I'm trying to figure out how to use basic control theory to design a controller that controls the amplitude of a harmonic oscillator at resonance. For example, I want to drive a spring mass damper with a sinusoidal force source at the resonant frequency. I want to adjust the amplitude of the force source based on the amplitude of the resonant motion.
Consider the resonator to be: 1/(m*s^2 + b*s + k)
In order to setup the situation as I've described above, I'm tempted to say that in front of the resonator, I need the LaPlace transform of the sinusoidal drive, and then after the resonator, I need another sine wave and a LPF that acts as the amplitude demodulation. So my overall open loop transfer function would look something like:
s/(s^2+k/m) * 1/(s^2 + b/m*s + k/m) * s/(s^2+k/m) * LPF
where the terms in order are: excitation sinusoidal force, the resonator itself, and the demodulation sine wave, followed by the final LPF. The input of the system is the magnitude of the sinusoidal force and the final output is the amplitude of the motion at the resonant frequency.
When I try to perform a root locus on this I get all sorts of poles and zeros on the positive side of the complex plane. Yet in practice (or at least a Simulink simulation) I'm able to control the thing to some degree of stability with just a simple P controller.
I'm clearly not a controls expert and am in over my head here. Is the method that I'm using have any legitimacy to it? Is the reason that the simple P controller sort of works due to the fact that the actual gain is so very small?
I've tried looking on the web for help on this type of probably but haven't been able to find much. The standard control problem of a 2nd order system is the position, not the amplitude of resonant motion. Any recommended further reading would be greatly appreciated.
Thanks! Mike