# Stochastic Air-Fuel Ratio Control of Compressed Natural Gas Engines Using State Observer

Authors: Jun Yang; Jian Wang; Xuesheng Zhou; Yanxiao Li
Journal: Mathematical Problems in Engineering
 (2020)

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## Abstract

In this paper, the air-fuel ratio regulation problem of compressed natural gas (CNG) engines considering stochasticL2 disturbance attenuation is researched. A state observer is designed to overcome the unmeasurability of the total air mass and total fuel mass in the cylinder, since the residual air and residual fuel that are included in the residual gas are unmeasured and the residual gas reflects stochasticity. With the proposed state observer, a stochastic robust air-fuel ratio regulator is proposed by using a CNG engine dynamic model to attenuate the uncertain cyclic fluctuation of the fresh air, and the augmented closed-loop system is mean-square stable. A validation of the proposed stochastic robust air-fuel ratio regulator is carried out by the numerical simulation of two working conditions. The accuracy control of the air-fuel ratio is realized by the proposed stochastic robust air-fuel ratio regulator, which in turn leads to an improvement in fuel economy and emission performance of the CNG engines.


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## Body

## 1. Introduction

Natural gas is an acknowledged appropriate optional fuel, since it is characterized by clean combustion, a preferable fuel economy, richness, and low cost among various alternative fuels. Consequently, researchers have paid much attention to natural gas engines to achieve fuel economy and low emissions.A gasoline engine with the multipoint injection system was modified to a CNG engine by [1], from which the results of the experiment had been developed. To calculate the air-fuel ratio in the chamber of precombustion, a model reflecting various fuel flow rates and mixtures was established in [2]. The emission and combustion performance affected by compression ratio of natural gas engines with excess hydrogen in the case of various air-fuel ratios was investigated in [3] through experiments. The results on the performance of the emission and combustion of natural gas engines, which were affected by the composition of natural gas, were reviewed in [4]. The emissions of particles were researched in [5] for a natural gas engine, and experiments were conducted with a natural gas engine that was modified from a gasoline-fuelled engine. The cyclic variation in the combustion in natural gas engines with premixed and lean burn characteristics was discussed in [6]. Experimental research on improving the thermal efficiency by enhancing the compression ratio was carried out in [7], to achieve a greater ratio of expansion. Integrated effects of injection timing and boost pressure on combustion and performance of a direct injection CNG engine were researched in [8], and the experimental results demonstrated that an improvement was achieved when the boost pressures were larger than 7.5 Kpa at every operating point. To discover the potential application of a CNG engine to transport, the technology of a lean mixture ignited by a laser was investigated in [9], considering various excess air ratios and compression ratios. Current research results on the efficiency, emission, and combustion performances of CNG engines were reviewed in [10]. A summary of the injection technologies to improve gas engine performance was given in [11], and the injection timing and compression ratio were assessed to be the major factors affecting the combustion and performance of CNG engines. Experimental results on the emission characteristics and performance of a CNG engine that was modified from a port injection SI engine were presented in [12], and the results showed that despite the low-power and high energy consumption, the emissions were improved dramatically. A adaptive technology for the fuel-air ratio regulation of CNG engines was researched in [13] to attenuate the uncertain variation in the methane content. The combustion and emission performance of a stratified air-fuel mixture was investigated for a CNG engine in [14], and an enhancement of the overall efficiency of the engine was realized compared to the premixed case.However, it is known that, the fuel economy and emissions of CNG engines are heavily affected by the control accuracy of the in-cylinder air-fuel ratio, which is defined as the ratio of the total air mass to the total fuel mass in the cylinder. Unfortunately, the masses of the total air and the total fuel in the cylinder are unmeasured, since the so-called residual gas trapped in the cylinder at the end of the exhaust stroke contains the residual air and residual fuel, which are unmeasured, and the cyclic variation of the residual gas has stochasticity. Moreover, the fresh air mass exhibits an uncertain cyclic fluctuation and also affects the control accuracy of the in-cylinder air-fuel ratio. In this paper, a state observer-based stochastic robust air-fuel ratio regulator is presented to overcome the errors in the estimation of the masses of the total air and the total fuel, and the fluctuation of the fresh air mass, which prominently influence the accuracy of the air-fuel ratio control of CNG engines. Consequently, by using the designed stochastic robust air-fuel ratio regulator, the fuel economy and emission performance of the CNG engines will be enhanced. The regulator is designed based on the dynamic model of CNG engines which represents the cyclic transient characteristics of the air path and the fuel path. A numerical simulation is given to demonstrate the performance of the designed stochastic robust air-fuel ratio regulator.


## 2. Control Problem Formulation

The exchange process of the in-cylinder gas in CNG engines is depicted in Figure1.Figure 1
Schematic of the gas exchange system of the CNG engines.Since CNG engines have similar dynamics to gasoline engines, a dynamic model with discrete-time characteristics is given with the assumption that the mixture flowing to the manifold during the pressure equalization process breaths into the cylinder in the intake stroke [15]:(1)Mak+1=Mak−λdμMfkrk+Mank,Mfk+1=Mfk1−μrk+Mfnk,where Mak denotes the total mass of the in-cylinder air, Mfk denotes the total mass of the in-cylinder fuel, λd denotes the ideal air-fuel ratio, μ∈0,1 denotes the combustion efficiency, rk denotes the fraction of the residual gas, Mank denotes the mass of the intake air, and Mfnk denotes the mass of the intake fuel. The regulation error of the air-fuel ratio yk is defined as(2)yk=Mak−λdMfk,(3)Mank=ΔMank+Man0,where the cyclic fluctuation of the intake air ΔMank is modeled as a disturbance pertaining to L2⌈0,∞⌉, and Man0 denotes the constant part of the intake air mass Mank. From (1), (2), and (3), we have(4)yk+1=rkyk+ΔMank−λdΔMfnk,where the fresh fuel cyclic variation ΔMfnk is defined as(5)ΔMfnk=Mfnk−Mfn0,and Mfn0 denotes the constant part of the intake fuel mass Mfnk and can be calculated by(6)Mfn0=Man0λd,rk is modeled as a Markov chain with discrete-time characteristics, and the prediction model of rk in one step is shown as(7)r^k+1=∑j=1Nsjpij,where r^k+1 denotes the output of the model, sj denotes the state, and pij denotes the transition probability in one step between si and sj. For system (4), we design a state observer and a feedback regulator:(8)y^k+1=fy^k,ΔMfnk,λk,λ^k,(9)Mfnk=gy^k,Man0,λd,rk,such that the following inequality,(10)E∑k=0∞y2k<γ∑k=0∞ΔMan2k,holds for all ΔMan2k. With the stochastic robust air-fuel ratio regulator (9), the augmented system that consists of (4) and (8) is mean-square stable.


### 2.1. Stochastic Robust Air-Fuel Ratio Controller

The design process of the observer-based stochastic robust air-fuel ratio regulator is described in this part. For the design of the observer, approximate linearization is used as follows:(11)λk=MakMfk≈Man0Mfn0+Mak−Man0Mfn0−Man0Mfk−Mfn0Mfn02,since the air-fuel ratio is a nonlinear function of the total air mass and total fuel mass in the cylinder. Substituting (6) into (11), we have(12)λk−λd≈MakMfn0−Man0MfkMfn02=ykMfn0,from which we can obtain that the air-fuel ratio is a linear function of yk.Theorem 1.
For system (4), setting a constant γ>0, if a set χsj>0,si∈Ω exists, satisfying(13)γ−2∑j=1NpijχsjMfn022∑j=1NpijχsjG2−χsiMfn02+Mfn02Φ>0,where(14)2∑j=1NpijχsjG2−χsiMfn02+Mfn02Φ<0,(15)Φ=χ2siMfn022χsjMfn02−2∑j=1NpijχsjsjMfn0−G2>0,and G is a design parameter, then there exist state observer and stochastic robust regulator:(16)y^k+1=rky^k−λdΔMfnk+Gλk−λ^k,(17)ΔMfnk=siλd−GλdMfn0y^k,such that system (4) exhibits L2-gain performance γ over 0,∞.Proof.
Select the stochastic Lyapunov function as(18)Vk,rk=si=χsiy^2k+χsiε2k,where ε is the observer error of the regulation error of the air-fuel ratio yk and is defined as(19)εk=yk−y^k.
Substituting (4), (12), and (16) into (19), we can obtain(20)εk+1=yk+1−y^k+1=rkyk+ΔMank−λdΔMfnk−rky^k+λdΔMfnk−Gλk−λ^k=rkεk−GMfn0εk+ΔMank.
From (16), (18), and (20), we have(21)ΔVk,rk=si=EVk+1,rk+1rk=si−Vk,rk=si=∑j=1Npijχsjy^2k+1−χsiy^2k+∑j=1Npijχsjε2k+1−χsiε2k=∑j=1Npijχsjsiy^k−λdΔMfnk+GMfn0yk−GMfn0y^k2−χsiy^2k+∑j=1Npijχsjsiεk−GMfn0εk+ΔMank2−χsiε2k.
Rearrange (21) as follows:(22)ΔVk,rk=sj≤2∑j=1Npijχsjsi−GMfn0y^k−λdΔMfnk2+2∑j=1NpijχsjG2Mfn02y2k−χsiyk−εk2+2∑j=1Npijχsjsiεk−GMfn0εk2+2∑j=1NpijχsjΔMan2k−χsiε2k≤2∑j=1Npijχsjsi−GMfn0y^k−λdΔMfnk2−χsiMfn02−2∑j=1NpijχsjG2−Mfn02ΦMfn02y2k−Φy2k+2χsiykεk−χ2siΦε2k+χsiMfn02−2∑j=1NpijχsjG2−Mfn02ΦMfn02γΔMan2k−χsiMfn02−2∑j=1NpijχsjG2−Mfn02Mfn02γ−2∑j=1NpijχsjΔMan2k.
Based on (13)–(15) and (17), we obtain(23)ΔVk,rk=si≤−χsiMfn02−2∑j=1NpijχsjG2−Mfn02ΦMfn02y2k+χsiMfn02−2∑j=1NpijχsjG2−Mfn02ΦMfn02γΔMan2k.
For everysi, (23) is suitable, then,(24)EVk+1,rk+1−EVk,rk=∑i=1NEVk+1,rk+1∣rk=si−Vk,sipsi≤∑j=1N−χsiMfn02−2∑j=1NpijχsjG2−Mfn02ΦMfn02y2k+χsiMfn02−2∑j=1NpijχsjG2−Mfn02ΦMfn02γΔMan2kpsi=−DEy2k+γDΔMan2k,where(25)D=∑j=1NχsiMfn02−2∑j=1NpijχsjG2−Mfn02ΦMfn02psi>0.
Summing both sides from 0 to∞, we get(26)EV∞,r∞−EV0,r0≤−DE∑k=0∞y2k+γD∑k=0∞ΔMan2k.
Hence,(27)E∑k=0∞y2k≤V0,r0D+γ∑k=0∞ΔMan2k.
Therefore, system (4) exhibits L2-gain performance γ over 0,∞.


### 2.2. Stability Analysis

A stability analysis of the augmented closed-loop system of (4), (16), and (17) is presented. The definition of mean-square stable and the corresponding stability theorem that will be used in the design process is given in the following:(28)xk+1=Aξkxk+Bξkωk,x0=x0,r0=r0,where xk denotes the state of the system, ξk denotes a Markov chain with discrete-time characteristics, ωk denotes second-order Gaussian white noise with Eωk=ω0 and EωkωTk=Ω, and ωTk is the transpose of ωk.Definition 1 (see [16]).
System (28) is said to be mean-square stable, if for every initial value x0 and initial distribution ξ0, the constants q and Q which are independent of x0 exist, so that(29)qk−q⟶0,ask⟶∞,(30)Qk−Q⟶0,ask⟶∞,where(31)qk=∑i=1NExkIξk=si,(32)Qk=∑i=1NExkxTkIξk=si,and I⋅ is the Dirac measure.Lemma 1 (see [16]).
If for every set of symmetric matricesWsi>0,i=1,…,N, a set of appropriate dimension matrices χsi>0,i=1,…,N exists, so that(33)∑j=1NpijATsiχsjAsi−χsi=−Wsi,where Asi means ξk=si in matrix A, and then system (28) is mean-square stable.Theorem 2.
For every initial valuey0,y^0T and fluctuation ΔMank, the augmented closed-loop system of (4), (16), and (17) is mean-square stable.Proof.
From (4), (16), and (17), we can obtain(34)yk+1y^k+1=rkrk−GMfn0GMfn00yky^k+ΔMank0.
Defining(35)A=rkrk−GMfn0GMfn00,and choosing(36)χsi=χ=1λd00λd,i=1,…,N,we have(37)∑j=1NpijATsiχAsi−χ=−Wsi,(38)Wsi=W11siW12siW21siW22si,where(39)W11si=1λd−si2λd−G2λd2Mfn02,W12=−siλdsi−GMfn0,W21=−siλdsi−GMfn0,W22=λd−1λdsi−GMfn02.
From (14) and (15) and noting that rk∈0,0.5 [10], we obtain(40)1λd−si2λd−G2λd2Mfn02>0,detWsi>0,i.e., W is a matrix with symmetric positive-definite characteristics. Therefore, the augmented system of (4), (16), and (17) is mean-square stable.


### 2.3. Simulation Verification

A numerical simulation of CNG engines is established by (1) and (2) and mean-value models [17–19] as follows:(41)yk=Mak−Mfkλd,Mak+1=Mak−λdμMfkrk+Mank,Mfk+1=Mfk1−μrk+Mfnk,M˙an=ρaVdηv4πPaωePm,Te=HuVdηiηvPm4πRTmλ,Jω˙e=Te−Tl,P˙m=RTmVmM˙i−M˙an,M˙i=s01−cosϕPaRTaψPaPm,where Tm is 298.15K, Vm is 5.897E − 03 m3, Vd is 5.76E − 04 m3, s0 is 3.5E − 03 m2, Ta is 298.15K, λd is 17.4, and(42)ψs=s2/k2kk−11−sk−1/k,ifs≥2k+1k/k−1,k2k+1k+1/k−1,if otherwise.The numerical simulation is conducted under working conditionsW1 and W2. For working condition W1, the engine revolution is 1200 rpm and the external load is 60 Nm. For working condition W2, the engine revolution is 1600 rpm and the external load is 90 Nm. The measurement of the residual gas fraction of each cycle rk is shown with the assumptions that the exhaust stroke is an adiabatic polytropic process, and the ideal gas constant of the exhaust stroke is unvaried [15]:(43)rk=1εPTDCekPBDCek1/n,where ε denotes the ratio of effective compression, n denotes the polytropic value, PTDCek denotes the cylinder pressure at the end of the exhaust stroke, and PBDCek denotes the cylinder pressures at the end of the combustion stroke.The samples of the residual gas fraction underW1 and W2 are separated into four segments, and the state spaces are given by calculating the average values of each segment, which are shown in the following equations:(44)SW1=0.085,0.082,0.080,0.076,(45)SW2=0.087,0.081,0.075,0.072.The probability matrices of the transition in one-step (46) and (47) are obtained by computing the value of the one-step transition frequency:(46)PW1=0.100.330.390.270.110.140.390.360.310.140.140.410.410.310.100.18,(47)PW2=0.250.020.190.540.510.240.040.210.210.520.230.040.030.210.550.21.Based on (44)–(47), the prediction model in one-step of the residual gas fraction for the two working conditions is obtained. The parameter G is set to 4.55 × 10−7, γ is set to 6.2, and χsi is set to 1 for all of the working conditions. Figures 2 and 3 show the performances of the air-fuel ratio for W1 and W2, where USORk denotes the proposed stochastic observer-based robust air-fuel ratio regulator and Uopenk denotes the open-loop regulator which is given as follows:(48)Uopenk=Man0λd.Figure 2
Air-fuel ratio ofW1.Figure 3
Air-fuel ratio ofW2.From Figures2 and 3, we can observe that both USORk and Uopenk regulate the air-fuel ratio into the neighborhood of the ideal value 17.4, and the range of the air-fuel ratio fluctuation of Uopenk is larger than that of USORk under W1 and W2. In contrast, the range of the fresh fuel mass fluctuation of Uopenk is smaller than that of USORk under W1 and W2, which is found in Figures 4 and 5, since Uopenk does not focus on the errors of the estimation of the total air and the total fuel and the fluctuation of the fresh air mass. The performances of the observer values of W1 and W2 are exhibited in Figures 6 and 7, which show that the observed values of the regulation error of the air-fuel ratio can be regulated into neighborhoods of the corresponding steady value. The size of the steady value is determined by the parameter D, which is defined by (25). Indeed, the range of the air-fuel ratio of typical CNG engines is between 10 to 22 [1], and the CNG engines are stable in this range. Therefore, the stable operation of the CNG engines in this paper can be ensured, since the range of the air-fuel ratio variation shown in Figures 2 and 3 is only 17 to 17.8.Figure 4
Fresh fuel mass ofW1.Figure 5
Fresh fuel mass ofW2.Figure 6
Observer value ofW1.Figure 7
Observer value ofW2.Table1 shows the control performance over 20000 cycles of USORk and Uopenk. From Table 1, we find that robustness is achieved by USORk in the presence of a fluctuation of the fresh air mass and USORk achieves better performance than that of Uopenk, where(49)JN=∑k=1NyTkyk∑k=1NΔManTkΔMank.Table 1
Control performance.
J
U
SOR
k
20000
J
U
open
k
20000
γ
W
1
0.8666
0.9993
6.2
W
2
0.9144
1.0092
6.2


## 2.1. Stochastic Robust Air-Fuel Ratio Controller

The design process of the observer-based stochastic robust air-fuel ratio regulator is described in this part. For the design of the observer, approximate linearization is used as follows:(11)λk=MakMfk≈Man0Mfn0+Mak−Man0Mfn0−Man0Mfk−Mfn0Mfn02,since the air-fuel ratio is a nonlinear function of the total air mass and total fuel mass in the cylinder. Substituting (6) into (11), we have(12)λk−λd≈MakMfn0−Man0MfkMfn02=ykMfn0,from which we can obtain that the air-fuel ratio is a linear function of yk.Theorem 1.
For system (4), setting a constant γ>0, if a set χsj>0,si∈Ω exists, satisfying(13)γ−2∑j=1NpijχsjMfn022∑j=1NpijχsjG2−χsiMfn02+Mfn02Φ>0,where(14)2∑j=1NpijχsjG2−χsiMfn02+Mfn02Φ<0,(15)Φ=χ2siMfn022χsjMfn02−2∑j=1NpijχsjsjMfn0−G2>0,and G is a design parameter, then there exist state observer and stochastic robust regulator:(16)y^k+1=rky^k−λdΔMfnk+Gλk−λ^k,(17)ΔMfnk=siλd−GλdMfn0y^k,such that system (4) exhibits L2-gain performance γ over 0,∞.Proof.
Select the stochastic Lyapunov function as(18)Vk,rk=si=χsiy^2k+χsiε2k,where ε is the observer error of the regulation error of the air-fuel ratio yk and is defined as(19)εk=yk−y^k.
Substituting (4), (12), and (16) into (19), we can obtain(20)εk+1=yk+1−y^k+1=rkyk+ΔMank−λdΔMfnk−rky^k+λdΔMfnk−Gλk−λ^k=rkεk−GMfn0εk+ΔMank.
From (16), (18), and (20), we have(21)ΔVk,rk=si=EVk+1,rk+1rk=si−Vk,rk=si=∑j=1Npijχsjy^2k+1−χsiy^2k+∑j=1Npijχsjε2k+1−χsiε2k=∑j=1Npijχsjsiy^k−λdΔMfnk+GMfn0yk−GMfn0y^k2−χsiy^2k+∑j=1Npijχsjsiεk−GMfn0εk+ΔMank2−χsiε2k.
Rearrange (21) as follows:(22)ΔVk,rk=sj≤2∑j=1Npijχsjsi−GMfn0y^k−λdΔMfnk2+2∑j=1NpijχsjG2Mfn02y2k−χsiyk−εk2+2∑j=1Npijχsjsiεk−GMfn0εk2+2∑j=1NpijχsjΔMan2k−χsiε2k≤2∑j=1Npijχsjsi−GMfn0y^k−λdΔMfnk2−χsiMfn02−2∑j=1NpijχsjG2−Mfn02ΦMfn02y2k−Φy2k+2χsiykεk−χ2siΦε2k+χsiMfn02−2∑j=1NpijχsjG2−Mfn02ΦMfn02γΔMan2k−χsiMfn02−2∑j=1NpijχsjG2−Mfn02Mfn02γ−2∑j=1NpijχsjΔMan2k.
Based on (13)–(15) and (17), we obtain(23)ΔVk,rk=si≤−χsiMfn02−2∑j=1NpijχsjG2−Mfn02ΦMfn02y2k+χsiMfn02−2∑j=1NpijχsjG2−Mfn02ΦMfn02γΔMan2k.
For everysi, (23) is suitable, then,(24)EVk+1,rk+1−EVk,rk=∑i=1NEVk+1,rk+1∣rk=si−Vk,sipsi≤∑j=1N−χsiMfn02−2∑j=1NpijχsjG2−Mfn02ΦMfn02y2k+χsiMfn02−2∑j=1NpijχsjG2−Mfn02ΦMfn02γΔMan2kpsi=−DEy2k+γDΔMan2k,where(25)D=∑j=1NχsiMfn02−2∑j=1NpijχsjG2−Mfn02ΦMfn02psi>0.
Summing both sides from 0 to∞, we get(26)EV∞,r∞−EV0,r0≤−DE∑k=0∞y2k+γD∑k=0∞ΔMan2k.
Hence,(27)E∑k=0∞y2k≤V0,r0D+γ∑k=0∞ΔMan2k.
Therefore, system (4) exhibits L2-gain performance γ over 0,∞.


## 2.2. Stability Analysis

A stability analysis of the augmented closed-loop system of (4), (16), and (17) is presented. The definition of mean-square stable and the corresponding stability theorem that will be used in the design process is given in the following:(28)xk+1=Aξkxk+Bξkωk,x0=x0,r0=r0,where xk denotes the state of the system, ξk denotes a Markov chain with discrete-time characteristics, ωk denotes second-order Gaussian white noise with Eωk=ω0 and EωkωTk=Ω, and ωTk is the transpose of ωk.Definition 1 (see [16]).
System (28) is said to be mean-square stable, if for every initial value x0 and initial distribution ξ0, the constants q and Q which are independent of x0 exist, so that(29)qk−q⟶0,ask⟶∞,(30)Qk−Q⟶0,ask⟶∞,where(31)qk=∑i=1NExkIξk=si,(32)Qk=∑i=1NExkxTkIξk=si,and I⋅ is the Dirac measure.Lemma 1 (see [16]).
If for every set of symmetric matricesWsi>0,i=1,…,N, a set of appropriate dimension matrices χsi>0,i=1,…,N exists, so that(33)∑j=1NpijATsiχsjAsi−χsi=−Wsi,where Asi means ξk=si in matrix A, and then system (28) is mean-square stable.Theorem 2.
For every initial valuey0,y^0T and fluctuation ΔMank, the augmented closed-loop system of (4), (16), and (17) is mean-square stable.Proof.
From (4), (16), and (17), we can obtain(34)yk+1y^k+1=rkrk−GMfn0GMfn00yky^k+ΔMank0.
Defining(35)A=rkrk−GMfn0GMfn00,and choosing(36)χsi=χ=1λd00λd,i=1,…,N,we have(37)∑j=1NpijATsiχAsi−χ=−Wsi,(38)Wsi=W11siW12siW21siW22si,where(39)W11si=1λd−si2λd−G2λd2Mfn02,W12=−siλdsi−GMfn0,W21=−siλdsi−GMfn0,W22=λd−1λdsi−GMfn02.
From (14) and (15) and noting that rk∈0,0.5 [10], we obtain(40)1λd−si2λd−G2λd2Mfn02>0,detWsi>0,i.e., W is a matrix with symmetric positive-definite characteristics. Therefore, the augmented system of (4), (16), and (17) is mean-square stable.


## 2.3. Simulation Verification

A numerical simulation of CNG engines is established by (1) and (2) and mean-value models [17–19] as follows:(41)yk=Mak−Mfkλd,Mak+1=Mak−λdμMfkrk+Mank,Mfk+1=Mfk1−μrk+Mfnk,M˙an=ρaVdηv4πPaωePm,Te=HuVdηiηvPm4πRTmλ,Jω˙e=Te−Tl,P˙m=RTmVmM˙i−M˙an,M˙i=s01−cosϕPaRTaψPaPm,where Tm is 298.15K, Vm is 5.897E − 03 m3, Vd is 5.76E − 04 m3, s0 is 3.5E − 03 m2, Ta is 298.15K, λd is 17.4, and(42)ψs=s2/k2kk−11−sk−1/k,ifs≥2k+1k/k−1,k2k+1k+1/k−1,if otherwise.The numerical simulation is conducted under working conditionsW1 and W2. For working condition W1, the engine revolution is 1200 rpm and the external load is 60 Nm. For working condition W2, the engine revolution is 1600 rpm and the external load is 90 Nm. The measurement of the residual gas fraction of each cycle rk is shown with the assumptions that the exhaust stroke is an adiabatic polytropic process, and the ideal gas constant of the exhaust stroke is unvaried [15]:(43)rk=1εPTDCekPBDCek1/n,where ε denotes the ratio of effective compression, n denotes the polytropic value, PTDCek denotes the cylinder pressure at the end of the exhaust stroke, and PBDCek denotes the cylinder pressures at the end of the combustion stroke.The samples of the residual gas fraction underW1 and W2 are separated into four segments, and the state spaces are given by calculating the average values of each segment, which are shown in the following equations:(44)SW1=0.085,0.082,0.080,0.076,(45)SW2=0.087,0.081,0.075,0.072.The probability matrices of the transition in one-step (46) and (47) are obtained by computing the value of the one-step transition frequency:(46)PW1=0.100.330.390.270.110.140.390.360.310.140.140.410.410.310.100.18,(47)PW2=0.250.020.190.540.510.240.040.210.210.520.230.040.030.210.550.21.Based on (44)–(47), the prediction model in one-step of the residual gas fraction for the two working conditions is obtained. The parameter G is set to 4.55 × 10−7, γ is set to 6.2, and χsi is set to 1 for all of the working conditions. Figures 2 and 3 show the performances of the air-fuel ratio for W1 and W2, where USORk denotes the proposed stochastic observer-based robust air-fuel ratio regulator and Uopenk denotes the open-loop regulator which is given as follows:(48)Uopenk=Man0λd.Figure 2
Air-fuel ratio ofW1.Figure 3
Air-fuel ratio ofW2.From Figures2 and 3, we can observe that both USORk and Uopenk regulate the air-fuel ratio into the neighborhood of the ideal value 17.4, and the range of the air-fuel ratio fluctuation of Uopenk is larger than that of USORk under W1 and W2. In contrast, the range of the fresh fuel mass fluctuation of Uopenk is smaller than that of USORk under W1 and W2, which is found in Figures 4 and 5, since Uopenk does not focus on the errors of the estimation of the total air and the total fuel and the fluctuation of the fresh air mass. The performances of the observer values of W1 and W2 are exhibited in Figures 6 and 7, which show that the observed values of the regulation error of the air-fuel ratio can be regulated into neighborhoods of the corresponding steady value. The size of the steady value is determined by the parameter D, which is defined by (25). Indeed, the range of the air-fuel ratio of typical CNG engines is between 10 to 22 [1], and the CNG engines are stable in this range. Therefore, the stable operation of the CNG engines in this paper can be ensured, since the range of the air-fuel ratio variation shown in Figures 2 and 3 is only 17 to 17.8.Figure 4
Fresh fuel mass ofW1.Figure 5
Fresh fuel mass ofW2.Figure 6
Observer value ofW1.Figure 7
Observer value ofW2.Table1 shows the control performance over 20000 cycles of USORk and Uopenk. From Table 1, we find that robustness is achieved by USORk in the presence of a fluctuation of the fresh air mass and USORk achieves better performance than that of Uopenk, where(49)JN=∑k=1NyTkyk∑k=1NΔManTkΔMank.Table 1
Control performance.
J
U
SOR
k
20000
J
U
open
k
20000
γ
W
1
0.8666
0.9993
6.2
W
2
0.9144
1.0092
6.2


## 3. Conclusions

This paper addressed the problem of state observer-based stochastic robust air-fuel ratio regulation of CNG engines, since the existence of errors in estimation of the masses of the total air and the total fuel, and the uncertain cyclic fluctuation of the fresh air mass heavily affect the control accuracy of the air-fuel ratio. A stochastic robust air-fuel ratio regulator has been designed based on approximate linearization, and the augmented closed-loop system achieves mean-square stable. A validation of the proposed stochastic robust air-fuel ratio regulator has been verified through a numerical simulation under two working conditions. Based on the implementation of the proposed stochastic robust air-fuel ratio regulator, the control accuracy of the air-fuel ratio has been improved and then the effect of the control inaccuracy of the air-fuel ratio on the fuel economy and emission performance of the CNG engines is attenuated. Considering alternative fuels, the control problem of gas engines fuelled by biogas will be our future work.



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*Source: 2028398-2020-03-12.xml*