Prüfung Regelungstechnik I (Control Systems I) Prof. Dr. Lino Guzzella 9. 8. 2 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).
Question (modelling, linearization) 6 Points A ventilator produces an air mass flow S(t), which depends on the rotational speed ω(t). S(t) = c ω(t) () The resisting torque T w (t) of the ventilator is T w (t) = c 2 ω(t) 2 (2) The ventilator is driven by an electric motor, which produces a driving torque T m (t) T m (t) = k U(t) k k 2 ω(t) (3) Where U(t) is the armature voltage of the motor. The rotational inertia of the system ventilatormotor is given by Θ. Friction can be neglected. The introduced parameters (c,c 2,k,k 2,Θ) are constant and positive real numbers ( R + ). a) (2 points) The armature voltage U(t) is the input to the system, the rotational speed ω(t) is the state of the system and the air mass flow S(t) is the output of the system. Derive the system equations. The Result has to be written in the following form: d ω(t) = f(ω(t),u(t)) S(t) = g(ω(t),u(t)) (4) dt b) (2 points) Calculate the rotational speed ω e and the armature voltage U e such, that the system is in equilibrium at an air mass flow of S e. c) (2 points) Linearize the system around the equilibrium point (U e,ω e,s e ), a normalization of the system is not required. Use the following definitions: Give the result in the following form: ω(t) = ω e + δω(t) (5) U(t) = U e + δu(t) (6) S(t) = S e + δs(t) (7) d δω(t) = a δω(t) + b δu(t) δs(t) = c δω(t) + d δu(t) (8) dt /
Question 2 (frequency domain, time domain) 8 Points You are given 4 transfer functions of open loops (L (s),l 2 (s),l 3 (s),l 4 (s)), the corresponding Nyquist plots of the open loops (diagram A, diagram B, diagram C, diagram D), as well as the step response of the closed loops (step response, step response 2, step response 3, step response 4). For every transfer function, allocate the corresponding Nyquist plot as well as the corresponding step response. Use the prepared table for your solution. Pointing: for a correct allocation: + point for a wrong allocation: point minimum number of points for the whole question: points transfer function table for solution L (s) = L 2 (s) = L 3 (s) = L 4 (s) =.5 s+ s+ e.7 s (s+) (s 2 +.4 s+).3 s 2 (.6 s+) 2 s (.5 s+) nyquistplot step response Nyquist plot A Nyquist plot B.5.5.5.5 Im Im.5.5.5.5.5.5.5 Re.5.5.5.5.5 Re 2 /
Nyquist plot C Nyquist plot D.5.5.5.5 Im Im.5.5.5.5.5.5.5 Re.5.5.5.5.5 Re step response step response 2.8.8 amplitude [ ].6.4 amplitude [ ].6.4.2.2 2 3 4 5 6 7 8 9 time [s] 2 3 4 5 6 7 8 9 time [s] step response 3 step response 4.2.8 amplitude [ ].8.6.4 amplitude [ ].6.4.2.2 2 3 4 5 6 7 8 9 time [s] 2 3 4 5 6 7 8 9 time [s] 3 /
Question 3 (controller design) 7 Points The following plant is given P(s) = (s +.5) (s + 2) e T d s, T d = π 8 seconds. a) (4 points) A PI-controller of the form C(s) = k p + k i s has to be designed. The control system is required to have a cross-over frequency of rad s and a phase margin of 45. Determine the parameters (k p, k i ) of the controller. b) ( point) Now, a controller has to be realized following the design rules of Ziegler and Nichols. For this purpose, the critical gain k and the critical oscillation period T are required. Describe the experimental procedure in order to obtain k and T. c) ( point) From experiments the following values are obtained for the critical gain k and the critical oscillation period T : k = 7.36 T = 2.63 s. Determine the parameters k p, T i, T d and τ of a PID controller of the form ( C PID (s) = k p + ) + s T d s T i (s τ + ) 2 using the design rules of Ziegler and Nichols. d) ( point) Which specific property of a plant that contains a time delay needs to be considered for the controller design? 4 /
Question 4 (Laplace Transformation) 8 Points The following tasks can be solved independently. a) (4 Points) A plant with the transfer function Σ (s) is given: Σ (s) = s + s 2 + 5 s + 6 The plant is excited with the following input signal u(t): u(t) = 4 sin (3t) Calculate the resulting output signal y (t) in the time domain. b) (4 Points) A linear, time-invariant SISO system is being excited with the following signal: u(t) = sin (2t) The system response (in the time domain) is: y (t) = 4 9 te t + 52 45 e t + 5 26 e t + ( 7 cos (2t) 9 sin (2t)) 3 i) Calculate the transfer function Σ (s) of the system. ii) iii) iv) What is the static gain of the system? What are the poles of the system? Approximate the rise time t 9 of the system by neglecting the non-relevant dynamics of the system. 5 /
Question 5 (Constraints) 8 Points Figure shows a pendulum-cart-system that has to be controlled. You know already from the lecture that die transfer function of this system is Y (s) = P(s) U(s), P(s) = g s 2 (l M s 2 g (m + M)), where the input signal u(t) is the force acting on the cart and the output signal y(t) is the position of the top end of the pendulum. U a (t) u(t) << m y(t) l M C(s) r Figure : Regelsystem Pendel auf einem Wagen. You are responsible for designing a cost efficient controller that stabilizes the pendulum at its upper position (the design of the controller C(s) is not a part of the question). The electronic amplifier (in Figure indicated by <<) that is already purchased, has a very high bandwidth. Therefore, it is considered to be an ideal amplifier. For the electric motors you have the choice among three different types. The corresponding Bode-plots that are available from the manufacturer, are given in Figure 2 (input signal U a (t) is the electric voltage, output signal u(t) is the force acting on the cart). The prices of the three electric motors are: EM- costs CHF, EM-2 costs 2 CHF, and EM-3 costs 3 CHF. The parameters of the pendulum are known from the documentation of the pendulum: m = 3 kg, M = kg, l =. m. It is assumed that the gravitational acceleration is g m/s 2. a) (8 Punkte) Which one of the three electric motors would you chose? Provide quantitative arquments! 6 /
... db - EM-3-2 EM-2-3 EM- - 2 3 log ω (rad/s) arg{...} -5 - -5 EM- EM-2 EM-3-2 3 Figure 2: Bode-diagrams of the three available motors. 7 /
Question 6 (Bode diagram/nyquist criterion) 7 Points The following bode-diagram of an unknown plant is measured: Magnitude (db) Bode Diagram 3 2 2 3 4 8 Phase (deg) 9 9 3 2 2 3 Frequency (rad/sec) a) (3 Points) Determine the transfer function Σ (s) of this plant plant. b) ( Point) Sketch the Nyquist diagram of the analyzed plant in the prepared image: Nyquist Diagram 5 Imaginary Axis 5 2 8 6 4 2 2 4 Real Axis c) (3 Points) The plant will be stabilized with a P controller (gain k p ). Use the Nyquist criterion to determine the limits within which k p has to be chosen such that an asymptotically stable closed-loop system results. 8 /
Question 7 ( system analysis ) 8 Points A state-space representation is given ẋ(t) = A x(t) + b u(t) y(t) = c x(t) with A = 2, b =, c = ( 2 ). a) ( point) Is the control system completely controllable? Justify your answer mathematically. b) ( point) Is the control system completely observable? Justify your answer mathematically. c) (2 points) Compute the eigenvalues of the system. Is it stable? Justify your answer. d) ( point) Is the system completely stabilizable? Justify your answer. e) (2 points) Derive the transfer function Σ(s) of the system. Compute the poles and the zeros. f) ( point) The detailed flow chart of a different control system is given with the input signal u(t) and the output signal y(t) (Fig. 3). Derive the state-space matrices A,b,c,d for the given system! 7 u x x 2 y -2-5 x 3-4 Figure 3: signal flow chart 9 /
Question 8 (Problem 8) 6 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) {x e = π 3,u e = π 6 } is the equilibrium of the system ẋ = cos(x) x + 2sin(u) u. b) The differential equation δẋ = π 2 δx+ π 2 δu is the linearization of the non-linear system ẋ = x 2 + sin(2x) u 2 around the equilibrium point {x e = π 4,u e = π 4 }. c) The following differential equation represents the model of a system: d 3 d2 d2 dt3y(t) + 2 dt2y(t) 5y(t) = 3 dt2u(t) + 4u(t) Hence the transfer function of the system is P(s) = 3s2 +4 s 3 +2s 2 5s. d) An unstable system with the transfer function s (s 5) can be stabilized by a P-controller. e) The Bode-diagram of a fifth order linear system has been accurately plotted, although it has two unstable poles. Nevertheless, this Bode-plot can be utilized to determine the range of the gain factor k p of a P-controller for which an asymptotic stable control system results. f) A PI-controller is used to control an asymptotic stable system. The time constant of the integral part of the controller T i is set to.3s. Starting at k p =, the controller gain will be gradually increased up to the critical gain factor. At the critical gain factor the time constant of the integral part will also be readjusted to.4s. As a consequence, the control system will be unstable. Be aware of this fact! /
g) A plant with the transfer function G(s) = s 4 is stabilized by a P-controller. There is a disturbance signal d at the input of the plant (see figure below!). d r + + k ---------- P s 4 y With a gain factor k P > 4 it can be achieved that the maximum of the impulse response (for zero initial condition and r = ) remains less than 4. h) A plant with the transfer function G(s) = s 6 has to be stabilized by a controller. A s 2 +5s 6 cross over frequency of ω c = 3 rad/s represents a meaningful bandwidth for the specification of the control system. Falsch /