A method for designing a functional observer for failure detection in urban water systems

Abstract

The application discloses a kind of urban water system fault detection functional observer design methods.First, the method establishes the switching positive system state space model for urban water system.Then, the function state equation of functional observer and the estimation vector are designed, and the fault detection functional observer and error dynamic system of urban water system are constructed.Further, the positive and stability conditions of error dynamic system are optimized using linear programming method, so that the designed fault detection functional observer can provide a positive asymptotic estimate.Through simulation experiment, it is proved that the fault detection functional observer designed by the method can effectively track the estimated quantity and timely detect the occurrence of fault signal, improve the safety and applicability of water system.It has very important significance for control theory and practical application.

Description

Technical Field

[0001] This invention belongs to the field of automation control technology and relates to fault detection in urban zoned water supply systems, specifically to a design method for a functional observer for fault detection in urban water systems. Background Technology

[0002] Urban water systems are vital for urban residents' lives and businesses' production and development. However, with the increasing scale of modern cities, the demand for urban water is also gradually increasing, making it more difficult to ensure normal water supply. Urban water systems typically include components such as water sources, water tanks, and water pipes and valves. Due to the complexity and large number of components, many factors can affect fault detection. When a water system malfunctions, failure to promptly locate and address the source of the fault can lead to widespread water outages. Therefore, urban water system fault detection systems are required to select appropriate fault detection methods based on different needs, detecting faults as quickly as possible.

[0003] In urban water supply fault detection systems, since water consumption is always non-negative at different times, the fault detection system conforms to the characteristics of a positive system. Because the water demand of different user groups in different areas of the city varies, it is necessary to construct a water supply subsystem with multiple modes and corresponding fault detectors. Existing technologies typically use filters and observers for fault detection, which can promptly issue alarm signals for faults that may be caused by components such as water valves or water tanks, allowing staff to handle the fault. However, the design structure of filters and observers is relatively fixed, and they require estimation and detection of all system states, which is time-consuming, has poor alarm timeliness, and cannot cope with some special emergencies. Summary of the Invention

[0004] To address the shortcomings of existing technologies, this invention proposes a design method for a fault detection functional observer for urban water systems. This method enables fault detection in water systems by using a fault detection functional observer, solving the problem that general filters or observers are inflexible in fault detection due to their relatively fixed structure and the estimation of the entire system state, making them unable to cope with faults caused by various unforeseen circumstances.

[0005] A design method for a functional observer for fault detection in urban water systems, comprising the following steps:

[0006] Step 1: Establish the state-space model of the urban water system during switching:

[0007]

[0008] Where x(t)∈R n Indicates the system status. Let u(t) denote the first derivative of x, where u(t) ∈ R.w Let y(t) ∈ R represent the control input. m Indicates the control output, w(t)∈R r f(t)∈R represents the unmeasurable disturbance input affecting the opening of water valves in the water supply network. g This represents the fault input for water valves on the water utility's official website, where g represents the number of water valves. Define σ(t) = i to represent the activation of the i-th subsystem, where i ∈ {1, 2, ..., N}, N ∈ N. + . R is a known system matrix. n Represents an n-dimensional vector. Let N represent an m×n dimensional positive matrix. + Let x represent a positive integer. t≥0 represents time, and t0 and x0 represent the initial time and initial state, respectively.

[0009] Step 2: Design the functional state equations of the functional observer:

[0010] z(t) = L σ(t) x(t)

[0011] Where z(t)∈R p , 1≤p≤n, is a positive linear function of x(t) with sufficient degrees of freedom of choice. It is the constant matrix used to generate z(t).

[0012] Step 3: Construct a fault detection functional observer for the urban water system. The specific steps are as follows:

[0013] Step 3.1: Design a functional observer for fault detection in the water system.

[0014]

[0015] in, Let ξ(t) be the estimated vector of z(t), and let N be the state signal of the functional observer. i ∈R p×p J i ∈R p×m H i ∈R p×w and G i ∈R p×m It is the unknown functional observer parameter matrix. r(t) is the residual signal, Q 1i ∈R q×p and Q 2i ∈R q×m Let be the unknown matrix that needs to be solved.

[0016] Step 3.2: Design the residual evaluation function J r (T), and determine its corresponding threshold J. thThe details are as follows:

[0017]

[0018]

[0019] Wherein, the scalar value T is the evaluation step size, defined as follows: Let the vector r(t) ∈ R q The 1-norm, where r i (t) represents the i-th element of vector r(t). J represents r The supremum of the set of values ​​of (T).

[0020] The method to determine whether a fault has occurred is as follows:

[0021]

[0022]

[0023] Step 3.3: Define the estimation error vector e(t):

[0024]

[0025] The residual signal r(t) can be calculated as:

[0026]

[0027] And r(t) can be reduced to

[0028] r(t) = Q 1σ(t) e(t)

[0029] When condition Q is satisfied 1σ(t) L σ(t) +Q 2σ(t) C σ(t) When = 0, the residual signal r(t) has no dependence on the state vector x(t).

[0030] Step 4: Design the positive estimate and error dynamic system exponential stability conditions for the fault detection functional observer:

[0031] Design constant α>0, S∈Z + T i * ∈R + If there exists a p-dimensional vector ν i,s For any i, j ∈ {1, 2, ..., N}, and i ≠ j, s ∈ {1, 2, ..., S}, the following condition holds:

[0032]

[0033] J i C i +G i C i A i -N i G i C i ≥0

[0034] H i +G i C i B i ≥0

[0035] G i C i D i ≥0

[0036] G i C i F i ≥0

[0037] H i +G i C i B i -L i B i =0

[0038] G i C i D i -L i D i =0

[0039] J i C i +G i C i A i -L i A i -N i G i C i +N i L i =0

[0040]

[0041]

[0042]

[0043] ν i,0 -ν j,S ≤0

[0044] Therefore, the fault detection functional observer can be obtained to provide a positive estimate and have exponential stability, where Let S represent the Metzler matrix, S represent the mode, and T represent the mode. i * This represents the minimum dwell time for modal dependencies.

[0045] Step 5: Verify the positive estimate of the fault detection functional observer:

[0046] calculate First derivative:

[0047]

[0048] Based on the above, by and The augmenting system consists of:

[0049]

[0050] Among them, 0 n×p Let represent an n-row p-column matrix with all elements equal to 0. Combining this with the stability condition in step 4, we can conclude that under any non-negative initial conditions, the fault detection functional observer will provide a positive estimate.

[0051] Step 6: Verify the stability of the error dynamic system exponent. The specific steps are as follows:

[0052] Step 6.1: Select the discrete linear copositive Lyapunov function

[0053] V σ(t) (e(t))=e T (t)ν σ(t) (t)

[0054] Where, ν σ(t) (t) is a time-dependent function. Based on the modal dependency minimum residence time constraint, we define k∈Z, t k This represents the k-th switching time, within the interval [t]. k ,t k+1 ) can be divided into [t k ,t k +T i * ) and [t k +T i * ,t k+1 ), the interval [t k ,t k +T i * Divide into S parts, each part being The length of each piece is and ν i (t) in each part Si,s The interior is linear, and ν i (t) in the interval [t k ,t k +T i * The content within ) is piecewise linear, and can be represented as:

[0055]

[0056] After differentiation, we get:

[0057]

[0058] in

[0059] Step 6.2, consider t∈[t k ,t k +T i * Combining this with the stability condition in step 4, we can conclude that...

[0060]

[0061] Step 6.3, consider t∈[t k +T i * ,t k+1 Combining this with the stability condition in step 4, we can conclude that...

[0062]

[0063] Combining the results obtained in steps 6.2 and 6.3, we can conclude that:

[0064]

[0065] this means:

[0066]

[0067] Step 6.4, consider the switching time t k When σ(t) = i, t ∈ [t k ,t k+1 When σ(t) = j, t ∈ [t k-1 ,t k Combining this with the stability condition in step 4, we can conclude that:

[0068]

[0069] in, and They represent t respectively kThe right and left limits at time points are further derived by combining the results obtained in steps 6.3 and 6.4:

[0070]

[0071] Step 6.5, Define ε1 = min λ (ν i,s ), in λ (ν i,s )and Representing vectors ν respectively i,s The minimum and maximum terms can be obtained as follows:

[0072] V σ(t) (e(t))=e T (t)ν σ(t) (t)≥ε1||e(t)||1

[0073]

[0074] In summary:

[0075]

[0076] make It can be obtained This means that the error dynamic system is exponentially stable.

[0077] Step 7: Solve for the fault detection functional observer. The specific steps are as follows:

[0078] Step 7.1, Observer parameters G i and H i for:

[0079] G i =L i D i (C i D i ) -

[0080] H i =L i B i -G i C i B i

[0081] Step 7.2, the equation to be solved is:

[0082] X i Ψ i =Ω i

[0083] Among them, Xi =[N i J i [ is an unknown matrix] Ω i =(L i -G i C i A i .

[0084] Further solving yields the following results:

[0085] N i =M 1i Θ1+Γ i M 2i Θ1

[0086] J i =M 1i Θ2+Γ i M 2i Θ2

[0087] in also, Represents Ψ i The generalized inverse, I m This represents the m×m dimensional identity matrix.

[0088] Step 7.3: Existence conditions for fault detection functional observers that satisfy positive estimation and stability:

[0089] Design constant α>0, S∈Z + T i * ∈R + ,vector κ∈R + If a constant exists vector η i,s ∈R p+m η i,s,l ∈R p+m For any l∈{1,2,…,p}, a fault detection functional observer with positive estimation and stability can be obtained if the following conditions are met:

[0090]

[0091]

[0092]

[0093]

[0094]

[0095] νi,0 -ν j,S ≤0

[0096] η i,s,l ≤η i,s

[0097] Among them, 1 p This represents a p-dimensional column vector where each element is 1. Let represent a p-dimensional vector whose element in the l-th row is κ and all other elements are 0. Under the mode-dependent minimum residence time constraint, the functional observer parameter matrix N i and J i It is further represented as:

[0098]

[0099]

[0100]

[0101] The present invention has the following beneficial effects:

[0102] This method establishes a state-space model of an urban water system using a switching positive system and designs a fault detection functional observer to detect faults. The fault detection functional observer structural model is constructed, defining the estimation vector and error vector, and generating residual signals independent of the state vector. Then, a residual evaluation function is designed using the residual signal, and compared with a threshold value for the residual evaluation function when there is no fault (f(t) = 0). If the residual evaluation function is greater than the threshold, a fault occurs, and an alarm is triggered; if it is less than or equal to the threshold, no fault occurs. By utilizing a discrete linear copositive Lyapunov function, the designed fault detection functional observer can provide an exponentially positive estimate and can smoothly detect faults, ensuring the safety of residents' water supply. Attached Figure Description

[0103] Figure 1 This is a schematic diagram of the framework of a fault detection functional observer for a switching positive system;

[0104] Figure 2 It is the switching signal σ(t);

[0105] Figure 3 It is z1(t) and its estimated vector

[0106] Figure 4 It is z2(t) and its estimated vector

[0107] Figure 5It is the residual evaluation function J r (t). Detailed Implementation

[0108] The present invention will be further explained below with reference to the accompanying drawings;

[0109] The theoretical verification of the fault detection functional observer designed above is performed:

[0110] Step 8.1: First, provide a proof that the proposed fault detection functional observer can provide a positive estimate:

[0111] Because κ p ≥0 and ν i,s ≥0, have It is a positive real number, and combining this with the existence condition in step 7.3, we get:

[0112]

[0113] Therefore, M 1i Θ1+Γ i,s M 2i Θ1 is a Metzler matrix. Similarly, since... It is a positive real number, which, combined with the conditions in step 7, gives:

[0114] (M 1i Θ2+Γ i,s M 2i Θ2)C i +G i C i A i -(M 1i Θ1+Γ i,s M 2i Θ1)G i C i ≥0

[0115] Therefore, the designed fault detection functional observer can provide a positive estimate.

[0116] Step 8.2: Based on the conditions in Step 7, the following conditions can be derived:

[0117]

[0118] because It is a positive real number, and the condition in step 7 is equivalent to:

[0119]

[0120] Further conclusions can be drawn:

[0121]

[0122] Similarly, the other conditions in step 7 can also be obtained using this method.

[0123] Therefore, when the conditions in step 7 are met, the fault detection functional observer involved will be able to provide a positive estimate under exponential stability conditions.

[0124] The effectiveness of this method will be discussed from the perspective of experimental simulation:

[0125] like Figure 1 As shown, consider a switched positive system with two subsystems, with the following system parameters:

[0126]

[0127] D1 = [0.2 0.6 0] T F1 = [0.1 0.7 0] T ,

[0128]

[0129] D2 = [0.1 0.6 0] T F2 = [0.1 0.5 0] T

[0130] Design estimates for states x1(t) and x2(t), with state vector x(t) = [x1(t)x2(t)x3(t)], and:

[0131]

[0132] First, the parameters G of the fault detection functional observer are calculated using the method given in step 7. i and H i :

[0133]

[0134] Then the coefficient matrix of the residual signal is calculated:

[0135] Q 11 =[2.5 12],Q 12 =[4 24],

[0136] Q 21 =[0 -5],Q 22 =[0 -10]

[0137] That is, the residual signal can be expressed as:

[0138]

[0139]

[0140] Design constant α = 0.3, modal-dependent minimum residence time T1 * =1.0、 S = 2, κ = 0.01, thus obtaining The feasible solution obtained by solving the existence condition of the fault detection functional observer in step 7 is:

[0141]

[0142]

[0143] ∈1=4879.3, ∈2=4456.0

[0144] Then, the parameter matrix N of the assumed functional observer is calculated. i,s and J i,s :

[0145]

[0146]

[0147]

[0148]

[0149] In this embodiment, the initial value of the state vector is set to x(0) = [4 6 5]. T The initial value of the observer's estimation vector is The external disturbance signal is w(t) = 0.01e -0.2t The design fault signal is as follows:

[0150]

[0151] Simulation results are as follows Figures 2-5 As shown, where Figure 2 To satisfy the mode-dependent minimum dwell time constraint for the switching signal, Figure 3 and Figure 4 These are the curve trajectories of the estimated vector and its estimated vector, respectively. Figure 5 J is the residual evaluation function r (t) Evolution process, where the solid line represents the fault-free case and the dashed line represents the case with faults. When no faults occur, after time t = 2.78 seconds, the threshold of the evaluation function curve is J. th =28.64. When a fault occurs, after time t = 2.1 seconds, the threshold J of the evaluation function curve is... th=29.11>28.64, which means that after the fault signal is generated at t=0.6 seconds, the designed fault detection functional observer can detect the fault after 1.5 seconds. This demonstrates the timeliness and correctness of the fault detection functional observer.

Claims

1. A design method for a functional observer for fault detection in urban water systems, characterized in that: Step 1: Establish a state-space model of the urban water system using a switching positive system: in, Indicates the system status. express The first derivative, Indicates control input, Indicates control output. This represents the unpredictable disturbance input that affects the opening of water valves in the water supply network. This indicates a fault input for the water valve on the water utility's official website. Indicates the number of water valves; definition Representing the Each subsystem is activated. , ; , , , , The system matrix is ​​known; express dimensional vector, express 3D positive matrix Represents a positive integer; t≥0 represents time. and These represent the initial time and the initial state, respectively. Step 2: Design the functional state equations of the functional observer: in , It has a high degree of freedom of choice. A positive linear function; It is used to generate The constant matrix; Step 3: Construct a fault detection functional observer for the urban water system. The specific steps are as follows: Step 3.1: Design a functional observer for fault detection in the water system. in, yes The estimated vector, The state signal of the functional observer. , , and It is an unknown functional observer parameter matrix; It is a residual signal. and Let be the unknown matrix that needs to be solved; Step 3.2: Design the residual evaluation function And determine its corresponding threshold. The details are as follows: Among them, scalar value It is the evaluation step size, defined For vectors of norm, where Representing vectors The One element, express The supremum of the set of values; The method to determine whether a fault has occurred is as follows: Step 3.3: Define the estimation error vector : residual signal It can be calculated as: and It can be reduced to When the condition is met At that time, residual signal With state vector No dependencies; Step 4: Design the positive estimator and the exponential stability conditions of the error dynamic system for the fault detection functional observer; Step 5: Verify the positive estimation of the fault detection functional observer; Step 6: Select a discrete linear copositive Lyapunov function to verify the stability of the error dynamic system exponent; Step 7: Solve for the fault detection functional observer based on the existence conditions of the fault detection functional observer that satisfy positive estimation and stability.

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