torque management of gasoline...
TRANSCRIPT
TORQUE MANAGEMENT OF GASOLINE ENGINES
By
Daniel Michael Lamberson
BS (Illinois Institute of Technology) 1998 BA (Wheaton College) 1999
A report submitted in partial satisfaction of the Requirements for the degree of
Masters of Science, Plan II
in
Mechanical Engineering
at the
University of California at Berkeley
Committee in Charge: Professor J. Karl Hedrick, Chair
Professor Andrew Packard
Fall 2003
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Abstract
Torque Management of Gasoline Engines
by
Daniel M. Lamberson
Masters of Science in Mechanical Engineering
University of California at Berkeley
Professor J. Karl Hedrick, Chair A torque management strategy for gasoline engines is developed. The torque management concept is described in detail. A cylinder air flow observer is derived. Open loop estimation is shown as a starting point. Non-linear observer theory is used to derive an improved estimate for cylinder flow. An engine torque control strategy is developed using the commanded throttle position of the electronic throttle as the system input. Control of manifold pressure as a means of torque control and direct control of driveline torque are investigated and the results compared. Finally, a complete torque management strategy is derived. The derivation assumes a known estimate of engine torque, either by the use of a torque sensor or some type of torque estimation algorithm. The torque management strategy involves the coordinated control of the throttle, ignition timing, and the air to fuel ratio. Non-linear controllers are derived for ignition timing and air to fuel ratio setpoint. The controllers are designed to maintain engine speed through transient torque loadings. Simulation results are given for each observer and controller. Attention is given to the application of MoBIES (Model Based Integration of Embedded Systems) tools and methodology in the design and implementation process of such a controller.
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Table of Contents List of Figures iii List of Variable v 1 Introduction ................................................................................................... 1 2 Plant Model ................................................................................................... 8 3 Previous Control Development.................................................................... 12
3.1 Electronic Throttle Control.................................................................... 12 3.2 Air to Fuel Ratio Control ....................................................................... 12
4 Cylinder Air Flow Estimation........................................................................ 14 4.1 Background.......................................................................................... 14 4.2 Open Loop Estimation using a Manifold Pressure Sensor ................... 16 4.3 Open Loop Estimation Using a Mass Air Flow Sensor......................... 18 4.4 Sliding Observer using Multiple Sensors.............................................. 24
5 Engine Torque Management....................................................................... 27 5.1 Background.......................................................................................... 27 5.2 Sliding Mode Control Review ............................................................... 31 5.3 Engine Torque Control using Manifold Pressure Control ..................... 33 5.4 Engine Torque Control ......................................................................... 39 5.5 Torque Management Strategy.............................................................. 52
5.5.1 Derivation of the Control Laws ...................................................... 56 5.5.2 Selection of Pressure Control or Torque Control Strategy ............ 59 5.5.3 Results .......................................................................................... 60
6 Application to the MoBIES Project............................................................... 67 7 Future Work................................................................................................. 67 8 Summary..................................................................................................... 69 9 Acknowledgements ..................................................................................... 70 10 References.................................................................................................. 71
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List of Figures Figure 1. Schematic of a gasoline engine............................................................ 2 Figure 2. Pedal position to desired engine torque map for sporty vehicle feel.
(taken from [7]) .............................................................................................. 5 Figure 3. Pedal position to desired engine torque map for economical vehicle
feel. (taken from [7]) ...................................................................................... 5 Figure 4. Schematic of gasoline engine with modeled states. ........................... 11 Figure 5. Schematic of feed-forward plus P-I control of fuel flow....................... 14 Figure 6. Open loop estimate of cylinder flow using manifold pressure sensor -
perfect model............................................................................................... 17 Figure 7. Open loop estimate of cylinder flow using manifold pressure sensor -
10% error in volumetric efficiency................................................................ 18 Figure 8. Open loop estimate of cylinder flow using a throttle flow sensor -
perfect model............................................................................................... 20 Figure 9. Open loop estimate of cylinder flow using a throttle flow sensor – ..... 21 10% error in volumetric efficiency....................................................................... 21 Figure 10. Open loop estimate of cylinder flow using a throttle flow sensor – ... 22 10% error in volumetric efficiency. (detail of Figure 9)........................................ 22 Figure 11. Open loop estimate of cylinder flow using a throttle flow sensor – ... 23 10% error in volumetric efficiency and step input in throttle position. ................. 23 Figure 12. Open loop estimate of cylinder flow using throttle flow sensor – ....... 24 10% error in throttle flow measurement.............................................................. 24 Figure 13. Closed loop estimate of cylinder flow using throttle flow sensor and
manifold pressure sensor – 10% error in throttle flow measurement........... 25 Figure 14. Closed loop estimate of cylinder flow using throttle flow sensor and
manifold pressure sensor – 10% error in volumetric efficiency. .................. 26 Figure 15. Schematic of the torque management control strategy. ................... 27 Figure 16. Engine torque change with air to fuel ratio. ...................................... 30 Figure 17. Engine torque change with ignition timing. ....................................... 30 Figure 18. Pressure control results – throttle..................................................... 37 Figure 19. Pressure control results – pressure and torque................................ 38 Figure 20. Pressure control results – pressure and speed. ............................... 38 Figure 21. Pressure control results – control surfaces....................................... 39 Figure 22. Torque control results – throttle........................................................ 43 Figure 23. Torque control results – torque and pressure................................... 44 Figure 24. Torque control results – torque and speed....................................... 44 Figure 25. Torque control results – control surfaces.......................................... 45 Figure 26. Torque constant (CT) parameter estimate using adaptive control law.
.................................................................................................................... 51 Figure 27. Torque setpoint and engine speed used to test adaptive control law.
.................................................................................................................... 52 Figure 28. Engine response to accessory load with no torque control –............ 53 throttle and accessory. ....................................................................................... 53 Figure 29. Engine response to accessory load with no torque control –............ 54 torque and speed................................................................................................ 54
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Figure 30. Engine response to accessory load with pressure control only – throttle and accessory. ................................................................................ 55
Figure 31. Engine response to accessory load with pressure control only – pressure and speed..................................................................................... 55
Figure 32. Torque management with discontinuous ignition timing control law – throttle and accessory. ................................................................................ 61
Figure 33. Torque management with discontinuous ignition timing control law – pressure and speed..................................................................................... 61
Figure 34. Torque management with discontinuous ignition timing control law – ignition timing. ............................................................................................. 62
Figure 35. Torque management with smooth ignition timing control law – pressure and speed..................................................................................... 63
Figure 36. Torque management with smooth ignition timing control law – ........ 63 ignition timing. .................................................................................................... 63 Figure 37. Torque management results using only ignition timing to control
engine speed – pressure and speed. .......................................................... 64 Figure 38. Torque management results using only ignition timing to control
engine speed – ignition timing and air to fuel ratio. ..................................... 65 Figure 39. Torque management results using both ignition timing and air to fuel
ratio to control engine speed – pressure and speed.................................... 66 Figure 40. Torque management results using both ignition timing and air to fuel
ratio to control engine speed – ignition timing and air to fuel ratio............... 66
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List of Variables Variable Name Description Units
β Stoichiometric air to fuel ratio dimensionless
CT Torque constant N.m.rad/kg
ηvol Volumetric efficiency dimensionless
δ Ignition timing degrees BTDC
ε Fraction of fuel delivered as vapor in the fuel wall wetting model
dimensionless
Ieng Engine rotational inertia kg.m2
λ Air to fuel ratio dimensionless
cylm Cylinder flow kg/s
fcm Commanded fuel flow kg/s
fom Actual fuel flow kg/s
maxm Maximum throttle flow kg/s
throttlem Throttle flow kg/s
n Number of engine cylinders dimensionless
Neng Engine speed rpm
ωeng Engine speed rad/s
Pa Atmospheric pressure Pa
Pm Manifold pressure Pa
Pm,exh Exhaust manifold pressure Pa
R Gas constant for air J/kg.K
T Intake manifold temperature Kelvin
τf Time constant used in fuel wall wetting model
sec
τmaf_sensor Time constant of the mass air flow sensor sec
τth Time constant of the electronic throttle sec
θth Throttle angle degrees
TQacc Accessory torque N.m
TQcomb Engine combustion torque N.m
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Variable Name Description Units
TQfric Engine friction torque N.m
TQimp Impeller torque N.m
TQpump Engine pumping torque N.m
Vdispl Engine displacement m3
Vm Intake manifold volume m3
X Cylinder air charge kg/stroke
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1 Introduction In a conventional gasoline engine, shown in Figure 1, the driver controls the
pedal position. The pedal is mechanically linked to the engine throttle body. In
this way, the driver controls the throttle position and, in a way, the air flow to the
engine. In general, the amount of torque produced by an engine is directly
related to engine air flow. Thus, the engine performance and, to a great extent,
the vehicle performance is defined once the engine is selected. Advanced
engine technologies are currently being developed to improve engine fuel
economy and reduce engine emissions. Many of these technologies can be
separated into two categories. First, there are engine systems that divert engine
torque away from the driveline during portions of the drive cycle. These types of
engine systems include engine accessories (air conditioner, etc.) and hybrid
engine technologies. Second, there are engine systems that change engine
torque production, for the same air flow through the throttle, during portions of the
drive cycle. These types of engine systems include variable cam timing engines
and lean burn engine technologies. When a mechanical throttle is used, any
engine function that diverts engine torque away from the driveline or changes the
torque output of the engine during portions of the drive cycle adversely affects
vehicle performance. The current automotive market will not permit vehicle
performance degradation caused by the use of these advanced engine
technologies. In order to make these technologies an attractive option, vehicles
using these advanced engine technologies must perform as well or better than
those in the current market.
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AccessoriesDriveline
Pedal Throttle
MechanicalLinkage Engine
Cylinders
Exhaust Manifold
OxygenSensor
Fuel Injectors
Intake Manifold
Idle AirBypass Valve
Figure 1. Schematic of a gasoline engine.
With the advent of electronic throttle systems, the pedal is not mechanically
connected to the throttle. Instead, the throttle plate is driven by an electric motor.
In these systems, the pedal position is read as a voltage signal by the throttle
control system. Based on this pedal position signal, the controller determines the
actuating signal to send to the throttle motor. Since the engine control unit has
control of the throttle position, use of an electronic throttle allows for greater
flexibility on the control system. This flexibility can be used to improve engine
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performance and help meet legislative emissions requirements. In addition, use
of an electronic throttle system may be used to improve the performance of the
aforementioned advanced engine technologies. Current effort is being made to
develop control strategies that make use of electronic throttle systems and these
advanced engine technologies. The goal of these control strategies is to improve
overall vehicle performance and to maintain vehicle performance as the engine
system changes its mode of operation.
It is possible to implement the electronic throttle control system so that it behaves
in a similar manner to the mechanical throttle system. To do this, the pedal
position signal is interpreted as a desired throttle opening. The controller drives
the throttle to the desired throttle opening in closed loop using the throttle
position sensor signal as feedback. The only difference between this
implementation and the mechanical throttle are the dynamics associated with the
electronic throttle system. In this implementation of an electronic throttle system,
the time lag of the throttle position can be used to give the controller an
advantage in predicting the future position of the throttle and the future mass air
flow. Magner et. al. [14] developed an improved cylinder air charge algorithm
using the delta air charge anticipation based on the difference between the
commanded and actual throttle position of an electronic throttle.
However, with an electronic throttle system, the control system can interpret the
pedal position signal in any number of ways and drive the throttle based on that
interpretation. To better take advantage of the control flexibility available with an
electronic throttle system, an engine torque management strategy can be used.
A torque management strategy has two goals. First, a torque management
strategy uses vehicle performance as a parameter in the control design process.
The desired vehicle performance is used to develop a map that converts pedal
position to a desired torque delivered to the driveline. The position of the
electronic throttle is then controlled to achieve this desired engine torque.
Second, a torque management strategy tries to eliminate transients caused by
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the engine system changing its mode of operation. As the engine changes its
mode of operation, the torque delivered to the driveline changes. By using the
throttle to control the driveline torque, it is possible to design a control strategy
such that vehicle performance is maintained as the engine changes its mode of
operation.
In a torque management scheme, the pedal position signal is interpreted as a
commanded torque delivered to the driveline. Using this interpretation, the
vehicle response is no longer physically correlated to the pedal position. Instead,
the pedal position is passed into a map to determine the desired torque delivered
to the driveline. Possible pedal position to desired engine torque maps are
shown in Figures 2 and 3 for both a “sporty” vehicle feel and an “economical”
vehicle feel. These maps give desired engine torque as a function of pedal
position and engine speed. A “sporty” vehicle feel is achieved with a large
change in torque demand for a small change in pedal position at relatively low
pedal positions and low engine speeds. At high pedal positions and high engine
speeds, this “sporty” map is approximately the same as the “economical” map.
These maps are not unique and can be designed to fit the application.
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Figure 2. Pedal position to desired engine torque map for sporty vehicle feel. (taken from [7])
Figure 3. Pedal position to desired engine torque map for economical vehicle feel. (taken from [7])
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Under normal driving operation, the torque management strategy uses the
throttle position to achieve the desired driveline torque while the fuel and ignition
timing are controlled to minimize fuel consumption, emissions, etc. The engine is
controlled in this manner for changes in pedal position, road grade, and other
conditions where the resulting vehicle response is anticipated by the driver.
However, during the onset of an accessory load or the changing of the cam
timing, the amount of engine torque transmitted to the driveline is ‘immediately’
changed in a way that is unexpected by the driver. These mode changes of the
engine can be sudden (accessory loads) or occur over a short period of time
(variable cam timing). Also, once in a given mode, the engine tends to stay in
that mode for an extended period of time. Since the throttle is being used to
control the driveline torque, the throttle will adjust to maintain the desired
driveline torque regardless of the engine operating mode. However, for sudden
changes in the driveline torque, the throttle and intake manifold dynamics may be
too slow to compensate for the torque transients without an adverse effect on
vehicle performance.
The magnitude and the time of application of these torque loads are known ‘a
priori’ by the controller. Thus, if an actuator were fast enough to offset the torque
change, vehicle performance could be maintained until the throttle and manifold
dynamics have caught up. Under some restrictions, the ignition timing and the
air to fuel ratio can be adjusted to achieve the ‘instantaneous’ change in engine
torque required to maintain the driveline torque while the throttle is adjusted.
These actuators, while fast, have limited control authority and are limited by other
considerations such as emissions and component life.
The transient torque rejection goal of the torque management strategy is similar
to the engine idle speed control problem. In controlling the engine idle speed,
the mechanical throttle is closed. The air flow to the engine is controlled by the
idle air bypass valve, shown in Figure 1. Due to the throttle valve dynamics and
the manifold filling dynamics, this actuator is unable to robustly control idle speed
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in the presence of torque disturbances acting on the engine. The ignition timing
and, if necessary, the air to fuel ratio are used to control idle speed during these
disturbances. In order to give the ignition timing a ‘torque reserve’, the ignition
timing is retarded slightly from maximum brake torque. This gives both the
ignition timing and the air to fuel ratio sufficient control over the engine torque, in
both the positive and negative directions, that these actuators can maintain
engine idle speed during torque transients. In the torque management strategy,
the ignition timing and the air to fuel ratio are also used to reject torque
disturbances. However, in order to minimize fuel consumption, the ignition timing
is not retarded throughout the engine map. This further limits the control
authority of the ignition timing in the torque management strategy.
In the following, a full torque management strategy for gasoline engines is
derived. First a plant model is discussed as well as a short description of
previous work done in the areas of electronic throttle control and air to fuel ratio
control. Various methodologies are used to derive a cylinder air flow observer.
Non-linear control theory is then used to develop two engine torque controllers.
The first strategy controls intake manifold pressure as a means of torque control.
In the second strategy, driveline torque is controlled directly. Finally, a complete
torque management strategy is developed to control the throttle position, ignition
timing, and the air to fuel ratio in a coordinated manner.
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2 Plant Model
The engine model used for controller development is based on the model
developed by Cho et. al. [2]. This is a lumped parameter [1], mean value engine
model. The throttle flow is given by
( ) ( )amthmaxthrottle P,PPRIθTCmm ⋅⋅= (2.1)
where maxm is the maximum flow through the throttle, TC(θth) accounts for the
effect of throttle angle, and PRI(Pm, Pa) accounts for the effect of the pressure
ratio across the throttle. The functions TC(θth) and PRI(Pm, Pa) act to reduce the
flow through the throttle with decreasing throttle angle (θth) and increasing
manifold pressure (Pm). Using the ideal gas law, the intake manifold dynamics
are given by
( )m
cylthrottle
m
manifoldm
V
TRmm
V
TRmP
⋅⋅−=
⋅⋅= (2.2)
where cylm is the flow to the engine cylinders (out of the manifold), Vm is the
manifold volume, R is the gas constant for air, and T is the intake air
temperature. The derivative of temperature is neglected since the manifold
temperature is assumed constant. In general, the temperature derivative can be
neglected since it has only a minor effect on the manifold pressure dynamics
[11]. The engine flow is found from a speed density approximation
( )TR
Pωω,PηV4π1
m mengengmvoldisplcyl ⋅
⋅⋅⋅⋅= (2.3)
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where Vdispl is the volumetric displacement of the engine, ηvol is the volumetric
efficiency of the engine given as a function of manifold pressure and engine
speed, and ωeng is the engine speed. The engine combustion torque is given by
( ) ( )eng
cylTcomb ω
δSPIλAFImCTQ
⋅⋅⋅= (2.4)
where CT is an engine specific constant, AFI(λ) accounts for the air to fuel ratio
effect on engine torque, and SPI(δ) accounts for the ignition timing effect on
engine torque. The air to fuel ratio (λ) is given by
fo
cyl
m
mλ = (2.5)
where fom is the amount of fuel entering the cylinders in the air/fuel mixture. In
order to meet engine emissions standards, strict control of air to fuel ratio is
required in gasoline engines. Current air to fuel ratio controllers are designed to
control the amount of fuel delivered so that a stoichiometric air to fuel mixture
enters the cylinders. A stoichiometric air to fuel mixture is one in which the
amount of oxygen in the cylinders is completely burned in the combustion
process [12].
The engine acceleration is given by
( )impaccfricpumpcombeng
eng TQTQTQTQTQI1ω −−−−⋅= (2.6)
where Ieng is the engine inertia, TQpump is the torque required to pump the fluid
from the intake manifold pressure to the exhaust manifold pressure, TQfric is the
engine friction torque, TQacc is the sum of all accessory torques, and TQimp is the
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load torque from the driveline (i.e. the impeller torque if an automatic
transmission is used). The pumping torque is given by
( )mexhm,displ
pump PP4π
VTQ −= (2.7)
where Pm,exh is the exhaust manifold pressure. The friction torque can be
approximated using
2eng1fric CωCTQ +⋅= (2.8)
where C1 and C2 are engine specific constants.
Figure 4 shows a schematic of the engine with the modeled variables shown. As
shown, the air flows through the throttle and into the intake manifold. Flow from
the manifold mixes with fuel from the injectors before entering the engine
cylinders. Combustion of the air/fuel mixture produces an increase in cylinder
pressure and results in an applied torque about the engine crankshaft. Exhaust
gases pass out of the engine cylinders and into the exhaust manifold. In the
exhaust manifold, an oxygen sensor is used to measure the amount of oxygen in
the exhaust stream.
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OxygenSensor
FuelInjector
thm
mP
cylm
exh,mP
Combustion ofAir/Fuel Mixture
engω
Piston
Friction BetweenPiston and
Cylinder Wall
)TQ( fric
)TQ( comb
SparkPlug
thθ
Figure 4. Schematic of gasoline engine with modeled states.
This engine model does not include the effects of exhaust gas recirculation
(EGR). Exhaust gas recirculation is the process of passing hot exhaust gases
into the intake manifold. This process is becoming more widely used in gasoline
engines as a way to reduce emissions and fuel consumption. It is neglected here
to simplify the derivation of the torque management strategy. The model is
implemented in Matlab/Simulink and, for the following analysis, is parameterized
to represent a Ford Taurus engine.
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3 Previous Control Development
3.1 Electronic Throttle Control
A detailed model of an electronic throttle system and an associated control law
was developed by Griffiths [5]. This model accounted for the non-linearities of
the electronic throttle system including the Coulomb friction of the valve.
For implementation, the non-linearities of the throttle body must be accounted
for. However, of more importance for this application is the non-linear effect of
throttle position on the throttle flow. So, the following derivations use an
electronic throttle model of the form
cmdth
thth
th θτ1θ
τ1θ +−= (3.1)
This simplification will ease the control derivations while still allowing for an
overall torque management strategy to be developed and evaluated.
3.2 Air to Fuel Ratio Control
An air to fuel ratio controller was previously developed by Souder [17]. The
controller is designed to deliver fuel in order to achieve a stoichiometric air to fuel
ratio. The output of the oxygen sensor is used in feedback to control the amount
of fuel delivered. This model includes the non-linear effects of fuel wall wetting.
In gasoline engines, fuel is delivered to the air stream prior to entering the
cylinders. However, not all of the fuel vaporizes as it mixes with the air. Instead,
some of the fuel forms a puddle on the manifold wall. Over time, fuel from the
puddle is vaporized and enters the cylinders. Inclusion of this effect in the control
model leads to improved air to fuel ratio control.
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For this application, it is assumed that the air to fuel ratio control is working
correctly. That is, the air to fuel ratio tracks the setpoint specified by the torque
management strategy. Thus, the air to fuel ratio control is ignored in the
following discussion.
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4 Cylinder Air Flow Estimation
4.1 Background
Most production air to fuel ratio controllers consist of a feed forward term and a
proportional plus integral (P-I) closed loop portion as shown in Figure 5. The
feedback portion of the control scheme uses the output of the oxygen sensors in
the exhaust stream as an indication of the air to fuel ratio. The oxygen sensor is
a highly nonlinear sensor giving a non-saturated reading only in a narrow range
around the stoichiometric air to fuel ratio. In practice, this sensor can only be
used to determine if the mixture is rich or lean.
The P-I portion of the control cannot be relied on during transients and cold start.
This is due to two factors. First, fuel is injected in the intake ports, before the
intake valves are opened, while the oxygen sensor is placed in the exhaust
stream. This leads to a large time delay in the system that varies with engine
speed. Second, the oxygen sensor is only operational once it has reached its
operating temperature. Thus, the feed forward portion of the control is of
particular importance in maintaining precise air to fuel ratio control.
Oxygen SensorOutput
Mass Air Flow,Intake Temperature,
Engine Speed
Kp
Ki1
s
Feed-Forward
+
+
+
+ Desired FuelFlow
Figure 5. Schematic of feed-forward plus P-I control of fuel flow.
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A non-linear, sliding mode control for air to fuel ratio was developed by Souder
[17]. The sliding mode control was derived using a sliding surface defined by
focyl mβmS −= (4.1)
where fom is the mass of fuel entering the cylinder. The final control law was
given as
yηmτ1
mτβ
mmβε cylf
fcf
cylfc ⋅+−−= (4.2)
where fcm is the commanded fuel, and y is the output of the oxygen sensor
defined as
( ) ( ) ymβmsgnSsgn focyl =−= (4.3)
The output of the control is a desired mass flow rate. This requires equation 4.2
to be integrated. After this integration, the term cylm appears in the control law.
Thus, the same feed forward term required in the P-I controller is also needed in
this sliding control.
The feed forward portion of an air to fuel ratio controller requires an accurate
estimate of cylinder air flow ( cylm ). Since a cylinder flow sensor is currently not a
cost effective option, this estimate is made using either a mass air flow sensor
placed at the throttle body or an intake manifold pressure sensor.
In fuel control systems, the use of a throttle flow sensor, instead of a manifold
pressure sensor, is made for a variety of reasons as listed below
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• No need for ambient pressure measurement
• Unreliable accuracy of the throttle position sensor
• High mass flow leads to a high pressure drop across the air filter
• Controller able to distinguish between air and EGR
• Improved accuracy of the cylinder flow estimate
Most importantly, the accuracy of the estimate of the cylinder air charge is
improved when a throttle flow sensor is used. The throttle flow depends on the
pressure upstream and downstream of the throttle. The air pressure upstream of
the throttle (downstream of the air filter) and the barometric pressure are usually
not continuously sensed variables. In the case of barometric pressure, it is
usually sensed at engine startup and assumed to remain constant over the drive
cycle. Even if this were a valid assumption, relating the atmospheric pressure to
the pre-throttle/post-filter pressure is non-trivial. Use of a mass air flow sensor
eliminates this issue. However, the mass air flow sensor is more expensive and
has a lower bandwidth than a manifold pressure sensor.
The following sections will look at some possible cylinder flow estimators using
these sensors. In current production engines, only one of these sensors would
be used at a time. However, a scenario is investigated using both sensors
simultaneously. Use of redundant sensors can be used to improve the quality of
the flow estimate and allow for improved diagnostic functionality [8]. In section
4.4, non-linear observer theory is used to create a closed loop cylinder flow
observer. An extended Kalman filter [4] could also be used for this purpose.
4.2 Open Loop Estimation using a Manifold Pressure Sensor
A direct estimate of cylinder air flow can be made using a manifold pressure
sensor. In this case, the speed density approximation, given in equation 2.3, can
be used to calculate cylinder flow.
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Figure 6 shows the cylinder flow estimate using the speed density approximation
when the cylinder pumping model is perfect. In this example, the engine speed
is held constant. As expected, the estimate tracks the actual cylinder flow
perfectly.
Figure 6. Open loop estimate of cylinder flow using manifold pressure sensor - perfect model.
Due to engine to engine variations and engine deterioration over time, the
volumetric efficiency of the engine may be different than that of the engine used
to develop the volumetric efficiency map. Figure 7 shows the cylinder flow
estimate using the speed density approximation when the plant has a volumetric
efficiency 10% greater than the model. This model uncertainty leads to a
significant (10%) error in the cylinder flow estimate.
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Figure 7. Open loop estimate of cylinder flow using manifold pressure sensor - 10% error in volumetric efficiency.
4.3 Open Loop Estimation Using a Mass Air Flow Sensor
In steady state, the throttle flow is equal to the cylinder flow. However, due to the
manifold filling dynamics, a direct estimate of cylinder air flow cannot be made
using a mass air flow sensor when the engine is operating under transient
conditions. Instead, the estimate is based on a model of the flow sensor
dynamics and the manifold dynamics. This air flow estimate derivation is based
on work done by Grizzle et. al. [6].
First, by differentiating the ideal gas law, the rate of change of the manifold air
pressure is given by
( )( )mengcylthrottlem
manifoldm
m P,ωmmV
TRm
VTR
P −⋅=⋅= (4.4)
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where )P,(ωm mengcyl is an experimentally determined cylinder flow as a function
of engine speed and manifold pressure. Development of the cylinder flow
function [ )P,(ωm mengcyl ] is similar to that of the volumetric efficiency map of
equation 2.3. The air charge in the cylinder (per stroke) is given by
( )mengcyleng
P,ωmNn
120Χ⋅
= (4.5)
where n is the number of engine cylinders, Neng is the engine speed in rpm, and
the factor of 120 accounts for the necessary unit conversions. The dynamics of
the mass air flow sensor also should be taken into account and are given by
throttlemaf_sensor
measuredthrottle, m1sτ
1m
+= (4.6)
where τmaf_sensor is the time constant for the flow sensor and s is the Laplace
operator. In order to eliminate the derivative of measuredthrottle,m in the pressure
equation (4.4), a new variable, x, can be defined as
measuredthrottle,maf_sensorm
m mτV
TRPx ⋅⋅⋅−= (4.7)
Substituting equations 4.6 and 4.7 into equation 4.4 gives
⋅⋅⋅+−⋅= measuredthrottle,maf_sensor
mengcylmeasuredthrottle,
m
mτV
TRx,ωmm
VTR
x (4.8)
Equation 4.8 is numerically integrated, using the forward Euler method of
integration, leading to
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⋅⋅⋅+−
−
⋅⋅⋅+−=
(k)mτVT(k)R
1)x(k(k),ωm
(k)m
VT(k)R∆t1)x(kx(k)
measuredthrottle,maf_sensorm
engcyl
measuredthrottle,
m
(4.9)
Thus, the mass of air in the cylinders is given by
⋅⋅⋅+
⋅= (k)mτ
VT(k)R
x(k)(k),ωm(k)Nn
120Χ(k) measuredthrottle,maf_sensorm
engcyleng
(4.10)
Figure 8, shown below, shows the estimated cylinder flow assuming a perfect
model is available. In this case, the engine speed is held constant and the
sensor is assumed infinitely fast. As shown, the cylinder flow estimate is
identical to the actual cylinder flow.
Figure 8. Open loop estimate of cylinder flow using a throttle flow sensor - perfect model.
21
Figures 9 and 10 show the cylinder flow estimate given a sinusoidal input in
throttle position. In this case, the volumetric efficiency of the engine is 10% lower
than the engine pumping model in the estimator. As shown, the manifold
pressure estimate is significantly incorrect (10%). However, the cylinder flow
estimate is close to the actual cylinder flow (less than 1% error). This is a much
better result than the open loop observer using the manifold pressure sensor
developed in section 4.2.
Figure 9. Open loop estimate of cylinder flow using a throttle flow sensor – 10% error in volumetric efficiency.
22
Figure 10. Open loop estimate of cylinder flow using a throttle flow sensor – 10% error in volumetric efficiency. (detail of Figure 9)
One interesting point about this estimator is the fact that the steady state
estimate does not depend on an accurate model of the engine volumetric
efficiency. Using the pressure dynamics given in equation 2.2 and assuming the
system is in steady state, it is shown that
( )
( )m
mengengmvoldispthrottle
m
cylthrottlem
V
TRTR
Pωω,PηV4π1
m
V
TRmm0P
⋅⋅
⋅⋅⋅⋅⋅−
=
⋅⋅−==
(4.11)
Solving equation 4.11 for the manifold pressure (Pm) gives
( )engmvolengthrottle
dispm ω,Pηω
TRm
V4π
P⋅
⋅⋅⋅= (4.12)
23
As shown, the manifold pressure is inversely proportional to the volumetric
efficiency. Thus a decrease in the volumetric efficiency leads to an equal
percentage increase in manifold pressure. Thus, an inaccurate model of the
volumetric efficiency will lead to an equally incorrect pressure estimate (but in the
opposite direction). Since the cylinder flow is proportional to the product of
volumetric efficiency and manifold pressure (equation 4.12), the cylinder flow
estimate would be correct.
Figure 11 shows the cylinder flow estimate during a step input in throttle. As
shown, an error exists in the flow estimation during the transient. However, as
the system reaches steady state, the difference between the actual flow and the
flow estimate tends toward zero.
Figure 11. Open loop estimate of cylinder flow using a throttle flow sensor – 10% error in volumetric efficiency and step input in throttle position.
24
4.4 Sliding Observer using Multiple Sensors
If a manifold pressure sensor were used in conjunction with the throttle flow
sensor, a sliding observer could be used to improve the estimate of cylinder flow
derived in section 4.3. A sliding observer [15] would lead to a pressure estimate
of
( ) ( )mm1m
cylthrottlem PPk
V
TRmmP ˆˆ −⋅+
⋅⋅−= (4.13)
Use of these multiple measurements (throttle flow and manifold pressure) can be
used to improve the estimate in the presence of model uncertainty. For instance,
if the estimator of section 4.3 is used and an error exists in the throttle flow
measurement, an incorrect cylinder flow estimate would be calculated. These
results are shown in Figure 12.
Figure 12. Open loop estimate of cylinder flow using throttle flow sensor – 10% error in throttle flow measurement.
25
Using a manifold pressure sensor and the sliding observer defined by equation
4.13, an improved estimate can be made. These results are shown in Figure 13.
Figure 13. Closed loop estimate of cylinder flow using throttle flow sensor and manifold pressure sensor – 10% error in throttle flow measurement.
A more common error in the cylinder flow model would be in the volumetric
efficiency, ηvol. The results of the closed loop estimate with a 10% error in
volumetric efficiency are shown in Figure 14. This estimator improves the
estimate of the manifold pressure, but the estimate of cylinder flow is significantly
degraded.
26
Figure 14. Closed loop estimate of cylinder flow using throttle flow sensor and manifold pressure sensor – 10% error in volumetric efficiency.
As discussed in section 4.3, in the presence of a volumetric efficiency error, use
of the mass air flow sensor estimator leads to a zero steady state error in the
cylinder flow estimate although a significant error exists in the pressure estimate.
If a pressure sensor is used to try to improve the estimate, this would lead to an
improved pressure estimate but an incorrect estimate of cylinder flow. Thus, the
most likely type of model error must be determined before the application of this
type of observer.
The true benefit of closed loop observers is the estimation of unmeasured
system states. This closed loop observer is only used to improve on an estimate.
Implementation of this multiple measurement estimator would be most effective if
the pressure sensor could be used for more than just improving the cylinder flow
estimate (i.e. diagnostic functionality, etc.).
27
5 Engine Torque Management
5.1 Background
Torque management is a strategy that uses the individual throttle, fuel, and
ignition timing controllers and adds an additional layer of control which
determines the desired setpoints for each of these low level controllers. The
torque management control strategy consists of two modes, the first is used for
normal operation and the second used for short duration transient torque
rejection. Figure 15 shows a schematic of the torque management strategy.
Normal Operation
Throttle changed toachieve torquedemand
Ignition timing set to'optimal'
Desired AFR set to'optimal'
Transient Torque Rejection
Throttle changed to'eventually' achieve torquedemand
Ignition timing changed toInstantaneously meettorque demand
Desired AFR changed toinstantaneouly meet torquedemand (if necessary)
Throttle Control(Closed Loop)
Ignition Control(Open Loop)
Fuel Control(Closed Loop)
PedalPosition
Other KnownTorque
Demands
Short Term TorqueAddition/Subtraction
Required
Throttle PositionSetpoint
Ignition TimingCommand
Ignition TimingSetpoint
Throttle PositionCommand
Fuel FlowCommand
Air to Fuel RatioSetpoint
Desired Cylinder ChargeAttained
Figure 15. Schematic of the torque management control strategy.
28
The inputs to the high level controller are the pedal position and all known torque
demands acting on the engine. These other torque demands are known at some
higher level of the engine control strategy and are passed to the torque
management strategy. In the torque management strategy, the pedal position is
interpreted as a desired torque at the driveline. To this effect, the pedal position
signal is passed through a torque map to determine the desired driveline torque.
This map is not unique and can be made to fit the application or desired
drivability feel (e.g. a sport car feel versus an economy car feel). The desired
driveline torque is summed with all the other known torque demands acting on
the engine. This desired engine torque is compared to the “actual” engine
torque. The “actual” engine torque is found using a torque sensor, some open
loop approximation of engine torque, or using some more advanced torque
estimation algorithm. The difference between the ‘actual’ and the desired engine
torque is used to drive the position of the electronic throttle.
In ‘Normal Operation’, when the load demands are to be handled only by the
throttle, the throttle position is adjusted to achieve the desired driveline torque.
The strategy functions in this mode for conditions where the resulting vehicle
response is anticipated by the driver (changes in throttle position, road grade,
etc.). In this mode, the fuel and ignition timing are controlled to try to minimize
emissions and brake specific fuel consumption. This requires strict control of the
air to fuel ratio around stoichiometric and has motivated the previous derivations
for cylinder flow estimation.
Due to the throttle dynamics and the pressure dynamics of the intake manifold, a
near instantaneous change in engine torque is not achievable using only the
throttle. For small, instantaneous torque variations in engine loading or torque
production, the ignition angle and/or air to fuel ratio can be modified to change
the engine torque to compensate for the torque transient. The magnitude and
the time of application of these torque loads are known ‘a priori’ by the torque
management strategy. Once the strategy is signaled that a change in load
29
torque is coming, the strategy moves to ‘Transient Torque Rejection’ mode. In
this mode, the throttle is controlled to achieve the control objective (constant
driveline torque). The ignition timing and, if necessary, the air to fuel ratio are
controlled to maintain engine speed through the transient.
Under normal operation, the spark timing of the engine is retarded slightly from
the angle for maximum torque in an effort to reduce engine emissions. Also, the
air to fuel ratio is kept at stoichiometric to achieve good emissions. However, for
short periods of time, the ignition timing can be advanced/retarded and the air to
fuel ratio be increased/decreased to achieve a lower/ higher engine torque for the
same amount of air flow. These two systems have much faster dynamics than
the throttle valve and intake manifold. However, there are additional
considerations when using the ignition timing and the air to fuel ratio to adjust
engine torque. First, the upper and lower limits for ignition timing are governed
by pre-ignition of the air/fuel mixture and other component requirements. The
upper and lower limits for the air to fuel ratio are governed by emission
requirements and component requirements. Figures 16 and 17 show examples
of how engine torque varies with air to fuel ratio and ignition timing. As shown in
Figure 16, engine torque increases as the air/fuel mixture is rich and decreases
when the mixture is lean. As shown in Figure 17, the engine torque decreases
as the ignition timing is retarded (positive degrees) from the point of maximum
brake torque (MBT). Advancing the timing from MBT is not done due to the
possibility of pre-ignition of the air/fuel mixture. In the following analysis, an
ignition timing of five degrees from the point of maximum brake torque is selected
as the nominal operating point.
30
Figure 16. Engine torque change with air to fuel ratio.
Figure 17. Engine torque change with ignition timing.
31
In the following sections, two strategies to control engine torque will be
developed using only the throttle command as the system input. With these
strategies in place, a torque management strategy is developed using ignition
timing and air to fuel ratio as the system inputs. These strategies will use sliding
mode control methodologies to develop the control laws.
5.2 Sliding Mode Control Review
Sliding mode control [16] is a robust control methodology for non-linear systems.
The basic idea is to reduce a tracking control problem of a high order system to
an equivalent first order stabilization problem. This is done by defining a control
surface (S) in such a way that the control goal is achieved (or will be achieved
asymptotically) when the surface is zero (S=0). The control law is then designed
to push the system to the control surface and to keep it there. This is done by
selecting a Lyapunov function candidate in S. For instance,
2S21
V = or SV = (5.1)
Thus, the Lyapunov function candidate is a positive definite function in S. The
derivative of the Lyapunov function (choosing the first Lyapunov function
candidate) is
SSV ⋅= (5.2)
The system is stable if V is negative definite. This is achieved if
SηSS ⋅−≤⋅ (5.3)
32
where η is strictly positive. The control law is designed so that, using the worst
case model uncertainty,
sgn(S)ηS ⋅−= (5.4)
Once the control law has been derived, is can be shown that
SηSS ⋅−≤⋅ (5.5)
meaning that the derivative of the Lyapunov function candidate is negative
definite. Thus, the surface is stable and the Lyapunov function candidate is a
true Lyapunov function.
As stated this is a robust control strategy. The control gain, η, is used to account
for all known model uncertainty. Provided the parameter uncertainty in the model
can be bounded, η is selected to be large enough to ensure stability of the
surface. Once the system has reached the surface, the sliding condition is
achieved. This means that the system remains on the surface and travels toward
the equilibrium point in a manner defined by S. Using this method, the surface is
defined such that
• The control goal is achieved (or will be achieved asymptotically) at S=0
• The control input appears in the definition of S
In cases where the control input does not appear in the definition of S , methods
such as multiple surface control or dynamic surface control are available. These
control methodologies are most commonly used when a parameter uncertainty
appears in a system state that has no control input. This type of system is
termed a ‘mismatched’ system.
33
As stated, the robustness term, η, is used to account for model uncertainty. A
methodology known as adaptive control can be used to reduce the uncertainty in
the model. In addition to meeting the control objective, an adaptive controller
attempts to estimate an unknown, constant parameter in the system. This is
done by creating a parameter update law in such a way that the estimate
converges to the true value of the parameter. By adapting on an unknown
parameter, the model uncertainty is reduced and a smaller η can be used in the
control law.
To achieve perfect tracking, sliding mode control requires infinite sampling
frequency and high control activity. This is due to the use of the discontinuous
sgn() function in the control law. For implementation, smooth control laws can be
developed. Use of these smooth control laws eliminates the possibility of perfect
tracking but reduce the control activity.
In section 5.3 a sliding control is developed to control the manifold pressure as a
means of torque control. This control uses dynamic surface control, a variant of
sliding control. In section 5.4 an adaptive, dynamic surface control is developed
to control driveline torque directly.
5.3 Engine Torque Control using Manifold Pressure Control
When controlling engine torque, the control goal is to drive the engine torque to
the desired engine torque (TQengine TQdesired). As shown in equation 2.4, engine
combustion torque is a function of cylinder air flow. From equation 2.3, cylinder
air flow is a function of the manifold pressure. Thus, assuming constant air to
fuel ratio and ignition timing, the control goal can be redefined as Pm Pm,des. In
the following, a sliding mode control is developed to control manifold pressure.
First, a positive definite Lyapunov function candidate is chosen as
34
21S
21
V = (5.6)
where the control surface is defined as
des1 PPS −= (5.7)
Taking the derivative of the control surface gives
( ) ( )
( ) desm,mengengmvoldisp
amthmax
m
descylthrottlem
desm,m1
P
TRPωω,PηV
4π1
P,PPRIθTCm
VTR
P)mm(V
TR
PPS
−
⋅⋅⋅⋅⋅
−⋅⋅⋅=
−−⋅=
−=
(5.8)
The control surface is attractive if
111 SηSS ⋅−≤⋅ (5.9)
Using the worst case model uncertainty, the control law is derived so that
)sgn(SηS 11 ⋅−= (5.10)
From equation 5.8, the actual throttle angle appears in the derivative of the
sliding surface. However, since this engine is equipped with an electronic
throttle, the input to the system is a throttle command. Thus, a synthetic input
(θd) is defined and the control law is derived using this synthetic input. The
control law is given by
35
( ) ( )( )
( ) ( )
+
⋅⋅⋅⋅⋅⋅
⋅−
+
+−
⋅⋅
⋅−
= −
1TR
Pωω,PηV4π1
P,PPRIm1
PSsgnηTR
VP,PPRIm
1
cosθm
engengmvoldispammax
desm,11m
ammax1d (5.11)
Now a second control law must be derived to drive the throttle angle to the
desired throttle angle. Using a similar Lyapunov function candidate as that given
in equation 5.6, a second control surface is used to achieve the control goal,
θth θd. This second sliding surface is defined as
dth2 θθS −= (5.12)
Derivation of a control law using this sliding surface requires the time derivative
of θd to be known. In almost all cases, the rate of change of this synthetic input is
unknown. In order to eliminate this requirement, the control surface can be
redefined as
zθS th2 −= (5.13)
where z is the output of a first order filter defined by
d2 θzzτ =+ (5.14)
Taking the derivative of the second surface yields
( ) ( )zθτ1θθ
τ1
zθS
d2
thcmdth
th2
−−−=
−= (5.15)
36
Since θd is passed through a first order filter, knowledge of the derivative of θd in
the control law is no longer a requirement. As shown, the input to the system,
θcmd, appears in the derivative of this second surface. Again using the worst
case model uncertainty, the control law is given by
( ) ( ) thd2
22thcmd θzθτ1
Ssgnητθ +
−+−= (5.16)
The final control law is defined as
( ) ( )
( )
( ) ( )( )
( ) ( )
+
⋅⋅⋅⋅⋅⋅
⋅−
+
+−
⋅⋅
⋅−
=
−=
+
−+−=
−
1TR
Pωω,PηV4π1
P,PPRIm1
PSsgnηTR
VP,PPRIm
1
cosθ
zθτ1
z
θzθτ1
Ssgnητθ
mengengmvoldisp
ammax
desm,11m
ammax1d
d2
thd2
22thcmd
(5.17)
In this derivation, the derivative of the desired manifold pressure ( desm,P ) is
assumed known. In implementation, the derivative of the desired manifold
pressure ( desm,P ) would have to be calculated using the rate of change of the
pedal position.
Figures 18 to 21 show the results of the pressure control strategy for a sinusoidal
input in the pressure setpoint. Figure 18 shows the throttle position and throttle
flow required to achieve the control objective. As stated, the throttle actuator is
used to achieve the desired manifold pressure. Figure 19 shown the actual and
desired manifold pressure as well as the combustion torque. Figure 20 shows
the transient portion of the pressure control as the controller begins to track the
pressure setpoint. Engine speed is also shown in Figure 20. In this test, the
engine speed is not held constant, but is assumed known. Figure 21 shows the
37
two control surfaces defined in the control law. As shown, the system converges
to the two control surfaces. Once the system reaches the control surfaces, the
system stays on each surface as desired.
Figure 18. Pressure control results – throttle.
38
Figure 19. Pressure control results – pressure and torque.
Figure 20. Pressure control results – pressure and speed.
39
Figure 21. Pressure control results – control surfaces.
5.4 Engine Torque Control In section 5.3, a manifold pressure control is derived as a means of controlling
engine torque. The control goal, however, is to control the engine torque. With
the use of a torque sensor or an engine torque estimation algorithm, a feedback
control on driveline torque can be made. Control of engine torque is required if
the control is to be robust with respect to the effects of engine aging. Although
this type of torque control may not be required for automotive applications, it is
included here for completeness.
Again, starting with the Lyapunov function candidate
21S
21
V = (5.18)
where the control surface is defined as
40
accfricpumpcomb
des1
TQTQTQTQΤ
ΤΤS
−−−=
−= (5.19)
where T is the actual torque delivered to the driveline and Tdes is the desired
torque at the driveline. The derivative of the sliding surface is given by
( ) ( )deseng1m
deng
eng
cylcyl
eng
T
des1
ΤωCP4πVω
ωm
mω
δSPIλAFIC
ΤΤS
−⋅−⋅+
⋅−⋅⋅=
−=
(5.20)
where the exhaust manifold pressure is assumed to remain constant in the
pumping torque calculation and the accessory loads are assumed constant.
After simplification, the derivative of the sliding surface is given by
( ) ( )
( ) ( )
desmdispl
m
volmengengvol
displ
eng
T
eng
1eng
cyl
eng
volmengmvol
displ
eng
T
1
ΤP
4πV
Pη
Pωωη
TR
V
4π1
ωδSPIλAFIC
ωC
ωm
ωη
PωPηTR
V
4π1
ωδSPIλAFIC
S
−⋅
+
∂∂
⋅⋅+⋅
⋅⋅
⋅⋅⋅⋅
+⋅
−
−
∂∂
⋅⋅+⋅⋅⋅
⋅
⋅⋅⋅
=
(5.21)
where engω and mP are given by equations 2.6 and 2.2. For simplicity, two
functions are defined such that
41
( ) ( )
( ) ( )
4πV
Pη
PωωηTR
V
4π1
ωδSPIλAFIC
)h(
Cωm
ωη
PωPηTR
V
4π1
ωδSPIλAFIC
g()
displ
m
volmengengvol
displ
eng
T
1eng
cyl
eng
volmengmvol
displ
eng
T
+
∂∂
⋅⋅+⋅⋅⋅
⋅
⋅⋅⋅
=
−
−
∂∂
⋅⋅+⋅⋅⋅
⋅
⋅⋅⋅=
(5.22)
where g() and h() are functions of the system parameters and states, including
the torque constant, CT. Using these functions, equation 5.21 simplifies to
desmeng1 ΤPh()ωg()S −⋅+⋅= (5.23)
As in section 5.3, the commanded throttle position does not appear in the
derivative of the control surface. Instead, a synthetic input (θd) is used to derive
the control law. Using this synthetic input, the control law is defined as
( )
( ) ( )
+
⋅
⋅⋅⋅⋅⋅
−
+
⋅−+−⋅
⋅⋅
−
= −
1TR
Pωω,Pη4π
V
P,PPRIm1
)h(
ω)g(Τ)ηsgn(S
TRV
P,PPRIm1
cosθm
engengmvoldisp
ammax
engdes1m
ammax1d (5.24)
A second control law is then defined to drive θth to θd. As in section 5.3, dynamic
surface control is used to derive this second control law. To this end, a second
control surface is defined as
zθS th2 −= (5.25)
42
where
d2 θzzτ =+ (5.26)
Taking the derivative of the second surface yields
( ) ( )zθτ1θθ
τ1
zθS
d2
thcmdth
th2
−−−=
−= (5.27)
As shown, the input to the system appears in the derivative of this second
surface. The control law is then derived as
( ) ( ) thd2
22thcmd θzθτ1
Ssgnητθ +
−+−= (5.28)
The final control law is given by
( ) ( )
( )
( )
( ) ( )
+
⋅
⋅⋅⋅⋅⋅
−
+
⋅−+−⋅
⋅⋅
−
=
−=
+
−+−=
−
1TR
Pωω,Pη4π
V
P,PPRIm1
)h(
ω)g(Τ)ηsgn(S
TRV
P,PPRIm1
cosθ
zθτ1
z
θzθτ1
Ssgnητθ
mengengmvol
disp
ammax
engdes1m
ammax1d
d2
thd2
22thcmd
(5.29)
Figures 22 to 25 show the results of the torque control strategy for a sinusoidal
input in the driveline torque setpoint. Figure 22 shows the throttle position and
flow for this test. Figure 23 shows the actual and desired torque as well as the
manifold pressure. Figure 24 shows the transient portion of the torque control as
43
the controller begins to track the torque setpoint. Also shown, is the engine
speed. In this test, the engine speed was not held constant, but was assumed
known. Figure 25 shows the control surfaces for the system.
Figure 22. Torque control results – throttle.
44
Figure 23. Torque control results – torque and pressure.
Figure 24. Torque control results – torque and speed.
45
Figure 25. Torque control results – control surfaces.
This derivation has assumed the model for engine torque, throttle flow, and
engine flow are exactly correct. However, over time, the engine parameters may
change. For instance, over time, the engine will not produce as much torque as
when it was new. Therefore, the torque constant (CT) may change over time. In
some cases, it may be appropriate to account for this variation using the
robustness parameter, η, in the sliding control. However, this would lead to
increased control effort. Instead, an adaptive controller can be used to account
for an unknown, slowly varying parameter such as this. In addition to meeting
the control objective, an adaptive controller attempts to estimate an unknown,
constant or slowly varying parameter in the system. This is done by creating a
parameter update law in such a way that the estimate of the parameter
converges to the true value of the parameter. Use of an adaptive controller
reduces the uncertainty in the model. This allows for a lower gain, η, to be used
in the controller.
46
For the adaptive control strategy, the synthetic input (θd) of equation 5.24 is
defined using the estimate of CT (call this TC ). Solving for θd using the estimate
TC gives
( )( )
( ) ( )
+
⋅
⋅⋅⋅⋅⋅
−
+
⋅−+⋅−⋅
⋅⋅
−
= −
1TR
Pωω,Pη4π
V
P,PPRIm1
)Ch(
ω)Cg(ΤSsgnηTR
VP,PPRIm
1
cosθm
engengmvoldispl
ammax
T
engTdes1m
ammax1d
ˆ
ˆ
(5.30)
Substituting this control law into the derivative of the sliding surface leads to
des
T
engTdesadaptT
engTadapt
Τ)Ch(
ω)Cg(Τ)ηsgn(S)h(C
ω)g(CS
−
⋅−+−⋅
+=
ˆ
ˆ (5.31)
Defining ∆CT to be
TTT CC∆C ˆ−= (5.32)
and substituting into equation 5.31 yields
( ) ( )
( ) ( )( ) ( )
( )( )
⋅−+⋅−
⋅⋅⋅+⋅⋅⋅⋅
⋅⋅⋅
⋅
+⋅−⋅⋅⋅
⋅⋅⋅
⋅
+⋅−=
engTdesadapt
engmvolT
vol
T
eng
eng
cyl
mvol
displ
eng
T
adaptadapt
ωCgΤ)sgn(Sη
TRωPηδSPIλAFICmPηδSPIλAFI
∆C
ωω
mPη
TR
1
4π
V
ωδSPIλAFI
∆C
)sgn(SηS
ˆ
ˆ
(5.33)
47
The goal is to define a parameter update law for TC . Thus, the system must be
converted into a two state system consisting of the states Sadapt and TC .
Using a Lyapunov argument, a Lyapunov function candidate can be defined as
2∆CρS
21
V2
T2adaptadapt += (5.34)
where ρ defines the rate of parameter convergence and T∆C is the difference
between CT and TC . This Lyapunov function candidate is a positive definite
function in the two states S and TC . The derivative of the Lyapunov function
candidate is given by
TTadaptadaptadapt C∆∆CρSSV ⋅⋅+⋅= (5.35)
Since CT is assumed to be slowly varying, it can be treated as a constant. Thus,
adaptV reduces to
−⋅⋅+⋅= TTadaptadaptadapt C∆CρSSV ˆ (5.36)
By substituting the definition of adaptS , given in equation 5.33, into adaptV , it is
shown that
48
( ) ( )
( ) ( )( ) ( )
( )( )
⋅−+⋅−
⋅⋅⋅+⋅⋅⋅⋅
⋅⋅⋅
⋅
+
⋅
−⋅⋅
⋅⋅
⋅⋅
⋅
+⋅−
⋅
+
−⋅⋅=
engTdesadapt
engmvolT
mvol
T
engeng
cylmvol
displ
eng
T
adapt
adapt
TTadapt
ωCgΤ)sgn(Sη
TRωPηδSPIλAFIC
PηδSPIλAFI
∆C
ωωm
PηTR
14π
V
ωδSPIλAFI
∆C
)sgn(Sη
S
C∆CρV
ˆ
ˆ
ˆ
(5.37)
The system is stable if adaptV is negative definite. Towards this end, TC is
defined to be
( ) ( )
( ) ( )( ) ( )
( )( )
⋅−+⋅−
⋅⋅⋅+⋅⋅⋅⋅
⋅⋅⋅
+⋅
−⋅⋅
⋅⋅⋅⋅
⋅⋅=
engTdesadapt
engmvolT
mvol
engeng
cylmvol
displ
eng
adaptT
ωCgΤ)sgn(Sη
TRωPηδSPIλAFIC
PηδSPIλAFI
ωωm
PηTR
14π
V
ωδSPIλAFI
Sρ1
C
ˆ
ˆˆ (5.38)
where TC has been defined to cancel out the ∆CT terms in the derivative of the
Lyapunov function (equation 5.37). With this parameter update law, adaptV
reduces to
)sgn(SSηV adaptadaptadapt ⋅⋅−= (5.39)
which is a negative semi-definite function of the system states.
49
Since Tdes is not constant, Barbalat’s Lemma [16] must be used to determine the
stability of the time varying system. Barbalat’s Lemma has three conditions,
which, if satisfied, guarantee a negative semi-definite Lyapunov function
candidate has only a one solution. The conditions are
1. V is lower bounded Vadapt is positive definite. Thus, it is lower bounded.
2. V is negative semi-definite As stated in equation 5.39
3. V is uniformly continuous OR V is bounded The second derivative of
the Lyapunov function is bounded provided desΤ is bounded.
Since the three conditions of Barbalat’s Lemma are satisfied, the equilibrium
point of the Lyapunov function candidate is globally, asymptotically, stable.
The modified control law is given by
50
( ) ( )
( )
( )( )
( ) ( )
( ) ( )
( ) ( )( ) ( )
( )( )
⋅−+⋅−
⋅⋅⋅+⋅⋅⋅⋅
⋅⋅⋅
+⋅
−⋅⋅
⋅⋅⋅⋅
⋅⋅=
+
⋅
⋅⋅⋅⋅⋅
−
+
⋅−+⋅−⋅
⋅⋅
−
=
−=
+
−+−=
−
engTdesadapt
engmvolT
mvol
engeng
cylmvol
displ
eng
adaptT
mengengmvol
displ
ammax
T
engTdes1m
ammax1d
d2
thd2
22thcmd
ωCgΤ)sgn(Sη
TRωPηδSPIλAFIC
PηδSPIλAFI
ωωm
PηTR
14π
V
ωδSPIλAFI
Sρ1
C
1TR
Pωω,Pη4π
V
P,PPRIm1
)Ch(
ω)Cg(ΤSsgnηTR
VP,PPRIm
1
cosθ
zθτ1
z
θzθτ1
Ssgnητθ
ˆ
ˆˆ
ˆ
ˆ
(5.40)
The parameter update law requires Tdes to vary ( 0S ≠ ) in order for the parameter
estimate to converge. This is due to the persistency of excitation requirement of
adaptive control systems [16].
Figure 26 shows the convergence of the parameter estimate for a low and high
initial estimate. Parameter convergence has been demonstrated. However, a
rigorous proof of convergence is not given here.
Figure 27 shows the desired driveline torque used in the convergence test. The
desired torque is a single sinusoid with additive white noise. In implementation,
the normal drive cycle of a vehicle should provide enough excitation to the
system to ensure convergence.
51
It should be noted that holding engine speed constant in effect changes the order
of the system. Since the parameter CT only acts to change the engine speed,
adaptive control on this parameter requires that engine speed not be constant.
Figure 26. Torque constant (CT) parameter estimate using adaptive control law.
52
Figure 27. Torque setpoint and engine speed used to test adaptive control law.
5.5 Torque Management Strategy At this point, two control laws have been developed which control the engine
torque to a desired level using only the engine throttle. However, due to the
throttle and manifold dynamics, the engine throttle cannot compensate for
instantaneous changes in the driveline torque. These changes can occur due to
changes in the accessory loads acting on the engine or due to changes in the
actual engine torque production of the engine (such as when the cam timing
changes). In addition, this type of strategy also has applications in lean burn
engines and hybrid engine technologies. An example of the engine performance
during the onset of an accessory load is shown in Figures 28 and 29. With the
throttle position held constant, the addition of an accessory load causes the
system to reach a new steady state. This change in engine speed and driveline
torque directly effects the vehicle performance in a way that is unexpected by the
driver. As shown in Figures 28 and 29, the throttle position is held constant (as
with a mechanical throttle) and an accessory load is removed from the engine.
This decrease in engine load causes the engine to accelerate to a new steady
53
state engine speed. Due to the change in operating condition, the combustion
torque is decreased. For this analysis, the change in engine loading or engine
torque production will be achieved through the accessory load. However, the
following derivation would be similar for any known change in engine loading or
engine torque production.
Figure 28. Engine response to accessory load with no torque control – throttle and accessory.
54
Figure 29. Engine response to accessory load with no torque control – torque and speed.
If an electronic throttle is used in conjunction with a pressure control strategy or a
torque control strategy (as described in sections 5.3 and 5.4), the engine
achieves the same steady state engine speed after a short transient. This is
shown in Figures 30 and 31. At the time the accessory is removed from the
engine, the pressure setpoint changes. Over a short period of time, the throttle
adjusts to achieve the new pressure setpoint. However, at the time the
accessory is removed, the engine initially accelerates. As the manifold pressure
approaches the new setpoint, the engine decelerates and achieves the speed
prior to the accessory load change. The rate at which the system converges to
the new pressure setpoint is defined by the robustness term, η, in the sliding
control. This is a design parameter. However, increasing η also increases the
control effort.
55
Figure 30. Engine response to accessory load with pressure control only – throttle and accessory.
Figure 31. Engine response to accessory load with pressure control only – pressure and speed.
56
The goal of the torque management strategy is to maintain the torque to the
driveline through these short transients. This would have the effect of keeping
the engine speed constant.
Jankovic et. al [10] designed a feed forward control strategy for the electronic
throttle on an engine with variable cam timing. This control was designed to
cancel out the dynamics in cylinder flow during changes in cam timing. The
strategy used by Heintz et. al. [7] uses ignition timing and air to fuel ratio to
eliminate these types of torque transients. This strategy is more general and
applicable to a variety of applications requiring torque management. This
general framework has been used to develop the strategy discussed in the
following sections.
5.5.1 Derivation of the Control Laws
As shown in equation 2.4, the engine torque is a function of cylinder air flow, air
to fuel ratio, and ignition timing. In the previously derived torque controllers of
sections 5.3 and 5.4, it has been assumed that the low level air to fuel ratio
control and ignition timing control are working properly and in such a way that
their effects on engine torque can be assumed constant. In the torque
management strategy, the ignition timing and, if necessary, the air to fuel ratio
will be changed to maintain the torque to the driveline until the air flow can be
adjusted to produce the desired torque. The amount of torque change produced
by each of these actuators is limited by pre-ignition of the fuel, engine emissions,
and other component considerations. Also, the time spent away from the optimal
ignition timing and the optimal air to fuel ratio is limited due to engine emissions
and other component considerations.
From equation 2.4, the engine torque is given by
57
( ) ( )eng
cylTcomb ω
δSPIλAFImCTQ
⋅⋅⋅= (5.41)
One option for this problem is to split it into two parts. First, the throttle can be
controlled to achieve the air flow required to achieve the desired torque. For this
control, it can be assumed that the effects of air to fuel ratio and ignition timing
are constant. The difference between the actual driveline torque and the desired
driveline torque (δTQ) will be controlled to zero using the ignition timing and the
air to fuel ratio.
The desired effects of the ignition timing and the air to fuel ratio on engine torque
are calculated using the relationship
xTQδTQoptimalδ,λcomb ⋅=
= (5.42)
Using the known functions of SPI(δ) and AFI(λ), the desired ignition timing and
air to fuel ratio could be calculated to achieve the desired value for x.
However, it would seem that, as with the idle speed control problem, what is
really desired is to use δ and λ to maintain engine speed (or acceleration)
through the transient. Using the sliding mode control methodology discussed in
section 5.2, a sliding surface is defined as
desωωS −= (5.43)
where ωdes is taken to be the engine speed before the strategy is started. The
derivative of the sliding surface, assuming a constant desired engine speed, is
given by
58
( )( ) ( ) ( )
( )
−−+⋅
−−−⋅⋅⋅
⋅=
−−−−⋅=
=−=
impacc2eng1
mexhm,disp
eng
cylT
eng
impaccfricpumpcombeng
des
TQTQCωC
PP4π
V
ωδSPIλAFImC
I1
TQTQTQTQTQI1
ωωωS
(5.44)
In equation 5.44, the desired engine acceleration is assumed to be zero. This is
not a requirement for the strategy.
The control goal is to use the ignition timing, δ, and the air to fuel ratio, λ, to
control the engine speed. However, these control inputs will not be used in
tandem. First, the ignition timing, δ, will be used to control the engine speed. If δ
reaches some predefined limit, λ will be used to control engine speed with δ left
at the saturated value. Thus, this is not a true multi-input control problem, but a
system that can be decoupled. The individual controllers for ignition timing and
air to fuel ratio are derived using equation 5.44 and are given by
( )[ ]( )
−⋅⋅
⋅
++
++−⋅⋅−= 1
λAFImC
ωTQTQTQ
TQSηsgnI
0.000151δ
stoiccylT
eng
pumpfricacc
impeng (5.45)
and
( )[ ]( )
−⋅⋅
⋅
++
++−⋅⋅−+= 1
δSPImC
ωTQTQTQ
TQSηsgnI
0.01561
13.6λsatcylT
eng
pumpfricacc
impeng (5.46)
where λstoic is the stoichiometric air to fuel ratio, and δsat is the saturated value of
the ignition timing.
59
5.5.2 Selection of Pressure Control or Torque Control Strategy
Control strategies for manifold pressure and driveline torque were derived in
sections 5.3 and 5.4. Before the torque management strategy can be
implemented, a torque control strategy must be selected.
For the driveline torque control strategy, the engine torque sensor has been
assumed to measure crankshaft torque. This measurement includes the effects
of the engine combustion torque, friction torque, pumping torque, and all
accessory loads. As stated, in this derivation of the torque management
strategy, δ and λ are used to control the engine speed. Maintaining engine
speed is equivalent to maintaining constant torque at the driveline. So, if the
throttle is used to control the driveline torque and δ and λ are actuated to control
engine speed, the error between desired and actual driveline torque is
eliminated. Thus, there is no error to drive the throttle. This would lead to δ and
λ remaining away from the optimal ranges for long periods of time. This is
unacceptable performance. Thus, the torque control is not a legitimate option for
the torque management strategy derived here.
If instead, the throttle is used to control manifold pressure, the effect of ignition
timing and air to fuel ratio on combustion torque does effect the throttle control.
Thus, the pressure control strategy is proper for implementing the torque
management strategy. Also, use of this strategy will not require a torque sensor.
The disadvantage is a larger amount of calibration required to get a proper
conversion from desired torque to desired manifold pressure for all engine
operating conditions.
60
5.5.3 Results
Figures 32 to 34 show the results of the torque management strategy when a 7
N.m accessory load is removed from the engine. In this case, the ignition timing
has enough control authority to achieve the objective. At two seconds, the
accessory load is removed from the engine. The pressure setpoint immediately
changes. Over a third of a second, the throttle adjusts to achieve the desired
manifold pressure. As desired, the engine speed (shown in Figure 33) remains
approximately constant through the transient. However, Figure 34 shows the
control input required to achieve this response. This type of rapid chatter in the
control input is undesirable and unrealizable in implementation.
This chatter is due to the high gain used in the engine speed control. Since this
strategy requires strict control of engine speed, a high gain is used to rapidly
drive the system to the control surface. As shown in Figure 34, the ignition timing
saturates for a short period of time at 20 degrees. During this time, the system
has not reached the control surface. Once the control surface is reached, the
control input begins to chatter rapidly. A reduction of the control gain would
reduce the peak to peak value of the chatter, but would cause the system to take
longer to reach the control surface.
61
Figure 32. Torque management with discontinuous ignition timing control law – throttle and accessory.
Figure 33. Torque management with discontinuous ignition timing control law – pressure and speed.
62
Figure 34. Torque management with discontinuous ignition timing control law – ignition timing.
In order to eliminate the chatter, a smooth sliding mode control law is
investigated. In this case, a smooth control law is chosen such that
2SηSS ⋅−≤⋅ (5.48)
Although perfect tracking will not be achieved, the high frequency chatter will be
reduced with this control law.
Figures 35 and 36 show the results using the smooth control law. As shown, the
performance is similar to the discontinuous control law. However, the chatter of
the ignition timing is greatly reduced. Also, since the control gain is still high, the
time that it takes the system to reach the control surface remains the same.
63
Figure 35. Torque management with smooth ignition timing control law – pressure and speed.
Figure 36. Torque management with smooth ignition timing control law – ignition timing.
64
Figures 37 and 38 show the engine response to a 10 N.m accessory load being
removed from the engine. In this case, the ignition timing is unable to attain the
control goal of constant engine speed. In this case, air to fuel ratio can be
controlled to drive the system towards the control surface more rapidly.
Figure 37. Torque management results using only ignition timing to control engine speed – pressure and speed.
65
Figure 38. Torque management results using only ignition timing to control engine speed – ignition timing and air to fuel ratio.
Figures 39 and 40 show the response to a 10 N.m load being removed from the
engine. Again, smoothed control laws are used both for the ignition timing and
the air to fuel ratio control. As shown, during the portion of time the ignition
timing is saturated, the air to fuel ratio is controlled. Once the system has
approached the control surface to a point at which air to fuel ratio control is
unnecessary, the air to fuel ratio is set to its optimal value and ignition timing is
used to achieve the control goal. As shown, engine speed is maintained
approximately constant throughout the transient.
66
Figure 39. Torque management results using both ignition timing and air to fuel ratio to control engine speed – pressure and speed.
Figure 40. Torque management results using both ignition timing and air to fuel ratio to control engine speed – ignition timing and air to fuel ratio.
67
6 Application to the MoBIES Project
The goal of the Model Based Integration of Embedded Systems (MoBIES)
program is to develop tools to aid the control developer in moving from a
controller implemented in a simulation environment to a controller implemented
on a target platform. The end goals are to have greater re-use of code and a
decrease in implementation time. The tools under development for this program
include tools that perform hybrid system verification analysis, timing and
schedulability analysis, and automatic code generation tools. The Ford Taurus
engine was used as an open experimental platform on which to test the
generated control code before and after all analyses had been performed. This
was used as a way to test the usability and applicability of the developed tools.
The air to fuel ratio and torque management problems were used as test cases
for the MoBIES technologies. The MoBIES program and the model based
methodology for embedded system design are discussed in detail in [13].
7 Future Work
The sliding mode control laws have been derived using discontinuous control. If
this method of engine torque management is found to be useful, an investigation
into the use of “smoothed” control laws or other control methodologies may be
useful. In particular, some introductory work has been done in applying linear
robust control methodologies, such as HHHH∞ [3] to this problem. Initial work in this
area has been done by Ingram et. al. [9]. It is believed that use of these types of
linear control methodologies could provide excellent performance even in the
presence of model uncertainty.
As stated, the plant model used throughout this derivation does not include the
effects of EGR. Since EGR would act to displace air in the intake manifold, this
effect would significantly change the model, the estimators for cylinder flow, and
68
the controller derivations. Extending the torque management strategy to deal
with these types of engines would be beneficial.
This analysis has showed the benefits of torque management and introduced a
strategy for implementation. At this point the strategy should be expanded to
include the complete operating range of the engine. This would require
additional calibration and a map of pedal position to desired engine torque must
be selected. It is probable that the desired engine torque would be a function of
engine speed. This adds to the complexity, but does not change the form of the
control strategy.
As stated, most of the control derivations assume a perfect model of the plant
with only parameter uncertainty. While a complete uncertainty analysis can be
made for each control law, it is believed that the control strategy should be tested
on a more complete engine model and eventually an actual engine. The engine
model used in this control derivation is a simple model which eliminates some of
the complex behavior of the engine. While this type of model is useful for
implementation in real time, the engine has much more complex dynamics. The
performance of the controller will depend more on the unmodeled, or simplified,
dynamics than the parameter uncertainty of the model.
Finally, a complete test of the control logic should be made. If possible, hybrid
system verification tools (such as those developed in the MoBIES program)
should be used to guarantee proper operation of the control logic for all engine
operating conditions. At the very least, potential problems should be seen when
the complete strategy is run over the entire engine operating range.
69
8 Summary
It was shown that, in the presence of model uncertainty, an open loop estimate of
the cylinder air flow using a mass air flow sensor was more accurate than one
using a manifold pressure sensor. A closed loop observer for the cylinder air
flow, using both a throttle flow sensor and a manifold pressure sensor, was
developed to obtain a better estimate in the presence of model uncertainty. The
sliding observer achieved excellent tracking in the presence of some types of
uncertainty. However, error in the cylinder pumping model leads to a
degradation of the estimator performance as compared to the open loop
estimator using a throttle flow sensor.
Two torque control strategies were developed. These control strategies used the
electronic throttle actuator to control the manifold pressure and the driveline
torque. Control of torque would require either a torque sensor or some torque
estimation algorithm. Both control strategies were similar in operation although
the pressure control was more direct in that pressure is a system state and thus
its derivative is readily available. However, by controlling torque directly, model
uncertainty in the combustion torque model is directly accounted for in the
control. An adaptive controller was developed to adapt on the engine torque
constant.
A torque management strategy was developed using the ignition timing and the
air to fuel ratio. It was shown that using the pressure control as a means of
controlling engine torque is more easily implemented in the torque management
strategy. Also, smooth sliding control laws were derived for both ignition timing
and air to fuel ratio control. It was found that the smoothed controllers gave
acceptable performance in controlling engine speed.
70
9 Acknowledgements
This work was done in part for the Model Based integration of Embedded
Systems (MoBIES) project that was funded by DARPA/ITO (Contract #F33615-
00-C-1698, Smart Vehicles: An Open Platform for the Design, Testing, and
Implementation of Automotive Embedded Systems).
I would like to thank:
Jason Souder and Paul Griffiths – These guys did great work on the air to fuel
ratio control and the electronic throttle control.
Nicholas Teske and Carlos Zavala – Thank you for your work on the MoBIES
project. It has been great working with you.
Professors J. Karl Hedrick and Pravin Variaya (the principle investigators for the
MoBIES project) – Thank you for your help in completing this project and for the
continued funding throughout my time at Berkeley.
Jessica and Dalen Lamberson – Thank you for allowing me to take this time to
continue my education. Although this Master’s degree is great, Dalen ended up
being my biggest accomplishment over the last two years.
71
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