# ps6 - EE 429 PROBLEM SET 6 DUE 5 May 2008 Reading...

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EE 429 PROBLEM SET 6 DUE: 5 May 2008 Reading assignment: Chapter 7 Sections 7.1 through 7.6 and Chapter 8 Problem 25: (16 points) Consider a discrete-time plant with the state-space representation x ( k +1) = ± 0 . 50 . 5 01 . 5 ² x ( k )+ ± 0 2 ² u ( k ) y ( k )= ( 12 ) x ( k ) . 1. (2 points) Compute the open-loop plant transfer function G p ( z Y ( z ) U ( z ) , and determine whether or not the open-loop system is stable. What are the open-loop zero(s)? 2. (6 points) In this section of the problem you will design a closed-loop system using output feedback u ( k k o y ( k r ( k ) , where r ( k ) is a reference signal for the feedback system and k o is a scalar gain. (2 points) Determine the closed-loop transfer function G of ( z Y ( z ) R ( z ) in terms of the controller parameter k o . (1 point) Compare the zero(s) of the open-loop transfer function G p ( z ) and the closed-loop transfer function G of ( z ). Does output feedback aﬀect the location of the open-loop plant zero(s)? (2 points) Using the Jury stability test, determine for what range of controller gains k o , if any, the closed- loop system stable. (1 point) Is it possible to arbitrarily place all the poles of the closed-loop system using output feedback? Justify your answer using one or two short sentences. 3. (8 points) Now consider designing a controller using full-state feedback u ( k K s x ( k r ( k ) , where r ( k ) is a reference signal for the feedback system and K s is a 1-by-2 constant gain vector. (2 points) Compute the controllability matrix P and determine its rank. Is the open-loop system completely-controllable? (4 points) Using the gain vector K s = ( k 1 k 2 ) , determine the closed-loop transfer function G sf ( z Y ( z ) R ( z ) in terms of the gain parameters k 1 and k 2 . (1 point) Compare the zero(s) of the open-loop transfer function G p ( z ) and closed-loop transfer functions G sf ( z ). Does state feedback aﬀect the location of the open-loop plant zero(s)? (1 point) Based on the form of the closed-loop transfer function G sf ( z ), is it possible to arbitrarily place all the poles of the closed-loop system using full-state feedback? Justify your answer using one or two short sentences.

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ps6 - EE 429 PROBLEM SET 6 DUE 5 May 2008 Reading...

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