An Adaptive Event-Triggered Distributed Control Method for a Sewage Treatment System

Abstract

The present invention belongs to the field of automation technology and modern control, and discloses an adaptive event-triggered distributed control method for a sewage treatment system. The method of the present invention considers the problem of switching failures in the sewage treatment control system and utilizes a positive switching system to establish a state-space model of the sewage treatment system. With the help of a linear cosine Lyapunov function and an adaptive event-triggered law, an adaptive event-triggered output distributed controller is designed, enabling the system to operate safely and smoothly. The distributed control method can still ensure the stable operation of the system in the event of a switching failure, thereby improving the safety of the system. The adaptive trigger control strategy can effectively adapt to changes in system performance, reduce the conservatism of system resource utilization, and ensure production efficiency.

Description

Technical Field

[0001] The present invention belongs to the field of automation technology and modern control, and in particular relates to an adaptive event-triggered distributed control method for a sewage treatment system. Background Art

[0002] With the industrial upgrading in my country, the standards for sewage treatment are becoming increasingly stringent, and designing more efficient and environmentally friendly sewage treatment systems has become a research hotspot.

[0003] Urban sewage treatment is urgent. Adopting new sewage treatment processes and control system solutions can meet the sewage treatment needs brought about by rapid urban development. Rapid technological advancements have also ushered in a fully automated era of computer-based intelligent monitoring for sewage treatment control systems. Different sewage treatment control schemes are selected based on the composition of the sewage and the size of the influent flow rate. Since water volume is non-negative, this patent proposes to utilize a positive switching system to characterize the sewage treatment system. Given that system states are often unmeasurable or unknown in real systems, output feedback control is an effective control method for addressing these issues. Compared to centralized output feedback control, distributed output feedback control has a lower measurement and communication burden, resulting in a simpler and more reliable control approach, effectively improving the safety and operational efficiency of sewage treatment equipment. Among the commonly used triggering strategies, traditional time-triggered mechanisms often waste resources and increase design costs. While static event-triggered mechanisms can effectively reduce unnecessary signal transmission, their fixed trigger thresholds make them difficult to adapt to changes in system performance and can also lead to conservative system resource utilization. For these reasons, adaptive event-triggered mechanisms have been proposed. These triggering mechanisms are more flexible and well-suited for system signal transmission. From a theoretical perspective, static event-triggered mechanisms are a special case of adaptive event-triggered strategies. Furthermore, in switching systems, switching signals are implemented to coordinate subsystems. In practice, some subsystems switch early or late. Incorrect switching timing, especially premature switching, can lead to system instability. Summary of the Invention

[0004] The present invention aims to provide an adaptive event-triggered distributed control method for a sewage treatment system to address the aforementioned technical issues. This method utilizes modern control theory and techniques to establish a state-space model of the sewage treatment system. Combining an adaptive event-triggered mechanism with a distributed control method, the present invention designs an adaptive event-triggered distributed controller for the sewage treatment system, ensuring that the system remains positive and stable even in the event of switching errors.

[0005] To solve the above technical problems, the specific technical solutions of the adaptive event-triggered distributed control method for the sewage treatment system of the present invention are as follows:

[0006] An adaptive event-triggered distributed control method for a sewage treatment system comprises the following steps:

[0007] Step 1: Establish a positive switching system state space model of the sewage treatment control system;

[0008] Step 2: Establish an adaptive trigger mechanism for the sewage treatment control system;

[0009] Step 3: Construct a dynamic output feedback controller with a system state observer;

[0010] Step 4: Build a closed-loop system for the sewage treatment system;

[0011] Step 5: Design the conditions for the stable operation of the sewage treatment control system;

[0012] Step 6: Positive verification of the sewage treatment control system under normal switching conditions;

[0013] Step 7: Verify the positive performance of the sewage treatment control system under switching failure conditions;

[0014] Step 8: Verify the stability of the sewage treatment control system under normal switching conditions;

[0015] Step 9: Verify the stability of the sewage treatment control system under switching failure conditions.

[0016] Furthermore, the step 1 includes the following specific steps:

[0017] Step 1.1: Establish the state space model of the sewage treatment control system:

[0018] x(k+1)=A σ(k) x(k)+B σ(k) u(k)+D σ(k) ω(k),

[0019]

[0020] in, Z∈N + are the system state of the sewage treatment system, the control input, the input disturbance and the measurable input of the s-th sensor node respectively; the function σ(k) represents the switching law and is selected from the finite set S = {1,2,...,J},J∈N + Take the value; assume

[0021] when When the system matrix is defined as A i , B i ,

[0022] D i ,

[0023] Furthermore, the specific steps of step 2 are as follows:

[0024] Step 2.1: Define the sampling error of the event generator as:

[0025]

[0026] in, y s (k l ) is the event generator at the event triggering time k l , the output signal when l∈N;

[0027] Step 2.2: The output will be released according to the following adaptive event trigger conditions:

[0028]

[0029] Where β>0, θ>0 and when the initial condition η(k0)=η0 and When η(k) is an internal dynamic variable, it satisfies:

[0030]

[0031] Furthermore, the construction form of step 3 is as follows:

[0032]

[0033]

[0034] in, is the observer state, G i and L i is the observer gain matrix, K i represents the controller gain matrix, The error is defined as:

[0035]

[0036] Furthermore, the construction method of step 4 is as follows:

[0037] definition Then under the controller in step 3, the closed-loop system of the sewage treatment control system in step 1 is expressed as:

[0038]

[0039] in,

[0040]

[0041] Furthermore, the construction form of step 5 is as follows:

[0042] Design constant ε ι >0, ι=1,2,3, ε3<ε2, θ>0, γ>0, λ>1, 0≤β<1, 0< m <μ<1, and vector Make

[0043]

[0044]

[0045]

[0046]

[0047]

[0048]

[0049]

[0050]

[0051]

[0052]

[0053] For any and Established, where q it ,l=1,2,...,n, q i is the lth element, Then, in the case of normal switching, the switching condition is

[0054]

[0055] Under this condition, the observer in step 3 is positive and the sewage treatment control system is positive and stable, where the control gain matrix is

[0056]

[0057]

[0058]

[0059] When a switching fault occurs, the switching condition is as follows:

[0060]

[0061] Under this condition, the observer in step 3 is positive and the sewage treatment control system is positive and stable, where the fault controller gain matrix is:

[0062]

[0063] Furthermore, the positive verification process of step 6 is as follows:

[0064] Step 6.1: Given any initial state and output Combined with the adaptive event triggering conditions in step 2, we have

[0065]

[0066] roll out:

[0067]

[0068] Therefore, the dynamic output feedback controller in step 2 and the closed-loop system in step 3 satisfy and in,

[0069] Step 6.2: Due to and Using the conditions in step 5, we can deduce and therefore, Step 6.3: Combine the second condition in step 5:

[0070]

[0071] Step 6.4: Using the fifth condition in step 5, we have Combined with the third condition in step 5:

[0072]

[0073] Step 6.5: Based on the fourth condition in step 5:

[0074]

[0075] Step 6.6: Use the controller gain matrix from step 5 to derive From the conditions for the stable operation of the sewage treatment system proposed in step 5, we can see that Thus we get and for and have and Using recursive derivation, we get and That is, the dynamic output feedback controller and the closed-loop system of the sewage treatment system in step 3 are positive.

[0076] Furthermore, the positive verification process of step 7 is as follows:

[0077] Step 7.1: Average dwell time switching condition Indicates that a switching failure occurs. When a switching failure occurs, the fault gain matrix is proposed in step 5, ε3<ε2. Therefore, substituting the fault gain matrix into the second condition in step 5 is:

[0078]

[0079] Step 7.2: This means:

[0080]

[0081] Step 7.3: Substituting the fault gain matrix into the third condition in step 5, we get:

[0082]

[0083] Using the fault controller gain matrix we get Therefore, we get Using the same method as in step 6, the closed-loop system of the dynamic output feedback controller and the sewage treatment system in step 3 is positive;

[0084] According to step 6 and step 7, the sewage treatment control system is positive in both normal switching and switching failure cases, so the positivity of the sewage treatment control system is proved.

[0085] Furthermore, the stability verification process of step 8 is as follows:

[0086] Step 8.1: Choose the linear cosine Lyapunov function as:

[0087]

[0088] in, Step 8.2: Consider ω(k) = 0, then:

[0089]

[0090] in, Using the gain matrix in step 5, we have:

[0091]

[0092] Using the gain matrix in step 5, we have

[0093]

[0094] From the conditions and gain matrix in step 5, we can get:

[0095]

[0096] Then, combining the conditions in step 5 and the above formula, derive ΔV i (k)≤(μ-1)V i (k), which means V i (k)≤μV i (k-1); Therefore, for the switching instant Can get in According to the conditions in step 5:

[0097]

[0098] Thus, the above inequality is further transformed into:

[0099]

[0100] this means in and They are Minimum and maximum elements, using average dwell time switching conditions, Therefore, the sewage system is exponentially stable;

[0101] Step 8.3: Consider ω(k)≠0,

[0102]

[0103] in,

[0104]

[0105]

[0106]

[0107] Define Ξ(k) = γ‖ω(k)‖1 - ‖e(k)‖1. From the linear copositive Lyapunov function in step 8.1, the inequality is derived as follows:

[0108]

[0109] Using the conditions in step 5, we can get:

[0110]

[0111]

[0112]

[0113] Substituting the above formula into the inequality, we have V i (k)≤μV i (k-1)+Ξ(k-1), using a similar method as in step 8.2, gives Furthermore, the sewage treatment system in step 1 is l1-gain stable with a performance of γ.

[0114] Furthermore, the stability verification process of step 9 is as follows:

[0115] Step 9.1: Substitute the fault controller gain matrix from step 5 into the condition,

[0116]

[0117] Therefore, under the conditions in step 5 and the fault controller gain matrix, it is not difficult to conclude that there exists a real number m (0< m <μ) makes:

[0118]

[0119]

[0120] Furthermore, V can be obtained i (k)≤ m V i (k-1); Using the Lyapunov function in step 8.1, we get

[0121]

[0122] in and They are The minimum and maximum elements of the switch condition from the average residence time, This indicates that the wastewater treatment control system maintains gain stability under switching fault conditions;

[0123] Step 9.2: Combining steps 8 and 9.1, it is concluded that the sewage treatment control system is stable.

[0124] The adaptive event-triggered distributed control method for a sewage treatment system presented in this invention has the following advantages: First, the method utilizes a positive switching system to establish a state-space model of the sewage treatment control system. Leveraging linear cosine Lyapunov functions and adaptive event-triggered laws, an adaptive event-triggered output distributed controller is designed, ensuring safe and stable system operation. This distributed control method ensures stable system operation even in the event of a switching failure, improving system safety. The adaptive triggering control strategy effectively adapts to changes in system performance, reduces conservative resource usage, and ensures production efficiency. BRIEF DESCRIPTION OF THE DRAWINGS

[0125] Figure 1 This is the sewage treatment process flow chart;

[0126] Figure 2 Schematic diagram of the framework of the positive switching system based on the distributed control of the adaptive event triggering mechanism of the present invention; DETAILED DESCRIPTION

[0127] In order to better understand the purpose, structure and function of the present invention, the adaptive event-triggered distributed control method of a sewage treatment system of the present invention is further described in detail below with reference to the accompanying drawings.

[0128] The specific steps of the adaptive event-triggered distributed control method for a sewage treatment system according to the present invention are as follows:

[0129] Step 1: Establish a positive switching system state space model of the sewage treatment control system. The specific method is:

[0130] 1.1 Describe the sewage treatment control system

[0131] Urban sewage includes domestic sewage and industrial sewage. The pollutants in sewage are ever-changing. It is necessary to use sewage, physical, and various sewage online detection instruments to detect the content of important indicators in sewage, so as to reflect the water quality before and after sewage treatment. The sewage treatment process is complex, and the main processes include coarse screen, sewage lifting, fine screen, aeration and grit settling, primary sedimentation tank, biochemical tank, backwash, ultraviolet light, etc. (see Figure 1Sewage treatment process flow chart). Among them, the main function of the primary sedimentation tank is to remove some inorganic suspended matter in the sewage to facilitate subsequent treatment in the biochemical tank. Since the water quality of urban domestic industrial sewage is different, it may be acidic or alkaline, and the inlet and outlet water flows of the primary sedimentation tank are large and the speed is fast. In order to ensure the quality of the effluent, different modes can be set according to the pH value of the water. Subsequent biochemical tanks, secondary sedimentation tanks, fiber filters, and backwash pump rooms all need to adjust different modes according to the different water quality and water inflow of sewage. Taking into account the characteristic that the water volume is always non-negative, the present invention intends to use a positive switching system to characterize the sewage treatment control system. In addition, adaptive event triggering laws and distributed control are also incorporated (see Figure 2 Framework diagram of a positive switching system with distributed control based on an adaptive event-triggered mechanism), which ensures that the sewage treatment control system can still operate stably in the event of a switching failure.

[0132] 1.2 Using the above description to establish the state space model of the sewage treatment control system

[0133] x(k+1)=A σ(k) x(k)+B σ(k) u(k)+D σ(k) ω(k),

[0134]

[0135] in, Z∈N + are the system state of the sewage treatment system, the control input, the input disturbance and the measurable input of the sth sensor node respectively. Function σ(k) represents the switching law and is selected from the finite set S = {1,2,...,J},J∈N + Assume For simplicity, when When the system matrix is defined as A i , B i , D i ,

[0136] Step 2: Establish an adaptive trigger mechanism for the sewage treatment control system, which is constructed as follows:

[0137] 2.1 Define the sampling error of the event generator as

[0138]

[0139] in, y s (k l ) is the event generator at the event triggering time k l, the output signal when l∈N.

[0140] 2.2 Output will be released according to the following adaptive event trigger conditions:

[0141]

[0142] Where β>0, θ>0 and when the initial condition η(k0)=η0 and When η(k) is an internal dynamic variable, it satisfies

[0143]

[0144] Step 3: Construct a dynamic output feedback controller with a system state observer

[0145]

[0146]

[0147] in, is the observer state, G i and L i is the observer gain matrix, K i represents the controller gain matrix, Define the error as

[0148]

[0149] Step 4: Build a closed-loop system for sewage treatment

[0150] definition Then under the controller in step 3, the closed-loop system of the sewage treatment control system in step 1 can be expressed as

[0151]

[0152] in,

[0153]

[0154] Step 5: Design the conditions for the stable operation of the sewage treatment control system as follows:

[0155] Design constant ε ι >0, ι=1,2,3, ε3<ε2, θ>0, γ>0, λ>1, 0≤β<1, 0< m <μ<1, and vector Make

[0156]

[0157]

[0158]

[0159]

[0160]

[0161]

[0162]

[0163]

[0164]

[0165]

[0166] For any and Established, where q it ,i=1,2,,n, q i The i-th element is, Then, in the case of normal switching, the switching condition is

[0167]

[0168] Under this condition, the observer in step 3 is positive and the sewage treatment control system is positive and stable, where the control gain matrix is

[0169]

[0170]

[0171]

[0172] When there is a switching fault, the switching condition is

[0173]

[0174] Under this condition, the observer in step 3 is positive and the sewage treatment control system is positive and stable, where the fault controller gain matrix is

[0175]

[0176] Step 6: The positive verification process of the sewage treatment control system under normal switching conditions is as follows:

[0177] 6.1 Given any initial state and output Combined with the adaptive event triggering conditions in step 2, we have

[0178]

[0179] Then, we can deduce

[0180]

[0181] Therefore, the dynamic output feedback controller in step 2 and the closed-loop system in step 3 satisfy and in,

[0182]

[0183] 6.2 Due to and Using the conditions in step 5, we can deduce and therefore,

[0184] 6.3 Combine the second condition in step 5

[0185]

[0186] 6.4 Using the fifth condition in step 5, we have Combined with the third condition in step 5,

[0187]

[0188] 6.5 According to the fourth condition in step 5

[0189]

[0190] 6.6 Using the controller gain matrix from step 5, we can derive From the conditions for the stable operation of the sewage treatment system proposed in step 5, we can see that Thus we get and for and have and Using recursive derivation, we get and That is, the dynamic output feedback controller and the closed-loop system of the sewage treatment system in step 3 are positive.

[0191] Step 7: The positive verification process of the sewage treatment control system under switching failure is as follows:

[0192] 7.1 Average dwell time switching conditions This means that the switching fails. When the switching fails, the fault gain matrix is proposed in step 5. Note that ε3 < ε2. Therefore, substituting the fault gain matrix into the second condition in step 5, we have

[0193]

[0194] 7.2 This means

[0195]

[0196] 7.3 Substituting the fault gain matrix into the third condition in step 5, we have

[0197]

[0198] Using the fault controller gain matrix we get Therefore, we get Using a similar method in step 6, it can be obtained that the dynamic output feedback controller and the closed-loop system of the sewage treatment system in step 3 are positive.

[0199] According to step 6 and step 7, the sewage treatment control system is positive in both normal switching and switching failure cases, so the positivity of the sewage treatment control system is proved.

[0200] Step 8: The verification process of the stability of the sewage treatment control system under normal switching conditions is as follows:

[0201] 8.1 Choose the linear cosine Lyapunov function as

[0202]

[0203] in,

[0204] 8.2 Consider ω(k) = 0, then

[0205]

[0206] in,

[0207] Using the gain matrix in step 5, we have

[0208]

[0209] Similarly, using the gain matrix in step 5, we have

[0210]

[0211] From the conditions and gain matrix in step 5, we can get

[0212]

[0213] Then, combining the conditions in step 5 and the above formula, derive ΔV i (k)≤(μ-1)V i (k), which means V i (k)≤μV i (k-1. Therefore, for the switching instant Can get in According to the conditions in step 5

[0214]

[0215] Thus, the above inequality can be further transformed into

[0216]

[0217] This means in and They are Minimum element and maximum element. Use the average residence time to switch the condition. Therefore, the sewage treatment system is exponentially stable.

[0218] 8.3 Consider ω(k)≠0,

[0219]

[0220] in,

[0221]

[0222]

[0223]

[0224] Define Ξ(k) = γ‖ω(k)‖1-‖e(k)‖1. Based on the linear copositive Lyapunov function in step 8.1, the inequality is derived as

[0225]

[0226] Using the conditions in step 5, we can get

[0227]

[0228]

[0229]

[0230] Substituting the above formula into the inequality, we have V i(k)≤μV i (k-1)+Ξ(k-1). Using a similar method as in step 8.2 gives Furthermore, the sewage treatment system in step 1 is l1-gain stable with a performance of γ.

[0231] Step 9: The verification process of the stability of the sewage treatment control system under switching failure is as follows:

[0232] 9.1 Substitute the fault controller gain matrix from step 5 into the condition,

[0233]

[0234] Therefore, under the conditions in step 5 and the fault controller gain matrix, it is not difficult to conclude that there exists a real number m (0< m <μ) makes

[0235]

[0236]

[0237] Furthermore, V can be obtained i (k)≤ m V i (k-1). Using the similar Lyapunov function in step 8.1, we get

[0238]

[0239] in and They are The smallest and largest element of . Switching conditions from the average residence time, This means that the wastewater treatment control system maintains gain stability in the event of a switching fault.

[0240] 9.2 Therefore, combining steps 8 and 9.1, it can be concluded that the sewage treatment control system is stable.

[0241] It will be appreciated that the present invention is described with reference to certain embodiments, and those skilled in the art will appreciate that various modifications or equivalent substitutions may be made to these features and embodiments without departing from the spirit and scope of the present invention. Furthermore, under the teachings of the present invention, these features and embodiments may be modified to suit specific circumstances and materials without departing from the spirit and scope of the present invention. Therefore, the present invention is not limited to the specific embodiments disclosed herein, and all embodiments falling within the scope of the claims of this application are intended to be protected by the present invention.

Claims

1. An adaptive event-triggered distributed control method for a sewage treatment system, characterized in that: The steps include: Step 1: Establish a positive switching system state space model of the sewage treatment control system; Step 1.1: Establish the state space model of the sewage treatment control system: x(k+1)=A σ(k) x(k)+B σ(k) u(k)+D σ(k) ω(k), in, are the system state of the sewage treatment system, the control input, the input disturbance and the measurable input of the s-th sensor node respectively; the function σ(k) represents the switching law and is selected from the finite set S = {1,2,...,J},J∈N + Take the value in the middle; assume A σ(k) ≥0, when When the system matrix is defined as A i , B i , D i , Step 2: Establish an adaptive trigger mechanism for the sewage treatment control system; Step 2.1: Define the sampling error of the event generator as: in, y s (k l ) is the event generator at the event triggering time k l , the output signal when l∈N; Step 2.2: The output will be released according to the following adaptive event trigger conditions: Where β>0, θ>0 and when the initial condition η(k0)=η0 and When η(k) is an internal dynamic variable, it satisfies: Step 3: Construct a dynamic output feedback controller with a system state observer; in, is the observer state, G i and L i is the observer gain matrix, K i represents the controller gain matrix, The error is defined as: Step 4: Build a closed-loop system for the sewage treatment system; definition Then under the dynamic output feedback controller in step 3, the closed-loop system of the sewage treatment control system in step 1 is expressed as: in, Step 5: Design the conditions for the stable operation of the sewage treatment control system; Design constant ε ι >0, ι=1,2,3, ε3<ε2, θ>0, γ>0, λ>1, 0≤β<1, 0< μ <μ<1, and vector Make For any and Established, where q iι ,ι=1,2,...,n,q i yes No. elements, Then, in the case of normal switching, the switching condition is Under this condition, the observer in step 3 is positive and the sewage treatment control system is positive and stable, where the controller gain matrix is When there is a switching fault, the switching condition is Under this condition, the observer in step 3 is positive and the sewage treatment control system is positive and stable, where the fault controller gain matrix is: Step 6: Positive verification of the sewage treatment control system under normal switching conditions; Step 7: Verify the positive performance of the sewage treatment control system under switching failure conditions; Step 8: Verify the stability of the sewage treatment control system under normal switching conditions; Step 9: Verify the stability of the sewage treatment control system under switching failure conditions.

2. The adaptive event-triggered distributed control method for a sewage treatment system according to claim 1, characterized in that: The positive verification process of step 6 is as follows: Step 6.1: Given any initial state x(k0) ≥ 0 and output Combined with the adaptive event triggering conditions in step 2, we have roll out: Therefore, the dynamic output feedback controller in step 3 and the closed-loop system in step 4 satisfy and in, Step 6.2: Due to and Using the conditions in step 5, we can deduce and Therefore, -B i K i ≥0; Step 6.3: Combine the second condition in step 5: Step 6.4: Using the fifth condition in step 5, we have Combined with the third condition in step 5: Step 6.5: Based on the fourth condition in step 5: Step 6.6: Use the controller gain matrix from step 5 to derive From the conditions for the stable operation of the sewage treatment system proposed in step 5, we can see that Thus we get and For x(k0)≥0, and ω(k0)≥0, we have and Using recursive derivation, we get and That is, the dynamic output feedback controller and the closed-loop system of the sewage treatment system in step 4 are positive.

3. The adaptive event-triggered distributed control method for a sewage treatment system according to claim 2, characterized in that: The positive verification process of step 7 is as follows: Step 7.1: Average dwell time switching condition Indicates that a switching failure occurs. When a switching failure occurs, the fault controller gain matrix proposed in step 5, ε3 < ε2, therefore, the fault controller gain matrix is substituted into the second condition in step 5: Step 7.2: This means: Step 7.3: Substituting the fault controller gain matrix into the third condition in step 5, we get: A is given by the fault controller gain matrix i +B i K fi ≥0, Therefore, we get Using the same method as in step 6, the closed-loop system of the dynamic output feedback controller and the sewage treatment system in step 4 is positive; According to step 6 and step 7, the sewage treatment control system is positive in both normal switching and switching failure cases, so the positivity of the sewage treatment control system is proved.

4. The adaptive event-triggered distributed control method for a sewage treatment system according to claim 3, characterized in that: The stability verification process of step 8 is as follows: Step 8.1: Choose the linear cosine Lyapunov function as: in, Step 8.2: Consider ω(k) = 0, then: in, Using the gain matrix in step 5, we have: Using the gain matrix in step 5, we have From the conditions and gain matrix in step 5, we can get: Then, combining the conditions in step 5 and the above formula (1), ΔV is derived i (k)≤(μ-1)V i (k), which means V i (k)≤μV i (k-1); therefore, for the switching instant get in According to the conditions in step 5: Thus, the above inequality is further transformed into: this means in and They are v i The smallest and largest elements of Using the average dwell time switching condition, Therefore, the sewage system is exponentially stable; Step 8.3: Consider ω(k)≠0, in, Define Ξ(k) = γ‖ω(k)‖1 - ‖e(k)‖1. From the linear copositive Lyapunov function in step 8.1, the inequality is derived as follows: Using the conditions in step 5, we can get: Substituting the above formula into the inequality, we have V i (k)≤μV i (k-1)+Ξ(k-1), using a similar method as in step 8.2, gives Furthermore, the sewage treatment system in step 1 is The gain is stable and the performance is gamma.

5. The adaptive event-triggered distributed control method for a sewage treatment system according to claim 4, characterized in that: The stability verification process of step 9 is as follows: Step 9.1: Substitute the fault controller gain matrix from step 5 into the condition, Therefore, under the conditions in step 5 and the fault controller gain matrix, it follows that there exists a real number μ ,0< μ <μ, such that: Furthermore, V can be obtained i (k)≤ μ V i (k-1); Using the Lyapunov function in step 8.1, we get in and They are v i The smallest and largest elements of Switching conditions from average dwell time, This indicates that the wastewater treatment control system maintains gain stability under switching fault conditions; Step 9.2: Combining steps 8 and 9.1, it is concluded that the sewage treatment control system is stable.

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