- Email: [email protected]
Combustion Engine Air Intake Theoretical Modelling, Model-Verfication & Application to Optimal Valve Actuation
Combustion Engine Air Intake Theoretical Modelling, Model-Verfication & Application to Optimal Valve Actuation S. Studener ITK Engineering AG, Lochhamer Straße 15 D-82152 Martinsried, Germany (e-mail: [email protected]). Abstract: A theoretical model for the air intake of a four stroke internal combustion engine is derived from first principles, the conservation of mass and the conservation of energy. Full valve actuation is taken into account: The opening angle of both the intake as well as the exhaust valve, as well as the variable valve lift are considered as plant inputs, which may be manipulated to maximize the air intake rate and hence the power, the engine delivers. The theoretical model is verified by comparing the model output to steady state data collected at an engine test bench. Therefor a formal connection between state-of-the-art mean value models and the crank angle resolved physical model must be established. The pressure loss in the intake valve is considered a disturbance of the ideal cylinder filling dynamics, which is estimated using non-linear optimization techniques. The intended purpose of physical model building for the cylinder air intake is prediction of the air intake rate and computation of optimal valve control action before a lot of effort is put into a real experiment at the engine test bench. A significant reduction of operating time and cost consumed by the real engine test bench can be achieved, if a small subset of steady state measurements are used to compute a meta-model for the pressure loss. The combination of the black-box pressure loss model and the air intake model allows for prediction of the air intake rate at unknown operating points of the engine with sufficient accuracy.
Keywords: Automotive control, computer-aided control system design, engine modelling, engine control 1. INTRODUCTION Positive displacement machines with internal combustion are one of the most important technological developments of the past century. They have become the most frequently encountered source of propulsion energy in passenger cars. One reason for this is, that this source of propulsion energy does rely on the large energy density of liquid hydrocarbons, which is very hard to be be beaten for by alternative propulsion concepts. In spite of all the research efforts that have been made, internal combustion engines still have potential for improvements, which are hard to be realized without model based control and optimization. From a control engineer’s point of view, the positive displacement machine with internal combustion is a non-linear, time periodic plant. It is the the time periodicity, whose frequency varies with the number of revolutions, that makes the application of contemporary control techniques a challenge (see Guzzella and Onder (2010) for an introduction to control and estimation of the internal combustion engine). In this contribution, we propose a model for the air intake of a positive displacement machine, that is derived from first principles - the conservation of mass and energy in the cylinders. Moreover, we introduce a disturbance of the ideal cylinder filling dynamics and provide a way to rapidly estimate the disturbance for a given operating point of the engine. The
exhaust manifold 𝑚 ˙ 𝑐𝑦𝑙 𝑝𝑒𝑚 𝑇𝑒𝑚
intake manifold 𝑚 ˙ 𝑖𝑚 𝑝𝑖𝑚 𝑇𝑖𝑚
ℎ𝑉,𝑖 𝑚𝑎𝑖𝑟 𝑈𝑎𝑖𝑟
𝑠0 𝑠
𝑙 𝑟 crank, angle 𝜙, 𝜙˙ = 𝜔 Fig. 1. Illustration of the crank and piston kinematics and the thermodynamic states governing the air intake. internal combustion engine we have in mind takes four strokes to complete a cycle: Air admission, compression, fuel injection followed by combustion and rejection of the piston and finally air cleaning. The amount of air, that is aspirated by the engine’s cylinders, limits the amount of fuel, that can be burned inside and hence limits the power,
the engine is able to deliver (see Basshuysen and Schaefer (2010); Pischinger et al. (2009) for thermodynamics of the combustion engine). Many applications being processed on an electronic engine control unit resign to disclose those dynamics, that are associated with the variation of the crank angle and use so called mean value models to represent the low-frequency range behavior of key engine parts. An example for such a mean value model is the air intake rate of the engines cylinder, but also the engine characteristic map, giving the momentum delivered to the crank as a function of the number of revolutions, injection angle, pressure in the intake manifold and other physical quantities in closed loop (see Isermann and Muenchhof (2011)). Even most recent developments on motor control, such as the works of Jankovic and Magner (1999), Khiar et al. (2006), Leroy et al. (2007), Palladino et al. (2008) or Storset et al. (2000) use a high order polynomial in order to formally represent the amount of air, that is aspirated by the engine at a given operating point. The parametrization of these polynomials must be carried out at the engine test bench. Due to the fact, that these meta-models must represent the low-frequency behavior of the time-periodic plant, long measurement times must be accepted, which can be associated with high development cost. In the following section, the dynamics of the air intake are derived from first principles. A disturbance of the ideal cylinder filling dynamics is defined, which is estimated from experimental data using non-linear optimization techniques in section 3. The order of magnitude at which the estimated disturbance settles, proves to be plausible. In section 4, it will be shown, that a small subset of steady state operating points of the combustion engine is sufficient to derive (i) a meta-model for the disturbance and (ii) predict the air intake rate of the engine over a large range of operating conditions, if the theoretical air intake model and a proper black-box model for the disturbance is used. In this section, the act of proper meta-modeling of the disturbance is briefly discussed. The model allows for a priori computation of optimal valve actuation. This point is addressed in section 5. In the final section directions for future research are discussed. 2. THEORETICAL AIR INTAKE MODEL The state of the cylinder is completely defined by the mass of air and the internal energy of the air contained in the cylinder, denoted by 𝑚𝑎𝑖𝑟 and 𝑈𝑎𝑖𝑟 respectively. The filling and emptying dynamics of the cylinder are fast compared to the dynamics governing the intake and exhaust manifold, indicated by circles in figure 1. Hence, we can assume, that these temperatures and pressures do remain approximately constant during a cycle composed of air intake, compression, combustion followed by rejection of the piston and finally air-cleaning. Throughout this work, we assume, that the state of the air in the cylinder follows the ideal gas law, 𝑝𝑎𝑖𝑟 𝑝𝑎𝑖𝑟 ⋅ 𝑉𝑐𝑦𝑙 = 𝑚𝑎𝑖𝑟 ⋅ 𝑅 ⋅ 𝑇𝑎𝑖𝑟 , or 𝜌𝑎𝑖𝑟 = (1) 𝑅 ⋅ 𝑇𝑎𝑖𝑟 The state variables relate to the temperature as follows: 𝑅 ⋅ 𝑈𝑎𝑖𝑟 (2) 𝑈𝑎𝑖𝑟 = 𝑚𝑎𝑖𝑟 ⋅ 𝑐𝑣 ⋅ 𝑇𝑎𝑖𝑟 → 𝑝𝑎𝑖𝑟 = 𝑉𝑐𝑦𝑙 ⋅ 𝑐𝑣
In the Equation above, 𝑝𝑎𝑖𝑟 denotes the pressure in the cylinder, 𝑉𝑐𝑦𝑙 denotes the cylinder volume. 𝑅 is the gas constant and 𝑐𝑣 the heat capacity (constant volume). As illustrated in figure 1, 𝑚 ˙ 𝑖𝑚 is the mass flow from the intake manifold into the cylinder and 𝑚 ˙ 𝑐𝑦𝑙 is the mass flow from the cylinder into the exhaust manifold. A mass balance yields the following equation: 𝑑𝑚𝑎𝑖𝑟 ˙ 𝑖𝑚 − 𝑚 ˙ 𝑐𝑦𝑙 (3) =𝑚 ˙ 𝑎𝑖𝑟 = 𝑚 𝑑𝑡 The physical quantity of interest in this work, denoted by 𝑦, is the mass flow 𝑚 ˙ 𝑖𝑚 , that is aspirated by a single cylinder during the admission phase of the four stroke combustion engine cycle. It is given by the product of the density in the intake manifold, 𝜌𝑖𝑚 , the intake valve’s cross-section area, 𝑆𝑉,𝑖 , and the velocity at the intake valve, 𝑣𝑉,𝑖 : 𝑦=𝑚 ˙ 𝑖𝑚 = 𝜌𝑖𝑚 ⋅ 𝑆𝑉,𝑖 ⋅ 𝑣𝑉,𝑖 (4) The valve’s cross-section area depends on the intake valve 2
dead center
1.8 1.6
exhaust valve
intake valve
1.4
𝑢4
1.2
𝑢2
1 0.8 0.6
ℎ𝑉,𝑖 𝑢1
0.4 0.2 0 −6
−4
−2
0
𝜙 (in 𝑟𝑎𝑑)
2
4
6
Fig. 2. Illustration of the valve actuation, as defined in equations (6) and (9). 𝑢2 and 𝑢4 denote the phase shifts of the valves. The intake valve amplitude ℎ𝑉,𝑖 is displayed for various control inputs 𝑢1 . lift ℎ𝑉,𝑖 and the diameter of the intake valve 𝑑𝑉,𝑖 : (5) 𝑆𝑉,𝑖 = 𝜋 ⋅ 𝑑𝑉,𝑖 ⋅ ℎ𝑉,𝑖 In contemporary combustion engines the course of the valve lift ℎ𝑉,𝑖 = ℎ𝑉,𝑖 (𝑡) may be manipulated using electromechanical actuation (see Basshuysen and Schaefer (2010) for details on electro-mechanical valve actuation). The course of the intake valve lift is given as a function of phase shift, 𝑢2 , crank angle, 𝜙, and amplitude 𝑢1 : ℎ𝑉,𝑖 (𝑡) = 𝑢1 (𝑡) ⋅ 𝑓𝑖 (𝜙(𝑡) − 𝑢2 (𝑡)) , where 𝜙˙ = 𝜔 (6) 𝜔 is the engine’s rotational speed and 𝑓𝑖 is a 4 ⋅ 𝜋periodic, analytic function, illustrated in figure 2 1 . Applying Bernoullis Law, we obtain an expression for the velocity 𝑣𝑉,𝑖 : ⎧√ ⎨ 2 (Δ𝑝𝑖𝑚 − 𝑝𝑙𝑜𝑠𝑠,𝑖 ) , Δ𝑝𝑖 > 𝑝𝑙𝑜𝑠𝑠,𝑖 𝑣𝑉,𝑖 = (7) ⎩ 𝜌𝑖𝑚 0 , else 1 In the figure dimensionless quantities are displayed, the physical ranges of the inputs belong to confidential matter and cannot be published.
where Δ𝑝𝑖𝑚 = 𝑝𝑖𝑚 − 𝑝𝑎𝑖𝑟 . 𝑝𝑙𝑜𝑠𝑠,𝑖 is the pressure loss of the inlet valve. It may be given the interpretation of a disturbance of the ideal cylinder filling dynamics: The flow through any real valve is subject to a pressure loss, consequently 𝑑 = 𝑝𝑙𝑜𝑠𝑠,𝑖 = 0 characterizes the ideal cylinder filling dynamics, which yield the largest mass flow at the valve possible for a given initial state of the cylinder, set of inputs u and states of the manifolds. Any real cylinder shows a pressure loss 𝑑 > 0, depending on valve lift and other quantities, as we will see in the following. The mass flow into the exhaust manifold is given by the following equation: 𝑚 ˙ 𝑐𝑦𝑙 = 𝜌𝑎𝑖𝑟 ⋅ 𝑆𝑉,𝑒 ⋅ 𝑣𝑉,𝑒 , 𝑆𝑉,𝑒 = 𝜋 ⋅ 𝑑𝑉,𝑒 ⋅ ℎ𝑉,𝑒 (8) 𝑑𝑉,𝑒 is the diameter of the exhaust valve and the valve lift ℎ𝑉,𝑒 is given by the product of the input 𝑢3 and a 4 ⋅ 𝜋−periodic function, ℎ𝑉,𝑒 (𝑡) = 𝑢3 (𝑡) ⋅ 𝑓𝑒 (𝜙(𝑡) − 𝑢4 (𝑡)) (9) The exhaust valve velocity 𝑣𝑉,𝑒 is given by the following equation: ⎧√ ⎨ 2 (Δ𝑝𝑐𝑦𝑙 − 𝑝𝑙𝑜𝑠𝑠,𝑒 ) , Δ𝑝𝑐𝑦𝑙 > 𝑝𝑙𝑜𝑠𝑠,𝑒 (10) 𝑣𝑉,𝑒 = 𝜌 𝑎𝑖𝑟 ⎩ 0 , else where Δ𝑝𝑐𝑦𝑙 = 𝑝𝑎𝑖𝑟 − 𝑝𝑒𝑚 . The pressure loss of the exhaust valve 𝑝𝑙𝑜𝑠𝑠,𝑒 has an impact on the cylinder emptying dynamics. Hence, it determines, how much waste gas is trapped in the cylinder after combustion and air cleaning. The pressure loss of the exhaust valve can be computed once a combustion model is available and the pressure after combustion is known. The error, that is associated with setting the exhaust valve pressure loss equal to zero is considered to be minor, since the exhaust valve is nearly closed at top dead center, which is the point in time, at which the air intake simulation is initialized (𝜙(𝑡 = 0) = 0 and 𝜙˙ = 𝜔). An energy balance yields the following ordinary differential equation (ODE) for the internal energy of the gas: 𝑑𝑈𝑎𝑖𝑟 = 𝑈˙ 𝑎𝑖𝑟 = 𝐻˙ − 𝑝 ⋅ 𝑉˙ 𝑐𝑦𝑙 (11) 𝑑𝑡 where the flow of enthalpy is given by the following expression: ˙ 𝑐𝑦𝑙 ⋅ 𝑐𝑝 ⋅ 𝑇𝑎𝑖𝑟 (12) 𝐻˙ = 𝑚 ˙ 𝑖𝑚 ⋅ 𝑐𝑝 ⋅ 𝑇𝑖𝑚 − 𝑚 In the following, we assume, that the pressure and the temperature of the air in the cylinder at the beginning of the air intake (𝜙(𝑡 = 0) = 0, the top dead center) is equal to the pressure and temperature of the exhaust manifold. This assumption allows for definition of reasonable initial conditions (ICs) for the mass balance, eq. (3), and the energy balance, eq. (11): 𝑚𝑎𝑖𝑟 (𝑡 = 0) = 𝑈𝑎𝑖𝑟 (𝑡 = 0)/(𝑐𝑣 ⋅ 𝑇𝑒𝑚 ) and 𝑈𝑎𝑖𝑟 (𝑡 = 0) = 𝑝𝑒𝑚 ⋅ 𝑉𝑐𝑦𝑙 (𝑡 = 0) ⋅ 𝑐𝑣 /𝑅. In order to complete the model for the aspiration of air, the kinematics of the crank and the piston need to be described . These are well known, see e.g. Basshuysen and Schaefer (2010) or Pischinger et al. (2009). Let 𝑠(𝑡) be the position of the piston in the cylinder and 𝑠(𝑡 = 0) = 0 be the position at crank angle 𝜙 = 0 (see illustration 1). 𝑉0 is the volume of the cylinder at the top dead center. 𝑉𝑐𝑦𝑙 is given by the following relationship: (13) 𝑉𝑐𝑦𝑙 (𝑡) = 𝑠(𝑡) ⋅ 𝐴𝑝 + 𝑉0 → 𝑉˙ 𝑐𝑦𝑙 = 𝑠˙ ⋅ 𝐴𝑝 𝐴𝑝 is the surface of the piston and 𝑠(𝑡) is related to the crank angle as follows:
) ( 1 𝑟 (14) 𝑠(𝑡) = 𝑟 ⋅ 1 − cos (𝜙(𝑡)) + ⋅ ⋅ (sin (𝜙(𝑡)))2 2 𝑙 The ODEs for the mass of air and the internal energy may be written as a coupled, nonlinear, time-variant system of ODEs: T x˙ = f (x, u, p, 𝑑, 𝑡) , x = [ 𝑚𝑎𝑖𝑟 𝑈𝑎𝑖𝑟 ] (15a) T
T
u = [ 𝑢1 ... 𝑢4 ] , p = [ 𝜔 𝑝𝑖𝑚 𝑝𝑒𝑚 𝑇𝑖𝑚 𝑇𝑒𝑚 ] (15b) with an output equation 𝑦(𝑡) = 𝑔 (x, u, p, 𝑑) given by eq. (4). The set of ODEs must be integrated up to the point in time the intake valve is closed, i.e. ℎ𝑉,𝑖 = 0, in order to capture the whole mass of air, that is aspirated during a cycle. Definition: Mean Value Model. The positive displacement machine with internal combustion is a non-linear, time periodic plant. Consequently, as 𝑡 → ∞, we can expect the output to be time periodic, i.e. 𝑦(𝑡) = 𝑦(𝑡 + 𝑇 ), where 𝑇 = 4 ⋅ 𝜋/𝜔 (with the engine rotating at a constant pace equal to 𝜔). A mean value model is a relationship between the low frequency content of the time periodic signal 𝑦(𝑡) = 𝑦(𝑡 + 𝑇 ) and the inputs, parameters and disturbances of the plant or, more precisely, a mapping from the space of inputs, parameters and disturbances to the DC component of the signal 𝑦(𝑡) = 𝑦(𝑡 + 𝑇 ), denoted as 𝑦: ∫𝑇 ∫𝑇 1 1 𝑦 (u, p, 𝑑) = 𝑦(𝜏 ) 𝑑𝜏 = 𝑔 (x, u, p, 𝑑) 𝑑𝜏 (16) 𝑇 𝑇 0
0
Remark. In practice, the mean value of the aspirated air mass is determined by low pass-filtering the signal provided by the mass flow sensor, which comes along with a requirement for long measurement time and hence large operating cost of of the engine test bench. 3. DISTURBANCE ESTIMATION In this section, the theoretical model will be verified, by comparing the model output 𝑦 to steady state air intake rates of a six-cylinder combustion engine, which are measured at an engine test bench. As expected, the mean mass flow 𝑦 predicted by the nominal plant (𝑑 = 0) is larger than the mass flow, that is observed at the engine test bench. This is illustrated in figure 3. The real cylinder filling dynamics are subject to a pressure loss (𝑑 > 0). The pressure loss of the real cylinder is estimated from steady state operating points, by solving the following optimization problem: 1 1 2 min Φ(𝑑) = ⋅ 𝑒2 = ⋅ (𝑦 (u, p, 𝑑) − 𝑦 𝑚𝑒𝑎𝑠 ) (17) 2 2 Clearly, 𝑦(𝑑) depends on the engine rotational speed, the pressure and temperature in the manifolds and the valve actuation. These quantities are available and some are displayed in figure 3. State-of-the-art optimization techniques, such as the Levenberg-Marquardt Algorithm (LMA) require the gradient and the hessian of the objective function to be provided with sufficient accuracy (see More (1978) or Nocedal (2006)). Differentiation of the objective function w.r.t. 𝑑 allows for determination of the gradient and the hessian: ( )2 𝑑Φ ∂𝑦 𝑑2 Φ ∂𝑦 ∂2𝑦 =𝑗 =𝑒⋅ = ℎ = + 𝑒 ⋅ (18) 𝑑𝑑 ∂𝑑 𝑑𝑑2 ∂𝑑 ∂𝑑2 negligible
The partial derivative of the mean mass flow w.r.t. 𝑑 is ∫𝑇 ∫𝑇 [ ]T ∂𝑔 ∂𝑦 ∂𝑔 ∂x ∂𝑔 1 1 ⋅ = 𝑑𝜏 = + 𝑑𝜏 (19) ∂𝑑 𝑇 ∂𝑑 𝑇 ∂x ∂𝑑 ∂𝑑
ideal filling, i.e. 𝑑 = 0 real filling, i.e. 𝑑 = 𝑑∗ prediction error (KP)
0
The partial derivatives of the state variables w.r.t. 𝑑 are computed by solving the following, augmented system of ODEs for the cylinder filling dynamics (compare to eq. (15a) and (15b)): f (x, u, p, 𝑑, 𝑡) x˙ (20) = ∂f ∂f ∂x 𝑑 ∂x + ⋅ ⋅ ∂x x,u,p,𝑑,𝑡 ∂𝑑 ∂𝑑 x,u,p,𝑑,𝑡 𝑑𝑡 ∂𝑑
𝑦/𝑦 𝑟𝑒𝑓
0
1
0.5
0 50
4. META-MODELING THE PRESSURE-LOSS AND GENERALIZATION OF THE MODEL This section addresses the important point of reducing operating time of the engine test bench and hence development cost of engine control systems by using physical models and meta models for prediction of the air intake rate over a wide range of operating conditions. Without a reasonable pressure loss 𝑑, the theoretical intake model is of limited value, the mean intake rate may exceed the real intake rate by up to a hundred percent (depending on the operating point). Since it’s operation time is expensive, one may want to capture only a very small number of steady state operating points at the engine test bench and predict the remaining points using a model. In the following, twenty percent of the measurements available (see figure 3) are used to compute the pressure loss of the cylinder using the optimization techniques presented in section 3. These results are used to generate a meta-model for the pressure loss 𝑑 = 𝑑 (u, p). Since the prediction error is of sole deterministic nature (no stochastic disturbance, no noise), the authors in Simpson et al. (2001) suggest to train a neural network (NN) or try to find a suitable correlation function for the Kriging Predictor (KP). As opposed to NNs, the KP is an interpolatory meta-model, 2 Dimensionless quantities are displayed for reasons of confidentiality
150
200
250
300
150
200
250
300
2
T
with ICs x(𝑡 = 0) = [ 𝑚𝑎𝑖𝑟 (𝑡 = 0) 𝑈𝑎𝑖𝑟 (𝑡 = 0) ] and T ∂x 0 0 ] . The ODEs are solved using the Forward∂𝑑 𝑡=0 = [ Euler Approach (finite differences in time), with the step size chosen s.t. a single degree crank angle is resolved. The initial guess for the LMA is 𝑑 = 0. 𝑑 is increased in the following iterations, using local information about the hessian (ℎ in eq. (18)) and the gradient (𝑗 in eq. (18)) (see Nocedal (2006) for details), until the norm of the gradient settles below a prescribed tolerance. The optimization procedure is carried out for various operating points and the results are displayed in figure 3 2 ; it can be observed, that the pressure loss is not constant, but varies in a deterministic fashion with the control input 𝑢1 , the intake valve lift. Recalling Bernulli’s Law (pressure loss is proportional to the square of the velocity), one can expect the disturbance to depend on the remaining inputs and parameters as well: 𝑑 = 𝑑 (u, p). Predicting the air intake over a wide range of operating conditions using only a small subset of the measurements available will be the subject of the following section.
100
𝜔/𝜔𝑟𝑒𝑓
1.5
𝑢1
1 0.5 0
50
100
1
𝑑/𝑝𝑟𝑒𝑓 𝑒/𝑝𝑟𝑒𝑓 , prediction error of the KP 0.5
0 50
100
150
200
250
300
steady state operating point index
Fig. 3. Verification of the theoretical air intake model and estimated pressure loss. Top: Mean mass flow (ideal filling and real filling) and prediction error using a meta-model for the pressure loss for prediction of the whole set of operating points. Center: Valve control input 𝑢1 (𝑢3 is kept constant) and engine rotational speed. Bottom: Estimated pressure loss. the prediction error at a design site is equal to zero (see Lophaven et al. (2002) for fundamentals of the KP and an implementation). Using the KP in connection with a suitable correlation function pays respect to the fact, that the experimental acquisition of the air intake rate is expensive and one does want to avoid the creation of blurry information by using a regression model for predicting information that is already available. In this numerical experiment, a Gaussian correlation function is used and the prediction error of the pressure loss using the KP is documented in figure 3 (bottom). Every fifth data point of the available engine test bench data documented in figure 3 has been taken as a design site for the predictor. Consequently, there is a lot of room for improvement by employing a smarter design of experiment (DOE). The ultimate goal is not prediction of the pressure loss, but prediction of the air intake rate 𝑦. Using the meta-model for the pressure loss 𝑑 = 𝑑 (u, p) in the set of ODEs (15a), one is able to predict the air intake rate for any given
operating point defined by the pressures and temperatures in the manifolds, the engine rotational speed and the valve actuation. The resulting prediction error for the real air intake rate is documented in figure 3 (top).
80
5. APPLICATION TO OPTIMAL VALVE ACTUATION
6. CONCLUSIONS A theoretical model for the air intake of a four stroke internal combustion engine has been derived from first Note, that the amplitudes of the valve lifts, 𝑢1 and 𝑢3 belong to confidential matter and cannot be published.
3
𝜆 (in %)
60 50 40 30 6000
4000
2000
1
0
2
3
𝑢2 (in 𝑟𝑎𝑑)
𝜔 (in 𝑟𝑒𝑣 ′ 𝑠 ⋅ 𝑚𝑖𝑛−1 )
Fig. 4. Computer experiment illustrating the functional relationship 𝜆 = 𝜆 (𝜔, 𝑢2 ) for a set of operating conditions defined in table 1.
1.6 1.4 𝑢∗2 (in 𝑟𝑎𝑑)
The power, a combustion engine delivers is proportional to the amount of fuel, that can be burned during a cycle. The amount of fuel, that can be burned is limited by the amount of air, that can be delivered. In this section the issue of ramming as much air as possible into the cylinders of a four stroke combustion engine by means of optimal choice of the control input 𝑢2 is addressed. It is clear, that, in order to maximize the air intake, the amplitude of the valve lift must be chosen as large as possible, consequently, in the following studies 𝑢1 = 𝑢1,𝑚𝑎𝑥 . Clearly, the valve timing influences how much air is aspirated by the engine and must be optimized at the test bench. Here, the expensive experiment carried out at the test bench is replaced by a computer experiment using the theoretical air intake model. In order to proceed, the relative filling must be defined. Definition: Relative Filling. The relative filling 𝜆 ∈ [0 1] can be defined by considering an ideal pump, aspirating a fluid from the intake manifold every other revolution. The mass flow delivered by the ideal pump is 4⋅𝜋 𝑝𝑖𝑚 1 where 𝑇 = (21) 𝑦𝑝𝑢𝑚𝑝 = ⋅ 𝑉𝑑𝑖𝑠𝑝𝑙𝑎𝑐𝑒𝑑 ⋅ 𝑅 ⋅ 𝑇𝑖𝑚 𝑇 𝜔 and 𝑉𝑑𝑖𝑠𝑝𝑙𝑎𝑐𝑒𝑑 = max {𝑉𝑐𝑦𝑙 (𝑡) − 𝑉0 }. The relative filling is the mass of air aspirated by the engine related to the mass of air aspirated by the ideal pump: 𝑦 𝜆= (22) 𝑦𝑝𝑢𝑚𝑝 For any fixed set of parameters p and inputs 𝑢1 = 𝑢1,𝑚𝑎𝑥 , 𝑢3 and 𝑢4 , there exists a phase shift 𝑢2 , maximizing the relative filling (see figure 4 for illustration). An equivalent statement is, that there exists a phase shift 𝑢2 , which solves the following optimization problem: 1 2 (23) min Φ(𝑢2 ) = ⋅ (𝜆(𝑢2 ) − 1) 2 The problem is tackled using the optimization techniques presented in section 3 (differentiation must be carried out w.r.t. 𝑢2 ). A set of operating points of the engine characterized by varying engine rotational speed is defined in table 1 3 . For that set of operating conditions, the optimal choice of phase shift 𝑢2 = 𝑢∗2 (with regard to the optimization problem (23)) is found using the theoretical air intake model. As one can observe in figure 4, the optimal point is very flat, but the numerical results displayed in figure 5 point out, that when the engine rotational speed is large, it is wise to open later, in order to ram as much air as possible into the cylinder. This qualitative result has been affirmed by experimental results.
70
1.2 1 0.8
1000 2000 3000 4000 5000 6000 7000 𝜔 (in 𝑟𝑒𝑣 ′ 𝑠 ⋅ 𝑚𝑖𝑛−1 ) Fig. 5. Optimal valve timing 𝑢2 = 𝑢∗2 as a function of the revolutions of the engine and operating conditions defined in table 1. principles, the conservation of mass and the conservation of energy. The model takes into account, that the phase shift as well as the amplitude of the lift of both the intake and the exhaust valve can be manipulated. This is the case in many contemporary combustion engines used as primary propulsion system in conventional motor cars. Non-linear optimization techniques have been employed to match the model to a set of measurements, operating Table 1. Parameters and inputs for optimal valve actuation. Quantity 𝑑 𝑝𝑙𝑜𝑠𝑠,𝑒 𝜔 𝑝𝑖𝑚 𝑝𝑒𝑚 𝑇𝑖𝑚 𝑇𝑒𝑚 𝑢1 𝑢2 𝑢3 𝑢4
Unit 𝑏𝑎𝑟 𝑏𝑎𝑟 𝑚𝑖𝑛−1 𝑏𝑎𝑟 𝑏𝑎𝑟 𝐾 𝐾 𝑐𝑚 𝑟𝑎𝑑 𝑐𝑚 𝑟𝑎𝑑
Value 0.29 0 {400...6800} 1.1 1.15 303 1073 { n.a. } 0... 7⋅𝜋 9 𝑛.𝑎. 𝜋 2
Comment press. loss (intake valve) press. loss (exhaust valve) revolutions per min. manifold pressure manifold pressure manifold temperature manifold temperature amplitude (see eq. (6)) phase shift (see fig. 2) amplitude (see eq. (9)) phase shift (see fig. 2)
conditions and plant inputs, which have been collected at an engine test bench. The (mean value) model represents the real plants over a wide range of operating conditions and the disturbance, which has been estimated comes to rest in a reasonable order of magnitude (the pressure loss varies in between 0.1 - 0.6 𝑏𝑎𝑟). The computational effort required to give an estimate for the pressure loss bearable; usually, the optimization algorithm terminates after five steps and the absolute value of the error settles below one percent of the measured mass flow. The procedure presented at this point has got some potential for online parameter estimation, which may be run in parallel to the process of identification of an engine characteristic map at the engine test bench. To speed up the procedure, and to provide a basis for application of the model to hardware-in-the-loop simulation and engine control, the non-linear mapping 𝑦 (u, p, 𝑑) in equation (16) may be approximated by a suitable meta model. Since the errors of the meta model are of sole deterministic nature, the meta model structure, which is recommended in the literature on design and analysis of computer experiments is a NN or a KP (see Simpson et al. (2001)). The deficiency of the KP is, that the whole set of design sites (which may be large) must be kept in the memory section of the real time processor. Another direction of future research is a detailed numerical study of the optimal valve timing 𝑢∗2 . When the optimal valve timing does not vary much with the pressure loss, i.e. ∣∣∂𝑢∗2 /∂𝑑∣∣∞ ≪ 1 where 𝑑 ⊆ ℝ+ (24) then the optimal valve timing can be computed a priori, i.e. before any expensive experiment at the engine test bench has been carried out. In other words: The design of the valve control action 𝑢2 may be based on the ideal cylinder filling dynamics, knowing, that the ever present pressure loss of the real cylinder 𝑑 = 𝑝𝑙𝑜𝑠𝑠,𝑖 does not have a large effect on the optimality of the valve timing. Since the computational effort, that must be taken is deeply rooted in computation of the mapping 𝑦 (u, p, 𝑑) in equation (16), the synthesis of suitable meta models must be prioritized. ACKNOWLEDGEMENTS The author would like to thank Patrick Deisenhofer and Johannes Ruder at ITK-Engineering AG for their considerations, criticism and many fruitful discussions. REFERENCES Basshuysen, R. and Schaefer, F. (2010). Handbuch Verbrennungsmotor. Vieweg und Teubner (in German), Wiesbaden. Guzzella, L. and Onder, C. (2010). Introduction to Modeling and Control of Internal Combustion Engine Systems. Springer, Berlin. Isermann, R. and Muenchhof, M. (2011). Identification of Dynamic Systems. Springer, Berlin. Jankovic, M. and Magner, S.W. (1999). Air-charge estimation and predicition in spark ignition internal combustion engines. In Proceedings of the Amercian Control Conference. Khiar, D., .Lauber, J., Guerra, T., Floquet, T., Chamaillard, Y., and Colin, G. (2006). Nonlinear modeling and
control approaches for a turbocharged si engine. In Proceedings of the IEEE 32nd Annual Conference on Industrial Electronics (IECON). Leroy, T., Chauvin, J., Solliec, G.L., and Corde, G. (2007). Air path estimation with variable valve timing. In Proceedings of the 2007 Amercian Control Conference. Lophaven, S., Nielsen, H., and Soendergaard, J. (2002). Dace-a matlab kriging toolbox. Technical Report IMM-TR-2002-12, Informatics and Mathematical Modeling Departement, The Technical University of Denmark, available online @ http://www2.imm.dtu.dk/ hbn/dace/dace.pdf. More, J. (1978). The levenberg-marquardt algorithm: Implementation and theory. Numerical Analysis, 630. Nocedal, J. (2006). Numerical Optimization. Springer, Ney York. Palladino, A., Fiengo, G., Cristofaro, F., Casavola, A., and Glielmo, L. (2008). In cylinder air charge prediction for vva system: Experimental validation. In Proceedings of the 17th IEEE International Conference on Control Applications. Pischinger, R., Klell, M., and Sams, T. (2009). Thermodynamik der Verbrennungskraftmaschine. Springer, Berlin. Simpson, T., Peplinski, J., Koch, P., and Allen, J. (2001). Metamodels for computer-based engineering design: Survey and recommendations. Engineering with Computers, 17. Storset, O., Stefanopoulou, A., and Smith, R. (2000). Air charge estimation for turbocharged diesel engines. In Proceedings of the American Control Conference.
Keywords: Automotive control, computer-aided control system design, engine modelling, engine control 1. INTRODUCTION Positive displacement machines with internal combustion are one of the most important technological developments of the past century. They have become the most frequently encountered source of propulsion energy in passenger cars. One reason for this is, that this source of propulsion energy does rely on the large energy density of liquid hydrocarbons, which is very hard to be be beaten for by alternative propulsion concepts. In spite of all the research efforts that have been made, internal combustion engines still have potential for improvements, which are hard to be realized without model based control and optimization. From a control engineer’s point of view, the positive displacement machine with internal combustion is a non-linear, time periodic plant. It is the the time periodicity, whose frequency varies with the number of revolutions, that makes the application of contemporary control techniques a challenge (see Guzzella and Onder (2010) for an introduction to control and estimation of the internal combustion engine). In this contribution, we propose a model for the air intake of a positive displacement machine, that is derived from first principles - the conservation of mass and energy in the cylinders. Moreover, we introduce a disturbance of the ideal cylinder filling dynamics and provide a way to rapidly estimate the disturbance for a given operating point of the engine. The
exhaust manifold 𝑚 ˙ 𝑐𝑦𝑙 𝑝𝑒𝑚 𝑇𝑒𝑚
intake manifold 𝑚 ˙ 𝑖𝑚 𝑝𝑖𝑚 𝑇𝑖𝑚
ℎ𝑉,𝑖 𝑚𝑎𝑖𝑟 𝑈𝑎𝑖𝑟
𝑠0 𝑠
𝑙 𝑟 crank, angle 𝜙, 𝜙˙ = 𝜔 Fig. 1. Illustration of the crank and piston kinematics and the thermodynamic states governing the air intake. internal combustion engine we have in mind takes four strokes to complete a cycle: Air admission, compression, fuel injection followed by combustion and rejection of the piston and finally air cleaning. The amount of air, that is aspirated by the engine’s cylinders, limits the amount of fuel, that can be burned inside and hence limits the power,
the engine is able to deliver (see Basshuysen and Schaefer (2010); Pischinger et al. (2009) for thermodynamics of the combustion engine). Many applications being processed on an electronic engine control unit resign to disclose those dynamics, that are associated with the variation of the crank angle and use so called mean value models to represent the low-frequency range behavior of key engine parts. An example for such a mean value model is the air intake rate of the engines cylinder, but also the engine characteristic map, giving the momentum delivered to the crank as a function of the number of revolutions, injection angle, pressure in the intake manifold and other physical quantities in closed loop (see Isermann and Muenchhof (2011)). Even most recent developments on motor control, such as the works of Jankovic and Magner (1999), Khiar et al. (2006), Leroy et al. (2007), Palladino et al. (2008) or Storset et al. (2000) use a high order polynomial in order to formally represent the amount of air, that is aspirated by the engine at a given operating point. The parametrization of these polynomials must be carried out at the engine test bench. Due to the fact, that these meta-models must represent the low-frequency behavior of the time-periodic plant, long measurement times must be accepted, which can be associated with high development cost. In the following section, the dynamics of the air intake are derived from first principles. A disturbance of the ideal cylinder filling dynamics is defined, which is estimated from experimental data using non-linear optimization techniques in section 3. The order of magnitude at which the estimated disturbance settles, proves to be plausible. In section 4, it will be shown, that a small subset of steady state operating points of the combustion engine is sufficient to derive (i) a meta-model for the disturbance and (ii) predict the air intake rate of the engine over a large range of operating conditions, if the theoretical air intake model and a proper black-box model for the disturbance is used. In this section, the act of proper meta-modeling of the disturbance is briefly discussed. The model allows for a priori computation of optimal valve actuation. This point is addressed in section 5. In the final section directions for future research are discussed. 2. THEORETICAL AIR INTAKE MODEL The state of the cylinder is completely defined by the mass of air and the internal energy of the air contained in the cylinder, denoted by 𝑚𝑎𝑖𝑟 and 𝑈𝑎𝑖𝑟 respectively. The filling and emptying dynamics of the cylinder are fast compared to the dynamics governing the intake and exhaust manifold, indicated by circles in figure 1. Hence, we can assume, that these temperatures and pressures do remain approximately constant during a cycle composed of air intake, compression, combustion followed by rejection of the piston and finally air-cleaning. Throughout this work, we assume, that the state of the air in the cylinder follows the ideal gas law, 𝑝𝑎𝑖𝑟 𝑝𝑎𝑖𝑟 ⋅ 𝑉𝑐𝑦𝑙 = 𝑚𝑎𝑖𝑟 ⋅ 𝑅 ⋅ 𝑇𝑎𝑖𝑟 , or 𝜌𝑎𝑖𝑟 = (1) 𝑅 ⋅ 𝑇𝑎𝑖𝑟 The state variables relate to the temperature as follows: 𝑅 ⋅ 𝑈𝑎𝑖𝑟 (2) 𝑈𝑎𝑖𝑟 = 𝑚𝑎𝑖𝑟 ⋅ 𝑐𝑣 ⋅ 𝑇𝑎𝑖𝑟 → 𝑝𝑎𝑖𝑟 = 𝑉𝑐𝑦𝑙 ⋅ 𝑐𝑣
In the Equation above, 𝑝𝑎𝑖𝑟 denotes the pressure in the cylinder, 𝑉𝑐𝑦𝑙 denotes the cylinder volume. 𝑅 is the gas constant and 𝑐𝑣 the heat capacity (constant volume). As illustrated in figure 1, 𝑚 ˙ 𝑖𝑚 is the mass flow from the intake manifold into the cylinder and 𝑚 ˙ 𝑐𝑦𝑙 is the mass flow from the cylinder into the exhaust manifold. A mass balance yields the following equation: 𝑑𝑚𝑎𝑖𝑟 ˙ 𝑖𝑚 − 𝑚 ˙ 𝑐𝑦𝑙 (3) =𝑚 ˙ 𝑎𝑖𝑟 = 𝑚 𝑑𝑡 The physical quantity of interest in this work, denoted by 𝑦, is the mass flow 𝑚 ˙ 𝑖𝑚 , that is aspirated by a single cylinder during the admission phase of the four stroke combustion engine cycle. It is given by the product of the density in the intake manifold, 𝜌𝑖𝑚 , the intake valve’s cross-section area, 𝑆𝑉,𝑖 , and the velocity at the intake valve, 𝑣𝑉,𝑖 : 𝑦=𝑚 ˙ 𝑖𝑚 = 𝜌𝑖𝑚 ⋅ 𝑆𝑉,𝑖 ⋅ 𝑣𝑉,𝑖 (4) The valve’s cross-section area depends on the intake valve 2
dead center
1.8 1.6
exhaust valve
intake valve
1.4
𝑢4
1.2
𝑢2
1 0.8 0.6
ℎ𝑉,𝑖 𝑢1
0.4 0.2 0 −6
−4
−2
0
𝜙 (in 𝑟𝑎𝑑)
2
4
6
Fig. 2. Illustration of the valve actuation, as defined in equations (6) and (9). 𝑢2 and 𝑢4 denote the phase shifts of the valves. The intake valve amplitude ℎ𝑉,𝑖 is displayed for various control inputs 𝑢1 . lift ℎ𝑉,𝑖 and the diameter of the intake valve 𝑑𝑉,𝑖 : (5) 𝑆𝑉,𝑖 = 𝜋 ⋅ 𝑑𝑉,𝑖 ⋅ ℎ𝑉,𝑖 In contemporary combustion engines the course of the valve lift ℎ𝑉,𝑖 = ℎ𝑉,𝑖 (𝑡) may be manipulated using electromechanical actuation (see Basshuysen and Schaefer (2010) for details on electro-mechanical valve actuation). The course of the intake valve lift is given as a function of phase shift, 𝑢2 , crank angle, 𝜙, and amplitude 𝑢1 : ℎ𝑉,𝑖 (𝑡) = 𝑢1 (𝑡) ⋅ 𝑓𝑖 (𝜙(𝑡) − 𝑢2 (𝑡)) , where 𝜙˙ = 𝜔 (6) 𝜔 is the engine’s rotational speed and 𝑓𝑖 is a 4 ⋅ 𝜋periodic, analytic function, illustrated in figure 2 1 . Applying Bernoullis Law, we obtain an expression for the velocity 𝑣𝑉,𝑖 : ⎧√ ⎨ 2 (Δ𝑝𝑖𝑚 − 𝑝𝑙𝑜𝑠𝑠,𝑖 ) , Δ𝑝𝑖 > 𝑝𝑙𝑜𝑠𝑠,𝑖 𝑣𝑉,𝑖 = (7) ⎩ 𝜌𝑖𝑚 0 , else 1 In the figure dimensionless quantities are displayed, the physical ranges of the inputs belong to confidential matter and cannot be published.
where Δ𝑝𝑖𝑚 = 𝑝𝑖𝑚 − 𝑝𝑎𝑖𝑟 . 𝑝𝑙𝑜𝑠𝑠,𝑖 is the pressure loss of the inlet valve. It may be given the interpretation of a disturbance of the ideal cylinder filling dynamics: The flow through any real valve is subject to a pressure loss, consequently 𝑑 = 𝑝𝑙𝑜𝑠𝑠,𝑖 = 0 characterizes the ideal cylinder filling dynamics, which yield the largest mass flow at the valve possible for a given initial state of the cylinder, set of inputs u and states of the manifolds. Any real cylinder shows a pressure loss 𝑑 > 0, depending on valve lift and other quantities, as we will see in the following. The mass flow into the exhaust manifold is given by the following equation: 𝑚 ˙ 𝑐𝑦𝑙 = 𝜌𝑎𝑖𝑟 ⋅ 𝑆𝑉,𝑒 ⋅ 𝑣𝑉,𝑒 , 𝑆𝑉,𝑒 = 𝜋 ⋅ 𝑑𝑉,𝑒 ⋅ ℎ𝑉,𝑒 (8) 𝑑𝑉,𝑒 is the diameter of the exhaust valve and the valve lift ℎ𝑉,𝑒 is given by the product of the input 𝑢3 and a 4 ⋅ 𝜋−periodic function, ℎ𝑉,𝑒 (𝑡) = 𝑢3 (𝑡) ⋅ 𝑓𝑒 (𝜙(𝑡) − 𝑢4 (𝑡)) (9) The exhaust valve velocity 𝑣𝑉,𝑒 is given by the following equation: ⎧√ ⎨ 2 (Δ𝑝𝑐𝑦𝑙 − 𝑝𝑙𝑜𝑠𝑠,𝑒 ) , Δ𝑝𝑐𝑦𝑙 > 𝑝𝑙𝑜𝑠𝑠,𝑒 (10) 𝑣𝑉,𝑒 = 𝜌 𝑎𝑖𝑟 ⎩ 0 , else where Δ𝑝𝑐𝑦𝑙 = 𝑝𝑎𝑖𝑟 − 𝑝𝑒𝑚 . The pressure loss of the exhaust valve 𝑝𝑙𝑜𝑠𝑠,𝑒 has an impact on the cylinder emptying dynamics. Hence, it determines, how much waste gas is trapped in the cylinder after combustion and air cleaning. The pressure loss of the exhaust valve can be computed once a combustion model is available and the pressure after combustion is known. The error, that is associated with setting the exhaust valve pressure loss equal to zero is considered to be minor, since the exhaust valve is nearly closed at top dead center, which is the point in time, at which the air intake simulation is initialized (𝜙(𝑡 = 0) = 0 and 𝜙˙ = 𝜔). An energy balance yields the following ordinary differential equation (ODE) for the internal energy of the gas: 𝑑𝑈𝑎𝑖𝑟 = 𝑈˙ 𝑎𝑖𝑟 = 𝐻˙ − 𝑝 ⋅ 𝑉˙ 𝑐𝑦𝑙 (11) 𝑑𝑡 where the flow of enthalpy is given by the following expression: ˙ 𝑐𝑦𝑙 ⋅ 𝑐𝑝 ⋅ 𝑇𝑎𝑖𝑟 (12) 𝐻˙ = 𝑚 ˙ 𝑖𝑚 ⋅ 𝑐𝑝 ⋅ 𝑇𝑖𝑚 − 𝑚 In the following, we assume, that the pressure and the temperature of the air in the cylinder at the beginning of the air intake (𝜙(𝑡 = 0) = 0, the top dead center) is equal to the pressure and temperature of the exhaust manifold. This assumption allows for definition of reasonable initial conditions (ICs) for the mass balance, eq. (3), and the energy balance, eq. (11): 𝑚𝑎𝑖𝑟 (𝑡 = 0) = 𝑈𝑎𝑖𝑟 (𝑡 = 0)/(𝑐𝑣 ⋅ 𝑇𝑒𝑚 ) and 𝑈𝑎𝑖𝑟 (𝑡 = 0) = 𝑝𝑒𝑚 ⋅ 𝑉𝑐𝑦𝑙 (𝑡 = 0) ⋅ 𝑐𝑣 /𝑅. In order to complete the model for the aspiration of air, the kinematics of the crank and the piston need to be described . These are well known, see e.g. Basshuysen and Schaefer (2010) or Pischinger et al. (2009). Let 𝑠(𝑡) be the position of the piston in the cylinder and 𝑠(𝑡 = 0) = 0 be the position at crank angle 𝜙 = 0 (see illustration 1). 𝑉0 is the volume of the cylinder at the top dead center. 𝑉𝑐𝑦𝑙 is given by the following relationship: (13) 𝑉𝑐𝑦𝑙 (𝑡) = 𝑠(𝑡) ⋅ 𝐴𝑝 + 𝑉0 → 𝑉˙ 𝑐𝑦𝑙 = 𝑠˙ ⋅ 𝐴𝑝 𝐴𝑝 is the surface of the piston and 𝑠(𝑡) is related to the crank angle as follows:
) ( 1 𝑟 (14) 𝑠(𝑡) = 𝑟 ⋅ 1 − cos (𝜙(𝑡)) + ⋅ ⋅ (sin (𝜙(𝑡)))2 2 𝑙 The ODEs for the mass of air and the internal energy may be written as a coupled, nonlinear, time-variant system of ODEs: T x˙ = f (x, u, p, 𝑑, 𝑡) , x = [ 𝑚𝑎𝑖𝑟 𝑈𝑎𝑖𝑟 ] (15a) T
T
u = [ 𝑢1 ... 𝑢4 ] , p = [ 𝜔 𝑝𝑖𝑚 𝑝𝑒𝑚 𝑇𝑖𝑚 𝑇𝑒𝑚 ] (15b) with an output equation 𝑦(𝑡) = 𝑔 (x, u, p, 𝑑) given by eq. (4). The set of ODEs must be integrated up to the point in time the intake valve is closed, i.e. ℎ𝑉,𝑖 = 0, in order to capture the whole mass of air, that is aspirated during a cycle. Definition: Mean Value Model. The positive displacement machine with internal combustion is a non-linear, time periodic plant. Consequently, as 𝑡 → ∞, we can expect the output to be time periodic, i.e. 𝑦(𝑡) = 𝑦(𝑡 + 𝑇 ), where 𝑇 = 4 ⋅ 𝜋/𝜔 (with the engine rotating at a constant pace equal to 𝜔). A mean value model is a relationship between the low frequency content of the time periodic signal 𝑦(𝑡) = 𝑦(𝑡 + 𝑇 ) and the inputs, parameters and disturbances of the plant or, more precisely, a mapping from the space of inputs, parameters and disturbances to the DC component of the signal 𝑦(𝑡) = 𝑦(𝑡 + 𝑇 ), denoted as 𝑦: ∫𝑇 ∫𝑇 1 1 𝑦 (u, p, 𝑑) = 𝑦(𝜏 ) 𝑑𝜏 = 𝑔 (x, u, p, 𝑑) 𝑑𝜏 (16) 𝑇 𝑇 0
0
Remark. In practice, the mean value of the aspirated air mass is determined by low pass-filtering the signal provided by the mass flow sensor, which comes along with a requirement for long measurement time and hence large operating cost of of the engine test bench. 3. DISTURBANCE ESTIMATION In this section, the theoretical model will be verified, by comparing the model output 𝑦 to steady state air intake rates of a six-cylinder combustion engine, which are measured at an engine test bench. As expected, the mean mass flow 𝑦 predicted by the nominal plant (𝑑 = 0) is larger than the mass flow, that is observed at the engine test bench. This is illustrated in figure 3. The real cylinder filling dynamics are subject to a pressure loss (𝑑 > 0). The pressure loss of the real cylinder is estimated from steady state operating points, by solving the following optimization problem: 1 1 2 min Φ(𝑑) = ⋅ 𝑒2 = ⋅ (𝑦 (u, p, 𝑑) − 𝑦 𝑚𝑒𝑎𝑠 ) (17) 2 2 Clearly, 𝑦(𝑑) depends on the engine rotational speed, the pressure and temperature in the manifolds and the valve actuation. These quantities are available and some are displayed in figure 3. State-of-the-art optimization techniques, such as the Levenberg-Marquardt Algorithm (LMA) require the gradient and the hessian of the objective function to be provided with sufficient accuracy (see More (1978) or Nocedal (2006)). Differentiation of the objective function w.r.t. 𝑑 allows for determination of the gradient and the hessian: ( )2 𝑑Φ ∂𝑦 𝑑2 Φ ∂𝑦 ∂2𝑦 =𝑗 =𝑒⋅ = ℎ = + 𝑒 ⋅ (18) 𝑑𝑑 ∂𝑑 𝑑𝑑2 ∂𝑑 ∂𝑑2 negligible
The partial derivative of the mean mass flow w.r.t. 𝑑 is ∫𝑇 ∫𝑇 [ ]T ∂𝑔 ∂𝑦 ∂𝑔 ∂x ∂𝑔 1 1 ⋅ = 𝑑𝜏 = + 𝑑𝜏 (19) ∂𝑑 𝑇 ∂𝑑 𝑇 ∂x ∂𝑑 ∂𝑑
ideal filling, i.e. 𝑑 = 0 real filling, i.e. 𝑑 = 𝑑∗ prediction error (KP)
0
The partial derivatives of the state variables w.r.t. 𝑑 are computed by solving the following, augmented system of ODEs for the cylinder filling dynamics (compare to eq. (15a) and (15b)): f (x, u, p, 𝑑, 𝑡) x˙ (20) = ∂f ∂f ∂x 𝑑 ∂x + ⋅ ⋅ ∂x x,u,p,𝑑,𝑡 ∂𝑑 ∂𝑑 x,u,p,𝑑,𝑡 𝑑𝑡 ∂𝑑
𝑦/𝑦 𝑟𝑒𝑓
0
1
0.5
0 50
4. META-MODELING THE PRESSURE-LOSS AND GENERALIZATION OF THE MODEL This section addresses the important point of reducing operating time of the engine test bench and hence development cost of engine control systems by using physical models and meta models for prediction of the air intake rate over a wide range of operating conditions. Without a reasonable pressure loss 𝑑, the theoretical intake model is of limited value, the mean intake rate may exceed the real intake rate by up to a hundred percent (depending on the operating point). Since it’s operation time is expensive, one may want to capture only a very small number of steady state operating points at the engine test bench and predict the remaining points using a model. In the following, twenty percent of the measurements available (see figure 3) are used to compute the pressure loss of the cylinder using the optimization techniques presented in section 3. These results are used to generate a meta-model for the pressure loss 𝑑 = 𝑑 (u, p). Since the prediction error is of sole deterministic nature (no stochastic disturbance, no noise), the authors in Simpson et al. (2001) suggest to train a neural network (NN) or try to find a suitable correlation function for the Kriging Predictor (KP). As opposed to NNs, the KP is an interpolatory meta-model, 2 Dimensionless quantities are displayed for reasons of confidentiality
150
200
250
300
150
200
250
300
2
T
with ICs x(𝑡 = 0) = [ 𝑚𝑎𝑖𝑟 (𝑡 = 0) 𝑈𝑎𝑖𝑟 (𝑡 = 0) ] and T ∂x 0 0 ] . The ODEs are solved using the Forward∂𝑑 𝑡=0 = [ Euler Approach (finite differences in time), with the step size chosen s.t. a single degree crank angle is resolved. The initial guess for the LMA is 𝑑 = 0. 𝑑 is increased in the following iterations, using local information about the hessian (ℎ in eq. (18)) and the gradient (𝑗 in eq. (18)) (see Nocedal (2006) for details), until the norm of the gradient settles below a prescribed tolerance. The optimization procedure is carried out for various operating points and the results are displayed in figure 3 2 ; it can be observed, that the pressure loss is not constant, but varies in a deterministic fashion with the control input 𝑢1 , the intake valve lift. Recalling Bernulli’s Law (pressure loss is proportional to the square of the velocity), one can expect the disturbance to depend on the remaining inputs and parameters as well: 𝑑 = 𝑑 (u, p). Predicting the air intake over a wide range of operating conditions using only a small subset of the measurements available will be the subject of the following section.
100
𝜔/𝜔𝑟𝑒𝑓
1.5
𝑢1
1 0.5 0
50
100
1
𝑑/𝑝𝑟𝑒𝑓 𝑒/𝑝𝑟𝑒𝑓 , prediction error of the KP 0.5
0 50
100
150
200
250
300
steady state operating point index
Fig. 3. Verification of the theoretical air intake model and estimated pressure loss. Top: Mean mass flow (ideal filling and real filling) and prediction error using a meta-model for the pressure loss for prediction of the whole set of operating points. Center: Valve control input 𝑢1 (𝑢3 is kept constant) and engine rotational speed. Bottom: Estimated pressure loss. the prediction error at a design site is equal to zero (see Lophaven et al. (2002) for fundamentals of the KP and an implementation). Using the KP in connection with a suitable correlation function pays respect to the fact, that the experimental acquisition of the air intake rate is expensive and one does want to avoid the creation of blurry information by using a regression model for predicting information that is already available. In this numerical experiment, a Gaussian correlation function is used and the prediction error of the pressure loss using the KP is documented in figure 3 (bottom). Every fifth data point of the available engine test bench data documented in figure 3 has been taken as a design site for the predictor. Consequently, there is a lot of room for improvement by employing a smarter design of experiment (DOE). The ultimate goal is not prediction of the pressure loss, but prediction of the air intake rate 𝑦. Using the meta-model for the pressure loss 𝑑 = 𝑑 (u, p) in the set of ODEs (15a), one is able to predict the air intake rate for any given
operating point defined by the pressures and temperatures in the manifolds, the engine rotational speed and the valve actuation. The resulting prediction error for the real air intake rate is documented in figure 3 (top).
80
5. APPLICATION TO OPTIMAL VALVE ACTUATION
6. CONCLUSIONS A theoretical model for the air intake of a four stroke internal combustion engine has been derived from first Note, that the amplitudes of the valve lifts, 𝑢1 and 𝑢3 belong to confidential matter and cannot be published.
3
𝜆 (in %)
60 50 40 30 6000
4000
2000
1
0
2
3
𝑢2 (in 𝑟𝑎𝑑)
𝜔 (in 𝑟𝑒𝑣 ′ 𝑠 ⋅ 𝑚𝑖𝑛−1 )
Fig. 4. Computer experiment illustrating the functional relationship 𝜆 = 𝜆 (𝜔, 𝑢2 ) for a set of operating conditions defined in table 1.
1.6 1.4 𝑢∗2 (in 𝑟𝑎𝑑)
The power, a combustion engine delivers is proportional to the amount of fuel, that can be burned during a cycle. The amount of fuel, that can be burned is limited by the amount of air, that can be delivered. In this section the issue of ramming as much air as possible into the cylinders of a four stroke combustion engine by means of optimal choice of the control input 𝑢2 is addressed. It is clear, that, in order to maximize the air intake, the amplitude of the valve lift must be chosen as large as possible, consequently, in the following studies 𝑢1 = 𝑢1,𝑚𝑎𝑥 . Clearly, the valve timing influences how much air is aspirated by the engine and must be optimized at the test bench. Here, the expensive experiment carried out at the test bench is replaced by a computer experiment using the theoretical air intake model. In order to proceed, the relative filling must be defined. Definition: Relative Filling. The relative filling 𝜆 ∈ [0 1] can be defined by considering an ideal pump, aspirating a fluid from the intake manifold every other revolution. The mass flow delivered by the ideal pump is 4⋅𝜋 𝑝𝑖𝑚 1 where 𝑇 = (21) 𝑦𝑝𝑢𝑚𝑝 = ⋅ 𝑉𝑑𝑖𝑠𝑝𝑙𝑎𝑐𝑒𝑑 ⋅ 𝑅 ⋅ 𝑇𝑖𝑚 𝑇 𝜔 and 𝑉𝑑𝑖𝑠𝑝𝑙𝑎𝑐𝑒𝑑 = max {𝑉𝑐𝑦𝑙 (𝑡) − 𝑉0 }. The relative filling is the mass of air aspirated by the engine related to the mass of air aspirated by the ideal pump: 𝑦 𝜆= (22) 𝑦𝑝𝑢𝑚𝑝 For any fixed set of parameters p and inputs 𝑢1 = 𝑢1,𝑚𝑎𝑥 , 𝑢3 and 𝑢4 , there exists a phase shift 𝑢2 , maximizing the relative filling (see figure 4 for illustration). An equivalent statement is, that there exists a phase shift 𝑢2 , which solves the following optimization problem: 1 2 (23) min Φ(𝑢2 ) = ⋅ (𝜆(𝑢2 ) − 1) 2 The problem is tackled using the optimization techniques presented in section 3 (differentiation must be carried out w.r.t. 𝑢2 ). A set of operating points of the engine characterized by varying engine rotational speed is defined in table 1 3 . For that set of operating conditions, the optimal choice of phase shift 𝑢2 = 𝑢∗2 (with regard to the optimization problem (23)) is found using the theoretical air intake model. As one can observe in figure 4, the optimal point is very flat, but the numerical results displayed in figure 5 point out, that when the engine rotational speed is large, it is wise to open later, in order to ram as much air as possible into the cylinder. This qualitative result has been affirmed by experimental results.
70
1.2 1 0.8
1000 2000 3000 4000 5000 6000 7000 𝜔 (in 𝑟𝑒𝑣 ′ 𝑠 ⋅ 𝑚𝑖𝑛−1 ) Fig. 5. Optimal valve timing 𝑢2 = 𝑢∗2 as a function of the revolutions of the engine and operating conditions defined in table 1. principles, the conservation of mass and the conservation of energy. The model takes into account, that the phase shift as well as the amplitude of the lift of both the intake and the exhaust valve can be manipulated. This is the case in many contemporary combustion engines used as primary propulsion system in conventional motor cars. Non-linear optimization techniques have been employed to match the model to a set of measurements, operating Table 1. Parameters and inputs for optimal valve actuation. Quantity 𝑑 𝑝𝑙𝑜𝑠𝑠,𝑒 𝜔 𝑝𝑖𝑚 𝑝𝑒𝑚 𝑇𝑖𝑚 𝑇𝑒𝑚 𝑢1 𝑢2 𝑢3 𝑢4
Unit 𝑏𝑎𝑟 𝑏𝑎𝑟 𝑚𝑖𝑛−1 𝑏𝑎𝑟 𝑏𝑎𝑟 𝐾 𝐾 𝑐𝑚 𝑟𝑎𝑑 𝑐𝑚 𝑟𝑎𝑑
Value 0.29 0 {400...6800} 1.1 1.15 303 1073 { n.a. } 0... 7⋅𝜋 9 𝑛.𝑎. 𝜋 2
Comment press. loss (intake valve) press. loss (exhaust valve) revolutions per min. manifold pressure manifold pressure manifold temperature manifold temperature amplitude (see eq. (6)) phase shift (see fig. 2) amplitude (see eq. (9)) phase shift (see fig. 2)
conditions and plant inputs, which have been collected at an engine test bench. The (mean value) model represents the real plants over a wide range of operating conditions and the disturbance, which has been estimated comes to rest in a reasonable order of magnitude (the pressure loss varies in between 0.1 - 0.6 𝑏𝑎𝑟). The computational effort required to give an estimate for the pressure loss bearable; usually, the optimization algorithm terminates after five steps and the absolute value of the error settles below one percent of the measured mass flow. The procedure presented at this point has got some potential for online parameter estimation, which may be run in parallel to the process of identification of an engine characteristic map at the engine test bench. To speed up the procedure, and to provide a basis for application of the model to hardware-in-the-loop simulation and engine control, the non-linear mapping 𝑦 (u, p, 𝑑) in equation (16) may be approximated by a suitable meta model. Since the errors of the meta model are of sole deterministic nature, the meta model structure, which is recommended in the literature on design and analysis of computer experiments is a NN or a KP (see Simpson et al. (2001)). The deficiency of the KP is, that the whole set of design sites (which may be large) must be kept in the memory section of the real time processor. Another direction of future research is a detailed numerical study of the optimal valve timing 𝑢∗2 . When the optimal valve timing does not vary much with the pressure loss, i.e. ∣∣∂𝑢∗2 /∂𝑑∣∣∞ ≪ 1 where 𝑑 ⊆ ℝ+ (24) then the optimal valve timing can be computed a priori, i.e. before any expensive experiment at the engine test bench has been carried out. In other words: The design of the valve control action 𝑢2 may be based on the ideal cylinder filling dynamics, knowing, that the ever present pressure loss of the real cylinder 𝑑 = 𝑝𝑙𝑜𝑠𝑠,𝑖 does not have a large effect on the optimality of the valve timing. Since the computational effort, that must be taken is deeply rooted in computation of the mapping 𝑦 (u, p, 𝑑) in equation (16), the synthesis of suitable meta models must be prioritized. ACKNOWLEDGEMENTS The author would like to thank Patrick Deisenhofer and Johannes Ruder at ITK-Engineering AG for their considerations, criticism and many fruitful discussions. REFERENCES Basshuysen, R. and Schaefer, F. (2010). Handbuch Verbrennungsmotor. Vieweg und Teubner (in German), Wiesbaden. Guzzella, L. and Onder, C. (2010). Introduction to Modeling and Control of Internal Combustion Engine Systems. Springer, Berlin. Isermann, R. and Muenchhof, M. (2011). Identification of Dynamic Systems. Springer, Berlin. Jankovic, M. and Magner, S.W. (1999). Air-charge estimation and predicition in spark ignition internal combustion engines. In Proceedings of the Amercian Control Conference. Khiar, D., .Lauber, J., Guerra, T., Floquet, T., Chamaillard, Y., and Colin, G. (2006). Nonlinear modeling and
control approaches for a turbocharged si engine. In Proceedings of the IEEE 32nd Annual Conference on Industrial Electronics (IECON). Leroy, T., Chauvin, J., Solliec, G.L., and Corde, G. (2007). Air path estimation with variable valve timing. In Proceedings of the 2007 Amercian Control Conference. Lophaven, S., Nielsen, H., and Soendergaard, J. (2002). Dace-a matlab kriging toolbox. Technical Report IMM-TR-2002-12, Informatics and Mathematical Modeling Departement, The Technical University of Denmark, available online @ http://www2.imm.dtu.dk/ hbn/dace/dace.pdf. More, J. (1978). The levenberg-marquardt algorithm: Implementation and theory. Numerical Analysis, 630. Nocedal, J. (2006). Numerical Optimization. Springer, Ney York. Palladino, A., Fiengo, G., Cristofaro, F., Casavola, A., and Glielmo, L. (2008). In cylinder air charge prediction for vva system: Experimental validation. In Proceedings of the 17th IEEE International Conference on Control Applications. Pischinger, R., Klell, M., and Sams, T. (2009). Thermodynamik der Verbrennungskraftmaschine. Springer, Berlin. Simpson, T., Peplinski, J., Koch, P., and Allen, J. (2001). Metamodels for computer-based engineering design: Survey and recommendations. Engineering with Computers, 17. Storset, O., Stefanopoulou, A., and Smith, R. (2000). Air charge estimation for turbocharged diesel engines. In Proceedings of the American Control Conference.