hw5_solutions-3

hw5_solutions-3 - ECE 486 Assignment # 5 Issued: February...

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Unformatted text preview: ECE 486 Assignment # 5 Issued: February 24 Due: March 3, 2011 Reading Assignment: FPE , Sections 5.1 - 5.6. Begin reviewing frequency response plots (see your ECE 210 notes, or the first sections of FPE , Ch. 6). Problems: 3+5+7=15 pts. 11. This problem concerns the design of a PI compensator: On the lecture of Feb. 24, Prof. Meyn claimed that the transient specifications can be designed using lead or PD compensation, and then steady specifications can be met by slapping in a PI or lag compensator. Here is a justification of this claim. Consider the feedback architecture shown below: + +- G c ( s ) G PI ( s ) G p ( s ) H ( s ) r ( t ) y ( t ) where G PI denotes the PI compensator, G PI ( s ) = (1+ K I s- 1 ) = ( s- z ) /s , with z =- K I . The compensator G c has been designed so that the system is BIBO stable when K I = 0. That is, the solutions to 1 + H ( s ) = 0 lie in the strict LHP. (a) Obtain the closed loop transfer function Y/R (b) Consider the root locus with respect to the integral gain K I : 1 + (1 + K I s- 1 ) H ( s ) = 0 Express this in standard form 1 + K I G ( s ) = 0 for some transfer function G . (c) Provide an interpretation of the transfer function G . What can you say about stability of the closed loop for small and positive K I ? Are Prof. Meyns claims justified? Solution : (a) Y ( s ) R ( s ) = G PI ( s ) H ( s ) 1 + G PI ( s ) H ( s ) (b) 1 + (1 + K I 1 s ) H ( s ) = 0 (1 + H ( s )) + K I 1 s H ( s ) = 0 1 +...
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hw5_solutions-3 - ECE 486 Assignment # 5 Issued: February...

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