Exam January 23, 2017

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

Exam

Exam Duration: 135 minutes + 15 minutes reading time

Number of Problems: 45

Number of Points: 53

Permitted aids: 40 pages (20 sheets) of A4 notes, Appendix D ofthe book ”Lino Guzzella: Analysis and Synthesisof Single-Input Single-Output Control Systems”,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.

Good luck!

Corrected

Corrected

1 Control Architectures and Systems

Question 1 Mark all correct statements. (2 Points)

Feedforward control systems...

A ...rely on precise knowledge of the plant.

B ...can change the dynamics of the plant.

C ...reject external disturbances.

Question 2 Mark all correct statements. (2 Points)

Feedback control systems...

A ...can stabilize an unstable plant.

B ...can feed sensor noise into the system.

C ...can de-stabilize a stable plant.

Question 3 Mark all correct statements. (2 Points)

All signals are scalars. The system ddty(t) = (u(t+ 1))2 is:

A Causal

B Memoryless / Static

C Time-Invariant

D Linear

Question 4 Mark all correct statements. (2 Points)

All signals are scalars. The system ⌃(s) = e5·s

⌧ ·s+1

is:

A Causal

B Time-Invariant

C Memoryless / Static

D Linear

Question 5 Mark all correct statements. (2 Points)

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

2 · u(t� 1) is:

A Linear

B Time-Invariant

C Causal

D Memoryless / Static

Corrected

2 System Modeling

Box 1: Questions 6, 7

Description: Consider an electric motor which you would like to operate at a constant rotationalspeed !

0

. Applying a voltage U(t) results in a change in the circuit current I(t), which is governedby the di↵erential equation

L · d

dt

I(t) = �R · I(t)� · !(t) + U(t) , (1)

whereby L is the circuit inductance, R its resistance and a constant relating the motor speed!(t) to an electro motor-force (EMF). The dynamics of the motor speed are given by

⇥ · d

dt

!(t) = �d · !(t) + T (t) , (2)

where ⇥ represents its mechanical inertia, d a friction constant and T (t) = · I(t) the current-dependent motor torque.

Box 2: Question 6

r(t)C

u(t)

−P

y(t)e(t) x(t)

Question 6 Choose the correct answer. (1 Point)

Relate the variables in the block diagram above to the correct signals.

Au(t) = U(t), x(t) =

!(t)I(t)

�, y(t) = !(t), r(t) = !

0

Bu(t) = U(t), x(t) =

U(t)T (t)

�, y(t) = T (t), r(t) = !

0

Cu(t) = U(t), x(t) =

!(t)I(t)

�, y(t) = I(t), r(t) = !

0

Du(t) = I(t), x(t) =

!(t)U(t)

�, y(t) = I(t), r(t) = !

0

Eu(t) = I(t), x(t) =

!(t)T (t)

�, y(t) = !(t), r(t) = !

0

Corrected

Question 7 Choose the correct answer. (1 Point)

A colleague tells you that the circuit inductance is very small and can actually be neglected. Thearising motor model can now be represented as. . .

A ...a second-order system.

B ...a first order system.

C ...an integrator.

D ...a static system.

Question 8 Choose the correct answer. (1 Point)

Consider the nonlinear systemd

dt

x(t) =�x(t)� u(t)

�4

y(t) = x(t)� 20 .

You are measuring a constant output y(t) = 10.What is the value of x(t) and u(t)?

Ax(t) = 20 · t and u(t) = 20.

Bx(t) = 10 and u(t) = 30.

Cx(t) = 20 and u(t) = 20.

Dx(t) = 30 and u(t) = 30.

Question 9 Choose the correct answer. (1 Point)

Given the system

d

dt

x(t) = x(t)2 + 5 · u(t)� 10

y(t) =4 · x(t)� 12

u(t),

you have to linearize it around the equilibrium xe = 0, ue = 2.Which are the state-space matrices A, b, c and d?

AA = 0, b = 5, c = 4, d = �3.

BA = 5, b = 5, c = 2, d = �3.

CA = �10, b = 5, c = 2, d = 3.

DA = 0, b = 5, c = 2, d = 3.

Corrected

3 Linear Systems Analysis

Box 3: Question 10

The figure below shows the step response of a second-order system.

time

y

Question 10 Choose the correct answer. (1 Point)

Assess the bounded-input bounded-output (BIBO) stability.

A BIBO unstable

B BIBO stable

C No conclusion possible

Corrected

Box 4: Questions 11, 12

The figure below shows the response of an internal state x of a system as an impulse is applied tothe system’s input.

time

internal

statex

Question 11 Choose the correct answer. (1 Point)

A No conclusion about the stability is possible.

B The system is unstable.

C The system is asymptotically stable.

D The system is Lyapunov stable.

Question 12 Choose the correct answer. (1 Point)

A The system is BIBO stable.

B No conclusion about the BIBO stability is possible.

C The system is BIBO unstable.

Question 13 Choose the correct answer. (1 Point)

Given a system with state space representation

x(t) =

�1 10 · ↵2

0 ↵� 1

�· x(t) +

1↵

�· u(t)

y(t) = ↵ · x(t) + u(t) ,

where ↵ 2 R. For which values of ↵ is the system asymptotically stable?

A↵ < 1.

B↵ < 0.

C↵ 1.

D↵ � 1.

Corrected

Box 5: Question 14

Description: The figure below shows the step response of a first-order system of the form

x(t) = a · x(t) + b · u(t)y(t) = x(t) .

(3)

t [s]

y(t)

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

2

4

6

8

10

Question 14 Choose the correct answer. (1 Point)

Which are the values of a and b?

Aa = �2 and b = 20.

Ba = �0.5 and b = �10.

Ca = �0.5 and b = 20.

Da = �1 and b = 10.

Box 6: Question 15

Description: The figure below shows the impulse responses yA(t) and yB(t) of two systems A

and B. Both systems are linear time-invariant second order systems. In addition to the responses,an unknown function f(t) and its tangential line at time t = 0 is drawn in the plot to the right.

0 1 2 3 4 5 6�1

0

1

time in seconds

yA(t)

0 1 2 3 4 5 6�1

0

1

time in seconds

y(t) = f(t)yB(t)

y(t) = f(t = 0) · t+ f(t = 0)

Question 15 Choose the correct answer. (1 Point)

Which are the eigenvalues �A and �B of both systems?

A�A = 2⇡ and �B = � 1

2

.

B�A = 0± j and �B = � 1

2

± j.

C�A = 0± j · 2⇡ and �B = �2± j · 2⇡.

D�A = 0± j · 2⇡ and �B = � 1

2

± j · 2⇡.

Corrected

Box 7: Questions 16, 17

Description: A LTI system has the form

A =

0 01 0

�, b =

10

�, c =

⇥0 1

⇤, d = 0 .

Question 16 Choose the correct answer. (1 Point)

The system in the above equation is . . .

A only observable.

B only controllable.

C observable and controllable.

D neither observable nor controllable.

Question 17 Choose the correct answer. (1 Point)

For x1

(t = 0) = 1, x2

(t = 0) = 0 the initial condition response of the system is . . .

Ay(t) = 0.

By(t) = h(t).

Cy(t) = h(t) · t.

Dy(t) = h(t) · et.

Corrected

4 Frequency Response

Box 8: Questions 18, 19

Description: Given a linear time-invariant system in state-space description.

x(t) =

2

4�1 0 02 0 �81 0 �4

3

5 · x(t) +

2

4200

3

5 · u(t)

y(t) =⇥1 0 1

⇤· x(t) .

Question 18 Choose the correct answer. (1 Point)

The transfer function g(s) is . . .

Ag(s) = 2s+10

s(s+4)(s+1)

.

Bg(s) = 2s+10

s2+5s+4

.

Cg(s) = s+5

s2+5s+4

.

Dg(s) = 2(s+5)

s2+4s+5

.

Question 19 Choose the correct answer. (1 Point)

The system is . . .

A Asymptotically stable.

B Lyapunov stable.

C Unstable.

Question 20 Choose the correct answer. (1 Point)

Given the tranfer function g(s) of a system.

g(s) =3s+ 7

s(s+ 1)

A unit step is applied to the system’s input:

u(t) = h(t) .

The time response y(t) is . . .

Ay(t) = h(t) · (�4 + 7t+ 4e�t).

By(t) = h(t) · (2 + 7t+ 7e�t).

Cy(t) = h(t) · (7� 4t+ 4e�t).

Dy(t) = h(t) · (7� 4t+ 4e�t).

Corrected

Box 9: Question 21

Given the tranfer function g(s) of a system.

g(s) =s+ 3

s

2 + 0.7s+ 1

In addition the figure below shows the four time responses A, B, C and D.

0 5 10 151

0

1

2

3

4

time in seconds

y(t)

diagram 1

0 5 10 151

0

1

2

3

4

time in seconds

diagram 2

0 5 10 151

0

1

2

3

4

time in seconds

diagram 3

0 5 10 151

0

1

2

3

4

time in seconds

diagram 4

Question 21 Choose the correct answer. (1 Point)

Which of the four diagrams shows the correct step response of the system?

A Diagram 3.

B Diagram 4.

C Diagram 1.

D Diagram 2.

Corrected

Box 10: Question 22

Question 22 Choose the correct answer. (1 Point)

Select the transfer function L(s) which matches the Bode Plot given above:

AL(s) = 0.01 · (s+10)

2

s+0.1

BL(s) = 10

s+0.1

CL(s) = 10

0.1�s

DL(s) = s+100

(s+0.1)(s�100)

Question 23 Choose the correct answer. (1 Point)

What is the steady state output of the system

x(t) = �x(t) + u(t), y(t) = �x(t), (4)

when u(t) = sin(!t)?

Ayss(t) =

! cos(!t)�sin(!t)1+!2

Byss(t) =

1p1+!2 sin(!t� arctan(!))

Cyss(t) =

sin(!t)�!2cos(!t)

1+!2

Corrected

Box 11: Question 24

Consider the following Bode plot for a loop transfer function L(s):

Question 24 Choose the correct answer. (1 Point)

How many poles and zeros does L(s) have?

AL(s) has 2 poles and 0 zeros.

BL(s) has 4 poles and 2 zeros.

CL(s) has 3 poles and 2 zeros.

DL(s) has 3 poles and 0 zeros.

Corrected

Box 12: Question 25

Consider the following Bode plot of a transfer function g(s):

Question 25 Choose the correct answer. (1 Point)

Select the transfer function which matches the Bode plot.

Ag(s) = �20

s+1

Bg(s) = 10

s�1

Cg(s) = 20

s�1

Dg(s) = 10

s+1

Corrected

5 Polar Plot and Closed-Loop Stability

Box 13: Question 26

The Nyquist plot for a loop transfer function L(s) with two unstable poles is shown below on theleft. On the right is a feedback configuration for the closed loop system.

Question 26 Mark all correct statements. (2 Points)

Which of the following statements are true about the closed loop system T (s) = kL(s)1+kL(s)?

A The closed loop system T (s) is stable when k = 1.

B The closed loop system T (s) is unstable when k > 2.

C The closed loop system T (s) is unstable when k = 1.

D The closed loop system T (s) is stable when k > 2.

Corrected

Box 14: Question 27

The Bode plot for a plant transfer function P (s) is given below.

Question 27 Choose the correct answer. (1 Point)

Which of the controller designs will produce a stable closed loop system?

AC(s) = 0.01s+1

0.001s+1

BC(s) = 1

0.01s+1

CC(s) = 0.1

DC(s) = 10.0

Corrected

Box 15: Question 28

The Nyquist plot for a loop transfer function of the form

L(s) =k

(s+ p

1

)(s+ p

2

)(s+ p

3

)(5)

is given below.

Question 28 Choose the correct answer. (1 Point)

How many unstable poles does the closed loop system T (s) = L(s)1+L(s) have?

AT (s) has 2 unstable poles.

BT (s) has 0 unstable poles.

CT (s) has 1 unstable pole.

D Not enough information is provided.

Corrected

Box 16: Question 29

The bode plot for a system with transfer function g(s) is given below.

Question 29 Mark all correct statements. (2 Points)

Select all answers that correctly classify the stability of the system.

Ag(s) is unstable.

Bg(s) is asymptotically stable.

Cg(s) is Lyapunov stable.

Dg(s) is bounded input bounded output (BIBO) stable.

Corrected

Box 17: Question 30

Consider the following Nyquist plot of an open loop gain L(s):

Question 30 Choose the correct answer. (1 Point)

Select the transfer functions which match the Nyquist plot.

AL(s) = 10

(s�1)

3

BL(s) = 10

(s+1)

3

CL(s) = �10

(s+1)

3

DL(s) = 10

(1�s)3

Corrected

Box 18: Question 31

Consider the system described by the following block diagram

P (s)y

C(s)e

–

r

+

where

P (s) =1

10s+ 1, C(s) = k.

Question 31 Choose the correct answer. (1 Point)

Determine the smallest positive gain k such that the feedback system is stable and, when r(t)is a unit step, limt!1 |e(t)| 0.1.The smallest positive gain k is:

A 10

B 2

C 1

D 5

E 9

Corrected

Question 32 Choose the correct answer. (1 Point)

We are given the transfer function:

g(s) =(s+ 4)

(s4 � 9s2).

Which of the following is the root locus plot of g(s)?

A

-10 -8 -6 -4 -2 0 2 4 6 8 10-15

-10

-5

0

5

10

15Root Locus

Real Axis (seconds-1)

Imag

inar

y Ax

is (s

econ

ds-1

)

B

-10 -8 -6 -4 -2 0 2 4 6 8 10-15

-10

-5

0

5

10

15Root Locus

Real Axis (seconds-1)

Imag

inar

y Ax

is (s

econ

ds-1

)

C

-10 -8 -6 -4 -2 0 2 4 6 8 10-15

-10

-5

0

5

10

15Root Locus

Real Axis (seconds-1)

Imag

inar

y Ax

is (s

econ

ds-1

)

D

-10 -8 -6 -4 -2 0 2 4 6 8 10-15

-10

-5

0

5

10

15Root Locus

Real Axis (seconds-1)Im

agin

ary

Axis

(sec

onds

-1)

E

-10 -8 -6 -4 -2 0 2 4 6 8 10-3

-2

-1

0

1

2

3Root Locus

Real Axis (seconds-1)

Imag

inar

y Ax

is (s

econ

ds-1

)

Corrected

6 Feedback Control and Specifications

Question 33 Choose the correct answer. (1 Point)

For the following system, you have the choice between implementing two di↵erent sensors:y

1

(t) = x

1

(t) or y2

(t) = x

2

(t).

x(t) =

3 10 2

�x(t) +

01

�u(t)

Which sensor should you choose?

Ay

2

(t)

By

1

(t)

Corrected

Box 19: Questions 34, 35, 36

Consider the block diagram given below to answer the following three questions:

r

+C(s) P

1

(s)

P

2

(s)

y

�+

�

Question 34 Choose the correct answer. (1 Point)

Suppose P

1

(s) = 2

s�1

, P2

= 1

s�1

, and C(s) = 2. Which of the following choices is the correcttransfer function, G(s) = Y (s)/R(s), from r to y?

AG(s) = 4

s+1

BG(s) = 4(s�2)

(s+1)

CG(s) = 4(s+1)

(s�1)

2

DG(s) = 4(s�1)

(s+1)

2

EG(s) = 4

s�1

Question 35 Choose the correct answer. (1 Point)

Is the system defined in the above problem internally stable?

A No.

B Yes.

Question 36 Choose the correct answer. (1 Point)

Now assume instead that P1

(s) = P

2

(s) = P (s) for some P (s) and that C(s) is stable. Which ofthe following statements is true?

A The system is internally stable if and only if P (s) has no zeros with positive real part.

B The system is NOT internally stable for any choice of P (s).

C The system is internally stable for any choice of P (s).

D The system is internally stable if and only if P (s) is stable.

Corrected

Box 20: Question 37

Consider the state-space system below, where a is a real parameter:

x(t) =

1 01 �4

�x(t) +

a

1

�u(t)

y(t) =⇥2 a

⇤x(t)

Question 37 Choose the correct answer. (1 Point)

Is the system stabilizable for a = 0?

A No.

B Yes.

Question 38 Mark all correct statements. (2 Points)

Mark all of the following statements about control limitations that are true.

A The dominant right-half-plane pole puts a lower limit on the system’s bandwidth.

B The sensitivity function, S(s), plus the complementary sensitivity function, T (s), must equalone for all s.

C A plant’s right-half-plane zero limits the bandwidth of the closed-loop system.

D When a plant has a right-half-plane zero z, the sensitivity function at z, S(z), must equal 1.

Corrected

7 Control Design

Box 21: Question 39

The figure below shows the block diagram of a cascaded controller:

+ + A(s) B(s)r

e

1

e

2

y

�

�

Question 39 Choose the correct answer. (1 Point)

Derive the transfer function T (s) from r ! y for the system depicted in the block diagram above.

A B(s)1+A(s)+A(s)B(s)

B A(s)B(s)1+A(s)+B(s)+A(s)B(s)

C A(s)B(s)1+A(s)+A(s)B(s)

D A(s)B(s)1�A(s)+A(s)B(s)

E A(s)B(s)1+A(s)B(s)

Question 40 Choose the correct answer. (1 Point)

Let L(s) = (s+5)(s+3)

(s+2)

2 be an open loop transfer function in a standard feedback architecture. Letthe input to the system be a unit step input.What is the static error e1 of the system?

A 15

19

B 4

19

C 1

D 1

2

E 13

19

Question 41 Choose the correct answer. (1 Point)

To which of the following ideal systems can Ziegler/Nichols-tuning by using critical oscillation notbe applied?

A 7

(s+1)

5

B 3

s+1

C 4

s+1

e

�0.1s

D 3

(s+1)

3

Corrected

Question 42 Choose the correct answer. (1 Point)

Your system engineering counterpart gives you the system characteristics below:

• Noise above !n = 500 rad/s needs to be suppressed.

• Disturbances below !d = 5 rad/s need to be rejected.

• Modeling uncertainty reaches 100 percent at !2

= 40 rad/s.

• An unstable pole at s = 0.5 is present.

• The plant has a nonminimumphase zero at s = 30.

You want to design a control system where the crossover frequency satisfies

max{10 · !d, 2 · !⇡} < !c < min{0.5 · !2

, 0.5 · !delay, 0.5 · !⇣ , 0.1 · !n} (6)

Can you find a feasible crossover frequency !c in presence of the above specifications?

A Yes.

B No.

Box 22: Question 43

The plot below shows step responses of di↵erent variations of PID controllers (P, PI, PD, PID)

Time [s]0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

Uni

t ste

p re

spon

se [-

]

0

0.5

1

1.51234without controlreference

Question 43 Choose the correct answer. (1 Point)

Which step response in the figure above belongs to the PD-controller?

A 2

B 3

C 1

D 4

Corrected

Box 23: Question 44

Consider the following 4 implementations of an anti - windup scheme:

1.

2.

3.

4.

Corrected

Question 44 Choose the correct answer. (1 Point)

Which of the above is a correct anti-windup scheme?

A 1

B 3

C 4

D 2

Corrected

Box 24: Question 45

Consider the following Bode plot of a plant:

10-1 100 101

Ampl

itude

[dB]

-30

-20

-10

0

10

20

Bode plot

Frequency [rad/s]10-1 100 101

Phas

e [°

]

-180-150-120

-90

Question 45 Choose the correct answer. (1 Point)

Use the bode plot shown above to determine the approximate crossover frequency !c, phase margin' and gain margin �.

A!c ⇡ 2.5rad/s, ' ⇡ 30�, � ⇡ 1

B!c ⇡ 4.4rad/s, ' ⇡ 15�, � ⇡ 3

C!c ⇡ 2.5rad/s, ' ⇡ 30�, � ⇡ 3

D!c ⇡ 4.4rad/s, ' ⇡ 30�, � ⇡ 3