Multi-target fault detection method of time-varying delay non-linear fractional order system based on memory-free full-order observer

Abstract

The invention discloses a fault detection method for a time-varying delay nonlinear fractional order system based on a memory-free full-order observer, and belongs to the technical field of nonlinear fractional order continuous system fault detection. A nonlinear fractional order continuous system model including system interference, time-varying delay, actuator faults and sensor faults is established; a memory-free full-order observer is designed, and an error dynamic equation and performance indexes which need to be met when a fault detection target is achieved are given; a full condition of global asymptotic stability of the system is given, the full condition is converted into a linear matrix inequality, observer parameters are obtained, and a fault detection task is completed based on a residual calculation function. Compared with the prior art, the designed fault detection observer improves the fault sensitivity to the maximum extent and reduces the influence of interference as much as possible.

Description

Technical Field

[0001] The present invention belongs to the technical field of nonlinear fractional-order continuous system fault detection, and in particular relates to a multi-objective fault detection method for a time-varying delay nonlinear fractional-order system based on a memoryless full-order observer. Background Art

[0002] With the increasing demand for safety and reliability in industrial systems, fault diagnosis technology has become a hot research area in recent decades. Fault detection is an important part of the fault diagnosis process, which can achieve rapid detection of faults and eliminate them in the initial stage. Among the model-based fault detection methods, observer-based techniques have been proven to be an effective method by comparing the residual signal value with a pre-designed threshold. This is essential for online fault-tolerant control and real-time detection. Among the model-based fault detection methods, observer-based methods have been proven to be an effective method and have attracted widespread attention. Fault detection can be achieved by using various observer techniques, such as adaptive observers, sliding mode observers, PI observers, and augmented system observers.

[0003] In recent years, with the rise of fractional-order systems, research on fault detection and fault estimation has continued to emerge. However, when the system has the characteristic of time-varying delay nonlinearity, related research is still relatively lacking. To address this issue, this paper focuses on the design of multi-objective fault detection observers for fractional-order systems with time-varying delay nonlinearity. By adding time-varying delays and system disturbances to the nonlinear fractional-order system, we design a memoryless full-order observer. This observer not only ensures the asymptotic stability of the nonlinear fractional-order system, but also has H ∞ / H - performance. Summary of the invention

[0004] Purpose of the invention: In view of the problems existing in the current prior art, the present invention provides a multi-objective fault detection method for time-varying delay nonlinear fractional-order systems based on a memoryless full-order observer, and uses a new lemma for estimating the Caputo derivative of a quadratic function, combined with the model-related Lyapunov function technology and linear matrix inequalities, to obtain sufficient conditions for the existence of a fault detection observer, so that the system is globally asymptotically stable and has H - and H ∞ It can accurately measure fault sensitivity and interference robustness online and realize fault detection of nonlinear fractional-order systems.

[0005] Technical solution: The present invention provides a multi-objective fault detection method for a time-varying delay nonlinear fractional-order system based on a memoryless full-order observer, comprising the following steps:

[0006] Step 1: Establish a nonlinear fractional-order system model including system disturbance, time-varying delay, actuator fault and sensor fault;

[0007]

[0008] in, is the Caputo fractional derivative, α∈(0,1), represents the state vector, is the system output, represents the control input, Indicates a fault. Indicates interference; A, A d , B, C, D, F a and F s is a known constant matrix with appropriate dimension; φ(t) is a continuous initial condition, h(t) is a time-varying delay and satisfies:

[0009] 0<h(t)≤h1,t≥0

[0010] Nonlinear functions Right now:

[0011]

[0012] x1(t), x2(t) represent the two system states at the current time t, x1(th(t)), x2(th(t)) represent the two system states at the time after the delay h(t), for all Where E1 and E2 are known constant matrices with appropriate dimensions;

[0013] Step 2: Design a memoryless full-order observer:

[0014]

[0015] in, represents the state estimation vector, is the estimated output vector, r(t) represents the residual signal, L is the gain of the observer, is the design object, and the system state estimation error is

[0016] Step 3: Based on the memoryless full-order observer in step 2, convert it into the observer error equation and further determine the corresponding dynamic estimation error system;

[0017] Step 4: For the dynamic estimation error system, use the Lyapunov function to give the sufficient conditions for the exponential stability of the system λ is a given positive scalar, z(t)=[e T (t), x T (t)] T; Designing the parameters of the fault observer according to the sufficient conditions;

[0018] Step 5: Solve the linear matrix inequality in step 4 to obtain the parameters that need to be designed in the memoryless full-order observer. Based on the multi-target detection observer, effective fault detection is achieved through the residual evaluation function.

[0019] Furthermore, the memoryless full-order observer is transformed into the following observer error equation:

[0020]

[0021] Furthermore, let z(t)=[e T (t), x T (t)] T , we get the dynamic estimation error system, so that H ∞ / H - Performance is asymptotically stable:

[0022]

[0023] in,

[0024] Furthermore, in step 4, the Lyapunov function is:

[0025]

[0026] in, P and Z are symmetric matrices.

[0027] Furthermore, in step 5, the observer parameter design process is:

[0028] Use Matlab's LMI toolbox to solve the following four LMI equations for real symmetric matrices P, Q, Z and matrix X to obtain the optimal solution L for the observer parameters:

[0029]

[0030]

[0031] Among them, φ 11 =ZA+A T Z-XC-C T X T +(1+ε)Z+λI,

[0032] φ 12 =AP+PA T +(1+ε)P+(λ+1)I, ε>0, h1>0 and λ>0, L=Z -1 X,γ>0 is H∞ Performance index, ||r(t)||<γ||d(t)||, β>0 is H - Performance index, ||r(t)||>β||f(t)||.

[0033] Furthermore, the residual evaluation function is selected:

[0034]

[0035] Among them, t0 represents the initial evaluation time, t is the evaluation time step, and the threshold is:

[0036]

[0037] Therefore, based on the following logical rules, by comparing J(r(t)) and J th , effectively detecting the situation where a failure occurs:

[0038]

[0039] Beneficial effects:

[0040] 1. This paper proposes a novel actuator and sensor fault detection method for nonlinear hierarchical systems with time-varying delays. By using a new lemma for estimating the Caputo derivative of quadratic functions and combining the model-dependent Lyapunov function technology and linear matrix inequalities, sufficient conditions for the existence of a fault detection observer are obtained, making the system globally asymptotically stable and having H - and H ∞ It can accurately measure fault sensitivity and interference robustness online, realize fault detection of nonlinear fractional-order systems, maximize fault sensitivity and minimize the impact of interference.

[0041] 2. When establishing a general model of the system, the present invention uses an observer method for systems with time-varying delays, interference, nonlinearity, actuator and sensor failures, so that the system's estimated dynamic error system is asymptotically stable when there is no disturbance and failure, and multi-objective fault detection is completed. The fault detection method proposed by the present invention is relatively novel. BRIEF DESCRIPTION OF THE DRAWINGS

[0042] Figure 1 The flowchart of the method of the present invention is shown in FIG.

[0043] Figure 2 Schematic diagram of the fault signal f(t) of the present invention;

[0044] Figure 3 is the residual graph r(t) of the fault signal f(t) of the present invention;

[0045] Figure 4is the residual graph r(t) of the fault-free signal f(t) of the present invention;

[0046] Figure 5 The threshold diagram J(r(t)) of the fault signal f(t) and the fault-free signal f(t) of the present invention;

[0047] Figure 6 is the fault signal f of the present invention a (t) Schematic diagram;

[0048] Figure 7 is the fault signal f of the present invention s (t) Schematic diagram;

[0049] Figure 8 There is a fault signal f in the present invention a (t) and fault signal f s The residual graph r(t) of (t);

[0050] Fig. 9 There is a fault signal f in the present invention a (t) and f s (t) and no fault signal f a (t) and f s (t) is the threshold map J(r(t)). DETAILED DESCRIPTION

[0051] In order to make the purpose, technical solution and advantages of the present invention clearer, the technical solution of the present invention is further described below in conjunction with the accompanying drawings and experiments.

[0052] Figure 1 Flow chart of the multi-objective fault detection method for time-varying delay nonlinear fractional-order system based on memoryless full-order observer of the present invention. Figure 1 As shown, the present invention discloses a multi-objective fault detection method for a time-varying delay nonlinear fractional-order system based on a memoryless full-order observer, which specifically includes the following steps:

[0053] Step 1: Establish a nonlinear fractional-order system model that includes system disturbance, time-varying delay, actuator fault, and sensor fault. The nonlinear fractional-order system model with time-varying delay is as follows:

[0054]

[0055] in, is the Caputo fractional derivative, α∈(0,1), represents the state vector, is the system output, represents the control input, Indicates a fault. Represents interference. Among them, A, Ad , B, C, D, F a and F s is a known constant matrix of appropriate dimension; φ(t) is a continuous initial condition. h(t) is a time-varying delay and satisfies

[0056] 0 <h(t)≤h1,t≥0

[0057] Nonlinear functions Right now:

[0058]

[0059] For all Where E1 and E2 are known constant matrices with appropriate dimensions.

[0060] Step 2: Design a memoryless full-order observer for the nonlinear continuous system in step 1, transform it into an observer error equation, and further determine the corresponding dynamic estimation error system.

[0061]

[0062] in, represents the state estimation vector, To estimate the output vector, represents the control input, r(t) represents the residual signal, L is the gain of the observer as the design object, and the system state estimation error is

[0063] According to the system model and the memoryless full-order observer, the observer error equation is obtained as follows:

[0064]

[0065] Let z(t)=[e T (t), x T (t)] T , we can get the following augmented system:

[0066]

[0067] in,

[0068] Step 4: Using the Lyapunov function for the dynamic estimation error system Sufficient conditions for exponential stability of the system are given Where λ is a given positive scalar, and the parameters of the fault observer are designed according to the sufficient condition.

[0069] The Lyapunov function is:

[0070]

[0071] in, P and Z are symmetric matrices.

[0072] Step 5: Solve the linear matrix inequality in step 4 to obtain the parameters that need to be designed in the memoryless full-order observer. Based on the multi-target detection observer, effective fault detection is achieved through the residual evaluation function.

[0073] Use Matlab's LMI toolbox to solve the following four LMI equations for real symmetric matrices P, Q, Z and matrix X, and obtain the optimal solution L for the observer parameters:

[0074]

[0075] Among them, φ 11 =ZA+A T Z-XC-C T X T +(1+ε1)Z+λI,

[0076] φ 12 =AP+PA T +(1+ε)P+(λ+1)I, ε>0, h1>0, λ>0, L=Z -1 X,γ>0 is H ∞ Performance index, ||r(t)||<γ||d(t)||, β>0 is H - Performance index, ||r(t)||>β||f(t)||.

[0077] In order to effectively detect faults, we choose the following residual evaluation function:

[0078]

[0079] Among them, t0 represents the initial evaluation time, and t is the evaluation time step. Then the threshold is:

[0080]

[0081] Therefore, based on the following logical rules, by comparing J(r(t)) and J th , you can detect when a fault occurs:

[0082]

[0083] The specific simulation experiment is as follows:

[0084] The present invention considers a model of a fractional order circuit, wherein the system parameters are as follows:

[0085]

[0086] h(t)=1-sint, α=0.5, λ=0.5, γ=0.5, β=1, ε=1, h1=3;

[0087] Solving the LMI inequality, we can obtain:

[0088]

[0089] The fault detection observer gain matrix is:

[0090]

[0091] Assume that the system simulation period is 100s, the fractional derivative α is 0.1, the control input is a step signal, and the system interference is band-limited white noise. The following two fault modes occur:

[0092] (1) When the sensor fails s (t) and actuator fault f a (t) For the same fault:

[0093]

[0094] (2) When the sensor fails s (t) and actuator fault f a (t) for different faults:

[0095]

[0096] The method of the present invention is further described below by simulation. For the simulation, the fault signal f(t) is as follows: Figure 2 As shown; Figure 3 and Figure 4 is the residual signal with and without faults; the state response of the residual evaluation function J(r(t)) with and without faults is as follows Figure 5 We will consider the special case of the parameters of a nonlinear fractional-order system when f a (t)≠f s At (t), the actuator failure and sensor failure are as follows: Figure 6 and 7 As shown; Figure 8 The residual curves for actuator and sensor faults are shown; Fig. 9 The state responses of the residual evaluation function J(r(t)) with and without faults are given.

[0097] It can be seen from the simulation results that for the multi-objective fault detection method of time-varying delay nonlinear fractional-order system, the fault observer designed by the present invention can detect the fault of the system in time, the system is asymptotically stable and has robustness and sensitivity, and has practical reference value.

[0098] 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

1. A multi-objective fault detection method for time-varying delay nonlinear fractional-order systems based on a memoryless full-order observer, characterized in that: The steps include: Step 1: Establish a nonlinear fractional-order system model including system disturbance, time-varying delay, actuator fault and sensor fault; in, is the Caputo fractional derivative, α∈(0,1), represents the state vector, is the system output, represents the control input, Indicates a fault. Indicates interference; A, A d , B, C, D, F a and F s is a known constant matrix with appropriate dimension; φ(t) is a continuous initial condition, h(t) is a time-varying delay and satisfies: 0 <h(t)≤h1,t≥0 Nonlinear function g(t, x(t), x(th(t))), g(t, 0, 0) = 0, that is: x1(t), x2(t) represent the two system states at the current time t, x1(th(t)), x2(th(t)) represent the two system states at the time after the delay h(t), for all x1(t), x2(t), x1(th(t)), Where E1 and E2 are known constant matrices with appropriate dimensions; Step 2: Design a memoryless full-order observer: in, represents the state estimation vector, is the estimated output vector, r(t) represents the residual signal, L is the gain of the observer, is the design object, and the system state estimation error is Step 3: Based on the memoryless full-order observer in step 2, convert it into the observer error equation and further determine the corresponding dynamic estimation error system; Step 4: For the dynamic estimation error system, use the Lyapunov function to give sufficient conditions for the exponential stability of the system λ is a given positive scalar, V(t, z(t)) is a Lyapunov function, z(t) = [e T (t), x T (t)] T , designing the parameters of the fault observer according to the sufficient conditions; Step 5: Solve the linear matrix inequality in step 4 to obtain the parameters that need to be designed in the memoryless full-order observer. Based on the multi-target detection observer, effective fault detection is achieved through the residual evaluation function.

2. The multi-objective fault detection method for time-varying delay nonlinear fractional-order systems based on a memoryless full-order observer according to claim 1 is characterized in that: The memoryless full-order observer is transformed into the following observer error equation:

3. The multi-objective fault detection method for time-varying delay nonlinear fractional-order systems based on a memoryless full-order observer according to claim 2 is characterized in that: The dynamic estimation error system makes H ∞ / H - The performance is asymptotically stable, as follows: in, 4. The multi-objective fault detection method for time-varying delay nonlinear fractional-order systems based on a memoryless full-order observer according to claim 1, characterized in that: In step 4, the Lyapunov function is: in, P and Z are symmetric matrices.

5. The multi-objective fault detection method for time-varying delay nonlinear fractional-order systems based on a memoryless full-order observer according to claim 1, characterized in that: In step 5, the observer parameter design process is: Use Matlab's LMI toolbox to solve the following four LMI equations for real symmetric matrices P, Q, Z and matrix X to obtain the optimal solution L for the observer parameters: among them,f 11 =ZA+A T Z-XC-C T X T +(1+ε)Z+λI, φ 12 =AP+PA T +(1+ε)P+(λ+1)I, ε>0, h1>0 and λ>0, L=Z -1 X,γ>0 is H ∞ Performance index, ||r(t)||<γ||d(t)||, β>0 is H - Performance index, ||r(t)||>β||f(t)||.

6. The multi-objective fault detection method for time-varying delay nonlinear fractional-order systems based on a memoryless full-order observer according to claim 5, characterized in that: Select the residual evaluation function: Among them, t0 represents the initial evaluation time, t is the evaluation time step, and the threshold is: Therefore, based on the following logical rules, by comparing J(r(t)) and J th , effectively detecting the situation where a failure occurs:

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