exam - ethz.ch · exam february 13, 2019 control systems ii (151-0590-00) dr. jacopo tani exam exam...

Exam February 13, 2019

Control Systems II (151-0590-00) Dr. Jacopo Tani

Exam

Exam Duration: 120 minutes + 15 minutes reading time

Number of Problems: 28

Number of Points: 37

Permitted aids: 10 pages (5 sheets) A4. Sheets must be written byhand.

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 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 on the grading page.

Good luck!

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y +1/3/58+ yQuestion 1 Mark all correct statements. (2 Points)

Which of the following statements about the sensitivity function S and the complementary sensi-tivity function T are correct for a SISO system?

A At every frequency |S|+ |T | ≤ 1

B Reduction of |S| at low frequencies doesrestrict the ability to reduce |S| at highfrequencies.

C S can be shaped arbitrarily in the designphase if T is allowed to take any values.

D At every frequency ||S| − |T || ≤ 1

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y +1/4/57+ yQuestion 2 Choose the correct answer. (1 Point)

Consider a continuous-time LTI system:

x(t) =

[0 11 0

]x(t) +

[1−1

]u(t).

What is the system description resulting from forward Euler discretization with sampling time Ts?

A x[k + 1] =

[0 TsTs 0

]x[k] +

[Ts−Ts

]u[k].

B x[k + 1] =

[1 TsTs 1

]x[k] +

[Ts−Ts

]u[k].

C x[k + 1] =

[1 TsTs −1

]x[k] +

[TsTs

]u[k].

D x[k + 1] =

[1 Ts + 1

Ts + 1 1

]x[k] +

[−TsTs

]u[k].

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Using a sampling time of Ts = 13 seconds, which signals can be sampled without aliasing?

A∑4k=1 sin

(πk3 t+ π

)B∑1000k=1 cos

(3πk+2 t

)C∑5k=1 sin

(πk3 t)

D∑3k=1 cos

(πk3 t)

E cos(4π t+ π)

Question 4 Choose the correct answer. (1 Point)

To prevent aliasing when implementing a control scheme on a microcontroller, a dedicated anti-aliasing filter (AAF) is typically used. Which of the following is true?

An AAF is designed to...

A ...attenuate high frequency components ofthe continuous time signal entering theplant.

B ...attenuate low frequency components ofthe continuous time signal entering thecontroller block.

C ...attenuate low frequency components ofthe continuous time signal entering theplant.

D ...attenuate high frequency components ofthe continuous time signal entering thecontroller block.

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A minimum phase SISO LTI system is controlled such that lims→∞ sL(s) = 0, with L(s) beingthe loop transfer function. You are given a plot of the sensitivity function S(s):

log|S(jω)|

ω

A2

A1

0

Assuming the areas A1 and A2 are identical. Which of the following statements is true?

A The open-loop system is unstable.

B The open-loop system is not observable.

C The open-loop system is stable.

D The open-loop system is controllable.

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y +1/7/54+ yBox 1: Questions 6, 7

Consider the following matrix for the the discrete system x[k + 1] = Ax[k] with

A =

[c− 1 0− 2

3 c+ 12

]


For which values of c ∈ R is the system asymptotically stable?

A 12 < c < 2

3

B 0 < c < 12

C − 12 < c < 0

D 0 < c < 1


For c = 12 the system is:

A Stable

B Asymptotically stable if the initial state is negative

C Unstable

D Asymptotically stable if the initial state is bounded

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Consider the MIMO system described by the following transfer function matrix

P (s) =

[1

(s−1)(s+3)2(s+1)(s−1)3

0 1(s−1)2(s+3)

].

The poles p of the system are:

A {−3,−3,−3, 2, 2, 1, 1, 1}B {−3,−3,−3, 1, 1}

C {−3,−3,−3, 1, 1, 1}D {−3,−3,−3, 2, 1, 1, 1, 1}

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Given the matrix

A =

[1 0 10 1 0

]and its SVD decomposition A = UΣV T . Mark all correct matrices U, V that fulfill the SVDdecomposition.

A

U =

[1 00 1

], V =

1 0 −1− 1√

21√2

0

0 1√2

1√2

B

U =

[−1 00 1

], V =

1√2

0 − 1√2

1√2

1√2

0

0 0 1

C

U =

[1 00 −1

], V =

1√2

0 1√2

0 −1 01√2

0 − 1√2

D

U =

[1 00 1

], V =

1√2

0 1√2

0 1 01√2

0 − 1√2

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Consider the following dynamic system

x(t) =

[0 12 1

]x(t) +

[01

]u(t)

y(t) =[1 −2

]x(t),

which of the following statements about the system stability are correct?

A The system is internally stable

B The system is unstable

C The system is BIBO stable

D The system is not BIBO stable

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y +1/11/50+ yBox 2: Questions 11, 12

Consider this block diagram for the following two questions:

G(s)

K(s)

+

+

w1 e1

e2 w2


Consider the following transfer function matrices with a real parameter a:

G(s) =

[ s−1a+s 0

0 1s+1

]K(s) =

[0 10 0

]For which choice of a is the closed-loop system internally stable?

A a ≤ 1

B a > 0

C For any value of a.

D a > −1


Letting G(s) = 2s2+2s+2 and K(s) = −k with k ∈ R, what values of k guarantee closed loop

stability?

A |k| > 1.

B |k| < 1.

C None of the above.

D 0.5 < |k| < 2.

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What is the condition number of a MIMO system?

A None of the other options.

B A measure of its “directionality”.

C A measure of its “performance”.

D A measure of its “robustness”.

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Let C(s) = k with k ∈ R, k > 0 be a proportional controller for a SISO plant described by thenominal transfer function P0(s) = 1

s+1 . What values of k guarantee robust stability of the feedbacksystem in presence of multiplicative plant uncertainty, such that: P (s) = P0(s)(1 + W (s)∆(s)),|∆(jω)| ≤ 1,∀ω, and W (s) = 1?

A k > ω2 − 1.

B k > 0.

C k < ω2+12 .

D This system cannot be robustly stabilized.

E k > ω2+12 .

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What is the relative-gain-array (RGA) of the following plant P (s) at ω = 1 rad/s?

P (s) =

[ s+2(s+1)2

s+4s2+5s+4

0 s+1−s2−2s+1

]A RGA(P ) =

[1 00 1

]B RGA(P ) =

[1/3 2/32/3 1/3

] C RGA(P ) =

[1/2 1/21/2 1/2

]D RGA(P ) =

[0 11 0

]

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Consider the following MIMO system:

P (s) =

[ s+2s+1 −s1 s+1

s+2

]What is the frequency ω at which the input-output-pairing is maximally coupled? Hint: This isthe frequency, for which it is most difficult to decouple the system.

A ω = 2 rad/s

B ω =√

3 rad/s

C ω = 1 rad/s

D ω = 0 rad/s

Question 17 Mark all correct statements. (2 Points)

Which statements about the Q-parametrization are correct?

A The Q-parametrization ideally makes iteasier to tune a controller while ensuringinternal stability.

B The Q-parametrization is necessary if onewants to apply SISO control methods toMIMO systems.

C The Q-parametrization cannot be appliedto unstable plants.

D Non-minimum phase zeros in the plantlead to unstable Q-functions.

E Q-parametrization is necessary to controltime-delayed systems.

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Consider an LTI system:

x(t) =

[a b1 1

]x(t) +

[ab

]u(t)

For which of the values a, b ∈ R below is the system reachable?

A a = −1, b = 1

B a = 1, b = 1

C a =√

2, b =√

3

D a = 2, b = 2


Consider the following LTI-system:

x(t) =

[0 1−1 1

]x(t) +

[01

]u(t)

For which gain K =[k1 k2

]and controller u(t) = −Kx(t) does the closed-loop system oscillate

with 1 rad/s without decaying?

A k1 = 0, k2 = 1

B k1 = 3, k2 = 1

C For no possible K since the system is notreachable.

D k1 = 1, k2 = 0

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Consider the LTI-system:

x(t) =

[0 1−2 0

]x(t) +

[01

]u(t)

with cost matrices

R = 1 Q =

[5 00 2

]N =

[00

]And the Riccati equation:

Φ(BR−1BT )Φ− Φ(A−BR−1NT )− (A−BR−1NT )TΦ−Q = 0

Which Φ solves the Riccati equation above?

A Φ =

[6 −4−4 2

]B Φ =

[1 11 4

] C Φ =

[−6 11 −2

]D Φ =

[6 11 2

]

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An LQR controller with high-precision requirements must be designed. It is used to follow atrajectory that penalizes state x1 four times as much as state x2. Additionally, the deviations ofthe states and control-action should be penalized equally. Given the cost function∫ ∞

0

x(t)TQx(t) + u(t)TRu(t) dt

,which selection of weights Q,R is most suitable?

A Q =

[4 00 1

], R = 1

B Q =

[1 00 1

], R = 2

C Q =

[4 00 1

], R = 5

D Q =

[2 00 1

], R = 3

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Consider an LTI dynamic system with two states x1(t), x2(t) with initial conditions x1(0) ∈ R andx2(0) ∈ R for which the output is given by:

y(t) = c1x1(t) + c2x2(t)

=1

3((2c1 + 2c2)x1(0) + (c1 + c2)x2(0)) et +

1

3((c1 − 2c2)x1(0) + (−c1 + 2c2)x2(0)) e−2t

Which statements about the observability and detectability of the system are correct?

A The A matrix of the system must beknown to assess observability.

B The system is detectable for any choice ofc1, c2.

C If c1 = 2c2, the system is not observable

but detectable.

D If c1 = −c2, the system is not detectable.

E If c1 = −c2, the system is not detectablebut observable.


A dynamic system is described by the mathematical equations

x(t) =

[−1 10 −2

]x(t) +

[058

]u(t)

y(t) =[1 0

]x(t)

For a Luenberger observer, which gain L will place all observer eigenvalues at −3?

A L =[1 3

]TB L =

[3 2

]T C L =[2 3

]TD L =

[3 1

]T

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y +1/20/41+ yBox 3: Questions 24, 25

The following two exercises consider the duality of state feedback and state observers. Considerthe following diagrams that represent the dynamics of the system states and the dynamics of theobservation error:

+M1

∫M2

M3

+M4

∫M5

M6

system dynamics error dynamics

wanted

known

wanted known

Both flow charts are divided into the known quantities which are given by the LTI system charac-teristics and the wanted quantities which are specified by controller / observer design.


What matrices M1, ...,M6 inserted into above diagrams correctly represent the system state dy-namics of an LTI sytem (A,B,C,D) with state feedback K and of a Luenberger observer withgain L, respectively?

A M1 = B, M2 = −KT , M3 = A,M4 = −L, M5 = A, M6 = C

B M1 = B, M2 = A, M3 = −K,M4 = −L, M5 = A, M6 = C

C M1 = BT , M2 = A, M3 = −K,M4 = −L, M5 = AT , M6 = C

D M1 = B, M2 = A, M3 = −K,M4 = −L, M5 = AT , M6 = C


Fortunately, the problem of observer design is dual to the problem of state feedback. This meansthat the theoretic results used for the derivation of the state feedback gain K can be used tocompute the observer gain L. What is a necessary condition to bring the structure of errordynamics on the right hand side to the equivalent form of the structure of the state dynamics onthe left hand side?

A The LTI system has a symmetric matrixA and nonsingular matrices B and C.

B A matrix and its transpose have the sameeigenvalues.

C The LTI system is detectable.

D The LTI system has an equal number ofinputs and outputs.

E It holds that A−1 = AT

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When should H∞ be used to design a controller rather than the LQG method?

A When the LQR cannot be applied due tothe large number of system states.

B When the gains of the sensitivity functionmust be limited at some frequencies.

C When the energy of the control signal mustbe minimized.

D When the specifications in frequency spaceare not relevant and no feedforward con-trol is used.

E When the application requires that somerequirements are met for all modeled un-certainties.

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Consider a nonlinear system:

x = f(x, u)

y = g(x, u)

What statements about nonlinear systems and control of nonlinear systems are correct?

A Nonlinear systems are never fully observ-able, unless they are input-state lineariz-able.

B The system x = x(x2 − a2), a ∈ R is un-stable at x = 0 because its region of at-traction is not sufficiently large.

C If one equilibrium point of the nonlin-ear system is stable, all of its equilibriumpoints are stable.

D If f and / or g is a nonlinear function, thedynamic system is called nonlinear.

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A total of 6 nonlinear systems have state-space descriptions:

System 1: x = −cos(u+ x2)

System 2: x = cos(u)

System 3: x = −cos(u)x2

System 4: x = 2√x(x+ ux2

)System 5: x = cos2(x) + u

System 6: x = x2 + ux3

Additionally, 6 transformations are available to compute a control signal u for the nonlinear sys-tems:

Transformation 1: u = arccos(v + x)

Transformation 2: u = −x2 + arccos(−v − x)

Transformation 3: u = 1x4 − x2 + v

Transformation 4: u =(

1x2 − x+ v

)1

2√x

Transformation 5: u = arccos(− vx2

)Transformation 6: u = −1 + sin2(x) + x+ v

After the transformations are applied to the nonlinear systems, all of them can be described bythe linear ordinary differential equation z = z + v with control signal v.Which transformation was applied to which of the nonlinear systems?

A System 1→ Transformation 2System 2→ Transformation 1System 3→ Transformation 5System 4→ Transformation 4System 5→ Transformation 6System 6→ Transformation 3

B System 1→ Transformation 6System 2→ Transformation 5System 3→ Transformation 1System 4→ Transformation 4System 5→ Transformation 2System 6→ Transformation 3

C System 1→ Transformation 6System 2→ Transformation 1System 3→ Transformation 5System 4→ Transformation 3System 5→ Transformation 2System 6→ Transformation 4

D System 1→ Transformation 2System 2→ Transformation 5System 3→ Transformation 1System 4→ Transformation 4System 5→ Transformation 6System 6→ Transformation 3

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How to select answer B :

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Answers must be given exclusively on this sheet;answers given on the other sheets will not be counted.

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exam - ethz.ch · exam february 13, 2019 control systems ii (151-0590-00) dr. jacopo tani exam exam...

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