CN114169569A - Nonlinear water system fault quantitative estimation method based on event trigger mechanism - Google Patents

Nonlinear water system fault quantitative estimation method based on event trigger mechanism Download PDF

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CN114169569A
CN114169569A CN202111297588.8A CN202111297588A CN114169569A CN 114169569 A CN114169569 A CN 114169569A CN 202111297588 A CN202111297588 A CN 202111297588A CN 114169569 A CN114169569 A CN 114169569A
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张俊锋
张石涛
林枫雨
周晓月
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Abstract

The invention discloses a nonlinear water affair system fault quantitative estimation method based on an event trigger mechanism. The method comprises the following steps: step 1, establishing a state space model of an urban water supply system; step 2, establishing an event trigger condition of the urban water supply system; step 3, establishing a model for output signal quantization based on an event trigger strategy in the urban water supply system; step 4, establishing a nonlinear asynchronous filter model with output quantization; step 5, establishing an error system of the asynchronous switching filter; and 6, designing a nonlinear asynchronous filter based on event triggering aiming at the urban water supply system. The invention is helpful for making a more scientific scheduling scheme, thereby effectively relieving the conditions of unstable water supply, cut-off and the like during the peak period of water consumption. The invention can not only avoid the waste of system resources caused by periodic sampling of water flow, but also reduce the design cost.

Description

Nonlinear water system fault quantitative estimation method based on event trigger mechanism
Technical Field
The invention belongs to the technical field of automation, relates to a nonlinear water system fault quantitative estimation method based on an event trigger mechanism, and provides a nonlinear asynchronous filtering method based on an event trigger strategy, so that the water consumption demand of a user can be predicted.
Background
Water is a source of life, and people can not boil water in daily production and life. Particularly, in recent years, with the increasing level of urbanization, the number of urban population is becoming saturated and the demand for water is becoming larger and larger. The municipal water supply system is an integral part of the municipal utility and serves as an infrastructure for the development of cities, the role of which is self-evident. However, due to the continuous development of industrialization, the increasing environmental pollution and the lack of consciousness of saving, the problem of water shortage has attracted people's attention, and in addition, the utilization rate of water resource is not high, so that the urban water supply system faces a great pressure. Nowadays, the water supply scale is gradually enlarged, the formed system is more complex, the management and scheduling are more difficult, and the water supply system is abnormal if not reasonable. For example, in the peak period of water consumption, some high-rise residents can have the phenomena of insufficient water supply pressure, unstable water flow, even water cut-off and the like, which brings great inconvenience to the daily life of citizens. Therefore, it is very necessary to establish a more accurate model for the current water supply system, introduce a new scientific technology, optimize scheduling management, and maximize the benefit of the water supply system.
The optimized scheduling means that under the conditions that the water pressure is stable, the water quality reaches the standard and the water can be normally supplied, the water consumption in the next time period is predicted, and then a scientific scheduling scheme is formulated to realize the maximization of social and economic benefits. The precondition is to predict the water consumption. Therefore, establishing a proper water supply system model and designing a corresponding filter are key problems in estimating the water consumption. Considering the non-negativity of water flow in an actual water supply system, the positive system modeling is more accurate than a common system. In a multi-node water supply system, because different users have different water consumption in different time periods and the water consumption has the characteristic of nonlinearity, the process of water consumption change can be effectively described by using a nonlinear positive switching system.
Event triggering is an efficient signaling strategy. The problem of unreasonable water supply during peak water utilization periods can be effectively solved by adopting an event triggering strategy, and efficient operation and benefit maximization of a water supply system are realized by preferentially supplying water to certain areas or floors. In addition, in the event triggering mechanism, the triggering condition based on the current measurement value is continuously monitored, and when the condition is met, the event is triggered, so that the problems of system resource waste and high design cost caused by periodic sampling are effectively solved. In an ideal control system, it is expected that the switching of the system filter and the switching of the actual control system are synchronous, but due to the limited capability of the components, the sensor needs a certain time to identify the correct model, which causes the switching of the filter to delay for a period of time, so that an asynchronous filter based on event triggering is designed. In addition to this, sensors in practical systems may experience intermittent failures, resulting in the measured signal being quantified. To accommodate this process, we have designed event-triggered based quantizers to produce an output signal that matches the actual system.
The design method of the event-triggered asynchronous filter for switching the nonlinear positive system can predict the water consumption of an actual water supply system, and has important significance for solving the problems of unstable water supply and cutoff of an urban water supply system in a water peak period.
Disclosure of Invention
The invention provides a nonlinear water affair system fault quantitative estimation method based on an event trigger mechanism. The invention provides a design method for ensuring the high-efficiency water supply of a municipal water supply system based on a nonlinear positive switching model, an event triggering strategy and a design method of an asynchronous filter, which is used for acquiring the water supply quantity of the municipal water supply system and can well improve the performance of the municipal water supply system and improve the efficiency of the municipal water supply system even if the municipal water supply system has multiple interference factors, thereby effectively improving the problems of unstable municipal water supply, even cutoff and the like.
The technical scheme adopted by the invention for solving the problems comprises the following steps:
step 1, constructing a state space model of a city water supply system according to the city water supply system, wherein the specific method comprises the following steps:
1.1, the input and output data of the urban water supply system are collected to describe the actual operation process of the system:
considering the operation process of the urban water supply system, the urban water supply system is generally composed of a water source, a pump station, a valve and a water supply network, as shown in the schematic diagram of the urban water supply system in fig. 1 (see the attached drawing in the specification). Fig. 1 illustrates the relationship between the various components of a municipal water supply system. Blue arrows indicate a municipal water supply network and black line segments indicate a water distribution network that delivers water to consumers. The water stored in the water source is sent to a water plant through a water delivery pipe duct for water quality treatment, and the treated water is pressurized and then sent to users through a water distribution pipe network. When the urban water consumption peak period is reached, the conditions of unstable water flow, even water cut-off and the like generally occur to high-rise residents, so that the water consumption demand of the users in the next time period is estimated in advance, a scientific scheduling scheme is made, and the urban water consumption peak period water consumption scheduling method has important significance for solving the problem. Considering the characteristics of non-negativity of water flow in an actual system, nonlinearity presented by water consumption of a user and the like, the actual system is abstracted into a nonlinear switching positive system model. Because the sensor in the actual system can have intermittent faults, the output meeting a certain condition is quantized by a quantizer, so that the accuracy of the established model is improved. The estimated system state (water demand of the user for the next time period) is obtained through a nonlinear asynchronous filter with output quantization. Fig. 2 shows a framework of event-triggered nonlinear filters with output quantization.
1.2, carrying out data acquisition on the urban water supply system, and establishing a state space model of the flow of water of the urban water supply system by using the data:
x(k+1)＝Aσ(k)f(x(k))+Bσ(k)g(ω(k)),
y(k)＝Cσ(k)h(x(k))+Dσ(k)l(ω(k)),
z(k)＝Eσ(k)p(x(k))+Fσ(k)q(ω(k)),
wherein x (k) ═ x1(k),x2(k),...,xn(k)]T∈RnThe water consumption of users in the k moment area is obtained, and n represents the number of pump stations in the urban water supply system;
Figure BDA0003336929650000021
is the external disturbance of the urban water supply system (such as the surge of water consumption at a certain moment, the failure of components of the system, etc.); y (k) ε RmThe actual output is acquired by the sensors at the moment k, and m represents the number of the measurement output sensors; z (k) ε RsThe water consumption of the user at the moment k is estimated and output, and s represents the number of the simulation pump stations in the established model. The function σ (k) represents a switching signal, which is a mapping of the interval [0, ∞ ] to a finite set S ═ 1,2, …, N + }. When σ (k) is i, i ∈ S, the ith subsystem is activated, with its corresponding system matrix denoted as ai,Bi,Ci,Di,Ei,FiAnd matrix of
Figure BDA0003336929650000031
Rm,Rn,Rs,
Figure BDA0003336929650000032
N+Respectively representing m-dimensional, n-dimensional, s-dimensional, n-dimensional non-negative, m-dimensional non-negative vectors and positive integer sets.
1.3f (x), g (ω), h (x), l (ω), p (x), q (ω) are all nonlinear functions used in modeling the municipal water supply system, and f (x) ═ f (x)1(x1),......,fn(xn))T，g(ω)＝(g1(ω1),......,gm(ωm))T，h(x)＝(h1(x1),......,hn(xn))T，l(ω)＝(l1(ω1),......,lm(ωm))T；p(x)＝(p1(x1),......,pn(xn))T，q(ω)＝(q1(ω1),......,qm(ωm))T. For arbitrary xi∈R,ωtE R, i ═ 1, 2., n, i ═ 1, 2., m, R denote a real number set, each of which satisfies the following sector area condition:
Figure BDA0003336929650000033
Figure BDA0003336929650000034
Figure BDA0003336929650000035
Figure BDA0003336929650000036
Figure BDA0003336929650000037
Figure BDA0003336929650000038
wherein the content of the first and second substances,
Figure BDA0003336929650000039
0＜ε1＜ε2,0＜ε3＜ε4,0＜ε5＜ε6,fi(0)＝0。
step 2, establishing an event trigger condition of the urban water supply system:
||ey(k)||1＞β||y(k)||1,
where β ∈ [0,1) is the event trigger coefficient, i.e. a given constant. e.g. of the typey(k) Is the error in the sampling of the signal,
Figure BDA00033369296500000310
y(kl) Is that the urban water supply system triggers the moment k at the eventlIs output value of l ∈ N+Y (k) represents the output value of the urban water supply system at the moment k, | · | calculation1Represents the 1 norm of the vector, i.e., the sum of the absolute values of all the elements in the vector.
Step 3, establishing a model of output signal quantization based on an event trigger strategy in the urban water supply system:
3.1 quantizing the event-triggered output signal of the system of step 1.2 by means of a quantizer, the quantized output signal being of the form:
Figure BDA00033369296500000311
wherein the content of the first and second substances,
Figure BDA0003336929650000041
is an output signal of the event generator and,
Figure BDA0003336929650000042
to represent
Figure BDA0003336929650000043
The quantized signal of (a) is quantized,
Figure BDA0003336929650000044
representing a sub-quantizer.
3.2 sub-quantizers
Figure BDA0003336929650000045
Is described in the following set of forms:
uc＝{φc|φc＝κcφc0},
defining the specific sub-quantizer model to be built for the subsystem:
Figure BDA0003336929650000046
wherein the content of the first and second substances,0＜κc＜1,φc0＞0，κcis a constant value of phicDefining a quantization level phic0For the initial quantization level of the c-th sub-system,
Figure BDA0003336929650000047
representing a constant.
3.3 the output quantized signal received by the filter is of the form:
Figure BDA0003336929650000048
wherein the content of the first and second substances,
Figure BDA0003336929650000049
satisfying the quantization error sector area expression:
Figure BDA00033369296500000410
i is an identity matrix, Δ (k) is diag { Δ1(k),Δ2(k),...,Δm(k)},|Δc(k)|≤∈c。
Step 4, establishing a nonlinear asynchronous filter model with output quantization, wherein the structural form is as follows:
Figure BDA00033369296500000411
Figure BDA00033369296500000412
wherein x isf(k) Representing the state of the filter, zf(k0 denotes the filter's estimate of the system model output signal z (k) (. sigma.). sigma.f(k) Is the switching signal of the filter, noting σf(k) J denotes that the filter is asynchronous to the subsystem, σf(k) I denotes the filter is synchronized with the subsystem and for an arbitrary switching time kr,r＝0,1,2,...,σf(kr)＝σ(kr)+Δr,ΔrIs the lag time, Δ, of the filterr＜kr+1-kr。
Figure BDA00033369296500000413
Is the filter gain matrix that needs to be designed. At asynchronous time k ∈ [ k ]r,kr+Δr) In which the subsystem and filter are asynchronous, i.e. sigma (k) is if(k) J; at synchronization time k e kr+Δr,kr+1), the subsystem and filter are in synchronization, i.e. σ (k) ═ i, σf(k)＝i。
Figure BDA00033369296500000414
Is a sector-area bounded nonlinear function used to estimate the nonlinear function f (k), the sector-area bounded nonlinear function
Figure BDA00033369296500000415
Satisfies the following conditions:
Figure BDA00033369296500000416
and is
Figure BDA00033369296500000418
Step 5, establishing an error system of the asynchronous switching filter, which comprises the following specific steps:
5.1 order
Figure BDA00033369296500000417
e(k)＝zf(k) -z (k). From step 1.2 and step 4, the following error system can be obtained:
when k is equal to kr,kr+Δr) When there is
Figure BDA0003336929650000051
e(k)＝Efjp(xf(k))+Ffj(I+Δ(k))(Cih(x(k))+Dil(ω(k))+ey(k))-Eip(x(k))-Fiq(ω(k)),
When k is equal to kr+Δr,kr+1) When there is
Figure BDA0003336929650000052
e(k)＝Efip(xf(k))+Ffi(I+Δ(k))(Cih(x(k))+Dil(ω(k))+ey(k))-Eip(x(k))-Fiq(ω(k)),
Wherein x isT(k) Representing the transpose of the vector x (k).
5.2 let Λ ═ diag { ∈ {. E1,∈2,...,∈m}, then there are
Figure BDA0003336929650000053
Wherein, L is I-Lambda, and J is I + Lambda. As can be derived from step 1.3, the error systems in the synchronous and asynchronous states, respectively, can be further expressed as:
when k is equal to kr,kr+Δr) When there is
Figure BDA0003336929650000054
Figure BDA0003336929650000055
Figure BDA0003336929650000056
Figure BDA0003336929650000057
When k is equal to kr+Δr,kr+1) When there is
Figure BDA0003336929650000058
Figure BDA0003336929650000059
Figure BDA00033369296500000510
Figure BDA00033369296500000511
Wherein the content of the first and second substances,
Figure BDA0003336929650000061
Figure BDA0003336929650000062
Figure BDA0003336929650000063
Figure BDA0003336929650000064
Figure BDA0003336929650000065
Figure BDA0003336929650000066
Figure BDA0003336929650000067
Figure BDA0003336929650000068
step 6, designing a nonlinear asynchronous filter based on event triggering aiming at the urban water supply system:
6.1 the designed filter gain matrix is:
Figure BDA0003336929650000069
wherein the content of the first and second substances,
Figure BDA00033369296500000620
representing an n-dimensional column vector, deltaiι,
Figure BDA00033369296500000621
Representing an m-dimensional column vector; 1nAn n-dimensional column vector representing elements all 1,
Figure BDA00033369296500000611
an n-dimensional column vector representing that the iota-th element is 1 and the remaining elements are all 0,
Figure BDA00033369296500000612
is shown as
Figure BDA00033369296500000619
An s-dimensional column vector with 1 element and 0 elements;
Figure BDA00033369296500000613
respectively represent
Figure BDA00033369296500000618
1nTransposing; and Σ is the summation sign.
6.2 design constants
Figure BDA00033369296500000614
0＜ε1≤ε2,0＜ε3≤ε4,0＜ε5≤ε6,
Figure BDA00033369296500000615
γ＞0，λ＞1,0≤β＜1,0＜μ1＜1,μ2> 1, if R is presentn(Vector)
Figure BDA00033369296500000616
And Rm(Vector)
Figure BDA00033369296500000617
So that the following inequality holds:
Figure BDA0003336929650000071
Figure BDA0003336929650000072
Figure BDA0003336929650000073
Figure BDA0003336929650000074
Figure BDA0003336929650000075
Figure BDA0003336929650000076
Figure BDA0003336929650000077
Figure BDA0003336929650000078
Figure BDA0003336929650000079
Figure BDA00033369296500000710
Figure BDA00033369296500000711
Figure BDA00033369296500000712
Figure BDA00033369296500000713
Figure BDA00033369296500000714
Figure BDA00033369296500000715
Figure BDA00033369296500000716
Figure BDA00033369296500000717
then for any (i, j) ∈ S, i ≠ j, iota ═ 1,2, …, n,
Figure BDA00033369296500000720
…, s, the error system constructed in step 5.1 is positive and l is satisfied under the conditions of the filter gain matrix designed in step 6.1 and the switching rate designed in step 6.31The gain is stable.
6.3 the switching rate for a positive switching system is designed as follows:
Figure BDA00033369296500000718
Figure BDA00033369296500000719
wherein, gamma is-(k0，k)，Γ+(k0K) represents the total time, τ, for the synchronous and asynchronous operation of the system and filter, respectivelyaRepresenting the average dwell time, Δ, of the switching systemmaxRepresenting the maximum lag time of the filter.
6.4 combining step 2, step 5.2, step 6.1 and step 6.2 can obtain, to ensure that the error system is positive, the following conditions need to be satisfied:
when k is equal to kr，kr+Δr) When there is
Figure BDA0003336929650000081
Figure BDA0003336929650000082
When k is equal to kr+Δr,krWhen +1), there are
Figure BDA0003336929650000083
Figure BDA0003336929650000084
Wherein the content of the first and second substances,
Figure BDA0003336929650000085
Figure BDA0003336929650000086
Figure BDA0003336929650000087
Figure BDA0003336929650000088
6.5 according to step 2 and step 5, one wants to guarantee the error system l1Gain stability, needs to satisfy:
when k is equal to kr，kr+Δr) When there is
Figure BDA0003336929650000089
Figure BDA00033369296500000810
When k is equal to kr+Δr，krWhen +1), there are
Figure BDA00033369296500000811
Figure BDA00033369296500000812
Wherein the content of the first and second substances,
Figure BDA00033369296500000813
Figure BDA00033369296500000814
Figure BDA00033369296500000815
Figure BDA00033369296500000816
6.6 construct a piecewise redundant positive Lyapunov function as follows:
Figure BDA00033369296500000817
wherein the content of the first and second substances,
Figure BDA00033369296500000818
according to the step 6.5, the forward difference of the lyapunov function is obtained as follows:
Figure BDA0003336929650000091
wherein the content of the first and second substances,
Figure BDA0003336929650000092
Figure BDA0003336929650000093
Figure BDA0003336929650000094
Figure BDA0003336929650000095
obtaining an error system in step 5.1 according to step 6.3 that satisfies l1Conditions for gain stability:
Figure BDA0003336929650000096
therefore, the urban water supply system in the invention is l under the designed nonlinear event triggered asynchronous filter based on the switching positive system1The gain is stable.
The invention provides a filter design method for predicting water consumption of an urban water supply system. The invention provides a method for predicting the water consumption of users at the next moment based on a positive switching system model, an output quantization model, an event triggering strategy and an asynchronous nonlinear filtering method, aiming at carrying out data acquisition on the water consumption of the users in urban areas, so that a scientific scheduling scheme can be made according to the prediction, and the phenomenon that the water pressure of high-rise users is unstable and even cut off at the peak water consumption period is avoided. The model established by the method fully considers the characteristics of positive, nonlinear and asynchronous switching and the like of an actual system, and has higher practicability.
Drawings
FIG. 1 is a schematic diagram of a municipal water supply network system according to the invention;
fig. 2 is a framework of an event-triggered nonlinear filter with output quantization.
Detailed Description
The invention is further described below with reference to the accompanying drawings.
As shown in fig. 1, a dynamic model of a municipal water supply system is established by using the municipal water supply system as a research object, using water flow of a water supply network as an input, and using water flow delivered to a user actually measured by a sensor as an output.
Step 1, constructing a state space model of a city water supply system according to the city water supply system, wherein the specific method comprises the following steps:
1.1, the input and output data of the urban water supply system are collected to describe the actual operation process of the system:
considering the operation process of the urban water supply system, the urban water supply system is generally composed of a water source, a pump station, a valve and a water supply network, as shown in the schematic diagram of the urban water supply system in fig. 1 (see the attached drawing in the specification). Fig. 1 illustrates the relationship between the various components of a municipal water supply system. Blue arrows indicate a municipal water supply network and black line segments indicate a water distribution network that delivers water to consumers. The water stored in the water source is sent to a water plant through a water delivery pipe duct for water quality treatment, and the treated water is pressurized and then sent to users through a water distribution pipe network. When the urban water consumption peak period is reached, the conditions of unstable water flow, even water cut-off and the like generally occur to high-rise residents, so that the water consumption demand of the users in the next time period is estimated in advance, a scientific scheduling scheme is made, and the urban water consumption peak period water consumption scheduling method has important significance for solving the problem. Considering the characteristics of non-negativity of water flow in an actual system, nonlinearity presented by water consumption of a user and the like, the actual system is abstracted into a nonlinear switching positive system model. Because the sensor in the actual system can have intermittent faults, the output meeting a certain condition is quantized by a quantizer, so that the accuracy of the established model is improved. The estimated system state (water demand of the user for the next time period) is obtained through a nonlinear asynchronous filter with output quantization. Fig. 2 shows a framework of event-triggered nonlinear filters with output quantization.
1.2 carrying out data acquisition on the urban water supply system, and establishing the urban water supply system by using the data
State space model of the flow of the system water:
x(k+1)＝Aσ(k)f(x(k))+Bσ(k)g(ω(k))，
y(k)＝Cσ(k)h(x(k))+Dσ(k)l(ω(k))，
z(k)＝Eσ(k)p(x(k))+Fσ(k)q(ω(k))，
wherein x (k) ═ x1(k)，x2(k),...,xn(k)]T∈RnThe water consumption of users in the k moment area is obtained, and n represents the number of pump stations in the urban water supply system;
Figure BDA0003336929650000101
is the external disturbance of the urban water supply system (such as the surge of water consumption at a certain moment, the failure of components of the system, etc.); y (k) ε RmThe actual output is acquired by the sensors at the moment k, and m represents the number of the measurement output sensors; z (k) ε RsThe water consumption of the user at the moment k is estimated and output, and s represents the number of the simulation pump stations in the established model. The function σ (k) represents a switching signal, which is a mapping of the interval [0, ∞ ] to a finite set S ═ 1,2, …, N + }. When σ (k) is i, i ∈ S, the ith subsystem is activated, with its corresponding system matrix denoted as ai,Bi,Ci,Di,Ei,FiAnd matrix of
Figure BDA0003336929650000102
Rm,Rn,Rs,
Figure BDA0003336929650000103
N+Respectively representing m-dimensional, n-dimensional, s-dimensional, n-dimensional non-negative, m-dimensional non-negative vectors and positive integer sets.
1.3f (x), g (omega), h (x), l (omega), p (x), q (omega) are all nonlinear functions used in the modeling of the urban water supply system, and for any xi∈R,ωt∈R，i＝1,2,...,n,
Figure BDA0003336929650000104
.., m, R represent real number sets, which all satisfy the following sector area condition:
Figure BDA0003336929650000111
Figure BDA0003336929650000112
Figure BDA0003336929650000113
Figure BDA0003336929650000114
Figure BDA0003336929650000115
Figure BDA0003336929650000116
wherein
Figure BDA0003336929650000117
0＜ε1＜ε2,0＜ε3＜ε4,0＜ε5＜ε6,fi(0)＝0。
Step 2, establishing an event trigger condition of the urban water supply system:
||ey(k)||1＞β||y(k)||1,
where β ∈ [0,1) is the event trigger coefficient, i.e. a given constant. e.g. of the typey(k) Is the error in the sampling of the signal,
Figure BDA0003336929650000118
y(kl) Is that the urban water supply system triggers the moment k at the eventlIs output value of l ∈ N+Y (k) represents the output value of the urban water supply system at the moment k, | · | calculation1Represents the 1 norm of the vector, i.e., the sum of the absolute values of all the elements in the vector.
Step 3, establishing a model of output signal quantization based on an event trigger strategy in the urban water supply system:
3.1 quantizing the event-triggered output signal of the system of step 1.2 by means of a quantizer, the quantized output signal being of the form:
Figure BDA0003336929650000119
wherein the content of the first and second substances,
Figure BDA00033369296500001110
is an output signal of the event generator and,
Figure BDA00033369296500001111
to represent
Figure BDA00033369296500001112
The quantized signal of (a) is quantized,
Figure BDA00033369296500001113
representing a sub-quantizer.
3.2 sub-quantizers
Figure BDA00033369296500001114
Is described in the following set of forms:
uc＝{φc|φc＝κcφc0},
defining the specific sub-quantizer model to be built for the subsystem:
Figure BDA00033369296500001115
wherein, 0 < kappac＜1,φc0＞0，κcIs a constant value of phicDefining a quantization level phic0For the initial quantization level of the c-th sub-system,
Figure BDA00033369296500001116
representing a constant.
3.3 the output quantized signal received by the filter is of the form:
Figure BDA00033369296500001117
wherein the content of the first and second substances,
Figure BDA00033369296500001118
satisfying the quantization error sector area expression:
Figure BDA00033369296500001119
i is an identity matrix, Δ (k) is diag { Δ1(k),Δ2(k),...,Δm(k)},|Δc(k)|≤∈c。
Step 4, establishing a nonlinear asynchronous filter model with output quantization, wherein the structural form is as follows:
Figure BDA0003336929650000121
Figure BDA0003336929650000122
wherein x isf(k) Representing the state of the filter, zf(k) Representing the estimate, σ, of the filter on the system model output signal z (k)f(k) Is the switching signal of the filter, noting σf(k) J denotes that the filter is asynchronous to the subsystem, σf(k) I denotes the filter is synchronized with the subsystem and for an arbitrary switching time kr,r＝0,1,2,...,σf(kr)＝σ(kr)+Δr,ΔrIs the lag time, Δ, of the filterr＜kr+1-kr。
Figure BDA0003336929650000123
Is the filter gain matrix to be designed.
Figure BDA0003336929650000124
Is a bounded nonlinear function of a sector area used to estimate the nonlinear function f (k), the sector areaBounded nonlinear function
Figure BDA0003336929650000125
Satisfies the following conditions:
Figure BDA0003336929650000126
and is
Figure BDA0003336929650000127
Step 5, establishing an error system of the asynchronous switching filter, which comprises the following specific steps:
5.1 order
Figure BDA0003336929650000128
e(k)＝zf(k) -z (k). From step 1.2 and step 4, the following error system can be obtained:
when k is equal to kr，kr+Δr) When there is
Figure BDA0003336929650000129
e(k)＝Efjp(xf(k))+Ffj(I+Δk))(Cih(x(k))+Dil(ω(k))+ey(k))-Eip(x(k))-Fiq(ω(k)),
When k is equal to kr+Δr,kr+1) When there is
Figure BDA00033369296500001210
e(k)＝Efip(xf(k))+Ffi(I+Δk))(Cih(x(k))+Dil(ω(k))+ey(k))-Eip(x(k))-Fiq(ω(k)),
Wherein x isT(k) Representing the transpose of the vector x (k). In a time period k ∈ [ k ]r,kr+Δr) In the method, the system and the filter are in an asynchronous state, and the system is switched to the ith subsystem when the filter is still in the jth filtering stateA device state; in a time period k ∈ [ k ]r+Δr,kr+1) In the method, the system and the filter are in a synchronous state, the system is switched to the ith subsystem at the moment, and correspondingly, the ith filter also operates at the same time.
5.2 let Λ ═ diag { ∈ {. E1,∈2,...,∈m}, then there are
Figure BDA00033369296500001211
Wherein, L is I-Lambda, and J is I + Lambda. As can be derived from step 1.3, the error systems in the synchronous and asynchronous states, respectively, can be further expressed as:
when k is equal to kr,kr+Δr) When there is
Figure BDA00033369296500001212
Figure BDA00033369296500001213
Figure BDA0003336929650000131
Figure BDA0003336929650000132
When k is equal to kr+Δr,kr+1) When there is
Figure BDA0003336929650000133
Figure BDA0003336929650000134
Figure BDA0003336929650000135
Figure BDA0003336929650000136
Wherein the content of the first and second substances,
Figure BDA0003336929650000137
Figure BDA0003336929650000138
Figure BDA0003336929650000139
Figure BDA00033369296500001310
Figure BDA00033369296500001311
Figure BDA00033369296500001312
Figure BDA00033369296500001313
Figure BDA00033369296500001314
step 6, designing a nonlinear asynchronous filter based on event triggering aiming at the urban water supply system:
6.1 the designed filter gain matrix is:
Figure BDA00033369296500001315
wherein the content of the first and second substances,
Figure BDA00033369296500001323
representing an n-dimensional column vector and,
Figure BDA00033369296500001324
representing an m-dimensional column vector; 1nAn n-dimensional column vector representing elements all 1,
Figure BDA00033369296500001317
is shown as
Figure BDA00033369296500001326
An n-dimensional column vector having 1 as one element and 0 as the remaining elements,
Figure BDA00033369296500001318
is shown as
Figure BDA00033369296500001327
An s-dimensional column vector with 1 element and 0 elements;
Figure BDA00033369296500001319
respectively represent
Figure BDA00033369296500001325
1nTransposing; and Σ is the summation sign.
6.2 design constants
Figure BDA00033369296500001320
0＜ε1≤ε2,0＜ε3≤ε4,0＜ε5≤ε6，
Figure BDA00033369296500001321
γ＞0，λ＞1，0≤β＜1，0＜μ1＜1，μ2> 1, if R is presentn(Vector)
Figure BDA00033369296500001322
And Rm(Vector)
Figure BDA0003336929650000141
So that the following inequality holds:
Figure BDA0003336929650000142
Figure BDA0003336929650000143
Figure BDA0003336929650000144
Figure BDA0003336929650000145
Figure BDA0003336929650000146
Figure BDA0003336929650000147
Figure BDA0003336929650000148
Figure BDA0003336929650000149
Figure BDA00033369296500001410
Figure BDA00033369296500001411
Figure BDA00033369296500001412
Figure BDA00033369296500001413
Figure BDA00033369296500001414
Figure BDA00033369296500001415
Figure BDA00033369296500001416
Figure BDA00033369296500001417
Figure BDA00033369296500001418
then for any (i, j) ∈ S, i ≠ j, iota ≠ 1,2, …, n ═ 1,2, …, S, the error system constructed in step 5.1 is positive and l is satisfied under the conditions of the filter gain matrix designed in step 6.1 and the switching rate designed in step 6.31The gain is stable.
6.3 the switching rate for a positive switching system is designed as follows:
Figure BDA00033369296500001419
Figure BDA00033369296500001420
wherein, gamma is-(k0,k),Γ+(k0K) represents the total time, τ, for the synchronous and asynchronous operation of the system and filter, respectivelyaRepresenting the average dwell time, Δ, of the switching systemmaxRepresenting the maximum lag time of the filter.
6.4 combining step 2, step 5.2, step 6.1 and step 6.2 can obtain, to ensure that the error system is positive, the following conditions need to be satisfied:
when k is equal to kr,kr+Δr) When there is
Figure BDA0003336929650000151
Figure BDA0003336929650000152
When k is equal to kr+Δr,krWhen +1), there are
Figure BDA0003336929650000153
Figure BDA0003336929650000154
Wherein the content of the first and second substances,
Figure BDA0003336929650000155
Figure BDA0003336929650000156
Figure BDA0003336929650000157
Figure BDA0003336929650000158
6.5 according to step 2 and step 5, one wants to guarantee the error system l1Gain stability, needs to satisfy:
when k is equal to kr,kr+Δr) When there is
Figure BDA0003336929650000159
Figure BDA00033369296500001510
When k is equal to kr+Δr,krWhen +1), there are
Figure BDA00033369296500001511
Figure BDA00033369296500001512
Wherein the content of the first and second substances,
Figure BDA00033369296500001513
Figure BDA00033369296500001514
Figure BDA00033369296500001515
Figure BDA00033369296500001516
6.6 construct a piecewise redundant positive Lyapunov function as follows:
Figure BDA00033369296500001517
wherein the content of the first and second substances,
Figure BDA00033369296500001518
according to the step 6.5, the forward difference of the lyapunov function is obtained as follows:
Figure BDA0003336929650000161
wherein the content of the first and second substances,
Figure BDA0003336929650000162
Figure BDA0003336929650000163
Figure BDA0003336929650000164
Figure BDA0003336929650000165
obtaining an error system in step 5.1 according to step 6.3 that satisfies l1Conditions for gain stability:
Figure BDA0003336929650000166
therefore, the urban water supply system in the invention is l under the designed nonlinear event triggered asynchronous filter based on the switching positive system1The gain is stable.

Claims (7)

1. The nonlinear water system fault quantitative estimation method based on the event trigger mechanism is characterized by comprising the following steps of:
step 1, establishing a state space model of an urban water supply system, wherein the specific method comprises the following steps:
1.1, acquiring input and output data of the urban water supply system to describe the actual operation process of the urban water supply system;
1.2, carrying out data acquisition on the urban water supply system, and establishing a state space model of the flow of water of the urban water supply system by using the data;
1.3, determining that the nonlinear system meets the condition;
step 2, establishing an event trigger condition of the urban water supply system;
step 3, establishing a model for output signal quantization based on an event trigger strategy in the urban water supply system;
step 4, establishing a nonlinear asynchronous filter model with output quantization;
step 5, establishing an error system of the asynchronous switching filter;
and 6, designing a nonlinear asynchronous filter based on event triggering aiming at the urban water supply system.
2. The method for quantitatively estimating the fault of the nonlinear water system based on the event trigger mechanism according to claim 1, wherein the nonlinear system in step 1.3 satisfies the condition specifically as follows:
Figure FDA0003336929640000019
Figure FDA0003336929640000011
wherein the content of the first and second substances,
Figure FDA0003336929640000012
0＜ε1≤ε20＜ε3≤ε4,0＜ε5≤ε6,fi(0) 0; f (x), g (ω), h (x), l (ω), p (x), q (ω) are all nonlinear functions used in modeling the municipal water supply system, and f (x) is (f) (x)1(x1),......,fn(xn))T，g(ω)＝(g1(ω1),......,gm(ωm))T，h(x)＝(h1(x1),......,hn(xn))T，l(ω)＝(l1(ω1),......,lm(ωm))T；p(x)＝(p1(x1),......,pn(xn))T，q(ω)＝(q1(ω1),......,qm(ωm))T。
3. The method for quantitatively estimating the fault of the nonlinear water system based on the event trigger mechanism according to claim 2, wherein the step 3 is to establish a model for output signal quantization based on the event trigger strategy in the municipal water supply system, and specifically comprises the following steps:
3.1 quantizing the event-triggered output signal of the system in step 2 by using a quantizer, wherein the quantized output signal is in the form of:
Figure FDA0003336929640000013
wherein the content of the first and second substances,
Figure FDA0003336929640000014
is an output signal of the event generator and,
Figure FDA0003336929640000015
to represent
Figure FDA0003336929640000016
The quantized signal of (a) is quantized,
Figure FDA0003336929640000017
a representation sub-quantizer;
3.2 sub-quantizers
Figure FDA0003336929640000018
Is described in the following set of forms:
uc＝{φc|φc＝κcφc0},
defining the specific sub-quantizer model to be built for the subsystem:
Figure FDA0003336929640000021
wherein, 0 < kappac＜1,φc0＞0，κcIs a constant value of phicDefining a quantization level phic0For the initial quantization level of the c-th sub-system,
Figure FDA0003336929640000022
represents a constant;
3.3 the output quantized signal received by the filter is of the form:
Figure FDA0003336929640000023
wherein the content of the first and second substances,
Figure FDA0003336929640000024
satisfying the quantization error sector area expression:
Figure FDA0003336929640000025
i is an identity matrix, Δ (k) is diag { Δ1(k),Δ2(k),...,Δm(k)},|Δc(k)|≤∈c。
4. The method according to claim 3, wherein the step 4 is to establish a nonlinear asynchronous filter model with output quantization, and the structure form is as follows:
Figure FDA0003336929640000026
Figure FDA0003336929640000027
wherein x isf(k) Representing the state of the filter, zf(k) Representing the estimate, σ, of the filter on the system model output signal z (k)f(k) Is the switching signal of the filter, noting σf(k) J denotes that the filter is asynchronous to the subsystem, σf(k) I denotes the filter is synchronized with the subsystem and for an arbitrary switching time kr,r＝0,1,2,...，σf(kr)＝σ(kr)+Δr,ΔrIs the lag time, Δ, of the filterr＜kr+1-kr；
Figure FDA0003336929640000028
Is the filter gain matrix to be designed;
Figure FDA0003336929640000029
is a sector-area bounded nonlinear function used to estimate the nonlinear function f (k), the sector-area bounded nonlinear function
Figure FDA00033369296400000210
Satisfies the following conditions:
Figure FDA00033369296400000211
and 0 < theta1＜θ2。
5. The method for quantitatively estimating the fault of the nonlinear water system based on the event trigger mechanism according to claim 4, wherein the step 5 constructs an error system of the nonlinear asynchronous switching filter, which is specifically as follows:
order to
Figure FDA00033369296400000212
e(k)＝zf(k)-z(k)；
When k is equal to kr,kr+Δr) When there is
Figure FDA00033369296400000213
Figure FDA00033369296400000214
Figure FDA00033369296400000215
Figure FDA00033369296400000216
When k is equal to kr+Δr,kr+1) When there is
Figure FDA0003336929640000031
Figure FDA0003336929640000032
Figure FDA0003336929640000033
Figure FDA0003336929640000034
Wherein the content of the first and second substances,
Figure FDA0003336929640000035
Figure FDA0003336929640000036
Figure FDA0003336929640000037
Figure FDA0003336929640000038
Figure FDA0003336929640000039
Figure FDA00033369296400000310
Figure FDA00033369296400000311
Figure FDA00033369296400000312
6. the method according to claim 5, wherein step 6 is implemented by designing a nonlinear asynchronous event-triggered filter with output quantization for user water consumption estimation, and the state space model of the municipal water supply system is:
x(k+1)＝Aσ(k)f(x(k))+Bσ(k)g(ω(k)),
y(k)＝Cσ(k)h(x(k))+Dσ(k)l(ω(k)),
z(k)＝Eσ(k)p(x(k))+Fσ(k)q(ω(k)),
wherein x (k) ═ x1(k),x2(k),...,xn(k)]T∈RnThe water consumption of users in the k moment area is obtained, and n represents the number of pump stations in the urban water supply system;
Figure FDA00033369296400000313
is an external disturbance of the urban water supply system; y (k) ε RmThe actual output is acquired by the sensors at the moment k, and m represents the number of the measurement output sensors; z (k) ε RsEstimating and outputting the water consumption of the user at the moment k, wherein s represents the number of simulation pump stations in the established model; the function σ (k) represents the switching signal, which is the interval [0, ∞ ] to the finite set S ═ 1,2, …, N+A mapping of { C }; when σ (k) is i, i ∈ S, the ith subsystem is activated, with its corresponding system matrix denoted as ai,Bi,Ci,Di,Ei,FiAnd matrix of
Figure FDA00033369296400000314
Rm,Rn,Rs,
Figure FDA00033369296400000315
N+Respectively representing m-dimensional, n-dimensional, s-dimensional, n-dimensional non-negative, m-dimensional non-negative vectors and positive integer sets.
7. The method for quantitatively estimating the fault of the nonlinear water system based on the event trigger mechanism according to claim 5 or 6, wherein the specific steps of step 6 are as follows:
6.1 the designed event triggered asynchronous filter gain matrix is as follows:
Figure FDA0003336929640000041
wherein ξil,ρij,
Figure FDA0003336929640000042
Representing an n-dimensional column vector, deltail,θijRepresenting an m-dimensional column vector; 1nAn n-dimensional column vector representing elements all 1,
Figure FDA0003336929640000043
an n-dimensional column vector representing that the l-th element is 1 and the remaining elements are all 0,
Figure FDA0003336929640000044
an s-dimensional column vector representing that the jth element is 1 and the rest elements are all 0;
Figure FDA0003336929640000045
respectively represent xiil,δil,ρij,θij,1nTransposing; Σ is the summation symbol;
6.2 design constants
Figure FDA0003336929640000046
0＜ε1≤ε2，0＜ε3≤ε4,0＜ε5≤ε6,0＜θ1≤θ2,γ＞0,λ＞1,0≤β＜1,0＜μ1＜1,μ2> 1, if R is presentn(Vector)
Figure FDA0003336929640000047
Figure FDA0003336929640000048
And Rm(Vector)
Figure FDA0003336929640000049
So that the following inequality holds:
Figure FDA0003336929640000051
Figure FDA0003336929640000052
Figure FDA0003336929640000053
Figure FDA0003336929640000054
Figure FDA0003336929640000055
Figure FDA0003336929640000056
Figure FDA0003336929640000057
Figure FDA0003336929640000058
Figure FDA0003336929640000059
Figure FDA00033369296400000510
Figure FDA00033369296400000511
Figure FDA00033369296400000512
Figure FDA00033369296400000513
Figure FDA00033369296400000514
Figure FDA00033369296400000515
Figure FDA00033369296400000516
Figure FDA00033369296400000517
for any (i, j) ∈ S, i ≠ j, i ≠ 1,2, …, n, j ═ 1,2, …, S, the error system constructed is l under the filter gain matrix and the switching rate1The gain is stable;
6.3 the switching rate for a positive switching system is designed as follows:
Figure FDA00033369296400000518
Figure FDA00033369296400000519
wherein, gamma is-(k0K) and Γ+(k0K) represents the total time, τ, for the synchronous and asynchronous operation of the system and filter, respectivelyaRepresenting the average dwell time, Δ, of the switching systemmaxRepresents the maximum lag time of the filter;
6.4 combining step 2, step 5, step 6.1 and step 6.2 can obtain, to ensure the error system is positive, the following conditions need to be satisfied:
when k is equal to kr,kr+Δr) When there is
Figure FDA0003336929640000061
Figure FDA0003336929640000062
When k is equal to kr+Δr,krWhen +1), there are
Figure FDA0003336929640000063
Figure FDA0003336929640000064
Wherein the content of the first and second substances,
Figure FDA0003336929640000065
Figure FDA0003336929640000066
Figure FDA0003336929640000067
Figure FDA0003336929640000068
6.5 according to step 2 and step 5, one wants to guarantee the error system l1Gain stability, needs to satisfy:
when k is equal to kr,kr+Δr) When there is
Figure FDA0003336929640000069
Figure FDA00033369296400000610
When k is equal to kr+Δr,krWhen +1), there are
Figure FDA00033369296400000611
Figure FDA00033369296400000612
Wherein the content of the first and second substances,
Figure FDA00033369296400000613
Figure FDA00033369296400000614
Figure FDA00033369296400000615
Figure FDA00033369296400000616
6.6 construct a piecewise redundant positive Lyapunov function as follows:
Figure FDA00033369296400000617
wherein the content of the first and second substances,
Figure FDA00033369296400000618
according to the step 6.5, the forward difference of the lyapunov function is obtained as follows:
Figure FDA0003336929640000071
wherein the content of the first and second substances,
Figure FDA0003336929640000072
Figure FDA0003336929640000073
Figure FDA0003336929640000074
Figure FDA0003336929640000075
obtaining an error system in step 5 according to step 6.3 that satisfies l1Conditions for gain stability:
Figure FDA0003336929640000076
thus, the municipal water supply system is under the designed positive system based non-linear event triggered asynchronous filter is/1The gain is stable.
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Cited By (2)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN115225381A (en) * 2022-07-19 2022-10-21 海南大学 Asynchronous fault detection filter design method
CN116009392A (en) * 2022-11-07 2023-04-25 深圳大学 Quantizer-based asynchronous event trigger control method, quantizer-based asynchronous event trigger control device, quantizer-based asynchronous event trigger control equipment and medium

Cited By (4)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN115225381A (en) * 2022-07-19 2022-10-21 海南大学 Asynchronous fault detection filter design method
CN115225381B (en) * 2022-07-19 2023-05-12 海南大学 Asynchronous fault detection filter design method
CN116009392A (en) * 2022-11-07 2023-04-25 深圳大学 Quantizer-based asynchronous event trigger control method, quantizer-based asynchronous event trigger control device, quantizer-based asynchronous event trigger control equipment and medium
CN116009392B (en) * 2022-11-07 2023-08-08 深圳大学 Quantizer-based asynchronous event trigger control method, quantizer-based asynchronous event trigger control device, quantizer-based asynchronous event trigger control equipment and medium

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