CALIFORNIA INSTITUTE OF TECHNOLOGYControl and Dynamical SystemsCDS 101/110R. M. MurrayFall 2003Homework Set #8 Issued: 24 Nov 03Due: 3 Dec 03All students should complete the following problems:1. Consider the dynamics of the magnetic levitation system from lecture. The transfer function from theelectromagnet input voltage to the IR sensor output voltage is given byP (s) =ks2− r2with k = 4000 and r = 25 (these parameters are slightly different than those used in the MATLABfiles distributed with the lecture).(a) Design a stabilizing compensator for the process, assuming unity feedback. Compute the polesand zeros for the loop transfer function and for the closed loop transfer function between thereference input and measured output.(b) Plot the Nyquist plot corresponding to your compensator and verify that the Nyquist criterion issatisfied.(c) Plot the log of the magnitude of the sensitivity function, log |S(jω)|, versus ω on a linear scaleand numerically verify that the Bode integral formula is (approximately) satisfied. (Hint: youcan do the integration numerically in MATLAB, using the trapz function. Make sure to chooseyour frequency range sufficiently large.)2. Consider a second order system with transfer fuctionP (s) =s − 1(s + 10)2.(a) Plot the Bode plot for the system. Find another transfer function with the same magnitude butwhose phase lag is less than the phase lag of P (s).(b) Consider a proportional controller C(s) = Kp. Compute the range of gains for which the controllerstabilizes the system and show that as Kp→ ∞, one of the poles of the closed loop transferfunction approaches the zero at s = 1.Only CDS 110a students need to complete the following additional problems.3. For the control systems below, design a PID control law that stabilizes the sytem. You may youany method (loop shaping, Ziegler-Nichols, sisotool, etc). For the closed loop system, determine thatsteady state error, the maximum frequency for which the closed loop system can track with less than5% error, and the approximate bandwidth of the system.(a) Disk drive read head positioning system:P (s) =1s3+ 10s2+ 3s + 10(b) Second order system:P (s) =100(100s + 1)(s + 1)4. In this problem we will design a PID compensator for the pitch axis of the Caltech ducted fan. Usethe following transfer function to represent the vehicle dynamics:P (s) =rJs2+ bs + mglg = 9.8 m/sec2m = 1.5 kg b = 0.05 kg/secl = 0.05 m J = 0.0475 kg m2r = 0.25 m(a) Design a PID compensator that stabilizes the system. You can use any method that you choose.Plot the pole zero diagram, frequency response, and step response for the closed loop system.(b) Plot the root locus plot for the system. Mark the open and closed loop pole locations correspondingto the PID compensator at the default gain (from part (a)).(c) Use the root locus plot to choose a new gain such that the dominant poles have a settling time ofhalf of the value of the settling time for the original compensator designed in part (a). Show thelocation of the the poles with the new gains on your root locus
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