CN115508665B - Fault detection method for RLC (radio link control) fractional linear circuit - Google Patents

Abstract

The present invention discloses a fault detection method for an RLC fractional linear circuit, including: constructing a mathematical model of an RLC fractional linear circuit; considering fractional order, interference, nonlinearity and fault conditions, a general expression of a system state equation is given; a Luenberger-like observer is designed, and an error dynamic equation and a residual model are given; sufficient conditions for the system to be asymptotically stable and have good robustness and sensitivity are given, and the sufficient conditions are converted into linear matrix inequalities to obtain the Luenberger-like observer and residual model parameters; residual evaluation functions and threshold fault decision logic are designed, and detection rules that need to be met to complete the fault detection task are given. Compared with the prior art, the fault detection method designed in the present invention has good robustness to disturbances and good sensitivity to faults, and can realize fault detection of the fault detection method for RLC fractional order circuits.

Description

A Fault Detection Method for RLC Fractional Linear Circuits

Technical Field

The present invention relates to the technical field of fault diagnosis and detection, and in particular to a fault detection method for an RLC fractional linear circuit.

Background technique

With the development and progress of various industrial systems, people are paying more and more attention to the safety and reliability of industrial systems in the process of research and application, and fault detection technology has attracted widespread attention in recent decades. In fault diagnosis, fault detection is an important component, and fault detection is often used for rapid detection and early detection and elimination of faults. Among the model-based fault detection methods, the observer-based fault detection method compares the value of the residual signal generated by the fault detection observer with the pre-set evaluation threshold. This method has been proven to be a very effective method.

At the same time, the development of electronic technology is also advancing by leaps and bounds. The frequency of using electronic devices in daily life is getting higher and higher, and the functions of electronic devices are becoming more and more complete. Among them, different working modes correspond to different circuit requirements.

As an important branch of mathematical analysis, fractional calculus has received extensive attention in recent years. Its rich historical development process and excellent global relevance make it a bridge between mathematical models in various fields of science and engineering. With the introduction of fixed point theory, Lyapunov theory and Mittage-Leffler function, the proof of asymptotic stability of fractional-order systems can be completed. In the existing literature, some non-fragile robust observer design for Lipschitz nonlinear fractional-order systems is given based on iterative methods; some reduce-order observer design methods for fractional-order generalized nonlinear systems are given under the influence of unknown inputs; some study the design problem of observer-based state feedback controllers by introducing continuous frequency distribution equivalent models and indirect Lyapunov methods.

The system described by the fractional order model is simple, clear, and closer to the actual situation. However, compared with the model described by the integer order equation, the model described by the fractional order equation is very rare. This is an important reason why the fractional order system has been studied by many scholars in recent years. Therefore, the research on the control and observer design of the fractional order system has made great progress in recent years.

When used in practical applications, accurate circuit fault detection is particularly important for electronic equipment to maintain safe and efficient operation. Accurate circuit fault detection can achieve safer circuit operation by timely detecting faults in the circuit. The influence of nonlinearity, noise and interference of RLC circuits in practical applications cannot be ignored. At present, there are many studies on linear integer-order system fault detection for circuit modeling, but none of them can simultaneously consider the fault detection of fractional-order nonlinear systems with nonlinearity, noise and interference.

Summary of the invention

Purpose of the invention: In view of the problems existing in the current prior art, the present invention provides a fault detection method for RLC fractional linear circuits, which uses a Luenberger-like observer as a residual generator and can accurately realize fault detection online, thereby realizing fault detection of RLC fractional linear circuits.

Technical solution: The present invention provides a fault detection method for an RLC fractional linear circuit, comprising the following steps:

Step 1: Construct a mathematical model of the RLC fractional circuit based on Kirchhoff's current law (KCL) and Kirchhoff's voltage law (KVL), construct an augmented matrix, and transform the differential equation into a state equation in standard form;

Step 2: Based on the state equation in step 1, considering the variational fractional order, interference, nonlinearity and fault conditions, a general system model of the RLC fractional circuit with interference and fault is given;

Step 3: Design a Luenberger-like observer as a residual signal generator to obtain the corresponding dynamic estimation error system;

The Luenberger-like observer system is

in, represents the state estimation vector, To estimate the output vector, represents the residual signal, L is the gain of the observer, which is the design object; 0＜α＜1, represents the state vector, is the system output, represents the control input, A, B ₁ , B ₂ , C are known constant matrices with appropriate dimensions; Φ(x(t), u(t)) is a nonlinear vector function with a Lipschitz constant η, η＞0, that is:

Define the state error as:

The dynamic estimation error system is:

in:

Step 4: For the dynamic error system obtained in step 3, using the Lyapunov function, give the sufficient conditions for the system to be exponentially stable and satisfy the two performances of _H∞ and _{H_} , and design the parameters of the fault observer according to the sufficient conditions;

Step 5: According to the fault observer designed in step 3, set the threshold _Jth , construct the residual evaluation function J(r), and determine whether the system has a fault.

Furthermore, the circuit is a linear circuit composed of a resistor, a capacitor, a coil and a voltage source. According to Kirchhoff's current law (KCL) and Kirchhoff's voltage law (KVL), the mathematical model of the RLC fractional circuit is:

in, and are the currents flowing through the coil, resistor and inductor respectively, u _L , u _R and u _C are the voltages flowing through the coil, resistor and inductor respectively, L represents inductance, C represents capacitance and R represents load resistance;

because Where 0＜α, β＜1, then equation (4) can be written as

Among them, u _C (t) and i _L (t) are state variables, u(t) is the control input, if we define Then the state equation of the system model of the RLC fractional circuit is as follows:

In the formula,

Furthermore, the general system model of the RLC fractional circuit containing interference and faults in step 2 is:

Among them, 0＜α＜1, represents the state vector, is the system output, represents the control input, Indicates a fault. represents the external disturbance, where the fault can represent any type of actuator or sensor fault, and f(t) is defined as f _a (t) and f _s (t) represent actuator fault and sensor fault respectively; A, B ₁ , B ₂ , C, E ₁ , E ₂ , F ₁ , F ₂ are known constant matrices with appropriate dimensions; Φ(x(t), u(t)) is a nonlinear vector function with Lipschitz constant η, η＞0, that is:

Furthermore, in step 4, the sufficient conditions for the system to be exponentially stable and satisfy the two properties of _H∞ and _{H_} are:

1) The sufficient condition for the system to be exponentially stable and to satisfy the _H∞ performance index is: for a given positive scalar ε ₁ , η, γ, if there exists a positive symmetric matrix P ₁ , a matrix Z ₁ , satisfying

in, E ₁ and E ₂ are known constant matrices with appropriate dimensions;

When the positive symmetric matrix P ₁ and the matrix Z ₁ satisfy the above equation, the dynamic estimation error system is asymptotically stable and has H _∞ performance γ; in this case, the observer gain is

2) The sufficient condition for the system to be exponentially stable and to satisfy the _{H_performance} index is: for a given positive scalar ε ₂ , η, β, if there exists a positive symmetric matrix P ₂ , a matrix Z ₂ , satisfying

in, F ₁ and F ₂ are known constant matrices with appropriate dimensions;

When the positive symmetric matrix P ₂ and the matrix Z ₂ satisfy the above equation, the dynamic estimation error system is asymptotically stable and has _H performance β; in this case, the observer gain is

Furthermore, the residual evaluation function is:

In the formula, T is the total evaluation time, and its threshold is:

Furthermore, the following logic is used to determine whether a fault has occurred:

Beneficial effects:

The present invention proposes a novel fault detection method for RLC fractional linear circuits, which can accurately complete fault detection online, meet the asymptotic stability of the system, and simultaneously meet the two performances of H _∞ and H _{_} , thereby ensuring the robustness and sensitivity of the system. When establishing a general model of the system, the present invention adopts a Luenberger-like observer to realize the fault detection of RLC fractional linear circuits containing actuator faults, sensor faults and disturbances, and uses a Luenberger-like observer as a residual generator, so that the dynamic estimation error system satisfies: (1) the system is asymptotically stable in the absence of disturbances and faults; (2) when the system is fault-free and has disturbances, under zero initial conditions, it satisfies certain H _∞ performance indicators; (3) when the system is fault-free and has faults, under zero initial conditions, it satisfies certain H _{_} performance indicators, and obtains sufficient conditions for the existence of a fault detection observer by using Lyapunov functions and linear matrix inequalities, which can accurately realize fault detection online and realize fault detection of RLC fractional linear circuits.

BRIEF DESCRIPTION OF THE DRAWINGS

Fig. 1 is a RLC fractional linear circuit diagram of the present invention;

FIG2 is a schematic diagram of a state estimation error e(t) in the presence of a disturbance signal ω(t) according to the present invention;

FIG3 is a schematic diagram of a state error e(t) in the presence of a disturbance signal ω(t) and a fault signal f(t) according to the present invention;

FIG4 is a schematic diagram of a fault signal f(t) according to the present invention;

FIG5 is a schematic diagram of a residual signal r(t) in a fault-free situation of the present invention;

FIG6 is a schematic diagram of a residual signal r(t) with a fault signal f(t) according to the present invention;

FIG7 is a schematic diagram of a threshold J(r) curve of the system of the present invention;

Detailed ways

The present invention is further described below in conjunction with the accompanying drawings. The following embodiments are only used to more clearly illustrate the technical solution of the present invention, and cannot be used to limit the protection scope of the present invention.

The present invention takes an RLC fractional linear circuit as an implementation object. Aiming at the occurrence of faults in the system, a fault detection method for the RLC fractional linear circuit is proposed. A Luenberger-like observer is used as a residual generator, so that the dynamic estimation error system satisfies: (1) the system is asymptotically stable in the absence of disturbance and fault; (2) when the system is fault-free and has disturbance, it satisfies a certain H _∞ performance index under zero initial conditions; (3) when the system is fault-free and has faults, it satisfies a certain H _{_} performance index under zero initial conditions, and the Lyapunov function and linear matrix inequality are used to obtain sufficient conditions for the existence of a fault detection observer, so that fault detection can be accurately realized online, thereby realizing fault detection of the RLC fractional linear circuit.

The fault detection method for the RLC fractional linear circuit of the present invention comprises the following steps:

Step 1: Construct the mathematical model of the RLC fractional circuit according to Kirchhoff's current law (KCL) and Kirchhoff's voltage law (KVL), construct the augmented matrix, and transform the differential equation into the state equation in standard form.

Figure 1 shows an RLC fractional linear circuit. L represents inductance, C represents capacitance, and R represents load resistance. According to Kirchhoff's current law and Kirchhoff's voltage law, the mathematical model of the RLC fractional circuit is:

Among them, i _L , i _R and i _C are the currents flowing through the coil, resistor and inductor respectively (we stipulate that the circuit direction is clockwise), u _L , u _R and u _C are the voltages flowing through the coil, resistor and inductor respectively. L represents inductance, C represents capacitance, and R represents load resistance.

because Where 0＜α, β＜1, then equation (1) can be written as

Among them, u _C (t), i _L (t) are state variables, u(t) is the control input, if we define Then the state equation of the RLC fractional linear circuit system model is as follows:

In the formula,

In this embodiment, R = 1Ω, L = 1H, C = 1F, we can get

Step 2: Based on the state equation in step 1, considering the variational fractional order, interference, nonlinearity and fault conditions, a general system model of the RLC fractional circuit with interference and fault is given, which is;

Among them, 0＜α＜1, represents the state vector, is the system output, represents the control input, Indicates a fault. represents the external disturbance, where the fault can represent any type of actuator or sensor fault, and f(t) is defined as f _a (t) and f _s (t) represent actuator fault and sensor fault respectively. A, B ₁ , B ₂ , C, E ₁ , E ₂ , F ₁ , F ₂ are known constant matrices with appropriate dimensions; Φ(x(t), u(t)) is a nonlinear vector function with Lipschitz constant η, that is:

In order to achieve the purpose of the present invention, the following assumptions are made:

Assumption 1: (A, C) is observable.

System observability is the prerequisite for system fault detection. Assumption 1 ensures the observability of the system.

Step 3: Design a Luenberger-like observer as the residual signal generator and obtain the corresponding dynamic estimation error system;

The Luenberger-like observer system in step 3 is:

in, represents the state estimation vector, To estimate the output vector, represents the residual signal, L is the gain of the observer as the design object;

Define the state error as, Available:

The dynamic estimation error system is:

in:

In order to realize fault detection of RLC fractional linear circuits, the following definitions and lemmas are introduced before proceeding to the next step of research.

Definition 1: There exists a function f(t) defined on [t ₀ , +∞), and the fractional derivative of the α-order function h(t) starting from time t ₀ is:

For any t＞t ₀ , and α∈(0，1].

Definition 2: The fractional integral of the α-order function h(t) starting from time t ₀ is defined as follows, where α∈(0, 1],

Definition 3: For any s ≥ 0, the fractional exponential function is defined as:

Where α∈(0,1],

Given a system of fractional differential equations with derivatives 0＜α＜1:

in, is a given nonlinear function, and for any t＞0, h satisfies h(t,0)＝0.

Definition 4: (Fractional exponential stability) The origin of the system (13) is fractional exponentially stable if:

||x(t)||≤K||x ₀ ||E _α (-λ, tt ₀ ), t≥t ₀ (14)

Among them, λ＞0, K＞0.

Definition 5: When f(t) = 0 in the dynamic estimation error system (9), it can be said that the system is exponentially stable under the H _∞ performance γ and satisfies:

(a) At any time when ω(t) = 0, the system is exponentially stable;

(b) Under zero initial conditions, when ω(t)≠0

Definition 6: When ω(t) = 0 in system (9), it can be said that the system is exponentially stable under _H performance β and satisfies:

(c) At any time when f(t) = 0, the system is exponentially stable;

(d) Under zero initial conditions, when f(t)≠0

Definition 7: Consider the nonlinear fractional-order system (4). For two given positive scalars γ and β, if the dynamic estimation error system (9) is exponentially stable under the condition of ω(t) = 0, f(t) = 0 and satisfies conditions (15) and (16), then the observer (6) is called the _H∞ / _{H_fault} -proof observer of the system (4).

Lemma 1: Assume h(t) is a continuous function. For any t＞t ₀ , and α∈(0，1], we can obtain:

Lemma 2: Let h(t) be a continuous function on t∈[t ₀ ,+∞) such that exists in [t ₀ ，+∞), when When , for any t∈[t ₀ ,+∞), h(t) is a decreasing function. When , for any t∈[t ₀ ,+∞), h(t) is an increasing function.

Lemma 3: Let x(t) be a continuous function on t∈[t ₀ ,+∞) such that exists in [t ₀ ,+∞), P is a symmetric positive definite matrix, then exists in [t ₀ ,+∞), and

Lemma 4: (Schur's Complement Lemma) For a real matrix ∑ = ∑ ^T , the following three conditions are equivalent:

(a)

(b)

(c)

Lemma 5: If any real matrices H and E satisfy:

HF(t)E+E ^T F ^T (t)H ^T ＜0 (19)

Any F(t) satisfies F ^T (t)F(t)≤I, and there exists ε＞0.

εHH ^T +ε ^-1 ETE＜0 (20)

Lemma 6: Let x = 0 be an equilibrium point of system (13), and let V: is continuous, assuming that μ ₁ , μ ₂ , μ ₃ are any positive integers that satisfy the following conditions: x ₁

(a) μ ₁ ||x(t)|| ² ≤V(t,x(t))≤μ ₂ ||x(t)|| ² ;

(b) For 0≤t ₀ <t, V(t, x(t)) has a fractional derivative of order α;

(c)

Step 4: For the dynamic estimation error system obtained in step 3, the Lyapunov function is used to give sufficient conditions for the system to be exponentially stable and satisfy the two performances of _H∞ and _{H_} ; the parameters of the fault observer are designed according to the sufficient conditions. The specific process is as follows:

4.1 When the system satisfies the _H∞ performance, the present invention expresses the sufficient condition for the observer's asymptotic stability using the linear matrix inequality method.

Sufficient condition: For a given positive scalar ε ₁ , η, γ, if there exists a positive symmetric matrix P ₁ , matrix Z ₁ , satisfying

in,

When the positive symmetric matrix P ₁ and the matrix Z ₁ satisfy equation (21), the dynamic estimation error system is asymptotically stable and has H _∞ performance γ; in this case, the observer gain is

Proof: Given a robust fault detection dynamic estimation error system. When f(t) = 0, the dynamic estimation error system (9) can be transformed into

First, a system stability analysis model is established. When ω(t)=0, the present invention considers the Lyapunov function of the system (22):

V ₁ (t, e(t)) = e(t) ^T P ₁ e(t) (23)

From Lemma 3, we know that:

According to Lemma 5 and inequality (5), inequality

is equivalent to:

Substituting (26) into (24) we can obtain:

in,

According to Lemma 4, Ψ ₁ ＜0 can be equivalent to:

Wherein Z ₁ =P ₁ L.

Inequality (28) ensures that under the condition μ ₃ =λ _max (Ψ ₁ ), Therefore, Lemma 6 shows that system (22) has exponential stability.

Secondly, when w(t)≠0, the present invention sets the following performance index function:

According to Lemma 3, we can get

According to inequalities (25) and (26), we can get

Therefore, we can further get:

where ξ(t)＝[e ^T (t)w ^T (t)] ^T ,

According to Lemma 4, formula (33) is equivalent to:

Inequality (34) ensures that (28) holds.

Define the function g _w(t) =V ₁ (t, e(t))+J _w(t) . According to Theorem 1, we can get

According to Theorem 4, inequalities (21) and (32), we can get According to Lemma 2, we can get that g _w(t) is a decreasing function, so we can get that under zero initial conditions, for any t＞t ₀ , we have:

When V ₁ (t, e(t)) ≥ 0, for any t＞t ₀ , J _w(t) ＜ 0. Let t→+∞, we get

Formula (37) is equivalent to

From Definition 5, we can see that the error dynamic system (22) is asymptotically stable and has H _∞ performance γ. The proof is complete.

4.2 When the system satisfies _{H_performance} , the present invention expresses the sufficient condition for the observer's asymptotic stability using the linear matrix inequality method.

Sufficient condition: For a given positive scalar ε ₂ , η, β, if there exists a positive symmetric matrix P ₂ , matrix Z ₂ , satisfying

in,

When the positive symmetric matrix P ₂ and the matrix Z ₂ satisfy equation (39), the dynamic estimation error system is asymptotically stable and has _H performance β; in this case, the observer gain is

Proof: First, we give the dynamic estimation error system of fault detection with respect to sensitivity. When ω(t) = 0, the dynamic estimation error system (9) can be transformed into

First, a system stability analysis model is established. When f(t)=0, the present invention considers the Lyapunov function of the system (40):

V ₂ (t, e(t)) = e(t) ^T P ₂ e(t) (41)

From Lemma 3, we know that:

According to Lemma 5 and inequality (5), inequality

is equivalent to:

Substituting (44) into (42) we can obtain:

in,

According to Lemma 4, Ψ ₂ ＜0 can be equivalent to:

Wherein Z ₂ =P ₂ L.

Inequality (46) ensures that under the condition μ ₃ =λ _max (Ψ ₁ ), Therefore, Lemma 6 shows that system (40) has exponential stability.

Secondly, when f(t)≠0, the present invention sets the following performance index function:

According to Lemma 3, we can get

According to inequalities (43) and (44), we can get

Therefore, we can further get:

Where, v(t) = [e ^T (t)f ^T (t)] ^T ,

According to Lemma 4, formula (33) is equivalent to:

Inequality (52) ensures that (46) holds.

Define the function g _f(t) = V ₂ (t, e(t)) + J _f(t) . According to Theorem 1, we can get

According to Theorem 4, inequalities (39) and (45), we can get According to Lemma 2, we can get that g _w(t) is a decreasing function, so we can get that under zero initial conditions, for any t＞t ₀ , we have:

When V ₂ (t, e(t)) ≥ 0, for any t＞t ₀ , J _f(t) ＜ 0. Let t→+∞, we get

Formula (55) is equivalent to

From Definition 6, we can see that the dynamic estimation error system (20) is asymptotically stable and has _H performance β. The proof is complete.

In the present invention, ε ₁ =1.0, ε ₂ =1.2, η =0.5, γ =0.3, β =0.9, and define Applying the results of sufficient conditions, the parameters of the fault observer are as follows:

Step 5: Based on the fault observer designed in step 3, design the threshold _Jth , construct the residual evaluation function J(r), and determine whether the system has a fault. The specific process is as follows:

Before designing the threshold J _th , the present invention designs a composite fractional-order fault detection observer for the system (9).

Sufficient condition: According to the nonlinear fractional-order system (4), for two given positive scalars γ and β, there exists a nonlinear fractional-order fault detection observer (6) such that the dynamic estimation error system (9) is exponentially stable and satisfies _H∞ performance γ and _{H_performance} β. If there exists a matrix P＞0, Z, positive scalars _ε1 , _ε2 , η satisfying conditions (21) and (39), then the observer gain L＝P ^- 1Z.

Proof: In the proof of the first two sufficient conditions, let V ₁ (t, e(t)) = V ₂ (t, e(t)), and we can get P ₁ = P ₂ , Z ₁ = Z ₂ , and then we can get L = P ^-1 Z. The proof is complete.

In order to detect faults more accurately and quickly, it is necessary to set a suitable threshold _Jth and residual evaluation function J(r) to evaluate whether a fault has occurred. First, the residual evaluation function is selected as:

In the formula, T is the total evaluation time, and its threshold is:

Furthermore, the following logic is used to determine whether a fault has occurred:

Assuming a constant fault f(k) occurs in the RLC fractional linear circuit, the fault mode is as follows:

A preset threshold J _th =21.47 is selected, and the simulation results show that at t=39.15, J(r)>J _th , that is, the fault f(k) can be detected quickly.

The method of the present invention is further illustrated by simulation below. For the simulation, the state estimation error e(t) of the system with a disturbance signal ω(t) is shown in FIG2 ; the error e(t) in the undisturbed and faulty state is shown in FIG3 ; the fault signal f(t) is shown in FIG4 ; the residual signal r(t) in the non-faulty state is shown in FIG5 ; the residual signal r(t) with a fault signal f(t) is shown in FIG6 ; and the system threshold J(r) is shown in FIG7 .

It can be seen from the simulation results that, for the RLC fractional linear circuit fault detection method, the fault observer designed by the present invention can detect whether the system has a fault in a timely manner, and when there is no disturbance and no fault, the system is asymptotically stable. When there is disturbance and fault in the system, under zero initial conditions, the system meets certain _H∞ performance and _{H_} performance indicators, and has practical reference value.

The above embodiments are only for illustrating the technical concept and features of the present invention, and their purpose is to enable people familiar with the technology to understand the content of the present invention and implement it accordingly, and they cannot be used to limit the protection scope of the present invention. All equivalent changes and modifications made according to the spirit of the present invention should be included in the protection scope of the present invention.

Claims (4)

1. A fault detection method for an RLC fractional linear circuit, characterized in that it comprises the following steps: Step 1: Construct a mathematical model of the RLC fractional circuit based on Kirchhoff's current law (KCL) and Kirchhoff's voltage law (KVL), construct an augmented matrix, and transform the differential equation into a state equation in standard form; The circuit is a linear circuit composed of a resistor, a capacitor, a coil and a voltage source. According to Kirchhoff's current law (KCL) and Kirchhoff's voltage law (KVL), the mathematical model of the RLC fractional circuit is: Where i _L , i _R and i _C are the currents flowing through the coil, resistor and inductor respectively, u _L , u _R and u _C are the voltages flowing through the coil, resistor and inductor respectively, L represents inductance, C represents capacitance, and R represents load resistance; because Where 0＜α, β＜1, then equation (4) can be written as Among them, u _C (t) and i _L (t) are state variables, u(t) is the control input, if we define Then the state equation of the system model of the RLC fractional circuit is as follows: In the formula, C = [0 1]; Step 2: Based on the state equation in step 1, considering the variational fractional order, interference, nonlinearity and fault conditions, a general system model of the RLC fractional circuit with interference and fault is given; The general system model of the RLC fractional circuit with interference and faults in step 2 is: Among them, 0＜α＜1, represents the state vector, is the system output, represents the control input, Indicates a fault. represents the external disturbance, where the fault can represent any type of actuator or sensor fault, and f(t) is defined as f _a (t) and f _s (t) represent actuator fault and sensor fault respectively; A, B ₁ , B ₂ , C, E ₁ , E ₂ , F ₁ , F ₂ are known constant matrices with appropriate dimensions; Φ(x(t), u(t)) is a nonlinear vector function with Lipschitz constant η, η＞0, that is: Step 3: Design a Luenberger-like observer as a residual signal generator to obtain the corresponding dynamic estimation error system; The Luenberger-like observer system is in, represents the state estimation vector, To estimate the output vector, represents the residual signal, L is the gain of the observer, which is the design object; 0＜α＜1, represents the state vector, is the system output, represents the control input, A, B ₁ , B ₂ , C are known constant matrices with appropriate dimensions; Φ(x(t), u(t)) is a nonlinear vector function with a Lipschitz constant η, η＞0, that is: Define the state error as: The dynamic estimation error system is: in: Step 4: For the dynamic error system obtained in step 3, using the Lyapunov function, give the sufficient conditions for the system to be exponentially stable and satisfy the two performances of _H∞ and _H- , and design the parameters of the fault observer according to the sufficient conditions; Step 5: According to the fault observer designed in step 3, set the threshold _Jth , construct the residual evaluation function J(r), and determine whether the system has a fault. 2. The fault detection method for RLC fractional linear circuit according to claim 1, characterized in that the sufficient conditions for the system exponential stability and the satisfaction of the two performances of _H∞ and _H- in step 4 are: 1) The sufficient condition for the system to be exponentially stable and to satisfy the _H∞ performance index is: for a given positive scalar ε ₁ , η, γ, if there exists a positive symmetric matrix P ₁ , a matrix Z ₁ , satisfying in, Π ₁₂ ＝P ₁ E ₁ -Z ₁ E ₂ +C ^T E ₂ , E ₁ and E ₂ are known constant matrices with appropriate dimensions; When the positive symmetric matrix P ₁ and the matrix Z ₁ satisfy the above equation, the dynamic estimation error system is asymptotically stable and has H _∞ performance γ; in this case, the observer gain is 2) The sufficient condition for the system to be exponentially stable and to satisfy the H _- performance index is: for a given positive scalar ε ₂ , η, β, if there exists a positive symmetric matrix P ₂ , a matrix Z ₂ , satisfying in, Υ ₁₂ =P ₂ F ₁ -Z ₂ F ₂ -C ^T F ₂ , F ₁ and F ₂ are known constant matrices with appropriate dimensions; When the positive symmetric matrix P ₂ and the matrix Z ₂ satisfy the above equation, the dynamic estimation error system is asymptotically stable and has H _- performance β; in this case, the observer gain is 3. The fault detection method for an RLC fractional linear circuit according to claim 1, wherein the residual evaluation function is: In the formula, T is the total evaluation time, and its threshold is: 4. The fault detection method for the RLC fractional linear circuit according to claim 3 is characterized in that whether a fault occurs is determined according to the following logic: