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Unformatted text preview: Stability Design via RouthHurwitz Given the system below find the range of the gain K that will lead to stability, unstability and marginal stability. E(s) s(s+7)(s+11) K + R(s) C(s) Closing the loop: T ( s ) = K s 3 + 18 s 2 + 77 s + K (1) 1 s 3 1 77 s 2 18 K s 1 1386 K 18 s K If K is positive (assumed) all the elements in the first column are positive except that the s 1 entry may be positive/nagative. If K < 1386, the system will remain stable. If K > 1386, the system will remain unstable. If K = 1386, a row of zeros, so using P ( s ) = 18 s 2 + 1386 (2) 2 dP ( s ) ds = 36 s + 0 (3) s 3 1 77 s 2 18 K s 1 6 36 s 1386 Since there are no sign change from s 2 to s the even polynomial has no unstable roots, only j roots, therefore the system is marginally stable when K = 1386. 3 Root locus techniques Root locus, a graphical representation of the closed loop poles as a system parameter is var ied....
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 Fall '10
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