Use Recursive Least Square as an "Actual" Filter?

Question

I am currently studying recursive least square (RLS) and as far as I understand, the setup is (for example) that we are given a process with uncertain parameters/noise and we want to adapt parameters of a model of that process online.

To me, that sounds like a Luenberger observer, where the parameters of the underlying model are adapted online, depending on the model quality.

If this is true, a side question would be why this technique isn't usually called an observer, but a filter instead? However, my main question is:

Question: Can RLS also be used instead of a standard low-pass filter?

So, given only a signal from which I want to remove high-frequency components (like noise), can a RLS filter be used to do that as well? And if so, would it have any advantage over normal low-pass filters?

Answer 1

The problem that recursive least squares (RLS) can solve can be formulated as recursively solving for $\hat{\theta}$, such that it is the least squares solution to

$$ \hat{\theta}_n = \arg\min_x \sum_{k=0}^n w[k]\,\|z[k] - \phi[k]^\top x\|_2^2, $$

where $w[k]$ are weights, $z[k]$ and $\phi[k]$ are known and $z[k]$ is assumed to be generated by using $\phi[k]^\top \theta$ with $\theta$ constant and unknown to the user. The recursive part of RLS refers to that the solution $\hat{\theta}_n$, plus additional variables that encode the problem, are used to calculate the solution $\hat{\theta}_{n+1}$. Now, if $\phi[k]$ satisfies a persistence of excitation condition, then over time $\hat{\theta}$ can be shown to converge to the value of $\theta$. A more detailed formulation and derivation can be seen in Mathematics StackExchnage - What's the Idea Behind Recursive Least Squares (RLS)? How Could It Be Derived?

So this formulation initially suggests that it would only be suited for estimating constant variables. However, by using exponentially decaying weights (so weigh old data exponentially less and less) RLS can still behave similar to a low-pass filter. So that the RLS can still reasonably track a slowly varying $\theta$.

It can also be shown that RLS (with constant weights) is equivalent to a Kalman filter for the system with $F_k = I$, $B_k = 0$, $H_k = \phi[k]^\top$, $Q_k=0$ and $R_k=1$. So if you want to filter out noise from a time varying signal and you have a model of how you think that signal should change over time, you might be better of using a Kalman filter using that knowledge. Namely, RLS with exponentially decaying weights will always lag behind a changing signal, similar to a low-pass filter, while a Kalman filter corrects for this using the prediction step.

Finally, regarding RLS being called a filter and not an observer might be because the label observer is I believe usually reserved for estimating properties of dynamical processes, which the constant $\theta$ which RLS tries to estimate is lacking. However, I am often also confused about when to use the term filter, estimator or observer, so difference in meaning might be small.