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

In this paper, the air-fuel ratio regulation problem of compressed natural gas (CNG) engines considering stochastic L2 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. +e accuracy control of the air-fuel ratio is realized by the proposed stochastic robust airfuel ratio regulator, which in turn leads to an improvement in fuel economy and emission performance of the CNG engines.


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 airfuel ratio in the chamber of precombustion, a model reflecting various fuel flow rates and mixtures was established in [2]. e 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. e 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]. e 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. e 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. e 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. e 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.

Control Problem Formulation
e exchange process of the in-cylinder gas in CNG engines is depicted in Figure 1.
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]: where M a (k) denotes the total mass of the in-cylinder air, M f (k) denotes the total mass of the in-cylinder fuel, λ d denotes the ideal air-fuel ratio, μ ∈ (0, 1) denotes the combustion efficiency, r(k) denotes the fraction of the residual gas, M an (k) denotes the mass of the intake air, and M fn (k) denotes the mass of the intake fuel. e regulation error of the air-fuel ratio y(k) is defined as where the cyclic fluctuation of the intake air ΔM an (k) is modeled as a disturbance pertaining to L 2 ⌈0, ∞⌉, and M an0 denotes the constant part of the intake air mass M an (k).

From (1), (2), and (3), we have
where the fresh fuel cyclic variation ΔM fn (k) is defined as and M fn0 denotes the constant part of the intake fuel mass M fn (k) and can be calculated by r(k) is modeled as a Markov chain with discrete-time characteristics, and the prediction model of r(k) in one step is shown as where r(k + 1) denotes the output of the model, s j denotes the state, and p ij denotes the transition probability in one step between s i and s j . For system (4), we design a state observer and a feedback regulator: such that the following inequality, holds for all ΔM 2 an (k). With the stochastic robust air-fuel ratio regulator (9), the augmented system that consists of (4) and (8) is mean-square stable.

Stochastic Robust Air-Fuel Ratio Controller.
e 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: 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 from which we can obtain that the air-fuel ratio is a linear function of y(k).
and G is a design parameter, then there exist state observer and stochastic robust regulator: such that system (4) Proof. Select the stochastic Lyapunov function as where ε is the observer error of the regulation error of the air-fuel ratio y(k) and is defined as Substituting (4), (12), and (16) into (19), we can obtain From (16), (18), and (20), we have Rearrange (21) as follows: Based on (13)- (15) and (17), we obtain For every s i , (23) is suitable, then, where 4 Mathematical Problems in Engineering Summing both sides from 0 to ∞, we get Hence, erefore, system (4) exhibits L 2 -gain performance c over [0, ∞].

Stability Analysis.
A stability analysis of the augmented closed-loop system of (4), (16), and (17) is presented. e definition of mean-square stable and the corresponding stability theorem that will be used in the design process is given in the following: where x(k) denotes the state of the system, ξ(k) denotes a Markov chain with discrete-time characteristics, ω(k) denotes second-order Gaussian white noise with E(ω(k)) � ω 0 and E(ω(k)ω T (k)) � Ω, and ω T (k) is the transpose of ω(k).
Definition 1 (see [16]). System (28) is said to be meansquare stable, if for every initial value x(0) and initial distribution ξ(0), the constants q and Q which are independent of x(0) exist, so that where and I · { } is the Dirac measure.
Lemma 1 (see [16]). If for every set of symmetric matrices where A(s i ) means ξ(k) � s i in matrix A, and then system (28) is mean-square stable.

Simulation Verification.
A numerical simulation of CNG engines is established by (1) and (2) and mean-value models [17][18][19] as follows: where T m is 298.15K, V m is 5.897E − 03 m 3 , V d is 5.76E − 04 m 3 , s 0 is 3.5E − 03 m 2 , T a is 298.15K, λ d is 17.4, and e numerical simulation is conducted under working conditions W 1 and W 2 . For working condition W 1 , the engine revolution is 1200 rpm and the external load is 60 Nm. For working condition W 2 , the engine revolution is 1600 rpm and the external load is 90 Nm. e measurement of the residual gas fraction of each cycle r(k) 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]: where ε denotes the ratio of effective compression, n denotes the polytropic value, P TDC e (k) denotes the cylinder pressure at the end of the exhaust stroke, and P BDC e (k) denotes the cylinder pressures at the end of the combustion stroke.
Based on (44)-(47), the prediction model in one-step of the residual gas fraction for the two working conditions is obtained. e parameter G is set to 4.55 × 10 −7 , c is set to 6.2, and χ(s i ) is set to 1 for all of the working conditions. Figures 2 and 3 show the performances of the air-fuel ratio for W 1 and W 2 , where U SOR (k) denotes the proposed stochastic observer-based robust air-fuel ratio regulator and U open (k) denotes the open-loop regulator which is given as follows: From Figures 2 and 3, we can observe that both U SOR (k) and U open (k) regulate the air-fuel ratio into the neighborhood of the ideal value 17.4, and the range of the air-fuel ratio fluctuation of U open (k) is larger than that of U SOR (k) under W 1 and W 2 . In contrast, the range of the fresh fuel mass fluctuation of U open (k) is smaller than that of U SOR (k) under W 1 and W 2 , which is found in Figures 4 and 5, since U open (k) 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. e performances of the observer values of W 1 and W 2 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. e 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. erefore, 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. Table 1 shows the control performance over 20000 cycles of U SOR (k) and U open (k). From Table 1, we find that robustness is achieved by U SOR (k) in the presence of a 6 Mathematical Problems in Engineering

Conclusions
is 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.

Data Availability
No data were used to support this study.

Conflicts of Interest
e authors declare that there are no conflicts of interest regarding the publication of this paper.