Transient Air-to-Fuel Ratio Control of an Spark …...Transient Air-to-Fuel Ratio Control of an Spark Ignited Engine Using Linear Quadratic Tracking Modern spark ignited (SI) internal

Stephen PaceMem. ASME

Electrical and Computer Engineering,

Michigan State University,

East Lansing, MI 48824

e-mail: [email protected]

Guoming G. ZhuFellow ASME

Department of Mechanical Engineering,

Department of Electrical and

Computer Engineering,

Michigan State University,

1497 Engineering Research Court,

Room E148,

East Lansing, MI 48824

e-mail: [email protected]

Transient Air-to-Fuel RatioControl of an Spark IgnitedEngine Using Linear QuadraticTrackingModern spark ignited (SI) internal combustion engines maintain their air-to-fuel ratio(AFR) at a desired level to maximize the three-way catalyst conversion efficiency and du-rability. However, maintaining the engine AFR during its transient operation is quitechallenging due to rapid changes of driver demand or engine throttle. Conventional tran-sient AFR control is based upon the inverse dynamics of the engine fueling dynamics andthe measured mass air flow (MAF) rate to obtain the desired AFR of the gas mixturetrapped in the cylinder. This paper develops a linear quadratic (LQ) tracking controllerto regulate the transient AFR based upon a control-oriented model of the engine port fuelinjection (PFI) wall wetting dynamics and the air intake dynamics from the measured air-flow to the manifold pressure. The LQ tracking controller is designed to optimally trackthe desired AFR by minimizing the error between the trapped in-cylinder air mass andthe product of the desired AFR and fuel mass over a given time interval. The performanceof the optimal LQ tracking controller was compared with the conventional transient fuel-ing control based on the inverse fueling dynamics through simulations and showedimprovement over the baseline conventional inverse fueling dynamics controller. To vali-date the control strategy on an actual engine, a 0.4 l single cylinder direct-injection (DI)engine was used. The PFI wall wetting dynamics were simulated in the engine controllerafter the DI injector control signal. Engine load transition tests for the simulated PFIcase were conducted on an engine dynamometer, and the results showed improvementover the baseline transient fueling controller based on the inverse fueling dynamics.[DOI: 10.1115/1.4025858]

1 Introduction

Most modern SI internal combustion engines maintain theirAFR at a desired level to maximize the three-way catalyst effi-ciency and durability. Operating SI engines at a desired AFR isdue to the fact that the highest conversion efficiency of a three-way catalyst occurs around the stoichiometric AFR. The controlof AFR is an increasingly important control due to the federal andstate emission regulations. As the emission regulation gets tight,transient AFR control becomes an important topic. In recent years,there have been several fuel control strategies developed for auto-motive engines to improve the efficiency and to reduce exhaustemissions. These AFR research efforts include adaptive control[1], sliding mode control [2], linear parameter-varying control [3],and flex-fuel puddle compensation [4].

There have been significant studies through experimentalresearch in the last few decades to characterize the deviation ofAFR during engine transient operations. These studies include theuse of a transfer function to simulate the mixture-formation pro-cess by Stivender [5], a detailed study of the simple phenomeno-logical model for fuel transport in the intake port by Aquino [6],an investigation of fuel transfer characteristics during transientoperations including cold start by Hires [7] and Rose et al. [8–10]found that the durations of AFR deviations were shorter than whatwere previously discovered by other researchers. Currently, a fewresearchers have developed control strategies for gasoline SIengines that are designed specifically to address and improve thetransient AFR control problem. These research efforts include acontrol strategy that involves a model that takes account of

manifold filling and the delays in transport of fuel from the injec-tors to the cylinder by Hu [11], a simple linear control approachusing least squares estimation by Ye [12], and a control strategythat combined the modified Elman neural network and the tradi-tional PI (proportional and integral) controller by Yao [13]. Ulti-mately, there has not been much research dedicated to transientAFR control within the last a few years and this paper discussesthe application of the optimal LQ theory to transient AFR control.

Production AFR control is commonly achieved by combiningfeedback and feedforward control of the fuel injection for a givenair charge in the cylinder. The feedback control is based on theoxygen sensor measurement of the AFR. The feedforward controlis used to reduce transient effects of the PFI wall wetting dynam-ics and air intake dynamics. There is increasing literature onimproving transient AFR control, see Refs. [14–17]. Conventionalfeedforward AFR control is based on the inverse fueling dynam-ics, where it is primarily derived from the estimated cylinder aircharge divided by the desired AFR ratio of the air and fuel (Refs.[4,18–20]). Although the use of feedforward control can signifi-cantly improve the transient response of AFR, its control accuracyis based upon the fuel injection and air intake model and there isroom for improvement.

An alternative approach of designing a feedforward AFR con-trol is to use the measured air flow at the engine throttle and its aircharge dynamics from the engine throttle to cylinder to track thedesired AFR during engine transients [21]. This control isexpected to reduce the AFR tracking error during engine transientoperations.

In this paper, a control-oriented model of the fuel and air flowdynamics is used for AFR control design and simulation evalua-tion. The fuel flow model includes the wall wetting dynamics of aPFI injector; and the air flow model includes throttle dynamics,transport delays, and manifold filling dynamics. A finite horizon

Contributed by the Dynamic Systems Division of ASME for publication in theJOURNAL OF DYNAMIC SYSTEMS, MEASUREMENT, AND CONTROL. Manuscript receivedApril 20, 2012; final manuscript received October 24, 2013; published onlineDecember 9, 2013. Assoc. Editor: Eric J. Barth.

Journal of Dynamic Systems, Measurement, and Control MARCH 2014, Vol. 136 / 021008-1Copyright VC 2014 by ASME

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LQ tracking AFR controller is designed based on this model totrack the estimated airflow into the engine cylinder during tran-sient engine operations. The finite horizon LQ tracking controllaw, motivated by its tracking feature, for a given period isupdated at every control step and only the first sample of the cal-culated control signal is used for real-time control at current step.This control strategy is also commonly referred to model predic-tive control (receding horizon control). The developed controlstrategy was validated in simulations and compared with the con-ventional inverse fueling dynamics control law and showedimprovements in the AFR over the conventional controller duringengine transient operations. Next, the designed LQ transient AFRcontrol strategy was validated on a single cylinder engine throughdynamometer tests. The test results confirm the simulations.

The remainder of the paper is organized as follows: Section 2presents a control-oriented engine airflow and fuel flow model,followed by a section that describes an innovative LQ trackingcontroller that adjusts the fuel injection quantity during enginetransient operations. Section 4 provides the simulation and experi-mental results are presented in Sec. 5. Finally, conclusions arediscussed.

2 Air Flow and Fuel Flow Model

The control problem studied in this paper is to adjust the portfuel injection rate xPFI so that the engine AFR deviation from thedesired level (e.g., stoichiometry) is minimized during enginetransient operations.

A control-oriented hardware-in-the-loop (HIL) four-cylinderdual-fuel mean-value engine model, developed based upon Ref.[22] and modified from Ref. [23], was used to develop the air andfuel flow model used in this study. The term “mean-value” indi-cates that the previously developed engine model neglects thereciprocating behavior of the engine, assuming all processes andeffects are spread out over the engine cycle. During the HIL simu-lations, this model describes the input–output behavior of thephysical engine system with reasonable simulation accuracyusing relatively low computational throughput. Reference [24]provides a good overview of engine modeling, along with most ofdynamic equations used in the four-cylinder model. This enginemodel also includes all engine transient dynamics. Figure 1 showsthe overall mean-value engine model architecture, along withmain subsystem models, such as AFR, manifold air pressure, Pin,brake mean effective pressure, engine torque, exhaust tempera-ture, etc.

The subsystems from Fig. 1 that were used for the airflow andfuel model in this study were the throttle and intake manifold dy-namics model, fueling dynamics model, air charge model, andengine combustion AFR calculation. Note that the control-oriented model needs to be simple enough for online control

update but also includes necessary dynamics since the developedLQ control will be used for real-time engine control and the con-trol parameters are updated every control step. Therefore, theequations used to describe these dynamics and calculations weresimplified for control design purposes and a linear air and fuelflow model was created as follows.

The engine throttle dynamics, xthrottle, are modeled by the fol-lowing transfer function:

xthrottle ¼1

s1sþ 1xair (1)

where xair is the airflow into the throttle and s1 is the throttle timeconstant including both motor and throttle plate dynamics. Thestate space representation of the throttle plate dynamics is

_xthrottle ¼ �1

s1

xthrottle þ1

s1

xair

zthrottle ¼ xthrottle

(2)

The delay due to air travel from the engine throttle to manifold ismodeled as a transport delay. It is typically a function of enginespeed and is approximated as a pure transport delay sd. The iso-thermal model of the filling dynamics of an engine intake mani-fold, xmanifold, can be approximated by a first order transferfunction as

xmanifold

xthrottle

¼ 1

s2sþ 1(3)

where s2 is the time constant.The effective fuel flow for wall wetting dynamics from the port

fuel injector, xPFI_E, is modeled by the following transfer function

xPFI E

xPFI

¼ asþ 1

bsþ 1(4)

which is a continuous time approximation of the event-based portfuel injection process, where a portion of the injected fuel, xPFI, isinjected directly into the cylinder and portion of the fuel remainson the back of the intake valve and, at the same time, a portion ofthe residual fuel remained on the back of the intake valve isvaporized and flew into the cylinder. The fuel flow wall wettingmodel for discrete time dynamics was proposed by Aquino in Ref.[6]. Coefficients a and b are obtained to match the discrete timetransfer function. In addition to the wall wetting dynamics, thereis an average PFI fuel injection delay, s3, from updating the fuelinjection command to the time that fuel is injected, which is again

Fig. 1 Mean-value engine model architecture

021008-2 / Vol. 136, MARCH 2014 Transactions of the ASME


approximated by a unitary gain first order transfer function, simi-lar to Eqs. (2) and (3). The state space representation for the com-plete PFI fueling dynamics is

_xfuel ¼� 1

s3

0

1

b1� a

b

� �� 1

b

2664

3775xfuel þ

1

s3

0

24

35ufuel

yfuel ¼ab

1

� �xfuel

(5)

where ufuel is the input to the PFI injector and yfuel is the injectedfuel. Also note that yfuel¼xPFI.

Due to the three-way catalyst used for emission control,most engines are design to achieve a target A/F ratio aroundstoichiometric. For this study, we use a normalized target A/Fratio, ktarget, which is defined as the desired AFR divided by stoi-chiometric AFR. Note that at stoichiometry the normalized targetA/F ratio is equal to one. The normalized A/F ratio can be calcu-lated as

k ¼ xcyl

14:6 � xPFI(6)

A schematic diagram of the AFR control scheme is shown in Fig.2 and the details of the LQ tracking controller will be discussed indetail in Sec. 3.

3 Linear Quadratic Tracking Controller

In this section, a finite horizon LQ tracking AFR controller isdesigned based on the model previously described to track thedesired normalized AFR during engine transients. More specifi-cally, the control objective is for yfuel to optimally track ktarget

over a given time interval. Later, the optimal control problem willbe transferred to the equivalent optimal control problem that mini-mizes the tracking error between the measured airflow and theproduct of fuel flow and desired AFR.

The continuous time system in Eq. (5) is discretized at a samplerate of 10 ms with the engine parameters defined in Table 1, wherethese parameters were determined by matching the AFR responsesat the engine speed of 1500 RPM. Thus, it becomes a linear, time-invariant system described below

xfuelðk þ 1Þ ¼ AxfuelðkÞ þ BufuelðkÞyfuelðkÞ ¼ CxfuelðkÞ (7)

where yfuelðkÞ is the fuel flow into the cylinder. The values for thematrices of discrete time system (7) are

A ¼1:7664 �0:887

0:887 0

" #; B ¼

1

0

" #;

C ¼ 0:1383 �0:1545½ �

For notational simplicity, xk, uk, and yk will be used to denote thesystem state xfuelðkÞ, control ufuelðkÞ, output yfuelðkÞ at time k. Letzk be the airflow into the cylinder divided by the 14.6 and the nor-malized desired AFR at time k. Note that the detailed discussionsregarding how to generate zk can be found at the end of this section.Consider the discrete time, linear time-invariant system in Eq. (7)and define the performance cost function to be minimized as

Jðk0Þ ¼1

2ykf� zkf

� �TF ykf

� zkf

� �þ 1

2

Xkf�1

k¼k0

yk � zk½ �TQ yk � zk½ � þ uTk Ruk

n o(8)

where, F and Q are positive semidefinite symmetric matrices andR is a positive definite symmetric matrix. The initial state is xk0

and the final state xkf is free with kf fixed, thus this is a finite hori-zon LQ control problem. The selected performance output is

ek ¼ yk � zk (9)

such that the weighted cost function of the performance output ek

and control uk is minimized. To begin the methodology to obtainthe solution for the optimal tracking system, first define theLagrangian function

Lðxk; ukÞ ¼1

2

Xkf�1

k¼k0

Cxk � zk½ �TQ Cxk � zk½ � þ uTk Ruk

n o(10)

and formulate the Hamiltonian as

Hðxk; uk; pkþ1Þ ¼ Lðxk; ukÞ þ pTkþ1 Axk þ Buk½ �

¼ 1

2

Xkf�1

k¼k0

Cxk � zk½ �TQ Cxk � zk½ � þ uTk Ruk

n oþ pT

kþ1 Axk þ Buk½ � (11)

Fig. 2 Schematic of AFR control problem

Table 1 Engine model parameters at 1500 RPM

s1 s2 s3 sd a b

30 ms 50 ms 50 ms 250 ms 0.5 0.8

Journal of Dynamic Systems, Measurement, and Control MARCH 2014, Vol. 136 / 021008-3


Following the approach from Ref. [25], equations for the optimalstate x�k , optimal costate p�k , and optimal control u�k can be obtainedas follows:

@H

@p�kþ1

¼ x�kþ1 ¼ Ax�k þ Bu�k (12)

@H

@x�k¼ p�k ¼ ATp�kþ1 þ CTQCx�k � CTQzk (13)

@H

@u�k¼ 0) u�k ¼ �R�1BTp�kþ1 (14)

Note that in the above equations “*” denotes the optimal trajecto-ries of the corresponding vectors. The terminal condition becomes

pkf¼ CTFCxkf � CTFzkf (15)

Substituting the open-loop optimal control of Eq. (14) into Eqs.(12) and (13), the following system is obtained:

x�kþ1

p�k

" #¼

A �E

V AT

" #x�k

p�kþ1

" #þ

x�k

p�kþ1

" #zk (16)

where, E¼BR�1BT and V¼CTQC.Next we assume that the terminal condition in Eq. (15) is of the

form

p�k ¼ Kkx�k � gk (17)

see Ref. [25] for details. By substituting Eq. (17) into state equa-tion (16), the following can be obtained:

Kk ¼ ATKkþ1 I þ EKkþ1ð Þ�1Aþ V (18)

gk ¼ ATTgkþ1 þ CTQzk (19)

where

T ¼ I � K�1kþ1 þ E

� �1E (20)

The corresponding boundary conditions for Eqs. (18) and (19),respectively, are

Kkf¼ CTFC (21)

gkf¼ CTFzkf

(22)

Note that Eq. (18) is a nonlinear matrix difference Riccati equa-tion to be solved backward using the terminal condition (21), andthe linear vector difference equation (19) can also be solved back-ward using the terminal condition (22). Once these equations aresolved, the control equation (14) can be rewritten as

u�k ¼ �R�1BTKkþ1 Ax�k þ Bu�k�

þ R�1BTgkþ1 (23)

Multiplying Eq. (23) by R and solving for the optimal controlresults in

u�k ¼ �L1kx�k þ L2kgkþ1 (24)

where

L1k ¼ Rþ BTKkþ1B� �1

BTKkþ1A (25)

L2k ¼ Rþ BTKkþ1B� �1

BT (26)

Note that Eq. (24) requires the state x, which is used as feedbackfrom the fueling dynamics. Therefore, Eq. (24) is a closed loop

optimal control with feedforward control. Finally, the optimalstate trajectory in Eq. (12) can be rewritten using Eq. (24) as

x�kþ1 ¼ A� B � L1kð Þx�k þ B � L2kgkþ1 (27)

Since the optimal control used in the LQ tracking controllerrequires the state x, a state estimator is designed to obtain state in-formation in real-time from the available measurements. Theavailable measurements are assumed to be the MAF sensor, fuelinjection signal, and oxygen sensor. Recall the linear, time-invariant system described in Eq. (7) that models the wall wettingdynamics of the PFI injection process and the fuel injection delay.A Luenberger state observer [26] is used to estimate the states ofEq. (7) in real time and has the following form:

xkþ1 ¼ Axk þ Buk þ Lð~yk � ykÞyk ¼ Cxk

(28)

where matrix L is chosen such that the estimation error goes tozero as time goes to infinity. Note that the measurement ~yk is a vir-tual sensor signal of the discretized mass air flow at throttle zthrottle

divided by the AFR

~yk ¼zthrottleðk � nÞ

14:6kðkÞ (29)

where k is the measured normalized AFR signal and parameter nis used to compensate the air flow transportation delay from throt-tle to cylinder and it can be approximated by the transportationdelay sd divided by the sample period. The estimation systemequation can also be rewritten as

xkþ1 ¼ A� LCð Þxk þ B L½ �uk

yk

� �yk ¼ Cxk

(30)

and the error between the actual state xk and the estimated state xk

is governed by the following equation:

ekþ1 ¼ xkþ1 � xkþ1

¼ Axk þ Buk � ðA� LCÞxk � BL½ �uk

yk

" #

¼ ðA� LCÞðxk � xkÞ ¼ ðA� LCÞek (31)

Choosing the estimation gain matrix L as

L ¼2:8579

2:6013

" #(32)

to assign both poles of the state estimator system matrix, A� LC,to around 0.885 to guarantee that the state estimation error ek willbecome zero as time goes to infinity. Figure 3 shows the architec-ture of LQ tracking controller with the state estimator.

The implementation of the discrete time LQ tracking controllerbegins by discretizing the manifold filling dynamics in Eq. (3) at asample period of 10 ms, resulting in the following equation:

xmðk þ 1Þ ¼ AmxmðkÞ þ BmzthrottleðkÞy

mðkÞ ¼ Cmx

mðkÞ

(33)

where the values for the matrices of Eq. (33) are

Am ¼ 0:8187; Bm ¼ 0:009063;

Cm ¼ 20



Notice that the input of the system in Eq. (33) is the output of thethrottle plate and electrical actuator dynamics, zthrottle, approxi-mated in Eq. (1), discretized at a sample period of 10 ms, that is,zthrottle is the sampled airflow at the throttle location.

Figure 3 shows the details of the implementation of the LQtracking control scheme and will be described as follows: First,the estimated air flow measured by the mass airflow sensor, zthrot-

tle, is used to calculate the airflow to cylinder zND (see Fig. 3) withthe manifold filling dynamics of Eq. (3). Note that physically itwill take n steps for the airflow zND to get into the cylinder sincethe transportation delay sd was not included when calculating zND.Therefore, zND, is a sd-second forward prediction of the in-cylinder airflow.

Next, this predicted airflow, sampled every 10 ms, is divided bythe desired AFR 14.6kR(k) and pushed into a reversed buffer ofsize N after n-N step delay. The reversed buffer means that thenewest sample will stay at the end of the buffer and the oldest onewill be at the beginning of the buffer, see Fig. 4, where q is theone-step advance operator. The buffered signal forms the refer-ence vector z indicated in Fig. 3. Note that this works when N� nand the buffered signal z contains the future N step in-cylinderrequired (reference) fuel flow. This is done so that for every con-trol step, the finite horizon optimal tracking control can be solved.

As a summary, during each sample the Riccati difference equa-tions (18) and (19) are solved backward for N steps and areupdated every 10 ms, where k0¼ 0 and kf¼N. Similarly, the con-troller matrices (25) and (26) are calculated during the same con-trol step and the optimal control (24) is obtained. Consequently, atthe next control step only the first control signal out of all N calcu-lated control signals is used for controlling the PFI injector, and inthe next step, the N control signals will be recalculated by follow-ing the same procedure and using the required (reference) fuelflow. For this study, N was set equal to 10, and thus the controllerrequires ten samples of the required (reference) fuel flow to solvethe Riccati equations and to determine the controller matrices atevery control step, which is equal to 100 ms of real-time simula-tion. Note that the airflow transport delay of sd is 250 ms (seeTable 1) and n is 25, which is greater than N. Therefore, there issufficient time to calculate the optimal fueling based upon therequired (reference) fuel flow.

Although in this paper we are going to investigate the LQ track-ing control performance at one speed and load condition, to imple-ment the proposed control strategy for wide engine operationalconditions, the modeling parameters, including time constants anddelay, will be functions of the engine speed, load, variable valvetiming (VVT), etc. The control strategy, updated in real-time, will

Fig. 3 LQ tracking controller with state estimator

Fig. 4 Reference buffer formation



be calculated based upon the model parameter at current enginespeed, load, and valve timing so that the LQ tracking controlreflects current engine operational condition.

4 Simulation Results

Since the typical feedforward AFR control is achieved by usinginverse fueling and the estimated cylinder air charge divided bythe desired stoichiometric ratio of the air and fuel, a baselineinverse fueling dynamics controller was developed for compari-son purposes to the LQ tracking controller. The controller uses theinverse wall wetting dynamics of the PFI injector and the esti-mated airflow measured by the mass airflow sensor to determinethe desired stoichiometric ratio of the air and fuel. The baselinecontroller also includes the estimated manifold filling dynamicsand a pure time delay to match the transport delay of the airflowfrom the throttle to the engine cylinder, see Fig. 5, where 14.6 isthe desired stoichiometric (reference) AFR.

Simulations of the LQ tracking controller under different throt-tle plate opening changes were conducted and compared withthese of the inverse fueling dynamics controller. The performancecost function weighting matrices (8) were tuned for each simula-tion such that the transient response was acceptable and wereselected below as

F ¼ Q ¼1 0

0 1

" #; R ¼ 10�7

It was found that increasing the magnitude of Q is equivalent todecreasing R. Since minimizing tracking error (9) is the main

goal, selection of the weighting matrices F, Q, and R was focusedon matrices F and Q. That is, minimizing the tracking error (9) isour first priority regardless of the amount of control effort. This issuitable since the overall control problem is to minimize theengine AFR deviation from a desired level during engine transientoperations, which in general will improve engine fuel efficiencyand reduce engine emissions. Placing more emphasis on minimiz-ing the tracking error from the selection of these matrices wasfound to be acceptable because there was no singularity in the so-lution of the Riccati equations and also the fueling input was notsaturated.

The four-cylinder mean-value model discussed in the beginningof air flow and fuel flow modeling section was used in the simula-tions. A constant throttle opening of 40%, where this percentageis of the total throttle valve opening, was used to begin the firstsimulation. Figure 6 shows the response of the inverse fueling dy-namics controller compared with the LQ tracking controller forsimulation 1. This simulation also adds 2% sensor noise to themeasured airflow across the throttle, and adds a step throttleincrease of 25% at the 5th second. At the 10th second, a decreaseof 15% to the throttle opening was applied. The upper plot of Fig.6 shows the air flow (g/s) into the cylinder and its relative fuelflow, and the lower plot shows the AFR response to the throttlechanges. The LQ tracking controller maintains the maximumAFR deviation from stoichiometry to fewer than 4.5%, whereasthe inverse fueling dynamics controller only maintains the maxi-mum AFR deviation to 9%.

For the previous simulation, a constant target relative AFR of 1was used. In simulation 2, it begins with a target relative AFR of1 and a constant engine throttle opening of 40%, where this per-centage is of the total throttle valve opening. At the 5th second,

Fig. 5 Schematic of baseline inverse fueling controller

Fig. 6 Response of simulation 1



the simulation adds both a step decrease of 0.05 to the referenceAFR and a step increase of 25% to the throttle opening the. At the10th second, both an increase back to unity AFR and a decreaseof 15% to the throttle opening is achieved. This is to simulate thestep throttle engine operation where the reference AFR is reducedto improve engine knock tolerance. Figure 7 shows the responsesof simulation 2 under the throttle changes and engine relativeAFR changes. As seen in the lower plot of Fig. 7, the LQ trackingcontroller maintains the deviation of the AFR from stoichiometryto fewer than 5%. The inverse fueling dynamics controller has anAFR deviation of 8%.

5 Engine Test Results

5.1 Engine and Controller Setup. The LQ tracking control-ler was implemented and tested on a 0.4 l single cylinder direct-injection engine equipped with VVT, a “hot-wire” MAF sensor,an intake manifold pressure sensor, and a universal exhaust gasoxygen (UEGO) sensor. Since the engine was equipped with a DIinjector, wall wetting dynamics were added to the output of thefuel injection pulse width control signal to emulate a PFI injectorand 5% parameter error of a and b was later added to the controldesign model to simulate the modeling error. All engine testswere conducted at a fixed engine speed of 1500 RPM and theVVT position was fixed. Table 2 lists the engine specificationsand the engine setup is shown in Fig. 8.

The engine controller hardware that was used for the enginetests was an Opal-RT HIL engine controller [27], capable of sam-pling engine sensor signals and updating engine control signalsevery 1.0 ms, and also has the ability to send and receive crank-angle based signals. For the engine tests, only the engine oxygensensor and MAF sensor signals are used by Opal-RT engine

prototype controller, and similarly only the direct injector fuelpulse signal and ignition timing signal are determined andupdated. The MAF sensor signal and UEGO sensor signals areaveraged over one engine cycle to remove the large fluctuations inthe signals due to the single cylinder engine. The discretized LQtracking controller was implemented in Simulink and autocodedinto the Opal-RT controller and all controller matrices and signalswere updated every 10 ms.

5.2 Model Redesign. The models used for the intake mani-fold filling dynamics and throttle plate dynamics were acceptablein simulations but these simplified models were redesigned due tothe difference between the model and the physical engine. Toemulate a step increase and decrease in the engine airflow for thetest engine, a two-way normally closed solenoid valve was used,to allowing a small manually adjustable amount of airflow to enterthe engine when no voltage is applied to the actuator solenoid.When the voltage is applied to the solenoid, the valve openedallowing a larger amount of air to be added to the small manuallyadjustable amount. Since the response of the solenoid is very fastwhen powered, this emulates a step increase of airflow into theengine. Note that this created an engine step open operation muchworse than an actual one. In order to model the aforementionedphysical process of opening, the throttle to increase and decreasethe airflow, the first order model in Eq. (1) that approximates theengine throttle plate dynamics was redesigned. A second ordertransfer function of the form below was used

xthrottle ¼x2

n

s2 þ 21xnsþ x2n

xair (34)

where the standard equations for maximum percent overshoot andsettling time were used. The validation of the modeled dynamicsin simulation versus the physical system is shown in Fig. 9.

To improve the modeling accuracy of the intake manifold fill-ing dynamics, a step throttle increase of the air flow into the intakemanifold was measured by the intake manifold pressure sensor.From this signal, it was observed that the filling dynamics canindeed be approximated by a first order system with a time con-stant of s2¼ 0.16 s, where time constant was turned to fit themeasured response through visual inspection. It can be observed

Fig. 7 Response of simulation 2

Table 2 Engine specifications

Number of cylinders 1Total displacement 0.4 lBore 83 mmStroke 73.9 mmRod 225 mmCompression ratio 12:1



that due to nonlinearity the first order approximation cannot matchthe experimental data exactly. Note that in the simulation s2 was0.05 s. The large time constant of the intake filling dynamics isdue to the fairly large manifold volume for the test engine. Thevalidation of the modeled intake manifold filling dynamics in sim-ulation against the intake manifold filling dynamics of the physi-cal engine is shown in Fig. 10. It can be observed that the intakemanifold filling dynamics matched pretty well at the beginningand due to nonlinearity the actual filling dynamics is much fastthan the modeled one.

5.3 LQ Tracking Feedforward Control Results. For thefirst engine test of the LQ tracking feedforward controller, adesired AFR of stoichiometry was chosen. The wall wetting dy-namics for a PFI injector given in Eq. (4) were added to the con-trol output of the injector fuel pulse where a¼ 0.5 and b¼ 0.8. Astep increase and decrease in the airflow was applied and theresults are shown in Fig. 11. The LQ tracking controller maintainsthe deviation of the AFR during the step increase and decrease toless than 7.87%. Similar to the case when state estimation was notused, the deviation of the AFR during the step decrease in airflowis very minimal, at less than 3.11%.Fig. 8 Single cylinder engine setup

Fig. 9 Throttle dynamics model validation

Fig. 10 Intake manifold filling dynamics model validation



For the previous test, the LQ tracking controller uses the samemodeled wall wetting dynamics as the wall wetting dynamics thatare added to the control output of the fuel injector pulse signal.New wall wetting parameters a¼ 0.475 and b¼ 0.76 (a 5%

Fig. 11 Response of LQ tracking controller with state estimator

Fig. 12 Response of LQ tracking controller with new wall wetting parameters

Fig. 13 Response of inverse fueling dynamics feedforward controller

Table 3 PID feedback control gains

KP 0.01KI 0.0005KD 0.0001



parameter variation) were chosen to test the performance of thecontroller when the modeled wall wetting dynamics in the con-troller differ from the actual wall wetting dynamics of a PFI sys-tem. The response is shown in Fig. 12 and it shows that the LQtracking controller is able to maintain the AFR deviation fromstoichiometry to less than 9% despite the variation in the modeledwall wetting parameters. The transient control error is relativelarge and it could be due to modeling error and the fast step throt-tle solenoid actuator that provide a much fast throttle step openingthan an actual throttle driven by an electrical motor.

5.4 Inverse Fueling Dynamics Controller Results. The con-ventional inverse fueling dynamics feedforward controller strat-egy previously discussed, where the schematic is shown in Fig. 5,was implemented in the Opal-RT and its control signal wasupdated every 10 ms. The inverse fueling dynamics feedforwardcontrol was carefully tuned to achieve the best performance possi-ble. Engine tests were conducted to compare with the LQ trackingcontroller. The response is shown in Fig. 13 and shows that thedeviation of the AFR from stoichiometry is 10.2% for the stepincrease of the airflow and 6.13% for the step decrease. Note thatthe LQ tracking control utilizes both measured mass air flow andexhaust oxygen signals and while the inverse fueling dynamicsfeedforward control uses only the measured mass air flow infor-mation. It is expected that the LQ tracking control shall provedimproved performance.

5.5 LQ Tracking With PID Feedback Control Results. APID (proportional, integral, and derivative) controller that uses theoxygen sensor signal as feedback was designed and implementedwith the LQ tracking controller. The controller gains were tunedonline such that the system maintained stability with very littleoscillations, and the transient response was acceptable and thePID gains that were used for the engine tests are shown in Table3. The results are shown in Fig. 14 and it can be seen that theAFR deviation from stoichiometry for the step increase and

decrease in airflow is 4.67% and 5.78%, respectively, comparingwith 7.87% and 3.11% for the case without PID control, wherethe results are summarized in Table 4. The improvement on set-tling time for both step up and down is uniform. This improve-ment is most likely due to the closed loop compensation of theunmodeled nonlinear dynamics by the PID controller. Also notethat the LQ tracking control design provides both feedforward andfeedback control gains simultaneously for the given weightingmatrices. This makes both feedforward and feedback controldesign coupled. The PID controller adds additional degree of free-dom for tuning the feedback control gains, which could be the rea-son for the improved performance.

6 Conclusion and Future Work

Engine test were conducted on a single cylinder research engineto validate the ability of the LQ tracking control to reduce thedeviation of the AFR from a desired value during engine transientoperations. Wall wetting dynamics were added to the output ofthe DI control signal to emulate a PFI system. The proposed LQtracking feedforward controller was able to maintain the AFRdeviation to less than 8% for both step up and down engine tran-sient operations and to less than 5.6% when the PID feedback con-trol was added. Furthermore, the performance of the LQ trackingfeedforward controller reduced the AFR deviation during enginetransients much better than the baseline inverse fueling dynamicscontroller. This paper also shows the performance comparison ofthe LQ tracking controller, inverse fueling feedforward controller,and LQ tracking control with PID feedback control.

For the future work, it is proposed to improve the LQ trackingcontroller by accurately modeling the time delays of the fuelinjection, the intake manifold filling dynamics and the throttle dy-namics. Currently the fuel injection time delay is not modeled inthe LQ tracking controller directly and but was lumped to theintake airflow delay. Matching these models with the physical sys-tems should reduce the AFR tracking error significantly for tran-sient engine operations when the LQ tracking control is used.

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