Joint 48th IEEE Conference on Decision and Control and 28th Chinese Control Conference Shanghai, P.R. China, December 16-18, 2009
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Nonlinear Model Predictive Control of Lean NOx Trap Regenerations Ming Feng Hsieh and Junmin Wang* c
Abstract—This paper describes a diesel engine lean NOx trap (LNT) regeneration control system using a nonlinear model predictive control (NMPC) technique. A two-level NMPC architecture is proposed for the LNT regenerations. The control objective is to minimize the fuel penalty caused by LNT regenerations while keeping the tailpipe NOx emissions amount under the regulation level. A physically-based and experimentally-validated nonlinear LNT dynamic model was employed to construct the NMPC control algorithms. The NMPC control system was evaluated on a vehicle simulator, cX-Emissions, with a 1.9L diesel engine model through the FTP75 driving cycle. Compared with a conventional LNT control strategy, 26.6% of fuel penalty reduction was observed during a single regeneration event, and 36.4% fuel penalty reduction was achieved for an entire FTP75 test cycle.
phase. In the meantime, the released is converted to nontoxic gases such as nitrogen by the available reductants. During regenerations, extra fuel is dosed at the upstream of the LNT in order to generate a rich environment that can activate NOx desorption and produce reductants for NOx conversion [2][3]. After the regeneration is finished, LNT regains the NOx adsorption capacity and returns to the NOx adsorption phase. The LNT was demonstrated of being able to capture more than 90% of the diesel engine exhaust NOx. However, a common concern about LNT is that the associated fuel penalty during regenerations can lead to higher overall vehicle fuel consumption. Fuel penalty is the extra fuel mass used for the LNT regenerations and is defined here by the following equation.
I. INTRODUCTION engines have noticeable advantages in terms of D IESEL efficiency, reliability, and power density compared with gasoline counterparts. However, diesel engine emissions control, especially NOx reduction, is much more challenging than that for gasoline engines. The lean burn characteristic of diesel engine is notable of higher NOx emissions compared with gasoline stoichiometric engines. It is now impossible to achieve the tight NOx emission regulations by engine control and combustion improvement only [1]. A catalytic aftertreatment is widely considered necessary for diesel engine vehicles to achieve the stringent NOx emissions regulations. Catalytic aftertreatment systems have been introduced to significantly reduce the tailpipe emissions. Among a variety of selections, the most promising and commonly used NOx aftertreatment systems are lean NOx trap (LNT) for light-duty vehicles and selective catalytic reduction (SCR) for medium- and heavy-duty vehicles. Between these two choices, LNT is the objective to be studied in this paper. The main LNT operation process can be described by two steps. The first step, named NOx adsorption, is a NOx storage reaction in which LNT traps the engine exhaust NOx and stores it as solid state NOx compound. Because the LNT storage capacity is finite, it has to be periodically purged every time the surface coverage is close to full. The second step is called NOx desorption or regeneration. In this step, the solid state NOx compound is released as in gaseous Ming Feng Hsieh is a Ph.D. student in the Department of Mechanical Engineering, The Ohio State University, Columbus, OH 43210 USA (e-mail:
[email protected]). *Corresponding author. Junmin Wang is an assistant professor in the Department of Mechanical Engineering, The Ohio State University, Columbus, OH 43210 USA (e-mail:
[email protected]).
978-1-4244-3872-3/09/$25.00 ©2009 IEEE
.
(1)
: mass air flow rate (g/sec) : air to fuel ratio during regeneration t : engine air to fuel ratio Different LNT regeneration control strategies aiming at fuel penalty and emission reductions have been studied by researchers in industry and academia [3]-[9]. For instance, Nakagawa et al. proposed a model-based approach to control the rich pulse timing and its duration for LNT regenerations [4]. A LNT model adaptation mechanism was used to adjust the rich pulse timing and duration to reduce the tailpipe emissions. In [7], the authors proposed an adaptive LNT purge control strategy, in which on-line adaptation for the LNT capacity and LNT-in NOx flow rate was conducted with the assistance provided by an exhaust gas oxygen sensor. The estimated LNT capacity usage percentage was then used to trigger the LNT regeneration event. The adaptive control schemes for the regeneration trigger control provided many improvements to the NOx slip because the LNT dynamics do change with temperature and many other environmental factors. However, because the dynamics of LNT are essentially chemical reactions, system delay and nonlinearity are two important factors that should be considered when designing a LNT controller. Based on our observations, feed-forward control and linear model based optimal control may not be sufficient enough to compensate the large delay and the high nonlinearity of the LNT reactions. Nonlinear model predictive control (NMPC) approach is thus proposed
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in this paper to take these characteristics into account. Besides regeneration trigger control, due to the highly nonlinear NOx desorption efficiency, the air to fuel ratio (AFR) control during regenerations is also of importance for fuel penalty and emission reduction. However, less attention has been devoted to this aspect. In this paper, in addition to the regeneration trigger control, we also specifically study the optimal AFR profile control during LNT regenerations from the physical aspect using nonlinear model predictive control approach. Based on the simulation study using an experimentally-validated LNT model, it was observed that the designed NMPC AFR controller can increase 26.6% of NOx desorption efficiency comparing with a conventional LNT regeneration AFR PID controller. For the consideration of computational efficiency, a two-level NMPC LNT control scheme is proposed in this paper. While the mentioned lower-level NMPC AFR controller minimizes the fuel penalty during regenerations, a regeneration trigger higher-level NMPC controller considers a longer horizon to find an optimal regeneration point that can maximize the regeneration efficiency. At the same time, the higher-level MPC also forces the tailpipe NOx emissions to be under the desired limit through regeneration timing control. The two-level NMPC LNT control reduced 28.1% of tailpipe NOx and 36.4% of fuel penalty comparing to a conventional LNT controller at a simulated warm-up start FTP75 cycle. The rest of this paper is organized as follows. In section II, a physically-based LNT model is briefly described. An introduction of NMPC and the description of the proposed two-level NMPC LNT control architecture are presented in section III. Sections IV and V describe the NMPC design of the two-level control laws, respectively. Simulation results and analyses are presented in section VI. Conclusive remarks are summarized in section VII.
:
NOx release efficiency,
(2)
where the multipliers and depend linearly on the catalyst temperature with positive first order derivative. The efficiency term accounts for the oxygen available to promote the NOx storage reactions. The mass rate of NOx conversion to can be defined as: ,
,
,
(3)
where the conversion efficiency is a complex function that includes the effects of the mentioned variables. ,
(4)
where is a linear function of the catalyst temperature with a is the mass flow ratio of positive first-order derivative, reductant and NOx, and represents the mass of equivalent reductant ( ) that is required to convert the released NO , normalized to the corresponding stoichiometric value [13][20][26]. III. NMPC FOR LNT CONTROL A two-level NMPC control scheme for LNT control is proposed as shown in Figure 1.
II. LNT MODEL AND VALIDATION The dynamic equations of the LNT reactions, originally derived in [13] and validated in [26], will be briefly summarized in this section. Interested readers can refer [13] and [14] for details regarding the model. The dynamics associated with the NOx adsorption and release (desorption) are described by the following equations. , ,
,
,
,
,
.
,
(1)
is the LNT NOx fill ratio and is the LNT The NOx storage capacity. The NOx storage and release rates can be expressed by: 1
,
,
, ,
,
1 :
NOx storage efficiency,
Figure 1. Two-level MPC control scheme for LNT control
The NMPC controller follows the approach used in the authors’ work [27]. The LNT control is divided into two parts: regeneration trigger control and regeneration AFR profile control. The main objective of the LNT control is to minimize the fuel penalty, as defined by (1), and keep the tailpipe NOx emissions under the regulation. The model used in the NMPC is a simplified 3-state model based on the dynamics described in section II. The model utilizes the available measurements from a typical aftertreatment system as the inputs. The used measurements are from temperature sensor ( ), NOx sensor ( , ), ), and mass air flow sensor ( ) at oxygen sensor ( , upstream of the aftertreatment system. The same set of sensors is also available at the downstream of the LNT. These sensors are used for state estimation. The three states of the model are the oxygen storage, , NOx storage, , and LNT temperature, . The model can be expressed by the following equations:
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1
according to the initial state values. For the real application, the initial states can be obtained by either model prediction or estimator. Some good studies on state estimators in LNT control can be found in [4][7].
,
1
,
1
,
,
,
V. NMPC IN HIGHER-LEVEL CONTROL
,
,
,
,
, ,
(5)
where , is a combination of , , experimental functions and maps to approximate the reductant ( and ) mass flow rate by air to fuel ratio (AFR), oxygen concentration, and mass air flow rate. In addition, the is equal to the engine air to fuel ratio during NOx adsorption, but it is controlled by the lower-level NMPC with additional fuel injection during regenerations.
The objective of the higher-level control (HLC) NMPC is to minimize the overall fuel penalty and regulate the tailpipe NOx emission under a desired value by controlling the regeneration timing. Efficiencies that can be improved by regeneration timing control also affect the fuel penalty and NOx emissions. These efficiencies include LNT release efficiency, storage efficiency, and NOx conversion efficiency. The features of these efficiencies are utlized to design the HLC NMPC. The receding horizon optimization problem is formulated by the following equations. Cost function:
IV. NMPC IN LOWER-LEVEL CONTROL The objective of NMPC in the lower-level control (LLC) is to minimize the fuel penalty caused by regenerations while removing the stored NOx by proper AFR control. The optimization problem of the LLC NMPC is formulated as follows:
|
, |
| |
∑
|
,…,
·
|
,
| |
,
(6)
k: :
: ,
:
1| , , .
: minimum air to fuel ratio : stoichiometric air to fuel ratio NOx storage weighting factor : the simplified LNT model in (5) length of control and prediction horizon
Numerical optimization algorithm was implemented to minimize the cost function by specifying the optimal design | , variables 1| , … , 1| in the horizon. After that, the first design variable, the control | , is applied to the exhaust gas AFR output controller accordingly. This control is then applied to the plant, the initial state of the new receding horizon x k 1|k 1 is obtained from the model. At the same time, NMPC starts calculating the new optimal control sequence
1 ,
,
,
,
(7)
1
,
1| , … , 1
|
,
,
,
Subjected to:
Subjected to: 1 ,
|
∑
,
,
,
,
|
,…,
Constraint:
Constraint: Design variables: | ,
|
,
Design variable: ,
Cost function: |
|
,
1 ,
,
cumulated tailpipe NOx emission (g/mile) : maximum tailpipe NOx emission (g/mile) constraint weighting function with respect to temperature , : constants, , , ,
The AFR during regeneration is simplified as a constant in the prediction model in order to increase the computational speed. The constraint restricts the cumulate tailpipe NOx emission at the end of the receding horizon , 1 to be less than the minimum tailpipe NOx emission required by regulation plus a constraint weighting function, , which is a function of the horizon initial temperature . This function ( ) loosens the tailpipe NOx emissions restriction when the catalyst temperature is low to prevent inefficient regenerations for fuel saving. On the other hand, it tightens the NOx emission restriction ( ) when the catalyst temperature is suitable for regenerations. In such a way, the NMPC achieves the optimization with respect to the slow dynamics as mentioned before.
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VI. SIMULATION RESULTS AND ANALYSES A complete Diesel engine NOx aftertreatment simulator named cX-Emissions was developed in Matlab/Simulink by OSU Center for Automotive Research (CAR) [14]. The aftertreatment model was coupled with a quasi steady-state vehicle model that includes models for engine, transmission, and powertrain components. The simulator is calibrated with data obtained from a 1.9l Diesel engine. This allows the possibility of simulating the aftertreatment performance during steady-state and transient driving conditions. Figure 2 illustrates the combined vehicle-engine-aftertreatment simulation model structure.
penalty by minimizing the fuel-penalty/NOx-removed. As can be seen in Table I, the fuel penalty was reduced by 7% in Case 3 comparing to Case 2. However, the controller can only find a local minimum close to the initial value ( 12). To seek for the global minimum or a better local minimum, by the assumption that a faster NOx release rate can lead to lower fuel penalty, the NOx fill ratio term was added to the cost function in Case 4. As Figure 5 illustrates, the AFR was brought farther away from the initial value and a faster NOx release rate and lower unit NOx fuel penalty were achieved. From Table I, we can see the proposed LLC NMPC reduced 28.3% of unit NOx fuel penalty during this regeneration. 2 PID Control NMPC with k=0 NMPC with k=0.8
NOx Storage (g)
1.5
0.5
Figure 2. Combined vehicle-engine-aftertreatment system simulation model
.
364
364.5
365
365.5 366 Time (sec)
366.5
367
367.5
Figure 3. Comparison of the NOx storages with different AFR control strategies during the first LNT regeneration, the ‘PID control’ and ‘MPC with k=0.8’ have the same NOx storage level at the end in this figure. 291
Without Regeneration PID Control NMPC with k=0 NMPC with k=0.8
290 Fuel Consumption (g)
A. Lower-Level Control Only Four different cases are compared in this section. In the first case, no regeneration was applied to the LNT. This case was set as a reference to compare with other cases using different LNT regeneration controllers. In the second case, a PID controller from a previous work [15] was used to control the LNT regeneration AFR. In the third case, the proposed LLC NMPC control scheme was used. In this case, the weighting factor k was set to zero to minimize the fuel-penalty/NOx-removed ratio only. In the forth case, besides the fuel cost minimization, the NOx fill ratio at the end of the regeneration was taken into account in the cost function. The weighting factor k was set to 0.8 in this case. Figure 3 to Figure 5 show the LNT NOx storage, fuel consumption, and AFR of the four cases in the first regeneration of the FTP75 cycle. We compare the first regenerations because in all the three cases, these regenerations started at the same time and had the same initial conditions. Because different masses of NOx were removed in the three cases, the reference for fuel penalty judgment is the fuel penalty per NOx removal, which is named as “unit NOx fuel penalty” and defined by the following equation.
0
289 288 287 286 285 284 283
364 364.5 365 365.5 366 366.5 367 367.5 Time (sec)
Figure 4. Comparison of the fuel consumptions, the ‘PID control’ and ‘MPC with k=0.8’ have the same NOx storage level at the end in this figure. 16 15
AFR Command
In the following sections, the FTP75 cycle simulation results and analyses based on the physical insights into the LNT characteristics are presented.
1
(8)
PID Control NMPC with k=0 NMPC with k=0.8
14 13 12 11 10
Table I shows the comparisons of the released NOx, fuel penalty, and the unit NOx fuel penalty of the four cases. The results verify that the NMPC approach can reduce the fuel
9
364 364.5 365 365.5 366 366.5 367 367.5 368 368.5 Time (sec)
Figure 5. Comparison of the AFR commands during the first LNT regeneration.
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Speed (m/sec) /Regeneration Trigger
B. Higher-Level Control + Lower-Level Control In this section, instead of the conventional regeneration trigger approach that issues regenerations according to the NOx slip value, HLC NMPC was used for regeneration trigger control. Three cases were compared: the Case 1 and Case 2 mentioned before, and Case 5, which uses LLC NMPC and HLC NMPC for regeneration AFR control and regeneration trigger control, respectively. The tailpipe NOx emission constraint was set to 0.07 g/mile as the , , US EPA Tier II Bin 5 light-duty vehicle regulation. Table II compares the simulation results of the studied cases. One can see that the NMPC LNT control approach reduced 36.4% of fuel penalty compared with that of the conventional LNT controller. Moreover, the NOx emission of Case 5 is 0.061 g/mile, which is 28% less than that of Case 2 and the number is under the EPA regulation as expected.
fuel penalty with regard to the fast dynamics in a finite horizon. On the other hand, the conventional regeneration trigger controller that has no prediction ability can trigger the regeneration only when the considered values passed the thresholds. As can be seen in Table II, by implementing HLC NMPC as the regeneration trigger controller, with only 1 gram extra fuel penalty, the tailpipe NOx emissions were reduced by 24.3% compared with that obtained in Case 4.
0.25
Without Regeneration Conventional Control NMPC with HLC and LLC
0.2 0.15 0.1 0.05 0 0
500
1000 Time (sec)
1500
Figure 6. Comparison of the tailpipe NOx emissions during the FTP 75 cycle
Figure 7 shows the regeneration triggers with a reference to the vehicle speed and Figure 8 is the zoomed-in version of the first regeneration. Figure 8 is an example that shows the HLC NMPC can find the best regeneration point in the finite horizon. From Figure 6, we can see the tailpipe NOx emission of Case 5 just passed 0.05 g/mile when the first regeneration started. Intuitively, the regeneration should be triggered later than it was because the NOx emission was 28% lower than the emission limit. However, by looking at the speed curve, we can see the vehicle slowed down just after the regeneration, which means that the engine load started to decrease after the regeneration and regeneration will be less efficient after the point where regeneration was triggered. Figure 9 also explains the situation. The regeneration was triggered at the point where the minimum engine AFR and high temperature were presented. As explained before, low engine-out AFR can contribute to lower fuel penalty. And also high temperature can contribute to a high regeneration efficiency. This result confirms the NMPC’s ability of minimizing the
Vehicle Speed Trigger - Conventional Control Trigger - NMPC with HLC and LLC
30
20
10
0 0
500
1000 Time (sec)
1500
25 20 15 10 5 0 200
250
300 350 Time (sec)
400
450
Figure 8. Comparison of the first regeneration timing between the conventional regeneration trigger control and the MPC based regeneration trigger control. 5
7 Eng. Power (RPM*Nm) Reg. Trig. AFR MAF (g/sec)
Tailpipe NOx Emission (g/mile)
0.35 0.3
40
Figure 7. Comparison of the regeneration timings between the conventional regeneration trigger control and the MPC based regeneration trigger control based on the vehicle speed during the FTP 75 cycle. Speed (m/sec) /Regeneration Trigger
TABLE I COMPARISON OF FUEL CONSUMPTION AT THE FIRST REGENERATION Unit NOx Fuel Released NOx Fuel Penalty Controller Penalty (g) (g) Case 1 0 0 0 Case 2 1.361 4.7 3.453 Case 3 0.963 3.1 3.219 Case 4 1.263 3.2 2.533
x 10
6 5 4
Eng. Power Reg. Trigger AFR*104 MAF*3*103 Temp.*103
3 2 1 0 250
260
270 280 Time (sec)
290
300
Figure 9 Comparison of regeneration timing, engine output power, mass air flow rate (MAF), and air to fuel ration (AFR) in the first regeneration region.
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TABLE II COMPARISON OF FUEL CONSUMPTION AND NOX EMISSION AT THE END OF THE FTP 75 CYCLE Tailpipe NOx Fuel Consumption Fuel Penalty Controller Emission (g/mile) (g) (g) Case 1
1149.35
0
0.3125
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1164.71 1162.41 1158.16 1159.16
15.4 13.0 8.8 9.8
0.0847 0.0922 0.0807 0.0611
[7] [8]
VII. CONCLUSION
[9]
In this paper, a two-level nonlinear model predictive control approach was proposed for LNT control application. The higher-level NMPC administrates the regeneration trigger control, and the lower-level NMPC controls the air to fuel ratio profile during regenerations triggered by the higher-level NMPC. The control designs were based on the physical understanding of the LNT reactions. The prediction ability of NMPC makes the regenerations be triggered at the high efficiency points and allows the AFR control with respect to LNT NOx fill ratio and engine operating conditions. Different NMPC designs were compared with a conventional LNT control algorithm and analyzed base on a diesel engine aftertreatment simulator developed by OSU-CAR. Through the simulation studies, 26.6% reduction of the unit NOx fuel penalty was observed during a single regeneration. In addition, 28.1% of tailpipe NOx reduction and 36.4% of fuel penalty reduction were achieved through the entire FTP75 cycle. The LNT regeneration efficiency was greatly increased by using the proposed two-level NMPC control strategy. The results obtained in this paper show the effectiveness and benefits of model predictive control approach for diesel engine aftertreatment control applications. However, the computational issue of MPC was not solved in this study. Simulations indicated the computational requirement is above the capability of a real-time controller. A specific computationally-efficient MPC approach for Diesel aftertreatment control has been proposed in [28]. Implementation of this approach to LNT is left as a future work. ACKNOWLEDGMENT The authors would like to thank the members of the OSU CAR Industrial Consortium for providing financial support to this project. REFERENCES [1] [2] [3]
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