CN110705892A - Water flow state detection method for urban drainage pipeline - Google Patents
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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
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 graphsRepresenting the topology of the sensor network;
wherein the content of the first and second substances,a set of sensors representing the arrangement of the detection areas,represents a set of edges, C ═ Cij]N×NA weighted adjacency matrix representing the directed graph,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 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
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:
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;a real matrix representing n × m dimensions; superscript T represents the transpose of the 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), notedAndwherein E {. denotes the mathematical expectation of the random variable,andis 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 anyAt 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 anyAt 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: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;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,indicating the moment when the sensor node i next satisfies the triggering condition,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:wherein the content of the first and second substances,an estimation vector representing the sensor node i at the moment k, namely an estimation value of a state vector x (k);representing the nonlinear interference corresponding to the estimation vector of the sensor node i at the moment k;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:
(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 The system equation for the distributed estimation error is obtained as follows:
the system is rewritten into the following estimation error dynamic system by using the Kronecker product principle of the matrix:
in the formula (I), the compound is shown in the specification,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 vectorAnd (3) amplifying the estimation error dynamic system to obtain an estimation error amplification system:wherein:
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:
Assuming that the interference v (k) is 0, the mathematical expectation of the difference of the Lyapunov function is calculated, resulting in:
wherein the content of the first and second substances,in the formula (I), the compound is shown in the specification,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:
for the trigger condition inequality, it can be further rewritten as:
from the above derivation, we obtain:
defining an augmented vectorTwo inequalities are obtained by utilizing two conditions of a unilateral Lipschitz nonlinear function:andwherein 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:wherein the content of the first and second substances,
according to the Lyapunov stability theory, whenIn 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:
defining an augmented vector
The mathematical expectation of the Lyapunov function difference is rewritten asWherein the content of the first and second substances,
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
when in useAnd 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:
simultaneous left and right multiplication of a diagonal matrix for the inequality Ψ < 0And orderObtaining a linear matrix inequalityWherein the content of the first and second substances,
solving the linear matrix inequality by using a linear matrix inequality tool box in MATLABObtaining an unknown matrixAnda value of (d); byCalculating to obtain a matrixA value of (d); according toGet the bookGain of distributed estimator of urban drainage pipeline water flow detection system
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,a set of sensors representing the arrangement of the detection areas,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.
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:
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;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;
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), notedAndwherein E {. denotes the mathematical expectation of the random variable,andis 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 anyAt 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 anyAt 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: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;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,indicating the moment when the sensor node i next satisfies the triggering condition,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:wherein the content of the first and second substances,an estimation vector representing the sensor node i at the moment k, namely an estimation value of a state vector x (k);representing the nonlinear interference corresponding to the estimation vector of the sensor node i at the moment k;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:
(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:
the system is rewritten into the following estimation error dynamic system by using the Kronecker product principle of the matrix:
in the formula (I), the compound is shown in the specification,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 vectorAnd (3) amplifying the estimation error dynamic system to obtain an estimation error amplification system:wherein:
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:
Assuming that the interference v (k) is 0, the mathematical expectation of the difference of the Lyapunov function is calculated, resulting in:
wherein the content of the first and second substances,in the formula (I), the compound is shown in the specification,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:
for the trigger condition inequality, it can be further rewritten as:
from the above derivation, we obtain:
defining an augmented vectorTwo inequalities are obtained by utilizing two conditions of a unilateral Lipschitz nonlinear function:andwherein 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:wherein the content of the first and second substances,
according to the Lyapunov stability theory, whenIn 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:
defining an augmented vector
The mathematical expectation of the Lyapunov function difference is rewritten asWherein the content of the first and second substances,
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
when in useAnd 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:
simultaneous left and right multiplication of a diagonal matrix for the inequality Ψ < 0And orderObtaining a linear matrix inequalityWherein the content of the first and second substances,
solving the linear matrix inequality by using a linear matrix inequality tool box in MATLABEquation ofObtaining an unknown matrixAnda value of (d); byCalculating to obtain a matrixA value of (d); according toGain of the distributed estimator of the urban drainage pipeline water flow detection system is obtained
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,represents a set of edges, C ═ Cij]N×NThe weighted adjacency matrix representing the directed graph, i,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 allStipulating: 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:
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;a real matrix representing n × m dimensions; t represents the transpose of the 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), notedAndwherein E {. denotes the mathematical expectation of the random variable,andis 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 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,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: 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;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,indicating the moment when the sensor node i next satisfies the triggering condition,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: wherein the content of the first and second substances,an estimation vector representing the sensor node i at the moment k, namely an estimation value of a state vector x (k);representing the nonlinear interference corresponding to the estimation vector of the sensor node i at the moment k;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:
(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 The system equation for the distributed estimation error is obtained as follows:
the above system is adapted to the following estimation error dynamics system:
wherein:
in the formula (I), the compound is shown in the specification,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 vectorAnd (3) amplifying the estimation error dynamic system to obtain an estimation error amplification system:wherein:
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 functionWhereinP1,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:
wherein the content of the first and second substances,in the formula (I), the compound is shown in the specification,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 inequalityThus, the mathematical expectation of the Lyapunov function difference is:
the inequality for the trigger condition is rewritten as:
obtaining:
defining an augmented vectorTwo inequalities are obtained by utilizing two conditions of a unilateral Lipschitz nonlinear function:andwherein 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,
(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:
The mathematical expectation of the Lyapunov function difference is rewritten asWherein the content of the first and second substances,
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
wherein the content of the first and second substances,
when in useWhen 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 withEquivalent expansion is as followsWherein the content of the first and second substances,
simultaneous left and right multiplication of a diagonal matrix for the inequality Ψ < 0And orderObtaining a linear matrix inequality
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