Eksamen I Fag Sie3040 Reguleringsteknikk

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NTNU Fakultet for elektroteknikk

Norges teknisk-naturvitenskapelige og telekommunikasjon

universitet Institutt for teknisk kybernetikk

Exam in SIE3040 : “Reguleringsteknikk m/elektriske kretser”

(version for English-speaking students)

Thursday, May 16th 2002 9AM – 3-PM

Read the following carefully before starting on the exam.

Combination of allowed help remedies B1:

Certified calculator with empty memory. No printed or hand-written notes.

Percentages denote the weight assigned to each problem when grades are evalued.

For questions during teh exam, contact:

Pål Gisvold, phone 73 55 95 88 or Fredrik Sandquist, phone 73 59 43 62

We will be present for two help sessions, approx 1015 – 1045 and kl. 1315 - 1345

Examination results will be published by Friday June 7th.

Some questions may be answered by referring to the formulae and information given in the attachments (the last pages of the exam)

Some questions in this exam call for ”measurements” done on a diagram. In such cases some degree of inaccuracy is of course acceptable.

Problem 1 (10%)

a) (4%)

Attachment 1 shows the step-response for 4 systems ( A-D).

In each case a unit step is applied at t = 0.

Also shown are the frequency responses (Gain only !) for the same 4 systems ( E-H)

Which step-responses and frequency-responses belong together ?

State your answers clearly by using the letters assigned to each response.

Write a short but sufficient explanation for each answer.

If you wish you may tear out attachment 1 and use it as part of your exam-paper.

b) (6%)

Given the following system:

The system’s step-response is shown below in figure 1.

The input is a step signal with amplitude 40 at t = 0 . Find the system-parameters K, ζ og ω0 .

figure 1: Step-response for problem 1b

Problem 2 (14%)

a)(8%)

A system has the following loop-transferfunction (“sløyfetransferfunksjon”):

Sketch an asymptotic Bode-diagram (gain-frequency and opase-frequency) for this system.

Use ordinary exercise-paper with graph ruling. Remember to draw the 0-dB line.

Scale the frequency-axis as follows: one octave (frequency-doubling) equals 1,5 cm (3 squares)

one frequency-decade equals 5 cm (10 squares)

Scale the y-axis so that 1 square equals 2 dB for the gain and 10 degrees for the phase.

Give the formulae for the gain and phase-shift for this system ("plotteformlene").

Now we have as our system input:u(t) = 4∙sin(2t).

Write down the mathematical expression for the output-signal: y(t).

b)(6%)

The function above is the loop-transferfunction (“sløyfetransferfunksjon”) i a control loop.

Sketch the asymptotic Bode diagram for |N(jω)|, based on your previous sketch of |h0(jω)|.

Yoy may use the same diagram as in the preceding question.

Use these asymptotes to sketch the function |N(jω)| (no calculations necessary)

The resonance peak will be 3 dB.

Use this sketch of |N(jω)| to answer the following questions:

* What will be the steady state error fo the step response ?

* What is the system bandwidth ?

* Does the system have a good stability margin ?

* A disturbance has frequency 2 Hz. Will it be attenuated by the control system ?

Problem 3 (40%)

Given a two-tank system:

KV , TV qinn

v1 A1 q1

v2 A2

figure 2: To-tank system

The symbols have this interpretation (1 dm = 10cm ):

u ”pådrag” / control-signal [mA]

qinnliquid-flow through the control valve[liter/min]

q1liquid flow from tank 1 to tank 2[liter/min]

v1disturabnce / liquid pumped out from tank 1[liter/min]

v2disturabnce / liquid pumped out from tank 2[liter/min]

x1state1 / liquid level in tank 1[dm]

x2state2 / liquid level in tank 2[dm]

A1cross –section area of tank 11000[dm2]

A2cross –section area of tank 2 500[dm2]

KVvalve-constant (control-valve) 50 [liter/(min∙mA)]

TVtime-constant (control-valve) 2.0 [min]

K1valve-constant for the liquid flow q1 100[liter/(min∙dm)]

Vi assume that the control-valve is linear. The dynamics between q1 og u is specified by the following transferfunction:

We also assume that the flow q1out of tank 1 is proportional to the liquid level: q1 = K1∙x1

(This is a linearized relationship that is valid only for small deviations from the steady state)

a)(6%)

Use mass-balance relationships to find the differential equations describing the dynamics for this two-tank system. Use symbols for the time being, do not use the numerical values yet.

We assume that v1 and v2 are independant of liquid levels.

b)(2%)

Sketch a block-diagram for this process. Include the control-valve in the block-diagram.

Use symbols only ! Do not use numerical values.

c)(2%)

Use the numerical values given on the preceding page and reduce the block-diagram.

Show that we get the following process-model:

(Time constant in minutes, lengths in dm)

u _ v1 _ v2 x2 = y

figure 3: blokkdiagram for to-tank systemet uten regulering

d)(4%)

We want to control the level in tank 2 , so we set x2 = y and close the control-loop:

r (y0) _ v1 _ v2 x2 = y

figure 4: blokkdiagram for to-tank systemet med regulering

First we try to control the level with a P-controller : hr(s) = KP

Use Routh’s criterion to find the critical gain KK .

e)(4%)

What KP -value would you choose if you want 6 dB gain-margin ?

( Tip: You don’t need a Bode –diagram !)

f)(4%)

Set KP = 2

We apply a step-input r, with stepsize 2 mA . What will be the steady-state error ?

What will be the steady-state error if the disturbance v1 is a step with stepsize 50 liter/min ?

Attachment 2 concrning problems 3g /h may be torn out and used as part of your exam-paper

g)(4%)

Attachment 2 shows a Bode-diagram for the system. h0 = hr∙hu . Here we have hr = KP = 1

Is the closed system stable using this controller-gain ?

Explain your answer by referring to the Bode-plot.

How large is the crossover frequency ωC ? How large is ω180 ?

Find the gainmargin ΔK og phasemargin Ψ .

h)(6%)

We now wish to enhance the properties of this system with a controller hr(s).

These are the system-specifications:

crossover-frequency ωC ≥ 0,2 rad/min
gainmargin ΔK ≥ 14 dB
phasemargin Ψ ≥ 40o
Zero steady state error for step inputs in the reference-signal and the disturbances.

Which controller-type (P, PI, PD eller PID) do we need in order to meet the specifications ?

Explain your answer. Design all the necessary controller-parameters.

i)(4%)

It turns out that disturbance v1 is especially problematic. We would therefore like to compensate for this by using a feedforward regulator hf(s). Sketch a blokkdiagram for the complete control-system with feedback and feedforward. Disregard the sensor-dynamics.

j)(4%)

What is the ideal feedforward function? Why is this function physically unrealizable ?

Suggest a transferfunction based on the ideal feedforward function that is physically realizable

What controller-type is this ?

Problem 4 (12%)

a)(4%)

We want to control the pressure in a steam boiler. The steamflow delivered by the boiler supplies several units and is best regarded as an uneven disturbance. The boiler pressure is applied to a pressure sensor which outputs a current signal (mA). This signal is then transmitted to a current-pressure (I/P) transducer. The output from the I/P transducer is an air-pressure signal which is fed into a pneumatic controller. The controller output serves as input to the control-valve which is also pneumatic.

Sketch a block diagram for the control system. The diagram should be as complete as possible.

Instead of assigning transferfunctions to the blocks use a name that clearly designates what the block represents; for example ”boiler”, ”I/P-transducer”, etc. Also show clearly what each signal (”arrow”) represents. On each “arrow” supply 3 pieces of information:

general symbol (f.ex. y(s))
description (f.ex pressure)
physical dimension (f.ex. Bar)

b)(2%)

A process has the following transferfunction:

What is T63 for this process approximately ?

(T63 is defined as the time it takes for the output signal to reach 63% of its steady-state value,

measured from the time the step input occurs.)

Sketch the step-response for this system.

c)(2%)

A textbook suggests this transferfunction for a DC-motor:

φ(s) is the motor’s output angle and u(s) is the motor input voltage.

Tmog Te are mechanical and electrical timeconstants.

You suspect that there must be something wrong with this transferfunction.

What is it ?

Explain your answer.

d)(4%)

A process is controlled by a PI-controller. We would like a little better stability-margin, ie. somewhat calmer transients without degrading the system bandwidth. What should we do:

increase the gain (KP)
increase the integration-time (Ti)
decrease the gain (KP)
decrease the integration-time (Ti)

Choose clearly only one of the above alternatives and explain why this should work.

Explain also why the other alternatives will not work.

Your parameter change will also produce an undesirable side-effect . What side-effect ?

Problem 5 (24%)

a)(10%)

Given the following AC-circuit:

figure 7

Source voltage E = 220 V (rms), frequency 50 Hz

Parameters:

R1 = 10,0 Ω L = 19,1 mH

R2 = 8,0 Ω C = 318 μF

R3 = 20,0 Ω

Calculate all 4 currents in this circuit. Show their mathematical expression an sketch a phasor-diagram. The angles in the diagram should be reasonable, but accuracy is not expected.

Derive the power P deliverd by the source.

Also derive the reactive power Q and the apparent power S.

Explain the difference between active and reactive power.

b)(8%)

We would like to realize the following transferfunction with operational amplifiers, resistors, and capacitors:

Symbols:

u(s) output signal from the controller [V]

e(s) error-signal input to the controller [V]

r(s)reference voltage 3,0 [V]

y(s)sensor voltage from the process [V]

Controller parameters: KP = 3,25 Ti = 8,3 sec Td = 2,7 sec Tf = 0,27 sec

Design a circuit and sketch the circuit diagram for this controller. Design also the circuit necessary to provide the error-signal . Clearly show in the diagram where the reference-voltage and sensor-voltage are applied. Show the parameter-values clearly.

c1)(1%)

Write the numbers 3, 6, and 7 in binary format.

c2)(5%)

We want to design a decoder as shown in the figure below:

B decoder Y

figure 8

Decoders like this are useful in a variety of situations; signal-conditioning, lotteries, elections etc.

In this simple decoder the inputs A, B, and C represent a binary number. A is the most significant bit while C is the least significant bit. The output Y goes high (1) when the input number is 3 or 6 or 7 Otherwise the output goes low (0).

Design a decoder that accomplishes this, it should be as simple as possible.

Use AND-gates and OR-gates.

(If you like you can also use NAND-gates or NOR-gates)

Sketch the decoder circuit diagram.

Attachment 1: Step-responses and frequency-responses for problem 1a

Fagnr.: SIE 3040Student no. :

Dato: 16.mai 2002Page no. :

Step-responses:

Frequency-responses:

Attachment 2: Bode-diagram for problem 3f and 3g

Fagnr.: SIE 3040Student no. :

Dato: 16.mai 2002Page no. :

Attachment 3:

Formulae I: The Laplace Transform

Lineærkombinasjon :

Tidsforsinkelse:

Tidsskalering:

Frekvensskift (”dempningsregelen”):

Derivasjon:

Integrasjon:

Konvolusjon (folding):

Begynnelsesverditeoremet:

Sluttverditeoremet:

Attachment 4a:

Formulae II: Transformation pairs

Enhetsimpuls:

Enhetssprang:

Enhetsrampe:

Attachment 4b:

FormulaeII: Transformation pairs

Attachment 5:

Formulae III: Routh's Stability Criterion

Attachment 6:

FormulaeIV: General formulae