# Combined Estimation and Optimal Control of Batch Membrane Processes

^{1}

^{2}

^{3}

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## Abstract

**:**

## 1. Introduction

## 2. Process Description

- Concentration mode: During this mode, no diluent is added into the feed tank, i.e., $\alpha =0$.
- Constant-volume diafiltration mode: Here, the rate of inflow of the diluent is kept the same as the rate of permeate outflow, i.e., $\alpha =1$.
- (Pure) Dilution mode: In this mode, a certain amount of diluent can be added instantaneously into the feed tank. This can be represented as $\alpha \to \infty $.

## 3. Membrane Fouling

#### 3.1. Cake Filtration Model

#### 3.2. Intermediate Blocking Model

#### 3.3. Internal Blocking Model

#### 3.4. Complete Pore Blocking Model

## 4. Optimization Problem

#### 4.1. Problem Definition

#### 4.2. Nominal Optimal Operation

- In the first interval, the control is kept on the boundaries (e.g., minimum or maximum) until the singular curve is reached:$$\begin{array}{c}\hfill S(t,{c}_{1},{c}_{2},K,n)=J+{c}_{1}{\displaystyle \frac{\partial J}{\partial {c}_{1}}}+{c}_{2}{\displaystyle \frac{\partial J}{\partial {c}_{2}}}=0.\end{array}$$The minimum control action ($\alpha =0$) is applied if the initial concentrations lie to the left side of the singular surface ($S(0,{c}_{1,0},{c}_{2,0},K,n)<0$) in the state diagram. Conversely, the maximum ($\alpha \to \infty $) control action is applied when the initial concentrations are on the right of the singular curve ($S(0,{c}_{1,0},{c}_{2,0},K,n)>0$).
- Once the singular curve is reached, the singular control is applied, which forces the states to stay on the singular curve:$$\begin{array}{c}\hfill \alpha (t,{c}_{1},{c}_{2},K,n)={\displaystyle \frac{{\displaystyle \frac{\partial S}{\partial {c}_{1}}}{c}_{1}}{{\displaystyle \frac{\partial S}{\partial {c}_{1}}}{c}_{1}+{\displaystyle \frac{\partial S}{\partial {c}_{2}}}{c}_{2}}}+{\displaystyle \frac{{\displaystyle \frac{\partial S}{\partial t}}}{{\displaystyle \frac{{c}_{1}AJ}{{c}_{10}{V}_{0}}}\left({\displaystyle \frac{\partial S}{\partial {c}_{1}}}{c}_{1}+{\displaystyle \frac{\partial S}{\partial {c}_{2}}}{c}_{2}\right)}}.\end{array}$$This step is terminated once the ratio of the concentrations is equal to the ratio of the final concentrations or once the final concentration of the micro-solute is reached.
- In the last step, the control is kept on one of the boundaries similar to the first step with the difference that the operation mode to be applied is determined by the final time constraints. The concentration mode ($\alpha =0$) is applied if the final concentration of the micro-solute has been reached. The dilution mode with $\alpha \to \infty $ is applied once the ratio of the final concentrations is equal to the final one. Both steps are performed until the final concentrations are reached.

#### 4.3. Optimal Operation with Imperfect Knowledge of Fouling Model Parameters

- If the operation starts with the concentration mode, the a priori knowledge of unknown parameters K and n should be improved during this step by parameter estimation, such that the switching time can be determined accurately. If this estimation is successful, the optimal singular surface is followed thereafter.
- Should the optimal operation commence with a pure dilution step, the entry point to the singular surface would be known exactly, as it only depends on ${J}_{0}$. In other words, the amount of the diluent that should be added at the beginning of the operation does not depend on the parameters of the fouling model. It would then be again necessary to improve the knowledge about the unknown parameters in order to maintain the singular-surface condition.

## 5. Parameter Estimation Problem

#### 5.1. Problem Definition

#### 5.2. Identifiability Problem

**Definition**

**1.**

#### 5.2.1. Identifiability of the Corresponding Linearized System

#### 5.2.2. Generating Series Approach

#### 5.3. Software Tools for the Identifiability Test

#### Results and Discussion

#### 5.4. Parameter Estimation Methods

#### 5.4.1. Weighted Least-Squares Method

#### 5.4.2. Weighted Least-Squares Method with Moving Horizon

#### 5.4.3. Recursive Least-Squares Method

#### 5.4.4. Extended Kalman Filter

^{−}), and its linearized counterpart around the current estimate is exploited in the propagation of the covariance matrix P and in the subsequent correction step (denoted by the superscript

^{+}) with the discrete system measurements. Compared to the recursive least-squares method, one can additionally account for the nonlinearity and the noise that affects the system dynamics using EKF. This can prove to be highly advantageous if parametric uncertainty influences the measurements strongly. Similarly to the recursive least-squares method, the absence of the treatment of the constraints and severe nonlinearities can lead to the divergence of the estimator. On the other hand, the advantage of EKF lies in its simpler implementation, where no parametric sensitivities are required, and in increased tuning capabilities, which might yield faster and more robust convergence.

#### 5.4.5. General Assessment of the Presented Estimation Methods

## 6. Case Study

^{2}. The a priori unknown fouling rate is $K=2\phantom{\rule{0.166667em}{0ex}}\mathrm{units}$. As the degree of nonlinearity of the model is strongly dependent on the nature of the fouling behavior, represented by the a priori unknown parameter n (see Section 3), we will study the cases when $n=\{0,\phantom{\rule{0.166667em}{0ex}}1,\phantom{\rule{0.166667em}{0ex}}1.5\}$.

## 7. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**Figure 2.**Graphical representation of the four classical fouling models developed by Hermia. (

**a**) Cake filtration model ($n=0$); (

**b**) intermediate blocking model ($n=1$); (

**c**) internal blocking model ($n=3/2$); (

**d**) complete blocking model ($n=2$).

**Figure 3.**Estimation of the fouling parameters $K,n$ for the three chosen cases together with the optimal switching time. (

**a**) Estimation of the fouling parameter K; (

**b**) estimation of the fouling parameter n.

**Figure 4.**Concentration state diagram and optimal control profile for ideal and estimated fouling parameters ($K=2$ and $n=1.5$) by employing the EKF method. (

**a**) Measured, ideal and estimated concentrations; (

**b**) measured, ideal and estimated flux; (

**c**) concentration state diagram; (

**d**) optimal control profile.

**Figure 5.**Estimation of fouling parameters, concentration state diagram and optimal control profile for ideal and estimated fouling parameters ($K=2$ and $n=1$) using the WLS estimation method. (

**a**) Estimation of the fouling parameter K; (

**b**) estimation of the fouling parameter n; (

**c**) Concentration state diagram; (

**d**) optimal control profile.

**Table 1.**Comparison of the rate of the suboptimality of final processing times, the normalized root mean squared error for unknown parameters and concentrations and the computational time for different estimation techniques. MH, moving horizon; RLS, recursive least-squares.

Estimation method | $\mathit{n}=0$ | $\mathit{n}=1$ | $\mathit{n}=1.5$ | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

${\mathit{\delta}}_{\mathbf{tf}}^{*}$ (h) | ${\mathcal{R}}_{{\scriptstyle \widehat{\mathit{\theta}}}}$ | ${\mathcal{R}}_{{\scriptstyle \widehat{\mathit{x}}}}$ | ${\mathit{t}}_{\mathit{c}}$ (h) | ${\mathit{\delta}}_{\mathbf{tf}}^{*}$ (h) | ${\mathcal{R}}_{{\scriptstyle \widehat{\mathit{\theta}}}}$ | ${\mathcal{R}}_{{\scriptstyle \widehat{\mathit{x}}}}$ | ${\mathit{t}}_{\mathit{c}}$ (h) | ${\mathit{\delta}}_{\mathbf{tf}}^{*}$ (h) | ${\mathcal{R}}_{{\scriptstyle \widehat{\mathit{\theta}}}}$ | ${\mathcal{R}}_{{\scriptstyle \widehat{\mathit{x}}}}$ | ${\mathit{t}}_{\mathit{c}}$ (h) | |

WLS | 0.21 | 0.24 | 0.17 | 3.73 | 0.10 | 0.46 | 0.16 | 4.71 | 0.09 | 0.14 | 0.24 | 23.33 |

WLSMH | 0.21 | 0.48 | 0.23 | 0.55 | 0.10 | 0.64 | 0.46 | 1.28 | 0.09 | 0.29 | 0.74 | 8.52 |

RLS | 0.21 | 0.11 | 1.65 | 0.04 | 0.10 | 0.03 | 1.15 | 0.04 | 0.09 | 0.04 | 1.25 | 0.13 |

EKF | 0.21 | 0.09 | 5.26 | 0.01 | 0.10 | 0.05 | 3.18 | 0.01 | 0.09 | 0.07 | 2.82 | 0.01 |

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**MDPI and ACS Style**

Jelemenský, M.; Pakšiová, D.; Paulen, R.; Latifi, A.; Fikar, M.
Combined Estimation and Optimal Control of Batch Membrane Processes. *Processes* **2016**, *4*, 43.
https://doi.org/10.3390/pr4040043

**AMA Style**

Jelemenský M, Pakšiová D, Paulen R, Latifi A, Fikar M.
Combined Estimation and Optimal Control of Batch Membrane Processes. *Processes*. 2016; 4(4):43.
https://doi.org/10.3390/pr4040043

**Chicago/Turabian Style**

Jelemenský, Martin, Daniela Pakšiová, Radoslav Paulen, Abderrazak Latifi, and Miroslav Fikar.
2016. "Combined Estimation and Optimal Control of Batch Membrane Processes" *Processes* 4, no. 4: 43.
https://doi.org/10.3390/pr4040043