Exam January 23, 27 Control Systems I (5-59-L) Prof. Emilio Frazzoli Exam Exam Duration: 35 minutes + 5 minutes reading time Number of Problems: 45 Number of Points: 53 Permitted aids: Important: 4 pages (2 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:. 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. Good luck!
Control Architectures and Systems Question Mark all correct statements. (2 Points) Feedforward control systems... A...rely on precise knowledge of the plant. B...can change the dynamics of the plant. C...reject external disturbances. Question 2 Mark all correct statements. (2 Points) Feedback control systems... A...can stabilize an unstable plant. B...can feed sensor noise into the system. C...can de-stabilize a stable plant. Question 3 Mark all correct statements. (2 Points) All signals are scalars. The system d dt y(t) =(u(t + ))2 is: A Causal B Memoryless / Static C Time-Invariant D Linear Question 4 Mark all correct statements. (2 Points) All signals are scalars. The system (s) = e5 s s+ is: A Causal B Time-Invariant C Memoryless / Static D Linear Question 5 Mark all correct statements. (2 Points) All signals are scalars. The system y(t) =t 2 u(t A Linear B Time-Invariant ) is: C Causal D Memoryless / Static
2 System Modeling Box : Questions 6, 7 Description: Consider an electric motor which you would like to operate at a constant rotational speed!. Applying a voltage U(t) results in a change in the circuit current I(t), which is governed by the di erential equation L d I(t) = R I(t) apple!(t)+u(t), () dt whereby L is the circuit inductance, R its resistance and apple a constant relating the motor speed!(t) to an electro motor-force (EMF). The dynamics of the motor speed are given by d!(t) = d!(t)+t(t), (2) dt where represents its mechanical inertia, d a friction constant and T (t) =apple I(t) thecurrentdependent motor torque. Box 2: Question 6 r(t) e(t) C u(t) P x(t) y(t) Question 6 Relate the variables in the block diagram above to the correct signals. apple!(t) A u(t) =U(t),x(t) = I(t),y(t) =!(t),r(t) =! apple U(t) B u(t) =U(t),x(t) = T (t),y(t) =T (t),r(t) =! apple!(t) C u(t) =U(t),x(t) = I(t),y(t) =I(t),r(t) =! apple!(t) D u(t) =I(t),x(t) = U(t),y(t) =I(t),r(t) =! apple!(t) E u(t) =I(t),x(t) = T (t),y(t) =!(t),r(t) =!
Question 7 A colleague tells you that the circuit inductance is very small and can actually be neglected. The arising motor model can now be represented as... A...a second-order system. B...a first order system. C...an integrator. D...a static system. Question 8 Consider the nonlinear system You are measuring a constant output y(t) =. What is the value of x(t) and u(t)? d 4 x(t) = x(t) u(t) dt y(t) =x(t) 2. A x(t) = 2 t and u(t) = 2. B x(t) = and u(t) = 3. C x(t) = 2 and u(t) = 2. D x(t) = 3 and u(t) = 3. Question 9 Given the system d dt x(t) =x(t)2 +5 u(t) 4 x(t) 2 y(t) =, u(t) you have to linearize it around the equilibrium x e =, u e = 2. Which are the state-space matrices A, b, c and d? A A =, b = 5, c = 4, d = 3. B A = 5, b = 5, c = 2, d = 3. C A =, b = 5, c = 2, d = 3. D A =, b = 5, c = 2, d = 3.
3 Linear Systems Analysis Box 3: Question The figure below shows the step response of a second-order system. y time Question Assess the bounded-input bounded-output (BIBO) stability. A BIBO unstable B BIBO stable C No conclusion possible
Box 4: Questions, 2 The figure below shows the response of an internal state x of a system as an impulse is applied to the system s input. internal state x time Question A No conclusion about the stability is possible. B The system is unstable. C The system is asymptotically stable. D The system is Lyapunov stable. Question 2 A The system is BIBO stable. B No conclusion about the BIBO stability is possible. C The system is BIBO unstable. Question 3 Given a system with state space representation apple ẋ(t) = 2 y(t) = x(t)+u(t), x(t)+apple u(t) where 2 R. For which values of is the system asymptotically stable? A <. B <. C apple. D.
Box 5: Question 4 Description: The figure below shows the step response of a first-order system of the form ẋ(t) =a x(t)+b u(t) y(t) =x(t). (3) 8 6 y(t) 4 2.5.5 2 2.5 3 3.5 4 4.5 5 t [s] Question 4 Which are the values of a and b? A a = 2 and b = 2. B a =.5 and b =. C a =.5 and b = 2. D a = and b =. Box 6: Question 5 Description: The figure below shows the impulse responses y A (t) and y B (t) of two systems A and B. Both systems are linear time-invariant second order systems. In addition to the responses, an unknown function f(t) and its tangential line at time t = is drawn in the plot to the right. y A (t) 2 3 4 5 6 time in seconds y(t) =f(t) y B (t) y(t) = f(t = ) t + f(t = ) 2 3 4 5 6 time in seconds Question 5 Which are the eigenvalues A and B of both systems? A A =2 and B = 2. B A =± j and B = 2 ± j. C A =± j 2 and B = 2 ± j 2. D A =± j 2 and B = 2 ± j 2.
Box 7: Questions 6, 7 Description: A LTI system has the form apple apple A =, b =, c =, d =. Question 6 The system in the above equation is... A only observable. B only controllable. C observable and controllable. D neither observable nor controllable. Question 7 For x (t = ) =,x 2 (t = ) = the initial condition response of the system is... A y(t) =. B y(t) =h(t). C y(t) =h(t) t. D y(t) =h(t) e t.
4 Frequency Response Box 8: Questions 8, 9 Description: Given a linear time-invariant system in state-space description. 2 3 2 2 3 ẋ(t) = 4 2 8 5 x(t)+ 4 5 u(t) 4 y(t) = x(t). Question 8 The transfer function g(s) is... A g(s) = 2s+ s(s+4)(s+). B g(s) = 2s+ s 2 +5s+4. C g(s) = s+5 s 2 +5s+4. D g(s) = 2(s+5) s 2 +4s+5. Question 9 The system is... A Asymptotically stable. B Lyapunov stable. C Unstable. Question 2 Given the tranfer function g(s) of a system. A unit step is applied to the system s input: The time response y(t) is... g(s) = 3s +7 s(s + ) u(t) =h(t). A y(t) =h(t) ( 4+7t +4e t ). B y(t) =h(t) (2 + 7t +7e t ). C y(t) =h(t) (7 4t +4e t ). D y(t) =h(t) (7 4t +4e t ).
Box 9: Question 2 Given the tranfer function g(s) of a system. g(s) = s +3 s 2 +.7s + In addition the figure below shows the four time responses A, B, C and D. 4 diagram 4 diagram 2 4 diagram 3 4 diagram 4 3 3 3 3 y(t) 2 2 2 2 5 5 time in seconds 5 5 time in seconds 5 5 time in seconds 5 5 time in seconds Question 2 Which of the four diagrams shows the correct step response of the system? A Diagram 3. B Diagram 4. C Diagram. D Diagram 2.
Box : Question 22 Question 22 Select the transfer function L(s) which matches the Bode Plot given above: A L(s) =. (s+)2 s+. B L(s) = s+. C L(s) =. s D L(s) = s+ (s+.)(s ) Question 23 What is the steady state output of the system when u(t) =sin(!t)? ẋ(t) = x(t)+u(t), y(t) = x(t), (4) A y ss (t) =! cos(!t) sin(!t) +! 2 B y ss (t) = p +! 2 sin(!t arctan(!)) C y ss (t) = sin(!t)!2 cos(!t) +! 2
Box : Question 24 Consider the following Bode plot for a loop transfer function L(s): Question 24 How many poles and zeros does L(s) have? A L(s) has 2 poles and zeros. B L(s) has 4 poles and 2 zeros. C L(s) has 3 poles and 2 zeros. D L(s) has 3 poles and zeros.
Box 2: Question 25 Consider the following Bode plot of a transfer function g(s): Question 25 Select the transfer function which matches the Bode plot. A g(s) = 2 s+ B g(s) = s C g(s) = 2 s D g(s) = s+
5 Polar Plot and Closed-Loop Stability Box 3: Question 26 The Nyquist plot for a loop transfer function L(s) with two unstable poles is shown below on the left. On the right is a feedback configuration for the closed loop system. Question 26 Mark all correct statements. (2 Points) Which of the following statements are true about the closed loop system T (s) = A The closed loop system T (s) is stable when k =. B The closed loop system T (s) is unstable when k>2. C The closed loop system T (s) is unstable when k =. D The closed loop system T (s) is stable when k>2. kl(s) +kl(s)?
Box 4: Question 27 The Bode plot for a plant transfer function P (s) is given below. Question 27 Which of the controller designs will produce a stable closed loop system? A C(s) =.s+.s+ B C(s) =.s+ C C(s) =. D C(s) =.
Box 5: Question 28 The Nyquist plot for a loop transfer function of the form L(s) = k (s + p )(s + p 2 )(s + p 3 ) (5) is given below. Question 28 How many unstable poles does the closed loop system T (s) = A T (s) has 2 unstable poles. B T (s) has unstable poles. C T (s) has unstable pole. D Not enough information is provided. L(s) +L(s) have?
Box 6: Question 29 The bode plot for a system with transfer function g(s) is given below. Question 29 Mark all correct statements. (2 Points) Select all answers that correctly classify the stability of the system. A g(s) is unstable. B g(s) is asymptotically stable. C g(s) is Lyapunov stable. D g(s) is bounded input bounded output (BIBO) stable.
Box 7: Question 3 Consider the following Nyquist plot of an open loop gain L(s): Question 3 Select the transfer functions which match the Nyquist plot. A L(s) = (s ) 3 B L(s) = (s+) 3 C L(s) = (s+) 3 D L(s) = ( s) 3
Box 8: Question 3 Consider the system described by the following block diagram r + e C(s) P (s) y where P (s) = s +, C(s) =k. Question 3 Determine the smallest positive gain k such that the feedback system is stable and, when r(t) is a unit step, lim t! e(t) apple.. The smallest positive gain k is: A B 2 C D 5 E 9
Question 32 We are given the transfer function: g(s) = (s + 4) (s 4 9s 2 ). Which of the following is the root locus plot of g(s)? 5 Root Locus 5 Root Locus Imaginary Axis (seconds - ) 5-5 Imaginary Axis (seconds - ) 5-5 - - A -5 - -8-6 -4-2 2 4 6 8 Real Axis (seconds - ) D -5 - -8-6 -4-2 2 4 6 8 Real Axis (seconds - ) 5 Root Locus 3 Root Locus 2 Imaginary Axis (seconds - ) 5-5 Imaginary Axis (seconds - ) - - -2 B -5 - -8-6 -4-2 2 4 6 8 Real Axis (seconds - ) E -3 - -8-6 -4-2 2 4 6 8 Real Axis (seconds - ) 5 Root Locus Imaginary Axis (seconds - ) 5-5 - C -5 - -8-6 -4-2 2 4 6 8 Real Axis (seconds - )
6 Feedback Control and Specifications Question 33 For the following system, you have the choice between implementing two di erent sensors: y (t) =x (t) or y 2 (t) =x 2 (t). ẋ(t) = apple 3 2 x(t)+ apple u(t) Which sensor should you choose? A y 2 (t) B y (t)
Box 9: Questions 34, 35, 36 Consider the block diagram given below to answer the following three questions: r + C(s) P (s) y P 2 (s) + Question 34 Suppose P (s) = 2 s, P 2 = s, and C(s) = 2. Which of the following choices is the correct transfer function, G(s) = Y (s)/r(s), from r to y? A G(s) = 4 s+ B G(s) = 4(s 2) (s+) C G(s) = 4(s+) (s ) 2 D G(s) = 4(s ) (s+) 2 E G(s) = 4 s Question 35 Is the system defined in the above problem internally stable? A No. B Yes. Question 36 Now assume instead that P (s) =P 2 (s) =P (s) for some P (s) and that C(s) is stable. Which of the following statements is true? A The system is internally stable if and only if P (s) has no zeros with positive real part. B The system is NOT internally stable for any choice of P (s). C The system is internally stable for any choice of P (s). D The system is internally stable if and only if P (s) is stable.
Box 2: Question 37 Consider the state-space system below, where a is a real parameter: ẋ(t) = y(t) = 2 apple apple a 4 x(t)+ u(t) a x(t) Question 37 Is the system stabilizable for a =? A No. B Yes. Question 38 Mark all correct statements. (2 Points) Mark all of the following statements about control limitations that are true. A The dominant right-half-plane pole puts a lower limit on the system s bandwidth. B The sensitivity function, S(s), plus the complementary sensitivity function, T (s), must equal one for all s. C A plant s right-half-plane zero limits the bandwidth of the closed-loop system. D When a plant has a right-half-plane zero z, the sensitivity function at z, S(z), must equal.
7 Control Design Box 2: Question 39 The figure below shows the block diagram of a cascaded controller: r e e 2 y + + A(s) B(s) Question 39 Derive the transfer function T (s) from r! y for the system depicted in the block diagram above. A B C B(s) +A(s)+A(s)B(s) A(s)B(s) +A(s)+B(s)+A(s)B(s) A(s)B(s) +A(s)+A(s)B(s) D E A(s)B(s) A(s)+A(s)B(s) A(s)B(s) +A(s)B(s) Question 4 Let L(s) = (s+5)(s+3) (s+2) be an open loop transfer function in a standard feedback architecture. Let 2 the input to the system be a unit step input. What is the static error e of the system? A 5 9 B 4 9 C D 2 E 3 9 Question 4 To which of the following ideal systems can Ziegler/Nichols-tuning by using critical oscillation not be applied? A 7 (s+) 5 B 3 s+ C 4 s+ e.s D 3 (s+) 3
Question 42 Your system engineering counterpart gives you the system characteristics below: Noise above! n = 5 rad/s needs to be suppressed. Disturbances below! d =5rad/s need to be rejected. Modeling uncertainty reaches percent at! 2 = 4 rad/s. An unstable pole at s =.5 ispresent. The plant has a nonminimumphase zero at s = 3. You want to design a control system where the crossover frequency satisfies max{! d, 2! } <! c < min{.5! 2,.5! delay,.5!,.! n } (6) Can you find a feasible crossover frequency! c in presence of the above specifications? A Yes. B No. Box 22: Question 43 The plot below shows step responses of di erent variations of PID controllers (P, PI, PD, PID).5 2 3 4 without control reference Unit step response [-].5.5.5 2 2.5 3 3.5 4 4.5 5 Time [s] Question 43 Which step response in the figure above belongs to the PD-controller? A 2 B 3 C D 4
Box 23: Question 44 Consider the following 4 implementations of an anti - windup scheme:. 2. 3. 4.
Question 44 Which of the above is a correct anti-windup scheme? A B 3 C 4 D 2
Box 24: Question 45 Consider the following Bode plot of a plant: Amplitude [db] 2 - -2 Bode plot -3 - Phase [ ] -9-2 -5-8 - Frequency [rad/s] Question 45 Use the bode plot shown above to determine the approximate crossover frequency! c, phase margin ' and gain margin. A! c 2.5rad/s, ' 3, B! c 4.4rad/s, ' 5, 3 C! c 2.5rad/s, ' 3, 3 D! c 4.4rad/s, ' 3, 3