I want to know whether a luenberger observer or a modified form of it be effective for estimating the states of a system of the following forms:
1. x_dot = f(x,u,d); y = x
2. x_dot = f(x,u); y = x
where x_dot is the derivative of the state vector, u is the control input and d is the external disturbance input. f is a nonlinear function with the product of states, control inputs, and disturbances.
Please give some specific suggestions regarding the observers which are best suitable for state estimation of such models.
Dear Malyala Venkata Satya Sai Chandra,
I assume that the "y=x" in your system structure is a typo, since if you can measure the state x directly, there is no need for an observer. Hence, let's assume that you wanted to write something like "y=h(x)".
An answer then mainly depends on your notion of a "Luenberger Observer" in a nonlinear context. For linear systems the term can be considered well defined, while for the class of nonlinear systems you are referring to, this could mean e.g. an observer structure of the form
\hat x _dot = f(\hat x, u) + g(y,\hat y), \hat y = h(\hat x)
and the function g(. , . ) is sought for. Notice however that more general observer structures are possible.
There is a vast body of theoretical results regarding necessary and sufficient conditions under which a g can be found such that some convergence of the observer error is achieved. The result may still be of local character, or depend on your specific input signal u. Most importantly, the conditions do not have such a simple and easy to verify structure as in the linear case. If you are mainly after theoretical results you can start e.g. with a monography like this:
New Directions in Nonlinear Observer Design (Lecture Notes in Control and Information Sciences (244))
Many of the methods stated there - or in the cited references- will serve you for practical implementations. However, if your main interest is in practical implementations: I personally made in many applications very good experiences with using stochastic state estimators like Extended or Unscented Kalman Filters, even in the fully deterministic case. The downside of these methods is that you normally do not easily get a theoretical guarantee for error convergence.
Dear Malyala Venkata Satya Sai Chandra
See my work in this regard in the following link:
https://ieeexplore.ieee.org/abstract/document/9097397
Dear Malaya Venka Satya Sai Chandra,
Yes there is a nonlinear version of this observer, I suggest you to see links and attached files on topic.
https://www.collectionscanada.gc.ca/obj/s4/f2/dsk1/tape8/PQDD_0016/NQ45691.pdf
Article A Luenberger-Like Observer for Nonlinear Systems
Article Review and classification of recent observers applied in che...
Best regards
Thank you all for your answers. I would certainly look into them.
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