sliding mode control of both stephen pace air-to-fuel and ... · the control strategies, mentioned...

Stephen PaceDepartment of Electrical and

Computer Engineering,

Michigan State University,

East Lansing, MI 48824

e-mail: [email protected]

Guoming G. Zhu1

Department of Electrical and

Computer Engineering,


East Lansing, MI 48824;

Department of Mechanical Engineering,


East Lansing, MI 48824

e-mail: [email protected]

Sliding Mode Control of BothAir-to-Fuel and Fuel Ratiosfor a Dual-Fuel InternalCombustion EngineA multi-input-multi-output (MIMO) sliding mode control scheme was developed withguaranteed stability to simultaneously control air-to-fuel ratio (AFR) and fuel ratios todesired levels under various air flow disturbances by regulating the mass flow rates ofengine port-fuel-injection (PFI) and direct injection (DI) systems. The sliding mode con-trol performance was compared with a baseline multiloop proportional integral differen-tial (PID) controller through simulations and showed improvements. A four cylindermean value engine model and the proposed sliding mode controller were implementedinto a hardware-in-the-loop (HIL) simulator and a target engine control module, andHIL simulations were conducted to validate the developed controller for potential imple-mentation in an automotive engine. [DOI: 10.1115/1.4005513]

Keywords: sliding mode control, powertrain systems, air-to-fuel ratio, dual-fuel enginesystems, hardware-in-the-loop simulations

1 Introduction

Increasing concerns about global climate changes and ever-increasing demands on fossil fuel capacity call for reduced emis-sions and improved fuel economy. Vehicles equipped with DI(direct injection) fuel systems have been introduced to marketsglobally. In order to improve DI engine full load performance athigh speed, Toyota introduced an engine with a stoichiometricinjection system with two fuel injectors for each cylinder [1]. Oneis a DI injector generating a dual-fan-shaped spray with wide dis-persion, while the other is a port injector. The dual-fuel systemintroduces one additional degree of freedom for engine combus-tion optimization to reduce emissions with improved fueleconomy.

Using gasoline PFI (port-fuel-injection) and ethanol DI dual-fuel system to substantially increase gasoline engine efficiency isdescribed in Ref. [2]. The main idea is to use a highly boostedsmall turbocharged engine to match the performance of a muchlarger engine. Direct injection of ethanol is used to suppressengine knock at high engine load due to its substantial air chargecooling resulting from its high heat of vaporization [3]. Thus, themost important advantage of controlling both fuel injection sys-tems to achieve a desired air-to-fuel ratio (AFR) and fuel injectionratio is to further improve fuel economy with reduced emissions,while maintaining high engine output performance. This paperstudies how to precisely control both fuel systems to achieve thedesired air-to-fuel ratio and fuel ratio.

The control of AFR ratio is an increasingly important controlproblem due to federal and state emission regulations. Operatingthe spark ignited internal combustion engines at a desired AFR isdue to the fact that the highest conversion efficiency of a three-way catalyst occurs around stoichiometric AFR.

There have been several fuel control strategies developed forinternal combustion engines to improve the efficiency and to

reduce exhaust emissions. A key development in the evolutionwas the introduction of a closed-loop fuel injection control algo-rithm [4], followed by a linear quadratic control method [5], andan optimal control and Kalman filtering design [6]. Specific appli-cations of AFR control based on observer measurements in theintake manifold were developed in 1991 [7]. Another approachwas based on measurements of exhaust gases by an oxygen sensorand the throttle position [8]. Hedrick also developed a nonlinearsliding mode control of AFR based upon the oxygen sensor feed-back [9]. The conventional AFR control for automobiles uses bothclosed-loop feedback and feed forward control to achieve goodsteady state and transient responses.

The control strategies, mentioned above, were mainly forPFI systems. Mitsubishi began to investigate combustion controltechnologies for direct injection engines in 1996 [10]. Due to theintroduction of internal combustion engines with dual-fuel sys-tems (PFI and DI), control of both AFR and fuel ratio (ratio ofPFI fueling versus total fueling) became part of a combustionoptimization problem in Ref. [11].

In this paper, a control oriented model of a dual-fuel systemwas developed for control development and evaluation purposes,and an multi-input-multi-output (MIMO) sliding model controllerwas developed to regulate the engine AFR and fuel ratios to targetlevels. The sliding mode control method was adopted for this con-trol problem due to its ability of handling the MIMO nonlinearsystem and its robustness to matched uncertainties as described inRef. [11]. The control strategy was validated in simulation basedupon the developed dual-fuel system model. Furthermore, a morecomprehensive mean value engine model (than the control ori-ented model) and the proposed sliding mode controller wereimplemented into a hardware-in-the-loop (HIL) simulator and atarget engine control module [12], respectively, and HIL simula-tions were conducted to validate the developed controller forpotential implementation in an automotive engine.

This paper is organized as follows. It begins by presenting asimplified AFR and fuel ratio control oriented model in Sec. 2,followed by a section that describes an innovative sliding modecontroller with state feedback that regulates both AFR and fuelratios to desired nonzero targets, and its simulation results are

1Corresponding author.Contributed by the Dynamic Systems Division of ASME for publication in the

JOURNAL OF DYNAMIC SYSTEMS, MEASUREMENT AND CONTROL. Manuscript receivedJanuary 26, 2011; final manuscript received November 4, 2011; published onlineApril 3, 2012. Assoc. Editor: Warren E. Dixon.

Journal of Dynamic Systems, Measurement, and Control MAY 2012, Vol. 134 / 031012-1Copyright VC 2012 by ASME

Downloaded From: http://dynamicsystems.asmedigitalcollection.asme.org/ on 04/02/2015 Terms of Use: http://asme.org/terms

shown in Sec. 4. Next, a state estimator design with variable pa-rameter feedback gain is presented in Sec. 5 and the HIL simula-tion environment setup using the four cylinder mean value enginemodel, along with the HIL simulation results, is described inSec. 6. Finally, the conclusions are discussed.

2 An AFR and Fuel Ratio Model

The control problem of this work is to vary both PFI and DIfuel mass injection rates ( _mPFI and _mDI) so that the engine AFR isregulated at a desired level (e.g., stoichiometric) and the fuel ratioof effective PFI fueling, _mPFI E, to total fueling, _mtotal ¼ _mPFI E

þ _mDI E, is maintained at a desired value as shown in Fig. 1. Notethat the effective fueling for DI, _mDI E, is equal to the injected DIfuel _mDI .

A nonlinear model for this problem, using simplified engine dy-namics to model both the engine AFR and fuel ratios, is discussedbelow. The air flow, _mair , is modeled as

_mair ¼ x ¼ x0 þ Dx (1)

where x0 is the nominal air flow and Dx is the air flow disturb-ance due to the engine operational condition changes. The fuelflow for wall-wetting dynamics from the port fuel injector is mod-eled by the following transfer function:

_mPFI EðsÞ_mPFIðsÞ

¼ asþ 1

bsþ 1(2)

where a and b are selected to be 0.5 and 0.8, respectively, in thismodel based upon the existing engine experimental data. Notethat the PFI fueling, _mPFI , is the fuel mass injected per enginecycle. The fuel flow from the direct injector contains negligibledynamics.

Due to the three-way catalyst used for emission control, mostengines are design to achieve a target AFR around stoichiometry.For this research, the relative (normalized) target AFR, ktarget,which is defined as desired air-to-fuel ratio divided by stoichio-metric air-to-fuel ratio (14.6 for gasoline), is used. Note that atstoichiometry the relative target AFR is equal to one. The normal-ized AFR can be expressed as

k ¼ _mair

ls � _mtotal(3)

where ls represents the stoichiometry air-to fuel ratio. Now, theengine equivalence ratio / is defined as the inverse of relativeAFR k and can be approximated using Eqs. (1) and (3)

/ � ls � _mtotal1

x0

� 1

x20

Dx

� �(4)

where Eq. (1) is approximated by a first order Taylor expansion.For the rest of the paper, only equivalence ratio control instead ofAFR is considered. The fuel ratio of the dual-fuel system isdefined as the effective PFI fueling divided by total fueling, where

Rfuel ¼_mPFI E

_mPFI E þ _mDI¼ _mPFI E

_mtotal(5)

Similarly, fuel ratio, Rfuel, is approximated by substituting Eq. (4)into (5), replacing the engine equivalence ratio with the targetratio. Therefore,

Rfuel ¼ ls � _mPFI E1

x0

� Dxx2

0

� �=/target (6)

The equivalence and fuel ratio model, operating at a fixedengine speed (1500 rpm), includes wall-wetting dynamics of PFIfuel system, average DI fuel injection delay (50 ms), oxygen(AFR) sensor delay (40 ms), and air flow travel delay from enginethrottle to cylinder (200 ms). These time delays are approximatedby unitary gain first order transfer functions of the form

1

ssþ 1

where s corresponds to the physical time in seconds. Thecomplete model is divided into three subsystems, as shown inFig. 2, where the oxygen sensor dynamics are denoted as G1, theair flow dynamics as G2, and the fuel flow dynamics as G3.The state space realizations of the three subsystems are shown inEqs. (7)–(9)

_x1 ¼ a1x1 þ b1u1

y1 ¼ c1x1

�(7)

_x3 ¼ A3x3 þ B3u3

y3 ¼ C3x3 þ D3u3; C3 ¼ ½cT31cT

32�T

�(8)

_x2 ¼ a2x2 þ b2Dxy2 ¼ c2x2

�(9)

Note that

u1 ¼ ls � y2 � y31; u2 ¼ _mair; u3 ¼ _mPFI _mDI½ �T

The entire system can be expressed by the following state-spacemodel:

_x ¼a1

a2

A3

264

375xþ

b1

0

0

264

375lsc2c31x3x2

þ0

b2

0

264

375Dxþ

0

0

B3

264

375u3 (10)

where

x ¼ x1 x2 x31 x32 x33½ �T (11)

Fig. 1 Diagram of AFR and fuel ratio control problem

031012-2 / Vol. 134, MAY 2012 Transactions of the ASME


and the output equations for the equivalence and fuel ratios usingEq. (10) can be approximated by

yEqRatio ¼ ls � c2c31x2x3 (12)

yFuelRatio ¼ ls � c2x2ðc32x3 þ 0:625u1Þ=/target (13)

For the rest of the paper, ls ¼14.6 will be used, and for a blend ofgasoline and another fuel (such as ethanol), ls needs to bechanged correspondingly. The parameter matrices of the enginemodel are listed below with x0¼ 1, where system matrices, A3,B3, C3, and D3 are obtained from state space realization of G3 (seeFig. 2)

a1 ¼ �25; b1 ¼ 14:6; c1 ¼ 25

a2 ¼ �5; b2 ¼ �1; c2 ¼ 5

A3 ¼�20 0 0:4688

0 �20 0

0 0 �1:25

24

35; B3 ¼

0:625 0

0 1

1 0

24

35

C3 ¼20 20 0

0 0 0:46875

� �; D3 ¼

0 0

0:625 0

� �

The nonlinear state space engine model must be transformed intothe regular form (see Ref. [11]) below to apply sliding modecontrol

_g ¼ faðg; zÞ þ dðg; zÞ (14)

_z ¼ fbðg; zÞ þGðg; zÞu (15)

where the forcing term of Eq. (13), d(g,z), contains the disturb-ance input Dx and fa(g,z) contains the nonlinear portion ofEq. (10). A change of variables

g

z

� �¼ T � x (16)

transfers the original state vector

x ¼ x1 x2 x31 x32 x33½ �T (17)

into the regular form coordinates, where the transformation matrixT is chosen such that

g ¼ ½ x1 x2 ðx31 � 0:625x33 Þ�T (18)

z ¼ ½x33 x32�T (19)

Thus, the original system (represented in the regular form) isshown in Eqs. (20) and (21). It can be seen that the model containstwo control inputs (u1 and u2) corresponding to PFI and DI fuelingand one mass air flow disturbance input Dx

_g ¼

�25 0 0

0 �5 0

0 0 �20

2664

3775

g1

g2

g3

264

375þ

0

0

�11:25

264

375z1 þ

1

0

0

264375hðg; zÞ

|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}faðg;zÞ

þ0

�1

0

264

375Dx

|fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl}dðg;zÞ

(20)

_z ¼ �1:25 0

0 �20

� �z1

z2

� �|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

fbðg;zÞ

þ 1 0

0 1

� �|fflfflfflffl{zfflfflfflffl}

G

u1

u2

� �(21)

where

hðg; zÞ ¼ 1460g2ðg3 þ 0:625z1 þ z2Þ

3 Sliding Mode Control Strategy

Existing sliding mode AFR control applications in Refs.[13,14] utilized the binary nature of an HEGO (heated exhaust gasoxygen) sensor to reduce its oscillation resulting from the timedelay by using a dynamic, one dimensional sliding surface.

In this paper, a two dimensional sliding surface is selected tocontrol both equivalence and fuel ratios of the dual-fuel system.The sliding surface is defined as

s ¼ z� uðgÞ (22)

where the control objective is to regulate “s” to zero by designinga feedback u(g) such that it stabilizes Eq. (20). Choosing

z ¼ uðgÞ ¼ z1

z2

� �¼ 1:6g1

�g1 � g3

� �(23)

results in the asymptotic stability at the origin for Eq. (20) anddecouples the nonlinear dynamics contained in fa(g,z) from theremaining equation. The initial choice of Eq. (22) and conse-quently Eq. (23) was to linearize the system through feedback andachieve acceptable stabilization of Eq. (20). Also, note that theselection of u(g) does not contain the state g2 and removes it from

Fig. 2 Equivalence and fuel ratio model

Journal of Dynamic Systems, Measurement, and Control MAY 2012, Vol. 134 / 031012-3


having any affect on the system, which is significant since thestate g2 contained the input disturbance, Dx.

The selection of the control u to cancel the known terms of thedifferentiation of Eq. (22) and the control input v that guaranteesasymptotic stability and forces s toward zero are designed from[11], see the following:

u ¼ ueq þG�1ðg; zÞv; where

ueq ¼ G�1ðg; zÞ �fbðg; zÞ þ@u

@gfaðg; zÞ

� �and

v ¼ �b sgnðsÞ (24)

where b is a vector to be determined later. Note that to eliminatethe control signal chattering due to the signum function inEq. (24), it is replaced with the saturation function in the controlimplementation. The resulting control forces the system statesonto the sliding surface in finite time and eventually brings thesestates on the sliding surface to zero. To achieve the desired non-zero equivalence and fuel ratios, consider

Dg ¼ ðg� g0Þ ¼g1 � g10

g2 � g20

g3 � g30

24

35; where g ¼ Dgþ g0 (25)

Dz ¼ ðz� z0Þ ¼z1 � z10

z2 � z20

� �; where z ¼ Dzþ z0 (26)

where Dg and Dz go to zero, leading g and z converging to the tar-get states g0 and z0 as time goes to infinity since the sliding modecontroller regulates the states Dg1, Dg3, Dz1, and Dz2 to the slidingmanifold s¼ 0. Using these target states, g0 and z0 can bring theequivalence and fuel ratios to any desired value. To investigatethe stability of the system with these new target states, u(g) fromEq. (23) is used and the final system becomes

D _g ¼�25 �1460ðg30

þ 0:625z10þ z20

Þ 0

0 �5 0

�11:25 � 1:6 0 �20

264

375

�Dg1

Dg2

Dg3

264

375þ

0

�1

0

264

375d

þ1460g20

ðg30þ 0:625z10

þ z20Þ þ 25g10

5g20

20g30þ 11:25z10

264

375 (27)

D _z ¼�1:25 0

0 �20

� �Dz1

Dz2

� �

þ1 0

0 1

� �u1

u2

� �þ

1:25 0

0 20

� �z10

z20

� �(28)

It can be seen that the system is linear with constant matrices plusthe forcing d term. For entire system stability analysis, the charac-teristic equation of the composite linear system matrix Acomp is

detðkI� AcompÞ ¼ ðkþ 25Þðkþ 5Þðkþ 20Þðkþ 1:25Þðkþ 20Þ

with all eigenvalues in the left half plane, where

Acomp

¼

�25 �1460ðg30þ0:625z10

þz20Þ 0 0 0

0 �5 0 0 0

�11:25 �1:6 0 �20 0 0

0 0 0 �1:25 0

0 0 0 0 �20

26666664

37777775

Note that the target states g0 and z0 can be determined with thegiven input air disturbance and desired equivalence and fuelratios. The following output equations are the result of coordinatetransformation of Eqs. (12) and (13) using Eq. (16):

~yEqRatio ¼ 25g1 ¼ 5 � 20 � g2ðg3 þ 0:625z1 þ z2Þ (29)

~yFuelRatio ¼ 14:6 � 5 � g2ð0:46875z1 þ 0:625u1Þ=/target (30)

Consider the state Eqs. (20) and (21) at steady state by settingderivative terms equal to zero, leading to the system steady-statestates g and z expressed as a function of the desired output ratiosand input disturbance. The target states g0 and z0 can be obtainedwith given target ratios and disturbance

g10¼ 1

25~yEqRatio

g20¼ � 1

5Dx

g30¼ � 11:25

20z10

8>>>>><>>>>>:

(31)

and

z10¼

~yFuelRatio � /target

14:6 � 5 � g20ð0:46875þ 0:625 � 1:25Þ

z20¼

25g10

14:6 � 5 � 20 � g20

� g30� 0:625z10

8>><>>: (32)

The zero target state sliding mode control strategy is modifiedsuch that the closed-loop system converges to the desired targetstates, see Fig. 3.

To improve the performance of the sliding mode controller, thestabilizing function u(g) was generalized as follows:

z ¼ uðDgÞ ¼ 1:6Dg1 þ eDg3

�Dg1 � Dg3

� �(33)

Fig. 3 Schematic diagram of sliding mode control strategy



where e is a positive constant. Substituting Eq. (33) into (20)yields

D _g ¼�25 0 0

0 �5 0

�11:25 � 1:6 0 �20� 11:25e

264

375 Dg1

Dg2

Dg3

264

375

þdDg2

0

0

264

375Dg3e (34)

where d¼ 91.25. Note that Eq. (34) is in the following form:

D _g ¼ AðeÞDgþ gðDg; eÞ (35)

It can been seen that A(e) is Hurwitz if e> 0 and also

dDg2

0

0

24

35Dg3e

��

2

� d ek k2 Dgk k22 (36)

Let Q¼ I and define X(e)> 0 as the solution of the Lyapunovequation assuming e> 0

ATðeÞXðeÞ þ XðeÞAðeÞ ¼ �Q

The maximum and minimum eigenvalues of X(e) are plottedas function of e in Fig. 4. Define the Lyapunov functionV(Dg)¼DgT

X(e)Dg, leading to the following properties:

kminðXðeÞÞ Dgk k22 � VðDgÞ � kmaxðXðeÞÞ Dgk k2

2;

@V

@DgADD ¼ �DgTQDD � �kminðQÞ Dgk k2

2;

@V

@Dg

�� ¼ 2DgTX

�� 2� 2 Xk k2 Dgk k2¼ 2kmaxðXÞ Dgk k2

(37)

The derivative of V(Dg) satisfies

_VðDgÞ ¼ DgTðATðeÞXðeÞ þ XðeÞAðeÞÞDg

þ 2edDg3

Dg2

0

0

264

375

T

XðeÞDg

� �kminðQÞ Dgk k22þ2kmaxðXðeÞÞed Dg3j j � Dgk k2

2: (38)

Note that the origin is exponentially stable if _VðDgÞ < 0. There-fore, the system is exponentially stable if

e <1

2kmaxðXðeÞÞd Dg3j j : (39)

The stability condition will hold assuming

Dg3j j ¼ g3 � g30

� c

for all time, where c is a positive constant, and can be restated as

e <1

2kmaxðXðeÞÞd � c(40)

Figure 4 shows that kmax(X(e))¼ 0.1 for all e> 0, thus

e <1

2 � 0:1 � d � c <1

2 � 0:1 � 1460 � 0:625 � c ¼1

182:5 � c (41)

Equation (41) shows that if c is greater than zero, then there existsan e> 0 with guaranteed stability.

4 State Feedback Simulation Results

An MIMO PID controller was designed for comparisonpurposes to the nonlinear MIMO sliding mode controller. Thiscontroller cascaded two PID controllers such that the first PID isused to control the equivalence ratio and the second PI is used to

Fig. 4 Maximum and minimum eigenvalues of X(e)



reduce the fuel ratio error, see Fig. 5. The controller gains for bothPID and PI controllers are shown in Table 1. Note that the PIDcontrollers 1 and 2 have the same gains listed in Table 1.

All PID controller gains parameters were tuned such that theclosed-loop system is stable, and the equivalence ratio and fuelratio responses are as fast as possible with reasonable overshoots.The derivative and proportional gains in equivalence ratio control-ler were first tuned to reduce the transient response time withgood stability while the integration gain was set to zero. Next, theintegration gain was tuned to reduce the steady state error, andfinally, the proportional gain was tuned again to optimize the sys-tem performance. The same process was repeated for the fuel ratiocontroller. Simulations of the sliding mode controller under differ-ent air flow input disturbances were conducted and compared tothe PID controller.

The gain matrix b in the sliding mode controller for each simu-lation was tuned such that the transient response was acceptableand it was selected as

b ¼ 1:6 0

0 1:7

� �(42)

A unitary target equivalence ratio and 60% (0.6) target fuel ra-tio were chosen for each simulation. Figure 6 shows the closed-loop response of the PID controller and sliding mode controllerfor simulation #1. The simulation uses a constant input disturb-ance Dx of 0.1 plus 5 percent noise, adding a step input of 0.15 tothe constant disturbance at the 6th second. It also has a fuel ratioreduction from 0.6 to 0.4 at the 9th second and shows that the con-troller rejects the disturbance quickly. Figure 6 also shows the PFIand DI fuel control inputs for both PID and sliding mode control-lers. It can be observed that the sliding mode controller providesquick fueling inputs of both PFI and DI systems, leading to betterdisturbance rejection and transient response over the PIDresponses. Table 2 summarizes the overshoot and settling time forboth PID and sliding mode controllers.

Figure 7 shows the closed-loop response of both PID and slid-ing mode controllers for simulation #2. This simulation decreases

the target equivalence ratio from 1 to 0.9 at the 6th second andincreases it back to unity at the 9th second. Again, the constantinput disturbance Dx was 0.1 plus 5 percent noise. Figure 7 alsodisplays both fueling inputs of PID and sliding mode controllers.

Although the PID controller has a faster equivalence ratioresponse than the sliding mode one, it is under the penalty of thefuel ratio control accuracy (huge overshoot (29%) during the tran-sition). On the other hand, the sliding mode controller providessmooth transitions for both equivalence and fuel ratios. Therefore,the sliding mode control responses are more favorable over thePID ones. Table 3 summarizes the overshoot and settling time forboth PID and sliding mode controllers for simulation #2. It isworth to noting 29% fuel ratio overshoot for the PID controllerwhile the sliding mode controller only has 2.27%.

Figure 8 shows the closed-loop response of both PID and slid-ing mode controllers for Simulation #3. This simulation has anairflow disturbance of zero and x0 is equal to unity, which consti-tutes the wide open throttle (WOT) case. The simulation decreasesthe target fuel ratio from 0.5 to zero (100% DI fueling). Sincethe engine speed is fixed, WOT implies high engine load and the

Table 2 Comparison of controllers for simulation #1

% Overshoot (after 6th second) Settling time (after 9th second) Steady state error (after 12th second)

PID Sliding PID Sliding PID Sliding

Eq. ratio 5.19% 2.44% 1.6 (s) <0.5 (s) 0.016 0.0003Fuel ratio 2.30% 2.61% 3.29 (s) 2.31 (s) 0.033 0.0003

Fig. 5 Schematic diagram of PID controller

Table 1 PID and PI controller parameters

Kp1¼Kp2 0.000005 Kp 0.5Ki1¼Ki2 0.3 Ki 5Kp1¼Kp2 0.0207

Fig. 6 Closed loop response of simulation #1



simulation shows that complete direct injection fueling can beachieved by the controller and therefore can be used to suppressengine knock at high engine load.

5 State Estimator Design

The sliding mode control strategy was implemented using statefeedback, but not all state information will be available for closedloop control since in practice only limited sensors are available tomeasure certain states/outputs. In some cases, even though thesestates are measurable, the cost limitations prohibit utilizing statefeedback. Therefore, to implement the sliding mode controlscheme, a state estimator must be designed to obtain state infor-mation in real-time from the available measurements. Thesemeasurements are the equivalence and fuel ratio outputs, and thePFI and DI fueling inputs. Note that the equivalence ratio can bemeasured by a UEGO (universal exhaust gas oxygen) sensor; thefuel ratio can be estimated using a virtual sensor combiningUEGO signal, DI fueling quantity, and measured mass air flowrate. Both PFT and DI fueling inputs can be estimated based upon

fuel rail pressure and injection durations. Therefore, for state esti-mation, it was assumed that system control inputs (PFI and DIfueling), outputs (equivalence and fuel ratios), and mass air flowrate variation Dx are measurable.

Note that the fuel ratio estimation also works with the case thattwo different fuels are used for DI and PFI, for example, gasolinefor PFI and ethanol for DI. Assuming that _mDI in Eq. (5) can beestimated from fuel injection duration and pressure, PFI effectivefueling _mPFI E in Eq. (5) is required for estimating fuel ratio. Notethat the overall equivalence ratio can be expressed below

/ ¼ _mDI

_mairlDI þ

_mPFI E

_mairlPFI (43)

where / can be estimated from the oxygen sensor; _mair can beestimated from the mass-air-flow sensor; lDI and lPFI are stoichi-ometry air-to-fuel ratios for DI fuel and PFI fuel, respectively.Therefore, _mPFI E can be estimated and as well as the fuel ratio.

Consider the nonlinear system described in Eqs. (10), (12), and(13). Since state x2 can be estimated from the mass air flow sensor


Table 3 Comparison of controllers for simulation #2

% Overshoot (after 6th second) Settling time (after 9th second) Steady state error (after 12th second)

PID Sliding PID Sliding PID Sliding

Eq. ratio 0.99% 0.05% 0.45 (s) 1.29 (s) 0.0018 0.0007Fuel ratio 29% 2.27% 1.53 (s) <0.1 (s) 0.0042 0.0005




signal Dx equipped on the engine, assuming that x2 is known, thesystem can be rewritten as

_~x ¼

�25 1460x2 1460x2 0

0 �20 0 0:46875

0 0 �20 0

0 0 0 �1:25

26664

37775~x

þ

0 0

0:625 0

0 1

1 0

26664

37775u ¼ AS~xþ BSu (44)

~y ¼0 1460x2 1460x2 0

0 0 0 34:21875x2=/target

" #~x

þ0 0

45:625x2=/target 0

" #u ¼ CS~xþ DSu (45)

where ~x ¼ x1 x31 x32 x33½ �T and note that the system is nowin a linear parameter variation (LPV) system form. Note that thetime varying system is not observable since when x2 is zero, states2 and 3 of Eqs. (44) and (45) cannot be reconstructed from theoutput. Through the Luenberger state observer [12] design, it is tobe demonstrated that the proposed observer error reduces to zeroasymptotically. The state estimator has the following form:

_x ¼ ASxþ BSuþ Lð~y� yÞy ¼ CSxþ DSu (46)

where matrix L is a function of both x2 and /target such that theestimation error goes to zero as time goes to infinity. Note that theestimation system equation can be rewritten as

_x ¼ ðAS � LCSÞxþ BS � LDS L½ �u

~y

� �y ¼ CSxþ DSu (47)

and the error between the actual state ~xðtÞ and the estimated statexðtÞ is governed by the following equation:

_eðtÞ ¼ _~xðtÞ � _xðtÞ

¼ AS~xþ BSu� ðAS � LCSÞx� BS � LDS L½ �u

~y

� �¼ ðAS � LCSÞð~x� xÞ ¼ ðAS � LCSÞeðtÞ: (48)

By choosing the estimation gain matrix L as a function of both x2

and /target

LTðx2;/targetÞ ¼19 0 0 0

00:46875 � /target

34:21875x2

048:75 � /target

34:21875x2

24

35(49)

the error system matrix becomes

�Aðx2Þ ¼ AS � LCS ¼

�500 1460x2 1460x2 0

0 �20 0 0

0 0 �20 0

0 0 0 �50

2664

3775 (50)

The stability of �A when x2 is a constant is guaranteed since itseigenvalues have negative real parts. Since x2 varies as a functionof time, the stability of �A(x2) cannot be determined by the location

of its eigenvalues. To investigate the stability of the LPV errorsystem in Eq. (48), for a given range of x2, �A can be rewritten as

�Aðx2ðtÞÞ ¼ a1A1 þ a2A2 (51)

where

a1 � 0; a2 � 0; a1 þ a2 ¼ 1 (52)

Error system in Eq. (48) is now in the polytopic form [15], and itis quadratically stable if there exists fixed P¼PT> 0 such that forall x2 [16,17]

�ATPþ P�A < 0 (53)

and furthermore Eq. (53) is equivalent to

�ATPþ P�A ¼ a1AT1 þ a2AT

2

�Pþ P a1A1 þ a2A2ð Þ < 0 (54)

Define

A1 ¼

�500 0 0 0

0 �20 0 0

0 0 �20 0

0 0 0 �50

26664

37775 and

DA ¼

0 1460 1460 0

0 0 0 0

0 0 0 0

0 0 0 0

26664

37775 (55)

Let

A1 ¼ A0 þ cDA and A2 ¼ A0 � cDA; c ¼ 0:2 (56)

that allows x2 to vary between �0.2 and 0.2. Choosing

P ¼

0:01 0:001 0:001 0

0:001 0:25 0 0

0:001 0 0:25 0

0 0 0 0:1

2664

3775 > 0 (57)

and note that it satisfies Eq. (54), guarantees that the LPV errorsystem in Eq. (48) is stable for any x2 between �0.2 and 0.2. As aresult

limt!1

_~xðtÞ � _xðtÞ�� ¼ 0; 8 x2 2 �0:2; 0:2½ � (58)

Note that since the separation theory does not apply for thisnonlinear case, the closed loop system stability using the slidingmode controller with the state estimator is not guaranteed. Figure 9shows the comparison of the estimated states constructed bythe state estimator in Eq. (48) with the actual states used toachieve the desired target equivalence and fuel ratios. Note thatthe states are shown after the state transformation described inEqs. (18) and (19).

6 HIL Simulations

The engine model used for the HIL simulation is a control ori-ented four cylinder dual-fuel mean-value engine model developedbased upon Ref. [17]; see Ref. [18] for the mean value modeldeveloped at Michigan State University. This model is to be usedfor real-time simulation for developing and validating engine con-trol strategies. “Mean-value” indicates that the developed enginemodel neglects the reciprocating behavior of the engine, assumingall processes and effects are averaged over the entire engine cycle.



For the HIL simulation, the mean-value model describes a fourcylinder naturally aspired spark ignited engine. It provides theinput–output behavior of the physical engine systems with reason-able simulation accuracy using fairly low computational through-put. To enable the dual fuel system configuration for the HILsimulation, both PFI and DI fuel systems were modeled. Refer-ence [17] provides a good overview of engine modeling, and mostof dynamic equations used in this model are based upon this refer-ence book, including all engine transient dynamics. The mean-valve engine model contains subsystem models such as relativeair-to-fuel ratio calculation k, MAP (manifold air pressure),BMEP (brake mean effective pressure), engine torque, exhausttemperature, etc. The subsystems that were used in this study hada structure similar to Fig. 2.

The mean value engine model was implemented in MATLAB/SIMULINK and auto-coded into an Opal-RT (http://www.opal-rt.com/, 2010) based HIL simulation system. The engine model wasexecuted in the Opal-RT HIL simulator with a sample period of

1 ms. Similarly, the continuous time sliding mode controller,along with its state estimator, was discretized with a five-millisecond sample period (T¼ 0.005 s) and implemented in SIMU-

LINK. The discrete SIMULINK control model was then implementedinto a production Mototron Engine Control Module (ECU) (http://mcs.woodward.com/, 2010) sampled every 5 ms. The Opal-RTHIL simulator communicates with the Mototron ECU through thehigh speed controller-area network (CAN), where signals weresent and received with minimal delay. The HIL simulation schemeis shown in Fig. 10.

The Opal-RT simulation step size of 1 ms was chosen in orderto emulate a real-world continuous time engine system, while theMototron controller sampling period of 5 ms was close to that ofmany production engine control systems. The CAN communica-tion between Opal-RT and Mototron had a time delay between thetime when signals were sent from Mototron and the time whenthese signals were received by Opal-RT, and vice versa. Further-more, the total delay which includes the model/controller timing

Fig. 9 Comparison of state estimation and actual states in HIL simulation

Fig. 10 HIL simulation setup



http://www.opal-rt.com/

http://www.opal-rt.com/

http://mcs.woodward.com/

http://mcs.woodward.com/

synchronization delays, the delay from when signals were sentfrom the model to the controller, controller computation, and thedelay from signals sent from the controller back the model, wasfound to be an average of around 8 ms. This is acceptable for thissetup since one engine cycle at 1500 rpm is 80 ms. See the timingscheme in Fig. 11. Also in the HIL simulation, the gain b definedin Eq. (42) was tuned for minimal settling time without oscilla-tion. Figure 12 shows the responses of the equivalence and fuelratios for different b values. It turns out that when bHIL¼ 0.2b,the HIL simulation provides the best response for the discretizedsliding mode controller.

For the HIL simulation, a fixed engine speed of 1500 rpm alongwith a fixed throttle opening of 90% was used throughout theentire simulation. The air flow disturbance Dx was measured,resulting in the estimate of the state x2. A unitary target equiva-lence ratio and 60% (0.6) target fuel ratio were chosen for thesimulation. The equivalence ratio and fuel ratio HIL responses ofthe mean value engine model and the equivalence and fuel ratio

model simulations are shown in Fig. 13, which shows a step fuelratio reduction from 0.6 to 0.4 at the 10th second. Figure 13 alsoshows the pure simulation response of the mean value enginemodel. It can be observed that the fuel ratio response for the meanvalue model in HIL has very minimal overshoot compared withthe other output responses. Although the sliding mode controllergain of 0.2b was used for all three simulations, their responses areslightly different which is due to both the feedback and controltime delays between the HIL simulator and engine sliding modecontroller. The tuning of b for each model in its simulation setupis important in determining its acceptable output response.

Figure 14 shows the simulation results of an equivalence ratiostep increment from unity to 1.1 at the 30th second. Similarly, theHIL simulation responses show negligible steady state errors ofboth equivalence and fuel ratios due to cycle-to-cycle airflowdynamics. The mean value engine model HIL simulation equiva-lence ratio response achieves the target slightly quicker than theother responses, and its fuel ratio response also has minimal

Fig. 11 HIL timing scheme

Fig. 12 Response of ratios for different b



overshoot. In summary, both the fuel and equivalence ratio stepresponses demonstrate that the real-time sliding mode controllerimplemented in the HIL environment was able to achieve similarperformance comparing to those of the MATLAB simulations.

The possible issues associated with implementing and validat-ing the sliding mode control strategy on an engine dynamometerwill be addressed in the future work. The future work alsoincludes studying the sliding mode control within a known rangeof engine speed.

7 Conclusions

A multi-input-multi-output sliding mode controller, with stateestimator, was developed based upon a simplified equivalence andfuel ratio model to achieve nonzero desired equivalence and fuel

ratio targets. The state feedback sliding mode controller was vali-dated with the developed simple nonlinear dual-fuel system modelin MATLAB simulations, and the simulation results showed perform-ance improvement over that of the ad hoc multi-loop PID control-ler. A state estimator was developed with variable parameter gainand guaranteed convergence. Simulations were then conductedusing the simplified nonlinear model where the output feedbackcontroller with state estimator was implemented into an ECM in areal-time HIL environment and then progressed to a four cylindermean value dual-fuel system engine model. The latter of the HILsimulations achieved similar performance comparing to those ofthe pure MATLAB simulations, which demonstrated that the devel-oped output feedback sliding mode controller is feasible to beimplemented in a production engine control module controllerwith satisfactory performance.

Fig. 13 CL response of HIL simulation of fuel ratio step down

Fig. 14 CL response of HIL simulation for step equivalence ratio



References

[1] Ikoma, T., Abe, S., Sonoda, Y., Suzuki, H., Suzuki, Y., and Basaki, M., 2006,“Development of V-6 3.5-Liter Engine Adopting New Direct Injection Sys-tem,” SAE Paper No. 2006-01-1259.

[2] Bromberg, L., Cohn, D. R., and Heywood, J. B., 2007, “Optimized Fuel Man-agement System for Direct Injection Ethanol Enhancement of GasolineEngines,” U.S. Patent No. 7,225,787.

[3] Zhu, G., Hung, D., and Schock, H., 2010, “Combustion Characteristics of aSingle-Cylinder Spark Ignition Gasoline and Ethanol Dual Fueled Engine,”Proc. Inst. Mech. Eng., Part D (J. Automob. Eng.), 224, pp. 387–403.

[4] Rivard, J. G., 1973, “Closed-Loop Electronic Fuel Injection Control of the In-ternal Combustion Engine,” SAE Paper No. 730005.

[5] Cassidy, J. F., Athans, M., and Lee, W.-H., 1980, “On the Design of ElectronicAutomotive Engine Controls Using Linear Quadratic Control Theory,” IEEETrans. Control Syst. Technol., 25(5), pp. 678–687.

[6] Powers, W. E., 1983, “Applications of Optimal Control and Kalman Filtering toAutomotive Systems,” Int. J. Veh. Des., Vol. Applications of Control Theory inthe Automotive Industry, pp. 75–88.

[7] Benninger, N. F., and Plapp, G., 1991, “Requirements and Performance of EngineManagement Systems Under Transient Conditions,” SAE Paper No. 910083.

[8] Onder, C. H., and Geering, H. P., 1993, “Model-Based Multivariable Speed andAir-to-Fuel Ratio Control of an SI Engine,” SAE Paper No. 930859.

[9] Choi, S. B., and Hedrick, J. K., 1993, “An Observer-Based Controller DesignMethod for Automotive Fuel-Injection Systems,” Proceedings of 1993 Ameri-can Controls Conference, pp. 2567–2571.

[10] Kume, T., Iwamoto, Y., Iida, K., Murakami, M., Akishino, K., and Ando, H.,1996, “Combustion Technologies for Direct Injection SI Engine,” SAE PaperNo. 960600.

[11] Khalil, H., 2002, Nonlinear Systems, 3rd ed., Prentice Hall, Englewood Cliffs,Upper Saddle River, NJ.

[12] Luenberger, D. G., 1966, “Observers for Multivariable Systems,” IEEE Trans.Autom. Control, 11, pp. 190–197.

[13] Won, M., Choi, S. B., and Hedrick, J. K., 1995, “Air to Fuel Ratio Control ofSpark Ignition Engines Using Dynamic Sliding Mode Control and GaussianNeural Network,” Proceedings of 1995 American Controls Conference, pp.2591–2595.

[14] Souder, J., and Hedrick, J. K., 2004, “Adaptive Sliding Mode Control of Air-Fuel Ratio in Internal Combustion Engines,” Int. J. Robust Nonlinear Control,14, pp. 525–544.

[15] Soliman, M., Emara, H., Elshafei, A., Bahgat, A., and Malik, O., 2008,“Exponential Stabilization of LPV Systems: An LMI Approach,” Proceedingsof Canadian Conference on Electrical and Computer Engineering, pp.203–206.

[16] Boyd, S., Ghaoui, L., Feron, E., and Balakrishnan, V., 1994, Linear MatrixInequalities in System and Control Theory, SIAM, Philadelphia, PA.

[17] Guzzella, L., and Onder, C., 2004, Introduction to Modeling and Control of In-ternal Combustion Engine Systems, Springer, Berlin, Germany.

[18] Yang, X., and Zhu, G., 2010, “A Mixed Mean-Value and Crank-Based Modelof a Dual-Stage Turbocharged SI Engine for Hardware-in-the-Loop Simu-lation,” Proceedings of 2010 American Control Conference, Baltimore, MA,June, pp. 3791–3796.



http://dx.doi.org/10.1109/TAC.1980.1102457




http://dx.doi.org/10.1002/rnc.v14:6

sliding mode control of both stephen pace air-to-fuel and ... · the control strategies, mentioned...

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