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Reference Management for Aftertreatment Temperature Control of Automotive Diesel Engines
2012 Workshop on Engine and Powertrain Control, Simulation and Modeling The International Federation of Automatic Control Rueil-Malmaison, France, October 23-25, 2012
Reference Management for Aftertreatment Temperature Control of Automotive Diesel Engines Hayato Nakada ∗ Gareth Milton ∗∗ Peter Martin ∗∗ Akiyuki Iemura ∗ Akira Ohata ∗ ∗
Higashifuji Technical Center, Toyota Motor Corporation, 1200, Mishuku, Susono-city, Shizuoka 410-1193, Japan (e-mail: [email protected]) ∗∗ Shoreham Technical Centre, Ricardo UK Ltd, Shoreham-by-Sea, West Sussex, BN43 5FG, UK Abstract: This paper describes aftertreatment temperature control of automotive diesel engines based on reference governor (RG). The RG algorithm computes a modified reference of catalyst temperature so that the predicted output of the closed-loop system on a finite prediction horizon does not exceed the constraint of its maximal temperature. We apply a bisectional search algorithm to the RG where the search region is extended in order to take account of model uncertainties. We show the effectiveness of the present method through an experiment with a production vehicle. Keywords: Reference governor, bisectional search, automotive control, diesel engine, aftertreatment temperature, physical models. 1. INTRODUCTION Plants in real world have constraints that should be considered in the control. For example, in air path control of automotive diesel engines, boost pressure and turbine speed are constrained under their allowable maximum limits. In aftertreatment control of the diesel engines, there are temperature constraints under which performance of catalysts is guaranteed in the hardware design. Generally, if we neglect such constraints in control design, it is wellknown that the closed-loop behaviour leads to degraded control performance, or worse, to instability (Gilbert, 1992). In control systems developed by industry, constraints are satisfied through a great deal of experimental trial-anderror effort in adjusting many constants, maps and tables. Especially, the automotive control algorithms have a lot of switches among parameters, tables and sub-routines. Such control structure requires many man-hours and great effort to calibrate. Recently, regulations on emissions from automotive vehicles become more stringent, and then the control system including both hardware and software have become more complicated. Along with increasing complexity of automotive control systems, more constraints should be treated in the control design. For this reason, control methods fulfilling constraints based on model prediction receive a lot of attention in the automotive control community (del Re et al., 2010). One of the control design methods considering constraints is reference governors (RG) (Albertoni et al., 2003; Bemporad, 1998; Gilbert and Kolmanovsky, 2002; Hirata and Fujita, 1999; Kogiso and Hirata, 2003; Oh and Agrawal, 978-3-902823-16-8/12/$20.00 © 2012 IFAC
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2005; Oh-hara and Hirata, 2003; Vahidi et al., 2007). The RG modifies original references so that the closedloop system does not violate constraints, by predicting the future behaviour using plant models. Recent theoretical research works on RGs contain (i) an iterative search-based method (Gilbert and Kolmanovsky, 2002; Vahidi et al., 2007), (ii) a method based on constrained positive invariant set (Hirata and Fujita, 1999), and (iii) a reference modification method based on a piecewise affine function (Kogiso and Hirata, 2003). These works and related ones demand a lot of computational burden, so tend to go beyond the applicability to vehicle control. Moreover, recent application studies of RGs include a cable robot application (Oh and Agrawal, 2005), and a simulation study of compressor control in fuel cell system (Vahidi et al., 2007). As far as the authors know, there are not so many research results on applications of the RG methods to control systems requiring realtime and high-speed computations. Considering real-time executability and limited computation resources as in automotive controllers, we need to study a more efficiently computable algorithm of RGs. In this paper, we pursue a real-time RG algorithm applicable to automotive control. The RG is applied to aftertreatment temperature control of automotive diesel engines. Then, by exploiting ‘fast RG’ (Gilbert and Kolmanovsky, 2002; Vahidi et al., 2007), we propose an RG algorithm that is suitable to automotive control applications with limited computational resources. In order to predict future behavior in the RG, we need to derive a thermal model of the aftertreatment system. The model should be as 10.3182/20121023-3-FR-4025.00003
2012 IFAC E-CoSM (E-CoSM'12) Rueil-Malmaison, France, October 23-25, 2012
derive a control-oriented model rather than a complicated physical model so that the model can be repeatedly used in computation on a real-world embedded controller. Namely, we derive the model so that it can be used for computation of future prediction of constrained variables in the RG.
Fuel injector
Turbine
Engine block
DOC DPF
In defining the models, we put the following assumptions for simplicity.
@ @ Temperature sensors
Fig. 1. Aftertreatment system of a diesel engine simple as possible to compute the prediction of future outputs in a finite horizon for a number of candidates of modified reference on automotive computers. Rather than building a complicated physical model, we put emphasis on control-oriented modeling in this paper. Namely, we derive a lumped two-state thermal model for the aftertreatment system that is suitable in the online RG algorithm. While, in the conventional fast RG, a modified reference is calculated via a bisectional search algorithm, in the present method, we extend the search region to the opposite direction to a reference change in order to avoid constraint violation due to model uncertainties. The paper is organized as follows. In Section 2, we describe an aftertreatment system of automotive diesel engines, then in Section 3 the thermal model of the system is derived. In Section 4, we introduce catalyst temperature constraints and design an RG algorithm. We show the effectiveness of the present method through an experiment with a production vehicle in Section 5. Finally, we summarize the paper and mention a future topic in Section 6. 2. SYSTEM CONFIGURATION In this paper, we consider an aftertreatment system of an automotive diesel engine, shown in Fig. 1. This system is composed of a diesel oxidation catalyst (DOC) to reduce hydrocarbon and carbon monoxide, and a diesel particulate filter (DPF) to collect particulate matter (PM). Requirements for the control of this aftertreatment system are as follows. • In order to maintain conversion efficiency, the catalyst is required to be warmed up in a short time frame. • To maintain DPF performance, PM stored in DPF should be burned periodically.
• The DOC and DPF are respectively treated as lumped masses, and spatial temperature distributions are not considered. • Temperatures in DOC and DPF are uniform, respectively. • Influence of vehicle speed on temperature fluctuation is negligible. • The following disturbances are fixed. · Mass flow rate and temperature of exhaust gas. · Ambient temperature. In reality, the mass flow rate and the temperature of exhaust gas can vary. However, the assumption that these disturbances are constant is made, because the model is used for future prediction in the RG as described in Section 4. In the prediction in the RG, the mass flow rate and the temperature of exhaust gas are to be regarded as exogenous inputs, and the future prediction is done by assuming that such variables stay constant in the future due to the causality. 3.1 Modeling of DOC From the heat balance as shown in Fig. 2, we obtain dTdoc 1 = [Qexo,doc − Qair,doc − Qexh,doc ] , (1) dt Cdoc Mdoc where a nomenclature is shown in Appendix. The heat transfers from DOC to the atmosphere and to the exhaust gas are described by Qair,doc = Katm (Tdoc − Tatm ) Qexh,doc = hdoc Adoc [Tdoc − (Rdoc Tdoc,us + (1 − Rdoc ) Tdoc,ds )] , (3) where in the right-hand side of (3) the term Rdoc Tdoc,us + (1 − Rdoc ) Tdoc,ds implies the temperature at a representative point between upstream and downstream of DOC, given the assumption that the spatial distribution of temperature in DOC is not considered. Regarding the gas inside DOC, the heat exchange is expressed as
As an actuator for the aftertreatment control, a fuel injector is located at an exhaust port of the cylinder head. When the fuel is introduced into the catalyst, a chemical reaction occurs and the catalyst temperature increases. It should be noticed that there are constraints on maximal temperatures for both DOC and DPF, that should be considered in the fuel injection control. 3. PLANT MODELING In this section, we derive models of DOC and DPF temperatures. The objective of modeling in this paper is to
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(2)
Fig. 2. Heat transfer diagram in DOC
2012 IFAC E-CoSM (E-CoSM'12) Rueil-Malmaison, France, October 23-25, 2012
where the heat transfer from DPF to the atmosphere and to exhaust gas are respectively given by Qair,dpf = Katm (Tdpf − Tatm )
(8)
Qexh,dpf = hdpf Adpf [Tdpf − {Rdpf Tdpf,us +(1 − Rdpf )Tdpf,ds }] .
(9)
Regarding the exhaust gas inside DPF, the heat balance is expressed as Fig. 3. Heat transfer diagram in DPF dTgas,doc = W Cgas (Tdoc,us − Tdoc,ds ) dt +Qexh,doc . Assume that the mass of gas in DOC is quite small compared to the gas flow inside DOC, namely Mgas,doc ≈ 0. By substituting (3) into the above equation, we obtain
Cgas Mgas,dpf
Cgas Mgas,doc
W Cgas (Tdoc,ds − Tdoc,us )
NT,dpf,ds DT,dpf,ds NT,dpf,ds = hdpf Adpf Tdpf + Tdpf,us × (W Cgas − Rdpf hdpf Adpf )
NT,doc,ds W Cgas + (1 − Rdoc )hdoc Adoc NT,doc,ds = hdoc Adoc Tdoc + Tdoc,us × Tdoc,ds =
DT,dpf,ds = W Cgas + (1 − Rdpf )hdpf Adpf . Substituting Tdpf,ds into (9) gives
(W Cgas − Rdoc hdoc Adoc ). Substituting Tdoc,ds into (3) yields NQ,exh,doc DQ,exh,doc NQ,exh,doc = hdoc Adoc W Cgas (Tdoc − Tdoc,us )
The mass Mgas,dpf of exhaust gas inside DPF can be considered extremely small, compared to the exhaust gas flow W , namely Mgas,dpf ≈ 0. By eliminating Qexh,dpf from (9) and (10), Tdpf,ds is described by Tdpf,ds =
= hdoc Adoc [Tdoc − (Rdoc Tdoc,us + (1 − Rdoc ) Tdoc,ds )] . Solving this equation with respect to Tdoc,ds gives
Qexh,doc =
dTgas,dpf = W Cgas (Tdpf,us − Tdpf,ds ) dt +Qexh,dpf . (10)
NQ,exh,dpf DQ,exh,dpf = hdpf Adpf W Cgas
Qexh,dpf = (4)
NQ,exh,dpf
(11)
× [W Cgas (Tdpf − Tdoc,us ) +hdoc Adoc (Tdpf − Tdoc )
DQ,exh,doc = W Cgas + (1 − Rdoc )hdoc Adoc .
+Rdoc hdoc Adoc (Tdoc,us − Tdpf )]
The chemical release of heat from the gas to the catalyst brick surface is considered as a function of the fuel injection quantity, the DOC temperature and the mass flow of exhaust gas, namely Qexo,doc = Qexo,doc (Qinj , Tdoc , W ). (5) The detailed description of (5) is shown in Appendix A.1. By substituting (2),(4),(5) into (1), we obtain a differential equation with respect to the DOC temperature as dTdoc 1 = [Qexo,doc (Qinj , Tdoc , W ) dt Cdoc Mdoc ] NQ,exh,doc −Katm (Tdoc − Tatm ) − . (6) DQ,exh,doc It should be noted that the amount of fuel injection Qinj is the control input, and that a constraint is at the maximal limit of the DOC temperature Tdoc . 3.2 Modeling of DPF Similarly to DOC, a thermal model of DPF is derived in this sub-section. From the heat balance shown in Fig 3, the DPF brick temperature is obtained as dTdpf 1 = [Qexo,dpf − Qair,dpf −Qexh,dpf ] , (7) dt Cdpf Mdpf
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DQ,exh,dpf = (W Cgas + (1 − Rdpf )hdpf Adpf ) × (W Cgas + (1 − Rdoc )hdoc Adoc ) . We suppose that the exotherm generated by burned PM is a function of the exhaust gas mass flow, the mass of PM accumulated in DPF, the mass of fuel injection, the DOC and DPF temperatures, i.e., Qexo,dpf = Qexo,dpf (W, mpm , Qinj , Tdoc , Tdpf ).
(12)
Again, the detailed description of (12) is shown in Appendix A.2. Substituting (8), (11) and (12) into (7) yields the differential equation of the DPF temperature given by dTdpf 1 = × dt Cdpf Mdpf [Qexo,dpf (W, mpm , Qinj , Tdoc , Tdpf ) ] NQ,exh,dpf −Katm (Tdpf − Tatm ) − . DQ,exh,dpf
(13)
Note that the mass of PM mpm is assumed to be an available signal in the temperature control. Thus, we do not introduce the dynamics of the mass of PM here as in (Hiroyasu et al., 1983; Hirsch et al., 2010).
2012 IFAC E-CoSM (E-CoSM'12) Rueil-Malmaison, France, October 23-25, 2012
3.3 State-space model From (6) and (13), we obtain the state-space model of the plant x˙ p = fp (xp , u, w) c = xp
{
?? rg-+ e eRG −6
xp
v FB controller
w
u-
? Plant
yr-
(15)
y = Tdpf , T
(14)
x
r
(16)
where xp = [ Tdoc Tdpf ] ∈ R is the state of the system, T w = [ W Tatm mpm Tdoc,us ] ∈ R4 is the exogenous input, u = Qinj ∈ R is the control input, c ∈ R2 is the variables constrained and y ∈ R is the output. The function fp : R2 × R × R4 → R2 is inferred from (6) and (13). 2
3.4 Feedback controller In this paper, the tracking control of the DPF temperature ref Tdpf to the reference r = Tdpf ∈ R is considered. Here, we design a PI feedback controller described by u = KP e + KI v
(17)
Fig. 4. Block diagram of control system 4. DESIGN OF REFERENCE GOVERNOR 4.1 Definition of constraints We put constraints on DOC and DPF temperatures as shown below. For prescribed T¯doc , T¯dpf (> 0), it is necessary to ensure Tdoc ≤ T¯doc and Tdpf ≤ T¯dpf . Equivalently, for a set { [ ] [ ]} Tdoc T¯doc 2 C= c∈R : 0
v˙ = e e = r − y, where e is the tracking error, and v denotes the state of the integrated tracking error, and KP and KI are coefficients. Note that the integrator can be reset in the implementation of the PI feedback controller to prevent the integrator wind-up if u = Qinj = 0. 3.5 Closed-loop system By applying the feedback controller (17) to the plant (14), we obtain the continuous-time closed-loop system
This subsection considers the design of an RG algorithm that modifies the DPF reference to meet the DOC and DPF temperature constraints. As shown in Fig. 4, the RG modifies the original reference rk to gk using a parameter αk ∈ R via the equation described by gk = gk−1 + αk (rk − gk−1 ), where αk ∈ [αmin , 1] with a given αmin ∈ R. If αk = 1, then gk = rk , namely, the RG outputs the original reference. If αk = 0, then gk = gk−1 , equivalently, the RG returns the previous modified reference. The optimal αk is chosen so that the predicted closedloop behavior over a finite prediction horizon satisfies the constraints. Namely, the optimal αk can be obtained by solving the optimization problem
x˙ = fcl (x, w, r) c = h(x, w), T
where x = [ xp v ] is the state of the closed-loop system and fcl : R4 × R4 × R → R4 can be derived by fp . By approximating x˙ to (xk+1 −xk )/δ with sampling period δ(> 0), we obtain the discrete-time closed-loop model xk+1 = f (xk , wk , rk ) ck = h(xk , wk ), where the subscript k = 0, 1, . . . denotes the discrete time step, and is attached to continuous-time vectors to express their discrete-time version. The function f : R4 × R4 × R → R4 can be obtained from fcl . It should be noted that f is nonlinear due to the nonlinearities in (6) and (13). Though the sampling period δ is set in a trial-and-error manner in this paper, we need to consider the relation between the speed of system response and the prediction horizon in RG as described in Section 4.2. For example, if a small δ is chosen, the future estimation becomes precise, but the resulting computational cost becomes high because a large number of prediction horizon may be required.
380
Given rk , xk , wk , gk−1 , maximize αk such that
(19)
xk+i+1|k = f (xk+i|k , wk , gk )
(20)
ck+i|k = h(xk+i|k , wk , gk )
(21)
ck+i|k ∈ C
(22)
xk|k = xk
(23)
gk = gk−1 + αk (rk − gk−1 )
(24)
αk ∈ [αmin , 1]
(25)
i = 0, 1, . . . , Nh , where xk+i|k and ck+i|k denote the predicted state and constrained variables at time k + i given a measurement at time k, respectively. The nonnegative integer Nh is the prediction horizon. The objective function of the optimization problem (19) implies that the modified reference is chosen as close as possible to the original reference while fulfilling the constraints. It should be noted that the exogenous
In regards to the choice of prediction horizon Nh , it is preferable to take it longer than the settling time of the closed-loop system. However, there are limitations on computation resources in real-world controllers. We need to consider the compromise between such limitation and the length of the prediction horizon. The optimization problem (19) includes a nonlinear constraint (20). So, it is not easy to obtain its global optimum. Instead, we solve the optimization problem by the following iterative algorithm based on the bisectional search to acquire a sub-optimal solution. [Bisectional search algorithm] 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
DOC temperature K
input wk does not vary over the prediction horizon, since future exogenous values are not available.
Modified DPF temperature reference K
2012 IFAC E-CoSM (E-CoSM'12) Rueil-Malmaison, France, October 23-25, 2012
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j := 0 (j) (j) (j) (j) αmin := αmin , αmax := 1, αmid := αmax (j) (j) while αmin < αmax (j) αk := αmid solve (20), (21), (23) and (24) if (22) is satisfied ∀i = 0, 1, . . . Nh (j+1) (j) (j+1) (j) αmin := αmid , αmax := αmax (j+1) (j) (j+1) (j) else αmax := αmid , αmin := αmin end if j = jmax (j) αk∗ := αmin , and break end j := j + 1 ( ) (j) (j) (j) αmid := 12 αmin + αmax end
Original reference
900
25
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35
40 45 Time [sec]
50
Fig. 5. Modified reference without/with back-off reference modification the prediction horizon is set to Nh = 5 and the sampling period δ = 0.5 sec, therefore the behavior 2.5 sec into the future is predicted. In the case of αmin = 0, once the modified reference approaches the original reference, it can not go in the opposite direction and it remains large. This leads to violation of the DOC temperature constraint. In the case of αmin = −0.2, the reverse direction reference modification is made, and the constraint violation is averted.
In the literature (Gilbert and Kolmanovsky, 2002; Vahidi et al., 2007), by setting αmin = 0, gk is always selected in the interval [gk−1 , rk ]. But, once gk is chosen, in the following time step a selection of the next gk must be made toward rk , so a correction in the opposite direction is not permitted. For engine control, model uncertainties as well as measurement noise can be large. Without the ability to make this reverse direction correction, there is a high possibility that the constraints will be violated. To avoid this, we admit setting αmin ∈ [−1, 0) in this paper. Its calibration can be done through repeated simulation of the closed-loop system with the RG. We have applied a bisectional search algorithm to obtain a sub-optimal solution here, but other optimization methods such as the steepest descent algorithm (Vanderpl, 1984) may be applicable. In such a case, it is necessary to consider again their implementability and computational costs. 4.3 Simulation result In Fig. 5, we show a simulation example of the opposite direction reference modification of the RG. The plant model is parameterized with a vehicle data. The upper plot depicts the original DPF reference and the modified references, and the bottom plot represents the resulting DOC temperatures. The dotted curves correspond to αmin = 0 and the solid curves to αmin = −0.2. In both cases, as
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As the catalyst temperature characteristics are strongly nonlinear, the response time can vary significantly. Even if the prediction horizon is calibrated at a finite number of operating conditions, there is still a possibility of violating temperature constraints at an operating condition between adjacent calibration points. By admitting the negative αmin as a tuning parameter in the application of RG, the RG can modify the reference so that the constraints are satisfied. 5. EXPERIMENT RESULT In this section, we apply the RG algorithm described in Section 4 to the aftertreatment temperature control of a production vehicle with a 4-cylinder 2.0 l diesel engine. Operating in constant engine speed of 2200 rpm and the fuel quantity of 22.8 mm3 /stroke, the DPF temperature step reference is introduced. Here, as an example of experimental condition, we set the upper bound of DPF temperature to 890 K. Note that we need to know the feedback controller in predicting the closed-loop behavior in the RG, the coefficients KP and KI of the feedback controller (17) are tuned in a trial-and-error method. Therefore, this control strategy is not a production level. The parameters of the RG algorithm are set as follows. First, we set the sampling period for discretization of the plant model to 5 sec and the prediction horizon to Nh = 20, therefore, the RG predicts for 100 sec ahead. The lower bound of the search region for modified references is fixed as αmin = −0.05. The number of iterations in the bisectional search algorithm is jmax = 6. The controller including the RG runs at a period of 0.1 sec.
2012 IFAC E-CoSM (E-CoSM'12) Rueil-Malmaison, France, October 23-25, 2012
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Fig. 6. Comparison of DPF temperature with/without RG Fuel injection mm3/s
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We have considered temperature constraints in DOC and DPF in this paper in order to show the applicability of RG to real-world plants for establishing a fundamental result of RG applications. By extending the present method, more detailed constraints in practical control systems can be treated. Future topics include the extension of the present method to multiple references and its application to control in a production vehicle. ACKNOWLEDGEMENTS The authors gratefully acknowledge the contributions of Dr. Tony Truscott, Mr. Peter Fussey of Ricardo UK Ltd., Mr. Takeshi Iwanami and Mr. Martin Egginton of Ricardo Japan Ltd. for valuable comments in regards to vehicle testing.
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temperature control of automotive diesel engines. The plant model has been derived based on physical principles governing the DOC and DPF temperatures. Considering uncertainties in real-world control systems, the search region for modified references has been extended by allowing modification of the reference in the opposite direction. From the result of the vehicle testing, we have shown the effectiveness and applicability of the present method.
100 120 Time sec
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Fig. 7. Comparison of exhaust injection command and PM with/without RG The parameter Rdoc is calibrated using a as a function of Tdoc and W , that varies from 0.02 to 0.51 on the planar space. The experimental data is shown in Figs. 6 and 7. Fig. 6 depicts the original and modified references, and the maximum of actual DPF temperatures at four measurement points with thermocouples placed in line on the central axis of the DPF column. And in Fig. 7 the fuel injection Qinj and the PM in DPF mpm are plotted. From these figures, we see that, by modifying the original DPF temperature reference downward, the fuel injection quantity is reduced, therefore, the constraint on the DPF temperature is satisfied. The above experimental result shows the effectiveness of the present method and the applicability to the real-world control system. 6. CONCLUSION In this paper, we have focused on an RG method modifying a reference based on model predictions taking account of constraints. The RG has been applied to aftertreatment
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REFERENCES Albertoni, L., Balluchi, A., Casavola, A., Gambelli, C., Mosca, E., and Sangiovanni-Vincentelli, A. (2003). Hybrid command governors for idle speed control in gasoline direct injection engines. Proceedings of the 2003 American Control Conference, 773–778. Bemporad, A. (1998). Reference governor for constrained nonlinear systems. IEEE Transactions on Automatic Control, 43(3), 415–419. del Re, L., Allgower, F., Glielmo, L., Guardiola, C., and Kolmanovsky, I. (2010). Automotive Model Predictive Control: Models, Methods and Applications. SpringerVerlag. Gilbert, E. (1992). Linear control systems with pointwisein-time constraints: what do we do about them? Proceedings of the 1992 Americal Control Conference, 2595. Gilbert, E. and Kolmanovsky, I. (2002). Nonlinear tracking control in the presence of state and control constraints: a generalized reference governor. Automatica, 38(12), 2063–2073. Hirata, K. and Fujita, M. (1999). Set of admissible reference signals and control of systems with state and control constraints. Proceedings of the 38th IEEE Conference on Decision and Control, 1427–1432. Hiroyasu, H., Kadota, T., and Arai, M. (1983). Development and use of a spray combustion modelling to predict diesel engine efficiency and pollutant emission. Bulletin of The Japan Society of Mechanical Engineers, 26(214), 569–575. Hirsch, M., Oppenauer, K., and del Re, L. (2010). Dynamic engine emission models. In Automotive model predictive control, chapter 5, 73–87. Springer-Verlag. Kogiso, K. and Hirata, K. (2003). A reference governor in a piecewise state affine function. Proceedings of the 42nd IEEE Conference on Decision and Control, 1747–1752.
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Oh, S. and Agrawal, S. (2005). A reference governorbased controller for a cable robot under input constraints. IEEE Transactions on Control Systems Technology, 13(4), 639–645. Oh-hara, S. and Hirata, K. (2003). Experimental evaluation of on-line reference governors for constrained systems. Proceedings of the SICE Annual Conference 2003, 1458–1463. Vahidi, A., Kolmanovsky, I., and Stefanopoulou, A. (2007). Constraint handling in a fuel cell system: a fast reference governor. IEEE Transactions on Control Systems Technology, 15(1), 86–98. Vanderpl, G. (1984). Numerical optimization techniques for engineering design with applications. Mchraw-Hill book company. Appendix A. DERIVATION OF EXOTHERM TERMS
Table B.1. Nomenclature Variable
Quantity
Unit
Adoc , Adpf
Convective surface area of the DOC/DPF monolith DOC/DPF specific heat capacity Specific heat capacity of exhaust gas Convective heat transfer coefficient between DOC/DPF and gas Lower heating value of fuel Heat transfer constant between DPF and ambient Mass of PM DOC/DPF mass Mass of gas in DOC/DPF Heat transfer rate from DOC/DPF monolith to air Heat transfer rate from DPC/DPF monolith to exhaust gas DOC/DPF exothermic heat of reaction Fuel injection to catalyst Heat transfer rate of soot accumulated in DPF Temperature weighting constants in [0, 1] in DOC/DPF Ambient temperature DOC/DPF monolith temperature Exhaust gas temperature upstream/downstream of the DOC/DPF Temperature of gas in DOC/DPF Mass flow rate of exhaust gas
m2
Cdoc , Cdpf Cgas
A.1 Exotherm term in DOC
hdoc , hdpf
The heat conversion of Qinj depends mainly on the exhaust gas mass flow W and the DOC temperature Tdoc . By introducing the conversion efficiency ηexo,doc (Tdoc , W ) ∈ [0, 1], we describe Qexo,doc = ηexo,doc (Tdoc , W )Hv Qinj . (A.1)
Hv Katm
mpm Mdoc , Mdpf Mgas,doc , Mgas,dpf
A.2 Exotherm term in DPF The exotherm in DPF Qexo,dpf consists of two portions, namely, that from PM accumulated in DPF Qexo,dpf,pm and that of the fuel injected and slipping through DOC without burning Qexo,dpf,slip . Thus, Qexo,dpf is expressed as Qexo,dpf = Qexo,dpf,pm + Qexo,dpf,slip . (A.2) The exotherm of PM is supposed to be a function of DPF temperature and mass of PM, namely Qexo,dpf,pm = Qexo,dpf,pm (Tdpf , mpm ). (A.3) By considering (A.1), Qexo,dpf,slip is given by Qexo,dpf,slip = [1 − ηexo,doc (Tdoc , W )] Hv Qinj . (A.4) By substituting (A.3) and (A.4) into (A.2), we obtain Qexo,dpf = Qexo,dpf,pm (Tdpf , mpm ) + [1 − ηexo,doc (Tdoc , W )] Hv Qinj . Appendix B. NOMENCLATURE
Qair,doc , Qair,dpf
Qexh,doc , Qexh,dpf
Qexo,doc , Qexo,dpf Qinj Qsoot
Rdoc , Rdpf
Tatm Tdoc , Tdpf Tdoc/dpf,us/ds
We summarize the nomenclature in Section 3 in the following table. Tgas,doc , Tgas,dpf W
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J/(kg·K) J/(kg·K) W/(m2 ·K)
J/kg W/K
kg kg kg W
W
W kg/s W
K K K
K kg/s
Reference Management for Aftertreatment Temperature Control of Automotive Diesel Engines Hayato Nakada ∗ Gareth Milton ∗∗ Peter Martin ∗∗ Akiyuki Iemura ∗ Akira Ohata ∗ ∗
Higashifuji Technical Center, Toyota Motor Corporation, 1200, Mishuku, Susono-city, Shizuoka 410-1193, Japan (e-mail: [email protected]) ∗∗ Shoreham Technical Centre, Ricardo UK Ltd, Shoreham-by-Sea, West Sussex, BN43 5FG, UK Abstract: This paper describes aftertreatment temperature control of automotive diesel engines based on reference governor (RG). The RG algorithm computes a modified reference of catalyst temperature so that the predicted output of the closed-loop system on a finite prediction horizon does not exceed the constraint of its maximal temperature. We apply a bisectional search algorithm to the RG where the search region is extended in order to take account of model uncertainties. We show the effectiveness of the present method through an experiment with a production vehicle. Keywords: Reference governor, bisectional search, automotive control, diesel engine, aftertreatment temperature, physical models. 1. INTRODUCTION Plants in real world have constraints that should be considered in the control. For example, in air path control of automotive diesel engines, boost pressure and turbine speed are constrained under their allowable maximum limits. In aftertreatment control of the diesel engines, there are temperature constraints under which performance of catalysts is guaranteed in the hardware design. Generally, if we neglect such constraints in control design, it is wellknown that the closed-loop behaviour leads to degraded control performance, or worse, to instability (Gilbert, 1992). In control systems developed by industry, constraints are satisfied through a great deal of experimental trial-anderror effort in adjusting many constants, maps and tables. Especially, the automotive control algorithms have a lot of switches among parameters, tables and sub-routines. Such control structure requires many man-hours and great effort to calibrate. Recently, regulations on emissions from automotive vehicles become more stringent, and then the control system including both hardware and software have become more complicated. Along with increasing complexity of automotive control systems, more constraints should be treated in the control design. For this reason, control methods fulfilling constraints based on model prediction receive a lot of attention in the automotive control community (del Re et al., 2010). One of the control design methods considering constraints is reference governors (RG) (Albertoni et al., 2003; Bemporad, 1998; Gilbert and Kolmanovsky, 2002; Hirata and Fujita, 1999; Kogiso and Hirata, 2003; Oh and Agrawal, 978-3-902823-16-8/12/$20.00 © 2012 IFAC
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2005; Oh-hara and Hirata, 2003; Vahidi et al., 2007). The RG modifies original references so that the closedloop system does not violate constraints, by predicting the future behaviour using plant models. Recent theoretical research works on RGs contain (i) an iterative search-based method (Gilbert and Kolmanovsky, 2002; Vahidi et al., 2007), (ii) a method based on constrained positive invariant set (Hirata and Fujita, 1999), and (iii) a reference modification method based on a piecewise affine function (Kogiso and Hirata, 2003). These works and related ones demand a lot of computational burden, so tend to go beyond the applicability to vehicle control. Moreover, recent application studies of RGs include a cable robot application (Oh and Agrawal, 2005), and a simulation study of compressor control in fuel cell system (Vahidi et al., 2007). As far as the authors know, there are not so many research results on applications of the RG methods to control systems requiring realtime and high-speed computations. Considering real-time executability and limited computation resources as in automotive controllers, we need to study a more efficiently computable algorithm of RGs. In this paper, we pursue a real-time RG algorithm applicable to automotive control. The RG is applied to aftertreatment temperature control of automotive diesel engines. Then, by exploiting ‘fast RG’ (Gilbert and Kolmanovsky, 2002; Vahidi et al., 2007), we propose an RG algorithm that is suitable to automotive control applications with limited computational resources. In order to predict future behavior in the RG, we need to derive a thermal model of the aftertreatment system. The model should be as 10.3182/20121023-3-FR-4025.00003
2012 IFAC E-CoSM (E-CoSM'12) Rueil-Malmaison, France, October 23-25, 2012
derive a control-oriented model rather than a complicated physical model so that the model can be repeatedly used in computation on a real-world embedded controller. Namely, we derive the model so that it can be used for computation of future prediction of constrained variables in the RG.
Fuel injector
Turbine
Engine block
DOC DPF
In defining the models, we put the following assumptions for simplicity.
@ @ Temperature sensors
Fig. 1. Aftertreatment system of a diesel engine simple as possible to compute the prediction of future outputs in a finite horizon for a number of candidates of modified reference on automotive computers. Rather than building a complicated physical model, we put emphasis on control-oriented modeling in this paper. Namely, we derive a lumped two-state thermal model for the aftertreatment system that is suitable in the online RG algorithm. While, in the conventional fast RG, a modified reference is calculated via a bisectional search algorithm, in the present method, we extend the search region to the opposite direction to a reference change in order to avoid constraint violation due to model uncertainties. The paper is organized as follows. In Section 2, we describe an aftertreatment system of automotive diesel engines, then in Section 3 the thermal model of the system is derived. In Section 4, we introduce catalyst temperature constraints and design an RG algorithm. We show the effectiveness of the present method through an experiment with a production vehicle in Section 5. Finally, we summarize the paper and mention a future topic in Section 6. 2. SYSTEM CONFIGURATION In this paper, we consider an aftertreatment system of an automotive diesel engine, shown in Fig. 1. This system is composed of a diesel oxidation catalyst (DOC) to reduce hydrocarbon and carbon monoxide, and a diesel particulate filter (DPF) to collect particulate matter (PM). Requirements for the control of this aftertreatment system are as follows. • In order to maintain conversion efficiency, the catalyst is required to be warmed up in a short time frame. • To maintain DPF performance, PM stored in DPF should be burned periodically.
• The DOC and DPF are respectively treated as lumped masses, and spatial temperature distributions are not considered. • Temperatures in DOC and DPF are uniform, respectively. • Influence of vehicle speed on temperature fluctuation is negligible. • The following disturbances are fixed. · Mass flow rate and temperature of exhaust gas. · Ambient temperature. In reality, the mass flow rate and the temperature of exhaust gas can vary. However, the assumption that these disturbances are constant is made, because the model is used for future prediction in the RG as described in Section 4. In the prediction in the RG, the mass flow rate and the temperature of exhaust gas are to be regarded as exogenous inputs, and the future prediction is done by assuming that such variables stay constant in the future due to the causality. 3.1 Modeling of DOC From the heat balance as shown in Fig. 2, we obtain dTdoc 1 = [Qexo,doc − Qair,doc − Qexh,doc ] , (1) dt Cdoc Mdoc where a nomenclature is shown in Appendix. The heat transfers from DOC to the atmosphere and to the exhaust gas are described by Qair,doc = Katm (Tdoc − Tatm ) Qexh,doc = hdoc Adoc [Tdoc − (Rdoc Tdoc,us + (1 − Rdoc ) Tdoc,ds )] , (3) where in the right-hand side of (3) the term Rdoc Tdoc,us + (1 − Rdoc ) Tdoc,ds implies the temperature at a representative point between upstream and downstream of DOC, given the assumption that the spatial distribution of temperature in DOC is not considered. Regarding the gas inside DOC, the heat exchange is expressed as
As an actuator for the aftertreatment control, a fuel injector is located at an exhaust port of the cylinder head. When the fuel is introduced into the catalyst, a chemical reaction occurs and the catalyst temperature increases. It should be noticed that there are constraints on maximal temperatures for both DOC and DPF, that should be considered in the fuel injection control. 3. PLANT MODELING In this section, we derive models of DOC and DPF temperatures. The objective of modeling in this paper is to
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(2)
Fig. 2. Heat transfer diagram in DOC
2012 IFAC E-CoSM (E-CoSM'12) Rueil-Malmaison, France, October 23-25, 2012
where the heat transfer from DPF to the atmosphere and to exhaust gas are respectively given by Qair,dpf = Katm (Tdpf − Tatm )
(8)
Qexh,dpf = hdpf Adpf [Tdpf − {Rdpf Tdpf,us +(1 − Rdpf )Tdpf,ds }] .
(9)
Regarding the exhaust gas inside DPF, the heat balance is expressed as Fig. 3. Heat transfer diagram in DPF dTgas,doc = W Cgas (Tdoc,us − Tdoc,ds ) dt +Qexh,doc . Assume that the mass of gas in DOC is quite small compared to the gas flow inside DOC, namely Mgas,doc ≈ 0. By substituting (3) into the above equation, we obtain
Cgas Mgas,dpf
Cgas Mgas,doc
W Cgas (Tdoc,ds − Tdoc,us )
NT,dpf,ds DT,dpf,ds NT,dpf,ds = hdpf Adpf Tdpf + Tdpf,us × (W Cgas − Rdpf hdpf Adpf )
NT,doc,ds W Cgas + (1 − Rdoc )hdoc Adoc NT,doc,ds = hdoc Adoc Tdoc + Tdoc,us × Tdoc,ds =
DT,dpf,ds = W Cgas + (1 − Rdpf )hdpf Adpf . Substituting Tdpf,ds into (9) gives
(W Cgas − Rdoc hdoc Adoc ). Substituting Tdoc,ds into (3) yields NQ,exh,doc DQ,exh,doc NQ,exh,doc = hdoc Adoc W Cgas (Tdoc − Tdoc,us )
The mass Mgas,dpf of exhaust gas inside DPF can be considered extremely small, compared to the exhaust gas flow W , namely Mgas,dpf ≈ 0. By eliminating Qexh,dpf from (9) and (10), Tdpf,ds is described by Tdpf,ds =
= hdoc Adoc [Tdoc − (Rdoc Tdoc,us + (1 − Rdoc ) Tdoc,ds )] . Solving this equation with respect to Tdoc,ds gives
Qexh,doc =
dTgas,dpf = W Cgas (Tdpf,us − Tdpf,ds ) dt +Qexh,dpf . (10)
NQ,exh,dpf DQ,exh,dpf = hdpf Adpf W Cgas
Qexh,dpf = (4)
NQ,exh,dpf
(11)
× [W Cgas (Tdpf − Tdoc,us ) +hdoc Adoc (Tdpf − Tdoc )
DQ,exh,doc = W Cgas + (1 − Rdoc )hdoc Adoc .
+Rdoc hdoc Adoc (Tdoc,us − Tdpf )]
The chemical release of heat from the gas to the catalyst brick surface is considered as a function of the fuel injection quantity, the DOC temperature and the mass flow of exhaust gas, namely Qexo,doc = Qexo,doc (Qinj , Tdoc , W ). (5) The detailed description of (5) is shown in Appendix A.1. By substituting (2),(4),(5) into (1), we obtain a differential equation with respect to the DOC temperature as dTdoc 1 = [Qexo,doc (Qinj , Tdoc , W ) dt Cdoc Mdoc ] NQ,exh,doc −Katm (Tdoc − Tatm ) − . (6) DQ,exh,doc It should be noted that the amount of fuel injection Qinj is the control input, and that a constraint is at the maximal limit of the DOC temperature Tdoc . 3.2 Modeling of DPF Similarly to DOC, a thermal model of DPF is derived in this sub-section. From the heat balance shown in Fig 3, the DPF brick temperature is obtained as dTdpf 1 = [Qexo,dpf − Qair,dpf −Qexh,dpf ] , (7) dt Cdpf Mdpf
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DQ,exh,dpf = (W Cgas + (1 − Rdpf )hdpf Adpf ) × (W Cgas + (1 − Rdoc )hdoc Adoc ) . We suppose that the exotherm generated by burned PM is a function of the exhaust gas mass flow, the mass of PM accumulated in DPF, the mass of fuel injection, the DOC and DPF temperatures, i.e., Qexo,dpf = Qexo,dpf (W, mpm , Qinj , Tdoc , Tdpf ).
(12)
Again, the detailed description of (12) is shown in Appendix A.2. Substituting (8), (11) and (12) into (7) yields the differential equation of the DPF temperature given by dTdpf 1 = × dt Cdpf Mdpf [Qexo,dpf (W, mpm , Qinj , Tdoc , Tdpf ) ] NQ,exh,dpf −Katm (Tdpf − Tatm ) − . DQ,exh,dpf
(13)
Note that the mass of PM mpm is assumed to be an available signal in the temperature control. Thus, we do not introduce the dynamics of the mass of PM here as in (Hiroyasu et al., 1983; Hirsch et al., 2010).
2012 IFAC E-CoSM (E-CoSM'12) Rueil-Malmaison, France, October 23-25, 2012
3.3 State-space model From (6) and (13), we obtain the state-space model of the plant x˙ p = fp (xp , u, w) c = xp
{
?? rg-+ e eRG −6
xp
v FB controller
w
u-
? Plant
yr-
(15)
y = Tdpf , T
(14)
x
r
(16)
where xp = [ Tdoc Tdpf ] ∈ R is the state of the system, T w = [ W Tatm mpm Tdoc,us ] ∈ R4 is the exogenous input, u = Qinj ∈ R is the control input, c ∈ R2 is the variables constrained and y ∈ R is the output. The function fp : R2 × R × R4 → R2 is inferred from (6) and (13). 2
3.4 Feedback controller In this paper, the tracking control of the DPF temperature ref Tdpf to the reference r = Tdpf ∈ R is considered. Here, we design a PI feedback controller described by u = KP e + KI v
(17)
Fig. 4. Block diagram of control system 4. DESIGN OF REFERENCE GOVERNOR 4.1 Definition of constraints We put constraints on DOC and DPF temperatures as shown below. For prescribed T¯doc , T¯dpf (> 0), it is necessary to ensure Tdoc ≤ T¯doc and Tdpf ≤ T¯dpf . Equivalently, for a set { [ ] [ ]} Tdoc T¯doc 2 C= c∈R : 0
v˙ = e e = r − y, where e is the tracking error, and v denotes the state of the integrated tracking error, and KP and KI are coefficients. Note that the integrator can be reset in the implementation of the PI feedback controller to prevent the integrator wind-up if u = Qinj = 0. 3.5 Closed-loop system By applying the feedback controller (17) to the plant (14), we obtain the continuous-time closed-loop system
This subsection considers the design of an RG algorithm that modifies the DPF reference to meet the DOC and DPF temperature constraints. As shown in Fig. 4, the RG modifies the original reference rk to gk using a parameter αk ∈ R via the equation described by gk = gk−1 + αk (rk − gk−1 ), where αk ∈ [αmin , 1] with a given αmin ∈ R. If αk = 1, then gk = rk , namely, the RG outputs the original reference. If αk = 0, then gk = gk−1 , equivalently, the RG returns the previous modified reference. The optimal αk is chosen so that the predicted closedloop behavior over a finite prediction horizon satisfies the constraints. Namely, the optimal αk can be obtained by solving the optimization problem
x˙ = fcl (x, w, r) c = h(x, w), T
where x = [ xp v ] is the state of the closed-loop system and fcl : R4 × R4 × R → R4 can be derived by fp . By approximating x˙ to (xk+1 −xk )/δ with sampling period δ(> 0), we obtain the discrete-time closed-loop model xk+1 = f (xk , wk , rk ) ck = h(xk , wk ), where the subscript k = 0, 1, . . . denotes the discrete time step, and is attached to continuous-time vectors to express their discrete-time version. The function f : R4 × R4 × R → R4 can be obtained from fcl . It should be noted that f is nonlinear due to the nonlinearities in (6) and (13). Though the sampling period δ is set in a trial-and-error manner in this paper, we need to consider the relation between the speed of system response and the prediction horizon in RG as described in Section 4.2. For example, if a small δ is chosen, the future estimation becomes precise, but the resulting computational cost becomes high because a large number of prediction horizon may be required.
380
Given rk , xk , wk , gk−1 , maximize αk such that
(19)
xk+i+1|k = f (xk+i|k , wk , gk )
(20)
ck+i|k = h(xk+i|k , wk , gk )
(21)
ck+i|k ∈ C
(22)
xk|k = xk
(23)
gk = gk−1 + αk (rk − gk−1 )
(24)
αk ∈ [αmin , 1]
(25)
i = 0, 1, . . . , Nh , where xk+i|k and ck+i|k denote the predicted state and constrained variables at time k + i given a measurement at time k, respectively. The nonnegative integer Nh is the prediction horizon. The objective function of the optimization problem (19) implies that the modified reference is chosen as close as possible to the original reference while fulfilling the constraints. It should be noted that the exogenous
In regards to the choice of prediction horizon Nh , it is preferable to take it longer than the settling time of the closed-loop system. However, there are limitations on computation resources in real-world controllers. We need to consider the compromise between such limitation and the length of the prediction horizon. The optimization problem (19) includes a nonlinear constraint (20). So, it is not easy to obtain its global optimum. Instead, we solve the optimization problem by the following iterative algorithm based on the bisectional search to acquire a sub-optimal solution. [Bisectional search algorithm] 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
DOC temperature K
input wk does not vary over the prediction horizon, since future exogenous values are not available.
Modified DPF temperature reference K
2012 IFAC E-CoSM (E-CoSM'12) Rueil-Malmaison, France, October 23-25, 2012
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T
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j := 0 (j) (j) (j) (j) αmin := αmin , αmax := 1, αmid := αmax (j) (j) while αmin < αmax (j) αk := αmid solve (20), (21), (23) and (24) if (22) is satisfied ∀i = 0, 1, . . . Nh (j+1) (j) (j+1) (j) αmin := αmid , αmax := αmax (j+1) (j) (j+1) (j) else αmax := αmid , αmin := αmin end if j = jmax (j) αk∗ := αmin , and break end j := j + 1 ( ) (j) (j) (j) αmid := 12 αmin + αmax end
Original reference
900
25
30
35
40 45 Time [sec]
50
Fig. 5. Modified reference without/with back-off reference modification the prediction horizon is set to Nh = 5 and the sampling period δ = 0.5 sec, therefore the behavior 2.5 sec into the future is predicted. In the case of αmin = 0, once the modified reference approaches the original reference, it can not go in the opposite direction and it remains large. This leads to violation of the DOC temperature constraint. In the case of αmin = −0.2, the reverse direction reference modification is made, and the constraint violation is averted.
In the literature (Gilbert and Kolmanovsky, 2002; Vahidi et al., 2007), by setting αmin = 0, gk is always selected in the interval [gk−1 , rk ]. But, once gk is chosen, in the following time step a selection of the next gk must be made toward rk , so a correction in the opposite direction is not permitted. For engine control, model uncertainties as well as measurement noise can be large. Without the ability to make this reverse direction correction, there is a high possibility that the constraints will be violated. To avoid this, we admit setting αmin ∈ [−1, 0) in this paper. Its calibration can be done through repeated simulation of the closed-loop system with the RG. We have applied a bisectional search algorithm to obtain a sub-optimal solution here, but other optimization methods such as the steepest descent algorithm (Vanderpl, 1984) may be applicable. In such a case, it is necessary to consider again their implementability and computational costs. 4.3 Simulation result In Fig. 5, we show a simulation example of the opposite direction reference modification of the RG. The plant model is parameterized with a vehicle data. The upper plot depicts the original DPF reference and the modified references, and the bottom plot represents the resulting DOC temperatures. The dotted curves correspond to αmin = 0 and the solid curves to αmin = −0.2. In both cases, as
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As the catalyst temperature characteristics are strongly nonlinear, the response time can vary significantly. Even if the prediction horizon is calibrated at a finite number of operating conditions, there is still a possibility of violating temperature constraints at an operating condition between adjacent calibration points. By admitting the negative αmin as a tuning parameter in the application of RG, the RG can modify the reference so that the constraints are satisfied. 5. EXPERIMENT RESULT In this section, we apply the RG algorithm described in Section 4 to the aftertreatment temperature control of a production vehicle with a 4-cylinder 2.0 l diesel engine. Operating in constant engine speed of 2200 rpm and the fuel quantity of 22.8 mm3 /stroke, the DPF temperature step reference is introduced. Here, as an example of experimental condition, we set the upper bound of DPF temperature to 890 K. Note that we need to know the feedback controller in predicting the closed-loop behavior in the RG, the coefficients KP and KI of the feedback controller (17) are tuned in a trial-and-error method. Therefore, this control strategy is not a production level. The parameters of the RG algorithm are set as follows. First, we set the sampling period for discretization of the plant model to 5 sec and the prediction horizon to Nh = 20, therefore, the RG predicts for 100 sec ahead. The lower bound of the search region for modified references is fixed as αmin = −0.05. The number of iterations in the bisectional search algorithm is jmax = 6. The controller including the RG runs at a period of 0.1 sec.
2012 IFAC E-CoSM (E-CoSM'12) Rueil-Malmaison, France, October 23-25, 2012
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Fig. 6. Comparison of DPF temperature with/without RG Fuel injection mm3/s
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We have considered temperature constraints in DOC and DPF in this paper in order to show the applicability of RG to real-world plants for establishing a fundamental result of RG applications. By extending the present method, more detailed constraints in practical control systems can be treated. Future topics include the extension of the present method to multiple references and its application to control in a production vehicle. ACKNOWLEDGEMENTS The authors gratefully acknowledge the contributions of Dr. Tony Truscott, Mr. Peter Fussey of Ricardo UK Ltd., Mr. Takeshi Iwanami and Mr. Martin Egginton of Ricardo Japan Ltd. for valuable comments in regards to vehicle testing.
Without RG
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temperature control of automotive diesel engines. The plant model has been derived based on physical principles governing the DOC and DPF temperatures. Considering uncertainties in real-world control systems, the search region for modified references has been extended by allowing modification of the reference in the opposite direction. From the result of the vehicle testing, we have shown the effectiveness and applicability of the present method.
100 120 Time sec
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Fig. 7. Comparison of exhaust injection command and PM with/without RG The parameter Rdoc is calibrated using a as a function of Tdoc and W , that varies from 0.02 to 0.51 on the planar space. The experimental data is shown in Figs. 6 and 7. Fig. 6 depicts the original and modified references, and the maximum of actual DPF temperatures at four measurement points with thermocouples placed in line on the central axis of the DPF column. And in Fig. 7 the fuel injection Qinj and the PM in DPF mpm are plotted. From these figures, we see that, by modifying the original DPF temperature reference downward, the fuel injection quantity is reduced, therefore, the constraint on the DPF temperature is satisfied. The above experimental result shows the effectiveness of the present method and the applicability to the real-world control system. 6. CONCLUSION In this paper, we have focused on an RG method modifying a reference based on model predictions taking account of constraints. The RG has been applied to aftertreatment
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REFERENCES Albertoni, L., Balluchi, A., Casavola, A., Gambelli, C., Mosca, E., and Sangiovanni-Vincentelli, A. (2003). Hybrid command governors for idle speed control in gasoline direct injection engines. Proceedings of the 2003 American Control Conference, 773–778. Bemporad, A. (1998). Reference governor for constrained nonlinear systems. IEEE Transactions on Automatic Control, 43(3), 415–419. del Re, L., Allgower, F., Glielmo, L., Guardiola, C., and Kolmanovsky, I. (2010). Automotive Model Predictive Control: Models, Methods and Applications. SpringerVerlag. Gilbert, E. (1992). Linear control systems with pointwisein-time constraints: what do we do about them? Proceedings of the 1992 Americal Control Conference, 2595. Gilbert, E. and Kolmanovsky, I. (2002). Nonlinear tracking control in the presence of state and control constraints: a generalized reference governor. Automatica, 38(12), 2063–2073. Hirata, K. and Fujita, M. (1999). Set of admissible reference signals and control of systems with state and control constraints. Proceedings of the 38th IEEE Conference on Decision and Control, 1427–1432. Hiroyasu, H., Kadota, T., and Arai, M. (1983). Development and use of a spray combustion modelling to predict diesel engine efficiency and pollutant emission. Bulletin of The Japan Society of Mechanical Engineers, 26(214), 569–575. Hirsch, M., Oppenauer, K., and del Re, L. (2010). Dynamic engine emission models. In Automotive model predictive control, chapter 5, 73–87. Springer-Verlag. Kogiso, K. and Hirata, K. (2003). A reference governor in a piecewise state affine function. Proceedings of the 42nd IEEE Conference on Decision and Control, 1747–1752.
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Oh, S. and Agrawal, S. (2005). A reference governorbased controller for a cable robot under input constraints. IEEE Transactions on Control Systems Technology, 13(4), 639–645. Oh-hara, S. and Hirata, K. (2003). Experimental evaluation of on-line reference governors for constrained systems. Proceedings of the SICE Annual Conference 2003, 1458–1463. Vahidi, A., Kolmanovsky, I., and Stefanopoulou, A. (2007). Constraint handling in a fuel cell system: a fast reference governor. IEEE Transactions on Control Systems Technology, 15(1), 86–98. Vanderpl, G. (1984). Numerical optimization techniques for engineering design with applications. Mchraw-Hill book company. Appendix A. DERIVATION OF EXOTHERM TERMS
Table B.1. Nomenclature Variable
Quantity
Unit
Adoc , Adpf
Convective surface area of the DOC/DPF monolith DOC/DPF specific heat capacity Specific heat capacity of exhaust gas Convective heat transfer coefficient between DOC/DPF and gas Lower heating value of fuel Heat transfer constant between DPF and ambient Mass of PM DOC/DPF mass Mass of gas in DOC/DPF Heat transfer rate from DOC/DPF monolith to air Heat transfer rate from DPC/DPF monolith to exhaust gas DOC/DPF exothermic heat of reaction Fuel injection to catalyst Heat transfer rate of soot accumulated in DPF Temperature weighting constants in [0, 1] in DOC/DPF Ambient temperature DOC/DPF monolith temperature Exhaust gas temperature upstream/downstream of the DOC/DPF Temperature of gas in DOC/DPF Mass flow rate of exhaust gas
m2
Cdoc , Cdpf Cgas
A.1 Exotherm term in DOC
hdoc , hdpf
The heat conversion of Qinj depends mainly on the exhaust gas mass flow W and the DOC temperature Tdoc . By introducing the conversion efficiency ηexo,doc (Tdoc , W ) ∈ [0, 1], we describe Qexo,doc = ηexo,doc (Tdoc , W )Hv Qinj . (A.1)
Hv Katm
mpm Mdoc , Mdpf Mgas,doc , Mgas,dpf
A.2 Exotherm term in DPF The exotherm in DPF Qexo,dpf consists of two portions, namely, that from PM accumulated in DPF Qexo,dpf,pm and that of the fuel injected and slipping through DOC without burning Qexo,dpf,slip . Thus, Qexo,dpf is expressed as Qexo,dpf = Qexo,dpf,pm + Qexo,dpf,slip . (A.2) The exotherm of PM is supposed to be a function of DPF temperature and mass of PM, namely Qexo,dpf,pm = Qexo,dpf,pm (Tdpf , mpm ). (A.3) By considering (A.1), Qexo,dpf,slip is given by Qexo,dpf,slip = [1 − ηexo,doc (Tdoc , W )] Hv Qinj . (A.4) By substituting (A.3) and (A.4) into (A.2), we obtain Qexo,dpf = Qexo,dpf,pm (Tdpf , mpm ) + [1 − ηexo,doc (Tdoc , W )] Hv Qinj . Appendix B. NOMENCLATURE
Qair,doc , Qair,dpf
Qexh,doc , Qexh,dpf
Qexo,doc , Qexo,dpf Qinj Qsoot
Rdoc , Rdpf
Tatm Tdoc , Tdpf Tdoc/dpf,us/ds
We summarize the nomenclature in Section 3 in the following table. Tgas,doc , Tgas,dpf W
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J/(kg·K) J/(kg·K) W/(m2 ·K)
J/kg W/K
kg kg kg W
W
W kg/s W
K K K
K kg/s