Sage Journals: Discover world-class research

Abstract

In wastewater treatment processes, the concentration of dissolved oxygen affects the performance of wastewater treatment directly. It is one of the key factors that determines effluent quality of the wastewater treatment. However, a simple closed-loop control has a high-energy consumption, and it cannot guarantee the effluent quality due to large perturbations in wastewater treatment plants, such as the influent rate, the temperature, and the complex biochemical reactions. In this paper, a new disturbance rejection controller is designed to address those perturbations. Dynamics of dissolved oxygen is transformed into a controllable canonical form. Discrepancy between the dissolved oxygen dynamics and the controllable canonical form is estimated by a disturbance observer and compensated by a control law. Stability and the bound of tracking errors are obtained. Finally, numerical results on the benchmark simulation model number 1 are presented to confirm the proposed method.

Keywords

Wastewater treatment processes dissolved oxygen disturbance rejection benchmark simulation model number 1

Introduction

Shortage of the fresh water and increasing demand on it are a prominent contradiction. Therefore, the demand for reclaimed water, especially for the water-lacking areas, is increasing remarkably.¹ Wastewater treatment attracts extensive attention from all over the world.² Various kinds of new techniques have been proposed.³ However, a wastewater treatment process is not only a biochemical reaction. Actually, automatic control plays a vital role in maintaining the effluent quality and improving efficiency of the treatments.⁴ Simultaneously, it should be pointed out that control and optimization of a wastewater treatment is a challenge, due to the complexity of a biochemical reaction, diversity of the time constants involved in a biological process, and the variability of the influent patterns.⁵

To address those challenges in wastewater treatment plants (WWTPs), numerous control strategies have been put forward over the past years. Typical methods include proportional–integral–derivative (PID) control,^6,7 multi-objective optimal control,⁸ multivariable control,^9–11 and nonlinear model predictive control (NMPC).^12–14 The PID control, a commonly used approach in industrial processes, is also popular in wastewater treatment processes. However, nonlinearities and disturbances greatly degrade the performance of PID.⁶ Multivariable control⁹ can get satisfied effects. However, it largely depends on the accuracy of a neural network model, and the structure of neural network is determined by trial and error.¹⁵ NMPC is designed in wastewater treatments for its advantages. For example, it simply integrates constraints into an optimization problem.⁶ However, NMPC is a data-driven control, whose robustness may be deteriorated owing to uncertainties in the WWTPs.¹⁶

Actually, if nonlinearities, uncertainties, and external disturbances—such as coupling, influent temperature, influent rate, and constituents—are considered as time-varying disturbance, desired performance can be obtained by estimating and eliminating disturbance by an advanced control algorithm. Starting from this idea, a wide variety of disturbance rejection approaches have been proposed, such as active disturbance rejection control (ADRC)^17–19 and disturbance observer–based control (DOBC).^20,21 For successful applications in various industrial sectors, disturbance rejection control has become a hot topic in recent years. In this paper, we also focus on the disturbance rejection control and its application on the WWTPs.

For DOBC approach, internal disturbance from processes definitely degrades system performance. Although an extended state observer (ESO) designed in ADRC is able to estimate the internal and external disturbances, an ESO still has steady-state estimation errors for a time-varying disturbance. Motivated by improving system performance, a new disturbance rejection control (NDRC) is designed to achieve satisfied dissolved oxygen (DO) concentration in WWTPs.

In NDRC, a controllable canonical form is taken as the standard dynamics. Any discrepancy between the standard form and system dynamics is considered as the generalized disturbance. Observers designed in NDRC are able to estimate the generalized disturbance effectively. Rather than depending on a concrete model of a WWTP, estimating and eliminating the disturbance in real time make the DO concentration control more reliable. In other words, by choosing proper tunable parameters of the baseline controller and the observers, the NDRC can get satisfied level of the DO concentration. In addition, clear physical explanations of those parameters are also helpful for design and use of NDRC.

The rest of this paper is organized as follows. In section “System description and problem statements,” the dynamics and control problems of the WWTPs are described. In section “Disturbance rejection control,” a disturbance rejection control, stability, and its tracking error are presented. In section “Numerical studies,” simulation results are provided to demonstrate the NDRC. Finally, conclusions are drawn in section “Conclusion.”

System description and problem statements

System description

In order to evaluate a control approach in a WWTP objectively, a benchmark simulation model number 1 (BSM1) has been developed by the International Water Association (IWA), and the layout of the BSM1 is presented in Figure 1.²²

Figure 1.

Layout of the BSM1.

The BSM1 can be divided into two parts. One is a biochemical reactor, which is composed of a five-compartment activated sludge reactor consisting of two anoxic tanks (V₁ = V₂ = 1000 m³) followed by three aerobic tanks (V₃ = V₄ = V₅ = 1333 m³). The other part is a secondary clarifier (V = 6000 m³). It is divided into 10 layers evenly. Top layer outputs the water that satisfies the discharge standard, and the bottom layer outflows the waste.

The DO concentration in the fifth tank directly affects the effluent quality, and it is critical to regulate it at an appropriate level. Dynamics of the DO concentration in the fifth tank is²²

\frac{d S_{O, 5}}{d t} = \frac{1}{V_{5}} (Q_{4} S_{O, 4} + r_{Z, 5} + K_{L a 5} V_{5} (S_{O}^{*} - S_{O, 5}) - Q_{5} S_{O, 5})

(1)

where Q₄ is the effluent flow rate of the fourth biological reactor, S_O,₄ denotes the DO concentration of the fourth biological reactor, K_La₅ is the oxygen transfer coefficient of the fifth biological reactor, $S_{O}^{*}$ is the saturation concentration of DO (8 g m⁻³), Q₅ is the effluent flow rate of the fifth biological reactor, S_O,₅ is the DO concentration of the fifth biological reactor, and r_Z,₅ is the appropriate conversion rate of the fifth biological reactor²³

r_{Z, 5} = - \frac{1 - Y_{H}}{Y_{H}} ρ_{1} - \frac{4.57 - Y_{A}}{Y_{A}} ρ_{3}

(2)

ρ_{1} = μ_{H} (\frac{S_{S}}{K_{S} + S_{S}}) (\frac{S_{O}}{K_{O, H} + S_{O}}) X_{B, H}

(3)

ρ_{3} = μ_{A} (\frac{S_{N H}}{K_{N H} + S_{N H}}) (\frac{S_{O}}{K_{O, A} + S_{O}}) X_{B, A}

(4)

where X_B,A and X_B,H represent the active autotrophic biomass and the active heterotrophic biomass of the fifth biological reactor, respectively; S_S is the readily biodegradable substrate of the fifth biological reactor; and S_NH denotes the NH₄⁺ and NH₃ nitrogen of the fifth biological reactor. The other parameters are listed in Tables 1 and 2.²³

Table 1.

Stoichiometric parameters

Parameters	Units	Values
Y_A	g cell COD formed (g N oxidized)⁻¹	0.24
Y_H	g cell COD formed (g COD oxidized)⁻¹	0.67

COD: Chemical Oxygen Demand

Table 2.

Kinetic parameters.

Parameters	Units	Values
μ_H	d ⁻¹	4.0
μ_A	d ⁻¹	0.5
K_S	gCOD m⁻³	10.0
K_O,H	g(-COD) m⁻³	0.2
K_NH	gNH ₃-N m⁻³	1.0
K_O,A	g(-COD) m⁻³	0.4

Problem statements

The DO concentration in a WWTP, to a great extent, determines the effluent quality. From equation (1), it can be seen that DO concentration is affected by numerous factors, such as the flow rate, constituents of the wastewater, biological activities, and nitrate concentration. However, in practice, those factors are usually unknown, nonlinear, and strongly coupled. Therefore, how to design and realize an effective control on a WWTP is a big challenge. Therefore, the control objective is to maintain the DO concentration in the fifth tank at an ideal level, even if uncertainties—such as the flow rate, constituents of the wastewater, and biological activities—exist.

Disturbance rejection control

In order to address the challenge, an NDRC approach is designed.

Consider a general n-order nonlinear system

{\begin{matrix} {\overset{\cdot}{x}}_{i} = x_{i + 1} \\ {\overset{\cdot}{x}}_{n} = f (x, t) + d (t) + u (t) \\ y = x_{1} \end{matrix}

(5)

where $x = [x_{1}, x_{2}, \dots, x_{n}] \in R^{n}$ , $f (x, t) \in R$ , $d (t) \in R$ , $u (t) \in R$ , and $y \in R$ . f(x, t) represents the unknown nonlinear dynamics. It includes internal uncertainties and unmodeled dynamics. d(t) is the unknown external disturbance, u(t) is the control input, and y is the system output.

Let

\begin{matrix} A_{x} = [\begin{matrix} 0 & 1 & 0 & \dots & 0 & 0 \\ 0 & 0 & 1 & \dots & 0 & 0 \\ 0 & 0 & 0 & \dots & 0 & 0 \\ ⋮ & ⋮ & ⋮ & ⋱ & ⋮ & ⋮ \\ 0 & 0 & 0 & \dots & 0 & 1 \\ 0 & 0 & 0 & \dots & 0 & 0 \end{matrix}] \in R^{n \times n}, B = [\begin{matrix} 0 \\ 0 \\ 0 \\ ⋮ \\ 0 \\ 1 \end{matrix}] \in R^{n} \\ C^{T} = [\begin{matrix} 1 \\ 0 \\ 0 \\ ⋮ \\ 0 \\ 0 \end{matrix}] \in R^{n}, and D = f (x, t) + d (t) \in R \end{matrix}

then system (5) can be rewritten as

{\begin{matrix} \overset{\cdot}{x} = A_{x} x + B (u + D) \\ y = Cx \end{matrix}

(6)

By solving equation (6), we have

y = C \exp (A_{x} t) x (0) + C \int_{0}^{t} \exp (A_{x} (t - τ)) B (u + D) (τ) d τ

(7)

Clearly, the system output can be decoupled from both the internal and external disturbances, if u+D=0.

Control law design

In this paper, a disturbance rejection control law u is designed as²⁴

{\begin{matrix} u = u_{0} - \hat{D} \\ u_{0} = - a^{T} \hat{x} + a^{T} y_{r} \end{matrix}

(8)

where u₀ is designed to stabilize the system and track the reference; $\hat{D}$ denotes the output of a disturbance observer, which is utilized to estimate the unknown dynamics and external disturbance. $a = [a_{n}, a_{n - 1}, \dots, a_{1}]^{T} \in R^{n}$ is the parameter vector, $\hat{x} = [{\hat{x}}_{1}, {\hat{x}}_{2}, \dots, {\hat{x}}_{n}]^{T} \in R^{n}$ is the estimated state, and $y_{r} = [y_{r}, {\overset{\cdot}{y}}_{r}, \dots, {y_{r}}^{(n - 1)}]^{T} \in R^{n}$ is the vector consisted of a reference input y_r and its derivatives.

Remark 1

When substituting equation (8) into (6), we can get a controllable canonical form (9), and A = A_x − Ba ^T is the system matrix of the controllable canonical form. In other words, disturbance rejection control law (8) is used to linearize the nonlinear uncertain system (6) into a linear time invariant (LTI) system

{\begin{matrix} \overset{\cdot}{x} = Ax + B U \\ y = Cx \end{matrix}

(9)

where $U = a^{T} \tilde{x} + a^{T} y_{r} + \tilde{D} \in R$ , $\tilde{x} = x - \hat{x} \in R^{n}$ , $\tilde{D} = D - \hat{D} \in R$ , and

\begin{matrix} A = A_{x} - B a^{T} \\ = [\begin{matrix} 0 & 1 & 0 & \dots & 0 & 0 \\ 0 & 0 & 1 & \dots & 0 & 0 \\ 0 & 0 & 0 & \dots & 0 & 0 \\ ⋮ & ⋮ & ⋮ & ⋱ & ⋮ & ⋮ \\ 0 & 0 & 0 & \dots & 0 & 1 \\ - a_{n} & - a_{n - 1} & - a_{n - 2} & \dots & - a_{2} & - a_{1} \end{matrix}] \in R^{n \times n} \end{matrix}

The state observer can be designed as

{\begin{matrix} \overset{\cdot}{\hat{x}} = A_{x} \hat{x} + B u_{0} + L (y - \hat{y}) \\ \hat{y} = C \hat{x} \end{matrix}

(10)

where $L = [l_{1}, l_{2}, \dots, l_{n}]^{T} \in R^{n}$ is the parameter vector.

Meanwhile, the disturbance observer can be selected as²⁵

{\begin{matrix} \hat{D} = ξ + p (\hat{x}) \\ \overset{\cdot}{ξ} = - l_{n + 1} ξ - l_{n + 1} [p (\hat{x}) + u] \end{matrix}

(11)

where $p (\hat{x}) = l_{n + 1} \hat{x}$ , in which $l_{n + 1}$ is the tunable parameter.

Remark 2

The disturbance defined in NDRC is the difference between practical system dynamics and the controllable canonical form. The estimation error is bounded, and the control scheme of an NDRC system is shown in Figure 2.²⁴

Figure 2.

Control scheme of an NDRC system.

Stability and tracking error

Definition

A system is considered to be input-to-state stable if there are a class KL function β and a class K function γ so that for any initial state x(t₀) and any bounded input u(t), the solution x(t) exists for all t > t₀ and satisfies²⁶

‖ x (t) ‖ \leq β (‖ x (t_{0}) ‖, t - t_{0}) + γ (sup_{t_{0} \leq τ \leq t} ‖ u (τ) ‖)

Consequently, we have Lemma 1.

Lemma 1

If we choose a proper parameter vector a and $[L, l_{n + 1}]$ , the estimation error is bounded, and A is Hurwitz, then the closed-loop system (9) is input-to-state stable.²⁴

According to Lemma 1, closed-loop system (9) is input-to-state stable. Next, the tracking error is analyzed on the basis of an input-to-state stable closed-loop NDRC system. For the input-to-state stable system (9), let tracking error be e = y_r − y, we have

\begin{matrix} e = e_{1} = y_{r} - x_{1}, e_{2} = {\overset{\cdot}{y}}_{r} - {\overset{\cdot}{x}}_{1}, e_{3} = {\overset{\cdot\cdot}{y}}_{r} - {\overset{\cdot\cdot}{x}}_{1}, \dots, e_{n} \\ = {y_{r}}^{(n - 1)} - {x_{1}}^{(n - 1)} \end{matrix}

Thus

{\begin{matrix} {\overset{\cdot}{e}}_{1} = e_{2} \\ {\overset{\cdot}{e}}_{2} = e_{3} \\ \dots \\ {\overset{\cdot}{e}}_{n - 1} = e_{n} \\ {\overset{\cdot}{e}}_{n} = {y_{r}}^{(n)} + a^{T} \hat{x} - a^{T} y_{r} - \tilde{D} \end{matrix}

(12)

Here

\begin{matrix} {\overset{\cdot}{e}}_{n} = {y_{r}}^{(n)} + a^{T} \hat{x} - a^{T} y_{r} - \tilde{D} \\ = {y_{r}}^{(n)} + a_{n} {\hat{x}}_{1} + a_{n - 1} {\hat{x}}_{2} + \dots + a_{1} {\hat{x}}_{n} \\ - a_{n} y_{r} - \dots - a_{1} {y_{r}}^{n - 1} - \tilde{D} \end{matrix}

(13)

For $\tilde{x} = x - \hat{x}$ , then $\hat{x} = x - \tilde{x}$ .

Therefore

\begin{matrix} {\hat{x}}_{1} = y_{r} - e_{1} - {\tilde{x}}_{1}, {\hat{x}}_{2} = {\overset{\cdot}{y}}_{r} - e_{2} - {\tilde{x}}_{2}, \dots, {\hat{x}}_{n} \\ = {y_{r}}^{(n - 1)} - e_{n} - {\tilde{x}}_{n} \end{matrix}

(14)

Substituting equation (14) into (13), we obtain

\begin{matrix} {\overset{\cdot}{e}}_{n} = - a_{n} e_{1} - a_{n - 1} e_{2} - \dots - a_{1} e_{n} - a_{n} {\tilde{x}}_{1} \\ - a_{n - 1} {\tilde{x}}_{2} - \dots - a_{1} {\tilde{x}}_{n} - \tilde{D} + {y_{r}}^{(n)} \end{matrix}

(15)

If we define $e = [e_{1}, e_{2}, \dots, e_{n}]^{T} \in R^{n}$ and $\tilde{x} = [{\tilde{x}}_{1}, {\tilde{x}}_{2}, \dots, {\tilde{x}}_{n}]^{T} \in R^{n}$ , equation (15) can be rewritten as

{\overset{\cdot}{e}}_{n} = - a^{T} e - a^{T} \tilde{x} + {y_{r}}^{(n)} - \tilde{D}

(16)

From equations (12) and (16), we have

{\begin{matrix} {\overset{\cdot}{e}}_{1} = e_{2} \\ {\overset{\cdot}{e}}_{2} = e_{3} \\ \dots \\ {\overset{\cdot}{e}}_{n - 1} = e_{n} \\ {\overset{\cdot}{e}}_{n} = - a^{T} e - a^{T} \tilde{x} + {y_{r}}^{(n)} - \tilde{D} \end{matrix}

(17)

Equation (17) represents the closed-loop tracking error system, and it can be rewritten as

\overset{\cdot}{e} = Ae + ε (t)

(18)

where $ε (t) = [0, 0, \dots, 0, - a^{T} \tilde{x} + {y_{r}}^{(n)} - \tilde{D}]^{T} \in R^{n}$ .

Since $‖ [\begin{matrix} \tilde{x} \\ \tilde{D} \end{matrix}] ‖ \leq δ_{z}$ and ${y_{r}}^{(n)}$ is also bounded, without loss of generality, $‖ ε (t) ‖ \leq δ$ is assumed. Here, δ is a constant.

Lemma 2

For the tracking error system (18), if $λ_{max} (A) < - δ / ‖ e ‖_{2}$ , tracking error $‖ e ‖ = O (1)$ , then the bound is²⁴

- \frac{δ}{λ_{max} (A)} \sqrt{\frac{λ_{max} (P)}{λ_{min} (P)}}

where $P = (T^{- 1})^{T} (T^{- 1})$

T = [\begin{matrix} 1 & 1 & 1 & \dots & 1 \\ - λ_{1} & - λ_{2} & - λ_{3} & \dots & - λ_{2} \\ {(- λ_{1})}^{2} & {(- λ_{2})}^{2} & {(- λ_{3})}^{2} & \dots & {(- λ_{3})}^{2} \\ ⋮ & ⋮ & ⋮ & ⋱ & ⋮ \\ {(- λ_{1})}^{n - 1} & {(- λ_{2})}^{n - 1} & {(- λ_{3})}^{n - 1} & \dots & {(- λ_{n})}^{n - 1} \end{matrix}]

and −λ₁, −λ₂,&, −λ_n are the eigenvalues of matrix A .

From Lemma 2, we can see that, for the input-to-state stable closed-loop system (9), its tracking error is bounded. In addition, the following remarks have been made.

Remark 3

If controller parameters are selected properly, −λ₁, −λ₂,&, −λ_n are distinct. The system matrix A can be transformed to a diagonal matrix by a Vandermonde matrix T .

Remark 4

According to Wei,²⁴ the parameters L can be designed according to bandwidth of the state observer. For second-order systems, $L = [l_{1}, l_{2}]^{T} = [2 ω_{o}, {ω_{o}}^{2}]^{T}$ , and for third-order systems, $L = [l_{1}, l_{2}, l_{3}]^{T} = [3 ω_{o}, 3 {ω_{o}}^{2}, {ω_{o}}^{3}]^{T}$ . Here, ω_o is the bandwidth of the state observer.

Numerical studies

In this section, numerical simulations are performed on the BSM1 in dry, rainy, and stormy days.

Controller design

Based on section “Disturbance rejection control,” a disturbance rejection control law can be designed as

{\begin{matrix} u = u_{0} - \hat{D} \\ u_{0} = - a \hat{x} + a y_{r} \end{matrix}

(19)

A state observer can be designed as

{\begin{matrix} \overset{\cdot}{\hat{x}} = u_{0} + l_{1} (y - \hat{y}) \\ \hat{y} = \hat{x} \end{matrix}

(20)

The disturbance observer can be selected as

{\begin{matrix} \hat{D} = ξ + p (\hat{x}) \\ \overset{\cdot}{ξ} = - l_{2} ξ - l_{2} [p (\hat{x}) + u] \end{matrix}

(21)

where $p (\hat{x}) = l_{2} \hat{x}$ .

Simulation design

Two cases are considered to demonstrate the approach. The first case is the constant tracking response, and the second case is the varying setting value response. Based on the responses of the two cases, the difference among proportional–integral (PI) control, self-organizing fuzzy control (SOFC), and NDRC can be seen clearly. In both cases, the parameters of NDRC are set to be l₁ = 10,000, l₂ = 300, and a₁ = 1500. The parameters of PI are taken the same as in Copp’s²³ work, that is, k_p = 25 and k_i = 12,500. Closed-loop performance is evaluated by the integral of the absolute error (IAE), the integral of the squared error (ISE), and the maximal deviation from the set point (DEV^max)²³

IAE = \int_{7}^{14} | e | dt

(22)

ISE = \int_{7}^{14} e^{2} dt

(23)

De v^{\max} = \max {| e |}

(24)

where the tracking error e = y_r − y.

Simulation results

In this section, simulation results have been presented.

Case 1: constant DO set value

In this case, the DO concentration is set to be 2 mg/L and the simulation time is 14 days. PI is commonly used in the process control, and the SOFC proposed by Qiao et al.²⁷ is also considered in BSM1. In order to evaluate the closed-loop performance, the DO concentrations regulated by NDRC, PI, and SOFC²⁷ are compared.

As shown in Figure 3, in dry weather, it is obvious that NDRC has a better tracking response than PI. From K_La₅ curves given in Figure 4, we can see clearly that, with similar control energy, NDRC is able to achieve much better tracking response. In other words, NDRC needs much lower energy than PI when they achieve similar tracking performance. Figure 5 illustrates that disturbance estimator is able to estimate disturbance precisely.

Figure 3.

Tracking responses in dry days.

Figure 4.

Control signals of PI and NDRC in dry days.

Figure 5.

Disturbance and its estimation (I).

Performance indexes in dry weather are listed in Table 3. Compared to SOFC,²⁷ there is a 55.6% decrease in IAE, DEV^max has dropped by 55.5%, and the ISE is reduced by 84.7%. In addition, by comparison with PI, performance is improved more significantly. IAE, DEV^max, and ISE fall by 96.8%, 29.0%, and 99.9%, respectively. The data given in Table 3 show that NDRC is the best method.

Table 3.

Comparisons of DO tracking performance in dry days.

Weather	Controllers	IAE	ISE	DEV^max(mg/L)
Dry	NDRC	0.0080	2.2826e−05	0.0087
	SOFC²⁷	0.0180	1.4900e−04	0.0200
	PI	0.2510	2.2071e−02	0.2606

DO: dissolved oxygen; IAE: integral of the absolute error; ISE: integral of the squared error; DEV^max: maximal deviation from the set point; NDRC: new disturbance rejection control; SOFC: self-organizing fuzzy control; PI: proportional–integral.

As to the effluent, NDRC also satisfies the requirements. For instance, NH is 2.4 mg/L, N_tot is 16.8 mg/L, total suspended solids (TSS) is 13.0 mg/L, 5-day biochemical oxygen demand (BOD₅) is 2.7 mg/L, and COD is 48.2 mg/L. Therefore, NDRC is effective in WWTPs.

For rainy and stormy conditions, simulations are also performed. Numerical results are shown in Figures 6 –11.

Figure 6.

Tracking responses in rainy days.

Figure 7.

Control signals of PI and NDRC in rainy days.

Figure 8.

Disturbance and its estimation (II).

Figure 9.

Tracking responses in stormy days.

Figure 10.

Control signals of PI and NDRC in stormy days.

Figure 11.

Disturbance and its estimation (III).

As shown in Figures 6 and 9, in rainy and stormy days, NDRC has superior tracking responses. K_La₅ curves given in Figures 7 and 10 describe that NDRC needs much less energy than PI when they achieve similar tracking performance. Figures 8 and 11 present that disturbance estimator is able to estimate disturbance precisely. Performance indexes in rainy and stormy days are displayed in Table 4.

Table 4.

Comparisons of DO tracking performance in rainy and stormy days.

Weathers	Controllers	IAE	ISE	DEV^max(mg/L)
Rain	NDRC	0.0069	1.7209e−05	0.0082
	SOFC²⁷	0.0310	7.2600e−04	0.0360
	PI	0.2174	1.6738e−02	0.2462
Storm	NDRC	0.0079	2.1315e−05	0.0087
	SOFC²⁷	0.0250	8.6300e−04	0.0970
	PI	0.2472	2.0677e−02	0.2606

As shown in Table 4, IAE, ISE, and DEV^max of NDRC are better than those of PI and SOFC in rainy and stormy days. Therefore, compared to both PI and SOFC, NDRC is the most suitable approach.

Case 2: varying DO set value

Furthermore, to validate NDRC, varying DO set points are taken into consideration. According to Copp,²³ data in rainy weather contain 1 week’s dry weather influent data and a long rainfall event during the second week. Stormy weather is a combination of dry weather with two stormy events. Therefore, in this case, rainy and stormy conditions are taken to test the closed-loop performance.

Varying DO set points are taken the same as those utilized in Qiao et al.’s²⁷ work, and tracking performance is shown in Figure 12. Quantitative comparisons have been listed in Table 5.

Table 5.

Comparisons of DO tracking performance in rainy days (varying set points).

Weather	Controllers	IAE (mg/L)	ISE (mg/L)
Rain	NDRC	0.0075	5.6019e−05
	SOFC²⁷	0.0220	2.8600e−04
	PI	0.2202	1.7300e−02

DO: dissolved oxygen; IAE: integral of the absolute error; ISE: integral of the squared error; NDRC: new disturbance rejection control; SOFC: self-organizing fuzzy control; PI: proportional–integral.

Figure 12.

Tracking responses in rainy days (varying set points).

The results presented in Figure 12 show that NDRC can track the varying set points much better than PI. Indexes shown in Table 5 reveal that, by comparison, NDRC is the best approach among the three approaches listed in Table 5. It means that NDRC has much stronger disturbance rejection ability, and it is able to achieve superior performance in this case.

In Qiao et al.’s²⁷ work, only the control results in rain weather are provided. In order to illustrate the tracking ability of NDRC, stormy weather is also considered. The simulation results are given in Figure 13.

Figure 13.

Tracking responses in stormy days (varying set points).

The results shown in Figure 13 demonstrate that, compared to PI, NDRC can achieve better performance for the varying set value tracking in stormy days. Performance indexes are provided in Table 6. From those values, similar conclusion can be drawn that, in stormy days, NDRC still has much stronger disturbance rejection ability than PI.

Table 6.

Comparisons of DO tracking performance in stormy days (varying set points).

Weather	Controllers	IAE (mg/L)	ISE (mg/L)
Storm	NDRC	0.0085	5.7551e−05
	PI	0.2508	2.1600e−02

DO: dissolved oxygen; IAE: integral of the absolute error; ISE: integral of the squared error; NDRC: new disturbance rejection control; PI: proportional–integral.

Discussion

Generally speaking, the DO concentration of a WWTP is difficult to control due to its nonlinear, uncertainties, and couplings. Observers utilized in this paper are capable of estimating system states and disturbance more accurately, and the control law designed is able to compensate the disturbance more effectively. NDRC, by comparison, has much stronger disturbance rejection ability, and it is able to improve the closed-loop performance greatly. In addition, compared to the SOFC proposed by Qiao,²⁷ adjustable parameters of NDRC are clearer in physical backgrounds, which is useful to select the tunable parameters, and it is easier to obtain a group of suitable parameters.

Conclusion

In this paper, an NDRC has been designed for DO concentration control in a WWTP. With the help of observers, the time-varying disturbance in a WWTP can be estimated, and the system dynamics can be approximately transformed into an LTI system with controllable canonical form. Based on a commonly accepted platform BSM1, PI, SOFC, and NDRC are simulated under the same conditions to get unbiased comparison results. Both constant tracking responses and varying set values’ responses under different weathers have been presented. From the simulation results, it is obvious that the NDRC is able to obtain more satisfactory performance for its advantage on disturbance estimation and rejection. However, it still needs to be verified in a real WWTP, and it is our future work.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the Key Program of Beijing Municipal Education Commission (grant no. KZ201810011012) and the National Natural Science Foundation of China (grant no. 61873005).

ORCID iD

Min Zuo

References

Mahjouri

Pourmand

A social choice-based methodology for treated wastewater reuse in urban and suburban areas. Environ Monit Assess 2017; 189(7): 325.

Shannon

Bohn

Elimelech

, et al. Science and technology for water purification in the coming decades. Nature 2008; 452(7185): 301–310.

Fito

Alemu

Microalgae–bacteria consortium treatment technology for municipal wastewater management. Nanotechnol Environ Eng 2019; 4: 4.

Olsson

ICA and me – a subjective review. Water Res 2012; 46(6): 1585–1624.

Silvana

Ramon

Mario

, et al. PI Dissolved Oxygen control in wastewater treatment plants for plantwide nitrogen removal efficiency. IFAC PapersOnLine 2018; 51(4): 450–455.

Iratni

Chang

Advances in control technologies for wastewater treatment processes: status, challenges, and perspectives. IEEE/CAA J Automatica Sinica 2019; 6(2): 337–363.

Qiao

Luo

. Robust PID controller design using genetic algorithm for wastewater treatment process. In: Advanced information management, communicates, electronic and automation control conference, Xi’an, China, 3–5 October 2016.

Rainier

Nicolas

Brahim

, et al. Multi-objective optimal control of small-size wastewater treatment plants. Chem Eng Res Des 2015; 102(3): 345–353.

Yamamoto

Multi-input multi-output model-free predictive control and its application to wastewater treatment. IEEJ T Electr Electr 2017; 12(5): 753–758.

10.

Qiao

Han

Neural network online modeling and controlling method for multi-variable control of wastewater treatment processes. Asian J Control 2014; 16(3): 1213–1223.

11.

Nejjari

Dahhou

Benhammou

, et al. Nonlinear multivariable adaptive control of an activated sludge wastewater treatment process. Int J Adapt Control 2015; 13(5): 347–365.

12.

Francisco

Skogestad

Vega

Model predictive control for the self-optimized operation in wastewater treatment plants: analysis of dynamic issues. Comput Chem Eng 2015; 82(2): 259–272.

13.

Sadeghassadi

Macnab

CJB

Gopaluni

, et al. Application of neural networks for optimal-setpoint design and MPC control in biological wastewater treatment. Comput Chem Eng 2018; 155: 150–160.

14.

Han

Zhang

Qiao

Data-based predictive control for wastewater treatment process. IEEE Access 2018; 6(2): 1498–1512.

15.

Zhang

Qiao

Multi-variable direct self-organizing fuzzy neural network control for wastewater treatment process. Asian J Control 2018; 22(1): 1–13.

16.

Han

Qiao

A self-organizing sliding-mode controller for wastewater treatment processes. IEEE T Contr Syst T 2018; 27: 1480–1491.

17.

Madonski

Nowicki

Herman

. Application of active disturbance rejection controller to water supply system. In: Proceedings of the 33rd Chinese control conference, Nanjing, China, 28–30 July 2014.

18.

Wei

Zuo

, et al. Control of dissolved oxygen for a wastewater treatment process by active disturbance rejection control approach. Control Theory Appl 2018; 35(1): 24–30.

19.

Madonski

Nowicki

Herman

Practical solution to positivity problem in water management systems – an ADRC approach. In: 2016 American control conference, Boston, MA, 6–8 July 2016.

20.

Gao

On the centrality of disturbance rejection in automatic control. ISA T 2014; 53(4): 850–857.

21.

Chen

Yang

Guo

, et al. Disturbance-observer-based control and related methods – an overview. IEEE T Ind Electron 2016; 63(2): 1083–1095.

22.

Qiao

Hou

Han

Optimal control for wastewater treatment process based on an adaptive multi-objective differential evolution algorithm. Neural Comput Appl 2019; 31(7): 2537–2550.

23.

Copp

The COST simulation benchmark: description and simulator manual. Luxembourg: Official Publications of the European Community, 2001.

24.

Wei

On disturbance rejection for a class of nonlinear systems. Complexity 2018; 2018: 1212534.

25.

Chen

Ballance

Gawthrop

, et al. A nonlinear disturbance observer for robotic manipulators. IEEE T Ind Electron 2000; 47(4): 932–938.

26.

Khalil

HK.

Nonlinear systems. 3rd ed. Upper Saddle River, NJ: Prentice Hall, 2002.

27.

Qiao

Zhang

Han

Self-organizing fuzzy control for dissolved oxygen concentration using fuzzy neural network. J Intell Fuzzy Syst 2016; 30(6): 3411–3422.

Disturbance rejection control for the dissolved oxygen in a wastewater treatment process

Abstract

Keywords

Introduction

System description and problem statements

System description

Problem statements

Disturbance rejection control

Control law design

Remark 1

Remark 2

Stability and tracking error

Definition

Lemma 1

Lemma 2

Remark 3

Remark 4

Numerical studies

Controller design

Simulation design

Simulation results

Case 1: constant DO set value

Case 2: varying DO set value

Discussion

Conclusion

Footnotes

Declaration of conflicting interests

Funding

ORCID iD

References