I am playing around with an disturbance observer to estimate disturbance in a nonlinear system. (I had to screen snip the equations because the equation editor doesn't seem to work for me)
System prelims:
Basic estimator:
where the estimated disturbance is:
The basic estimator above is great for step or constant disturbances and has fast convergence, however when adding a ramp disturbance the estimator cant keep up and there is a delta.
Therefore the intermediate estimator is used:
Now the estimated disturbance is:
The intermediate one works well for step/constant and ramp disturbances and has good convergence. However when higher order disturbances are present such as sinusoidal disturbances it fails.
Therefore an advanced version is used to essentially integrate the remaining delta as many times as required, usually around 3 additional integrators works well:
The advanced one is the best version but comes at the cost of more complexity. I have simulated this and I can see how it converges to different attacks, but I do not know how to analytically/mathematically describe this. I am an engineer and not amazing at higher level mathematics.
I know there are methods for linear system where there are poles and zeros that indicate if something will converge, but this will be a nonlinear system.
How can I describe this in a meaningful way mathematically?