Event-Triggered Neural Network Multivariate Control for Wastewater Treatment Process

Abstract

Recently, the neural network control has been widely used in the field of wastewater treatment process (WWTP). However, most neural network (NN) control methods are time-driven, with a large number of transmissions and a large amount of neural network computation. To reduce the number of controller executions and save computational cost, the event-triggered neural network multivariate method is proposed to control WWTP. Firstly, different from the traditional NN-based control, the event-triggered mechanism based on sliding windows is designed to reduce the computation. Then, the multi-input and multi-output recurrent wavelet neural network (RWNN) controller is proposed for simultaneous control of dissolved oxygen and nitrate nitrogen. Furthermore, the stability of the RWNN controller is analyzed through the Lyapunov stability theorem. Experimental results demonstrate that the event-triggered RWNN delivers a significant 25% reduction in the number of executions without compromising control accuracy.

1. Introduction

Recently, frequent water shortages worldwide have affected economic development and society. To alleviate water shortages, the wastewater treatment process (WWTP) has attracted a lot of attention as an important way to reduce the need for natural water and weaken the pollution of the water environment [1,2]. Meanwhile, discharge standards for WWTPs have become stricter as water quality deteriorates, leading to an increase in the operating cost of WWTP [3]. On the other hand, the WWTP is a dynamic and nonlinear process due to its multiple biochemical reactions [4,5]. Therefore, it is a challenge to reduce the operating costs of WWTP while maintaining the desired control performance.

In WWTP, the activated sludge method (ASM) has become a widely used method for nitrogen removal and phosphorus removal with the development of technology [6,7]. During the process of ASM, the dissolved oxygen (DO) in the fifth tank and the nitrate nitrogen (NO₃⁻) in the second tank are two important control variables. The nitrification reaction and denitrification reaction, as the core of ASM, are affected by DO and NO₃⁻ concentrations [8,9]. Therefore, the precise control of DO and NO₃⁻ is important to improve the efficiency of nitrogen and phosphorus removal in WWTP. Effective removal of DO and NO₃⁻ can prevent water eutrophication and substandard water quality. Furthermore, since high concentrations of DO and NO₃⁻ can inhibit microbial activity, their efficient elimination is essential for maintaining the metabolic function of microorganisms in activated sludge and ensuring the stable operation of the treatment system.

Traditional control methods such as feedforward control, proportional integral (PI) control and proportional integral derivative (PID) control have been widely applied to the control of WWTP [10,11]. These methods are simple in structure, but the control accuracy is low due to the fact that the WWTP is a complex nonlinear system [12]. To address this issue, with the development of technology, some intelligent control methods have attracted attention [13,14,15]. For example, the data-driven adaptive control based on the deep belief network was proposed to improve the stabilization of process control in WWTP [16]. To solve the problem of disturbances in the laboratory-activated sludge bioreactor process, the boundary-based predictive controller was designed to control the DO concentration within the desired range [17]. Wei et al. combined the state observer and adaptive sliding mode control to compensate for disturbances, effectively enhancing the robustness of the control system [18]. These intelligent methods can effectively improve the control accuracy of WWTP. However, the WWTP is a typical multivariate process, and these univariate control methods can only control the DO concentration, which cannot ensure the efficient operation of WWTP.

To deal with the problem of univariate control [19,20,21], multivariate control methods have been proposed, such as the reinforcement learning control [22], the sliding mode control [23] and the neural network control [24]. Among these methods, neural network control can realize precise control due to its powerful self-learning capability. For example, the self-organizing fuzzy neural network was proposed to control the DO concentration and NO₃⁻ concentration simultaneously. The results demonstrate the proposed controller can achieve high-precision multivariable control [25]. To deal with the local optimization problem of the traditional gradient descent algorithm of the neural network, Bai et al. designed the multi-gradient recurrent reinforcement learning algorithm to train the radial basis function neural network to control uncertain nonlinear systems. Experimental results show that the proposed algorithm effectively improves the learning accuracy of the network [26]. The recurrent fuzzy neural network with two hidden layers was combined with the sliding mode control to improve the control accuracy of the nonlinear system [27]. However, most neural network controls are time-driven, with large amounts of transmitted data and high computational load.

Unlike time-driven controls, event-triggered controls operate only when a specific event occurs, reducing the computational load. Recently, event-triggered controls have been studied in many fields. For example, the switching-like event-triggered scheme was proposed to counter denial-of-service attacks, thereby improving the communication efficiency of network control systems [28]. Liu et al. constructed the event-triggered controller to ensure the stability of repeated scalar nonlinear systems [29]. Moreover, the event-triggered distributed hybrid control scheme was designed to realize the safe and economic operation of the integrated energy system [30]. Experimental results have demonstrated that these event-triggered mechanisms can be successfully applied. However, the event-triggered algorithm in the framework of neural network control has not been well addressed.

Based on the above analysis, to reduce the computational cost of WWTP control, the event-triggered recurrent wavelet neural network (ERWNN) multivariate controller is designed in this paper. Unlike the previous neural network controls, the control process is event-driven to reduce the computational load of the neural network. The main contributions of this paper can be summarized as follows:

(1) Based on the control errors within a sliding window, the event-triggered mechanism is designed. Unlike traditional control methods that execute periodically at fixed time intervals, the mechanism only initiates control actions when the error meets specific conditions. This methodology effectively reduces the computational burden while improving overall control performance.

(2) The recurrent wavelet neural network is employed as the controller for WWTP. Otherwise, the control performance of the ERWNN is evaluated using the Benchmark Simulation Model No. 1 (BSM1) platform.

2. Activated Sludge Process

The activated sludge process (ASP), as an aerobic biological treatment method, is the most widely used method for WWTP. A typical ASP is composed of the biochemical reaction tank, the secondary settling tank, the sludge return system and the residual sludge discharge system, as shown in Figure 1. Among them, the biochemical reactor is the core of the ASP. The red part and blue part are the anoxic and aerobic zones, respectively, where denitrification and nitrification reactions are carried out.

In the biochemical reaction tank, the DO concentration and NO₃⁻ concentration have a great influence on the nitrification and denitrification reactions. Moreover, when the DO concentration is too high or too low, it will have an inhibitory effect on the denitrification reaction, thus affecting the NO₃⁻ concentration. Coupling exists between the control loops of DO and NO₃⁻. Therefore, it is very important to design a high-performance multivariable controller for DO and NO₃⁻ concentration to improve the efficiency of WWTP.

According to the reaction mechanism of ASP, the kinetic equations of DO and NO₃⁻ concentration are as follows

$$\left\{\begin{array}{l}{\displaystyle \frac{d{S}_{NO,k}}{dt}}={\displaystyle \frac{1}{V_k}}\left(\left(Q_a+Q_r+Q_0\right)\left(Z_{k-1}-Z_k\right)+r_k{V}_k\right)\hfill \\ {\displaystyle \frac{d{S}_{O,k}}{dt}}={\displaystyle \frac{1}{V_k}}\left(Q_k{S}_{O,k-1}+r_k{V}_k+K_{Lak}{V}_k{S}_{O,sat}-K_{Lak}{V}_k{S}_{O,k}-Q_k{S}_{O,k}\right)\hfill \end{array}\right.$$

(1)

where $Q_k$ and $Q_a$ are the flow rate of kth tank and internal flow of the second tank, $V_k$ and $Z_k$ are the volume of the kth tank and component concentration, $K_{Lak}$ is the oxygen transfer coefficient of the kth tank, and $S_{O,k}$ and $S_{O,sat}$ are the DO concentration and DO saturation concentration of the kth tank. Based on Equation (1) and the mechanism of biochemical reaction, the main factors affecting the DO and NO₃⁻ are $Q_a$ and $K_{Lak}$. Thus, the $Q_a$ and $K_{La5}$ are chosen as the operating variables for NO₃⁻ of the second tank and DO of the fifth tank.

3. Event-Triggered Neural Network Control

In this section, to reduce the control cost while ensuring control accuracy, the event-triggered recurrent wavelet neural network (ERWNN) is proposed to control DO and NO₃⁻ concentrations in WWTP, as shown in Figure 2.

The structure of the RWNN controller is displayed in Figure 3, which contains four layers: input layer, mother wavelet layer, wavelet layer and output layer. It is assumed that the number of wavelet nodes is q, then the input to the controller at time t is

$$x(t)=[e_{DO}(t),\Delta e_{DO}(t),e_{NO}(t),\Delta e_{NO}(t)]$$

(2)

where $e_{DO}(t)=S_{DO,set}(t)-S_{DO}(t)$, $e_{NO}(t)=S_{NO,set}(t)-S_{NO}(t)$, $S_{DO,set}(t)$ and $S_{NO,set}(t)$ are the set-points of DO and NO₃⁻ concentration, $S_{DO}(t)$ and $S_{NO}(t)$ are the actual values of DO and NO₃⁻ concentration and $\Delta e_{DO}(t)$ and $\Delta e_{NO}(t)$ are the changes in error, respectively.

Then the output of the controller $y(t)=[y_1(t),y_2(t)]=[\Delta K_{La5}(t),\Delta Q_a(t)]$ can be calculated as

$$y_k(t)={\displaystyle \sum _{j=1}^m{w}_{j,k}^0(t)u_j^{(3)}(t)}+{\displaystyle \sum _{i=1}^n{a}_{i,k}(t)x_i(t)} (k=1,2)$$

(3)

where $a_{i,k}$ is the weight between input and output, $x_i(t)$ is the ith input, $w_{j,k}^0$ is the output weight and $u_j^{(3)}(t)$ is the output of jth wavelet neuron which is shown as follows

$$u_j^{(3)}(t)={\displaystyle \prod _{i=1}^n{u}_{i,j}^{(2)}(t)}={\displaystyle \prod _{i=1}^n\Phi \left(\frac{h_{i,j}(t)-b_{i,j}(t)}{c_{i,j}(t)}\right)}(j=1,2,\dots ,q)$$

(4)

$$\Phi (x)=-x\exp (-{\displaystyle \frac{1}{2}}{x}^2)$$

(5)

$$h_{i,j}(t)=x_i(t)+w_{i.j}(t)u_{i,j}^{(2)}(t-1)$$

(6)

where $u_{i,j}^{(2)}(t)$ is the output of the mother wavelet layer, $b_{i,j}(t)$ and $c_{i,j}(t)$ are the translation and dilation coefficients, respectively and $w_{i.j}(t)$ is the feedback weight.

Remark 1.

In practice, the WWTP is a dynamic and nonlinear process. The RWNN combines the dynamic performance of the recurrent neural network and the capability of wavelet analysis, which effectively improves the learning ability of neural networks. Therefore, the RWNN is selected as the controller in this paper. Furthermore, there is a coupling between the control loops of DO and NO₃⁻. Thus, a single controller is used to weaken the influence of the coupling problem on the control results.

3.1. Parameter Learning Mechanism

In this subsection, the parameter learning algorithm combined with adaptive learning rates is designed to obtain the desired control accuracy. The cost function of RWNN controller is defined as

$$J(t)={\displaystyle \frac{1}{2}}{e}^2(t)={\displaystyle \frac{1}{2}}{\left(\beta e_{DO}(t)+(1-\beta )e_{NO}(t)\right)}^2$$

(7)

where $e_{DO}(t)$ and $e_{NO}(t)$ are the control error of DO and NO₃⁻ concentration, respectively, and $\beta =\left|e_O(t)\right|/\left(\left|e_O(t)\right|+\left|e_{NO}(t)\right|\right)$ is the proportionality factor.

Then, according to the gradient descent mechanism, the parameters can be updated as follows

$$\left\{\begin{array}{l}W(t+1)=W(t)+{\eta }_w(-{\displaystyle \frac{\partial J(t)}{\partial W(t)}})=W(t)+{\eta }_{w}e(t){\displaystyle \frac{\partial e(t)}{\partial W(t)}}\hfill \\ W^0(t+1)=W^0(t)+{\eta }_{w^0}(-{\displaystyle \frac{\partial J(t)}{\partial W^0(t)}})=W^0(t)+{\eta }_{w^0}e(t){\displaystyle \frac{\partial e(t)}{\partial W^0(t)}}\hfill \\ b(t+1)=b(t)+{\eta }_b(-{\displaystyle \frac{\partial J(t)}{\partial b(t)}})=b(t)+{\eta }_{b}e(t){\displaystyle \frac{\partial e(t)}{\partial b(t)}}\hfill \\ \begin{array}{l}c(t+1)=c(t)+{\eta }_c(-{\displaystyle \frac{\partial J(t)}{\partial c(t)}})=c(t)+{\eta }_{c}e(t){\displaystyle \frac{\partial e(t)}{\partial c(t)}}\hfill \\ a(t+1)=a(t)+{\eta }_a(-{\displaystyle \frac{\partial J(t)}{\partial a(t)}})=a(t)+{\eta }_{a}e(t){\displaystyle \frac{\partial e(t)}{\partial a(t)}}\hfill \end{array}\hfill \end{array}\right.$$

(8)

where ${\eta }_{\Omega }=\left[{\eta }_w,{\eta }_{w^0},{\eta }_b,{\eta }_c,{\eta }_a\right]$ that can be defined as

$${\eta }_{\Omega }=2/\underset{k}{\min }{\left({\displaystyle \frac{\partial e_k(t)}{\partial \Omega (t)}}\right)}^2$$

(9)

where $\Omega (t)=\left[W(t),W^0(t),b(t),c(t),a(t)\right]$, which is described for clarity.

The RWNN transforms the input error into control signals $\Delta K_{La5}(t)$ and $\Delta Q_a(t)$ via Equations (3)–(6), directing the WWTP to update the DO and NO₃⁻ concentrations. Concurrently, parameters $\Omega (t)$ are updated in real-time by the online learning algorithm specified in Equations (7)–(9). This forms a closed-loop where the learning algorithm refines parameters $\Omega (t)$ using the feedback error, thereby forcing the output of RWNN to converge toward the desired values and consequently regulating the actual DO and NO₃⁻ concentrations to match their targets.

3.2. Event-Triggered Mechanism

The traditional neural network controls are time-driven, i.e., the controller executes at every time, resulting in a large computational load on the controller. To reduce the computational cost of WWTP, the event-triggered mechanism based on the control error is designed for the RWNN controller in this subsection. The procedure of the event-triggered mechanism is described as follows:

Step 1: Define a sliding window of window size m and the confidence level as $\alpha$.

Step 2: Calculate the confidence intervals of DO and NO₃⁻ errors as $[{\mu }_{DO,1},{\mu }_{DO,2}]$ and $[{\mu }_{NO,1},{\mu }_{NO,2}]$.

Step 3: Define the events which are shown as follows

$$\left\{\begin{array}{l}{E}_1:{\mu }_{DO,1}<{\mu }_{DO,\min } \& {\mu }_{DO,2}>{\mu }_{DO,\max }\hfill \\ E_2:{\mu }_{NO,1}<{\mu }_{NO,\min } \& {\mu }_{NO,2}>{\mu }_{NO,\max }\hfill \end{array}\right.$$

(10)

where ${\mu }_{DO,\min },{\mu }_{DO,\max },{\mu }_{NO,\min },{\mu }_{NO,\max }$ are the artificially set thresholds.

Step 4: Design an event-triggered learning strategy. When events E1 or E2 occur, the system is poorly controlled, which means that the controller is required to make adjustments to the DO and NO₃⁻ concentrations. Thus, we have the following rule

$$\left\{\begin{array}{l}if E_1 occur: K_{La5}(t+1)=K_{La5}(t)+\Delta K_{La5}(t)\hfill \\ Otherwise:K_{La5}(t+1)=K_{La5}(t)\hfill \end{array}\right.$$

(11)

$$\left\{\begin{array}{l}if E_2 occur: Q_a(t+1)=Q_a(t)+\Delta Q_a(t)\hfill \\ Otherwise:Q_a(t+1)=Q_a(t)\hfill \end{array}\right.$$

(12)

where $\Delta K_{La5}(t)$ and $\Delta Q_a(t)$ are the outputs of RWNN controller at time t.

Remark 2.

Based on the event-triggered mechanism, the controller executes only when event E₁ or E₂ occurs.

3.3. Control Process of WWTP

Based on the above introduction, the details of the ERWNN-based multivariate control of WWTP are described in

Table 1. The control process of WWTP.

Initialization:
for $t=1:t_{\max }$
Calculate the outputs of ERWNN $\Delta K_{La5}(t)$ and $\Delta Q_a(t)$ Calculate the DO concentration and NO3− concentration Update $W(t)$, $W^0(t)$, $b(t)$, $c(t)$ and $a(t)$ as () If the event E1 occurs $K_{La5}(t+1)=K_{La5}(t)+\Delta K_{La5}(t)$ If the event E2 occurs $Q_a(t+1)=Q_a(t)+\Delta Q_a(t)$ Otherwise $K_{La5}(t+1)=K_{La5}(t)$, $Q_a(t+1)=Q_a(t)$ End End

and Figure 4.

3.4. Stability Analysis

In practice, the stability of the controller is very important for successful application. Thus, the stability of the ERWNN controller is analyzed based on the Lyapunov stability theorem in this subsection.

Theorem 1.

Assuming that the wavelet node of ERWNN is q, if the adaptive learning rates satisfy Equation (9), the stability of the ERWNN controller can be guaranteed.

Proof. 

Define the Lyapunov function as

$$V_1(t)={\displaystyle \frac{1}{2}}\left(e_1^2(t)+e_2^2(t)\right)$$

(13)

where $e_1(t)$ and $e_2(t)$ are the control errors of the DO and NO₃⁻ concentrations, respectively.

Then the change in Lyapunov function is as follows

$$\begin{array}{ll}\Delta V(t)& =V(t+1)-V(t)={\displaystyle \frac{1}{2}}{\displaystyle \sum _{k=1}^2(e_k^2(t+1)-e_k^2(t))}\\ & =\Delta V_1(t)+\Delta V_2(t)\end{array}$$

(14)

According to [25], the change in control error can be calculated as

$$\Delta e_k(t)=e_k(t+1)-e_k(t)\approx \left[{\displaystyle \frac{\partial e_k(t)}{\partial \Omega (t)}}\right]\Delta \Omega (t)$$

(15)

Based on the update rule of Equation (8), we have

$$\Delta e_k(t)=-{\eta }_{\Omega }{e}_k(t){\left[{\displaystyle \frac{\partial e_k(t)}{\partial \Omega (t)}}\right]}^T\left[{\displaystyle \frac{\partial e_k(t)}{\partial \Omega (t)}}\right]$$

(16)

Then Equation (14) can be further described as

$$\begin{array}{ll}\Delta V_k(t)& ={\displaystyle \frac{1}{2}}\left(2e_k(t)\Delta e_k(t)+{\left(\Delta e_k(t)\right)}^2\right)\\ & ={\displaystyle \frac{1}{2}}\left(-2{\eta }_{\Omega }{e}_k^2(t){\left[{\displaystyle \frac{\partial e_k(t)}{\partial {\Omega }_k(t)}}\right]}^2+{\eta }_{\Omega }^2{e}_k^2(t){\left[{\displaystyle \frac{\partial e_k(t)}{\partial {\Omega }_k(t)}}\right]}^4\right)\\ & =-{\displaystyle \frac{1}{2}}{e}_k^2(t){\eta }_{\Omega }{\left[{\displaystyle \frac{\partial e_k(t)}{\partial {\Omega }_k(t)}}\right]}^2\left(2-{\eta }_{\Omega }{\left[{\displaystyle \frac{\partial e_k(t)}{\partial {\Omega }_k(t)}}\right]}^2\right)\end{array}$$

(17)

It can be seen that when ${\eta }_{\Omega }<2/{\left({\displaystyle \frac{\partial e_k(t)}{\partial \Omega (t)}}\right)}^2$, $\Delta V_k(t)<0$. Then when the learning rates satisfy Equation (9), we have $0<V(t+1)<V(t)$, i.e., Theorem 1 is proved. □

4. Simulation Results and Discussion

Based on the BSM1 platform, the performance of the ERWNN controller is analyzed in this section. The set-points of DO and NO₃⁻ concentrations are chosen as fixed and variable values to measure the control performance more comprehensively. The parameters of ERWNN are determined by the trail-and-error method in the experiment. The control accuracy of different methods is measured by the maximal derivation from set point (Dev^max), integral of squared error (ISE) and integral of absolute error (IAE), as shown in the following formulas.

$${\mathrm{Dev}}^{\max }=\max \left\{\left|d(t)-y(t)\right|\right\}$$

(18)

$$\mathrm{ISE}={\displaystyle \frac{1}{N}}{\displaystyle \sum _{t=1}^{t_{\max }}{\left(d(t)-y(t)\right)}^2}$$

(19)

$$\mathrm{IAE}={\displaystyle \frac{1}{N}}{\displaystyle \sum _{t=1}^{t_{\max }}\left|d(t)-y(t)\right|}$$

(20)

where N is the total number of samples, d(t) and y(t) are the desired output and actual output. The smaller IAE, ISE and Devmax represent better control performance.

4.1. Constant Set-Points of DO and NO₃⁻ Concentration

In this subsection, the control performance of the ERWNN controller is fist measured at the constant set-points, i.e., $S_{DO,set}=2 \mathrm{mg}·{\mathrm{L}}^{-1}$ and $S_{NO,set}=1 \mathrm{mg}·{\mathrm{L}}^{-1}$.

Under the dry condition, the control errors of DO and NO₃⁻ concentrations are shown in Figure 5. Obviously, the control results of ERWNN are better than RWNN. Most of the control errors of ERWNN are contributed in the range of [−0.008, 0.008] and [−0.01, 0.01], while most of the control errors of RWNN are in the range of [−0.02, 0.02] and [−0.04, 0.02]. As shown in Figure 6 and Figure 7, the ERWNN can also obtain good control effects in rainy and rainstorm weather. The control performance of ERWNN on the DO concentration is better than that of RWNN. Moreover, the control results of ERWNN and RWNN are comparable in NO₃⁻, which is due to the coupling between DO and NO₃⁻.

To further measure the control performance of ERWNN under the dry condition, the control results are compared with the RWNN, NNOMC, RARFNNC and DRFNNC. Moreover, under two other operating conditions, the control performances are compared with the RFNN and RBFNN. The comparison results are summarized in

Table 2. Comparison of different control methods for fixed set-point.

Condition	Controller	DO			NO3−			Events
		IAE	ISE	Devmax	IAE	ISE	Devmax
Dry	ERWNN	5.97 × 10−4	1.62 × 10−6	0.0069	0.0016	3.01 × 10−5	0.0413	19,555
	RWNN	0.0016	3.08 × 10−5	0.0526	0.0034	7.20 × 10−4	0.0514	26,880
	NNOMC	0.0390	5.31 × 10−4	0.0725	0.0490	7.18 × 10−4	0.1630	26,880
	RARFNNC	0.0073	1.61 × 10−4	0.0104	0.0126	2.83 × 10−4	0.1050	26,880
	DRFNNC	0.0079	1.82 × 10−4	0.0154	0.0085	3.25 × 10−4	0.0176	26,880
Rainy	ERWNN	0.0031	4.64 × 10−5	0.042	0.0093	9.86 × 10−4	0.3847	22,354
	RWNN	0.0048	2.08 × 10−4	0.1434	0.0110	0.0014	0.4244	26,880
	RFNN	0.0184	2.01 × 10−4	0.2749	0.3950	5.33 × 10−2	1.9522	26,880
	RBFNN	0.0198	2.43 × 10−3	0.3742	0.4613	6.36 × 10−2	1.9978	26,880
Rainstorm	ERWNN	0.0040	1.21 × 10−4	0.0867	0.0068	4.78 × 10−4	0.1835	23,811
	RWNN	0.0074	3.73 × 10−4	0.1310	0.0108	0.0015	0.3975	26,880
	RFNN	0.0081	6.31 × 10−4	0.1637	0.3715	5.18 × 10−2	1.1077	26,880
	RBFNN	0.0096	7.41 × 10−4	0.2913	0.4303	3.96 × 10−2	1.1977	26,880

, which include the IAE, ISE and Dev^max of the DO and NO₃⁻ concentrations. Otherwise, the Events column in Table 2 refers to the number of executions of the controller. It can be seen that the control error of ERWNN is smaller than other methods under any of the operating conditions. In practical applications, the event-triggered mechanism offers a significant reduction in controller executions, improving overall control efficiency without compromising performance.

4.2. Changed Set-Points of DO and NO₃⁻ Concentrations

In this subsection, to further analyze the dynamic characteristics of the ERWNN controller, the set-points of DO and NO₃⁻ are set to step values as follows,

$$S_{DO,set}=\left\{\begin{array}{ll}1.8 \mathrm{mg}·{\mathrm{L}}^{-1}\hfill & 8\le t<9\hfill \\ 2.2 \mathrm{mg}·{\mathrm{L}}^{-1}& 9\le t<10\hfill \\ 2 \mathrm{mg}·{\mathrm{L}}^{-1}& otherwise\hfill \end{array}\right.$$

(21)

$$S_{NO,set}=\left\{\begin{array}{ll}0.9 \mathrm{mg}·{\mathrm{L}}^{-1}& 11\le t<12\hfill \\ 1.1 \mathrm{mg}·{\mathrm{L}}^{-1}& 12\le t<13\hfill \\ 1 \mathrm{mg}·{\mathrm{L}}^{-1}& otherwise\hfill \end{array}\right.$$

(22)

where $t=[0,14]$.

Under the dry condition, the control results of ERWNN and RWNN are shown in Figure 8. It can be seen that the proposed ERWNN can obtain good control results when the set-point is dynamically changed. Moreover, the simulation results of the DO concentration and NO₃⁻ concentration under the other two conditions are displayed in Figure 9 and Figure 10. Obviously, the actual value of DO and NO₃⁻ concentrations can track the set-points well.

The corresponding control errors under the rainy and rainstorm conditions are displayed in Figure 11 and Figure 12, respectively. As shown in Figure 11, the control errors of ERWNN are contributed in the range of [−0.02, 0.02] and [−0.09, 0.06], where the control errors of RWNN are in the range of [−0.07, 0.05] and [−0.1, 0.3]. Furthermore, under the rainstorm condition, most of the errors of ERWNN are contributed in [−0.02, 0.02] and [−0.05, 0.05], where most of the errors of RWNN are in [−0.05, 0.05] and [−0.1, 0.1]. It can be concluded that the control errors of ERWNN are smaller than that of RWNN, which indicates that the ERWNN can obtain better control performance than RWNN.

To further measure the dynamic performance of ERWNN, the control indicators are compared with the RWNN, PID, RFNN, and RBFNN, respectively. The comparison results are shown in

Table 3. Comparison of different control methods for changed set-points.

Condition	Controller	DO			NO3−			Events
		IAE	ISE	Devmax	IAE	ISE	Devmax
Dry	ERWNN	0.0045	1.78 × 10−4	0.0693	0.0040	8.17 × 10−5	0.0583	21,532
	RWNN	0.0063	2.48 × 10−4	0.1156	0.0075	2.21 × 10−4	0.0612	26,880
	PID	0.0127	2.38 × 10−3	0.1038	0.0271	4.90 × 10−3	0.2184	26,880
Rainy	ERWNN	0.0025	3.01 × 10−5	0.0275	0.0078	3.34 × 10−4	0.1335	23,399
	RWNN	0.0065	1.81 × 10−4	0.0644	0.0083	6.57 × 10−4	0.3335	26,880
	RFNN	0.0084	5.64 × 10−4	0.1778	0.3585	4.69 × 10−2	1.6809	26,880
	RBFNN	0.0751	1.71 × 10−2	0.1714	0.0347	1.84 × 10−2	0.9457	26,880
Rainstorm	ERWNN	0.004	1.35 × 10−4	0.0764	0.0110	0.0028	0.3696	24,313
	RWNN	0.0052	1.98 × 10−4	0.0959	0.0110	0.0029	0.5759	26,880
	RFNN	0.0081	5.66 × 10−4	0.0921	0.4189	4.24 × 10−2	1.2759	26,880
	RBFNN	0.0141	8.31 × 10−3	0.1204	0.4516	8.32 × 10−2	1.9388	26,880

. It can be seen that the ERWNN can ensure the control accuracy while reducing the number of executions in the three operating conditions. Among them, the control error of ERWNN is minimized under both dry and rainy conditions. Moreover, the control performance of the DO concentration is better than other methods under the rainstorm condition, while the control error of the NO₃⁻ concentration is about the same as that of RWNN. It can be concluded that the ERWNN can also achieve better control performance than other control methods when the set-points change.

Experimental results demonstrate that the ERWNN achieves superior control performance under both fixed and varied set-points. This effectiveness stems from three key design elements. First, the incorporation of wavelet functions enhances the nonlinear approximation capability of the model, which is crucial for handling the inherent nonlinearity of WWTPs. Second, the recurrent structure enables the network to dynamically adapt to process changes. Furthermore, the event-triggered mechanism reduces unnecessary controller updates, thereby improving control efficiency without compromising accuracy.

5. Conclusions

In this paper, the event-triggered recurrent wavelet neural network (ERWNN) control method is proposed to address the computational cost issue in controller execution. First, the event-triggered mechanism is designed to reduce the computational burden. Then, a RWNN controller is introduced to simultaneously regulate the concentrations of DO and NO₃⁻. Based on this, the stability of the closed-loop system is analyzed using the Lyapunov stability theory. Finally, the control performance of ERWNN is evaluated through two sets of experiments on the BSM1 platform. Experimental results demonstrate that the proposed method significantly reduces the number of controller executions while maintaining high control accuracy, effectively balancing computational efficiency and control performance.