What are the Kalman filter capabilities for the state estimation in presence of the uncertainties in the system input?

Question

I have a question regarding the capabilities of the discrete Kalman filter for estimation of the unmeasurable state variables of a dynamic system.

In the time being I have been using a discrete Luenberger observer for estimation of the unmeasurable state variables of a dynamic system. The observer provides pretty good estimates but I have encountered that there exists a certain range of the operating points of the dynamic system where there is serious discrepancy between the input of the observed dynamic system and the observer input. Due to this fact there is a pretty serious error in the estimated state variables in this range of the operating points.

My question is whether usage of the Kalman filter instead of the Luenberger observer can help me achieve better estimates in presence of the uncertainties in the system input? Thanks in advance for explanation.

Answer 1

The Luenberger observer uses the following as its signal model:

\begin{align} \mathbf{x}_{k+1} &= \mathbf{A}\mathbf{x}_k + \mathbf{B}\mathbf{u}_k\\ \mathbf{y}_k &= \mathbf{C}\mathbf{x}_k + \mathbf{D}\mathbf{u}_k \end{align} where $\mathbf{x}_k$ is the system state, $\mathbf{y}_k$ is the system output, $\mathbf{u}_k$ is the system input all at time $k$.

The Kalman filter signal model is \begin{align} \mathbf{x}_{k+1} &= \mathbf{A}\mathbf{x}_k + \mathbf{B}\mathbf{u}_k + \mathbf{w}_k\\ \mathbf{y}_k &= \mathbf{C}\mathbf{x}_k + \mathbf{D}\mathbf{u}_k + \mathbf{v}_k \end{align} where we've added the process noise term $\mathbf{w}_k$ and the measurement noise term $\mathbf{v}_k$ to give a measure of uncertainty about the input and the output.

If there's no "real" process noise in your system, then one way to think about $\mathbf{w}_k$ is as a way to say "I'm not very certain about the input".

So I'd answer your question:

[Can] usage of the Kalman filter instead of the Luenberger observer ... help me achieve better estimates in presence of the uncertainties in the system input?

with $\color{green}{\large\bf \mbox{Yes}}$.

How much better will be highly dependent on the signal model and the type of uncertainty you wish to deal with.