CN115327899B - Filtering estimation method of sewage treatment system under switching signal fault - Google Patents
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- 239000010865 sewage Substances 0.000 title claims abstract description 154
- 238000000034 method Methods 0.000 title claims abstract description 35
- 238000001914 filtration Methods 0.000 title claims abstract description 14
- 230000008569 process Effects 0.000 claims abstract description 16
- 230000001960 triggered effect Effects 0.000 claims abstract description 8
- 239000011159 matrix material Substances 0.000 claims description 29
- 239000013598 vector Substances 0.000 claims description 27
- XLYOFNOQVPJJNP-UHFFFAOYSA-N water Substances O XLYOFNOQVPJJNP-UHFFFAOYSA-N 0.000 claims description 13
- 238000004062 sedimentation Methods 0.000 claims description 11
- 238000005070 sampling Methods 0.000 claims description 8
- 238000004065 wastewater treatment Methods 0.000 claims description 7
- 230000003416 augmentation Effects 0.000 claims description 6
- 238000005259 measurement Methods 0.000 claims description 4
- QVGXLLKOCUKJST-UHFFFAOYSA-N atomic oxygen Chemical compound [O] QVGXLLKOCUKJST-UHFFFAOYSA-N 0.000 claims description 3
- 230000001419 dependent effect Effects 0.000 claims description 3
- 229910052760 oxygen Inorganic materials 0.000 claims description 3
- 239000001301 oxygen Substances 0.000 claims description 3
- SLXKOJJOQWFEFD-UHFFFAOYSA-N 6-aminohexanoic acid Chemical compound NCCCCCC(O)=O SLXKOJJOQWFEFD-UHFFFAOYSA-N 0.000 claims description 2
- 238000010276 construction Methods 0.000 claims description 2
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- 235000013601 eggs Nutrition 0.000 description 2
- 239000003344 environmental pollutant Substances 0.000 description 2
- 239000013505 freshwater Substances 0.000 description 2
- 238000009776 industrial production Methods 0.000 description 2
- 239000002957 persistent organic pollutant Substances 0.000 description 2
- 231100000719 pollutant Toxicity 0.000 description 2
- 230000004083 survival effect Effects 0.000 description 2
- 241000894006 Bacteria Species 0.000 description 1
- 241000238631 Hexapoda Species 0.000 description 1
- OAICVXFJPJFONN-UHFFFAOYSA-N Phosphorus Chemical compound [P] OAICVXFJPJFONN-UHFFFAOYSA-N 0.000 description 1
- NINIDFKCEFEMDL-UHFFFAOYSA-N Sulfur Chemical compound [S] NINIDFKCEFEMDL-UHFFFAOYSA-N 0.000 description 1
- 241000700605 Viruses Species 0.000 description 1
- 238000005660 chlorination reaction Methods 0.000 description 1
- 238000010586 diagram Methods 0.000 description 1
- 238000012851 eutrophication Methods 0.000 description 1
- 230000002550 fecal effect Effects 0.000 description 1
- 229910001385 heavy metal Inorganic materials 0.000 description 1
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- 229910052751 metal Inorganic materials 0.000 description 1
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- 229910052698 phosphorus Inorganic materials 0.000 description 1
- 239000011574 phosphorus Substances 0.000 description 1
- 150000003839 salts Chemical class 0.000 description 1
- 239000013049 sediment Substances 0.000 description 1
- 239000007787 solid Substances 0.000 description 1
- 238000004659 sterilization and disinfection Methods 0.000 description 1
- 239000000126 substance Substances 0.000 description 1
- 229910052717 sulfur Inorganic materials 0.000 description 1
- 239000011593 sulfur Substances 0.000 description 1
- 231100000167 toxic agent Toxicity 0.000 description 1
- 239000002699 waste material Substances 0.000 description 1
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- G05B—CONTROL OR REGULATING SYSTEMS IN GENERAL; FUNCTIONAL ELEMENTS OF SUCH SYSTEMS; MONITORING OR TESTING ARRANGEMENTS FOR SUCH SYSTEMS OR ELEMENTS
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Abstract
The invention discloses a filtering estimation method of a sewage treatment system under a switching signal fault, which utilizes a positive switching system to model a sewage treatment process of the sewage treatment system, designs an event trigger filter of the sewage treatment system and estimates an execution mode and an operation state of the sewage treatment system. Firstly, a direct-conversion system model of sewage treatment is established by data acquisition of a sewage treatment system. Secondly, an event-triggered filter of the sewage treatment system is designed. Then, an execution mode error evaluation function is constructed to evaluate an execution mode of the event-triggered filter in the event of a switching signal failure. Compared with the filtering estimation technology of the existing sewage treatment system, the method can effectively solve the problems of system operation state and execution model estimation when the switching signal of the sewage treatment system is in fault, avoid unstable operation and high energy consumption of the system when the switching signal is in fault in the sewage treatment process, and ensure that the sewage treatment system is operated safely and stably.
Description
Technical Field
The invention belongs to the field of filtering estimation methods of sewage treatment systems, and particularly relates to a filtering estimation method of a sewage treatment system under a switching signal fault.
Background
Water is one of important matters for human survival and development, and is mainly reflected in metabolism, industrial production and other aspects of human body. With the continuous increase of the urban level and the industrialized level, the demand for fresh water resources is increasing, and naturally, the sewage quantity is also increasing with the increase of the fresh water demand. The sewage in China mainly comprises three types of production sewage, domestic sewage and polluted rainwater, wherein the production sewage can be divided into industrial sewage, agricultural sewage and medical sewage, and the industrial sewage is the main part. As shown in fig. 1, for sewage treatment, a two-stage treatment scheme is generally adopted, and the one-stage treatment is carried out in a primary sedimentation tank, so that suspended solid pollutants such as sediment and the like which are not easily dissolved in water are mainly removed; the secondary treatment is carried out in a secondary sedimentation tank, and mainly removes organic pollutants in the sewage; however, the sewage containing the insect eggs and other pollutants needs to be subjected to advanced treatment such as chlorination and disinfection so as to achieve the effect of not affecting the ecological environment. At present, industrial sewage and domestic sewage treatment in production sewage is an important treatment object in sewage treatment industry. Heavy metals, toxic compounds and the like in the production sewage seriously affect the water environment. Domestic sewage mainly refers to sewage produced by human life, and because the sewage contains organic pollutants such as sulfur, phosphorus and the like, water eutrophication is easy to cause, so that water quality is malodorous, and the sewage also easily contains microorganisms such as worm eggs, various viruses, bacteria and the like, various metal matters and salt substances, if the sewage is improperly treated and carelessly used by human beings, the sewage can cause great harm to human bodies. Therefore, sewage treatment has great significance for survival and development of human beings.
There is a class of systems in the field of automatic control that are non-negative for any initial state and input, all states and outputs of the system are non-negative, and such systems are referred to as positive systems. The state and output of the positive system have non-negative characteristics, so that the change process of non-negative actual variables such as sewage quantity, COD, BOD 5, fecal coliform number and the like in the modeling sewage treatment process of the positive system has obvious advantages, and modeling redundancy can be avoided. Because the change condition of the sewage amount flowing into the sewage treatment system is greatly influenced by factors such as time, weather, seasons and the like, for example, the domestic sewage amount to be treated flowing in the morning and evening time period is higher than that in other time periods, and the demand of the product produced in industrial production for the sewage amount to be treated flowing in is greatly influenced. The busy time mode and the idle time mode are generally used for respectively representing the operation condition of the sewage treatment system, namely, when the inflow sewage to be treated is large, the sewage treatment system is in the busy time mode, and the rest time is in the idle time mode, so that the sewage treatment system is actually a switching process, and the sewage treatment process can be described by modeling a direct-swing system model. Whether the switching signal can normally run or not determines whether the sewage treatment system can accurately realize free switching between a busy hour mode and an idle hour mode. When the mode of the sewage treatment system is required to be switched to a busy hour mode and the switching signal fails due to the remarkably increased amount of the sewage to be treated, namely, the sewage treatment system is used for treating the sewage which can be treated only in the busy hour mode by using the idle hour mode but the switching signal cannot be switched normally, the sewage treatment system can have the problems that the water level of a sewage treatment tank is too high, the overflow of the sewage treatment tank is caused, the quality of the treated sewage is reduced, and the like, so that the sewage treatment effect of the sewage treatment system is reduced slightly, and the sewage treatment system is crashed seriously. In addition, the corresponding filter established based on the time triggering mechanism (periodic sampling mechanism) can frequently acquire the state variable information of the sewage treatment system in estimating the running state of the sewage treatment system, and the system can cause high energy consumption and development and implementation costs. Therefore, based on the theory of the positive switching system and an event triggering mechanism, the filtering estimation method of the sewage treatment system under the fault of the switching signal has important economic value.
In summary, we propose a filtering estimation method of a sewage treatment system under a switching signal fault, which can accurately estimate the operation state of the sewage treatment system when the switching signal fails. When the sewage treatment is in a busy hour mode, the problems of sewage overflow, unstable sewage treatment system and unsatisfactory treatment effect caused by overlarge inflow sewage can be avoided for each sewage treatment pool, so that the estimation effect of the sewage treatment system when the switching signal fails is improved. Compared with the existing filtering estimation method of the sewage treatment system, the method can reduce the redundancy of the model in the sewage treatment process based on the theory modeling of the positive switching system, can reduce the development and implementation cost in the sewage treatment process based on the event triggering mechanism, solves the problems of resource waste and high development and implementation cost generated in the high-frequency sampling process of the mechanical time triggering mechanism, accurately estimates the mode of the switching signal, ensures that the sewage treatment system can safely and stably operate on the premise of not reducing the sewage treatment quality, and is a better solution.
Disclosure of Invention
The invention aims to solve the problem of filtering estimation of the running state of a sewage treatment system when a switching signal is in fault, avoid the problems of instability of the sewage treatment system and high energy consumption of the sewage treatment system in the sewage treatment process under the switching signal fault, and provides a filtering estimation method of the sewage treatment system under the switching signal fault. Based on an event triggering strategy, an observer and a filter design technology, by designing a state observer and an event triggering filter, the execution mode of the sewage treatment system is estimated when the switching signal fails, the reliability of the sewage treatment system is improved, the safe and stable operation of sewage treatment is realized, the accurate estimation performance under the switching signal failure is improved, and the sewage treatment quality is improved. The specific technical scheme is as follows:
a filtering estimation method of a sewage treatment system under a switching signal fault comprises the following steps:
Step 1, establishing a state space model of a sewage treatment system;
Step 2, designing an event trigger filter of the sewage treatment system;
and 3, estimating an execution mode of the event trigger filter under the fault of the switching signal.
Further, the step 1 specifically comprises the following steps:
by collecting water quantity data of the sewage treatment system, a state space model of the sewage treatment system is established:
x(k+1)＝Aσ(k)x(k)+Bσ(k)ω(k)
y(k)＝Cσ(k)x(k)+Dσ(k)ω(k)
z(k)＝Eσ(k)x(k)+Fσ(k)ω(k)
Wherein sigma (k) represents a switching signal of the sewage treatment system, controlling an operation mode of the sewage treatment system, and generally consists of a busy hour mode and an idle hour mode; Representing the water quantity of each secondary sedimentation tank of the sewage treatment system, wherein n represents the number of the secondary sedimentation tanks of the sewage treatment system; a variable which affects the sigma (k) th subunit of the sewage treatment system at the time k, for example, an external environment variable which affects the sewage treatment process of the sewage treatment system, such as the pH value, the temperature, the dissolved oxygen amount of a secondary sedimentation tank of the sewage treatment system, and the like, and the number of the external environment variables represented by m; An output representing a measurement of the wastewater treatment system at time k; an estimated output representing an operating state of the wastewater treatment system at time k; And The system matrix representing proper dimension can be acquired by the actual sewage treatment process; for each sigma (k) epsilon S, A σ(k)≥0,Bσ(k) is more than or equal to 0, more than or equal to the sum of all elements in matrix A σ(k),Bσ(k) are non-negative; sigma (k): [0, infinity) →s 1 = {1,2,., N } is a function dependent on switch point k l, is a switch signal of the system; let σ (k) =i, i e S 1, then there is a σ(k)＝Ai,Bσ(k)＝Bi; representing an n-dimensional vector and an n x n-dimensional euclidean matrix space, respectively.
Further, the step2 specifically comprises the following steps:
event trigger control framework for establishing sewage treatment system
‖ey(k)‖1＞β‖y(k)‖1,k∈[k't,k't+1)
Where the constant beta > 0, e y (k) is the sampling error, Is a valid sampling output; the 1-norm operator for the vector is adopted by 1, and the requirements are satisfiedY l (k) is the first element of y (k); k' t is the time of the t-th event trigger.
Further, the step 3 specifically comprises the following steps:
Step 3.1 designing event triggered Filter as
Wherein the method comprises the steps of Is the firstAt the moment of the secondary handover,
AndIs the gain matrix of the event triggered filter of the sigma f (k) th subsystem to be designed; sigma f (k) is a switching signal of the sewage treatment system;
step 3.2 designing a state observer of the sewage treatment system to be
Wherein,To the system state x (k)The seed estimation is performed in a way that, Is the first to be designedGain matrices for the individual observers;
Order the An observation error system can be established as
Then
Wherein,When the switching signal to be estimated is consistent with the actual switching signal, i.eSetting the convergence time of the system equation (1) as T c;
step 3.3 establishing a switching signal estimation framework for the fault of the switching signal of the sewage treatment system as
Wherein,And is also provided withReams theex(k)＝xf(k)-x(k),ez(k)＝zf(k)-z(k)；
Step 3.4 obtaining the error augmentation system as
Wherein,The corresponding gain matrix satisfies
When (when)In this case, the available error augmentation system is
Wherein,
Step 3.5 establishing an event trigger filter positive condition under a switching signal fault of the sewage treatment system as follows
Wherein,The constant satisfies 0 < beta <1, lambda > 1; Vector quantity ξiι≥0,ρiι≥0；Vector delta iι is more than or equal to 0,Theta iι is more than or equal to 0; matrix m=i- β1 m×m,H＝I+β1m×m;
Step 3.6 construction of a Linear Yu Zhengli Ubbunov function
Wherein,Is a3 n-dimensional real column vector and each element in the column is a positive number; the differential equation for calculating the Lyapunov function is
Wherein,
Step 3.7 establishing an event trigger filter stability condition under a switching signal fault of the sewage treatment system as follows
Wherein,The constant satisfies 0 < beta < 1,0 < mu 1＜1, μ2 > 1, lambda > 1; Vector quantity ξi≥0；Vector delta iι≥0,δi is more than or equal to 0,Matrix m=i- β1 m×m,H＝I+β1m×m;
step 3.8 establishment The gain performance index function is
Γ(k)＝γ‖ω(k)‖1-‖y(k)‖1
Thereby having the following characteristics
Step 3.9 establishmentThe gain performance index condition is
Wherein,The constant satisfies 0 < beta < 1, gamma > 0; Vector quantity Vector delta i is more than or equal to 0,Matrix h=i+β1 m×m;
step 3.10 is obtained from step 3.5 to step 3.9,
When (when)In the time-course of which the first and second contact surfaces,
When (when)In the time-course of which the first and second contact surfaces,
Step 3.11 the gain matrix of the observer and filter can be obtained as follows from the conditions established in steps 3.5, 3.6 and 3.9
The corresponding average residence time condition is satisfied
Aiming at the problem of unstable operation of the current sewage treatment system when a switching signal fails, the invention provides an event triggering filtering estimation method for estimating the mode of the switching signal, namely an execution mode of the sewage treatment system; firstly, a state space model is established based on a positive switching system theory, then, an observer and an event triggering filter of a state variable of the sewage treatment system are designed by adopting a Lyapunov function method, the expected performance index of the sewage treatment system is achieved, and finally, safe and stable operation and accurate estimation of the operation state of the sewage treatment system in the case of switching signal faults are realized.
Drawings
FIG. 1 is a schematic diagram of a prior art wastewater treatment system;
FIG. 2 is a sewage treatment system to which the method of the present invention is applied.
Detailed Description
The invention is further described below with reference to the accompanying drawings.
As shown in fig. 2, the state space model is established by taking the water quantity flowing into and out of each sewage treatment tank by the sewage treatment system as a state variable and taking the measurement of a sensor as an output.
Step 1, firstly, establishing a state space model of a sewage treatment system by collecting water quantity data of the sewage treatment system:
x(k+1)＝Aσ(k)x(k)+Bσ(k)ω(k)
y(k)＝Cσ(k)x(k)+Dσ(k)ω(k)
z(k)＝Eσ(k)x(k)+Fσ(k)ω(k)
Wherein sigma (k) represents a switching signal of the sewage treatment system, controlling an operation mode of the sewage treatment system, and generally consists of a busy hour mode and an idle hour mode; representing the water quantity of each secondary sedimentation tank of the sewage treatment system, wherein n represents the number of the secondary sedimentation tanks of the sewage treatment system; a variable which affects the sigma (k) th subunit of the sewage treatment system at the time k, for example, an external environment variable which affects the sewage treatment process of the sewage treatment system, such as the pH value, the temperature, the dissolved oxygen amount of a secondary sedimentation tank of the sewage treatment system, and the like, and the number of the external environment variables represented by m; An output representing a measurement of the wastewater treatment system at time k; an estimated output representing an operating state of the wastewater treatment system at time k; And The system matrix representing the proper dimension can be acquired by the actual sewage treatment process. For each σ (k) ∈S, there is A σ(k)≥0,Bσ(k) +.0 (the relational notation ε.gtoreq.is valid for each element in matrix A σ(k),Bσ(k), i.e., all elements within the matrix are non-negative). Sigma (k): [0, infinity) →s 1 = {1,2,..n } is a function dependent on the switching point k l, which is the switching signal of the system. For convenience, note σ (k) =i, i e S 1, then there is a σ(k)＝Ai,Bσ(k)＝Bi; representing an n-dimensional vector and an n x n-dimensional euclidean matrix space, respectively.
Step 2, establishing an event trigger control frame of the sewage treatment system
‖ey(k)‖1＞β‖y(k)‖1,k∈[k't,k't+1)
Where the constant beta > 0, e y (k) is the sampling error, Is a valid sampling output; the 1-norm operator for the vector is adopted by 1, and the requirements are satisfiedY l (k) is the first element of y (k); k' t is the time of the t-th event trigger.
And 3, designing an event trigger filter of the sewage treatment system, wherein the event trigger filter mainly comprises the following substeps.
Step 3.1 designing event triggered Filter as
Wherein the method comprises the steps of Is the firstAt the moment of the secondary handover, AndIs the gain matrix of the event triggered filter of the sigma f (k) th subsystem to be designed; σ f (k) is a switching signal of the sewage treatment system.
Step 3.2 designing a state observer of the sewage treatment system to be
Wherein,To the system state x (k)The seed estimation is performed in a way that, Is the first to be designedGain matrix of each observer. Order theAn observation error system can be established as
Thus there is
Wherein,When the switching signal to be estimated is consistent with the actual switching signal, i.eLet the convergence time of system equation (1) be T c.
Step 3.3 establishing a switching signal estimation framework for the fault of the switching signal of the sewage treatment system as
Wherein,And is also provided withReams theex(k)＝xf(k)-x(k),ez(k)＝zf(k)-z(k)。
Step 3.4 obtaining the error augmentation system as
Wherein,The corresponding gain matrix satisfies
When (when)In this case, the available error augmentation system is
Wherein,
Step 3.5 establishing an event trigger filter positive condition under a switching signal fault of the sewage treatment system as follows
Wherein,The constant satisfies 0 < beta <1, lambda > 1; Vector quantity ξiι≥0,ρiι≥0；Vector delta iι is more than or equal to 0,Theta iι is more than or equal to 0; matrix m=i- β1 m×m,H＝I+β1m×m.
Step 3.6 constructing a Linear Yu Zhengli Ubbunov function
Wherein,Is a3 n-dimensional real column vector and each element in the column is a positive number. The differential equation for calculating the Lyapunov function is
Wherein,
Step 3.7 establishing an event trigger filter stability condition under a switching signal fault of the sewage treatment system as follows
Wherein,The constant satisfies 0 < beta < 1,0 < mu 1＜1, μ2 > 1, lambda > 1; Vector quantity ξi≥0；Vector delta iι≥0,δi is more than or equal to 0,Matrix m=i- β1 m×m,H＝I+β1m×m.
Step 3.8 establishmentThe gain performance index function is
Γ(k)＝γ‖ω(k)‖1-‖y(k)‖1
Thereby having the following characteristics
Step 3.9 establishmentThe gain performance index condition is
Wherein,The constant satisfies 0 < beta < 1, gamma > 0; Vector quantity Vector delta i is more than or equal to 0,Matrix h=i+β1 m×m.
Step 3.10 is obtained from step 3.5 to step 3.9,
When (when)In the time-course of which the first and second contact surfaces,
When (when)In the time-course of which the first and second contact surfaces,
Step 3.11 the gain matrix of the observer and filter can be obtained as follows from the conditions established in steps 3.5, 3.6 and 3.9
The corresponding average residence time condition is satisfied
Claims (1)
1. A filtering estimation method of a sewage treatment system under a switching signal fault is characterized by comprising the following steps:
Step 1, establishing a state space model of a sewage treatment system;
Step 2, designing an event trigger filter of the sewage treatment system;
step 3, estimating an execution mode of an event trigger filter under the fault of a switching signal;
The step 1 is specifically as follows:
by collecting water quantity data of the sewage treatment system, a state space model of the sewage treatment system is established:
x(k+1)＝Aσ(k)x(k)+Bσ(k)ω(k)
y(k)＝Cσ(k)x(k)+Dσ(k)ω(k)
z(k)＝Eσ(k)x(k)+Fσ(k)ω(k)
Wherein sigma (k) represents a switching signal of the sewage treatment system, controlling an operation mode of the sewage treatment system, and generally consists of a busy hour mode and an idle hour mode; Representing the water quantity of each secondary sedimentation tank of the sewage treatment system, wherein n represents the number of the secondary sedimentation tanks of the sewage treatment system; a variable which affects the sigma (k) th subunit of the sewage treatment system at the time k, for example, an external environment variable which affects the sewage treatment process of the sewage treatment system, such as the pH value, the temperature, the dissolved oxygen amount of a secondary sedimentation tank of the sewage treatment system, and the like, and the number of the external environment variables represented by m; An output representing a measurement of the wastewater treatment system at time k; an estimated output representing an operating state of the wastewater treatment system at time k; And The system matrix representing proper dimension can be acquired by the actual sewage treatment process; for each sigma (k) epsilon S, A σ(k)≥0,Bσ(k) is more than or equal to 0, more than or equal to the sum of all elements in matrix A σ(k),Bσ(k) are non-negative; sigma (k): [0, infinity) →s 1 = {1,2,., N } is a function dependent on switch point k l, is a switch signal of the system; let σ (k) =i, i e S 1, then there is a σ(k)＝Ai,Bσ(k)＝Bi; Respectively representing n-dimensional vectors and n multiplied by n-dimensional Euclidean matrix spaces;
The step 2 is specifically as follows:
event trigger control framework for establishing sewage treatment system
‖ey(k)‖1＞β‖y(k)‖1,k∈[k't,k't+1)
Where the constant beta > 0, e y (k) is the sampling error, (K) Is a valid sampling output; the 1-norm operator for the vector is adopted by 1, and the requirements are satisfiedY l (k) is the first element of y (k); k' t is the time of the t time event trigger;
The step 3 is specifically as follows:
Step 3.1 designing event triggered Filter as
Wherein the method comprises the steps of Is the firstAt the moment of the secondary handover, AndIs the gain matrix of the event triggered filter of the sigma f (k) th subsystem to be designed; sigma f (k) is a switching signal of the sewage treatment system;
step 3.2 designing a state observer of the sewage treatment system to be
Wherein,To the system state x (k)The seed estimation is performed in a way that, Is the first to be designedGain matrices for the individual observers;
Order the An observation error system can be established as
Then
Wherein,When the switching signal to be estimated is consistent with the actual switching signal, i.eSetting the convergence time of the system equation (1) as T c;
step 3.3 establishing a switching signal estimation framework for the fault of the switching signal of the sewage treatment system as
Wherein,And is also provided withReams theex(k)＝xf(k)-x(k),ez(k)＝zf(k)-z(k)；
Step 3.4 obtaining the error augmentation system as
Wherein,The corresponding gain matrix satisfies
When (when)In this case, the available error augmentation system is
Wherein,
Step 3.5 establishing an event trigger filter positive condition under a switching signal fault of the sewage treatment system as follows
Wherein,Iota = 1,2, n, j = 1,2, …, m; the constant satisfies 0 < beta < 1, lambda > 1; vector p i＞0,qi＞0,hi＞0,ξiι≥0,ρi is more than or equal to 0; Vector delta iι is more than or equal to 0; Theta iι is more than or equal to 0; matrix m=i- β1 m×m,H＝I+β1m×m;
Step 3.6 construction of a Linear Yu Zhengli Ubbunov function
Wherein,Is a 3 n-dimensional real column vector and each element in the column is a positive number; the differential equation for calculating the Lyapunov function is
Wherein,
Step 3.7 establishing an event trigger filter stability condition under a switching signal fault of the sewage treatment system as follows
ξi-μ1hi＜0
ξi-μ2hj,i＜0
pi≤λpi,j,pi≤λpj,i,pi,j≤λpi,pj,i≤λpi,qi≤λqi,j,qi≤λqj,i
qi,j≤λqi,qj,i≤λqi,hi≤λhi,j,hi≤λhj,i,hi,j≤λhi,hj,i≤λhi
Wherein,l＝1,2,...,n,The constant satisfies 0 < beta < 1,0 < mu 1＜1,μ2 > 1, lambda > 1; vector p i＞0,pj,i＞0,qi＞0,qj,i＞0,hi＞0,hj,i＞0,ξi is more than or equal to 0; vector delta i≥0,δi is more than or equal to 0, Matrix m=i- β1 m×m,H＝I+β1m×m;
Step 3.8 establishing the l 1 gain performance index function as
Γ(k)＝γ‖ω(k)‖1-‖y(k)‖1
Thereby having the following characteristics
Step 3.9 establishing the L 1 gain performance index condition as follows
Wherein,ι＝1,2,...,n,The constant satisfies 0 < beta < 1, gamma > 0; Vector p i＞0,pj,i＞0,hi＞0,hj,i > 0; vector delta i is more than or equal to 0, Matrix h=i+β1 m×m;
step 3.10 is obtained from step 3.5 to step 3.9,
When (when)In the time-course of which the first and second contact surfaces,
When (when)In the time-course of which the first and second contact surfaces,
Step 3.11 the gain matrix of the observer and filter can be obtained as follows from the conditions established in steps 3.5, 3.6 and 3.9
The corresponding average residence time condition is satisfied
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