stability3 - s 5 8 s 3 –16 s place these coefficients in...

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Stability Q UESTION 3 For the system shown in Figure 3 with ( 29 ( 29 6 5 4 3 2 8 2 2 4 8 4 G s s s s s s s s = - - + + - - , create the Routh table and determine how many closed–loop poles are in the left s–plane, in the right s– plane, and on the imaginary axis. Without the use of the Routh table, what can be said about the stability of this closed–loop system? Explain. Numerically determine the location of the closed– loop poles. + - R(s) G(s) Y(s) E(s) Figure 3 The closed–loop transfer function is ( 29 ( 29 ( 29 7 6 5 4 3 2 8 1 2 2 4 8 4 8 G s T s G s s s s s s s s = = + - - + + - - + . The closed–loop characteristic polynomial is 7 6 5 4 3 2 2 2 4 8 4 8 s s s s s s s - - + + - - + . Since the signs of the coefficients are not the same, there is at least one unstable pole. Therefore, the closed–loop system is unstable.
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Stability The Routh table is 7 6 5 4 3 2 1 0 1 1 4 4 2 2 8 8 0 12 08 0 16 0 0.6667 5.333 8 0 88 128 0 0 4.364 8 0 0 33.33 0 0 0 8 0 0 0 s s s s s s s s - - - - - - - - - - . Since there is a row of zeros in the s 5 row, the characteristic equation contains a 6 th order even polynomial. The even polynomial is in the s 6 row and is –2 s 6 +2 s 4 –8 s 2 +8 = 0. Differentiate this polynomial to obtain –12
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Unformatted text preview: s 5 +8 s 3 –16 s , place these coefficients in the s 5 row, and continue. Investigate sign changes from the s 5 row to the s row to determine the stability of the 6 th order even polynomial. Since there are three sign changes and the roots of an even polynomial are symmetric about the origin, three poles are in the right s–plane and three poles are in the left s– plane. To determine the location of the other pole, investigate sign changes from the s 6 row to the s 7 row. Since there is one sign change, the remaining pole is in the right s–plane. Therefore, there are four poles in the right s–plane and three poles in the left s–plane. The Matlab code for computing the closed–loop pole locations is roots([1 –2 –1 2 4 –8 –4 8]) The closed–loop poles are located at –1, 1, 2, –1 ± j , and 1 ± j . 2...
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