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Unformatted text preview: ECE 421, Spring 2006 Solutions to HW Assignment #7 1. Zero located at s = − 8 G 1 ( s ) H 1 ( s ) = K ( s + 8) s ( s + 1) ( s + 2) ( s + 3) (1) (a) K > : For small values of K, the closed-loop pole on each of the branches of the root locus is in the left-half plane, so the closed-loop system is stable. Since two of the branches are also in the right-half plane, the closed-loop poles on those branches will be unstable for certain values of K. Since closed-loop poles move from open-loop poles to open-loop zeros or to zeros located at ∞ , those parts of the branches that are in the right-half plane correspond to larger values of K. Therefore, for this system, the closed-loop system is stable for small K > and unstable for large K > , and there is one value of K that places two of the closed-loop poles on the jω axis. (b) K < : One branch of the root locus starts at the open-loop pole located at s = 0 , and it moves along the positive real axis toward in f nity. Therefore, this closed-loop pole is in the right-half plane for all K < , and the closed-loop system is unstable for all K < . 2. Zero located at s = − 4 G 2 ( s ) H 2 ( s ) = K ( s + 4) s ( s + 1) ( s + 2) ( s + 3) (2) (a) K > : Similar to the f rst system. The closed-loop system is stable for small K > and unstable for large K > , and there is one value of K that places two of the closed-loop poles on the...
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This note was uploaded on 10/25/2010 for the course ECE 421 taught by Professor Cook,g during the Spring '08 term at George Mason.
- Spring '08