Proceedings of the 2006 IEEE International Conference on Control Applications Munich, Germany, October 4-6, 2006
FrB05.4
Nonlinear Model Predictive Control of a Turbocharged Diesel Engine Martin Herceg1 , Tobias Raff2 , Rolf Findeisen2 , and Frank Allg¨ower2 1
Department of Information Engineering and Process Control, Slovak University of Technology, Slovakia
[email protected] 2 Institute for Systems Theory and Automatic Control (IST), University of Stuttgart, Germany {raff,findeise,allgower}@ist.uni-stuttgart.de
Abstract— Control of turbocharged diesel engines is a challenging task due to system nonlinearities and constraints on the inputs and process variables. In this paper nonlinear model predictive control is applied to control a diesel engine with a variable geometry turbocharger and an exhaust gas recirculation valve. The overall control objective is to regulate the setpoints of the air-fuel ratio and the amount of recirculated exhaust gas in order to obtain low exhaust emission values and low fuel consumption without smoke generation. Simulation results are presented to study the advantages and disadvantages of nonlinear model predictive control. The achieved performance is compared in simulations with a linear state feedback controller and an input-output linearization based control method. As shown, nonlinear model predictive control achieves good overall control performance and constraint satisfaction.
I. INTRODUCTION Diesel engines have gained high interest in the automobile market over the last years because they provide higher fuel efficiency than gasoline engines. However, the price to pay for the better fuel efficiency is the rather complicated exhaust gas treatment of diesel engines if compared to gasoline engines. Steadily increasing requirements in exhaust emission standards make it necessary to permanently improve the exhaust gas system in diesel engines. One possibility to reduce emissions in diesel engines, in particular nitrogen oxide (N Ox ), is to recirculate the exhaust gas through the exhaust gas recirculation (EGR) valve in the intake manifold [13, 15]. The recirculated exhaust gas decreases the temperature of combustion in the cylinders which results in a N Ox reduction. However, the amount of exhaust gas that can be recirculated depends on the operating conditions because the exhaust gas reduces the amount of fresh air in the intake manifold that is needed to provide the required engine torque and to avoid smoke generation. Thus, one important control objective in turbocharged diesel engine control is to supply the driver’s requested engine torque while keeping the N Ox concentration in the exhaust gas as well as the smoke generation as low as possible. This can be achieved by controlling the air-fuel ratio in the cylinders and the amount of recirculated exhaust gas using the EGR valve position and the position of the variable geometry turbocharger (VGT). The turbocharger is used to increase the power density in the cylinders by forcing additional fresh air in the intake manifold in order to be able to burn more fuel without smoke generation. Much 0-7803-9796-7/06/$20.00 ©2006 IEEE
attention has been paid to this challenging control problem. The main problems involved in designing a suitable control strategy are the nonlinear multivariable nature of the problem and the presence of constraints on inputs and process variables [11, 15]. By now various control methods have been applied to the control of diesel engines. Examples are gain scheduled linear parameter-varying control [15, 16], robust H∞ control [27], linear model predictive control [26], backstepping based control [10], nonlinear control Lyapunov function based control [12, 13], adaptive control approaches [3], and nonlinear passivity based control [17, 18]. All these control methods share one common disadvantage, namely, they cannot directly incorporate performance specifications and input and state constraints, or nonlinear system models. One possible control strategy to overcome this limitation is nonlinear model predictive (NMPC). NMPC is especially suited for the control of nonlinear systems subject to input and state constraints [1, 2, 21, 24, 25]. The basic idea of NMPC is to solve at each time instant a finite horizon optimal control problem for the current state. The first part of the resulting open loop optimal control input is applied to the system until the next sampling instant, at which the finite horizon optimal control problem is solved again for the new state. Traditional application areas for NMPC are limited to control problems with rather slow dynamics, like in process control [7]. The reason for this is the rather high computational load, since a finite horizon optimal control problem must be solved in real-time. Both, the above outlined advantages as well as the limitations of NMPC are the main points of motivation for the current work. Particularly, the purpose of this paper is twofold. First, to study the theoretically achievable performance by applying NMPC for diesel engine control. This may be considered as benchmark results for other control methods. Second, to consider diesel engine control also as a benchmark problem for NMPC itself, motivated by the long term goal to apply NMPC to control problems with fast dynamics. The remainder of the paper is organized as follows: In Section II, a third order nonlinear model of a diesel engine with EGR and VGT is described. In Section III, the basic idea of NMPC is reviewed. The NMPC setup for the diesel engine control problem is described in Section IV and the simulation results are presented in Section V. Finally, a summary and an outlook are given in Section VI.
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II. M ODEL OF A D IESEL E NGINE In this section a mean-value model of the airpath of a turbocharged diesel engine with EGR is described. For a detailed derivation of the mean-value model see [13, 15] and the references quoted therein. A schematic diagram of the considered turbocharged diesel engine including the EGR and VGT is shown in Figure 1. A third order nonlinear model can be derived using the conservation of mass and energy, the ideal gas law for modeling the intake and exhaust manifold pressure dynamics, and a first order differential equation with time constant τ for modeling the power transfer dynamics of the VGT. Under the assumption that the intake and exhaust manifold temperatures, the compressor and turbine efficiencies, the volumetric efficiency and the time constant τ of the turbocharger are constant, this modeling approach results in the nonlinear model [13, 15] RTi (Wci + Wxi − Wie ) Vi RTx p˙x = (Wie − Wxi − Wxt + Wf ) Vx 1 P˙c = (−Pc + ηm Pt ) . τ p˙i =
(1)
Here pi is the intake manifold pressure, px the exhaust manifold pressure, Pc the power transfered by the turbocharger, τ = 0.11s the time constant, ηm = 0.98 the mechanical efficiency, and Vi = 0.006m3 , Vx = 0.001m3 the volumes of the intake and exhaust manifold. In the diesel engine model (1), Wci describes the relation between the flow through the compressor and the power. This relation is modeled as Wci =
ηc Pc , cp Ta pi μ − 1
(2)
pa
where ηc = 0.61 is the compressor efficiency, Ta = 298K J J , cv = 727.4 kgK the ambient temperature, cp = 1014.4 kgK cp −cv heat at constant pressure and volume, μ = cp = 0.286 a constant, and pa = 101.3kP a the ambient pressure. Wci Tic , pc
Intercooler
Wxi xr
Ti , pi, mi Intake Manifold EGR
Tr , px
Cooler
Tc, pc
Tx , px, mx
Wex
Exhaust Manifold
Wxt Compressor
Nt VGT
Ta, pa
Fig. 1.
Turbine
xv
Tt , pt
The flow Wxi in (1) describes the flow through the EGR valve. It is given by pi Aegr (xegr )px 2pi (3) √ 1− Wxi = px px RTx where Aegr (xegr ) is the effective area of the EGR valve, Tx = 509K is the exhaust manifold temperature, and R = J is the gas constant. Moreover, the flow Wie from the 287 kgK intake manifold into the cylinders is modeled by the speeddensity equation [15] pi N V d (4) Wie = ηv 120Ti R with the volume efficiency ηv = 0.87, the engine speed N , the intake manifold pressure Ti = 313K, and the displacement volume Vd = 0.002m3 . The turbine flow Wxt in kg/s is given by px px Wxt = (axvgt + b) c −1 +d pa pref (5) pa Tref 2pa 1− × Tx px px with the parameters a = −0.136, b = 0.176, c = 0.4, d = 0.6, the reference pressure pref = 101.3kP a, and the reference temperature Tref = 298K. Finally, the turbine power Pt is modeled by the equation μ pa (6) Pt = Wxt cp Tx ηt 1 − px with the turbine efficiency ηt = 0.76. Furthermore, the engine speed N and the fueling rate Wf are considered as known external parameters. As shown in Figure 1 the exhaust gas recirculation and the turbocharger introduce feedback paths in the turbocharged diesel engine. Furthermore, since the exhaust gas recirculation and the turbocharger are both driven by the exhaust gas, the turbocharged diesel engine with EGR is a coupled nonlinear system. The control inputs of the diesel engine model (1) are the EGR position xegr and the VGT position xvgt . The EGR position and the VGT position range between 0 and 1. However, to simplify the considerations, the effective areas Aegr and Avgt = axvgt + b are directly used as control inputs instead of the EGR valve position xegr and the VGT position xvgt . Since the effective area Aegr as well as the effective area Avgt are monotonically increasing functions in their variables [15], the positions xegr and xvgt can be uniquely determined from the effective areas. Furthermore, the control inputs Aegr and Avgt are constrained due to the minimal and maximal EGR and VGT positions. In particular, the input constraints of the nonlinear diesel engine model are given by Aegr ∈ [0m2 , 1.8 × 10−4 m2 ] Avgt ∈ [0.04, 0.176].
Oxicat
Schematic diagram of a turbocharged diesel engine with EGR.
(7)
In the next sections, a NMPC controller is designed in order to control the diesel engine described by the model (1). 2767
III. N ONLINEAR M ODEL P REDICTIVE C ONTROL In the following a brief review of NMPC is given. Consider the stabilization problem of the nonlinear system x˙ = f (x, u)
(8)
subject to the input and state constraints u ∈ U, and x ∈ X , where x ∈ Rn is the system state and u ∈ Rm the control input. For simplicity of presentation it is assumed that the equilibrium point of the system (8) is at the origin, i.e. the vector field f satisfies f (0, 0) = 0. Suppose that the full state x of the system (8) can be measured. In NMPC the control input applied to the system (8) over the interval [ti , ti+1 ) is given by the repeated solution of the finite horizon optimal control problem ti +TP
x(ti + TP )) + min E(¯
F (¯ x(τ ), u ¯(τ ))dτ
u(·)
ti
s.t. x ¯˙ (τ ) = f (¯ x(τ ), u ¯(τ )), x ¯(ti ) = x(ti ) u ¯(τ ) ∈ U, ∀ τ ∈ [ti , ti + TP ]
(9)
x ¯(τ ) ∈ X , ∀ τ ∈ [ti , ti + TP ] x ¯(ti + TP ) ∈ E. In the optimal control problem (9), ti denotes the sampling instants, TP the control/prediction horizon, and F the stage cost which can for example arise from economical and ecological considerations. Furthermore the bar denotes the predicted variables, i.e. x ¯ denotes the solution of the system driven by the input u ¯ with the initial condition x(ti ). The distinction between the real state x of the system (8) and the predicted state x ¯ in the NMPC controller (9) is necessary due to the moving horizon nature in NMPC. In general, the predicted state x ¯ differs from the real state x at least after one sampling instant. This difference is the reason why the closed loop in NMPC is not naturally stable. Therefore, the so called terminal region E and the so called terminal penalty term E of the optimal control problem (9) are used to enforce closed loop stability and to increase the performance in NMPC. A wide variety of approaches have been developed in the literature to achieve asymptotic closed loop stability [8, 21]. All these approaches are based, implicitly or explicitly, on three ingredients: a terminal cost E, a terminal region E, and a locally stabilizing control law ϕ. Many of these approaches are covered by the following theorem [8, 21]:
Theorem 1: The closed loop is asymptotically stable if the finite horizon optimal control problem (9) is feasible at the first sampling instant and the following assumptions are satisfied for a terminal cost E ∈ C 1 , a terminal region E ⊆ X , closed, 0 ∈ E, and a stabilizing control law ϕ: (i) E(x) ≥ 0 ∀x ∈ Rn \ {0} (ii) ϕ(x) ∈ U ∀x ∈ E (iii) E is positively invariant for x˙ = f (x, ϕ(x)) (iv) ∂E ∂x f (x, ϕ(x)) + F (x, ϕ(x)) ≤ 0 ∀x ∈ E.
A proof of Theorem 1 can be found [8, 21]. In the following, the quasi-infinite horizon NMPC approach with guaranteed closed loop stability is described which is used in the simulation study in the next section. In the so called quasiinfinite horizon NMPC approach [4] the terminal cost E and the terminal region E are chosen as E(x) = xT P x, E = {x ∈ Rn |xT P x ≤ α}
(10)
with the state feedback ϕ(x) = Kx considering a quadratic stage cost F (x, u) = xT Qx + uT Ru with Q ≥ 0 and R > 0. The underlying idea of the quasi-infinite horizon NMPC approach is to approximate the infinite horizon optimal control problem, i.e. the optimal control problem (9) with TP → ∞, by the terminal cost E. The following procedure [4, 8] describes a procedure to compute the parameters P and α of the quasi-infinite horizon NMPC controller, assuming that the Jacobian linearization (A, B) ∂f of (8) is stabilizable, where A = ∂f ∂x (0, 0) and B = ∂u (0, 0). Step 1: Solve the linear control problem based on the Jacobian linearization (A, B) of (8) to obtain a locally stabilizing linear state feedback ϕ(x) = Kx. Step 2: Define AK = A + BK, and choose a constant κ ∈ [0, ∞) satisfying κ ≤ − maxi (Re(λi (AK ))) and solve the Lyapunov equation (AK + κI)T P + P (AK + κI) = −(Q + K T RK)
(11)
to get a positive definite and symmetric matrix P . Step 3: Find the largest possible α1 specifying a region E1 = {x ∈ Rn |xT P x ≤ α1 },
(12)
such that Kx ∈ U, for all x ∈ E1 . Step 4: Find the largest α ∈ (0, α1 ] specifing a region E = {x ∈ Rn |xT P x ≤ α},
(13)
such that the optimal value of the following optimization problem with ξ(x) = f (x, Kx) − AK x is non positive: max{xT P ξ(x) − κxT P x|xT P x ≤ α}. x
(14)
If the parameters P and α are computed as described above then the closed loop is asymptotically stable. Note that there exists further NMPC approaches with guaranteed closed loop stability. For a detailed and more rigorous treatment of NMPC, see e.g. [8, 21] and the references quoted therein. Furthermore, for practical applications NMPC should be able to cope with uncertainties and disturbances. As shown in [9, 19, 20, 22] NMPC provides some inherent robustness properties due to the close relation of NMPC to optimal control and inverse optimality. In summary, NMPC is suited to control nonlinear multivariable systems subject to input and state constraints. Furthermore, closed loop stability and some inherent robustness is guaranteed in NMPC.
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IV. NMPC OF THE D IESEL ENGINE The control objective is to control the air-fuel ratio and the amount of recirculated exhaust gas to their setpoints AFs and EGRs . The setpoints AFs and EGRs depend on the operating conditions of the diesel engine, i.e. the engine speed N and the fueling rate Wf , and are optimized with respect to N Ox emission, fuel consumption, and smoke generation [13]. The setpoints AFs and EGRs are determined from a lookup table scheduled on engine speed N and fueling rate Wf . The determination of these setpoints is not an easy task that is not considered in this paper. The control objective, i.e. the control of the air-fuel ratio and the amount of recirculated exhaust, is in the following achieved by regulating the intake manifold pressure pi , the exhaust manifold pressure px , and the compressor power Pc to their setpoints psi , psx , and Pcs . The setpoints psi , psx , and Pcs can be determined from the setpoints AFs and EGRs as described in [13]. Therefore, the objective of the NMPC controller is to regulate the diesel engine to the setpoints psi , psx , and Pcs while improving the transient performance and preserving the input and state constraints. In the following the finite horizon NMPC optimal control problem (9) for the control problem described above is set up. In the setpoint centered and normalized coordinates, pi − psi px − psx Pc − Pcs , x = , x = x1 = 2 3 psi psx Pcs (15) s s Aegr − Aegr Avgt − Avgt u1 = , u = , 2 Asegr Asvgt where setpoint variables are in the following denoted by a superscript s, e.g. Asvgt denotes the setpoint value for Avgt , the diesel engine model (8) can be rewritten as x˙ 1 = k1 (φ1 (x1 , x2 ) + φ2 (x1 , x2 , u1 ) − φ3 (x1 )) x˙ 2 = k2 (φ3 (x1 ) − φ2 (x1 , x2 , u1 ) − φ4 (x2 , u2 )) x˙ 3 = k3 (φ5 (x2 , u2 ) − (x3 Pcs + Pcs )) with the nonlinearities and constants ηc x3 P s + P s s c s μ c , φ1 = cp Ta x1 pi +pi −1 pa
(u1 Asegr + Asegr )(x2 psx + psx ) √ φ2 = RTx s s x1 psi + psi x1 pi + pi 1− , × x2 psx + psx x2 psx + psx N Vd (x1 psi + psi ), φ3 = η v 120Ti R x2 psx + psx −1 +d φ4 = (u2 Asvgt + Asvgt ) c pa s s 2p2a Tref 2pa (x2 px + px ) × − , Tx p2ref p2ref μ pa , φ5 = φ4 cp Tx ηt 1 − x2 psx + psx RTx 1 RTi k1 = , k2 = , k3 = . s s V i pi V x px τ Pcs
Wf
Wf
Asvgt Asegr s s u1 Asegr Lookup pi , px NMPC Table Controller u2 As Pcs vgt
pi , px , P c
N
Fig. 2.
Diesel Engine
Control structure of the diesel engine.
In the new coordinates the control objective is to regulate the variables x1 , x2 and x3 to zero. This can be achieved, for example, with the stage cost F (x, u) = x21 + x22 + x23 + u21 + u22 .
(17)
Note that additional performance specifications can be directly considered in NMPC. For example, in order to have a fast torque response and thus a good acceleration, the flow into the intake manifold Wci , which is a measure of this performance specification, could be directly considered in the stage cost F . Furthermore, input and state constraints are considered in the NMPC setup. The input and state constraints stem for example from actuator limits, mechanical limits or exhaust gas limits of the diesel engine [11]. Specifically the following constraints on the inputs are considered: Aegr ∈ [0m2 , 1.8 × 10−4 m2 ] Avgt ∈ [0.04, 0.176],
(18)
and state constraints pi ∈ [102kP a, 155kP a] px ∈ [102kP a, 175kP a].
(16)
(19)
The input constraints stem from the minimal and maximal EGR and VGT positions and the state constraints on the intake and exhaust manifold pressure are imposed in order to protect the diesel engine from damage due to overboost. Note that before the input and state constraints (18), (19) are used in the NMPC controller, the constraints must be transformed into the new coordinates (15). Hence, the finite horizon NMPC optimal control problem that has to be solved at each time instant ti in the control structure shown in Figure 2 is given by ti +TP x(ti + TP )) + (¯ x21 + x ¯22 + x ¯23 + u ¯21 + u ¯22 )dτ min E(¯ u(·)
ti
s.t. (16), (18), (19) x ¯(ti + TP ) ∈ E, where the terminal penalty E and the terminal region E are determined by the quasi-infinite horizon NMPC approach. In particular, the NMPC parameters P and α are computed via the procedures described in [4, 5]. Furthermore, the sampling time Δt = ti+1 − ti and the control/prediction horizon TP are chosen as Δt = 0.1s and TP = 0.9s respectively. 2769
Setpoint 1 12.87 70 4 1900 123.17 131.37 932.0
Setpoint 2 72.14 10 6 2100 107.44 108.97 239.6
Setpoint 3 3.07 86 5 2000 146.86 171.59 2480.8
Setpoint 4 12.87 70 4 1900 123.17 131.37 932.0
ACKNOWLEDGMENTS The authors acknowledge financial support by REGINS through the project Predictive Control of Combustion Engines (PREDIMOT). M. Herceg acknowledges the financial support of the Scientific Grant Agency of the Slovak Republic under grants No. 1/3081/06 and No. 1/1046/04. 160
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Fig. 3. Simulation results of pi and px without slew rate constraints: NMPC (black) controller, linear (gray) and nonlinear (dark gray) controller. 3000
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Fig. 4. Simulation results of Pc and Wc without slew rate constraints: NMPC (black) controller, linear (gray) and nonlinear (dark gray) controller. 100
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xvgt [%]
Variable xegr [%] xvgt [%] Wf [kg/h] N [1/min] pi [kP a] px [kP a] Pc [W ]
VI. C ONCLUSIONS In this paper NMPC is applied to control the airpath of a turbocharged diesel engine with EGR. As shown, NMPC is a well suited control approach to regulate a diesel engine due to the nonlinear multivariable structure of the process and the present constraints on inputs and states. In simulations studies it was shown that NMPC can improve the transient behavior of a diesel engine. Future work involves the implementation of an explicit NMPC controller on a diesel engine.
xegr [%]
The simulation results of the developed NMPC controller are shown in Figure 3 to Figure 5. For comparison also results for a linear and a nonlinear state feedback controller are presented. The linear state feedback controller is a LQR controller designed via the Jacobian linearization of (8) for the considered setpoints and the nonlinear state feedback controller is based on the input-output linearization method described in [13]. The control inputs of the linear state feedback controller and of the nonlinear state feedback controller are saturated in order to satisfy the input constraints (18). The controllers are tested based on a sequence of setpoint changes as given in Table I. As can be seen in Figure 3 to Figure 5, the NMPC controller, the linear state feedback controller, and the nonlinear state feedback controller regulate the diesel engine to the desired setpoints. Furthermore, Figure 3 shows that the NMPC controller satisfies the state constraint on the exhaust manifold pressure while the linear state feedback controller does not. In addition, Figure 5 shows that the NMPC controller is less aggressive than the linear and the nonlinear state feedback controller while having the same performance in the variables pi , px , and Pc . Especially during the third setpoint change at time t = 5s, the NMPC controller immediately employs the steady state control input xsegr without much additional control action. This behavior underpins that the calculation of the steady stepoints is very important in the overall performance of the diesel engine. However, the control inputs of the developed controllers are unrealistic fast, so that the rate of change of the control inputs must be taken into account. The slew rate constraints ˙ ∀t ≥ 0 on the control inputs can be considered in u(t) ˙ ∈ U, the NMPC setup by adding integrators in the system before the control inputs. Note, however, that this transforms the input constraint (18) to constraints on the integrator states in the NMPC setup. Since the slew rate constraints on the control inputs cannot be directly considered in the design of the linear and nonlinear state feedback controller, the slew rates of the control inputs of these controllers are limited by a slew rate limiter. Figure 6 to Figure 9 show the simulation results of the NMPC controller with a control/prediction horizon TP = 1.5s and the linear and nonlinear controllers with slew rate constraints on the control inputs. It can be seen that the controllers regulate the diesel engine to the setpoints while satisfing the state constraints (19). Additional, the consideration of the slew rate constraints reduces the inverse response behavior in the the variable Wc . Summarizing, the NMPC controller can achieve good transient performance of a diesel engine while satisfying the input and state constraints
in a systematic way. However, on the downside, currently it is not possible to implement NMPC in real-time due to the limited computational power available on todays embedded control systems, which are not suitable for the required fast solution of the resulting finite-time optimal control problem. Possibilities to circumvent this problem are explicit formulations of NMPC, like for example presented in [14] and strategies based on approximated solutions [6, 23].
Pc [W ]
V. S IMULATION RESULTS
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t [s]
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Fig. 5. Simulation results of xegr and xvgt without slew rate constraints: NMPC (black) controller, linear (gray) and nonlinear (dark gray) controller.
TABLE I
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px [kP a]
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Fig. 6. Simulation results of pi and px with slew rate constraints: NMPC (black) controller, linear (gray) and nonlinear (dark gray) controller. 3000
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Fig. 7. Simulation results of Pc and Wc with slew rate constraints: NMPC (black) controller, linear (gray) and nonlinear (dark gray) controller.
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Fig. 8. Simulation results of xegr and xvgt with slew rate constraints: NMPC (black) controller, linear (gray) and nonlinear (dark gray) controller.
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Fig. 9. Simulation results of x˙ egr and x˙ vgt with slew rate constraints: NMPC (black) controller, linear (gray) and nonlinear (dark gray) controller.
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