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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).

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3 Sessionsprüfung Regelungstechnik I Page Question (Modeling, Linarization) 8 Points g (Gravitation) x Water Droplet h Wind (System Input) Figure : Water Droplet Hovering in the Air In this exercise, a linearized model of the system presented in Figure has to be created. The System is a water droplet which is kept from falling by air streaming in vertically upwards direction. The droplet has a diameter of d = 2 [mm] and the water has a density of ρ W = [kg/m 3 ]. The droplet is accelerated by the gravitational force towards the ground with 9.8 [m/s 2 ]. Evaporation effects as well as movements of the droplet in horizontal direction can be neglected. The velocity of the air flow (wind) s(t) at ground level (at h = ) can be arbitrarily adjusted and is denoted with u(t) [m/s]. The magnitude of the flow velocity decreases as the height h [m] increases, the functional relationship is as follows: s(t) = u(t) α h [m/s], where α = [/s] is given as a constant parameter. The vertical position of the droplet is denoted by x. The variable h denotes the general coordinate in vertical direction. The force that pushes the water droplet upwards can be modeled as follows: F a = 2 ρ L c a A v 2 [N], where A [m 2 ] is the area of the droplet, c a =.445 [ ] is the drag coefficient, ρ L =.2 [kg/m 3 ] is the density of air, and v denotes the velocity with which the air flows towards the droplet. a) (3 points) Choose the state vector z(t) = [x(t), ẋ(t)] T and derive the nonlinear state space description of the form dz(t) dt = f(z(t), u(t)), w(t) = g(z(t), u(t)). Use the variable names z (t), z 2 (t), u(t), and w(t). b) (2 points) Calculate the flow velocity u e at ground level, which is necessary for the droplet to stay at the positon z,e = 3 [m] in a state of equilibrium.

4 Page 2 Sessionsprüfung Regelungstechnik I c) (3 points) Linearize the system equations around this equilibrium point (no normalization is required). Express the system equations in the standard form (state space description with the matrices {A, b, c, d}). Express the matrices in general form, i.e. use the variables of the system and do not use their corresponding numerical values.

5 Sessionsprüfung Regelungstechnik I Page 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 (see table for solution). Furthermore, the Nyquist plots (see below the diagrams A, B, C and D; plotted for positive frequencies only) of these transfer functions, and the resulting step responses (see on the next page the step responses to 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 on the solution page of this question for your solution. You do not need to justify your answers..5 Nyquist Plot A.5 Nyquist Plot B Im Im Re Re.5 Nyquist Plot C.5 Nyquist Plot D Im Im Re Re

6 Page 4 Sessionsprüfung Regelungstechnik I 2 Step Response 2 Step Response 2 Amplitude [ ].5 Amplitude [ ] Time [s] Time [s] 2 Step Response 3 2 Step Response 4 Amplitude [ ].5 Amplitude [ ] Time [s] Time [s]

7 Sessionsprüfung Regelungstechnik I Page 5 Question 3 (Controller Synthesis) 8 Points The department of modeling at your company has created a very accurate model of a system to be controlled. The corresponding dynamics are given by the following transfer function: P (s) = (s + 2 ) (s + 3 ) Your job is to control this system. All tasks of this question can be solved independently of each other. a) (3 points) You have to design a PI-controller C P I (s) = k p ( + T i s ), with the following specifications on the contol system: The crossover frequency must be at ω c =.85 [rad/s]. The phase marigin must be 45. Calculate the values of the parameters {k p, T i } which lead to a control system that fulfills these specifications. b) (3 points) Your colleague suggests the following PD-controller as an alternative controller: ( 5 C P D (s) = 2 s + ) 3 Your boss says she wants the one controller which leads to a faster rise time t 9 in a step response analysis. Which controller do you suggest? c) (2 points) Another colleague suggests two different P-controllers C P (s) = k p with the following specifications Controller Crossover Frequency ω c Phase Margin C P, = k p,.85 [rad/s] 45 C P,2 = k p,2.5 [rad/s] 45 Your boss supports his intentions as she wants to keep the structures of the controllers as simple as possible. What do you think of your colleagues suggestions? Justify your answer.

8 Page 6 Sessionsprüfung Regelungstechnik I Question 4 (Laplace-Transformation) Points The following subtasks a), b) and c) can be solved independently. a) The two systems Σ and Σ 2 are connected in series. u(t) x(t) y(t) Σ Σ 2 Figure 2: System overview. The output x(t) of Σ is characterized by the following differential equation: ẍ(t) = 4 ẋ(t) 4 x(t) + u(t) with ẋ(t) = x(t) = u(t) =, t. The transfer function of Σ 2 (s) = Y (s) X(s) is given by: Σ 2 (s) = 3 s s + i) ( point) Determine the transfer function of the entire system Σ a (s) = Y (s) ii) iii) U(s). (2 points) The system Σ a (s) is subjected to a step excitation u(t) = h(t), calculate the time domain response y(t). (2 points) Illustrate the time response of ii) in the associated template on the solution page qualitatively. By doing that, also think about the following characteristics: What is the system s static gain? Does the system response overshoot? b) Consider the block diagram in figure 3. x x 2 x 3 y s u 5 s s 2 2 Figure 3: Block diagram. i) ( point) Determine the associated state space description A, B, C, D. ii) (2 points) Determine the transfer function Σ b = Y (s) U(s). c) The time response of another time-invariant SISO system is given as: y(t) = ( e (t T ) cos (ω(t T ))) h(t T ) with T = ms, ω = π 3 rad/s. i) (2 points) Calculate the transfer function of the system Σ c (s).

9 Sessionsprüfung Regelungstechnik I Page 7 Question 5 (Stabilization / Performance & Robustness) 9 Points You would like to develop a controller for a plant with the following state space representation of its model. [ d 3 dt x(t) = y(t) = [ 4 4 ] [ ] a + 3 x(t) + u(t), x() = (a) a ] x(t). (b) The parameter a specifies the actuator. The larger you choose a in the permissible interval 2 a 2 the more expensive is the actuator. Figure 4 shows the set-up of the control system. The control system is used in an environment where it is disturbed by a noise signal n(t) with a frequency of ω n 3 rad /s. r(t) C(s) u(t) P (s) y(t) n Figure 4: Control system with input and output signals. Remark: Solution of a) is required for the solutions of the subsequent questions b)-d). But the questions b)-e) can be partly solved indepedently from each other. a) (2 points) Determine the transfer function P (s) of the plant with the input signal u(t), output signal y(t) and state vector x(t) in function of the actuator parameter a. Determine also the pole(s) and zero(s) of the plant. b) (3 points) In which range the cross over frequency ω c should be selected such that an appropriate controller C(s) may be designed? Which cross over frequency do you choose if at the same time the actuator costs have to be minimized? Determine also the corresponding actuator parameter a that minimizes the actuator costs. Important remark: Based on a special offer you decide to purchase an actuator with a =. Use this value to solve the following questions c) to e). c) (2 points) You would like to stabilize the control system with a P-controller C(s) = k p. Justify why it is surely possible to stabilize the control system for negative values of k p in the range 2 < k p <.3. d) ( point) You use a P-controller according to point c) that stabilizes the control system. What is the amplification of a high frequent sensor noise (ω n ) at the output of the control system? e) ( point) Determine the steady state error of the control system for k p =.3.

10 Page 8 Sessionsprüfung Regelungstechnik I Question 6 (Bode-Diagram/Nyquist Criterion) Points The following subtasks a) and b) can be solved independently. a) The bode-diagram of a critically damped plant with the transfer function P (s) was measured. Figure 5: Bode-diagram of the plant with the corresponding transfer function P (s). i) (2 points) Determine the transfer function of the plant P (s) with the aid of the measured bode-diagram in figure 5. ii) ( point) The system outlined in figure 6 is controlled using a proportional controller with the transfer function C (s) = k p =. The control system is subjected to a disturbance (unit step) on input w. Will the output of the system return to the original value, without a steady state error? Assume that the system was at equilibrium with r = and y =, prior to the disturbance. r + - C (s) + + w P (s) y Figure 6: Control system for the plant P (s). iii) iv) ( point) Determine the phase margin in case the plant P (s) of exercise i) is controlled by the controller C (s) = k p = according to the structure presented in figure 6 (w = ). ( point) Which structural changes would you consider to apply to C (s) firstly, in order to eliminate the steady state error?

11 Sessionsprüfung Regelungstechnik I Page 9 b) Another control system consists of the plant P 2 (s) and the controller C 2 (s). The transfer function P 2 (s) is known and the Nyquist-diagram of P 2 (s) is given in figure 8. P 2 (s) = s (s + ) 3 (2) Figure 8: Nyquist-diagram of the plant P 2 (s) Figure 7: Block-diagram of the control system. r + - C 2 (s) P 2 (s) y i) (2 points) The Bode-diagram of the controller C 2 (s) is given. Illustrate the according Nyquist-diagram as accurately as possible in the associated template on the solution page. It is not demanded to derive the exact transfer function. Figure 9: Bode-diagram of the controller C 2 (s). ii) (3 points) Assume now that a simple P-controller C 2 (s) = k p is used. Make use of the Nyquist-criterion in order to calculate the range of k p that leads to an asymptotically stable system.

12 Page Sessionsprüfung Regelungstechnik I Question 7 (System Analysis) 7 Points Consider the model of a geosynchronous satellite as shown in Figure. The gravitational force of the earth is approximately equal to F g = MG r 2 m, where M > is the mass of the earth and G > is its gravitational constant. The mass of the satellite is described by m >. The distance from the satellite to the earth s center of mass is expressed with the radius r >. The centrifugal force counteracts to the gravitational force and is given by (3) F z = m r ω 2, (4) where ω is the rotational velocity of the satellite around the earth. F t ω m F z F g r M Figure : Geosynchronous satellite Assume a thruster with the force F t R is mounted tangential onto the satellite. As a result, the equations of motion for the geosynchronous satellite yield r = r ω 2 MG r 2, ω = F t r m. The linearization about the equilibria, r >, ω >, F t, =, leads to the system of linear equations, i.e. ω MG 2 r r x 3 ω ṙ = x + F t with x = r. (7) r m ω (5) (6)

13 Sessionsprüfung Regelungstechnik I Page a) ( point) State the system matrices {A, b, c, d} for the system of linear equations (7) with the assumption that the radius r and the rotational velocity ω are measured. b) (2 points) In the sense of Lyapunov, is the given system stable, asymptotically stable or unstable? Justify your answer mathematically. c) Given the tangential thruster, is the system controllable? i) (2 points) Make a point about the controllability of the system and justify your answer mathematically. ii) iii) ( point) Assume that the satellite has slightly approached the earth. Is the tangential thruster able to bring the satellite back onto its original orbit? ( point) If yes, does the thruster have to accelerate or decelerate the satellite? If no, why is the thruster not able to do so? d) ( point) How would the system matrices look like, if the tangential thruster was replaced by a radial thruster F r R that acts in the same direction than the centrifugal force? e) (2 points) Which statements hold about the Lyapunov stability, the observability, and the controllability of the new system?

14 Page 2 Sessionsprüfung Regelungstechnik I Question 8 (Multiple-Choice) 8 Points Decide whether the following statements are true or false and check the corresponding check box with an X ( ) on the solution page of this question. 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 = δx + 4 δu is the linearization of the non-linear system ẋ = 4 x + + 3x + 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 s 2 +6s+5 c) An unstable system with the transfer function s (s 2) C(s) = k p (k p R). can be stabilized by a P-controller d) The state space representation {A, b, c, d} of a second order system has the transfer function P (s) = s+3. The system is completely controllable and observable. s 2 +s 6 e) The following state space model {A, b, c, d} represents a realization for a system with the transfer function Σ(s) = s s 3 : [ ] [ ] 2 A =, b =, 3 2 c = [ ], D = [ ] f) The Matlab instruction P = zpk(,[-i +i 3],-2) defines in Matlab a system with the transfer function P (s) = 2s (s 3)(s 2 2s+2). g) The system with the transfer function Σ(s) = s+7 + s 4 has no system zeros. h) The transfer function of a closed loop system from the reference signal to the output y is T (s) = s+ (complementary sensitivity). The loop gain of the control system is s 2 +4s+ L(s) = s+ s(s+3). Be aware of this fact!

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