A sewage treatment process sliding mode control method based on predetermined time performance

Abstract

This invention discloses a sliding mode control method for wastewater treatment processes based on predetermined time performance, belonging to the field of wastewater treatment and nonlinear control technology. This invention is the first to combine second-order integral sliding mode control with a second-order command filter and apply it to the control system of a wastewater treatment plant (WWTP). It considers the requirement that the control coefficient must be greater than 0, and the control effect is superior to existing results. To ensure the timely operation of the wastewater treatment plant, predetermined time performance is used to keep the overall control performance within a specified range and achieve faster convergence. An improved integral sliding surface is constructed, and a novel adaptive controller is designed, achieving stable control of the wastewater treatment plant under three different weather conditions.

Description

Technical Field

[0001] This invention relates to the field of wastewater treatment and nonlinear control technology, specifically to a sliding mode control method for wastewater treatment processes based on predetermined time performance. Background Technology

[0002] With the acceleration of urbanization, sewage discharge is increasing year by year, and urban water pollution problems are becoming increasingly prominent. To improve the resource utilization rate of sewage and achieve a virtuous cycle of water resources, countries are actively building urban sewage treatment plants. The sewage treatment process mainly includes multiple stages such as grit chambers, primary sedimentation tanks, biological reaction tanks, secondary sedimentation tanks, and filters. Among them, the biological reaction tank involves complex biochemical reactions, characterized by time delays and nonlinearity, posing a significant challenge to its stable control.

[0003] To achieve stable control of wastewater treatment processes, researchers have proposed various control methods. For example, one study proposed a high-precision control method for wastewater treatment processes based on artificial intelligence asymmetric constraints. This method can solve the actuator saturation problem and uses fuzzy neural networks to estimate uncertainties. Another study designed a self-organizing fuzzy terminal sliding mode control strategy to address the need for accurate tracking of a preset dissolved oxygen concentration within a finite time. This strategy constructs a terminal sliding mode controller to formulate the control law, ensuring finite-time tracking of the desired dissolved oxygen concentration. However, none of these control methods consider the impact of time delays on the control process. In actual wastewater treatment systems, push flow phenomena exist, which cause the control process to exhibit time delay characteristics, leading to a decrease in the original system's control performance and even affecting the stability of the control process.

[0004] Wastewater treatment dynamic systems play an increasingly crucial role in environmental protection and resource recycling, helping to alleviate related challenges faced by modern urban development. However, these systems are complex nonlinear systems with continuous reaction processes such as nitrification and denitrification, which poses a significant obstacle to the implementation of control strategies in practical dynamic treatment scenarios. To address this issue, constructing a benchmark simulation model for control system verification is an effective approach. Currently, the mainstream benchmark simulation models include Benchmark Simulation Model 1 (BSM1) and Benchmark Simulation Model 2 (BSM2), with BSM1 having the widest application. Some studies have designed a sliding mode controller with adaptive switching gain coefficients based on BSM1, achieving stable control of dissolved oxygen and nitrate nitrogen affected by time delays. This research abstracts BSM1 as a nonlinear system to simplify the design process of the sliding mode controller. However, this research overlooks a crucial condition: in the simplified nonlinear system, the control coefficients must be strictly greater than 0. Therefore, by adding constraints to the control coefficients and deriving reasonable boundary information, and then using the new boundary information to design a suitable control law and sliding surface, the system can be stabilized while eliminating external disturbances. This has become a pressing technical problem, and it is one of the research motivations of this invention.

[0005] It is important to note that wastewater treatment process control is highly time-sensitive. Some existing research focuses on ensuring system stability through the design of suitable controllers, but neglects the core issue of whether the controller can ensure system stability within a finite time frame. To address this deficiency, researchers have proposed various improvement methods applicable to wastewater treatment systems. For example, some studies have designed finite-time convergence observers to estimate unmeasurable variables such as biomass concentration in bioreactors and sedimentation tanks, ensuring good estimation performance. Another study proposed a sulfate reduction observer based on a mass balance model of a stirred intermittent anaerobic bioreactor. This observer has a simple structure, and by selecting appropriate gain, it can offset upper bound disturbances under different kinetic mechanisms, achieving the required finite-time convergence effect. Yet another study designed a proportional-integral controller for the finite-time extended state observer of dissolved oxygen to maintain the dissolved oxygen concentration at the desired level. However, finite-time control theory has an inherent limitation: it requires initial state information of the system, which greatly restricts its application in scenarios where the initial state is unknown. To overcome this limitation, pre-set time control theory has been introduced into related research fields. The core advantage of this theory lies in the fact that the system convergence time is not only independent of the initial state, but can also be used as a definite, user-defined parameter. Furthermore, within the framework of the preset-time control theory, the calculation of the upper bound of the system convergence time is simpler, and the conservatism of the results is lower than that of the finite-time control theory. Although the preset-time control theory has been widely applied in the field of nonlinear system control, its application in nonlinear control of wastewater treatment remains very rare, which is another research motivation for this invention.

[0006] To address this, a sliding mode control method for wastewater treatment processes based on predetermined time performance is proposed. Summary of the Invention

[0007] The technical problem to be solved by this invention is: how to improve the stability and convergence speed of a wastewater treatment system under time delay, nonlinearity and external disturbance by using an adaptive sliding mode control method and a predetermined time theory, and provides a sliding mode control method for wastewater treatment process based on predetermined time performance.

[0008] The present invention solves the above-mentioned technical problems through the following technical solution, and the present invention includes the following steps:

[0009] S1: Transform the No. 1 benchmark simulation model of wastewater treatment into a nonlinear system mathematical model, and use a fuzzy logic system to approximate the unmodeled dynamics of the nonlinear system mathematical model.

[0010] S2: Design a second-order integral sliding surface, and design a controller and update laws for weights, disturbances and approximation error upper bounds based on Lyapunov function analysis;

[0011] S3: Using Lyapunov's stability theorem and combining the controller and second-order integral sliding surface designed in step S2, give sufficient conditions to ensure the stability of the closed-loop system.

[0012] S4: By providing parameters, the controller is used to control wastewater treatment under different weather conditions.

[0013] Furthermore, in step S1, the specific processing procedure is as follows:

[0014] S11: Consider the first baseline simulation model as follows:

[0015] ;

[0016] ;

[0017] in, and These represent the hysteresis time constants for dissolved oxygen and nitrate nitrogen, respectively. This indicates that the fourth chamber of the biochemical reaction vessel is in Dissolved oxygen concentration at any given time; This indicates that the first chamber of the biochemical reaction vessel is in The nitrate nitrogen concentration at time; This represents the oxygen transfer coefficient of the fifth chamber at the current moment; Indicates component concentration The reaction rate; Indicates component concentration The reaction rate; , and It's traffic; Indicates internal circulation flow; This indicates the flow rate from the first compartment to the second compartment;

[0018] S12: Order By transforming the first-line baseline simulation model, the mathematical model of the nonlinear system is obtained as follows:

[0019] ;

[0020] ;

[0021] in:

[0022] and , and The nonlinear function and control coefficients are known. and It is an unmodeled dynamic; and It is a bounded and unknown time-varying disturbance caused by the environment;

[0023] S13: The simplified mathematical model in step S12 is expressed as follows:

[0024] ;

[0025] Unmodeled dynamics The fuzzy logic system approximates it as follows:

[0026] ;

[0027] in, These are basis functions; These are weighting coefficients; Is it satisfied by the inequality The approximate error and It is a constant.

[0028] Furthermore, in step S2, the specific processing procedure is as follows:

[0029] S21: To achieve the control objectives of the wastewater treatment system, design the following predetermined time function:

[0030] ;

[0031] Its differential form is as follows:

[0032] ;

[0033] in, as well as It is a constant and ;

[0034] The tracking error is defined as ,in, Let the expected value be denoted as follows, and define the error conversion formula as follows:

[0035] ;

[0036] in, It is unconstrained tracking error; It is a transformation function;

[0037] S22: When there is no constraint tracking error When it is bounded, we get ;when At that time, In other words, obtain the state. Then, calculate the state. The largest interval to which it belongs is as follows:

[0038] ;

[0039] For dissolved oxygen concentration, the control coefficient It should be greater than zero, that is ,get Thus we obtain ,Right now:

[0040] ;

[0041] Therefore, we get:

[0042] ;

[0043] For nitrate nitrogen concentration, the control coefficient It should be less than zero, that is Then, we can obtain Therefore, we can conclude that:

[0044] ;

[0045] Therefore, we get:

[0046] ;

[0047] Based on the error transformation formula and transformation function in step S21, we obtain:

[0048] ;

[0049] as well as:

[0050] ;

[0051] in:

[0052] ,when hour, , representing variables It is reversible;

[0053] S23: To enhance the robustness of the controller, the following sliding mode variables are designed:

[0054] ;

[0055] in, as well as It is a constant;

[0056] Due to variables Including unknown, unmodeled dynamics and time-varying disturbances, the following second-order command filter is used to obtain the variables. :

[0057] ;

[0058] in, as well as It is a constant;

[0059] Then, the sliding surface is modified as follows:

[0060] ;

[0061] Combining the unmodeled dynamics of the fuzzy logic system approximation in step S13 The derivative of the unconstrained tracking error in step S22 and the second-order command filter yield:

[0062] ;

[0063] in, It is a known function;

[0064] S24: To verify the effectiveness of the constructed second-order integral sliding surface, the following Lyapunov function is constructed:

[0065] ;

[0066] Its differential calculation is as follows:

[0067] ;

[0068] in, It is the weight estimation error; This is the perturbation estimation error. Furthermore, based on Young's inequality, we obtain:

[0069] ;

[0070] Further results were obtained:

[0071] ;

[0072] in, It is the estimation error of the upper limit of the approximation error;

[0073] S25: Design the following controller structure:

[0074] ;

[0075] Therefore, we get:

[0076] ;

[0077] in, It is a constant;

[0078] The update laws for designing weights, perturbations, and upper bounds of approximation errors are as follows:

[0079] ;

[0080] ;

[0081] ;

[0082] in, and It is a positive definite symmetric matrix. It is a constant. .

[0083] Furthermore, in step S21, the transformation function Specifically as follows:

[0084] ;

[0085] in, , ; , .

[0086] Furthermore, in step S3, the specific processing procedure is as follows:

[0087] S31: Based on the simplified model and fuzzy logic system in step S13, the predetermined time function in step S21, the state constraints and unconstrained tracking error in step S22, the second-order command filter and sliding surface in step S23, and the controller, weights, perturbation, and update law of the approximate error upper bound in step S25, for i∈{1,2}, select the following parameters:

[0088] Unconstrained tracking error It is bounded, and the tracking error Constrained within the prescribed scope; furthermore, when At that time, ;

[0089] S32: The specific proof is as follows:

[0090] The Lyapunov function is chosen as follows:

[0091] ;

[0092] Combining the update law of the controller, weights, perturbation, and upper bound of the approximation error in step S25, we obtain:

[0093] ;

[0094] Therefore, according to Young's inequality, we get:

[0095] ;

[0096] Through derivation, the following inequality holds:

[0097] ;

[0098] in:

[0099] ;

[0100] ;

[0101] Furthermore, it was learned that the sliding mode variable and estimation error Both are bounded; as can be seen from the second-order command filter, the approximation error is... Bounded; based on the sliding surface structure in step S23, the unconstrained tracking error is known. Bounded tracking error Constrained within a given range, ,Right now ;when Sometimes, .

[0102] The present invention has the following advantages over the prior art:

[0103] 1. This paper presents the first application of a combination of second-order integral sliding mode control and a second-order command filter to the control system of a wastewater treatment plant (WWTP). The method takes into account the requirement that the control coefficients must be greater than 0, and its control performance is superior to existing results.

[0104] 2. To ensure the timely operation of the wastewater treatment plant, a predetermined time performance is adopted to keep the overall control performance within a specified range and achieve a faster convergence speed.

[0105] 3. An improved integral sliding surface was constructed, and a novel adaptive controller was designed to achieve stable control of the wastewater treatment plant under three different meteorological conditions. Attached Figure Description

[0106] Figure 1 This is a schematic diagram of the adaptive fuzzy second-order integral sliding mode control method in the wastewater treatment process according to an embodiment of the present invention;

[0107] Figure 2 These are the dissolved oxygen concentration, nitrate nitrogen concentration and their reference values ​​in dry weather in the embodiments of the present invention, wherein (a) is the dissolved oxygen concentration and its reference value, and (b) is the nitrate nitrogen concentration and its reference value;

[0108] Figure 3 These are the tracking errors and boundaries of dissolved oxygen concentration and nitrate nitrogen concentration in dry weather in the embodiments of the present invention, wherein (a) is the tracking error and boundary of dissolved oxygen concentration, and (b) is the tracking error and boundary of nitrate nitrogen concentration.

[0109] Figure 4 This is a dry weather control input diagram in an embodiment of the present invention, wherein (a) is the oxygen transfer coefficient and (b) is the internal circulation velocity;

[0110] Figure 5 These are the dissolved oxygen concentration, nitrate nitrogen concentration and their reference values ​​during rainstorm weather in the embodiments of the present invention, wherein (a) is the dissolved oxygen concentration and its reference value, and (b) is the nitrate nitrogen concentration and its reference value;

[0111] Figure 6 These are the tracking errors and boundaries of dissolved oxygen concentration and nitrate nitrogen concentration during rainstorms in this embodiment of the invention, wherein (a) is the tracking error and boundary of dissolved oxygen concentration, and (b) is the tracking error and boundary of nitrate nitrogen concentration.

[0112] Figure 7 This is a control input diagram for heavy rain weather in an embodiment of the present invention, wherein (a) is the oxygen transfer coefficient and (b) is the internal circulation velocity;

[0113] Figure 8 These are the dissolved oxygen concentration, nitrate nitrogen concentration and their reference values ​​during rainfall in the embodiments of the present invention, wherein (a) is the dissolved oxygen concentration and its reference value, and (b) is the nitrate nitrogen concentration and its reference value;

[0114] Figure 9 These are the tracking errors and boundaries of dissolved oxygen concentration and nitrate nitrogen concentration during rainfall in the embodiments of the present invention, wherein (a) is the tracking error and boundary of dissolved oxygen concentration, and (b) is the tracking error and boundary of nitrate nitrogen concentration.

[0115] Figure 10 This is a control input diagram for rainy weather in an embodiment of the present invention, where (a) is the oxygen transfer coefficient and (b) is the internal circulation velocity. Detailed Implementation

[0116] The embodiments of the present invention are described in detail below. These embodiments are implemented based on the technical solution of the present invention, and provide detailed implementation methods and specific operation processes. However, the scope of protection of the present invention is not limited to the following embodiments.

[0117] like Figure 1 As shown, this embodiment provides a technical solution: an adaptive fuzzy second-order integral sliding mode control method for wastewater treatment processes based on predetermined time performance, comprising the following steps:

[0118] Step S1: Transform the No. 1 baseline simulation model of wastewater treatment into a nonlinear system mathematical model. Use a fuzzy logic system to approximate the unmodeled dynamics in the constructed nonlinear system mathematical model;

[0119] Step S2: Design a second-order integral sliding surface, and design the controller and update law for weights, disturbances and upper bounds of approximation error based on Lyapunov function analysis;

[0120] Step S3: Using Lyapunov's stability theorem and combining it with the adaptive controller and integral sliding surface designed in step S2, give sufficient conditions to ensure the stability of the closed-loop system;

[0121] Step S4: Implement wastewater treatment control under three different weather conditions using given system parameters;

[0122] In this embodiment, the specific processing procedure in step S1 is as follows:

[0123] S11: Consider the first baseline simulation model as follows:

[0124]

[0125]

[0126] in, and These represent the hysteresis time constants for dissolved oxygen and nitrate nitrogen, respectively. This indicates that the fourth chamber of the biochemical reaction vessel is in Dissolved oxygen concentration at any given time; This indicates that the first chamber of the biochemical reaction vessel is in The nitrate nitrogen concentration at time; This represents the oxygen transfer coefficient of the fifth chamber at the current moment; Indicates component concentration The reaction rate; Indicates component concentration The reaction rate; , and It's traffic; Indicates internal circulation flow; This indicates the flow rate from the first compartment to the second compartment.

[0127] To facilitate analysis and design, the above model is transformed. Let... The following mathematical model for control is obtained:

[0128]

[0129] in:

[0130] and , and The nonlinear function and control coefficients are known. and It is an unmodeled dynamic; and It is a bounded and unknown time-varying disturbance caused by the environment.

[0131] S12: The simplified representation of the control-oriented mathematical model in step S11 is as follows:

[0132] ;

[0133] Unknown function The fuzzy logic system approximates it as follows:

[0134]

[0135] in, These are basis functions; These are weighting coefficients; Is it satisfied by the inequality The approximate error and It is a constant.

[0136] In this embodiment, the specific processing procedure in step S2 is as follows:

[0137] S21: To achieve the control objectives of the wastewater treatment system, the following predetermined time function was designed:

[0138]

[0139] Its differential form is as follows:

[0140]

[0141] in, as well as It is a constant and .

[0142] Tracking error is defined as ,in, Let be the expected value. To achieve the predetermined performance, the following error transformation formula is given:

[0143]

[0144] in, It is unconstrained tracking error; It is a transformation function, as follows:

[0145]

[0146] in, , ; as well as .

[0147] S22: When there is no constraint tracking error When it is bounded, we get .when At that time, In other words, we can obtain Then, the state. The maximum interval is calculated as follows:

[0148]

[0149] Therefore, we can conclude the following two possibilities:

[0150] 1) For dissolved oxygen concentration, control coefficient It should be greater than zero, that is This means Therefore, we can conclude that... ,Right now:

[0151]

[0152] Therefore, we can conclude that:

[0153]

[0154] 2) For nitrate nitrogen concentration, control coefficient It should be less than zero, that is Then, we can obtain Therefore, we can conclude that:

[0155]

[0156] Therefore, we can conclude that:

[0157]

[0158] Based on the error transformation formula and transformation function in step S21, we can obtain:

[0159]

[0160] as well as:

[0161]

[0162] in:

[0163] .when hour, This means that the variable It is reversible.

[0164] S23: To enhance the robustness of the controller, the following sliding mode variables were designed:

[0165]

[0166] in, as well as It is a constant.

[0167] However, due to variables Including unknown, unmodeled dynamics and time-varying disturbances, the following second-order command filter is used to obtain the variables. :

[0168]

[0169] in, as well as It is a constant. Then, the sliding surface is modified as follows:

[0170]

[0171] Combining the unknown nonlinear function approximated by the fuzzy logic system in step S12, the derivative of the unconstrained tracking error in step S22, and the second-order command filter, we obtain:

[0172]

[0173] in, It is a known function.

[0174] S24: To verify the effectiveness of the constructed second-order integral sliding surface, the following Lyapunov function is constructed:

[0175]

[0176] Its differential calculation is as follows:

[0177]

[0178] in, It is the weight estimation error; This is the perturbation estimation error. Furthermore, based on Young's inequality, we can obtain:

[0179]

[0180] Furthermore, we can obtain:

[0181]

[0182] in, It is the estimated error of the upper limit of the approximation error.

[0183] S25: Design the controller structure as shown below:

[0184]

[0185] Therefore, we can conclude that:

[0186]

[0187] in, It is a constant.

[0188] The update laws for designing weights, perturbations, and upper bounds of approximation errors are as follows:

[0189]

[0190] in, and It is a positive definite symmetric matrix. It is a constant. .

[0191] In this embodiment, the specific processing procedure in step S3 is as follows:

[0192] S31: Based on the simplified model and fuzzy logic system in step S12, the predetermined time function in step S21, the state constraints and unconstrained tracking error in step S22, the second-order command filter and sliding surface in step S23, and the controller, weights, perturbation, and update law of the upper bound of the approximate error in step S25, for i∈{1,2}, select the following parameters:

[0193] Unconstrained tracking error It is bounded, and the tracking error It is confined to a prescribed range. Furthermore, when At that time, it can be obtained .

[0194] S32: The specific proof is as follows:

[0195] The Lyapunov function is chosen as follows:

[0196]

[0197] Combining the update law of the controller, weights, perturbation, and upper bound of the approximation error in step S25, we can obtain:

[0198]

[0199] Therefore, according to Youngs' inequality, we can obtain:

[0200]

[0201] Through simple derivation, the following inequality holds:

[0202]

[0203] in:

[0204] .

[0205] Therefore, the sliding mode variable and estimation error Both are bounded. As can be seen from the second-order command filter, the approximation error... Bounded.

[0206] Based on the sliding surface structure in step S23, it can be seen that the unconstrained tracking error... Bounded, therefore the tracking error is known. Constrained within a given range, .Right now, .when Sometimes, .

[0207] At this point, simulation software can be used to calculate the controller gain matrix using the given matrix parameters. The value was determined and its control effect was verified.

[0208] Step S4: Unmodeled dynamic and time-varying disturbances are set as follows:

[0209]

[0210] The parameters in the designed control method are as follows:

[0211]

[0212] The basis functions of the fuzzy logic system are chosen as follows .

[0213] The input variables, center point, and width are as follows:

[0214]

[0215] The expected values ​​for dissolved oxygen concentration and nitrate nitrogen concentration are respectively and The initial values ​​for the state are set as follows: The initial values ​​for all other states are set to zero. Figure 2-10 Simulation results are presented under dry, heavy rain, and precipitation conditions. The figures show that the controller based on the second-order integral sliding mode model can achieve the desired concentrations of dissolved oxygen and nitrate nitrogen within the predetermined time, thus meeting the expected performance. The oxygen transfer coefficient and internal circulation velocity are both within reasonable ranges. However, on the one hand, the controller based on the first-order integral sliding mode model performs poorly; on the other hand, the controller for nitrate nitrogen concentration fails to function properly.

[0216] Although embodiments of the present invention have been shown and described above, it is understood that the above embodiments are exemplary and should not be construed as limiting the present invention. Those skilled in the art can make changes, modifications, substitutions and variations to the above embodiments within the scope of the present invention.

Claims

1. A sliding mode control method for a wastewater treatment process based on predetermined time performance, characterized in that, Includes the following steps: S1: Transform the No. 1 benchmark simulation model of wastewater treatment into a nonlinear system mathematical model, and use a fuzzy logic system to approximate the unmodeled dynamics of the nonlinear system mathematical model. S2: Design a second-order integral sliding surface, and design a controller and update laws for weights, disturbances and approximation error upper bounds based on Lyapunov function analysis; S3: Using Lyapunov's stability theorem and combining the controller and second-order integral sliding surface designed in step S2, give sufficient conditions to ensure the stability of the closed-loop system. S4: By providing parameters, the controller is used to control wastewater treatment under different weather conditions.

2. The sliding mode control method for a wastewater treatment process based on predetermined time performance according to claim 1, characterized in that, In step S1, the specific processing procedure is as follows: S11: Consider the first baseline simulation model as follows: ； ； in, and These represent the hysteresis time constants for dissolved oxygen and nitrate nitrogen, respectively. This indicates that the fourth chamber of the biochemical reaction vessel is in Dissolved oxygen concentration at any given time; This indicates that the first chamber of the biochemical reaction vessel is in The nitrate nitrogen concentration at time; This represents the oxygen transfer coefficient of the fifth chamber at the current moment; Indicates component concentration The reaction rate; Indicates component concentration The reaction rate; , and It's traffic; Indicates internal circulation flow; This indicates the flow rate from the first compartment to the second compartment; S12: Order By transforming the first-line baseline simulation model, the mathematical model of the nonlinear system is obtained as follows: ； ； in: and , and The nonlinear function and control coefficients are known. and It is an unmodeled dynamic; and It is a bounded and unknown time-varying disturbance caused by the environment; S13: The simplified mathematical model in step S12 is expressed as follows: ； Unmodeled dynamics The fuzzy logic system approximates it as follows: ； in, These are basis functions; These are weighting coefficients; Is it satisfied by the inequality The approximate error and It is a constant.

3. The sliding mode control method for a wastewater treatment process based on predetermined time performance according to claim 2, characterized in that, In step S2, the specific processing procedure is as follows: S21: To achieve the control objectives of the wastewater treatment system, design the following predetermined time function: ； Its differential form is as follows: ； in, as well as It is a constant and ; The tracking error is defined as ,in, Let the expected value be denoted as follows, and define the error conversion formula as follows: ； in, It is unconstrained tracking error; It is a transformation function; S22: When there is no constraint tracking error When it is bounded, we get ;when At that time, In other words, obtain the state. Then, calculate the state. The largest interval to which it belongs is as follows: ； For dissolved oxygen concentration, the control coefficient It should be greater than zero, that is ,get Thus we obtain ,Right now: ； Therefore, we get: ； For nitrate nitrogen concentration, the control coefficient It should be less than zero, that is Then, we can obtain Therefore, we can conclude that: ； Therefore, we get: ； Based on the error transformation formula and transformation function in step S21, we obtain: ； as well as: ； in: ,when hour, , representing variables It is reversible; S23: To enhance the robustness of the controller, the following sliding mode variables are designed: ； in, as well as It is a constant; Due to variables Including unknown, unmodeled dynamics and time-varying disturbances, the following second-order command filter is used to obtain the variables. : ； in, as well as It is a constant; Then, the sliding surface is modified as follows: ； Combining the unmodeled dynamics of the fuzzy logic system approximation in step S13 The derivative of the unconstrained tracking error in step S22 and the second-order command filter yield: ； in, It is a known function; S24: To verify the effectiveness of the constructed second-order integral sliding surface, the following Lyapunov function is constructed: ； Its differential calculation is as follows: ； in, It is the weight estimation error; This is the perturbation estimation error. Furthermore, based on Young's inequality, we obtain: ； Further results were obtained: ； in, It is the estimation error of the upper limit of the approximation error; S25: Design the following controller structure: ； Therefore, we get: ； in, It is a constant; The update laws for designing weights, perturbations, and upper bounds of approximation errors are as follows: ； ； ； in, and It is a positive definite symmetric matrix. It is a constant. .

4. The sliding mode control method for a wastewater treatment process based on predetermined time performance according to claim 3, characterized in that, In step S21, the transformation function Specifically as follows: ； in, , ; , .

5. The sliding mode control method for a wastewater treatment process based on predetermined time performance according to claim 3, characterized in that, In step S3, the specific processing procedure is as follows: S31: Based on the simplified model and fuzzy logic system in step S13, the predetermined time function in step S21, the state constraints and unconstrained tracking error in step S22, the second-order command filter and sliding surface in step S23, and the controller, weights, perturbation, and update law of the approximate error upper bound in step S25, for i∈{1,2}, select the following parameters: Unconstrained tracking error It is bounded, and the tracking error Constrained within the prescribed scope; furthermore, when At that time, ; S32: The specific proof is as follows: The Lyapunov function is chosen as follows: ； Combining the update law of the controller, weights, perturbation, and upper bound of the approximation error in step S25, we obtain: ； Therefore, according to Young's inequality, we get: ； Through derivation, the following inequality holds: ； in: ； ； Furthermore, it was learned that the sliding mode variable and estimation error Both are bounded; as can be seen from the second-order command filter, the approximation error is... Bounded; based on the sliding surface structure in step S23, the unconstrained tracking error is known. Bounded tracking error Constrained within a given range, ,Right now ;when Sometimes, .

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