# Prüfung Regelungstechnik I (Control Systems I) Übersetzungshilfe / Translation aid (English) To be returned at the end of the exam!

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1 Prüfung Regelungstechnik I (Control Systems I) Prof. Dr. Lino Guzzella Übersetzungshilfe / Translation aid (English) To be returned at the end of the exam! Do not mark up this translation aid - The German exam is the only valid version! All answers must be written on the regular exam sheets (which are in German).

2 Question (Modeling and Linearization) 8 Points The figure below illustrates a capacitor with variable distance between the plates. The lower plate of the capacitor is fixed. The upper plate of mass m is movable and connected to two springs and a damper. The springs are equal and their string stiffness is k. At the position y = y, the springs are in their zero force position. The damper force is proportional to the velocity of the plate. The damping coefficient is b. The upper plate is attracted by the lower plate with the force F EF = c U2 4 y 2, where U is the voltage across the capacitor. The position y is measured by an appropriate sensor. g U E-field m y a) (3 points) Determine the differential equations which describe the vertical motion of the upper plate. Select appropriate input, output and state variables. Give the system equations in the standard form, i.e., as a system of nonlinear first-order differential equations of the form ż(t) = f(z(t),v(t)), w(t) = h(z(t),v(t)), z(t) R 2, v(t),w(t) R. b) (2 points) Determine the voltage U e which keeps the upper plate at equilibrium at the position y e = y /2. Note: y > m g/k. c) (3 points) Linearize the system equations at the position y e = y /2 (a normalization is omitted). Give the system equations in the standard form, i.e., in state-space representation {A,b,c,d}. / 2

3 Question 2 (Frequency domain, time domain) 8 Points The open-loop transfer functions (loop gain) (L (s),l 2 (s),l 3 (s),l 4 (s)) of 4 control systems are given. Further the Nyquist plots (diagrams A, B, C und D) of these transfer functions, and the resulting step responses (step responses bis 4) of the corresponding closed loop systems are given. Assign the correct nyquist plot and the correct step response to each of the open loop transfer functions. Use the table provided below for your solution. You do not need to justify your answers. Credits: Per correct assignment: + credit Per incorrect assignment: credit Minimum amount of credits for the whole question: credits Transfer functions Table for solution L (s) = L 2 (s) = L 3 (s) = L 4 (s) = s 2 +s+ (3s) (s 2 +s+).5s+ s 2 +s+ 2s+ s 2 +s+ Nyquist plot (open loop) Step response (closed loop) Nyquist plot A Nyquist plot B Im Im Re Re 2 / 2

4 Nyquist plot C Nyquist plot D.5.5 Im Im Re Re step response step response amplitude [-].6.4 amplitude [-] time [s] 5 5 time [s] step response 3 step response amplitude [-].6.4 amplitude [-] time [s] 5 5 time [s] 3 / 2

5 Question 3 (Controller Design) Points The following plant is given P(s) = s (s + ) (s + 2) Your supervisor asks you and your colleague to design a controller for the given plant. You suggest to use a PID controller and to design its parameters with the Ziegler/Nichols method. a) (4 points) Determine the critical gain k p and the critical frequency ω. b) ( point) Now calculate according to the Ziegler/Nichols method the parameters k p, T i and T d of the following PID controller [ C PID (s) = k p + ] + s T d s T i Afterwards, your colleague computes with MATLAB the phase margin of the control system of your controller. This poor margin turns out to be Since your colleague wants to have more phase margin, he suggests to use instead of C PID (s) the following controller C 2 (s) = 2 s (2 s) c) (5 points) Since your supervisor wants to be sure that the phase margin is larger with C 2 (s), she asks you to compute the phase margin of the control system C 2 (s) P(s). 4 / 2

6 Question 4 (Laplace-Transformation) 9 Points The subtasks a) and b) can be solved independently. a) (4 points) Determine the time-domain output signal y(t) of the system P(s) P(s) = s + 2 (s s + ) (s + 3) for the following input signal u(t) u(t) = h(t). b) A linear time-invariant SISO system is excited by u(t) = h(t). Its step response in the time-domain is y(t) = ( e t (cos(2 t) + 2 ) sin(2 t)) h(t). i) (3 points) Calculate the transfer function Σ(s) of the SISO system. ii) iii) ( point) Determine the poles of the system. ( point) Approximate the rise time t 9 as well as the overshoot ˆǫ of the step response by respecting only the relevant system dynamics. 5 / 2

7 Question 5 (Constraints) Points Figure shows a sketch of the system to be analyzed. A ferromagnetic solid sphere is positioned in the magnetic field of an electromagnet. The magnitude of this field can be adjusted very rapidly by an amplifier. Therefore, the force acting on the solid sphere is assumed to be proportional to the input u(t). u(t) y(t) Figure : Floating ferromagnetic solid sphere in a magnetic field. A controller C(s) (whose design is not a part of this question) has to keep the ferromagnetic sphere levitating in the magnetic field. The position y(t) of the sphere is measured by a photo cell. The measurement y(t) is corrupted by the electromagnetic radiation that is produced by the amplifier. The lowest frequency of the noise signal is 5 Hz 3 rad/s. Figure 2 shows the set-up of the control system. r(t) C(s) u(t) P(s) y(t) n Figure 2: Control system with input and output signals. The dynamic behavior of the plant can be represented perfectly by the following state space model: [ ] [ ] d dt x(t) = x(t) + u(t) () 9 9 where x (t) = y(t) is the position and x 2 (t) is the velocity of the ferromagnetic solid sphere. a) ( point) Determine the transfer function P(s) of the plant with the input signal u(t) and the output signal y(t). b) (4 points) Sketch in the Bode-diagram provided on next page the magnitude P(j ω) and phase plot P(j ω) of the plant transfer function. Tip: Determine the asymptotes for very low and very high frequencies. c) ( point) Indicate in the Bode-diagram the frequency range where the noise signal n(t) is present. d) (4 points) Before starting with the design of the controller C(s), answer first the question, whether a controller exists that can full fill the requirements. What are your considerations? What is your answer? Justify your answer mathematically. 6 / 2

8 P(j ω) db ω rad/s P(j ω) ω rad/s Figure 3: Bode-Diagram for representing the solution. 7 / 2

9 Question 6 (Nyquist-plot, Nyquist-theorem) 8 Points For a plant P(s) the following Bode plot was measured. 4 Bode Diagram Magnitude [db] Phase [deg] Frequency [rad/s] Figure 4: Bode plot of the system a) (2 points) Draw the Nyquist plot of the plant P(s) qualitatively in the provided figure. Thereby, use the information that lim P(jω) = 4 j. ω b) (2 points) Identify the transfer function of the plant P(s) using the Bode plot. c) ( point) You want to control the plant using a P-controller C(s) = k p mit k p =. What are the gain and the phase margins of the resulting closed loop system. d) ( point) Use the Nyquist theorem in order to find the values of k p, for which the system is stable. e) (2 points) You want to improve the step response of the closed loop system. Therefore, you use a PD-controller with the transfer function C(s) =.5 +.5s. What is the gain margin of the resulting closed loop system? 8 / 2

10 4 Nyquist Plot 3 2 Imaginary Axis Real Axis Figure 5: Nyquist plot of the plant P(s) 9 / 2

11 Question 7 (System Analyses, Block Diagram) 8 Points A state-space representation is given ẋ(t) = A x(t) + b u(t) y(t) = c x(t) + d u(t) with A = 2, b =, c = [ ], d = a) (2 points) In the sense of Lyapunov, is the system stable, asymptotically stable, or unstable? Justify your answer mathematically. b) ( point) Is the control system completely controllable? Justify your answer mathematically. c) ( point) Is the control system completely observable? Justify your answer mathematically. d) (2 points) The detailed flow chart of a different control system is given with the input signal u(t) and the output signal y(t) (Fig. 6). Derive the state-space matrices {A, b, c, d} for the given system! -3 y(t) 2 u(t) x (t) x 2 (t) x 3 (t) 5 4 Figure 6: Signal flow chart, u(t) = input signal, y(t) = output signal e) (2 points) The transfer function Σ(s) is given, describing the input/output behavior of another linear dynamic system Σ(s) = s2 + 3s + 2 s 3 + 4s 2 + s + 5 Determine the system matrices {A,b,c,d} of an accordant state-space representation. / 2

12 Question 8 (Multiple-Choice) 8 Points Decide whether the following statements are true or false and check the corresponding check box with an X ( ). You are not required to justify your answers. All questions are equally weighted ( point). There will be a reduction of one point for a wrong answer. Unanswered questions will get points. The minimum sum for all questions is points. a) The differential equation δẋ = 9 δx+2 δu represents the linearization of the non-linear system ẋ = 7 x 2 5 x + 3 u 2 around the equilibrium point {x e =,u e = 2}. b) A constant signal u(t) = at the input of a system with the transfer function Σ(s) = produces for t a constant output signal of 2. s 4 s 2 +3s+2 c) The following state space model {A, b, c, d} represents a realization for a system with the transfer function Σ(s) = s+3 A = [ 7 5 s 2 5s 7 : ], b = [ ], c = [ 3 ], D = [ ] d) An unstable system with the transfer function P(s) = 2 s(s 3) can be stabilized in a closed loop control system by a PD-controller C(s) = k p + k d s with the parameters k p = 2 and k d =.5. e) A plant with the transfer function P(s) = s(s+5) (an integrator s and a first-order element (LP-) s+5 in series) is controlled by a P-controller (k p > ). There is a disturbance signal w at the input of the plant (see figure below). w r k p + + P s y The integrator of the plant will guarantee that a constant disturbance w(t) = h(t) will be completely rejected for r(t) = and t, i.e. lim t y(t) =. Be aware of this fact! / 2

13 f) A PI-controller C(s) = k P (+ T i s ) is used to control an asymptotic stable system. For the adjusted controller parameters k p (gain of the proportional part) and T i (time constant of the integral part), the output signal of the control system shows a harmonic oscillation (control system is critically stable). By increasing the time constant T i of the integral part, the control system will become asymptotically stable. g) An asymptotically stable control system has at least a guaranteed gain margin of k < 2 if its sensitivity function fulfills the condition max S(jω) < 2. ω h) The open loop gain of a control system L(s) = C(s) P(s) has two unstable poles. The Nyquist-plot L(jω) is shown below for ω von bis +. It can be cocluded that he closed loop control system is asymptotically stabel. 3 2 L(jω) Im Re 2 / 2

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