ASPECTS REFERRING TO THE INTERNAL COMBUSTION ENGINE’S FUNCTIONALITY

7/30/2019 ASPECTS REFERRING TO THE INTERNAL COMBUSTION ENGINES FUNCTIONALITY

1/10

ASPECTS REFERRING TO THE

INTERNAL COMBUSTION ENGINES FUNCTIONALITY

Univ. Prof. eng. Ion COPAE PhDMilitary Technical Academy, Bucharest email: [email protected]

Abstract

Certain procedures and methodologies will be presented in this paper, which can be

used in studying the internal combustion engine. These procedures are useful to identify,

manage and diagnose the behaviour of the internal combustion engine.

So we are going to use the Artificial Intelligence (AI) algorithms such as: neural

network, fuzzy logic, neuro-fuzzy algorithms, and vector support machine. Also we are going

to highlight the importance of the entropic, complex theories and their specific procedures

that find their use in studying the internal combustion engine. We are also going to explain

how the extremal and symbolic analysis and multivariable statistics analysis will find their

way into better understanding the processes of the internal combustion engine.

We are going to present procedures that investigate the internal combustion engine

using fractals and fractal calculus, time analysis, frequency analysis, and also time-frequency

analysis. We will use in the paper elements of time-scale calculus, vector calculus, matrix

calculus and also elements of tensor calculus. All these methods are used in this paper to

investigate the engines dynamics and the fuel saving methods available for the todays

engines. Also the immense possibilities offered by MatLab software will be presented. This

software is used to investigate the engines functionality based on the three inseparablecomponents: identifying controlling diagnosing the engine.

Keywords

car engine, information theory, multivariate statistics, tensor calculus, extreme value

analysis, on-board computer

Since the new technologies were implemented into the construction of the

automobiles, the nowadays engines have become in their own turn, complex systems. In

order to study these engines you need to turn to new algorithms and methods. The ECU

(Engine Control Unit) installed on-board offers important information which can be read and

stored and afterwards used to study the internal combustion engines processes [4].
mailto:[email protected]:[email protected]


2/10

The automobiles engines were designed, at the beginning, based only on theoretical

knowledge available at that time. We had limited practical data to use. The technical

approaches in studying classical or electronically controlled engine were conditioned by the

technical level available. Most improvements were made only in theoretical certain areas that

relate to the engines processes. These improvements paved the way towards innovating and

powerful new tools, that permit us to communicate with the cars ECU and even control theway it functions. We can now make important breakthroughs in both theoretical and practical

areas. Turning to new methodologies in studying the internal combustion engine we will also

turn to knowledge which is tackled by many other disciplines.

Studying the internal combustion engine has become much more systemic and also it

has reached a very high scientific level. In order to enhance the automobiles performances or

to reduce the vehicles emissions, the engines fast dynamic processes were studied in much

more detail.

Today, studying the internal combustion engine involves the following:

the studies are based on practical data and lead to relevant results; starting from the

experimental data we are establishing mathematical models that describe the internal

processes which, in their turn, are used as base for theoretical approaches that explain thephenomena in detail. Of course the end purpose is to enhance the vehicles overall

performance.

the engines functionality is studied in an interdisciplinary manner. There are

concepts, procedures and algorithms specific for different disciplines such as mathematics,

system and signal theories, functional analysis, electronic control, mathematical statistics,

computer programming, information theory etc.

the engines functionality is also studied in a systemic way, considering the terrains

and drivers influences, using concepts and algorithms from systemic and electronic control

theories. The mathematical patterns that describe the processes allow us to set the values for

maximum performance (described by the engine when it functions at maximum load) but

most important it allows us to set the values for the partial regimes, which are most common

during the engines exploitation.

tackle issues that never before were discussed in the engines usual area of

discussions. These problems were mainly discussed when investigating different systems

dynamic and they refer to: variance analysis, sensibility analysis, correlation analysis,

coherent analysis, bispectral frequency analysis, robust analysis, spectral dynamic,

regressions, neuronal networks, genetic algorithms, fuzzy assemblage, neuro-fuzzy

algorithms, bootstrap techniques, Bayesian statistics, fractals and fractional calculus,

extremal analysis, symbolic analysis, vectorial analysis, matrix and tensor analysis etc.

apply all three techniques of data processing: frequency analysis, time analysis and

time-frequency analysis. On each case the processing algorithms diversified and becamemore and more accurate. There are very many ways from where we can chose an

investigating procedure that allows us to understand and explain the details of the

phenomena.

use high scientific level algorithms and analysis procedures revealed by new

mathematical approaches of systems and signal theories. The fast improvement in numeric

calculus and computers also raises the technical scientific level. This level is also completed

by methods of investigation from AI and genetics areas.

to study dynamics and engines fuel consumption we can use new procedures and

algorithms that allow us to make these studies at the same time.

it uses a unitary approach of the engines functionality involving three inseparable

elements: identifying (establishing the mathematical patterns based on experimental data),controlling and diagnosis.


3/10

Certain examples will be presented in the following pages, that best explain the

possibilities offered by todays theoretical and experimental internal combustion engine

study. So, from the vectorial calculus we know that the scalar product of two vectors is given

by the relation:

cosu v u v

(1)

where is the smallest angle between vectors ( 0 ); if we know the two vectors then,resulting from the anterior relation, we can determine the angle between these two.

cos ; 1 cos 1u v

u v

(2)

We can notice that the cosine value of the scalar product is situated in the same area

as the correlation coefficient ( 1 1 ); for this reason the scalar product is frequentlyused to evaluate the indirect measurement of the linear correlation between vectors. To this

purpose, fig.1 shows the values of the scalar product between two vectors, described in thefigure, specific for Tacuma car. The vectors were obtained following a 50 experimental data

recording tests.

Fig. 1. Vector calculus

The graphs from fig.1 clearly show, among other things, the existence of some

unlinear correlations between certain values (best revealed by the differences obtained in the

7th and 9th tests). This has implications on setting the mathematical patterns that best describe

the internal combustion engine processes.

Information Theory is also applied to study the engines functionality; information

represents the fundamental concept in prediction and it is characterised by a probability


4/10

distribution p [2;6]. This way, Hartley has defined the information contained in n events

ix X with the help of the following relation:

2( ) log ( )i iI x p x (3)

As we can see from this relation, if ( ) 1p x , than ( ) 0I x ; this means that theevent that occurs once contains no information. In order to characterise the uncertainty

occurred during an event we can use the entropy concept: Shannon himself, the man who first

introduced this concept, used the term of uncertainty. The entropy represents the product

between probability and information throughout the all the n events ix X :

1

( ) ( ) ( )n

i i

i

H X p x I

x

i

(4)

or considering relation (3):

2

1

( ) ( )log ( )n

i

i

H X p x p

x(5)

As mentioned before, the entropy represents the measurement of uncertainty; in

thermodynamics the entropy defines a systems disorder. We deduct from here that when the

entropy has a higher value, the uncertainty as well has a higher value, and this way the

prediction is situated at a lower value. Observing a systems evolution in time, we can say

that its entropy is maximum when the system is at a static level. So, a systems dynamics

described by a reduced entropy, ensures a better prediction then its static state.

Lets take two variables Xand Y that have a joint probability density p(x,y). In this

case, the joint entropy of the two variables (also called comentropy) is calculated by a similarrelation with (5):

2( , ) ( , ) log ( , )x y

H X Y p x y p x y (6)

Another useful concept is mutual information, described by the following relation:

( ; ) ( ) ( ) ( , )I X Y H X H Y H X Y (7)

It represents a measurement of how much does the uncertainty of variable X is

reduced if we know variable Y. This is a concept that allows us to measure how does theprediction level increases by evaluating how the uncertainty level decreases. The higher

values for mutual information, the uncertainty level is lower thus the prediction level is

higher.

For exemplification in fig.2 are presented the entropy H values and the mutual

informationIxy for the observed parameters mentioned in the 50 tests which were carried out

on a Tacuma car. We can see, from fig.2a, that the engine speed n has the highest entropy.

The throttles position has the lowest entropy. This always happened for all 50

experimental tests. So we can say that the first parameter offers a better prediction of

outcome results if we use a self regressive mathematical model (using only the engines

speed and different regressors).

Fig.2b illustrates engine torque and engine speed parameters, which explains themutual information concept. We can see that the mutual information offered by the pair of


5/10

parameters which were mentioned earlier has higher values than the mutual information

between throttles position and engine torque. If we establish two mathematical models for

engines functionality likeMe=f(n) andMe=f(), we can say that the first model will have a

better prediction over the outcome of the engines torque than the second model.

Fig. 2. Values of entropy and mutual information

So if we want to establish mathematical models that best describe the engines

functionality and at the same time offer a high degree of precision, we need to choose those

parameters that have the highest mutual information. These parameters are called relevant

parameters.

As an example, lets establish a mathematical model that will give us the horary fuel

consumption Ch (as a resulting/dependeble parameter) taking into account two factorial

parameters (independent parameters); the factorial parameters are spark timing ; inlet air

pressurepa, injection duration ti, engine speed n and the throttles position . Fig.3 shows in

intersections the parameters name and also entropys value and on the arches we can reed

the mutual informationIxy.As we can see from fig.3, the first two relevant parameters are engine speed (mutual

information is 2,3547 bits) and throttles position (mutual information is 1,3152 bits); so we

can now say that the highest precision is assured by Ch=f(n, ) mathematical model.

Also we can see from the same fig.3 that if we wish to use a mathematical model

based on three independent parameters we need to turn to the Ch=f(n, , pa) model because

the mutual information between inlet air pressure and horary fuel consumption is 1,2249 bits.

Finally, from the same figure we can see that the highest mutual information value is shared

by the spark timing and the engine speed (3.9093 bits) and the lowest mutual information is

shared by injection duration and spark timing (0,82094 bits).

Accordingly to what weve established, the highest precision is ensured by a

mathematical model which gives us the horary fuel consumption and takes into account theengines speed and throttles position.


6/10

Fig. 3. Graph with values of entropy and mutual information

Indeed, from fig.4a we can see that the description offered by the mathematical value

is almost identical when compared with the actual data collected from the ECU.

Fig. 4. Generalized nonlinear mathematical models


7/10

As we can see from fig.4a the mathematical model is:

5 2 7 20,156 0,0889 0,00012 3,11 10 3,6 10hC n n (8)

This model is valid for all 50 tests weve had available (obviously nonlinear, we can

say that this is a generalised mathematical model).In fig.4b, there was a mathematical model, as a comparison, that took into account the

throttles position and the inlet air pressure; as expected the mathematical model gives an

error of 10,4% when compared to the real values collected from ECU. This error is an

unacceptable value.

We can study the functionality of the internal combustion engine using concepts from

multivariable statistics. Multivariable statistics, also known as spatial statistics or

geostatistics, is a part of statistics that deals with large sets of data that vary during time

classic statistics - but also in space this is where the two last concepts were introduced. The

first concept, used more frequently, comes from the fact that it deals with different type of

data or multiple parameters [3;7].

Multivariable statistics began being applied in the automotive industry when the

electronic control was first introduced. This was almost a necessity because the on-board

computer had to operate with large sets of data collected from all sensors installed by the

manufacturer. Multivariable statistics turns to multiple correlation and regressions, cluster

(group) analysis, discriminant analysis, principal components and factor analysis and it uses a

classification and establishes datas pattern [1;5;8].

Te emphasise this last statement, we can see in fig.5 how the classification algorithms

and datas pattern establishment is used when studying the internal combustion engine; the

graphic shows how some inlet pressure losses in the case of a Cielo Executive engine, divides

into classes. It reveals that, specific inlet pressure losses can be established by classification

and by data pattern recognition; in other words we can recognize and isolate different enginemalfunctions, in this case the range of engines speed (intentionally provoked pressure losses

that lead to engines speed alteration).

Fig. 5. Data pattern and classification


8/10

Tensor analysis also lets us emphasize certain aspects of engines functionality. This

analysis (also known as multimodal analysis) is an extension from matrix analysis which in

its turn is an extension of vector analysis. In fact their names come from the fact that vector

analysis deals with vectors, matrix analysis deals with matrix and tensor analysis deals with

tensors [9;10;11].

As an example, the dynamic study is shown in fig.6 which took into account theposition of the air dumper plates position, the engines speed and its power. The parameters

were measured during 10 tryouts which were carried out on a Tacuma vehicle. The criteria

for analysis is the 2 norm, marked as L2 (because the 2 norm square it represents the energy).

In fig.6a we have the results from tensor analysis (all parameters were taken into account at

the same time) and in fig.6b we have the results from vector analysis (the parameters were

taken into account separately). From these two representations we can see that tensor analysis

gives us a different classification from vector analysis. The most suitable classification is that

of tensors analysis (T6, T5, T8, T1, T2 etc.), because it is analysing the systems behaviour

considering the variation of all three parameters at same time. This fact isnt the case for

vector analysis.

Fig. 6. Dynamic study

In many practical situations, the average values for the engines parameters have less

importance then their extreme values. So we are more interested in the engines maximum

power, maximum fuel consumption or maximum noise it produces etc. In other words we are

much more interested in the maximum and minimum values that our investigated parameters

will show.

Classic statistics can not be applied in the extreme cases we have mentioned because

probability theory relies on operating with statistical averageand this is the reason we need to

turn to extreme value analysis. This theory is based not on the usual classic distribution but

on the probability asymptotic distribution. Because the extreme values are particular cases,

the theory relies on very rare events. The German mathematician Gumbel set the basis forthis theory.


9/10

To exemplify, in the fig.7 are presented the experimental values and the extreme

values for a Tacuma engines level of noise.

Fig.8 shows a detail with all three anterior sets of data.

Fig. 7. Experimental data and extreme values of the noise level

Fig. 8. Experimental data and extreme values of the noise level, detail


10/10

Extreme value theory relies on the generalised distribution defined by probability

density, for a certain parameterx:

1 1/ 1/1

( , , , ) 1 exp 1x x

f x

(9)

where is the positioning parameter, is the scale parameter and is the shape

parameter.

To study the engine today means that we can turn to different processes and

algorithms that are useful in dynamic systems and signal theories. The scientific levels are

very high and it allows us to better understand the intimacy of the fast dynamic processes that

occur inside the internal combustion engine.

References

1. Balakrishnama S. Linear Discriminant Analysis. Institute for Signal and

Information Processing, Mississippi State University, 19952. Blahut R. Principles and Practice of Information Theory. Addison-Wesley,

Cambridge MA, 1988

3. Clarke B. Analysis of Multivariate Time Series Data. Colorado State University,

2003

4. Copae I., Lespezeanu I., Cazacu C. Dinamica autovehiculelor. Editura ERICOM,

Bucureti, 2006

5. Duda R. Pattern Classification, Wiley-Interscience, New York, 2001

6. Gray R.Entropy and information theory. Stanford University, New York, 2007

7. Hytyniemi H. Multivariate Statistical Methods in Systems Engineering. Report

112, Helsinki University of Technology, 1998

8. Jain A. Statistical Pattern Recognition. Departament of Computer Science and

Engineering, Michigan State University, 1999

9. Martin Carla D. The Rank of a Tensor. James Madison University, 2006

10. Moravitz Martin Carla D. Tensor Decomposition Workshop Discussion Notes

American Institute of Mathematics (AIM) Palo Alto, CA. Cornell University, 2004

11. Smirnov A.V.Introduction to Tensor Calculus. McGraw-Hill, New York, 2004

12. Welling M. Extreme Components Analysis. Department of Computer Science,

University of Toronto, 2002

ASPECTS REFERRING TO THE INTERNAL COMBUSTION ENGINE’S FUNCTIONALITY

Documents