Exam August 15, 2017 Control Systems I (151-0591-00L) Prof Emilio Frazzoli Exam Exam Duration: 135 minutes, 15 minutes reading time Number of Problems: 44 Number of Points: 52 Permitted aids: Important: 40 pages (20 sheets) of A4 notes, Appendix D of the book Lino Guzzella: Analysis and Synthesis of Single-Input Single-Output Control Systems, simple calculators will be provided upon request. 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: 1. 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. Good luck!
1 Systems and Control Architecture Box 1: Questions 1, 2 A system is described by the equation y(t) = f(u(t)) where f is the function describing the system, u(t) is an input and y(t) is an output signal. u(t) f y(t) Question 1 Mark all correct statements. (2 Points) Which of these systems is not causal? A y(t) = u 2 (t) + 10 B y(t σ) = 2e u(t) σ > 0 C y(t) = u(t + σ) σ > 0 D y(t σ) = 1 σ > 0 Question 2 Which one of these systems is linear? A y(t) = u(t σ) + u(t + σ) σ > 0 B y(t) = 1 C y(t) = (u(t)) 2 D y(t) = au(t) + b a, b R, a 0 Question 3 Your friend and you want to design a control system for an unstable plant that has to operate under various disturbances. A perfect model of the plant is at hand and you have perfect knowledge about it, however you do not have perfect knowledge of the disturbances that might be affecting the plant. Your friend proposes the design of a feedback controller to stabilize the plant. What do you think? A Both a feedforward and feedback controller designs would work. Hence we will choose whichever is easier to implement. B Since we know everything about the plant, we should use the feedforward controller as in this case we will achieve the perfect reference tracking. C I agree with him, we need a feedback controller.
2 System Modeling and Analysis Box 2: Questions 4, 5 You are enthusiastic about control systems and decide to learn more about it by building a seesaw as shown in the figure below. You connect two motors with propellers on either side of the seesaw to control the angle α of the seesaw. The propellers have angular velocities ω 1 and ω 2 as shown in the figure and have a distance l from the center of the seesaw. An additional point mass with weight m w and distance to the center of the seesaw l w is present. ω 1 m w, l w ω 2 α α Question 4 Identify which signals are the input and output of the system. A Input: α; Output: ω 1, ω 2 B Input: ω 1, ω 2 ; Output: α C Input: α; Output: ω 1, ω 2 D Input: ω 1 ; Output: ω 2 Question 5 Let J be the angular inertia (including the point mass m w ) of the system. Furthermore, let F 1, F 2 be the forces generated by the propeller on the respective sides of the seesaw. Hint: E kin = 1 2 J α2, E pot = mgh, where m is a mass, g is earths gravitation and h represents height. Then looking at the kinetic and potential energy reservoirs of the system d dt (reservoir content) = inflows outflows, results in which equation of motion: A B α = α = l(f2 F1) mwglw cos α 2J mwglw cos α J C D α = α = (F2 F1)+mwg cos α J l(f2 F1) mwglw cos α J
Question 6 You have to design a speed sensor for an electric motor of a new locomotive. First, you need to normalize the physical state variable of the system, i.e. the motor speed z(t) = ω(t), such that the normalized state signal has its maximum value at 1 under regular operating conditions. The gearing is chosen, such that the electric motor has the highest efficiency at the maximum train speed of 200kmph. The gear transmission ratio is γ = 14rpm/kmph. From the data sheet of the electric motor you see that it can reach a speed of 4 000rpm without load. How do you choose the normalizing value z 0? A 2800 rpm. B 200 kmph. C 4000 rpm. D 286 kmph.
Box 3: Questions 7, 8, 9, 10, 11 You have passed the Control Systems 1 exam and now you are very bored. To do something more exciting, you decide to build a jet-kart, as shown in the figure below. However, before the fun starts you need to model it and design a controller for it. Assume that the cart moves in one direction only (1D motion). To control your vehicle you use thrust from the jet engine (this is your control input) and you are interested in controlling the kart s position. Assume that there are only three kinds of forces acting on the kart: Thrust force F T H (t) = k T H T (t) where T (t) is the thrust from the jet engine (can be both positive or negative) and K th is a constant. High velocity drag force F DH = k DH v 2 (t) where k DH is a constant and v(t) is the linear velocity of the vehicle. Viscous drag force F D = k D v(t) where k D is a constant and v(t) is the velocity. Question 7 Let m is the mass of the vehicle and x(t) its position. Which differential equation models your system? A mẍ k DH ẋ 2 k D ẋ = k T H T (t) B mẍ + k DH ẋ 2 + k D ẋ = k T H T (t) C mẍ k DH ẋ 2 k D ẋ = k T H T (t) D mẍ + k DH ẋ 2 + k D ẋ = 0 Question 8 If you represent your differential equation of the jet kart in state space representation ẋ = f(x, u), y = g(x, u), what is the dimension of the state vector, i.e. if x R n, what is n? A 3 B Cannot be determined from the information given C 1 D 2
Question 9 Assume that your jet kart is in a standstill and you d like to accelerate, so you hit the throttle. Unfortunately, something went wrong and instead of producing a steady thrust, the thrust produced by the jet engine is oscillatory. Jet engine s thrust can be described with T (t) = sin(ωt), where ω = 1 rad/s. Other parameters are m = 1000 kg, K TH = 100, k DH = 0 Ns 2 /m 2 and k D = 100 Ns/m. What is the value of jet-kart s velocity response at time t = 20 s? A v(20) 0.11 m/s B v(20) 0.017 m/s C v(20) 0.079 m/s D v(20) 0.018 m/s Question 10 Assume that the jet kart is moving at the constant velocity v = 100 m/s and you turn off the jet engine to slow down. How much time until you can drive through the 30 Zone here in Zurich? (i.e. how much time until you reach a velocity of v = 30 km/h?). For your calculations use m = 1000 kg, k DH = 0 Ns 2 /m 2 and k D = 100 Ns/m. A t 2.5 s B t 250 s C I will never reach that speed since since my jet engine is off D t 25 s Question 11 Assuming that k DH = 0 Ns 2 /m 2 and the system s output is the position x(t) of the jet kart. Is the jet kart BIBO stable? A Cannot be determined from the data given B Yes C No Question 12 Consider a linear time-invariant SISO system. stability criteria. Pick a correct logical relation between different A Asymptotically stable BIBO stable B Asymptotically stable = BIBO stable C Asymptotically stable BIBO stable = Lyapunov stable D Asymptotically stable = BIBO stable Lyapunov stable
Question 13 Recall the definition of Lyapunov stability from the lecture: A system is Lyapunov stable if for any bounded initial condition and zero input, the state remains bounded. Let the discrete update equation of state x R for k = 0, 1, 2,... be given by x k+1 = ax k + bu k For which of the following intervals of a is the state x Lyapunov stable? A a [ 0.5, 0.5] B a [ 1, 2] C a (, 0.5] D a (, 0] Question 14 For which values of a is the state BIBO stable (assuming b 0)? A a ( 1, 2) B a (, 0) C a [ 1, 1] D a ( 1, 1) Question 15 Which of the following transfer functions is not causal? A g(s) = s2 +2 s B g(s) = 1000 C g(s) = s3 s 3 +s+2 D g(s) = s+2 s 2 +4s+3 Box 4: Question 16 Consider the state-space system below, where a is a real parameter: ẋ(t) = y(t) = [ 2 [ ] 1 0 x(t) + 1 4 a ] x(t) [ ] a u(t) 1 Question 16 For which values of a is the state-space realization minimal? A a {0, 10} B a = {0, 10} C a = {0, 5, 10} D a = {0, 5} E a {0, 5, 10} F a {0, 5}
Question 17 You have derived the nonlinear dynamics of a simple seesaw system to be α = 0.5 cos(α) + 2u, where α(t) denotes the seesaw angle and u(t) denotes the input used to turn the seesaw. You linearize the equation around the equilibrium point α e = 0 to yield ẍ = 2u, where ẍ represents the linearized angular acceleration. For which angle α does linearized angular acceleration deviate not more then ε = 0.2 rad s 2 nonlinear (correct) acceleration? from the A α < 55.57 ẍ α < ε B α < 53.13 C α < 78.46 D α < 60.87
3 Frequency Domain Box 5: Questions 18, 19, 20, 21, 22 To keep your annoying younger sibling away, you gave them the seesaw with propellers on its endpoint in the picture below to play with. However, your younger sibling is not an expert in control systems so they need help with the controller design. The seesaw is fitted with two propellers that can produce thrust, a spring and a damper (both located in the seesaw s joint in the middle). You can assume that the joint is located exactly in the middle of the seesaw. Parameters describing the seesaw are its moment of inertia around the center of mass I [kgm 2 ], spring constant k [N m/rad] and damping coefficient d [N ms/rad]. α F 1 α F 2 Question 18 For which values of the spring parameters I, d and k is the seesaw asymptotically stable? A d 0 and k > 2 B I > 0, d > 0 and k > 0 C d 0 and I > 2 k D d 2I 0 Question 19 Assume that I = 1 [kgm 2 ], k = 1 Nm/rad, l = 1 [m] and d = 2 [Nms/rad] and that the input to the system is a difference between forces F 1 and F 2 : u = F 1 F 2 (note that this is a single input and not two inputs). In this question you can observe the output α of the system. What is the value of system s transfer function evaluated at s = i where i = 1? A g(i) = i 2 B g(i) = 1 C g(i) = 1 2i D g(i) = 1 2 Question 20 Unlike in Question 19, you can now observe the output α of the system. What is the value of systems s transfer function evaluated at s = 2i using the same numerical parameter values as in Question 19? A g(2i) = 1 4i+3 B g(2i) = 1 i 1 C g(2i) = 1 4i 3 D g(2i) = 2i 4i 3
Question 21 Assume that I = 0.1 [kgm 2 ] and d = 2 [Nms/rad]. For which values of spring constant k you expect to see oscillations in the time response to a step input? A k < 10 B k > 10 C k 10 D k 10 Question 22 Assume that I = 1 [kgm 2 ], k = 1 Nm/rad and d = 2 [Nms/rad]. What is the mode associated with the eigenvalue 1? A B 1 2 [ 1 1 [ 1 0] ] [ 0 C 1] [ ] D 1 1 2 1 Box 6: Question 23 Given an LTI system ẋ(t) = Ax(t) + Bu(t) y(t) = Cx(t) + Du(t) and the general solution y(t) = Ce At x(0) + C t 0 e A(t τ) Bu(τ)dτ + Du(t). Let the input of the system be an exponential function of time u(t) = e st. In addition the initial condition is zero and there is no feedthrough. x(0) = 0 D = 0 Question 23 Mark all correct statements. (2 Points) What function is the output y(t) converging to, in case the system is asymptotically stable and what function, in case it is unstable? Hint: Determine the steady-state response. A y unstable (t) = (C(sI A) 1 B)e st. B y stable (t) = C(sI A) 1 e At B + (C(sI A) 1 B)e st. C y unstable (t) = C(sI A) 1 e At B + (C(sI A) 1 B)e st. D y stable (t) = (C(sI A) 1 B)e st.
Magnitude (db) 125 100 75 50 25 0 Magnitude (db) 100 50 0 0.001 0.01 0.1 1 10 0.001 0.01 0.1 1 10-100 0 Phase(deg) -120-140 -160-180 0.001 0.01 0.1 1 10 Phase(deg) -100-200 -300 0.001 0.01 0.1 1 10 Frequency (rad/sec) Frequency (rad/sec) System 1 System 2 Magnitude (db) 100 50 0-50 Magnitude (db) 100 50 0-50 -100 0.001 0.1 10 1000-100 0.001 0.1 10 1000-100 0 Phase(deg) -120-140 -160-180 0.001 0.1 10 1000 Phase(deg) -50-100 -150 0.001 0.1 10 1000 Frequency (rad/sec) Frequency (rad/sec) System 3 System 4 Figure 1: Bode plots of four different systems.
Box 7: Questions 24, 25 Consider the Bode plots of four different systems are shown in figure 1. Question 24 Consider the open-loop transfer function L(s) = 1000 is the Bode plot corresponding to it? s s+1/2 2 (s+1000). Which of the plots in Figure 1 A System 1 B System 4 C System 2 D System 3 Question 25 Mark all correct statements. (2 Points) Now assume that the Bode plots shown in in Figure 1 show open loop gains of four different systems and that the plant has the transfer function P (s) = 2 s. 2 A System 4 has a nonminimumphase zero. B Out of the four systems shown, system 1 is with the best realizable controller for P (s) = 2 s 2. C The response of system 2 to a unit step is unstable. D System 3 has a nonminimumphase zero. E System 1 has the best phase margin. F Out of the four systems shown, system 3 is with the best realizable controller for P (s) = 2 s 2.
Imaginary Axis Corrected 0.2 Nyquist Diagram 0.15 0.1 0.05 0-0.05-0.1-0.15-0.2-0.2-0.15-0.1-0.05 0 0.05 0.1 0.15 0.2 0.25 0.3 Real Axis Figure 2: Nyquist plot of second-order system Question 26 You are given the Nyquist plot of a second-order open-loop transfer function L(s) shown in figure 2. What is the closed-loop system s steady-state error e to a step input of magnitude 1? A 1 6 B 0 C 4 3 D 5 6 Question 27 You are given the plant transfer function P (s) = k jω+2 e jωt with time delay T. Use the 1st order Padé approximation to calculate the frequency at the angle at which the gain margin is usually calculated. Choose the correct calculation below A 3 arctan( ω T 2 ) = π B arctan( ω 2 ) 2 arctan( ω T 2 ) = π C arctan( ω 2 ) + 2 arctan( ω T 2 ) = π D arctan( ω 2 ) 2 arctan( ω T 2 ) = 0
4 Controller Design Question 28 Mark all correct statements. (2 Points) Take a system with open-loop gain L(s) = 1 s and consider its positive root-locus curve (k > 0). 2 A For k = 0 both poles of the closed-loop system are identical. B A controller C(s) = k (s+2) will stabilize the closed-loop system for k > 0. C Increasing k to a sufficiently large number will stabilize the resulting closed-loop system and bring satisfactory system performance. D A controller C(s) = s+2 will stabilize the closed-loop system for k > 0. E For k > 0 the absolute value of both poles of the closed-loop system is 2 k. F For k > 0 the absolute value of both poles of the closed-loop system is k. k Question 29 Let P (s) = 1 (s+5) 2, which statement about the controller design for P (s) is correct? A There is no need to apply feedback control to the plant because it is stable and so it will reject any disturbances. B Any P I-Controller with sufficiently high gain can stabilize the plant. C The plant has two stable poles which is why any P -Controller with arbitrary high gain can stabilize the plant and result in sufficiently good control performance. D A P -Controller with a sufficiently small gain is a possible first approach for the design. Box 8: Question 30 Consider the system described by the following block diagram r + e C(s) P (s) y where P (s) = 1 (s + 4)(s + 0.5). We want to design the controller C(s) to satisfy the following specifications: The closed-loop system is stable. The phase margin is about 50 degrees. The cross-over frequency is at about 20 rad/sec. Question 30 Only one of the following controllers satisfies these specifications. Which one is it?
A C(s) = 110 s+16 s+25 B C(s) = 816 s+10 s+40 C C(s) = 225 s+33 s+12 D C(s) = 435 s+5 s(s+20) E C(s) = 315 s+27 s(s+15) Question 31 Mark all correct statements. (2 Points) In many cases, second-order system approximations are used to infer useful information about a control system. Which of the following systems should not be approximated as a second-order system? A g(s) = 4 s+1 e st B g(s) = 1 (s+1) 3 C g(s) = 3 (s+1)(s+2) 2 D g(s) = 3 (s+1) 2 (s 1) E g(s) = 0.5 s 1 (s+2) 4 Question 32 The phase of the frequency response of a control system can be decreased by applying a lead element. A True. B False. Box 9: Question 33 Let α(t) = 0.5cos(α(t)) + 2u(t) denote the dynamics of a system. equilibrium point α e = 0 yields the linearized system equations Linearization around the ẍ(t) = 2u(t) Question 33 What is the correct transfer function g(s) of the linearized system if y(t) = x(t)? A g(s) = 2 s 2 B g(s) = 1 2s 2 C g(s) = 2 s 2 + 1 D g(s) = 2 s 2 Question 34 Mark all correct statements. (2 Points) Which statements about the transfer function g(s) = 2 s are correct? 2 0 1 0 0 B The transfer function is proper. A A = 0 0 0, B = 2 3, 1 1 0 0 C = [ 3 0 0 ] C The transfer function has two zeros., D = 0 is a state space representation of the transfer function. D The transfer function is strictly proper.
Question 35 Consider the four pole/zero diagrams in figure 3 indication the pole, zero locations of 4 transfer functions: Im Im +3 Im Im 1 Re 1 3 Re 6 2 Re 6 2.1 2 Re (a) (b) (c) (d) Figure 3: Pole/zero plots of four different transfer functions. Your colleague asks you to order these transfer functions with increasing t 90 response time to a unit step input. What is the correct ordering? A c, d, b, a B c, d, a, b C d, c, a, b D d, b, c, a Box 10: Questions 36, 37 Let α(t) = 0.5cos(α(t))+2u(t) be the dynamics of a system. Linearization around the equilibrium point α e = 0 yields the linearized system equations Furthermore let y(t) = x(t). ẍ(t) = 2u(t) (1) Question 36 Mark all correct statements. (2 Points) Mark all correct statements. A The system is stabilizable. B The system is detectable. C The system is fully observable. D The system is fully controllable. Question 37 Mark all correct statements. (2 Points) Let L(s) denote the loop gain, S(s) the sensitivity and T (s) the complementary sensitivity of the system in equation 1. A The sensitivity is asymptotically stable. B The poles of the sensitivity and the complementary sensitivity are identical. C The magnitude of the open loop gain response to a unit step input is bounded. D The complementary sensitivity is BIBOstable.
Question 38 Your colleague has a system with one unstable open-loop pole. He used mathematica to find a stabilizing controller but has unfortunately mixed up his scripts. Figure 4 shows the plots that he presents to you. Help him to assign the correct Nyquist plots to the respective step responses. A 1 c, 2 b, 3 a, 4 d B 1 a, 2 b, 3 d, 4 c C 1 b, 2 c, 3 d, 4 a D 1 c, 2 b, 3 d, 4 a
1.5 200 400 600 1.0 0.5 Nyquist plot of system 1. 2 4 6 8 10 Step response a. -750-500 -250 Nyquist plot of system 2. -20-40 -60-80 -100-120 -140 0.5 1.0 1.5 2.0 Step response b. 2 1-2 -1 1 2-1 -2 Nyquist plot of system 3. -2 10 13-4 10 13-6 10 13-8 10 13 5 10 15 Step response c. 40 20-40 -20 20 40-20 -40 1.0 0.8 0.6 0.4 0.2 Nyquist plot of system 4. 0.1 0.2 0.3 0.4 0.5 Step response d. Figure 4: Bode plots of four different systems.
Im(jω) Im(jω) 1.5 0.3 0.2 1.0 0.1 0.5 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Re(jω) 1.5 2.0 2.5 3.0 3.5 4.0 Re(jω) -0.1-0.5-0.2-1.0-0.3 Im(jω) 0.2 Nyquist plot A -1.5 Im(jω) 1.0 Nyquist plot B 0.1 0.5 0.5 0.6 0.7 0.8 0.9 1.0 Re(jω) 2.5 3.0 3.5 4.0 Re(jω) -0.1-0.5-0.2-1.0 Nyquist plot C Nyquist plot D Figure 5: Figure illustrating different possible Nyquist diagrams for a lead compensator C lead (s) Question 39 Your colleague has designed a lead compensator. C lead = s/2 + 1 s/8 + 1 She has given you several Nyquist plots of and asks you which one corresponds to the lead compensator transfer function she designed. Pick the correct Nyquist plot in Figure 5 corresponding to the above lead compensator. A Nyquist plot D B Nyquist plot C C Nyquist plot A D Nyquist plot B
Amplitude Corrected 0.7 Step Response 0.6 0.5 0.4 0.3 0.2 0.1 0 0 0.5 1 1.5 2 2.5 3 Time (seconds) Figure 6: Step response of plant P (s) Question 40 You are tasked with designing a lag compensator C lag = k s/a+1 s/b+1 to increase the gain of your system at low frequencies. Currently you have L(0) = 5, you wish to achieve L(0) = 200. Furthermore, you know that your crossover frequency is ω c = 100 rad/s and that you do not want your lag compensator to substantially influence the system s phase margin. A C lag (s) = 40 s/0.25+1 s/10+1 B C lag (s) = 40 s/10+1 s/0.25+1 C C lag (s) = 0.025 s/2.5+1 s/100+1 D C lag (s) = 40 s/100+1 s/2.5+1 Question 41 Your coworker designed a controller for plant P (s), however the system does not track a step response as shown in figure 6. Which of the following controllers might ensure a correct tracking of the step input with zero steady state error (e = 0)? A PD B C lead (s) = s/a+1 s/b+1 with 0 < a < b C PI D P k (s+p)(s+r) s+q. Choose the correct coeffi- Question 42 You are given a Padé approximation of the form e st cients: A k = T, p = 2 T ± j 2 T, r = 2 T j 2 T, q = 6 T B k = 0.5T, p = 1 T, r = 3 T, q = 3 T C k = 0.5T, j 1 T, q = 3 T D k = 0.5T, j 2 T, q = 3 T p = 2 T ± j 1 T, r = 2 T p = 2 T ± j 2 T, r = 2 T
Phase [ ] Amplitude [db] Corrected 20 0-20 -40-60 10-1 10 0 10 1-60 -90-120 -150-180 -210-240 10-1 10 0 10 1 Frequency [rad/s] Figure 7: Bode plot of transfer function L(s) Question 43 You are given the Bode plot of the open-loop gain L(s) in Figure 7. Which magnitude can the multiplicative model disturbance W 2 (s) reach at the crossover frequency ω c while still ensuring stability? A 16.9 db B 6.29 db Question 44 Which of the following statements is true about the Anti-Windup scheme? A The Anti-windup reduces the total error multiplied by a gain parameter when the input saturates. B The Anti-windup reduces the proportional error term multiplied by a gain parameter when the input saturates. C The Anti-windup restricts the input to the maximal input when the input saturates. D The Anti-windup reduces the integral error multiplied by a gain parameter when the input saturates.
Answer sheet: student number 0 0 0 0 0 0 0 0 please encode your student number, and write your first and last name below. 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 Firstname and lastname:........................................... 4 5 4 5 4 5 4 5 4 5 4 5 4 5 4 5 How to select answer B : 6 7 8 6 7 8 6 7 8 6 7 8 6 7 8 6 7 8 6 7 8 6 7 8 Answer B chosen. Corrected, answer B chosen. Double corrected, answer B chosen. 9 9 9 9 9 9 9 9 No choice made. Answers must be given exclusively on this sheet; answers given on the other sheets will not be counted. 1 Systems and Control Architecture Q1: A B C D Q2: A B C D Q3: A B C 2 System Modeling and Analysis Q4: A B C D Q5: A B C D Q6: A B C D Q7: A B C D Q8: A B C D Q9: A B C D Q10: A B C D Q11: A B C Q12: A B C D Q13: A B C D Q14: A B C D Q15: A B C D Q16: A B C D E F Q17: A B C D 3 Frequency Domain Q18: A B C D Q19: A B C D Q20: A B C D Q21: A B C D Q22: A B C D Q23: A B C D Q24: A B C D Q25: A B C D E F Q26: A B C D Q27: A B C D 4 Controller Design Q28: A B C D E F Q29: A B C D Q30: A B C D E Q31: A B C D E Q32: A B Q33: A B C D Q34: A B C D Q35: A B C D Q36: A B C D Q37: A B C D Q38: A B C D
Q39: A B C D Q40: A B C D Q41: A B C D Q42: A B C D Q43: A B Q44: A B C D