Analysis of algorithms to parameter estimationFor to estimate the oxygen uptake rate (OUR) by a real time Kalman Filter, we have to develop a mathematical model to mimic, relatively well, the dissolved oxygen (DO) dynamics on activated sludge processes.This mathematical model consists of a set of equations, whose variables are the processes states and the coefficients are the processes parameters.For use these equations on Kalman Filter, we have to find out the best way to estimate the processes parameters. In this report, we compare some algorithms to parameters estimation.Main dynamic equations:(a) Dissolved oxygen dynamics: describes the dissolved oxygen variation in activated sludge reactor.(1)

Where:c(t)dissolved oxygen concentration;KLa(t)oxygen transfer function;csatsaturation of DO;R(t)oxygen uptake rate.(b) Aerator dynamics: describes the presumable first order behavior of the air bubbles when the aerator turns on/off.(2)

Where:tkair bubbles time constant.(c) Sensor dynamics: describes the DO probe as a first order system (assumed in early researches).(3)

Where:tyDO sensor time constant.Equations used in the algorithmsModel 1: take into account only the dissolved oxygen dynamics. The system can be modeled by

Figure 1 DO model 1R(s) acts like a noise input to the system and U(s) is a step input. Assuming OUR=cte (endogenous phase, this can be obtained experimentally) and KLa=Kmax (no aerator dynamics), one can obtain:(4)

Where: c(0) is the initial DO concentration.Model 2: take into account the DO dynamics and the DO sensor dynamics.

Figure 2 DO model 2.With the same considerations done in DO model 1, one can obtain:(5)

Where y(0) is the initial measurement of the DO sensor.Analysis of the estimation algorithms with simulation dataNon-linear least squares algorithm was used to evaluate the models and the applicability of the method in parameter estimation. Parameters used in simulation

200 [1/h]8 [mg/l]8 [s]5 [s]0, 10 or a rwm[footnoteRef:1] [mg/l/h] [1: Randon Walk Model.]

In the simulation we have considered the aerator dynamics, despite in the models we have neglected its effect.Three different behavior of OUR have been considered: zero (clean water), constant (endogenous phase) and random walk model (regular operation). The results are summarized in the tables below.(a) Clean water (R = 0): using the model 2 we can obtain a better fit to the data with respect to model 1. However the confidence interval to parameter estimation gets worst using the model 2.Algorithm [1/h] [mg/l]

[s][mg/l/h]95% confidence interval

Model 1111.488XXKLa = [109.55 113.4]csat = [8 8]

Figure 3 Step response

Figure 4 Estimation error.

Figure 5 Histogram.

Algorithm [1/h] [mg/l]

[s][mg/l/h]95% confidence interval

Model 2239815.06XKLa = [-1.7 +2.2]e+3csat = [8 8]ty = [-112.7 +142.9]

Figure 6 Step response.

Figure 7 Estimation error.

Figure 8 Histogram.

(b) Endogenous phase: assuming OUR constant, the model 2 provides better results with respect to model 1. R and csat can be estimated relatively well, while KLa and ty are overestimated by the algorithm.Algorithms [1/h] [mg/l] [s] [mg/l/h]95% confidence interval

Model 1 with R=10

110.567.98X2KLa = [108.61, 112.53]csat = [-3.5, +3.5]e+6R = [-1, +1]e+8

Figure 9 Step response.

Figure 10 Estimation error.

Figure 11 Histogram.Algorithms [1/h] [mg/l]

[s] [mg/l/h]95% confidence interval

Model 2 with R=10

238.757.9915.0810.85KLa = [237.67, 239.83]csat = [7.99, 7.99]ty = [15.08, 15.08]R = [10.47, 11.23]

Figure 12 Step response.

Figure 13 Estimation error.

Figure 14 Histogram.

(c) Regular operation: similar consideration to endogenous case.Algorithms [1/h] [mg/l]

[s] [mg/l/h]95% confidence interval

Model 1 with R=rwm

110.388XXKLa = [108.43, 112.32]csat = [-3.5, +3.5]e+6

Figure 15 Step response.

Figure 16 Estimation error.

Figure 17 Histogram.Algorithms [1/h] [mg/l]

[s] [mg/l/h]95% confidence interval

Model 2 with R=rwm238.15815.11XKLa = [237.06, 239.24]csat = [8, 8]ty = [15.11, 15.11]

Figure 18 Step response.

Figure 19 Estimation error.

Figure 19 Histogram.

Conclusion and remarksThe model 2 yielded better results when compared to model 1, however it has produced overestimates to KLa and ty.The nonlinear least squares algorithm presented convergence problems due to the initial choices of the parameters. The start values of the parameters have an important role in the algorithm.The confidence interval is affected by the way in which the equations are inserted in the estimation algorithm.Using the model 2, in endogenous and regular operation mode, a warning about ill-conditioned Jacobian has appeared frequently.