CALIFORNIA INSTITUTE OF TECHNOLOGYControl and Dynamical SystemsCDS 101/110R. M. MurrayFall 2003Homework Set #7 Issued: 17 Nov 03Due: 24 Nov 03All students should complete the following problems:1. For the control systems below, determine that steady state error, the maximum frequency for which theclosed loop system can track with less than 5% error, and the approximate bandwidth of the system.(a) Disk drive read head positioning system, using lead compensator:P (s) =1s3+ 10s2+ 3s + 10C(s) = 1000s + 1s + 10(b) Second order system with PD compensator:P (s) =100(100s + 1)(s + 1)C(s) = s + 102. In this problem we will design a lead 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 mDesign a unity feedback controller with the following specifications• Steady state error of less than 1%• Tracking error of less than 5% from 0 to 1 Hz (remember to convert this to rad/sec).• Closed loop step response with maximum overshoot of 20%.• Closed loop frequency response with no more than 3 dB gain at all frequencies.• High frequency disturbance rejection from reference to output (Hyr) of at least 10X above 100Hz.If you cannot meet all of the specifications, you should prioritize them in the order listed.(a) Plot the open loop Bode plot for the system and mark on the plot the various frequency domainconstraints in the above specification.(b) Design a compensator for the system that satisfies the specification. You should include plots toshow that all specifications are met.(c) Optional: Write down the state space representation of your controller. (I.e., write down a statespace system whose transfer function is the control law you designed.)Hint: start with the controller given in class and see if you can modify it to satisfy the specification.Only CDS 110 students need to complete the following additional problems:3. Continuing the previous problem, we will now investigate the stability of your control law underperturbations.(a) Determine whether your control law is robustly stable with respect to added sensor dynamicsG(s) =1s + 1.These dynamics should be inserted in the feedback loop, as we did in Lecture 7.1. If yourcontroller is not stable with for the perturbed system, redesign your control law to provide asmuch performance as you can for the nominal plant and still maintain stability for the perturbedplant.(b) For the control law that provided robust stability in (a), check to see if the performance spec-ifications are satisfied for the perturbed plant. If your controller is not robust, redesign yourcontrol law to provide robust stability and robust performance to the specified sensor dynamics.If necessary, relax the specifications according to the priorities.(c) Consider now the case where we have a time delay of 1 second instead of G(s). Determine whetheryour control law from part (b) provides robust stability and/or robust performance in the presenceof the specified time delay. (You just need to check to see if it works; you don’t need to
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