MAE13A Signals & Systems, Final Exam - Wednesday March 16, 5 Instructions This quiz is open book. You may use whatever written materials you choose including your class notes and the textbook. You may use whatever calculator you desire, provided it has no messaging communications capability, including infrared, radio and other wireless technology. These permissions should be taken to indicate the limited help that either written material or computational assistance is likely to provide please do not spend significant time looking up books or calculating. That is not what is being tested here. Marks are awarded for concepts, methods and the expression of ideas. You should attempt to answer all four questions. They are equally valued. You have 18 minutes. Please mark your papers with your name and student number. HINTS: - Read the questions very carefully. Answer the questions asked and do not second guess some presumed meaning. - Marks are awarded for methods, concepts and expression. Do not devote too much time to complicated algebraic manipulations. Question 1 Elevator Dynamics Figure 1: Elevator diagram indicating mass and position for car and counterweight. The figure above shows a diagram of an elevator system in which a motor applies a torque measured clockwise onto the cable suspending the elevator car, with mass M and position x past a nominal value l. A counterweight is suspended from the other end of the cable. Our aim is to study the dynamical properties of the elevator motor control system, with a view to bringing the elevator car to a halt precisely at the correct floor level. The distance x measures the distance below the target car position l, where l accounts for the natural length plus extension of the cable due to gravity, Kl = M g. The dynamics of this system are given by Mẍ = Mg K(x l) + F = Kx + F, (1) 1
where g is the acceleration due to gravity and F(t) is the force exerted on the cable by the motor torque. Take the nominal values M = 1Kg, K = 1Kg/s, and measure force F in units of Newtons and length x in meters. Part (i): Using Laplace transforms and (1) develop the following time-domain responses; the response to initial condition x() one meter below the target position and velocity ẋ() = 1 m/s, the step response of the system to a 1N step force input. Describe the quality of passenger ride in each case. Part (ii): We now wish to explore the application of three different feedback control signals; [P-control]: F(t) = 1[ g P x(t) + f(t)], known as proportional feedback, [PD-control]: F(t) = 1[ g D ẋ(t) g P x(t) + f(t)], proportional-plus-derivative, t [PID-control]: F(t) = 1[ g D ẋ(t) g P x(t) g I x(σ)dσ +f(t)], proportional-plus-integral-plusderivative, where f(t) is an additive signal to capture disturbances to the elevator and to model additional control actions. For the PID-controlled system compute the transfer function from f(t) to x(t). Specialize this for the PD-controlled system and for the P-controlled system by taking g I = and g D = g I = respectively. For the PD-controlled system describe for what values of gains g P and g D this transfer function is stable. [Note that this necessarily includes the limits of stability for the P-controlled system. But the PID-controlled system, with a third-order denominator polynomial, is harder to analyze by hand/head.] Part (iii): Using the transfer function for PID-control above and the Final-value Theorem, determine the steady-state response to a step input in f(t) in terms of control gains g P, g D and g I for each class of controlled system. If we model the exit of a large number of people from the elevator car as a step change to the applied force a reasonable thing what does this tell us about the response of the elevator to this event when the elevator is P-controlled, PD-controlled and PID-controlled? Question - Discrete Data Analysis The figures following show sampled data recorded during an experimental test of a gas turbine engine combustor operating in very lean fuel mode. During this operation, the engine emits an audible howl which indicates the presence of a combustion instability. A large number of researchers around the world are looking to stabilize this process in order to achieve much lower pollution from gas turbine engines in aircraft and power plants. [This is actual data.] The instability is described by a nonlinear interaction between the combustion chamber acoustics and the flame heat-release function and appears as a persistent oscillation. Overall, 16 data were recorded at a sampling rate of 5KHz. Figure shows the 1 data from 3 to 3999 in the left plot. This was produced using the matlab command plot([3:3999],p1(3:3999)) The right plot was produced by zooming in.
1 Pressure variation data from combustion experiment 8 Pressure variation data from combustion experiment 8 6 6 pressure variation KPa pressure variation KPa 6 6 8 3 31 3 33 3 35 36 37 38 39 sample number 38 33 33 33 336 338 3 sample number Figure : Sampled pressure fluctuations from combustion instability experiment. 18 Absolute value of FFT(pressure) Absolute value of FFT(pressure) 16 3 1 5 Component magnitude 1 1 8 6 Component magnitude 15 1 5 6 8 1 1 1 16 frequency bin number 5 1 15 5 3 frequency bin number Figure 3: DFT of sampled pressure fluctuations from combustion instability experiment. Figure 3 shows the magnitude of the DFT of all of the data. The left plot was created using plot(abs(fft(p1))) and the right plot was created by zooming in on this. Part (i): Create calibrated time and frequency axes for all four plots above. The two time axes should be in seconds (of course milliseconds are fine too) and the frequency axes should be in Hertz. Here is a table of the notes in a well-tempered musical scale by frequency rounded up to the nearest Hertz. C C D E E F F G G A B B 6 77 9 311 33 39 37 39 15 66 9 Part (ii): An octave up (down) is a multiple of (or division by) a factor of two. Identify the dominant audible tone on this scale by picking the nearest frequency. Part (iii): How much of the zoomed and correctly scaled frequency domain picture can you explain by looking at the zoomed and scaled time-domain plot? There are four peaks (excluding the dc term) to be explained. 3
Question 3 Fourier Series and Filtering! "# Figure : One period of square wave x(t) (left) and of triangular wave y(t) (right). The left-hand graph of Figure shows one period of the periodic square wave x(t) which has period T, amplitude A. This function has Fourier Series x(t) = k= where the Fourier coefficients {c k } are given by the integral c k e j πk T t, () c k = 1 T T πk j x(t)e T t dt. (3) The right-hand graph of Figure shows one period of the triangular wave y(t), which has the same period T and amplitude AT/. This function has Fourier Series y(t) = k= where the Fourier coefficients {d k } are given by the integral d k e j πk T t, () d k = 1 T T πk j y(t)e T t dt. (5) Part (i): Show, by evaluating the integral in (3) above, that {, k even, c k = ja πk, k odd. Part (ii): Noting that y(t) = t x(σ) dσ, integrate () and use () to show that AT, k =, d k =, k non-zero even, AT π k, k odd. [Hint: do not evaluate the integral in (5) for more than one specific value of k.]
Part (iii): The transfer function of an integrator is G(s) = 1 s. Sketch the Bode diagram of this system s frequency response. That is, plot the magnitude of the frequency response in db and the phase in degrees versus a logarithmic frequency (ω) axis. [Try frequencies of 1, 1, 1 radians per second.] Part (iv):derive the connection between {c k } and the formula for the {d k } above by arguing that y(t) is the output signal from an integrator when x(t) is the input signal and using the frequency response. Question Discrete-time Systems Question 3 looked at an integrator system. In discrete time, an integrator is a system which with input sequence x[n] yields output sequence y[n] = n l= x[l]. (6) (i) Using this definition, show that this system s output also satisfies the difference equation y[n] = y[n 1] + x[n]. (7) (ii) Accordingly, by taking z-transforms of (7), show that this system has z-domain transfer function G(z) = z z 1. (iii) Show by computing the impulse response of (7) that G(z) is the z-transform of the impulse response. (iv) Compute the discrete convolution of this impulse response with with sequence x[n] and show that this yields directly the sum (6). (v) Using the expression for G(z), write the discrete frequency response of the discrete integrator system in terms of discrete frequency ω ( π, π]. Now, using the globally convergent expansion e a = 1 + a + a! + a3 3! +..., establish that, for ω small, the discrete frequency response approximately equals, 1 jω (vi) What does the result from part (v) tell us about using a discrete integrator as a (computer based) substitute for a continuous-time one? Can you think of where the continuous-time integrator might have problems? 5