A double non-perturbing switching control method for a sewage treatment system

Abstract

The application provides a double-jam-free switching control method of a sewage treatment system. The application establishes a double-jam-free control method of the sewage treatment system through data acquisition, modeling, optimization control and other means, and the method can effectively solve the problem of inaccurate control precision caused by external uncertain disturbance factors and controller switching fluctuation. Based on the sewage treatment system with the tangent switching system modeling of the state and the disturbance observer, a kind of jam-free controller is designed, the sewage treatment is accurately controlled in stages, and the efficient operation of the sewage treatment system is ensured.

Description

Technical Field

[0001] This invention belongs to the field of automation technology and modern control, and relates to a dual-uninterrupted switching control method for a wastewater treatment system. Background Technology

[0002] In recent years, with the rapid development of industrialization and urbanization in my country, the discharge of urban sewage has been increasing day by day. If not effectively controlled, large amounts of industrial wastewater and domestic sewage will enter the water environment or soil, inevitably causing pollution of urban water bodies and soil, and even damaging the ecological environment. Therefore, urban water pollution control has become an urgent issue. At present, the sewage treatment capacity of most cities in my country is far from meeting the actual needs. With the continuous improvement of sewage discharge standards, there are higher requirements for pollutant removal. However, the technologies used in early sewage treatment systems were relatively backward, with low automation and low treatment efficiency, and there is still much room for improvement in sewage treatment methods. To achieve effective urban sewage treatment, it is not only necessary to build a large number of sewage treatment plants, but also to improve the control efficiency of sewage treatment systems. Therefore, it is particularly important to propose a more advanced and efficient sewage treatment system and control method.

[0003] Urban wastewater treatment systems are a crucial component of urban infrastructure, playing a vital role in controlling water pollution. Accelerating the construction and promoting the efficient operation of urban wastewater treatment systems is essential. The process flow of urban wastewater treatment systems generally includes multiple stages such as primary treatment, secondary treatment, and advanced secondary treatment. Different control measures are needed at each stage based on the water quality. Therefore, constructing a wastewater treatment system with multiple modes is appropriate, and since the water volume in the wastewater treatment system is always non-negative, this invention patent proposes to use a positive switching system to model the wastewater treatment system. Summary of the Invention

[0004] The purpose of this invention is to provide a dual-disturbance-free switching control method for wastewater treatment systems based on positive switching system modeling, addressing external disturbances and controller switching fluctuations in wastewater treatment systems. This invention adopts the following technical solution:

[0005] Step 1: Establish a state-space model of the wastewater treatment system during positive switching. The specific method is as follows:

[0006]

[0007] Where x(t)∈R n , u(t)∈R m , y(t)∈R s These represent the system state, control input, external disturbance input, and system output of the wastewater treatment system, respectively. The function σ(t) represents the switching law, and is derived from a finite set... Take a value from the middle. Assume A σ(t) It is a Metzler matrix, B σ(t) 0, C σ(t) 0, D σ(t) 0, E σ(t) 0. For simplicity, when When, the system matrix is ​​defined as A p B p C p D p E p .

[0008] Step 2: Establish a disturbance observer for the wastewater treatment system, which is constructed as follows:

[0009] 2.1 Define the external disturbance system as

[0010]

[0011] in, It is the state of the disturbed system, Υ σ(t) ∈R r×r It is a Metzler matrix, Γ σ(t) 0, Γ σ(t) ∈R r ×r .

[0012] 2.2 The state observer for the wastewater treatment system is designed as follows:

[0013]

[0014] in, It is the state of the state observer. It is an estimation of the disturbance signal, G σ(t) ∈R o×o M σ(t) ∈R o×r L cσ(t) ∈R o×s This is the state observer gain matrix, specifically designed as follows:

[0015]

[0016] Where, the symbol represents the transpose of a vector or matrix, 1 o This represents an o-dimensional column vector where all elements are 1. Let I represent an o-dimensional column vector where the i-th element is 1 and all other elements are 0. o Let v1 and z represent the o×o identity matrix. g and z c It is an o-dimensional column vector, and δ1 is a constant.

[0017] 2.3 Design the disturbance observer for the wastewater treatment system.

[0018]

[0019] in, It is the state of the perturbation observer, H σ(t) ∈R r×r F σ(t) ∈R r×r L dσ(t) ∈R r×r It is the perturbation observer gain matrix, and its specific design form is as follows:

[0020]

[0021] Step 3: Construct a disturbance-free controller with a system disturbance observer. The specific method is as follows:

[0022] 3.1 The controller for the wastewater treatment is designed as follows:

[0023]

[0024] Among them, K 1σ(t) K 2σ(t) K 3σ(t) It is a controller gain matrix, and its specific design is as follows:

[0025]

[0026] in, z k1 z k1 and z k1 It is an m-dimensional column vector.

[0027] 3.2 Define the disturbance-free switching condition for the controller as follows:

[0028]

[0029] Where α≥0, β≥0, and γ≥0 are the non-perturbation performance indices, u j and These are the j-th row elements of the control input and the expected control input, respectively.

[0030] Step 4: Construct the error system and closed-loop system of the wastewater treatment system disturbance observer, as detailed below:

[0031] Define error Therefore, under the disturbance observer in step 2, the error system of the wastewater treatment control system in step 1 can be expressed as:

[0032]

[0033] Furthermore, under the disturbance-free controller in step 3.1, the closed-loop system of the wastewater treatment control system in step 1 can be represented as follows:

[0034]

[0035] Step 5: The conditions for the stable operation of the wastewater treatment system are as follows:

[0036] Design constants μ1>0, μ2>0, λ1>1, λ2>1, α1>0, α2>0, δ1>0, δ2>0, δ3>0, δ4>0, R o Vector v1 > 0, z c >0, R r Vector v2 > 0, z d >0, z f >0, R m vector z k1 >0, z k2 >0, z k3 >0, Make

[0037]

[0038]

[0039]

[0040]

[0041]

[0042]

[0043]

[0044]

[0045] z g +z f -α1v1<0,

[0046] z k1 +z g +z f -α2v1<0,

[0047]

[0048]

[0049]

[0050]

[0051]

[0052]

[0053]

[0054]

[0055] v (p) λ1v (q) ,

[0056] For any If this holds true, then all observers are positive, the error system is stable, and the mean dwell time condition is... The wastewater treatment system is non-invasive and meets the following conditions.

[0057]

[0058]

[0059]

[0060]

[0061]

[0062]

[0063] Step 6: The positive verification process for the error system and closed-loop control system of the wastewater treatment system is as follows:

[0064] 6.1 From the gain matrix designed in step 2 and the equation in step 5, we can see that...

[0065] A p -L cp C p -G p =0,

[0066] E p -L cp D p -M p =0,

[0067] Υ p -L dp D p Γ p -H p =0,

[0068] B p K 2p -B p K 3p C p +E p =0.

[0069] 6.2 Further, the error system and closed-loop control system in step 4 are represented as follows:

[0070]

[0071] 6.3 Combining the gain matrix in step 2 and the positive constraint condition in step 5, we have: B p K 1p ≥0, B p K 2p Γ p ≥0,

[0072]

[0073] M p ≥0 and F p ≥0, further we can obtain A p +B p K 1p -B p K 3p C p G p and H p All are Metzler matrices.

[0074] 6.4 For a given non-negative initial value x(0), e x (0) and e ξ (0), according to the system in step 6.2, we can deduce that x(t)≥0, e x (t)≥0 and e ξ Since (t)≥0, the error system and closed-loop control system of the sewage treatment system are positive.

[0075] Step 7: The stability verification process of the wastewater treatment system is as follows:

[0076] 7.1 Choosing the linear copositive Lyapunov function as

[0077]

[0078] in, Then, for a given switching sequence 0≤t0≤t1≤…≤t m ≤t,t∈[t m ,t m+1), m∈N, according to the error system in step 6.2, we can obtain

[0079]

[0080] 7.2 From the gain matrix in step 2 and the constraints in step 5, we can see that...

[0081]

[0082]

[0083] 7.3 Based on the conditions in step 5 and step 7.2, we can conclude that...

[0084]

[0085]

[0086] 7.4 Combining steps 7.1 and 7.3, we can deduce...

[0087]

[0088] 7.5 Integrating both sides of step 7.4 simultaneously, we can see that...

[0089]

[0090] 7.6 Combining the Lyapunov function switching conditions in steps 7.5 and 5, we can obtain...

[0091]

[0092] in, Using the average residence time condition from step 5, we know that: ≤0. Therefore, the error system of the wastewater treatment system is exponentially stable.

[0093] Step 8: The stability verification process of the closed-loop control system of the wastewater treatment system is as follows:

[0094] 8.1 Choosing the linear copositive Lyapunov function as

[0095]

[0096] in, Then, for a given switching sequence 0≤t0≤t1≤t m ≤t,t∈[t m ,t m+1 ), m∈N, according to the closed-loop control system in step 6.2, we can obtain

[0097]

[0098] 8.2 From the gain matrix in step 3.1 and the constraints in step 5, we can see that...

[0099]

[0100] 8.3 Based on the conditions in step 5 and step 8.2, we can conclude that...

[0101]

[0102]

[0103]

[0104] 8.4 Combining steps 8.1 and 8.3, we can deduce...

[0105]

[0106] 8.5 Integrating both sides of step 8.4 simultaneously, we can see that...

[0107]

[0108] 8.6 Based on step 7.6, it can be similarly concluded that the error system of the sewage treatment system is exponentially stable.

[0109] Step 9: The non-intrusiveness verification process for the controller of the wastewater treatment system is as follows:

[0110] 9.1 Based on the unperturbed condition in step 3.2, the following inequality holds:

[0111]

[0112] in, This represents the j-th row of the controller gain matrix K. This represents the j-th row of the controller gain matrix K.

[0113] 9.2 Due to It can be known that y(t) ≥ 0. Furthermore, according to the unperturbed condition in step 5, it can be known that...

[0114]

[0115]

[0116]

[0117] 9.3 Based on the inequality in step 9.2, we can derive...

[0118]

[0119] Therefore, the controller of the wastewater treatment system is disturbance-free.

[0120] The beneficial effects of this invention are as follows:

[0121] In response to the current control methods and disturbance factors in wastewater treatment systems, this paper proposes a method to establish a state-space model of the wastewater treatment system using modern control theory. By designing a disturbance observer and a disturbance-free controller, the influence of disturbance factors in the wastewater treatment process can be effectively reduced and the control accuracy can be improved. This avoids repeated detection and treatment of wastewater components, reduces the operating cost of the wastewater treatment system, and ensures the stable and effective operation of the wastewater treatment system. Attached Figure Description

[0122] Figure 1 This is a flow chart of the wastewater treatment process;

[0123] Figure 2 This is a schematic diagram of the framework of a positive switching system based on a disturbance observer and a disturbance-free controller. Detailed Implementation

[0124] The present invention will be further described below with reference to specific embodiments, but the invention is not limited to these specific embodiments. Those skilled in the art should recognize that the present invention covers all alternatives, improvements, and equivalents that may be included within the scope of the claims.

[0125] like Figure 1 As shown, considering the complex composition of wastewater and the significant impact of external disturbances such as rainfall, discharge volume, and organic matter concentration on the quality and quantity of treated wastewater, the wastewater treatment process cannot ignore these external disturbances; otherwise, the quality of the treated wastewater will be degraded. Establishing disturbance observers is an effective method to address this issue. The switching processes between different stages of wastewater treatment require high control precision, and general controllers are insufficient to meet the control requirements of wastewater treatment systems, failing to achieve ideal control results. Therefore, employing more precise controllers is particularly necessary. Figure 2 As shown, this invention introduces constraints on the controller to design a disturbance-free controller for wastewater treatment systems, reducing controller fluctuations and thus eliminating disturbances during controller switching. Compared to direct switching control, the disturbance-free control method offers more stable control, is simpler and more reliable, and effectively improves the safety and operational efficiency of wastewater treatment systems.

[0126] Based on the above analysis, this invention utilizes modern control theory to establish a state-space model of a dual-disturbance-free switching control system. It designs a disturbance observer and a disturbance-free controller for the wastewater treatment system, ensuring the positivity and stability of the closed-loop system. In summary, designing a dual-disturbance-free switching control method for a wastewater treatment system based on positive switching system modeling has significant scientific research and practical application value.

[0127] This embodiment provides a dual-barrier-free switching control method for a wastewater treatment system, the specific steps of which are as follows:

[0128] Step 1: Establish a state-space model of the wastewater treatment system during positive switching. The specific method is as follows:

[0129]

[0130] y(t)=C σ(t) x(t)+D σ(t) w(t),

[0131] Where x(t)∈R n , u(t)∈R m , y(t)∈R s These represent the system state, control input, external disturbance input, and system output of the wastewater treatment system, respectively. The function σ(t) represents the switching law, and is derived from a finite set... Take a value from the middle. Assume A σ(t) It is a Metzler matrix, B σ(t) ≥0, C σ(t) ≥0, D σ(t) ≥0, E σ(t) ≥0. For simplicity, when When, the system matrix is ​​defined as A p B p C p D p E p .

[0132] Step 2: Establish a disturbance observer for the wastewater treatment system, which is constructed as follows:

[0133] 2.1 Define the external disturbance system as

[0134]

[0135] w(t)=Γ σ(t) ξ,

[0136] in, It is the state of the disturbed system, Υ σ(t) ∈R r×r It is a Metzler matrix, Γ σ(t)≥0, Γ σ(t) ∈R r×r .

[0137] 2.2 The state observer for the wastewater treatment system is designed as follows:

[0138]

[0139] in, It is the state of the state observer. It is an estimation of the disturbance signal, G σ(t) ∈R o×o M σ(t) ∈R o×r L cσ(t) ∈R o×s This is the state observer gain matrix, specifically designed as follows:

[0140]

[0141] Where, the symbol represents the transpose of a vector or matrix, 1 o This represents an o-dimensional column vector where all elements are 1. Let I represent an o-dimensional column vector where the i-th element is 1 and all other elements are 0. o Let v1 and z represent the o×o identity matrix. g and z c It is an o-dimensional column vector, and δ1 is a constant.

[0142] 2.3 Design the disturbance observer for the wastewater treatment system.

[0143]

[0144] in, It is the state of the perturbation observer, H σ(t) ∈R r×r F σ(t) ∈R r×r L dσ(t) ∈R r×r It is the perturbation observer gain matrix, and its specific design form is as follows:

[0145]

[0146] Step 3: Construct a disturbance-free controller with a system disturbance observer. The specific method is as follows:

[0147] 3.1 The controller for the wastewater treatment is designed as follows:

[0148]

[0149] Among them, K 1σ(t) K 2σ(t) K3σ(t) It is a controller gain matrix, and its specific design is as follows:

[0150]

[0151] in, z k1 z k1 and z k1 It is an m-dimensional column vector.

[0152] 3.2 Define the disturbance-free switching condition for the controller as follows:

[0153]

[0154] Where α≥0, β≥0, and γ≥0 are the non-perturbation performance indices, u j and These are the j-th row elements of the control input and the expected control input, respectively.

[0155] Step 4: Construct the error system and closed-loop system of the wastewater treatment system disturbance observer, as detailed below:

[0156] Define error Therefore, under the disturbance observer in step 2, the error system of the wastewater treatment control system in step 1 can be expressed as:

[0157]

[0158]

[0159] Furthermore, under the disturbance-free controller in step 3.1, the closed-loop system of the wastewater treatment control system in step 1 can be represented as follows:

[0160]

[0161] Step 5: The conditions for the stable operation of the wastewater treatment system are as follows:

[0162] Design constants μ1>0, μ2>0, λ1>1, λ2>1, α1>0, α2>0, δ1>0, δ2>0, δ3>0, δ4>0, R o Vector v1 > 0, z c >0, z g >0, R r Vector v2 > 0, z d >0, z f >0, R m vector z k1 >0, z k3 >0, Make

[0163]

[0164]

[0165]

[0166]

[0167]

[0168]

[0169]

[0170]

[0171] z g +z f -α1v1<0,

[0172] z k1 +z g +z f -α2v1<0,

[0173]

[0174]

[0175]

[0176]

[0177]

[0178]

[0179]

[0180]

[0181] v (p) λ1v (q) ,

[0182] For any If this holds true, then all observers are positive, the error system is stable, and the mean dwell time condition is... The wastewater treatment system is non-invasive and meets the following conditions.

[0183]

[0184]

[0185]

[0186]

[0187]

[0188]

[0189] Step 6: The positive verification process for the error system and closed-loop control system of the wastewater treatment system is as follows:

[0190] 6.1 From the gain matrix designed in step 2 and the equation in step 5, we can see that...

[0191] A p -L cp C p -G p =0,

[0192] E p -L cp D p -M p =0,

[0193] Υ p -L dp D p Γ p -H p =0,

[0194] B p K 2p -B p K 3p C p +E p =0.

[0195] 6.2 Further, the error system and closed-loop control system in step 4 are represented as follows:

[0196]

[0197] 6.3 Combining the gain matrix in step 2 and the positive constraint condition in step 5, we have: B p K 1p 0, B p K 2p Γ p 0,

[0198]

[0199] M p ≥0 and F p ≥0, further we can obtain A p +B p K 1p -B p K 3p C p G p and H p All are Metzler matrices.

[0200] 6.4 For a given non-negative initial value x(0), e x (0) and e ξ (0), according to the system in step 6.2, we can deduce that x(t)≥0, e x (t)≥0 and e ξ Since (t)≥0, the error system and closed-loop control system of the sewage treatment system are positive.

[0201] Step 7: The stability verification process of the wastewater treatment system is as follows:

[0202] 7.1 Choosing the linear copositive Lyapunov function as

[0203]

[0204] in, Then, for a given switching sequence 0≤t0≤t1≤…≤t m ≤t,t∈[t m ,t m+1 ), m∈N, according to the error system in step 6.2, we can obtain

[0205]

[0206] 7.2 From the gain matrix in step 2 and the constraints in step 5, we can see that...

[0207]

[0208]

[0209] 7.3 Based on the conditions in step 5 and step 7.2, we can conclude that...

[0210]

[0211]

[0212] 7.4 Combining steps 7.1 and 7.3, we can deduce...

[0213]

[0214] 7.5 Integrating both sides of step 7.4 simultaneously, we can see that...

[0215]

[0216] 7.6 Combining the Lyapunov function switching conditions in steps 7.5 and 5, we can obtain...

[0217]

[0218] in, Using the average dwell time condition from step 5, we know that: Therefore, the error system of the wastewater treatment system is exponentially stable.

[0219] Step 8: The stability verification process of the closed-loop control system of the wastewater treatment system is as follows:

[0220] 8.1 Choosing the linear copositive Lyapunov function as

[0221]

[0222] in, Then, for a given switching sequence 0≤t0≤t1≤…≤t m ≤t,t∈[t m ,t m+1 ), m∈N, according to the closed-loop control system in step 6.2, we can obtain

[0223]

[0224] 8.2 From the gain matrix in step 3.1 and the constraints in step 5, we can see that...

[0225]

[0226] 8.3 Based on the conditions in step 5 and step 8.2, we can conclude that...

[0227]

[0228]

[0229]

[0230] 8.4 Combining steps 8.1 and 8.3, we can deduce...

[0231]

[0232] 8.5 Integrating both sides of step 8.4 simultaneously, we can see that...

[0233]

[0234] 8.6 Based on step 7.6, it can be similarly concluded that the error system of the sewage treatment system is exponentially stable.

[0235] Step 9: The non-intrusiveness verification process for the controller of the wastewater treatment system is as follows:

[0236] 9.1 Based on the unperturbed condition in step 3.2, the following inequality holds:

[0237]

[0238] in, This represents the j-th row of the controller gain matrix K. This represents the j-th row of the controller gain matrix K.

[0239] 9.2 Due to It can be seen that y(t)≥0. Furthermore, according to the unperturbed condition in step 5, it can be seen that...

[0240]

[0241]

[0242]

[0243] 9.3 Based on the inequality in step 9.2, we can derive...

[0244]

[0245] Therefore, the controller of the wastewater treatment system is disturbance-free.

Claims

1. A dual-disruptionless switching control method for a wastewater treatment system, characterized in that... The method includes the following steps: Step 1: Establish a state-space model of the wastewater treatment system during positive switching. Step 2: Establish a disturbance observer for the wastewater treatment system; Step 3: Construct a disturbance-free controller with a system disturbance observer; Step 4: Construct the error system and closed-loop system of the sewage treatment system disturbance observer; Step 5: Design the conditions for stable operation of the wastewater treatment system; Step 6: Positive verification process of the error system and closed-loop control system of the wastewater treatment system; Step 7: Verification of the stability of the wastewater treatment system; Step 8: Stability verification process of the closed-loop control system of the wastewater treatment system; Step 9: Verification of the non-intrusiveness of the closed-loop control system of the wastewater treatment system; The state-space model of the wastewater treatment system in step 1, which is in the positive switching state space, is as follows: in, , , , These represent the system state, control input, external disturbance input, and system output of the wastewater treatment system, respectively; (function) Represents the switching law, and from a finite set , Take a value from; assume It is a Metzler matrix. , , , ;when When, the system matrix is ​​defined as , , , , ; A disturbance observer for a wastewater treatment system is established, and its construction is as follows: Define the external disturbance system as in, It is the state of the disturbed system. It is a Metzler matrix. , ; Design a state observer for the wastewater treatment system. in, It is the state of the state observer. It is an estimation of the disturbance signal. , , This is the state observer gain matrix, specifically designed as follows: Among them, symbols Represents the transpose of a vector or matrix. This represents an o-dimensional column vector where all elements are 1. Let represent an o-dimensional column vector where the i-th element is 1 and all other elements are 0. express The identity matrix, , and It is an o-dimensional column vector. It is a constant; Design a disturbance observer for a wastewater treatment system. in, It is the state of the perturbation observer. , , It is the perturbation observer gain matrix, and its specific design form is as follows: ； In step 3, a disturbance-free controller with a system disturbance observer is constructed, and its construction form is as follows: Design a controller for wastewater treatment. in, , , It is a controller gain matrix, and its specific design is as follows: in, , , and It is an m-dimensional column vector; Define the controller's disturbance-free switching condition as follows in, , , For non-disruptive performance indicators, and These are the j-th row elements of the control input and the expected control input, respectively.

2. The dual-uninterrupted switching control method for a wastewater treatment system according to claim 1, characterized in that: The error system and closed-loop system for constructing the disturbance observer of the wastewater treatment system in step 4 are constructed as follows: Define error , Therefore, under the disturbance observer in step 2, the error system of the wastewater treatment control system in step 1 can be expressed as: Furthermore, under the disturbance-free controller in step 3.1, the closed-loop system of the wastewater treatment control system in step 1 can be represented as follows: 。 3. The dual-disruptionless switching control method for a wastewater treatment system according to claim 2, characterized in that: The conditions for the stable operation of the wastewater treatment system in step 5 are constructed as follows: Design constants , , , , , , , , , , vector , , , , , vector , , , , , vector , , , , , , , making For any If this holds true, then all observers are positive, the error system is stable, and the mean dwell time condition is... The wastewater treatment system is non-disruptive and meets the conditions. 。 4. The dual-uninterrupted switching control method for a wastewater treatment system according to claim 3, characterized in that: The positive verification process for the error system and closed-loop control system of the wastewater treatment system in step 6 is as follows: From the gain matrix designed in step 2 and the equation in step 5, we can see that... Furthermore, the error system and closed-loop control system in step 4 can be represented as follows: Combining the gain matrix from step 2 and the positive constraint conditions from step 5, we have: , , , , , and Further, we can obtain , and All are Metzler matrices; For a given nonnegative initial value , and Based on the system described in step 6.2, it can be deduced that... , and Therefore, the error system and closed-loop control system of the sewage treatment system are positive.

5. The dual-uninterrupted switching control method for a wastewater treatment system according to claim 4, characterized in that: The stability verification process of the wastewater treatment system in step 7 is as follows: Choose the linear copositive Lyapunov function as in, , Therefore, for a given switching sequence According to the error system in step 6.2, we can obtain... From the gain matrix in step 2 and the constraints in step 5, we can know that Based on the conditions in step 5 and step 7.2, we can conclude that... Combining steps 7.1 and 7.3, we can deduce... Integrating both sides of step 7.4 simultaneously, we can see that... Combining the Lyapunov function switching conditions in steps 7.5 and 5, we can obtain... in, , Using the average dwell time condition from step 5, we know that: Therefore, the error system of the wastewater treatment system is exponentially stable.

6. The dual-uninterrupted switching control method for a wastewater treatment system according to claim 5, characterized in that: The stability verification process of the closed-loop control system of the wastewater treatment system in step 8 is as follows: Choose the linear copositive Lyapunov function as in, , Therefore, for a given switching sequence According to the closed-loop control system in step 6.2, we can obtain... From the gain matrix in step 3.1 and the constraints in step 5, we can see that... Based on the conditions in step 5 and step 8.2, we can conclude that... Combining steps 8.1 and 8.3, we can deduce... Integrating both sides of step 8.4 simultaneously, we can see that... Based on step 7.6, it can be similarly concluded that the error system of the wastewater treatment system is exponentially stable.

7. The dual-uninterrupted switching control method for a wastewater treatment system according to claim 6, characterized in that: The non-disruption verification process of the closed-loop control system of the wastewater treatment system in step 9 is as follows: According to the unperturbed condition in step 3.2, the following inequality holds: in, This represents the j-th row of the controller gain matrix K. This represents the j-th row of the controller gain matrix K; because , It can be known that Furthermore, according to the non-disturbance condition in step 5, it can be known that... According to the inequality in step 9.2, we can obtain... Therefore, the controller of the wastewater treatment system is disturbance-free.

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