Exam August 16, 2019

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

Exam

Exam Duration: 120 minutes + 15 minutes reading time

Number of Problems: 32

Number of Points: 45

Permitted aids: Summary sheet of 2 pages A4 (=1 sheet A4). Nocalculators. Sheets must be written by hand.

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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Question 1 Mark all correct statements. (2 Points)

Consider a linear time-invariant system of the following form:

x(t) = Ax(t) +Bu(t)

y(t) = Cx(t) +Du(t)

with initial condition x(0) = x0. A state feedback gain K should be computed such that it placesthe closed-loop system poles at the locations γ1, γ2, ..., γn. Which statements are correct?

A A suitable K can be computed in any casebut the Ackermann formula is only appli-cable if the system is reachable.

B The Ackermann formula allows to com-pute K using only A, B and γ1, γ2, ..., γn.

C A suitable K can be computed without theAckermann formula by analyzing the char-acteristic polynomial of the closed loopsystem.

D The Ackermann formula allows to com-pute K using only A,B and C.

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Question 2 Choose the correct answer. (1 Point)

−x = Ax+Bu

K C˙x = (A− LC)x

+Lyyx

xu

Consider the proposed observer scheme depicted above. The difference to the Luenberger observerdiscussed in class is that the signal u is not fed to the state observer as it is assumed to be verysmall. Does the separation principle hold for this observer structure for general systems withB 6= 0 ?

A Only for asymptotically stable A.

B No.

C Only for controllable A,B.

D Yes.

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Box 1: Questions 3, 4

The design of a state observer for a dynamic system with a, b ∈ R and c > 0 and the followingform is considered:

x(t) = a x(t) + b u(t)

y(t) = c x(t).

The predicted output error is subject to noise n(t) = β(x− x), β > 0 as shown in the followingdiagram:

b∫

c

a

l

β(x− x)y

−+

+u x y

The evolution of the estimation error is characterized by the equation e = x− ˙x.

Question 3 Choose the correct answer. (1 Point)According to specifications, the error should decay at least 99% within the first second.How should l be chosen to fulfill this requirement?

A l ≤ a+β−log(0.01)c

B l ≥ a−β−log(0.01)c

C l ≥ a−log(0.01)c

D l ≤ a−log(0.1)c

E l ≥ a−log(0.1)c

F l ≥ a+β−log(0.1)c

Question 4 Choose the correct answer. (1 Point)

Consider the case in which β = 0, a = 1, b = 10, l = 2, and c = 1. A stabilizing controller needsto be designed according to sound design principles taking into account the speed of the observer.Which of the following k is suitable for the control law u(t) = −kx(t)?

A None of the others.

B k = 110 .

C k = 1110 .

D k = − 110 .

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Question 5 Choose the correct answer. (1 Point)

Which of the following statements about the duality of estimation and control are correct?

A LQR controllers can only be implementedif duality is given.

B Duality is a concept that allows to deriveconditions on observability based on thosefor reachability.

C Reachability of the control problem im-plies observability of the estimation prob-lem.

D For LTI-systems, the estimation problemis identical to the control problem.

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Box 2: Questions 6, 7, 8

Consider the linear system derived for your robot[x1x2

]=

[1/2 −1/2−3/2 −1/2

] [x1x2

]+B•u

y =[1 1

] [x1x2

].

You want your robot’s actuators to be able to control all its states. For this purpose you haveaccess to different actuators configurations denoted by B• ∈ {B1, B2, B3}:

B1 =

[1−1

], B2 =

[11

], B3 =

[1 −11 1

].

Question 6 Mark all correct statements. (2 Points)

Mark all actuator configurations for which the system is reachable.

A With B3.

B With B2.

C None of the configurations make the sys-tem controllable.

D With B1.

Question 7 Mark all correct statements. (2 Points)

We now relax the requirement for reachability to stabilizability.Mark all actuator configurations for which the system becomes stabilizable.

A With B2.

B None of the configurations make the sys-tem stabilizable.

C With B3.

D With B1.

Question 8 Choose the correct answer. (1 Point)

We now turn our attention to observability. There is one unobservable mode. The mode ofwhich of the eigenvalues of A is unobservable?

A√

3

B −1

C 1

D −√

2

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Question 9 Choose the correct answer. (1 Point)

Consider the following system:

[x1x2

]=

[1/2 −1/2−3/2 −1/2

] [x1x2

]+

[1−1

]u

Which of the following transformations T decomposes the system into reachable and non-reachablesubspaces?

A T = 1/√

5

[1 22 1

]B T = 1/

√2

[1 −11 1

] C T = 1/√

2

[1 11 −1

]D T = 1/

√2

[1 1−1 1

]

Question 10 Mark all correct statements. (2 Points)Consider the following system: [

x1x2

]=

[1 −14 −1

] [x1x2

]+

[01

]u

y =[0 1

] [x1x2

]Which of the following controller/observer combinations stabilize the system asymptotically?

Observers x(t) = (A− L•C)x(t) +Bu(t) + L•y(t):

1. Observer 1: L1 =[0 1

]ᵀ, 2. Observer 2: L2 =

[0 2

]ᵀ.

Controllers u(t) = −K•x(t):

1. Controller 1: K1 =[0 1

], 2. Controller 2: K2 =

[0 3

].

A Controller 2, Observer 1

B Controller 1, Observer 1

C Controller 2, Observer 2

D Controller 1, Observer 2

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Question 11 Choose the correct answer. (1 Point)

Algorithm 1 Pseudo code of a controller

1: procedure Controller implementation2: errorprev ← 03: outputprev ← 04: setpoint← r5: ∆T ← sampling time6: loop:7: error ← setpoint−measured8: output← outputprev + k1 ∗ (error − errorprev) + k2 ∗∆T ∗ error9: errorprev ← error

10: outputprev ← output

Which controller and which discretization is used in the above pseudo-code?

A Euler-forward PI controller.

B Euler-backward PI controller.

C Tustin-foward PID controller.

D Tustin-foward PI controller.

Question 12 Choose the correct answer. (1 Point)

You uniformly and slowly accelerate a turbine from 0 to 2000 Hz and film the process witha camera with a frame rate of 200 Hz. How many times will the turbine appear not to turn atall between two consecutive frames? You only start filming (i.e. first frame capture) once theacceleration begins and immediately stop filming once you reach 2000 Hz. Assume there is ablack point on the turbine that you can track.Hint: Carefully evaluate the lowest and highest frequency of the range.

A 9 times.

B 5 times.

C 0 times.

D 11 times.

E 10 times.

F 1 time.

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Question 13 Choose the correct answer. (1 Point)

Σ2

Σ1

Σ3

r e1 e2− + y

−

You are given the block diagram. What is the transfer function from input r to output y?

A Σr→y = Σ2(I+Σ1Σ2)−1(I−Σ3)−1+Σ3(I+Σ3)−1

B Σr→y = Σ2(I+Σ3)−1(I+Σ1Σ2)−1+Σ3(I+Σ3)−1

C Σr→y = Σ2(I+Σ1Σ2)−1(I+Σ3)−1+Σ3(I+Σ3)−1

D Σr→y = Σ2(I+Σ2Σ1)−1(I+Σ3)−1+Σ3(I+Σ3)−1

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Question 14 Choose the correct answer. (1 Point)

Consider the system described by the following transfer function:

P (s) = 2s+ 1

s2 + 3s+ 2e−2s +

1

s+ 2e−s

Which of the following is the correct first-order Pade approximation of the provided system?

A s+2s+1 ( −1s+1 + −1

2+s )

B 1s+2 ( 2+2s

s−1 + 2+s2−s )

C 1s+2 ( 2−2s

s+1 + 2−s2+s )

D 1s+3 ( 2−2s

s+1 + 2−s2+s )

Question 15 Mark all correct statements. (2 Points)Mark all correct poles (π) and multiplicity (m) pairs of the system below:

P (s) =

1s+1 11s+1

s−1s+1

1s+1

1s+3

.A π = −1, m = 1

B π = −3, m = 1

C π = −3, m = 2

D π = −1, m = 2

E π = −2, m = 1

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Question 16 Mark all correct statements. (2 Points)Consider the matrix

A =

1 00 21 0

and its SVD decomposition A = UΣV ∗. Mark all possibles choices of U and V .

A

U =

0 1√2

1 00 − 1√

2

, V =

[0 1−1 0

]

B

U =

− 1√2

1√2

0 10 − 1√

2

, V =

[1 00 −1

]

C

U =

− 1√2

0

0 −1− 1√

20

, V =

[−1 00 −1

]

D

U =

0 − 1√2

1 00 − 1√

2

, V =

[0 −11 0

]

Question 17 Choose the correct answer. (1 Point)

Consider the SVD A = UΣV ∗, where A ∈ R2×2. We now consider a rotation matrix

R =

[cos θ sin θ− sin θ cos θ

],

and define B = RA. Which of the following statements is true about the SVD of B = U ′Σ′V ′∗?Hint: Recall that rotation matrices are unitary, i.e., RR∗ = I.

A None of the others is correct.

B It corresponds to a rotation of the eigen-vectors but does not scale the magnitudeof the singular values, thus only Σ′ = Σholds for all values of θ.

C The singular values might change depend-ing on the value of θ.

D Only V ′ = V and Σ′ = Σ holds for allvalues of θ.

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Question 18 Choose the correct answer. (1 Point)

Which of the following statements about the condition number of a generic MIMO system iscorrect?

A The condition number does not depend onthe pre-multiplication of the system ma-trix with a constant matrix.

B The condition number γ can take values inthe interval (−∞,−1] ∧ [1,∞).

C It is generally true that a small condition

number indicates that the multivariable ef-fects of uncertainty are not likely to be se-rious.

D It is always true that a large conditionnumber indicates high sensitivity to uncer-tainty.

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Box 3: Questions 19, 20, 21, 22

Consider the control loop as depicted below.

G(s)

K(s)

w1 e1

-

e2 w2

+

Question 19 Choose the correct answer. (1 Point)

For G(s) =[

(1−s)(1+s)(1+0.01s)

1s

]and K(s) =

[µ1

µ2

]which values of µ1, µ2 guarantee internal stabil-

ity?

A µ1 < 1 and µ2 > 0

B None of the others.

C µ1 < 0 and µ2 > 0

D µ1 > 0 and µ2 < 0

Question 20 Mark all correct statements. (2 Points)Mark all the correct statement referring to the system in the grey box above.

A If K(s) is minimum phase (in case of a MIMO system, each element is minimum phase), wecannot conclude anything about the internal stability of the system.

B The system is internally stable if and only if So(s) = (I +GK)−1 is stable and there are noRHP cancellations in G(s)K(s).

C If G(s) and K(s) contain a cancellation we can conclude that the system is not internallystable.

D If G(s) has a right half plane zero the system cannot be internally stable.

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Question 21 Choose the correct answer. (1 Point)Consider G(s) = 2

s2+3s−1 and K(s) = k, what is a sufficient condition for internal stability?Note: use the infinity norm if you need to use a norm.

A k > 12

B k < − 12 ∧ k >

12

C − 12 < k < 1

2

D k > −1

E k > 1

Question 22 Choose the correct answer. (1 Point)Consider G(s) = s+2

s−1 and K(s) = s−1s+1 , what can be concluded about the stability of the system?

A The system is BIBO stable and internally stable.

B The system is neither BIBO stable nor internally stable.

C The system is not BIBO stable but internally stable.

D The system is BIBO stable but not internally stable.

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Box 4: Questions 23, 24

Consider the MIMO system depicted below,

C(s) ∆(s) G(s)e y

−

G(s)

where

G(s) =

[ 1s+1

2s+1

−1s+1

−1s+1

].

Question 23 Mark all correct statements. (2 Points)

Fix ∆11 = 1 and ∆22 = 1, for which values of ∆12 and ∆21 the resulting system G(s) is completelydecoupled?

A ∆12 = −1 and ∆21 = − 12 .

B ∆12 = 2 and ∆21 = −1.

C ∆12 = −2 and ∆21 = −1.

D ∆12 = 2 and ∆21 = 2.

E ∆12 = −2 and ∆21 = 1.

F ∆12 = 12 and ∆21 = −1.

Question 24 Choose the correct answer. (1 Point)If you are not given the possibility to introduce a decoupling block, but you still want to keep thecontrol action decentralized, which of the following choices would be more sensible?Assume the current input-output pairing to be u1 → y1 and u2 → y2.

A Switching the input-output coupling,hence relabeling u′1 = u2 and u′2 = u1.

B Leaving the current input-output pairing,thus u′1 = u1 and u′2 = u2.

C Both pairings are equally sensible.

D It cannot be determined from the providedinformation.

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Question 25 Mark all correct statements. (2 Points)You are given a plant modeled as P (s) = e−s. Your task is to design a controller for the givenplant, using a Smith predictor. The structure of the Smith predictor is reported in the figure below.The closed-loop behavior of the system should be Σ(s)r→y = 1

2s+1e−s. You can assume that the

model of the plant and the real plant are the same. Which of the following statements is correct?

Q(s)

Pr(s)

e−s·τ

Pr(s) e−s·τr e u

P (s)

y

−

C(s)

−

−

A Q(s) = e−s

s .

B C(s) = Q(s)1−Q(s)Pr(s)(1−e−sτ ) .

C Q(s) is a P-Controller.

D Q(s) is an I-Controller.

E Q(s) = C(s)1−C(s)Pr(s)(1−e−sτ ) .

F Q(s) = 12s .

G C(s) = Q(s)1+Q(s)Pr(s)(1−e−sτ ) .

Question 26 Mark all correct statements. (2 Points)

C(s) P (s)r e y

−

Consider the standard negative feedback loop represented above with a stable plant P (s) and theassociated Q-parametrization Q(s). Which of the following statements is correct?

A All stable Q(s) can guarantee an internally stable closed-loop system.

B Not all stable Q(s) can guarantee an internally stable closed-loop system.

C If Q(s) maps to an unstable controller C(s), the closed-loop system is not internally stable.

D A negative feedback system is always internally stable.

E Let’s consider a proper controller C(s). As a consequence, Q(s) has to be proper.

F If Q(s) is stable, the closed-loop system is internally stable.

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Box 5: Questions 27, 28, 29

You are given the system

x1(t) = −x1(t)− x2(t) + u1(t) + u2(t)

x2(t) = −x2(t) + u1(t),(1)

and the LQR cost

J1(x(t), u(t)) =

∫ ∞0

(u(t)ᵀRu(t) + x(t)ᵀQx(t) + 2x(t)ᵀNu(t)) dt

=

∫ ∞0

(x1(t)2 + x2(t)2 + u1(t)2 + u2(t)2 + 2x1(t)u2(t)− 2x2(t)u1(t)

)dt.

(2)

Assume you are solving the LQR problem with the Hamiltonian method.

Question 27 Mark all correct statements. (2 Points)Consider in addition to J1, the alternative cost function

J2(x(t), u(t)) =

∫ ∞0

(x1(t)2 + x2(t)2 + u1(t)2 + u2(t)2

)dt. (3)

Which of the following statements is correct?

A LQR with J1(x(t), u(t)) provides infinite positive gain margin.

B LQR with J1(x(t), u(t)) provides a phase margin of 60 degrees.

C Suppose that you add uncertainty to your model, i.e. x1 → x1 + n1(t) and x2 → x2 + n2(t)and you consider J1(x(t), u(t)). Robust stability is provided.

D Suppose that you add uncertainty to your model, i.e. x1 → x1 + n1(t) and x2 → x2 + n2(t)and you consider J2(x(t), u(t)). Robust stability is provided.

E LQR with J2(x(t), u(t)) provides infinite positive gain margin.

Corrected

Question 28 Mark all correct statements. (2 Points)Can one solve the optimization problem described in Box 5 with the Hamiltonian method?

A Yes.

B No, because the system (A,Q) is not detectable.

C No, even if the system (A,B) is stabilizable.

D No, because the system (A,B) is not stabilizable and the system (A − BR−1Nᵀ, Q −NR−1Nᵀ) is not detectable.

E No, because the system (A,B) is not stabilizable.

F No, because the system (A−BR−1Nᵀ, Q−NR−1Nᵀ) is not detectable.

G Always, independently from the system.

Question 29 Choose the correct answer. (1 Point)Assume that you compute the Hamiltonian anyways, independently from whether it is a good ideaor not. Then, the eigenvalues of the Hamiltonian matrix are:

A λ1,2 = −2, λ3 = 0, λ4 = 2.

B λ1,2 = 0, λ3,4 = 0.

C λ1,2 = −2, λ3,4 = 0.

D λ1 = −2, λ2,3 = 0, λ4 = 2.

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Question 30 Choose the correct answer. (1 Point)You are given the SISO dynamics

x(t) = x(t) + 2u(t),

and the cost function

J(x(t), u(t)) =

∫ ∞0

(3x(t)2 + 2u(t)2 + x(t)u(t)

)dt.

How would the control law look like if you solve the associated LQR problem?

A u(t) = − 12

(1−√

5)x(t).

B u(t) = −√25 x(t).

C u(t) = − 12

(1 +√

5)x(t).

D u(t) = 12

(1 +√

5)x(t).

E It is impossible to find a positive definite solution P for the Riccati equation, hence, nooptimal control can be found.

Question 31 Mark all correct statements. (2 Points)You are given the dynamic system

x1(t) = −x1(t)− 2x2(t)

x2(t) = −x2(t).

Which of the following statements is correct?

A The system is Lyapunov stable.

B Consider V (x1, x2) = 12 (x21 + x22), then V (x1, x2) ≥ 0 ∀x1, x2 and hence the system is not

Lyapunov stable.

C Consider V (x1, x2) = 12 (x21 + x22), then V (x1, x2) ≤ 0 ∀x1, x2 and hence the system is

Lyapunov stable.

D Consider V (x1, x2) = x21 + 2x22, then V (x1, x2) ≥ 0 ∀x1, x2 and hence the system is notLyapunov stable.

E Consider V (x1, x2) = x21 + 2x22, then V (x1, x2) ≤ 0 ∀x1, x2 and hence the system is notLyapunov stable.

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Question 32 Choose the correct answer. (1 Point)Consider the dynamic system

x1(t) = x2(t),

x2(t) = u(t),

y(t) = x1(t).

(4)

You want to design a LQG controller. Consider the cost function

J(x(t), u(t)) =

∫ ∞0

(x2(t)2 + 2u(t)2

)dt. (5)

Choose the correct observer gain L.

A L =[1√

2].

B L =[√

2 −1]ᵀ

.

C L =[√

2 1].

D L =[1√

2]ᵀ

.

E L =[√

2 −1].

F L =[√

2 1]ᵀ

.

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