I encountered a question when using Bode stability criterion to analyze the closed-loop stability of a system. In a word, the Bode stability criterion says the system is stable but it turns out to be not. Here's the description. Thank you in advance for reading!
Given a feedback control system with a proportional controller $k$ and the open-loop transfer function $L(s) = \frac{1}{s(s+1)^2}$. I would like to analyze the system's stability using the Bode stability criterion.
The criterion reads: Consider an open-loop transfer function $L(s)$ controlled by a proportional controller $k$ and negative feedback. Assume that the open-loop system is stable and the amplitude and phase of the Bode plot are monotonically decreasing. Then, the closed-loop system is asymptotically stable if and only if $|kL(iω)| < 1$ where $∠L(iω) = −180◦$.
If I consider $k<0$, the bode plot from MATLAB shows (Bode plot of $k<0$ can be plotted using Bode($-L(s)$), namely, inverting $k$ is the same as inverting $L(s)$):
The bode plots are monotonely decreasing and the phase plot never hit $-180◦$, MATLAB also says its gain margin is infinite. However, the system is not stable using the proportional gain $k<0$.
It can be shown using the Nyquist plot and the Nyquist stability criterion that this system can not be stable when $k<0$. Because when $k<0$, $(-\frac{1}{k},0)$ lays on the RHP, which is always encircled by the Nyquist plot, which makes it unstable.
I wonder what makes this discrepency, namely, why the Bode criterion says it's stable but it's unstable however? There is one catch I found is that: Bode criterion applies when the feedback is negative. If $k<0$, it becomes a positive feedback so the Bode criterion no longer works. But I am not sure if this is correct. If so, does Nyquist criterion have this caveat? Are there any other caveats when using these criterion for stability, such as Bode, Nyquist?
Thank you so much for your time and help!
Best Answer
Considering $k<0$ is the same as considering $L = \frac{-1}{s(s+1)^2}$ with $k>0$. Multiplying a transfer function by $-1$ is the same as introducing a phase shift of $-180^\circ$. Thus, the initial phase shift of $L$ with the negative gain should be $-90^\circ-180^\circ = -270^\circ$. However, MATLAB does not help you here: the phase $-270^\circ$ is the same as $+90^\circ$. Subtracting $360^\circ$ from the phase plot you got from Matlab shows that your system is unstable for any negative $k$.