Klausurunterlagen - uni-due.de...Page 3 Attention: Give your answers to ALL problems directly below the questions in the exam question sheet. You are NOT allowed to use a pencil and

UNIVERSITÄT DUISBURG-ESSEN Fakultät für Ingenieurwissenschaften, Abt. Maschinenbau und Verfahrenstechnik Lehrstuhl Steuerung, Regelung und Systemdynamik System Dynamics February 11th , 2011

60 Minutes Page 1

Reading-up-time

For reviewing purposes of the problem statements, there is a "reading-up-time" of 10 minutes prior to the official examination time. During this period it is not allowed to start solving the problems. This means explicitly that during the entire "reading-up-time" no writing utensils, e.g. pen, pencil, etc. at all are allowed to be kept on the table. Furthermore the use of carried documents, e.g. books,· (electronic) translater, ·(electronic) dictionaries, etc. is strictly forbidden. When the supervisor refers to the end of the "reading-up-time" and thus the beginning of the official examination time, you are allowed to take your utensils and documents. Please then, begin with filling in the complete information on the titlepage and on page 3.

Good Luck!

LAST NAME

FIRST NAME

MATRIKEL-NO.

TABLE-NO.

Klausurunterlagen'~-~ )

Ich versichere hiermit, dass ich sämtliche für die Durchführung der Klausur vorgesehenen Unterlagen erhalten, und dass ich meine Arbeit ohne fremde Hilfe und ohne Verwendung unerlaubter Hilfsmittel und sonstiger unlauterer Mittel angefertigt ijabe. Ich weiß, dass ein Bekanntwerden solcher Umstände auch nachträglich zum Ausschluss von der Prüfung führt. Ich versichere weiter, dass ich sämtliche mir überlassenen Arbeitsunterlagen sowie meine Lösung vollständig zurück gegeben habe. Die Abgabe meiner Arbeit wurde in der Teilnehmerliste von Aufsichtsführenden schriftlich vermerkt.

Duisburg, den ____

(Unterschrift der/des Studierenden)

Falls Klausurunterlagen vorzeitig abgegeben:

Bewertungstabelle

Aufgabe 1

Aufgabe 2

Gesarntpunktzahl

Angehobene Punktzahl

%

Bewertung gern. PO

in Ziffern -

(Datum und Unterschrift 1. Prüfer, Univ.-Prof. DrAng. Dirk Söffker)

(Datum und Unterschrift 2. Prüfer, Yan Liu, M.Eng.)

(Datum und Unterschrift des für die Prüfung verantwortlichen Prüfers, Söffker)

Fachnote gemäß PfÜfungsordnung:

0 0 0 I 0 I [] 0 0 I 0 0 0 0 1,0 1,3 1,7 I 2,0 I 2,3 2,7 3,0 I 3,3 3,7 4,0 5,0

sehr gut gut befriedigend ausreichend mangelhaft

Bemerkung:


Page 3

Attention: Give your answers to ALL problems directly below the questions in the exam question sheet.

You are NOT allowed to use a pencil and also NOT red color (red color is used for corrections).

This exam "System Dynamics" is taken by me as a o mandatory (Pflichtfach)

elective (Wahlfach)

prerequsite (Auflage)

subject (cross ONE option according to your own situation).

Maximum achievable points: 40 Minimum points for the grade 1,0: 95% Minimum for the 50%

80) (3 points) St80te the difference between 80 SISO-system and a MIMO-system.

~1.~Q.:) -~~~\~~ ~ A Il\yJt A CU rpJ (I "

Fakultät fUr Ingenieurwissenschaften, Abt. Maschinenbau und Verfahrenstechnik

Lehrsi~~rSteueriiiili~]1(;gehing ,und Systemdynamik ' '

, Sy~teIT1~yIi&mj(j~i~~:~X' -'~;;:~.Bi'@;flCXth,2Ql1 '

.e,~M.JL ... 'cri'3"p'QiilJs) " , ,"~

'" 'Tlre-j!genvalue-=distributronc-ofac systemis-givel1iIi 'FigureLl: ,~- ' " ',"". "_~:-.,.._;.' ,.h . " ,. ,_ -. ," , ". "

Im

x x X

X X 0,001 +10,01)

-------&--'----+Re x x'

XX X

Figure 1.1: Eigenvalue distribution

Is the system asymptocally stable? Justify your statement.

~e ~b\.€~ \~ '{.\.et~e\ ~~~ le ,",,0' ~ \.~VV'\rto:AII ~~~\e ~e~e ~~ ~f\~tu.e ~ "cu. ,o~\iu·Q ~~ .

D

_._.f.~~ulJ;~t(iir IngeIJ,i~urwisse~chaf.t~Ib_Apt~.MaschiJ!E'ln1JaulJIl(LVerja.br(lm;technik ..•.•::..~J~e~[s.t1Jltl§~euerung,Regelung undSYlltelD.dyl,lJ!"Q)ilc

_._:_.L::~~B~~rß.lem~dynamics".. .. .. . .."

3"-,,-c-=="":~':C''''"(fj=f:rpo1Dtsr=oc-=~'='"'"' . . . .;;-.... ... .. ... . .. "~ ··;::~·.·"Asystß~Ir-iniequ:i1iOfium.caIidititirilsmexcltet::I~-:f;;"O witna·n.armönicruslgnäLwlth"arnplitude

_._._"A.=-2_and;:ftequel1cy~w_= O.5:..The measuroo:olltpuLsignru..oUhe..system-is-a.har.monicsignal... with amplitude A = 4 and frequency w=O.5. Classify·thestabiIity of·the system and justify JOur statemant.

o

Fakll!!.~~ für Ing~.!li~llrwiss~n~sch~t~I1.•. i\bt. Mas(!hinenbau und. V~rrahrenst~hnik .

Lehrstuhl Steuer.t~Jlg, ..• Reg~l1ffig.~uI!


Page 8

Aufgabe 2 (25 Points) In Fig. 2.1, a block diagram of a system with four transfer elements is given. The input is denoted as u and the output as y.

Element 1 Element 2

U Xl .. ... k"C X2 }zr( Y To .... ..

"!'-' : k2

," I ,- t TI,2 t

Element 3

X3 ~ .... .... TI,3 t

X5

, Element 4 ~

X4 ~ .... t

Figure 2.1: Block diagram of the given system

2a) (4 Points) Classify the transfer behavior of the four elements of the system and determine the four corresponding differential equations in a suitable form for classification under consideration of the given inscriptions.

'/-"tl\-\ =l~ \-y;;;\\~ +1~ i~~lc'

'~;;;~~~-rr:~J;;'~ .-_=tafu:~~n).~;1i~~~fn-t~!1~j~~;

EI~~e~-~- 3: PI )(3l\, =l3l'1 l\\ ,. ~J"I l\) .1.\)

I fJ~u\ JiI,c"

UNIVERSITÄT DUISBURG-ESSEN Fakultät für Ingenieurwissenschaften, Abt. Maschinenbau und Verfahrenstechnik Lehrstuhl Steuerung, Regelung und Systemdynamik System Dynamics February 11t h, 2011

Page 9

o 2b) (4 Points)

Give the differential equation of the whole system with respect to the input u and the output y as given in task a). The parameters are given as TD = T2 = T1,2 11,3 = 11,4 = k1 k2 = k3 1.


Page 10


Page 11

The mathematical model of an electrical system is described by the equations

Xl(t) + Xl(t) = KX2(t),

JXl(t) dt = T1y(t) + Jy(t) dt, and

X2(t) = u(t) + ;1 Ju(t) dt.

2c) (4 Points) Utilize the given equations to determine the scalar differential equation of the whole system (with input u and output y) and classify the resulting transfer behavior.

Which kind of transfer behavior of the controller is suitable for realizing stationary accuracy of the whole system (state reasons)?

-;. ,.....

. ..-::~ .

t.. =>

=- '/ 1,Ql\\ -I- lH1,,) ~ l\\ ~ '1(\\ ~ \


Page 12

D 2d) (5 Points)

In Fig. 2.2, the eigenvalues of four different systems are illustrated graphically. Furthermore, the step response behaviors of two of the four systems are measured and given in Fig. 2.3. Assign the two measurements to the related systems and state the reasons for your decisions (why or why not for each of the four systems).

-)1/t ~Sv;,c.W\eV\ 't "\ '" ~ [.'" J.'~I(')

~'fte"$",,,_..~I -'-;> '5'1~L;)

l\,+ LI"'~1 ,~s~~ "( ~ "o~ s\J\< -> (o-~ ",J bc \Yl('",-s~,( J ~'isk"" ~! J9..""f~"'~ ~~ ~c:>o ~~ ~r s*rp~~ o~\lQ.~·o~

(~O\)I\Q.\ 'Mi.~F.sc;.\\~~Cv\)


Page 13

x

-2 /.,-2 ... ··· 2 Re2 Re .....····-2 ............/·-2

........... X..... System 1 System 2

Im X

............ 2

.........

-2 ...... .·····-2

..........

.' X...

Im X" ....

" . .........~.. -2 ......2 Re 2 Re .·····-2

.?G//"

System 3 System 4

Figure 2.2: Eigenvalue distribution of four different systems

y

1

o.

-1~------~------~------~~------~------+ o 2 4 6 8 t [5] y Mea5urement 1

1

-1~------~------~------~~------~------+ o 2 4 6 , 8 t [5] Mea5urement 2

Figure 2.3: Measurement of the dynamic behaviors of two different systems


Page 14

D

UNIVERSITÄT DUISBURG-ESSEN Fakultät für Ingenieurwissenschaften, Abt. Maschinenbau und Verfahrenstechnik Lehrstuhl Steuerung, Regelung und Systemdynamik System Dynamics February 11th, 2011

Page 15

2e) (3 Points) In the following, the block diagram describing a vehicle's automatie air condition system (ac) and a driver's behavior (adjusting the ac) has to be given.

The driver senses the temperature inside the vehicle. The difference between the temperature convenient for the driver and the real temperature within the vehicle causes that the driver ehanges the desired value of the automatie ac to a new one. Depending on the differenee between the desired and the real temperature inside the vehicle, the automatie ac heats or eools the air flowing inside the vehicle by affecting humidity.

Mark additionally the part of the block diagram, whieh does not exist if an air eonditioner does not support automatie eontrol regarding the temperature inside the ear.

----N

(Q

deslred· dlfference air tempo temp.2convenlent dlfference flow current " temperature temp.1

air condltlon

vehlcle Intenor

tempo driver.,

r"1"

'.,L

" i wlthout automatie control -


Page 16

•

D

• 2f) (5 Points)

The system matrix of a dynamic system is given by

-2 0 k1]A = 0 -1 2 .

[ k2 0 0

Determine the characteristic polynomial p(>.) of the system. For which values of the parameter kl and k2 is the system asymptotically stable? Use the Hurwitz criterion for your calculation.

. . . ,...-~'-. ~., . .

. ',' . . .' '. ~_~ __ ~ _____~. ___ L_M___ ~~_~ • __•• _.. • .....___.... , ••

~ j :::> 0 . f.. ~ ; "'. --I") ~ ~ \:. 3\ \ ~o

. . "

\\ *\

. .

\::3 -k1kl\ _. I ~ L '..,t\1. '::. . ' - > 0 _/ ~ ~ ..J_

," 1 l-t.,~ =

H3= {-k1 k'l.)"H.l/o --:-7> \('1r2 k.'~2 c..~

3 ··-1("1k'2 0

:::> ":: . 1 2-- t1 t4 0 o .3 ·-~t'2

I

Documents

Klausurunterlagen - uni-due.de...Page 3 Attention: Give your answers to ALL problems directly below the questions in the exam question sheet. You are NOT allowed to use a pencil and