Circuit fault diagnosis method based on fractional order observer

Abstract

The invention discloses a circuit fault diagnosis method based on a fractional order observer, and the method comprises the steps: constructing a fractional order circuit model based on the Kirchhoff's law, and describing a capacitor end voltage and an inductive current as a Caputo fractional order derivative form; a double-disturbance decoupling mechanism is designed, total disturbance is decomposed into a decoupling disturbance component and a non-decoupling disturbance component, physical isolation of the decoupling disturbance is achieved through a projection matrix, and the influence of the non-decoupling disturbance is eliminated through a dynamic suppression coefficient. Constructing a state observer to generate a state estimation error and a fault error vector, and dynamically compensating a fault estimation value in combination with an adaptive gain matrix; and ensuring the system stability through a Lyapunov function, and proving that an error state is converged to a preset boundary domain under the condition of satisfying a characteristic value correlation matrix inequality. Compared with the prior art, rapid and accurate diagnosis of faults such as inductance saturation and capacitance aging can be realized.

Description

Technical Field

[0001] This patent relates to the field of real-time fault diagnosis technology, specifically a circuit fault diagnosis method based on a fractional-order observer. Background Technology

[0002] Current fault diagnosis methods for power electronic equipment are mainly based on integer-order calculus models, which are difficult to accurately describe physical processes with memory characteristics such as the relaxation effect of capacitor dielectric layer and eddy current loss of inductor core, resulting in a modeling error of 3% to 7% for state observers.

[0003] Traditional disturbance decoupling techniques are poorly adapted to the nonlocal characteristics of fractional-order systems, and the observation residuals are easily contaminated by high-frequency switching noise. Under complex operating conditions such as wind power converters and electric vehicle drive systems, the false alarm rate exceeds 8%, and the fault response delay is generally greater than 120ms, which cannot meet the real-time monitoring requirements of high-reliability power equipment. Summary of the Invention

[0004] Purpose of the invention: To address the above-mentioned problems, the present invention aims to solve the deficiency that traditional integer-order fault diagnosis models cannot accurately characterize the fractional-order dynamic characteristics of power electronic devices. By establishing a Caputo fractional-order circuit model based on Kirchhoff's laws and designing an observer with disturbance hierarchical decoupling capability, the invention enables rapid and accurate diagnosis of faults such as inductor saturation and capacitor aging.

[0005] Technical solution: This invention discloses a circuit fault diagnosis method based on a fractional-order observer, comprising the following steps:

[0006] Step S1: Establish a mathematical model of the RLC fractional circuit based on Kirchhoff's current law and voltage law, define the Caputo fractional derivative, and construct the state-space expression of the fractional system based on the circuit physical model. The state vector includes inductor current and capacitor voltage.

[0007] Step S2: Decompose the total disturbance into a decoupled disturbance component d1(t) and a non-decoupled disturbance component d2(t), satisfying d2(t)≤M d The boundedness condition of M d The observer parameter matrix represents the decoupled disturbance correlation, further determining the system state equations; a fractional-order state observer is designed, and the decoupled disturbance components of the system are decoupled by constraining the projection matrix T, where the projection matrix T = I. n -MC, configuring fractional-order state observer gain matrices K and M to decouple the decoupled perturbation components, where C is a constant matrix of appropriate dimension; I n It is an n-order identity matrix;

[0008] Step S3: Define the state estimation error and the fault estimation error, and derive the error including the non-decoupled disturbance d2(t) and the fault error. The fractional differential equations are used to dynamically adjust the fault estimates through an adaptive update law;

[0009] Step S4: Construct a Lyapunov function that includes state estimation error and fault estimation error. By configuring the positive definite matrix parameter ε and the disturbance suppression coefficient η, ensure that the stability criterion matrix Θ<0 holds.

[0010] Furthermore, the mathematical model of the RLC fractional circuit in step S1 includes inductor L, capacitor C1, load resistor R, and the currents flowing through the coil, resistor, and capacitor are respectively represented by i. L i R and i C The voltages flowing through the coil, resistor, and capacitor are respectively represented by u. L u R and u C express.

[0011] Furthermore, the Caputo fractional derivative in step S1 is defined as:

[0012]

[0013] Where, x(t)=[i L (t),u C (t)] T Represents the state vector, i L (t) represents the inductor current, u C (t) represents the capacitor voltage, Γ is the gamma function, t0 represents the starting time point of the fractional derivative, t represents the current time point of the fractional derivative, and α represents the order of the fractional derivative.

[0014] Furthermore, the fractional-order system state-space expression constructed based on the circuit physical model in step S1 is expressed as follows:

[0015]

[0016] In the formula, x(t)=[i L (t),u C (t)] T Represents the state vector, i L (t) represents the inductor current, u C (t) represents the capacitor voltage, u(t) is the control input, and y(t)∈ p Indicates system output, A, B, and C are constant matrices with appropriate dimensions. C =

[01] .

[0017] Furthermore, in S2, the total system disturbance is decomposed into a decoupled disturbance component d1(t) and a non-decoupled disturbance component d2(t), satisfying d2(t)≤M. d The boundedness condition determines the system state equation as follows:

[0018]

[0019] Where D, E1, E2 are constant matrices with appropriate dimensions, and f(t) ∈ q Describing faults, d1(t), d2(t)∈ d Indicates a disturbance.

[0020] Furthermore, the dynamic equations of the fractional-order state observer are as follows:

[0021]

[0022] in, Represents the state estimation vector. This represents the internal auxiliary state vector of an unknown input fractional-order observer, with parameter matrices N = TA - KC, L = K + NM, and T = I. n -MC, K∈ n×p and M∈ n×p The gain matrix of the fractional-order state observer is given by constraint TE1 = (I n -MC)E1=0 can ensure decoupling and isolation of decoupled disturbances.

[0023] Furthermore, step S3 includes non-decoupling disturbance d2(t) and fault error. The fractional differential equation is:

[0024]

[0025] Where, e(t)∈ n Represents the state error vector. Represents the fault estimation vector. This represents the fault error vector.

[0026] Furthermore, the adaptive update law in step S3 is:

[0027]

[0028] The fault error is dynamically represented as follows:

[0029]

[0030] in, e y (t)∈ pThe output error vector is represented by G, which is the adaptive gain matrix.

[0031] Furthermore, step S4 is specifically as follows:

[0032] The Lyapunov function is:

[0033]

[0034] Considering the system state equations in step S2, the fractional-order state observer in step S2, and the adaptive update law in step S3, if K, M, G and P = P exist... T >0 and the positive definite matrix parameter ε and the perturbation suppression coefficient η make:

[0035] KD=0

[0036] TE1 = (I n -MC)E1=0

[0037] PTE2 = H T

[0038]

[0039] This ensures that the state estimation error and the fault estimation error are consistent and eventually bounded, where A, B, C, D, E1, E2 are constant matrices of appropriate dimensions, N = TA - KC, L = K + NM, and T = I. n -MC, K∈ n×p and M∈ n×p This is the gain matrix of the fractional-order state observer.

[0040] Beneficial effects:

[0041] 1. Significantly improves fault diagnosis accuracy. This invention innovatively extends the traditional integer-order circuit model to a Caputo fractional-order description, accurately characterizing physical processes with memory characteristics such as capacitor dielectric relaxation effect and inductor core eddy current loss. This fundamentally solves the problem of misjudgment caused by dynamic characteristic mismatch in traditional models under complex operating conditions. The innovatively designed dual-perturbation decoupling architecture effectively eliminates the contamination of observation residuals by high-frequency switching noise through a combination of hardware-level isolation and dynamic suppression, significantly reducing the probability of false alarms.

[0042] 2. Enhance system disturbance rejection stability. Based on Lyapunov stability theory, the observer parameter optimization mechanism incorporates fault sensitivity control and disturbance suppression into a unified mathematical framework. By adjusting the fault tracking rate in real time through an adaptive gain matrix, the system maintains robustness while ensuring rapid error convergence, significantly improving its adaptability to complex electromagnetic environments such as sudden load changes and power grid fluctuations.

[0043] 3. Traditional disturbance decoupling methods rely on the idealized assumption that "unknown inputs can be completely decoupled" and are not suitable for the fractional-order characteristics of capacitor dielectric relaxation and inductor eddy current losses in circuits. They can only suppress non-decoupling disturbances coupled with high-frequency switching noise in the entire frequency domain, resulting in software lag and easy false suppression of fault signals, leading to contamination of observation residuals and a high false alarm rate. In contrast, the dual-disturbance decoupling architecture is based on the characteristics of fractional-order circuit disturbances and divides the total disturbance into decoupling and non-decoupling components: decoupling disturbances are isolated by a hardware-level projection matrix T to avoid software lag, and non-decoupling high-frequency switching noise is handled by a dynamic suppression coefficient (adjusted according to the noise amplitude) and an adaptive gain matrix G (compensated for noise influence by fault estimation update law). The Lyapunov stability criterion is used to ensure fault signal sensitivity, achieving "hardware isolation + software dynamic cancellation" synergistic management, effectively eliminating residual "false fault" contamination, reducing the false alarm rate, and adapting to fractional-order characteristics without requiring ideal decoupling conditions, which is more in line with actual circuit conditions. Attached Figure Description

[0044] Figure 1 This is the RLC fractional linear circuit diagram of the present invention;

[0045] Figure 2 This invention relates to the constant fault signal f(t) and its estimated signal. Schematic diagram;

[0046] Figure 3 This invention provides a state signal x1 and its estimated signal for a constant fault signal f(t). Schematic diagram;

[0047] Figure 4 This invention provides the state signal x2 and its estimated signal, which are constant fault signals f(t). Schematic diagram;

[0048] Figure 5 This invention relates to the time-varying fault signal f(t) and its estimated signal. Schematic diagram;

[0049] Figure 6 This invention relates to the state signal x1 and its estimated signal, which are time-varying fault signals f(t). Schematic diagram;

[0050] Figure 7 This invention relates to the state signal x2 and its estimated signal, which are time-varying fault signals f(t). Schematic diagram. Detailed Implementation

[0051] The present invention will be described in detail with reference to the accompanying drawings. The following embodiments are used to illustrate the technical solution of this patent, but the scope of protection of this patent is not limited thereto.

[0052] Before describing the detailed technical solution of this invention, the following definitions and lemmas are introduced:

[0053] Definition 1: The Riemann–Liouville fractional derivative with respect to α∈(0,1) is defined as:

[0054]

[0055] Here, Γ is the gamma function, which extends the concept of factorial to non-integer parameters. Let x(τ) denote the Riemann–Liouville fractional integral operator, x(τ) denote the objective function being integrated, and τ denote the integration variable.

[0056] Definition 2: Let a be the integer part of α+1, the Caputo fractional derivative is defined as:

[0057]

[0058] Where a⁻¹ < α < a. Setting a = 1, when 0 < α < 1: the Caputo fractional derivative α of x(t) simplifies to:

[0059]

[0060] Theorem 1: Let α∈(0,1), P∈ n×n Let be a constant square symmetric positive definite matrix. Then, the following relationship holds:

[0061]

[0062] Theorem 2: (Young's Inequality) For any non-negative real numbers a and b, p and q > 1, So

[0063]

[0064] If and only if a p =b q When, the equation holds true.

[0065] Theorem 3: Assume matrices TE2 and H T For a matrix to have full column rank, there exists a positive definite matrix P such that PTE2 = H. T It is true if and only if:

[0066] (HTE2) T =HTE2 (7)

[0067] The circuit fault diagnosis method based on a fractional-order observer of the present invention is described in detail below, including the following steps:

[0068] Step S1: Establish the circuit physical model

[0069] Based on Kirchhoff's Current Law (KCL) and Voltage Law (KVL), the original equations of the circuit are established, including resistors (R), inductors (L), and capacitors (C). The mathematical model of the RLC fractional circuit is as follows:

[0070]

[0071] The current flowing through the coil, resistor, and capacitor (with the circuit direction defined as clockwise) is represented by i. L i R and i C The voltages flowing through the coil, resistor, and capacitor are respectively represented by u. L u R and u C express.

[0072] because Where 0 < α < 1, then equation (8) can be written as:

[0073]

[0074] Among them, u C (t), i L (t) is the state variable, u(t) is the control input, defined as follows: The state equations for the RLC fractional linear circuit system model can be obtained as follows:

[0075]

[0076] In the formula, C =

[01] . In this embodiment, taking R = 1Ω, L = 1H, C1 = 1F, we can obtain: B =

[10] , C =

[01] ,

[0077] We consider the following fractional system with unknown input and derivative order α∈(0,1):

[0078]

[0079] Where, x(t)∈ n Let u(t) represent the state vector. m Represents the control input, y(t)∈ p Represents the system output, f(t)∈ q Denotes a fault, d(t)∈ d This represents the perturbation. A, B, C, D, and E are constant matrices with appropriate dimensions.

[0080] Step S2: Perturbation decomposition and observer construction.

[0081] Decomposing the disturbance d(t) into a decoupled part d1(t) and a non-decoupled part d2(t), we obtain the following system:

[0082]

[0083] f(t)∈ q Describes the fault, where the constant fault is:

[0084]

[0085] Time-varying faults are:

[0086]

[0087] Constructing an observer for unknown inputs:

[0088]

[0089] in

[0090] N = TA - KC (14)

[0091] L=K+NM (15)

[0092] T = I n -MC (16)K∈ n×p and M∈ n×p The matrix is ​​designed.

[0093] Step S3: Dynamic Error Generation and Adaptive Compensation

[0094]

[0095] in, Represents the state estimation vector. This indicates the output estimated vector. Let e(t) represent the fault estimation vector. n Represents the state error vector, e y (t)∈ p This represents the output error vector. This represents the fault error vector.

[0096] Derivation of the state estimation error equation:

[0097] Combining (11)(12)(13)(14) yields

[0098] Transforming the obtained state estimation error (20), we get

[0099] z(t)=Tx(t)-e(t)-MDf(t) (21)

[0100] Taking the derivative of the state estimation error (20), we get

[0101]

[0102] Substituting equations (11) and (21) into equation (22) yields...

[0103]

[0104] Simplify formula (23) using equations (14), (15), and (16):

[0105] coefficient of the x(t) term:

[0106]

[0107] coefficient of the f(t) term:

[0108]

[0109] Substituting (24) and (25) into formula (23) yields:

[0110]

[0111] because The observer's adaptive compensation mechanism changes the residual effects to:

[0112] Substituting (27) into (26) yields:

[0113]

[0114] Step S4: Stability guarantee mechanism.

[0115] Given the following adaptive rates:

[0116]

[0117] The fault error is dynamically represented as follows:

[0118]

[0119] Step S5: Construct a Lyapunov function that includes state estimation error and fault estimation error. By configuring the positive definite matrix parameter ε and the disturbance suppression coefficient η, ensure that the stability criterion matrix Θ<0 holds.

[0120] Consider a fractional-order system (12), under an observer (13) and an adaptive law (29), if there exist K, M, G and P = P T >0 and positive scalars ε and η make:

[0121] KD = 0 (31)

[0122] TE1 = (I n -MC)E1=0 (32)

[0123] PTE2 = H T (33)

[0124]

[0125] This allows the state estimation error and the fault estimation error to become consistent and eventually bounded.

[0126] The stability criterion described above is then proven using a theorem:

[0127] Combining (28), (31), and (32), we can obtain

[0128]

[0129] Consider the following Lyapunov functions

[0130]

[0131] Using Theorem 1, we can obtain that at any time t≥t0:

[0132]

[0133] For the disturbance term 2e(t) T Simplify PTE2d2(t) using Theorem 2:

[0134] set up B = d2, which can be obtained using (33).

[0135]

[0136] Substituting (38) into (37) and using it, we get:

[0137]

[0138] Step S6: Convergence Proof and Parameter Optimization. Prove that when the error state vector ζ(t)... 2 >δ / β, the fractional derivative of the Lyapunov function satisfies C D a V(t)≤0, where Based on the minimum eigenvalue β=λ of the matrix min (Θ) Dynamically optimize the convergence domain boundary to ensure that the error eventually converges to ζ(t). 2 The region ≤δ / β is bounded.

[0139] Cause disturbance:

[0140]

[0141] make

[0142]

[0143] β=λ min (Θ) (43)

[0144] Where ζ(t) is the error state vector, β is the minimum eigenvalue of the matrix, and δ is the perturbation suppression boundary.

[0145] Substituting (40)(41)(42)(43) into (39) yields:

[0146]

[0147] when hour:

[0148]

[0149] That is, ζ(t) converges to an interval It can be concluded that the state estimation error and the fault estimation error are consistent and eventually bounded, thus completing the proof.

[0150] The method of the present invention will be further illustrated below through simulation. For the simulation, the present invention has a constant fault signal f(t) and its estimated signal. like Figure 2 As shown; the state signal x1 and its estimated signal exist for a constant fault signal f(t). like Figure 3 As shown; the state signal x2 and its estimated signal exist for a constant fault signal f(t). like Figure 4 As shown; there exists a time-varying fault signal f(t) and its estimated signal. like Figure 5 As shown; the state signal x1 and its estimated signal exist for a time-varying fault signal f(t). like Figure 6 As shown; the state signal x2 and its estimated signal exist in the presence of a time-varying fault signal f(t). like Figure 7 As shown.

[0151] Simulation results show that, for the fault estimation method of RLC fractional linear circuits, the fault observer designed in this invention can estimate the system fault in a timely manner, and the system is asymptotically stable, which has practical reference value.

[0152] The embodiments described above are merely illustrative of several implementation methods of this application, and while the descriptions are specific and detailed, they should not be construed as limiting the scope of this application. It should be noted that those skilled in the art can make various modifications and improvements without departing from the concept of this application, and these modifications and improvements all fall within the protection scope of this application. Therefore, the protection scope of this application should be determined by the appended claims.

Claims

1. A circuit fault diagnosis method based on a fractional order observer, characterized by, The method comprises the following steps: Step S1: establishing a mathematical model of the RLC fractional circuit according to the Kirchhoff current law and the voltage law, defining Caputo fractional derivative, and constructing a fractional order system state space expression based on a circuit physical model, wherein a state vector comprises an inductor current and a capacitor voltage; Step S2: decompose the total disturbance into a decouplable disturbance component d1(t) and a non-decouplable disturbance component d2(t), satisfying d2(t)≤M d The boundedness condition of M d The observer parameter matrix associated with the decouplable disturbance is represented by further determining the system state equation; a fractional order state observer is designed, and the decouplable disturbance component is decoupled by constraining the projection matrix T, T=I n -MC, the fractional order state observer gain matrix K and M are configured to decouple the decouplable disturbance component, and C is a constant matrix with appropriate dimensions; I n is an n-order unit matrix; Step S3: define state estimation error and fault estimation error, derive the fractional differential equation containing the non-decoupled disturbance d2(t) and fault error of the adaptive update law dynamically adjusts the fault estimation value; Step S4: constructing a Lyapunov function comprising a state estimation error and a fault estimation error, and ensuring that a stability criterion matrix Θ<0 is established by configuring a positive definite matrix parameter ε and a disturbance suppression coefficient η.

2. The circuit fault diagnosis method based on a fractional-order observer according to claim 1, characterized in that, The mathematical model of the RLC fractional circuit in the step S1 includes inductance L, capacitance C1, load resistance R, the current flowing through the coil, the resistance and the capacitance are respectively represented by i L , i R , and i C , the voltage across the coil, the resistance and the capacitance are respectively represented by u L , u R , and u C .

3. The circuit fault diagnosis method based on fractional order observer according to claim 1, characterized in that, The Caputo fractional derivative in the step S1 is defined as: where x(t) = [i L (t), u C (t)] T represents a state vector, i L (t) represents an inductor current, u C (t) represents a capacitor voltage, Γ is a gamma function, t0represents a starting time point of a fractional derivative, t represents a current time point of the fractional derivative, and α represents an order of the fractional derivative.

4. The circuit fault diagnosis method based on fractional order observer according to claim 1, characterized in that, The fractional order system state space expression constructed based on the circuit physical model in the step S1 is expressed as: where x(t) = [i L (t), u C (t)] T is the state vector, i L (t) is the inductor current, u C (t) is the capacitor voltage, u(t) is the control input, and y(t) e p is the system output, A, B, C are constant matrices of appropriate dimensions, C = [0 1].

5. The circuit fault diagnosis method based on fractional order observer according to claim 4, characterized in that, The total disturbance of the system in S2 is decomposed into a decouplable disturbance component d1(t) and a non-decouplable disturbance component d2(t), satisfying the boundedness condition d2(t)≤M d The determined system state equation is: where D, E1, E2 are constant matrices of appropriate dimensions, f(t) e q denote faults, d1(t), d2(t) e d denote disturbances.

6. The circuit fault diagnosis method based on fractional order observer according to claim 5, characterized in that, The dynamic equation of the fractional order state observer is specifically as follows: wherein, denotes the state estimation vector, denotes the internal auxiliary state vector of the unknown input fractional order observer, the parameter matrix is N = TA - KC, L = K + NM, T = I n - MC, K ∈ n×p and M ∈ n×p is the fractional order state observer gain matrix, the constraint TE1 = (I n - MC)E1 = 0 can ensure decoupled disturbance decoupling isolation.

7. The circuit fault diagnosis method based on fractional order observer according to claim 6, characterized in that, The step S3 comprises a non-decouplable disturbance d2(t) and a fault error The fractional differential equation of the step S3 is: where e(t) e n denotes the state error vector, denotes the fault estimation vector, denotes the fault error vector.

8. The circuit fault diagnosis method based on fractional order observer according to claim 7, characterized in that, The adaptive update law in the step S3 is as follows: The fault error dynamics is expressed as: wherein e y (t)∈ p where e is the output error vector and G is the adaptive gain matrix.

9. The circuit fault diagnosis method based on fractional order observer according to claim 1, characterized in that, The step S4 is specifically as follows: The Lyapunov function is as follows: Consider that the system state equation in step S2 is satisfied, the fractional order state observer in step S2 and the adaptive update law in step S3, if there exist K, M, G and P = P T > 0 and positive definite matrix parameters ε and disturbance suppression coefficient η such that: KD=0 TE1 = (I n -MC)E1 = 0 PTE2 = H T such that the state estimation error and the fault estimation error are consistent and ultimately bounded, A, B, C, D, E1, E2 are constant matrices with appropriate dimensions, N = TA - KC, L = K + NM, T = I n - MC, K ∈ n×p and M ∈ n×p are the fractional order state observer gain matrices.

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