What would be the effect of using a Kalman filter to estimate the state of a system that is under active control? In an activated sludge system the air flow rate is adjusted to maintain a constant concentration of dissolved oxygen. If I use a Kalman filter to estimate the rate of oxygen transfer and the rate of oxygen uptake, what risk is there that the active control will prevent the Kalman filter converging on to a true answer?
I'm not an expert in this specific field you're working on but I know a linear quadratic Gaussian (LQG) control problem uses the separation principle to split the tasks of control and state estimation. There are many variations of this topic.
State Estimation does not assume anywhere absence of driving inputs to the system. It is only for mathematical convenience that it is removed in the analysis (WLOG). So, controlling a system should not prevent one from estimating the system's state simultaneously - in principle.
However, since the system you are working with is non-linear, and I am guessing that you are using the EKF (by the tag in the question). This can cause divergence problems, especially if you initialise it poorly, or move far away from your point of linearisation. The accuracy of the model will also affect the estimation algorithm.
So if you are confident about the model parameters and the initial conditions, the fact that you are controlling the system actively should definitely not prevent you from estimating its state.
The system is nonlinear. The model has been linearised so that one parameter (oxygen uptake) is regarded as a time-varying constant (i.e., no relation to dissolved oxygen) and the other as a (time-varying) constant multiplier to the dissolved oxygen concentration. This is partly as a convienence; and partly because the normal operational point is where the dissolved oxygen has a weak effect on the reaction rate.
So from what you both say the Kalman filter should be applicable despite the effect of active control on the system - any problems will lie with the assumptions used in the state space model, and would apply with or without active control. That's good to know!
Hi Jeremy,
In case of non-linear systems. You can use UKF which was developed after KF and EKF to overcome the shortcomings of these linear-based methods. As others have mentioned control input should not have effect on how you get your estimations, however, how you formulate your problem affects the outcomes. This paper is a great resource for you to look into.
-Afshin
Article Application of a Dual Unscented Kalman Filter for Simultaneo...
I have studied adaptive control and energetic optimization of aerobic fermenters taking air flow and agitation speed as manipulated variables. The parameters from the usual KLa correlation were estimated using sinusoidal excitation of air flow and agitation speed through the recursive least squares algorithm with forgetting factor. The control algorithm used compared favourably to PID: cf. C.M.G.A. Queirós, "Controlo do Oxigénio Dissolvido em Fermentadores para Minimização de Energia Consumida", MSc Thesis, Instituto Superior Técnico, Universidade Técnica de Lisboa (Lisbon), 1997 (in Portuguese).
Thesis Controlo do Oxigénio Dissolvido em Fermentadores para Minimi...