exam - eth z · answers given in the booklet will not be considered. ... exist multiple versions of...

Exam 30th of January, 2018

Control Systems I (151-0591-00L) Prof Emilio Frazzoli

Exam

Exam Duration: 120 minutes, +15 minutes reading time

Number of Problems: 45

Number of Points: 56

Permitted aids: 2 pages (1 sheet) of A4 notes, translation (e.g. En-glish to German) table for control systems terms,simple calculators will be provided upon request.

Important: Questions must be answered on the provided answer sheet;answers given in the booklet will not be considered.

There exist multiple versions of the exam, where the orderof the answers has been permuted randomly.

There are two types of questions:

1. One-best-answer type questions: One unique cor-rect answer has to be marked. The question is worth onepoint for a correct answer and zero points otherwise. Giv-ing multiple answers to a question will invalidate the an-swer. These questions are marked”Choose the correct answer. (1 Point)”

2. True / false type questions: All true statementshave to be marked and multiple statements can be true.If all statements are selected correctly, the full number ofpoints is allocated; for one incorrect answer half the num-ber of points; and otherwise zero points. These questionsare marked”Mark all correct statements. (2 Points)”.

No negative points will be given for incorrect answers.

Partial points (Teilpunkte) will not be awarded.

You do not need to justify your answers; your calculationswill not be considered or graded.

Use only the provided paper for your calculations; addi-tional paper is available from the supervisors. Use pensproducing a dark, solid and permanent line. The use ofpencils is not allowed.

Good luck!

Corrected

Question 1 Choose the correct answer. (1 Point)

Your thesis supervisor hands you a quadrocopter and asks you to design a control algorithm for it.Is the quadrocopter system causal or not?

A Causal B Not causal

Question 2 Choose the correct answer. (1 Point)You are in the shower and thinking about your control systems exam. There is a knob that canbe rotated and pulled/pushed to set a temperature T and water pressure p. You start wonderingwhat the input(s) u and output(s) y of the system are. Hint: In this scenario you are both thecontroller and measurement device of the system.

A Inputs: Temperature perception, water pressure perception, Outputs: Heat setting, pressuresetting of knob

B Inputs: Heat setting, pressure setting of knob, Outputs: Temperature perception, waterpressure perception

C Inputs: Temperature perception, Heat setting, Outputs: water pressure reading, pressuresetting of knob

Question 3 Mark all correct statements. (2 Points)

All signals are scalars. The system y(t) = n

3

t

3

u(t), with n 2 R>0

is:

A Causal

B Linear

C Memoryless/Static

D Time-invariant


All signals are scalars. Input u(t), output y(t), state x(t).

The system x(t) = u(t)

t

, y(t) = 3x(t)� x

2(t), with t > 0 is:

A Memoryless/Static

B Causal

C Linear

D Time-invariant

Corrected


⌃2

⌃1

⌃3

⌃4

r e

+ y

�

You are given the above system diagram. What is the associated transfer function from r ! y.

A ⌃r!y

= (⌃1

+ ⌃2

⌃4

+ ⌃3

)

B ⌃r!y

= (⌃1+⌃2⌃4+⌃3)

1+⌃2⌃4

C ⌃r!y

= (⌃1+⌃2⌃4+⌃3)

1+⌃1+⌃2⌃4+⌃3

D ⌃r!y

= (⌃1+⌃2⌃4+⌃3)

1+⌃3


You are designing a speed controller for a car. You can measure the current speed v of the car andcan input the position of the gas pedal p

pedal

. Which type of controller solution would not workin this scenario?

A Feedforward controller

B Feedforward and feedback controller

C All other mentioned options would notwork.

D Feedback controller

Corrected

Box 1: Questions 7, 8

You have decided to brew some beer and you want to go big. So you bought a tank system (shownin the Figure) that you will use for processing and storage. However, you need to model it first.Input into your system is the flow of beer into the first tank Q

in

, h1

and h

2

are heights of thebeer in tank 1 and 2, respectively. Surface area of tank 1 is A

1

and the surface area of tank 2 isA

2

. Each valve produces a flow through it that is dependent on the relative di↵erence of heightsof the fluid on both of its sides. The flow through the valve i is then described as Q

i

= K

i

p2g�h

(in m

3

/s) where K

i

is a constant that is associated to valve i (assumed to be given in the correctunit), g = 9.81m/s

2 refers to the gravitational acceleration and �h is the height di↵erence onboth sides of the valve.

Question 7 Choose the correct answer. (1 Point)Which set of equations correctly models the tank system?

A

A

1

dh

1

dt

= Q

in

�Q

1

�Q

12

A

2

dh

2

dt

= Q

12

�Q

2

Q

1

= K

1

p2gh

1

Q

2

= K

2

p2gh

2

Q

12

= K

12

p2g|h

1

� h

2

| sign(h1

� h

2

)

B

A

1

dh

1

dt

= Q

in

�Q

1

�Q

12

A

2

dh

2

dt

= Q

12

�Q

2

Q

1

= K

1

p2gh

1

Q

2

= K

2

p2gh

2

Q

12

= K

12

p2g(h

1

� h

2

)

C

dh

1

dt

= Q

in

�Q

1

�Q

12

dh

2

dt

= Q

12

�Q

2

Q

1

= K

1

p2gh

1

Q

2

= K

2

p2gh

2

Q

12

= K

12

p2g|h

1

� h

2

| sign(h1

� h

2

)

D

dh

1

dt

= Q

in

�Q

1

�Q

12

dh

2

dt

= Q

12

�Q

2

Q

1

= K

1

p2gh

1

Q

2

= K

2

p2gh

2

Q

12

= K

12

p2g(h

1

� h

2

)

Corrected

Question 8 Choose the correct answer. (1 Point)Now that you have figured out how to model your beer tank, you want to control the height ofthe beer in tank 2 since that is how you make good beer. You are an expert in linear controllerdesign and you have decided to linearize the system. Which system matrices A,B,C,D are thelinearization that you want? The operating point that is guaranteed to produce nice Porter beeris Q

in

= 10 m

3

/s, h

1

= 8 m,h

2

= 2 m. Parameters of the system are A

1

= 10 m

2

, A

2

=100 m

2

,K

1

= 0.25,K2

= 1,K12

= 0.577. Assume that the valve constants K

i

have the correctunits.

A

A =

0.052 �0.072�0.021 0.005

�, B =

0.10.0

�,

C =⇥0 1

⇤, D = 0

B

A =

�0.072 0.0520.005 �0.021

�, B =

0.10.0

�,

C =⇥0 1

⇤, D = 0

C

A =

�0.072 0.0520.005 �0.021

�, B =

0.10.0

�,

C =⇥1 0

⇤, D = 0

D

A =

0.052 �0.072�0.021 0.005

�, B =

0.00.1

�,

C =⇥0 1

⇤, D = 0


The Lotka-Volterra equations, also known as the predator-prey equations, are a set of di↵erentialequations frequently used to describe the dynamics of biological systems in which two speciesinteract, one as a predator and the other as prey. The populations change through time accordingto the pair of equations:

dx

dt

= ↵x� �xy

dy

dt

= �xy � �y

where {↵,�, �, �} 2 R>0

. These parameters model the interactions between the prey and predatorpopulation. x is the size of the prey population and y is the size of the predator population.


How would you classify the equilibrium points introduced by the Lotka-Volterra model?

A One is stable, the other is marginally sta-ble.

B One is stable, the other is unstable.

C Both are unstable.

D Both equilibria are stable.

E Both are marginally stable.

F One is unstable, the other one ismarginally stable.

Corrected


What are the equilibrium points of the Lotka-Volterra system and what do they correspond to?

A (x0

, y

0

) = (0, 0), (x0

, y

0

) = (��

,

↵

�

). Oneequilibrium point is an unstable equilib-rium where the population grows to infinityover time.

B (x0

, y

0

) = (0, 0), (x0

, y

0

) = (��

,

↵

�

). Oneequilibrium point corresponds to the ex-tinction of both species.

C (x0

, y

0

) = (0, 0), (x0

, y

0

) = (��

,

↵

�

). Thereis no special interpretation for either of the

equilibrium points.

D (x0

, y

0

) = (0, 0), (x0

, y

0

) = (��

,

↵

�

). Thesystem has non-zero derivatives in one ofthe equilibrium points.

E (x0

, y

0

) = (0, 0), (x0

, y

0

) = (��

,

↵

�

). One ofthe equilibria represents a state where bothpredator and prey sustain non-zero popu-lation number.


0 1 20

0.2

0.4A

time (seconds)

Am

plitu

de

0 1 2 3-0.4

-0.2

0B

time (seconds)

Am

plitu

de

0 10 20 300

5

10 10 25 C

time (seconds)

Am

plitu

de

0 5 10 15-5

0

5 10 11 D

time (seconds)

Am

plitu

de

0 1 20

0.05

0.1E

time (seconds)

Am

plitu

de

0 0.5 1 1.5-1

0

1F

time (seconds)

Am

plitu

de

In the Figure above variuos time responses are shown. However, only some of them correspond toa second order system; which ones are those?

A F

B E

C A

D C

E B

F D


Corrected

What additional property has to hold for an LTI system such that the BIBO stability is equivalentto the asymptotic stability?

A System has to be controllable.

B System has to be observable and control-lable.

C System has to be observable.

D This is always true since we have a linearsystem.

E This is never true.


There are 8 systems given by their state space representations:

1.

A =

10 00 �3

�, B =

01

�, C =

⇥1 1

⇤

2.

A =

0 00 �2

�, B =

11

�, C =

⇥1 1

⇤

3.

A =

�3 00 20

�, B =

11

�, C =

⇥1 0

⇤

4.

A =

�20 30 �5

�, B =

01

�, C =

⇥1 0

⇤

5.

A =

�5 00 1

�, B =

01

�, C =

⇥1 1

⇤

6.

A =

1 01 1

�, B =

10

�, C =

⇥1 1

⇤

7.

A =

0 �21 0

�, B =

10

�, C =

⇥1 0

⇤

8.

A =

0 101 0

�, B =

10

�, C =

⇥1 1

⇤

Question 13 Mark all correct statements. (2 Points)Which ones of these systems are BIBO stable from a purely theoretical perspective, i.e. no noise,no disturbances on x, u, y?

A 8

B 3

C 2

D 6

E 7

F 1

G 5

H 4

Corrected


Which ones of these systems are BIBO stable from a practical perspective, i.e. noise, disturbanceson x, u, y?

A 3

B 6

C 7

D 1

E 4

F 8

G 2

H 5


0 2 40

0.05

0.1Impulse Response

time (seconds)

Am

plitu

de

0 2 4-0.5

0

0.5Impulse Response

time (seconds)

Am

plitu

de

0 2 40

0.05

0.1

0.15

0.2Impulse Response

time (seconds)

Am

plitu

de

0 2 4-0.2

-0.1

0

0.1

0.2Impulse Response

time (seconds)

Am

plitu

de

In the figure above, impulse responses belonging to di↵erent transfer functions are shown. Howmany of those systems are asymptotically stable? You can assume that the transfer functions donot have any common factors in their numerator and denominator.

A 3

B 4

C 1

D 2

E 0

Corrected


Consider the following Linear Time Varying system:

x

1

= �x1

+ sin(t)x2

x

2

= 2x1

� !x

2

For which values of the parameter ! are the poles of the system in the left half-plane?

A! 2

B! � �1

C! �1

D! � 2


Consider the 4th order system, given by its state space matrices:

A =

2

664

0 1 0 00 0 1 00 0 0 �

↵ 0 0 1

3

775 , C =⇥� 0 0 �

⇤

For which values of the parameters ↵, �, � and � is the system observable/detectable?

A 2

664

↵

�

�

�

3

775 =

2

664

0001

3

775

B 2

664

↵

�

�

�

3

775 =

2

664

1001

3

775

C 2

664

↵

�

�

�

3

775 =

2

664

1010

3

775

D 2

664

↵

�

�

�

3

775 =

2

664

0100

3

775

E 2

664

↵

�

�

�

3

775 =

2

664

0110

3

775

F 2

664

↵

�

�

�

3

775 =

2

664

0101

3

775

Corrected


Consider the 4th order system, given by its state space matrices:

A =

2

664

0 1 0 00 0 1 00 0 0 10 0 0 �

3

775 , B =

2

664

↵

001

3

775

For which values of the parameters ↵ and � is the system controllable / stabilizable?

A↵ 6= 0, 6= 0

B↵ 2 R, � 6= 0

C↵ 6= 0, � 2 R

D↵ 2 R, � 2 R


A system is given by its transfer function. g(s) = s�1

s

2+s�2


Which one of the following state space representations is equivalent to the system given by itstransfer function ?

A

A =

0 12 �1

�, B =

1�2

�, C =

⇥1 0

⇤, D = 1

B

A =

0 12 �1

�, B =

1�2

�, C =

⇥0 1

⇤, D = 0

C

A =

0 12 �1

�, B =

1�2

�, C =

⇥1 0

⇤, D = 0

D

A =

0 12 �1

�, B =

1�2

�, C =

⇥0 1

⇤, D = 1


Consider the following system:

A =

1 20 �1

�, B =

10

�, C =

⇥1 0

⇤, D = 0

Can you design a feedback controller of the form u = �Ky such that the poles of the closed loopsystem are placed at arbitrary locations in the complex plane? Can you stabilize this system?

A Poles can be placed at arbitrary locations.The system can be stabilized.

B Poles cannot be placed at arbitrary loca-tions. The system can be stabilized.

C It cannot be deduced without seeing the ac-

tual controller design.

D Poles can be placed at arbitrary locations.The system cannot be stabilized.

E Poles cannot be placed at arbitrary loca-tions. The system cannot be stabilized.


Corrected

This time your boss asks you to design a controller for the following system:

A =

1 20 �2

�, B =

01

�, C =

⇥1 0

⇤, D = 0

The controller has to have the structure as shown in the Figure above, where g is the plant and K

is the gain. Your boss would like that the closed loop is asymptotically stable and that it has anaperiodic response. What should the gain be?

A Cannot be done. It will always have oscil-lations.

B 1 < K 1.125

C 1 � K

D 1 K 1.125

E 1 < K

F 1.125 � K

G Cannot be done. It cannot be stabilized.


Which di↵erential equation corresponds to the the following transfer function:

g(s) =y

u

=s+ 2

s

2 + 3s� 4

Ay + 3y � 4 = u+ 2

B �y � 3y + 4 = u+ 2

Cy + 3y � 4y = u+ 2u

D �y � 3y + 4y = u+ 2u

Corrected


-4 -2 0 2 4y

-3

-2

-1

0

1

2

dy

1

-5 0 5y

-3

-2

-1

0

1

2

3

dy

2

-2 -1 0 1y

-3

-2

-1

0

1

2

dy

3

-2 -1 0 1y 10 10

-10

-8

-6

-4

-2

0

2

4dy

10 9 4

In the figure above, some phase phase plots are shown. Each of these phase plots corresponds toone of the following systems:

A g(s) = 1

s

2+5

B g(s) = 1

s

2+2s+1

C g(s) = 1

s

2+s+5

D g(s) = 1

s

2�5

Which of the following is the correct assignment?

AA! 3, B ! 2, C ! 1, D ! 4

BA! 2, B ! 3, C ! 4, D ! 1

CA! 4, B ! 3, C ! 1, D ! 2

DA! 2, B ! 3, C ! 1, D ! 4

Corrected


Sodium nitroprusside is a substance with potent vasodilating e↵ects in arterioles and venules, i.e.,it can be used to temporarily lower the blood pressure of a patient when injected intravenously.For this reason, sodium nitroprusside is regularly used in intensive and intermediate care unitsto regulate the blood pressure of severely ill patients, for instance when they have a hypertensivecrisis. Let n be the concentration of sodium nitroprusside in the venous blood in mol

l

and p be thepressure of the arterial blood in kPa. A dosing valve with control signal u 2 [0, 1] can be used tocontrol the inflow of saline solution containing sodium nitroprusside to the patient’s arteries. Thedosing valve is fully closed at position u = 0 and fully opened at position u = 1. The followingfigure depicts the situation:

u

p

n

A careful analysis of the dynamics has revealed the the following system of dynamic equations thatdescribe the blood pressure of the patient:

n

p

�=

a

1

0a

2

a

3

�·n

p

�+

b

1

b

2

�· u+

k

1

0

�(1)

Where a

1

, a

2

, a

3

, b

1

, b

2

and k

1

are constants in R.Assume that you can measure p and that b

2

6= 0. For which values of the parameters a1

, a

2

, a

3

, b

1

, b

2

and k

1

do you need to consider the possibility of a pole-zero cancellation?

Aa

2

= 0

Bb

2

= �1 and a

2

= a

1

· b1

Ca

2

· b1

= b

2

· (a1

� a

3

)

Db

2

= 1 and a

2

= �a1

· b1

Eb

1

= 0

Fb

2

= ±1 and a

2

= a

1

· b1

Corrected


We consider the previous transfer function for the blood pressure regulation with the parametersa1 = 1

20

, a

2

= 1

10

, a

3

= 1

20

, b

2

= 1

10

, b

1

= 1. The numeric transfer function is:

P (s) =1

10·

s+ 21

20

(s+ 1

20

) · (s+ 1

20

)

Consider the following four root-locus curves:


Out of the four root-locus plot sketches displayed, which one is corresponding to the previoustransfer function and values k 2 {0, 20}?

a) b)

c) d)

Select the correct:

A c

B b

C a

D d

Corrected


Select all correct statements:

A Adding a zero at � 25

20

to the transfer func-tion yields an unstable closed-loop transferfunction for large values of k > 0.

B The open-loop transfer function L(s) =1

(s+1)·(s+2)

has its center of mass in its pos-

itive root-locus plot at s = � 3

2

.

C The angle rule for the open-loop trans-fer function L(s) = 1

(s+1)·(s+2)

states that

�\(s+1)�\(s+2) = 0�+360� ·m, m =±1, 2, ... for the negative root-locus.

D If a pole at � 24

20

is added to P (s), twoclosed-loop poles have real part � 1

8

forlarge values of k > 0.


Except the constant transfer function g(s) = 0 there is no other rational transfer function for whichthe positive and the negative root locus curve is identical.

A True. B False.


If a Bode plot of a rational transfer function is plotted for ! 2 {�1,1} then both the magnitudeplot and the phase plot are symmetrical with respect to the y-axis.

A True. B False.

Corrected


We consider the following transfer function:

L(s) =1

10·

s+ 21

20

(s+ 1

20

) · (s+ 1

20

)

Furthermore we consider a Bode plot:

0.01 0.1 1 10

-40

-20

0

20

|L(jω)|dB

0.01 0.1 1 10

-125

-100

-75

-50

-25

0

∠ L(jω)


Which statements are correct:

A The phase margin of L(s) is 1.

B The closed-loop system for L(s) will be ableto track a reference signal r : t ! sin(t)without steady-state error.

C The gain margin of L(s) is 1.

D At crossover frequency, the system showsa gradient of �40dB/dec.

E The displayed Bode plot is the correct Bodeplot for the transfer function L(s).


Corrected

A reference signal r : t! sin(t) is applied to the closed-loop system of k·L(s) with k = {�1, 1, 100}and the output responses are recorded and shown in the figure below.

10 20 30 40 50t [s]

-1.0

-0.5

0.5

1.0

y(t)

r(t)

(a)

10 20 30 40 50t [s]

-1.5×106

-1.0×106

-500 000

y(t)

r(t)

(b)

10 20 30 40 50t [s]

-1.0

-0.5

0.5

1.0

y(t)

r(t)

(c)

Assign the correct step response to the values of k:

Ak = �1! (a), k = 1! (c), k = 100! (b)

Bk = �1! (b), k = 1! (a), k = 100! (c)

Ck = �1! (a), k = 1! (b), k = 100! (c)

Dk = �1! (c), k = 1! (a), k = 100! (b)


Consider the Bode plots of two rational transfer functions g(s) given below:Both transfer functions must be identical.

Corrected

1 10 100

-45

-40

-35

-30

-25

-20

|L(jω)|dB

1 10 100

-160

-140

-120

-100

∠ L(jω)

1 10 100

-45

-40

-35

-30

-25

-20

|L(jω)|dB

1 10 100

-160

-140

-120

-100

∠ L(jω)

A False. B True.


Consider the Nyquist plots of two rational transfer functions g(s) given below:

-0.08 -0.06 -0.04 -0.02 0.00

-0.04

-0.02

0.02

0.04

-0.08 -0.06 -0.04 -0.02 0.00

-0.04

-0.02

0.02

0.04

Both transfer functions must be identical.

A True. B False.

Corrected


Consider an open-loop transfer function L(s) = k · s�z

s�p

with p 2 R, z 2 R<0

. The left Nyquistplot corresponds to the value k

1

and yields an unstable closed-loop system, the right Nyquist plotcorresponds to the value k

2

and yields a stable closed-loop system.

-0.5 -0.4 -0.3 -0.2 -0.1 0.1 0.2

-0.3

-0.2

-0.1

0.1

0.2

0.3

-20 -15 -10 -5 5 10

-15

-10

-5

5

10

15

What range of values k yields an asymptotically stable closed-loop system?

Ak >

1

2

Bk < 10

Ck >

1

4

Dk > 10

Ek <

1

4

Fk <

1

2


You are considering the following transfer function:

L(s) = e

�0.001·s ·(s+ 1)

�s+ 1

100

��s+ 1

2

�(s+ 3)(s+ 4)(s+ 30)


To simplify the calculations, you approximate the transfer function with a simpler model. Select theapproximation which can reasonably be assumed to model the behavior of L(s) for low frequencies! with negligible error.

AL(s) ⇡ 1

3000

· (s+1)

(s+ 12 )(s+3)(s+4)

BL(s) ⇡ 1

30

· (s+1)(s+ 1100 )

(s+ 12 )(s+3)(s+4)

CL(s) ⇡ e

�0.001·s · (s+1)(s+ 1100 )

(s+ 12 )(s+3)(s+4)

DL(s) ⇡ 1

30

· (s+1)

(s+ 12 )(s+3)(s+4)

EL(s) ⇡ (s+1)(s+ 1

100 )(s+ 1

2 )(s+3)(s+4)


Corrected

Next, you are interested in the reference tracking behavior of your open-loop transfer functionL(s). Select the correct statements.

A An additional pole at the origin is neededto track unit step references withoutsteady-state error.

B The system is of type 0.

C If L(s) had an additional pole at the ori-gin, it could track first order unit rampswith a steady-state error of �85dB.

D The system can track a unit step referenceswithout steady-state error but fails to tracka first-order ramp reference.

E If L(s) had an additional pole at the ori-gin, it could track second order unit rampswith a steady-state error of �85dB.


You are considering the following plant:

P (s) =s+ 12

10

(s� 1

10

) · (s+ 1)2

The requirements on the settling time are stringent and so the 2%-settling time has to be below0.1 s for safety reasons. Yet cost is also an important consideration in the choice of the controllerand a simple P -controller would be ideal from a system design point of view. Can you meet therequirements with such a controller? You may approximate the 2% -settling time with the formula:T

s

= �ln(2%)

�

, where � is the real part of the largest eigenvalue of L(s).

A No. B Yes.

Corrected


Background: cross-over frequency !

c

, rise time t

90

, phase margin ', overshoot M

p

!

c

⇡ 1.7

t

90

' ⇡ 71� � 117� ·Mp

Mag

nitu

de (d

B)

-40

-20

0

20

40

60P(s)C(s)

10-1 100 101 102 103

Phas

e (d

eg)

-90-80-70-60-50-40-30-20-10

0P(s)C(s)

Bode Diagram

Frequency (rad/s)

The bode plot of plant P (s) is given above. You decide to design an I-only controller of the formC(s) = 100

s

. What is the expected overshoot and rise time?

At

90

= 0.055 s, M

p

⇡ 22%

Bt

90

= 0.055 s, M

p

⇡ 45%

Ct

90

= 0.17 s, M

p

⇡ 56%

Dt

90

= 0.017 s, M

p

⇡ 22%

Corrected


Table 1: Ziegler/Nichols design rules

type kp Ti Td

P 0.50 · k⇤p 1 · T ⇤ 0 · T ⇤

PI 0.45 · k⇤p 0.85 · T ⇤ 0 · T ⇤

PD 0.55 · k⇤p 1 · T ⇤ 0.15 · T ⇤

PID 0.60 · k⇤p 0.50 · T ⇤ 0.125 · T ⇤

Mag

nitu

de (d

B)

-60

-50

-40

-30

-20

-10

0

10

20P(s)

10-1 100 101

Phas

e (d

eg)

-180

-135

-90

-45

0P(s)

Bode Diagram

Frequency (rad/s)

You design a PD-controller using the Ziegler/Nichols design rules. Which k

p

and T

d

do you obtain?

Ak

p

⇡ 5.4, T

d

⇡ 0.4

Bk

p

⇡ 10.8, T

d

⇡ 0.7

Ck

p

⇡ 0.6, T

d

⇡ 0.15

Dk

p

⇡ 3.1, T

d

⇡ 0.4

Corrected


You write a PID-controller in the following form:

PID(s) = k

⇤p

(s+ a)(s+ b)

(s)(s+ d)

For which condition on a, b, c does the PID-controller contain a lead-element? (_ is a logical OR)

Ab < a _ b < d

Ba > d _ b > d

Ca > d _ b < d

Da < d _ b < d


Your friend created Nyquist plots of a P, a PD and a PI controller for a second-order stable plantwith a delay and only real-poles. Which Nyquist plot refers to which controller?

-1 0 1 2 3 4 5-3

-2

-1

0

1

2

3FIGURE 1

Real Axis

Imag

inar

y Ax

is

-1 0 1 2 3 4 5-4

-3

-2

-1

0

1

2

3

4FIGURE 2

Real Axis

Imag

inar

y Ax

is

-2 -1.5 -1 -0.5 0 0.5-4000

-3000

-2000

-1000

0

1000

2000

3000

4000FIGURE 3

Real Axis

Imag

inar

y Ax

is

A P - Figure 1, PI - Figure 2, PD - Figure 3

B P - Figure 2, PI - Figure 1, PD - Figure 3

C P - Figure 2, PI - Figure 3, PD - Figure 1

D P - Figure 1, PI - Figure 3, PD - Figure 2

Corrected


Assume your open-loop system L(s) is of type 0 and that you are using a P-controller. You wantto increase the closed-loop tracking performance of your open-loop system L(s) at low frequencies.Currently your closed-loop tracking error is 10%. You would like this closed-loop error to bereduced to 2%. How does the low-frequency gain of your controller have to change to make thispossible?

Ak

new

= 6.33 · kold

Bk

new

= 4.66 · kold

Ck

new

= 5.44 · kold

Dk

new

= 5 · kold


You want to increase the phase at the crossover frequency !

c

= 1 rad/s by 30� by using a lead-

element C(s) = s/a+1

s/b+1

. Which parameters a, b should you choose?

Aa = 1, b = 1

Ba = 1/

p3, b =

p3

Ca =p3, b = 1/

p3

Da =p3/2, b = 2/

p3


You are given the task to discretize the controller

C(s) =s+ 3

s

2

Let e[k] be the input to the controller, x[k] be the internal state of the controller and u[k] be theoutput of the controller.What is a valid forward euler approximation of the above controller?

Ax x+ dt · (

0 10 0

�x+

01

�e), u =

⇥1 3

⇤x+

⇥0⇤e

Bx x+ dt · (

0 01 0

�x+

01

�e), u =

⇥1 3

⇤x+

⇥1⇤e

Cx x+ dt · (

0 10 �1

�x+

01

�e), u =

⇥3 1

⇤x+

⇥0⇤e

Dx x+ dt · (

0 10 0

�x+

01

�e), u =

⇥3 1

⇤x+

⇥0⇤e

Corrected


Let P (s) = e

�sT

(s+0.1)

2 .

You are given the frequency !

⇤ = 1 rad/s at which \P (j!⇤) = �180�. What is the time delay T

of the system?

AT ⇡ 0.2 s

BT ⇡ 2.9 s

CT ⇡ 1.7 s

DT ⇡ 0.1 s


You are given the Bode plot of a plant however the time delay has not been included in the model.You have a time delay of T

d

= ⇡

40

s. What is the new phase margin/ phase reserve of your system?

Mag

nitu

de (d

B)

-60

-50

-40

-30

-20

-10

0

10

20P(s)

100 101 102 103

Phas

e (d

eg)

-225

-180

-135

-90

-45

0P(s)

Bode Diagram

Frequency (rad/s)

A'

new

= 45�

B'

new

= 30�

C'

new

= 60�

D'

new

= 90�

Corrected

Answer sheet:

student number

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� please encode your student number, andwrite your first and last name below.

Firstname and lastname:

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

How to select answer B :

3 Answer B chosen.

3 Corrected, answer B chosen.

3 Double corrected, answer B chosen.

7 No choice made.

Answers must be given exclusively on this sheet;

answers given on the other sheets will not be counted.

Q1: A B

Q2: A B C

Q3: A B C D

Q4: A B C D

Q5: A B C D

Q6: A B C D

Q7: A B C D

Q8: A B C D

Q9: A B C D E F

Q10: A B C D E

Q11: A B C D E F

Q12: A B C D E

Q13: A B C D E F G H

Q14: A B C D E F G H

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