I want to estimate the bias r(k) of the following system: x(k+1)=Ak*x(k)+Bk*u(k)+Ek*r(k),+wk, y(k+1)=C*x(k+1)+vk. If the dimension of r(k) is greater than that of x(k), does the Two-Stage Kalman Filter works well or can I get the right r(k)?
You can consider r(k) either as a part of states (by augmenting with x vector) or treat it as an unknown input. Please refer to the literature on estimation of unknown inputs.
Here r(k) seems to be an additive term to the system dynamics, hence the name bias. You can use either the joint state/parameter or dual state/parameter approaches for this purpose of estimating r(k). Look into what is available in the literature and also take a look at some valuable resources like Optimal Estimation of Dynamic Systems book by John L. Crassidis, John L. Junkins.
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