CN110705892A - Water flow state detection method for urban drainage pipeline - Google Patents

Water flow state detection method for urban drainage pipeline Download PDF

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CN110705892A
CN110705892A CN201910963450.3A CN201910963450A CN110705892A CN 110705892 A CN110705892 A CN 110705892A CN 201910963450 A CN201910963450 A CN 201910963450A CN 110705892 A CN110705892 A CN 110705892A
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matrix
water flow
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朱萌泽
陈云
陈张平
赵晓东
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Hangzhou Dianzi University
Hangzhou Electronic Science and Technology University
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Abstract

The invention relates to a method for detecting the water flow state of a municipal drainage pipeline. Due to the complex condition of the underground pipe network and the multiple interference factors of the detection system, the detection of the water flow state of the urban drainage pipeline is relatively difficult. The distributed state estimator based on the sensor network adopts an event-triggered communication protocol to relieve the communication congestion phenomenon and reduce the energy consumption, uses a unilateral Lipschitz function to describe the nonlinear disturbance of the drainage pipe network, and solves the distributed state estimator based on the sensor network through a linear matrix inequality method. The method comprises the steps of firstly constructing a multi-sensor network model and a topological structure, then establishing a state space model of a water flow state detection system of the urban drainage pipeline, establishing a distributed state estimator and an error system model of the water flow state detection system of the urban drainage pipeline, and finally solving the distributed state estimator of the water flow detection system of the urban drainage pipeline based on an event trigger mechanism. The invention provides a timely and effective method for detecting the water flow state of the urban drainage pipeline.

Description

Water flow state detection method for urban drainage pipeline
Technical Field
The invention belongs to the technical field of automatic control, and relates to a method for detecting the water flow state of a urban drainage pipeline.
Background
In recent years, the urbanization of China is rapidly developed, the population of the urban permanent population is continuously increased, and the discharge amount of industrial water and domestic sewage is increasingly increased. The safe and effective operation of the urban drainage system is an important guarantee for the safe and healthy life of industrial production and urban residents. However, the current domestic urban drainage pipeline system is not ideal in operation. Due to the fact that the rainfall suddenly increases in extreme climates, the drainage capacity is reduced due to the fact that a built urban drainage pipeline system is incomplete, and the pipeline is blocked, sewage leakage, road ponding and urban waterlogging are caused very easily, and serious influences are brought to urban environment, healthy life and personal safety. Therefore, real-time detection and estimation of the water flow state of the urban drainage pipeline become the basis of safe production and healthy life of modern cities.
Although some simple methods are used for detecting the water flow change condition of the urban drainage pipeline at present, the condition of an underground pipe network drainage pipeline system is very complex, a plurality of sensors for detecting are deeply buried underground, a large amount of data are easily lost when being transmitted simultaneously, and interference factors of the detection system are increased due to the old parts because the working environment of the underground urban drainage pipeline is poor, which brings difficulty and challenge to the detection of the water flow state of the urban drainage pipeline. Therefore, a new method is urgently needed, which not only can deal with various interferences occurring in the detection of the urban drainage pipeline system, but also can solve the problem of data loss in the transmission of a large number of detection signals, and realizes effective detection and estimation of the water flow state of the urban drainage pipeline.
Disclosure of Invention
The invention aims to provide a distributed estimation method for the water flow state of urban drainage pipelines based on an event trigger mechanism, aiming at the problem that the current water flow state detection system of urban drainage pipelines in China cannot timely and accurately detect and estimate the water flow state.
The method is based on the wireless sensor network, communication congestion is relieved and energy consumption is saved by adopting a communication protocol of an event trigger mechanism, meanwhile, nonlinear disturbance of a drainage pipe network is described by utilizing a unilateral Lipschitz function, and the method is wider in application range compared with the common Lipschitz nonlinearity. By designing the distributed state estimator and solving the distributed state estimator by using a linear matrix inequality method, a timely and effective method is provided for detecting the water flow state of the urban drainage pipeline.
The method comprises the following specific steps:
step (1), constructing a multi-sensor network model and a topological structure:
arranging N sensors in the area needing water flow state detection, wherein the N sensors are used for respectively measuring the state information of the water level height, the water pressure, the flow speed and the flow of the drainage pipeline;
the N sensors form a sensor network with a topological structure, wherein the number of nodes is N; using directed graphs
Figure BDA0002229724730000021
Representing the topology of the sensor network;
wherein the content of the first and second substances,
Figure BDA0002229724730000022
a set of sensors representing the arrangement of the detection areas,
Figure BDA0002229724730000023
represents a set of edges, C ═ Cij]N×NA weighted adjacency matrix representing the directed graph,
Figure BDA0002229724730000024
cijrepresenting the strength of the coupling between sensor node i and node j [. ]]N×NRepresenting a matrix of N × N elements; c. CijIf the value is more than 0, the sensor node j transmits information to the sensor node i at the moment; for all
Figure BDA0002229724730000025
Stipulating: if i is j, c is notedii1 means that the sensor network is self-contained in communication.
The set of all sensor nodes connected to sensor node i is denoted as
Step (2), establishing a state space model of a water flow state detection system of the urban drainage pipeline:
establishing a dynamic equation of a water flow state detection system of the urban drainage pipeline as follows:
Figure BDA0002229724730000027
whereinRepresenting the water flow state vector, x, of the drainage pipe at time k1(k)、x2(k)、x3(k)、x4(k) Respectively representing the water flow, the water flow speed, the water level and the water pressure of the water drainage pipeline at the moment k;
Figure BDA0002229724730000029
a real matrix representing n × m dimensions; superscript T represents the transpose of the matrix;
Figure BDA00022297247300000210
representing the water flow state value measured by the sensor node i at the moment k;
Figure BDA00022297247300000211
representing the output signal to be estimated at time k;
Figure BDA00022297247300000212
an external perturbation that is energy-bounded;
Figure BDA00022297247300000213
and
Figure BDA00022297247300000214
is a known constant matrix;
αi(k)∈[0,1]to a obey a known random distributionThe random sequence of (2) is used for describing a random packet loss phenomenon occurring when the sensor node i transmits the measurement data;
obtaining alpha by using experimental and statistical analysis methodi(k) Mean and variance of (1), noted
Figure BDA00022297247300000215
And
Figure BDA00022297247300000216
wherein E {. denotes the mathematical expectation of the random variable,
Figure BDA00022297247300000217
and
Figure BDA00022297247300000218
is a known scalar;represents the nonlinear interference of urban industrial production and daily life sewage discharge on the water flow in the detection area, and the nonlinear interference meets the following unilateral Lipschitz condition:
condition 1 for any
Figure BDA0002229724730000031
At an arbitrary time k, a scalar ρ exists, and the nonlinear function f (k, x (k)) satisfies<f(k,u)-f(k,v),u-v>≤ρ||u-v||2；
Condition 2 for any
Figure BDA0002229724730000032
At an arbitrary time k, scalars α, β exist, and a nonlinear function f (k, x (k)) satisfies (f (k, u) -f (k, v))T(f(k,u)-f(k,v))≤β||u-v||2+α<u-v,f(k,v)>(ii) a Wherein the content of the first and second substances,<·>represents the inner product of a vector or matrix in euclidean space; i | · | represents the euclidean norm of a vector or matrix.
Step (3), establishing a distributed state estimator and an error system model of a water flow state detection system of the urban drainage pipeline:
(3-1) setting an event trigger mechanism of sensor network data transmission:
in order to relieve the communication congestion phenomenon caused by the simultaneous transmission of a large number of sensor measurement data and save energy consumption, the invention adopts a communication protocol of an event trigger mechanism.
Setting the event triggering conditions as follows:
Figure BDA0002229724730000033
wherein the content of the first and second substances,
Figure BDA0002229724730000034
representing the difference between the measurement output of the sensor node i at the moment k and the measurement output of the sensor node when the triggering condition is met for the last time;
Figure BDA0002229724730000035
is a scalar known to be greater than 0; min {. cndot.) represents the minimum value of the function value;indicating the moment when the sensor node i last satisfied the trigger condition,
Figure BDA0002229724730000037
indicating the moment when the sensor node i next satisfies the triggering condition,
Figure BDA0002229724730000038
representing the measured output of the sensor the last time the sensor node i satisfied the trigger condition, s e {0,1,2, … } representing the trigger sequence.
(3-2) establishing a distributed state estimator for flow state detection:
according to the established urban drainage pipeline water flow state dynamic equation, a distributed state estimator model is established:
Figure BDA0002229724730000039
wherein the content of the first and second substances,
Figure BDA00022297247300000310
an estimation vector representing the sensor node i at the moment k, namely an estimation value of a state vector x (k);
Figure BDA00022297247300000311
representing the nonlinear interference corresponding to the estimation vector of the sensor node i at the moment k;
Figure BDA00022297247300000312
representing an output signal to be estimated of an estimator corresponding to the sensor node i at the moment k;
Figure BDA00022297247300000313
representing a state estimator gain matrix to be designed; the symbol Σ represents a summation operation in mathematics.
In combination with the system dynamic equation, the distributed state estimator is rewritten as:
Figure BDA0002229724730000041
(3-3) establishing a distributed estimation error system for water flow state detection:
defining estimation error of water flow state detection of sensor node i at time k
Figure BDA0002229724730000042
Figure BDA0002229724730000043
The system equation for the distributed estimation error is obtained as follows:
Figure BDA0002229724730000044
Figure BDA0002229724730000045
the system is rewritten into the following estimation error dynamic system by using the Kronecker product principle of the matrix:
wherein:
Figure BDA0002229724730000047
in the formula (I), the compound is shown in the specification,
Figure BDA0002229724730000048
representing the Kronecker product of matrix a and matrix B; diag { … } represents a diagonal matrix; i isNAn identity matrix having a number of dimensions N × N; i denotes an identity matrix of appropriate dimensions.
Defining an augmented vector
Figure BDA0002229724730000049
And (3) amplifying the estimation error dynamic system to obtain an estimation error amplification system:
Figure BDA0002229724730000051
wherein:
Figure BDA0002229724730000052
and (4) solving a distributed state estimator of the urban drainage pipeline water flow detection system:
(4-1) stability analysis of the estimation error augmentation system:
defining a Lyapunov functionWherein
Figure BDA0002229724730000054
The positive definite diagonal matrix to be solved.
Assuming that the interference v (k) is 0, the mathematical expectation of the difference of the Lyapunov function is calculated, resulting in:
for inclusion of random variable alphai(k) Item of
Figure BDA0002229724730000056
And calculating to obtain:
Figure BDA0002229724730000057
wherein the content of the first and second substances,
Figure BDA0002229724730000058
in the formula (I), the compound is shown in the specification,
Figure BDA0002229724730000059
i.e. eiIs a column block matrix whose ith matrix block is an identity matrix I of appropriate dimensions.
For the event trigger item in the water flow state detection system of the urban drainage pipeline, the trigger condition inequality is obtained by the event trigger mechanism in the step (3)Thus, the mathematical expectation of the Lyapunov function difference is written as:
Figure BDA0002229724730000061
for the trigger condition inequality, it can be further rewritten as:
Figure BDA0002229724730000062
wherein, the matrix
Figure BDA0002229724730000063
Figure BDA0002229724730000064
The forms are respectively:
Figure BDA0002229724730000065
from the above derivation, we obtain:
Figure BDA0002229724730000066
defining an augmented vector
Figure BDA0002229724730000067
Two inequalities are obtained by utilizing two conditions of a unilateral Lipschitz nonlinear function:
Figure BDA0002229724730000068
and
Figure BDA0002229724730000069
wherein epsilon1And ε2Any scalar greater than 0; the prime in the formula represents a symmetric term in the matrix, i.e., a transposed element of a symmetric position in the matrix.
Therefore, the mathematical expectation of the difference of the Lyapunov function is written as:
Figure BDA00022297247300000610
wherein the content of the first and second substances,
Figure BDA0002229724730000071
according to the Lyapunov stability theory, when
Figure BDA0002229724730000072
In time, it is known that the estimation error augmentation system is stable in mean square when v (k) is 0.
(4-2) disturbance rejection performance analysis:
for any non-zero perturbation v (k), calculating the mathematical expectation of the difference of the Lyapunov function, namely:
Figure BDA0002229724730000073
defining an augmented vector
The mathematical expectation of the Lyapunov function difference is rewritten as
Figure BDA0002229724730000075
Wherein the content of the first and second substances,
Figure BDA0002229724730000076
defining performance indicatorsWhere the scalar γ is a given disturbance rejection performance indicator, and γ > 0.
Under zero initial conditions and the mean square stability conditions of the first step described above, there are V (0) ═ 0 and V (∞) ═ 0, and
Figure BDA0002229724730000081
obtaining:
Figure BDA0002229724730000082
wherein the content of the first and second substances,
Figure BDA0002229724730000083
when in use
Figure BDA0002229724730000084
And J is less than 0, namely the mean square of the estimation error augmentation system is stable, and the estimation error augmentation system is ensured to have a given disturbance suppression performance index gamma which is more than 0.
(4-3) solving for distributed state estimator gain:
will be provided with
Figure BDA0002229724730000085
Equivalent expansion is as follows
Figure BDA0002229724730000086
Wherein the content of the first and second substances,
Figure BDA0002229724730000087
Figure BDA0002229724730000091
simultaneous left and right multiplication of a diagonal matrix for the inequality Ψ < 0
Figure BDA0002229724730000092
And order
Figure BDA0002229724730000093
Obtaining a linear matrix inequality
Figure BDA0002229724730000094
Wherein the content of the first and second substances,
Figure BDA0002229724730000095
Figure BDA0002229724730000096
solving the linear matrix inequality by using a linear matrix inequality tool box in MATLAB
Figure BDA0002229724730000097
Obtaining an unknown matrix
Figure BDA0002229724730000098
And
Figure BDA0002229724730000099
a value of (d); by
Figure BDA00022297247300000910
Calculating to obtain a matrix
Figure BDA00022297247300000911
A value of (d); according to
Figure BDA00022297247300000912
Get the bookGain of distributed estimator of urban drainage pipeline water flow detection system
Figure BDA00022297247300000913
The invention provides a distributed state estimation method based on an event trigger mechanism, aiming at the problem that the current urban drainage pipeline water flow state detection system in China cannot accurately detect and early warn in time. The invention is based on a wireless sensor network method, adopts a communication protocol of an event trigger mechanism to relieve communication congestion and save energy consumption, and simultaneously considers a unilateral Lipschitz nonlinear function in a more general form. By designing the distributed state estimator and using a linear matrix inequality method to solve the distributed state estimator, the water flow state of the urban drainage pipeline is estimated, so that a timely and effective method is provided for detecting the water flow state of the urban drainage pipeline, and the safety and accuracy requirements of actual state estimation are met.
Detailed Description
A method for detecting the water flow state of an urban drainage pipeline comprises the following specific steps:
step (1), constructing a multi-sensor network model and a topological structure:
arranging N sensors in the area needing water flow state detection, wherein the N sensors are used for respectively measuring the state information of the water level height, the water pressure, the flow speed and the flow of the drainage pipeline;
the N sensors form a sensor network with a topological structure, wherein the number of nodes is N; using directed graphsRepresenting the topology of the sensor network;
wherein the content of the first and second substances,
Figure BDA0002229724730000102
a set of sensors representing the arrangement of the detection areas,
Figure BDA0002229724730000103
representing a set of edges, C ═[cij]N×NA weighted adjacency matrix representing the directed graph,wherein c isijRepresenting the strength of the coupling between sensor node i and node j [. ]]N×NRepresenting a matrix of N × N elements; c. CijIf the value is more than 0, the sensor node j transmits information to the sensor node i at the moment; for allStipulating: if i is j, c is notedii1 means that the sensor network is self-contained in communication.
The set of all sensor nodes connected to sensor node i is denoted as
Figure BDA0002229724730000106
Step (2), establishing a state space model of a water flow state detection system of the urban drainage pipeline:
establishing a dynamic equation of a water flow state detection system of the urban drainage pipeline as follows:
Figure BDA0002229724730000107
wherein
Figure BDA0002229724730000108
Representing the water flow state vector, x, of the drainage pipe at time k1(k)、x2(k)、x3(k)、x4(k) Respectively representing the water flow, the water flow speed, the water level and the water pressure of the water drainage pipeline at the moment k;a real matrix representing n × m dimensions; superscript T represents the transpose of the matrix;
representing a sensor node at time ki measured water flow state values;
Figure BDA00022297247300001011
representing the output signal to be estimated at time k;
Figure BDA00022297247300001012
an external perturbation that is energy-bounded;
and
Figure BDA00022297247300001014
is a known constant matrix;
αi(k)∈[0,1]the random sequence obeys known random distribution and is used for describing a random packet loss phenomenon occurring when the sensor node i transmits measurement data;
obtaining alpha by using experimental and statistical analysis methodi(k) Mean and variance of (1), noted
Figure BDA00022297247300001015
Andwherein E {. denotes the mathematical expectation of the random variable,
Figure BDA00022297247300001017
and
Figure BDA00022297247300001018
is a known scalar;represents the nonlinear interference of urban industrial production and daily life sewage discharge on the water flow in the detection area, and the nonlinear interference meets the following unilateral Lipschitz condition:
condition 1 for any
Figure BDA0002229724730000111
At an arbitrary time k, a scalar ρ exists, and the nonlinear function f (k, x (k)) satisfies<f(k,u)-f(k,v),u-v＞≤ρ||u-v||2；
Condition 2 for any
Figure BDA0002229724730000112
At an arbitrary time k, scalars α, β exist, and a nonlinear function f (k, x (k)) satisfies (f (k, u) -f (k, v))T(f(k,u)-f(k,v))≤β||u-v||2+α<u-v,f(k,v)>(ii) a Wherein the content of the first and second substances,<·>represents the inner product of a vector or matrix in euclidean space; i | · | represents the euclidean norm of a vector or matrix.
Step (3), establishing a distributed state estimator and an error system model of a water flow state detection system of the urban drainage pipeline:
(3-1) setting an event trigger mechanism of sensor network data transmission:
in order to relieve the communication congestion phenomenon caused by the simultaneous transmission of a large number of sensor measurement data and save energy consumption, the invention adopts a communication protocol of an event trigger mechanism.
Setting the event triggering conditions as follows:
Figure BDA0002229724730000113
wherein the content of the first and second substances,representing the difference between the measurement output of the sensor node i at the moment k and the measurement output of the sensor node when the triggering condition is met for the last time;
Figure BDA0002229724730000115
is a scalar known to be greater than 0; min {. cndot.) represents the minimum value of the function value;
Figure BDA0002229724730000116
indicating the moment when the sensor node i last satisfied the trigger condition,
Figure BDA0002229724730000117
indicating the moment when the sensor node i next satisfies the triggering condition,
Figure BDA0002229724730000118
which represents the measurement output of the sensor when the sensor node i last satisfied the trigger condition, s ∈ {0,1, 2.
(3-2) establishing a distributed state estimator for flow state detection:
according to the established urban drainage pipeline water flow state dynamic equation, a distributed state estimator model is established:
Figure BDA0002229724730000119
wherein the content of the first and second substances,
Figure BDA00022297247300001110
an estimation vector representing the sensor node i at the moment k, namely an estimation value of a state vector x (k);
Figure BDA00022297247300001111
representing the nonlinear interference corresponding to the estimation vector of the sensor node i at the moment k;
Figure BDA00022297247300001112
representing an output signal to be estimated of an estimator corresponding to the sensor node i at the moment k;representing a state estimator gain matrix to be designed; the symbol Σ represents a summation operation in mathematics.
In combination with the system dynamic equation, the distributed state estimator is rewritten as:
Figure BDA0002229724730000121
(3-3) establishing a distributed estimation error system for water flow state detection:
defining water of sensor node i at time kEstimation error of flow state detection The system equation for the distributed estimation error is obtained as follows:
Figure BDA0002229724730000124
Figure BDA0002229724730000125
the system is rewritten into the following estimation error dynamic system by using the Kronecker product principle of the matrix:
Figure BDA0002229724730000126
wherein:
Figure BDA0002229724730000127
in the formula (I), the compound is shown in the specification,
Figure BDA0002229724730000128
representing the Kronecker product of matrix a and matrix B; diag { … } represents a diagonal matrix; i isNAn identity matrix having a number of dimensions N × N; i denotes an identity matrix of appropriate dimensions.
Defining an augmented vector
Figure BDA0002229724730000129
And (3) amplifying the estimation error dynamic system to obtain an estimation error amplification system:
Figure BDA0002229724730000131
wherein:
Figure BDA0002229724730000132
and (4) solving a distributed state estimator of the urban drainage pipeline water flow detection system:
(4-1) stability analysis of the estimation error augmentation system:
defining a Lyapunov function
Figure BDA0002229724730000133
Wherein
Figure BDA0002229724730000134
The positive definite diagonal matrix to be solved.
Assuming that the interference v (k) is 0, the mathematical expectation of the difference of the Lyapunov function is calculated, resulting in:
Figure BDA0002229724730000135
for inclusion of random variable alphai(k) Item of
Figure BDA0002229724730000136
And calculating to obtain:
Figure BDA0002229724730000137
wherein the content of the first and second substances,
Figure BDA0002229724730000138
in the formula (I), the compound is shown in the specification,
Figure BDA0002229724730000139
i.e. eiIs a column block matrix whose ith matrix block is an identity matrix I of appropriate dimensions.
For the event trigger item in the water flow state detection system of the urban drainage pipeline, the trigger condition inequality is obtained by the event trigger mechanism in the step (3)
Figure BDA00022297247300001310
Thus, the mathematical expectation of the Lyapunov function difference is written as:
Figure BDA0002229724730000141
for the trigger condition inequality, it can be further rewritten as:
Figure BDA0002229724730000142
wherein, the matrix The forms are respectively:
Figure BDA0002229724730000145
from the above derivation, we obtain:
Figure BDA0002229724730000146
defining an augmented vector
Figure BDA0002229724730000147
Two inequalities are obtained by utilizing two conditions of a unilateral Lipschitz nonlinear function:
Figure BDA0002229724730000148
and
Figure BDA0002229724730000149
wherein epsilon1And ε2Any scalar greater than 0; the prime in the formula represents a symmetric term in the matrix, i.e., a transposed element of a symmetric position in the matrix.
Therefore, the mathematical expectation of the difference of the Lyapunov function is written as:
Figure BDA00022297247300001410
wherein the content of the first and second substances,
Figure BDA0002229724730000151
according to the Lyapunov stability theory, when
Figure BDA0002229724730000152
In time, it is known that the estimation error augmentation system is stable in mean square when v (k) is 0.
(4-2) disturbance rejection performance analysis:
for any non-zero perturbation v (k), calculating the mathematical expectation of the difference of the Lyapunov function, namely:
Figure BDA0002229724730000153
defining an augmented vector
The mathematical expectation of the Lyapunov function difference is rewritten as
Figure BDA0002229724730000155
Wherein the content of the first and second substances,
Figure BDA0002229724730000156
defining performance indicators
Figure BDA0002229724730000157
Where the scalar γ is a given disturbance rejection performance indicator, and γ > 0.
Under zero initial conditions and the mean square stability conditions of the first step described above, there are V (0) ═ 0 and V (∞) ═ 0, and
obtaining:
Figure BDA0002229724730000162
wherein the content of the first and second substances,
Figure BDA0002229724730000163
when in use
Figure BDA0002229724730000164
And J is less than 0, namely the mean square of the estimation error augmentation system is stable, and the estimation error augmentation system is ensured to have a given disturbance suppression performance index gamma which is more than 0.
(4-3) solving for distributed state estimator gain:
will be provided with
Figure BDA0002229724730000165
Equivalent expansion is as follows
Figure BDA0002229724730000166
Wherein the content of the first and second substances,
Figure BDA0002229724730000167
Figure BDA0002229724730000171
simultaneous left and right multiplication of a diagonal matrix for the inequality Ψ < 0
Figure BDA0002229724730000172
And order
Figure BDA0002229724730000173
Obtaining a linear matrix inequality
Figure BDA0002229724730000174
Wherein the content of the first and second substances,
Figure BDA0002229724730000175
Figure BDA0002229724730000176
solving the linear matrix inequality by using a linear matrix inequality tool box in MATLABEquation of
Figure BDA0002229724730000177
Obtaining an unknown matrixAnda value of (d); by
Figure BDA00022297247300001710
Calculating to obtain a matrixA value of (d); according to
Figure BDA00022297247300001712
Gain of the distributed estimator of the urban drainage pipeline water flow detection system is obtained
Figure BDA00022297247300001713

Claims (1)

1. A method for detecting the water flow state of a municipal drainage pipeline is characterized by comprising the following specific steps:
step (1), constructing a multi-sensor network model and a topological structure:
arranging N sensors in the area needing water flow state detection, wherein the N sensors are used for respectively measuring the state information of the water level height, the water pressure, the flow speed and the flow of the drainage pipeline;
the N sensors form a sensor network with a topological structure, wherein the number of nodes is N; using directed graphsRepresenting the topology of the sensor network; wherein the content of the first and second substances,indicating the arrangement of the detection areasThe set of sensors of (a) is,
Figure FDA0002229724720000013
represents a set of edges, C ═ Cij]N×NThe weighted adjacency matrix representing the directed graph, i,
Figure FDA0002229724720000014
cijrepresenting the strength of the coupling between sensor node i and node j [. ]]N×NRepresenting a matrix of N × N elements; c. CijIf the value is more than 0, the sensor node j transmits information to the sensor node i at the moment; for all
Figure FDA0002229724720000015
Stipulating: if i is j, c is notedii1, indicating that the sensor network is self-contained in communication;
the set of all sensor nodes connected to sensor node i is denoted as
Step (2), establishing a state space model of a water flow state detection system of the urban drainage pipeline:
establishing a dynamic equation of a water flow state detection system of the urban drainage pipeline as follows:
Figure FDA0002229724720000017
wherein
Figure FDA0002229724720000018
Representing the water flow state vector, x, of the drainage pipe at time k1(k)、x2(k)、x3(k)、x4(k) Respectively representing the water flow, the water flow speed, the water level and the water pressure of the water drainage pipeline at the moment k;
Figure FDA0002229724720000019
a real matrix representing n × m dimensions; t represents the transpose of the matrix;
Figure FDA00022297247200000110
representing the water flow state value measured by the sensor node i at the moment k;
Figure FDA00022297247200000111
representing the output signal to be estimated at time k;
Figure FDA00022297247200000112
an external perturbation that is energy-bounded;
Figure FDA00022297247200000113
and
Figure FDA00022297247200000114
is a known constant matrix;
αi(k)∈[0,1]the random sequence obeying known random distribution is used for describing a random packet loss phenomenon occurring when the sensor node i transmits measurement data;
obtaining alphai(k) Mean and variance of (1), noted
Figure FDA0002229724720000021
And
Figure FDA0002229724720000022
wherein E {. denotes the mathematical expectation of the random variable,
Figure FDA0002229724720000023
andis a known scalar;
Figure FDA0002229724720000025
represents the nonlinear interference of urban industrial production and daily life sewage discharge on the water flow in the detection area, and the nonlinear interference meets the following unilateral Lipschitz condition:
condition 1. for any u,at an arbitrary time k, a scalar ρ exists, and the nonlinear function f (k, x (k)) satisfies<f(k,u)-f(k,v),u-v>≤ρ||u-v||2；
Condition 2. for any u,
Figure FDA0002229724720000027
at an arbitrary time k, scalars α, β exist, and a nonlinear function f (k, x (k)) satisfies (f (k, u) -f (k, v))T(f(k,u)-f(k,v))≤β||u-v||2+α<u-v,f(k,v)>(ii) a Wherein the content of the first and second substances,<·>represents the inner product of a vector or matrix in euclidean space; | | · | represents the euclidean norm of a vector or matrix;
step (3), establishing a distributed state estimator and an error system model of a water flow state detection system of the urban drainage pipeline:
(3-1) setting an event trigger mechanism of sensor network data transmission:
setting the event triggering conditions as follows:
Figure FDA0002229724720000028
Figure FDA0002229724720000029
representing the difference between the measurement output of the sensor node i at the moment k and the measurement output of the sensor node when the triggering condition is met for the last time;
Figure FDA00022297247200000210
is a scalar known to be greater than 0; min {. cndot.) represents the minimum value of the function value;
Figure FDA00022297247200000211
indicating the moment when the sensor node i last satisfied the trigger condition,
Figure FDA00022297247200000212
indicating the moment when the sensor node i next satisfies the triggering condition,
Figure FDA00022297247200000213
representing the measurement output of the sensor when the sensor node i meets the triggering condition for the last time, and s belongs to {0,1, 2. } represents a triggering sequence;
(3-2) establishing a distributed state estimator for flow state detection:
establishing a distributed state estimator model:
Figure FDA00022297247200000214
wherein the content of the first and second substances,
Figure FDA00022297247200000216
an estimation vector representing the sensor node i at the moment k, namely an estimation value of a state vector x (k);
Figure FDA00022297247200000217
representing the nonlinear interference corresponding to the estimation vector of the sensor node i at the moment k;
Figure FDA00022297247200000218
representing an output signal to be estimated of an estimator corresponding to the sensor node i at the moment k;
Figure FDA00022297247200000219
representing a state estimator gain matrix to be designed; the symbol Σ represents a summation operation in mathematics; in combination with the system dynamic equation, the distributed state estimator is rewritten as:
Figure FDA0002229724720000031
(3-3) establishing a distributed estimation error system for water flow state detection:
defining estimation error of water flow state detection of sensor node i at time k
Figure FDA0002229724720000033
The system equation for the distributed estimation error is obtained as follows:
the above system is adapted to the following estimation error dynamics system:
Figure FDA0002229724720000035
wherein:
in the formula (I), the compound is shown in the specification,
Figure FDA0002229724720000037
representing the Kronecker product of matrix a and matrix B; diag { … } represents a diagonal matrix; i isNAn identity matrix having a number of dimensions N × N; i represents an identity matrix with appropriate dimensions;
defining an augmented vector
Figure FDA0002229724720000038
And (3) amplifying the estimation error dynamic system to obtain an estimation error amplification system:
Figure FDA0002229724720000039
wherein:
Figure FDA0002229724720000041
and (4) solving a distributed state estimator of the urban drainage pipeline water flow detection system:
(4-1) stability analysis of the estimation error augmentation system:
defining a Lyapunov function
Figure FDA0002229724720000042
Wherein
Figure FDA0002229724720000043
P1,P2,…,PN+1Determining a diagonal matrix for the positive matrix to be solved;
assuming that the interference v (k) is 0, the mathematical expectation of the difference of the Lyapunov function is calculated, resulting in:
for inclusion of random variable alphai(k) Item of
Figure FDA0002229724720000045
And calculating to obtain:
Figure FDA0002229724720000046
wherein the content of the first and second substances,
Figure FDA0002229724720000047
in the formula (I), the compound is shown in the specification,
Figure FDA0002229724720000048
i.e. eiIs a column block matrix;
event in water flow state detection system for urban drainage pipelineThe triggering item is triggered by the event triggering mechanism in the step (3) to obtain a triggering condition inequality
Figure FDA0002229724720000049
Thus, the mathematical expectation of the Lyapunov function difference is:
the inequality for the trigger condition is rewritten as:
Figure FDA0002229724720000051
wherein, the matrixThe forms are respectively:
Figure FDA0002229724720000053
obtaining:
Figure FDA0002229724720000054
defining an augmented vector
Figure FDA0002229724720000055
Two inequalities are obtained by utilizing two conditions of a unilateral Lipschitz nonlinear function:
Figure FDA0002229724720000056
and
Figure FDA0002229724720000057
wherein epsilon1And ε2Any scalar greater than 0; the number in the formula represents a symmetric item in the matrix, namely a transposed element at a symmetric position in the matrix;
the number of the difference of the Lyapunov functionThe mathematical expectation formula is written as:wherein the content of the first and second substances,
Figure FDA0002229724720000061
when in use
Figure FDA0002229724720000062
When v (k) is 0, the estimation error augmentation system is stable in mean square;
(4-2) disturbance rejection performance analysis:
for any non-zero perturbation v (k), calculating the mathematical expectation of the difference of the Lyapunov function, namely:
Figure FDA0002229724720000063
defining an augmented vector
Figure FDA0002229724720000064
The mathematical expectation of the Lyapunov function difference is rewritten asWherein the content of the first and second substances,
Figure FDA0002229724720000066
defining performance indicatorsWherein the scalar gamma is a given disturbance suppression performance index, and gamma is more than 0;
v (0) ═ 0 under the zero initial condition, V (∞) ═ 0 under the mean square stability condition, and
Figure FDA0002229724720000071
obtaining:
wherein the content of the first and second substances,
when in use
Figure FDA0002229724720000074
When J is less than 0, namely the mean square of the estimation error augmentation system is stable, and the estimation error augmentation system is ensured to have a given disturbance suppression performance index gamma which is more than 0;
(4-3) solving for distributed state estimator gain:
will be provided with
Figure FDA0002229724720000075
Equivalent expansion is as follows
Figure FDA0002229724720000076
Wherein the content of the first and second substances,
Figure FDA0002229724720000078
simultaneous left and right multiplication of a diagonal matrix for the inequality Ψ < 0
Figure FDA0002229724720000082
And order
Figure FDA0002229724720000083
Obtaining a linear matrix inequality
Figure FDA0002229724720000084
Wherein the content of the first and second substances,
Figure FDA0002229724720000085
solving the linear matrix inequality by using a linear matrix inequality tool box in MATLABObtaining an unknown matrixAnd
Figure FDA0002229724720000088
a value of (d); by
Figure FDA0002229724720000089
Calculating to obtain a matrixA value of (d); according to
Figure FDA00022297247200000811
Gain of distributed estimator of urban drainage pipeline water flow detection system is obtained
Figure FDA00022297247200000812
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