exam - ethz.ch€¦ · exam august 24, 2018 control systems i (151-0591-00l) prof emilio frazzoli...
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Exam August 24, 2018
Control Systems I (151-0591-00L) Prof Emilio Frazzoli
Exam
Exam Duration: 120 minutes + 15 minutes reading time
Number of Problems: 37
Number of Points: 50
Permitted aids: 2 pages (1 sheet) of A4 notes, List of translationsof German technical words to English, simple cal-culators 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
Question 1 Mark all correct statements. (2 Points)What statement is true about a feedback control system?
A It can feed sensor noise into the system.
B It can destabilize a stable plant.
C It can reject external disturbances.
D It always relies on precise knowledge of the plant.
Question 2 Mark all correct statements. (2 Points)
All signals are scalars. The system y(t) = (t2 + 1)u(t2)sin(πt2
5) with t ∈ R with input signal u(t)
and output signal y(t) is:
A Linear
B Memoryless
C Causal
D Time-invariant
Question 3 Mark all correct statements. (2 Points)
Three input signals u(t) and the corresponding output signals y(t) of a system are shown in thefigure below. Assume u(t) = 0 for all t < 0. The system is:
A Non-causal
B Nonlinear
C Memoryless
D Linear time-invariant
Corrected
Question 4 Mark all correct statements. (2 Points)
Consider a system whose input-output relation is described as y(t) + y(t) = u(t). The system is:
A Time-invariant
B Causal
C Memoryless
D Linear
Question 5 Choose the correct answer. (1 Point)
Given the system
x1(t) = x21(t)− sin(3x2(t)) + u3(t)
x2(t) = x2(t)− u(t) + x1(t)e−x2(t),
you have to linearize it around the equilibrium point xe =
[00
], ue = 0. What are the state space
matrices A and B?
A A =
[−3 0−1 −1
], B =
[0−1
]B A =
[0 −31 1
], B =
[0−1
] C A =
[3 01 1
], B =
[−10
]D A =
[0 31 −1
], B =
[01
]
Question 6 Choose the correct answer. (1 Point)
Consider a system with the following dynamics,
x(t) =
[0 10 0
]x(t) +
[01
]u(t) .
If u(t) = e−t, t ≥ 0, and x(0) =
[00
], find x(t) for t ≥ 0 .
A x(t) =
[−1 + t+ e−t
1− e−t]
B x(t) =
[1 + t+ e−t
−1 + e−t
] C x(t) =
[−1 + e−t
1 + t− e−t]
D x(t) =
[−1− e−t−1 + t+ e−t
]
Corrected
Question 7 Choose the correct answer. (1 Point)
Consider the two systems with output signal y(t) and input signal u(t), described below:
1. y(t) = sin(t)u(t)
2. y(t) =∫ t
0sin(τ)u(τ)dτ
Which system is BIBO stable?
A None of the systems
B System 2
C Both of the systems
D System 1
Box 1: Questions 8, 9, 10
You are given a linear time-invariant system in state-space form.
x(t) =
0 1 20 0 30 0 −1
x(t) +
001
u(t)
y(t) =[0 0 1
]x(t)
Question 8 Choose the correct answer. (1 Point)
The transfer function g(s) is:
A g(s) = 3+2ss2(s+1)
B g(s) = 1s+1
C g(s) = 3−2ss(s+1)
D g(s) 3s(s+1)
Question 9 Choose the correct answer. (1 Point)
The system is:
A Only observable
B Controllable and observable
C Only controllable
D Neither controllable nor observable
Corrected
Question 10 Choose the correct answer. (1 Point)
Consider an LTI system with the following dynamics with input u(t) and state x(t).
x(t) =
[0 1−2 −3
]x(t) +
[02
]u(t),
y =[1 0
]x(t)
Find a controller of the form u(t) = −Kx(t), K ∈ R1×2 such that the eigenvalues of the closed-loopsystem are -3 and -5.
A u = −[6.5 2.5
] [x1
x2
]B u = −
[4 3
] [x1
x2
] C u =[4 3
] [x1
x2
]D u =
[6.5 2.5
] [x1
x2
]
Corrected
Question 11 Choose the correct answer. (1 Point)
You are given the following set of input to output transfer functions:
1.
G(s) =s+ 0.5
s2 + 0.5s+ 1
2.
G(s) =1− s
s2 + 0.5s+ 1
3.
G(s) =s
s2 + 0.5s+ 1
4.
G(s) =1
s2 + 0.5s+ 1
0 5 10 15 20 25
Time [s]
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
Uni
t ste
p re
spon
e ou
tput
Figure A
0 5 10 15 20 25
Time [s]
0.2
0.0
0.2
0.4
0.6
Uni
t ste
p re
spon
e ou
tput
Figure B
0 5 10 15 20 25
Time [s]
0.25
0.00
0.25
0.50
0.75
1.00
1.25
1.50
Uni
t ste
p re
spon
e ou
tput
Figure C
0 5 10 15 20 25
Time [s]
0.0
0.2
0.4
0.6
0.8
1.0
Uni
t ste
p re
spon
e ou
tput
Figure D
Choose the correct assignment of transfer functions to unit step responses.
A 1→ D, 2→ A, 3→ B, 4→ C
B 1→ D, 2→ C, 3→ B, 4→ A
C 1→ A, 2→ C, 3→ D, 4→ B
D 1→ A, 2→ B, 3→ C, 4→ D
Corrected
Question 12 Choose the correct answer. (1 Point)
What are the initial y0 and final values y∞ of an impulse response for the following input to outputtransfer function?
G(s) =s+ 1
2s2 + 0.5s+ 1
A y0 = 0.5, y∞ = 0
B y0 = 1, y∞ = 0
C y0 = 0, y∞ = 1
D y0 = 0, y∞ = 0.5
Question 13 Mark all correct statements. (2 Points)
You are given the following LTI system dynamics.
A =
0 1 00 0 10 12 −1
B =
001
C =
[−3 1 0
]D = 0
Which of the following system descriptions have the same input-output behavior as the followingsystem dynamics?
A G(s) = s−3s2−s+12
B G(s) = 1/4s(s/4+1)
C A =
[0 10 −4
]B =
[01
]C =
[1 0
]D = 0
D G(s) = 1/4s+4 −
1/4s
Question 14 Mark all correct statements. (2 Points)
Which of the following statements are correct?
A From the poles and Nyquist plot of a givenopen loop system L(s) one can determinewhether the closed-loop system T (s) is sta-ble.
B The root locus method has a built-in way
of dealing with uncertainty.
C The root locus method can quickly sketchcontrol design feasibility.
D Bode plots are commonly used for sophis-ticated control system design.
Corrected
Question 15 Mark all correct statements. (2 Points)
6 4 2 0 2 4
Real
300
200
100
0
100
200
300
Imag
inar
y
Which points are on the root locus?
You are given the open-loop transfer function L(s).
L(s) =s+ 2
s(s+ 1)(s+ 4)
Which of the following points (marked in the figure) p are on the positive root-locus for the aboveL(s)?
A p ≈ −1.5 + 0.0j
B p ≈ −3 + 0.0j
C p ≈ −1 + 0.0j
D p ≈ −5 + 0.0j
E p ≈ −1.5− 300j
Question 16 Mark all correct statements. (2 Points)
Which of the following statements are true?
A For k = 0 the open-loop poles are equal to the closed-loop zeros.
B The closed-loop poles are symmetric with respect to the real axis.
C For the root locus of a causal system, the number of closed-loop poles is equal to the numberof open-loop poles.
D The angle rule of the positive root-locus states that∑zi∠(s − zi) −
∑pi∠(s − pi) =
0◦(±q360◦), q ∈ Z
Question 17 Choose the correct answer. (1 Point)
Your coffee-enthusiast friend experimentally determined the second-order closed-loop poles a coffeemachine should have. Both poles should have real-part Re(π1,2) = −2.5 and there should be nooscillations.Your open-loop coffee machine transfer function is P (s) = 1
(s+2)(s+3) .
Can this be achieved? If yes, what is the P-controller gain k needed to reach the desired poles?
A k = 12
B k = 14
C k = 1
D It is not possible to reach the desiredclosed-loop poles.
Corrected
Question 18 Choose the correct answer. (1 Point)
You are given the following Bode plot
102
101
100
101
101
100
Mag
nitu
de
102
101
100
101
Frequency (rad/sec)
90
45
0
Pha
se (d
eg)
Which of the following transfer functions corresponds to the above Bode plot?
1.
G(s) =1− s
s2 + 0.5s+ 1
2.
G(s) =s+ 0.5
s2 + 0.5s+ 1
3.
G(s) =s
s2 + 0.5s+ 1
4.
G(s) =1
s2 + 0.5s+ 1
A 4
B 2
C 1
D 3
Corrected
Question 19 Choose the correct answer. (1 Point)
You are given the following magnitude plot of a plant. Furthermore you know that the plant doesnot have any unstable poles.
101
100
101
102
103
Frequency (rad/sec)
103
102
101
100
101
102
Mag
nitu
de
Magnitude plot of a transfer function
Which of the following phase plots corresponds to the above plant?
101
100
101
102
103
Frequency (rad/sec)
100
75
50
25
0
25
50
75
100
Pha
se (d
eg)
Phase A
101
100
101
102
103
Frequency (rad/sec)
100
75
50
25
0
25
50
75
100
Pha
se (d
eg)
Phase B
101
100
101
102
103
Frequency (rad/sec)
100
75
50
25
0
25
50
75
100
Pha
se (d
eg)
Phase C
101
100
101
102
103
Frequency (rad/sec)
100
75
50
25
0
25
50
75
100
Pha
se (d
eg)
Phase D
A Phase plot B
B Phase plot C
C Phase plot D
D Phase plot A
Corrected
Question 20 Mark all correct statements. (2 Points)
You are given the following Bode plot of plant P (s).
101
100
101
102
103
104
102
100
102
Mag
nitu
deBode controller design
101
100
101
102
103
Frequency (rad/s)
225
180
135
90
45
0
Pha
se (d
eg)
Mark all of the following controllers that are able to control the plant with a crossover frequencyof ωc = 10 rad/s and phase margin of at least ϕ = 30◦.
A PID controller
B PI controller
C P controller
D Lead/lag controller
Corrected
Question 21 Choose the correct answer. (1 Point)
You are given the closed-loop T (s) Bode plot of a transfer function.
101
100
101
102
104
103
102
101
Mag
nitu
de
101
100
101
102
Frequency (rad/s)
180
135
90
45
0
Pha
se (d
eg)
Which of the following open-loop L(s) transfer functions corresponds to the closed-loop T (s) bodeplot depicted above?
1.
L(s) =1
s+ 1
2.
L(s) =s+ 1
s+ 10
3.
L(s) =s
s2 + 2s+ 1
4.
L(s) =1
s2 + 2s+ 1
A 2
B 4
C 3
D 1
Corrected
Question 22 Mark all correct statements. (2 Points)
You are given an open-loop transfer function of the following form.
L(s) = ks+ a
s+ b, k, a, b ∈ R
The transfer function is parametrized with unspecified constants k, a, b.Which of the following Nyquist plots are possible plots of the above transfer function?
1 0 1 2 3 4
1.5
1.0
0.5
0.0
0.5
1.0
1.5
Nyquist plot 1
1.0 0.5 0.0 0.5 1.0 1.5 2.0
1.0
0.5
0.0
0.5
1.0
Nyquist plot 2
1.2 1.0 0.8 0.6 0.4 0.2 0.01.5
1.0
0.5
0.0
0.5
1.0
1.5Nyquist plot 3
1 0 1 2 3
2
1
0
1
2
Nyquist plot 4
A 2
B 3
C 1
D 4
Corrected
Question 23 Choose the correct answer. (1 Point)
You are given the transfer function
L(s) = ks+ a
(s+ b)(s+ c), k, a, b, c ∈ R
and its associated Nyquist plot:
2.0 1.5 1.0 0.5 0.0 0.5 1.0
1.5
1.0
0.5
0.0
0.5
1.0
1.5
Nyquist plot with undetermined parameters
You furthermore know that the zero is minimumphase a > 0.How many unstable poles does the closed-loop system have?
A 1
B 2
C 0
D This cannot be determined from the aboveinformation.
Corrected
Question 24 Choose the correct answer. (1 Point)
You are given the following Bode plot.
102
101
100
101
102
104103102101100101102103104
Mag
nitu
de
Bode plot
102
101
100
101
102
Frequency (rad/s)
90
135
180
Pha
se (d
eg)
What is the associated Nyquist plot to this Bode plot?
3.5 3.0 2.5 2.0 1.5 1.0 0.5 0.0
1.5
1.0
0.5
0.0
0.5
1.0
1.5
Nyquist plot 1
1 0 1 2 31.5
1.0
0.5
0.0
0.5
1.0
1.5Nyquist plot 2
35 30 25 20 15 10 5 0
15
10
5
0
5
10
15
Nyquist plot 3
3.5 3.0 2.5 2.0 1.5 1.0 0.5 0.0 0.5
2
1
0
1
2
Nyquist plot 4
A 3
B 4
C 1
D 2
Corrected
Box 2: Questions 25, 26, 27
You have designed and built a submarine to offer tours in lake Zurich. The propulsive force Fpand the drag Fd were supposed to be exactly aligned with the center of mass S but unfortunatelythere is a small error d such that there is a momentum acting on the submarine’s angle ϕ whichmakes it more difficult for the commander to keep a constant level of depth during the tour. Thesubmarine has moment of inertia Θ.
d
mg − FB
Fd
Fp
Θ
ϕ
αFs
S
Lβϕ
When the states x1 = ϕ and x2 = ϕ are introduced, the dynamics describing the angle ϕ are:
[x1
x2
]=
[0 1
0 − βΘ
] [x1
x2
]+
[0
(Fd + Fp)dΘ − Fs
LΘ sin(α)
]A controller to track a certain trajectory for ϕ should be implemented that acts solely on α and isindependent of Fd and Fp. You can directly measure ϕ and the assumption can be made that α issmall.
Question 25 Choose the correct answer. (1 Point)
What condition must be satisfied to simplify the dynamics with a dominant-pole approximation?Which statement is true?
A Θ must be sufficiently large.
B βΘ must be sufficiently large.
C The dominant pole would be s = − βΘ .
D None of the other answers are correct.
Question 26 Mark all correct statements. (2 Points)
Assume that βΘ = 0.01. Which of the following statements regarding the closed-loop control system
is correct?
A A controller of the form C(s) = k(1 + sz ), k ∈ R>0, z ∈ R>0 would produce good results for
big z and big k but cannot practically be implemented.
B A P controller C(s) = k, k ∈ R>0 will result in poor settling times.
C A P controller C(s) = k, k ∈ R>0 with a high gain will produce a stable controller but resultin oscillatory behavior.
D When a P controller C(s) = k, k ∈ R>0 is used, the settling times can be improved bychoosing a bigger k.
Corrected
Question 27 Mark all correct statements. (2 Points)
You are using a controller C(s) = k, k ∈ R>0. Which statements are correct?
A The system can track reference signals which are constant.
B The submarine can track reference signals which rise quadratically in time.
C The system can track a reference signals which rise linearly in time.
D The dynamics of the submarine are a system of type 2.
Question 28 Choose the correct answer. (1 Point)
A PID controller C(s) = k(
1 + 1Tis
+ Tds), k ∈ R>0, Ti ∈ R>0, Td ∈ R>0 can be expressed with
serially connected lead and lag elements.
A True. B False.
Question 29 Choose the correct answer. (1 Point)
A plant P (s) = k s−zs+p , z ∈ R>0, p ∈ R>0, k ∈ R>0 is closed-loop stable independently of the chosengain k.
A False. B True.
Corrected
Question 30 Choose the correct answer. (1 Point)
A plant P (s) with non-minimum phase poles or zeros has been reformulated in order to design acontroller in the following way:
P (s) = Pmp(s)D(s)
Where D(jω) = 1, ∀ω. All transfer functions P (s), Pmp(s) and D(s) are displayed in the Bodeplot below:
0.1 1 10 100
-12.5
-10.0
-7.5
-5.0
-2.5
0
ω
|L(jω)|(d
B)
0.1 1 10 1000
50
100
150
ω
∠L(jω)
What is the transfer function of P (s)?
A P (s) = 10 s−2s+10
B P (s) = s+5(s−10)2
C P (s) = s−12s+60
D P (s) = s−2s+10
E P (s) = s+2(s−10)2
F P (s) = s−5(s+10)2
Corrected
Box 3: Questions 31, 32
You are faced with the task of designing a controller C(s) for the plant:
P (s) =1
500(s2 + s
100
)and decide to use loop-shaping to solve the problem. You face the following constraints:
• Disturbances at frequencies ω ≤ 0.01 should be reduced by at least 60dB.
• Noise which occurs at frequencies ω ≥ 100 should be attenuated by at least 100dB.
The Bode plot of P (s) is displayed below:
0.001 0.01 0.1 1 10
-50
0
50
100
ω
|L(jω)|(d
B)
0.001 0.01 0.1 1 10
-150
-100
-50
0
ω
∠L(jω)
Question 31 Choose the correct answer. (1 Point)How can the disturbance rejection and noise reduction specifications be represented in the Bodeplot?
A |L(jω)| ≈ 60dB for ω < 0.01 and |L(jω)| > −100dB for ω > 100
B |L(jω)| > 60dB for ω < 0.01 and |L(jω)| > −100dB for ω > 100
C |L(jω)| > 60dB for ω < 0.01 and |L(jω)| < −100dB for ω > 100
D |L(jω)| < 60dB for ω < 0.01 and |L(jω)| > −100dB for ω > 100
E |L(jω)| < 60dB for ω < 0.01 and |L(jω)| < −100dB for ω > 100
Corrected
Question 32 Mark all correct statements. (2 Points)
Which of the following control designs will not fulfill the specifications?
A C(s) = 100 s+1s
0.005 +1
B C(s) = 100
C C(s) = 1
D C(s) = 100s
0.005 +1
s+1
Question 33 Choose the correct answer. (1 Point)
P controllers will always show a steady-state error for first order unit ramps while PI controllerscan eliminate steady-state errors.
A True. B False.
Question 34 Choose the correct answer. (1 Point)
The transfer function of a time delay is...
A linear and rational.
B nonlinear and rational.
C nonlinear and not rational.
D linear and not rational.
Question 35 Choose the correct answer. (1 Point)
We consider a nonlinear system with real parameters p1, p2 and p3 and a delay T > 0 acting onthe control signal u(t):
[x1
x2
]=
[0 10 p1
] [x1
x2
]+
[0
p2 − p3 sin(u(t− T ))
]Which of the following expressions describes equilibria (x1,e, x2,e, ue) of the system?
A x1,e = x2,e = 0 and ue = arcsin(p2p3 − T ) for p2p3− T ∈ [−1, 1].
B The system does not have equilibria.
C x1,e = const., x2,e = u3 = 0.
D x1,e = const., x2,e = 0 and ue = arcsin(p2p3 ) for p2p3∈ [−1, 1].
Corrected
Question 36 Choose the correct answer. (1 Point)
The Nyquist plot below shows a nominal open-loop system L(s) and measurements taken on thephysical plant L(s) indicated with crosses. The corresponding frequency of each measurement forL(s) is indicated with a small dot.
-1
Im
Re
The uncertain open-loop function is L(s) = L(1 + ∆(s)W (s)). The uncertainty has unit norm|∆(s)| = 1∀s.You would like to design an uncertainty function W (s) which allows for all measurement pointsmarked by crosses above but which is the least conservative possible. What W (s) do you choose?
A W (s) = 0.14
B W (s) = 0.71
C W (s) = 0.51
D W (s) = 0.21
Corrected
Question 37 Choose the correct answer. (1 Point)
Algorithm 1: Control Algorithm
Data: k1, k2
1 x← 0;2 tPrev ← system time;3 while true do4 y ← current measurement;5 r ← current reference;6 e← y − r;7 t← system time;8 if measurement not corrupt then9 dt← t− tPrev;
10 x← x+ e ∗ dt;11 u← k1 ∗ x+ k2 ∗ e;12 tPrev ← t;
13 else14 u← 0;15 end16 apply u;
17 end
What does the algorithm implement?
A A P controller.
B A PID controller.
C A lead/lag controller.
D A PI controller.
Corrected
Corrected
Corrected
Corrected
Answer sheet:
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←− please encode your student number, andwrite your first and last name below.
Firstname and lastname:
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How to select answer B :
3
3 (Unwanted answer clearly removed)
3 (Desired answer clear)
7 (Cannot distinguish B and C)
Answers must be given exclusively on this sheet;answers given on the other sheets will not be counted.
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Q30: A B C D E F
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Corrected