Exam. 135 minutes, 15 minutes reading time

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1 Exam August 6, 208 Control Systems II ( ) Dr. Jacopo Tani Exam Exam Duration: 35 minutes, 5 minutes reading time Number of Problems: 35 Number of Points: 47 Permitted aids: 0 pages (5 sheets) A4. Sheets must be written by hand. 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 order of the answers has been permuted randomly. There are two types of questions:. One-best-answer type questions: One unique correct answer has to be marked. The question is worth one point for a correct answer and zero points otherwise. Giving multiple answers to a question will invalidate the answer. These questions are marked 2. True / false type questions: All true statements have to be marked and multiple statements can be true. If all statements are selected correctly, the full number of points is allocated; for one incorrect answer half the number of points; and otherwise zero points. These questions are 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 calculations will not be considered or graded. Use only the provided paper for your calculations; additional paper is available from the supervisors. Use pens producing a dark, solid and permanent line. The use of pencils is not allowed on the grading page. Good luck!

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3 Question Consider a control system in which S is the transfer function mapping disturbances to the output. T is the transfer function that maps noise to the output. Which statement is not correct? A One of the fundamental principles of feedback control states that S(s) + T (s) =, s B Reduction of S at low frequencies does not restrict the ability to reduce the influence of noise at high frequencies. C Even a well-designed control system will not manage to have a small value of S across all frequencies. D It is not possible to apply the plant inversion method to a plant with nonminimumphase zeros.

4 Box : Questions 2, 3 0 Image to sample pixels [px] pixels [px] You have vertical and horizontal uniform image samplers for which you need to know the maximal sampling period with units px (pixel). The image pattern follows a cos( ) function in vertical and another additive cos( ) function in horizontal direction. Question 2 The image above is vertically uniformly spatially sampled with sampling period T v. What is the maximal T v such that no aliasing occurs (only consider the vertical direction)? A T v = 25 px B T v = 00 px C T v = 50 px D T v = 2.5 px Question 3 The image above is horizontally spatially sampled with sampling period T h. What is the maximal T h such that no aliasing occurs (only consider the horizontal direction)? A T h = 50 px B T h = 2.5 px C T h = 00 px D T h = 25 px

5 Question 4 You are given the continuous-time LTI system equations 2 A = B = C = [ ] D = 0 with sampling time T s = s. What is the exact discrete time system dynamics matrix A d? [ ] e A A d = 2 (e e 3 ) 2 (e e 3 ) e 3 e 3 B A d = 2 (e e 3 ) 0 e 2e C A d = 2 (e e 3 ) 0 3e 3 e D A d = 2 (e e 3 ) 0 e 3 Question 5 Mark all correct statements. (2 Points) You are given the continuous time system dynamics P (s) = 2 (s+)(s+3). For T s = s, mark for which of the following discretization schemes, the resulting discrete time dynamics P d (z) are stable. A Euler forward. B Euler backward. C Tustin transform. Question 6 Mark all correct statements. (2 Points) You are reading a research article on discrete systems and are trying to remember the terminology. Which of the following discrete dynamical equations are described correctly? A x[k] = 0.5x[k ]: Convergent, aperiodic system B x[k] = 2 k x[0]: Divergent, aperiodic system C x[k] = 0.5x[k ]: Convergent, alternating system D x[k] = ( 0.5) k x[0]: Convergent, aperiodic system

6 Question 7 r + + P 4 (s) + y + P (s) P 2 (s) + P 3 (s) Assume both r and y are multi-dimensional and R(s) and Y (s) are their associated frequency space representations. Which MIMO transfer function Σ r y corresponds to the illustration above. A None of the above. B Y (s) = ((I P 4 (s)) P 4 (s) + (I + P 3 (s)p 2 (s)p (s)) )P 2 (s)p (s))r(s) C Y (s) = (P 4 (s)(i P 4 (s)) + P 2 (s)p (s)(i + P 3 (s)p 2 (s)p (s)) ))R(s) D Y (s) = (P 4 (s)(i P 4 (s)) + P (s)p 2 (s)(i + P (s)p 2 (s)p 3 (s)) ))R(s) E Y (s) = (P 4 (s)(i + P 4 (s)) + P 2 (s)p (s)(i + P 3 (s)p 2 (s)p (s)) ))R(s) Question 8 Mark all correct statements. (2 Points) Mark all discrete systems that diverge. A 0.25x[k]+x[k ]+x[k 2] = u[k]+2u[k ] B x[k] 0.5x[k ] = u[k] u[k ] C x[k + ] = x[k] 0 2 D x[k + ] = x[k] Question 9 Mark all correct statements. (2 Points) You are given the following system plant. P (s) = s+2 s 2 +2s+ s+ 0 s+2 s+3 s 2 +3s+2 Which of the following are correct poles π i and multiplicities m i of the transfer function P (s) above? s+3 A π = 2, m = 2 B π =, m = C π = 3, m = D π = 2, m = E π =, m = 3 F π =, m = 2

7 Question 0 You are given the controller C(s) = 2 (βs + )( s α + ) For which relation between α, β and sampling time T > 0 is the above controller transfer function asymptotically stable when using a forward Euler discretization? A C 0 < α < 2 T T 2 < β 0 < α < 2 T β < 0 B D 0 < α < T 0 < β < T 2 T 2 < α 0 < β < 2 T

8 Question Find the singular values of the system at the frequency ω = rad/s. P (s) = [ 2 ] s 2 s 2 +2 A σ = 6, σ 2 = B σ = 6, σ 2 = C σ = 3, σ 2 = D σ = 2, σ 2 = 3 Question 2 You have already computed the G (s)g(s) to be G (s)g(s) = What are input directions v i of this system? 5 A V = B V = 2 [ ] C V = D V = [ 5 ] Question 3 Mark all correct statements. (2 Points) Assume that the output of the system y(t) has the form m cos(ωt + φ y(t) = ) m 2 cos(ωt + φ 2 ) with m 2 + m 2 2 = 2 Furthermore assume that the maximal and minimal singular values of the system are σ max = 3 σ min = 0.5 Which of the following signals are possible inputs to the system that may lead to the output above? cos(ωt) 5 sin(ωt + π/2) A u(t) = C u(t) = 4 sin(ωt) 5 cos(ωt) sin(ωt) 3 cos(ωt 3π/2) B u(t) = D u(t) = cos(ωt + π) 3 sin(ωt)

9 Question 4 Let Σ(s) and Γ(s) be distinct rational transfer functions. The statement is... A True. B False. ΣΓ Σ Γ Question 5 You are designing a robotic eye surgery system. Which of the following signal norms should you use to evaluate the error e(t) of your surgery controller? A e(t) C e(t) B e(t) 2

10 Question 6 Mark all correct statements. (2 Points) You are given the dynamic system ẋ (t) = ax (t) + x 2 (t) + x 3 (t) + u (t) + u 2 (t) ẋ 2 (t) = x 2 (t) + 2x 3 (t) + u (t) + au 2 (t) ẋ 3 (t) = x 3 (t) + bu (t) y (t) = bx (t) + x 2 (t) y 2 (t) = bx 2 (t) + x 3 (t), () with a, b R. Mark all the correct statements A Let a = b =. The system is not controllable, but observable. B Let a = b =. The system is stabilizable and detectable. C Let a R, b = 0. The system is not controllable, not observable. D Let a R, b = 0. The system is controllable, not observable. E Let a = b =. The system is controllable and observable.

11 Question 7 Mark all correct statements. (2 Points) You are given the control loop depicted in the Figure below. w e h P (s) P 2 (s) h 2 C(s) Which of the following statements are correct? e 2 w 2 A For C(s) = 2, P (s) = s+2, P 2(s) = s+ the system is internally stable. B Assume the signals h and h 2 have already proven to be bounded. In order for the system to be internally stable, the following transfer functions must be stable. (I CP 2 P ) (I CP 2 P ) C (I P 2 P C) P 2 P (I P 2 P C) (2) C Assume the signals h and h 2 have already proven to be bounded. In order for the system to be internally stable, the following transfer functions must be stable. (I CP CP 2 P ) (I CP CP 2 P ) C (I P C P 2 P C) (P + P 2 P ) (I P C P 2 P C) (3) D Assume the signals h and h 2 have already proven to be bounded. In order for the system to be internally stable, the following transfer functions must be stable. (I CP CP 2 P ) (I CP CP 2 P ) C (I CP CP 2 P ) (P + P 2 P ) (I CP CP 2 P ) (4) E For C(s) = 2, P (s) = s+, P 2(s) = s 2 F For C(s) =, P (s) = s+2, P 2(s) = s+ the system is internally stable. the system is internally stable.

12 Box 2: Questions 8, 9 You are given the plant the controller P (s) = C(s) = ( s+ s+2 0 s+3 ( 2s 2s+ (s+)(s+2) s+3 0 ), (5) ), (6) and their singular values plots. For this case, disturbances occur at low frequencies. Question 8 The system shows good disturbance rejection. A True. B False. Question 9 It holds σ(c(jω)p (jω)), ω R >0. A False. B True.

13 Question 20 Given the feedback system, with k R: w e L (s) L 2 (s) e 2 w 2 with: [ L (s) = s 2 +2s L 2 (s) = k What is the condition for closed loop stability derived from the small gain theorem? ] A The system is stable k B k < 4 C k < 0 D k < 2 E The system is unstable k F k < 8

14 Question 2 Mark all correct statements. (2 Points) When is the use of an additive model uncertainty parameterization particularly appropriate? In case of: A Neglected or uncertain high frequency dynamics B Uncertainty on the input equations of the plant (due to, e.g., badly modeled actuators) C Uncertainty on the parameters describing the material properties of the plant D Neglected or uncertain low frequency dynamics E Uncertain RHP zeros Question 22 Which of the following propositions is true regarding the stability of an uncertain system when the inputs and outputs are scaled? A An uncertain system with scaled input and outputs is more likely to be stable if the inputs and outputs are magnified B An uncertain system with scaled input and outputs is more likely to be stable if the inputs and outputs are attenuated C Scaling does not affect stability Question 23 A control system designed to guarantee stability for an uncertain SISO system and guarantee performance for the nominal plant always guarantees robust performance as well. A True B False Question 24 Why is it important for a control system to be robust? A To save time and money on the preliminary charachterization B Because even the best of mathematical models will never completely capture the input-output relationship of a physical phenomenon C To prevent physical failures of plants operating at the edge of its performance envelope D None of the provided answers.

15 Question 25 You are given the dynamic system: ẋ (t) = 6x (t) + u (t) ẋ 2 (t) = x (t) + 2x 2 (t) + u 2 (t) ẋ 3 (t) = 7x 2 (t) 3x 3 (t) + u (t) y (t) = x (t) y 2 (t) = x 2 (t). (7) You would like to decentralize the system and, if possible, control is as two SISO systems. Which input-output pairing should you choose? A Given the state space description of this systen, it is not possible to conclude something about the correct input-output pairing. B u y 2 and u 2 y at all frequencies. C u y and u 2 y 2 at high frequencies, while u y 2 and u 2 y at low frequencies. D u y and u 2 y 2 at all frequencies. E u y and u 2 y 2 at low frequencies, while u y 2 and u 2 y at high frequencies.

16 Question 26 Mark all correct statements. (2 Points) You are given the system ẋ (t) = ax (t) + bx 2 (t) + u (t) ẋ 2 (t) = cx (t) + dx 2 (t) + 3u 2 (t). (8) You want to find a state feedback controller such that the error of the controller decreases as e 2t. Mark all the correct statements. A a =, b = 2, c = 3, d = 4. The state feedback matrix is K = ( 3 2 ). (9) B a =, b = 0, c =, d = 0. The state feedback matrix is C a =, b = 0, c = 0, d =. formulation. D a =, b = 2, c = 3, d = 4. The state feedback matrix is K = ( 4 3 2). (0) The Ackermann formula cannot be used for this problem K = ( 2 3 ). ()

17 Question 27 You are designing an LQR controller for a regulation problem. The dynamic system and objective function for the LQR design are: ẋ = Ax + Bu J = 0 ( x T Qx + u T Ru ) dt Your system has two states x and x 2. x is a distance in m on which you can tolerate approximately cm error. x 2 is an angle in radian on which you should not have more than π 0 of deviation. As the system is connected to the power grid, the cost of actuation is of limited concern with respect to the performance objectives. Which choice of weights will likely lead to the fastest performance and least violation of your constraints? A Q = diag(, 000), R = B Q = diag(, 000), R = 0 C Q = diag(00, 0.), R = D Q = diag(000, ), R = 0 E Q = diag(000, ), R =

18 Box 3: Questions 28, 29 You are designing an LQG controller to solve the regulation problem for the following fully controllable LTI-system: 8 4 ẋ = u 4 3 y = [ ] x The state feedback law u = Kx has already been designed, K = leads to a stable closed-loop behavior and closed loop poles (.82,.,.35). To implement the LQR state feedback control law, an observer estimating ˆx still needs to be designed. Question 28 Your colleague proposes to implement a Luenberger observer with gain L = A The proposed observer will result in asymptotically stable estimation error dynamics and is not suitable for the task. B The proposed observer will result in asymptotically stable estimation error dynamics and is suitable for the task. C The proposed observer will result in unstable estimation error dynamics. Question 29 Mark all correct statements. (2 Points) Which of the following observations are correct? A {Re(λ) < 0, λ such that det(λi A) = 0} The estimation error dynamics are asymptotically stable for L = 0. B It is possible to place the poles of the Luenberger observer at arbitrary locations in the complex plane. C It is not possible to design an LQG controller for this system. D The system is observable.

19 Question 30 The separation principle states that if the poles of A BK for some control gain K and the poles of A LC for some observer gain L are sufficiently different, then the control problem does not interact with the estimation problem and they can be solved independently. A Correct. B False.

20 Question 3 You are designing a MIMO controller using H control design. One row of the transfer function of your extended system T zw that maps the exogenous input w to the performance output z is W e S. Which statement is correct? A If the value T zw is less than, S(jω) We (jω), ω B If the value T zw is less than, S(jω) W e (jω), ω C The H approach guarantees that S(jω) We (jω), ω. Question 32 Mark all correct statements. (2 Points) You are designing a control system and would like to use a combined approach of feedforward and feedback control. Due to strict requirements on robustness and strong cross-couplings between input and output channels, you have decided to use the H control method. You design the following flowchart describing your extended system: w = r W (s) W 2 (s) z w 2 = d e F (s) C(s) u y P (s) W 3 (s) W 4 (s) z 2 z 3 Where F (s) is the part of the controller which is directly applied to the reference and C(s) is the term responsible to attenuate disturbances. Which statements about the H scheme are correct? A The choice W 3 (s) = 00 + will help to ensure a reasonably small control signal at s 0 frequencies below 0 rad/s. B W influences z and z 3. C A possibility to enforce an upper bound ˆT yr on the transfer function T yr : r y is to set W 4 (s) = ˆT yr. D F (s) cancels in the formulation and does not influence the magnitude of the control signal u and correspondingly z 2. E Small gains of W 2 at frequencies below 2 of the crossover frequency will ensure that the disturbance d has little effect on the tracking error e.

21 Question 33 Mark all correct statements. (2 Points) Which of the following statements about system equilibria are correct for a dynamical system ẋ(t) = f(x(t))? [ ] T A f = x 2, x + x3 2 6 x 2 has two equilibria at (0, 6) and (0, 6) B f linear x = 0 is the only equilibrium of f. C f linear x = 0 is an equilibrium of f. D f = sin(x) has more than one equilibrium.

22 Box 4: Questions 34, 35 We consider an underwater robot in which the buoyancy dynamics as well as friction of the water was modeled. u(t) g y(t) d(t) V (t), n(t), p(t) ε S m The system can be described by the following nonlinear equations: ẋ ẋ 2 = ẋ 3 c x 2 x c 2 c 3 3 x c 4 x α u With c, c 2, c 3, c 4 being real-valued constants. The state x is the water pressure at the depth of the robot, x 2 is the velocity in direction of water depth, x 3 amount of air in the robot buoyancy compensator. Only the variable x can be measured directly, i.e., y = x. Question 34 Which statement is not correct? A The output equation of the system is linear. B The system has infinitely many equilibria which depend on the pressure at the equilibrium point p e and are x e = (p e, 0, c2 c 3 p e ), u e = 0 C The system is globally asymptotically stable. D The system has a nonlinear state equation. Question 35 Which transformation z = T (x) brings the system into a form on which input-state feedback linearization can be applied? A The system is not input-state linearizable. B z = x,z 2 = x + x 2,z 3 = x3 x + c2 c 3 + z2 c 4 C z = x,z 2 = c x 2,z 3 = x3 x + c2 c 3 + z 2 ) x D z = x,z 2 = c x 2,z 3 = c (c 2 c 3 3 x c 4 x 2