Modeling and Control of a Diesel Engine With Regenerative … · 2019-08-28 · system results in four control inputs and two control outputs. However, the nonlinear, multivariable,

Tao ZengMechanical Engineering,

Michigan State University,

East Lansing, MI 48824

e-mail: [email protected]

Guoming Zhu1

Fellow ASME

Mechanical Engineering,

Michigan State University,

East Lansing, MI 48824

e-mail: [email protected]

Modeling and Control of a DieselEngine With RegenerativeHydraulic-Assisted TurbochargerDiesel engines are of great challenges due to stringent emission and fuel economyrequirements. Compared with the conventional turbocharger system, regenerativeassisted system provides additional degrees-of-freedom for turbocharger speed control.Hence, it significantly improves control capability for exhaust-gas-recirculation (EGR)and boost pressure. This paper focuses on modeling and control of a diesel engine air-path system equipped with an EGR subsystem and a variable geometry turbocharger(VGT) coupled with a regenerative hydraulic-assisted turbocharger (RHAT). The chal-lenges lie in the inherent coupling among EGR, turbocharger performance, and highnonlinearity of the engine air-path system. A control-oriented nonlinear RHAT systemmodel is developed; and a linear quadratic (LQ) control design approach is proposed inthis paper to regulate the EGR mass flow rate and boost pressure simultaneously and theresulting closed-loop system performance can be tuned by properly selecting the LQ con-trol weighting matrices. Multiple LQ controllers with integral action are designed basedon the linearized system models over a gridded engine operational map and the finalgain-scheduling controller for a given engine operational condition is obtained by inter-preting the neighboring LQ controllers. The gain-scheduling LQ controllers for both tra-ditional VGT-EGR and VGT-EGR-RHAT systems are compared with the in-housebaseline controller, consisting of two single-input and single-output (SISO) controllers,against the nonlinear plant. The simulation results show that the designed multi-inputand multi-output LQ gain-scheduling controller is able to manage the performancetrade-offs between EGR mass flow and boost pressure tracking. With the additionalassisted and regenerative power available on the turbocharger shaft for the RHAT sys-tem, engine transient boost pressure performance can be significantly improved withoutcompromising the EGR tracking performance, compared with the baseline control.[DOI: 10.1115/1.4041932]

Introduction

A regenerative hydraulic-assisted turbocharger (RHAT) systemis proposed in Refs. [1–3], where a hydraulic system is placedbetween turbocharger center housing as shown in Fig. 1. In thisnew system, a hydraulic turbine is used to spin the turbochargershaft via high-pressure supply fluid from a tank; and a turbo-pumpis used to absorb excessive power from the turbocharger shaftwhile pressurizing the fluid and pumping back into the tank. Adriveline pump is also used to recover vehicle kinetic energy dur-ing vehicle deceleration mode and pump the fluid into the high-pressure tank. Both the hydraulic turbine and the turbo-pump arepackaged inside the turbocharger center housing. The RHAT con-cept fundamentally changes the operation of a turbochargedengine with reduced turbo lag and engine pumping loss, as well asimproved surge margin. Compared to traditional electric-assistedand regenerative turbocharger, RHAT has a much higher assistingand regenerative capability, and it is also more durable and costeffective [1].

Engine downsizing and down-speeding are essential to meetfuture U.S. government fuel economy mandates. Further pushingthe envelope for even better fuel economy improvement withoutcompromising vehicle drivability via a turbocharged engine willrun into a major constraint, that is, limitation on transient responseof the turbocharger. When high torque is demanded during accel-eration, it will not be available for up to a few seconds with con-ventional turbocharger technologies, which is called “turbo lag.”The key for market acceptance of downsized turbocharged

engines is to bring the torque to the demand level without anoticeable delay. A regenerative hydraulic-assisted turbochargersystem is proposed in Refs. [1] and [3]. A hydraulic system is pro-posed to be placed at the turbocharger center housing as shown inFig. 1. For this new system, a hydraulic turbine is used to spin theturbocharger shaft via high-pressure fluid out of a supply tank;and a turbo pump is used to absorb excessive power from the tur-bocharger shaft while pressurizing the fluid and pumping it backinto the tank. A driveline pump is also used to recover vehiclekinetic energy during vehicle deceleration and pump the fluid intothe high-pressure tank. Both the hydraulic turbine and turbo pumpare packaged inside the turbocharger center housing. The RHATconcept fundamentally changes the operation of a turbocharged

Fig. 1 Diesel engine equipped with RHAT system

1Corresponding author.Manuscript received September 26, 2017; final manuscript received October 29,

2018; published online November 28, 2018. Assoc. Editor: Stani Bohac.

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engine with improved engine surge margin, reduced turbo lag andengine pumping loss. Compared to traditional electric-assistedand regenerative turbocharger, RHAT has a much higher assistand regenerative capability, and it is also more durable and costeffective [1]. The abundant hydraulic energy, recovered duringvehicle deceleration, can be used to assist the turbine so that thevariable geometry turbine (VGT) can operate at the most efficientopening positions in order to meet the compressor power demand,rather than at inefficient positions with a small opening. With twoextra actuators on the turbocharger shaft, the VGT could poten-tially be replaced with fixed geometry turbocharger without com-promising engine efficiency and durability but with much lowercost.

By considering only the air-path system, the VGT-EGR-RHATsystem results in four control inputs and two control outputs.However, the nonlinear, multivariable, and natural couplingbetween exhaust-gas-recirculation (EGR) and boost pressureloops of the diesel engine air-path system and dual control objec-tives make the closed-loop control design problem arduous. Tradi-tional control strategy for EGR-VGT control treats these twocontrol actions as two single-input and single-output (SISO) sys-tem by controlling EGR and VGT actuators independently.Reported in Ref. [4] is an overview of the control strategies usedin heavy-duty diesel engines for the EGR-VGT system. Duringtip-ins, the control system closes the EGR valve and uses aproportional-integral-derivative (PID) controller to regulate theboost pressure by rapidly increasing the air supply to the intakemanifold through the VGT vane control. Under steady-state oper-ations, the controller fully opens the EGR valve and regulates theEGR flow rate by controlling the VGT vane in an open loop man-ner. This strategy is satisfactory for the vehicle with prolongedsteady-state operations such as commercial vehicles. For passen-ger car applications, transient operations are more frequent andoften more severe. For instance, the natural coupling effect estab-lished by the VGT can be compromised by the high-pressure EGRvalve action. Especially, negative coupling effect of the compres-sor mass flow due to the step change of EGR introduces additionalchallenge in control design. With additional assisted power on theTC shaft, intake manifold pressure can be decoupled from theexhaust manifold pressure. This also decouples the EGR massflow rate from exhaust pressure. VGT control can be used forboost pressure control or exhaust gas recirculation control. There-fore, to fully exploit the potential of these devices, the controldesign needs to consider the problem in the context of multivari-able control to achieve improved performance.

A good control solution for production application must berobust to system uncertainty with low computational throughputand is simple to implement and calibrate. Many papers addressthe stability and robustness of control design for diesel engine airflow regulation. A control Lyapunov function based controldesign is introduced in Ref. [5]. This method is constructed for asimplified model using input–output linearization and robustnessis achieved through domination redesign. In Ref. [6], a multivari-able controller is designed based on input and output linearizationusing sliding model control. It provides a systematic method toregulate the EGR mass flow rate and intake manifold pressure bychoosing different sliding surfaces. A three-input-three-output(3I3O) multivariable control structure is proposed for electric-assisted and regenerative turbochargers in Ref. [7]. It shows goodtracking performance through experiment. However, few papersprovide a systematic approach for closed-loop control design withrespect to engine performance and emissions under transient oper-ations, specifically, not only considering stability and robustnessbut also accounting trade-off of engine transient performance dur-ing control design process.

The linear quadratic regulator (LQR) is a well-known designtechnique that provides practical feedback gains with knownrobustness of gain margin greater than 6 dB and phase margingreater than 60 deg. LQR technique has been widely used in othernonlinear systems [8–11]. In the LQR design routine, weighting

matrices can be directly used for control design for different per-formance targets. It eventually provides a multi-input and multi-output (MIMO) controller to regulate the EGR valve, VGT vaneposition, and RHAT power to their targets. However, LQR con-troller designs are only applicable for linear systems. Gain-scheduling is a natural approach to extend the linear controldesign to nonlinear systems by designing a family of linear con-trollers, where each individual controller is designed based on alinearized model at a specific operational condition in terms ofengine speed and load.

In this paper, linearized models are obtained from the nonlinearmodel over the gridded engine operational conditions and LQ con-trollers with integral action are designed based on the model line-arized from the high fidelity reduced-order nonlinear dieselengine model for both EGR-VGT and EGR-VGT-RHAT systemsunder number of engine operational conditions. The designed LQcontrollers regulate the tracking errors of both EGR mass flowrate and boost pressure down to zero. Control design for differentweighting matrices of cost functions is used to study the trade-offcharacteristics between engine performance and emissions. Theproposed MIMO controller is gain-scheduled using engine speedand load (fuel mass) and validated using the high fidelity nonlin-ear engine plant. By comparing with in-house baseline controllers,the proposed controller improves engine performance over thebaseline controller.

Control Objective and Problem Formulation

For the diesel engine air-path system, the control objective is toregulate the fresh air charge and oxygen concentration in theintake manifold to the desired levels provided by an optimizedengine static calibration process. These static maps are generatedbased on an optimized trade-off between fuel economy and NOx

emissions without violating the constraints on soot formation.Demanded fresh air and oxygen concentration can be transferredto desired air fuel ratio and EGR mass flow fraction. Hence, theset-point for air-to-fuel ratio determines the engine performanceand prevents smoke, and the EGR flow fraction is used to controlin-cylinder NOx generation. If the fueling rate is known (fromdrivers pedal position), the set-point for air-to-fuel ratio can be

transformed into a set-point for compressor flow rate _mdC. Simi-

larly, the set-point for EGR flow fraction can be expressed in

terms of the desired EGR flow rate _mderg and compressor flow rate

_mdC [6]. Furthermore, these set-points can be reformulated to

intake manifold pressure P2, exhaust manifold pressure P3, TCspeed x, and intake manifold oxygen concentration Oim as shownin Fig. 2. These are system states to be regulated to the desiredvalues in the control design (Fig. 3).

In order to compare with the in-house baseline controller, thecontrol objective of this study is to regulate the boost pressure(P2) and EGR mass flow rate ( _megr) to their desired values. Withdriver’s pedal position as the input, the desired engine torque isused to define desired fuel injection amount. Target boost pressureand target EGR mass flow rate are determined by the currentengine speed and fuel injection quantity. Meanwhile, optimalsteady-state set-points of VGT vane position (uvgt) and EGR valveposition (uegr) are obtained based on the steady-state performanceand emission requirement. The optimal reference trajectories aregenerated through intensive calibration work, which are notdiscussed in this study. Note that, for each set of ðNe; _mfuel;uvgt; uegr; urhatÞ, there exists a unique equilibrium for the dieselengine. Hence, with a given engine speed and fuel injection quan-tity, the diesel engine operational condition can be determined. Asa result, it is natural to select engine speed and fuel injection quan-tity as the gain-scheduling parameters. Since the hydraulic-assisted and regenerative power is only active during transientoperation, the nominal value for urhat is zero.

In this study, the closed-loop control design is formulated as aregulation problem [12–14]. Its target is to regulate both boost

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pressure and EGR mass flow rate tracking errors down to zero andat the same time to keep the steady-state errors to zero. For a regu-lation problem, the control design is based on the error system.Linearization of the nonlinear plant is investigated first.

System Modeling and Validation

The nonlinear model of the regenerative hydraulically assistedturbocharged engine has three major subsystems, engine system,turbocharger system, and hydraulic system, as shown in Fig. 4. Asimplified three-state diesel engine air-path model is adopted inearly literature [15–17] and also used in this paper; see Eq. (1).The nonlinear air-path and hydraulic system model has eightstates as indicated in Eq. (1). The eight states are the engine intakeand exhaust manifold pressures (P2;P3), the turbocharger speed(x), the hydraulic accumulator pressure (Pacc), the preturbinehydraulic pressure (Pt), the pump discharge pressure ðPp), thehydraulic accumulator piston position (x), and piston speed (v).These eight states are highlighted in double-solid blocks in Fig. 4.The system control inputs are the VGT vane position (uvgt), the

EGR valve position ðuegrÞ, the hydraulic turbine inlet valve posi-tion (uturbine), and the hydraulic pump discharge valve position(upump). Given the complexity of system dynamics, various cou-pling relationships shown in Fig. 4 are helpful, where each blockrepresents a modeled subcomponent. Arrows indicate couplingsbetween blocks via the indicated coupling variables. For instance,the inputs of “VGT Turbine” block are temperature and pressuresourced from the exhaust pressure block P3, and the turbochargerspeed is sourced from the “InertiaþFric” block, which makes itpossible to calculate the turbine mass flow rate and turbine power.Figure 4 can also be represented in the form of dynamic equationsshown in Eq. (1).

Five control volumes are identified, and they are engine intakemanifold, engine exhaust manifold, pipe volume (Vt) between thehydraulic turbine and hydraulic turbine inlet valve, pipe volume(Vp) between the hydraulic pump and hydraulic pump valve, andhigh pressure accumulator displacement (V0 þ xA), where V0 isthe initial displacement, and A is the piston area surface. The TCshaft dynamic model links the hydraulic loop to the engine air-path loop. The shaft dynamic equation represents the power

Fig. 2 Set-points for diesel engine air-path control

Fig. 3 Tracking reference generation in production controller

Fig. 4 System model architecture and calculating loops

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balance between the four components on the turbocharger shaft:turbine, compressor, hydraulic turbine, and hydraulic pump. Thepower balance is adjusted to compensate for shaft friction. Thehydraulic loop interacts with air-path loop directly through turbo-charger shaft by transferring assist power from hydraulic turbineand extracting regeneration power from the hydraulic pump. Thehydraulic turbine and pump are controlled by linear valves.Details of the system model can be found in Ref. [18]

Engine _P2 ¼RT2

V2

_mc � _min þ _megrð Þ

Air path _P3 ¼RT3

V3

_mout � _megr � _mTð Þ

Model x _x ¼ _WT � _W C � _WLoss þ _W turbine � _W pump

_PP ¼bVp

_mpump � _mvalve pð Þ

_Pt ¼bVt

_mvalve t � _mturbineð Þ

Hydraulic _Pacc ¼b

V0 þ xA_mpump � _mturbine � vAð Þ

System model _x ¼ v

_v ¼ Pacc � Preturnð ÞA� F0 � Ff vð Þ � cv� kx

xqA

(1)

Each of the nonlinear functions in Eq. (1) is investigated and vali-dated using experimental data. The developed engine submodelsare calibrated using the steady-state data from a medium duty die-sel engine and is further validated using transient driving cycletest data shown in Fig. 5. The developed model shows adequateaccuracy and is good for model-based control design.

Model Linearization of Regenerative Hydraulic-

Assisted Turbocharger Diesel Engine Air-Path System

Model Linearization. The diesel engine air path nonlinearplant (1) can be expanded with more details in Eq. (2); see Refs.[5], [6], and [19]. The turbine power model using VGT control asdirectly input can be found in Refs. [15] and [16]; and compressorpower modeling in Ref. [17]. The exhaust and intake manifoldsare modeled as volumes with ideal gas and constant specific heats.The EGR valve is modeled using valve flow equations for flowthrough orifices, where the effective area is determined experi-mentally. Volumetric efficiency and temperature rise are modeledas static nonlinearities. Each of the nonlinear functions in Eq. (2)is investigated and validated using experimental data. Developedengine submodels are calibrated using steady-state mapping datafrom a medium duty diesel engine. The model is further validatedusing the transient test data

_P3 ¼RfT3

P2; _mfuelð ÞV3

f _m inNe;P2ð Þ þ _mfuel � f _mt

uvgt;P3

P4

� �� f _megr

uegr;P3

P2

� ��

_P2 ¼RfT2

x;P2

P1

� �V2

f _mcx;

P2

P1

� �� f _m in

Nengine;P2ð Þ þ f _megruegr;

P3

P2

� ��

Jx _x ¼ f _W Tuvgt;

P3

P4

;x; T3

� �� f _W c

P2

P1

;x

� �� f _W Loss

� urhat

(2)

Fig. 5 Model validation results using the FTP 75 driving cycle


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A linearized model about a given equilibrium point of nonlineardiesel engine air-path system, described by Eq. (2), was obtainedanalytically and is in the following form:

d _xp tð Þ ¼ AP tð Þdxp tð Þ þ BP tð Þdu tð Þdyp tð Þ ¼ CP tð Þdxp tð Þ þ DPðtÞdu tð Þ

z tð Þ ¼ Fdxp tð Þ þWðtÞdu tð Þ(3)

where dy tð Þ ¼ d _megr dP2

� �T; xp tð Þ 2 R3 is the plant state vector

(dP3; dP2; dx); and du tð Þ 2 R3 is the control input vector(duvgt; d~uegr; durhat). The measurements for feedback are d _megr anddP2, and z tð Þ is the performance output vector to be regulated toits desired reference. In this study, it is assumed that both outputsare measurable and yp ¼ z tð Þ. The state, input, and output vectorsin Eq. (3) are all deviations of the corresponding trajectories ofthe nonlinear system from their equilibrium operation condition.For a diesel engine air-path system with regenerative hydraulic-assisted turbocharger, the linearized model is described in thestate-space form as in the below equation:

d _P3

d _P2

d _x

2664

3775¼

A11 A12 0

A21 A22 A13

A31 A32 A33

264

375

dP3

dP2

dx

264

375þ

B11 B12

0 B22

B31 0

0

0

B33

264

375

duvgt

d~uegr

durhat

264

375

dyp tð Þ ¼C11 C12 0

0 C22 0

" # dP3

dP2

dx

264

375þ 0 D21

0 0

0

0

" # duvgt

duegr

durhat

264

375

(4)

Augmented With Actuator Dynamics. Simple actuatordynamics has been augmented into the plant model and the aug-mented system is

d _~u egr tð Þ ¼ Ad tð Þd~uegr tð Þ þ Bdduegr tð Þ (5)

In this case, only EGR actuator dynamics is modeled. The param-eters of the actuator dynamics are chosen to approximate these foran actual EGR valve. In this case, EGR actuator response time isabout 0.03 s, and the model is shown below:

d _~u egr tð Þ ¼ � 1

0:03d~uegr tð Þ þ 1

0:03duegr tð Þ (6)

With the augmented actuator dynamics, the new state-space sys-tem model is

d _xs tð Þ ¼ Asdxs tð Þ þ Bsdus tð Þdys tð Þ ¼ Csdxs tð Þ

(7)

where

xs tð Þ ¼dxp tð Þd~u tð Þ

" #; dus tð Þ ¼

duvgt

duegr

durhat

264

375

As ¼Ap BPegr

0 Ad

" #; Bs ¼

BPvgt

0

0

Bd

" #; Cs ¼

CTp

DTPegr

" #T

where Ad and Bd are the matrices for the actuator dynamics; Ap,BP, and CP are the plant matrices in Eq. (4) associated with non-EGR inputs; and BPegr

and DPegrare associated with the EGR input

in Eq. (4). Note that only EGR control input in matrix Dp for plant(4) is the direct control for the EGR mass flow rate. With the aug-mented actuator dynamics, system (7) is strictly proper, making itsuitable for designing LQR controller.

Integral Action. It is desirable to include integral control intothe state feedback control to eliminate the steady-state error. Withmodel uncertainties presented in the system, the control designmust be able to compensate uncertainties with zero steady-stateerrors. By augmenting the system with the integral action, it ispossible for LQR design to choose the integral gain automatically[20]. Define the system with integral action using time derivativesof outputs as follows:

d

dt

d _xs tð Þdys tð Þ

� �¼ As tð Þ 0

Cs tð Þ 0

� �d _xs tð Þdys tð Þ

� �þ Bs tð Þ

0

� �dv tð Þ (8)

The new augmented system is

d

dtmðtÞð Þ ¼ ~A tð Þm tð Þ þ ~B tð Þui tð Þ (9)

where

ui tð Þ ¼ dv tð Þ ¼ d dus tð Þð Þdt

; m tð Þ ¼d _xs tð Þdys tð Þ

" #;

~A tð Þ ¼As tð Þ 0

Cs tð Þ 0

" #; ~B tð Þ ¼

Bs tð Þ0

" #

Using Eqs. (7)–(9), the final augmented system can be formed asfollows:

~A ¼

A11 A12 0

A21 A22 A13

A31

0

C11

0

A32

0

C12

C22

A33

0

0

0

B12 0 0

B22 0 0

0

Ad

D21

0

0

0

0

0

0

0

0

0

26666666664

37777777775; ~B

� ¼

B11

0

B31

0

0

0

0

0

0

Bd

0

0

0

0

B33

0

0

0

26666666664

37777777775

(10)

For instance, the linearized plant model at engine speed Ne ¼1200 rpm with fuel injection _mfuel ¼ 35 mg/stroke is shown in thebelow equation:

A¼

�73:94 77:13 0

2:48 �78:94 1:15�103

0:04

0

2:8�10�07

0

0:023

0

�2:6�10�07

�1

�5:1246

00

0

�1:61�105 0 0

2:03�104 0 00

�33:33�0:0024

0

0

00

0

0

00

0

2666666664

3777777775

B¼

1:129�105

0�20

00

0

00

00:0003

0

0

00

100

0

2666666664

3777777775

(11)

Linear Quadratic Regulator With Integral Action

Control Design. The goal for this control design is to find a lin-ear optimal controller that minimizes the tracking errors of bothEGR mass flow rate and boost pressure. The cost function is givenbelow:

J ¼ð1

0

mTðtÞ ~Qm tð Þ þ uTi ðtÞRui tð Þ

� dt; ~Q ¼ 0 0

0 Q

� �2 R6

(12)


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where ~Q ¼ diag 0 Q� �

2 R6�6. To keep the tracking error small,

the cost expressions mTðtÞ ~Qm tð Þ and uTi ðtÞRui tð Þ should be non-

negative and small, where matrix Q 2 R2�2 shall be positive semi-

definite and matrix R 2 R3�3 positive definite. Note that Q is theweighting matrix associated with tracking errors for EGR massflow rate and boost pressure; and R is for the derivative controlinputs. Now the problem is formulated as a standard LQR prob-lem. In this case, we are dealing with the infinite horizon LQR

design. In order to guarantee the solution to exist, pair ~A; ~B �

shall be stabilizable and pair ~A;

ffiffiffiffi~Q

q �detectable. Note that

under the operational condition, P1;P2;P3;P4ð Þ;P3 > P4;�

P2 > P1;P3 > P2Þg, the engine air-path system is stabilizable anddetectable for both VGT-EGR and VGT-EGR-RHAT systems,and furthermore, both pairs in Eq. (13) are controllable andobservable, respectively,

~A tð Þ;Q12 0nIm½ �

� and ~A tð Þ; ~B tð Þ� �

(13)

Assuming that both observability and controllability conditionsare satisfied, the feedback law, dv tð Þ ¼ ui tð Þ, is given by

dv tð Þ ¼ ui tð Þ ¼ �R�1 ~B tð Þ ~Pm tð Þ ¼ �Kxd _xs tð Þ � Kydys tð Þ (14)

where Kx;Ky; and ~P are given by

KxKyb c ¼ ½R�1 ~B tð Þ ~PxxR�1 ~B tð Þ ~Pxy� (15)

Note that ~P ¼~Pxx

~Pxy

~PT

xy~Pyy

" #is obtained by solving the following

matrix algebraic Riccati equation:

~A tð ÞT ~P þ ~P ~A tð Þ � ~P ~B tð ÞR�1 ~B tð ÞT ~P þ 0 0

0 Q

� �¼ 0 (16)

or in the following form:

A tð Þ 0

C tð Þ 0

" #T~Pxx

~Pxy

~Pyx~Pyy

" #þ

~Pxx~Pxy

~Pyx~Pyy

" #A tð Þ 0

C tð Þ 0

" #

�~Pxx

~Pxy

~Pyx~Pyy

" #B1 tð Þ

0

" #R�1 B1 tð Þ

0

" #T~Pxx

~Pxy

~Pyx~Pyy

" #þ

0 0

0 Q

" #¼ 0

(17)

that results in the following three algebraic equations:

2A ~Pxx þ 2C ~Pxy � ~PxxBR�1B ~Pxx ¼ 0

A ~Pxy þ C ~Pyy � ~PxxBR�1B ~Pxy ¼ 0

Q� ~PxyBR�1B ~Pxy ¼ 0

(18)

As a result, the control law can be obtained by integrating dv tð Þfrom time 0 to t. This control is in the form of proportional andintegral control law based on the state error defined in the belowequation:

du tð Þ ¼ðt

0

dv tð Þð Þdt ¼ðt

0

�Kxd _xs tð Þ � Kydys tð Þ� �

dt

¼ �Kxdxs tð Þ � Ky

ðt

0

dys tð Þð Þds (19)

where

KxKy½ � ¼ ½R�1 ~B tð Þ ~PxxR�1 ~B tð Þ ~Pxy�

In order to implement the LQI control strategy for the nonlinearplant, the equilibrium state values must be subtracted from thecurrent ones for the nonlinear system and the control inputs arethe combination of the linear control output and the equilibriumcontrol; see the below equation:

dxs tð Þ ¼ xs tð Þ � �xs; dys tð Þ ¼ Cs xs tð Þ � �xsð Þus tð Þ ¼ �us þ dus tð Þ

(20)

Then, the composite control for nonlinear plant is

us tð Þ ¼ �us � Kx xs tð Þ � �xsð Þ � Ky

ðt

0

dys tð Þð Þds (21)

where �us ¼�uvgt

�uegr

�urhat

24

35; �xs ¼

Pd2

Pd3

xd

udegr

26664

37775

Specifically, steady-state control gains have the followingstructure:

Kx¼Kx11 Kx12 Kx13 Kx14

Kx21

Kx31

Kx22

Kx32

Kx23

Kx33

Kx24

Kx34

24

35; Ky¼

Ky11 Ky12

Ky21

Ky31

Ky22

Ky32

24

35 (22)

where Kx is the proportional gain, and Ky is the integral gain. Thefirst row of matrix Kx and Ky is control gains for VGT vane posi-tion (duvgt); the second row of Kx and Ky is for EGR valve posi-tion (duegr); and the third row of Kx and Ky is for RHAT hydraulicpower (durhat). The control actions for both VGT vane, EGR valveposition, and hydraulic demanded power are coupled through statedeviations and output errors. In this study, the overall controlarchitecture is shown in Fig. 6, where xs tð Þ is the nonlinear plantstates, and �xs is the equilibrium values for the state vector associ-ated with the regulating trajectory (Pd

2;Pd3;x

d; udegr). Boost track-

ing error (dP2Þ and EGR mass flow rate tracking error ðd _megrÞ areused to drive the integral controller. The feedforward controls(�uvgt and �uegr) are generated from feedforward calibration mapbased on the current engine operational condition (Ne and _mfuel),which are used to move nonlinear plant close to the steady-stateoperational condition. Note that, for assisted power and regenera-tive power, there is no power set-point, hence, �urhat ¼ 0.

Observability and Controllability. Consider the systemdefined in Eqs. (10)–(12) with Q matrix as below:

~Q ¼ 0 0

0 Q

� �¼

0 0 0

0 0 00

00

0

0

00

0

0

00

0

0 0 0

0 0 00

00

0

0

0Q11

0

0

00

Q22

2666664

3777775 (23)

For standard LQR controller design, the solution relies on solvingRiccati equation (16). In this case, we are dealing with infiniteLQR design. In order to have a finite solution for cost function

(11), ~A; ~B �

needs to be controllable, and ~A;

ffiffiffiffi~Q

q �needs to be

observable. The associated observability and controllability Gra-mians are

O ¼ ~Q12 I ~A tð Þ ~A tð Þ2 � � � ~A tð Þ5h iT

O 2 R36�6 (24)

and

C ¼ I ~A tð Þ ~A tð Þ2 � � � ~A tð Þ5h i

~B tð Þ C 2 R6�36 (25)


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The condition for controllability Gramian (25) to lose rank iswhen P2 ¼ P3 and/or P3 ¼ P4. For instance, B12, B22, and D21 areblock parameters for EGR control input with respect to theexhaust pressure, boost pressure, and EGR mass flow rate. WhenP2 ¼ P3, B12 ¼ B22 ¼ D21 ¼ 0, EGR valve loses its controllabil-ity over EGR mass flow rate and controllability Gramian (C) rankequals to 4. This can be explained through system physics sincewhen HP EGR valve upstream pressure is equal to EGR valvedownstream pressure, EGR valve position could not control theEGR flow. For example, B12 is for EGR valve control input withrespect to EGR mass flow rate; see Eq. (26). WhenP2 ¼ P3;B12 ¼ 0; and when P2 > P3, EGR valve needs to befully closed to prevent inverse flow. The same happens under con-dition P3 < P4. Since there is no turbine mass flow, the systemcontrollability Gramian is no longer full rank

B12 ¼ �287

34 � T2 �P3 � 2 � P2

P3

� �57

Cp � 1� P2

P3

� �27

!14

T3Vim

(26)

Considering the system operational limits that TC shaft cannot bestalled ðP2 � P1Þ or over speed P3 � P4ð Þ and practical controltarget is to keep the exhaust pressure higher than intake manifoldpressure P2 < P3ð Þ, under normal operational conditions the sys-tem is always controllable. As a result, under the normal opera-tional conditions w ¼ P1;P2;P3;P4ð Þ;P3 > P4;P2 > P1;P3 >

�P2Þg, the system will be observable and controllable. It also needsto be noticed that control inputs, VGT and EGR positions, arebounded by their physical hardware actuator position. Forinstance, VGT position is 0–100% and EGR valve position0–100%. However, if control design is targeted for the best engineperformance, the EGR valve might need to be fully closed due toaggressive boost demand during transient tip-in.

Weighting Selection. The diagonal entries of weighting matrixQ are denoted by Q11 and Q22, and they are related to the trackingerrors of EGR mass flow rate and boost pressure. In order to tunematrices Q and R for different closed-loop system performancetargets, three different evaluation indexes are defined for engineperformance, emissions, and RHAT energy; see Eqs. (27)–(29).The performance index (Indperf) is defined as the normalized

accumulated boost pressure tracking error between 0 and 1 in Eq.(27), where the tracking error is normalized to its maximum andminimum range of a given operational condition for different Q11

and Q22. The same normalization process is also applied to theemission index (Indem) (28) used to evaluate the accumulatedtracking error of the EGR mass flow rate. Note that after normal-ization, both indexes are between 0 and 1. For the RHAT system,since the assisted and regenerative power is related to externalenergy usage, an additional RHAT energy index (IndRHAT) (29) isdefined as the accumulated assisted and regenerative power. Thisenergy index is used to evaluate the RHAT control performance.For both performance and emission indexes, the highest value rep-resents the best tracking results (minimum tracking errors). Thehigher the energy index, the lesser the hydraulic actuation energyis used. Frequent hydraulic actuation leads to lower energy costindex

Indperf Qi� �

¼ 1�

ðt2

t1

Pd2 � Pi

2

�� dt� min1<i<n

ðt2

t1

Pd2 � Pi

2

�� dt

max1<i<n

ðt2

t1

Pd2 � Pi

2


ðt2

t1

Pd2 � Pi

2

�� !dt

(27)

Indem Qi� �

¼ 1�

ðt2

t1

_mdegr � _mi

egr


ðt2

t1

_mdegr � _mi

egr

�� dt

max1<i<n

ðt2

t1

_mdegr � _mi

egr


ðt2

t1

_mdegr � _mi

egr

�� dt

(28)

IndRHAT Qi� �

¼ 1�

ðt2

t1

uirhat


ðt2

t1

uirhat

�� dt

max1<i<n

ðt2

t1

uirhat


ðt2

t1

uirhat

�� dt

(29)

for i ¼ 1; 2;…; n, where Pi2, _mi

egr, and uirhat are the time responses

associated the LQI controller designed using the weighting matrixwith Qi. For the control weighting matrix R, the ratio betweenVGT vane and EGR valve positions is chosen as 10:1 (R11 ¼ 10

Fig. 6 Proposed LQI regulator for engine EGR–VGT air-path system


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and R22 ¼ 1) since slow VGT action leads to slow exhaust pres-sure dynamics that makes the EGR mass flow rate easy to regu-late. The RHAT energy unit is in kW and its weighting is chosento be 1000 in this study; see equation below:

R ¼10 0 0

0 1 0

0 0 100

24

35 (30)

Entries Q11 and Q22 of weighting matrix Q for boost pressuretracking (Q11) and EGR mass flow rate (Q22) are tuned using asweep study. This sweep study is used to achieve different con-troller performance as defined in Eqs. (27)–(29). The identifiedrange for normalized block value of Q matrix is shown in Table 1and simulation study for different Q is conducted based on smallstep perturbations around a baseline engine speed and fuel injec-tion quantity. For example, small step change of 100 rpm and2 mg/cc is simulated around the low speed and light load engineoperational condition of 800 rpm engine speed and 20 mg/cc fuelinjection quantity as shown in Fig. 7. Linearized models areobtained at this local operational condition for both VGT-EGRand VGT-EGR-RHAT plants.

First, the conventional VGT-EGR control design is investi-gated. With the defined performance and emission indexes, simu-lation results for different weighting matrix Q in terms of Q11 andQ22 are shown in Fig. 8. The simulation results clearly indicate atrade-off relationship between the two design targets. Note thatQ11 and Q22 are penalty coefficients for tracking errors of EGR

flow rate and boost pressure, respectively. Large Q11 leads to lowEGR tracking error, and in other words, results in a tight EGRtracking error bound. The same works for Q22, that is, large Q22

leads to higher control effort used to correct boost pressure errorthan EGR mass tracking error. The combination of Q11 and Q22

leads to different EGR rates and boost pressure control perform-ance. Second, control design for VGT-EGR-RHAT is performedusing the proposed method at engine operational condition of800 rpm and 20 mg/cc fuel mass. With the assisted power on theTC shaft, extra energy is used to drive the compressor forimproved boost pressure response performance. Since the VGT

Fig. 7 A step load test profile for engine operated at 800 rpm with 20 mg/cc

Table 1 Normalized range for matrix Q

Q11 Q22

Min value 1 1Max value 10 10

Fig. 8 Normalized VGT–EGR performance indexes as a func-tion of Q


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vane position does not need to be further closed to build up highexhaust pressure for turbine power extraction, it can be usedmainly for boost pressure regulation, leading to reduced couplingbetween EGR flow rate and boost pressure dynamics. As shown inFig. 9, the performance index is mainly dependent on boost pres-sure weighting Q22 since the external RHAT power can be used tocontrol boost pressure instead of using VGT position. In otherwords, the boost pressure control can be almost independent ofexhaust manifold property. As a result, this reduces the influenceof EGR weighting Q11 to boost pressure tracking, compared withthe VGT-EGR case. For large Q11, EGR control improves thetracking performance of the EGR mass flow rate as expected.Also, large Q22 results in large energy input from RHAT. BothFigs. 8 and 9 show the trade-off relationship of the VGT-EGR-RHAT system. The simulation results also show that the bestEGR and boost pressure tracking cannot be achieved at the sametime for the conventional VGT-EGR system.

Gain-Scheduling for Linear Controller. In order to imple-ment the designed linear controllers for the nonlinear engine plant,gain-scheduling approach is used in this study. A bilinear interpo-lation is used to schedule the local linear controller gains inside agiven operating envelop. The gain-scheduling is based on enginespeed and fuel injection quantity as shown in Fig. 10. Withdesigned controller gains for the four boundary points as shown inTable 2 (VGT-EGR case), each controller gain inside the operat-ing range can be obtained as shown below:

k x; yð Þ �1

x2 � x1ð Þ y2 � y1ð Þx2 � x x� x1½ � k1 k2

k3 k4

� �y2 � yy� y1

� �(31)

where x is the engine speed (Ne), y is the fuel injection quantity( _mfuel), and k is the scheduled control gain entry of gain matricesKx and Ky. The final control law for the nonlinear plant is

us tð Þ ¼ �us � KxðNe; _m fuelÞ xs tð Þ � �xsð Þ � KyðNe; _m fuelÞ

ðt

0

dys tð Þð Þds

(32)

Control Design Validation Through Simulation

Baseline control is chosen as a two SISO PID controller devel-oped inside Ford. Note that the PID controller is widely used inindustry for nonlinear systems due to its simplicity and easy cali-bration. The current in-house baseline controller has two inde-pendent control loops for tracking EGR mass flow rate and boostpressure, respectively, where EGR valve position is used to tracktarget EGR mass flow rate and VGT vane position is for the boostpressure tracking. All controllers have been carefully calibratedunder their associated steady-state engine operational conditions;

and the control gains are scheduled based on the engine speed andfuel injection quantity. The major drawback is that the decoupledcontrol strategy is applied to the coupled system, leading to unco-ordinated control actions for VGT and EGR channels. BecauseEGR mass flow rate and boost pressure are strongly coupledtogether since they share the same exhaust flow to drive turbineand EGR. In some extreme cases, the performance of two control-lers need to be compromised each other.

Multi-input and multi-output controllers are designed for bothVGT-EGR and VGT-EGR-RHAT systems at four operationalconditions shown in the boundary locations of Fig. 11. Thedesigned control gains are scheduled based on engine speed andfuel injection quantity. For the VGT-EGR system, tracking EGRmass flow rate and boost pressure is the main target. The control-lers designed for the best performance (Q11 ¼ 10 and Q22 ¼ 10)and controllers designed for the minimal emissions (Q11 ¼ 10 andQ22 ¼ 3) are simulated using the nonlinear plant. For simplicity,we call these two controllers the performance controller and emis-sion controller, respectively. For the VGT-EGR-RHAT system,special attention is paid to the hydraulic actuation energy, andhence, two sets of controllers are designed for the VGT-EGR-RHAT system to have high and low energy indices under thesame operational conditions as the VGT-EGR case. The highenergy index controller uses relatively low RHAT energy for bothhydraulic turbine and hydraulic pump with (Q11 ¼ 2 andQ22 ¼ 2), and the low energy index controller uses relatively highRHAT energy with (Q11 ¼ 10 and Q22 ¼ 2). The same gain-scheduling approaches for both EGR-VGT and VGT-EGR-RHATsystems are implemented in the nonlinear engine plant. A

Fig. 9 Normalized VGT–EGR–RHAT performance indexes as a function of Q

Fig. 10 Gain-scheduling for local linear controllers


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transient profile as shown in Fig. 11 within the envelop of lightload operations is simulated for different control designs. The fivedifferent control algorithms, evaluated with the same set-pointstrategy, show comparable results on performance and emissions.The simulation results for five different control designs are shownin Fig. 12.

The VGT-EGR control design for emissions has better EGRmass flow rate tracking (less tracking error) compared to theVGT-EGR control design for performance and baseline control asshown in Fig. 12(b). However, the emission controller for VGT-EGR shows poor tracking results for the boost pressure comparedto performance control and baseline as shown in Fig. 12(a). Forthe emission controller, the VGT is less aggressive for boost pres-sure tracking, leading to high boost pressure tracking error. Sinceboth large weightings are used for the boost pressure and EGRtracking, the VGT-EGR performance controller shows betterboost pressure tracking over the emission and baseline controllers.

For VGT-EGR-RHAT low energy index controller with highassisted and regenerative power, RHAT provides aggressive assis-tance during the initial tip-in stage as shown in Fig. 12(f) and tur-bine mass flow rate increases due to increased TC speed, causingexhaust pressure dropping quickly. This leads to aggressive VGTclosing action to build up exhaust pressure to maintain the EGRmass flow rate; meanwhile, EGR closing action is used to keepEGR mass flow rate error small. The exhaust manifold pressurealso increases due to the increased air mass filled in the exhaustmanifold. Since less energy is required from gas turbine for thelow energy index case, VGT opens to reduce the exhaust pressure.With the increased boost pressure and exhaust pressure, bothVGT vane and EGR valve open, leading to a constant pressuredrops between exhaust pressure and intake pressure as shown inFig. 12. The small pressure different across EGR valve keepsEGR mass flowing through EGR valve, which minimizes thepumping loss. Hence, the low energy index controller has the best

fuel efficiency by reducing the pumping loss. With less assistedpower (high energy index), VGT action is to build up the exhaustpressure to have enough energy to drive the compressor for boostpressure tracking. With the increased exhaust pressure, EGR valvecloses due to high pressure across the EGR valve to keep the rightamount of EGR mass flow rate during tip-in. With coordinatedcontrol for the VGT-EGR-RHAT, engine boost pressure trackingperformance is improved as well as EGR mass flow rate trackingperformance.

Fig. 11 Gain-scheduling route for controller validation

Table 2 Controller designs for different engine operational conditions

EngineSpeed (rpm)

Fuel injection(mg/stroke)

Proportional gain Kx Integral gain Ky

Controller gain — — Kx11 Kx12 Kx13 Kx14

Kx21 Kx22 Kx23 Kx24

� �Ky11 Ky12

Ky21 Ky22

� �

Controller design for emission 800 20 2:19 � 10�5 5:25 � 10�5 4:66 � 10�3 6:6 � 10�2

4:34 � 10�4 �2:43 � 10�4 �7:5 � 10�3 1:60

" #5:06 � 103 3:49 � 10�4

11:06 � 104 �1:60 � 10�4

� �

1200 20 1:08 � 10�5 1:3881 � 10�5 4:3 � 10�3 8:07 � 10�2

2:96 � 10�4 �1:09 � 10�4 �2:9 � 10�3 3:13

" #3:03 � 103 3:52 � 10�4

11:06 � 104 �9:58 � 10�5

� �

800 35 1:83 � 10�5 1:48 � 10�5 5:4 � 10�3 6:4 � 10�2

3:30 � 10�4 �1:7154 � 10�4 �5:6 � 10�3 1:65

" #4:74 � 103 3:50 � 10�4

1:10 � 104 �1:49 � 10�4

� �

1200 35 1:32 � 10�5 1:76 � 10�5 4:61 � 10�3 9:3 � 10�2

2 � 10�4 �1 � 10�4 �3 � 10�4 2:71

" #4:13 � 104 3:51 � 10�4

11:10 � 104 �1:30 � 10�4

� �

Controller designfor performance

800 20 3:44 � 10�5 7:9 � 10�5 2:53 � 10�2 6:54 � 10�2

4:29 � 10�4 �2:7 � 10�4 �1:71 � 10�2 1:60

" #5:07 � 103 2:5 � 10�3

11:10 � 104 �1:1 � 10�3

� �

1200 20 2:28 � 10�5 8:82 � 10�5 2:19 � 10�2 6:54 � 10�2

2:92 � 10�4 �1:31 � 10�4 �0:8 � 10�2 3:13

" #3:11 � 103 2:5 � 10�3

11:10 � 104 �6:95 � 10�4

� �

800 35 3:37 � 10�5 9:58 � 10�5 2:28 � 10�2 6:32 � 10�2

3:28 � 10�4 �2:08 � 10�4 �1:35 � 10�2 1:65

" #4:93 � 103 2:5 � 10�3

11:10 � 104 �1:1 � 10�3

� �

1200 35 2:61 � 10�5 1:08 � 10�4 2:24 � 10�2 9:25 � 10�2

2:7 � 10�4 �1:4 � 10�4 �1:05 � 10�2 2:71

" #4:21 � 103 2:51 � 10�3

11:10 � 104 �9:41 � 10�4

� �


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Investigating the five control designs shown in Fig. 13 showsthat the designed MIMO controller is able to manage the trade-offs between EGR mass flow and boost pressure tracking, com-pared to the baseline controller (two SISO controllers). With addi-tional assisted and regenerative power on the TC shaft, transientperformance can be further improved without compromising EGRtracking, compared to baseline control. These results are quiteuseful to evaluate the benefit of additional assisted devices as wellas control and calibration of the whole system with respect to thesystem capability. For instance, under certain engine operationalconditions, the control design needs be turned to meet emissionrequirement.

Figure 13 shows a performance trade-off relationship of thefive controllers, the baseline control (dual-PID control) is locatedat the origin of the EGR and boost pressure tracking performance.One can see that the VGT-EGR emission controller provides thebest EGR tracking performance with the worst boost tracking; andthe VGT-EGR performance control improves both EGR and boostpressure tracking performance. The high energy VGT_EGR_-RHAT controller provides the best performance improvement forEGR and boost tracking. However, the low energy

VGT_EGR_RHAT controller leads to improved EGR and boosttracking performance with relatively low required hydraulicenergy and is suitable for practical applications.

Conclusions

In this paper, a systematic approach for diesel engine air-pathcontrol with regenerative and assisted turbocharger is proposed.Gain-scheduling control is designed based on the linear quadraticregulator with integral action (LQI). Scheduled linear controllersare implemented for the nonlinear system. Compared to the base-line VGT-EGR control design, the proposed approach provides anew method for designing controllers to meet different perform-ance target through LQI weighting selection. By benchmarkingwith the baseline (dual-loop SISO) controller, the designed multi-input and multi-output controller shows improved tracking resultsfor both EGR mass flow rate and boost pressure and is flexible formultivariable control design. The simulation results show that theclosed loop system design targets in terms of performance, emis-sions, and RHAT energy can be easily achieved through properweighting selection. Transient performance can be further

Fig. 12 Simulation results for VGT–EGR–RHAT control design: (a) boost pressure tracking error (hPa), (b) EGR MFR trackingerror (kg/h), (c) VGT position, (d) EGR valve position, (e) pressure difference across EGR valve, and (f) hydraulic actuationpower (kW)


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improved without compromising EGR tracking with additionalpower on turbocharger shaft, compared to baseline control. Futurework is to extend controller design over the entire engine operat-ing map and validate it against experiments.

Funding Data

Ford Motor Company (No. 14427).

References[1] Zeng, T., Upadhyay, D., Sun, H., Curtis, E., and Zhu, G. G., 2017, “Regenerative

Hydraulic Assisted Turbocharger,” ASME Paper No. GT2017-64927.[2] Kolmanovsky, I., Morall, P., Van Nieuwstadt, M., and Stefanopoulou, A., 1999,

“Issues in Modelling and Control of Intake Flow in Variable Geometry Turbo-charged Engines,” Research Notes in Mathematics, Chapman and Hall, BocaRaton, FL, pp. 436–445.

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[6] Upadhyay, D., 2001, “Modeling and Model Based Control Design of the VGT-EGR System for Intake Flow Regulation in Diesel Engines,” Ph.D. dissertation,The Ohio State University, Columbus, OH.

[7] Yang, Z., Winward, E., Zhao, D., and Stobart, R., 2016, “Three-Input-Three-Output Air Path Control System of a Heavy-Duty Diesel Engine,” IFAC-Paper-sOnLine, 49(11), pp. 604–610.

[8] Kapasouris, P., Athans, M., and Spang, H. A., 1985, “Gain-ScheduledMultivariable Control for the GE-21 Turbofan Engine Using the LQG/LTRMethodology,” IEEE American Control Conference (ACC), Boston, MA, June19–21, pp. 109–118.

[9] Masar, I., and St€ohr, E., 2011, “Gain-Scheduled LQR-Control for an Autono-mous Airship,” 18th International Conference on Process Control, Tatransk�aLomnica, Slovakia, June 14–17, pp. 197–204.

[10] Brasel, M., 2014, “A Gain-Scheduled Multivariable LQR Controller for Perma-nent Magnet Synchronous Motor,” 19th IEEE International Conference onMethods and Models in Automation and Robotics (MMAR), Miedzyzdroje,Poland, Sept. 2–5, pp. 722–725.

[11] Ostergaard, K. Z., Brath, P., and Stoustrup, J., 2007, “Gain-Scheduled LinearQuadratic Control of Wind Turbines Operating at High Wind Speed,” IEEEInternational Conference on Control Applications, Singapore, Oct. 1–3, pp.276–281.

[12] Chen, S., and Yan, F., 2015, “Control of a Dual-Loop Exhaust Gas Recircula-tion System for a Turbocharged Diesel Engine,” Int. J. Automot. Technol.,16(5), pp. 733–738.

[13] Yan, F., Haber, B., and Wang, J., 2009, “Optimal Control of Complex Air-PathSystems for Advanced Diesel Engines,” ASME Paper No. DSCC2009-2537.

[14] Naidu, D. S., 2003, Optimal Control Systems, CRC Press, New York.[15] Zeng, T., Upadhyay, D., Sun, H., and Zhu, G. G., 2016, “Physics-Based Turbine

Power Models for a Variable Geometry Turbocharger,” American Control Con-ference (ACC), Boston, MA, July 6–8, pp. 5099–5104.

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[18] Zeng, T., Upadhyay, D., and Zhu, G. G., 2017, “A RegenerativeHydraulically Assisted Turbocharger System Model,” ASME Paper No.DSCC2017-5101.

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Fig. 13 Comparing different control designs


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http://dx.doi.org/10.1115/GT2017-64927

https://patents.google.com/patent/US8915082

http://dx.doi.org/10.4271/960871

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https://www.uiam.sk/pc11/data/papers/097.pdf

http://dx.doi.org/10.1109/MMAR.2014.6957443

http://dx.doi.org/10.1109/CCA.2007.4389243

http://dx.doi.org/10.1007/s12239-015-0074-6

http://dx.doi.org/10.1115/DSCC2009-2537


http://dx.doi.org/10.1177/0954407017702996

https://www.egr.msu.edu/zhug/Publications/Conference%20Articles/A%20generalized%20compressor%20power%20model%20for%20turbocharged%20internal%20combustion%20engines%20with%20reduced%20complexity.pdf

http://dx.doi.org/10.1115/DSCC2017-5101

http://dx.doi.org/10.1177/0954407011398177

http://dx.doi.org/10.1177/0954407011398177

http://dx.doi.org/10.1115/1.3426416

http://dx.doi.org/10.1115/1.3426416

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