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I'm new to control systems, and I'm attempting to analyze the stability of an open-loop transfer function. I know that by checking the locations of the poles (and ensuring all poles are in the LHP) I can ensure that my transfer function is stable.

However, I'd love to know how stable this transfer function is. Reading textbooks like "Modern Control Engineering" by Ogata and watching Brian Douglas YouTube videos, I've learned about the concept of gain and phase margins, and their application to determining how resistant a system is to changes in the gain or phase. This would be helpful in my design, as there are many elements of my system that are approximations, and I want to ensure that the TF will be robust even with these errors. From my understanding though, the math behind using Bode/Nyquist plots to determine stability margins relies on the fact that you're analyzing the stability of an open-loop system in the case that it was closed with unity feedback.

Is there any similar concept for strictly open loop transfer functions, if I truly don't care about ever closing the loop? Or do I have some fundamental misunderstanding of the concept of stability?

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  • $\begingroup$ Instability is created by feedback. No feedback, no instability. But of course your idealized model of the system may not include some real feedback loops. $\endgroup$
    – alephzero
    Sep 29 '20 at 0:36
  • $\begingroup$ What about the transfer function 1/(s-2), for example? That has a pole at 2 + 0i, and explodes to infinity when given a step input. @alephzero $\endgroup$
    – jgholder
    Sep 29 '20 at 1:40
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The transient response (or homogeneous solution) of a linear ODE is $$y_h(t) = \sum_{i=1}^N C_i e^{p_i\cdot t} $$ where $p_i$ is the i-th pole of your system.

Assume you have a pole at location $p_i = \lambda + i\omega$, with $i = \sqrt{-1}$. The real part ($\lambda$) of this pole will determine the convergence rate towards zero, while the imaginary part $\omega$ represents the oscillation frequency. This is visualized below

Pole zero map

The pole with the largest real-part ($\lambda$) has the slowest convergence towards 0 if $\lambda<0$ or will blow up $\lambda>0$. Hence, the pole with the largest real part is dominant and will tell you something about 'how stable the system is'.

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