Prüfung Regelungstechnik I (Control Systems I) Prof. Dr. Lino Guzzella 5. 2. 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 (Modeling and Linearization) 8 Points An object of mass m is attached to a spring as shown in the figure below. From the bottom an airflow is blown onto the object. The object s position is denoted by x(t). The guidances on the sides are frictionless. The air flows with the velocity v(t), which can be adjusted. The lifting force of the airflow object can be modeled as: F a = 2 c a ρ A v 2 rel = k v2 rel. This force depends on the air density ρ, on the coefficient of lift c a, on the object s cross section area A, and on the relative flow velocity v rel. In this exercise k can be assumed to be constant. The spring is linear (spring stiffness k s ) and can be assumed to be massless. The rest position of the spring is at x =. The equilibrium position of the object is denoted by w e. All parameters are positive, i.e., k,k s,m,w e >. a) (3 Points) Give the differential equations which describe the vertical motion of the object. Use the air velocity v(t) as input and the object s position w(t) as output of the system. Formulate the equations in standard form, i.e. as a system of nonlinear first order differential equations 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) Find the air velocity v e, which keeps the object in equilibrium at position w e. Remark: w e m g/k s. c) (3 Points) Linearize the system equations around this equilibrium point (normalization is omitted). Give the linearized system equations in the standard form (state space representation with matrices {A, b, c, d}). / 2
Question 2 (Frequency domain, time domain) 8 Points The open-loop transfer functions (L (s),l 2 (s),l 3 (s),l 4 (s)), the open-loop Nyquist diagrams (Diagram A, Diagram B, Diagram C, Diagram D), and the closed-loop step responses (Step response, Step response 2, Step response 3, Step response 4) of 4 systems are given. Identify the corresponding open-loop Nyquist diagram and the corresponding closed loop step response of every transfer function. Note your solution in the table below. Grading: Per correct identification: + point Per false identification: point Minimum amount of points: points Transfer function Table for the solution L (s) = L 2 (s) = L 3 (s) = L 4 (s) = s 2 +2s+2 e s 3 (s 2 s+6) 4 (s 3 +2s 2 +4s) ( 2s+2) (s 2 +3s+4) Nyquist diagram Step response Nyquist diagram A Nyquist diagram B.5.5 Im Im.5.5.5.5 Re.5.5 Re 2 / 2
Nyquist diagram C Nyquist diagram D.5.5 Im Im.5.5.5.5 Re.5.5 Re 5 Step response Step response 2 4.2 3 2 amplitude [ ] 2 3 amplitude [ ].8.6.4.2 4 5 2 3 4 5 6 7 8 9 time [s] 2 3 4 5 6 7 8 9 time [s] Step response 3.2 Step response 4.8.8 amplitude [ ].6.4 amplitude [ ].6.4.2.2.2 2 3 4 5 6 7 8 9 time [s].4 2 3 4 5 6 7 8 9 time [s] 3 / 2
Question 3 (Controller Design) 7 Points The following plant is given P(s) = s (s + ) The plant output is corrupted by high-frequency measurement-noise. a) (3 points) For the given plant a PD-controller has to be designed. C(s) = k p ( + T d s) The control system is required to have a cross-over frequency of 3 rad s of 6. Determine the parameters (k p,t d ) of the controller. and a phase margin b) (3 points) The controller has to be extended by a second order low-pass filter. The transfer function of the filter is given by: F(s) = k F (τ s + ) 2 It is required that the cross-over frequency of the control systems remains 3 rad s. The phase margin has to be 45. Determine the parameters of the filter (k F,τ). The controller parameters (k p,t d ) must not be changed. c) ( point) What is the reason to extend the controller with a low-pass filter, why is it especially important in this example? (explain - 2 sentences) 4 / 2
Question 4 (Laplace Transformation) Points The following two sub-tasks can be solved independently. a) (4 Points) The transfer function P(s) is given. Calculate the time-response y(t) of the given system. The input signal is u(t), where ω = 2 holds. s + 2 P(s) = (s + ) (s + 3) u(t) = h(t) sin(ω t) b) (6 Points) The output of a system Σ(s) is measured using a sensor that can be described by the transfer function Σ s (s). The block diagram of the whole system looks as follows : The following additional information for the system Σ(s) (for a step response) is available: The rise time t 9 is 2.5 seconds The maximum overshoot ˆǫ is 25% i) What is the transfer function of the system Σ(s) (approximately)? ii) The response of the system on a unit step function was measured. The following time-function was fitted to the measured data: u(t) = h(t) y(t) = h(t T) [ + a e b (t T) + e b 2 (t T) (a 2 cos(ω (t T)) + a 3 sin(ω (t T)) )] The corresponding numerical values are as follows: a a 2 a 3 b b 2 ω T -.74 -.8286 -.86-2 -.359.6932. Determine the time-constant τ s and the time-delay T of the sensor. Note: The system does not contain any finite zeros. 5 / 2
Question 5 (Stabilizing) Points The new Sauber C3 vehicle has unfortunately the disadvantage of large oversteering (uncontrolled increasing of the yaw angle). Before sending Kamui Kobayashi to the test track, the team leader Peter Sauber asks you whether it is at all possible for Kobayashi to keep the car on the track. You first request the mathematical model of the racing car from the department of vehicle dynamics Θ d2 dt2γ(t) = c γ(t) + c α(t) () where γ is the yaw angle of the car (output of the system) and α is the steering angle (input of the system). The model parameters have already been identified experimentally and the values are: Θ 2 kg m 2 and c 5 N m/rad. For the driving behavior of Kobayashi you assume that his transfer function can be described by the following equation α(s) = C(s) e(s), C(s) = k b s +, k >, a >, b > a s + where the error signal e(t) is the difference of the actual yaw angle from the desired yaw angle. Kobayashi and the car build together a classical control system as shown in Figure (the disturbances d(t) are e.g. the forces that are caused by the curbs. d γ soll e Kobayashi α C3 γ Figure : Structure of the control system. a) ( Punkt) Find the transfer function P(s) of C3. b) ( Punkt) Find the frequency response P(jω) of C3. c) ( Punkt) Draw this frequency response in the empty Nyquist-Diagram provided. d) (3 Punkte) What shape must the frequency response of the open loop gain L(jω) = P(jω) C(jω) qualitatively have, such that the closed loop control system is asymptotically stable? (Sketch the Nyquist-plot in the Nyquist-Diagram provided.) Justify you answer! e) (3 Punkte) Specify numerical values for the parameters k, a, b that will ensure the control system to be asymptotically stable. f) ( Punkt) What do you think: Does Kobayashi meet these specifications? 6 / 2
Im + + Re Figure 2: Nyquist-Diagram. 7 / 2
Question 6 (Nyquist-Plot, Nyquist Theorem) Points a) (3 points) The transfer function of a system with two parameters k and a (a >!) is given: P(s) = k s s a s + a. In the following Figure the Nyquist-Diagram of the system P(jω) is plotted Nyquist Diagram of the system 3 2 Im axis 2 3 5 4 3 2 2 Re axis Use the information below in order to identify the values for the parameters k and a from the Nyquist-plot: P(j ) = + j lim P(jω) = 2 j w + b) (2 points) Sketch the frequency response of the system for ω [, ] in the empty Bode- Diagram provided on the next page. In your sketch the following quantities have to be specified: The slope(s) in db/dec in the Bode-plot for the magnitude, intersection point of the Nyquist-plot with the unit circle, the following two limits in the Phase-plot: w + and w. 8 / 2
Bode Diagram Phase (degree) magnitude (db) frequency (rad/s) c) (2 points) Now you want to find a stabilizing P-controller C(s) = k p for the system P(s). Use Nyquist Theorem to specify for which values of the gain factor k P the system can be stabilized by a P-controller. d) ( point) Determine the phase margin of the control system for k p =? e) (2 points) Since you are not satisfied with the behavior of the control system, you extend the P-controller to a PD-controller with the transfer function C(s) =.5 +.5 s. Draw qualitatively the step response in the figure provided below. The step occurs at t = sec. Hint: If you could not answer question a), your may use k = and a = to answer this question. 2 step response signal [ ] 2 2 3 4 5 6 7 8 9 time [sec] 9 / 2
Question 7 (System Analysis) 8 Points In biology, the difference in voltage between the interior and exterior of a cell is known as the membrane potential. Its nonlinear dynamics are approximated by the FitzHugh-Nagumo model. Linearized around its equilibrium this is given by ] [ẋ (t) ẋ 2 (t) [ ] [ ].45 x (t) = +.8.5 x 2 (t) y(t) = [ ][ ] x (t), x 2 (t) [ ] u(t) where y(t) denotes the difference of the membrane potential to its equilibrium and u(t) an external input current, e.g. applied via an electrode. a) ( Point) In terms of Lyapunov, is the system stable, asymptotically stable, or unstable? b) (2 Points) i) Determine the transfer function of the system. Simplify your result as much as possible. ii) Is the system minimum-phase? c) ( Point) Is the system completely controllable? d) ( Point) Is the system completely observable? e) (3 Points) Discuss following questions based on the given model and your results from the previous questions. i) The membrane potential is slightly deflected from its equilibrium by short-time application of a small external current. Will it return to its equilibrium? ii) iii) An external current in form of a step is applied to the system. Will there be a point in time where the difference of the membrane potential to its equilibrium changes sign? Can the relationship of external current to the difference of membrane potential to its equilibrium be described by a first-order system? / 2
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 2. Unanswered questions will get points. The minimum sum for all questions is points. a) The differential equation δẋ = 3 δx+6 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 for t a constant output signal of 5. s+3 s 2 7s+6 c) The poles of the system with the transfer function Σ(s) = s+ eigenvalues. (s+2)(s 2 +4s+3) coincide with its d) An unstable system with the transfer function (s ) 2 can be stabilized by a PD-controller C(s) = k p + k d s with the parameters k p and k d. e) A plant with the transfer function P(s) = s 5 is stabilized by a P-controller. There is a disturbance signal w at the input of the plant (see figure below!). w r + + k ---------- P s 5 y With a gain factor k P = 5 it can be achieved that the maximum of the impulse response (for zero initial condition and r = ) will not not become greater that (i.e.y(t), t ). 2 Be aware of this fact! / 2
f) A PI-controller 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 of the integral part T i, 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 control system is asymptotically stable because the Nyquist-plot L(jω) encircles the point twice in the clockwise direction. 2 / 2