Proceedings of the Institution of Mechanical Engineers, Part D: Journal of Automobile Engineering http://pid.sagepub.com/
A torque-based nonlinear predictive control approach of automotive powertrain by iterative optimization Lin He, Liang Li, Liangyao Yu, Enrong Mao and Jian Song Proceedings of the Institution of Mechanical Engineers, Part D: Journal of Automobile Engineering 2012 226: 1016 originally published online 16 February 2012 DOI: 10.1177/0954407011434444 The online version of this article can be found at: http://pid.sagepub.com/content/226/8/1016
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Original Article
A torque-based nonlinear predictive control approach of automotive powertrain by iterative optimization
Proc IMechE Part D: J Automobile Engineering 226(8) 1016–1025 Ó IMechE 2012 Reprints and permissions: sagepub.co.uk/journalsPermissions.nav DOI: 10.1177/0954407011434444 pid.sagepub.com
Lin He1, Liang Li1, Liangyao Yu1, Enrong Mao2 and Jian Song1
Abstract A torque-based nonlinear predictive control approach is developed to manipulate the automotive powertrain in this paper. This approach consists of three components, i.e. torque demand control, nonlinear predictive controller and torque load estimation. A control scheme of torque demand with proportional–derivative (PD) compensator is proposed to meet the torque requirement of drivers. Based on a mean value model of the internal combustion (IC) engine, a torque-based nonlinear predictive controller is designed by use of iterative optimization. A proportional–integral (PI) observer is employed to estimate the torque load of the powertrain. Through experimental validation, it can be concluded that the torque-based nonlinear predictive control approach is a significant candidate for the automotive powertrain to implement dynamic torque control.
Keywords Nonlinear predictive control, torque demand, automotive powertrain, dynamic torque control
Date received 1 May 2011; accepted 7 December 2011
Introduction As for the conventional control scheme of automotive powertrain, the engine management system (EMS) manipulates the throttle angle to control the angular speed of the powertrain based on the position of the acceleration pedal resulting from the speed demand of the driver. During the process of operation, the torque state of the internal combustion (IC) engine is unknown. In fact, the rotation of the powertrain is derived from the output torque of the IC engine. Simultaneously, the transient response of the output torque of the engine has a direct impact on vehicular performance, such as drivability and fuel economy. Therefore, it is important for automotive engineers to develop a torque-based control approach for the powertrain. Regarding the architecture of torque-based control powertrain, Gerhardt et al. introduced an architecture of torque-based control for the IC engine, which is named after the EMS ME7 of Bosch Inc.1 Heintz et al. describe an approach of torque-based EMSs of Ricardo Inc.2 Gerhardt focuses on description of the function of the software module, and Heintz gives an algorithm of the functional module. Yutaro et al. give a method to estimate the intake air mass from the correlation
between intake valve lift characteristics and air quantity entering the engine, which makes it possible to fully utilize the potential of a variable valve actuation system for engine torque control.3 Solliec et al. present the development torque-based engine control strategies for a downsized spark ignition (SI) engine from simulation design to final validation on a demonstration car.4 Livshiz et al. describe the role of engine torque control in a torque-based control powertrain and proposes a validation and calibration approach of the mean-value model that can be employed to implement the torque control of the engine.5 Lee et al. develop a new PCbased simulation platform of hardware in the loop to design an automotive engine control system.6 The previous literature focuses on the introduction of modular functions and implementary means for the torquebased control. In this paper, a formulated approach is 1
The State Key Laboratory of Automotive Safety and Energy, Tsinghua University, Beijing, People’s Republic of China 2 College of Engineering, China Agricultural University, Beijing, People’s Republic of China Corresponding author: Lin He, The State Key Laboratory of Automotive Safety and Energy, Tsinghua University, Beijing, 100084, People’s Republic of China. Email:
[email protected]
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proposed to realize the torque-based control of automotive powertrain. As for the torque-based control methods, Satou et al. use a feedback control algorithm and an airflow sensor to improve torque control accuracy for a SI engine.7 Wehrwein describes an analytical methodology for calibrating the engine torque, considering uncertainty, in order to minimize the clunk disturbance, while still meeting throttle response constraints.8 Nagata and Tomizuka apply an iterative learning control algorithm to realize fast-response engine torque control.9 With respect to the application of model predictive control (MPC), Giorgetti et al. illustrate the application of hybrid modeling and model predictive control techniques to the management of air/fuel ratio and torque in advanced technology gasoline DISC engines by simulation.10 Saerens et al. illustrate the capabilities of MPC for the control of automotive powertrains, and considers the minimization of the fuel consumption through dynamic optimization based on the tracking of optimal operation speed of the gasoline engine.11 Vermillion et al. presented some results of design and simulation for a off-idle engine torque control strategy that uses MPC to manipulate desired air/ fuel ratio, engine air charge, spark advance, and variable valve timing.12 Based on the adaptive neural network model, Wang et al. develop an MPC strategy to control the crankshaft speed.13 From the preceding review, it can be seen that the torque-based control is realized by many control methods but the MPC. Simultaneously, MPC also is employed to control air/ fuel ratio, crankshaft speed and so on. In this paper, the MPC is utilized to control the torque generation of IC engines. On the basis of the research mentioned above, a torque-based nonlinear predictive control approach is developed by iterative optimization to manipulate the automotive powertrain in this paper. This approach includes three components: torque demand control, nonlinear predictive controller and torque load estimation. The torque demand of the automotive powertrain arises from the torque requirements of driver including entertainment system, air conditioner in the vehicle, and so on. A proportional–derivative (PD) compensator is employed to compute the torque error between the actual output torque of the IC engine and torque demand of the powertrain so that the output torque of the IC engine is able to meet the torque demand from the driver. The MPC algorithm has many merits: it can handle nonlinear and multi-variable control problems, take account of actuators limitations, handle nonminimal phase, unstable processes and structural changes, and so on. It is hence useful to try it to control the torque generation of IC engines. Based on the mean value model of IC engines, a nonlinear predictive controller is designed by iterative optimization to implement the preceding torque demand. The shortcoming of nonlinear predictive control (NPC) for SI engines is that the load or output torque
of the engine needs to be known. As for the torque estimation of the engine, Park and Sunwoo make use of cylinder pressure to calculate combustion torque.15 This methods of torque estimation needs large online computation so that it is not fit for the real-time control. Stotsky deduces a trigonometric polynomial for real-time torque estimation of SI engines.16,17 But it is complicated for every engine to find a suitable trigonometric polynomial. Wang and Chu use high gain, sliding mode and second-order sliding mode observers to implement the online estimation of indicated torque, respectively.18 These observers are sensitive to signal noise. In terms of the estimation of load torque Tload , Potenza et al. constructed a two-degree-of-freedom dynamic model to simulate the instantaneous crank kinematics and total mechanical losses arising in a multi-cylinder gasoline engine coupled to a dynamometer.14 Likewise, it is difficult apply the dynamic model to every engine. In this paper, a proportional– integral (PI) observer is employed to estimate the torque load, with the merits of simple structure, anti-noise, and little online computation. Then, for validation of the torque-based NPC approach, some experiments are carried out in the automotive powertrain test bench including rapid control prototype (RCP) with dSPACE. The paper is organized as follows. The next section illustrates a torque control strategy of automotive powertrain. The subsequent section designs a nonlinear predictive controller by iterative optimization and PI observer of torque load to implement the dynamic torque control of the powertrain. Then, some experimental results are analyzed. Finally, concluding remarks are given.
Torque control strategy of powertrain Based on an idea of torque demand control, a torquebased control strategy of IC engine powertrain is illustrated in this section.
Schematic architecture of automotive powertrain Figure 1 schematically illustrates the automotive powertrain. From this figure, it can be seen that the hardware of the automotive powertrain consists of IC engine, clutch, transmission, final drive, differential, tire, and so on. The software of automotive powertrain is composed of torque demand module, NPC controller and torque load estimation as depicted in this figure. As a driver manipulates the accelerator pedal during the driving, their objective is to achieve a desired output torque from the IC engine so that the vehicle can be driven by wheels based on their requirement. Through a torque demand module, both the position fa of accelerator pedal and the engine speed ve are transformed into a torque demand (Td ½k + i, i = 0, 1, . . . ) of the powertrain. Then, the torque demand together with the estimated load torque are delivered to the NPC
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NPC Controller
Td [k + i ]
φ
TE
c
Torque Demand
Tload Torque Load Estimation
Wheel
Transmission
ωe
φa
ω
Torque Engine
Accelerator
Differential
Clutch
Figure 1. Schematic diagram of automotive powertrain. NPC: nonlinear predictive control.
Tˆload Tˆload
φd Tv
ωe
Load Estimation 1 1 + Tω I d ωˆ e S ωˆ e ω -
+ -
Td [k ]
TE NPC Controller φ c
Torque Td [k + i ] Demand Td [k + ]
ωe
PI Observer
e
Engine
TE
ωe pm φa
Powertrain Dynamics
Figure 2. Flowchart of dynamic torque control for automotive powertrain. PI: proportional–integral; NPC: nonlinear predictive control.
Tcomb [k ]
PD Compensator
φd ωe
Tp
+ Td Engine Tφ + + Map Tv
T%d -
Td [k − 1] 1 + z Td [k ]
Torque Distributor
Td [k + i ] Td [k + L]
Figure 3. Control scheme of torque demand. PD: proportional–derivative.
controller. Consequently, the NPC controller computes the signal of throttle angle fc to the actuator of engine. In fact, except for throttle angle fc , some other input variables of engine are also able to control the torque generation, e.g. spark advance bSA , variable valve time uVVT , air/fuel ratio lAFR . All of the latter control variables of the engine have a similar characteristic of low efficiency for fuel consumption to use them to control the output torque of engine. In this paper, therefore, it is supposed that all of the latter variables are manipulated at the optimal value during driving. And the throttle angle fc is considered as a unique control variable for the IC engine.
Torque-based control approach On the basis of the architecture of the automotive powertrain mentioned above, a torque-based control approach is illustrated in Figure 2. From this figure, it
can be seen that the approach mainly includes torque demand module of the driver, torque-based NPC of the IC engine, and torque load estimation of the powertrain. In this figure, it is obvious that the function of torque demand module is to translate the position of acceleration pedal fd from drivers and engine speed ve into the value (Td ½k + i, i = 0, 1, . . . ) of torque demand. The computation of torque demand is illustrated in Figure 3 and described in the following context. Then, using this value of torque demand, the torque-based NPC controller computes the value of throttle angle fc for EMS. Consequently, the actuator of throttle angle receives the signal fc from the EMS such that the torque demand of the driver is realized completely. In terms of the estimation of load torque Tload , in this paper, a PI observer is designed to estimate the load torque during the experiment. From this figure, it can be seen that the load estimator utilizes the output torque of engine TE and powertrain moment of inertia Id to compute the derivative of the estimated value of ^ e is engine speed v ^_ e so that the estimated engine speed v found by the straightforward integral. Through the comparison between the actual engine speed ve and the estimated speed v ^ e , the engine speed error v e e is sent to the PI observer so that the estimated value of torque load T^load is calculated and transferred to the NPC controller. The detailed formulation of PI observer is described in the following section.
Control scheme of torque demand The control idea of torque demand is illustrated in Figure 3. From this figure, it can be seen that the control scheme of torque demand consists of three control modules, i.e. engine map, PD compensator and torque distributor. The engine map is a three-dimensional map that is able to output a torque value based on two inputs of throttle angle and engine speed. The function of PD compensator is to compute the unrealized torque demand Tp . The function of torque distributor is to calculate the torque demand sequence (Td ½k + i, i = 0, 1, . . . ) as the input variables of the NPC controller. From Figure 3, it is evident that the torque reference Tr is composed of three kinds of torque: pedal torque Tf , compensated torque Tp , and torque demand of vehicular accessories Tv . The control scheme of torque demand hence can be described by three parts: torque feedforward, torque compensation and torque distribution. Torque feedforward. Torque feedforward computes the pedal torque Tf that is from the torque requirement of driver due to the input of accelerator pedal. In this research, the value of pedal torque Tf is simply achieved from the engine map that is a threedimensional table with two inputs, i.e. throttle angle and speed of engine.
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Torque compensation. Due to some nonlinear characteristics of engine dynamics, the torque demand is usually not able to be realized completely by the engine in engineering practice. Based on the actual output torque of the IC engine, therefore, a PD compensator is employed to compute the value of compensated torque Tp that is a torque error between the actual combustion torque Tcomb and the torque demand Td ½k 1. Based on the PD algorithm, the compensated torque Tp can then be given as follows ( e Tp = kpc Ted + kdc ddtT d ð1Þ Ted = Td Tcomb where kpc , kdc are the proportional and differential gains, respectively. Torque distribution. In addition to the torque demand from the accelerator pedal, it is possible for the driver to operate some vehicular accessories, e.g. air conditioner, brake system, entertainment system, and so on. In fact, the torque demand of vehicular accessories Tv is also a direct requirement of the driver. Thus, the torque reference Tr of powertrain is given as follows Tr = Tu + Tp + Tv
ð2Þ
The torque reference Tr is delivered to the torque distributor. Based on the operated states of powertrain, the distributor will produce a subsequence of torque demand, i.e. Td ½k, Td ½k + 1, Td ½k + at time k. Finally, this subsequence is transferred to the NPC controller. Note. In essence, the torque-based control approach can be divided into two hierarchies. The high level is of torque demand control, and a subsequence of torque demand is generated for the IC engine at time k. The low level is of the NPC algorithm, and the throttle angle is produced for actuator of IC engines based on the preceding subsequence of torque demand.
Design of controller After introduction of the mean value model, the torque-based nonlinear predictive control law will be developed to manipulate the IC engine by throttle angle. Then, a PI observer will be described, which is employed to estimate the load torque T^L of the powertrain.
Problem formulation In this research, the engine model used for the design of controller is based on the mean value model, which has two subsystems: the air intake and torque production. Since this paper focuses only on torque control of the engine, the exhaust system is not modeled and the fuel delivery is taken up to the air intake.
Dynamical model of engine. The mean value model can be employed to compute the output torque of IC engines by the intake manifold pressure pm . The intake manifold dynamics, crankshaft dynamics and combustion torque production are given as follows, respectively20 8 < v_ e = I1d (Tcomb Tloss Tload ) ð3Þ h V h Q RT : p_ m = vol d ve pm + th th uth Vm 4pVm hl hD hvol CT Vd pm Tcomb = ð4Þ 4pRT where uth = 1 cos uth , uth is throttle angle, and the meaning of the other parameters can be found in the Appendix. Simultaneously, if there is no specific declaration, all quantities adopt SI units in the context. Note. In the above dynamic model of the engine, the derivative of temperature is neglected since it has only a minor effect on the manifold pressure dynamics in general so that the intake manifold temperature can be assumed constant. State-space model. As depicted in equation (3), we can choose the state variables as x1 = ve , x2 = pm and the control input as u = uth . Then, the engine model equation (3) can be rewritten as the following state space equation x_ 1 = a1 x1 + a2 x2 d ð5Þ x_ 2 = a3 x1 x2 + bu where the parameters are given by he h h h CT Vd a2 = l d vol Id 4pId RT h Vd Tload h Qth RT d= b = th a3 = vol Vm 4pVm Id
a1 =
Usually, the torque Tcomb generated by combustion of air–fuel mixture is balanced by two kinds of torque: engine loss Tloss and load torque Tload . As is well known, the engine resistant torque is from the friction, air pump and so on. Stotsky proposes an algorithm to estimate the engine friction torque via the crankshaft speed fluctuations at the fuel cut-off and idle state.19 But, in this research, it is supposed that the torque Tloss of engine loss within the small range of engine speed is similarly proportional with the engine speed ve based on a suitable factor he , i.e. Tloss = he ve . Discrete-time model. Assume that the torque demand for the engine is delivered in discrete time fashion with interval Ts , i.e. the desired reference signal for torque generation given at time t = kTs is yd ½k + 1jk, yd ½k + 2jk, . . . , where k denotes the sampling index with period Ts . In order to design a desired control law in the discrete time framework, we first make a
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one-step prediction of system (5) by use of a control input increment Du½k at time k, which is equivalent to discretizing equation (5) into the following form (
x^1 ½k + 1jk = a7 x1 ½k + a8 x2 ½k d0 x^2 ½k + 1jk = a9 x1 ½kx2 ½k + x2 ½k + h1 (u½k 1 + Du½k)
ð6Þ y½k = cx2 ½k
ð7Þ
where system output y½k is the combustion torque Tcomb depicted in equation (4), the other parameters are given by a7 = 1 a1 Ts , a8 = a2 Ts , a9 = a3 Ts , d0 = dTs , h h h CT Vd , and c = l d vol 4pRT ð8Þ
h1 = bTs
From equation (6), it can be seen that the control input u½k at time k is computed by backward difference, which is equivalent to an embedded integrator for the control input as follows u½k = u½k 1 + Du½k
subject to equations (6), (7), (9), (10) and jDu½kj 4 1 cos uD ju½kj 4 1 cos umax vmin 4 x1 ½k 4 vmax pmin 4 x2 ½k 4 pmax
J(Du½k) = (r + w1 c2 h21 + w2 c2 h22 ))Du2 ½k 2c(w1 h1 e^½k + 1jk + w2 h2 e^½k + 2jk)Du½k
ð9Þ
+ e^2 ½k + 1jk + e^2 ½k + 2jk
ð15Þ
where the pseudo-errors e^½k + 1jk, e^½k + 2jk of system output are given by as follows
Nonlinear predictive controller First of all, a two-step predictive controller is deduced from a cost function. Then, the two-step NPC algorithm is extended to the multi-step one by use of an iterative method approach. Two-step predictive controller. Using this approach iteratively for two-step-ahead prediction, under the condition of the control input keeping the value u½k + 1 = u½k, i.e. Du½k + 1 = 0, yields 8 2 > < x^1 ½k + 2jk = a8 a9 x1 ½kx2 ½k + a7 x1 ½k + p1 x2 ½k p2 + a8 b0 u2 ½k x^2 ½k + 2jk = p3 x1 ½kx2 ½k + p4 x2 ½k + a7 a29 x21 ½kx2 ½k > : + a8 a29 x1 ½kx22 ½k + h2 (u½k 1 + Du½k)
ð10Þ
where p1 = a8 (a7 + 1), p2 = a7 (d0 + 1), p3 = a9 (a7 + a9 d0 + 1), p4 = a9 d0 + 1, and h2 = a7 a9 b0 x1 ½k a8 a9 b0 x2 ½k + a9 b0 d0 + 2b0 ð11Þ
The design problem of the control system can then be presented as follows: for given torque demand subsequence yd ½k + 1jk, yd ½k + 2jk, find a feedback control law for even increment Du½k of every predictive step. The control input solves the following moving horizontal optimization problem min Du½k
i=1
ð14Þ
where uD , umax , vmin , vmax , pmin , pmax are some constants of engine control system, p, m is the number of predictive step for system states and control inputs, respectively. rj . 0, wi . 0 is the weighting coefficients, y^½k + ijk are the predictive values of system outputs based on the predictive states x^½k + ijk. Now, we adopt the two-step prediction of system states and one-step prediction of control input, i.e. p = 2, m = 1. Substituting equations (6), (10) and (9) into equation (12), the cost function can be rewritten as follows
where u½k 1 is the known last-step control input at time k.
p X
ð13Þ
wi (^ y½k + ijk yd ½k + ijk)2 +
m X
rj Du½k2
j=1
ð12Þ
8 > < e^½k + 1jk = yd ½k + 1 c( a9 x1 ½kx2 ½k + x2 ½k + h1 u½k 1) e^½k + 2jk = yd ½k + 2 c( p3 x1 ½kx2 ½k + p4 x2 ½k > : + a7 a29 x21 ½kx2 ½k + a29 a8 x1 ½kx22 ½k + h2 u½k 1)
ð16Þ
As denoted in equation (15), it is easy to see that the problem is to find the least value of quadratic equation. Then, a straightforward calculation of the derivative of equation (15) yields the unique optimal solution of cost function equation (12) as follows Du½k =
w1 ch1 e^½k + 1jk + w2 ch2 e^½k + 2jk r1 + w1 c2 h21 + w2 c2 h22
ð17Þ
Thus based on the mean-value model, a torque-based two-step NPC algorithm is to be found, which is depicted in equation (17). From the above deduction, it can be seen that the NPC law has some characteristics as follows: firstly, as shown in equation (5), the mean value model used is nonlinear. Secondly, as depicted in equation (17), the optimal solution of the cost function is an analytical expression, i.e. the control law is analytical. Finally, the control law as denoted in equation (17) is a unique solution of the cost function equation (12). Multi-step predictive controller. If using the previous predictive way to proceed to the three-step or more-step prediction of system states, the degree of cost function will exceed two so that the cost function (12) will have multi-solution. Due to the largely online computation
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from multi-solution to optimal solution, the previous idea of predictive deduction is not suitable for the realtime control as exceeding two-step prediction of system states. As for the preceding issue, if consider the value x^½k + 2jk of No. 2 step prediction as a new origin to proceed to a same two-step predictive computation based on the achieved u½k, i.e. the x^½k + 3jk and x^½k + 4jk are deduced from the x^½k + 2jk. Hence, in accordance with equations (6) and (10), adopting one-step prediction of control input u½k + 1 = u½k + Du½k yields 8 > < x^1 ½k + 3jk = a7 x^1 ½k + 2jk + a8 x^2 ½k + 2jk d0 x2 ½k + 2jk x^2 ½k + 3jk = a9 x^1 ½k + 2jk^ > : + x^2 ½k + 2jk + h3 u½k + 1 ð18Þ
8 > x^ ½k + 4jk = a8 a9 x1 ½k + 2jkx2 ½k + 2jk + a27 x1 ½k + 2jk > > 1 > > > + p1 x2 ½k + 2jk p2 + a8 b0 u2 ½k + 2jk > < x^2 ½k + 4jk = a8 a29 x1 ½k + 2jkx22 ½k + 2jk + p4 x2 ½k + 2jk > > > > p3 x1 ½k + 2jkx2 ½k + 2jk > > > : + a a2 x2 ½k + 2jk x ½k + 2jk + h u½k + 1 7 9 1
2
4
ð19Þ
where 8 > < h3 = h1 h4 = a7 a9 b0 x^1 ½k + 2jk a8 a9 b0 x^2 ½k + 2jk > : + a9 b0 d0 + 2b0 ð20Þ
Adopting a four-step prediction of system states and two-step prediction of control input, i.e. p = 4, m = 2, therefore, a control law by iterative optimization can be achieved by use of the same cost function equation (12) and solution method as follows P4 wi chi e^½k + ijk Du½k = P2 i = 1 ð21Þ P4 2 2 j = 1 rj + i = 1 wi c hi where the pseudo-errors e^½k + 3jk, e^½k + 4jk of system output are as follows 8 x2 ½k + 2jk e^½k + 3jk = yd ½k + 3 c( a9 x^1 ½k + 2jk^ > > > > > ^ ½k + 2jk + h u½k) + x 2 3 > < x2 ½k + 2jk e^½k + 4jk = yd ½k + 4 c( p3 x^1 ½k + 2jk^ > > 2 2 > + p4 x^2 ½k + 2jk + a7 a9 x^1 ½k + 2jk^ x2 ½k + 2jk > > > : 2 2 + a9 a8 x^1 ½k + 2jk^ x2 ½k + 2jk + h4 u½k)
ð22Þ
Therefore, using the same idea of iterative optimization is able to achieve the control law of multi-step prediction for p = 2n and m = n as follows P2n wi chi e^½k + ijk Du½k = Pn i = 1 ð23Þ P2n 2 2 j = 1 rj + i = 1 wi c hi where
8 hi = h1 , as i = 3, 5, 7, . . . > > > < h = a a b x^ ½k + i 2jk a a b x^ ½k + i 2jk i 7 9 0 1 8 9 0 2 > + a b d + 2b , 9 0 0 0 > > : as i = 4, 6, 8, . . .
ð24Þ 8 > x2 ½k + i 1jk e^½k + ijk = yd ½k + i c( a9 x^1 ½k + i 1jk^ > > > > > + x^2 ½k + i 1jk + hi u½k) as i = 3, 5, 7, . . . > > > > > ^½k + ijk = yd ½k + i c( p3 x^1 ½k + i 2jk^ e x2 ½k + i 2jk > < + p4 x^2 ½k + i 2jk > > > x2 ½k + i 2jk + a7 a29 x^21 ½k + i 2jk^ > > > > > 2 > + a9 a8 x^1 ½k + i 2jk^ x22 ½k + i 2jk + hi u½k) > > > : as i = 4, 6, 8, . . .
ð25Þ
Up to now, the multi-step NPC law equation (23) has been designed by iterative optimization. Iterative optimization means that using the last-step predictive value of the system states as the nominal value of this step proceeds to a two-step prediction of system states. By use of iterative optimization, the cost function equation (12) can be held as a quadratic equation as the number of predictive steps is extended to the finite quantity n(n = 1, 2, . . . ). An analytic and unique optimal control law can be solved from the quadratic cost function equation (12) instead of increasing the online computation. Hence, it is easy to deduce that this multi-step predictive algorithm can be applied to real-time control. Note. As described for equation (9), the Du½k is equivalent to an embedded integrator that accumulates the increment Du½k of control input to control variable of system u½k. Then, it can be seen from the control law of Du½k (23) that the weighting coefficient rj is similar to the proportional factor and wi is similar to the integral factor. Furthermore, the predictive behavior of system states is able to advance the system phase, which is similar to the function of the differentiator. Simultaneously, as depicted in equation (24), the coefficient hi of control law (23) varies at every time k, which is equivalent to updating the weighting coefficient wi at every time k. From the viewpoint of feedback control, therefore, the NPC is similar to the adaptive PID controller. But, between the PID and NPC, there is also a different feature: the former uses the one-order backward difference, whereas the latter uses the resultant value of twoorder forward difference.
Torque load estimation In fact, the load Tload is an unmeasurable variable in engineering practice. For the practical application of NPC controller of the IC engine, a PI observer is employed to estimate the load torque of automotive powertrain.
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Let the engine output torque TE = Tcomb Tloss , then the crankshaft dynamics of equation (3) can be rewritten as follows Id v_ e = TE Tload
ð26Þ
Assumption I: At time k 1, the load torque is TL0 . Then, at time k, the actual load Tload can be described as follows Tload = TL0 + D( v e e)
200
a. Engine Torque [Nm]
T1 T2 T3 T4
180 160 140
T5 T6 T7
120 x104 5.6
b. Intake Manifold Pressure [Pa]
P1 P2
c. Load Torque [Nm]
L1 L2
5.2
ð27Þ
4.8
where the error of engine speed is v ee = v ^ e ve , v ^ e is the estimated speed of engine, D( v e e ) is unknown function, which is bounded by a known function z( v e e ), i.e. e e ). jD( v e e )j 4 z( v Based on the principle of PI observer in Figure 2, the load observer can be formulated as follows ( R T^load = kpt v e e + kit v e e )sgn( v e e) e e dt + z( v ð28Þ v ^_ e = I1d (TE T^load )
4.4 4 125 115 105 95 3000
where the kpt , kit are the proportional and integral gains, respectively. The sgn is a signum function.
2500
Note. When the observer is applied to engineering practice by digital implementation, the estimated error may not tend to zero due to the limited sampling rate and quantization. For the proof of its stability, see Hong et al.20
1500 13
S1 S2 S3
2000 d. Engine Speed [rpm]
A1 A2
11 9
Experimental validation
e. Throttle Angle [deg]
In order to verify the torque-based NPC controller, some experimental maneuvers are conducted on a test bench of automotive powertrain.
Experimental scheme The test engine was provided by Toyota Motor Corporation in Japan, and its specification is given in the Appendix. The engine was installed in a test platform and connected to a dynamometer. The EMS operates as a standard commercial controller and receives the signal of throttle angle from dSPACE (DS1103) by controller area network (CAN) bus. dSPACE also receives required quantities from EMS and some sensor signals by the input/output (I/O) interface, e.g. engine speed, intake manifold pressure, dynamometer torque. Under the condition of the test bench mentioned above, some speed tracking maneuvers will be carried out to verify the NPC controller. In order to perform the engine speed tracking, a PI controller is employed to generate the torque reference by use of the speed error between the desired speed and the actual. Then, the torque reference is transferred to the torque distributor as the control objective of engine. In terms of the parameters of model (5), a1 = 0:2709, a2 = 0:0055, a3 = 0:51, b = 3:633108 , c = 0:00301, respectively. The gross moment of inertia including crankshaft, transmission and dynamometer, is
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Figure 4. Experimental results of accelerated maneuver for the tracking of desired speed. Where the desired speed steps from 1500 rpm to 2500 rpm, and Tload ’100 Nm. T4, T5, T6, T7 and T1 are the torque demand of 1st, 3rd, 5th, 7th and 8th step, respectively. T2 and T3 are the estimated, 1st-step predictive torque of engine, respectively. P1 and P2 are the actual, 1st-step predictive intake manifold pressure, respectively. L1 and L2 are the estimated and actual load torque, respectively. s1, s2 and s3 are the desired, the actual, 1st-step predictive engine speed, respectively. A1 and A2 are the desired and actual throttle angle, respectively.
Id = 0:55. The sampling period is Ts = 0:01 s. The gear ratio of transmission is ig = 1. In the following experiments, The number of predictive steps for system states is eight, and the number of control inputs is four, i.e. p = 8 and m = 4. Then, the weighting coefficients of the control inputs are ri = 23106 , i = 1, 2, 3, 4. And the weighting coefficients for the errors of system output are w1 = w2 = 1, w3 = 0:9, w4 = 0:8, w5 = 0:7, w6 = 0:6, w7 = 0:5, w8 = 0:4, respectively.
Experimental results Three experimental maneuvers are applied to the test bench mentioned above, i.e. accelerated and decelerated
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maneuver as Tload ’ 100 Nm, accelerated maneuver as Tload ’150 Nm. Case I: Accelerated maneuver as Tload ’ 100 Nm. In Figure 4, there are some experimental results of accelerated maneuver by use of the torque-based NPC approach, where the desired angular speed of powertrain steps from 1500 rpm to 2500 rpm, and the load torque is about Tload ’100 Nm. In Figure 4(a), the inset title ‘Engine Torque’ means that this subplot shows some experimental results of engine torque. The same naming convention is applied to the following subplots. In addition, the estimated torque of engine is regarded as the actual torque generated from the combustion of air–fuel mixture. Simultaneously, for clear illustration, the experimental results of the desired torque of the 2nd, 4th and 6th steps are neglected in Figure 4(a) during the plotting. In Figure 4(e), the desired throttle angle is the output value of the NPC controller and the actual throttle angle is from EMS by CAN bus. From Figure 4(d), it can be seen that the desired engine speed steps from 1500 rpm to 2500 rpm near 1.5 s. Then, the torque demands of engine also change with the step of engine speed in Figure 4(a). From 2 to 4 seconds in Figure 4(a), it also can be seen that, when the torque error between the torque demand and the actual torque is larger than 10 Nm, the NPC controller shows a sequence of torque demand. The torque demands of all predictive step are equal to each other, however, when the torque error is less than 10 Nm. From Figure 4(d), it can be seen that the actual speed of powertrain is able to track the desired speed. In Figure 4(a), the actual output torque of the engine is also able to track the torque demand exactly. Simultaneously, the estimated load torque is approximately equal to the value of the dynamometer in Figure 4(c). But, from 2 to 4 seconds in this figure, it can be seen that the tracking is not good. The reason is that the engine output torque TE used by PI observer is from the real-time computation of combustion torque Tcomb by the mean value model based on the engine speed ve and intake manifold pressure pm . In reality, the measured values of the preceding signals are always in great fluctuation during the dynamic course so that the corresponding computed value of combustion torque Tcomb is also fluctuant. At the same time, the computed value of combustion torque Tcomb by the mean value model is not accurate so that the estimated load torque is different with the actual during the transient course. Comparing Figure 4(a) and (d) with Figure 4(c), it can be seen that the trend of estimated load torque is closed to the actual engine torque and speed. The reason is that the estimated load torque is derived from the actual engine torque and speed as formulated in equation (28). Case II: Decelerated maneuver as Tload ’ 100 Nm. As shown in Figure 5, there are some experimental results
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Figure 5. Experimental results of decelerated maneuver for the tracking of desired speed. Where the desired speed steps from 2500 rpm to 1500 rpm, and Tload ’100Nm. The naming convention of curves is the same as in Figure 4.
of decelerated maneuver by use of the torque-based NPC approach, where the desired speed of powertrain steps from 2500 rpm to 1500 rpm at about 1.5 s, and the load torque is Tload ’100 Nm. Likewise, from Figure 5(d), it can be seen that the actual speed of powertrain is able to trace the desired speed by the use of the torque-based NPC approach. In Figure 5(a), the actual output torque of engine is also able to trace the torque demand exactly. Simultaneously, the estimated load torque is close to the measured value of dynamometer in Figure 5(c). But, from 2 to 4 seconds in this figure, it is obvious that the tracking is not good in the decelerated maneuver, for the same reason as with the previous accelerated maneuver. This indicates that the dynamics performance of the PI observer needs to be improved due to the lack of the differential. Case III : Accelerated maneuver as Tload ’ 150 Nm. As illustrated in Figure 6, there are some experimental results of accelerated maneuver by use of the torque-based NPC approach, where the desired speed of powertrain
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Proc IMechE Part D: J Automobile Engineering 226(8)
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NPC controller can be used to manipulate an engine to make the powertrain reach the desired speed on the test bench. From the discussion of these three case studies, it can be concluded that the torque-based NPC controller is a significant candidate to manipulate the powertrain directly by torque in engineering practice.
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Conclusion
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Figure 6. Experimental results of accelerated maneuver for the tracking of desired speed. Where the desired speed steps from 1500 rpm to 2500 rpm, and Tload ’150 Nm. The naming convention of curves is the same as in Figure 4.
steps from 1500 rpm to 2500 rpm at about 1.5 s, and the load torque is Tload ’ 150 Nm. From Figure 6(d), it can be seen that the actual speed of powertrain is able to trace the desired speed by the use of the torque-based NPC approach. In Figure 6(a), the actual output torque of engine is also able to trace the torque demand exactly. Simultaneously, the estimated load torque is able to catch the measured value of the dynamometer in Figure 6(c). From 2 to 4 seconds in Figure 6(c), it is also easy to see that there is better tracking in the accelerated maneuver with 150 Nm load than the previous accelerated with 100 Nm load. The reason is that the trend of actual load is close to the engine torque and speed so that the trend of estimated load torque computed by the engine torque and speed is also close to the actual values. In summary, for the load estimation of the PI observer, the estimated load can track the actual during the dynamic operation of powertrain, and the tracking performance for the steady state is good. From these experimental results of the desired speed tracking, it can be concluded that the torque-based
A torque-based nonlinear predictive control approach of automotive powertrain by iterative optimization was developed in this paper. Simultaneously, the control scheme of torque demand is proposed to meet the torque requirement of drivers. From the experimental results and analysis, an NPC controller like adaptive PID can be used to control the automotive powertrain directly by torque in the test bench. By use of the iterative optimization, the NPC algorithm can be extended from a two-step to a multi-step prediction of system states instead of increasing the online computation, which makes it possible to realize the application of the multi-step NPC in the real-time control of engineering practice. Through the experimental analysis, it can be concluded that the dynamic performance of the PI observer needs to be improved due to the lack of the differential. In summary, the estimated load can track the actual load during the dynamic operation of the powertrain, and the tracking performance of the steady state is good. After validation in the test bench, the NPC approach will be applied to a real vehicle in the next step. Simultaneously, except for the NPC approach, other control methods can also be used to implement the torque-based control of powertrain, e.g. adaptive PID and other ordinary optimization. Comparative experiments between NPC and other control methods hence should be conducted in the future. Funding The authors are grateful to the Toyota Motor Corporation in Japan. The authors also greatly appreciate the support from the National Natural Science Foundation of China (grant no. 50905092) and the National Basic Research Program of China (grant no. 2011CB711205). References 1. Gerhardt J, Henninger H and Bischof H. A new approach to functional and software structure for engine management systems – BOSCH ME7. SAE paper 980801, 1998. 2. Heintz N, Mews M and Stier G. An approach to torquebased engine management systems. SAE paper 2001-010269, 2001. 3. Yutaro M, Hiroshi I, Hiraku O and Naonori O. A study of engine torque control by variable valve actuation. In: JSAE annual congress, Japan, 2005. 4. Solliec GL, Berr FL, Corde G and Colin A. Downsized SI engine control: a torque-based design from simulation to vehicle. SAE paper 2007-01-1506, 2007.
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5. Livshiz M, Kao M and Will A. Engine torque control variation analysis. SAE paper 2008-01-1016, 2008. 6. Lee W, Yoon M and Sunwoo M. A cost- and timeeffective hardware-in-the-loop simulation platform for automotive engine control systems. Proc IMechE Part D: J Automobile Engineering 2003; 217: 41–52. 7. Satou S, Nakagawa S, Kakuya H, et al. An accurate torque-based engine control by learning correlation between torque and throttle. SAE paper 2008-01-1015, 2008. 8. Wehrwein D. Optimization of engine torque management under uncertainty for vehicle driveline clunk using timedependent metamodels. J Mech Des 2009; 131: 1–11. 9. Nagata T and Tomizuka M. Robust engine torque control by iterative learning control. In: Proceedings of the American control conference, St Louis, MO, June 2009, pp.2064–2069. 10. Giorgetti N, Ripaccioli G and Bemporad A. Hybrid model predictive control of direct injection stratefied charge engines. IEEE/ASME Trans Mechatron 2006; 11: 499–506. 11. Saerens B, Diehl M and Swevers J. Model predictive control of automotive powertrains – first experimental results. In: IEEE conference on decision and control, December 2008, pp.5692–5697. 12. Vermillion C, Butts K and Reidy K. Model predictive engine torque control with real-time driver-in-the-loop simulation results. In: American control conference, Baltimore, MD, June 2010, pp.1459–1464. 13. Wang SW, Yu DL, Gomm JB, Page GF and Douglas SS. Adaptive neural network model based predictive control of an internal combustion engine with a new optimization algorithm. Proc IMechE Part D: J Automobile Engineering 2006; 220: 195–208. 14. Potenza R, Dunne JF, Vulli S and Richardson D. A model for simulating the instantaneous crank kinematics and total mechanical losses in a multicylinder in-line engine. Int J Engine Res 2007; 8: 379–397. 15. Park S and sunwoo M. Torque estimation of spark ignition engines via cylinder pressure measurement. Proc IMechE Part D: J Automobile Engineering 2003; 7: 809–817. 16. Stotsky A. Fast algorithm for real-time torque estimation. Int J Engine Res 2008; 9: 239–247. 17. Stotsky A. Computationally efficient filtering algorithms for engine torque estimation. Proc IMechE Part D: J Automobile Engineering 2005; 219: 1099–1107. 18. Wang Y and Chu F. Application of non-linear observers to online estimation of indicated torque in automotive engines. Proc IMechE Part D: J Automobile Engineering 2005; 219: 65–75. 19. Stotsky A. Data-driven algorithms for engine friction estimation. Proc IMechE Part D: J Automobile Engineering 2007; 221: 901–909. 20. Hong M, Shen T, Ouyang M and Kako J. Torque observers design for spark ignition engines with different intake air measurement sensors. IEEE Trans Contr Syst Technol 2011; 19: 229–237.
pm Qth
Tp Tr Ts Tv Tf Vm Vd bSA hth hvol hD hl uVVT lAFR fa fc ve v ^e v ee
gas constant of air intake air temperature combustion torque of IC engine torque demand of powertrain output torque of IC engine load torque of powertrain loss torque of IC engine including friction, air pump and so on compensated torque of IC engine torque reference of powertrain sampling period torque demand of vehicular accessories pedal torque of IC engine from accelerator intake manifold volume displacement of engine spark advance effect of pressure ratio across the throttle volumetric efficiency of engine effect of ignition timing air–fuel ratio variable valve time air–fuel ratio actual throttle angle control signal of throttle angle engine angular speed estimated engine angular speed speed error between the estimated engine angular speed and the actual
Abbreviations CAN DISC EMS IC I/O MPC NPC PC PD PI PID RCP SI
controller area network direct-injection stratified-charge engine management system internal combustion input/output model predictive control nonlinear predictive control personal computer proportional–derivative controller proportional–integral controller proportional–integral–derivative controller rapid control prototype spark ignition
Engine specifications
Appendix Notation CT Id
R T Tcomb Td TE Tload Tloss
specific constant of engine squivalent moment of inertia of powertrain intake manifold pressure maximum flow rate of throttle
Type Injection Bore Stroke Displacement Maximum power Maximum torque
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V6 port and direct-injection 94 mm 83 mm 3.456 L 232 kW/(6400 rpm) 377 Nm/(4800 rpm)
