A sensor network-based distributed leakage estimation method for urban sewer networks

Abstract

The application discloses a kind of distributed leakage estimation method of urban drainage pipe network based on sensor network.The method constructs the topological structure of sensor network and the state space model of urban drainage pipe network, describes the nonlinear random variation in drainage pipe network by using Bernoulli random sequence, describes the measurement attenuation phenomenon randomly occurred when sensor measures by using Rice attenuation model, and adopts Round-Robin communication protocol for data transmission scheduling;Then design distributed state estimator, then establish error augmented system, obtain the sufficient condition of mean square exponential stability and H ∞ performance;Finally, the gain matrix of distributed estimator is obtained by convex optimization method.It provides a real-time and effective method for accurate remote estimation and monitoring of urban drainage pipe network.

Description

Technical Field

[0001] This invention belongs to the field of automatic control technology and relates to a method for estimating leakage in drainage pipe networks, specifically a distributed leakage estimation method for urban drainage pipe networks based on sensor networks. Background Technology

[0002] Urban drainage networks bear the heavy responsibility of flood control, drainage, and sewage collection and transportation, and are an important component of municipal infrastructure. How to fully utilize the efficiency of urban drainage networks is one of the most pressing issues to be addressed, and it directly relates to urban safety. Especially during the rainy season, cities are prone to varying degrees of flooding, posing a serious threat to urban traffic and the safety of people's lives and property.

[0003] Currently, due to outdated urban drainage network monitoring technology, unreasonable design, and harsh operating environments, serious leakage in urban drainage networks cannot be effectively detected, failing to meet the needs of rapid urban development. With the continuous integration of new-generation information technologies such as big data, cloud computing, and mobile internet into traditional industries, modern urban water systems are also increasingly integrating with emerging technologies, applying new information technologies to solve problems and promote the automation and intelligentization of urban water systems.

[0004] To achieve accurate and real-time remote monitoring of leakage in urban drainage networks, distributed estimation based on sensor networks is a highly effective and feasible method. However, the operating environment of sensors varies randomly during data acquisition, necessitating consideration of measurement attenuation phenomena that occur randomly during sensor measurements. Furthermore, the bandwidth of public communication networks transmitting sensor measurement data is limited, and the simultaneous transmission of large amounts of sensor data through these networks can easily lead to network congestion and reduced transmission performance. Therefore, maintaining data transmission and leakage estimation performance under limited network bandwidth is a challenging problem. Summary of the Invention

[0005] To address the shortcomings of existing technologies, this invention proposes a distributed leakage estimation method for urban drainage pipe networks based on sensor networks. It uses the Rice attenuation model to describe the random measurement attenuation phenomenon that occurs during sensor measurements, utilizes Bernoulli sequences to describe the nonlinear random changes of the system, and applies the Round-Robin protocol in information transmission scheduling to save network resource usage, thus solving the problem of real-time and accurate remote estimation and monitoring of urban drainage pipe networks.

[0006] A distributed leakage estimation method for urban drainage pipe networks based on sensor networks specifically includes the following steps:

[0007] Step 1: Construct the sensor network topology

[0008] N sensors are deployed in the urban drainage network. Each sensor independently measures the water resource status within the network and uses a directed graph. This represents the network topology consisting of N sensors.

[0009] The above directed graph middle, Represents the set of all sensor nodes. Describe the set of edges. This represents the weighted adjacency matrix of the directed graph, where the sensor nodes... a ij a represents the connection strength between sensor nodes i and j. ij >0 indicates that sensor node j has transmitted information to sensor node i. When i = j, let a be the value of a. ij =a ii =1, indicating that the sensor network contains itself; [·] N×N This represents a matrix consisting of N×N elements; This represents the set of all sensor nodes that transmit information to sensor node i.

[0010] Step 2: Establish the state-space model of the drainage pipe network system.

[0011] Based on the hydraulic equations and measured water data of a city's drainage network, the system's dynamic equations are established:

[0012]

[0013] in, Let x1(t) represent the state vector of the drainage network at time t, x2(t) represent the water flow rate in the drainage network at time t, x3(t) represent the water pressure in the drainage network at time t, and x4(t) represent the water velocity in the drainage network at time t. y represents the measured output value of sensor node i at time t; i1 (t) represents the water flow rate in the drainage network measured by sensor node i at time t. i2 (t) represents the water temperature and y measured by sensor node i in the drainage network at time t. i3 (t) represents the water pressure measured by sensor node i in the drainage network at time t. This represents the output vector of the drainage network to be estimated at time t. It is non-zero noise belonging to l2[0,∞). and It is a known constant matrix. Nonlinear function. Used to describe nonlinear phenomena in drainage pipe network systems, satisfying sector bounded constraints, i.e., for a known vector... exist:

[0014] [f(a)-f(b)-U1(ab)] T [f(a)-f(b)-U2(ab)]≤0

[0015] [g(a)-g(b)-U3(ab)] T [g(a)-g(b)-U4(ab)]≤0

[0016] in, All are known matrices. β(t) is a Bernoulli random sequence taking values ​​of 0 or 1. Where Prob{} represents the probability of a random event occurring. It is a known positive scalar.

[0017] Considering the attenuation phenomenon that occurs during sensor measurement, the Rice attenuation model is used to describe the measured output value of sensor node i at time t as...

[0018]

[0019] in It is a known constant matrix. Based on experimental and theoretical analysis, the signal acquired by the sensor forms three attenuation channels due to refraction, reflection, and obstruction, so s = 0, 1, 2, and the number of channels l is 2. The attenuation coefficient of each channel of a single sensor can be represented by a set of independent random sequences α. i,s (t)∈[0,1] represents a value whose mean and variance are denoted as t ∈ [0,1] . in and Both are known positive scalars. Also, assume that y = ... when t ∈ [-∞, 0]. i (t) = 0, ν(t) = 0.

[0020] Transmitting large amounts of measurement data from sensors over public networks consumes significant network resources. Therefore, to conserve communication resources, the Round-Robin communication protocol is used, transmitting only one component of all measurement data at any given time to avoid network congestion.

[0021] Each sensor node acquires measurement data with a dimension of 3. According to the Round-Robin communication protocol, at each moment, only one component of the measurement output from all sensors is transmitted; that is, only when... hour, Only then will it be transmitted, where mod represents the remainder. Therefore, in the λth transmission cycle, the information sequence transmitted from sensor node i to its neighboring nodes is y.i1 (3λ+1), y i2 (3λ+2), y i3 (3λ+3). Therefore, the information transmitted by sensor node i to its neighboring nodes... It can be represented as in

[0022] Step 3: Establish a distributed state estimator and error system model for drainage network leakage monitoring.

[0023] The following distributed state estimator is designed for the state-space model of a drainage pipe network system:

[0024]

[0025] in, It is the estimated state of node i. These are the estimated values ​​of x1(t), x2(t), x3(t), and x4(t) at sensor node i, respectively. This is the estimated value of z(t) at sensor node i; and It is the gain matrix of the distributed estimator to be designed.

[0026] Rewrite formula (3) in the form of the Kronecker product as shown in formula (4):

[0027]

[0028] in: X T X -1 These represent the transpose or inverse of X, respectively; diag N {…} denotes an N-order block diagonal matrix; 0 and I denote the zero matrix and identity matrix, respectively. m I represents a diagonal matrix whose m diagonal blocks are appropriately dimensional zero matrices. m Let I represent a diagonal matrix consisting of m identity matrices I; This represents the Kronecker product of matrices A and B.

[0029] definition Combining formulas (1) and (4), we can obtain:

[0030]

[0031] in:

[0032] definition Combining formulas (1) and (5), we construct an estimation error augmentation system:

[0033]

[0034] in,

[0035] Step 4: Solve for the distributed estimator

[0036] s4.1 Stability Analysis of Systems with Augmented Estimation Errors

[0037] To analyze the stability of the error-amplified system, we first assume that the noise ν(t) = 0. We define the Lyapunov function as V(t) = V1(t) + V2(t):

[0038]

[0039] in, W s It is a positive definite symmetric matrix W1=diag{W 11 W 12}、W2=diag{W 21 W 22}, W 11 W 12 W 21 W 22 All are positive definite symmetric matrices.

[0040] Calculate the expected value of the Lyapunov function difference above:

[0041]

[0042] According to the augmented matrix From the definition, we can obtain:

[0043]

[0044] in,

[0045] Using a similar method, we can obtain:

[0046]

[0047] in,

[0048] Simultaneously calculable

[0049] E{△V2(t)}=E{V2(t+1)-V2(t)}

[0050] =E{ηT (t)W1η(t)+η T (t-1)W2η(t-1)

[0051] -η T (t-1)W1η(t-1)-η T (t-2)W2η(t-2)} (11)

[0052] Therefore, there is

[0053]

[0054] Based on the sector bounded constraints of the nonlinear functions f(x(t)) and g(x(t)), the following inequalities can be obtained.

[0055]

[0056] Define augmented vectors in According to E{△V(t)} and formula (13), we can obtain:

[0057] E{V(t+1)-V(t)}≤E{ε T (t)S1ε(t)} (14)

[0058] In the formula,

[0059]

[0060] According to Lyapunov stability theory, when S1 < 0, the estimation error augmentation system is mean square exponentially stable when the noise is equal to 0.

[0061] s4.2, H of the estimation error augmentation system ∞ Performance Analysis

[0062] Next, we will discuss the H of the estimation error augmentation system. ∞ Performance is analyzed. For non-zero noise ν(t)∈l2[0,∞), a performance index function is selected.

[0063]

[0064] Define vector Choosing the Lyapunov function shown in formula (7), and solving for the expected value of its difference, we can see that:

[0065]

[0066]

[0067] in,

[0068] Therefore, for non-zero noise ν(t), when Given that V(0) = 0 under zero initial conditions and the system stability V(∞) = 0, we can obtain:

[0069]

[0070] Right now

[0071] Therefore, when At that time, the error augmentation system satisfies H ∞ Performance-constrained, and H ∞ The performance metric is γ.

[0072] s4.3 Solving for distributed H ∞ The gain matrix of the estimator

[0073] Based on the analysis of s4.1 and s4.2, the estimation error augmentation system stability and H are obtained. ∞ Performance, solving for the gain matrix of the distributed estimator:

[0074] For matrix inequalities Using Schur's supplementary lemma:

[0075]

[0076] in,

[0077]

[0078]

[0079] For any matrix χ, according to the inequality It can be calculated that:

[0080]

[0081] Define two matrices and Make Consider the linear matrix inequality shown in equation (20). Solving this inequality is a convex optimization problem:

[0082]

[0083] Combining formulas (19) and (20), we can obtain:

[0084]

[0085] Multiply both sides of formula (21) by the left side. And right multiplication Matrix inequalities can be obtained

[0086] Based on the derivations in 4.1 and 4.2, the following conditions are met. It can be concluded that the estimation error augmented system satisfies mean square exponential stability and H ∞ Performance constraints, and formula (18) can be derived from the matrix Applying Schur's complement lemma, we obtain that as long as S² < 0, the mean square exponential stability and H of the estimation error augmented system are satisfied. ∞ Performance can then be guaranteed. Based on the above derivation, formula (20) can be equivalently derived to S2<0. Therefore, as long as the inequality shown in formula (20) is satisfied, it can be concluded that the estimation error augmented system is mean square exponentially stable and has H ∞ Performance constraints. and Substitute into matrix In the middle, solving the convex optimization problem (20) yields the following results: and Finally, according to L=diag N {L ii} and K = [a ij K ij ] N×N The gain matrix L of the estimator can be obtained. ii and K ij That is, the gain matrix of the estimator for the urban drainage network designed in this invention.

[0087] The present invention has the following beneficial effects:

[0088] 1. In the distributed estimation of leakage monitoring of drainage pipe network, the random nonlinear interference that exists in the urban drainage pipe network system in actual application is considered, and the Rice attenuation model is used to describe the measurement attenuation phenomenon in sensor measurement.

[0089] 2. A scheduling mechanism based on the Round-Robin communication protocol is proposed to minimize the occupation of communication resources and the energy consumption of sensor nodes while ensuring system performance.

[0090] 3. The mean square exponential stability of the estimation error augmented system was analyzed and H was performed. ∞ Performance analysis reveals that a novel, real-time, and effective distributed estimation method for monitoring leakage in drainage networks is proposed, which solves the distributed estimator using convex optimization methods, meeting the requirements for remote monitoring of leakage in actual drainage networks. Detailed Implementation

[0091] A distributed leakage estimation method for urban drainage pipe networks based on sensor networks specifically includes the following steps:

[0092] Step 1: Construct the sensor network topology

[0093] N sensors are deployed in the urban drainage network. Each sensor independently measures the water resource status within the network and uses a directed graph. This represents the network topology consisting of N sensors. Represents the set of all sensor nodes. Describe the set of edges. This represents the weighted adjacency matrix of the directed graph, where the sensor nodes... a ij This indicates the connection strength between sensor nodes i and j; [·] N×N This represents a matrix consisting of N×N elements; This represents the set of all sensor nodes that transmit information to sensor node i.

[0094] Step 2: Establish the state-space model of the drainage pipe network system.

[0095] Based on the hydraulic equations and measured water data of a city's drainage network, the system's dynamic equations are established:

[0096]

[0097] in, Let x1(t) represent the state vector of the drainage network at time t, x2(t) represent the water flow rate in the drainage network at time t, x3(t) represent the water pressure in the drainage network at time t, and x4(t) represent the water velocity in the drainage network at time t. y represents the measured output value of sensor node i at time t; i1 (t) represents the water flow rate in the drainage network measured by sensor node i at time t. i2 (t) represents the water temperature and y measured by sensor node i in the drainage network at time t. i3 (t) represents the water pressure measured by sensor node i in the drainage network at time t. This represents the output vector of the drainage network to be estimated at time t. It is non-zero noise belonging to l2[0,∞). and It is a known constant matrix. Nonlinear function. Used to describe nonlinear phenomena in drainage pipe network systems, satisfying sector bounded constraints, i.e., for a known vector... exist:

[0098] [f(a)-f(b)-U1(ab)] T[f(a)-f(b)-U2(ab)]≤0

[0099] [g(a)-g(b)-U3(ab)] T [g(a)-g(b)-U4(ab)]≤0

[0100] in, All are known matrices. β(t) is a Bernoulli random sequence taking values ​​of 0 or 1. Where Prob{} represents the probability of a random event occurring. It is a known positive scalar.

[0101] Considering the attenuation phenomenon that occurs during sensor measurement, the Rice attenuation model is used to describe the measured output value of sensor node i at time t as...

[0102]

[0103] in It is a known constant matrix. Based on experimental and theoretical analysis, the signal acquired by the sensor forms three attenuation channels due to refraction, reflection, and obstruction, so s = 0, 1, 2, and the number of channels l is 2. The attenuation coefficient of each channel of a single sensor can be represented by a set of independent random sequences α. i,s (t)∈[0,1] represents a value whose mean and variance are denoted as t ∈ [0,1] . in and Both are known positive scalars. Also, assume that y = ... when t ∈ [-∞, 0]. i (t) = 0, ν(t) = 0.

[0104] Each sensor node acquires measurement data with a dimension of 3. According to the Round-Robin communication protocol, only when... hour, Only then will it be transmitted, where mod represents the remainder. Therefore, in the λth transmission cycle, the information sequence transmitted from sensor node i to its neighboring nodes is y. i1 (3λ+1), y i2 (3λ+2), y i3 (3λ+3). Therefore, the information transmitted by sensor node i to its neighboring nodes... It can be represented as in

[0105] Step 3: Establish a distributed state estimator and error system model for drainage network leakage monitoring.

[0106] The following distributed state estimator is designed for the state-space model of a drainage pipe network system:

[0107]

[0108] in, It is the estimated state of node i. These are the estimated values ​​of x1(t), x2(t), x3(t), and x4(t) at sensor node i, respectively. This is the estimated value of z(t) at sensor node i; and It is the gain matrix of the distributed estimator to be designed.

[0109] Rewrite formula (3) in the form of the Kronecker product as shown in formula (4):

[0110]

[0111] in: X T X -1 These represent the transpose or inverse of X, respectively; diag N {…} denotes an N-order block diagonal matrix; 0 and I denote the zero matrix and identity matrix, respectively. m I represents a diagonal matrix whose m diagonal blocks are appropriately dimensional zero matrices. m Let I represent a diagonal matrix consisting of m identity matrices I; This represents the Kronecker product of matrices A and B.

[0112] definition Combining formulas (1) and (4), we can obtain:

[0113]

[0114] in:

[0115] definition Combining formulas (1) and (5), we construct an estimation error augmentation system:

[0116]

[0117] in,

[0118] Step 4: Solve for the distributed estimator

[0119] Analysis of the stability conditions and H of the system with augmented estimation error ∞Performance, solving for the gain matrix L of the distributed estimator ii and K ij That is, the gain matrix of the estimator for the urban drainage network designed in this invention.

Claims

1. A distributed leakage estimation method for urban drainage pipe networks based on sensor networks, characterized in that: Specifically, the following steps are included: Step 1: Construct the sensor network topology N sensors are deployed in the urban drainage network. Each sensor independently measures the water resource status within the network and uses a directed graph. This represents the network topology consisting of N sensors; Step 2: Establish the state-space model of the drainage pipe network system. Based on the hydraulic equations and measured water data of a city's drainage network, the system's dynamic equations are established: in, Let x1(t), x2(t), x3(t), and x4(t) represent the state vector of the drainage network at time t, respectively. y represents the measured output value of sensor node i at time t; i1 (t), y i2 (t), y i3 (t) represent the water flow rate, water temperature, and water pressure in the drainage network measured by sensor node i at time t, respectively. This represents the output vector of the drainage network to be estimated at time t; It is non-zero noise belonging to l2[0,∞). and It is a known constant matrix; nonlinear function It satisfies the sector bounded constraint; β(t) is a Bernoulli random sequence taking values ​​of 0 or 1. Where Prob{} represents the probability of a random event occurring. Given a positive scalar; The measured output value of sensor node i at time t is described using the Rice decay model. in It is a known constant matrix; the signal collected by the sensor forms three attenuation channels due to refraction, reflection, and obstruction, so s = 0, 1, 2, and the number of channels l = 2; a set of independent random sequences α i,s (t)∈[0,1] represents the attenuation coefficient of each channel of a single sensor, and its mean and variance are denoted as . in and Both are known positive scalars; at the same time, assume that when t∈[-∞,0], y i (t) = 0, ν(t) = 0; The measurement data obtained by each sensor node has a dimension of 3. According to the Round-Robin communication protocol, in the λ-th transmission cycle, the information sequence transmitted from sensor node i to its neighboring nodes is y. i1 (3λ+1), y i2 (3λ+2), y i3 (3λ+3), information transmitted by sensor node i to its neighboring nodes. Represented as in Step 3: Establish a distributed state estimator and error system model for drainage network leakage monitoring. The following distributed state estimator is designed for the state-space model of a drainage pipe network system: in, It is the estimated state of node i. These are the estimated values ​​of x1(t), x2(t), x3(t), and x4(t) at sensor node i, respectively. This is the estimated value of z(t) at sensor node i; and It is the gain matrix of the distributed estimator to be designed; a ij This indicates the connection strength between sensor nodes i and j; Rewrite formula (3) in Kronecker product form to construct an estimation error augmentation system; Step 4: Solve for the distributed estimator Define a Lyapunov function, and then evaluate the stability and H of the system with augmented estimation error for the cases of noise ν(t) = 0 and noise ν(t) ∈ l2[0,∞), respectively. ∞ The performance constraints are analyzed, and the system satisfies mean square exponential stability and H under the condition of increased estimation error. ∞ Under performance constraints, the gain matrix L of the distributed estimator is obtained by solving the problem using convex optimization methods. ii and K ij .

2. The distributed leakage estimation method for urban drainage pipe networks based on sensor networks as described in claim 1, characterized in that: The directed graph in, Represents the set of all sensor nodes. Describe the set of edges. This represents the weighted adjacency matrix of the directed graph, where the sensor nodes... a ij a represents the connection strength between sensor nodes i and j. ij >0 indicates that sensor node j has transmitted information to sensor node i. When i = j, let a be the value of a. ij =a ii =1, indicating that the sensor network contains itself; [·] N×N This represents a matrix consisting of N×N elements; This represents the set of all sensor nodes that transmit information to sensor node i.

3. The distributed leakage estimation method for urban drainage pipe networks based on sensor networks as described in claim 1, characterized in that: For a known vector exist: [f(a)-f(b)-U1(a-b)] T [f(a)-f(b)-U2(a-b)]≤0 [g(a)-g(b)-U3(a-b)] T [g(a)-g(b)-U4(a-b)]≤0 in, All of them are known matrices.

4. The distributed leakage estimation method for urban drainage pipe networks based on sensor networks as described in claim 1, characterized in that: The Kronecker product form of the distributed state estimator shown in equation (3) is: in: X T X -1 These represent the transpose or inverse of X, respectively; diag N {…} denotes an N-order block diagonal matrix; 0 and I denote the zero matrix and identity matrix, respectively. m I represents a diagonal matrix whose m diagonal blocks are appropriately dimensional zero matrices. m Let I represent a diagonal matrix consisting of m identity matrices I; This represents the Kronecker product of matrices A and B. definition Combining formulas (1) and (4), we can obtain: in: definition Combining formulas (1) and (5), we construct an estimation error augmentation system: in, 5. The distributed leakage estimation method for urban drainage pipe networks based on sensor networks as described in claim 4, characterized in that: The specific process of solving the distributed estimator is as follows: s4.1 Stability Analysis of Systems with Augmented Estimation Errors To analyze the stability of the error-amplified system, we first assume that the noise ν(t) = 0; and define the Lyapunov function as V(t) = V1(t) + V2(t): in, W s It is a positive definite symmetric matrix j=1,2,…,N+1, s={1,2}, W1=diag{W 11 W 12 }、W2=diag{W 21 W 22 }, W 11 W 12 W 21 W 22 All are positive definite symmetric matrices; Calculate the expected value of the Lyapunov function difference above: According to the augmented matrix From the definition, we can obtain: in, therefore: in, Simultaneously calculable E{△V2(t)}=E{V2(t+1)-V2(t)} =E{η T (t)W1η(t)+η T (t-1)W2η(t-1)-η T (t-1)W1η(t-1)-η T (t-2)W2η(t-2)} (11) Therefore, there is Based on the sector bounded constraints of the nonlinear functions f(x(t)) and g(x(t)), the following inequalities can be obtained. Define augmented vectors in According to E{△V(t)} and formula (13), we can obtain: E{V(t+1)-V(t)}≤E{ε T (t)S1ε(t)} (14) In the formula, According to Lyapunov stability theory, when S1 < 0, the estimation error augmentation system is mean square exponentially stable when the noise is equal to 0. s4.2, H of the estimation error augmentation system ∞ Performance Analysis Next, we will discuss the H of the estimation error augmentation system. ∞ Performance analysis is performed; for non-zero noise ν(t)∈l2[0,∞), a performance index function is selected. Define vector Choosing the Lyapunov function shown in formula (7), and solving for the expected value of its difference, we can see that: in, Therefore, for non-zero noise ν(t), when Given that V(0) = 0 under zero initial conditions and the system stability V(∞) = 0, we can obtain: Right now Therefore, when At that time, the error augmentation system satisfies H ∞ Performance-constrained, and H ∞ The performance index is γ; s4.3 Solving for distributed H ∞ The gain matrix of the estimator Based on the analysis of s4.1 and s4.2, the estimation error augmentation system stability and H are obtained. ∞ Performance, solving for the gain matrix of the distributed estimator: For matrix inequalities Using Schur's complement lemma, we get: in, For any matrix According to the inequality It can be calculated that: Define two matrices and Make Consider the linear matrix inequality shown in equation (20). Solving this inequality is a convex optimization problem: Combining formulas (19) and (20), we can obtain: Multiply both sides of formula (21) by the left side. And right multiplication Matrix inequalities can be obtained As can be seen from the above derivation, as long as the inequality shown in formula (20) is satisfied, the estimation error augmentation system is mean-square exponentially stable and has H ∞ Performance-constrained; will and Substitute into matrix In the middle, solving the convex optimization problem (20) yields the following results: and Finally, according to L=diag N {L ii } and K = [a ij K ij ] N×N The gain matrix L of the estimator can be obtained. ii and K ij That is, the gain matrix of the estimator for the urban drainage network.

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