Auto-Control Technique Using Gradient Method Based on Radial Basis Function Neural Networks to Control of an Activated Sludge Process of Wastewater Treatment

The wastewater treatment process is a very complex process; it presents strong nonlinearity and uncertainties regarding to its parameters. The wastewater treatment comprises various steps used for reducing the contaminants in the wastewater which are: pretreatment, primary treatment, secondary treatment [1, 2]. The pretreatment has the objective of removing solid objects, and to skimming off floating greases and oils. Without passing the wastewater through the pretreatment, these objects may cause block and damage the equipment and the other steps of treatments. The primary treatment removes the remaining suspended and dissolved solids.

The secondary or biological treatment is the most important step of wastewater treatment. It aims to add microorganisms to reduce the organic matter, nitrogen and phosphorus from the wastewater. There are different methods used in the wastewater treatment process, but the most used and popular one is the activated sludge process (ASP) [3, 4].

In the last years, varieties of researches have been conducted about the control of the level of dissolved oxygen to enhance the process. In general, improvements are related to the controlling techniques. A linear proportional integral (PI) controller with feedforward from the flow rate and the respiration rate has been shown as a basic strategy [4]. Because of the PID controller limitation, the classical method of proportional integral derivative (PID) has been attempted, and the controlling effect of the dissolved oxygen is not satisfactory enough. However, the controllers are normally designed for the particular operational conditions because of the scarcity of sufficient hard or soft sensors and the nonlinear features of the bioprocesses [5]. In recent times, scholars have begun to study the artificial intelligence (AI) technologies which can be widely implemented in numerous areas, including chemical and biochemical processes and. Due to its great capabilities and adaptabilities of nonlinear modelling, the most prevalent AI controlling strategies are neural network and fuzzy network, which are usually integrated with the PID control. A fuzzy method to the control of dissolved oxygen in the process of aeration was studied by Man et al. [6]. An adaptive fuzzy control strategy for dissolved oxygen concentration was used to control of an activated sludge process Liu et al. [7]. In the paper [8] an adaptive fuzzy neural network-based model predictive control (AFNN-MPC) is proposed for the control problem of DO concentration. Piotrowski proposed a supervisory heuristic fuzzy control system applied to a Sequencing Batch Reactor (SBR) in the Wastewater Treatment Plant (WWTP) [9]. Lin and Luo [10] developed an adaptive neural technique using a disturbance observer to solve the dissolved oxygen concentration control problem. An improved multi objective optimal control (MOOC) strategy is developed to improve the operational efficiency, satisfy the effluent quality (EQ) and reduce the energy consumption (EC) in wastewater treatment process [11]. Mirghasemi proposed a robust adaptive neural network control strategy and used it to control the dissolved oxygen in activated sludge process application [12].

In this paper, a control strategy based on Euler method and gradient method with RBF neural network is proposed to control the dissolved oxygen concentration in an activated sludge process of wastewater treatment. The Euler is a numerical method that is used to approximate the solution of a nonlinear differential equation (nonlinear system), and the gradient descent algorithm and the RBF neural network are used to find the values of a function's parameters (coefficients) that minimize a performance function as possible. The performance of the proposed control strategies laws is illustrated with numerical simulations comparing their results with RBFNN-PI controller.

The remainder of this paper is organized as follows: in Section 2, Euler method is clearly discussed, in Section 3, presents a nonlinear control strategy based on gradient method accompanied with neural network method. In Section 4, RBF neural network PI control is discussed. The Section 5, a mathematical model of wastewater treatment process is given. In Section 6, the simulation results are presented. Finally, it is ended by a conclusion.

The proposed control strategy is depicted in Figure 4.

4.png

Figure 4. Curve of Principal of the proposed control strategy

Consider the following nonlinear system:

$\frac{\partial {{y}_{{{u}_{0}}}}\left( t \right)}{\partial t}=f\left( t,{{y}_{{{u}_{0}}}}\left( t \right),{{u}_{0}} \right)$    (14)

The numerical solution of Eq. (14) by using Euler's method is:

${{y}_{{{u}_{0}}}}\left( {{t}_{1}} \right)={{y}_{{{u}_{0}}}}\left( {{t}_{0}} \right)+h.f\left( {{t}_{0}},{{y}_{{{u}_{0}}}}\left( {{t}_{0}} \right),{{u}_{0}} \right)$    (15)

At time t₁, we have to find the value of the input control u₁ to get: ${{y}_{{{u}_{0}}}}\left( {{t}_{1}} \right)=r\left( {{t}_{1}} \right)$.

${{y}_{{{u}_{1}}}}\left( {{t}_{1}} \right)={{y}_{{{u}_{0}}}}\left( {{t}_{0}} \right)+h.f\left( {{t}_{0}},{{y}_{{{u}_{0}}}}\left( {{t}_{0}} \right),{{u}_{1}} \right)$    (16)

The input control u₁ is adjusted by using the gradient descent algorithm by minimizing the performance function with respect to u₀.The performance function is the squared error $E({{t}_{1}})$ between ${{y}_{{{u}_{1}}}}({{t}_{1}})$ and ${{y}_{{{u}_{0}}}}({{t}_{1}})$.

$E\left( {{t}_{1}} \right)=\frac{1}{2}{{\left( e\left( {{t}_{1}} \right) \right)}^{2}}=\frac{1}{2}{{\left( {{y}_{{{u}_{1}}}}\left( {{t}_{1}} \right)-{{y}_{{{u}_{0}}}}\left( {{t}_{1}} \right) \right)}^{2}}$    (17)

$E\left( {{t}_{1}} \right)=\frac{1}{2}{{\left( r\left( {{t}_{1}} \right)-{{y}_{{{u}_{0}}}}\left( {{t}_{1}} \right) \right)}^{2}}$    (18)

Input control u₁ updating by using the gradient descent algorithm:

${{u}_{1}}={{u}_{0}}+\Delta {{u}_{0}}$    (19)

$\Delta {{u}_{0}}=-\lambda .\frac{\partial E\left( {{t}_{1}} \right)}{\partial {{u}_{0}}}$    (20)

Then:

${{u}_{1}}={{u}_{0}}-\lambda .\frac{\partial E\left( {{t}_{1}} \right)}{\partial {{u}_{0}}}$    (21)

where, $\lambda $: learning rate.

${{u}_{1}}={{u}_{0}}-\lambda .\frac{\partial E\left( {{t}_{1}} \right)}{\partial {{y}_{{{u}_{0}}}}\left( {{t}_{1}} \right)}.\frac{\partial {{y}_{{{u}_{0}}}}\left( {{t}_{1}} \right)}{\partial f\left( {{t}_{0}},{{y}_{{{u}_{0}}}}\left( {{t}_{0}} \right),{{u}_{0}} \right)}.\frac{\partial f\left( {{t}_{0}},{{y}_{{{u}_{0}}}}\left( {{t}_{0}} \right),{{u}_{0}} \right)}{\partial {{u}_{0}}}$    (22)

${{u}_{1}}={{u}_{0}}+\lambda .e\left( {{t}_{1}} \right).h.\frac{\partial f\left( {{t}_{0}},{{y}_{{{u}_{0}}}}\left( {{t}_{0}} \right),{{u}_{0}} \right)}{\partial {{u}_{0}}}$    (23)

${{u}_{1}}={{u}_{0}}+\lambda .e\left( {{t}_{1}} \right).\frac{\partial {{y}_{{{u}_{0}}}}\left( {{t}_{1}} \right)}{\partial {{u}_{0}}}$    (24)

In general case, it is difficult to calculate  the expression of $\frac{\partial {{y}_{{{u}_{0}}}}\left( {{t}_{1}} \right)}{\partial {{u}_{0}}}$ , therefore, it has been known that the neural network can approximate any nonlinear function, then, it can be used to approximate $\frac{\partial {{y}_{{{u}_{0}}}}\left( {{t}_{1}} \right)}{\partial {{u}_{0}}}$.

3.1 RBF neural network algorithm

5.png

Figure 5. RBF neural network structure

A radial basis function neural network (RBFNN) is introduced into neural network in the paper [13], it has three layers: an input layer, a nonlinear hidden layer that uses Gaussian function as activation function, and a linear output layer [14]. The RBF networks have many uses, including function approximation, classification, and system control. It has the advantage of fast learning speed and is able to avoid the problem of local minimum.

The structure of the RBF neural network is illustrated in Figure 5.

The output of the ${{j}^{th}}$ hidden neurons is given by the following equation:

${{h}_{j}}=\exp \left( -\frac{{{\left\| X-{{C}_{i,j}} \right\|}^{2}}}{2b_{j}^{2}} \right)$    (25)

where, $X=\left[x_{1}, x_{2}, \ldots, x_{n}\right]^{T}$ is the input vector of the neural network.

The RBF neural network output is described in the following equation:

${{y}_{RBFNN}}=\sum\nolimits_{j=1}^{m}{{{W}_{l,j}}{{h}_{j}}}=\sum\nolimits_{j=1}^{m}{{{W}_{l,j}}}\exp \left( -\frac{{{\left\| X-{{C}_{i,j}} \right\|}^{2}}}{2b_{j}^{2}} \right)$    (26)

where, W_l,j is the weight between the hidden layer and the output layer. The center C_i,j, the basis width parameter b_j and the weights W_l,j of the RBF neuron network are adjusted by using the gradient descent algorithm to minimize the sum of  square error E_RBFNN.

The expression of E_RBFNN is given as follow:

${{E}_{RBFNN}}=\frac{1}{2}{{\sum\nolimits_{{}}^{{}}{\left( {{e}_{RBFNN}}\left( k \right) \right)}}^{2}}=\frac{1}{2}{{\sum\nolimits_{k=1}^{r}{\left( {{y}_{{{u}_{k}}}}\left( k \right)-{{y}_{RBFNN}}\left( k \right) \right)}}^{2}}$    (27)

${{C}_{i,j}}\left( k \right)={{C}_{i,j}}\left( k-1 \right)+\Delta {{C}_{i,j}}+\alpha \left( {{C}_{i,j}}\left( k-1 \right)-{{C}_{i,j}}\left( k-2 \right) \right)$    (28)

${{b}_{j}}\left( k \right)={{b}_{j}}\left( k-1 \right)+\Delta {{b}_{j}}+\alpha \left( {{b}_{j}}\left( k-1 \right)-{{b}_{j}}\left( k-2 \right) \right)$    (29)

${{W}_{l,j}}\left( k \right)={{W}_{l,j}}\left( k-1 \right)+\Delta {{W}_{l,j}}+\alpha \left( {{W}_{l,j}}\left( k-1 \right)-{{W}_{l,j}}\left( k-2 \right) \right)$    (30)

The corresponding modifier formulas are as follows:

$\Delta {{C}_{i,j}}=-\eta \frac{\partial {{E}_{RBFNN}}}{\partial {{C}_{i,j}}}=-\eta \frac{\partial {{E}_{RBFNN}}}{\partial {{y}_{RBFNN}}}\frac{\partial {{y}_{RBFNN}}}{\partial {{h}_{j}}}\frac{\partial {{h}_{j}}}{\partial {{C}_{i,j}}}$    (31)

$\Delta {{C}_{i,j}}=\eta .{{e}_{RBF}}{{W}_{l,j}}{{h}_{j}}.\left( \frac{X-{{C}_{i,j}}}{b_{j}^{2}} \right)$    (32)

$\Delta {{b}_{j}}=-\eta \frac{\partial {{E}_{RBFNN}}}{\partial {{b}_{j}}}=-\eta \frac{\partial {{E}_{RBFNN}}}{\partial {{y}_{RBFNN}}}\frac{\partial {{y}_{RBFNN}}}{\partial {{h}_{j}}}\frac{\partial {{h}_{j}}}{\partial {{b}_{j}}}$    (33)

$\Delta {{b}_{j}}=\eta .{{e}_{RBFNN}}{{W}_{l,j}}{{h}_{j}}.\frac{{{\left\| X-{{C}_{i,j}} \right\|}^{2}}}{b_{j}^{3}}$    (34)

$\Delta {{W}_{l,j}}=-\eta \frac{\partial {{E}_{RBFNN}}}{\partial {{W}_{l,j}}}=-\eta \frac{\partial {{E}_{RBFNN}}}{\partial {{y}_{RBFNN}}}\frac{\partial {{y}_{RBFNN}}}{\partial {{W}_{l,j}}}$    (35)

$\Delta {{W}_{l,j}}=\eta .{{e}_{RBFNN}}.{{h}_{j}}$    (36)

where, a is the momentum factor, and h is the learning rate.

From Eq. (24), we have:

${{u}_{1}}={{u}_{0}}+\lambda .e\left( {{t}_{1}} \right).\frac{\partial {{y}_{{{u}_{0}}}}\left( {{t}_{1}} \right)}{\partial {{u}_{0}}}$    (37)

The RBF neural network will be used to calculate the expression of $\frac{\partial {{y}_{{{u}_{0}}}}\left( {{t}_{1}} \right)}{\partial {{u}_{0}}}$.

The schema of RBF neural network identification is depicted in Figure 6.

6.png

Figure 6. Schema of RBF neural network identification

If the RBF neural network output y_RBFNN is equal to the system output ${{y}_{{{u}_{0}}}}$ [15, 16], then, we can use the expression of y_RBFNN to calculate the expression of $\frac{\partial {{y}_{{{u}_{0}}}}\left( {{t}_{1}} \right)}{\partial {{u}_{0}}}$.

$\frac{\partial {{y}_{{{u}_{0}}}}\left( {{t}_{1}} \right)}{\partial {{u}_{0}}}\approx \frac{\partial {{y}_{RBFNN}}}{\partial {{u}_{0}}}=\frac{\partial }{\partial {{u}_{0}}}\sum\nolimits_{j=1}^{r}{{{W}_{1,j}}}{{h}_{j}}$    (38)

$\frac{\partial {{y}_{{{u}_{0}}}}\left( {{t}_{1}} \right)}{\partial {{u}_{0}}}=\frac{\partial }{\partial {{u}_{0}}}\sum\nolimits_{j=1}^{r}{{{W}_{1,j}}}\exp \left( -\frac{{{\left\| X-{{C}_{i,j}} \right\|}^{2}}}{2b_{j}^{2}} \right)$    (39)

$\frac{\partial {{y}_{{{u}_{0}}}}\left( {{t}_{1}} \right)}{\partial {{u}_{0}}}=\sum\nolimits_{j=1}^{r}{{{W}_{1,j}}\frac{\partial }{\partial {{u}_{0}}}}\exp \left( -\frac{{{\left\| X-{{C}_{i,j}} \right\|}^{2}}}{2b_{j}^{2}} \right)$    (40)

$\begin{align}  & \frac{\partial {{y}_{{{u}_{0}}}}\left( {{t}_{1}} \right)}{\partial {{u}_{0}}}=\sum\nolimits_{j=1}^{r}{{{W}_{1,j}}\frac{\partial }{\partial {{x}_{1}}}} \\  & \exp \left( -\frac{{{X}^{T}}X-{{X}^{T}}{{C}_{i,j}}-C_{i,j}^{T}X+C_{i,j}^{T}{{C}_{i,j}}}{2b_{j}^{2}} \right) \\ \end{align}$    (41)

With: $X={{\left[ \begin{matrix}   {{u}_{0}} & {{y}_{{{u}_{0}}}}  \\\end{matrix} \right]}^{T}}={{\left[ \begin{matrix}   {{x}_{1}} & {{x}_{2}}  \\\end{matrix} \right]}^{T}}$

${{C}_{i,j}}=\left[ \begin{matrix}   {{C}_{1,1}} & \begin{matrix}   {{C}_{1,2}} & ... & {{C}_{1,m}}  \\\end{matrix}  \\   {{C}_{2,1}} & \begin{matrix}   {{C}_{2,2}} & ... & {{C}_{2,m}}  \\\end{matrix}  \\\end{matrix} \right]$

$\begin{align}  & \frac{\partial {{y}_{{{u}_{0}}}}\left( {{t}_{1}} \right)}{\partial {{u}_{0}}}=\sum\nolimits_{j=1}^{r}{{{W}_{1,j}}\frac{\partial }{\partial {{x}_{1}}}}\exp  \\  & \left( -\frac{\left[ \begin{matrix}   {{x}_{1}} & {{x}_{2}}  \\\end{matrix} \right]\left[ \begin{matrix}   {{x}_{1}}  \\   {{x}_{2}}  \\\end{matrix} \right]-\left[ \begin{matrix} {{x}_{1}} & {{x}_{2}}  \\\end{matrix} \right].{{C}_{i,j}}-C_{i,j}^{T}\left[ \begin{matrix}   {{x}_{1}}  \\   {{x}_{2}}  \\\end{matrix} \right]+C_{i,j}^{T}{{C}_{i,j}}}{2b_{j}^{2}} \right) \\ \end{align}$    (42)

$\begin{align}  & \frac{\partial {{y}_{{{u}_{0}}}}\left( {{t}_{1}} \right)}{\partial {{u}_{0}}}=\sum\nolimits_{j=1}^{r}{{{W}_{1,j}}\frac{\partial }{\partial {{x}_{1}}}}\exp  \\  & \left( -\frac{x_{1}^{2}+x_{2}^{2}-2{{x}_{1}}{{C}_{1,j}}-2{{x}_{2}}C_{2,j}^{T}+C_{i,j}^{T}{{C}_{i,j}}}{2b_{j}^{2}} \right) \\ \end{align}$    (43)

$\begin{align}  & \frac{\partial {{y}_{{{u}_{0}}}}\left( {{t}_{1}} \right)}{\partial {{u}_{0}}}=\sum\nolimits_{j=1}^{r}{{{W}_{1,j}}\left( \frac{{{C}_{1,j}}-{{x}_{1}}}{b_{j}^{2}} \right)}.\exp  \\  & \left( -\frac{x_{1}^{2}+x_{2}^{2}-2{{x}_{1}}{{C}_{1,j}}-2{{x}_{2}}C_{2,j}^{T}+C_{i,j}^{T}{{C}_{i,j}}}{2b_{j}^{2}} \right) \\ \end{align}$    (44)

So:

$\frac{\partial {{y}_{{{u}_{0}}}}\left( {{t}_{1}} \right)}{\partial {{u}_{0}}}=\sum\nolimits_{j=1}^{r}{{{W}_{1,j}}\left( \frac{{{C}_{1,j}}-{{x}_{1}}}{b_{j}^{2}} \right)}.{{h}_{j}}$    (45)

We substitute Eq. (45) in Eq. (37) to get the control law:

${{u}_{1}}={{u}_{0}}+\lambda .e\left( {{t}_{1}} \right).\sum\nolimits_{j=1}^{r}{{{W}_{1,j}}\left( \frac{{{C}_{1,j}}-{{x}_{1}}}{b_{j}^{2}} \right)}.{{h}_{j}}$    (46)

The steps of the proposed control strategy are as follows:

Step1. We start by u=u₀ (the choice of u₀ depend on the study system).

Step2. Initializing the network parameters: the number of nodes in input, hidden and output layer, the center C_i,j, the basis width parameter b_j and the weights W_l,j, learning rate.

Step3. for k³0.

- Calculating the output ${{y}_{{{u}_{k}}}}\left( {{t}_{k+1}} \right)$ and network output y_RBFNN(t_k+1), calculating the error e_RBFNN(t_k+1).

- Adjusting of neural network parameters: C_i,j, b_j and W_l,j from Eq. (32), Eq. (34) and Eq. (36).

- Calculating error e(t_k+1) between ${{y}_{{{u}_{k}}}}\left( {{t}_{k+1}} \right)$ and r(t_k+1).

- Calculating the new value of u_k+1 via the following expression:

${{u}_{k+1}}={{u}_{k}}+\lambda .e\left( {{t}_{k+1}} \right).\sum\nolimits_{j=1}^{r}{{{W}_{1,j}}\left( \frac{{{C}_{1,j}}-{{u}_{k}}}{b_{j}^{2}} \right)}.{{h}_{j}}$   (47)

Step 4. We replace the found value of u_k+1 on the system.

${{y}_{{{u}_{0}}}}\left( {{t}_{k+1}} \right)={{y}_{{{u}_{0}}}}\left( {{t}_{k}} \right)+h.f\left( {{t}_{k}},{{y}_{{{u}_{0}}}}\left( {{t}_{k}} \right),{{u}_{k+1}} \right)$    (48)

According to this algorithm, we can obtain: ${{y}_{{{u}_{0}}}}\left( {{t}_{k+1}} \right)=r\left( {{t}_{k+1}} \right)$.

The structure of the proposed method is illustrated in Figure 7.

7.png

Figure 7. Schema of the proposed control strategy

8.png

Figure 8. Adaptive RBF neural network PI structure

The schematic of the wastewater treatment process is represented in Figure 9.

9.png

Figure 9. Schema of activated sludge process

The aeration tank is a biological reactor in which the microorganisms are developed by removing the organic substrate in the presence of the dissolved oxygen concentration. In the settler tank the solids are separated from the wastewater. The growth of micro-organisms (biomass) in the aeration tank is not sufficient, then, a part of the removed sludge (the sludge contains the biomass) is recycled back tothe aeration tank while the other part is removed from the system.

The mathematical model considered in this paper contains four differential equations, the biomass concentrationX, the substrate concentration S, the dissolved oxygen concentration DO and the recycled biomass concentration X_r. The model is given by the following equations [17, 18].

$\frac{\partial X\left( t \right)}{\partial t}=\mu \left( t \right).X\left( t \right)-D.\left( 1+r \right).X\left( t \right)+r.D.{{X}_{r}}\left( t \right)$    (58)

$\frac{\partial S\left( t \right)}{\partial t}=-\frac{1}{Y}\mu \left( t \right).X\left( t \right)-D.\left( 1+r \right).S\left( t \right)+D.{{S}_{in}}$    (59)

$\frac{\partial DO\left( t \right)}{\partial t}=-\frac{{{K}_{0}}}{Y}\mu \left( t \right).X\left( t \right)-D.\left( 1+r \right).DO\left( t \right)+KLa.\left( D{{O}_{\max }}-DO\left( t \right) \right)+D.D{{O}_{in}}$    (60)

$\frac{\partial {{X}_{r}}\left( t \right)}{\partial t}=D.\left( 1+r \right).X\left( t \right)-D.\left( \beta +r \right).{{X}_{r}}\left( t \right)$    (61)

With:

$\mu \left( t \right)={{\mu }_{\max }}.\frac{S\left( t \right)}{S\left( t \right)+{{k}_{s}}}.\frac{DO\left( t \right)}{DO\left( t \right)+{{k}_{DO}}}$    (62)

$KLa=\alpha .W\left( k \right)$.

W: the aeration flow rate.

The aeration rate W is considered in this paper as the variable control for controlling the oxygen concentration in the aeration tank. Table 2 and Table 3 below show the parameters and initial values of the model respectively.

Table 2. Model parameters

Table 3. Initial values