I have a basic question about stability of a system whose phase
approaching DC is more negative than -180 degrees. For example,
consider a the transfer function (s^2 + 2s +25)/(s^3). It is obvious
looking at a root locus for this system that it is conditionally
stable (i.e. stable for sufficiently large gain). The Bode plot of
the open loop system has phase that starts at -270 and eventually
approaches -90 degrees. Picking a large feedback gain, the system is
stable and notice that the open loop gain can be greater than 1 when
the phase passes through -180 degrees.

Clearly this system can be made stable for sufficient gain, so my question is: what is wrong with the following thought experiment? Let say the open loop gain at the -180 phase crossing is 1.5 (which is clearly possible). Then I inject a sinusoid into the closed loop system at the frequency which corresponds to the -180 phase crossing. This signal will be amplified and will then be a larger input at the same frequency and phase at the input to the system. This will continue and lead to unbounded oscillation. Obviously this does not happen. Why? That is, why is the 180 phase crossing (with larger than unity gain) stable for an increasing crossing, but not for a decreasing phase crossing.

Many thanks for any thoughts on this.

Clearly this system can be made stable for sufficient gain, so my question is: what is wrong with the following thought experiment? Let say the open loop gain at the -180 phase crossing is 1.5 (which is clearly possible). Then I inject a sinusoid into the closed loop system at the frequency which corresponds to the -180 phase crossing. This signal will be amplified and will then be a larger input at the same frequency and phase at the input to the system. This will continue and lead to unbounded oscillation. Obviously this does not happen. Why? That is, why is the 180 phase crossing (with larger than unity gain) stable for an increasing crossing, but not for a decreasing phase crossing.

Many thanks for any thoughts on this.