Extremum seeking-based optimal EGR set-point design for combustion engines in lean-burn mode

Haoyun Shi·Yahui Zhang·Tielong Shen

Abstract In lean combustion mode,exhaust gas ratio (EGR) is a significant factor that affects fuel economy and combustion stability.A proper EGR level is beneficial for the fuel economy;however,the combustion stability (coefficient of variation (COV)in indicated mean effective pressure (IMEP)) deteriorated monotonously with increasing EGR.The aim of this study is to achieve a trade-off between the fuel economy and combustion stability by optimizing the EGR set-point.A cost function (J)is designed to represent the trade-off and reduce the calibration burden for optimal EGR at different engine operating conditions.An extremum-seeking (ES) algorithm is adopted to search for the extreme value of J and obtain the optimal EGR at an operating point.Finally,a map of optimal EGR set-value is designed and experimentally validated on a real driving cycle.

Keywords Extremum seeking·EGR optimization·Lean burn mode

1 Introduction

An internal combustion engine generates vehicle propulsion power as an energy conversation device by burning the mixture of air with fuel in the cylinder.Thus,the efficiency of the energy conversation and the emission performance are influenced by the in-cylinder state of the gas mixture and its component.In the last 3 decades,many actuation factors have been developed to improve efficiency and emission by controlling the state of the in-cylinder gas mixture.For example,the exhaust gas recirculation (EGR) system is one of the effective technologies developed to mitigate the engine-out NOxand improve efficiency.However,managing the EGR fraction of inlet gas in a cylinder is challenging due to the complicated dynamics of the air intake and the fueling path,especially for the engines with an EGR loop.

Indeed,in the past two decades,various strategies have been proposed to model the dynamics or control the EGR rate with the actuator of the EGR valve.A survey on the EGR loop control has been given in [1].A nonlinear dynamic model for describing the gas flow dynamics is proposed for the intake,and exhaust manifold,enabling the transient control for optimization of pumping loss and emissions [2].A multi-variable control scheme is also proposed in [3,4].Focusing on the coupling of the EGR loop with the other intake path,a decoupling control approach is presented in [5].A model predictive control scheme is developed in [6],which focused on the torque generation and fuel consumption by introducing a balanced cost function.Meanwhile,modeling the dynamical behavior of the intake and the exhaust manifold pressure,which leads to a much simpler model structure,is challenged in [7].With the proposed simple model,a nonlinear feedback regulation scheme is further developed in [8,9].However,almost literature above targeted the transient control of the engines with the EGR system.How to decide the set-point for the EGR fraction with coordination with the other actuation factors is still an open issue.

When only limited plant knowledge is available,the method of self-optimizing control,extremum-seeking (ES)control,has been an effective method to address the online optimization problem [10].In the previous research,the common method of utilizing zero-mean periodic dithers such as sinusoids to obtain a gradient estimation has proven to be effective and robust in extremum determination of static maps and dynamic systems.However,the periodic dither may lead to corrupted control inputs that are undesirable for some online optimization applications such as tracking control [11].Moreover,the fact that the dithers are uniformly bounded may restrict the algorithm’s region of application.One prior approach is a stochastic spark advance self-optimizing control algorithm using a probabilistic guaranteed gradient learning method [12].Using noisy observations,the gradient is estimated by stochastic approximation methods,such as the Kiefer–Wolfowitz finite-difference stochastic approximation (FDSA)[13,14],random direction stochastic approximation [15],and simultaneous perturbation stochastic approximation[16,17].

Numerous applications of SOC methods for engine calibration and control have recently been published,including SA calibration [18–20],diesel engine exhaust gas recirculation control [21],fuel injection control [22],homogeneous charge compression ignition engine combustion timing control [23],and multi-variable calibrations [24,25].Some studies use the performance indicator,and combustion phase information besides [26,27].Corti and Forte [26] propose an SA proportion–integration–differentiation controller to maximize the indicated mean effective pressure (IMEP) for SA calibration based on the IMEP-CA50 distribution over the pastNcycles,and they suggest that the acceptable range forNshould be between 50 and 100.

This paper will address this issue with the extremum seeking method.First,the potential in improving the fuel efficiency by the EGR set-point decision is shown based on the observation of the experiment.Then,an extremumseeking algorithm is constructed to seek the optimal setvalue for the EGR valve.However,the feasible domain is constraint according to the limitation of the combustion variation,represented by the coefficient of variation (COV)of the indicated mean effective pressure (IMEP).With the extremum-seeking algorithm,a map-based EGR control system is designed.Finally,experimental validation results of the proposed scheme are given to show the effectiveness of the proposed approach.

2 Engine system and motivation

We start in this section with a brief explanation of the physical background of combustion engines and the experimental set-up,which enable us to understand the engine properties and the control design problem.

2.1 Combustion engine with EGR loop

A systematic configuration of the engine control system is sketched in Fig.1,where the attention is focused on the control actuation signals and the measured signals for feedback control.The well-known combustion engine is a device for transforming the fuel energy into mechanical power by managing the four strokes.More precisely,it depends on the ignition timing and the state of the in-cylinder gas mixture,and the combustion process,which is usually represented by the in-cylinder pressure profile.One more critical factor that determines the energy efficiency and the amount of generated thermal energy during one cycle is the amount of mass of the fuel and the fresh gas charged to the cylinder,if we suppose the ignition timing and the fuel mass injection are decided for keeping the maximum brake torque (MBT),and the equivalence ratio to the desired value [28].In this case,the total mass of the fuel,the fresh gas,and the burnt gas will be determined by the operation of the fuel injection,throttle angle,and the opening of the EGR valve.

Fig.1 Engine control system

Meanwhile,an alternative way to evaluate the efficiency of the fuel to the mechanical power might be the fuel mass injected during a cycle at a given operating point with a certain torque and rotational speed of the crankshaft.Hence,at the given engine operating point,the efficiency can be improved through minimizing the engine loss per cycle,and at the same time,tuning the actuators such as the ignition timing,fuel injection,and EGR valve,etc.EGR indicates the burnt gas fraction of the total gas charged into the cylinder per cycle.The mechanism of increasing engine effi-ciency can suppress the occurrence of knock by slowing down the speed of combustion,leading the ignition timing of the engine closer to MBT [29].On the other hand,leanburn mode (The ratio of air to fuel is greater than stoichiometry) also improves the efficiency,which can employ higher compression ratios and thus provide better performance[30].Usually,the throttle opening is decided according to the load,and the fuel injection mass is used to control the air–fuel ratio.The spark timing is typically set to the MBT value.In this paper,the EGR valve is used to optimize efficiency.

2.2 Experiment set-up

To analyze the physical process of the engine and verify the controller,the experiment is implemented on a real engine test bench.Figs.2 and 3 shows the engine photo and rapid prototype system,respectively.A 1.8 L 4-cylinder engine is equipped in this test bench and coupled with a low-inertia alternating-current (AC) electrical dynamometer.The specification of the gasoline engine is listed in Table 1.The dynamometer is used for emulating the external load in real time.The real-time control system dSPACE 1006 isequipped to fully control the engine via the bypass connection with the engine prototype ECU.The engine ECU can control the engine itself and provides the whole engine sensor signals and actuator signals back to the dSPACE,while the control algorithm can be programmed and downloaded in dSPACE to control the engine through enabling the control channel of the prototype ECU.

Table 1 Specification of the engine test bench

2.3 Motivation

In particular,EGR has provided significant benefits in fuel consumption and emissions.However,it affects the combustion kinetics by reducing the flame growth rate and increasing the combustion variation,which is usually measured by the coefficient of variation (COV) in indicated mean effective pressure (IMEP) [31].Since the fresh gas corresponds to the engine torque generation when the equivalence ratio is fixed,the burnt gas fraction is mainly determined by opening the EGR valve.In practice,the set-point of the EGR valve is previously calibrated as a map that provides a set-value according to the engine operating conditions.However,effi-ciency is influenced by many factors,such as the thermal environment,combustion quality,etc.It is not easy to handle the relations between the factors and efficiency.This implies that at a given engine operating point,usually coordinated by the speed and the torque.There is still freedom in the EGR valve set,which is effective for efficiency.

The causality of EGR and fuel consumption,EGR,and combustion variation are shown in the experiment data in Fig.4.It can be seen from the experiment that under the constraint of required torque and speed,the fuel consumptionand the IMEP variation is changed according to different values of the EGR and air–fuel ratioλ.This is the potential in minimizing the fuel.It is clear that when the EGR valve open or increase the value ofλ,the fuel consumption rate (BSFC) tends to decrease,but the combustion variation(COVIMEP) significantly increases.Whenλ=1.4,the fuel consumption and the operation state of the engine become worse,which means the opening of the EGR valve is not recommended (see the black line).Here,COVIMEPover 2%can be considered as violent combustion variation.

In this way a whole year passed; and then one night she vanishedagain, and was not to be found. The whole of the next day was spent in a useless search after her.

Fig.2 Engine test bench

Fig.3 Rapid prototype system

Fig.4 Fuel consumption and combustion variation characteristic under different EGR and λ

To handle the trade-off of fuel consumption and benefit from this potential and the observation,the following two issues will be solved in the next section:(1) decide the optimal opening of the EGR valve that minimizes the fuel consumption under the constraint of IMEP variation;and (2)design a map-based feedback control system that is applied through a driving cycle.

3 Proposed design approach

3.1 Extremum-seeking algorithm for EGR set-point

Extremum seeking (ES) is an iterative optimization process performed on a target system in real time.The static relationship between the system’s input parameter and its performance output is the function being optimized.The function in this study is denoted byJ(?),which is a combination of both fuel economy and combustion variation.SinceJ(?) is unknown,the ES algorithm relies only on its measurements to search for the optimum.Starting from an initial value,the ES algorithm iteratively perturbs the inputs,monitors the performance output,and adjusts the input toward improved performance.

First,we give some definitions or calculations of physical variables used for the ES algorithm in this study.

The following equation defines the fuel consumption index BSFC (g·kWh?1) as

wheremfis the fuel consumption per second (g·s?1),τethe engine brake torque (Nm),andωethe engine speed (r·min?1).

The combustion variation is measured by the coefficient of variation in IMEP,COVIMEP,which is the standard deviation in IMEP divided by the mean IMEP expressed as a percentage

where the standard deviationσIMEPand the meanμIMEPare evaluated overNconsecutive combustion cycles.Existing works of the literature suggest that the numberNis usually set to 100–300.The IMEP is calculated by the following equation using in-cylinder pressure

In this research,a cost function that consists of the fuel economy and combustion stability is constructed as follows:

whereuEGRis the control input,EGR step.The functionσ(?) is the penalty term for combustion instability,γ>0 the weighting factor for the penalty term.Letthe functionσ(?) is defined as follows:

The constructed functionJ(?) is chosen as the cost function of the extremum-seeking algorithm for the EGR set-point.Then,the optimization problem can be formulated as follows:

The ES algorithm iteratively searches for the optimal inputusing a recursion of the form

wherek=1,2,… is the iteration number,uEGR,kis the current estimate of the optimal control input,the positive numberαkis a step size or gain,and ?J(uEGR,k) is calculated from gradient estimate ofJ(?) atuEGR,k.The algorithm uses symmetric two-sided finite-differences to estimate gradient

where the perturbationckis a small non-zero constant.Computation of one finite difference requires two new measurements off(?) .Then update the estimates by,first,calculatingwith Eqs.(7) and (8).Then,projectingonto the constrained set

where the projection operatorπ{?}∶?→Ω∶[0,50] is a simple truncation.

Several studies recommend thatuEGR,kupdated from Eq.(7) converges toalmost surely ask→∞ if the step sizesαk→0,ck→0 and satisfy specified conditions.Asαk→0 and cck→0,the gradient descent-based recursive form may be acceptable for static problems,but may have limited applicability for real-time optimization applications in which the optimalmay slowly shift.The goal thatuEGR,ktracks a shiftingis beyond the capacity of the convergentαkandck,because the exploratory capability of the slowly time-varying unknown functionJ(uEGR)and the slowly shifted optimalis depressed with convergent step size sequences.Hence,the SOC algorithm for the problem addressed in this paper will adopt constant step sizes.

The extremum-seeking algorithm for EGR set-point optimization is summarized in Algorithm 1.

3.2 Proposed control scheme

The proposed control scheme shown in Fig.5 is a simple feed-forward map-based controller.Besides the basic ECU control loop includes the fuel injection control,spark advance control,and throttle control,the set-value of the EGR valve is determined by a three-dimensional map corresponding to the torque demand and the engine speed.

Fig.5 The general structure of the proposed control scheme

This map stores the optimal EGR set-values of full engine operating conditions obtained by the extremum-seeking algorithm.In particular,the map is switched according to differentλ.Using this map,the engine can be operated at a minimum fuel consumption point meanwhile satisfy the combustion variation constraint.In the next section,this controller will be validated through a driving cycle.

4 Experiment validation

4.1 Extremum-seeking algorithm validation

The control experiment applying extremum seeking by directly using the calculated BSFC,COVIMEPsignals as the feedback variables of the seeking scheme is conducted.Two groups of experimental results are shown in Figs.6 and 7.

These two groups of the experiment were performed under the following steady conditions:engine speed is 1200 r/min and 1600 r/min,water temperature is 80–85?C,oil temperature is 85–90?C,and room temperature is 25–30?C .Experiment results of group 1 are shown in Fig.6.In group 1,the references of ECU control logic are set as follows:the volume charge efficiencythe crank angle position where 50% of the heat is released CA50?=3?,the air–fuel ratioλ?=1.2 .The opening of the EGR valve starts from 2 step .The extremum-seeking algorithm maintains the opening of EGR valve around the optimal value≈10 step after 1800 cycles where the combustion variation COVIMEPis limited under 2%.

Fig.6 Group 1 experimental extremum seeking results

In group 2 which is shown in Fig.7,the ECU control logic references are set as the volume charge efficiencyCA50?=3?,the air–fuel ratioλ?=1.4 .The opening of the EGR valve starts from 4 step .In this case,the effect of EGR on BSFC is insignificant.The reason is the combustion speed becomes very slow whenλ?=1.4,expand EGR make combustion unstable,leading the fuel consumption increase.The extremum-seeking algorithm catches the phenomenon that BSFC does not change obviously when open EGR valve at this operating condition,so the algorithm maintains the opening of EGR valve around the optimal value 6–8 step after 1800 cycles.In these cycles,COVIMEPis limited under the threshold.It should be noted that when the local minimum of the cost function is not obvious,the extremum-seeking algorithm can not make the EGR set-value converge to a constant.In this case,the step size should be carefully adjusted,but it will lead to a timeconsuming extremum-seeking process.

Fig.7 Group 2 experimental extremum seeking results

Once all the optimal opening of the EGR valve is found using the extremum-seeking algorithm,we can obtain a map of the EGR set-value.To validate the effectiveness of the EGR set-valued map,a driving cycle called the UDDS cycle is used to check the performance of torque tracking and BSFC.The experiment is conducted on an engine test bench,in which the dynamometer controls the driving force generated by a real engine and the engine speed corresponding to the specified driving cycle.Fig.8 shows the comparison of 3 differentλperformances of the UDDS cycle (40–120 s).With the increase ofλ,the throttle valve needs a larger opening to achieve the same engine torque output,while EGR has to reduce the opening to avoid severe combustion variation.In particular,compared to the stoichiometry (λ=1),the BSFC is significantly reduced by increasing theλalthough the opening of the EGR valve decrease (see the red line).From this experiment,a simple controller with an EGR set-valued map is validated on a real driving cycle.The extremum-seeking algorithm can be applied to different torque,speed,and air–fuel ratio conditions.

Fig.8 The validation of the extremum-seeking algorithm by UDDS driving cycle (40–120 s)

5 Conclusions

This paper proposed a map-based EGR control design system under the different air–fuel ratio by the extremumseeking method.The design process involves two phases,EGR set-value seeking and map-based EGR control system designing.The former is formulated as a static optimization problem for the cost function of fuel consumption rate with the constraint of combustion variation.The latter is a map-based EGR control design that tracks the engine torque demand and reduces the fuel consumption rate.The effectiveness of the presented control scheme can be claimed with the demonstrated experiment results.

From the experiment,we can get the following two conclusions.First,the extremum-seeking algorithm can search the optimal EGR set-value while satisfying the combustion variation threshold.Second,the map-based EGR controller can be applied on a real driving cycle,the fuel consumption improvement of lean-burn mode is validated through the deriving cycle.

Moreover,the single control input seeking is investigated in this paper;however,the increase of control inputs make the need for multivariate optimization.This issue will be continued as further research of the next stage.