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Prof. Dr.-Ing. F. Keller Control Theory CSexss12.docKarlsruhe University of Examination Summer 2012 page 1 / 7Applied Sciences

Name:............................

Formulas:

Laplace Transform Table:

( )t 1

s

e a t 1

s a

t 12

s

1

1a

e at

1

s s a

sin at as a

2 2

cos at ss a

2 2

for d1: 11 2

2,1

21

2

21

1 21

ddTTwhereeTT

TeTT

T Tt

T

t

1

2 12 2( )T s d T s s

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Prof. Dr.-Ing. F. Keller Control Theory CSexss12.docKarlsruhe University of Examination Summer 2012 page 2 / 7Applied Sciences

Problem 1: (6 points)

Given a system with input signal x(t) and output signal y(t) which is described by

xTyTy

a) Obtain the transfer function G s Y s

X s( )

( )

( ) .

b) The input is x(t)=(t). What is the Laplace-transform of x(t) ? What is the Laplace-transform of y(t) ? Obtain y(t).

c) Make a sketch of the Nyquist plot of G(s).The starting point and the end point must be clearly shown.

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Prof. Dr.-Ing. F. Keller Control Theory CSexss12.docKarlsruhe University of Examination Summer 2012 page 3 / 7Applied Sciences

Problem 2: (7 points)

Given a system which is the series connection of an integrator (transfer func-

tion: 1

T si

) and a lowpass-system (transfer function:

sT1

1).

a) Make a sketch of the Nyquist plot for T=Ti=1s.Rough sketch is sufficient; the starting point and the end point must be clearlyshown.

b) Obtain the step response (as a formula) of the system for Ti= T.

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Prof. Dr.-Ing. F. Keller Control Theory CSexss12.docKarlsruhe University of Examination Summer 2012 page 4 / 7Applied Sciences

Problem 3: (10 points)

Given the following control loop. Theprocess is described by

TsTsK

sG

1

)( .

a) Obtain the transfer function G(s) = C(s)/R(s) of the closed loop, if a P-controllerwith gain V is used.

b) What is the steady state error (in %) for K=4, V=1 and T=1s ?

c) Is it possible, that the control loop is unstable with P-control ? Why ?

d) If the strategy of cancelling poles is to be used, which controller is to be used(P, PI, PD, PID or I).? Why?

controller process

-

r ce y

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Prof. Dr.-Ing. F. Keller Control Theory CSexss12.docKarlsruhe University of Examination Summer 2012 page 5 / 7Applied Sciences

Problem 4: (7 points)

Given the following Nyquist plot of anopen control loop. A P-controller withgain 1 is used.

a) What is the order of the process?

b) What is the maximum gain of thecontroller, so that the control loop is

still stable.Approximate value issufficient; explain the way how theresult is obtained.

c) What is the steady state error (in %)?

-2 -1 0 1 2 3 4 5-5

-4

-3

-2

-1

0

1

real part

imaginarypart

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Prof. Dr.-Ing. F. Keller Control Theory CSexss12.docKarlsruhe University of Examination Summer 2012 page 6 / 7Applied Sciences

Problem 5: (9 points)

A process defined by sTwithe

Ts

G sT

process 11

is controlled by a P-Controller (gain

V).

a) Draw the block diagram for the closed loop.

b) Make a sketch of the Nyquist plot of the open loop for V=1.

c) What is the maximum value for V for which the control loop is still stable?

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Prof. Dr.-Ing. F. Keller Control Theory CSexss12.docKarlsruhe University of Examination Summer 2012 page 7 / 7Applied Sciences

Problem 6: (9 points)

We consider a liquid level control system. The liquid level h (dimension: m) in a con-tainer is to be controlled. The cross section area of the container is A. The controlleroutput is q

controller(flow rate, dimension: m

3/s).

a) What is the transfer function of the process (input: qcontroller, output: h)

b) We use a P-controller with gain K. Obtain the transfer function of the closed loop.Is it possible that the control loop is unstable? Why?

c) Make a drawing of the following signals: reference input, controller output, con-trolled variable if the container is empty for t=0. The reference input is a step from

0 to 1m. Parameters:m

s

m

KmA

3

2

1

,1