H(jω) does not exist for unstable systems, but we still use it when designing controllers - contradiction?
According to Signal processing and linear systems by Lathi, the transfer function $H(j\omega)$ does not exist for systems with poles in the RHP: -
This makes sense to me, since $H(j\omega) = \frac{Y(j\omega)}{X(j\omega)} $. However, since the system is unstable $Y $ is unbounded (and growing) and the fourier transform doesn't exist for such functions. So $Y(j\omega)$ doesn't exist and therefore, $H(j\omega) $ must be meaningless for unstable systems.
BUT! we use $H(j\omega) $ when designing controllers for unstable systems anyway - and it works. We look at the Bode Plot, we look at the Nyquist Plot both of which you need to know $H(j\omega)$ and design a controller based on what we see - and the controller actually works!
How can this be? How can there be this contradiction between systems and signals theory and control theory? It seems that concepts like region of convergence and existence of fourier integral are only dealt with on coursework and once that's done, you don't hear from them ever again.
2 answers
I think this is partly semantics.
For example, consider designing a compensator for a power supply. The transfer function under consideration is essentially the open loop impulse response of the system. If that goes nuts, then you have other problems to fix first. You are right in that it needs to be stable (not oscillate or grow exponentially or something) in response to a single blip.
But again, that's the open loop response. You can certainly make a mess and cause closed loop instability with the wrong feedback, but that's not the transfer function being quantified. In the end, of course, we do care about the closed loop transfer function. By that time, we've designed the compensator (feedback) to make sure the system is stable.
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Please note that the text refers to the integral, or the mathematical evaluation through integration which, indeed, cannot be obtained. But that doesn't mean you can't obtain the Laplace transfer function directly. A hypothetical RLC filter with a negative resistor is very much possible. In fact, it can be made in practice with emulated elements, and its transfer function will give a pole in the RHP:
$$H(s)=\dfrac{\dfrac{1}{LC}}{s^2-\dfrac{1}{RC}s+\dfrac{1}{LC}}$$
This can be obtaind by considering the raw Laplace equivalents of the elements. Is it unstable? Yes. Can it be done practically? Yes, here is the concept of it:
You can clearly see the phase going up. Does that violate the textbook? No -- the textbook only picks on the mathematical aspect (Olin says it better: semantics).
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