TITLE: What is the physical meaning of Bode plot in case of unstable system?
QUESTION [5 upvotes]: I know that from the mathematical point of view it doesn't matter if we plot Bode diagram of stable or unstable system. It's just a function of complex value.
However from the physical point of view, Bode plot shows steady response to a sinusoidal input. If the system is unstable there is no steady response, am I right? So is it only a "magic" of mathematics that although there is no meaning, it's still useful to use it for design?

REPLY [6 votes]: The 'frequency response' interpretation of a Bode Plot only holds for stable (or marginally stable) systems. If you choose to draw it for an unstable system, then it does not have the usual physical interpretation. Specifically, it is NOT a frequency response (the steady-state amplitude and phase response to a sinusoidal input).
There is a simple mathematical reason why. The Bode plot at a particular frequency, $\omega$, is defined as being the system transfer function $G(s)$, evaluated at $s = j\omega$. However, for an unstable system, this evaluation violates the assumptions under which the original transfer function was derived using the Laplace Transform. Remember that Laplace Transforms always have allowable values of $s$, called the "Region of Convergence". The Region of Convergence for the Laplace Transform of an unstable system is $Re(s) \geq a$, where $a>0$. Therefore, when you use the $s = j\omega$ substitution, you are violating the Region of Convergence condition. What you get is no longer the Fourier Transform of a time-domain signal and should not be interpreted as such.
Certain control analysis techniques use the $s = j\omega$ substitution for unstable systems, but it is just mathematical trickery and the result is never interpreted as a Fourier Transform. The Nyquist Criterion is an example: this is a mathematical 'trick' that just helps visualise the stability of a closed-loop system. The Nyquist Criterion can be checked using a Bode Plot, which is where you may come across the Bode Plot of an unstable system. But when handling such a Bode Plot, always bear in mind that it is NOT a frequency response.