Prüfung Regelungstechnik I (Control Systems I) Prof. Dr. Lino Guzzella 3.. 24 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) 8 Points Figure shows a system consisting of a balloon and a pressure cylinder with a moving piston. Your task is to derive a linearized model of this system. The balloon and the volume of the cylinder that is not taken up by the piston are completely filled with oil. The oil is assumed to be incompressible. The total oil volume is V tot ; the oil volume in the balloon is V b (t). The cylinder and the balloon are connected through a large opening, i.e., the oil pressure is equal in both chambers. This pressure is determined by the filling of the balloon and it is computed by p(t) = e a V b(t) + b, where a and b are known parameters. The piston has a mass m and a frontal area F and it moves axially inside the cylinder. The position of the piston is described by r(t). Only two forces act on the piston, namely the pressure force of the oil and the external force K(t) = v(t), which is the input of the system. Friction is neglected. The output of the system is volume of the balloon w(t) = V b (t). Figure : Balloon and pressure cylinder. a) (3 Points) The state vector is defined as z(t) = [ r(t) ṙ(t) ] T. Derive the nonlinear statespace description of the form dz(t) dt = f ( z(t), v(t) ) w(t) = g ( z(t), v(t) ). Use the variable names z (t), z 2 (t), v(t), and w(t). b) (3 Points) Calculate the force v e that is necessary to keep the system in an equilibrium at z,e =. c) (2 Points) Linearize the system around this equilibrium and calculate the system matrices A, b, c, and d. /
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 below and on the solution page of this question). 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 below 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. Credits: Per correct assignment: + point Per incorrect assignment: point Minimum amount of credits for the whole question: points Table for solution L (s) = L 2 (s) = L 3 (s) = L 4 (s) = Transfer functions ( ) +. s.2 s+ ( ) (.2 s+) + 2. s (.2 s+) 2 (.2 s+) Nyquist plot (open loop) Step response (closed loop) Nyquist plot A Nyquist plot B.5.5 Im Im -.5 -.5 - - -.5.5 Re - - -.5.5 Re 2 /
Nyquist plot C Nyquist plot D.5.5 Im Im -.5 -.5 - - -.5.5 Re - - -.5.5 Re Step response Step response 2.5.5 Amplitude [-].5 Amplitude [-].5 2 3 4 5 Time [s] 2 3 4 5 Time [s] Step response 3 Step response 4.5.5 Amplitude [-].5 Amplitude [-].5 2 3 4 5 Time [s] 2 3 4 5 Time [s] 3 /
Question 3 (Controller Design) Points You are asked to design a controller for the position control of a helicopter in hover. For the sake of simplicity, you can assume that both the vertical and lateral positions are perfectly controlled. Your task is to design the remaining controller for the longitudinal position x(t). The relevant tilt angle for the longitudinal motion is the tilt angle around the lateral axis, denoted by γ(t). γ(t) x(t) Figure 2: Helicopter in hover The linearised equations of motion have already been derived: ẍ(t) = a γ(t) γ(t) = u(t) Where u(t) is the control input and a is a positive constant. The position of the helicopter is measured by GPS. The transfer function of the plant P (s) has already been derived: P (s) = a s 4 You should use a controller C(s) with the following transfer function: C(s) = k s 3 k 2 (τs + ) 4 All sub-questions can be solved independently! a) (2 Points) Is it possible to control the system by a PD-Controller? Please give reasons for your answer. b) (3 Points) In a first step, the low-pass filter is neglected, the controller is therefore given by C(s) = k s 3. i) Calculate the controller gain k such that a crossover frequency of 2 rad s ii) Calculate the resulting phase margin. is reached. c) (3 Points) In a next step, the low-pass filter of the controller F (s) should be designed. F (s) = k 2 (τs + ) 4 The phase-margin at the cross-over frequency of ω c = 2 rad s should only be reduced by 3. Further on, the cross-over frequency should not be changed by the filter. Calculate k 2 and τ. d) (2 Points) Your controller does not work properly in the real helicopter. Even though the controller contains a low-pass filter, the 3-times differentiation causes problems. How could you augment the plant to avoid the 3-times differentiation? 4 /
Question 4 (Laplace-Transformation) Points The following subtasks can be solved independently. a) (4 Points) Consider the system realization of the system Σ. u ẋ - - ẋ 2 + - s 3 - s + + y 2 Figure 3: System realization of Σ. i) ( Point) Determine the associated state space description {A, b, c, d}. ii) ( Point) Determine the transfer function of Σ. iii) (2 Points) Determine the time domain system response of the system Σ( s) = 2s 2 (s+2) (s+). The system is excited by u(t) = h(t) sin(t). b) (2 Points) The following figure shows the step response of a linear, time invariant system Σ 2. y(t) [-] 2..8.6.4.2..8.6.4.2-2 - 2 3 4 5 6 7 t [s] Figure 4: Step response of Σ 2. Determine the parameters of the system under the assumption, that it is a second order system. c) (2 Point) Determine the step response of the following second order system Σ 3 in the time domain. Σ 3 (s) = 6 s 2 + 5s + 6. 5 /
d) (2 Points) Your colleague claims that it is possible to connect two first order systems in series such that a second order system results that can oscillate when a step change in the reference value occurs. i) ( Point) Illustrate the system responses y (t) and y 2 (t) to a step in the reference value h(t) schematically. Assume that both systems have a static gain of. Hint: Keep in mind the tangent at the step time and the rise time. h(t) y (t) y 2 (t) Σ A Σ B Figure 5: Series connection of two first order systems Σ A and Σ B. ii) ( Point) Justify without calculation that a series connection of two first order systems cannot oscillate when a step in the reference value occurs. 6 /
Question 5 (Stabilization / Performance & Robustness) 9 Points You want to design a controller for a plant of which you already know the following state space description. ẋ(t) = x(t) + u(t) y(t) = ( a) x(t) + u(t) () The parameter a (physical unit rad /s) represents a characteristic of the sensor and can be selected in the range a 2. The higher the value of a is selected, the more expensive is the sensor used. The sensor is supplied with alternating current from the public electricity network. Therefore it shows distinctive noise at the network frequency of 5 Hz ( 3 rad /s). Note: The solution of the sub-question a) is a prerequisite for the subsequent sub-questions b)-e). The sub-questions b)-e) can be solved partly independently of each other. a) (2 Points) Determine the transfer function P (s) of the given plant with input u(t), output y(t) and state x(t) as a function of the sensor parameter a. Additionally, calculate the poles and zeros of this plant. b) (3 Points) In which range should the crossover frequency ω c of the control system lie, such that a reasonable controller exists? Which crossover frequency do you select, if the costs of the sensor should be minimized? Please also determine the resulting sensor parameter a which minimizes the sensor costs. Important: Due to a special offer of the sensor manufacturer you decide to buy a sensor with a = 5 rad /s. Use this value for the sub-questions c) to e). c) (2 Points) You want to stabilize the plant with a P-controller C(s) = k p. Which values of k p are possible? d) ( Point) The absolute value of the amplification of the high-frequency sensor noise should be exactly equal to. Which value of the controller parameter k p do you have to select? e) ( Point) Use the value of k p found in d). How large is the steady state error with this value? 7 /
Question 6 (Bode-Diagram/Nyquist Criterion) Points You want to control a plant P (s) with a transfer function P (s) = K(s2 + 3s + 7)(s 2 8s + 4) (s + 7)(s 2 s + )(s 2 + 3s + 7) where K is a constant. The plant is in a closed-loop system with a controller C(s) as shown below. Due to using a wireless sensor for measuring the output, a delay of τ seconds is present in the feedback path. Also, a Nyquist-diagram of the plant P (s) is measured and shown below (the plus indicates the point + j). Nyquist-diagram of the plant P (s) 5 Block diagram of the closed-loop system. Imaginary Axis 5-5 ω = + ω = - - -5-5 5 Real Axis All sub-questions can be solved independently. a) (3 Points) Assume that C(s) = and τ =. Investigate the stability of the closed-loop system using the Nyquist criterion. b) (2 Points) Identify the constant K of the plant P (s), using the Bode-diagram of the plant shown in Figure 6 (This Figure is given also on the solution page of this question). c) (3 Points) Assume that C(s) =. Find the maximum value of sensor delay τ (in seconds) for which the closed-loop system remains asymptotically stable. Hint: Determine the phase-margin of the system and relate it to the delay τ. d) (2 Points) Assume that τ =. For a P-controller C(s) = k p, find the value of gain k p for which the closed-loop system is asymptotically stable. Hint: Determine the gain-margin of the system and relate it to the gain k p. 8 /
Bode-diagram of P(s) 2.2 2 Magnitude (db) - -2 27 Phase (deg) 8 9-9 - 2 3 Frequency (rad/s) Figure 6: Bode-diagram of the plant P (s). 9 /
Question 7 (System Analysis) 7 Points The vehicle dynamics of a tractor semi-trailer combination during forward travel can be described by.5 5 d dt x(t) = x(t) + u(t), y(t) = ( 2.5 ) x(t), 6 22 and during reverse travel by.5 5 d dt x(t) = 6 x(t) + u(t), 22 8 y(t) = ( 2.5 ) x(t). 8 The system input u(t) is the steering angle of the tractor and the system output y(t) is the articulation angle between tractor and semi-trailer, as defined in Fig. 7. u(t) y(t) tractor semi-trailer Figure 7: Tractor semi-trailer combination a) (4 Points) During forward travel, initially moving straight-ahead, a step is applied to the steering angle, u(t) = k h(t), k >. i) What is the sign of the articulation angle for t? ii) In which direction (positive/negative) does the articulation angle change immediately after the step input is applied? Justify your answers mathematically. b) (3 Points) During reverse travel, initially moving straight-ahead, the articulation angle is steered to and maintained at a reference value y ref. Explain why the steering angle must change its sign during the maneuver. Use for your explanation one of the following system properties and show this property mathematically: asymptotically stable / unstable completely controllable / not completely controllable completely observable / not completely observable /
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 = 3 δx+2 δu is the linearization of the non-linear system ẋ = x 3 3x + u 2 around the equilibrium point {x e = 3, u e = 6}. b) A constant signal u(t) = at the input of a system with the transfer function Σ(s) = produces a constant output signal of 4 for t. s+2 s 2 4s+3 c) An unstable system with the transfer function s (s 2) can be stabilized by a P-controller. d) A plant with the transfer function G(s) = s 2 has to be stabilized by a controller. A s 2 s 2 cross over frequency of ω c = 6 rad/s represents a meaningful bandwidth for the specification of the control system. e) The following state space model {A, b, c, d} represents a realization for a system with the transfer function Σ(s) = s+5 [ A = 3 2 s 2 2s+3 : ], b = [ ], c = [ 5 ], D = [ ] f) The Matlab instruction tf([ -7],[ -2 5]) plots the frequency response of a system s 7 with the transfer function s 2 2s+5. g) The output of a system with the transfer function 2s+5 s+5 exceed a value of if the input is the step function h(t). (initial condition = ) will not h) The transfer function of a closed loop system from the reference signal r to the output y is T (s) = 2s+. The sensitivity of the control system is S(s) = s(s+5) s 2 +7s+ s 2 +7s+. Be aware of this fact! /