BULETINUL INSTITUTULUI POLITEHNIC DIN IAŞI
Publicat de
Universitatea Tehnică „Gheorghe Asachi” din Iaşi
Tomul LIX (LXIII), Fasc. 3, 2013
SecŃia
AUTOMATICĂ şi CALCULATOARE
REAL-TIME SIMULATION OF A
NETWORK-CONTROLLED VEHICLE DRIVETRAIN
BASED ON MODEL PREDICTIVE CONTROL
BY
CONSTANTIN F. CĂRUNTU*
“Gheorghe Asachi” Technical University of Iaşi,
Faculty of Automatic Control and Computer Engineering
Received: September 1, 2013
Accepted for publication: October 15, 2013
Abstract. This paper proposes a real-time implementation of a networked
predictive controller designed to damp driveline oscillations, which is crucial in
improving drivability and passenger comfort, while compensating the time-varying
delays that appear due to sending the control commands and the measurements
from the sensors through Controller Area Network (CAN). Firstly, the designed
real-time structure integrated with CAN test-bench is described and then the model
of the drivetrain is derived. Secondly, considering that the CAN-induced timevarying delays are bounded, a method of modelling the physical plant (vehicle
drivetrain) including the delays is proposed. Then, a predictive control strategy,
which makes use of the previously developed model, is designed in order to damp
the driveline oscillations. The proposed control scheme is tested using the designed
test-bench and the experiments based on realistic scenarios show that the proposed
controller can outperform classical controllers, e.g., PI.
Key words: real-time simulation, networked control systems, drivetrain
oscillations damping, model predictive control.
2010 Mathematics Subject Classification: 93C10, 93D05, 00A72.
*
Corresponding author; e-mail: caruntuc@ac.tuiasi.ro
44
Constantin F. Căruntu
1. Introduction
The testing of control algorithms in their real environment represents an
important phase in the development of control units for drivetrains. The major
feature of a real-time simulation is that the simulation can be carried out as
quickly as the real system would actually run, thereby allowing to combine the
simulation and the real Controller Area Network (CAN). The absence of the
real device in the simulation particularly simplifies the development, since it
allows systematically and effectively validation of the closed-loop control
system, even if the controller is not hardware implemented.
The use of real-time simulation can replace significantly the
construction of expensive prototypes to test drivetrain systems. Due to the
utilization of the model in the closed loop, the simulation is cheaper, easier and
less time consuming than experiments on the real device. Moreover, the risks to
damage the real device are eliminated.
The computing power required by real-time simulation highly depends
on the characteristics of the simulated model: if it contains very demanding
calculations you have to provide a lot of computing power because the timing
cannot be satisfied otherwise. A simulation performed in Simulink takes as
much computer time as needed to compute the control command and then to
calculate the behaviour of the system. If the model is complex, much more time
is needed to carry out the necessary calculations. Because a strict time
requirement for Simulink simulation does not need to be fulfilled, the
complexity of the model does not need to be reduced.
Recently, the need for increased drivability and passenger comfort lead
to the design of proper models for vehicle drivetrains and to the development of
different control strategies to minimize the effects of drivetrain oscillations.
When a vehicle is subjected to acceleration, the elasticity of the various
components in the driveline may cause torsional vibrations or disturbances. The
oscillations give rise, apart from material stress, to noticeable reduced
drivability and comfort.
Various research has been carried on understanding and damping the
driveline oscillations of a conventional vehicle and numerous control strategies
were reported in the literature: robust pole placement (Richard et al., 1999),
(Stewart et al., 2005), H∞ optimization (Lefebvre et al., 2003), linear quadratic
gaussian control design with loop transfer recovery (LQG/LTR) (Petterson,
1997; Fredriksson et al., 2002; Berriri et al., 2008) and model predictive control
(MPC) (Rostalski et al., 2007).
All of the above control solutions assume that the sensors, controllers
and actuators are directly connected, which is not realistic. Rather, in modern
vehicles, the control signals from the controllers and the measurements from the
sensors are exchanged using a communication network, e.g., CAN or Flexray,
among control system components. This brings up a new challenge on how to
Bul. Inst. Polit. Iaşi, t. LIX (LXIII), f. 3, 2013
45
deal with the effects of the network-induced delays and packet losses in the
control loop. The delays may be unknown and time-varying and may degrade
the performances of control systems designed without considering them and can
even destabilize the closed-loop system.
As such, the problem considered in this paper is the development of a
real-time simulation test-bench designed to validate a networked predictive
control strategy proposed to minimize the oscillations of a vehicle drivetrain,
while compensating the time-varying delays introduced by CAN. The proposed
solution consists of three steps: firstly, a standard continuous-time model of the
vehicle drivetrain is derived and a technique to model the physical plant
including the network-induced time-varying delays is proposed; secondly, a
predictive control scheme is presented and designed using the previously
developed model. Thirdly, the networked predictive controller and the plant
model are implemented on a real-time simulation test-bench and the designed
experiments validate the proposed approach and indicate that the proposed
scheme can outperform other types of controllers, e.g., PI.
The remainder of the paper is organized as follows. Section II illustrates
the control architecture for the network-controlled vehicle drivetrain and then
the real-time simulation setup is described. In Section III the mathematical
model describing the vehicle drivetrain is introduced, including the modelling
of the time-varying delays induced by CAN. In Section IV the predictive
control strategy is presented and then the experimental results are discussed in
Section V. The concluding remarks are given in Section VI.
2. Drivetrain Control Architecture
The structure of a vehicle drivetrain has following parts: engine, clutch,
transmission, final drive, driveshafts and wheels, as it can be seen in Fig. 1.
Fig. 1 – Schematic vehicle drivetrain structure.
2.1. Network-Controlled Architecture
As it can be seen in Fig. 2 the components of the closed-loop vehicle
drivetrain control system, e.g., sensors, controllers, actuators, communicate
46
Constantin F. Căruntu
through a communication network, i.e., the engine torque (control signal), and
also the engine and wheel speeds are sent through CAN. Many of the messages
transmitted on CAN, have real-time constraints associated with them.
The complete network-controlled architecture considered in this paper,
which is graphically depicted in Fig. 2, consists of the following operations:
− the sensor measures the output of the system and sends the sample to
the controller through CAN;
− the cruise controller receives the measurement from the sensor and
the desired velocity reference vvr = rwωwr and computes the required torque,
while handling the physical constraints and the delays, where rw is the wheel
radius and ωwr is the desired wheel speed;
− the control signal, i.e., the torque computed by the controller, is sent
to the engine controller through CAN;
− the engine controller actuates the spark timing and airflow ( ua ) as
requested for driveline control.
In Fig. 2 the dashed lines represent the direction of the messages sent to
and from the controllers.
Fig. 2 – Drivetrain control architecture.
In the study case considered in this paper, the actuator is the internal
combustion engine and it is assumed that the request is accomplished
instantaneously. For the engine torque this can be achieved by creating a torque
reserve by changing the spark timing in order to have quickly available torque
modifications (Di Cairano et al., 2010). The torque reserve can be maintained
by appropriately actuating the fuel injection.
2.2. Real-Time Simulation Setup
The test-bench setup for real-time simulation of the closed-loop
drivetrain control system, which is designed to validate the control algorithm,
consists of: dSPACE MicroAutoBox, Host PC, CAN network, power supply, a
Simulink model of the drivetrain and controller and a Graphical User Interface
(GUI) as it can be seen in Fig. 3.
Bul. Inst. Polit. Iaşi, t. LIX (LXIII), f. 3, 2013
47
MicroAutoBox consists of the base board DS1401 and it performs the
drivetrain plant-model and controller calculations. It has an Ethernet interface
for direct connection to the host PC, for tasks such as loading models and
reading or adjustig parameters in GUI. The HostPC also runs the simulator’s
console software (GUI), designed in ControlDesk, to communicate with the
real-time Simulator (MicroAutoBox). The application allows to download and
control the real time process such as download, start, stop, variable observation
and data collection.
Real-Time Interface (RTI) is the link between dSPACE MicroAutoBox
hardware and the MATLAB/Simulink models. Real-Time Workshop (RTW)
generates the model code which is compiled and downloaded in the real-time
hardware. The RTI CAN MultiMessage Blockset is an extension for Real-Time
Interface and it is used for configuring the CAN network. CAN configurations
are read from a database container (DBC) file.
Fig. 3 – Real-time test-bench.
Note that the Simulink model of the drivetrain and controller exchange
data via a CAN bus. Also, the Host PC has access to the CAN bus through the
developed ControlDesk application.
3. Drivetrain Model
In order to develop a controller, an accurate drivetrain model is required
to predict the vehicle's response to a torque input. The model can then be used
to design and simulate the closed-loop control system.
A piece-wise affine (PWA) three inertias model, which takes into
consideration the clutch flexibility together with the driveshaft flexibility, was
48
Constantin F. Căruntu
derived from the laws of motion (Kiencke & Nielsen, 2005; Grotjahn et al.,
2006; Van Der Heijden et al., 2007), in which the first inertia corresponds to the
engine, the second one includes the inertia of the gearbox and the inertia of the
final drive, and the last inertia corresponds to the wheel and vehicle mass, as it
is represented in Fig. 4.
The clutch torsional flexibility is a result of an arrangement with
smaller stiffness springs in series with springs with higher stiffness. The reason
for this arrangement is vibration insulation. Unlike in the above mentioned
modeling approaches, where only two or three working modes are considered
for the clutch dynamics, in this paper four working modes for the clutch are
introduced, i.e., open, closing, closed and locked, which yields a more accurate
model of the clutch.
Fig. 4 – Three inertias drivetrain model including the clutch.
In the open mode, there is no mechanical connection between the
engine and the rest of the driveline, so no torque is transmitted towards the
wheels. In the closing mode, the smaller springs in the clutch are compressed,
which means that the engine torque is gradually transmitted to the driveline. In
the closed mode, the stiffer springs in the clutch are compressed and the gradual
transmission of torque to the driveline continues. The locked mode corresponds
to the phase when the springs in the clutch cannot be compressed any further,
i.e., the clutch hits a mechanical stop, and the maximum amount of torque is
transmitted from the engine to the wheels through the driveline.
3.1. Continuous-Time Model
Consider as state variables: the torsional angle between engine and
transmission, the torsional angle between final drive and wheel, the angular
speed of the engine, the angular speed of the transmission and the angular speed
of the wheel, and as control input of the system, the engine torque, i.e.,
Bul. Inst. Polit. Iaşi, t. LIX (LXIII), f. 3, 2013
x1 = θ e − θ t it , x2 =
θt
if
− θ w , x3 = ωe , x4 = ωt , x5 = ω w , u = Te ,
49
(1)
where θ e , θt and θ w are angles of the engine, transmission and wheel,
respectively, ω e , ωt and ω w are the angular speeds of the engine, transmission
and wheel, respectively, Te is the torque produced by the engine and it and i f
are the transmission and final drive gear ratios, respectively.
The following PWA state-space model is obtained:
xɺ (t ) = A ci x(t ) + b c u (t ) + fc
if
x (t ) ∈ Ω i
y (t ) = [ 0 0 0 0 rw ] x(t )
(2)
where x := ( x1 ,…, x5 )⊤ ∈ ℝ5 and i ∈ I := ℤ [1,4] . Here i denotes the active mode
at time t ∈ ℝ + , Aci ∈ ℝ5×5 , bc ∈ ℝ5×1 are the system matrices and fc ∈ ℝ5×1 is the
affine term. Notice that although there are 3 angles, only two states are
introduced as only the angle difference is relevant. The collection of sets
{Ωi ∣ i ∈ I } defines a partition of the state-space X ⊆ ℝ 5 as follows:
Ω1 := {x ∈ ℝ5 | x3 ≤ ωeclosing },
Ω 2 := {x ∈ ℝ5 | x3 > ωeclosing & | x1 |≤ θ1},
Ω3 := {x ∈ ℝ 5 | x3 > ωeclosing & θ1 <| x1 |≤ θ 2 },
(3)
Ω 4 := {x ∈ ℝ5 | x3 > ωeclosing & θ 2 <| x1 |},
where ωeclosing is the engine closing speed and θ1 , θ 2 are threshold values for the
torsional angle between the engine and the transmission, which determine the
working mode of the clutch. Note that when a transition from the open mode to
the closing mode occurs, the following reset condition must be imposed:
∀t1 ∈ ℝ+ , ∀t2 ∈ ℝ>t1 , if x(τ ) ∈Ω1 , ∀τ ∈ ℝ[ t1 ,t2 ) and x(t2 ) ∈Ω2 , set x1 (t2 ) := 0
(4)
As the engine angle θ e tends to infinity in the open mode, so the state
x1 tends to infinity, synchronization of the engine angle and the transmission
angle must be attained at the moment the clutch switches from the open mode to
the closing mode. Notice that the reset condition does not apply to initial
conditions, as the model cannot be initialized in the closing mode. The matrices
A ci , b c and fc can be obtained by deriving the whole dynamical model of the
drivetrain using the generalized Newton's second law of motion as in (Căruntu
et al., 2011). The output of the system is the vehicle speed y (t ) = rwω w .
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Constantin F. Căruntu
The engine torque (i.e., the control input) is restricted by lower and
upper bounds:
0 ≤ u(t ) ≤ Temax , ∀t ∈ ℝ+ ,
(5)
where Temax is the maximum torque that can be generated by the internal
combustion engine.
3.2. Drivetrain Model Including the Delays
In order to apply the predictive control strategy based on input-output
models that will be described in the next section, consider the plant described by
the CARIMA (Controlled AutoRegressive Integrated Moving Average) model
(Camacho & Bordons, 2004)
( )
( )
A z −1 y ( k ) = z − d B z −1 u ( k − 1) +
( )
D(z )
e ( k ) C z −1
−1
(6)
where d is the delay introduced by the communication network, i.e., CAN,
e ( k ) is white noise with zero mean value, A z −1 and B z −1 are the system
( )
( )
polynomials
( )
B( z ) = b
A z −1 = 1 + a1 z −1 + ... + anA z − nA ,
−1
0
(7)
+ b1 z −1 + ... + bnB z − nB ,
( )
( )
where n A and nB represent the polynomials degrees and C z −1 and D z −1
are the disturbances polynomials
( )
D(z ) =1− z
C z −1 = 1,
−1
−1
(8)
.
The equation developed in (Klehmet et al., 2008) it is used to determine
the upper bound of the communication delays that appear on CAN in
automotive applications
dj ≤
( j + 2) ⋅ l
j −1
R − ∑ i = 0 ( l / ci )
(9)
where l = 136 bits denotes the maximum frame length including the 6 bit CS
time, R = 500 kbps is the rate of a high-speed CAN, ci is the cycle length of
the i − th priority message and j is the priority of the node for which the upper
Bul. Inst. Polit. Iaşi, t. LIX (LXIII), f. 3, 2013
51
bound is being calculated. A cycle length cn , corresponding to a message of
priority n , represents the period after which the message is repeated.
Now, consider that the delay introduced by the communication network
d c is time-varying, but bounded
dm ≤ dc ≤ d M
(10)
where d m is the minimum and d M is the maximum delay, determined using (9),
that appear in the communication network.
The delay to be considered by the predictive strategy is taken equal to
the minimum delay that can appear in the communication network:
d = dm
(11)
and instead of the polynomial B , another polynomial Bɶ , identified in order to
model the system including the delays between d m and d M is introduced
(Căruntu et al., 2010)
( )
−n
Bɶ z −1 = bɶ0 + bɶ1 z −1 + ... + bɶnBɶ z Bɶ ,
(12)
with:
nBɶ = nB + d M − d m ,
bɶ0 = bɶ1 = ... = bɶn ɶ =
B
b0 + b1 + ... + bnB
nBɶ + 1
,
(13)
B (1) = Bɶ (1).
4. Networked Predictive Control Strategy
Predictive control techniques have been introduced mainly in order to
deal with plants that have complex dynamics (unstable inverse systems, timevarying delay, etc.) and plant model mismatch. They are of a particular interest
from the point of view of both broad applicability and implementation
simplicity, being applied on large scale in industry processes, having good
performances and being robust at the same time.
The prediction model is given by
( ) ( )
( z ) D ( z ) u ( k − 1) + F ( z ) y ( k )
C(z )
C(z )
yˆ ( k + j | k ) = G j − d z −1 D z −1 z − d −1u ( k + j ) +
+
H j −d
−1
−1
−1
j
−1
−1
(14)
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Constantin F. Căruntu
with j = hi, hp , where hi is the minimum prediction horizon and hp is the
prediction horizon. u ( k + j − 1 k ), j = 1, hc is the future control, computed at
time k and yˆ (k + j k ) are the predicted values of the output, hc being the
control horizon.
For determining the polynomials Fj z −1 , G j z −1 and H j z −1 the
( )
( )
( )
two well known Diophantine equations (Camacho & Bordons, 2004) are used.
Considering as inputs D z −1 u ( k ) and collecting the j-step predictors
( )
in a matrix notation, the prediction model can be written as
yˆ = Gu d + yˆ 0
(15)
where yˆ 0 represents the free response and the matrix G is given in (Camacho &
Bordons, 2004).
The objective function is based on the minimization of the tracking
error and on the minimization of the controller output, the control weighting
factor λ being introduced in order to make a trade-off between these objectives
J = ( Gu d + yˆ 0 − w ) ( Gu d + yˆ 0 − w ) + λuTd u d
T
(16)
subject to: D( z −1 )u (k + i) = 0 for i ∈ [ hc, hp − d − 1] , where w is the reference
trajectory vector with the components w( k + j k ), j = hi, hp . Minimizing the
objective function ( ∂J ∂u d = 0 ), the optimal control sequence yields as
(
u∗d = G T G + λ I hc
)
−1
G T [ w − yˆ 0 ]
(17)
Using the receding horizon principle and considering that γ j , j = [ hi , hp ]
(
are the elements of the first row of the matrix G T G + λ I hc
)
−1
G T , the following
control algorithm results
( )
D z −1 u ( k ) =
hp
∑γ
j = hi
j
 w ( k + j k ) − yˆ 0 ( k + j k ) 


(18)
5. Experimental Results
This section presents the validations of the proposed networked
predictive control strategy investigated on the vehicle drivetrain model using the
designed real-time simulation test-bench. The simulator's console software
(GUI), designed in ControlDesk, is represented in Fig. 5, including the
messages sent on CAN.
Bul. Inst. Polit. Iaşi, t. LIX (LXIII), f. 3, 2013
53
Fig. 5 – ControlDesk console.
Using typical values for the parameters in (9), it yields that the sum of
the delays ( τ sc + τ ca ) is randomly distributed in the interval [ 0,4Ts ] , where
Ts = 5 ms is the sampling period of the system. Furthermore, being components
of the same powertrain subsystem it can be considered that the communication
delays from sensor to controller and from controller to actuator have the same
values and they are uniformly distributed.
In order to apply the predictive control strategy, a CARIMA model for
the drivetrain system was identified, using as input the engine torque Te and as
output the vehicle speed vv and the parameters given in (Căruntu et al., 2011).
Using a pseudo-random binary sequence (PRBS) as the system input of the
PWA state-space model (2), the drivetrain was identified with an ARX
equivalent model employing the design model, which is given by the following
system polynomials
( )
B ( z ) = −0.77 ⋅ 10
A z −1 = 1 − 1.962 z −1 + 0.96203z −2 ,
−1
−6
+ 0.143 ⋅ 10−4 z −1 .
(19)
It was considered that d m = 0 and d M = 4 from (9) and the polynomial
( ) was determined using (12) and (13) which yielded
Bɶ z
−1
( )
Bɶ z −1 = 0.3383 + 0.3383z −1 + 0.3383z −2 + 0.3383z −3
(20)
The predictive control strategy was designed using the following
parameters: hc = n A + nD = 3, hi = d + 1, hp = hc + d , with d from (11). The
control action was then applied to the initial model of the vehicle drivetrain.
54
Constantin F. Căruntu
Also, a PI controller was designed using (O’Dwyer, 2006) and it was
tuned to have a fast response, which yielded the proportional and integral terms:
K R = 0.01 and K i = 0.001 .
A sequence of stairs was applied as the reference for the vehicle
velocity and it was desired that the system tracks the reference signal as fast as
possible, the following figures showing the results obtained in the simulations.
Fig. 6 illustrates the reference vehicle velocity value and the response of
the system with communication delay when both control strategies are applied.
It can be seen that the closed-loop control system tracks the reference signal,
having no steady state error.
Fig. 6 – Vehicle velocity.
The working modes of the clutch are presented in Fig. 7, where it can
be seen that the clutch starts from open mode and after it reaches the engine
closing speed ωeclosing , the clutch enters in closing mode. In order to put the
vehicle in motion, the load torque has to be defeated, so the engine speed varies
around the threshold value, which results in the switching between the open and
closing mode. Note that when the clutch springs are being compressed, the
clutch enters the closed mode.
Fig. 7 – Clutch mode of operation.
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55
As it can be seen in Fig. 8 the engine torque (control signal) for both
control methods have smooth behaviours so the stiffer springs do not fully
compress. As a result, the first state x1 does not pass the θ 2 threshold so the
clutch cannot reach the looked mode.
Fig. 8 – Engine torque (control signal).
Fig. 9 illustrates the driveshaft torque in which it can be seen that the
switching between the clutch modes results in very high increases/decreases in
the driveshaft torque. Note that when the vehicle velocity reference is switched
from 30 km/h to 10 km/h, the driveshaft torque reaches negative values
corresponding to the engine breaking.
Fig. 9 – Driveshaft torque.
The axle wrap angular speed ( ωd = ωe itot − ωw , where itot = it ⋅ i f ) is
represented in Fig. 10 (and a detail in Fig. 11) as a measure of driveline
oscillations that appear when the clutch switches through the operating modes.
56
Constantin F. Căruntu
Fig. 10 – Speed difference.
In the figures it can be seen that, due to the undershoot of the PI
response (represented in Fig. 6), the clutch opens (Fig. 7), which results in an
increased driveshaft torque (Fig. 9) and driveline oscillations (Figs. 10 and 11).
6. Conclusions
This paper considered the design of a real-time simulation test-bench in
order to validate and to test the performances of the control algorithm developed
with the aim of damping driveline oscillations, while decreasing the influence of
the communication time-varying delays on the closed-loop control
performances over the CAN network. The designed predictive strategy, which
uses an identification method to model the physical plant (vehicle drivetrain)
including the delays, was tested on the real-time simulation test-bench and the
experiments designed based on realistic scenarios verify the better performances
of the proposed method when comparing the results with the ones obtained with
classical controllers (PI).
Fig. 11 – Speed difference - detail.
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Constantin F. Căruntu
SIMULAREA ÎN TIMP REAL A UNUI LANł DE TRANSMISIE
A PUTERII CONTROLAT PRIN REłEA UTILIZÂND
REGLAREA PREDICTIVĂ
(Rezumat)
Această lucrare propune o implementare în timp real a unui regulator predictiv
proiectat pentru a amortiza oscilaŃiile care apar în lanŃul de transmisie a puterii la
autovehicule, ceea ce este esenŃial în îmbunătăŃirea manevrabilităŃii şi confortului
pasagerilor. De asemenea, regulatorul permite compensarea întârzierilor variabile în
timp care apar ca urmare a transmiterii mărimilor de comandă şi a mărimilor măsurate
de senzori prin reŃeaua de comunicaŃii Controller Area Network (CAN). Metodologia
are la bază mai mulŃi paşi: în primul rând, este descrisă structura de timp real cu reŃea
CAN integrată şi apoi este prezentat modelul lanŃului de transmisie a puterii la
autovehicule. Apoi, Ńinând cont de faptul că întârzierile variabile în timp introduse de
reŃeaua CAN sunt mărginite, este propusă o metodă de a modela procesul fizic (lanŃul
de transmisie a puterii) care să includă şi întârzierile. În final, este proiectată o strategie
de control predictiv care utilizează modelul dezvoltat anterior şi care este concepută
pentru a amortiza oscilaŃiile lanŃului de transmisie a puterii. Sistemul de control este
testat utilizând structura de timp real şi experimentele bazate pe scenarii realiste
ilustrează faptul că regulatorul propus poate avea performanŃe mai bune faŃă de
regulatoarele clasice (PI).