CALIFORNIA INSTITUTE OF TECHNOLOGYControl and Dynamical SystemsCDS 101/110R. M. MurrayFall 2003Homework Set #5 Issued: 3 Nov 03Due: 10 Nov 03All students should complete the following problems:1. Consider the block diagram for the following second order system− −++z1s1scabr y(a) Compute the transfer function Hyrbetween the input r and the output y.(b) Show that the following state space system has the same transfer function, with the appropriatechoice of parameters:ddt·x1x2¸=·0 1−a2−a1¸·x1x2¸+·01¸ry =£b2b1¤·x1x2¸+ drGive the values of ai, bi, and d that correspond to the transfer function you computed in (a).(c) Compute the transfer function Hzrbetween the input r and the output z. (Hint: it is notHzr= 1.)2. Consider the cruise control system from homework #2. The equations of motion for the system weregiven bym ˙v = −bv + Kτ + Fhill˙τ = −aτ + ue,(1)where m = 1000 kg is the mass of the vehicle, b = 50 N sec/m is the viscous damping coefficient,K = 5 is the conversion factor between engine torque and force applied to the vehicle, a = 0.2 is thelag coefficient, and Fhillrepresents the external effect of going up or down hills on the vehicle dynamics.The simplest controller for this system is a proportional control, ue= −Kpe, where e = (v − r) (r isthe reference speed).(a) Draw a block diagram for the system, with the engine dynamics and the vehicle dynamics inseparate blocks and represented by transfer functions. Label the reference input to the closedloop system as r, the disturbance due to the hill as d, and the output as y (= v).(b) (MATLAB) Construct the transfer functions Herand Hydfor the closed loop system and useMATLAB to generate the step response and frequency response for the each. Make sure to usethe transfer function computation.(c) Consider a more sophisticated control law of the form˙xc= eyc= Kixc+ Kpe.This control law contains an “integral” term, which uses the controller state xcto integrate theerror. Compute the transfer functions for this control law and redraw your block diagram frompart (a) with the default controller replaced by this one.(d) (MATLAB) Using the default gains from previous homeworks (Ki= 100, Kp= 500), use MAT-LAB to compute the transfer function from r to y and plot the step response and frequencyresponse for the system. This should match your answers in previous homework sets, but makesure to use the transfer function computation.Only CDS 110a students need to complete the following additional problems:3. In this exercise we will explore the use of state feedback to generate a cruise controller for the system (1)in problem 2.(a) Construct the linear state space system corresponding to the open loop system, in the form˙x = Ax + Bu. Verify that the system is controllable.(b) Design a pole placement control law that places the poles of the closed loop system at −1 and−2. Plot the step response and frequency response of the closed loop system with this controllaw. Hint: MATLAB’s place command may be usful.4. Consider the following transfer function for a second order system:P (s) =Ks2+ b1s + b2,where b1and b2are positive real numbers.Sketch the Bode diagram for this system for the cases where the poles of P (s) are real and complex (soyou should have two sets of plots). Calculate and label on your plot (when applicable) the maximumgain and where it occurs, where “roll off” in gain begins, and where the gain crosses 1 (0 dB), all asfunctions of K, b1and
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