Model Predictive Engine Air-Ratio Control Using Online Sequential Relevance Vector Machine

Abstract

Engine power, brake-specific fuel consumption, and emissions relate closely to air ratio (i.e., lambda) among all the engine variables. An accurate and adaptive model for lambda prediction is essential to effective lambda control for long term. This paper utilizes an emerging technique, relevance vector machine (RVM), to build a reliable time-dependent lambda model which can be continually updated whenever a sample is added to, or removed from, the estimated lambda model. The paper also presents a new model predictive control (MPC) algorithm for air-ratio regulation based on RVM. This study shows that the accuracy, training, and updating time of the RVM model are superior to the latest modelling methods, such as diagonal recurrent neural network (DRNN) and decremental least-squares support vector machine (DLSSVM). Moreover, the control algorithm has been implemented on a real car to test. Experimental results reveal that the control performance of the proposed relevance vector machine model predictive controller (RVMMPC) is also superior to DRNNMPC, support vector machine-based MPC, and conventional proportional-integral (PI) controller in production cars. Therefore, the proposed RVMMPC is a promising scheme to replace conventional PI controller for engine air-ratio control.

1. Introduction

Engine power, brake-specific fuel consumption, and emissions relate closely to air ratio among all the engine variables [1]. Air ratio is also called lambda. It indicates the amount that the actual available air-fuel ratio mixture differs from the stoichiometric air-fuel ratio of the fuel being used. Manzie et al. [2, 3] mentioned that if the air-fuel ratio is 1% lower than its stoichiometric ratio (e.g., 14.7 : 1 for gasoline), carbon monoxide (CO) and hydrocarbon (HC) emissions will be significantly increased. An air-fuel ratio that is 1% higher than the stoichiometric ratio produces more nitrogen oxides (NO_x), up to 50%. Modern automotive engines are controlled by the electronic control unit (ECU) which usually uses look-up tables with compensation of a proportional-integral (PI) closed-loop controller for lambda regulation. Since the nature of engine combustion is multivariable, time-varying, time-delay, and chaotic, look-up tables with PI controller cannot produce desirable and accurate lambda control [2, 3].

So far, there are only a few papers focusing on air-ratio control, but some control strategies were developed for air-fuel ratio (AFR) control in the past decade, including the sliding mode control [4], radial basis function neural-network feed-forward feedback control [5], and model predictive control (MPC) using neural network-based models [6–8]. In the aforementioned researches, the most appropriate and the latest technique is MPC based on diagonal recurrent neural network (DRNN) [8] because of its fast computational time. The MPC is very robust and suitable for a multivariable, time-varying, and delay system that matches the characteristic of modern engine AFR control systems [9]. A reliable engine performance model is a core component of the MPC. However, the engine models developed in [5–8] were surrogate models, which were trained from the data generated by empirical equations. Moreover, there were many assumptions in the empirical equations. In fact, many coefficients in the empirical equations are also difficult to determine for a real engine [10]. Therefore the neural-network prediction models derived from the data generated by empirical equations cannot reflect the actual performance of the controller in real engines. Meanwhile, their control target and engine models only focused on AFR instead of air ratio. AFR control is ineffective for engine performance control because it does not consider the fuel variation whereas air-ratio is a fuel independent index. Furthermore, in [5–8], only simulation tests were presented and the testing on real car engines has never been done. Besides, the inherent drawbacks of the neural network (NN) would make itself difficult put into practice including multiple local minima, user burden on selection of optimal NN structure, and overfitting. Another practical challenge to the modelling of engine lambda performance is that the lambda model is required to be updated for any changes in engine performance such as the engine aging or fair user modification on it. Therefore, the current research objective is to develop a reliable and nonlinear time-series prediction model for chaotic engine lambda behaviour and the model should have ability of online update as well.

Least-squares support vector machine (LS-SVM) is an alternative technique of nonlinear modelling [11, 12], which combines the advantages of NNs (handling large amount of highly nonlinear data) and nonlinear regression (high generalization). In recent years, LS-SVM has been successfully applied to a wide range of engineering applications [10, 13–17]. Hence it is believed that LS-SVM can also be applied to estimate the lambda model. However, LS-SVM is just an offline algorithm which cannot continually update the lambda prediction models with the subsequent samples for correction. Although there is an online version of LS-SVM [18] based on pruning, namely, decremental least-squares support vector machine (DLSSVM), it still suffers from two drawbacks. The first one is that there are two user-defined hyperparameters in DLSSVM which seriously affect the model accuracy and generalization. The optimization of these two hyperparameters is time-consuming and the optimality of these hyperparameter values is not guaranteed after decremental update. Since the number of support vectors typically grows linearly with the size of the training dataset, DLSSVM makes unnecessarily liberal use of basis functions which results in long computational time for real-time control applications.

To overcome the deficiencies of neural networks and LS-SVM, Tipping and Faul [19] proposed an advanced modelling technique, namely, relevance vector machine (RVM). RVM is an online machine learning technique which utilizes more flexible and sparser model without setting additional regularization parameters. RVM takes shorter training time than NN. In addition, the model updating time of RVM is also typically fast so that the RVM model can be continually updated whenever new lambda samples arrive.

In view of the deficiencies of the existing work and the chaotic nature of engine combustion [20] as well as the advantages of RVM, a promising avenue of research is to apply relevance vector machine and model predictive controller together to complicated engine systems and test it on a real car engine for practical examination of the MPC. This paper presents a nonlinear MPC algorithm for air-ratio control based on an RVM model for a real car engine. The method is new for real-time engine lambda control. The RVM model can be updated continually to model the engine performance variation and severely nonlinear and chaotic natures. Based on the multiple-step-ahead prediction of the lambda, an optimal control signal is obtained to regulate the lambda to the desired value upon the change of engine operating condition. In order to show the advantages of the presented control algorithm over the existing algorithms, the control performance of the proposed relevance vector machine model predictive controller (RVMMPC) is compared with the latest neural network-based method for air-fuel ratio control, diagonal recurrent neural network MPC (DRNNMPC) [8] and conventional PI controller in production cars. Besides, it is interesting in extending the decremental least-squares support vector machine (DLSSVM) [18] to MPC and making comparison with the RVMMPC for engine air-ratio. To the best knowledge of the authors, this research is the first attempt at extending online LS-SVM and RVM to the domain of automotive engine air-ratio modelling and control.

2. Relevance Vector Machine

In the first part of this section, the algorithm of RVM modelling is briefly reviewed. In order to handle the online update of RVM model, an online sequential algorithm is adopted and described in the last part of this section.

2.1. RVM Modelling

Given a training dataset D of 𝑁 input vectors 𝐱_𝑛, 𝑛=1 to 𝑁, along with 𝑁 corresponding scalar-valued output 𝑦_𝑛. The input vector 𝐱_𝑛∈ℝ^𝑚 contains the previous measured engine time-series parameters including fuel injection time, throttle position, and air ratio at a specific time instant. The corresponding air ratio at that time is defined as the output 𝑦_𝑛∈ℝ which is assumed to contain zero-mean Gaussian noise with variance 𝜎². Hence, the probability of prediction error 𝜀_𝑛 for 𝑦_𝑛 is a Gaussian distribution of zero mean and variance 𝜎², that is, 𝑝(𝜀_𝑛∣𝜎²)=𝑁(0,𝜎²), with 𝑝𝑦𝑦_𝑛=𝑓𝐱_𝑛,𝐰+𝜀_𝑛,(1)_𝑛∣𝐱_𝑛,𝐰,𝜎²=𝑁𝑦_𝑛̂,𝜎²,(2) where 𝑦_𝑛̂=𝑓(𝐱_𝑛,𝐰) is the output prediction of the true value 𝑦_𝑛, and 𝐰=[𝑤₀,…,𝑤_𝑁] is the weight vector for the RVM model.

Generally, the prediction can be represented by𝑦̂==𝑤𝑓₀(𝐱,𝐰)+^𝑁_𝑛=1=𝐰𝝓(𝐱),𝑤_𝑛𝐾𝐱,𝐱_𝑛(3) where 𝐾(𝐱,𝐱_𝑖) is a basis function and 𝜙(𝐱)=[1𝐾(𝐱,𝐱₁)⋯𝐾(𝐱_𝑛,𝐱_𝑁)]^T. In the current research, radial basis function (RBF) is chosen as the basis function 𝐾 because it is commonly used for modelling problem [10].

Under the aforementioned formulation, the likelihood function of the output vector 𝐲=𝑦_𝑛,𝑛=1 to 𝑁, is constructed by a multivariate Gaussian:𝑝𝐲∣𝐰,𝜎²=(2𝜋)^−𝑁/2𝜎^−𝑁exp−‖𝐲−𝐲̂‖²2𝜎²,(4) where 𝐲̂=Φ𝐰^T and Φ represents an 𝑁×(𝑁+1) design matrix with the notation Φ=[𝜙(𝐱₁),…,𝜙(𝐱_𝑁)]^T, and 𝜙(𝐱_𝑛)=[1𝐾(𝐱_𝑛,𝐱₁)⋯𝐾(𝐱_𝑛,𝐱_𝑁)]^T,𝑛=1to𝑁.

The likelihood function, (4), is complemented by a prior over the weights, 𝐰={𝑤_𝑝},𝑝=0 to 𝑁, to control the complexity of the model and avoid overfitting. The prior is a zero-mean Gaussian distribution and is defined over every weight 𝑤_𝑛 as follows: 𝐩(𝐰∣𝜶)=(2𝜋)^{𝑁−𝑁/2}_𝑝=0𝛼^𝑝1/2exp−𝛼^𝑝𝐰^2𝑝2.(5)

The hyperparameter vector 𝜶=[𝛼₀⋯𝛼_𝑁]^T that controls how far each weight 𝑤_𝑝 is allowed to deviate form zero. Given 𝜶, using Bayes’ rule, the posterior over 𝐰 is expressed as𝑝𝐰∣𝐲,𝜶,𝜎²=𝑝𝐲∣𝐰,𝜎²𝑝(𝐰∣𝜶)𝑝𝐲∣𝜶,𝜎²=𝑁(𝐰∣𝝁,𝚺).(6)

The posterior mean 𝝁 and covariance 𝚺 for 𝐰 are given as follows [19]:𝚺𝝁==𝐀𝐰+=𝜎𝜎⁻²₋₂𝚽^T𝚽⁻¹,ΣΦ^T𝐲,(7) where 𝐀 is defined as diag (𝛼₀⋯𝛼_𝑁).

The posterior mean 𝝁 is an estimation of the weight vector 𝐰 for prediction. The only unknowns in (7) are the hyperparameters 𝜶 which can be estimated via a type-II maximum likelihood procedure [19]. It is called sparse Bayesian learning which is formulated as the local maximization with respect to 𝜶 of the marginal likelihood or, equivalently, its logarithm 𝐿(𝜶):𝐿(𝜶)==−log1𝑝𝐲∣𝜶,𝜎²=log^∞−∞𝑝𝐲∣𝐰,𝜎²𝑝(𝐰∣𝜶)𝑑𝐰2𝑁log2𝜋+log||𝐂||+𝐲^T𝐂⁻¹𝐲,(8) where 𝐂=𝜎²𝐈+Φ𝐀Φ^T.

By maximizing (8) over 𝜶, the most probable values 𝜶_MP can be generated. Then, 𝐀 can be obtained by substituting 𝜶_MP into (7). Afterwards, the covariance 𝚺 and posterior mean 𝝁=[𝜇₀,𝜇₁,…,𝜇_𝑁]^T can be estimated. Therefore, the RVM model 𝑓 can be found by setting 𝐰=𝝁:𝑦𝑦̂=𝑦_𝑝=𝑓(𝐱,𝐰)=𝑓(𝐱,𝝁),_𝑝=𝜇₀+^𝑁_𝑛=1𝑦_𝑝𝜇_𝑛𝐾𝐱,𝐱_𝑛,=𝜇₀+^𝑁_𝑛=1𝜇_𝑛exp−‖‖𝐱−𝐱^𝑛‖‖𝜎²,(9) where 𝑦̂ is the prediction of the output lambda with the unseen input data 𝐱 containing the recent measured engine data series, which is explained in Section 4.1. One crucial observation is that the optimal values of many hyperparameters 𝛼_𝑝 are typically infinite. Considering that 𝑤_𝑝=𝜇_𝑝∝(1/𝛼_𝑝), this leads to a parameter posterior infinitely peaked at zero for many weights 𝑤_𝑝. In consequence, the posterior mean 𝝁 consists of very few nonzero elements. This results in good sparseness for RVM whereas LS-SVM and NN do not have this advantage. A good sparseness implies that the computational time for prediction can be shortened.

2.2. Online RVM Modelling

In order to train and update the RVM model continually, an online algorithm for RVM [19] is employed. The algorithm starts with an empty set of basis function for the RVM model and sequentially adds basis functions 𝜙_𝑛=[1𝐾(𝐱_𝑛,𝐱₁)⋯𝐾(𝐱_𝑛,𝐱_𝑁)]^T to increase the marginal likelihood and modify their corresponding weights 𝑤_𝑛. Within the same principal framework, the likelihood can also be updated by deleting those basis functions which subsequently become redundant. Removing a basis function 𝜙_𝑛 implies that the corresponding 𝐱_𝑛 is no longer important and can be excluded from the design matrix Φ in constructing the RVM model f, which can be simply done by setting the corresponding hyperparameter 𝛼_𝑝 equal to infinity (practically a very large value). In this way, the corresponding weight value 𝑤_𝑝=𝜇_𝑝∝(1/𝛼_𝑝) becomes zero. Since 𝑤₀ is a bias corresponding to no training data, only the weights 𝑤₁,𝑤₂,…,𝑤_𝑁 are considered. The data 𝐱_𝑛 with nonzero weights 𝑤_𝑛, 𝑛=1 to 𝑁, are referred to relevance vectors. Since the basis functions are sequentially added to or deleted from the lambda model by RVM, the likelihood can be continually updated and hence this mechanism makes online update of the lambda model feasible.

In the following discussions, the constraints to add and delete basis function from the RVM lambda model are given, and then the detailed online RVM algorithm is presented.

2.2.1. Constraints for the Basis Functions Update

In (8), the covariance matrix 𝐂 includes all basis functions in the RVM prediction model. When it is necessary to remove a basis function 𝜙_𝑛 from 𝐂, the new covariance matrix 𝐂_−𝑛 with the influence of removed basis function 𝜙_𝑛 can be expressed as follows:𝐂_−𝑛(𝜶)=𝜎²𝐈+_𝑗≠𝑛𝛼^𝑗−1𝜙_𝑗𝜙^T_𝑗.(10) It was shown in [21] that 𝐿(𝜶) has a unique maximum with respect to 𝜶:𝛼_𝑛=𝑆^2𝑛𝑄^2𝑛−𝑆_𝑛𝛼,if𝑄^2𝑛>𝑆_𝑛,(11)_𝑛=∞,if𝑄^2𝑛≤𝑆_𝑛,(12) where𝑆𝑄_𝑛=𝜙^T_𝑛𝐂^−1−𝑛𝜙_𝑛,(13)_𝑛=𝜙^T_𝑛𝐂^−1−𝑛𝐲,(14) with 𝜙_𝑛=[1𝐾(𝐱_𝑛,𝐱₁)⋯𝐾(𝐱_𝑛,𝐱_𝑁)]^T,𝑛=1 to 𝑁.

The results of (11) and (12) imply that if 𝜙_𝑛 is currently included in the lambda model (i.e., 𝛼_𝑛<∞) and 𝑄^2𝑛≤𝑆_𝑛, then 𝜙_𝑛 can be deleted by setting 𝛼_𝑛 to ∞. On the other hand, if 𝜙_𝑛 is currently excluded from the lambda model (i.e., 𝛼_𝑛 = ∞) and 𝑄^2𝑛>𝑆_𝑛, then 𝜙_𝑛 can be added by setting 𝛼_𝑛 to some optimal finite values. With these constraints, an online RVM algorithm can be implemented here in after.

2.2.2. Online Sequential Algorithm of RVM

The steps for the implementation of the online RVM algorithm are described as follows.(1) Initialize 𝜎² to some sensible values (e.g., var[𝐲]×0.1) and all 𝛼_𝑛 are notionally set to infinity.(2) Initialize 𝑆_𝑛 and 𝑄_𝑛 with a single basis function 𝜙_𝑛 from (13) and (14) and compute new 𝛼_𝑛 from (11) which can be simplified as follows.𝛼_𝑛=‖‖𝝓^𝑛‖‖²‖‖𝝓^T_𝑛𝐲‖‖/‖‖𝝓_𝑛‖‖²−𝜎².(15)(3) Explicitly compute 𝝁 and 𝚺 (which are scalars initially), along with initial values of 𝑆_𝑛 and 𝑄_𝑛 for all 𝑁 basis functions 𝜙_𝑛 using (7) and (10)–(14).(4) Select a candidate basis function 𝜙_𝑛 from the set of all 𝑁 basis functions.(5) Compute 𝜃_𝑛=𝑄^2𝑛−𝑆_𝑛.(6) If 𝜃_𝑛>0 and 𝛼_𝑛<∞ (i.e., 𝜙_𝑛 is included in the model), then reestimate 𝛼_𝑛 using (11).(7) If 𝜃_𝑛=0 and 𝛼_𝑛=∞, then add 𝜙_𝑛 to the model with updated 𝛼_𝑛.(8) If 𝜃_𝑛<0 and 𝛼_𝑛<∞, then delete 𝜙_𝑛 from the model and set 𝛼_𝑛=∞.(9) Estimate the noise level, and update 𝜎² as follows:𝜎²=‖𝐲−𝐲̂‖²∑^𝑁𝑛=1𝛼_𝑛𝚺_𝑛𝑛.(16)(10) Recompute or update 𝚺, 𝝁 and all 𝑆_𝑛 and 𝑄_𝑛 using (7) and (10)–(14).(11) If converged then terminate, otherwise go to Step 4.

Remarks 2.2.2. For the initialization in Step (2), a potential basis function could be the one with the largest normalized projection onto the output vector which gives the largest initial likelihood. In Step (4), a candidate basis function 𝜙_𝑛, both included in and excluded from the current RVM model, must be selected for updating but the selection is purely at random. In Step (10), 𝚺, 𝝁 and all 𝑆_𝑛 and 𝑄_𝑛 are recomputed in full forms of (7) and (10)–(14). In Step (11), attaining a local maximum of the marginal likelihood must be judged. The online algorithm terminates when the changes in 𝐿(𝜶) for all basis functions in the RVM model are smaller than 10⁻⁶ and all other 𝜃_𝑛≤0 [19].
The aforementioned online sequential algorithm for RVM ensures to increase the marginal likelihood at each step until a local maximum is attained. Although adding and deleting basis functions appear notionally to be independent, posterior statistics for all basis functions are being maintained concurrently; that is, all elements of 𝝁 and 𝚺 corresponding to removed basis functions are trivially zero.

3. MPC with RVM Model

The structure of the proposed RVMMPC is shown in Figure 1. The controller consists of the RVM engine lambda model and the optimizer based on Brent’s method [22]. The RVM engine lambda model predicts the engine response over a specified time horizon. The predictions are used by the optimizer to determine the tentative fuel injection time 𝑢^, that minimizes the following performance criterion over the specified time horizon, and then the optimal fuel injection time signal 𝑢 is sent to the engine:min𝐽𝑢^=^𝑁2_𝑗_=𝑁1𝑦_𝑟(𝑡+𝑗)−𝑦_𝑝(𝑡+𝑗)²+𝜌^𝑁𝑢_𝑗=1𝑢^(𝑡+𝑗−1)−𝑢^(𝑡+𝑗−2)²,(17) where 𝑁₁ and 𝑁₂ define the prediction horizon. 𝑡 is the time step. 𝑁_𝑢 is the control horizon. 𝜌 is a user-defined control weighting factor which penalizes excessive movement of the control signal (i.e., the fuel injection time). The variables 𝑢′(𝑡+𝑗−1) and 𝑢′(𝑡+𝑗−2) in the second part of (17) are the tentative fuel injection time at the time step 𝑡+𝑗−1 and 𝑡+𝑗−2, respectively. The second part of (17) ensures the stability of the controller output. 𝑦_𝑟(𝑡+𝑗) is the target lambda at the time step 𝑡+𝑗, and 𝑦_𝑝 is the predicted lambda by the RVM model which is computed from (9) at the time step 𝑡+𝑗, from which the input vector 𝐱 consists of three time series of engine control parameters: 𝑢′(𝑡+𝑗−1),𝑢′(𝑡+𝑗−2),…, throttle position TP(𝑡+𝑗−1),TP(𝑡+𝑗−2)…, and previous measured lambda 𝑦(𝑡+𝑗−1),𝑦(𝑡+𝑗−2)…. In other words,𝑦_𝑝(𝑡+𝑗)=𝝁₀+^𝑁_𝑛=1𝝁_𝑛exp−‖‖𝐱−𝐱^𝑛‖‖𝜎².(18)

Figure 1 

Structure of RVMMPC for engine lambda control.

3.1. Single-Dimension Optimization Approach

The original optimization problem involved in this paper is multidimensional and constrained with the tentative control signals over the control horizon 𝑁_𝑢 and tentative fuel injection times 𝑢′(𝑡),𝑢′(𝑡+1),…,𝑢′(𝑡+𝑁_𝑢−1), which can minimize the objective function 𝐽(𝑢′) of (17). Then the predicted lambda values, 𝑦_𝑝(𝑡+𝑁₁),𝑦_𝑝(𝑡+𝑁₁+1),…,𝑦_𝑝(𝑡+𝑁₂), can trace the target lambda values, 𝑦_𝑟(𝑡+𝑁₁),𝑦_𝑟(𝑡+𝑁₁+1),…,𝑦_𝑟(𝑡+𝑁₂), by using the optimized fuel injection time series. Each fuel injection time is normally bounded within the range from 3 ms to 60 ms. However, the multi-dimensional optimization always requires heavy computation, especially when constraints exist. Real-time control applications often put emphasis on computational speed. The research of [7] also showed that the one-dimensional approach is efficient for real time AFR control and the overall tracking error is similar to that using multidimensional optimization approach. Therefore, the optimization problem to be solved is reduced to onedimensional. In this paper, the control signal 𝑢 is assumed to remain constant over the control horizon. Therefore, the tentative control signal in the objective function is also constant over the control horizon, that is, 𝑢′(𝑡)=𝑢′(𝑡+1),…,=𝑢′(𝑡+𝑁_𝑢−1). In this way, only one parameter 𝑢′(𝑡) is needed to determine, and the final fuel injection time at each time step 𝑢 is set to be the optimal value of 𝑢′(𝑡).

3.2. Brent’s Method

There are many optimization methods available in the literature; it is impossible to examine all of them. For illustrative purpose, a well-known technique—Brent’s method—is examined in this research. Brent’s method is a robust and efficient optimization method. It combines the typical parabolic interpolation and golden-section search. The objective function in each iteration is approximated by an interpolating parabola through three existing points. The minimum point of the parabola is taken as a guess for the minimum point if certain criteria are met. Otherwise, golden-section search is carried out. The advantage of this method is that the high convergence rate of parabolic interpolation can be maintained without losing the robustness of golden-section search [22]. According to this advantage, Brent’s method was selected as the MPC optimizer in this study. The general working principle of Brent’s method is shown in Figure 2. The detail optimization procedure of Brent’s method was presented in [22] and is not presented herein. There are three parameters of Brent’s method which are the initial interval of the input variable, [𝑎,𝑏], that is the limit of the fuel injection time, as well as the tolerance, tol, for stopping the optimization procedure. The three variables are set at 3, 60, and 0.05, respectively, because the fuel injection time varies within 3 to 60 ms from 0 to 100% throttle.

Figure 2 

General working principle of Brent’s method.

4. Implementation and Evaluation of RVM Engine Air-Ratio Model

4.1. RVM Engine Air-Ratio Model Implementation

The objective of the RVM engine lambda model is to predict the future lambda 𝑦_𝑝 from three time series of inputs: fuel injection time u, throttle position TP, and previous measured lambda 𝑦. The structure of the RVM model was chosen to be second order (i.e., 2 past time steps) which gives the minimum prediction error [7, 8], and the structure is shown in Figure 3.

Figure 3 

Structure of RVM engine lambda model.

To obtain the engine data for building the engine lambda model, 3000 data samples including lambda, fuel injection time, and throttle positions were collected. These 3000 lambda samples were collected over a Honda Type-R K20A i-VTEC engine controlled by a MoTeC M800 programmable electronic control unit with nonfactory calibration data. A dyno test was done for collecting the data samples. In the dyno test, the engine speed was ranged from 1500 r/min to 6500 r/min with random throttle positions bounded between 5% and 65%. The lambda data samples were collected using an in-car wide-band lambda sensor at a sampling frequency of 200 Hz. The first 2000 data samples were used as training dataset 𝐃 to build the lambda model. The last 1000 data samples were used as test dataset TEST and the first 900 data samples of TEST were regarded as update dataset UPDATE for real-time updates. The lambda model was updated every 100 measured lambda data samples during prediction. With respect to the training dataset 𝐃, the test dataset TEST is unseen cases for testing the generalization of the built lambda model.

4.2. Evaluation Criteria for Engine Air-Ratio Model

To illustrate the accuracy, superiority, and online update ability of the proposed RVM model, its prediction result was compared with those obtained from the latest methods, DLSSVM [18] and DRNN [8].

After obtaining the lambda model 𝑅𝑀(𝐱) with RVM through the training algorithm presented in Section 2 over 𝐃., 𝑅𝑀(𝐱) was then updated for 9 times with UPDATE, 100 data samples each time, to build the updated RVM lambda model 𝑅𝑀^∗(𝐱). Apart from the RVM model, classical LS-SVM was employed to build the lambda model 𝐷𝑀(𝐱) over 𝐃. After that, 𝐷𝑀(𝐱) was updated with UPDATE but employing the updating algorithm presented in [18] to build the decremental LS-SVM model 𝐷𝑀^∗(𝐱). Moreover, DRNN, the same modelling algorithm for AFR model predictive control presented in [8], was applied to estimate the lambda model, 𝑁𝑁(𝐱). The network was trained for 60 epochs. After that, 𝑁𝑁(𝐱) was updated with UPDATE but employing the dynamic back-propagation with automatic differentiation technique presented in [8] to build the online model 𝑁𝑁^∗(𝐱).

After constructing all 𝑅𝑀^∗(𝐱),𝐷𝑀^∗(𝐱), and 𝑁𝑁^∗(𝐱), the performance of the three lambda models can be evaluated in terms of accuracy. Since the range of lambda for combustible mixture is very narrow, the prediction errors of the above three models are presented by logarithmic mean absolute error (LMAE), and they were evaluated one by one against the test dataset, TEST, using (19):𝐿𝑀𝐴𝐸=−log1𝑇^𝑇_𝑘=1||𝑦_𝑘−𝑓^∗𝐱_𝑘||,(19) where 𝑓^∗(𝐱_𝑘) represents either 𝑅𝑀^∗(𝐱_𝑘), 𝐷𝑀^∗(𝐱_𝑘), or 𝑁𝑁^∗(𝐱_𝑘),𝐱_𝑘 is the 𝑘𝑡ℎ new input vector for lambda prediction, 𝑦_𝑘 is the corresponding actual lambda value of 𝑓^∗(𝐱_𝑘), and 𝑇 is the total number of predictions. The value of 𝑇 is equal to 1000 in this case study. Besides LMAE, the accuracies of the three models were also evaluated by the linear regression 𝑅² value. The prediction results between the predicted lambda values and the corresponding actual lambda values over TEST are shown in Figure 4.

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Figure 4 

(a) Comparison between predicted lambda and the corresponding actual lambda; (b) in detail.

Table 1 shows the LMAE and 𝑅² values of the three lambda models. According to the LMAE values in Table 1, the RVM lambda model 𝑅𝑀^∗(𝐱) outperforms 𝐷𝑀^∗(𝐱) and 𝑁𝑁^∗(𝐱) by approximately 4% and 29%, respectively. It is believed that the accuracy of the RVM lambda model can be significantly increased if the model update is continually carried out. Moreover, as compared with DLSSVM, RVM has better sparseness. This property is very important for online control and system identification because the computational time and the size of memory are fewer.

Table 1 

LMAE and 𝑅² of different lambda models.

In addition to the improvement on prediction accuracy, RVM can also reduce the time for training and updating the lambda model significantly. Table 2 shows the model training time, average model updating time, and cumulative model updating time after nine times of updates. In Table 2, the lambda model training and average model updating time of RVM are 1.52 s and 0.08 s, respectively. The lambda model training time of 𝑅𝑀^∗(𝐱) is the shortest among the three lambda models. The average model updating time of 𝑅𝑀^∗(𝐱), 0.08 s, is only about one-sixth of 𝐷𝑀^∗(𝐱), 0.52 s. It is because in RVM, the updating procedure is applied to every basis function individually while DLSSVM has to work on a large matrix for its estimated inversion. Moreover, the average model updating time of 𝑅𝑀^∗(𝐱) is also about one-eleventh of 𝑁𝑁^∗(𝐱). This achievement may be accomplished by the effective online algorithm of RVM. The total cumulative model updating time saved by RVM from DLSSVM and DRNN after 9 update iterations is 3.99 s and 7.56 s, respectively. However, if the lambda model is often updated, the cumulative model updating time saved by RVM will be very significant. Obviously, RVM has more advantage than DLSSVM and DRNN algorithms. As a whole, the high accuracy and short updating time of the lambda model using RVM make online model predictive air-ratio control more feasible.

Table 2 

Training and updating time of different lambda models.

Experimental results show that the model accuracy, model training time, and updating time of RVM are the best among the aforementioned algorithms. Therefore, RVM was confidently selected to implement the model predictive engine air-ratio controller.

5. Implementation and Evaluation of RVMMPC

5.1. Experimental Setup

The proposed RVMMPC algorithm was implemented and tested on a Honda Type-R K20A i-VTEC engine with MoTeC M800 programmable ECU and National Instrument (NI) USB-6259. The model predictive control algorithm was implemented using MATLAB. MoTeC M800 is mainly used for engine control, whereas NI USB-6259 is used for sending control signal to the MoTeC ECU via a LabVIEW interface program according to the MATLAB MPC program embedded. In other words, NI USB-6259 serves as an interface between the MATLAB program and the MoTeC ECU. Apart from fuel injector control, the MoTeC ECU also contains many control maps, such as ignition map and valve timing map, to maintain the engine operation. The experimental setup and the LabVIEW interface program are shown in Figures 5 and 6, respectively.

Figure 5 

Experimental setup.

Figure 6 

LabVIEW interface program.

The initial offline training data for building the RVM model were obtained using a wide-band lambda sensor subject to random throttle positions and are discussed in the pervious section. There were three pilot tests in this study. The tests evaluate the tracking error, robustness, and adaptability of the controllers respectively.

5.2. Pilot Test 1: System Tracking Ability

In pilot test 1, the test cycle is shown in Figure 7 where the throttle position changes from 0 to 50% throttle (i.e., partial throttle). In this test, the lambda value is needed to track the target values from the stoichiometric value (1.00) to a value for the best brake-specific fuel consumption (1.05) as the throttle position is changed. This lambda value for such partial throttle is a normal requirement for automobiles. After choosing the sampling time to be 0.005 s, the tracking ability of the RVMMPC can be examined. By testing many values around the setting used in [8], the parameters of the optimizer were chosen as 𝑁₁=1,𝑁₂=10,𝜌=0.5,and𝑁_𝑢=3. With the test cycle shown in Figure 7 and the parameters chosen, the lambda control result and the corresponding fuel injection time are shown in Figures 8(a) and 9(a), respectively. In order to show the advantage of the RVMMPC, its control result is compared with other two latest model predictive controllers, decremental LS-SVM model predictive controller (DLSSVMMPC) and DRNN model predictive controller (DRNNMPC), as well as the PI control algorithm used in the existing automotive ECU. The engine lambda models used in the DLSSVMMPC and DRNNMPC are the models mentioned in Section 4.2. The parameters of nonlinear optimization for the DLSSVMMPC, and DRNNMPC were the same as those of RVMMPC. The PI gains of the PI controller were obtained by the Ziegler-Nichols method. The lambda control results and the corresponding fuel injection time of the DLSSVMMPC, DRNNMPC, and PI controller are shown in Figures 8(b)–8(d) and 9(b)–9(d), respectively.

Figure 7 

Throttle position against time in pilot test 1.

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Figure 8 

Lambda control results of pilot test 1 of (a) RVMMPC, (b) DLSSVMMPC, (c) DRNNMPC, and (d) PI controller.

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Figure 9 

Fuel injection time of pilot test 1 of (a) RVMMPC, (b) DLSSVMMPC, (c) DRNNMPC, and (d) PI controller.

Figure 8(a) shows that the RVMMPC can control the lambda to follow the target lambda with the smallest tracking error and overshoot among all the controllers. The control performances of the four controllers are shown in Table 3. As the range of lambda for combustible mixture is very narrow, logarithmic mean absolute error is chosen as the logarithmic tracking error (LTE) to evaluate the tracking ability of the controllers and is defined by 𝐿𝑇𝐸=−log1𝑇^𝑇_𝑡=1||𝑦_𝑡−𝑦_𝑟(𝑡)||,(20) where 𝑡 is time step, 𝑇 is the total number of time step, 𝑦_𝑡 is the actual lambda at each time step, and 𝑦_𝑟(𝑡) is the corresponding target lambda at each time step. Table 3 reveals that the LTE of the RVMMPC outperforms the DLSSVMMPC, DRNNMPC and PI controller by approximately 5%, 8%, and 38%, respectively.

Table 3 

Control performance of pilot test 1 of different controllers.

5.3. Pilot Test 2: System Robustness

In pilot test 2, the test cycle is shown in Figure 10 where the throttle position changes from 0 to 50% throttle. This change can be viewed as a disturbance. In this test, the lambda value is needed to control within the ±5% bounds of the stoichiometric value (1.0). After choosing the sampling time to be 0.005 s, the effectiveness of the RVMMPC can be examined. The parameters of the optimizer were chosen as 𝑁₁=1,𝑁₂=10,𝜌=0.5,and𝑁_𝑢=3 again. The lambda control result and the fuel injection time of the RVMMPC are shown in Figures 11(a) and 12(a), respectively. Similar to pilot test 1, the control result is compared with the two model predictive controllers, DLSSVMMPC and DRNNMPC, as well as typical PI controller. The control results and the corresponding fuel injection time of the DLSSVMMPC, DRNNMPC, and PI controller are shown in Figures 11(b)–11(d) and 12(b)–12(d), respectively. Figure 11(a) shows that the RVMMPC can regulate the lambda to the target lambda with the smallest deviation (i.e., minimum LTE) in the steady state among all the controllers. Moreover, the RVMMPC can also achieve the smallest overshoot in the transient state among all the controllers. These results show the superior robustness of the RVMMPC. The control performances of the four controllers are shown in Table 4, and the control performance of the RVMMPC in terms of LTE is superior to the DLSSVMMPC, DRNNMPC, and PI controller by 3%, 6%, and 35%, respectively.

Table 4 

Control performance of pilot test 2 of different controllers.

Figure 10 

Throttle position against time in pilot test 2.

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Figure 11 

Lambda control results of pilot test 2 of (a) RVMMPC, (b) DLSSVMMPC, (c) DRNNMPC, and (d) PI controller.

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Figure 12 

Fuel injection time of pilot test 2 of (a) RVMMPC, (b) DLSSVMMPC, (c) DRNNMPC, and (d) PI controller.

5.4. Pilot Test 3: Online Update Ability

In the pervious two tests, the “bypass air valve” of the test engine is 60% opening. In order to test the update ability of the RVMMPC, the “bypass air valve” was set from 60% to 30%. It is equivalent to the clogging of the engine intake filter as the engine aging. Normally, the lambda value must decrease under the same throttle position as lack of intake air. The testing procedure and the parameter setting are exactly the same as those in pilot test 2 except the change of the “bypass air valve” position. The variation of throttle position in the test cycle is shown in Figure 13. The lambda control result and the fuel injection time of the RVMMPC are shown in Figures 14(a) and 15(a), respectively. Similar to pilot tests 1 and 2, the control result is compared with the two model predictive controllers, DLSSVMMPC and DRNNMPC, as well as typical PI controller. The control results and the corresponding fuel injection time of the DLSSVMMPC, DRNNMPC and PI controller are shown in Figures 14(b)–14(d) and 15(b)–15(d) respectively. Figures 14(a)–14(d) depict the lambda values decrease after changing the position of the “bypass air valve” due to fuel rich. Figures 14(a)–14(c) illustrate that the three controllers, RVMMPC, DLSSVMMPC, and DRNNMPC, can regulate the lambda with obviously less deviation from the stoichiometric value and overshoot than that of the PI controller because the engine lambda model can be self-updated for any changes in engine condition. The control performances of the four controllers are shown in Table 5, and the control performance of the RVMMPC in terms of LTE outperforms the DLSSVM, DRNNMPC, and PI controller by approximately 6%, 10%, and 62%, respectively. This promising result indicates that the RVMMPC can regulate air ratio very well even the engine ages and undergoes external disturbance simultaneously.

Table 5 

Control performance of pilot test 3 of different controllers.

Figure 13 

Throttle position against time in pilot test 3.

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Figure 14 

Lambda control results of pilot test 3 of (a) RVMMPC, (b) DLSSVMMPC, (c) DRNNMPC, and (d) PI controlle.

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Figure 15 

Fuel injection time of pilot test 3 of (a) RVMMPC, (b) DLSSVMMPC, (c) DRNNMPC, and (d) PI controller.

5.5. Discussion of Results

All the experimental results show that the overall lambda control performance of the RVMMPC is better than those of DLSSVMMPC, DRNNMPC, and conventional PI controller. There are two important factors affecting the control performance of the model predictive controllers which are the lambda model accuracy, and computational time. As presented in Sections 2.1 and 4.2, the sparseness, model accuracy and model updating time of RVM are better than those of DLSSVM and DRNN. Therefore, the model accuracy, and computational time of the RVMMPC is the best among the three controllers. Overall, the RVMMPC is the most suitable method for engine air-ratio control. Although Tables 3 to 5 show that the logarithmic tracking errors of the RVMMPC have a small improvement only, and the lambda is a very delicate value, a small change can result in a big change of engine performance, so the LTE is not very large. For instance, the air ratio is changed from 1 to 1.05; an increment of 0.05 results in changing from the best emission performance to the best brake-specific fuel consumption. Besides, [2, 3] stated that if AFR is 1% lower than its stoichiometric ratio, CO and HC emissions are significantly increased. An AFR that is 1% higher than the stoichiometric ratio produces more NO_x, up to 50%. As a result, the actual improvement achieved by the RVMMPC is very significant.

6. Conclusions

This research is the first attempt at developing RVMMPC for engine air-ratio control. This study is also the first to extend DLSSVM to MPC for engine air-ratio control. The RVM is trained by sequential learning algorithm and real engine data, which can let the engine lambda model update continually so as to maintain the lambda model accuracy for any changes in engine performance as the engine aging or fair user modification on it. With the highly accurate engine lambda model, the new MPC strategy can perform lambda control effectively. The proposed intelligent control algorithm was successfully implemented and tested on a real automotive engine, whereas many previous researches were simulation tests only. Experimental results show that the lambda control performance of the RVMMPC is better than of DLSSVMMPC, DRNNMPC and conventional PI controller. Tables 3 to 5 reveal that the RVMMPC can effectively reduce the lambda deviation and overshoot from target lambda value up to 62% and 67%, respectively. Thus, the RVMMPC is a potential control scheme to replace conventional PI controller in the automotive ECU for engine lambda control. In the future, some advanced optimization algorithms for the RVMMPC and system stability will be studied.

Acknowledgments

This work is supported by the University of Macau Research Grant, Grant no. MYRG149(Y1-L2)-FST11-WPK. The authors would also like to thank the support from Mr. Adrian Kowalski.