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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 1.1
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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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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 COURSE ORGANIZATION ORGANIZATION ........................................................... .......... 5 Objective ............................................... ................................................... ....... 5 Assignments Assignments and examinations examinations .......................................... ............................. 5 What do we mean by “System Dynamics”? Dynamics”? ................................................... 6 What this course is not ............................................... ..................................... 6 Structure of the course ............................................ ........................................ 6 Required background ............................................... ....................................... 7 1.2.
BACKGROUND BACKGROUND AND MOTIVATION MOTIVATION ................................................ 8
Fuel efficiency improvement improvement .......................................................................... 8 Emission Emission standards ...................................................................... ................... 9 Emission Emission controls for spark-ignition spark-ignition engines ............................................ ... 12 The importance of engine control systems .......................................... ......... 14 1.3. INTRODUCTION TO POWERTRAIN DYN DYNAMICS AMICS ........... ...... ........... ............ ............ ...... 16 1.4. MODELING MODELING FOR POWERTRAIN POWERTRAIN CONTROL CONTROL ..................................... 20 Fundamental Fundamental equations for modeling fluid systems dynamics dynamics ..................... 25 Model coupling techniques ................................................ ........................... 29 Modeling guidelines guidelines ................................................................ ..................... 31 1.5.
TWO IMPORTANT ENGINE CONTROL PROBLEMS ............ ...... ........... ....... .. 32
The AFR Control Problem............................................ ................................ 32 Open-loop Open-loop AFR Control ............................................................................... .......................................... ..................................... 32 Closed-Loop Closed-Loop AFR Control ................................................ ............................ 37 1.3
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The Idle Speed Control Problem .................................................................. 39 Idle Air Control................................................. ............................................ 41 Ignition Timing Control .......................................... ...................................... 47 Summary Summary ....................................................... ....................................................................................................... ................................................ 50 1.6. REFERENCES REFERENCES ...................................................... ......................................................................................... ................................... 51 51 1.7. EXAMPLES EXAMPLES .................................................... ............................................................................................ ........................................ 52 Example Example 1.1: Water tank filling dynamics model ........................................ 52 Example 1.2: Compressible Compressible flow through an isentropic nozzle ................... 54 Example Example 1.3: Torsional system dynamics 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 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 1.6
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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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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, lightduty 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, NO X, 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 NO X, 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 CO 2 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 1.14
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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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Fuel Injector fuel in Fuel Dynamics
Tload
Vehicle
fuel in cyl. air in
air in cyl. Intake Combustion Manifold
Throttle Body
engine speed
Tind
A/F exh
Crankshaft
Transmission
EGO Sensor
Spark Controller
Vo 2
Figure 1.5: More detailed view of powertrain dynamics
Open-loop fuel control
fuel Canister purge from ECM
Closed-loop fuel control from ECM
Fuel Inject ion
can Fuel Dynamics (Well Wetting)
Canister
Drive Train Vehicle Load
Fuel
Intake Manifold
Throttle Body
Combustion Air
sp
from ECM
from ECM Electronic Throttle Control
Exhaust Manifold
EGR from ECM
Inertia
engine torque
engine speed
Damping/Friction Ignition Control
EGR HEGO
EGR Control
Catalytic Converter Exhaust
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
2.
Fuel Injector
Fast Dynamics: 200 -400 Hz Slow Dynamics: 1 - 2 Hz Time constant: 0.5 - 3 ms
3.
Fuel Dynamics
4.
Combustion
Evaporation: 0.5 s Mixing: 1 - 10 ms 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 highorder 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 cycle1.23
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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 1.24
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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.
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a) Finite control volume fixed in space
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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: 1.26
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• 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:
dm dt
m
in i
i
m out j
(1)
j
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:
d met dt
m in hi m out h j Q W i
j
i
(2)
j
where et is the total energy per unit of mass comprising the thermodynamic internal energy (u), as well as the kinetic (
1 2
2 c ) and potential energy ( ). The latter includes all
the forms of potential energy associated to conservative fields, such as gravitational or electrical.
et u
1 2
c 2
(3)
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Moreover , h represents the enthalpy of the flow, Q is the total heat exchanged through the is the mechanical power generated by the system. surface of the system and W
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:
d M dt
M
in
M
out
F i
(4)
i
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:
f p, v, T 0
(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 v const . Moreover, for most of the applications involving liquids, constant temperature
is an acceptable assumption.
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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:
pV mRT
(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:
u u T c v T
(7)
h hT c p T
where cv is the specific heat at constant volume and c p 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
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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. 1.30
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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.
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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 1.32
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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 1
for the fuel dynamics due to the wall wetting phenomenon . 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.
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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 nonsynchronous 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 socalled transient fuel compensation on throttle variation is based on the above mentioned engine speed, pressure and coolant temperature.
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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 the extreme case
in
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 1.35
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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), antiknocking, 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 openloop 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.
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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 Icorrection 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.
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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 PIcorrection 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 selfadaptive 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
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(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. WOT Engine Load
Min Speed
Area 2
Area 1 Area 3
Max speed
Min Load
Engine Speed
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:
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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.
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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.
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Open-loop Base Idle Duty Cycle
Duty Battery Correction Vehicle Condition Based Corrections
Bat Voltage Engine Load Correction
Duty
Load
Dash Pot Correction
Final Idle Duty Command
Closed-loop PI Correction
Self-adaptive Correction
Others
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:
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Leakage Air
Duty
Coolant Temp. Base Terms
Duty
Air Temp. Factor Altitude Correction Ambient Press. Nominal Engine Speed Correction
Duty
Coolant Temp.
Cooling Fan Correction
Transmission Neutral Drive Shift Vehicle
Switch
Duty
ATF Temp. Or Coolant Temp.
Event Related Terms
Open-Loop Base Idle Duty Cycle
Switch
A/C Switch Others
Switch
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 1.43
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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
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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
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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 wallwetted 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 steadstate engine RPM error. For a new, well-tuned engine system, the mean value of PIcorrection 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.
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Idle Air
Closed-loop
Open-loop
Controllable Air
Self-adaptation Leakage
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.
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Normalized Torque
MBT
1
0.9
0
10
20
35 Ignition Timing (BTDC) [Degree]
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 1.48
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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
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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.
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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 multicylinder 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.
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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. in m
ht
F
out m
y (t ) Figure 1.15: Cylindrical water tank system, mt mass of water in the tank, and corresponding height ht , 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 mt of water in the tank (state variable), which is proportional to the water height
ht
in , i.e. the inflowing water mass flow, (output variable). The input for the system is m
out depends on the exit valve opening area A (control while the out flowing water m
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: d dt
in t m out t mt m
(e-1)
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where the water mass in the tank can be written as mt Fh t , 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: out t A vt , vt m
2 pt ,
pt ght
(e-2)
which results in the following expression: out t A 2 ght m
(e-3)
Combining (e-1) with (e-3), the model equation for the system in Figure 1.15 is obtained: d dt
ht
1 F
m
in
t A
2 ght
which represents a first order nonlinear model.
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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
p t 0 and temperature T in t 0 are known, as well as the geometric data of the vessel and the nozzle. The outlet pressure pout 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 in observe that there is no input mass flow ( m
0 ) and the state variable of the system is
the mass mt of fluid in the vessel. Applying the continuity equation to the system, the resulting differential equation is: d dt
out t mt m
(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]).
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C Ap m out d in RT in m out
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p out p in
1 p out 2 1 p 1 in 1
C d Ap in RT in
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2 1 1
if
p out 2 1 p 1 in
if
p out 2 1 1 p in
(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), C d 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 (e5) to determine the pressure in the vessel: d dt
mt
d p t V
dt RT
m out t
(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:
dp RT C d A dt V RT
2 1 p 1 1
which is a first order linear differential equation with constant coefficients.
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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,
d V 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. vehicle crankshaft and equivalent inertia flywheel inertia stiffness and damping of clutch coupling TE
E JE
BC
K
C
B
V
TL
JV
E damping due to bearing friction at flywheel
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: T k ( ) B ) J E E E C E V E E BC ( E V
(e-10)
T L k C ( E V ) BC ( E V )
(e-11)
J V V
Equations (e-10 – e-11) become
BC B E s K C E ( s ) BC s K C V ( s) T E BC s K C E ( s) J V s 2 BC s K C V ( s ) T L J E s
2
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(e-12)
