1.
Semesterarbeit, Oktober 1997- April 1998, Portugal
Inhalt
Table of contents
1 Introduction
2 PID control
2.1 Demystifying PID control
2.2 The different components of PID
2.3 How to change the parameters KP, KI and
KD
2.4 Choice of the Parameters KP, KI and
KD with the Ziegler-Nichols technics
3 Fuzzy control
3.1 Motivations for the Fuzzy control
3.2 Proceeding of a human being
3.3 Simulated human behaviour with Fuzzy Logic
4 How to apply Fuzzy Logic to PID control
5 Equations of movement of aircraft
5.1 General Equations
5.2 Simplified Equations in the X-Z plane
6 Simulating the plane / Controlling the plane
6.1 Creating a simulator with Matlab
6.1.1 Non-linear Simulator Equaru3 with constant a
6.1.2 Non-linear Simulator Zutrep, a not
constant
6.1.3 Linear Simulator Charrun
6.1.4 Comparison between the simulators and conclusion
6.2 Creating a PID controller and Simulation
6.2.1 Equaru3 with PID control, a is the
control variable
6.2.2 Zutrep with PID control, h is the
control variable
6.2.3 Charrun with PID control
6.2.4 Conclusion
7 Simulating the climb
7.1 Usual climb definitions
7.2 Simulated climb
8 Application to Sagres with conventional PID control
9 Application with Fuzzy PID control
9.1 The way it works
9.2 Simulation
9.3 Comparison with normal PID control
10 Conclusion and discussion
Appendix I: Bibliography
Appendix II: Matlabâ
programs
Equaru3
Eqmov3
Equdatd2
Zutrep
Zutreda
Statgl
Charrun
Charfun
Sagruf2
Sagrfp2
Stasagf
Sagrdatf
Sagrdatc
Ms_test4
Rg
Gr
Appendix III: Graphs
1. Introduction
The object of this project is to study how Fuzzy Logic can be used to
optimise PID automatic aircraft control during the climb flight.
However, an important part of the project is to program simulation
software for longitudinal aircraft motion. This simulator is then used to see
if the built controller works and if it has some advantages over other
controllers.
Building such a software is necessary because even after long search
there could not be found already existing software that is easy to understand
and to adapt to control problems.
The software that will be programmed should be easy to understand, to
handle and to work with. The interfaces between controller and simulator and
between simulator and aircraft data should be well explained in order to make
it possible for other people to use the created software for their own control
problems.
That is the reason why the software programmed will be included in the
project in form of 3,25" floppy disks.
For the control it is intended to use a normal
Proportional-Integral-Derivative-controller (= PID controller) as basic and to
improve its performance by means of Fuzzy Logic. Fuzzy control has been
motivated by the needs of controlling incompletely known dynamic systems as for
example aircraft.
Ordinary PID-controllers operate with constant gains KP, KI and KD, even if the
environment of the system to be controlled changes, that means for example if
the reference values change.
Improved PID-controllers change the gains depending on each situation.
Up to date, these changes are made by means of gain scheduling (see fig.2.4).
The disadvantage of the gain scheduling is that the variation of the gains is
not smooth, but discontinuously. Gain scheduling is complicated due to the
functions that have to be made in order to determine the gains.
By means of Fuzzy Logic, the gains of the controller could be changed
continuously, i.e. softly, depending on the flight condition (climb, cruise,
approach) for example.
Some of the disadvantages of PID-controllers as overshoot, large rise
time and oscillations may be improved, thus providing better control. Fuzzy
Logic is easy to understand and to program and provides good control results.
Later we will see why.
First, I describe the function of a pure PID, then I try to improve it
by using Fuzzy means.
Context of work
The project work was made in the "Universidade da Beira Interior" (UBI),
in Covilhã (mid Portugal) within the scope of the Erasmus program. I
have studied in Covilhã from the end of October 1997 till the end of
April 1998.
Although good work environment (supervising by Prof. K. Bousson and
Prof. O. Wagner, place in an office, own computer) it was not easy to fulfil
the aims of the project.
First of all it was a language problem. The language of the project
report is English as usual for Erasmus projects, but the work has been done in
Portuguese, French and German. With Prof. K. Bousson I spoke French due to the
facts that he is from France and I speak quite well the French language. The
classes of the subject "Automação e Controlo II" which I
frequented in order to improve my knowledge of control technics were held in
Portuguese, which is quite normal for a lecture in a Portuguese university.
With the colleagues in the office, in the secretariat and in the laboratory of
the department, whom I consulted to solve problems of flight mechanics and
computers and who gave me information about the ultra light aircraft SAGRES I
had to speak Portuguese as well, which was not easy at the beginning. My native
language is German, for that reason I also worked with German literature and
spoke German with my supervising Professor in Munich, Prof. O. Wagner.
So the project has been made in four languages which was interesting,
sometimes funny and certainly helpful for my knowledge in at least three
languages. But this was not very helpful for the project.
The second problem was the subject. While there is an enormous number of
projects which can be chosen in the TUM, I had only the choice between two
projects proposed by Prof. Bousson. This is due to the fact that I had to fix
the subject before I left Munich, and from Munich it was not possible to get
information about more subjects proposed at the UBI.
That is why I had to choose a subject that I wouldn’t have chosen
under normal conditions, because I prefer to do more practical work as
experimental or constructive projects.
So I first had to learn the principles of PID control, Fuzzy Logic and
the Matlab program before I could begin with the main work.
Finally the work became as interesting as the life in Covilhã has
been from the beginning.
In spite of long days in the university I appreciated my stay in
Portugal and the different way of life of the friendly people.
The present report is not only a report on results, but also a report on
the work that has been done to reach the aims, even if not all the intermediate
steps were important for the final results.
Acknowledgements
I would like to thank the following people for their help during my
project work:
Professor Kouamana Bousson, Professor Otto Wagner, Sofia Fonseca,
Ricardo Gaito das Neves, Carlos, Egas, Mario, Pedro Gamboa, Miguel da Silva,
Ten. Cor. Rui Carvalho, Cmt. José Monteiro, Prof. Eduardo Bauzer,
Gislinde Binder.
I also thank all the other people not mentioned for their
help.
Hauptteil
(...)
Zusammenfassung
10 Conclusion and
discussion
The main aims of the project have been reached. An aircraft simulator
for longitudinal motion simulation has been programmed with Matlab, using the
simple but effective Euler-method for the calculation of the differential
equations. A PID controller has been programmed and tested with the different
simulators. A climb program has been made to simulate the climb of the
ultra-light aircraft SAGRES. The created PID controller has been improved by
varying the gains KP and KD with Fuzzy Logic. Both the
normal and the improved PID controller have been tested on the climb program,
and I was able to prove that the Fuzzy Gains controller has some advantages on
the ordinary PID controller with constant gains.
However it was not so simple to reach the aims of the project.
Programming the simulation program was not considered to be the most important
part of the project. But finally this part took the most of the time, due to
the knowledge in flight mechanics and programming that had to be refreshed or
acquired first.
For example, three different programs had to be made in chapter 6.1
before the best simulating method could be found. The programs use different
simplifications: The non-linear simulator Equaru3 works with constant
a , a simplification which is only allowed for
regular phugoid movements where a keeps nearly
constant. The simulator Charrun works with linear equations and is
therefor very fast, but less realistic. Finally, the simulator used for the
further work is the program Zutrep, which works with the non-linear
equations and has a non-constant a . This is the
simulator on which the main program of the project, Sagruf, is based.
Should all the existing PID controllers now be replaced by the Fuzzy
Gains controller that I created?
No. There are very good PID controllers, which work with gains that are
changed by gain scheduling (see fig. 2.4).
The proposed controller changes the gains using Fuzzy Logic. This is a
different solution for the problem of the gains, that for effective control
should be adapted to the different flight conditions.
According to the precision of the control required, the engineer who
implements the flight controller can decide to take a normal PID controller or
a Fuzzy Gains PID controller, whereas in the latter case it has not yet been
shown how this controller can be implemented in the hardware of the flight
computer.
Outlook:
Unfortunately programming the flight simulator took so long, that it was
not possible to try even more improvements of the PID controller. The created
Fuzzy Gains controller is quite good, but further improvements are imaginable:
one big advantage of Fuzzy controllers is the possibility to calculate an exact
system output with more than one system input, without the necessity to invent
complicated mathematical models.
My Fuzzy Gains controller only works with one system input, the actual
error, that means the difference between the actual value of the controlled
variable and the reference value. More system inputs are thinkable: The
changerate of the error, the flap position, the engine thrust and the actual
height. All these inputs could influence the fuzzyfication of the PID gains and
would improve the automatic control of the plane in a wide range of flight, not
only for the climb.
An completely other idea would be to control the aircraft with a normal
Fuzzy controller without the PID part. Programming and calculating time could
so be reduced, the results could be quite encouraging.
Personally I am quite satisfied with the results obtained in the
project. The implemented Fuzzy Gains controller is working better than a normal
PID controller with constant gains. The program Sagrfp includes the
simulation and the control part for both controllers. With this software, it is
possible to simulate the longitudinal movement of different aircraft and to
implement different flight controllers. So I would be happy if other students
could profit from this software to make own simulations of other kinds of
flight controllers.
Note that these results where hard work. During six months here in
Portugal I did not have other work to do but the project. About 900 man-hours
have been invested in this project. However they are not wasted: In addition to
the results obtained, I improved my knowledge in flight mechanics, programming,
PID control and Fuzzy Logic. And last but not least I improved my language
skills and widened my personal horizon. Thanks to all the people that made this
possible.
Michael Binder, Covilhã, 29.04.1998
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Semesterarbeit: Konzeption und Konstruktion der Außenhaut und
Dimensionierung der Klimaversorgung für eine
Unterflur-On-Board-Meßanlage eines Airbus A340-600/500
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