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ps3 - EE 482 Reading assignment Ch 3 PROBLEM SET 3 DUE 26...

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EE 482 PROBLEM SET 3 DUE: 26 Feb 2008 Reading assignment: Ch 3 Problem 11: (15 points) Consider an n th -order system with state-space representation x ( k + 1) = Ax ( k ) + Bu ( k ) y ( k ) = Cx ( k ) + Du ( k ) . 1. (10 points) Assuming that the initial conditions y (0) , · · · , y ( n - 1) and input u ( k ) are known for k 0, find an expression for the initial state-vector x (0) and place your answer in the form Y = Qx (0) + G, and specify the elements of Y , Q , and G . 2. (5 points) What condition must hold so that we can solve for a unique value of x (0)? Problem 12: (25 points) Consider a phase-lead compensator with transfer function representation G c ( s ) = K s/ω o + 1 s/aω o + 1 , where K, ω o > 0 and a > 1. In this problem you will design a series of sampled-data systems, each with sample period T , that approximates the dynamic behavior of the continuous-time compensator. 1. (5 points) Using an appropriate substitution from the handout for Laboratory #1, find the transfer function representation G BR c ( z ) using the backward rectangular integration method, and express your answer in the form G BR c ( z ) = ¯ K z + ¯ b o z + ¯ a o . Specify the parameters ¯ K, ¯ b o and ¯ a o in terms of T, K, ω o and a . 2. (5 points) Using an appropriate substitution from the handout for Laboratory #1, find the transfer function representation G TR c (
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