Hi All,
I am a bit confused between the stability of a second order system as
predicted by Bode plots and by the Root Locus technique.
I have a system with just two real overlapping poles at s , and the
gain of the system is 10000:

G(s)000/(s+10)*(s+10)

A. Bode Plot: The Bode plot will start at 10k(80dB) and will have a -6dB at s. After roughly 2 decades ( |jw|00 ) the gain will drop to 1 ( B). The phase contribution of both the poles at the unity gain will be 2*89.48.5 . So if the loop is closed in a negative unity feedback fashion, the system for all practical purpose will oscillate due to insufficient phase margin (1.5 degrees) However, if I reduce the open loop gain of the system to 10, the closed loop system will always be stable and never oscillate.

B. Root Locus: The root locus plot will have two branches originating on the real axis at sigma. The branches will then move away along the asymptotes located at sigma and at angles 90deg and 270deg. Clearly the roots of the closed loop system, as a function of gain, will have fixed real part while the imaginary part of the roots will change with gain. Hence the distance between the imaginary axis and the location of roots (as a function of gain) is INDEPENDENT of the gain. Now this distance of roots from the imaginary axis determines the stability of the closed loop system. In other words, the system described above is 'equally' stable or unstable for all the values of gain between 0 to infinity.

Thanks in advance.

-- Alex.

G(s)000/(s+10)*(s+10)

A. Bode Plot: The Bode plot will start at 10k(80dB) and will have a -6dB at s. After roughly 2 decades ( |jw|00 ) the gain will drop to 1 ( B). The phase contribution of both the poles at the unity gain will be 2*89.48.5 . So if the loop is closed in a negative unity feedback fashion, the system for all practical purpose will oscillate due to insufficient phase margin (1.5 degrees) However, if I reduce the open loop gain of the system to 10, the closed loop system will always be stable and never oscillate.

B. Root Locus: The root locus plot will have two branches originating on the real axis at sigma. The branches will then move away along the asymptotes located at sigma and at angles 90deg and 270deg. Clearly the roots of the closed loop system, as a function of gain, will have fixed real part while the imaginary part of the roots will change with gain. Hence the distance between the imaginary axis and the location of roots (as a function of gain) is INDEPENDENT of the gain. Now this distance of roots from the imaginary axis determines the stability of the closed loop system. In other words, the system described above is 'equally' stable or unstable for all the values of gain between 0 to infinity.

******* To my confused mind the Root Locus and the Bode plots are giving contradictory results. Please help me in clearing the misunderstanding.Thanks in advance.

-- Alex.