CN114035548B - Fault detection method of T-S fuzzy control system based on kernel characterization - Google Patents
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Abstract
The invention discloses a fault detection method of a T-S fuzzy control system based on kernel characterization. Then, the kernel representation and the image representation of the controller are constructed using the observer and the controller gain matrix, respectively, and their input-output gains are calculated. Finally, an evaluation function is designed by utilizing the residual signals of the system, the dynamic threshold is designed by combining the kernel representation and the image representation of the residual signals of the controller, and the fault detection system design is realized by selecting proper decision logic. The invention can effectively utilize the online measurable system and the control residual signal to detect the change of the stability of the fuzzy control system affected by the fault in real time.
Description
Technical Field
The invention relates to the technical field of fault diagnosis of modern complex industrial systems based on models, in particular to a fault diagnosis technology of a nonlinear system which can adopt infinite approximation of a Takagi-Sugeno (T-S) fuzzy model. The fault detection method of the T-S fuzzy control system based on the kernel characterization is a specific application of the complex nonlinear closed loop control system in the aspects of fuzzy model-based control and fault diagnosis.
Background
As one of the important means for ensuring safe and stable operation of industrial systems, the problem of fault detection of the systems has received extensive attention in the academia and industry over the past decades. Because the actual industrial system inevitably has nonlinearity and uncertainty, the fault detection method of the linear system is not suitable for the actual nonlinear industrial system any more, and the design of the fault detection method for the nonlinear system has important theoretical and practical significance for monitoring whether the system has faults or not in real time. On the other hand, the T-S fuzzy dynamic modeling technology provides a powerful and effective mathematical tool for analyzing and designing a nonlinear system. The complex nonlinear system can be approximated by a convex combination of fuzzy membership functions and fuzzy submodels, in such a way that the analysis and design problems of the nonlinear system can be converted into a series of analysis and solution problems of linear matrix inequalities. Therefore, the fault detection method provided for the T-S fuzzy system has very important theoretical and application values.
At present, the fault detection method of the T-S fuzzy system based on the model comprises a self-adaptive method, a sliding mode method and the like. Most of the fault detection methods are proposed for open-loop fuzzy systems, and the fault detection system is designed by taking the residual signal of the system as an evaluation function and taking the upper bound of the evaluation function as a threshold value, but the problems that the fault may cause the change of dynamic parameters of the system and even the closed-loop control performance of the system is reduced are not considered. Although studies have been made to design an evaluation function using the variation represented by the system core caused by the fault and to design a threshold using the stability margin of the closed-loop control system, a closed-loop fault detection method based on stability is designed for the T-S fuzzy system, the evaluation function is calculated by using a least squares estimation algorithm suitable for the linear system, which makes the calculation of the evaluation function very conservative. In addition, since the design of the controller is also an important design unit of the fault detection system, the existing fault detection scheme of the fuzzy closed-loop control system designs the stabilizing controller of the fuzzy system according to the asymptotic stability of the closed-loop control system, which makes the design of the controller and the core characterization of the fuzzy closed-loop control system separate, and cannot fully reveal the core characterization of the fuzzy system and the stability association existing between the core characterization of the controller and the internal stability of the fuzzy closed-loop control system. Therefore, the fault detection method of the T-S fuzzy closed loop control system based on the kernel characterization needs to be further improved.
Disclosure of Invention
In order to overcome the defects in the prior art, the invention provides a fault detection method of a T-S fuzzy control system based on core characterization. Firstly, constructing a nuclear characterization state space model of a system based on a T-S fuzzy system, and generating a system residual signal. Then, constructing a controller based on the observer, and establishing a nuclear characterization model of the controller to generate a residual signal of the controller. And constructing an inverse state space model of the fuzzy closed-loop control system core representation by using the system and the controller core representation, and solving an observer and a controller gain matrix by solving a series of matrix inequalities according to the inverse input-output stability of the fuzzy closed-loop control system core representation, thereby realizing the controller design. Further, constructing an image representation and a core representation of the controller by using an observer and controller gain matrix, and solving the input-output gains of the image representation and the core representation of the controller according to an input-output stability theory. And finally, designing an evaluation function by using a residual signal of the system, and designing decision logic by using an image representation and a core representation input-output gain design threshold value of a controller residual signal combined controller according to whether the stability of the closed-loop control system is affected by a fault or not, thereby realizing the design of a fault detection system of the fuzzy control system. According to the method, through establishing the input-output stability relation between the residual signals of the system and the residual signals of the controller, the evaluation function, the threshold value and the decision logic of the fault detection system are designed, so that the conservation of the evaluation function is reduced by on-line calculation, the real-time monitoring of the stability performance reduction of the fuzzy control system caused by faults can be realized, and the fault detection performance of the fuzzy closed-loop control system is improved.
The invention adopts the following technical scheme and implementation steps:
A. industrial system controller design:
1) By introducing the input variables of the system into the precursor variables, a class of nonlinear systems of general form is approximated with a continuous-time T-S fuzzy system of the form:
Ruleζ:IF q 1 (t)isand q 2 (t)is/>and…and q j (t)is/>THEN
where ζ=1, 2, …, λ represents ζ -th local system, λ represents the number of fuzzy system rules; t represents the time when the system is running; q (t) = [ q ] 1 (t) q 2 (t) … q j (t)]Representing a measurable front-piece variable; j represents the total number of front-piece variables under the zeta rule;representing a fuzzy set under a zeta rule, iota representing a iota fuzzy set; x (t), y (t), u (t) represent the state variables of the fuzzy system, the output variable and the input variable respectively; />Representing the first derivative of the state variable x (t) with respect to time t; a is that ζ ，B ζ ，C ζ ，D ζ Is a system matrix of the zeta-th local system with proper dimension; ΔA ζ ，ΔB ζ ，ΔC ζ ，ΔD ζ The model uncertainty matrix of the ζ -th local system is represented, and the change of the system matrix caused by the fault is also represented. By convex combinations of local subsystems, the global fuzzy system is expressed as:
wherein,
h ζ (q (t)) represents a normalized fuzzy membership function,representing fuzzy set +.>Front piece variable q in (a) ι Membership of (t), H ζ (q (t)) represents the product of membership of all the precursor variables under the zeta rule. Meanwhile, for any time t, the fuzzy membership function h ζ Product H of membership of all precursor variables under the (q (t)) and zeta rule ζ (q (t)) satisfies the following condition:
2) And constructing a nuclear characterization of the T-S fuzzy system. Based on the T-S fuzzy system model (1), the kernel of the construction system is characterized as follows:
Ruleζ:IF q 1 (t)isand q 2 (t)is/>and…and q j (t)is/>THEN
z P (t)＝y(t)-ψ(t),
where ζ (t) represents the state estimate of the system, ψ (t) =c ζ ξ(t)+D ζ u (t) represents the output estimate of the system, G ζ Representing the observer gain matrix that needs to be designed. The local sub-models of each core characterization system are subjected to convex combination by adopting a membership function, and the global system of the system core characterization is obtained as follows:
wherein,since equation (3) can be effectively regarded as an observer-based residual generator of system (2), z P And (t) represents a residual signal of the system.
3) Observer-based controllers and core characterizations of the controllers that construct T-S fuzzy systems. An observer-based controller is employed having the form of a state space:
wherein F is ζ A controller gain representing the ζ -th local model, and havingBased on controller model (4)The core that constructs the controller is characterized as
Where β (t) represents the system state characterized by the controller core, z K And (t) represents a residual signal of the controller.
4) And solving a gain matrix of the observer and the controller to realize the design of the controller. According to a core representation formula (3) of the fuzzy system and a core representation formula (5) of the controller, establishing an inverse state space model of the core representation of the closed-loop control system:
defining the state error as e (t) =ζ (t) - β (t), obtaining a state space model of the following augmentation system:
wherein phi (t) = [ e ] T (t) ξ T (t)] T Representing the state of the augmentation system, v (t) = [ u ] T (t) y T (t)] T Representing the output of the augmentation system,the inputs to the augmentation system are represented and the system matrices of the augmentation system are represented as follows:
in order to amplify the input signal z of the system PK (t) and the output signal v (t) satisfy the following inequality relationship:
the following matrix inequality is obtained:
wherein ζ, m, l=1, 2, …, λ; sigma > 0 is a constant, G ζ ，Respectively a variable matrix; and is also provided with
Ω 11 ＝Sym{X 1 (A ζ -G ζ C l )-X 2 B ζ F m },
Wherein, for matrix Y, the symbol Sym { Y } represents Y+Y T And the symbol T is the transposed symbol of the matrix. By solving the matrix inequality (7), the observer gain matrix G is obtained ζ And a controller gain matrix F m 。
B. And (3) designing a fault detection system:
1) Solving L between input and output signals of controller image representation and kernel representation 2 Gain. Using an observer gain matrix G obtained by solving the matrix inequality (7) ζ Gain matrix F with controller ζ ζ=1, 2, …, λ, build controllerThe image of (a) is characterized by:
wherein I is an identity matrix. Input-output stability characterised by the controller image, i.e. signal z P (t) and v (t) satisfy the inequalityWherein->For a constant, W is a positive definite matrix, resulting in the following linear matrix inequality:
where Λ=sym { W (a ζ +B ζ F m ) ζ, m=1, 2, …, λ. By solving the linear matrix inequality (9), variables can be obtainedIs a value of (2). Thus, according to system L 2 Gain definition, obtaining L of the controller image representation type (8) 2 Gain less than or equal to
Similarly, the input-output stability of equation (5), i.e., signals v (t) and z, is characterized by the controller core K (t) satisfy the inequalityWhere κ > 0 is a constant and Q is a positive definite matrix, the following linear matrix inequality is obtained:
wherein,by solving the linear matrix inequality (10), the value of the variable k is obtained.
Further, according to system L 2 Gain definition, obtaining the L of the controller core representation formula (5) 2 Gain less than or equal to
2) The evaluation function, threshold and decision logic design of the fault detection system. The residual signal z of the system is used in a framework of considering whether the stability of the fuzzy control systems (2) and (4) is affected by faults P L of (t) 2 The norm design evaluation function is:
and utilizes the residual signal z of the controller K L of (t) 2 The norm design threshold is:
wherein τ > 0 is the time window constant and γ is the maximum L of the controller image representation and the kernel representation 2 Reciprocal of the product of the gainsr 0 And > 0 is a constant customized by an operator according to the actual running condition of the system. Thus, the evaluation function (11) is expressed as
The threshold value (12) is expressed as
In order to realize the fault detection system design of the fuzzy control systems (2) and (4), the following decision logic is adopted to judge whether the stability of the fuzzy control system is affected by faults or not:
compared with the prior art, the invention designs the observer-based controller based on the nuclear characterization of the system and the controller by solving a series of matrix inequalities, and fully reveals the relationship between the stability of the fuzzy control system and the inverse input-output stability of the nuclear characterization of the fuzzy closed-loop control system while obtaining the residual signals of the system and the controller. Meanwhile, unlike the traditional open loop fault detection method, the invention realizes the real-time detection of the stability performance degradation of the fuzzy closed loop control system caused by faults by respectively designing an evaluation function, a threshold value and corresponding decision logic by utilizing the input-output stability relation between the system residual signal and the controller residual signal, effectively reduces the conservation of the evaluation function calculation and improves the fault detection performance.
Drawings
FIG. 1 is a flow chart of the method of the present invention.
FIG. 2 is a graphical representation of membership functions for a three-tank fuzzy system.
Fig. 3 is a graphical representation of the variation of the system residual signal in the case of no fault.
Fig. 4 is a graphical representation of the variation of the system residual signal in case of a fault.
Fig. 5 is a graphical representation of fault detection performance.
Detailed Description
The main body of the three-tank system consists of three tanks, two water pumps and a water tank, is a typical nonlinear system and is often used for simulation experiment verification of a control and fault detection method of the nonlinear system. In the experiment, a three-volume water tank system with a model DTS200 is taken as a simulation object, and a dynamic model of the system is described as follows:
wherein s is 0 ＝s 13 ＝s 23 ＝0.5cm 2 Representing three-volume water tank system parameters; g=981 cm/s 2 Representing the gravitational constant; x is x 1 (t)，x 2 (t) and x 3 (t) the water level heights of the water tank 1, the water tank 2 and the water tank 3 are respectively represented, and the maximum height of the water level of the water tank is 62cm; u (u) 1 (t) and u 2 (t) the inflow rates of the water pump 1 and the water pump 2, respectively, and the maximum value of the inflow rate is 100cm 3 /s；Λ＝154cm 2 Representing the cross-sectional area of the tank; a, a 1 ＝0.475，a 2 =0.6 and a 3 =0.475 is the flow coefficient of three pipes connecting tank 1, tank 2 and tank 3, respectively; y (t) = [ x ] 1 (t) x 2 (t)] T Is the output of a three-tank system. By being operated at three operating points [20 8 14 ]] T ，[30 8 19] T And [40 8 24 ]] T Linearizing the three-capacity water tank system, and selecting state variable x of the three-capacity water tank system 1 And (T) is a front piece variable, and a membership function shown in fig. 2 is adopted to obtain a T-S fuzzy system model (2) of the three-capacity water tank system. The system matrix of the fuzzy system model is as follows:
and, a system matrix ΔA ζ ，ΔB ζ ，ΔC ζ ，ΔD ζ ζ=1, 2,3 represent unknown changes in the dynamics of the fuzzy system caused by the fault, respectively.
The method is applied to the T-S fuzzy model of the three-capacity water tank system, and comprises two steps of designing a controller of the three-capacity water tank system and designing a fault detection system, and the specific steps are as follows:
A. three-volume water tank system controller design:
step one: according to a fuzzy system matrix of the three-capacity water tank, solving a matrix inequality (7) to obtain an observer and controller gain matrix as follows:
step two: using the observer and controller gain matrix, an observer-based controller (4) is constructed, and further, a core representation (3) and a core representation (5) of the system and an image representation (8) of the controller are constructed. The initial condition of the three-capacity water tank system is [30 8 19 ]] T And sets the reference input signal of the system to [ 37.16.3.16 ]] T . In the case of no fault occurrence, fig. 3 shows the residual signal of the system, and it can be seen that the residual signal of the system in the case of no fault is in a stable state.
B. And (3) designing a fault detection system:
step three: solving the linear matrix inequality (9) to obtain the maximum L of the controller image representation formula (8) 2 Gain ofThen solving the linear matrix inequality (10) to obtain the controller core representation type(5) Maximum L of (2) 2 Gain->And then calculate to getHas a value of 0.0092.
Step four: selecting a time window constant τ=10s and a constant r 0 =0.001. The threshold value (14) is calculated in real time using the online measurable controller residual signal and simultaneously the value of the evaluation function (13) is calculated in real time using the online measurable system residual signal.
Step five: simulating a pipe blockage fault of the three-tank system when the running time of the three-tank system is between 420s and 430s, so that the pipe coefficient a 2 The changes were as follows:
fig. 4 shows the variation of the system residual signal in case of a failure. By using decision logic (15), the results of fault detection can be obtained as shown in fig. 5. As can be appreciated from fig. 5, when the value of the evaluation function exceeds the designed threshold, the stability performance of the three-tank control system is degraded by the fault.
Claims (6)
1. A fault detection method of a T-S fuzzy control system based on core characterization is characterized by comprising the following steps: the method comprises two parts of industrial system controller design and fault detection system design, and comprises the following specific steps:
A. industrial system controller design:
1) The input variables of the industrial system are introduced into the front-end variables, and a continuous time T-S fuzzy system with the following form is utilized to approximate a nonlinear system with the general form, and the model of the T-S fuzzy system is as follows:
Ruleζ:
THEN
where ζ=1, 2, …, λ represents ζ -th local system, λ represents the number of fuzzy system rules; t represents the time when the system is running; q (t) = [ q ] 1 (t) q 2 (t)…q j (t)]Representing a measurable front-piece variable; j represents the total number of front-piece variables under the zeta rule;indicate->A fuzzy set under the rule, i representing the iota-th fuzzy set; x (t), y (t), u (t) represent the state variables of the fuzzy system, the output variable and the input variable respectively; />Representing the first derivative of the state variable x (t) with respect to time t; a is that ζ ，B ζ ，C ζ ，D ζ Is a system matrix of the zeta-th local system with proper dimension; ΔA ζ ，ΔB ζ ，ΔC ζ ，ΔD ζ Representing a model uncertainty matrix of the zeta-th local system, and simultaneously representing a change of the system matrix caused by a fault; by convex combinations of local subsystems, the global fuzzy system is expressed as:
wherein,
h ζ (q (t)) represents a normalized fuzzy membership function,representing fuzzy set +.>Front piece variable q in (a) ι Membership of (t), H ζ (q (t)) represents the product of membership of all the precursor variables under the zeta rule; meanwhile, for any time t, the fuzzy membership function h ζ Product H of membership of all precursor variables under the (q (t)) and zeta rule ζ (q (t)) satisfies the following condition:
2) Constructing a nuclear characterization of a T-S fuzzy system; based on the T-S fuzzy system model (1), the kernel of the construction system is characterized as follows:
Ruleζ:
THEN
z P (t)＝y(t)-ψ(t),
where ζ (t) represents the state estimate of the system, ψ (t) =c ζ ξ(t)+D ζ u (t) represents the output estimate of the system, G ζ An observer gain matrix representing the need for design; the local sub-models of each core characterization system are subjected to convex combination by adopting a membership function, and the global system of the system core characterization is obtained as follows:
wherein,since equation (3) is actually the observer-based residual generator of system (2), z P (t) represents a residual signal of the system;
3) Constructing an observer-based controller and a core characterization of the controller of the T-S fuzzy system; an observer-based controller is employed having the form of a state space:
wherein F is ζ A controller gain representing the ζ -th local model, and havingBased on the controller model (4), the core of the build controller is characterized as
Where β (t) represents the system state characterized by the controller core, z K (t) represents a residual signal of the controller;
4) Solving a gain matrix of the observer and the controller to realize the design of the controller; according to a core representation formula (3) of the fuzzy system and a core representation formula (5) of the controller, establishing an inverse state space model of the core representation of the closed-loop control system:
defining the state error as e (t) =ζ (t) - β (t), obtaining a state space model of the following augmentation system:
wherein phi (t) = [ e ] T (t) ξ T (t)] T Representing the state of the augmentation system, v (t) = [ u ] T (t) y T (t)] T Representing the output of the augmentation system,the inputs to the augmentation system are represented and the system matrices of the augmentation system are represented as follows:
in order to amplify the input signal z of the system PK (t) and the output signal v (t) satisfy the following inequality relationship:
the following matrix inequality is obtained:
wherein ζ, m, l=1, 2, …, λ; sigma > 0 is a constant, G ζ ， Respectively a variable matrix; and is also provided with
Ω 11 ＝Sym{X 1 (A ζ -G ζ C l )-X 2 B ζ F m },
Wherein, for matrix Y, the symbol Sym { Y } represents Y+Y T And the symbol T is the transposed symbol of the matrix; by solving the matrix inequality (7), the observer gain matrix G is obtained ζ And a controller gain matrix F m ；
B. And (3) designing a fault detection system:
1) Solving controller image tableL between input and output signals of sign and core characterization 2 Gain; using an observer gain matrix G obtained by solving the matrix inequality (7) ζ Gain matrix F with controller ζ ζ=1, 2, …, λ, the image of the build controller is characterized by:
wherein I is an identity matrix; input-output stability characterised by the controller image, i.e. signal z P (t) and v (t) satisfy the inequalityWherein->For a constant, W is a positive definite matrix, resulting in the following linear matrix inequality:
where Λ=sym { W (a ζ +B ζ F m ) ζ, m=1, 2, …, λ; by solving the linear matrix inequality (9), the variables are obtainedIs a value of (2); according to system L 2 Gain definition, obtaining L of the controller image representation type (8) 2 Gain less than or equal to%>
The input-output stability of equation (5), i.e. signals v (t) and z, is characterized by the controller core K (t) satisfy the inequalityWherein the method comprises the steps ofKappa > 0 is a constant and Q is a positive definite matrix, resulting in the following linear matrix inequality:
ζ, l=1, 2, …, λ; obtaining the value of a variable kappa by solving a linear matrix inequality (10);
2) An evaluation function, a threshold value and a decision logic design of the fault detection system; the residual signal z of the system is used in a framework of considering whether the stability of the fuzzy control systems (2) and (4) is affected by faults P L of (t) 2 The norm design evaluation function is:
and utilizes the residual signal z of the controller K L of (t) 2 The norm design threshold is:
wherein τ > 0 is the time window constant and γ is the maximum L of the controller image representation and the kernel representation 2 Reciprocal of the product of the gainsr 0 The value of > 0 is a constant customized by an operator according to the actual running condition of the system; the evaluation function (11) is expressed as
The threshold value (12) is expressed as
2. The fault detection method for a T-S fuzzy control system based on a kernel characterization of claim 1, wherein: and constructing an inverse state space model (6) of the fuzzy closed-loop control system core representation according to the system core representation formula (3) and the controller core representation formula (5).
3. The fault detection method for a T-S fuzzy control system based on a kernel characterization of claim 1, wherein: obtaining an observer gain matrix G of an observer-based controller (4) by solving a matrix inequality (7) according to the system input-output stability theory ζ And a controller gain matrix F m 。
4. The fault detection method for a T-S fuzzy control system based on a kernel characterization of claim 1, wherein: according to system L 2 Gain definition, obtaining the L of the controller core representation formula (5) 2 Gain less than or equal toObtaining the L of the controller image representation formula (8) 2 Gain less than or equal to%>
5. The fault detection method for a T-S fuzzy control system based on a kernel characterization of claim 1, wherein: designed for realizing fault detection system of fuzzy control systems (2) and (4) by L of system residual error signal 2 A norm design evaluation function (13) utilizing the controller residual signal and the maximum L of the controller kernel representation and image representation 2 A gain design threshold (14).
6. The fault detection method for a T-S fuzzy control system based on a kernel characterization of claim 1, wherein: in order to realize the fault detection system design of the fuzzy control systems (2) and (4), the following decision logic is adopted to judge whether the stability of the fuzzy control system is affected by faults or not:
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