Stochastic stability for a model representing the intake manifold pressure of an automotive engine

Abstract: The paper presents conditions to assure stochastic stability for a nonlinear model. The proposed model is used to represent the input-output dynamics of the angle of aperture of the throttle valve (input) and the manifold absolute pressure (output) in an automotive spark-ignition engine. The automotive model is second moment stable, as stated by the theoretical result—data collected from real-time experiments supports this finding.


Introduction
In combustion engines, the correct adjustment of the air-fuel (A/F) ratio of the gas mixture used into the combustion chamber is of foremost importance (Guzzella & Onder, 2010, Sec. 2.7;Stotsky, 2009, Sec. 2.3). The ratio chosen for the mixture must be set to the stoichiometric ratio of the corresponding fuel as it influences, for instance, the efficiency of the engine's catalytic converter and sets the level of emission pollutants, see the monograph (Heywood, 1988) for further details.
Adjustment of the A/F ratio is done in closed loop mode via an oxygen sensor installed at the exhaust duct. For a given air mass, the controller calculates the fuel amount based on stoichiometry (Guzzella & Onder, 2010, Sec. 4.3). Thus the A/F ratio depends primarily on the air mass flow in the intake manifold, see the diagram in Figure 1. The air entering into the intake manifold crosses the ABOUT THE AUTHORS Alessandro N. Vargas

PUBLIC INTEREST STATEMENT
The paper presents a nonlinear model for representing the relation between the angle of aperture of the throttle valve (input) and the manifold absolute pressure (output) in sparkignition engines. Besides, the paper shows that the model is stable provided that some mild conditions are satisfied. Experimental data support the main findings.
throttle, a device with a circular plate that is used to adjust the mass air flow. The air flows into the intake manifold and then through the cylinder intake runners where the fuel injectors are installed. The up and down movement of the pistons creates a vacuum in the intake manifold and the A/F mixture flows into the cylinders. This vacuum can be measured through a sensor called Manifold Absolute Pressure (MAP). The pressure value indicated by the MAP sensor is of key importance, since one can easily calculate the mass air flow through the well-known speed-density Equation (Guzzella & Onder, 2010, Ch. 2;Stotsky, 2009, Equation (2.1), p. 16). This paper contributes towards this problem by presenting a stochastic nonlinear model that aims to characterize the input-output relation between the angle of aperture of the throttle device and the MAP value, as detailed in the sequence.
The literature is rich for models that aim to characterize the dynamics of spark-ignition combustion engines. Nowadays the most accepted model comes from the seminal work (Hendricks & Sorenson, 1990), which was the first to coin the term mean value engine model (MVEM). Although suffering from a certain level of empiricism (Hendricks, 1997, p. 389), the MVEM model is quite complete and general for modeling the three main spark-ignition engine subsystems, i.e. intake manifold, crank shaft, and fuel supply subsystems (Balluchi, Benvenuti, Di Benedetto, Pinello, & Sangiovanni-Vincentelli, 2000;Casavola, Famularo, & Gagliardi, 2013;Guzzella & Onder, 2010, Ch. 2;Hendricks, 1997;Hendricks & Sorenson, 1990;Tang, Weng, Dong, & Yan, 2009). But this general model imposes a drawback for the analysis of any subsystem in particular because it requires all subsystems working together in a strongly nonlinear, coupled fashion. Consequently determining the specific throttle-MAP relation through the MVEM model requires the knowledge of all subsystems, which is a difficult task-identifying the MVEM model requires simultaneous measurements of many sensors (Fleming, 2001) and can be time-consuming (Hendricks, 1997, p. 391). An advantage of our approach is that it aims to simplify this task, i.e. the throttle-MAP relation is studied here in the viewpoint of a unique single-input single-output relation, and for this reason our approach is completely detached from the ones in the literature. Finding a model for the throttle-MAP relation represents the main practical contribution of this paper.
Indeed, our main practical contribution is to propose a stochastic nonlinear model for the throttle-MAP relation. To account this relation, we adopt a structure borrowed from the nonlinear ARX model (Ljung, 1999, Ch. 5;Pearson, 2003;Sjöberg et al., 1995), as follows. Let y k be a real-valued variable representing the output, measured by an appropriate device attached to the system, corresponding to the input values (u 0 , u 1 , … , u k ) applied in the system until the k-th stage. If we let for some integer d ≥ 0, then the identification problem consists of finding a function f (⋅) such that (1) where {W k } represents some finite-dimensional stochastic process.
The exact format of the function f (⋅) is usually unknown. As a first attempt, one may resort to linear maps (Ortner & del Re, 2007). Nonlinear maps, though, expand the throttle-MAP representation-one may check whether the real-time measured data resembles the simulated one produced by some benchmark some nonlinear functions (Ljung, 1999, Ch. 5;Sjöberg et al., 1995). This strategy is accounted in our study, as detailed next.
From the theoretical point of view, the main contribution of this paper is to present a sufficient condition to the second moment stability of the stochastic nonlinear system in (2), under some as- For sake of completeness, let us recall the definition of such stability concept (see also Li, Sui, Tong, 2016;Wu, Cui, Shi, & Karimi, 2013;Yin, Khoo, Man, & Yu, 2011;Zhao, Feng, Kang, 2012 for further details).
Definition 1.1 Arnold (1974, p. 188), Vargas and do Val (2010) We say the nonlinear stochastic system in (2) is second moment stable if there exists a constant c > 0 (which may depend on the initial values (y 0 , … , y d )) such that where E[⋅] represents the expected value operator.
A conclusion drawn from our findings is that the identified throttle-MAP stochastic model, obtained from real-time experiments and written as in (2), is second moment stable. This fact sets the practical benefit of this paper.
The paper is organized as follows. Section 2 quotes the notation, definitions, and presents the stability result. Section 3 presents the experiments that were carried out to obtain a stochastic model for the throttle-MAP nonlinear relation. Finally, Section 4 presents some concluding remarks.

Basic definitions and main result
Let us denote the n-dimensional Euclidean space by ℝ n and the corresponding norm by ‖ ⋅ ‖; the set made up by matrices of dimension n × m is denoted by ℝ n×m . An element x from ℝ n is denoted by Denoting the symbol ⊗ by the Kronecker product, we recall that ( where M, N, and V are matrices of compatible dimensions (Brewer, 1978).
Hereafter, we assume that the system in (2) is governed by (Ω,  , { k }, P), a fixed filtered probability space, and that the input sequence {u k } takes values in ℝ. In addition, we assume that {W k } on ℝ represents an independent and identically distributed (i.i.d.) stochastic process with null mean and covariance matrix identical to the identify. (2) The authors of (Sjöberg et al. 1995, Sec. 4.1) suggest the use of both sigmoid and Gaussian bell functions as candidates for representing the nonlinear term in (2). This idea motivated us to sum these two functions in order to obtain the function g:ℝ s ↦ ℝ as where i and i are s-dimensional constant row vectors; a i , b i , c i , d i , e i are real-valued constants; and n is some finite integer.
Remark 1 It follows from the definition of g in (3) that Thus the function g in (3) is bounded.
Setting some row vectors H i ∈ ℝ 1×s , i = 1, … , , and A ∈ ℝ 1×s , we can define the function f :ℝ s × ℝ ↦ ℝ as in (2) as follows: With f as in (4), the nonlinear stochastic system (2) now reads at the k-th stage as Before presenting the main theoretical contribution of this paper, let us introduce some additional notation. Consider the matrices Theorem 2.1 Assume that the input sequence {u k } on ℝ is bounded. Then the stochastic nonlinear system (5) is second moment stable if all the eigenvalues of the matrix lie inside the unit circle.
Remark 2 The result of Theorem 2.1 allows us to check whether a nonlinear stochastic system as in (5) is second moment stable through a simple numerical evaluation. The result can then be used to check whether an identification procedure is successful on generating a stable model. This property is illustrated in this paper for a real-time automotive engine, as described in Section 3.

Proof of Theorem 2.1
Proof Before presenting the main argument for the proof of Theorem 2.1, we need some preliminary results. Consider the identify Similarly, for each i = 1, … , , we can write Summing up for i the elements inside the rightmost curly brackets of (8), and adding in this evaluation the term the rightmost curly brackets of (7), we obtain a random value, say (Y k , W k ). We have immediately from (5) that Applying in (9) the matrices ,  i , i = 1, … , , as defined in (6), we obtain the next identity: Notice that (5) and (10) are equivalent. Now, for sake of notational simplicity, let us fix With this notation, (10) is identical to z k+1 =  k z k +  k , which in turn is identical to Let us now introduce a result, necessary here to continue with the argument.  (11); this assures that the stochastic nonlinear system (5) is second moment stable according to Definition 1.1.
Applying the Euclidean norm on both sides of (11), and passing the expected value operator, we have

Model for the intake manifold pressure: experimental approach
The laboratory testbed used in the experiments was equipped with an automotive engine mock-up, Model Volkswagen AT EA-111 RSH 1.0 Total Flex, with four cylinders and eight valves, used in the cars manufactured by Volkswagen models VW Gol and VW Fox (see Figure 2). This engine can be fueled with gasoline or ethanol, or if needed with any mixture of them, but the experiments were carried out with ethanol only.
In our experiments, we were interested in finding a single-input single-output model to characterize the relation between the throttle valve (input) and pressure MAP sensor (output). To obtain measurements for both input and output, we used a data acquisition card model NI-USB 6008 to measure the voltages informed by the two corresponding sensors. The engine has built-in sensors for both variables, that is, the pressure is measured by the MAP sensor ( Figure 1) and the angle of aperture of the throttle is measured by its corresponding sensor assembled in the valve plate. The sampling time of the data acquisition card was kept fixed at 5 milliseconds. For this investigation, the engine was operated at no load condition.
By pressing manually the gas pedal, we generated a movement in the angle of the throttle valve and this induced a dynamics for the intake manifold pressure, see Figure 3 for a pictorial representation. Part of this experimental data was selected to identify the model as in (5) whereas the remaining data was used for model validation. In this study, we used a least square routine with no noise input in the model in (5) The values of the identified parameters of (5) are omitted here for sake of brevity, but for an account we mention that d = 5 in (1) (quantity of regressors) and n = 8 in (3).
The identified model as in (5) with no noise input presented a fit value of 67% when compared with real data. However, although the combustion engine is a highly complex system, the fit of 67% is a positive indication that confirms the difficulty on characterizing precisely the throttle-MAP relation; such lack of precision is recurrent in the literature (Alberer, Hirsch, & del Re, 2010;Ortner & del Re, 2007;Wahlström & Eriksson, 2011).    As an attempt to improve the throttle-MAP relation, we assumed that {W k } in (5) is a standard Gaussian white noise processes. And to illustrate the influence of such noise in the proposed throttle-MAP model, we generated a Monte-Carlo simulation with eight hundred realizations. Interestingly, the real data lies within the feasible region covered by the simulated data, as can be seen in Figure 4. This evidence suggests that (5) can be a candidate for representing the throttle-MAP dynamics.
Finally, a question of practical interest is to determine whether the identified throttle-MAP model as in (5) is second moment stable. The identified parameters to be checked are (recall that d = 5) and each value in H [j],i , j = 1, … , 13, i = 1, … , 6, assumes 0.05 when i = j and zero elsewhere.
Using these values to generate the matrix  ⊗  + ∑ i=1  i ⊗  i (e.g. (6)), we can conclude that its eigenvalues are located inside the unit disc ( Figure 5). Hence, Theorem 2.1 guarantees that the identified throttle-MAP model is second moment stable. This illustrates the practical usefulness of Theorem 2.1.

Concluding remarks
We have presented a simple numerical condition to check the second moment stability of singleinput single-output stochastic nonlinear systems. Such a stochastic system can be accounted to represent the throttle-MAP dynamics of a spark-ignition engine fueled with ethanol.
Data taken from real-time experiments suggest that throttle-MAP process is second moment stable, an evidence that corroborates the theoretical result.
As further investigation goes on, control of the MAP can be a useful for vehicle platooning (Liang, Måartensson, Johansson, 2016).