module 1 part 1 engine dynamics

ME 7236 Module 1 Autumn 2012

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ME 781 Powertrain Dynamics

Giorgio Rizzoni

Krishnaswamy (Cheena) Srinivasan

The Ohio State University

Department of Mechanical Engineering

Autumn Semester 2012

The Ohio State University

Center for Automotive Research


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Part I Engine Dynamics

Giorgio Rizzoni

Jonathan Dawson (The Mathworks)

Yong-Wha Kim (Ford Motor Company)

Byungho Lee (General Motors)

Qi Ma (General Motors)

Amr Radwan (Detroit Diesel Corporation)

Devesh Upadhyay (Ford Motor Company)

Inkwang Yoo (Ford Motor Company)

Marcello Canova (The Ohio State University)

Fabio Chiara (The Ohio State University)


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Module 1

Introduction to Powertrain Dynamics

1.1. COURSE ORGANIZATION ..................................................................... 5

Objective ......................................................................................................... 5

Assignments and examinations ....................................................................... 5

What do we mean by “System Dynamics”? ................................................... 6

What this course is not .................................................................................... 6

Structure of the course .................................................................................... 6

Required background ...................................................................................... 7

1.2. BACKGROUND AND MOTIVATION ................................................ 8

Fuel efficiency improvement .......................................................................... 8

Emission standards ......................................................................................... 9

Emission controls for spark-ignition engines ............................................... 12

The importance of engine control systems ................................................... 14

1.3. INTRODUCTION TO POWERTRAIN DYNAMICS ............................. 16

1.4. MODELING FOR POWERTRAIN CONTROL ..................................... 20

Fundamental equations for modeling fluid systems dynamics ..................... 25

Model coupling techniques ........................................................................... 29

Modeling guidelines ..................................................................................... 31

1.5. TWO IMPORTANT ENGINE CONTROL PROBLEMS ................... 32

The AFR Control Problem ............................................................................ 32

Open-loop AFR Control ............................................................................... 32

Closed-Loop AFR Control ............................................................................ 37


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The Idle Speed Control Problem .................................................................. 39

Idle Air Control ............................................................................................. 41

Ignition Timing Control ................................................................................ 47

Summary ....................................................................................................... 50

1.6. REFERENCES ......................................................................................... 51

1.7. EXAMPLES ............................................................................................ 52

Example 1.1: Water tank filling dynamics model ........................................ 52

Example 1.2: Compressible flow through an isentropic nozzle ................... 54

Example 1.3: Torsional system dynamics .................................................... 55


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1.1. COURSE ORGANIZATION

Objective

The objective of this course – the first in a sequence of two - is to introduce

practicing engineers and engineering students (the latter typically in the first or second

year of a M.S. program in electrical or mechanical engineering) to the essential aspects of

modeling and control of automotive powertrains. The course recognizes the significance

of this growing area of engineering and its central role in the automotive industry. The

lectures place emphasis on the integration of many different aspects of powertrain

engineering, including: the dynamics of mechanical, fluid, and thermodynamic systems;

sensor and actuator technology; and feedback controls. Primary emphasis will be given to

dynamics and control of fuel-injected, spark-ignited, internal combustion engines, while

the integration with the complete powertrain (torque converter and transmission) will

also be addressed. The course will present an overview of the major dynamic phenomena

that characterize powertrain behavior: intake and exhaust air flow dynamics; fuel system

dynamics; combustion and emissions; crankshaft dynamics; and air-fuel ratio control.

Emphasis will be placed on explaining the interaction between subsystems, and the

importance of considering the entire vehicle system when assessing the impact of the

performance of a subsystem on overall system performance. Modeling and computer

simulation will be integral part of the course, showing how to build numerical models of

engine/powertrain systems and components and apply them to solve problems pertaining

to powertrain dynamics.

Assignments and examinations

A homework assignment will be handed out each week. Each homework

assignment will include the following components:

a) theoretical: understanding the nature of the problem and identify the path

for solution;


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b) analytical: development of a mathematical formulation of the problem,

based on a set of equations and a number of physical constants;

c) computational: develop numerical procedures for solving the equations,

formulate criteria for verification and interpretation of the results.

In addition to the homework assignments, the students will complete a modeling,

simulation and system identification project based on data from an actual powertrain.

What do we mean by “System Dynamics”?

System dynamics is a discipline that studies the mathematical representation of

systems, focusing on the dynamic behavior. Typically, the system-dynamic

representation of a complex system such as an automotive powertrain consists of a set of

coupled, nonlinear, ordinary differential equations. Linear and nonlinear system analysis

and computer simulation methods are used to analyze the properties and characteristics of

the describing equations. System dynamics is a discipline that is often presented in the

junior or senior year of the undergraduate mechanical engineering curriculum, focusing

on the dynamics of electrical, mechanical, electro-mechanical, hydraulic, pneumatic, and

thermal systems [1]. In this course it is expected that students have a good working

knowledge of modeling and analysis of these families of systems, and of linear analysis

methods. In addition, the use of computer-aided tools such as Matlab/Simulink is also

necessary to complete many of the assignments in this course [2]. A summary of systems

dynamics concepts is available for those students who are in need of a review [3].

What this course is not

a) Modeling of IC engine thermo-fluid processes.

b) Modeling of mechanical dynamics of engines.

c) Design of transmission elements or subsystems.

Structure of the course

Module 1: Introduction to powertrain dynamics

Module 2: Intake and exhaust dynamics


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Module 3: Fuel injection and fueling dynamics

Module 4: Combustion, Ignition control and knock

Module 5: Crankshaft dynamics

Module 6: Overview of engine control systems

Module 7: Fluid couplings and torque converters

Module 8: Transmission mechanical system and vehicle longitudinal dynamics

Module 9: Transmission shift hydraulic system

Module 10: Open loop transmission control

Module 8: Shift schedules and continuously variable transmissions

Each of the above modules is presented according to the following sequence:

Present physical phenomenon in intuitive terms

Show physical components, where possible, and illustrate performance curves

Define physical laws

Derive equations of motion

Interpret equations of motion

Discuss computer simulation

Discuss model identification experiments (where appropriate)

Required background

o Undergraduate level linear system theory

o System Dynamics (dynamics of electrical, electro-mechanical, fluid, thermal, and

mechanical systems) - review available in the technical presentation “Signal and

System Dynamics Integration”, by Profs. Rizzoni, Srinivasan and Yurkovich.

o Basic IC engine processes - review available in the technical presentation

“Internal Combustion Engine Fundamentals”, by Prof. Guezennec.

o Familiarity with the Matlab/Simulink environment - review available in the

GMTEP technical presentation “Matlab/Simulink - Introduction”.


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1.2. BACKGROUND AND MOTIVATION

Emissions, fuel efficiency and safety requirements have steadily become more

stringent over the last few decades. These regulations can be considered the driving force

behind most of the advancements in the technology of internal combustion engines,

which have been improving at increasing rate with multi-disciplinary effort at all stages

of development.

Fuel efficiency improvement

The recent surge in oil prices has dramatically increased the awareness on

improving the fuel efficiency of passenger vehicles. The problem becomes increasingly

alarming when the entire chain of oil supply, refinement, transportation and utilization is

considered (well-to-wheel analysis).

Figure 1: U.S. Energy Flow Trends in 2005 (Units in quadrillion BTUs)


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Figure 1 summarizes the energy utilization in the U.S. based on data collected in

2005. Focusing on the bottom part of the chart, it is possible to notice that the petroleum

supplies, which rely mostly on foreign sources (62%), are used almost exclusively for

transportation (out of which, 43% is used for fueling light-duty vehicles, such as

passenger cars and light duty trucks). Furthermore, the energy used for transportation

shows the worst overall efficiency, with values around 20%. This scenario suggests the

strong priority of the automotive industry to design fuel efficient vehicles, as well as the

efforts in improving the overall well-to-wheel efficiency and diversifying the fuel sources

for transportation (i.e., biofuels and natural gas).

Emission standards

Another important motivation for the development of modern and efficient

engines and powertrain systems is dictated by the emission legislations, which have

become increasingly stringent in the past years in the major world automotive markets

(U.S. and Europe), as well as in the emerging economies, such as Asia and Latin

America. These regulations can be considered the driving force behind most of the

advancements in the technology of internal combustion engines.

In the United States, emission standards are managed by the Environmental

Protection Agency (EPA) (www.epa.gov). Few state governments, however, implement

own regulations. This is the case of California, where the California Air Resource Board

(CARB) has applied some of the strictest standards in the world.

Within the Clean Air Act Amendments (CAAA) of 1990, the EPA defined two

sets of federal standards for light-duty vehicles (similar rules have been defined for

heavy-duty vehicles and trucks):

Tier 1 Standards, which were published on 1991 and were effective from

1994 until 2003;

Tier 2 Standards, which were initially adopted in 1999 with an

implementation schedule from 2004 to 2009.


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Currently, vehicles sold in the United States must meet Tier 2 standards, which

are characterized by lower emission limits and a number of additional changes that made

the standards more stringent for larger vehicles. Although the former Tier 1 standards

were different between automobiles and light trucks, the current Tier 2 regulation apply

the same emission standards to all vehicle weight categories (i.e., cars, minivans, light-

duty trucks, SUV), regardless of the fuel they use. With these regulations, large engines

(such as those used in light trucks or SUV) are forced to use more advanced emission

control technologies than smaller engines in order to meet the standards.

Within the Tier 2 standards, there is a ranking of 8 different emission levels,

named Certification Bins, ranging from BIN 1 (corresponding to zero emissions) to BIN

8. Vehicle manufacturers are allowed to certify their vehicles to any of the 8 categories.

At the same time, the average NOx emissions of the entire vehicle fleet sold by each

manufacturer has to meet the average NOx standard of 0.07 g/mi. Temporary emission

standard, less restrictive, have been set as a transitional step until the full implementation

or Tier 2 Standards in 2007.

The EPA Bins cover California LEV 2 emission categories, to uniform vehicles

certification to the Federal and California standards. The emission standards for all

pollutants, are shown in the following figure.

Figure 2.1: Tier 2 Federal Emission Standards (units in [g/mile]) (source: http://www.dieselnet.com)


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California emission standards have been traditionally more stringent than the EPA

requirements, but their evolution and structure is similar to that of the federal legislation.

The California Air Resource Board (CARB) adopted a former emission standard

program, applied until 2003, where six vehicle types are defined:

Tier 1;

Transitional Low Emission Vehicle (TLEV);

Low Emission vehicle (LEV);

Ultra Low Emission Vehicle (ULEV);

Super Ultra Low Emission Vehicle (SULEV);

Zero Emission Vehicle (ZEV).

Each category was characterized by more stringent emission restrictions. Tier 1

was the baseline used to determine the standards. Car manufacturers were required to

produce a percentage of vehicles certified to increasingly more stringent emission

categories.

In 1998 CARB defined the LEV 2 emission standards, adopted from 2004 to

2010. Under the LEV 2 standard, NOx and PM standards for all emission categories are

significantly tightened, and the same emission levels apply to both gasoline and Diesel

vehicles. Specific emission standards are defined for passenger cars (including light-duty

trucks and medium-duty vehicles below 8500 lbs gross weight) and for heavy-duty

vehicles. As a result, most pick-up trucks and sport utility vehicles are required to meet

the passenger car emission standards The TLEV emission category has been eliminated.

It is, therefore, believed that the LEV 2 emission standards can only be met by

vehicles fitted with advanced emission control technologies, such as particulate filters

and NOx catalysts, or by hybrid-electric vehicles. The following figure reports an excerpt

of the LEV 2 emission standards for passenger cars.


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Figure 1.2: California LEV 2 Emission Standards for passenger cars (units in [g/mile])

(source: http://www.dieselnet.com)

Emission controls for spark-ignition engines

To comply with the emission regulations, spark ignition (S.I.) engines require the

presence of a three-way catalytic converter to reduce emission levels of the three major

pollutants (HC, NOX, and CO). The catalytic converter operates most efficiently in a

narrow window around the stoichiometric value of the air-fuel ratio. Closed-loop

regulation of the air-fuel ratio is therefore required to achieve this goal; the feedback

signal used to close the loop in the air-fuel ratio controller is provided by an exhaust gas

oxygen, or lambda, sensor. The oxygen sensor contains a ceramic material that has an

electrical response to changes in the oxygen partial pressure in the exhaust stream

relative to ambient; this sensor becomes active when it reaches a temperature T > 250° C.

The concentration of the amount of oxygen relative to ambient contained in the exhaust is

related to the air/fuel ratio. The voltage output of the lambda sensor is processed by the

Engine Control Unit (ECU), a microcontroller, which outputs a signal to the fuel

injectors (pulse width) adjusting the amount of fuel injected according to the duration of

the input voltage pulse. In this way the air-fuel ratio is maintained as close as possible to

the stoichiometric value to take advantage of the higher conversion efficiency of the

catalytic converter.

To further reduce the formation of NOX, it is possible to recirculate a fraction of

the exhaust gas into the intake manifold through a controlled valve. This practice,

conventionally known as Exhaust Gas Recirculation (EGR), is nowadays in decline for

S.I. engines, but is largely adopted in Diesel engines, where it can reach up to 50 percent

of the total trapped charge. The use of EGR corresponds to adding an inert gas to the


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mixture and thereby reducing the combustion temperature, leading to the reduction of

NOX formation. In a typical automotive S.I. engine, 5 to 15 percent of the exhaust gas is

routed back to the intake as EGR.

The control of EGR is typically accomplished using an open-loop structure. The

engine ECU contains maps specifying the opening position of the valve across the engine

operating range. In modern EGR systems, the valve is electrically operated (through a

solenoid actuator), providing a feedback signal on the valve position. Although EGR

does measurably slow combustion, this can be compensated for by advancing the spark

timing.

Open-loop control using other available sensors (throttle position, manifold

pressure and temperature, mass air flow, etc.) is typically used at operating conditions

which require richer or leaner mixtures than in the case of closed-loop control; cold start

and heavy acceleration are two conditions among others which require the open-loop

control strategy to take over.

Additional actuation mechanisms may also be employed to aid in the control of

the engine exhaust emissions; these may include: heated catalysts; electronic throttle

control; auxiliary air handling during starting; idle air bypass actuator; and other systems.

Further, HC emissions which originate from evaporative sources (i.e. evaporation

of the fuel in the tank) are reduced by the use of an evaporative emission control system

which is designed to store and to dispose of the fuel tank vapors. This evaporative system

consists of an active-charcoal canister that stores the HC vapors; these vapors are then

purged into the intake manifold to be burned with the mixture at operating conditions that

require additional mixture enrichment. This system has a strong interaction with the fuel

control system because during the purge cycle air and fuel vapors are introduced into the

intake system in unknown quantities, disturbing the regulation of the lambda control.

The combination of the above described control strategies constitutes what is

generally known as an automotive engine emissions control system. The implementation

of such a control system with the performance required to satisfy emission regulations

requires the use of event-based control (i.e. the control inputs are not computed on a


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fixed time increment but are dependent on the position of the engine crankshaft), so as to

manage the individual fuel injection and spark ignition events, which occur at different

rates as the engine speed varies. Further, the complete control strategy requires the

measurement of all the signals necessary to obtain information of the engine operating

condition and the parallel implementation of several strategies, including: i) spark timing;

ii) fuel injection (quantity and timing); iii) exhaust gas recirculation (EGR); iv) canister

purge cycle; v) on-board diagnostics.

The importance of engine control systems

The directives for fuel efficiency and environmental protection are followed both

by the Industry and Research Centers focusing mostly on two key areas: design

optimization and control.

Figure 1.3 compares the main tasks related to design optimization and control for

mid class and luxury segment vehicles. In response to both environmental and

acceptability challenges, significant improvements are required to fulfill the targets

imposed in each vehicle segment. For mid class segment vehicles, a large effort is

dedicated to fuel economy improvement through engine downsizing, i.e. reducing the

displacement. This trend is also related to the upcoming regulation regarding CO2

emissions. Moreover, in order to conquer a market share, the consumer acceptability

requirements (acoustic refinement and driving performance) must be improved.

Figure 1.3: Future engine development trends for mid class and luxury segment vehicles


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Slightly different effort must be dedicated to the development of luxury segment

vehicles. Here, the consumer acceptability implies a higher priority for driving

performance, combustion and structural refinements.

The tasks delineated must be met in very short time due to increased

competitiveness of the market. The most direct and effective way of improving engines

performance relies on design, either with the development of innovative technical

solutions or the refinement of existing products. The research effort in this field is

generally oriented in two directions. First of all, enhancements are sought for all the

various phenomena that affect the energy conversion process, such as combustion, gas

exchange processes, thermal management, etc… Then it is also required to accommodate

physical constraints that are a result of the improvements sought, for example higher

peak pressures or temperatures.

On the other hand, control design improvement seeks to optimize the various

processes in order to deliver power most efficiently while meeting constraints on

emissions, safety and reliability. Generally, new design and control concepts are tested

extensively prior to prototype applications, a task that constitutes a labor intensive and

expensive process.

In this sense, one of the priorities of industry is to shorten the development time

as much as possible, therefore striving to reduce the time-consuming and expensive tests

and calibration phases typically associated to the engine control development. This can

be accomplished by relying more on mathematical modeling tools to assist both the

engine design phase and the development of its control systems. Such modeling

capabilities are the key to reduce time and costs related to traditional experimental tests

and calibration efforts.

Even though not able to replace experimental investigations in full, simulation

models are capable of shortening consistently the development time from the definition

of the control system tasks to final tests and prototyping.

For internal combustion engines, a wide range of models is available to assist all

the design phases, from the simulation of flow and combustion processes, to the


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implementation and test of the control system. For example, Computational Fluid

Dynamics (CFD) and Finite Elements Method (FEM) are largely adopted in the

mechanical and fluid dynamic design of engines. Even though computationally intensive,

these tools are extremely powerful in assisting engineers and designers in the

development of individual components, such as manifolds, runners, ports and cylinder

parts. Their use can also replace part of the experimental analysis, which would normally

require complex testing environments and equipment.

However, when the final objective of the study is the design of a controller, it is

essential to have a global understanding of the behavior of the entire engine system,

rather than focusing on the details of a single component. In addition, computation time

becomes a significant constrain.

A typical engine system model, conceived for engine and powertrain control, will

be designed to predict only certain features, namely the dynamic response to

environmental and control inputs, avoiding an overly complex description of phenomena

that are not relevant for the analysis.

1.3. INTRODUCTION TO POWERTRAIN DYNAMICS

Modern engines and powertrain systems have become complex units, their

behavior influenced by the interaction of several components. Consequently, the number

of variables that must be considered for control and diagnostics has dramatically grown.

As stated above, this forces the control engineers to rely more on models from the

early development phases of the control systems. Even though models are not able to

replace experimental investigations, they contribute to shorten development time from

the definition of design specifications to final road tests.

The number and the complexity of design requirements are now forcing a shift

from map-based control systems towards model-based algorithms able to support control

and diagnostics functions.


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The approach commonly adopted for the analysis of steady-state and transient

behavior of engine and powertrain systems has its foundations in the system dynamics

theory, a common practice in the study of mechanical, electrical and hydraulic systems.

The general idea is to divide the system studied in elementary components which are

interconnected. The connections, namely the inputs and outputs of each element, become

then the focus of the analysis.

The following block diagrams are intended to illustrate in increasing order of

complexity the relationships among components and subsystems in a typical automotive

powertrain.

Figure 1.4 depicts the simplest overview, in which the engine is interpreted as a

subsystem that converts air and fuel flows to torque and exhaust gases; the developed

torque is what provides the tractive force for the vehicle through a transmission and

driveline.

Figure 1.4: Overview of basic engine function

A more detailed view of basic engine functions, illustrating the difference

between ideal and actual external inputs, is shown in Figure 1.5. In the figure, it is

shown that the flows of air and fuel are actually regulated by ”valves” or “flow restriction

devices” (throttle body and fuel injectors), and are affected by internal dynamics. In

particular, two specific subsystems are source of dynamic behavior, namely the intake

manifold for air and the evaporation dynamics for the injected fuel.


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Throttle Body

Intake Manifold

Combustion

Fuel Injector

Crankshaft

EGO Sensor

Fuel Dynamics

Tind

Tload

A/Fexh

Vo2

Spark Controller

air in

fuel in

air in cyl.

fuel in cyl.

engine speed

Transmission

Vehicle

Figure 1.5: More detailed view of powertrain dynamics

IntakeManifold

Fuel Dynamics(Well Wetting)

Combustion

ExhaustManifold

Fuel Injection

EGR

Canister

HEGO

CatalyticConverter

fuel

can

EGR

Throttle Body Inertia

Damping/Friction

Drive TrainVehicleLoad

sp

Exhaust

engine speed

engine torque

Air

Fuel

Electronic Throttle Control

from ECM

from ECM

Canister purge

from ECM

from ECM

from ECM

Open-loop fuel control

Closed-loop fuel control

EGR Control

Ignition Control

Figure 1.5: Complete block diagram of automotive powertrain.


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Figure 1.5 shows a third block diagram which depicts a more complete system

model, including closed-loop air-fuel ratio control, and including disturbances such as

EGR and canister purge flows. This representation allows one to better understand the

interactions between components, defining the inputs, outputs of each subsystem and the

related controlled variables.

The characterization of the engine and powertrain dynamics involves studying the

dynamic response of each subsystem, as well as the interactions between the different

components. Further complexity is added by the characteristic frequency associated to

the dynamic response of each component, as summarized in Table 1.1.

Table 1.1 Powertrain dynamics time constants.

Subsystem Bandwidth or Time Constant

1. Intake Manifold Fast Dynamics: 200 -400 Hz Slow Dynamics: 1 - 2 Hz

2. Fuel Injector Time constant: 0.5 - 3 ms

3. Fuel Dynamics Evaporation: 0.5 s Mixing: 1 - 10 ms

4. Combustion Delay: 1/2 engine cycle

5. Crankshaft 0 - 2000 Hz

6. Transmission/Vehicle 3 - 5 Hz

7. O2 Sensor Time constant: of the order of a few ms (speeds up with aging)

The engine system, as an assembly of mechanical, hydraulic, thermal and

electrical components, is characterized by a number of different phenomena that typically

occur at different time scales. For example, the typical response of a fuel injector

(electro-hydraulic actuator) is rather fast, in the order of 0.1 – 3 ms. Conversely, the fuel

evaporation dynamics, being mostly dependent on the slowly-varying intake manifold

temperature and the fuel vapor partial pressure in the port, is typically characterized by a

larger time constant.


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The separation of time constants is a typical problem that occurs when modeling

the dynamics of engines and powertrain systems, which may resolve in numerical issues

(i.e., stiffness) and complications when the model is used for control design.

It is important then to realize that no system can be modeled exactly in all its

detail, but that a good control-oriented model will capture with reasonable accuracy the

behavior of the system in a bandwidth sufficiently limited to allow for a simple but

reliable control design.

Powertrain models aimed at the development of control strategies have been

developed since 1980, following the advent of digital simulations of internal combustion

engine processes. To this extent, it is worth mentioning the early works of Dobner [1.7],

Powell and Cook [1.8], Moskwa and Hedrick [1.9], Hendricks and Sorenson [1.10], Turin

and Geering [1.11]. In recent years, various control approaches have been proposed to

use such models for control purposes, and model based control methods are gaining

wider acceptance in the research community. Relevant works in the field have been

presented by Amstutz et al. [1.12], Ault et al. [1.13], Azzoni et al. [1.14], Chang et al.

[1.15], Cho and Hedrick [1.16], Grizzle and Cook [1.17-1.19], Hendricks et al. [1.20],

Powell et al. [1.21], Kao and Moskwa [1.22, 1.23], Turin and Geering [1.11].

1.4. MODELING FOR POWERTRAIN CONTROL

The application of the system dynamics principles to the study of powertrain

systems allows one to operate a deconstruction of the plant into a series of interconnected

components.

Once each component is determined in its relevant inputs/states/outputs, it is

necessary to develop a mathematical model that allows to predict the states and outputs in

relation with the inputs and (if any) the control variables. In this process it is important to

maintain a ”system viewpoint”, hence not focusing on the details of the individual

component, but considering it as part of a more complex entity. This approach, even with

the loss of details for each individual component, is necessary to develop a description of


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the system capable of dynamically accounting for the most relevant phenomena, while

maintaining a reasonable computational ease.

In general, a preliminary stage for the analysis of a system or component consists

of operating a classification, namely making assumptions on the categories of models that

would best address the problem. Although there are neither standard nor accepted

methods of classifying system models, it is possible to define several categories in

relation with the characteristics of the system and the scopes of the analysis.

The operation of an internal combustion engine can be seen as the result of

interactions between several phenomena that occur at different temporal and spatial

scales. When modeling a specific engine component, a fundamental choice has to be

made to determine the bandwidth of the model, i.e., the maximum temporal and spatial

resolution. This choice leads to implicitly operate a spatial and temporal average

(~lowpass filter), removing spatial and temporal scales from the model, which will be

unresolved. Therefore, the capabilities of a model to capture specific engine phenomena

will be affected.

Following the principles above, two possible classifications of engine models can

be made:

Classification by Space Scale (Characteristic Length):

• Micro-length (multi-D) – the boundaries of the systems are very small,

allowing for detailed characterization of scalar and vector fields;

• Small length (1D) – the boundaries are set to characterize the field in typically

one direction; properties are assumed constant on any plane orthogonal to the

chosen direction;

• Large length (0D lumped, high-order) – the boundaries are assumed equal to a

component (e.g., valve, receiver,…), wherein the properties are considered

uniformly distributed;

• Very large length (0D lumped, low-order) – several components are included

within the boundaries.


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Classification by Time Scale:

• Very short time scales - captures evolution of internal system states at micro-

scale (e.g.: sub-millisecond).

• Short time scales - captures dynamics much faster than the excitation time

scale (e.g.: tens of milliseconds);

• Medium time scales - captures input-output dynamic behavior of comparable

time scale relative to excitation (e.g.: fractions of one second);

• Long time scales (quasi steady) - system reaches equilibrium very quickly

relative to the time scale of excitation (e.g.: seconds, minutes, hours).

This classification concept can be better explained through an example. A very

important engine subsystem is the intake manifold. Inside this component, several

thermodynamic, fluid dynamic and heat transfer processes occur, and the “breathing

performance” of the engine (i.e., the ability to draw fresh air/fuel charge at each cycle)

results from complex interactions.

Figure 1.6: Example of model classification applied to intake manifold modeling


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Several models can be used to study the flow through the intake manifold. Figure

1.6 illustrates the classification based on the spatial and temporal scales reporting the

most relevant model that are generally adopted for studying the system at different levels

of accuracy and computation time:

3D Computational Fluid Dynamics (CFD) Models: the system is studied

in four dimensions (space and time), adopting a very high resolution that

allows to characterize in great detail the velocity field. Such models are

typically used for design optimization;

1D Wave-Action Models (WAM): these models remove two spatial

dimension from the problem, assuming that all the properties (pressure,

temperature, etc…) vary only with respect to time and one spatial

coordinate (length). The time resolution and discretization length are

typically small, allowing for the characterization of the high-frequency

pressure fluctuations (wave propagation dynamics) that are very relevant

for engine tuning. Wave-action models are also used for optimizing the

manifold design, with respect to tuning and volumetric efficiency. From a

numerical standpoint, WAM are typically based on nonlinear partial

differential equations, which are solved using numerical approximations;

0D Filling-and-Emptying (F&E) Models: these models approximate all

the spatial resolution of the system, assuming that the thermodynamic

properties are uniformly distributed. The time resolution is typically set to

provide an adequate characterization of the system properties during one

engine cycle (typically, one degree of crank angle). For these reasons,

such models are often named “thermodynamic, crank-angle based

models”. Models pertaining to this category are considered a fair

compromise between accuracy (most of the high-frequency variations in

pressures and flow rates due to the alternative motion of the piston are

captured) and computation time. These models typically result in a high-

order system of nonlinear ordinary differential equations;

0D , Low-Order Models: these models capture only the low-frequency

dynamics of the system, considering all the system properties as cycle-


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averaged. This approach is often known as Mean-Value Modeling

(MVM). The result of such approximations is a low-order system of

nonlinear ordinary differential equations, as typically occurs in system

dynamics theory. This facilitates the applications of these models in

control system design.

Figure 1.6 illustrates also that the choice of a specific model category implies that

all the phenomena occurring at higher spatial and temporal scales will not be considered.

However, in order to respect the conservation principles and the agreement with the

behavior of the physical component, the effects of unresolved scales must be

approximated on the resolved scales. This can be done by introducing corrective

coefficients (in engine modeling, these are typically known as, discharge coefficients,

friction coefficients, heat transfer coefficients…). Such parameters, requiring

experimental calibration, approximate the unresolved physics that would be otherwise

removed from the model, generating errors.

As introduced above, mean-value models capture only the low frequency

spectrum of a system input/output behavior. For control oriented modeling approach, the

bandwidth of interest is typically associated to the transients that result from variations of

the engine load torque, which can be translated into correspondent variations of the

throttle position, intake manifold pressure and spark timing. With this approach, the

models developed have a time resolution which is adequate to capture the desired details

of the engine “throttle-to-torque” dynamics, but not the high-frequency behavior (which

can be associated to the fuel system dynamics, the engine cyclical behavior, or to the

tuning effects in the engine manifolds and runners). Such high frequency modes are

typically time-averaged.

The flexibility of these models and their capability of representing the

input/output behavior of the system with reasonable precision but low computational

complexity, have made them a very powerful tool in the analysis, simulation and design

of internal combustion engines.

The MVM technique is applied to each engine component to obtain a

mathematical representation of the input/output behavior. The characterization can be


1.25

made with different modeling criteria as well as different levels of detail. This process

requires knowledge of all the system components behavior, hence knowledge of different

engineering fields. Focusing on the throttle-to-torque dynamics, all the necessary

principles and information concerning thermodynamics, fluid mechanics, heat transfer,

chemistry and combustion have to be used together. Often, notions of applied mechanics,

electrical and electromechanical systems, and automatic control are also required.

Once all components have been individually modeled, they are coupled together

in relation with the system block diagram representation. The result is a set of differential

and algebraic equations (DAE) describing the dynamic of the entire system. Models

created following this methodology can serve a number of applications which, depending

on the model complexity, range from feasibility and design studies, to optimization,

control design, calibration and validation.

Fundamental equations for modeling fluid systems dynamics

An important step for formulating a MVM of an engine (or any fluid system), is

stating the fundamental equations that represent the time evolution of the system

variables. The equations are based on conservation principles that stem from the

fundamental notions of thermodynamic and fluid mechanics.

The formulation of the fundamental equations for a fluid system assumes the

definition of a suitable model of the flow. Focusing on a macroscopic viewpoint, based

on the continuum scheme commonly adopted in fluid mechanics, two methods of

analysis are available. The first step in both cases consists of drawing an arbitrary closed

volume within a region of the flow field studied. This defines a control volume V, whose

boundary is defined by a control surface A.


1.26

a) Finite control volume fixed in space

b) Finite control volume moving with the fluid

Figure 1.7: Overview of flow modeling approaches

The Eulerian approach states that the control volume is fixed in space and the

fluid moves through it, as depicted in Figure 1.7(a). Alternatively, as Figure 1.7(b)

shows, in the Lagrangian approach the control volume is moving with the fluid such that

the same fluid particles are always contained in the volume. The conservation laws are

applied to the fluid inside the control volume and, if the Eulerian approach is chosen, to

the mass and energy flows across the control surface.

Most of the practical cases in fluid systems dynamics deal with the study of the

flows into and out of components, an Eulerian approach is usually chosen, defining the

control volume as the physical volume of the component. Hence, the conservation laws

are applied to a control volume fixed in space, where mass, energy and momentum can

flow across its boundary. In this context, a general conservation equation for an extensive

system property can be written in the following form:

{net change in time} = {flow in through boundary}-{flow out through boundary} + {net

generation}-{net consumption}

The conservation laws, whose validity is independent of the nature of the

particular fluid or problem, are a summary of theoretical analyses and experimental

observations and rely on the assumption of nuclear and relativity effects being absent.

The conservation principles are usually expressed in the form of equations:


1.27

• mass conservation equation;

• energy conservation equation;

• momentum equation.

The principle of mass conservation, referred to an open thermodynamic system of

finite extension and fixed boundary, states that the overall mass of the system is constant,

hence the rate of accumulation of mass within the control volume is equal to the excess

of the incoming mass flow rate over the outgoing mass flow rate:

j

outi

in jimm

dt

dm (1)

In steady-state (hence, with the left-hand side equal to zero), the equation is also

known as the continuity equation

The energy conservation equation is the expression of the first law of

thermodynamics for open systems, i.e. capable of exchanging mass and/or energy with

the boundary. Considering a flow model of a finite control volume fixed in space such as

in Figure 1.7(a), the principle states that rate of change of the total energy of the system

is equal to the difference between the rate of energy flowing into the system and the rate

of energy flowing out of the system:

j

jouti

iint WQhmhm

dt

medji

(2)

where et is the total energy per unit of mass comprising the thermodynamic internal

energy (u), as well as the kinetic ( 2

2

1c ) and potential energy ( ). The latter includes all

the forms of potential energy associated to conservative fields, such as gravitational or

electrical.

2

2

1cuet (3)


1.28

Moreover, h represents the enthalpy of the flow, Q is the total heat exchanged through the

surface of the system and W is the mechanical power generated by the system.

The momentum equation, unlike the previous two, is not the result of a

conservation principle but rather the application of Newton’s second law of motion to an

open thermodynamic system. For this reason, the equation is actually a vector equation

(depending on three directions and time). This law states that the rate of change of the

system momentum in any direction is equal to difference between the incoming and the

outgoing rate of momentum flow and the sum of the external forces acting on the system:

i

ioutin FMMdt

Md (4)

The fundamental equations allow one to form a coupled system of nonlinear

ordinary differential equations in terms of several unknown variables. Therefore, it is

necessary to introduce further assumptions in order to create a model that can be solved

analytically or numerically. To this extent, constitutive relations can be introduced to

complete the equations set.

Unlike the conservation equations, whose validity is completely general, the

constitutive relations are a specific characteristic of the problem. Usually they are related

to the fluid considered and involve its properties, in the following form:

0,, Tvpf (5)

where p is the fluid pressure, v the specific volume (per unit of mass) and T the

temperature.

The expression of a constitutive relation generally leads to a set of algebraic

equations, even though ordinary differential equations are possible in some cases.

Constitutive relations often derive from assumptions made on the nature of the fluid. For

instance, if the fluid can be considered incompressible, the relation will simply become

.constv Moreover, for most of the applications involving liquids, constant temperature

is an acceptable assumption.


1.29

Since internal combustion engines operate with gaseous fluids, another important

constitutive relation that will often be used is the equation of state for a perfect gas:

mRTpV (6)

where R is the specific gas constant. The perfect gas model is a reasonable approximation

of the behavior of all compressible fluids, provided their thermodynamic state is far

enough from the saturation conditions (for example, during evaporation or condensation).

The perfect gas model implies further assumptions that allow one to close the

entire equations set. For example, the thermodynamic internal energy and enthalpy (used

in the energy equation) are functions of the sole gas temperature:

TcThh

TcTuu

p

v

(7)

where cv is the specific heat at constant volume and cp the specific heat at constant

pressure. Equation (7), sometimes known as the caloric equation of state, will be further

applied in the following modules.

Model coupling techniques

One of the advantages of using the input/output representation is that this

approach emphasizes modularity. Hence, large and complex powertrain systems can be

decomposed into elementary components interconnected. If each component is modeled

to be easily interfaced at the input and output ports with other subsystems, it is possible

to use “standard” component to assemble complex systems in a simple and

straightforward procedure. At the same time the calibration effort is reduced, because it

can be made separately for the model of each component and then only minor

adjustments are required on the final assembly.

In order to achieve these benefits, a suitable coupling concept must be defined for

multiport interconnected systems. Several approaches have been proposed in literature,

the most important ones relying on transfer functions, transfer matrices and bond graphs.

However, in the analysis of internal combustion engines (or any fluid system), a simple


1.30

and immediate representation can be obtained by dividing the system input and output

variables in two categories:

Level Variables: in general they are differential variables provided by the

fundamental equations that indicate the amount of thermodynamic

properties stored inside a component;

Flow Variables: they usually relate to the flow of a specific variable

through the control surfaces.

Examples of level variables may be mass (either total mass or mass of individual

components in a mixture), internal energy, or kinetic energy; flow variables may be mass

flow rate or enthalpy flow rate.

Likewise, when modeling any physical system there are two main classes of

objects that must be considered:

Reservoirs: these components are characterized by one or more states that

represent the ”stored” amount of level variables (state determined system);

Flow Control Devices: they determine the amount of properties that flow

through the component itself, typically as a result of differences between reservoir levels

(purely algebraic system).

Reservoirs receive flow variables as inputs and their outputs are level variables.

Conversely, flow control devices receive level variables and determine the flows

associated. Figure 1.8 briefly summarizes the concept.

Figure 1.8: Classifications of systems components and signals

The representation adopted facilitates the connection between components,

solving typical causality problems that are associated with dynamical systems modeling.

As Figure 1.9 shows, alternating reservoirs (state determined systems) to flow control

devices allows one to respect the cause and effect priorities between the input and output

signals of each block, allowing to immediately identify the driving and driven variables.


1.31

Figure 1.9: Example of connection between components

Modeling guidelines

The modeling approach described so far can be summarized with a sequence of

procedures:

1. apply the system dynamics approach to deconstruct the engine and/or

powertrain system into a series of fundamental components, therefore

determining the system boundaries, inputs and outputs);

2. identify the relevant sources of dynamics by choosing a number of

reservoirs and their corresponding “level variables”. The number of state

determined components will influence the order of the model (i.e., the

number of state equations);

3. using the fundamental equations and the fluid properties, formulate

differential equations for all the state determined components;

4. using the fundamental equations in quasi-static conditions (i.e., without

the time derivatives) and the fluid properties, formulate algebraic relations

for the flow resistance components, relating the flows between the

reservoirs as functions of the state variables;

5. as a result of the quasi-static approximations, the algebraic equations will

be characterized by a number of unknown parameters that need to be

identified from experimental data or other available information;

6. once the calibration is done, assemble the components into the overall

system model;

7. validate the complete model on a set of data points that have not been used

to identify the parameters.


1.32

1.5. TWO IMPORTANT ENGINE CONTROL PROBLEMS

This section introduces representative control strategies used in production

vehicles for air-fuel ratio (AFR) and idle speed control. These control strategies use both

look-up table based control schemes and dynamic feedback or feedforward control

schemes. The engine control unit (ECU) has limited computational capability; however,

a large memory space is available through the use of Read Only Memory (ROM). The

use of sensors is limited due to cost factors. In keeping with the relatively slow

computational speed and large memory of the ECU, current look-up table based control

strategies work well. However the calibration processes associated with such tables

require a significant amount of time and effort. The development period of a new vehicle

therefore is significantly affected by the time duration of these processes. The use of

dynamic models may be useful in reducing this development time by reducing the overall

calibration effort.

The AFR Control Problem

The AFR control problem for an internal combustion engine is not limited

to the control of AFR about stoichiometry. There are multiple objectives that need to be

achieved via AFR control. These are essentially dependent on the various engine

operating conditions, including engine load engine speed, coolant temperature,

acceleration, deceleration, etc. However, we may generally classify the AFR control

problems into two groups: open-loop control and closed-loop control. Modern ECU’s

have the capability to decide whether the control of AFR is to be closed-loop or open

loop. This is achieved by using information from various sensors installed on the engine

subsystems. These two types of strategies are discussed next.

Open-loop AFR Control

Open loop AFR control is essentially the control of the AFR at or about a desired

value through the use of (often elaborately) constructed look up tables that encompass

various engine-operating conditions. When the ignition key is turned on and the engine

cranks, a rich air-fuel mixture is required to guarantee the initiation and sustenance of


1.33

stable combustion. The amount of fuel to be injected is primarily dependent upon the

temperature of the intake manifold with the assumption that the engine motoring RPM

remains constant, as does the intake pressure at each cranking instant. Since most of the

current mass-production engines do not have an intake manifold wall temperature sensor,

engine coolant temperature information is used. This is necessary in order to compensate

for the fuel dynamics due to the wall wetting phenomenon1. Air flow rate information,

which also plays an important part in these estimations, is supplied via a mass air flow

meter or other kinds of air metering sensors. Too much fuel, however, is not always

“good” for engine cranking. If too much fuel is injected during engine cranking, it may

cause flooding of the spark plug electrode gap that will in most cases results in no spark.

Therefore, when a repeated engine-cranking situation is detected by the ECU within a

relatively short time period, it is easily implied that the previously injected liquid fuel has

not had sufficient time to evaporate. The ECU should therefore have an algorithm to limit

the amount of fuel injected such that the overall liquid fuel present in the intake system is

compatible with optimum engine performance under a cranking scenario.

At the onset of engine cranking, all the injectors start to inject fuel simultaneously

(full group injection) upon the detection of crankshaft tooth wheel signal. Normally, full

group injection is performed only once, and the amount of fuel to be injected is

dependent on the coolant temperature, and not on the air flow rate. This is due to the

inherent delay of the air-metering sensor not allowing sufficient time to both sense and to

calculate the amount of air entering the engine. During the engine cranking, the fuel

amount from the injector solenoid valve is heavily dependent on the battery voltage,

because the starter motor draws large amounts of current from the battery, causing a

temporary drop in battery terminal voltage. The battery voltage may drop to around 7

Volts in the worst case (at extreme cold engine conditions), thus decreasing the amount

of current that passes through the injector solenoid coil. The less the current at the coil

1 I.e.: the tendency of injected fuel to condense in a puddle in the inlet port, and then to evaporate

at a rate dependent on local pressure and temperature conditions. A more detailed account is given in

Module 3.


1.34

the lower the lift of injector plunger, thereby injecting less fuel. Therefore, compensation

of battery voltage drop via increasing injection pulse width at cranking phase is another

important consideration. The detection of the specific cylinder that is to be fired after the

first firing is decided upon by the combination of the crankshaft sensor signal, the

number 1 cylinder Top Dead Center (#1 TDC) sensor signal, and the engine firing order.

Since full group injection causes excessive fuel at the beginning of cranking, this results

in a high HC emission immediately after the engine cranking. To reduce the cold HC

emission, some car manufacturers are adopting a static #1 TDC sensor, typically mounted

on the camshaft, thereby avoiding full group injection. Half group injection, which

injects fuel on one bank of the cylinders, is thus a good alternate technique for firing

initiation at cranking.

In addition, the AFR is controlled to be rich at Wide-Open Throttle (WOT)

condition. Since the primary objective at WOT acceleration is to get enough engine

power for better driveability performance, AFR control remains in a rich state. When

there is an abrupt acceleration command from the driver, the engine experiences a “lean

spike” mainly due to the time lag of the air-metering sensor and ECU calculation time

delay. Thus it usually results in under fuelling during sudden accelerations. One typical

strategy to compensate the lean spike at the sudden acceleration situation is the non-

synchronous injection timing scheme. Under normal operating conditions, fuel is

injected based on predefined engine events, that is the firing order. During instances of

sudden acceleration the ECU drives additional injection through an interrupt signal. This

is non-synchronous injection. Non-synchronous injection is applied to the injector of the

next firing cylinder only on the detection of a hard acceleration scenario by the ECU.

This situation is detected through the use of the TP sensor signal. The ECU is able to

discriminate between normal acceleration rates and hard accelerations by looking at the

rates of the throttle position change. The amount of fuel to be injected in these

conditions is dependent on the engine speed, the pressure at the intake runner, and

temperature of the intake subsystem. In most of the mass-production engines, the so-

called transient fuel compensation on throttle variation is based on the above mentioned

engine speed, pressure and coolant temperature.


1.35

The injection timing of each cylinder will be optimum if injection is terminated

before the start of the intake valve opening. This provides sufficient time for the liquid at

the intake port and valve to evaporate, thereby ensuring a homogeneous mixture of air

and fuel. The homogeneity of mixture plays an important part in the reduction of exhaust

emissions over the entire engine running range.

Not only does the lean spike worsen driveability and exhaust emissions but it also

tends to induce engine knock. A rich AFR has other advantages. Traditionally the

engine-knocking problem at the beginning of a sudden acceleration was rectified mainly

by spark timing control. In most mass-production vehicles today, enrichment of air-fuel

mixture is also introduced to reduce knocking tendencies in the engine at the initiation of

a sudden acceleration. This is because a rich AFR condition improves flame propagation

speed in the combustion chamber.

On the other hand, fuel cut-off (or reduction) is necessary for deceleration

condition. The purpose of fuel cut-off is to improve fuel economy. Among the factors

which affect the fuel cut-off are: coolant temperature; engine speed; and air conditioner

compressor engagement status. The status of the compressor engagement is an important

factor in deciding the fuel cut-off. A fuel cut-off with the compressor engaged, which

can be considered an external load, could result in an unstable engine operation or in

the extreme case engine stall. Hence it becomes necessary to consider the compressor

engagement status even when looking at speed ranges for fuel cut-off. Several other

considerations come into play. For example, when the engine is below the base engine

warm-up temperature, there will be relatively large engine torque loss due to increased

engine friction as a result of high lubricant viscosity. Therefore, if the fuel-cut is

executed at low engine temperature conditions while decelerating, such undesirable states

may be achieved as discussed above.

The basic injection look-up table, which is obtained through steady state engine

dynamometer test, has two independent variables: engine speed and engine load. By

breaking up the engine speed and load into several points, various steady state test

condition-operating points are determined. These operating conditions are referred to as

break points. By running the engine at these break point conditions, the injection pulse


1.36

width for the desired AFR (stoichiometry) is thus calibrated. The fuel injection look-up

table uses units of milliseconds as the time scale. The basic fuel injection look-up table,

therefore, implicitly includes such static engine parameters as charge efficiency, residual

gas, fuel evaporation at the intake ports, injector dead time and air metering sensor’s

(pressure sensor, MAF, etc) static characteristics.

Since the basic fuel injection look-up table is constructed for standard conditions,

there are many other factors relating to ambient conditions and powertrain system that

must be taken into account as correction terms for open-loop control. These correction

groups consist of battery voltage, coolant temperature, ambient air temperature, altitude

compensation, transient compensation (throttle and engine load variations), anti-

knocking, automatic transmission compensation, etc. The corrections are added or

subtracted from the basic fuel injection look-up table, some of them are even multiplied

to calculate open-loop injection pulse width. Increasing the fuel pulse from the open-

loop value is very important for robust AFR control of an engine. The open-loop fuel

and closed-loop fuels are in a trade-off relationship with each other. That is to say that if

for a total fuel amount to be injected, if the open-loop related fuel quantity is increased,

then the closed-loop portion of the fuel has to be decreased. By reducing the portion of

fuel commanded in closed-loop, a faster and more stable AFR control can be achieved.

The ECU being used today still often consists of 16-bit microcomputers with an

increasing number of applications employing 32-bit architecture. Automotive

microcontrollers are characterized by fixed-point arithmetic, relatively slow processor

speeds, and substantial amounts of memory. For this reason, the use look-up tables for -

loop control has been historically a preferred choice for control applications. With the

introduction of 32-bit controllers, it is conceivable that in some instance dynamic control

strategies might replace some of today’s look-up tables, and that the calibration load

might be subsequently reduced.


1.37

Closed-Loop AFR Control

Closed-loop AFR control corrects the quantity of fuel injected as a function of the

signal sensed by the EGO sensor. In most gasoline engine applications, binary EGO

sensors have been used as the exhaust gas AFR sensing device. Due to the cost and

insufficient reliability of the Universal Exhaust Oxygen (UEGO) sensors, these sensors

are seldom used for the AFR control except for lean-burn engine applications.

The basic AFR control strategy that is commonly applied to mass production

vehicles is a Proportional and Integral (PI) control algorithm with gain scheduling

dependent on how long the EGO sensor signal stays at one level (lean or rich). The

closed-loop correction term oscillates as a function of the state of the EGO sensor. The

ECU employs fuel compensation by using a P-correction that intervenes when the EGO

sensor switches, and an I-correction that intervenes when the EGO sensor stays at the

same level. Therefore, the P-gain comes into play instantaneously with the EGO sensor

switching and the I-gain compensates progressively while the EGO sensor stays at one

level (lean or rich). The PI gains are not always the same; instead, these gains change to

adjust according to the engine operating conditions, and on the number of steps of I-

correction already applied. The duration of the EGO signal staying at on one level also

has an impact in gain scheduling. This variable gain concept affords a fast control of

AFR to stoichiometry in the presence of large AFR deviations.

For a well-tuned engine system with an appropriately calibrated ECU, the mean

value of PI-correction will oscillate about a zero value. However, there are various

factors that keep the mean PI-correction from being zero. These factors include the

following:

Fuel system degradation and tolerance: degradation of fuel pressure regulator, fuel

pump deterioration, injector clogging, and the use of gasoline other than that used for

calibration may result in the PI control behaving as if for a lean system. This

misinterpretation by the control scheme will result in a shift of the mean correction

value from zero.


1.38

Leakage of air into the intake system leading to an unmeasured quantity of air being

added to the engine, thus leading to a lean excursion. The effect of leakage on MAF

sensor-based systems might be more critical than on speed-density systems.

Aging and tolerance of air measurement sensors: degradation of not only the air

measurement sensors but also of such sensors as those that are used for fuel

calibration, air temperature and coolant temperature will affect the mean PI-

correction shift.

Air density change: the altitude effects on air density will affect fuelling over the

entire engine operating range.

If no closed-loop AFR control compensation is introduced, the previously listed

factors may force the engine to have a steady-state fuelling error. Therefore, a self-

adaptive AFR control algorithm is necessary to effectively control for each of the engine

specific circumstances. When the listed factors occur, the closed-loop PI-correction

strategy comes into play and compensates for the AFR deviation, thus resulting in the

mean PI-correction value to shift from zero. If the PI-correction deviation remains for a

specified time duration, the ECU may add/subtract fuel corresponding to the amount of

PI deviation from the open-loop fuel calculation sum (additive compensation). Thereby,

keeping the AFR at stoichiometry with the mean PI-correction being forced back to a

zero value. The “self-adaptive additive compensation” is especially effective when an

engine is experiencing intake air leakage or changes of injector delay time. If the shift of

mean PI-correction value is caused due to an air density change, a multiplication of scale

factors is more effective to cover the whole engine operating range (multiplicative

compensation).

The engine and ambient conditions that introduce the deviation of mean PI -

orrection have different magnitude of influence depending on the engine speed and load.

Therefore, it is necessary to distinguish the engine operating areas into several sections to

find which factors are dominant in that particular area. Figure 1.10 specifies these areas.

Intake leaks need to be compensated at low-load conditions as an additive term to the

open-loop injection map, because leakage air might be magnified at low engine load


1.39

(area 2: high vacuum condition). Injector delay, which affects each engine stroke, has to

be compensated as an additive term at high engine speed and low-load (area 3). Finally,

fuel system degradation and air density changes might effect the entire engine operation

range. So the open-loop fuel map has to be multiplied by scaling factors to compensate

across the whole engine operating range (area 1) except WOT condition where the AFR

is not controlled to stoichiometry.

Area2

Area1

Area3

WOT

Maxspeed

Min Load

MinSpeed

EngineLoad

EngineSpeed

Figure 1.10: Areas for additive and multiplicative correction.

The Idle Speed Control Problem

The objective of idle speed control is to maintain a smooth and comfortable

driving condition while minimizing fuel consumption rates. To get good fuel efficiency,

the engine idle speed should be set to a very low RPM. If we set the idle RPM too low,

however, then the idle speed control tends to become very unstable as the engine may be

producing insufficient torque at this engine RPM. Therefore in deciding the idle RPM

one has to consider an operating speed where torque production is robust enough to reject

disturbances from various sources. Moreover for drive smoothness and comfort it is

necessary to control the idle speed within a very narrow range. This would imply

maintaining the engine speed at almost a constant level. This would be a trivial task if

there were no unexpected disturbances. However, in a real life scenario there are various

kinds of disturbances present during an engine idle condition; these are:


1.40

Engagement of automatic transmission: shifting the transmission from neutral to the

drive range adds a torque disturbance to the engine through the torque converter.

The reverse procedure (drive to neutral shift) also introduces the same torque

amount as a subtractive disturbance to the engine. Moreover, the amount of this

disturbance is not always same; instead it is dependent on the temperature of the

torque converter fluid because torque transmissibility is related to the viscosity of

Automatic Transmission Fluid (ATF).

Electrical disturbance: due to the operation (on/off) of such electrical devices as

defroster, head lamp and direction lamps, the engine speed is affected. The engine

speed drop is dependent in the electrical capacity of such devices.

Direct engine torque disturbances: the power steering pump and air conditioning

compressor is good examples of direct torque intervention.

Canister purge valve on and off at engine idle: as discussed in the Chapter 1, the

opening of CPV causes unmeasured air and fuel introduction to the engine system,

thereby prompting an increase engine speed. In contrast, closing of the CPV will

induce an engine speed drop.

For engine applications with small displacement volumes, the above mentioned

disturbance effects are more dominant as compared to engines with a larger

displacement volume. This necessitates the use of more sophisticated idle speed control

strategies for small displacement volume engines. Driver comfort is a strong motivation

for controlling the idle speed to the nominal value. It is common knowledge that a

change in the nominal idle speed affects driver comfort, as discussed above. However,

it needs to be mentioned that a drop in the nominal idle speed can affect the comfort

level more drastically than a similar increase in the idle speed. Researchers in the

automotive industry have therefore been concentrating on investigating methods of

avoiding engine speed decrease during idle conditions. Ignition timing and idle air

control strategies are two methods popularly adopted for controlling idle speed.


1.41

Idle Air Control

Idle air control is often accomplished through the use of either a stepper motor or

a rotary solenoid valve actuating the air by-pass valve. This affords control of the section

area of the air passage thereby allowing a control over the amount of air entering the

engine. The airflow through the Idle Air Control Valve (IACV) can be managed through

a control over the valve flow area; this can be accomplished by varying the actuator duty

cycle. The factors affecting the calculation of duty cycle vary with engine running

conditions.

During engine cranking and warm-up phase, the IACV is opened as a function of

coolant temperature corrected for altitude. The nominal engine speed is then decided

upon by relying on the engine coolant temperature; for cold engine conditions, the

nominal engine speed is maintained high so as to make up for the high friction torque

loss. The idle RPM is then gradually reduced to the idle speed normal for a warmed up

engine. This procedure can be achieved using a two-dimensional look-up table (coolant

temperature vs. idle valve duty cycle). Altitude compensation is implemented by means

of a multiplying factor for the look-up table, thereby ensuring sufficient airflow into the

engine at high altitudes. Thus, the altitude compensation factor should be larger than

unity. Detection of altitude other than the level of calibration can be easily done using

the Manifold Absolute Pressure (MAP) sensor. When the ignition key is turned on

without the engine running, the MAP sensor senses the ambient pressure. With the

engine running, ambient pressure can be continuously updated by sensing the maximum

intake pressure for low RPM at high engine loads. This is so because the pressure

existing in the intake manifold during such a condition closely approximates ambient

pressure. At the engine warmed-up condition, the idle speed duty cycle calculation is

dependent on engine events and conditions. The calculation of idle duty cycle can be

subdivided into four main parts; these are open-loop duty cycle, vehicle event based

corrections, closed-loop, and self-adaptive compensations.


1.42

Open-loop Base IdleDuty Cycle

Battery Correction

Engine LoadCorrection

Dash Pot Correction

Closed-loop PI Correction

Self-adaptiveCorrection

Others

Bat Voltage

Duty

Load

Duty

Final Idle

DutyCommand

VehicleCondition

BasedCorrections

Figure 1.11: Final idle air bypass valve duty command calculation.

Figure 1.11 shows how these correction terms are added to calculate the final idle duty

cycle. The detection of vehicle on or off events such as air/con, cooling fan, electric load

and transmission neutral to drive shift can be done via various sensory systems of a

vehicle. Each of the above mentioned four-correction terms are calculated respectively

and added to finalize the total idle duty cycle to be output. Following is a brief discussion

of how each of the corrections are calculated and calibrated:


1.43

Leakage Air

Altitude Correction

Nominal EngineSpeed Correction

Cooling FanCorrection

Transmission Neutral Drive Shift

A/C

Others

Coolant Temp.

Duty

Air Temp.

Duty

Ambient Press.

Factor

Coolant Temp.

Duty

ATF Temp.OrCoolant Temp.

Duty

Switch

Switch

Switch

Switch

Open-LoopBase Idle

Duty Cycle

BaseTerms

Vehicle

EventRelatedTerms

Figure 1.12: Schematic of Open-loop Base Duty Cycle.

Open loop duty cycle calculation: the open-loop duty cycle is calibrated by

referring to the coolant and air temperatures, and the ambient pressure along with the

corrections associated with vehicle events that inject torque disturbances to the engine.

During the engine idle condition, the intake manifold pressure level is low (high vacuum)

hence the airflow through idle valve can be modeled as a one-dimensional choked flow.

In addition, most of the idle speed valves are designed to allow a linear relationship

between the duty cycle and airflow rate. The airflow rate is primarily dependent on

ambient pressure. Therefore, air temperature and ambient pressure are base factors that


1.44

decide the amount of air that passes through the idle valve. As is shown in Figure 1.12,

all the open-loop duty can be sub-grouped either as base terms or as vehicle events

related terms. The base terms consist of leakage air, nominal engine speed correction and

corrections for air and coolant temperatures. Air leakage defines the air that leaks past

throttle valve seating circumferentially while the throttle is closed. The leakage occurs

through an annular aperture because the valve can not complete seal the throttle body.

Selection of such valves is therefore necessarily based on the valve satisfying the

specifications on leakage limits at standard operating conditions. Air leaking past the

throttle valve will therefore add to the idle airflow, hence throttle valves with small

leakage are preferred for idle air control purposes. This allows the idle air control valve

to have a larger control span. The corrections based on air and coolant temperature

contribute to the idle duty calculation. Since the coolant temperature level determines

nominal idle RPM, this correction factor is added to compensate the nominal idle speed

differences. The vehicle event related terms are added to the base terms as a correction

factor. Turning on the A/C compressor upon the driver’s command will result in a steady

state engine speed error if the system lacks compensation. Thus increment of idle duty

cycle is required to maintain the nominal engine idle speed. The neutral to drive shift of

the transmission also introduces a similar kind of steady state error and thus needs to be

compensated. Each of the vehicle events related terms have to be decided upon by

empirical means. This is done by subjecting the engine to each of these conditions. The

calibration procedure thus requires a lot of time and effort. However, this open-loop

calibration procedure reduces the closed-loop contribution thus leading to a fast and

robust idle speed control. In most of the engine applications, the nominal idle speed

between A/C on and off are set differently. The nominal engine speed level for A/C on

condition is higher than that of A/C off condition to provide more engine torque to

compensate the disturbance from the A/C compressor.

Anticipating Disturbance Torque: One of the most effective ways of rejecting a

torque disturbance is anticipating it before it occurs. Due to the large delay between the

opening of the idle valve to the production of engine torque, an engine will experience

abrupt engine speed fluctuation whenever accessory loads disturbance are applied or


1.45

removed. Many accessory load disturbances can be anticipated to allow for air

compensation via the idle valve. For instance, if an ECU receives a command to turn on

A/C from driver, then it begins to compensate air via the idle air valve before the

compressor is actually turned on. After the torque production time delay the engine will

have developed the torque lost to the compressor related disturbance. The ECU will then

send a command to turn on the compressor without affecting the engine RPM.

Therefore, anticipating the disturbance torque is a problem that determines how much air

needs to be compensated. The time period between the start of idle valve compensation

and the instant of compressor turning on also should be calibrated. The neutral to drive

shift of the transmission can also be anticipated through the detection of transmission

lever movements. However, the amount of idle duty compensation required is based on

the temperature of the ATF. The viscosity differences, due to the various possible

temperatures of the ATF, result in different levels of torque disturbances acting on the

engine. If the ECU does not have access to the ATF temperature information, engine

coolant temperature may be used instead. Using the procedures explained above, almost

all of the torque disturbances can be effectively anticipated. As long as the vehicle’s

electrical components are under the control hierarchy of the ECU, the electrical

disturbances to the idle speed control can be anticipated too. These electrical loads

include radiator cooling fans, headlamps, defroster, etc. The anticipating control scheme

becomes more essential for small displacement volume engines.

Vehicle Events Based Corrections: There are other vehicle-running conditions,

however, that must also be considered. These consist of corrections that compensate for

battery voltage, engine load and dashpot function, etc. The idle valve is essentially a

solenoid valve or stepper motor hence the opening section area is effected by the battery

voltage. If a vehicle’s battery voltage deviates from the standard calibration condition,

then the voltage needs to be compensated. The engine load, if it is different from the

value at the time of calculation, needs to be compensated too. Different engine loads

other than those during calibration may effect the amount of air that passes through idle

valve. In addition, the load difference also effects the charge efficiency, which

eventually changes the air that enters the engine. The dashpot correction plays an


1.46

important role at the moment of throttle valve closing. When the throttle valve abruptly

closes during deceleration, the intake manifold will be in a high vacuum, presenting a

possible risk of damaging the rubber hoses and connections attached to the intake

manifold. Moreover, the Positive Crankcase Ventilation (PCV) system can leak

lubricating oil into the intake manifold. Another contribution of the dashpot function is

to reduce the “rich spike” of the AFR during deceleration. At this condition, the wall-

wetted fuel from the previous cycle continues to enter the cylinder although deceleration

fuel cut-off occurs. Thus leading to a rich AFR spike. However, by opening the idle

valve at a pre-defined mode, above listed high vacuum and AFR rich excursion problems

can be solved.

Closed-loop Idle Air Control: The closed-loop idle control strategies rely on the

conventional PI-control scheme with various gain-scheduling concepts. Each gain is

tuned according to the difference of the specific target RPM and current engine RPM.

The Crankshaft Position Sensor (CPS) usually performs the measurement of engine

RPM. As in the same case of the injection pulse width control, the P-gains intervene as

a correction for instantaneous engine RPM fluctuation while I-gains compensate stead-

state engine RPM error. For a new, well-tuned engine system, the mean value of PI-

correction is expected to be centered at zero. However, as the engine intake system ages,

deposits of dust and oil mixtures can be found on the throttle and idle valves. This

process leads to among other effects a decrease in the idle airflow rate owing to a

constriction of the air passages. Variations among engines attributable to manufacturing

processes may also force the closed-loop controller to come into play. The amount of

shift can be separately compensated through self-adaptive correction, thus returning the

mean PI correction value zero. The self-adaptive correction, therefore, provides more

room for closed-loop idle air control.


1.47

Leakage

Self-adaptation

Open-loop

Closed-loop

Idle Air

ControllableAir

Figure 1.13: Idle Air Control.

Strategies for idle valve control have been explored. The contributions of

leakage, open loop, closed-loop and self-adaptive air correction have been reviewed.

Figure 1.13 illustrates how these corrections are linked together. Elaborate calibration

work makes it possible to reject almost all disturbances that are expected for various

engine conditions. However, for unanticipated disturbances this scheme suffers in its

effectiveness due to the inherent delay of the idle speed air control strategy.

Ignition Timing Control

Ignition timing control schemes are used to effectively compensate for the

inherent delay of the idle air control method. The torque production delay associated

with the airflow dynamic is substantially larger than that for ignition. It is also known

that changes in ignition timing will affect the engine torque production instantaneously.

This characteristic can therefore be used as an idle speed control method under dynamic

engine conditions.


1.48

Ignition Timing (BTDC)[Degree]

MBT

NormalizedTorque

1

0.9

3520100

Figure 1.14: Schematic of Engine Torque vs. Ignition.

Engine torque production is dependent on ignition timing. Figure 1.14 shows the

effect of variations in ignition timing on brake torque for typical spark-ignition engine

[6]. Ignition timings can typically be varied to be advanced or retarded. An ignition

advance implies increasing the angular position of ignition initiation with respect to Top

Dead Center (TDC). Retarding the ignition timing would mean moving the initiation of

ignition closer to TDC. The engine torque increases as the ignition timing is advanced

until it reaches to the Maximum Brake Torque (MBT) point. Further advancing the

ignition timing may result in the engine knocking. Substantially retarding the ignition

timing on the other hand will cause the engine to misfire. Hence the importance placed

by automotive manufactures on efficient calibrations for optimal ignition tuning.

Calibration of MBT or optimum ignition timing at various engine loads and

speeds forms a basis for ignition control. To find the MBT point, an engine is set up on a

dynamometer. Running the engine at various predefined load and RPM breakpoints, the

ignition timing is adjusted to for MBT. Thus the MBT ignition timing provides the best

torque at each of the engine running conditions tested. Knocking points exist for all

engines; it is preferred to have knock points after the MBT point. However, hardware

design may force the ideal MBT point to lie after the knock point. Continuous knock,

however, needs to be avoided since it can seriously damage the combustion chamber


1.49

including valves and piston. Once the knocking ignition timing is found, the optimum

ignition timing should be decided by retarding a few degrees away from the knocking

point. This provides a margin of safety to allow for engine to engine production

variations. If an engine has a knock detection sensor installed, the ignition margin of

safety can be reduced, thus advancing the ignition timing by a few degrees. This will

result in the production of more engine torque. Ignition timing relies heavily on engine

RPM when the engine load is constant, the ignition timing needs to be advanced as the

engine RPM increases. This is done to allow the combustion process similar time

duration at all engine speeds. The basic ignition map for part load and full load

conditions is found by considering the previously discussed factors. This leads to the

steady-state ignition map for part load and full load.

Retarding the ignition timing from MBT or optimum torque ignition timing

should set the basic ignition map for the idle condition. Idle ignition map is necessary if

the engine were to be able to operate within a torque range. For instance, selecting the

basic ignition timing at, 10 degrees before TDC, the ECU can have room for controlling

the torque through ignition timing variation. The ECU may therefore retard or advance

the ignition timing through control algorithms to reject unknown disturbances.

Generally, the ignition timing can vary by about ±10 degrees centered at the calibrated

basic idle ignition timing. In most cases, the map of idle ignition timing relies on the

measured coolant temperature. The basic ignition timing is usually set to be advanced,

with the coolant temperature low, to provide more torque. Then as the coolant

temperature increases, the basic idle ignition timing is retarded gradually. To

accommodate different disturbances like A/C or transmission shift, several maps will be

useful for compensation.

Ignition control at idle is done mostly by proportional correction with various

gains. Since the prime aim for the use of ignition timing is to compensate dynamic

engine rpm fluctuation, integral control is seldom used. Instead the steady-state error is

corrected by idle air control. Knowing that the engine torque is linearly proportional to

the ignition advance in the vicinity of idle ignition control range, the amount of ignition

timing compensation can be calculated simply by multiplying a gain to the difference of


1.50

the filtered and the current engine RPM. In addition, since the shape of the engine torque

curve is different when the coolant temperature changes, gain scheduling is needed to

take into account the coolant temperature variation with the engine running. Therefore,

the proportional gains are determined based on the different coolant temperatures.

Although the ignition timing can reject unknown and high frequency disturbances

effectively, it is not appropriate to use the ignition timing to compensate steady-state

engine RPM deviation. Consequently, the idle ignition map provides control capability

by sacrificing engine torque.

Summary

Typical production control strategies for both AFR and idle speed control have

been explored. Most of these control strategies use event-based methods rather than

using model-based control method. Thus lots of experimental works are involved in

calibration procedure. Also, since there are too many calibration factors involved each

other like a web, changing one calibration factor may influence to the other control

performance that is difficult to predict. Although all the current production control

strategies may not be replaced by the model-based control schemes, engine subsystem

models may replace part of them; thus, providing a more simple and systematic approach

to the control problems.


1.51

1.6. REFERENCES

[1.1] J. Lowen Shearer, Bohdan T. Kulakowski, and John F. Gardner “Dynamic Modeling and Control of Engineering Systems", 2nd Ed., Prentice-Hall, 1997.

[1.2] " Matlab/Simulink Introduction", GMTEP Technical Presentation.

[1.3] G. Rizzoni, K. Srinivasan, S. Yurkovich, "Signal and System Dynamics Integration", Notes for GMTEP Technical Presentation GM 1989.

[1.4] M. Ross, R. Goodwin, R. Watkins, M. Wang, and T. Wenzel, “Real-world emissions from Model Year 1993, 2000 and 2010 passenger cars”, distributed by the American Council for an Energy-efficient Economy, USA, November 1995.

[1.5] PNGV

[1.6] H Heywood, J. R., “Internal Combustion Engine Fundamentals,” McGraw Hill Publishing Company, 1988.

[1.7] D. J. Dobner, “An engine model for dynamic engine control development,” ASME Winter annual meeting, Paper No. WA4-11:15. 1986.

[1.8] B. K. Powell and J. A. Cook, “Nonlinear low frequency phenomenological engine modeling and analysis,” Proceedings of the American Control Conference, pp.332-340, Minneapolis, MN, June, 1987.

[1.9] J. J. Moskwa and J. K Hedrick, “Modeling and validation of automotive engines for control algorithm development,” Advanced Automotive Technologies-1989, pp.237-247, ASME DSC-vol.13.

[1.10] E. Hendricks and S. C. Sorenson, “SI engine controls and mean value engine modeling,” SAE Technical Paper No. 910258, 1991.

[1.11] R. C. Turin and H. P. Geering, “Model based adaptive fuel control in an SI engine,” SAE Technical Paper No. 940374. 1994.

[1.12] A. Amstutz, N. P. Fekete and J. D. Powell, “Model-based air-fuel ratio control is SI engines with a switching type EGO sensor,” SAE Technical Paper NO. 940972, 1994.

[1.13] B. A. Ault, V. K. Jones, J. D. Powell and G. F. Franklin, “Adaptive air-fuel ratio control of a spark-ignition engine,” SAE Technical Paper No. 940373, 1994.

[1.14] P. Azzoni, D. Moro, F. Ponti and G. Rizzoni, “Engine and load torque estimation with application to electronic throttle control,” SAE Technical Paper No. 980795, 1998.

[1.15] C. F. Chang, N. P. Fekete, A. Amstutz, and J. D. Powell, “Air-fuel ratio control in spark ignition engine using estimation theory,” IEEE Transactions on Control System technology, Vol. 3, pp. 22-31. March, 1995.

[1.15] D. Cho, D. and J. K. Hedrick, “A nonlinear controller design Method for fuel-injected automotive engines,” ASME Journal of Engineering for Gas Turbines and Power, Vol. 110, pp. 313-320, July, 1988.

[1.16] J. W. Grizzle, J. A. Cook and W. P. Milam, “Improved cylinder air charge estimation for transient air fuel ratio control,” Proceedings of the American Control Conference, Baltimore, MD, June, 1994.

[1.17] J. W. Grizzle, K. L. Dobbins and J. A. Cook, “Individual cylinder air-fuel ratio control with a single EGO sensor,” IEEE Transactions on Vehicular Technology, Vol. 40, No.1, pp. 280-286, February, 1991.

[1.18] A. G. Stefanopoulou, J. A. Cook and J. W. Grizzle, “Modeling and control of a spark ignition engine with variable cam timing,” Proceedings of the American Control Conference, pp. 2576-2581, Seattle, WA, June, 1995.

[1.19] E. Hendricks, T. Vesterholm and S. C. Sorenson, “Nonlinear, closed loop, SI engine control observers,” SAE Technical Paper No. 920237, 1992.

[1.20] N. P. Fekete, U. Nester, I. Gruden and J. D. Powell, “Model-based air-fuel ratio control of a lean multi-cylinder engine,” SAE Technical Paper No. 950846, 1995.

[1.21] M. Kao and J. J. Moskwa, “Turbocharged diesel engine modeling for nonlinear engine control and state estimation,” Journal of Dynamic Systems, Measurements, and Control, Vol. 117, pp. 20-30. 1995.

[1.22] M. Kao and J. J. Moskwa, “Nonlinear diesel engine control and cylinder pressure,” Journal of Dynamic Systems, Measurements, and Control, Vol. 117, pp. 183-192. 1995.


1.52

1.7. EXAMPLES

Example 1.1: Water tank filling dynamics model

This example shows how to derive a model of a cylindrical water tank as shown

in Figure 1.15. The scope of the problem is to determine a mathematical model for the

water level in the tank, assuming the entering mass flow rate of water as an independent

variable (control variable). An exit valve is connected to the tank.

inm

outm th

F

)(ty

Figure 1.15: Cylindrical water tank system, tm mass of water in the tank, and

corresponding height th , F tank-floor area, A exit orifice area

The model of the system can be built following the general modeling guidelines

summarized in this module. In particular, the only relevant "reservoir" is the mass tm

of water in the tank (state variable), which is proportional to the water height th

(output variable). The input for the system is inm , i.e. the inflowing water mass flow,

while the out flowing water outm depends on the exit valve opening area A (control

variable).

Assuming the reservoir effects in the measuring device can be neglected (as

typically very fast), and the water temperature (and therefore its density) changes very

slowly such that it may be assumed constant, it is possible to apply the mass conservation

equation (1) to the mass of water contained in the tank:

tmtmtmdt

doutin (e-1)


1.53

where the water mass in the tank can be written as tFhtm , being F the tank area.

Assuming the incoming flow as the independent variable (control variable), the outgoing

mass flow can be determined considering the exit valve as a flow control device. For this

component, the steady-state continuity equation for an incompressible fluid (Bernoulli's

law) is applied:

tghtptptvtvAtmout ,2, (e-2)

which results in the following expression:

tghAtmout 2 (e-3)

Combining (e-1) with (e-3), the model equation for the system in Figure 1.15 is

obtained:

tghAtmF

thdt

din 2

1

(e-4)

which represents a first order nonlinear model.


1.54

Example 1.2: Compressible flow through an isentropic nozzle

This example is similar to the one discussed in the previous problem and shows

how to utilize the equation describing the isentropic flow of a compressible fluid through

a nozzle to determine the pressure in a vessel filled with gas. The initial pressure

0tp and temperature 0tTin are known, as well as the geometric data of the

vessel and the nozzle. The outlet pressure outp is assumed to be constant and equal to

ambient conditions.

Figure 1.16: Cylindrical, pressurized vessel with exit orifice

Following the general modeling guidelines summarized above, it is possible to

observe that there is no input mass flow ( 0inm ) and the state variable of the system is

the mass tm of fluid in the vessel. Applying the continuity equation to the system, the

resulting differential equation is:

tmtmdt

dout (e-5)

The mass flow rate of gas exiting the system can be calculated using the equation

of the flow through an isentropic nozzle (see [1.6]).


1.55

11

1

1

11

1

2

1

2

1

21

1

2

in

out

in

indout

in

out

in

out

in

out

in

indout

p

pif

RT

ApCm

p

pif

p

p

p

p

RT

ApCm

(e-6)

where p is the pressure inside the tank, R is the universal gas constant, T is the

temperature of the fluid in the tank, A is the area of the restriction and is the ratio of the

specific heat coefficients. In (e-6), Cd is known as discharge coefficient: this model

parameter is typically introduced to account for the real gas flow effects and requires

calibration with experimental data.

Assuming the fluid as a perfect gas and that the temperature of the fluid in the

tank does not change significantly, the ideal gas law can be used in conjunction with (e-

5) to determine the pressure in the vessel:

tmRT

Vtp

dt

dtm

dt

dout

(e-7)

Which, combined with (e-6), allows one to estimate the instantaneous pressure in the

vessel.

The expression (e-6) is nonlinear and particularly complex to model. Its

formulation can, however, be considerably simplified if the pressure ratio between the

tank and the external ambient is high (above the critical value) and the tank is sufficiently

large compared to the area of the nozzle. Under these approximations, it is possible to

assume that the nozzle operates in choked conditions. Therefore, by combining equations

(e-6) and (e-7), the differential equation that governs the dynamics of the pressure inside

the vessel can be derived as follows:

pRT

AC

V

RT

dt

dp d

1

1

1

2

(e-9)

which is a first order linear differential equation with constant coefficients.


1.56

Example 1.3: Torsional system dynamics

A simple model of an engine drivetrain is shown in Figure 1.17, where the input

to the system is the torque generated by a multicylinder engine and the desired output is

the engine angular velocity, d E

dt E . Please note that the damping term BE is

proportional to the flywheel speed, while the clutch damping is proportional to the

difference between vehicle speed, dV

dt V , and flywheel speed.

Using the method of determinants, find the transfer function between torque and

engine speed in symbolic form. Expand and group numerator and denominator to obtain

the transfer function in the form of a ratio of polynomials.

crankshaft andflywheel inertia

vehicle equivalent inertia

stiffness and damping of clutch coupling

JEJ V

KCBC

TE

EV

BE

damping due tobearing friction

at flywheel

TL

Figure 1.17: Simplified model of engine cranktrain dynamics

The equations of motion can be obtained by applying Newton’s law to the system

represented in Figure 1.17:

)()( VECEEVECEEE BBkTJ

)()( VECVECLVV BkTJ

(e-10)

(e-11)

Equations (e-10 – e-11) become

LVCCVECC

EVCCECECE

TsKsBsJsKsB

TsKsBsKsBBsJ

)()(

)()(2

2

(e-12)


1.57

or, in matrix form,

L

E

V

E

CCVCC

CCCECE

T

T

s

s

KsBsJKsB

KsBKsBBsJ

)(

)(2

2

(e-13)

E (s)

detTE BCs KC TL JV s2 BCs KC

detJEs 2 BC BE s KC BCs KC

BCs KC JV s2 BCs KC

JV s 2 BCs KC TE BCs KC TL

JEs 2 BC BE s KC JV s2 BCs KC BCs KC 2

E (s)TE (s)

TL (s)0

JV s 2 BCs KC

J Es2 BC BE s KC JVs 2 BCs KC BCs KC 2

Since we desire the transfer function from TE to E, we multiply both sides by s

to obtain

sE (s)

TE (s)TL (s )0

E (s)

TE (s)

s JVs 2 BCs KC JEs2 BC BE s KC JVs 2 BCs KC BCs KC 2

JV s3 BCs 2 KCs

JE JV s4 BCBE JV JEBC s3 JEKC BCBE BC KC JV s2KC2 BC

2 s2 2BC KCsKC2

JV s2 BCs KC

J E JV s3 BC BE JV JE BC s 2 JE KC BE BC KC JV s 2BCKC

Now we can substitute numerical values.

module 1 part 1 engine dynamics

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