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A Nonlinear Model Predictive Control Strategy with a Disturbance Observer for Spark Ignition Engines with External EGR.

INTRODUCTION

Engine manufacturers need to optimize their control strategies to achieve balance between performance and fuel economy. For traditional engine control strategies (e.g. map-based), that are designed to provide engine torque demanded by the driver, achieving one of the two objectives is often accompanied by the sacrifice of the other. Balancing between fuel economy and torque response can be handled by Model Predictive Control (MPC) strategies if future engine torque demand previews are available [1], [2], [3], [4]. Within each sampling time, the MPC engine control is able to find the optimal actuator set points that minimize fuel consumption and torque (or Indicated Mean Effective Pressure, IMEP) tracking error for the preview horizon, using a control oriented engine model. During this real time optimization process, constraints of actuator saturation, engine knock and combustion variability can be imposed to guarantee the feasibility of computed control actions. While the development of driver modelling and autonomous driving technology enables the preview of future torque demands, this topic is not investigated by this research. Considering the future IMEP demand is available, an Economic Nonlinear Model Predictive Control (E-NMPC) strategy for SI engines with external EGR is discussed in this research. Engine states are estimated by an Extended Kalman Filter (EKF) enhanced with disturbance observation. The proposed control system is evaluated in terms of closed loop stability, IMEP reference tracking performance, fuel economy gain and computation time.

The engine models required to formulate the proposed E-NMPC include air path, torque generation, knock and combustion variation. Control oriented engine air path and torque generation models are well established [5]. The SI engine system is modeled and controlled in the engine cycle domain in this paper. While this approach agrees with the discrete nature of both MPC and IC engines, it also benefits from the fact that most control oriented knock and combustion variation models were constructed in the engine cycle domain. Previous researchers have proposed that knock can be predicted by integrating the Arrhenius function output for the end gases [6,7]. Coefficient of variation (COV) of IMEP is commonly used as an indicator of combustion variability for IC engines. Previous literature has illustrated that the stability of turbulent combustion can be correlated to the time scale ratio between turbulence motion and flame propagation [8,9]. Lee et al. [10] suggested that the COV of IMEP has strong correlation with combustion phasing, differentiating the modeling of COV of IMEP from traditional combustion stability research. Finally, regression models of COV of IMEP have been proposed by [11] and [12].

The proposed E-NMPC penalizes fuel consumption for the preview horizon. This penalty differentiates the current work from a tracking MPC, which is designed to minimize least square error from a reference only. The non-convexity of this problem results in multiple local optimal solutions. Global Nonlinear Programming (NLP) solvers, like dynamic programming and particle swarm, can be employed on MPC applications [13,14] and guarantee the closed loop stability [15,16]. However, these global NLP solvers require numerous evaluations of the system model, making them infeasible for MPC applications requiring a fast sampling time. Most model predictive engine control researchers have selected sub-optimal strategies to reduce the computational demand [17,18,19]. Linear parameter variant (LPV) MPC is a widely adopted sub-optimal predictive controller for nonlinear systems [20,21]. However, the LPV MPC is not suitable for the investigated engine control problem due to the high nonlinearity of the system. In this case, the optimization cannot converge to a local minimum or guarantee the feasibility of computed control actions. The later situation is more undesirable since it could lead to misfire or knock.

This research considers Sequential Quadratic Programming (SQP) to solve the real time NLP problem and obtain desired control actions. SQP is a continuous NLP algorithm based on Newton's method [22]. It is proven to converge to a local optimal solution near the starting point [23,24,25], Previous research has discussed the possibility of applying SQP to NMPC [19]. The most important advantage of SQP is that it transforms complex NLP into a sequence of sub-level quadratic programming (QP) problems. Computing the Hessian matrix for each sub-QP is challenging, especially if the system model is complex and implicit. Numerical differentiation methods like algorithmic differentiation and finite difference are often necessary to linearize the model and obtain the Hessian matrix. This research exploits the Gauss-Newton-like structure of the investigated ENMPC to simplify the Hessian matrix computation. Compared to the Broyden - Fletcher - Goldfarb - Shanno (BFGS) rank-two update method [26,27] and disaggregated Hessian approximation [28], the Hessian of the proposed SQP strategy is inherently positive definite so well-developed convex QP algorithms can be applied.

The sub-QP problem can be solved using existing algorithms. Since the development of active set QP algorithms in the 1980s, this solver has been the fastest option for online operation [30], [31], [32], 33], This style of active set algorithm is based on the fact that QP problems have closed form solutions if the active constraint set of the optimal solution is known. More recently developed parametric active set QP algorithms have been shown to exploit the benefits of warm start techniques [34,35]. They have better degeneracy handling capabilities and faster convergence rates if the initial guess of the active constraint set is not far from the optimal solution. For each SQP iteration of the proposed MPC strategy, the sequence of QPs share similar structure and active constraint sets, making the parametric active set algorithm favorable for this application. In addition to active set methods, the primal-dual (or interior point) method is another option to solve QP for MPCs [36,37]. It is not widely considered for fast MPCs since it requires more computational effort to complete each iteration. Furthermore, the difficulty in finding a "warm" start point also makes it not suitable for fast online operation.

Engine state(s) estimation is required for the E-NMPC to compute the control actions. Most previous research discusses engine state(s) estimation separately in terms of different subsystems. Many publications can be found in air path system estimation [38], [39]. [40], [41], Cylinder composition estimation is extensively discussed in terms of air-to-fuel ratio control [42,43]. This research proposes a model based observer that utilizes sensors of different engine subsystems to generate a more informed estimation of multiple states simultaneously. This 'fusing' of sensors from the air path and in-cylinder reduces the impact of inaccuracy from individual sensors. It also provides possibilities to estimate states that are originally un-observable using decentralized estimators (e.g. residual gas mass).

Many estimation algorithms can be applied to this nonlinear observer application. The Extended Kalman Filter is considered in this research for its ability to optimally choose estimation gain based on the power of sensor noise and actuator disturbances. Furthermore, the engine model is already linearized within the execution of the sequential quadratic programming based MPC controller. The EKF can leverage this advantage to reduce the computation load. However, the performance of the EKF degrades if non-white-noise disturbances are present in the system. These situations occur due to modelling error, sensor/actuator fault and failure of a low level controller to track properly, so a disturbance observer is proposed to address these issues [44,45,46]. Similar to the concept of integral control that eliminates steady state tracking error, the disturbance observer approach augments the original state(s) estimation with integration states of potential error sources. With inputs from sensors the EKF can identify which error source(s) cause the disturbances instead of lowering confidence in the sensors.

This paper is organized as follows. First, an overview of the investigated engine system and proposed control structure is provided, and the control oriented engine model is formulated. The proposed SQP MPC strategy is then described in detail. The EKF engine state estimation routine with disturbance observer is then derived, followed by a discussion of simulation and experimental results. Experimental results describing how the E-NMPC tracks a step changing IMEP reference without violating constraints is then provided. The computational efficiency of the proposed algorithm is then quantified. Simulation results are then used to compare the fuel economy of-a map-based engine control strategy and the proposed E-NMPC using the same engine model. Finally, conclusions are drawn related to the contribution of this research.

SYSTEM OVERVIEW

This research focuses on IMEP control of SI engines with external EGR. Fuel injection control is assumed to maintain a stoichiometric air-to-fuel ratio (AFR), maximizing catalyst efficiency. Manifold temperature is assumed to be constant since the EGR is cooled with a heat exchanger. Finally, the gases in the air-path system are considered incompressible. Figure 1 shows the engine configuration with labels of sensors, control and modeling variables.

The proposed MPC is designed in the engine cycle domain. The MPC manipulates throttle air mass flow per cycle, [m.sub.[alpha]], EGR valve mass flow per cycle, [m.sub.[epsilon]], and combustion phasing C450 (crank angle at which 50% of total heat release occurs). These variables are sent to lower level controllers, with faster update frequencies, as references. These lower level controllers can be map-based or low-dimensional optimal control strategies that have low computation load (e.g. an optimal CA50 control strategy [47]). The two-layer supervisory control structure (shown in Figure 2) exploits the frequency separation of different system dynamics and removes nonlinearities from upper level controllers, making it favorable for fast MPC applications (e.g. [48]).

The 5 ms throttle and EGR valve controllers are based on the inversion of the orifice flow model. The spark timing (SPKT) is controlled by directly inverting an empirically calibrated CA50 map. The proposed engine IMEP control hierarchy transfers the nonlinear orifice valve flow and combustion phasing models to the faster sub-level controllers. This process reduces model complexity for the MPC and lowers computation load. Another advantage of this control structure is that feedback control and lookup tables can be applied to the sub-level controllers since they only deal with low order nonlinear dynamics. These controllers can have faster update frequencies than the MPC, better exploiting the available bandwidth of the control actuators. This structure also allows for fine tuning of individual actuator responses to compensate for dynamics ignored by the MPC loop.

CONTROL ORIENTED ENGINE MODEL

The control oriented engine model consists of intake manifold and cylinder models. The intake manifold model computes the amount of air and EGR flow into the cylinder, using a lumped volume method and speed-density for cylinder charge. The cylinder model computes IMEP and exhaust temperature using an energy balance method. The Residual Gas Mass (RGM) is also computed by the cylinder model using the camshaft phasing, engine speed, MAP and exhaust temperature result from the energy balancing model. Figure 3 shows the block diagram of the control oriented engine model.

In the engine cycle domain, cylinder air charge per cycle can be computed according to intake manifold air density and volumetric efficiency as:

[mathematical expression not reproducible] (1)

where:

[mathematical expression not reproducible] is air mass in the intake manifold:

[[eta].sub.V] is volumetric efficiency;

[V.sub.d] is engine displacement

[V.sub.m] is intake manifold volume.

The air mass balance of the intake manifold can be expressed as:

[mathematical expression not reproducible] (2)

Re-arranging equation (1) and substituting it into equation (2):

[mathematical expression not reproducible] (3)

where:

K = ([[eta].sub.V](k)[V.sub.d])/[V.sub.m]

The cylinder EGR flow can be modeled similar to the air mass flow as the follows:

[mathematical expression not reproducible] (4)

The manifold pressure is then computed by rearranging the speed density equation, as:

[mathematical expression not reproducible] (5)

It can be observed from equations (3), (4), and (5) that manifold dynamics are independent of engine speed (regardless of the slowly varying volumetric efficiency [[eta].sub.V]), unlike most time domain models. In-cylinder gas composition includes air, exhaust gas, fuel and other minor species that are neglected in this research. The amount of air and fuel can be determined by [mathematical expression not reproducible], assuming a stoichiometric AFR. The fuel mean effective pressure ([P.sub.f]) can be computed as:

[mathematical expression not reproducible] (6)

where:

[[sigma].sub.0] is stoichiometric AFR

LHV is low heating value of the fuel

The amount of in-cylinder exhaust gas is the summation of [mathematical expression not reproducible] and residual gas mass (RGM). This research utilizes a modified semi-empirical model originally proposed by Fox et al. [49] that separates RGM into two parts; (1) trapped residual at exhaust valve closing (EVC) due to un-swept cylinder volume, and (2) exhaust gas backflow into the cylinder and intake runner during the valve overlap period. After adding terms [DELTA][P.sub.exh] and [DELTA][P.sub.m] to account for wave tuning dynamics to the original Fox model [50], the residual gas mass for each engine cycle can be calculated according to:

[mathematical expression not reproducible] (7)

where:

[P.sub.exh] [approximately equal to] 110 kPa is the exhaust pressure.

[T.sub.exh] is the exhaust temperature.

R is gas constant.

[V.sub.c] is the cylinder clearance volume.

[A.sub.flow] is effective flow area during valve overlap period.

OLV is overlap volume which is the cylinder volume difference between EVC and IVO.

[C.sub.1], [C.sub.2] are calibration factors.

[[omega].sub.e] is engine speed.

The total fraction of in-cylinder exhaust gas [c.sub.eg] can be generated as:

[mathematical expression not reproducible] (8)

With the information of cylinder composition, IMEP can be modeled using the Willans approximation method [5]:

IME[P.sub.k] = [e.sub.k][([P.sub.f]).sub.k]-[P.sub.0] (9)

where:

e is the 'slope' factor.

[P.sub.0] = [P.sub.exh] - [P.sub.m] is pumping effective pressure (PMEP).

The slope factor, e, is related to engine speed ([[omega].sub.e]), CA50 and [c.sub.eg]. The final value of e is the product of multiple slope factors with dependence on fewer inputs:

e = [e.sub.[omega]]([[omega].sub.e])[e.sub.[zeta]](CA50)[e.sub.eg]([c.sub.eg], [[omega].sub.e]) (10)

Each slope factor in equation (10) can be approximated with a low order polynomial function, as demonstrated by Figure 4, Figure 5. and Figure 6 for the engine utilized for this research.

An energy balance approach is utilized to calculate exhaust temperature, [T.sub.exh], for the residual gas mass model. The IC engine transforms chemical energy of the injected fuel into mechanical work (IMEP) and rejected heat, which is the summation of heat transfer to the coolant and exhaust enthalpy. Thus, the exhaust gas temperature can be calculated according to:

[mathematical expression not reproducible] (11)

where:

[c.sub.p] is constant pressure gas heat capacity.

v is the ratio of transferred heat to coolant (in terms of the total rejected heat). It can be estimated with engine speed, CA50 and load [52][53]

Figure 7 shows IMEP and [T.sub.exh] validation results for the proposed engine model against experimental results. The relative error is less than 5% for more than 98% of the test points.

COV of IMEP is utilized as an indication of combustion variability. The proposed model predictive IMEP control should maintain the COV of IMEP below a certain threshold. COV of IMEP is correlated to the cylinder air mass flow [mathematical expression not reproducible] and CA90 [10]. CA90 is computed with an Artificial Neural Network (1 hidden layer and 10 neurons) with CA50, RPM and [mathematical expression not reproducible] as inputs. The knock model is a fully empirical model (4-D lookup table) as a function of RPM, CA50, [mathematical expression not reproducible] and RGM. The output of this model is the normalized knock intensity, KI, which indicates engine knock is likely if KI [greater than or equal to] 1. Both the COV of IMEP and knock models are able to achieve less than 10% RSME with negligible computation time.

In summary, the proposed engine model is a 4th order nonlinear state-space model:

[x.sub.k+1] = [f.sub.x]([x.sub.k], [u.sub.k]) [y.sub.k] = [f.sub.y]([x.sub.k]) [z.sub.k] = [f.sub.z]([x.sub.k]) [[zeta].sub.k] = [f.sub.[zeta]]([x.sub.k]) (12)

where:

control inputs u [member of] [R.sup.3], u = [[[m.sub.[alpha]], [m.sub.[epsilon]], CA50].sup.T]

system states x [member of] [R.sup.4], x = [mathematical expression not reproducible]

performance y [member of] [R.sup.1], y = IMEP

measurement z [member of] [R.sup.4], z = [[[P.sub.m], [m.sub.[alpha]], IMEP, CA50].sup.T]

constraints [zeta] [member of] [R.sup.3], [zeta] = [[[P.sub.m], COV, KI].sup.T]

[([x.sub.CA50]).sub.k] = [CA50.sub.k-1].

The CA50 output of the MPC is the target value for the next engine cycle, which induces a unit step delay.

Remark: The slowest dynamics of the investigated engine system are the manifold filling dynamics. Therefore, the characteristic "time" [tau] (in engine cycles of 4[pi] crank angle radians) can be calculated as:

[tau] [approximately equal to] [[V.sub.m]/[V.sub.d]] = 2 cycles with [[eta].sub.v] = 1 (13)

Hence, the control and preview horizon of the MPC can be as short as 2 steps without significant loss of optimality and stability [2].

E-NMPC ALGORITHM BASED ON SQP

The E-NMPC algorithm used for this research is described in detail by Zhu et al. in [51], and only a brief summary of some important concepts and equations is provided in this document. The objective of the proposed model predictive IMEP control is to track an IMEP reference with minimum fuel consumption. This determines that the stage cost of the objective function should penalize the least squared error of IMEP tracking, control effort and fuel consumption. Fuel consumption is calculated with engine air mass flow, [mathematical expression not reproducible] (we assume the engine operates under stoichiometric AFR). A terminal state penalty is also included for stability considerations.

[mathematical expression not reproducible] (14)

where:

[mathematical expression not reproducible]

The proposed model predictive engine control has upper bounds on COV of IMEP (denoted as CO[V.sub.ub]) and knock intensity (denoted as [KI.sub.ub]). The manifold pressure must be constrained to be less than the ambient pressure [P.sub.a] since the engine is naturally aspirated. The air mass flow through the throttle, [m.sub.[alpha]], and EGR valve, [m.sub.[epsilon]], are also non-negative to be physically reasonable. Finally, the following equation shows the complete NLP that is solved every engine cycle to obtain the optimal control sequence for the N steps into the future horizon.

[mathematical expression not reproducible] (15)

where:

i = k,k + 1,... k + N - 1;

[b.sub.[zeta]] = [[[P.sub.a], CO[V.sub.ub], [KI.sub.ub]].sup.T] is the upper bounds on [zeta].

[b.sub.u] = [[0, 0, -[infinity]].sup.T] is the lower bounds on u.

The equality constraints of system dynamics can be transferred to the objective function, leading to a new objective function. The new NTP problem only has inequality constraints, and can be written in a more compact form:

[mathematical expression not reproducible] (16)

where:

l: [R.sup.4+3w] [right arrow] [R.sup.5N] after eliminating the infinite lower bound on CA50.

With a given initial guess of U (represented by [U.sub.0]), the SQP computes the searching direction [DELTA]U by solving a sub-quadratic programing problem as following:

[mathematical expression not reproducible] (17)

The Hessian matrix [mathematical expression not reproducible] is computed as:

[mathematical expression not reproducible] (18)

where:

[mathematical expression not reproducible]

The Jacobian matrix [mathematical expression not reproducible] of the objective function is computed as:

[mathematical expression not reproducible] (19)

where:

R = [[r,r,... r].sup.T] [member of] [R.sup.3 x N]

The search step size in the direction [DELTA]U is scaled by a factor a*, which was generated by solving a one-dimensional search problem of a merit function of the original NTP, g(U). The following equations show the line search problem and formulation of the merit function:

[mathematical expression not reproducible] (20)

[mathematical expression not reproducible];

[sigma] is the penalty on the constraints violation.

For each major iteration, j (whereas the iterations solving the sub-QP problems are referred to as minor iterations), the updated solution is calculated as:

U* = [([U.sub.0]).sub.j+1] = ([U.sub.0]); + [[alpha].sub.j][DELTA][U.sub.j] (21)

The SQP is considered as converged if the search step [[alpha].sub.j][DELTA][U.sub.j] is smaller than a certain threshold. In this situation, the algorithm terminates, outputting U* as the final solution.

An initial guess of the control actions, [([U.sub.0]).sub.k+1], is generated from the optimal control sequence solution of the previous control sampling, [U*.sub.k]:

[mathematical expression not reproducible] (22)

The LQR solution of the linearized model at the last step of the prediction horizon can be a good option for [u.sup.+]. With the initial guess, [U.sub.0](k +1), closed loop stability can be proven (with the existence of the Lyapunov function) if the sub-optimal solution satisfies [15]:

l([x.sub.k+1], [U*.sub.k+1]) [less than or equal to] 0 J([x.sub.k+1][U*.sub.k+1]) [less than or equal to] J([x.sub.k+1], [([U.sub.0]).sub.k+1]) (23)

Figure 9 shows the flow chart of the entire proposed SQP model predictive IMEP control strategy.

EKF ENGINE STATE ESTIMATION WITH DISTURBANCE OBSERVER

The engine states, x(k), are estimated using an Extended Kalman Filter (EKF). The EKF is a recursive estimation technique that updates states, x, considering the error between the output measurement, y, and the output estimation, y. The state space representation of this process is:

[x.sub.k+1] = [f.sub.x]([x.sub.k],[u.sub.k]) + [L.sub.k]([y.sub.k]-[y.sub.k]) (24)

[y.sub.k] = [f.sub.y]([x.sub.k]) (25)

The estimation gain, [L.sub.k], is computed recursively as the following:

[mathematical expression not reproducible] (26)

[mathematical expression not reproducible] (27)

[P.sub.k|k] = [I - [L.sub.k][C.sub.k]][P.sub.k|k-1] (28)

where:

[[sigma].sup.2.sub.w] and [[sigma].sup.2.sub.v] are co variance of noise v and actuator disturbance [w.sub.a] respectively.

A, B and C are state space matrices from real time linearization.

P = E[(x-x)[(x- x).sup.T]]

The computed estimation gain minimizes J = trace(P). Figure 10 shows the block diagram of the EKF-based engine state(s) estimation algorithm.

The unmeasured disturbances, [w.sub.d], which can be modeling errors and/or sensor/actuator offsets, enter the system as:

[mathematical expression not reproducible] (29)

where:

[mathematical expression not reproducible]

Since the unmeasured disturbances, [mathematical expression not reproducible], have non-zero mean value and relatively slow rate of change, it is assumed that they stay the same between consecutive EKF samplings. Define the disturbance state, d [member of] [mathematical expression not reproducible] as:

[d.sub.k+1] = [d.sub.k] (30)

The augmented system state space model for the linearized state observer is given by:

[mathematical expression not reproducible] (31)

[mathematical expression not reproducible] (32)

From equation (31) and (32). it can be seen that the augmented disturbance states, d, actually integrate the effect of the unmeasured disturbances, [w.sub.d]. This characteristic gives the proposed MPC an integral-control like behavior.

In order to maintain detectability of the augmented system, the following conditions need to hold:

[mathematical expression not reproducible] (33)

where:

[([B.sub.d]).sub.k] and [([C.sub.d]).sub.k] are linearized state space matrices of disturbance d:

[n.sub.d] is the number of disturbance states.

This requirement suggests that the number of disturbance states has to be less than or equal to the number of sensors, in this case [n.sub.d] [less than or equal to] 4. Furthermore, the selected disturbance states have to be independent from each other.

The augmented disturbance states have to guarantee their estimation convergence, i.e. [d.sub.k+1] = [d.sub.k] as k [right arrow] [infinity] as k [right arrow] [infinity] [54]. The following equations show the update of the disturbance state(s).

[mathematical expression not reproducible] (34)

or:

[d.sub.k+1] = [d.sub.k] + [([L.sub.d]).sub.k][e.sub.k] (35)

where:

[([L.sub.d]).sub.k] [member of] [mathematical expression not reproducible] is the portion of [L.sub.k] corresponding to the disturbance states.

It is required that [n.sub.d] [greater than or equal to] p for the observer to drive [e.sub.k] to 0, so that [d.sub.k+1] = [d.sub.k]. Combining the above conclusions, the number of disturbance states must equal to the number of sensors, i.e. [n.sub.d] = p = 4. While the number of disturbance states is determined by the number of available sensors, there is freedom in the selection of these states to accommodate different system design objectives. This research assigns the four disturbance states to the possible errors of the four state update equations, [f.sub.x].

Figure 11 shows the CA50 disturbance rejection performance of the proposed EKF engine state estimator during an steady state engine dyno test. A known CA50 disturbance is added to the engine during 40 ~ 80th engine cycles. The EKF identifies this disturbance after approximately 10 engine cycles. It can also be observed from Figure 11 that the EKF successfully suppressed the noise in both IMEP and CA50 measurements. The noise reduction in MAP and MAF is not as significant due to the high confidence of these two sensors (small [[sigma].sup.2.sub.v]).

SIMULATION AND EXPERIMENTAL RESULTS

The spark ignited V6 engine investigated in this paper has port fuel injection and a displacement of 3.6L. The engine uses an EGR system with a cooler and post-throttle delivery to the intake manifold, as shown in Figure 1. The control system is validated for a mid-sized SUV over a FTP driving cycle. Parameters of the engine and vehicle are given in Table 1.

The experiment validation for the engine models was carried out on an AC dynamometer. The test engine was controlled using an ETAS ES910 system overriding the stock ECU. CA50 and IMEP measurements for the engine state(s) estimation are computed with a Cylinder Pressure Development Controller (CPDC) unit and AVL GH12D piezoelectric cylinder pressure transducers [55].

The proposed E-NMPC was evaluated experimentally on an engine dynamometer using a prototype engine controller. During the test, the E-NMPC tracked the step-changes in IMEP reference while maintaining COV [less than or equal to] 0.06, MAP [less than or equal to] 100 kPa md KI [less than or equal to] 1. Figure 12 and Figure 13 show the test results at 1500 RPM. It can be observed from the first plot in Figure 12 that the EKF based engine state(s) observer successfully suppresses IMEP measurement noise, and provides reliable state estimation to the E-NMPC. For a "tip in" situation (around 10th, 40th, 70th and 130th engine cycle), the throttle air mass flow, [m.sub.[alpha]], overshoots during IMEP reference steps. This maneuver is to compensate for the manifold delay and quickly increase the IMEP output. During "tip out" situations (around the 20th, 50th, 120th and 140th engine cycle) the throttle air mass is reduced to zero initially to compensate for manifold delay, then it converges to a steady state value without oscillation. It can be observed that the EGR flow is shut down before the throttle, in order to prevent excessive dilution and meet the COV of IMEP constraint. When the IMEP demand is not high (140-210th engine cycles), the MPC requests MBT combustion phasing and maximum EGR, as permitted by the COV of IMEP constraint. If the IMEP demand is high (10~20th and 130~140th engine cycles), the MPC reduces EGR to maximize engine air mass flow Combustion phasing is retarded if the knock constraint is active (10~20th and 40~50th engine cycles). These behaviors are in agreement with calibration objectives for traditional map based IMEP control.

In order to compare the performance of the proposed E-NMPC strategy against traditional engine control system, a vehicle running with a map-based engine control strategy was tested on a chassis dynamometer over a FTP drive cycle. The engine actuator set-points were recorded, and applied to the proposed engine model through an offline simulation to generate a IMEP target. Finally, the proposed ENMPC strategy was applied in a closed loop simulation with the same engine model (as the baseline) tracking the same IMEP as the baseline engine calibration created. In this way, drivability is matched between the baseline calibration and E-NMPC strategies. Fuel consumption was evaluated for both simulation runs, using the same engine model. Figure 14 shows results of the closed loop simulation using the E-NMPC. The second plot in Figure 14 shows that the E-NMPC follows the IMEP reference closely. It can be observed from Figure 14 that the E-NMPC commands more advanced CA50 and higher EGR concentration, compared to the default engine calibration. Previous analysis shows that the E-NMPC is able to overshoot throttle and EGR valve and alter cylinder air and EGR flow within less than two engine cycles, given the future IMEP reference. Therefore, it does not need delay CA50 to build up torque reserve. The E-NMPC also enables much more aggressive EGR dilution since it can foresee incoming tip out situations and close EGR valve ahead of time. However, it is noticed that the E-NMPC commands high EGR concentration and MBT combustion phasing during idle situations. The reason for this observation is that the current E-NMPC does not consider catalyst temperature. Furthermore, the proposed E-NMPC IMEP control strategy neglects the coupling between engine torque and speed. This assumption is not valid during the idle situation where the engine is decoupled from rest of the powertrain. Combustion variation during idle can cause significant engine speed fluctuations during one engine cycle. This research has not yet validated the proposed E-NMPC strategy for idle operation. After eliminating the idle situations from the closed loop FTP drive cycle simulation, the E-NMPC reduces fuel consumption by 6.68% compared to a map-based engine control strategy.

Table 2 summarizes execution time statistics of the proposed SQP MPC during experimental validation. The ETAS ES910 prototype engine controller has a double precision floating CPU with 800 MHz clock. The memory is DDR2-RAM with 512 megabytes of space and a 400 MHz clock speed. It can be observed that the control algorithm is computationally efficient for cyclic engine control considering the duration of every engine cycle is 200 to 20 ms for engine speeds from 600 to 6000 RPM. Table 2 also reveals that QP computation takes approximately 1/3 of the overall computational time, while the rest of the execution time is spent on evaluating the engine model. It can be concluded that simplification of the engine models can significantly reduce computational time.

CONCLUSIONS

This research proposes an economic nonlinear model predictive control strategy and EKF engine state estimation for SI engine IMEP control. The control objective is to track an IMEP reference and minimize fuel consumption, while respecting knock and COV of IMEP constraints. The proposed E-NMPC is designed to function in the engine cycle domain, which reduces the engine speed dependence on air-path dynamics. The real time NTP problem is solved by a sub-optimal SQP algorithm, which is tailored for this application to improve the computational efficiency. It exploits the Gauss-Newton like structure of the NLP formulated for MPC to simplify computation of the Hessian matrix. Warm start and merit function techniques are applied to ensure closed loop stability. The engine states are accurately estimated by an EKF based observer, augmented with disturbance states. The EKF conveniently utilizes the linearized system model from the E-NMPC, reducing the computational load of the overall system.

Simulation and experimental results demonstrate that the proposed model predictive IMEP control strategy achieves its design objectives, in terms of tracking torque reference, minimizing fuel consumption and respecting combustion constraints. Compared to a conventional engine control strategy, the looking-ahead E-NMPC has better fuel economy. It can operate the engine with higher percentage of optimal combustion phasing and EGR flow set points, resulting in approximately 6.5% of fuel economy improvement during a FTP driving cycle simulation. Finally, the computational time analysis of the proposed control system with prototype engine controllers demonstrates high potential for real time implementation with future production ECUs.

REFERENCES

[1.] Ali A. and Blath P. "Application of modern techniques to SI-engine torque control", in Proc. of 2006 IEEE International Conference on Control Applications, pp. 2405-2410, Oct., 2006, Munich, Germany.

[2.] Lee T.K. and Filipi Z.S. "Nonlinear model predictive control of a dual-independent variable valve timing engine with electronic throttle control", in Proc. of the Institution of Mechanical Engineers, Part D: Journal of Automobile Engineering, vol. 225, no. 9, pp.1221-1234, Sep., 2011.

[3.] Feru E., Lazar M., Gielen R.H., Kolmanovsky I.V., Cairano S.Di, "Lyapunov-based constrained engine torque control using electronic throttle and variable cam timing", in Proc. of 2012 American Control Conference, pp. 2866 ~ 2871, Jun., 2012, Fairmont Queen Elizabeth, Canada.

[4.] Kang M. and Shen T., "Nonlinear model predictive torque control for IC engines", in Proc. of the 11th World Congress on Intelligent Control and Automation, pp. 804~809, Jun., 2014, Shenyang, China.

[5.] Guzzella L. and Onder C.H., Introduction to modeling and control of internal combustion engine systems, 2nd ed., Springer, 2009.

[6.] Livengood J.C. and Wu P.C., "Correlation of auto-ignition phenomena in internal combustion engines and rapid compression machines", Symposium (International) on Combustion, vol. 5, no. 1, pp. 347-355, 1955.

[7.] Xiao, B., Wang, S., and Prucka, R., "Virtual Combustion Phasing Target Correction in the Knock Region for Model-Based Control of Multi-Fuel SI Engines," SAE Int. J. Engines 6(1):228-236, 2013, doi:10.4271/2013-01-0307.

[8.] Ozdor, N., Dulger, M., and Sher, E., "Cyclic Variability in Spark Ignition Engines A Literature Survey," SAE Technical Paper 940987, 1994, doi:10.4271/940987.

[9.] Lacour, C. PhD and Pera, C., "An Experimental Database Dedicated to the Study and Modelling of Cyclic Variability in Spark-Ignition Engines with LES," SAE Technical Paper 2011-01-1282, 2011, doi:10.4271/2011-01-1282.

[10.] Lee, T.K., Prucka R.G. and Filipi Z.S., "Real-time estimation of combustion variability for model-based control and optimal calibration of spark ignition engines", in Proc. of the Institution of Mechanical Engineers, Part D (Journal of Automobile Engineering), vol. 223, no. D11, pp. 1361-1372, Nov., 2009.

[11.] Dai, W., Trigui, N., and Lu, Y., "Modeling of Cyclic Variations in Spark-Ignition Engines," SAE Technical Paper 2000-01-2036, 2000, doi:10.4271/2000-01-2036.

[12.] Galloni E., "Analyses about parameters that affect cyclic variation in a spark ignition engine", Applied Thermal Engineering, vol. 29, pp. 1131-1137, 2009

[13.] Diehl M., "Robust dynamic programming for min-max model predictive control of constrained uncertain systems", IEEE Transactions on Automatic Control, vol. 49, no. 12, pp. 2253~2257, Dec., 2004.

[14.] Meadows E.S., "Dynamic programming and model predictive control", in Proc. of 1997 American Control Conference, pp. 1635~1639, Jun., 1997, Albuquerque, USA.

[15.] Rawlings J.B. and Mayne D.Q., Model Predictive Control: Theory and Design, Nob Hill Publishing, Madison, WI, 2009.

[16.] Mayne D.Q., Rawlings, J.B., Rao C.V. and Scokaert P.O.M., "Constrained model predictive control: Stability and optimality", Automatica, vol. 36, no. 6, pp. 789~814, 2000.

[17.] Ohtsuka T., "A continuation/GMRES method for fast computation of nonlinear receding horizon control", Automatica, vol. 40, no. 4, pp. 563~574, 2004.

[18.] Lazar M., "Flexible Control Lyapunov Functions", in Proc. of 2009 American Control Conference, pp. 102~107, Jun. 2009, St. Louis, USA.

[19.] Quirynen R., Houska B., Vallerio M., Telen D., Logist F., Impe J. Van and Diehl M., "Symmetric algorithmic differentiation based exact Hessian SQP method and software for Economic MPC", in Proc. of 2014 IEEE 53rd Annual Conference on Decision and Control, pp. 2752~2757, 2014.

[20.] Zhu Q., Onori S. and Prucka R., "Pattern recognition technique based active set QP strategy applied to MPC for a driving cycle test", in Proc. of 2015 American Control Conference, pp. 4935~4940. Jul., 2015, Chicago, USA.

[21.] Sharma, R., Nesic, D. and Manzie, C. "Idle speed control using linear time varying model predictive control and discrete time approximations", in Proc. of 210 IEEE International Conference on Control Applications, pp. 1140-1145, Yokohama, Japan, Sep., 2010.

[22.] Bertsekas D.P., Nonlinear Programming, 2nd Ed. 1999, Athena Scientific, pp. 431~448.

[23.] Bertsekas D.P., Constrained Optimization and Lagrange Multiplier Methods, Academic Press, London and New York, 1982.

[24.] Fletcher R., "Methods related o Lagrangian functions", in Gill P.E. and Murray W., editors, Numerical Methods for Constrained Optimization, pp. 228-234. Academic Press, London and New York, 1974.

[25.] Goodman J., "Newton's method for constrained optimization", Math. Prog., vol.33, pp. 162-171, 1985.

[26.] Broyden C.G., "The convergence of a class of double-rank minimization algorithms, parts I and II", J. Inst. Math. Applics., vol. 6, pp. 76-90 and 222-236, 1970.

[27.] Nocedal J. and Wright S., Numerical Optimization, Springer - Verlag, 2000.

[28.] Goldsmith M.J., Sequential Quadratic Programming Methods Based on Indefinite Hessian Approximations, PhD dissertation, Stanford University, 1999

[29.] Han S.P., "A globally convergent method for nonlinear programming", J. Optimization Theory and Applications, vol. 22, no. 3, pp. 297, 1977.

[30.] Gill P.E., Gould N.IM., Murray W., Saunders M.A., and Wright M.H., "A weighted Gram-Schmidt method for convex quadratic programming", Mathematical Programming, vol. 30, pp. 176-195, 1984.

[31.] Gill P.E., Murray W., Saunders M.A., and Wright M.H., "Procedures for optimization problems with a mixture of bounds and general linear constraints", ACM Transactions on Mathematical Software, vol. 10, no. 3, pp. 282-298, 1984.

[32.] Goldfarb D., and Idnani A., "A numerically stable method for solving strictly convex quadratic programs", Mathematical Programming, vol. 27, pp. 1-33, 1983.

[33.] Schmid C., and Biegler L.T., "Quadratic programming methods for reduced hessian SQP", Computers & Chemical Engineering, vol. 18, no. 9, pp. 817-832, 1994

[34.] Ferreau H.J., Bock H.G. and Diehl M., "An online active set strategy to overcome the limitations of explicit MPC", Int. J Robust Nonlinear Control, vol. 18, pp. 816-830, 2007.

[35.] Ferreau H.J., Ortner P., Langthaler P., Re L. del and Diehl M., "Predictive control of a real-world diesel engine using an extended online active set strategy", Annual Review in Control, vol. 31, no. 2, pp. 293-301, 2007.

[36.] Goncalves A.S., "A primal-dual method for quadratic programming with bounded variables", Numerical Methods for Nonlinear Optimization, Academic Press, London, 1972, pp. 255-263.

[37.] Wright S.J., Primal-dual Interior-point Methods, SIAM: Philadelphia, PA, 1997.

[38.] Wang J., "Air fraction estimation for multiple combustion mode diesel engines with dual-loop EGR systems", in Proc. of 46th IEEE conference on Decision and Control, pp. 2862-2867, New Orleans, USA, Dec., 2007.

[39.] Kolmanovsky I., Sun J., and Druzhinina M., "Charge Control for Direct Injection Spark Ignition Engines with EGR," in Proc. of the 2000 American Control Conference, pp. 34 - 38, 2000.

[40.] Chauvin J., Corde G., and Petit N., "Constrained Motion Planning for the Airpath of a Diesel HCCI Engine," in Proc. of the 45th IEEE Conference on Decision & Control, pp. 3589 - 3596, 2006.

[41.] Wang, Z., Zhu, Q., and Prucka, R., "A Review of Spark-Ignition Engine Air Charge Estimation Methods," SAE Technical Paper 2016-01-0620, 2016, doi:10.4271/2016-01-0620.

[42.] Chen X., Wang Y., Haskara I. and Zhu G., "Optimal air-to-fuel ratio control with adaptive estimation of biofuel content for diesel engine LNT regeneration", IEEE Transactions on Control Systems Technology, vol. 22, no. 2, pp. 428-439, 2014.

[43.] Ahn K., Stefanopoulou A.G. and Jankovic M., "AFR-based fuel ethanol content estimation in flex-fuel engines tolerant to MAF sensor drifts", IEEE Transactions on Control Systems Technology, vol. 21, no. 3, pp. 590-603, 2013.

[44.] Pannocchia G. and Rawlings J. B.. "Disturbance models for offset-free MPC control", AIChE J., vol. 49, no. 2, pp. 426-437, 2003.

[45.] Rajamani M. R., Rawlings J.B., and Qin S. J.. "Achieving state estimation equivalence for misassigned disturbances in offset-free model predictive control". AIChE J., vol. 55, no. 2, pp. 396-407, February 2009.

[46.] Rawlings J. B. and Mayne D. Q.. Model Predictive Control: Theory and Design. Nob Hill Publishing, Madison, WI, 2009, pp. 45-59.

[47.] Zhu, Q., Prucka, R., Wang, S., Prucka, M. et al., "Model-Based Optimal Combustion Phasing Control Strategy for Spark Ignition Engines," SAE Int. J. Engines 9(2):1170-1179, 2016, doi:10.4271/2016-01-0818.

[48.] Zhu Q. and Ayalew B., "Predictive roll, handling and ride control of vehicles via active suspensions", in Proc. of 2014 American Control Conference, pp. 2102~2107, Portland, USA.

[49.] Fox, J., Cheng, W., and Heywood, J., "A Model for Predicting Residual Gas Fraction in Spark-Ignition Engines," SAE Technical Paper 931025, 1993, doi:10.4271/931025.

[50.] Wang S., Prucka R., Prucka M., and Dourra H., "Control -oriented residual gas mass prediction for spark ignition engines", International Journal of Engine Research, vol. 16, no. 7, pp. 897-907, October 19, 2015.

[51.] Zhu Q., Onori S. and Prucka R., "Nonlinear economic model predictive control for SI engines based on sequential quadratic programming", in Proc. of 2016 ACC, pp. 1802-1807, Boston, MA, 2016.

[52.] Arici, O., Johnson, J., and Kulkarni, A., "The Vehicle Engine Cooling System Simulation Part 1 - Model Development," SAE Technical Paper 1999-01-0240, 1999, doi:10.4271/1999-01-0240.

[53.] Cortona E., Onder Ch.H., and Guzzella L., "Engine thermal management with components for fuel consumption reduction", International Journal of Engine Research, IMechE, vol. 3, no. 3, pp. 157-170, 2002.

[54.] Pannocchia G. and Rawlings J.B., "Disturbance models for offset-free model-predictive control", AIChE Journal, vol.49, no.2, pp.426-437, Feb., 2003.

[55.] Schten, K., Ripley, G., Punater, A., and Erickson, C., "Design of an Automotive Grade Controller for In-Cylinder Pressure Based Engine Control Development," SAE Technical Paper 2007-01-0774, 2007, doi:10.4271/2007-01-0774.

Qilun Zhu and Robert Prucka Clemson University

Michael Prucka and Hussein Dourra FCA US LLC

CONTACT INFORMATION

Qilun Zhu is with the Clemson University-International Center for Automotive Research, Greenville, SC, 29607 (

qilun@g.clemson.edu; Phone: 864-283-7220, Fax: 864-283-7208)

NOMENCLATURE

AFR - Air-to-Fuel Ratio

C450 [deg aTDC] - Crank angle of 50% mass fraction burned

[c.sub.eg] [%] - Cylinder exhaust gas fraction

[c.sub.p] [J/kg * K] - Constant pressure heat capacity

COV- Coefficient of variation

CO[V.sub.ub] - Upper bound of COV

e - Slop factor

ECU - Engine Control Unit

EGR - Exhaust Gas Recirculation

EKF - Extended Kalman Filter

EVC - Exhaust valve closing

[[eta].sub.V] - Volumetric efficiency

[gamma] - Ratio of heat capacity

IC - Internal Combustion

IMEP - Indicated Mean Effective Pressure

IVO - Intake valve opening

KI - Normalized knock intensity

[KI.sub.ub] - Upper bound of KI

[gamma] - Lagrange Multiplier

LHV [J/kg] - Lower heating value

LPV- Linear Parameter Varying

[m.sub.[alpha]] [kg] - Throttle air flow per engine cycle

[mathematical expression not reproducible] [kg] - Cylinder air flow per engine cycle

[mathematical expression not reproducible] [kg] - Cylinder exhaust flow per engine cycle

[m.sub.s] [kg] - EGR valve flow per engine cycle

MAF - Mass Air Flow

MAP - Manifold Absolute Pressure

MEP - Mean Effective Pressure

MPC - Model Predictive Control

[m.sub.m] [kg] - Intake manifold mass

NLP - Nonlinear Programming

NMPC - Nonlinear Model Predictive Control

ODE - Ordinary Differential Equation

OLV- Overlap volume between EVC and IVO

[[omega].sub.e] [rad/s] - Engine speed

[P.sub.0] [Pa] - Pumping effective pressure

[P.sub.a] [Pa] - Ambient pressure

[P.sub.exh] [Pa] - Exhaust pressure

[P.sub.f] [Pa] - Fuel mean effective pressure

[P.sub.i] [Pa] - Orifice input pressure

[P.sub.m] [Pa] - Intake manifold pressure

[P.sub.o] [Pa] - Orifice output pressure

QP - Quadratic Programming

R [J/kg * K] - Gas constant

RGM - Residual gas mass

SI - Spark ignition

[[sigma].sub.0] - Stoichiometric AFR

SPKT [deg bTDC] - Spark timing

SQP - Sequential Quadratic Programming

[T.sub.a] [K] - Ambient temperature

[tau] - Time constant

TDC - Top dead center

[T.sub.exh] [K] - Exhaust temperature

[T.sub.i] [K] - Orifice input temperature

[[theta].sub.EGR] [deg] - EGR valve angle

[[theta].sub.T] [deg] - Throttle angle

v - Ratio of heat transfer to coolant

[T.sub.m] [K] - Intake manifold temperature

[V.sub.c] [[m.sup.3]] - Cylinder clearance volume

[V.sub.d] [[m.sup.3]] - Engine displacement

[V.sub.m] [[m.sup.3]] - Intake manifold volume

doi:10.4271/2017-01-0608
Table 1. Engine and vehicle parameters

Engine

Fuel                    Gasoline (87 Pump Octane)
Compression Ratio       10.2
Bore                    96 mm
Stroke                  83 mm
Connecting Rod Length   156.5 mm
Intake Valve Diameter   39 mm
Exhaust Valve Diameter  30 mm

Vehicle

Gross weight            2948 kg
Wheelbase x Track       2916 mm x 1633 mm
Gear ratio              4.71/3.14/2.11/1.67/1.29/1.00/0.84/0.67
Axle ratio              3.45

Table 2. Statistics of the proposed SQP model predictive IMEP controller

                                      Mean  Max    Min

Number of major iterations            5     15     1
Number of QP iterations               8     50     1
Execution time per engine cycle (ms)  1.07   9     0.11
Execution time per major iteration    0.21   0.76  0.09
(ms)
Time for model evaluation per major   0.15   0.42  0.07
iteration (ms)
Time for QP per major iteration (ms)  0.06   0.32  0.02
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Author:Zhu, Qilun; Prucka, Robert; Prucka, Michael; Dourra, Hussein
Publication:SAE International Journal of Commercial Vehicles
Article Type:Technical report
Date:May 1, 2017
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