emissions to the level imposed by the legislation Engine Bis a light-duty engine that is used mainly in light commercialvehicles It has an EGR system but no SCR Both engines areequipped with a common-rail injection system and a dieselparticulate filter (DPF) The soot emissions are low enoughsuch that no active regeneration of the DPF is necessaryThe optimisation thus has to maintain a similar level of sootemissions and should not produce large instantaneous sootpeaks

22 Engine Modelling During an optimisation a vast num-ber of model evaluations are performed Either the modelfunction itself is evaluated for example during static calibra-tion or when employing simultaneous methods for optimalcontrol [24] or a forward simulation of the model is used forexample for parametric studies or in the context of shootingmethods for optimal control [25 Sections 32ndash34]Thereforethe model has to be simple to enable a fast execution Inboth cases described partial derivatives need to be calculatedas well If only standard mathematical operations are usedautomatic differentiation is applicable which enables a fastand accurate evaluation of partial derivatives [26 27]

Besides being computationally efficient models apt foroptimisation have to be smooth and quantitatively accuratecapture all relevant qualitative trends and provide plausibleextrapolation These properties qualify the model itself or itssimulation as an ldquoaccurate and consistent function generatorrdquo[25 Section 38] As a fundamental requirement the modeloutputs have to be predicted using the control signalsknown parameters and the ambient conditions only Ideally


mfcc

prail120593SOI

Air-pathmodel


model Tload


Neng

uVGT

uEF

uEGRx





119909emissions


119909emissions





119889119901IM119889119905

=

119877120581

119881IM(

lowast



119898cyl120599IM) (1a)

119889120599IM119889119905

=

119877120599IM119901IM119881IM119888V

[119888

119901(

lowast



119898cyl120599IM)

minus 119888V120599IM (

lowast

119898CP +lowast


119898cyl)]

(1b)

119889119909BGIM

119889119905

=

119877120599IM119901IM119881IM

(

lowast



119898CP)

(1c)


119909BGEM =

119909BGIM sdot

lowast

119898cyl + (1 + 120590

0) sdot

lowast

119898fuellowast

119898cyl +lowast

119898fuel

(2)



lowast

119898EGR = 119860EGR sdot119901EM


119901EM119901IM

) (3a)

Ψ (Π) =radic

2

119896

Πsdot Π

sdot (1 minus

1

119896

Πsdot Π

)(3b)


(3c)

The factor 119896Π




(4)


120578

120585O2

= 1 minus 119886

120585O2

(119873eng 119898fcc) sdot1003817

1003817

1003817

1003817

1003817

120585O2

minus

120585O2

1003817

1003817

1003817

1003817

1003817

119887120585O2 (5a)


2



119886

120585O2

= 119886

0+ 119886

1sdot 119873eng + 119886




120591

120585O2

= 1 minus 119896lin sdot (120585O2

minus

120585O2

)


minus

120585O2

)

2

(6a)



+ 119896quad3 sdot 1198732

eng + 119896quad4 sdot 1198982


(6b)




119909O2

= 120585

119896O2

sdot119873engO2

(7)






minx(sdot)u(sdot)

int

119879

0

119871 (x (119905) u (119905) 120587 (119905)) 119889119905 (8a)


(8b)

int

119879

0

g (x (119905) u (119905) 120587 (119905)) 119889119905 minus g le 0(8c)

c (x (119905) u (119905) 120587 (119905)) le 0 119905 isin [0 119879] (8d)

x (119905) le x (119905) le x (119905) 119905 isin [0 119879] (8e)

u (119905) le u (119905) le u (119905) 119905 isin [0 119879] (8f)



































initialisation)







min120596

119865 (120596) (9a)

st g (120596) = 0 g isin R119899119892

(9b)

h (120596) le 0 h isin R119899ℎ

(9c)


L (120596120582119892120582ℎ) = 119865 (120596) + 120582

119879

119892g (120596) + 120582119879

ℎh (120596) (10)


nabla

120596L (120596lowast

120582lowast

119892120582lowast

ℎ) = 0 (11a)

nabla

120582119892

L (120596lowast

120582lowast

119892120582lowast

ℎ) = g (120596lowast) = 0 (11b)

nabla

120582ℎ

L (120596lowast

120582lowast

119892120582lowast

ℎ) = h (120596lowast) le 0 (11c)


120582lowast

ℎge 0 (11d)

120582

lowast

ℎ119894sdot ℎ

119894(120596lowast

) = 0 119894 = 1 119899

ℎ (11e)


120596L(120596120582

119892120582ℎ)



(

nabla119865 (120596119896)

g (120596119896)

) + [

nabla

2

120596L (120596119896120582119892119896) nablag (120596

119896)

nablag(120596119896)

119879

0

] sdot (

120596 minus 120596119896

120582119892

) = 0

(12)


minp

1

2

p119879nabla2120596L (120596119896120582119892119896) p + nabla119865(120596

119896)

119879p (13a)

st nablag(120596119896)

119879p + g (120596119896) = 0

(13b)









minp

1

2

p119879nabla2120596L (120596119896120582119892119896120582ℎ119896) p + nabla119865(120596

119896)

119879p (14a)

st nablag(120596119896)

119879p + g (120596119896) = 0

(14b)

nablah(120596119896)

119879p + h (120596119896) le 0

(14c)


min120596

119865 (120596) minus 120591

119899ℎ

sum

119894=1


st g (120596) = 0 (15b)













119891(119909) on the interval [1199050 119905



0lt 120591

1lt 120591

2sdot sdot sdot lt

120591

119873= 119905


0= 119905


The notation 119909

119895= 119909(120591


equations reads

(

1

2

119904

) asymp [d0

D]⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

=Dsdot(

119909

0

119909

1

119909

2

119909

119904

)

= (

119891(119909

1)

119891 (119909

2)

119891 (119909

119904)

) (16)


119889

0119894sdot 119909

0+

119904

sum

119895=1

119863

119894119895sdot 119909

119895minus 119891 (119909

119894) = 0 (17)


119891minus 119905



119892(119905)119889119905 asymp sum

119904

119895=1119908

119895119892(120591

119895)


D sdot [

x0

X] = F (X) (18)




119905


119905

0lt 119905



119896can be


119897(119898 119904



119897 = 119897 (119896 119894) = 119894 +

119896minus1

sum

120572=1

119904

120572 (19)


minxsdotusdot

119872

sum

119897=1

119882

119897sdot 119871 (x119897 u119897) (20a)

st D(119896) sdot [[


X(119896)]

]

minus F (X(119896)U(119896)) = 0 119896 = 1 119898

(20b)

119872

sum

119897=1

119882


c (x119897 u119897) le 0 119897 = 1 119872 (20d)



119909sdot 119904




1 12 13 13 23 33 4


Pathconstraint

4 1




119897 x119879119897)

119879






119888119897


119894= (x(119896)


(119896)

119894at


L(119896)

119894= 119882

119897sdot 119871 (120596119897)

+ x(119896)119894

sdot (

119873119896

sum

119895=1

119863

(119896)

119895119894120582(119896)

119889119895+ 120575

(119896)

119894

119873119896+1

sum

119895=1

119889

(119896+1)

0119895120582(119896+1)

119889119895)

minus f (120596(119896)119894)120582(119896)

119889119894+119882

119897sdot 120582119879

119892g (120596119897) + 120582119879

119888119897c (120596119897)

(21)


119906+ 119899

119909)





x = x(1) + 120591(1) (22a)

120591(1)




119894119896+1119888

119894119896asymp 119888

119894119896119888

119894119896minus1 In



x = x(119901) + 120591(119901) (23a)

120591(119901)

= 119901 sdot c sdot ( ℎ119901

)

119904+1

=

c sdot ℎ119904+1

119901

119904

=

120591(1)

119901

119904

(23b)


120591(1)

asymp

x(2) minus x(1)

1 minus 2

minus119904

120591

(1)

119894=

120591

(1)

119894

119909

(1)

119894

for 119894 = 1 119899

119909 (24)


120591

(119901)

119894asymp

120591

(119901)

119894

119909

(1)

119894

asymp

1

119901

119904

sdot

120591

(1)

119894

119909

(1)

119894

=

120591

(1)

119894

119901

119904

(25)


119901

119894=

[

[

[

[

119896 sdot (

1003816

1003816

1003816

1003816

1003816

120591

(1)

119894

1003816

1003816

1003816

1003816

1003816

120576rel)

1119904

]

]

]

]

(26)


119894







119871 reg (119906sdot) = 119888

119872sdot

Var2119905119873

(119906

sdot) = 119888

119872sdot

1

2

119872minus1

sum

119897=3

1003816

1003816

1003816

1003816

119904

119897+1minus 119904

119897

1003816

1003816

1003816

1003816

2

(27)

where 119904119897= (119906

119897minus 119906

119897minus1)(119905

119897minus 119905



119872is chosen as

119888

119872=

119888reg

(119872 minus 3) (119872 minus 1)

2 (28)







lowast



x =

(

(

(

(

(

(

(

(

(

(

119901IM119901EM120596TC119901

1

119901

4

120599IM119909BGIM120599EMC120599IMC

)

)

)

)

)

)

)

)

)

)

u =

(

(

(


)

)

)

120587 = (

119873eng

119873eng

119879load

)

(29)



lowast

119898NO119909

lowast

119898soot)119879









0 200 400 600 800 1000 1200 1400 1600 18000

1000

2000

Time (s)

583671

0

1000

2000

Time (s)49 55

1 6

654 670 690 708

3

900 920 940 958

5

1629 1650 1670 1690 1710 1730 1746


353 375 400 425 450 475 500 525 550 575 600 625 6480

1000

2000

Time (s)

24







































20

40

60

80

uVG

T(

)Re

lativ

e tru

nca-

tion

erro

r (mdash

)

10minus4

10minus3

10minus2

10minus1

49 50 51 52 53 54 55012

New

poi

nts

Time (s)








20

40

60

80

100

uVG

T(

)

0

05

1

(nor

mal

ised)

120593SO

I

05

354 355 356 357 358 359 3600

1

Time (s)

pra

il(n

orm

alise

d)

creg = 0

creg = 10

creg = 100

Bounds














12502000

0 110

220

0

05

1

500m

fcc (mg) Neng (rpm)


125020000 110 220

0

05

1

500

prail (normalised)


+25 NOx

minus35 NOx

Reference NOx level





119909emissions of





119909levels are


mfcc

mfcc lt 1mgNeng

uVGTdrag

uVGTFF

uVGTFB

pIMAirpathuVGT

+

+

+

minusPI

pIMref

dNeng

dt





119906VGTFB = 119896

119901sdot 119890

119901IM(119905) +

119896

119894

10

sdot int

119905

0

119890

119901IM(120591) 119889120591

(30)

with 119890119901IM

= 10

minus4







minus3 minus

2

minus1

minus0

5

minus0

5

1

10

20

30

40


2

0

minus1

minus2

minus102

10

1

10

20

30

40

50


07 0

4

03

025

025 0

3 04

02

18

1412

11

11

12


ki(mdash

)

ki(mdash

)

1

10

20

30

40

50

1 5 10 15

ki(mdash

)

kp (mdash)

1

10

20

30

40

50

1 5 10 15

ki(mdash

)

kp (mdash)









119909emissions


119909emissions to



119909



119909emissions






119909emissions








0

1000

2000


0

05

1

ReferenceOptimal

Causal

ReferenceOptimal

Causal

Dragphases

pIM

(nor

mal

ised)

Time (s)388 395 400 405 410 415 420 425 430 435 440 447

30

60

90

uVG

T(

)


1470 1472 1474 1476

500

1000

1500

2000

2500

Time (s)


025 05

(a) (b)

075 1

1

102

104

106

108

Reference no EGR

Sweep with EGR


Fuel

with

resp

ect t

o re

fere

nce (

mdash)






119879load (119905) = 120578

119882(119905) sdot 119898fcc (119905)

minus 119879

0(119873eng (119905) 119901EM (119905) minus 119901IM (119905))

(31)





100

200

300

400

Tlo

ad(N

m)

70

80

90

100

uVG

T(

)

0

05

1

(nor

mal

ised)

120593SO

I

1470 1471 1472 1473 1474 1475 14760

25

50

75

100

Time (s)



uEG

R(

)




119898fccNEW (119905) = 119898fcc (119905) sdot

119879load (119905) minus 119879

0(119905)

119879load (119905) minus 119879

0(119905)

(32)



119909







NO119909








119909emissions




119896ntf = minus

Δ119898NO119909

119898NO119909ref

sdot

119898fuelref

Δ119898fuel (33)



119909emissions is





4 Conclusion



Time (s)

Opa

city

(1m

)

388 400 410 420 430 440 4470

005

01

015

02




119909tradeoff



Nomenclature




119909-to-fuel


Latin Symbols


Greek Symbols





Acknowledgment


References































































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










mfcc

prail120593SOI

Air-pathmodel


model Tload


Neng

uVGT

uEF

uEGRx





119909emissions


119909emissions





119889119901IM119889119905

=

119877120581

119881IM(

lowast



119898cyl120599IM) (1a)

119889120599IM119889119905

=

119877120599IM119901IM119881IM119888V

[119888

119901(

lowast



119898cyl120599IM)

minus 119888V120599IM (

lowast

119898CP +lowast


119898cyl)]

(1b)

119889119909BGIM

119889119905

=

119877120599IM119901IM119881IM

(

lowast



119898CP)

(1c)


119909BGEM =

119909BGIM sdot

lowast

119898cyl + (1 + 120590

0) sdot

lowast

119898fuellowast

119898cyl +lowast

119898fuel

(2)



lowast

119898EGR = 119860EGR sdot119901EM


119901EM119901IM

) (3a)

Ψ (Π) =radic

2

119896

Πsdot Π

sdot (1 minus

1

119896

Πsdot Π

)(3b)


(3c)

The factor 119896Π




(4)


120578

120585O2

= 1 minus 119886

120585O2

(119873eng 119898fcc) sdot1003817

1003817

1003817

1003817

1003817

120585O2

minus

120585O2

1003817

1003817

1003817

1003817

1003817

119887120585O2 (5a)


2



119886

120585O2

= 119886

0+ 119886

1sdot 119873eng + 119886




120591

120585O2

= 1 minus 119896lin sdot (120585O2

minus

120585O2

)


minus

120585O2

)

2

(6a)



+ 119896quad3 sdot 1198732

eng + 119896quad4 sdot 1198982


(6b)




119909O2

= 120585

119896O2

sdot119873engO2

(7)






minx(sdot)u(sdot)

int

119879

0

119871 (x (119905) u (119905) 120587 (119905)) 119889119905 (8a)


(8b)

int

119879

0

g (x (119905) u (119905) 120587 (119905)) 119889119905 minus g le 0(8c)

c (x (119905) u (119905) 120587 (119905)) le 0 119905 isin [0 119879] (8d)

x (119905) le x (119905) le x (119905) 119905 isin [0 119879] (8e)

u (119905) le u (119905) le u (119905) 119905 isin [0 119879] (8f)



































initialisation)







min120596

119865 (120596) (9a)

st g (120596) = 0 g isin R119899119892

(9b)

h (120596) le 0 h isin R119899ℎ

(9c)


L (120596120582119892120582ℎ) = 119865 (120596) + 120582

119879

119892g (120596) + 120582119879

ℎh (120596) (10)


nabla

120596L (120596lowast

120582lowast

119892120582lowast

ℎ) = 0 (11a)

nabla

120582119892

L (120596lowast

120582lowast

119892120582lowast

ℎ) = g (120596lowast) = 0 (11b)

nabla

120582ℎ

L (120596lowast

120582lowast

119892120582lowast

ℎ) = h (120596lowast) le 0 (11c)


120582lowast

ℎge 0 (11d)

120582

lowast

ℎ119894sdot ℎ

119894(120596lowast

) = 0 119894 = 1 119899

ℎ (11e)


120596L(120596120582

119892120582ℎ)



(

nabla119865 (120596119896)

g (120596119896)

) + [

nabla

2

120596L (120596119896120582119892119896) nablag (120596

119896)

nablag(120596119896)

119879

0

] sdot (

120596 minus 120596119896

120582119892

) = 0

(12)


minp

1

2

p119879nabla2120596L (120596119896120582119892119896) p + nabla119865(120596

119896)

119879p (13a)

st nablag(120596119896)

119879p + g (120596119896) = 0

(13b)









minp

1

2

p119879nabla2120596L (120596119896120582119892119896120582ℎ119896) p + nabla119865(120596

119896)

119879p (14a)

st nablag(120596119896)

119879p + g (120596119896) = 0

(14b)

nablah(120596119896)

119879p + h (120596119896) le 0

(14c)


min120596

119865 (120596) minus 120591

119899ℎ

sum

119894=1


st g (120596) = 0 (15b)













119891(119909) on the interval [1199050 119905



0lt 120591

1lt 120591

2sdot sdot sdot lt

120591

119873= 119905


0= 119905


The notation 119909

119895= 119909(120591


equations reads

(

1

2

119904

) asymp [d0

D]⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

=Dsdot(

119909

0

119909

1

119909

2

119909

119904

)

= (

119891(119909

1)

119891 (119909

2)

119891 (119909

119904)

) (16)


119889

0119894sdot 119909

0+

119904

sum

119895=1

119863

119894119895sdot 119909

119895minus 119891 (119909

119894) = 0 (17)


119891minus 119905



119892(119905)119889119905 asymp sum

119904

119895=1119908

119895119892(120591

119895)


D sdot [

x0

X] = F (X) (18)




119905


119905

0lt 119905



119896can be


119897(119898 119904



119897 = 119897 (119896 119894) = 119894 +

119896minus1

sum

120572=1

119904

120572 (19)


minxsdotusdot

119872

sum

119897=1

119882

119897sdot 119871 (x119897 u119897) (20a)

st D(119896) sdot [[


X(119896)]

]

minus F (X(119896)U(119896)) = 0 119896 = 1 119898

(20b)

119872

sum

119897=1

119882


c (x119897 u119897) le 0 119897 = 1 119872 (20d)



119909sdot 119904




1 12 13 13 23 33 4


Pathconstraint

4 1




119897 x119879119897)

119879






119888119897


119894= (x(119896)


(119896)

119894at


L(119896)

119894= 119882

119897sdot 119871 (120596119897)

+ x(119896)119894

sdot (

119873119896

sum

119895=1

119863

(119896)

119895119894120582(119896)

119889119895+ 120575

(119896)

119894

119873119896+1

sum

119895=1

119889

(119896+1)

0119895120582(119896+1)

119889119895)

minus f (120596(119896)119894)120582(119896)

119889119894+119882

119897sdot 120582119879

119892g (120596119897) + 120582119879

119888119897c (120596119897)

(21)


119906+ 119899

119909)





x = x(1) + 120591(1) (22a)

120591(1)




119894119896+1119888

119894119896asymp 119888

119894119896119888

119894119896minus1 In



x = x(119901) + 120591(119901) (23a)

120591(119901)

= 119901 sdot c sdot ( ℎ119901

)

119904+1

=

c sdot ℎ119904+1

119901

119904

=

120591(1)

119901

119904

(23b)


120591(1)

asymp

x(2) minus x(1)

1 minus 2

minus119904

120591

(1)

119894=

120591

(1)

119894

119909

(1)

119894

for 119894 = 1 119899

119909 (24)


120591

(119901)

119894asymp

120591

(119901)

119894

119909

(1)

119894

asymp

1

119901

119904

sdot

120591

(1)

119894

119909

(1)

119894

=

120591

(1)

119894

119901

119904

(25)


119901

119894=

[

[

[

[

119896 sdot (

1003816

1003816

1003816

1003816

1003816

120591

(1)

119894

1003816

1003816

1003816

1003816

1003816

120576rel)

1119904

]

]

]

]

(26)


119894







119871 reg (119906sdot) = 119888

119872sdot

Var2119905119873

(119906

sdot) = 119888

119872sdot

1

2

119872minus1

sum

119897=3

1003816

1003816

1003816

1003816

119904

119897+1minus 119904

119897

1003816

1003816

1003816

1003816

2

(27)

where 119904119897= (119906

119897minus 119906

119897minus1)(119905

119897minus 119905



119872is chosen as

119888

119872=

119888reg

(119872 minus 3) (119872 minus 1)

2 (28)







lowast



x =

(

(

(

(

(

(

(

(

(

(

119901IM119901EM120596TC119901

1

119901

4

120599IM119909BGIM120599EMC120599IMC

)

)

)

)

)

)

)

)

)

)

u =

(

(

(


)

)

)

120587 = (

119873eng

119873eng

119879load

)

(29)



lowast

119898NO119909

lowast

119898soot)119879









0 200 400 600 800 1000 1200 1400 1600 18000

1000

2000

Time (s)

583671

0

1000

2000

Time (s)49 55

1 6

654 670 690 708

3

900 920 940 958

5

1629 1650 1670 1690 1710 1730 1746


353 375 400 425 450 475 500 525 550 575 600 625 6480

1000

2000

Time (s)

24







































20

40

60

80

uVG

T(

)Re

lativ

e tru

nca-

tion

erro

r (mdash

)

10minus4

10minus3

10minus2

10minus1

49 50 51 52 53 54 55012

New

poi

nts

Time (s)








20

40

60

80

100

uVG

T(

)

0

05

1

(nor

mal

ised)

120593SO

I

05

354 355 356 357 358 359 3600

1

Time (s)

pra

il(n

orm

alise

d)

creg = 0

creg = 10

creg = 100

Bounds














12502000

0 110

220

0

05

1

500m

fcc (mg) Neng (rpm)


125020000 110 220

0

05

1

500

prail (normalised)


+25 NOx

minus35 NOx

Reference NOx level





119909emissions of





119909levels are


mfcc

mfcc lt 1mgNeng

uVGTdrag

uVGTFF

uVGTFB

pIMAirpathuVGT

+

+

+

minusPI

pIMref

dNeng

dt





119906VGTFB = 119896

119901sdot 119890

119901IM(119905) +

119896

119894

10

sdot int

119905

0

119890

119901IM(120591) 119889120591

(30)

with 119890119901IM

= 10

minus4







minus3 minus

2

minus1

minus0

5

minus0

5

1

10

20

30

40


2

0

minus1

minus2

minus102

10

1

10

20

30

40

50


07 0

4

03

025

025 0

3 04

02

18

1412

11

11

12


ki(mdash

)

ki(mdash

)

1

10

20

30

40

50

1 5 10 15

ki(mdash

)

kp (mdash)

1

10

20

30

40

50

1 5 10 15

ki(mdash

)

kp (mdash)









119909emissions


119909emissions to



119909



119909emissions






119909emissions








0

1000

2000


0

05

1

ReferenceOptimal

Causal

ReferenceOptimal

Causal

Dragphases

pIM

(nor

mal

ised)

Time (s)388 395 400 405 410 415 420 425 430 435 440 447

30

60

90

uVG

T(

)


1470 1472 1474 1476

500

1000

1500

2000

2500

Time (s)


025 05

(a) (b)

075 1

1

102

104

106

108

Reference no EGR

Sweep with EGR


Fuel

with

resp

ect t

o re

fere

nce (

mdash)






119879load (119905) = 120578

119882(119905) sdot 119898fcc (119905)

minus 119879

0(119873eng (119905) 119901EM (119905) minus 119901IM (119905))

(31)





100

200

300

400

Tlo

ad(N

m)

70

80

90

100

uVG

T(

)

0

05

1

(nor

mal

ised)

120593SO

I

1470 1471 1472 1473 1474 1475 14760

25

50

75

100

Time (s)



uEG

R(

)




119898fccNEW (119905) = 119898fcc (119905) sdot

119879load (119905) minus 119879

0(119905)

119879load (119905) minus 119879

0(119905)

(32)



119909







NO119909








119909emissions




119896ntf = minus

Δ119898NO119909

119898NO119909ref

sdot

119898fuelref

Δ119898fuel (33)



119909emissions is





4 Conclusion



Time (s)

Opa

city

(1m

)

388 400 410 420 430 440 4470

005

01

015

02




119909tradeoff



Nomenclature




119909-to-fuel


Latin Symbols


Greek Symbols





Acknowledgment


References































































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











120591

120585O2

= 1 minus 119896lin sdot (120585O2

minus

120585O2

)


minus

120585O2

)

2

(6a)



+ 119896quad3 sdot 1198732

eng + 119896quad4 sdot 1198982


(6b)




119909O2

= 120585

119896O2

sdot119873engO2

(7)






minx(sdot)u(sdot)

int

119879

0

119871 (x (119905) u (119905) 120587 (119905)) 119889119905 (8a)


(8b)

int

119879

0

g (x (119905) u (119905) 120587 (119905)) 119889119905 minus g le 0(8c)

c (x (119905) u (119905) 120587 (119905)) le 0 119905 isin [0 119879] (8d)

x (119905) le x (119905) le x (119905) 119905 isin [0 119879] (8e)

u (119905) le u (119905) le u (119905) 119905 isin [0 119879] (8f)



































initialisation)







min120596

119865 (120596) (9a)

st g (120596) = 0 g isin R119899119892

(9b)

h (120596) le 0 h isin R119899ℎ

(9c)


L (120596120582119892120582ℎ) = 119865 (120596) + 120582

119879

119892g (120596) + 120582119879

ℎh (120596) (10)


nabla

120596L (120596lowast

120582lowast

119892120582lowast

ℎ) = 0 (11a)

nabla

120582119892

L (120596lowast

120582lowast

119892120582lowast

ℎ) = g (120596lowast) = 0 (11b)

nabla

120582ℎ

L (120596lowast

120582lowast

119892120582lowast

ℎ) = h (120596lowast) le 0 (11c)


120582lowast

ℎge 0 (11d)

120582

lowast

ℎ119894sdot ℎ

119894(120596lowast

) = 0 119894 = 1 119899

ℎ (11e)


120596L(120596120582

119892120582ℎ)



(

nabla119865 (120596119896)

g (120596119896)

) + [

nabla

2

120596L (120596119896120582119892119896) nablag (120596

119896)

nablag(120596119896)

119879

0

] sdot (

120596 minus 120596119896

120582119892

) = 0

(12)


minp

1

2

p119879nabla2120596L (120596119896120582119892119896) p + nabla119865(120596

119896)

119879p (13a)

st nablag(120596119896)

119879p + g (120596119896) = 0

(13b)









minp

1

2

p119879nabla2120596L (120596119896120582119892119896120582ℎ119896) p + nabla119865(120596

119896)

119879p (14a)

st nablag(120596119896)

119879p + g (120596119896) = 0

(14b)

nablah(120596119896)

119879p + h (120596119896) le 0

(14c)


min120596

119865 (120596) minus 120591

119899ℎ

sum

119894=1


st g (120596) = 0 (15b)













119891(119909) on the interval [1199050 119905



0lt 120591

1lt 120591

2sdot sdot sdot lt

120591

119873= 119905


0= 119905


The notation 119909

119895= 119909(120591


equations reads

(

1

2

119904

) asymp [d0

D]⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

=Dsdot(

119909

0

119909

1

119909

2

119909

119904

)

= (

119891(119909

1)

119891 (119909

2)

119891 (119909

119904)

) (16)


119889

0119894sdot 119909

0+

119904

sum

119895=1

119863

119894119895sdot 119909

119895minus 119891 (119909

119894) = 0 (17)


119891minus 119905



119892(119905)119889119905 asymp sum

119904

119895=1119908

119895119892(120591

119895)


D sdot [

x0

X] = F (X) (18)




119905


119905

0lt 119905



119896can be


119897(119898 119904



119897 = 119897 (119896 119894) = 119894 +

119896minus1

sum

120572=1

119904

120572 (19)


minxsdotusdot

119872

sum

119897=1

119882

119897sdot 119871 (x119897 u119897) (20a)

st D(119896) sdot [[


X(119896)]

]

minus F (X(119896)U(119896)) = 0 119896 = 1 119898

(20b)

119872

sum

119897=1

119882


c (x119897 u119897) le 0 119897 = 1 119872 (20d)



119909sdot 119904




1 12 13 13 23 33 4


Pathconstraint

4 1




119897 x119879119897)

119879






119888119897


119894= (x(119896)


(119896)

119894at


L(119896)

119894= 119882

119897sdot 119871 (120596119897)

+ x(119896)119894

sdot (

119873119896

sum

119895=1

119863

(119896)

119895119894120582(119896)

119889119895+ 120575

(119896)

119894

119873119896+1

sum

119895=1

119889

(119896+1)

0119895120582(119896+1)

119889119895)

minus f (120596(119896)119894)120582(119896)

119889119894+119882

119897sdot 120582119879

119892g (120596119897) + 120582119879

119888119897c (120596119897)

(21)


119906+ 119899

119909)





x = x(1) + 120591(1) (22a)

120591(1)




119894119896+1119888

119894119896asymp 119888

119894119896119888

119894119896minus1 In



x = x(119901) + 120591(119901) (23a)

120591(119901)

= 119901 sdot c sdot ( ℎ119901

)

119904+1

=

c sdot ℎ119904+1

119901

119904

=

120591(1)

119901

119904

(23b)


120591(1)

asymp

x(2) minus x(1)

1 minus 2

minus119904

120591

(1)

119894=

120591

(1)

119894

119909

(1)

119894

for 119894 = 1 119899

119909 (24)


120591

(119901)

119894asymp

120591

(119901)

119894

119909

(1)

119894

asymp

1

119901

119904

sdot

120591

(1)

119894

119909

(1)

119894

=

120591

(1)

119894

119901

119904

(25)


119901

119894=

[

[

[

[

119896 sdot (

1003816

1003816

1003816

1003816

1003816

120591

(1)

119894

1003816

1003816

1003816

1003816

1003816

120576rel)

1119904

]

]

]

]

(26)


119894







119871 reg (119906sdot) = 119888

119872sdot

Var2119905119873

(119906

sdot) = 119888

119872sdot

1

2

119872minus1

sum

119897=3

1003816

1003816

1003816

1003816

119904

119897+1minus 119904

119897

1003816

1003816

1003816

1003816

2

(27)

where 119904119897= (119906

119897minus 119906

119897minus1)(119905

119897minus 119905



119872is chosen as

119888

119872=

119888reg

(119872 minus 3) (119872 minus 1)

2 (28)







lowast



x =

(

(

(

(

(

(

(

(

(

(

119901IM119901EM120596TC119901

1

119901

4

120599IM119909BGIM120599EMC120599IMC

)

)

)

)

)

)

)

)

)

)

u =

(

(

(


)

)

)

120587 = (

119873eng

119873eng

119879load

)

(29)



lowast

119898NO119909

lowast

119898soot)119879









0 200 400 600 800 1000 1200 1400 1600 18000

1000

2000

Time (s)

583671

0

1000

2000

Time (s)49 55

1 6

654 670 690 708

3

900 920 940 958

5

1629 1650 1670 1690 1710 1730 1746


353 375 400 425 450 475 500 525 550 575 600 625 6480

1000

2000

Time (s)

24







































20

40

60

80

uVG

T(

)Re

lativ

e tru

nca-

tion

erro

r (mdash

)

10minus4

10minus3

10minus2

10minus1

49 50 51 52 53 54 55012

New

poi

nts

Time (s)








20

40

60

80

100

uVG

T(

)

0

05

1

(nor

mal

ised)

120593SO

I

05

354 355 356 357 358 359 3600

1

Time (s)

pra

il(n

orm

alise

d)

creg = 0

creg = 10

creg = 100

Bounds














12502000

0 110

220

0

05

1

500m

fcc (mg) Neng (rpm)


125020000 110 220

0

05

1

500

prail (normalised)


+25 NOx

minus35 NOx

Reference NOx level





119909emissions of





119909levels are


mfcc

mfcc lt 1mgNeng

uVGTdrag

uVGTFF

uVGTFB

pIMAirpathuVGT

+

+

+

minusPI

pIMref

dNeng

dt





119906VGTFB = 119896

119901sdot 119890

119901IM(119905) +

119896

119894

10

sdot int

119905

0

119890

119901IM(120591) 119889120591

(30)

with 119890119901IM

= 10

minus4







minus3 minus

2

minus1

minus0

5

minus0

5

1

10

20

30

40


2

0

minus1

minus2

minus102

10

1

10

20

30

40

50


07 0

4

03

025

025 0

3 04

02

18

1412

11

11

12


ki(mdash

)

ki(mdash

)

1

10

20

30

40

50

1 5 10 15

ki(mdash

)

kp (mdash)

1

10

20

30

40

50

1 5 10 15

ki(mdash

)

kp (mdash)









119909emissions


119909emissions to



119909



119909emissions






119909emissions








0

1000

2000


0

05

1

ReferenceOptimal

Causal

ReferenceOptimal

Causal

Dragphases

pIM

(nor

mal

ised)

Time (s)388 395 400 405 410 415 420 425 430 435 440 447

30

60

90

uVG

T(

)


1470 1472 1474 1476

500

1000

1500

2000

2500

Time (s)


025 05

(a) (b)

075 1

1

102

104

106

108

Reference no EGR

Sweep with EGR


Fuel

with

resp

ect t

o re

fere

nce (

mdash)






119879load (119905) = 120578

119882(119905) sdot 119898fcc (119905)

minus 119879

0(119873eng (119905) 119901EM (119905) minus 119901IM (119905))

(31)





100

200

300

400

Tlo

ad(N

m)

70

80

90

100

uVG

T(

)

0

05

1

(nor

mal

ised)

120593SO

I

1470 1471 1472 1473 1474 1475 14760

25

50

75

100

Time (s)



uEG

R(

)




119898fccNEW (119905) = 119898fcc (119905) sdot

119879load (119905) minus 119879

0(119905)

119879load (119905) minus 119879

0(119905)

(32)



119909







NO119909








119909emissions




119896ntf = minus

Δ119898NO119909

119898NO119909ref

sdot

119898fuelref

Δ119898fuel (33)



119909emissions is





4 Conclusion



Time (s)

Opa

city

(1m

)

388 400 410 420 430 440 4470

005

01

015

02




119909tradeoff



Nomenclature




119909-to-fuel


Latin Symbols


Greek Symbols





Acknowledgment


References































































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra






























initialisation)







min120596

119865 (120596) (9a)

st g (120596) = 0 g isin R119899119892

(9b)

h (120596) le 0 h isin R119899ℎ

(9c)


L (120596120582119892120582ℎ) = 119865 (120596) + 120582

119879

119892g (120596) + 120582119879

ℎh (120596) (10)


nabla

120596L (120596lowast

120582lowast

119892120582lowast

ℎ) = 0 (11a)

nabla

120582119892

L (120596lowast

120582lowast

119892120582lowast

ℎ) = g (120596lowast) = 0 (11b)

nabla

120582ℎ

L (120596lowast

120582lowast

119892120582lowast

ℎ) = h (120596lowast) le 0 (11c)


120582lowast

ℎge 0 (11d)

120582

lowast

ℎ119894sdot ℎ

119894(120596lowast

) = 0 119894 = 1 119899

ℎ (11e)


120596L(120596120582

119892120582ℎ)



(

nabla119865 (120596119896)

g (120596119896)

) + [

nabla

2

120596L (120596119896120582119892119896) nablag (120596

119896)

nablag(120596119896)

119879

0

] sdot (

120596 minus 120596119896

120582119892

) = 0

(12)


minp

1

2

p119879nabla2120596L (120596119896120582119892119896) p + nabla119865(120596

119896)

119879p (13a)

st nablag(120596119896)

119879p + g (120596119896) = 0

(13b)









minp

1

2

p119879nabla2120596L (120596119896120582119892119896120582ℎ119896) p + nabla119865(120596

119896)

119879p (14a)

st nablag(120596119896)

119879p + g (120596119896) = 0

(14b)

nablah(120596119896)

119879p + h (120596119896) le 0

(14c)


min120596

119865 (120596) minus 120591

119899ℎ

sum

119894=1


st g (120596) = 0 (15b)













119891(119909) on the interval [1199050 119905



0lt 120591

1lt 120591

2sdot sdot sdot lt

120591

119873= 119905


0= 119905


The notation 119909

119895= 119909(120591


equations reads

(

1

2

119904

) asymp [d0

D]⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

=Dsdot(

119909

0

119909

1

119909

2

119909

119904

)

= (

119891(119909

1)

119891 (119909

2)

119891 (119909

119904)

) (16)


119889

0119894sdot 119909

0+

119904

sum

119895=1

119863

119894119895sdot 119909

119895minus 119891 (119909

119894) = 0 (17)


119891minus 119905



119892(119905)119889119905 asymp sum

119904

119895=1119908

119895119892(120591

119895)


D sdot [

x0

X] = F (X) (18)




119905


119905

0lt 119905



119896can be


119897(119898 119904



119897 = 119897 (119896 119894) = 119894 +

119896minus1

sum

120572=1

119904

120572 (19)


minxsdotusdot

119872

sum

119897=1

119882

119897sdot 119871 (x119897 u119897) (20a)

st D(119896) sdot [[


X(119896)]

]

minus F (X(119896)U(119896)) = 0 119896 = 1 119898

(20b)

119872

sum

119897=1

119882


c (x119897 u119897) le 0 119897 = 1 119872 (20d)



119909sdot 119904




1 12 13 13 23 33 4


Pathconstraint

4 1




119897 x119879119897)

119879






119888119897


119894= (x(119896)


(119896)

119894at


L(119896)

119894= 119882

119897sdot 119871 (120596119897)

+ x(119896)119894

sdot (

119873119896

sum

119895=1

119863

(119896)

119895119894120582(119896)

119889119895+ 120575

(119896)

119894

119873119896+1

sum

119895=1

119889

(119896+1)

0119895120582(119896+1)

119889119895)

minus f (120596(119896)119894)120582(119896)

119889119894+119882

119897sdot 120582119879

119892g (120596119897) + 120582119879

119888119897c (120596119897)

(21)


119906+ 119899

119909)





x = x(1) + 120591(1) (22a)

120591(1)




119894119896+1119888

119894119896asymp 119888

119894119896119888

119894119896minus1 In



x = x(119901) + 120591(119901) (23a)

120591(119901)

= 119901 sdot c sdot ( ℎ119901

)

119904+1

=

c sdot ℎ119904+1

119901

119904

=

120591(1)

119901

119904

(23b)


120591(1)

asymp

x(2) minus x(1)

1 minus 2

minus119904

120591

(1)

119894=

120591

(1)

119894

119909

(1)

119894

for 119894 = 1 119899

119909 (24)


120591

(119901)

119894asymp

120591

(119901)

119894

119909

(1)

119894

asymp

1

119901

119904

sdot

120591

(1)

119894

119909

(1)

119894

=

120591

(1)

119894

119901

119904

(25)


119901

119894=

[

[

[

[

119896 sdot (

1003816

1003816

1003816

1003816

1003816

120591

(1)

119894

1003816

1003816

1003816

1003816

1003816

120576rel)

1119904

]

]

]

]

(26)


119894







119871 reg (119906sdot) = 119888

119872sdot

Var2119905119873

(119906

sdot) = 119888

119872sdot

1

2

119872minus1

sum

119897=3

1003816

1003816

1003816

1003816

119904

119897+1minus 119904

119897

1003816

1003816

1003816

1003816

2

(27)

where 119904119897= (119906

119897minus 119906

119897minus1)(119905

119897minus 119905



119872is chosen as

119888

119872=

119888reg

(119872 minus 3) (119872 minus 1)

2 (28)







lowast



x =

(

(

(

(

(

(

(

(

(

(

119901IM119901EM120596TC119901

1

119901

4

120599IM119909BGIM120599EMC120599IMC

)

)

)

)

)

)

)

)

)

)

u =

(

(

(


)

)

)

120587 = (

119873eng

119873eng

119879load

)

(29)



lowast

119898NO119909

lowast

119898soot)119879









0 200 400 600 800 1000 1200 1400 1600 18000

1000

2000

Time (s)

583671

0

1000

2000

Time (s)49 55

1 6

654 670 690 708

3

900 920 940 958

5

1629 1650 1670 1690 1710 1730 1746


353 375 400 425 450 475 500 525 550 575 600 625 6480

1000

2000

Time (s)

24







































20

40

60

80

uVG

T(

)Re

lativ

e tru

nca-

tion

erro

r (mdash

)

10minus4

10minus3

10minus2

10minus1

49 50 51 52 53 54 55012

New

poi

nts

Time (s)








20

40

60

80

100

uVG

T(

)

0

05

1

(nor

mal

ised)

120593SO

I

05

354 355 356 357 358 359 3600

1

Time (s)

pra

il(n

orm

alise

d)

creg = 0

creg = 10

creg = 100

Bounds














12502000

0 110

220

0

05

1

500m

fcc (mg) Neng (rpm)


125020000 110 220

0

05

1

500

prail (normalised)


+25 NOx

minus35 NOx

Reference NOx level





119909emissions of





119909levels are


mfcc

mfcc lt 1mgNeng

uVGTdrag

uVGTFF

uVGTFB

pIMAirpathuVGT

+

+

+

minusPI

pIMref

dNeng

dt





119906VGTFB = 119896

119901sdot 119890

119901IM(119905) +

119896

119894

10

sdot int

119905

0

119890

119901IM(120591) 119889120591

(30)

with 119890119901IM

= 10

minus4







minus3 minus

2

minus1

minus0

5

minus0

5

1

10

20

30

40


2

0

minus1

minus2

minus102

10

1

10

20

30

40

50


07 0

4

03

025

025 0

3 04

02

18

1412

11

11

12


ki(mdash

)

ki(mdash

)

1

10

20

30

40

50

1 5 10 15

ki(mdash

)

kp (mdash)

1

10

20

30

40

50

1 5 10 15

ki(mdash

)

kp (mdash)









119909emissions


119909emissions to



119909



119909emissions






119909emissions








0

1000

2000


0

05

1

ReferenceOptimal

Causal

ReferenceOptimal

Causal

Dragphases

pIM

(nor

mal

ised)

Time (s)388 395 400 405 410 415 420 425 430 435 440 447

30

60

90

uVG

T(

)


1470 1472 1474 1476

500

1000

1500

2000

2500

Time (s)


025 05

(a) (b)

075 1

1

102

104

106

108

Reference no EGR

Sweep with EGR


Fuel

with

resp

ect t

o re

fere

nce (

mdash)






119879load (119905) = 120578

119882(119905) sdot 119898fcc (119905)

minus 119879

0(119873eng (119905) 119901EM (119905) minus 119901IM (119905))

(31)





100

200

300

400

Tlo

ad(N

m)

70

80

90

100

uVG

T(

)

0

05

1

(nor

mal

ised)

120593SO

I

1470 1471 1472 1473 1474 1475 14760

25

50

75

100

Time (s)



uEG

R(

)




119898fccNEW (119905) = 119898fcc (119905) sdot

119879load (119905) minus 119879

0(119905)

119879load (119905) minus 119879

0(119905)

(32)



119909







NO119909








119909emissions




119896ntf = minus

Δ119898NO119909

119898NO119909ref

sdot

119898fuelref

Δ119898fuel (33)



119909emissions is





4 Conclusion



Time (s)

Opa

city

(1m

)

388 400 410 420 430 440 4470

005

01

015

02




119909tradeoff



Nomenclature




119909-to-fuel


Latin Symbols


Greek Symbols





Acknowledgment


References































































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










120582lowast

ℎge 0 (11d)

120582

lowast

ℎ119894sdot ℎ

119894(120596lowast

) = 0 119894 = 1 119899

ℎ (11e)


120596L(120596120582

119892120582ℎ)



(

nabla119865 (120596119896)

g (120596119896)

) + [

nabla

2

120596L (120596119896120582119892119896) nablag (120596

119896)

nablag(120596119896)

119879

0

] sdot (

120596 minus 120596119896

120582119892

) = 0

(12)


minp

1

2

p119879nabla2120596L (120596119896120582119892119896) p + nabla119865(120596

119896)

119879p (13a)

st nablag(120596119896)

119879p + g (120596119896) = 0

(13b)









minp

1

2

p119879nabla2120596L (120596119896120582119892119896120582ℎ119896) p + nabla119865(120596

119896)

119879p (14a)

st nablag(120596119896)

119879p + g (120596119896) = 0

(14b)

nablah(120596119896)

119879p + h (120596119896) le 0

(14c)


min120596

119865 (120596) minus 120591

119899ℎ

sum

119894=1


st g (120596) = 0 (15b)













119891(119909) on the interval [1199050 119905



0lt 120591

1lt 120591

2sdot sdot sdot lt

120591

119873= 119905


0= 119905


The notation 119909

119895= 119909(120591


equations reads

(

1

2

119904

) asymp [d0

D]⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

=Dsdot(

119909

0

119909

1

119909

2

119909

119904

)

= (

119891(119909

1)

119891 (119909

2)

119891 (119909

119904)

) (16)


119889

0119894sdot 119909

0+

119904

sum

119895=1

119863

119894119895sdot 119909

119895minus 119891 (119909

119894) = 0 (17)


119891minus 119905



119892(119905)119889119905 asymp sum

119904

119895=1119908

119895119892(120591

119895)


D sdot [

x0

X] = F (X) (18)




119905


119905

0lt 119905



119896can be


119897(119898 119904



119897 = 119897 (119896 119894) = 119894 +

119896minus1

sum

120572=1

119904

120572 (19)


minxsdotusdot

119872

sum

119897=1

119882

119897sdot 119871 (x119897 u119897) (20a)

st D(119896) sdot [[


X(119896)]

]

minus F (X(119896)U(119896)) = 0 119896 = 1 119898

(20b)

119872

sum

119897=1

119882


c (x119897 u119897) le 0 119897 = 1 119872 (20d)



119909sdot 119904




1 12 13 13 23 33 4


Pathconstraint

4 1




119897 x119879119897)

119879






119888119897


119894= (x(119896)


(119896)

119894at


L(119896)

119894= 119882

119897sdot 119871 (120596119897)

+ x(119896)119894

sdot (

119873119896

sum

119895=1

119863

(119896)

119895119894120582(119896)

119889119895+ 120575

(119896)

119894

119873119896+1

sum

119895=1

119889

(119896+1)

0119895120582(119896+1)

119889119895)

minus f (120596(119896)119894)120582(119896)

119889119894+119882

119897sdot 120582119879

119892g (120596119897) + 120582119879

119888119897c (120596119897)

(21)


119906+ 119899

119909)





x = x(1) + 120591(1) (22a)

120591(1)




119894119896+1119888

119894119896asymp 119888

119894119896119888

119894119896minus1 In



x = x(119901) + 120591(119901) (23a)

120591(119901)

= 119901 sdot c sdot ( ℎ119901

)

119904+1

=

c sdot ℎ119904+1

119901

119904

=

120591(1)

119901

119904

(23b)


120591(1)

asymp

x(2) minus x(1)

1 minus 2

minus119904

120591

(1)

119894=

120591

(1)

119894

119909

(1)

119894

for 119894 = 1 119899

119909 (24)


120591

(119901)

119894asymp

120591

(119901)

119894

119909

(1)

119894

asymp

1

119901

119904

sdot

120591

(1)

119894

119909

(1)

119894

=

120591

(1)

119894

119901

119904

(25)


119901

119894=

[

[

[

[

119896 sdot (

1003816

1003816

1003816

1003816

1003816

120591

(1)

119894

1003816

1003816

1003816

1003816

1003816

120576rel)

1119904

]

]

]

]

(26)


119894







119871 reg (119906sdot) = 119888

119872sdot

Var2119905119873

(119906

sdot) = 119888

119872sdot

1

2

119872minus1

sum

119897=3

1003816

1003816

1003816

1003816

119904

119897+1minus 119904

119897

1003816

1003816

1003816

1003816

2

(27)

where 119904119897= (119906

119897minus 119906

119897minus1)(119905

119897minus 119905



119872is chosen as

119888

119872=

119888reg

(119872 minus 3) (119872 minus 1)

2 (28)







lowast



x =

(

(

(

(

(

(

(

(

(

(

119901IM119901EM120596TC119901

1

119901

4

120599IM119909BGIM120599EMC120599IMC

)

)

)

)

)

)

)

)

)

)

u =

(

(

(


)

)

)

120587 = (

119873eng

119873eng

119879load

)

(29)



lowast

119898NO119909

lowast

119898soot)119879









0 200 400 600 800 1000 1200 1400 1600 18000

1000

2000

Time (s)

583671

0

1000

2000

Time (s)49 55

1 6

654 670 690 708

3

900 920 940 958

5

1629 1650 1670 1690 1710 1730 1746


353 375 400 425 450 475 500 525 550 575 600 625 6480

1000

2000

Time (s)

24







































20

40

60

80

uVG

T(

)Re

lativ

e tru

nca-

tion

erro

r (mdash

)

10minus4

10minus3

10minus2

10minus1

49 50 51 52 53 54 55012

New

poi

nts

Time (s)








20

40

60

80

100

uVG

T(

)

0

05

1

(nor

mal

ised)

120593SO

I

05

354 355 356 357 358 359 3600

1

Time (s)

pra

il(n

orm

alise

d)

creg = 0

creg = 10

creg = 100

Bounds














12502000

0 110

220

0

05

1

500m

fcc (mg) Neng (rpm)


125020000 110 220

0

05

1

500

prail (normalised)


+25 NOx

minus35 NOx

Reference NOx level





119909emissions of





119909levels are


mfcc

mfcc lt 1mgNeng

uVGTdrag

uVGTFF

uVGTFB

pIMAirpathuVGT

+

+

+

minusPI

pIMref

dNeng

dt





119906VGTFB = 119896

119901sdot 119890

119901IM(119905) +

119896

119894

10

sdot int

119905

0

119890

119901IM(120591) 119889120591

(30)

with 119890119901IM

= 10

minus4







minus3 minus

2

minus1

minus0

5

minus0

5

1

10

20

30

40


2

0

minus1

minus2

minus102

10

1

10

20

30

40

50


07 0

4

03

025

025 0

3 04

02

18

1412

11

11

12


ki(mdash

)

ki(mdash

)

1

10

20

30

40

50

1 5 10 15

ki(mdash

)

kp (mdash)

1

10

20

30

40

50

1 5 10 15

ki(mdash

)

kp (mdash)









119909emissions


119909emissions to



119909



119909emissions






119909emissions








0

1000

2000


0

05

1

ReferenceOptimal

Causal

ReferenceOptimal

Causal

Dragphases

pIM

(nor

mal

ised)

Time (s)388 395 400 405 410 415 420 425 430 435 440 447

30

60

90

uVG

T(

)


1470 1472 1474 1476

500

1000

1500

2000

2500

Time (s)


025 05

(a) (b)

075 1

1

102

104

106

108

Reference no EGR

Sweep with EGR


Fuel

with

resp

ect t

o re

fere

nce (

mdash)






119879load (119905) = 120578

119882(119905) sdot 119898fcc (119905)

minus 119879

0(119873eng (119905) 119901EM (119905) minus 119901IM (119905))

(31)





100

200

300

400

Tlo

ad(N

m)

70

80

90

100

uVG

T(

)

0

05

1

(nor

mal

ised)

120593SO

I

1470 1471 1472 1473 1474 1475 14760

25

50

75

100

Time (s)



uEG

R(

)




119898fccNEW (119905) = 119898fcc (119905) sdot

119879load (119905) minus 119879

0(119905)

119879load (119905) minus 119879

0(119905)

(32)



119909







NO119909








119909emissions




119896ntf = minus

Δ119898NO119909

119898NO119909ref

sdot

119898fuelref

Δ119898fuel (33)



119909emissions is





4 Conclusion



Time (s)

Opa

city

(1m

)

388 400 410 420 430 440 4470

005

01

015

02




119909tradeoff



Nomenclature




119909-to-fuel


Latin Symbols


Greek Symbols





Acknowledgment


References































































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra















119891(119909) on the interval [1199050 119905



0lt 120591

1lt 120591

2sdot sdot sdot lt

120591

119873= 119905


0= 119905


The notation 119909

119895= 119909(120591


equations reads

(

1

2

119904

) asymp [d0

D]⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

=Dsdot(

119909

0

119909

1

119909

2

119909

119904

)

= (

119891(119909

1)

119891 (119909

2)

119891 (119909

119904)

) (16)


119889

0119894sdot 119909

0+

119904

sum

119895=1

119863

119894119895sdot 119909

119895minus 119891 (119909

119894) = 0 (17)


119891minus 119905



119892(119905)119889119905 asymp sum

119904

119895=1119908

119895119892(120591

119895)


D sdot [

x0

X] = F (X) (18)




119905


119905

0lt 119905



119896can be


119897(119898 119904



119897 = 119897 (119896 119894) = 119894 +

119896minus1

sum

120572=1

119904

120572 (19)


minxsdotusdot

119872

sum

119897=1

119882

119897sdot 119871 (x119897 u119897) (20a)

st D(119896) sdot [[


X(119896)]

]

minus F (X(119896)U(119896)) = 0 119896 = 1 119898

(20b)

119872

sum

119897=1

119882


c (x119897 u119897) le 0 119897 = 1 119872 (20d)



119909sdot 119904




1 12 13 13 23 33 4


Pathconstraint

4 1




119897 x119879119897)

119879






119888119897


119894= (x(119896)


(119896)

119894at


L(119896)

119894= 119882

119897sdot 119871 (120596119897)

+ x(119896)119894

sdot (

119873119896

sum

119895=1

119863

(119896)

119895119894120582(119896)

119889119895+ 120575

(119896)

119894

119873119896+1

sum

119895=1

119889

(119896+1)

0119895120582(119896+1)

119889119895)

minus f (120596(119896)119894)120582(119896)

119889119894+119882

119897sdot 120582119879

119892g (120596119897) + 120582119879

119888119897c (120596119897)

(21)


119906+ 119899

119909)





x = x(1) + 120591(1) (22a)

120591(1)




119894119896+1119888

119894119896asymp 119888

119894119896119888

119894119896minus1 In



x = x(119901) + 120591(119901) (23a)

120591(119901)

= 119901 sdot c sdot ( ℎ119901

)

119904+1

=

c sdot ℎ119904+1

119901

119904

=

120591(1)

119901

119904

(23b)


120591(1)

asymp

x(2) minus x(1)

1 minus 2

minus119904

120591

(1)

119894=

120591

(1)

119894

119909

(1)

119894

for 119894 = 1 119899

119909 (24)


120591

(119901)

119894asymp

120591

(119901)

119894

119909

(1)

119894

asymp

1

119901

119904

sdot

120591

(1)

119894

119909

(1)

119894

=

120591

(1)

119894

119901

119904

(25)


119901

119894=

[

[

[

[

119896 sdot (

1003816

1003816

1003816

1003816

1003816

120591

(1)

119894

1003816

1003816

1003816

1003816

1003816

120576rel)

1119904

]

]

]

]

(26)


119894







119871 reg (119906sdot) = 119888

119872sdot

Var2119905119873

(119906

sdot) = 119888

119872sdot

1

2

119872minus1

sum

119897=3

1003816

1003816

1003816

1003816

119904

119897+1minus 119904

119897

1003816

1003816

1003816

1003816

2

(27)

where 119904119897= (119906

119897minus 119906

119897minus1)(119905

119897minus 119905



119872is chosen as

119888

119872=

119888reg

(119872 minus 3) (119872 minus 1)

2 (28)







lowast



x =

(

(

(

(

(

(

(

(

(

(

119901IM119901EM120596TC119901

1

119901

4

120599IM119909BGIM120599EMC120599IMC

)

)

)

)

)

)

)

)

)

)

u =

(

(

(


)

)

)

120587 = (

119873eng

119873eng

119879load

)

(29)



lowast

119898NO119909

lowast

119898soot)119879









0 200 400 600 800 1000 1200 1400 1600 18000

1000

2000

Time (s)

583671

0

1000

2000

Time (s)49 55

1 6

654 670 690 708

3

900 920 940 958

5

1629 1650 1670 1690 1710 1730 1746


353 375 400 425 450 475 500 525 550 575 600 625 6480

1000

2000

Time (s)

24







































20

40

60

80

uVG

T(

)Re

lativ

e tru

nca-

tion

erro

r (mdash

)

10minus4

10minus3

10minus2

10minus1

49 50 51 52 53 54 55012

New

poi

nts

Time (s)








20

40

60

80

100

uVG

T(

)

0

05

1

(nor

mal

ised)

120593SO

I

05

354 355 356 357 358 359 3600

1

Time (s)

pra

il(n

orm

alise

d)

creg = 0

creg = 10

creg = 100

Bounds














12502000

0 110

220

0

05

1

500m

fcc (mg) Neng (rpm)


125020000 110 220

0

05

1

500

prail (normalised)


+25 NOx

minus35 NOx

Reference NOx level





119909emissions of





119909levels are


mfcc

mfcc lt 1mgNeng

uVGTdrag

uVGTFF

uVGTFB

pIMAirpathuVGT

+

+

+

minusPI

pIMref

dNeng

dt





119906VGTFB = 119896

119901sdot 119890

119901IM(119905) +

119896

119894

10

sdot int

119905

0

119890

119901IM(120591) 119889120591

(30)

with 119890119901IM

= 10

minus4







minus3 minus

2

minus1

minus0

5

minus0

5

1

10

20

30

40


2

0

minus1

minus2

minus102

10

1

10

20

30

40

50


07 0

4

03

025

025 0

3 04

02

18

1412

11

11

12


ki(mdash

)

ki(mdash

)

1

10

20

30

40

50

1 5 10 15

ki(mdash

)

kp (mdash)

1

10

20

30

40

50

1 5 10 15

ki(mdash

)

kp (mdash)









119909emissions


119909emissions to



119909



119909emissions






119909emissions








0

1000

2000


0

05

1

ReferenceOptimal

Causal

ReferenceOptimal

Causal

Dragphases

pIM

(nor

mal

ised)

Time (s)388 395 400 405 410 415 420 425 430 435 440 447

30

60

90

uVG

T(

)


1470 1472 1474 1476

500

1000

1500

2000

2500

Time (s)


025 05

(a) (b)

075 1

1

102

104

106

108

Reference no EGR

Sweep with EGR


Fuel

with

resp

ect t

o re

fere

nce (

mdash)






119879load (119905) = 120578

119882(119905) sdot 119898fcc (119905)

minus 119879

0(119873eng (119905) 119901EM (119905) minus 119901IM (119905))

(31)





100

200

300

400

Tlo

ad(N

m)

70

80

90

100

uVG

T(

)

0

05

1

(nor

mal

ised)

120593SO

I

1470 1471 1472 1473 1474 1475 14760

25

50

75

100

Time (s)



uEG

R(

)




119898fccNEW (119905) = 119898fcc (119905) sdot

119879load (119905) minus 119879

0(119905)

119879load (119905) minus 119879

0(119905)

(32)



119909







NO119909








119909emissions




119896ntf = minus

Δ119898NO119909

119898NO119909ref

sdot

119898fuelref

Δ119898fuel (33)



119909emissions is





4 Conclusion



Time (s)

Opa

city

(1m

)

388 400 410 420 430 440 4470

005

01

015

02




119909tradeoff



Nomenclature




119909-to-fuel


Latin Symbols


Greek Symbols





Acknowledgment


References































































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










1 12 13 13 23 33 4


Pathconstraint

4 1




119897 x119879119897)

119879






119888119897


119894= (x(119896)


(119896)

119894at


L(119896)

119894= 119882

119897sdot 119871 (120596119897)

+ x(119896)119894

sdot (

119873119896

sum

119895=1

119863

(119896)

119895119894120582(119896)

119889119895+ 120575

(119896)

119894

119873119896+1

sum

119895=1

119889

(119896+1)

0119895120582(119896+1)

119889119895)

minus f (120596(119896)119894)120582(119896)

119889119894+119882

119897sdot 120582119879

119892g (120596119897) + 120582119879

119888119897c (120596119897)

(21)


119906+ 119899

119909)





x = x(1) + 120591(1) (22a)

120591(1)




119894119896+1119888

119894119896asymp 119888

119894119896119888

119894119896minus1 In



x = x(119901) + 120591(119901) (23a)

120591(119901)

= 119901 sdot c sdot ( ℎ119901

)

119904+1

=

c sdot ℎ119904+1

119901

119904

=

120591(1)

119901

119904

(23b)


120591(1)

asymp

x(2) minus x(1)

1 minus 2

minus119904

120591

(1)

119894=

120591

(1)

119894

119909

(1)

119894

for 119894 = 1 119899

119909 (24)


120591

(119901)

119894asymp

120591

(119901)

119894

119909

(1)

119894

asymp

1

119901

119904

sdot

120591

(1)

119894

119909

(1)

119894

=

120591

(1)

119894

119901

119904

(25)


119901

119894=

[

[

[

[

119896 sdot (

1003816

1003816

1003816

1003816

1003816

120591

(1)

119894

1003816

1003816

1003816

1003816

1003816

120576rel)

1119904

]

]

]

]

(26)


119894







119871 reg (119906sdot) = 119888

119872sdot

Var2119905119873

(119906

sdot) = 119888

119872sdot

1

2

119872minus1

sum

119897=3

1003816

1003816

1003816

1003816

119904

119897+1minus 119904

119897

1003816

1003816

1003816

1003816

2

(27)

where 119904119897= (119906

119897minus 119906

119897minus1)(119905

119897minus 119905



119872is chosen as

119888

119872=

119888reg

(119872 minus 3) (119872 minus 1)

2 (28)







lowast



x =

(

(

(

(

(

(

(

(

(

(

119901IM119901EM120596TC119901

1

119901

4

120599IM119909BGIM120599EMC120599IMC

)

)

)

)

)

)

)

)

)

)

u =

(

(

(


)

)

)

120587 = (

119873eng

119873eng

119879load

)

(29)



lowast

119898NO119909

lowast

119898soot)119879









0 200 400 600 800 1000 1200 1400 1600 18000

1000

2000

Time (s)

583671

0

1000

2000

Time (s)49 55

1 6

654 670 690 708

3

900 920 940 958

5

1629 1650 1670 1690 1710 1730 1746


353 375 400 425 450 475 500 525 550 575 600 625 6480

1000

2000

Time (s)

24







































20

40

60

80

uVG

T(

)Re

lativ

e tru

nca-

tion

erro

r (mdash

)

10minus4

10minus3

10minus2

10minus1

49 50 51 52 53 54 55012

New

poi

nts

Time (s)








20

40

60

80

100

uVG

T(

)

0

05

1

(nor

mal

ised)

120593SO

I

05

354 355 356 357 358 359 3600

1

Time (s)

pra

il(n

orm

alise

d)

creg = 0

creg = 10

creg = 100

Bounds














12502000

0 110

220

0

05

1

500m

fcc (mg) Neng (rpm)


125020000 110 220

0

05

1

500

prail (normalised)


+25 NOx

minus35 NOx

Reference NOx level





119909emissions of





119909levels are


mfcc

mfcc lt 1mgNeng

uVGTdrag

uVGTFF

uVGTFB

pIMAirpathuVGT

+

+

+

minusPI

pIMref

dNeng

dt





119906VGTFB = 119896

119901sdot 119890

119901IM(119905) +

119896

119894

10

sdot int

119905

0

119890

119901IM(120591) 119889120591

(30)

with 119890119901IM

= 10

minus4







minus3 minus

2

minus1

minus0

5

minus0

5

1

10

20

30

40


2

0

minus1

minus2

minus102

10

1

10

20

30

40

50


07 0

4

03

025

025 0

3 04

02

18

1412

11

11

12


ki(mdash

)

ki(mdash

)

1

10

20

30

40

50

1 5 10 15

ki(mdash

)

kp (mdash)

1

10

20

30

40

50

1 5 10 15

ki(mdash

)

kp (mdash)









119909emissions


119909emissions to



119909



119909emissions






119909emissions








0

1000

2000


0

05

1

ReferenceOptimal

Causal

ReferenceOptimal

Causal

Dragphases

pIM

(nor

mal

ised)

Time (s)388 395 400 405 410 415 420 425 430 435 440 447

30

60

90

uVG

T(

)


1470 1472 1474 1476

500

1000

1500

2000

2500

Time (s)


025 05

(a) (b)

075 1

1

102

104

106

108

Reference no EGR

Sweep with EGR


Fuel

with

resp

ect t

o re

fere

nce (

mdash)






119879load (119905) = 120578

119882(119905) sdot 119898fcc (119905)

minus 119879

0(119873eng (119905) 119901EM (119905) minus 119901IM (119905))

(31)





100

200

300

400

Tlo

ad(N

m)

70

80

90

100

uVG

T(

)

0

05

1

(nor

mal

ised)

120593SO

I

1470 1471 1472 1473 1474 1475 14760

25

50

75

100

Time (s)



uEG

R(

)




119898fccNEW (119905) = 119898fcc (119905) sdot

119879load (119905) minus 119879

0(119905)

119879load (119905) minus 119879

0(119905)

(32)



119909







NO119909








119909emissions




119896ntf = minus

Δ119898NO119909

119898NO119909ref

sdot

119898fuelref

Δ119898fuel (33)



119909emissions is





4 Conclusion



Time (s)

Opa

city

(1m

)

388 400 410 420 430 440 4470

005

01

015

02




119909tradeoff



Nomenclature




119909-to-fuel


Latin Symbols


Greek Symbols





Acknowledgment


References































































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













119871 reg (119906sdot) = 119888

119872sdot

Var2119905119873

(119906

sdot) = 119888

119872sdot

1

2

119872minus1

sum

119897=3

1003816

1003816

1003816

1003816

119904

119897+1minus 119904

119897

1003816

1003816

1003816

1003816

2

(27)

where 119904119897= (119906

119897minus 119906

119897minus1)(119905

119897minus 119905



119872is chosen as

119888

119872=

119888reg

(119872 minus 3) (119872 minus 1)

2 (28)







lowast



x =

(

(

(

(

(

(

(

(

(

(

119901IM119901EM120596TC119901

1

119901

4

120599IM119909BGIM120599EMC120599IMC

)

)

)

)

)

)

)

)

)

)

u =

(

(

(


)

)

)

120587 = (

119873eng

119873eng

119879load

)

(29)



lowast

119898NO119909

lowast

119898soot)119879









0 200 400 600 800 1000 1200 1400 1600 18000

1000

2000

Time (s)

583671

0

1000

2000

Time (s)49 55

1 6

654 670 690 708

3

900 920 940 958

5

1629 1650 1670 1690 1710 1730 1746


353 375 400 425 450 475 500 525 550 575 600 625 6480

1000

2000

Time (s)

24







































20

40

60

80

uVG

T(

)Re

lativ

e tru

nca-

tion

erro

r (mdash

)

10minus4

10minus3

10minus2

10minus1

49 50 51 52 53 54 55012

New

poi

nts

Time (s)








20

40

60

80

100

uVG

T(

)

0

05

1

(nor

mal

ised)

120593SO

I

05

354 355 356 357 358 359 3600

1

Time (s)

pra

il(n

orm

alise

d)

creg = 0

creg = 10

creg = 100

Bounds














12502000

0 110

220

0

05

1

500m

fcc (mg) Neng (rpm)


125020000 110 220

0

05

1

500

prail (normalised)


+25 NOx

minus35 NOx

Reference NOx level





119909emissions of





119909levels are


mfcc

mfcc lt 1mgNeng

uVGTdrag

uVGTFF

uVGTFB

pIMAirpathuVGT

+

+

+

minusPI

pIMref

dNeng

dt





119906VGTFB = 119896

119901sdot 119890

119901IM(119905) +

119896

119894

10

sdot int

119905

0

119890

119901IM(120591) 119889120591

(30)

with 119890119901IM

= 10

minus4







minus3 minus

2

minus1

minus0

5

minus0

5

1

10

20

30

40


2

0

minus1

minus2

minus102

10

1

10

20

30

40

50


07 0

4

03

025

025 0

3 04

02

18

1412

11

11

12


ki(mdash

)

ki(mdash

)

1

10

20

30

40

50

1 5 10 15

ki(mdash

)

kp (mdash)

1

10

20

30

40

50

1 5 10 15

ki(mdash

)

kp (mdash)









119909emissions


119909emissions to



119909



119909emissions






119909emissions








0

1000

2000


0

05

1

ReferenceOptimal

Causal

ReferenceOptimal

Causal

Dragphases

pIM

(nor

mal

ised)

Time (s)388 395 400 405 410 415 420 425 430 435 440 447

30

60

90

uVG

T(

)


1470 1472 1474 1476

500

1000

1500

2000

2500

Time (s)


025 05

(a) (b)

075 1

1

102

104

106

108

Reference no EGR

Sweep with EGR


Fuel

with

resp

ect t

o re

fere

nce (

mdash)






119879load (119905) = 120578

119882(119905) sdot 119898fcc (119905)

minus 119879

0(119873eng (119905) 119901EM (119905) minus 119901IM (119905))

(31)





100

200

300

400

Tlo

ad(N

m)

70

80

90

100

uVG

T(

)

0

05

1

(nor

mal

ised)

120593SO

I

1470 1471 1472 1473 1474 1475 14760

25

50

75

100

Time (s)



uEG

R(

)




119898fccNEW (119905) = 119898fcc (119905) sdot

119879load (119905) minus 119879

0(119905)

119879load (119905) minus 119879

0(119905)

(32)



119909







NO119909








119909emissions




119896ntf = minus

Δ119898NO119909

119898NO119909ref

sdot

119898fuelref

Δ119898fuel (33)



119909emissions is





4 Conclusion



Time (s)

Opa

city

(1m

)

388 400 410 420 430 440 4470

005

01

015

02




119909tradeoff



Nomenclature




119909-to-fuel


Latin Symbols


Greek Symbols





Acknowledgment


References































































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 200 400 600 800 1000 1200 1400 1600 18000

1000

2000

Time (s)

583671

0

1000

2000

Time (s)49 55

1 6

654 670 690 708

3

900 920 940 958

5

1629 1650 1670 1690 1710 1730 1746


353 375 400 425 450 475 500 525 550 575 600 625 6480

1000

2000

Time (s)

24







































20

40

60

80

uVG

T(

)Re

lativ

e tru

nca-

tion

erro

r (mdash

)

10minus4

10minus3

10minus2

10minus1

49 50 51 52 53 54 55012

New

poi

nts

Time (s)








20

40

60

80

100

uVG

T(

)

0

05

1

(nor

mal

ised)

120593SO

I

05

354 355 356 357 358 359 3600

1

Time (s)

pra

il(n

orm

alise

d)

creg = 0

creg = 10

creg = 100

Bounds














12502000

0 110

220

0

05

1

500m

fcc (mg) Neng (rpm)


125020000 110 220

0

05

1

500

prail (normalised)


+25 NOx

minus35 NOx

Reference NOx level





119909emissions of





119909levels are


mfcc

mfcc lt 1mgNeng

uVGTdrag

uVGTFF

uVGTFB

pIMAirpathuVGT

+

+

+

minusPI

pIMref

dNeng

dt





119906VGTFB = 119896

119901sdot 119890

119901IM(119905) +

119896

119894

10

sdot int

119905

0

119890

119901IM(120591) 119889120591

(30)

with 119890119901IM

= 10

minus4







minus3 minus

2

minus1

minus0

5

minus0

5

1

10

20

30

40


2

0

minus1

minus2

minus102

10

1

10

20

30

40

50


07 0

4

03

025

025 0

3 04

02

18

1412

11

11

12


ki(mdash

)

ki(mdash

)

1

10

20

30

40

50

1 5 10 15

ki(mdash

)

kp (mdash)

1

10

20

30

40

50

1 5 10 15

ki(mdash

)

kp (mdash)









119909emissions


119909emissions to



119909



119909emissions






119909emissions








0

1000

2000


0

05

1

ReferenceOptimal

Causal

ReferenceOptimal

Causal

Dragphases

pIM

(nor

mal

ised)

Time (s)388 395 400 405 410 415 420 425 430 435 440 447

30

60

90

uVG

T(

)


1470 1472 1474 1476

500

1000

1500

2000

2500

Time (s)


025 05

(a) (b)

075 1

1

102

104

106

108

Reference no EGR

Sweep with EGR


Fuel

with

resp

ect t

o re

fere

nce (

mdash)






119879load (119905) = 120578

119882(119905) sdot 119898fcc (119905)

minus 119879

0(119873eng (119905) 119901EM (119905) minus 119901IM (119905))

(31)





100

200

300

400

Tlo

ad(N

m)

70

80

90

100

uVG

T(

)

0

05

1

(nor

mal

ised)

120593SO

I

1470 1471 1472 1473 1474 1475 14760

25

50

75

100

Time (s)



uEG

R(

)




119898fccNEW (119905) = 119898fcc (119905) sdot

119879load (119905) minus 119879

0(119905)

119879load (119905) minus 119879

0(119905)

(32)



119909







NO119909








119909emissions




119896ntf = minus

Δ119898NO119909

119898NO119909ref

sdot

119898fuelref

Δ119898fuel (33)



119909emissions is





4 Conclusion



Time (s)

Opa

city

(1m

)

388 400 410 420 430 440 4470

005

01

015

02




119909tradeoff



Nomenclature




119909-to-fuel


Latin Symbols


Greek Symbols





Acknowledgment


References































































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra




































20

40

60

80

uVG

T(

)Re

lativ

e tru

nca-

tion

erro

r (mdash

)

10minus4

10minus3

10minus2

10minus1

49 50 51 52 53 54 55012

New

poi

nts

Time (s)








20

40

60

80

100

uVG

T(

)

0

05

1

(nor

mal

ised)

120593SO

I

05

354 355 356 357 358 359 3600

1

Time (s)

pra

il(n

orm

alise

d)

creg = 0

creg = 10

creg = 100

Bounds














12502000

0 110

220

0

05

1

500m

fcc (mg) Neng (rpm)


125020000 110 220

0

05

1

500

prail (normalised)


+25 NOx

minus35 NOx

Reference NOx level





119909emissions of





119909levels are


mfcc

mfcc lt 1mgNeng

uVGTdrag

uVGTFF

uVGTFB

pIMAirpathuVGT

+

+

+

minusPI

pIMref

dNeng

dt





119906VGTFB = 119896

119901sdot 119890

119901IM(119905) +

119896

119894

10

sdot int

119905

0

119890

119901IM(120591) 119889120591

(30)

with 119890119901IM

= 10

minus4







minus3 minus

2

minus1

minus0

5

minus0

5

1

10

20

30

40


2

0

minus1

minus2

minus102

10

1

10

20

30

40

50


07 0

4

03

025

025 0

3 04

02

18

1412

11

11

12


ki(mdash

)

ki(mdash

)

1

10

20

30

40

50

1 5 10 15

ki(mdash

)

kp (mdash)

1

10

20

30

40

50

1 5 10 15

ki(mdash

)

kp (mdash)









119909emissions


119909emissions to



119909



119909emissions






119909emissions








0

1000

2000


0

05

1

ReferenceOptimal

Causal

ReferenceOptimal

Causal

Dragphases

pIM

(nor

mal

ised)

Time (s)388 395 400 405 410 415 420 425 430 435 440 447

30

60

90

uVG

T(

)


1470 1472 1474 1476

500

1000

1500

2000

2500

Time (s)


025 05

(a) (b)

075 1

1

102

104

106

108

Reference no EGR

Sweep with EGR


Fuel

with

resp

ect t

o re

fere

nce (

mdash)






119879load (119905) = 120578

119882(119905) sdot 119898fcc (119905)

minus 119879

0(119873eng (119905) 119901EM (119905) minus 119901IM (119905))

(31)





100

200

300

400

Tlo

ad(N

m)

70

80

90

100

uVG

T(

)

0

05

1

(nor

mal

ised)

120593SO

I

1470 1471 1472 1473 1474 1475 14760

25

50

75

100

Time (s)



uEG

R(

)




119898fccNEW (119905) = 119898fcc (119905) sdot

119879load (119905) minus 119879

0(119905)

119879load (119905) minus 119879

0(119905)

(32)



119909







NO119909








119909emissions




119896ntf = minus

Δ119898NO119909

119898NO119909ref

sdot

119898fuelref

Δ119898fuel (33)



119909emissions is





4 Conclusion



Time (s)

Opa

city

(1m

)

388 400 410 420 430 440 4470

005

01

015

02




119909tradeoff



Nomenclature




119909-to-fuel


Latin Symbols


Greek Symbols





Acknowledgment


References































































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra

















20

40

60

80

uVG

T(

)Re

lativ

e tru

nca-

tion

erro

r (mdash

)

10minus4

10minus3

10minus2

10minus1

49 50 51 52 53 54 55012

New

poi

nts

Time (s)








20

40

60

80

100

uVG

T(

)

0

05

1

(nor

mal

ised)

120593SO

I

05

354 355 356 357 358 359 3600

1

Time (s)

pra

il(n

orm

alise

d)

creg = 0

creg = 10

creg = 100

Bounds














12502000

0 110

220

0

05

1

500m

fcc (mg) Neng (rpm)


125020000 110 220

0

05

1

500

prail (normalised)


+25 NOx

minus35 NOx

Reference NOx level





119909emissions of





119909levels are


mfcc

mfcc lt 1mgNeng

uVGTdrag

uVGTFF

uVGTFB

pIMAirpathuVGT

+

+

+

minusPI

pIMref

dNeng

dt





119906VGTFB = 119896

119901sdot 119890

119901IM(119905) +

119896

119894

10

sdot int

119905

0

119890

119901IM(120591) 119889120591

(30)

with 119890119901IM

= 10

minus4







minus3 minus

2

minus1

minus0

5

minus0

5

1

10

20

30

40


2

0

minus1

minus2

minus102

10

1

10

20

30

40

50


07 0

4

03

025

025 0

3 04

02

18

1412

11

11

12


ki(mdash

)

ki(mdash

)

1

10

20

30

40

50

1 5 10 15

ki(mdash

)

kp (mdash)

1

10

20

30

40

50

1 5 10 15

ki(mdash

)

kp (mdash)









119909emissions


119909emissions to



119909



119909emissions






119909emissions








0

1000

2000


0

05

1

ReferenceOptimal

Causal

ReferenceOptimal

Causal

Dragphases

pIM

(nor

mal

ised)

Time (s)388 395 400 405 410 415 420 425 430 435 440 447

30

60

90

uVG

T(

)


1470 1472 1474 1476

500

1000

1500

2000

2500

Time (s)


025 05

(a) (b)

075 1

1

102

104

106

108

Reference no EGR

Sweep with EGR


Fuel

with

resp

ect t

o re

fere

nce (

mdash)






119879load (119905) = 120578

119882(119905) sdot 119898fcc (119905)

minus 119879

0(119873eng (119905) 119901EM (119905) minus 119901IM (119905))

(31)





100

200

300

400

Tlo

ad(N

m)

70

80

90

100

uVG

T(

)

0

05

1

(nor

mal

ised)

120593SO

I

1470 1471 1472 1473 1474 1475 14760

25

50

75

100

Time (s)



uEG

R(

)




119898fccNEW (119905) = 119898fcc (119905) sdot

119879load (119905) minus 119879

0(119905)

119879load (119905) minus 119879

0(119905)

(32)



119909







NO119909








119909emissions




119896ntf = minus

Δ119898NO119909

119898NO119909ref

sdot

119898fuelref

Δ119898fuel (33)



119909emissions is





4 Conclusion



Time (s)

Opa

city

(1m

)

388 400 410 420 430 440 4470

005

01

015

02




119909tradeoff



Nomenclature




119909-to-fuel


Latin Symbols


Greek Symbols





Acknowledgment


References































































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










20

40

60

80

100

uVG

T(

)

0

05

1

(nor

mal

ised)

120593SO

I

05

354 355 356 357 358 359 3600

1

Time (s)

pra

il(n

orm

alise

d)

creg = 0

creg = 10

creg = 100

Bounds














12502000

0 110

220

0

05

1

500m

fcc (mg) Neng (rpm)


125020000 110 220

0

05

1

500

prail (normalised)


+25 NOx

minus35 NOx

Reference NOx level





119909emissions of





119909levels are


mfcc

mfcc lt 1mgNeng

uVGTdrag

uVGTFF

uVGTFB

pIMAirpathuVGT

+

+

+

minusPI

pIMref

dNeng

dt





119906VGTFB = 119896

119901sdot 119890

119901IM(119905) +

119896

119894

10

sdot int

119905

0

119890

119901IM(120591) 119889120591

(30)

with 119890119901IM

= 10

minus4







minus3 minus

2

minus1

minus0

5

minus0

5

1

10

20

30

40


2

0

minus1

minus2

minus102

10

1

10

20

30

40

50


07 0

4

03

025

025 0

3 04

02

18

1412

11

11

12


ki(mdash

)

ki(mdash

)

1

10

20

30

40

50

1 5 10 15

ki(mdash

)

kp (mdash)

1

10

20

30

40

50

1 5 10 15

ki(mdash

)

kp (mdash)









119909emissions


119909emissions to



119909



119909emissions






119909emissions








0

1000

2000


0

05

1

ReferenceOptimal

Causal

ReferenceOptimal

Causal

Dragphases

pIM

(nor

mal

ised)

Time (s)388 395 400 405 410 415 420 425 430 435 440 447

30

60

90

uVG

T(

)


1470 1472 1474 1476

500

1000

1500

2000

2500

Time (s)


025 05

(a) (b)

075 1

1

102

104

106

108

Reference no EGR

Sweep with EGR


Fuel

with

resp

ect t

o re

fere

nce (

mdash)






119879load (119905) = 120578

119882(119905) sdot 119898fcc (119905)

minus 119879

0(119873eng (119905) 119901EM (119905) minus 119901IM (119905))

(31)





100

200

300

400

Tlo

ad(N

m)

70

80

90

100

uVG

T(

)

0

05

1

(nor

mal

ised)

120593SO

I

1470 1471 1472 1473 1474 1475 14760

25

50

75

100

Time (s)



uEG

R(

)




119898fccNEW (119905) = 119898fcc (119905) sdot

119879load (119905) minus 119879

0(119905)

119879load (119905) minus 119879

0(119905)

(32)



119909







NO119909








119909emissions




119896ntf = minus

Δ119898NO119909

119898NO119909ref

sdot

119898fuelref

Δ119898fuel (33)



119909emissions is





4 Conclusion



Time (s)

Opa

city

(1m

)

388 400 410 420 430 440 4470

005

01

015

02




119909tradeoff



Nomenclature




119909-to-fuel


Latin Symbols


Greek Symbols





Acknowledgment


References































































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










12502000

0 110

220

0

05

1

500m

fcc (mg) Neng (rpm)


125020000 110 220

0

05

1

500

prail (normalised)


+25 NOx

minus35 NOx

Reference NOx level





119909emissions of





119909levels are


mfcc

mfcc lt 1mgNeng

uVGTdrag

uVGTFF

uVGTFB

pIMAirpathuVGT

+

+

+

minusPI

pIMref

dNeng

dt





119906VGTFB = 119896

119901sdot 119890

119901IM(119905) +

119896

119894

10

sdot int

119905

0

119890

119901IM(120591) 119889120591

(30)

with 119890119901IM

= 10

minus4







minus3 minus

2

minus1

minus0

5

minus0

5

1

10

20

30

40


2

0

minus1

minus2

minus102

10

1

10

20

30

40

50


07 0

4

03

025

025 0

3 04

02

18

1412

11

11

12


ki(mdash

)

ki(mdash

)

1

10

20

30

40

50

1 5 10 15

ki(mdash

)

kp (mdash)

1

10

20

30

40

50

1 5 10 15

ki(mdash

)

kp (mdash)









119909emissions


119909emissions to



119909



119909emissions






119909emissions








0

1000

2000


0

05

1

ReferenceOptimal

Causal

ReferenceOptimal

Causal

Dragphases

pIM

(nor

mal

ised)

Time (s)388 395 400 405 410 415 420 425 430 435 440 447

30

60

90

uVG

T(

)


1470 1472 1474 1476

500

1000

1500

2000

2500

Time (s)


025 05

(a) (b)

075 1

1

102

104

106

108

Reference no EGR

Sweep with EGR


Fuel

with

resp

ect t

o re

fere

nce (

mdash)






119879load (119905) = 120578

119882(119905) sdot 119898fcc (119905)

minus 119879

0(119873eng (119905) 119901EM (119905) minus 119901IM (119905))

(31)





100

200

300

400

Tlo

ad(N

m)

70

80

90

100

uVG

T(

)

0

05

1

(nor

mal

ised)

120593SO

I

1470 1471 1472 1473 1474 1475 14760

25

50

75

100

Time (s)



uEG

R(

)




119898fccNEW (119905) = 119898fcc (119905) sdot

119879load (119905) minus 119879

0(119905)

119879load (119905) minus 119879

0(119905)

(32)



119909







NO119909








119909emissions




119896ntf = minus

Δ119898NO119909

119898NO119909ref

sdot

119898fuelref

Δ119898fuel (33)



119909emissions is





4 Conclusion



Time (s)

Opa

city

(1m

)

388 400 410 420 430 440 4470

005

01

015

02




119909tradeoff



Nomenclature




119909-to-fuel


Latin Symbols


Greek Symbols





Acknowledgment


References































































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










minus3 minus

2

minus1

minus0

5

minus0

5

1

10

20

30

40


2

0

minus1

minus2

minus102

10

1

10

20

30

40

50


07 0

4

03

025

025 0

3 04

02

18

1412

11

11

12


ki(mdash

)

ki(mdash

)

1

10

20

30

40

50

1 5 10 15

ki(mdash

)

kp (mdash)

1

10

20

30

40

50

1 5 10 15

ki(mdash

)

kp (mdash)









119909emissions


119909emissions to



119909



119909emissions






119909emissions








0

1000

2000


0

05

1

ReferenceOptimal

Causal

ReferenceOptimal

Causal

Dragphases

pIM

(nor

mal

ised)

Time (s)388 395 400 405 410 415 420 425 430 435 440 447

30

60

90

uVG

T(

)


1470 1472 1474 1476

500

1000

1500

2000

2500

Time (s)


025 05

(a) (b)

075 1

1

102

104

106

108

Reference no EGR

Sweep with EGR


Fuel

with

resp

ect t

o re

fere

nce (

mdash)






119879load (119905) = 120578

119882(119905) sdot 119898fcc (119905)

minus 119879

0(119873eng (119905) 119901EM (119905) minus 119901IM (119905))

(31)





100

200

300

400

Tlo

ad(N

m)

70

80

90

100

uVG

T(

)

0

05

1

(nor

mal

ised)

120593SO

I

1470 1471 1472 1473 1474 1475 14760

25

50

75

100

Time (s)



uEG

R(

)




119898fccNEW (119905) = 119898fcc (119905) sdot

119879load (119905) minus 119879

0(119905)

119879load (119905) minus 119879

0(119905)

(32)



119909







NO119909








119909emissions




119896ntf = minus

Δ119898NO119909

119898NO119909ref

sdot

119898fuelref

Δ119898fuel (33)



119909emissions is





4 Conclusion



Time (s)

Opa

city

(1m

)

388 400 410 420 430 440 4470

005

01

015

02




119909tradeoff



Nomenclature




119909-to-fuel


Latin Symbols


Greek Symbols





Acknowledgment


References































































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0

1000

2000


0

05

1

ReferenceOptimal

Causal

ReferenceOptimal

Causal

Dragphases

pIM

(nor

mal

ised)

Time (s)388 395 400 405 410 415 420 425 430 435 440 447

30

60

90

uVG

T(

)


1470 1472 1474 1476

500

1000

1500

2000

2500

Time (s)


025 05

(a) (b)

075 1

1

102

104

106

108

Reference no EGR

Sweep with EGR


Fuel

with

resp

ect t

o re

fere

nce (

mdash)






119879load (119905) = 120578

119882(119905) sdot 119898fcc (119905)

minus 119879

0(119873eng (119905) 119901EM (119905) minus 119901IM (119905))

(31)





100

200

300

400

Tlo

ad(N

m)

70

80

90

100

uVG

T(

)

0

05

1

(nor

mal

ised)

120593SO

I

1470 1471 1472 1473 1474 1475 14760

25

50

75

100

Time (s)



uEG

R(

)




119898fccNEW (119905) = 119898fcc (119905) sdot

119879load (119905) minus 119879

0(119905)

119879load (119905) minus 119879

0(119905)

(32)



119909







NO119909








119909emissions




119896ntf = minus

Δ119898NO119909

119898NO119909ref

sdot

119898fuelref

Δ119898fuel (33)



119909emissions is





4 Conclusion



Time (s)

Opa

city

(1m

)

388 400 410 420 430 440 4470

005

01

015

02




119909tradeoff



Nomenclature




119909-to-fuel


Latin Symbols


Greek Symbols





Acknowledgment


References































































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










100

200

300

400

Tlo

ad(N

m)

70

80

90

100

uVG

T(

)

0

05

1

(nor

mal

ised)

120593SO

I

1470 1471 1472 1473 1474 1475 14760

25

50

75

100

Time (s)



uEG

R(

)




119898fccNEW (119905) = 119898fcc (119905) sdot

119879load (119905) minus 119879

0(119905)

119879load (119905) minus 119879

0(119905)

(32)



119909







NO119909








119909emissions




119896ntf = minus

Δ119898NO119909

119898NO119909ref

sdot

119898fuelref

Δ119898fuel (33)



119909emissions is





4 Conclusion



Time (s)

Opa

city

(1m

)

388 400 410 420 430 440 4470

005

01

015

02




119909tradeoff



Nomenclature




119909-to-fuel


Latin Symbols


Greek Symbols





Acknowledgment


References































































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Time (s)

Opa

city

(1m

)

388 400 410 420 430 440 4470

005

01

015

02




119909tradeoff



Nomenclature




119909-to-fuel


Latin Symbols


Greek Symbols





Acknowledgment


References































































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












Acknowledgment


References































































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra

































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Documents

Research Article Optimal Control of Diesel Engines ...downloads.hindawi.com/journals/mpe/2014/286538.pdf · Research Article Optimal Control of Diesel Engines: Numerical Methods,