stability3

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

This preview shows pages 1–2. Sign up to view the full content.

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.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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
This is the end of the preview. Sign up to access the rest of the document.

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...
View Full Document

{[ snackBarMessage ]}

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern