A quantitative estimation method for faults in nonlinear water supply systems based on event-triggered mechanism

Abstract

The present invention discloses a method for quantitative estimation of faults in a nonlinear water system based on an event trigger mechanism. The steps of the present invention are as follows: Step 1, establishing a state space model of an urban water supply system; Step 2, establishing event trigger conditions for an urban water supply system; Step 3, establishing a model for quantizing output signals in an urban water supply system based on an event trigger strategy; Step 4, establishing a nonlinear asynchronous filter model with output quantization; Step 5, establishing an error system for an asynchronous switching filter; Step 6, designing a nonlinear asynchronous filter based on event triggering for an urban water supply system. The present invention is helpful in formulating a more scientific scheduling scheme, thereby effectively alleviating the unstable water supply and interruption of water flow during peak water use periods. The present invention can not only avoid the waste of system resources caused by periodic sampling of water flow, but also reduce the design cost.

Description

Technical Field

[0001] The present invention belongs to the field of automation technology, relates to a nonlinear water system fault quantitative estimation method based on an event trigger mechanism, and proposes a nonlinear asynchronous filtering method based on an event trigger strategy, thereby realizing the prediction of user water consumption demand. Background Art

[0002] Water is the source of life, and people's daily production and life are inseparable from water. Especially in recent years, with the continuous improvement of urbanization level, the number of urban population is becoming increasingly saturated, and the demand for water is also increasing. The urban water supply system is an integral part of urban public utilities. As the infrastructure of urban development, its role is self-evident. However, due to the continuous development of industrialization, the intensified environmental pollution and people's lack of conservation awareness, the problem of water shortage has attracted widespread attention. Coupled with the low utilization rate of water resources, the urban water supply system is under great pressure. Nowadays, the scale of water supply is gradually expanding, the system formed is more complex, and the management and scheduling are more difficult. Any unreasonableness will lead to abnormal water supply system. For example, during the peak water use period, some high-rise residents will experience insufficient water pressure, unstable water flow and even water outage, which brings great inconvenience to the daily life of citizens. Therefore, it is very necessary to establish a more accurate model of the current water supply system, introduce new science and technology, optimize scheduling management, and maximize the benefits of the water supply system.

[0003] Optimal scheduling refers to predicting the water consumption in the next time period when the water pressure is stable, the water quality meets the standard and the water supply is normal, and then formulating a scientific scheduling plan to maximize social and economic benefits. The prerequisite is to predict the water consumption. Therefore, it is a key issue to establish an appropriate water supply system model, design a corresponding filter, and estimate the water consumption. Considering the non-negativity of water flow in the actual water supply system, the positive system modeling is more accurate than the general system. In a multi-node water supply system, since different users have different water consumption in different time periods and the water consumption shows nonlinear characteristics, the nonlinear positive switching system can effectively describe the process of water consumption changes.

[0004] Event triggering is an efficient signal transmission strategy. The event triggering strategy can effectively solve the problem of unreasonable water supply during peak water consumption. By giving priority to supplying water to certain areas or floors, the efficient operation and maximum benefit of the water supply system can be achieved. In addition, in the event triggering mechanism, the triggering conditions based on the current measurement value are continuously monitored. When the conditions are met, the event is triggered, which effectively improves the problem of system resource waste and high design cost caused by periodic sampling. In an ideal control system, people expect the switching of the system filter to be synchronized with the switching of the actual control system. However, due to the limited capabilities of components, the sensor needs a certain amount of time to identify the correct model, which causes a delay in the switching of the filter. Therefore, we designed an asynchronous filter based on event triggering. In addition, the sensors in the actual system may have intermittent failures, resulting in the quantization of the measured signal. In order to adapt to this process, we designed an event-triggered quantizer to generate an output signal that matches the actual system.

[0005] The event-triggered asynchronous filter design method for switched nonlinear positive systems proposed by us can predict the water consumption of actual water supply systems, which is of great significance for solving the problems of unstable water supply and interruption of water supply in urban water supply systems during peak water consumption periods. Summary of the invention

[0006] The present invention proposes a method for quantitatively estimating faults of nonlinear water systems based on an event trigger mechanism. Based on a nonlinear positive switching model, an event trigger strategy, and a design method for an asynchronous filter, the present invention collects data on the water supply of an urban water supply system and proposes a design method for ensuring efficient water supply of an urban water supply system. Even if there are multiple interference factors in the urban water supply system, the design method can well improve the performance of the urban water supply system and improve the efficiency of the urban water supply system, thereby effectively improving problems such as unstable urban water supply or even water interruption.

[0007] The technical solution adopted by the present invention to solve the problem includes the following steps:

[0008] Step 1: According to the urban water supply system, construct its state space model. The specific method is:

[0009] 1.1 Collect the input and output data of the urban water supply system to describe the actual operation process of the system:

[0010] Considering the operation process of urban water supply system, urban water supply system usually consists of water source, pumping station, valve and water supply network, such as Figure 1 Schematic diagram of the urban water supply system (see attached figure in the specification). Figure 1The connection between the various components of the urban water supply system is shown. The blue arrows represent the urban water supply network, and the black line segments represent the distribution network that delivers water from the source to the user. The stored water from the water source is sent to the water plant through the water transmission pipeline for water quality treatment. The treated water is pressurized and delivered to the user through the distribution network. During the peak period of urban water use, high-rise residents generally experience unstable water flow or even water outages. Therefore, it is of great significance to estimate the user's water demand in the next time period in advance and then make a scientific scheduling plan to solve this problem. Considering the non-negative nature of water flow in the actual system and the nonlinearity of user water consumption, we abstract the actual system into a nonlinear switching positive system model. Since the sensors in the actual system may have intermittent faults, we quantize the output that meets certain conditions through a quantizer to improve the accuracy of the established model. The estimated system state (user's water demand in the next time period) is obtained through a nonlinear asynchronous filter with output quantization. Figure 2 Shown is a framework for event-triggered nonlinear filters with output quantization.

[0011] 1.2 Collect data from the urban water supply system and use the data to establish a state space model of the water flow of the urban water supply system:

[0012] x(k+1)=A σ(k) f(x(k))+B σ(k) g(ω(k)),

[0013] y(k)=C σ(k) h(x(k))+D σ(k) l(ω(k)),

[0014] z(k)=E σ(k) p(x(k))+F σ(k) q(ω(k)),

[0015] where x(k)=[x1(k),x2(k),...,x n (k)] T ∈R n is the water consumption of users in the area at time k, and n represents the number of pumping stations in the urban water supply system; is the external disturbance of the urban water supply system (such as a surge in water consumption at a certain moment, failure of system components, etc.); y(k)∈R m is the actual output collected by the sensor at time k, m represents the number of sensors measuring the output; z(k)∈R sis the estimated output of the user's water consumption at time k, and s represents the number of simulated pumping stations in the constructed model. Function σ(k) represents the switching signal, which is a mapping from the interval [0,∞) to the finite set S = {1,2,…,N+}. When σ(k) = i, i∈S, the i-th subsystem is activated, and its corresponding system matrix is ​​expressed as A i ,B i ,C i ,D i ,E i ,F i , and the matrix R m ,R n ,R s , N + They represent m-dimensional, n-dimensional, s-dimensional, n-dimensional non-negative, m-dimensional non-negative vectors and positive integer sets respectively.

[0016] 1.3f(x),g(ω),h(x),l(ω),p(x),q(ω) are all nonlinear functions used in the modeling of the city water supply system, f(x)=(f1(x1),......,f n (x n )) T ,g(ω)=(g1(ω1),......,g m (ω m )) T ,h(x)=(h1(x1),......,h n (x n )) T ,l(ω)=(l1(ω1),......,l m (ω m )) T ; p(x)=(p1(x1),......,p n (x n )) T ,q(ω)=(q1(ω1),......,q m (ω m )) T For any x i ∈R,ω t ∈R, i=1,2,...,n, ι=1,2,...,m, R represents a set of real numbers, all of which satisfy the following fan-shaped region conditions:

[0017]

[0018]

[0019]

[0020]

[0021]

[0022]

[0023] in,

[0024] 0<ε1<ε2,0<ε3<ε4,0<ε5<ε6,f i (0)=0.

[0025] Step 2: Establish event trigger conditions for the urban water supply system:

[0026] ||e y (k)||1>β||y(k)||1,

[0027] Among them, β∈[0,1) is the event trigger coefficient, which is a given constant. y (k) is the sampling error, y(k l ) is the urban water supply system at the event triggering time k l The output value, l∈N + , y(k) represents the output value of the urban water supply system at time k, and ||·||1 represents the 1-norm of the vector, that is, the sum of the absolute values ​​of all elements in the vector.

[0028] Step 3: Establish a model for quantifying output signals based on event-triggered strategies in urban water supply systems:

[0029] 3.1 Use a quantizer to quantize the event-triggered output signal of the system in step 1.2. The form of the quantized output signal is as follows:

[0030]

[0031] in, is the output signal of the event generator, express The quantized signal, Represents a subquantizer.

[0032] 3.2 Sub-quantizer The quantization level of is described in the following set form:

[0033] u c ={φ c |φ c =κ c φ c0},

[0034] Define a specific sub-quantizer model for the subsystem:

[0035]

[0036] Among them, 0<κ c <1,φ c0 >0,κ c is a constant, φ c Defines the quantization level, φ c0 is the initial quantization level of the cth subsystem, Represents a constant.

[0037] 3.3 The output quantized signal received by the filter is in the following form:

[0038]

[0039] in, Satisfies the quantization error fan-shaped area expression: I is the identity matrix, Δ(k)=diag{Δ1(k),Δ2(k),...,Δ m (k)},|Δ c (k)|≤∈ c .

[0040] Step 4: Establish a nonlinear asynchronous filter model with output quantization, the structure of which is as follows:

[0041]

[0042]

[0043] Among them, x f (k) represents the state of the filter, z f (k0 represents the filter’s estimate of the system model output signal z(k), σ f (k) is the switching signal of the filter, denoted by σ f (k) = j means the filter is asynchronous with the subsystem, σ f (k) = i means that the filter is synchronized with the subsystem, and for any switching time k r ,r=0,1,2,...,σ f (k r )=σ(k r )+Δ r ,Δ r is the filter lag time, Δ r <k r+1 -k r . is the filter gain matrix to be designed. At asynchronous time k∈[k r ,kr +Δ r ), the subsystem and the filter are in an asynchronous state, that is, when σ(k)=i, σ f (k) = j; at synchronization time k∈[k r +Δ r ,k r +1), the subsystem and the filter are in synchronization, that is, when σ(k)=i, σ f (k) = i. is a fan-shaped bounded nonlinear function used to estimate the nonlinear function f(k). satisfy: and

[0044] Step 5: Establish the error system of the asynchronous switching filter, as follows:

[0045] 5.1 Order e(k)=z f (k)-z(k). According to steps 1.2 and 4, the following error system can be obtained:

[0046] When k∈[k r ,k r +Δ r )

[0047]

[0048] e(k)=E fj p(x f (k))+F fj (I+Δ(k))(C i h(x(k))+D i l(ω(k))+e y (k))-E i p(x(k))-F i q(ω(k)),

[0049] When k∈[k r +Δ r ,k r+1 )

[0050]

[0051] e(k)=E fi p(x f (k))+F fi (I+Δ(k))(C i h(x(k))+D i l(ω(k))+e y (k))-Ei p(x(k))-F i q(ω(k)),

[0052] Among them, x T (k) represents the transpose of vector x(k).

[0053] 5.2 Let Λ=diag{∈1,∈2,...,∈ m}, then Where L = I-Λ, J = I+Λ. According to step 1.3, the error system in synchronous and asynchronous states can be further expressed as:

[0054] When k∈[k r ,k r +Δ r )

[0055]

[0056]

[0057]

[0058]

[0059] When k∈[k r +Δ r ,k r+1 )

[0060]

[0061]

[0062]

[0063]

[0064] in,

[0065]

[0066]

[0067]

[0068]

[0069]

[0070]

[0071]

[0072]

[0073] Step 6: Design an event-triggered nonlinear asynchronous filter for the urban water supply system:

[0074] The filter gain matrix designed in 6.1 is:

[0075]

[0076] in, represents an n-dimensional column vector, δ iι , represents an m-dimensional column vector; 1 n represents an n-dimensional column vector whose elements are all 1, represents an n-dimensional column vector whose ιth element is 1 and the rest of the elements are all 0. Indicates An s-dimensional column vector with 1 element and all other elements being 0; Respectively 1 n is the transpose of ; ∑ is the summation symbol.

[0077] 6.2 Design constants

[0078] 0<ε1≤ε2,0<ε3≤ε4,0<ε5≤ε6, γ>0,λ>1,0≤β<1,0<μ1<1,μ2>1,if there exists R n vector and R m vector So that the following inequality holds:

[0079]

[0080]

[0081]

[0082]

[0083]

[0084]

[0085]

[0086]

[0087]

[0088]

[0089]

[0090]

[0091]

[0092]

[0093]

[0094]

[0095]

[0096] Then for any (i,j)∈S,i≠j,ι=1,2,…,n, …, s, the error system constructed in step 5.1 is positive and l1-gain stable under the conditions of satisfying the filter gain matrix designed in step 6.1 and the switching rate designed in step 6.3.

[0097] 6.3 Design the switching rate of the positive switching system as follows:

[0098]

[0099]

[0100] Among them, Γ - (k0, k), Γ + (k0, k) represent the total time of synchronous and asynchronous operation of the system and filter, respectively, τ a represents the average dwell time of the switching system, Δ max Indicates the maximum lag time of the filter.

[0101] 6.4 Combining steps 2, 5.2, 6.1 and 6.2, we can obtain that to ensure the positivity of the error system, the following conditions need to be met:

[0102] When k∈[k r , k r +Δ r )

[0103]

[0104]

[0105] When k∈[k r +Δ r ,kr +1)

[0106]

[0107]

[0108] in,

[0109]

[0110]

[0111]

[0112]

[0113] 6.5 According to steps 2 and 5, to ensure the l1 gain stability of the error system, it is necessary to satisfy:

[0114] When k∈[k r , k r +Δ r )

[0115]

[0116]

[0117] When k∈[k r +Δ r , k r +1), there is

[0118]

[0119]

[0120] in,

[0121]

[0122]

[0123]

[0124]

[0125] 6.6Construct a piecewise redundant positive Lyapunov function as follows:

[0126]

[0127] in, According to step 6.5, the forward difference of the above Lyapunov function can be obtained as:

[0128]

[0129] in,

[0130]

[0131]

[0132]

[0133]

[0134] According to step 6.3, the error system in step 5.1 satisfies the condition of l1 gain stability:

[0135]

[0136] Therefore, the urban water supply system in the present invention is l1-gain stable under the designed nonlinear event-triggered asynchronous filter based on the switched positive system.

[0137] The present invention proposes a filter design method for predicting water consumption in urban water supply systems. Based on the positive switching system model, output quantization model, event triggering strategy and asynchronous nonlinear filtering method, the present invention collects data on water consumption of users in urban areas and proposes a method for predicting water consumption of users at the next moment, so that a scientific scheduling plan can be made based on this prediction to avoid the phenomenon of unstable water pressure or even interruption of water flow for high-rise users during peak water consumption. The model established in this paper fully considers the characteristics of the actual system such as positivity, nonlinearity and asynchronous switching, and has greater practicality. BRIEF DESCRIPTION OF THE DRAWINGS

[0138] Figure 1 The urban water supply network system of the present invention;

[0139] Figure 2 Framework for event-triggered nonlinear filters with output quantization. DETAILED DESCRIPTION

[0140] The present invention will be further described below in conjunction with the accompanying drawings.

[0141] like Figure 1 As shown, the dynamic model of the urban water supply system is established by taking the urban water supply system as the research object, the water flow of the water supply network as the input, and the water flow delivered to the user actually measured by the sensor as the output.

[0142] Step 1: According to the urban water supply system, construct its state space model. The specific method is:

[0143] 1.1 Collect the input and output data of the urban water supply system to describe the actual operation process of the system:

[0144] Considering the operation process of urban water supply system, urban water supply system usually consists of water source, pumping station, valve and water supply network, such as Figure 1 Schematic diagram of the urban water supply system (see attached figure in the specification). Figure 1 The connection between the various components of the urban water supply system is shown. The blue arrows represent the urban water supply network, and the black line segments represent the distribution network that delivers water from the source to the user. The stored water from the water source is sent to the water plant through the water transmission pipeline for water quality treatment. The treated water is pressurized and delivered to the user through the distribution network. During the peak period of urban water use, high-rise residents generally experience unstable water flow or even water outages. Therefore, it is of great significance to estimate the user's water demand in the next time period in advance and then make a scientific scheduling plan to solve this problem. Considering the non-negative nature of water flow in the actual system and the nonlinearity of user water consumption, we abstract the actual system into a nonlinear switching positive system model. Since the sensors in the actual system may have intermittent faults, we quantize the output that meets certain conditions through a quantizer to improve the accuracy of the established model. The estimated system state (user's water demand in the next time period) is obtained through a nonlinear asynchronous filter with output quantization. Figure 2 Shown is a framework for event-triggered nonlinear filters with output quantization.

[0145] 1.2 Collect data on the urban water supply system and use the data to establish a

[0146] The state space model of water flow in the system:

[0147] x(k+1)=A σ(k) f(x(k))+B σ(k) g(ω(k)),

[0148] y(k)=C σ(k) h(x(k))+D σ(k) l(ω(k)),

[0149] z(k)=E σ(k) p(x(k))+F σ(k) q(ω(k)),

[0150] where x(k)=[x1(k),x2(k),...,x n (k)] T ∈R n is the water consumption of users in the area at time k, and n represents the number of pumping stations in the urban water supply system; is the external disturbance of the urban water supply system (such as a surge in water consumption at a certain moment, failure of system components, etc.); y(k)∈R m is the actual output collected by the sensor at time k, m represents the number of sensors measuring the output; z(k)∈R s is the estimated output of the user's water consumption at time k, and s represents the number of simulated pumping stations in the constructed model. Function σ(k) represents the switching signal, which is a mapping from the interval [0,∞) to the finite set S = {1,2,…,N+}. When σ(k) = i, i∈S, the i-th subsystem is activated, and its corresponding system matrix is ​​expressed as A i ,B i ,C i ,D i ,E i ,F i , and the matrix R m ,R n ,R s , N + They represent m-dimensional, n-dimensional, s-dimensional, n-dimensional non-negative, m-dimensional non-negative vectors and positive integer sets respectively.

[0151] 1.3f(x),g(ω),h(x),l(ω),p(x),q(ω) are all nonlinear functions used in modeling the city water supply system. For any x i ∈R,ω t ∈R,i=1,2,...,n, ..., m, R represents a set of real numbers, which all satisfy the following fan-shaped region conditions:

[0152]

[0153]

[0154]

[0155]

[0156]

[0157]

[0158] in

[0159] 0<ε1<ε2,0<ε3<ε4,0<ε5<ε6,f i (0)=0.

[0160] Step 2: Establish event trigger conditions for the urban water supply system:

[0161] ||e y (k)||1>β||y(k)||1,

[0162] Among them, β∈[0,1) is the event trigger coefficient, which is a given constant. y (k) is the sampling error, y(k l ) is the urban water supply system at the event triggering time k l The output value, l∈N + , y(k) represents the output value of the urban water supply system at time k, and ||·||1 represents the 1-norm of the vector, that is, the sum of the absolute values ​​of all elements in the vector.

[0163] Step 3: Establish a model for quantifying output signals based on event-triggered strategies in urban water supply systems:

[0164] 3.1 Use a quantizer to quantize the event-triggered output signal of the system in step 1.2. The form of the quantized output signal is as follows:

[0165]

[0166] in, is the output signal of the event generator, express The quantized signal, Represents a subquantizer.

[0167] 3.2 Sub-quantizer The quantization level of is described in the following set form:

[0168] u c ={φ c |φ c =κ c φ c0},

[0169] Define a specific sub-quantizer model for the subsystem:

[0170]

[0171] Among them, 0<κ c <1,φ c0 >0,κ c is a constant, φ c Defines the quantization level, φ c0 is the initial quantization level of the cth subsystem, Represents a constant.

[0172] 3.3 The output quantized signal received by the filter is in the following form:

[0173]

[0174] in, Satisfies the quantization error fan-shaped area expression: I is the identity matrix, Δ(k)=diag{Δ1(k),Δ2(k),...,Δ m (k)},|Δ c (k)|≤∈ c .

[0175] Step 4: Establish a nonlinear asynchronous filter model with output quantization, the structure of which is as follows:

[0176]

[0177]

[0178] Among them, x f (k) represents the state of the filter, z f (k) represents the filter’s estimate of the system model output signal z(k), σ f (k) is the switching signal of the filter, denoted by σ f (k) = j means the filter is asynchronous with the subsystem, σ f (k) = i means that the filter is synchronized with the subsystem, and for any switching time k r ,r=0,1,2,...,σ f (k r )=σ(k r )+Δ r ,Δ r is the filter lag time, Δ r <k r+1 -k r . is the filter gain matrix to be designed. is a fan-shaped bounded nonlinear function used to estimate the nonlinear function f(k). satisfy: and

[0179] Step 5: Establish the error system of the asynchronous switching filter, as follows:

[0180] 5.1 Order e(k)=z f (k)-z(k). According to steps 1.2 and 4, the following error system can be obtained:

[0181] When k∈[k r , k r +Δ r )

[0182]

[0183] e(k)=E fj p(x f (k))+F fj (I+Δk))(C i h(x(k))+D i l(ω(k))+e y (k))-E i p(x(k))-F i q(ω(k)),

[0184] When k∈[k r +Δ r ,k r+1 )

[0185]

[0186] e(k)=E fi p(x f (k))+F fi (I+Δk))(C i h(x(k))+D i l(ω(k))+e y (k))-E i p(x(k))-F i q(ω(k)),

[0187] Among them, x T (k) represents the transpose of vector x(k). In time segment k∈[k r ,k r +Δ r ), the system and the filter are in an asynchronous state. At this time, the system switches to the i-th subsystem, and the filter is still in the j-th filter state; in the time period k∈[k r +Δ r ,k r+1 ), the system and the filter are in synchronization. At this time, the system switches to the i-th subsystem, and the i-th filter is also running at the same time.

[0188] 5.2 Let Λ=diag{∈1,∈2,...,∈ m}, then Where L = I-Λ, J = I+Λ. According to step 1.3, the error system in synchronous and asynchronous states can be further expressed as:

[0189] When k∈[k r ,k r +Δ r )

[0190]

[0191]

[0192]

[0193]

[0194] When k∈[k r +Δ r ,k r+1 )

[0195]

[0196]

[0197]

[0198]

[0199] in,

[0200]

[0201]

[0202]

[0203]

[0204]

[0205]

[0206]

[0207]

[0208] Step 6: Design an event-triggered nonlinear asynchronous filter for the urban water supply system:

[0209] The filter gain matrix designed in 6.1 is:

[0210]

[0211] in, represents an n-dimensional column vector, represents an m-dimensional column vector; 1 n represents an n-dimensional column vector whose elements are all 1, Indicates An n-dimensional column vector with 1 element and all other elements being 0. Indicates An s-dimensional column vector with 1 element and all other elements being 0; Respectively 1 n is the transpose of ; ∑ is the summation symbol.

[0212] 6.2 Design constants

[0213] 0<ε1≤ε2,0<ε3≤ε4,0<ε5≤ε6, γ>0,λ>1,0≤β<1,0<μ1<1,μ2>1,if there exists R n vector and R m vector So that the following inequality holds:

[0214]

[0215]

[0216]

[0217]

[0218]

[0219]

[0220]

[0221]

[0222]

[0223]

[0224]

[0225]

[0226]

[0227]

[0228]

[0229]

[0230]

[0231] Then for any (i,j)∈S,i≠j,ι=1,2,…,n,=1,2,…,s, the error system constructed in step 5.1 is positive and l1-gain stable under the conditions of satisfying the filter gain matrix designed in step 6.1 and the switching rate designed in step 6.3.

[0232] 6.3 Design the switching rate of the positive switching system as follows:

[0233]

[0234]

[0235] Among them, Γ - (k0,k),Γ + (k0, k) represent the total time of synchronous and asynchronous operation of the system and filter, respectively, τ a represents the average dwell time of the switching system, Δ max Indicates the maximum lag time of the filter.

[0236] 6.4 Combining steps 2, 5.2, 6.1 and 6.2, we can obtain that to ensure the positivity of the error system, the following conditions need to be met:

[0237] When k∈[k r ,k r +Δ r )

[0238]

[0239]

[0240] When k∈[k r +Δ r ,k r +1), there is

[0241]

[0242]

[0243] in,

[0244]

[0245]

[0246]

[0247]

[0248] 6.5 According to steps 2 and 5, to ensure the l1 gain stability of the error system, it is necessary to satisfy:

[0249] When k∈[k r ,k r +Δ r )

[0250]

[0251]

[0252] When k∈[k r +Δ r ,k r +1), there is

[0253]

[0254]

[0255] in,

[0256]

[0257]

[0258]

[0259]

[0260] 6.6Construct a piecewise redundant positive Lyapunov function as follows:

[0261]

[0262] in, According to step 6.5, the forward difference of the above Lyapunov function can be obtained as:

[0263]

[0264] in,

[0265]

[0266]

[0267]

[0268]

[0269] According to step 6.3, the error system in step 5.1 satisfies the condition of l1 gain stability:

[0270]

[0271] Therefore, the urban water supply system in the present invention is l1-gain stable under the designed nonlinear event-triggered asynchronous filter based on the switched positive system.

Claims

1. A nonlinear water system fault quantification estimation method based on event triggering mechanism, characterized by Follow these steps: Step 1: Establish a state space model of the urban water supply system. The specific method is: 1.1 Collect input and output data of the urban water supply system to describe the actual operation process of the urban water supply system; 1.2 Collect data from the urban water supply system and use the data to establish a state space model of the water flow of the urban water supply system; 1.3 Determine whether the nonlinear system meets the conditions; Step 2: Establish event triggering conditions for the urban water supply system; ||e y (k)||1>β||y(k)||1, Among them, β∈[0,1) is the event trigger coefficient, that is, a given constant; e y (k) is the sampling error, y(k l ) is the urban water supply system at the event triggering time k l The output value, l∈N + , y(k) represents the output value of the urban water supply system at time k, ||·||1 represents the 1-norm of the vector, that is, the sum of the absolute values ​​of all elements in the vector; Step 3: Establish a model for quantifying output signals based on event-triggered strategies in urban water supply systems; Step 4: Establish a nonlinear asynchronous filter model with output quantization; Step 5, establishing an error system of the asynchronous switching filter; Step 6: Design an event-triggered nonlinear asynchronous filter for the urban water supply system; The model for quantifying the output signal based on the event-triggered strategy in the urban water supply system described in step 3 is as follows: 3.1 Use the quantizer to quantize the event-triggered output signal of the system in step 2. The form of the quantized output signal is: in, is the output signal of the event generator, express The quantized signal, represents a subquantizer; 3.2 Sub-quantizer The quantization level of is described in the following set form: Define a specific sub-quantizer model for the subsystem: Among them, 0<κ c <1,φ c0 >0,κ c is a constant, φ c Defines the quantization level, φ c0 is the initial quantization level of the cth subsystem, represents a constant; 3.3 The output quantized signal received by the filter is in the following form: in, Satisfies the quantization error fan-shaped area expression: I is the identity matrix, Δ(k)=diag{Δ1(k),Δ2(k),...,Δ m (k)},|Δ c (k)|≤∈ c ; The nonlinear asynchronous filter model with output quantization described in step 4 is constructed as follows: Among them, x f (k) represents the state of the filter, z f (k) represents the filter’s estimate of the system model output signal z(k), σ f (k) is the switching signal of the filter, denoted by σ f (k) = j means the filter is asynchronous with the subsystem, σ f (k) = i means that the filter is synchronized with the subsystem, and for any switching time k r ,r=0,1,2,...,σ f (k r )=σ(k r )+Δ r ,Δ r is the filter lag time, Δ r <k r+1 -k r ; is the filter gain matrix to be designed; is a fan-shaped bounded nonlinear function used to estimate the nonlinear function f(k). satisfy: and Step 5 constructs the error system of the nonlinear asynchronous switching filter, as follows: Let e(k) = z f (k) - z(k); When k∈[k r ,k r +Δ r ) When k∈[k r +Δ r ,k r+1 ) in, Step 6 describes the design of a nonlinear asynchronous event-triggered filter with output quantization for estimating user water consumption. The state space model of the urban water supply system is: x(k+1)=A σ(k) f(x(k))+B σ(k) g(ω(k)), y(k)=C σ(k) h(x(k))+D σ(k) l(ω(k)), z(k)=E σ(k) p(x(k))+F σ(k) q(ω(k)), where x(k)=[x1(k),x2(k),...,x n (k)] T ∈R n is the water consumption of users in the area at time k, and n represents the number of pumping stations in the urban water supply system; is the external disturbance of the urban water supply system; y(k)∈R m is the actual output collected by the sensor at time k, m represents the number of sensors measuring the output; z(k)∈R s is the estimated output of the user's water consumption at time k, s represents the number of simulated pumping stations in the constructed model; the function σ(k) represents the switching signal, which is the interval [0,∞) to the finite set S={1,2,…,N + }; when σ(k)=i,i∈S, the i-th subsystem is activated, and its corresponding system matrix is ​​expressed as A i ,B i ,C i ,D i ,E i ,F i , and the matrix A i ≥0,B i ≥0,C i ≥0,D i ≥0,E i ≥0; R m ,R n ,R s , N + Respectively represent m-dimensional, n-dimensional, s-dimensional, n-dimensional non-negative, m-dimensional non-negative vectors and positive integer sets; Step 6 The specific steps are as follows: The event-triggered asynchronous filter gain matrix designed in 6.1 is as follows: Among them, ξ i , represents an n-dimensional column vector, δ iι , represents an m-dimensional column vector; 1 n represents an n-dimensional column vector whose elements are all 1, represents an n-dimensional column vector whose ιth element is 1 and the rest of the elements are all 0. represents an s-dimensional column vector whose j-th element is 1 and the rest are all 0; Respectively represent ξ it ,δ iiι , 1 n is the transpose of ; ∑ is the summation symbol; 6.2 Design constants 0<ε1≤ε2,0<ε3≤ε4,0<ε5≤ε6, γ>0,λ>1,0≤β<1,0<μ1<1,μ2>1,if there exists R n vector ξ i ≥0,ξ i ≥0,ξ j ≥0, and R m Vector δ i ≥0, δ j ≥0, So that the following inequality holds: For any (i, j)∈S, i≠j, ι=1,2,...,n, The constructed error system is l1-gain stable under the filter gain matrix and switching rate; 6.3 Design the switching rate of the positive switching system as follows: Among them, Γ - (k0,k) and Γ + (k0, k) represent the total time of synchronous and asynchronous operation of the system and filter, respectively, τ a represents the average dwell time of the switching system, Δ max Indicates the maximum lag time of the filter; 6.4 Combining steps 2, 5, 6.1 and 6.2, we can obtain that to ensure the positivity of the error system, the following conditions need to be met: When k∈[k r ,k r +Δ r ) When k∈[k r +Δ r ,k r +1) in, 6.5 According to steps 2 and 5, to ensure the l1 gain stability of the error system, it is necessary to satisfy: When k∈[k r ,k r +Δ r ) When k∈[k r +Δ r ,k r +1) in, 6.6Construct a piecewise redundant positive Lyapunov function as follows: in, According to step 6.5, the forward difference of the above Lyapunov function can be obtained as: in, According to step 6.3, the error system in step 5 satisfies the condition of l1 gain stability: Therefore, the urban water supply system is l1-gain stable under the designed positive system-based nonlinear event-triggered asynchronous filter.

2. The method for quantitatively estimating faults in a nonlinear water system based on an event-triggered mechanism according to claim 1, characterized in that The nonlinear system described in step 1.3 satisfies the following conditions: in, 0<ε1≤ε20<ε3≤ε4,0<ε5≤ε6,f i (0)=0; f(x), g(ω), h(x), l(ω), p(x), q(ω) are all nonlinear functions used in the modeling of the city water supply system, f(x)=(f1(x1),......,f n (x n )) T ,g(ω)=(g1(ω1),......,g m (ω m )) T ,h(x)=(h1(x1),......,h n (x n )) T ,l(ω)=(l1(ω1),......,l m (ω m )) T ; p(x)=(p1(x1),......,p n (x n )) T ,q(ω)=(q1(ω1),......,q m (ω m )) T .

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