If the signal of interest is stationary, you can use a steady-state Kalman filter. This would be a time invariant filter. Think of an ARIMA(p,d,q) model, where d is the difference parameter. The data is nonstationary but by taking differences, it is converted to stationary. The same happen with the Kalman filter when the data is stationary. There would be no need to update the parameters.

Good day César,

if your system is observable, then you can effectively estimate the state vector of the system, no matter if it's in stationary or non-stationary phase. It is not even necessary to use KF in your case, you might neglect all the dynamics and just consider a static relationship between your inputs and outputs.

Certaiinly, Kalman filters are observer that can be applied whenever observability is satisfied. As a matter of fact, the key problem when dealing with no stationary systems is observability. In other situations, you should consider the computational cost of the observers that you want to apply. Indeed, in some cases is better to use ARMA and ARIMA models.

Regarding observability and adaption, this book is very useful for the design of observers:

http://ebooks.cambridge.org/ebook.jsf?bid=CBO9780511973437

Jose, thank you for your reply.

I've read about Steady-State Kalman filters but, from what I've understood, the "steady-state" refers to the evolution and observation models (typically, the matrices F and H are time-invariant). I was wondering if the same concept applies for the case when not only these matrices are time-invariant, but also the observations.

Pavel thank you for your reply.

Could you indicate me some references on considering this static relationship between the inputs and outpus?

David, thank you for your reply and for your reference/suggestions. I'll definitely check them.

César,

as far as I understand, you mean estimation of the system's state when the output does not change with time. So in simple terms, observability means the ability to unambiguously compute the state vector from the given output. If the derivative happens to be zero, you arrive at some set of (non-linear) equations. Since they may be non-linear, you can use some optimization technique to find an approximate solution like, say, RLS:

https://www.cs.iastate.edu/~cs577/handouts/recursive-least-squares.pdf

If this is really your situation, I cannot see why you have to use a KF.

Remark: steady-state does not refer to the evolution and observation models. This is wrong. What you mean by that is time varying or time invariant systems. In the former case, you can use the Kalman-Bucy filter. Steady-state refers to critical points.

Regards

Dear Pavel,

Thank you for your reply. As for the confusion between steady-state and time invariant, you are absolutely right. Sorry about that.

Regards,

Hi César, 

It seems you mean time varying or time invariant systems, and not steady-state as critical points... If you are asking about steady-state, you may check this paper, where I used EKF with five error reduction methods: http://omicsonline.com/open-access/pedestrian-indoor-navigation-system-using-inertial-measurement-unit-2090-4886-115.pdf?aid=27918

Best

Hello, Maryam,

Actually, my original idea was indeed to investigate steady state problems. Thank you for suggesting me this paper, I'll definitely take a look into it.

Regards,

The Kalman filter is defined as an extension of the Wiener filter to non-stationary signals characterized by a dynamic model. So the implementation of the Kalman filter is based on a dynamic model that describes the transformation of the state vector over time. In your case, where steady-state measurements are considered, it is more appropriate to use the Wiener filter whose purpose is to reduce the amount of a noise in a signal by comparing the received signal with an estimation of a desired noiseless signal. The Wiener filter assumes the inputs to be stationary.

Naouel, thank you very much for your input. I was hoping to get something in this direction. I will check the Wiener filter for this.

Regards,

If you mean that the measurement noise is zero, and there is no process (state) noise, you have a deterministic estimation (observabiilty) problem, where you just invert the system and solve for the state (parameter) vector. You need an observability condition to be satisfied. If you have no process noise and just measurement noise, you have a least squares estimation problem for a constant state vector. If you have no measurement noise and just process noise, I'm not sure what that is. At any rate, the Kalman filter will still apply in all these cases and you just look at the covariance (Riccati) equation to figure out what's going on. For a stationary system this shouldn't be too hard. I'm not sure what you mean by stopping criterion. If you mean stop when the error is small enough, you could use least squares or get the estimation error covariance matrix as a function of the number of measurements.

Actually, process noise is also introduced to address uncertainties in the model. It's not always literally "noise" like in radio signal.

There is a book called Fundamental of Kalman Filtering:A practical Approach .  Which maybe helpful . 

good morning,

My subject of my PhD was part of dynamic stochastic modeling of hydrological phenomena for real-time prediction purposes, which requires adaptive mode, which involves the use of a model with a retroactive structure which makes the output of the model This is a case study of the Kalman filter (FK), it is a new approach in the field of Hydrology, It is an online prediction that not only provides optimal multi-site predictions but also predictions that take into account the dynamic nature of the rainfall itself, and apparently the results were very acceptable,

thank you for your question

the best

Article Multi-site modeling and prediction of annual and monthly pre...

The recursive least squares algorithm (RLS) is the recursive application of the well-known least squares (LS) regression algorithm, so that each new data point is taken in account to modify (correct) a previous estimate of the parameters from some linear (or linearized) correlation thought to model the observed system. The method allows for the dynamical application of LS to time series acquired in real-time. As with LS, there may be several correlation equations and a set of dependent (observed) variables.

Years ago, while investigating adaptive control and energetic optimization of aerobic fermenters, I have applied the RLS algorithm with forgetting factor (RLS-FF) to estimate the parameters from the KLa correlation, used to predict the O2 gas-liquid mass-transfer, while giving increased weight to most recent data. Estimates were improved by imposing sinusoidal disturbance to air flow and agitation speed (manipulated variables). Simulations, carried in Excel 5.0 with Visual Basic for Applications (VBA) macros, assessed the effect of numerically generated white Gaussian noise (2-sigma truncated) and of first order delay. This investigation was reported at (MSc Thesis):

Thesis Controlo do Oxigénio Dissolvido em Fermentadores para Minimi...