UC Berkeley EE128 Homework 8 Due on 12 12 05 It is helpful to learn the MATLAB function c2d You can use it for verifying your answers You should not however reply on it to do the problem unless the problem indicates that it is ok to do so 1 Textbook problem 8 5 For part b use the following MATLAB commands for plotting the frequency responses numc 1 1 denc 0 1 1 Hc tf numc denc Ht c2d Hc 0 25 tustin Ht num Ht den tfdata Ht v Hm c2d Hc 0 25 matched Hm num Hm den tfdata Hm v W logspace 1 2 200 hold on bode Hc W dbode Ht num Ht den 0 25 W dbode Hm num Hm den 0 25 W Problem 2 a 2 f below are all based on the system in textbook problem 8 8 2 a Textbook problem 8 8 Hint i Use textbook Equation 8 33 to find the equivalent ii Use the following MATLAB commands to draw a root locus on the z plane Note that the rlocus command is the same for continuous time systems and discrete time systems The only difference is the grid hold on zgrid rlocus NUM DEN iii Find the k that renders the 0 7 closed loop poles Hint k should be very close to 1 2 b With the k found in 2 a what is the natural frequency in rad sec of the closed loop poles 2 c With the k found in 2 a would the closed loop system remain stable for all T If not what is the range of the sample frequency for a stable closed loop system Hint Use the following MATLAB script to find the closed loop pole for a range of value of T for T start step end plant tf 1 1 2 0 plantd c2d plant T cloop feedback plantd K cpoles pole cloop T cpoles end 2 d Express the system in control canonical state variable form and find its discrete time equivalent in state variable form Use T 1 2 e Find the discrete time transfer function using the formula H z c zI A 1b 2 f Design a discrete time state feedback controller i e find the feedback vector K such that frequency is about half of the value found in 3 b 3 Textbook problem 8 9 You may use MATLAB to the fullest extent for this problem 0 7 and the natural