CN115933604B - Fault detection and prediction control method based on data driving - Google Patents
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Abstract
The invention discloses a fault detection and prediction control method based on data driving, which comprises the following steps: the Hankel matrix is built by constructing input and output data, so that a state space model of the system is built, the identification of a linear time-invariant system, the model prediction of state space recursion, the realization of a Gap measurement and a stability margin under the condition of nuclear space and image space solving data based on data driving, and a simulation experiment are built. The invention adopts the fault detection and prediction control method based on data driving, based on a subspace identification based method (SIM), the fault detection based on a model can be replaced by using a Gap technology driven by data, and the characteristics of the data driving method can reconstruct the state of the system by using offline or online data instead of directly using the model.
Description
Technical Field
The invention relates to a fault diagnosis technology, in particular to a fault detection and prediction control method based on data driving.
Background
In modern industrial control system fault diagnosis method research, a model-driven fault diagnosis method framework has been formed. The Gap measurement is taken as an important tool for measuring the system variability in a model driving method, and has been widely applied, particularly in the field of robust control. The most primitive definition of Gap metric is the distance between two control objects on the hilbert space, which is also widely studied and applied in fault detectability analysis, fault isolation analysis, and threshold design of systems. The Gap metric, also combined with the optimal stability margin, has achieved extensive research in the area of multi-modal methods.
At present, model-based fault detection consists of a residual error generator, an evaluator and decision logic with threshold values. Fault Detection (FD) is achieved by examining a residual signal representing the difference between a measured process variable and its estimate. However, these methods generally require the creation of accurate mathematical models, which is often difficult to do in practical industrial systems, so Gap metrics are mainly used for offline analysis, process monitoring, fault-tolerant control of the system, and rarely used for calculation of online metrics. And the traditional Gap measurement fault diagnosis based on the model is the whole offset of the measurement system. However, in a real situation, an accurate model is difficult to build, and meanwhile, on-line identification of parameters in a changed model is difficult to realize.
Disclosure of Invention
In order to solve the above technical problems, the present invention provides a data-driven fault detection and prediction control method, based on a subspace identification-based method (SIM), a data-driven Gap technique may be used instead of a model-based fault detection, and the characteristics of the data-driven method may not directly use a model, but reconstruct the state of the system using offline or online data.
In order to achieve the above object, the present invention provides a data-driven based fault detection and prediction control method, comprising the steps of:
s1, building a Hankel matrix by constructing input and output data, so as to build a state space model of the system;
s2, identifying a linear time-invariant system;
s3, model prediction of state space recursion;
s4, realizing a kernel space and an image space based on data driving;
s41, introducing definition of a kernel space and an image space of data driving;
s42, calculating stability margins under a kernel space and an image space;
s43, realizing data driving under an open loop control system;
s5, solving Gap measurement and stability margin under the data;
s51, realizing data driving of Gap measurement;
s52, a data driving implementation method of the system stability margin;
s6, a simulation experiment;
s61, verifying consistency of data driving and model results;
s62, verifying on-line fault diagnosis based on Gap measurement under data driving;
and S63, obtaining an experimental conclusion.
Preferably, the step S1 specifically includes the following steps:
a state space model of a linear time-invariant system is given as follows:
in the above formula, A, B, C and D are corresponding system matrixes, A epsilon R n×n ,B∈R n×m ,C∈R l×m ,D∈R l×m ，x(k)∈R n Y (k) εR, which is a system state vector m An observation vector that is a state; w (w) 1 (l) And v 1 (l) Process noise and observation noise of gaussian type respectively;
and assuming that the system satisfies the following conditions:
1) The formula (1) satisfies the generalized energy control matrix and the generalized energy observation matrix full rank;
2) The characteristic value of A is positioned in the left half plane, and the condition of gradual stability of the system is met;
3) Covariance matrix of Hankel matrix formed by input and output data is satisfied
rank(Ω[U 0|2i-1 ,U 0|2i-1 ])＝2mi；
4) The input sequence is uncorrelated with the noise sequence.
Preferably, the step S2 specifically includes the following steps:
s21, decomposing the formula (1) into a determination part and a random part:
X＝X d +X s ，Y＝Y d +Y s (2)；
through iteration, obtain
Wherein the matrix is augmented
S22, determining a system Toeplitz matrix:
s23, defining a Hankel matrix composed of input and output:
where i is a positive integer, matrix U past Representing past inputs, the subscript past represents past data, U fu Representing future inputs, subscript fu representing future data, U past And U fu Hankel matrix U 0,2i-1 Divided into U 0,i-1 And U i,2i-1 Two parts, and U for distinguishing characteristics of past data from future data past And U fu There is no intersection of columns of (a), the past and future inputs are of the same amount of overlapping data, i.e. in U past And U fu Can find the input u i ；
Similarly, define the output Hankel matrix Z past ,Z fu
A state sequence X defined as follows i ∈R n×j ：
X i ＝(x i x i+1 … x i+j-2 x i+j-1 ) (10)
Wherein,
definition matrix E i The following are provided:
wherein,
s24, performing oblique projection on the (4) to obtain:
at the same time, the method comprises the steps of,the expression is as follows:
wherein,is->Orthogonal complement of (2)Matrix, symbol->Representing the pseudo-inverse of the matrix;
s25, solving E through singular value decomposition technology i Is a characteristic value of (a):
wherein the weight matrixZ 2 ∈R j×j And satisfies rank (Z past )＝rank(Z past Z 2 )；
S26, calculating an augmentation matrix:
calculating system matrices A and C by combining (6) and (15), and updating the augmentation matrix T in real time i ；
S27, defining matrixes L, K, P and Q:
s28, calculating N through (17) - (20) i ：
S29, combining (18), and solving a system matrix
Preferably, the step S3 specifically includes the following steps:
s31, assume w 1 (l+i)＝0,v 1 (l+i) =0, i=1, 2,3,4 … n, recursively the equation of state:
and similarly, recursion is carried out on a system observation equation;
the following expression is obtained by arrangement:
wherein n is the length of the system for prediction, and the output vector and the input vector of the system are respectively represented by the following formulas:
Y(l)＝[y(l) y(l+1) … y(l+m-1)] (25)
U(l)＝[u(l) u(l+1) … u(l+m-1)] (26)；
s32, U when the length of the Hankel matrix is m past And Y is equal to past Is known to U fu And Y is equal to fu Is calculated by the equations (25) and (26), assuming that the range of predicted data for the future l time instant is [ l, l+P+m-2 ]]Then the input/output data of the system is includedThe definition is as follows:
s33, in the formula (27)And performing LQ decomposition to obtain image space representation based on data driving, thereby obtaining Gap measurement and system margin under the data driving.
Preferably, the step S41 specifically includes the following steps:
s411, defining the following signals:
s412, defining one: assume that (1) satisfies w 1 (k)＝0，v 1 (k)＝0，When->Sample m is such that:
then the matrixIs a data-driven implementation of the kernel space, when +.>Satisfy->When it is calledIs standardized and is marked as->
S413, defining two steps: assuming that formula (1) satisfies and defines the same conditions, whenSample m is such that:
at this time, the matrixIs a data-driven implementation like space, if +.>Then call->Is a data-driven implementation of standardized image space, denoted +.>
Preferably, the step S42 specifically includes the following steps:
s421, consider an LTI system G form as shown in formula (1), define three: when the formula (1) respectively satisfiesWhen in use, then->And is->Nuclear space and image space, respectively, standardized for the system, in order not to be confused +.>And->A kernel space representation and an image space representation in a standardized form;
s422, according to the mutual mass decomposition technique, the system standardized kernel space representationAnd image space representation +.>The expression is as follows:
s423, defining four: the system G shown in the formula (1) and the corresponding SKR are given and shapedThe stability margin is calculated by:
wherein I H A Hankel norm of formula (1);
when the closed loop control circuit finds a stable controller H (z), the following holds:
preferably, the step S43 specifically includes the following steps:
s431, decomposing the defined input and output data by using an LQ technology to obtain the following formula:
wherein L is 11 、L 21 、L 32 The quadratic matrix is represented by a real symmetric matrix;
s432, theorem one: given LTI system as shown in formula (1), measurement input and output data of the system are given, and LQ decomposition as shown in formula (33) is performed on the system, so that the kernel space under data driving is as follows:
wherein the method comprises the steps of
In the method, in the process of the invention,representing a pseudo-inverse;
further standardized is obtained
Wherein V is 1 Is obtained by singular value decomposition as follows:
s433, theorem II: similarly, LQ decomposition is performed on input and output data, and then the obtained image space based on data driving is:
wherein,
further standardized is obtained
Preferably, the step S51 specifically includes the following steps:
s511, theorem three: given two systems G 1 ，G 2 The state space expressions of both are shown as formula (1), then the following holds when going to infinity for m:
and (3) proving: when m tends to infinity, i.e. τ m Trend I, at τ m G tends to G, z i,m Also tend to z i The Gap metric at this time is defined in the most primitive hilbert space, i.e., equation (42) holds;
furthermore, the directed Gap metric δ d,m The data driving implementation of (1) is obtained by calculating the singular values of the following formula:
where the symbol sigma is a singular value of the system,and->Represents G 1 ,G 2 Corresponding to the image space under the standardized condition;
s512, under the precondition of theorem three, it is assumed that the system G exists 1 And G 2 The Gap metric between the two is calculated through the directed Gap of the corresponding graph space, and the result is that:
s513, defining four: the truncating operator τ is defined as follows m ：
Wherein,the meaning of the above formula means that the time signal t is switched off after m+1 sampling times, and therefore, two systems G (i) ，G (j) The Gap metric under the i, j=1, 2 data driven framework is defined as:
system G 1 And G 2 The directional Gap metric in between can also be calculated by:
wherein z is 1,m ＝τ m z 1 ，z 2,m ＝τ m z 2 。
Preferably, the step S52 specifically includes the following steps:
s521, setting the kernel space as a residual error generator of the system, and giving a Hankel norm of the standardized kernel space:
s522, defining five: optimal stability margin for a systemThe data driving implementation process is as follows:
wherein,is defined as:
s523, similarly analogizing the truncating operator τ m Definition of kappa m Expressed as:
s524, theorem four: the system G represented by formula (1) is given, wherein w 1 (l)＝0,v 1 (l)＝0,When m tends to infinity, then there are:
calculated using the following formula:
wherein,i.e. < ->Is->Is a sub-matrix of the matrix.
Preferably, the following two discretized LTI systems G are considered in step S61 1 ,G 2 The formula is as follows:
in order to verify that the Gap metric value under the data driving framework can still be close to the result of the Gap metric value of the model under the condition that the system model parameters are changed, consider G 2 The parameter in (z) is changed to G i (z), wherein i=3, 4,5,6:
at the moment, the measurement value of the model is deviated, the length m of the Hankel matrix is set, and Gap measurement values based on data driving are calculated respectively;
in step S62, two system models of equation (1) continue to be employed, assuming that the process noise satisfies w 1 (l)～N[0,0.3]The observation noise satisfies v 1 (l)～N[0,0.1]Setting sampling time point, and comparing G from nth sampling time 2 (z) predicting a state space model by using a system matrix identified by subspace, driving a Gap metric value by using the solved data of each moment after n sampling points, and comparing the Gap metric value with a Gap metric value of a nominal system to further achieve the purpose of real-time fault diagnosis;
in step S63, for the nominal system and the fault system, the Gap metric is calculated by the obtained data, and from the simulation result, the Gap metric value under the data driving frame approximates the correlation value based on the model, in addition, the new system matrix is updated in real time by subspace identification, and then the Gap metric between the system in monitoring operation and each time of the nominal system is calculated by using the data driving Gap metric calculation method, so as to obtain the data driving Gap metric value of the system at each time, and the time deviated from the normal value of the system can be regarded as the fault by comparing with the reference value of the Gap metric of the nominal system model.
The invention aims to provide a data-driven fault detection and prediction control method, which is based on a subspace identification-based method (SIM), and can use a data-driven Gap technology to replace model-based fault detection, wherein the characteristics of the data-driven method can not directly use a model, but reconstruct the state of a system by using offline or online data.
The technical scheme of the invention is further described in detail through the drawings and the embodiments.
Drawings
FIG. 1 is a flow chart of the calculation of the data driven Gap metric algorithm of the present invention;
fig. 2 is a diagram of the real-time fault detection result of Gap measurement based on data driving in the present invention.
Detailed Description
The present invention will be further described with reference to the accompanying drawings, and it should be noted that, while the present embodiment provides a detailed implementation and a specific operation process on the premise of the present technical solution, the protection scope of the present invention is not limited to the present embodiment.
The fault detection and prediction control method based on data driving comprises the following steps:
s1, building a Hankel matrix by constructing input and output data, so as to build a state space model of the system;
preferably, the step S1 specifically includes the following steps:
a state space model of a linear time-invariant system is given as follows:
in the above-mentioned method, the step of,for the corresponding system matrix->x(k)∈R n Y (k) εR, which is a system state vector m An observation vector that is a state; w (w) 1 (l) And v 1 (l) Process noise and observation noise of gaussian type respectively;
and assuming that the system satisfies the following conditions:
1) The formula (1) satisfies the generalized energy control matrix and the generalized energy observation matrix full rank;
2) The characteristic value of A is positioned in the left half plane, and the condition of gradual stability of the system is met;
3) Covariance matrix of Hankel matrix formed by input and output data is satisfied
rank(Ω[U 0|2i-1 ,U 0|2i-1 ])＝2mi；
4) The input sequence is uncorrelated with the noise sequence.
S2, identifying a linear time-invariant system;
preferably, the step S2 specifically includes the following steps:
s21, decomposing the formula (1) into a determination part and a random part:
X＝X d +X s ，Y＝Y d +Y s (2)；
through iteration, obtain
Wherein the matrix is augmented
S22, determining a system Toeplitz matrix:
s23, defining a Hankel matrix composed of input and output:
where i is a positive integer of sufficient magnitude, matrix U past Representing past inputs, the subscript past represents past data, U fu Representing future inputs, subscript fu representing future data, U past And U fu Hankel matrix U 0,2i-1 Divided into U 0,i-1 And U i,2i-1 Two parts, and U for distinguishing characteristics of past data from future data past And U fu There is no intersection of columns of (a), the past and future inputs are of the same amount of overlapping data, i.e. in U past And U fu Can find the input u i ；
Similarly, define the output Hankel matrix Z past ,Z fu
The state sequence plays an important role in the SIM algorithm and is defined as state sequence X below i ∈R n×j ：
X i ＝(x i x i+1 … x i+j-2 x i+j-1 ) (10)
Wherein,
definition matrix E i The following are provided:
wherein,
s24, performing oblique projection on the (4) to obtain:
at the same time, the method comprises the steps of,the expression is as follows:
wherein,is->Orthogonal complement matrix of (2), symbol->Representing the pseudo-inverse of the matrix;
s25, solving E through singular value decomposition technology i Is a characteristic value of (a):
wherein the weight matrixZ 2 ∈R j×j And satisfies rank (Z past )＝rank(Z past Z 2 )；
S26, calculating an augmentation matrix:
combined meter (6) and (15)Calculating system matrices A and C, and updating the augmentation matrix T in real time i ；
S27, defining matrixes L, K, P and Q:
s28, calculating N through (17) - (20) i ：
S29, combining (18), and solving a system matrix
S3, model prediction of state space recursion;
preferably, the step S3 specifically includes the following steps:
s31, assume w 1 (l+i)＝0,v 1 (l+i) =0, i=1, 2,3,4 … n, recursively the equation of state:
and similarly, recursion is carried out on a system observation equation;
the following expression is obtained by arrangement:
wherein n is the length of the system for prediction, and the output vector and the input vector of the system are respectively represented by the following formulas:
Y(l)＝[y(l) y(l+1) … y(l+m-1)] (25)
U(l)＝[u(l) u(l+1) … u(l+m-1)] (26)；
s32, U when the length of the Hankel matrix is m past And Y is equal to past Is known to U fu And Y is equal to fu Is calculated by the equations (25) and (26), assuming that the range of predicted data for the future l time instant is [ l, l+P+m-2 ]]Then the input/output data of the system is includedThe definition is as follows:
s33, in the formula (27)And performing LQ decomposition to obtain image space representation based on data driving, thereby obtaining Gap measurement and system margin under the data driving.
S4, realizing a kernel space and an image space based on data driving;
s41, introducing definition of a kernel space and an image space of data driving;
preferably, the step S41 specifically includes the following steps:
s411, defining the following signals:
s412, defining one: assume that (1) satisfies w 1 (k)＝0，v 1 (k)＝0，When->Sample m is such that:
then the matrixIs a data-driven implementation of the kernel space, when +.>Satisfy->When it is calledIs standardized and is marked as->
S413, defining two steps: assuming that formula (1) satisfies and defines the same conditions, whenSample m is such that:
at this time, the matrixIs a data-driven implementation like space, if +.>Then call->Is a data-driven implementation of standardized image space, denoted +.>
S42, calculating stability margins under a kernel space and an image space;
preferably, the step S42 specifically includes the following steps:
s421, consider an LTI system G form as shown in formula (1), define three: when the formula (1) respectively satisfiesWhen in use, then->And is->Nuclear space and image space, respectively, standardized for the system, in order not to be confused +.>And->A kernel space representation and an image space representation in a standardized form;
s422, according to the mutual mass decomposition technique, the system standardized kernel space representationAnd image space representation +.>The expression is as follows:
s423, defining four: the system G shown in the formula (1) and the corresponding SKR are given and shapedThe stability margin is calculated by:
wherein I H A Hankel norm of formula (1);
when the closed loop control circuit finds a stable controller H (z), the following holds:
s43, realizing data driving under an open loop control system;
preferably, the step S43 specifically includes the following steps:
s431, decomposing the defined input and output data by using an LQ technology to obtain the following formula:
wherein L is 11 、L 21 、L 32 The quadratic matrix is represented by a real symmetric matrix;
s432, theorem one: given LTI system as shown in formula (1), measurement input and output data of the system are given, and LQ decomposition as shown in formula (33) is performed on the system, so that the kernel space under data driving is as follows:
wherein the method comprises the steps of
In the method, in the process of the invention,representing a pseudo-inverse;
further standardized is obtained
Wherein V is 1 Is obtained by singular value decomposition as follows:
s433, theorem II: similarly, LQ decomposition is performed on input and output data, and then the obtained image space based on data driving is:
wherein,
further standardized is obtained
S5, solving Gap measurement and stability margin under the data;
s51, realizing data driving of Gap measurement;
preferably, the step S51 specifically includes the following steps:
s511, theorem three: given two systems G 1 ，G 2 The state space expressions of both are shown as formula (1), then the following holds when going to infinity for m:
and (3) proving: when m tends to infinity, i.e. τ m Trend I, at τ m G tends to G, z i,m Also tend to z i The Gap metric at this time is defined in the most primitive hilbert space, i.e., equation (42) holds;
furthermore, the directed Gap metric δ d,m The data driving implementation of (1) is obtained by calculating the singular values of the following formula:
/>
where the symbol sigma is a singular value of the system,and->Represents G 1 ,G 2 Corresponding to the image space under the standardized condition;
s512, under the precondition of theorem three, it is assumed that the system G exists 1 And G 2 The Gap metric between the two passes through the corresponding graph spaceDirectional Gap to calculate, get:
s513, defining four: the truncating operator τ is defined as follows m ：
Wherein,the meaning of the above formula means that the time signal t is switched off after m+1 sampling times, and therefore, two systems G (i) ，G (j) The Gap metric under the i, j=1, 2 data driven framework is defined as:
system G 1 And G 2 The directional Gap metric in between can also be calculated by:
wherein z is 1,m ＝τ m z 1 ，z 2,m ＝τ m z 2 。
S52, a data driving implementation method of the system stability margin;
preferably, the step S52 specifically includes the following steps:
s521, setting the kernel space as a residual error generator of the system, and giving a Hankel norm of the standardized kernel space:
S522definition five: optimal stability margin for a systemThe data driving implementation process is as follows:
wherein,is defined as:
s523, similarly analogizing the truncating operator τ m Definition of kappa m Expressed as:
s524, theorem four: the system G represented by formula (1) is given, wherein w 1 (l)＝0,v 1 (l)＝0,When m tends to infinity, then there are:
calculated using the following formula:
wherein,i.e. < ->Is->Is a sub-matrix of the matrix. />
S6, a simulation experiment;
s61, verifying consistency of data driving and model results;
preferably, the following two discretized LTI systems G are considered in step S61 1 ,G 2 The formula is as follows:
assuming that the input and output data pair of n=3000 is obtained from the system and the sampling period is set to t=100 ms, different matrix lengths of the input and output data Hankel are set given the sufficient input excitation of the system, and the noiseless and noisy cases are considered. Assuming that the process noise satisfies w (k) to N [0,0.3], the observed noise satisfies v (k) to N [0,0.1]. The Gap metric and stability margin calculation results under the data driving framework are shown in table 1:
table 1 Gap metrics and under data driven frameworkIs calculated by the computer
As is evident from the data in table 1, the Gap metric and stability margin achieved under the data driven framework are equivalent to the results under the model as the length of m is continually increased. This shows that it is meaningful to replace the metrics of the model with those of the data, i.e., the model-driven fault diagnosis can be replaced by the variation of the Gap metrics and the system stability margin under the data-driven framework.
In order to verify that the Gap metric value under the data driving framework can still be close to the result of the Gap metric value of the model under the condition that the system model parameters are changed, consider G 2 The parameter in (z) is changed to G i (z), wherein i=3, 4,5,6:
at this time, the measurement value of the model is shifted, the length m of the Hankel matrix is set to be 500, gap measurement values based on data driving are calculated respectively, and the calculation results are shown in Table 2;
table 2 shows a Gap metric value table based on data driving under the condition of changing model parameters
As can be seen from Table 2, when the model parameters are changed, the data-driven Gap metric values are changed accordingly, and the Gap metric values of the model can be approximated. Therefore, the experimental result shows that the model is the abstract representation of the data change, and the data is the online implementation of the abstract model.
S62, verifying on-line fault diagnosis based on Gap measurement under data driving.
In step S62, two system models of equation (1) continue to be employed, assuming that the process noise satisfies w 1 (l)～N[0,0.3]The observation noise satisfies v 1 (l)～N[0,0.1]The sampling time point is set to 3000 in the present embodiment, and G is measured from the nth sampling time 2 And (z) predicting a state space model by using a system matrix identified by subspace, wherein n=1350, driving a Gap metric value by using the data of each moment after the solved n sampling points, and comparing the Gap metric value with a Gap metric value of a nominal system to further achieve the purpose of real-time fault diagnosis. As can be seen from fig. 2, with the increase of the sampling time, the online fault diagnosis can be performed on the system according to the difference between the data driving Gap metric value and the Gap metric value of the model.
S63, obtaining an experimental conclusion
In step S63, for the nominal system and the fault system, the Gap metric is calculated by the obtained data, and from the simulation result, the Gap metric value under the data driving frame approximates the correlation value based on the model, in addition, the new system matrix is updated in real time by subspace identification, and then the Gap metric between the system in monitoring operation and each time of the nominal system is calculated by using the data driving Gap metric calculation method, so as to obtain the data driving Gap metric value of the system at each time, and the time deviated from the normal value of the system can be regarded as the fault by comparing with the reference value of the Gap metric of the nominal system model.
Therefore, the fault detection and prediction control method based on the data driving is inspired by a big data technology, and in the framework of the data driving, the operation information is directly extracted from a large amount of off-line and on-line data. By means of statistical analysis techniques, appropriate test statistics are established for indicating the occurrence of faults. The overall offset of the model is also measured by a data-driven method, so that the overall offset of the model-based system is replaced, and the problems of on-line identification and difficult model establishment are overcome.
Finally, it should be noted that: the above embodiments are only for illustrating the technical solution of the present invention and not for limiting it, and although the present invention has been described in detail with reference to the preferred embodiments, it will be understood by those skilled in the art that: the technical scheme of the invention can be modified or replaced by the same, and the modified technical scheme cannot deviate from the spirit and scope of the technical scheme of the invention.
Claims (2)
1. A fault detection and prediction control method based on data driving is characterized in that: the method comprises the following steps:
s1, building a Hankel matrix by constructing input and output data, so as to build a state space model of the system;
s2, identifying a linear time-invariant system;
s3, model prediction of state space recursion;
s4, realizing a kernel space and an image space based on data driving;
s41, introducing definition of a kernel space and an image space of data driving;
s42, calculating stability margins under a kernel space and an image space;
s43, realizing data driving under an open loop control system;
s5, solving Gap measurement and stability margin under the data;
s51, realizing data driving of Gap measurement;
s52, a data driving implementation method of the system stability margin;
s6, a simulation experiment;
s61, verifying consistency of data driving and model results;
s62, verifying on-line fault diagnosis based on Gap measurement under data driving;
s63, obtaining an experimental conclusion;
the step S1 specifically comprises the following steps:
a state space model of a linear time-invariant system is given as follows:
in the above-mentioned method, the step of,for the corresponding system matrix->x(k)∈R n Y (k) εR, which is a system state vector m An observation vector that is a state; w (w) 1 (l) And v 1 (l) Process noise and observation noise of gaussian type respectively;
and assuming that the system satisfies the following conditions:
1) The formula (1) satisfies the generalized energy control matrix and the generalized energy observation matrix full rank;
2) The characteristic value of A is positioned in the left half plane, and the condition of gradual stability of the system is met;
3) Covariance matrix of Hankel matrix formed by input and output data is satisfied
rank(Ω[U 0|2i-1 ,U 0|2i-1 ])＝2mi；
4) The input sequence is uncorrelated with the noise sequence;
the step S2 specifically comprises the following steps:
s21, decomposing the formula (1) into a determination part and a random part:
X＝X d +X s ，Y＝Y d +Y s (2)；
through iteration, obtain
Wherein the matrix is augmented
S22, determining a system Toeplitz matrix:
s23, defining a Hankel matrix composed of input and output:
where i is a positive integer, matrix U past Representing past inputs, the subscript past represents past data, U fu Representing future inputs, subscript fu representing future data, U past And U fu Hankel matrix U 0,2i-1 Divided into U 0,i-1 And U i,2i-1 Two parts, and U for distinguishing characteristics of past data from future data past And U fu There is no intersection of columns of (a), the past and future inputs are of the same amount of overlapping data, i.e. in U past And U fu Can find the input u i ；
Similarly, define the output Hankel matrix Z past ,Z fu
A state sequence X defined as follows i ∈R n×j ：
Wherein,
definition matrix E i The following are provided:
wherein,
s24, performing oblique projection on the (4) to obtain:
at the same time, the method comprises the steps of,the expression is as follows:
wherein,is->Orthogonal complement matrix of (2), symbol->Representing the pseudo-inverse of the matrix;
s25, solving E through singular value decomposition technology i Is a characteristic value of (a):
wherein the weight matrixAnd satisfies rank (Z) past )＝rank(Z past Z 2 )；
S26, calculating an augmentation matrix:
calculating system matrices A and C by combining (6) and (15), and updating the augmentation matrix T in real time i ；
S27, defining matrixes L, K, P and Q:
s28, calculating N through (17) - (20) i ：
S29, combining (18), and solving a system matrix
The step S3 specifically comprises the following steps:
s31, assume w 1 (l+i)＝0,v 1 (l+i) =0, i=1, 2,3,4 … n, recursively the equation of state:
and similarly, recursion is carried out on a system observation equation;
the following expression is obtained by arrangement:
wherein n is the length of the system for prediction, and the output vector and the input vector of the system are respectively represented by the following formulas:
s32, U when the length of the Hankel matrix is m past And Y is equal to past Is known to U fu And Y is equal to fu Is calculated by the equations (25) and (26), assuming that the range of predicted data for the future l time instant is [ l, l+P+m-2 ]]Then the input/output data of the system is includedThe definition is as follows:
s33, in the formula (27)Performing LQ decomposition to obtain image space representation based on data driving, so as to obtain Gap measurement and system margin under the data driving;
the step S41 specifically includes the following steps:
s411, defining the following signals:
s412, defining one: assume that (1) satisfies w 1 (k)＝0，v 1 (k)＝0，When->Sample m is such that:
then the matrixIs the coreSpatially data-driven implementation, when +.>Satisfy->When in use, we call->Is standardized and is marked as->
S413, defining two steps: assuming that formula (1) satisfies and defines the same conditions, when Sample m is such that:
at this time, the matrixIs a data-driven implementation like space, if +.>Then call->Is a data-driven implementation of standardized image space, denoted +.>
The step S42 specifically includes the following steps:
s421, consider an LTI system G form as shown in formula (1), define three: when the formula (1) respectively satisfiesWhen in use, then->And is->Nuclear space and image space, respectively, standardized for the system, in order not to be confused +.>And->A kernel space representation and an image space representation in a standardized form;
s422, according to the mutual mass decomposition technique, the system standardized kernel space representationAnd image space representation +.>The expression is as follows:
s423, defining four: the system G shown in the formula (1) and the corresponding SKR are given and shapedThe stability margin is calculated by:
wherein I H A Hankel norm of formula (1);
when the closed loop control circuit finds a stable controller H (z), the following holds:
the step S43 specifically includes the following steps:
s431, decomposing the defined input and output data by using an LQ technology to obtain the following formula:
wherein L is 11 、L 21 、L 32 The quadratic matrix is represented by a real symmetric matrix;
s432, theorem one: given LTI system as shown in formula (1), measurement input and output data of the system are given, and LQ decomposition as shown in formula (33) is performed on the system, so that the kernel space under data driving is as follows:
wherein the method comprises the steps of
In the method, in the process of the invention,representing a pseudo-inverse;
further obtain the markNormalization of
Wherein V is 1 Is obtained by singular value decomposition as follows:
s433, theorem II: similarly, LQ decomposition is performed on input and output data, and then the obtained image space based on data driving is:
wherein,
further standardized is obtained
The step S51 specifically includes the following steps:
s511, theorem three: given two systems G 1 ，G 2 The state space expressions of both are shown as formula (1), then the following holds when going to infinity for m:
and (3) proving: when m tends to infinity, i.e. τ m Trend I, at τ m G tends to G, z i,m Also tend to z i The Gap metric at this time is defined in the most primitive hilbert space, i.e., equation (42) holds;
furthermore, the directed Gap metric δ d,m The data driving implementation of (1) is obtained by calculating the singular values of the following formula:
where the symbol sigma is a singular value of the system,and->Represents G 1 ,G 2 Corresponding to the image space under the standardized condition;
s512, under the precondition of theorem three, it is assumed that the system G exists 1 And G 2 The Gap metric between the two is calculated through the directed Gap of the corresponding graph space, and the result is that:
s513, defining four: the truncating operator τ is defined as follows m ：
Wherein,the meaning of the above formula means that the time signal t is switched off after m+1 sampling times, and therefore, two systems G (i) ，G (j) The Gap metric under the i, j=1, 2 data driven framework is defined as:
system G 1 And G 2 The directional Gap metric in between can also be calculated by:
wherein z is 1,m ＝τ m z 1 ，z 2,m ＝τ m z 2 ；
Step S52 specifically includes the following steps:
s521, setting the kernel space as a residual error generator of the system, and giving a Hankel norm of the standardized kernel space:
s522, defining five: optimal stability margin for a systemThe data driving implementation process is as follows:
wherein,is defined as:
s523, similarly analogizing the truncating operator τ m Definition of kappa m Expressed as:
s524, theorem four: the system G represented by formula (1) is given, wherein w 1 (l)＝0,v 1 (l)＝0,When m tends to infinity, then there are:
calculated using the following formula:
wherein,i.e. < ->Is->Is a sub-matrix of the matrix.
2. The data-driven based fault detection and prediction as claimed in claim 1The control method is characterized in that: in step S61, the following two discretized LTI systems G are considered 1 ,G 2 The formula is as follows:
in order to verify that the Gap metric value under the data driving framework can still be close to the result of the Gap metric value of the model under the condition that the system model parameters are changed, consider G 2 The parameter in (z) is changed to G i (z), wherein i=3, 4,5,6:
at the moment, the measurement value of the model is deviated, the length m of the Hankel matrix is set, and Gap measurement values based on data driving are calculated respectively;
in step S62, two system models of equation (1) continue to be employed, assuming that the process noise satisfies w 1 (l)～N[0,0.3]The observation noise satisfies v 1 (l)～N[0,0.1]Setting sampling time point, and comparing G from nth sampling time 2 (z) predicting a state space model by using a system matrix identified by subspace, driving a Gap metric value by using the solved data of each moment after n sampling points, and comparing the Gap metric value with a Gap metric value of a nominal system to further achieve the purpose of real-time fault diagnosis;
in step S63, for the nominal system and the fault system, the Gap metric is calculated by the obtained data, and from the simulation result, the Gap metric value under the data driving frame approximates the correlation value based on the model, in addition, the new system matrix is updated in real time by subspace identification, and then the Gap metric between the system in monitoring operation and each time of the nominal system is calculated by using the data driving Gap metric calculation method, so as to obtain the data driving Gap metric value of the system at each time, and the time deviated from the normal value of the system can be regarded as the fault by comparing with the reference value of the Gap metric of the nominal system model.
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