CN113065465B - Structure state/parameter/load joint identification method based on extended GDF - Google Patents
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Abstract
The invention discloses a structure state/parameter/load joint identification method based on an extended GDF, which comprises the following steps of 1, introducing modal coordinate transformation, constructing an extended state vector containing a structure state and an uncertainty structure parameter, and establishing a state transfer equation and an observation equation of a linear system; step 2, establishing a modal state transfer equation and an observation equation of a linear system time discretization containing process noise; step 3, establishing an extended GDF filter for fusing the strain response and the acceleration response; step 4, identifying a structure augmentation state and an unknown load through an extended GDF filter integrating a strain response and an acceleration response; step 5, recognizing the modal state as a structural state in the physical space. The invention can carry out the combined recognition of the structural state/parameter/load only by individual measurement signals; the strain response and acceleration response fusion strategy is simple and convenient to operate in practical application, the false low-frequency drift problem of displacement and load can be effectively solved, and the recognition accuracy is improved.
Description
Technical Field
The invention belongs to the technical field of structural health monitoring, and particularly relates to a structural state/parameter/load joint identification method based on an extended GDF.
Background
The external load on the structure plays a very important role in structural optimization design, fault diagnosis, health monitoring and the like. Because the external load is difficult to directly measure through an instrument, a plurality of deterministic load inverse methods are developed, namely, load identification is carried out according to system characteristics by adopting easily-measured structure dynamic response (such as displacement, speed, acceleration or strain signals), and the second type of inverse problem of structure dynamics is also called. In engineering practice, however, the structural model parameters used to implement the load back-finding are often also unknown or ambiguous, which tends to result in the load back-finding result being unreliable, while the parameter identification is also known as the first-class inverse problem of structural dynamics.
In addition, in the technical field of structural health monitoring, it is often required to obtain structural dynamic responses at all important positions, however, due to limitations of factors such as monitoring equipment, the number of sensors, arrangement and the like, real-time responses at all positions cannot be mastered. Therefore, the combined identification of the structural state (displacement, speed)/parameter/load by adopting a small amount of measurement signals has important practical significance.
Disclosure of Invention
The invention aims to provide a structural state/parameter/load combined identification method based on an extended GDF, which solves the inherent defects of displacement and load false low-frequency drift phenomenon in the identification method in the prior art, improves implementation operability and improves calculation precision and efficiency.
In order to solve the technical problems, the invention adopts the following technical scheme:
the structure state/parameter/load joint identification method based on the extended GDF comprises the following steps:
step 1, introducing modal coordinate transformation, constructing an augmented state vector containing structural states and uncertainty structural parameters, and establishing a state transfer equation and an observation equation of a linear system;
step 2: establishing a modal state transfer equation and an observation equation of a linear system time discretization containing process noise;
step 3, setting an initial value and a variance value of an augmentation state vector, and establishing an extended GDF filter for fusing strain response and acceleration response based on a first-order linearization idea of extended Kalman filtering;
step 4, identifying a structure augmentation state and an unknown load through an extended GDF filter integrating the strain response and the acceleration response according to the structure dynamic acceleration response and the strain response which are measured in real time;
and 5, recognizing the modal state as a structural state in the physical space.
Further optimizing, in the step 1, introducing a modal coordinate transformation p (t) =Φq (t), wherein Φ is a modal shape matrix, q (t) is a node displacement vector with respect to time t, and p (t) is a modal displacement vector with respect to time t;
building an augmented state vector containing structural states and uncertainty structural parametersWherein the structural state comprises displacement and velocity, alpha= [ alpha ] 1 α 2 … α α ] T An uncertainty parameter representing a structure;
and establishing a state transfer equation and an observation equation of the linear system, wherein the state transfer equation and the observation equation are shown in the following formula:
wherein,is a modal velocity vector with respect to time t, +.>For modal acceleration vector with respect to time t, Λ is the normalized modal frequency matrix, Γ is the modal damping matrix, u (t) is the external load excitation with respect to time t, B u Is the position influence matrix of the external load vector, y (t) represents the acceleration measurement response with respect to time t, h= [ -H 0 ΦΛ -H 0 ΦΓ]，D＝H 0 ΦΦ T B u ，H 0 A position influence matrix representing acceleration measurement signals for system identification, the matrix D being a reversible matrix, the superscript "T" representing a transpose of the matrix or vector;
in the step 2, a modal state transfer equation and an observation equation of a linear system time discretization containing process noise are established, wherein the equation is as follows:
z k+1 ＝f k (z k ,u k )+w k ；k＝1,2…N；
y k ＝h k (z k )+D k u k +v k ；k＝1,2…N；
wherein the subscript k represents the kth sampling time, N is a positive integer, z k An augmented state vector representing the kth sample time, u k Represents the external load excitation at the kth sampling instant, y k Representing acceleration measurement response, w, at the kth sample time k The mean and variance of the system noise representing the kth sampling time are set to 0 and G, respectively k The mean and variance of which are assumed to be 0 and G, respectively k ；v k The mean and variance of the observation noise representing the kth sampling time are set to 0 and R, respectively k ；f(z k ,u k ) Representing the vector z in the modal state transfer equation k 、u k Is a nonlinear function of h (z k ) Representing the vector z in the observation equation k Is a non-linear function of (2).
Further preferably, in the step 3, the step of establishing an extended GDF filter that fuses the strain response and the acceleration response includes the following four steps:
step S301: definition vector z k|k 、Respectively the true value z k 、u k In the observation vector (y) 0 ,y 1 ,y 2 ,…,y k ) The state variance matrix is assumed to be +.>Initial value z of given augmented state vector 1|0 Sum of variance value->
Step S302, load identification: the method comprises the following formula:
wherein,variance matrix for load at kth sampling instant, +.>For the sensitivity matrix at the kth sampling instant, D k For the load position influence matrix at the kth sampling instant, R εk Strain observation noise vector v for the kth sample time εk Is calculated as follows:
；
wherein the subscript l represents the amount of measured strain and the subscript udof represents the amount of unknown load;
step S303 measures the update step: the method comprises the following formula:
wherein,variance matrix for the augmentation state at the kth sampling instant, +.>Covariance matrix of the amplified state and load at the kth sampling time, H ε A strain-displacement transfer matrix for the kth sampling moment;
step S304, a time updating step: the method comprises the following formula:
wherein,for the sensitivity matrix, the following formula is calculated:
wherein,
B c ＝[[0] Φ T B u [0]] T 。
wherein f is based on mathematical writing specifications k And A is a f The following table f is the same.
Further preferably, in the step 4, the structure augmentation state vector { z ] is identified by an extended GDF filter fusing the strain response and the acceleration response according to the structure dynamic acceleration response and the strain response measured in real time 1|1 ,…,z k|k ,…,z N|N Sum of unknown loadThe structure augmentation state vector contains modal state and uncertainty structural parameters; vector z k|k Is the true value z k In the observation vector (y) 0 ,y 1 ,y 2 ,…,y k ) The posterior estimate below; />Is the true value u k In the observation vector (y) 0 ,y 1 ,y 2 ,…,y k ) The posterior estimate below;
further preferably, in the step 5, the transformation p is transformed according to the modal coordinate k|k ＝Φq k|k The modal state is identified as a structural state in physical space.
Compared with the prior art, the invention has the following beneficial effects:
1. the invention can perform the combined recognition of the structural state/parameter/load only by individual measurement signals.
2. The strain response and acceleration response fusion strategy is simple and convenient to operate in practical application, the false low-frequency drift problem of displacement and load can be effectively solved, and the recognition accuracy is improved.
3. The application of the modal subtraction method improves the calculation efficiency and is suitable for engineering practical application.
Drawings
FIG. 1 is a flow chart of a method for combined structural state/parameter/load identification based on an extended GDF filter and a data fusion strategy according to the present invention;
fig. 2 is a schematic view of a truss according to an embodiment of the present invention: wherein, fig. 2 (a) is a schematic view of a planar truss structure, and fig. 2 (b) is a schematic view of a truss finite element model and a sensor arrangement;
FIG. 3 shows the expanded GDF filter versus load u using step 3 of comparative example 1 in accordance with the present invention 1 The recognition result of (t); wherein the load u of FIG. 3 (a) 1 FIG. 3 (b) is a partial enlarged view of 10-10.5s in FIG. 3 (a);
FIG. 4 shows the load u of the invention using the step 3 extended GDF filter of comparative example 1 2 Is a result of the recognition of (a); wherein FIG. 4 (a) is a graph of the load u 2 FIG. 4 (b) is a partial enlarged view of 19-19.1s in FIG. 4 (a) showing the recognition result of (t);
FIG. 5 is a graph showing theoretical and identification values of the vertical displacement of node 8 obtained by expanding the GDF filter in step 3 of comparative example 1; fig. 5 (a) is a theoretical value and an identification value of the vertical displacement of the node 8, and fig. 5 (b) is a theoretical value and an identification value of the vertical velocity of the node 8;
FIG. 6 shows the load u of an extended GDF filter of the present invention employing the fusion of the strain response and the acceleration response of step 3 of example 1 1 The recognition result of (t); FIG. 6 (a) is a load u 1 FIG. 6 (b) is a partial enlarged view of 10-10.5s in FIG. 6 (a);
FIG. 7 is a graph showing the loading of the extended GDF filter of the present invention using the fusion of strain response and acceleration response of step 3 of example 1Lotus u 2 (t) identifying the result; FIG. 7 (a) shows the load u 2 FIG. 7 (b) is a partial enlarged view of 19-19.1s in FIG. 7 (a);
FIG. 8 is a theoretical and identification value of the vertical displacement of node 8 obtained by using the extended GDF filter of step 3 of example 1 to fuse the strain response and acceleration response; fig. 8 (a) shows the theoretical value and the identification value of the vertical displacement of the node 8, and fig. 8 (b) shows the theoretical value and the identification value of the vertical velocity of the node 8.
Detailed Description
In order to make the objects and technical solutions of the present invention more clear, the technical solutions of the present invention will be clearly and completely described below in connection with the embodiments of the present invention.
Example 1:
as shown in fig. 1, a method for identifying structural state/parameter/load combination based on extended GDF includes the following steps:
step 1, introducing a modal coordinate transformation p (t) =Φq (t), wherein Φ is a modal shape matrix, q (t) is a node displacement vector with respect to time t, and p (t) is a modal displacement vector with respect to time t;
building an augmented state vector containing structural states and uncertainty structural parametersWherein the structural state comprises displacement and velocity, alpha= [ alpha ] 1 α 2 … α α ] T An uncertainty parameter representing a structure;
and establishing a state transfer equation and an observation equation of the linear system, wherein the state transfer equation and the observation equation are shown in the following formula:
wherein,is a modal velocity vector with respect to time t, +.>For modal acceleration vector with respect to time t, Λ is the normalized modal frequency matrix, Γ is the modal damping matrix, u (t) is the external load excitation with respect to time t, B u Is the position influence matrix of the external load vector, y (t) represents the acceleration measurement response with respect to time t, h= [ -H 0 ΦΛ -H 0 ΦΓ]，D＝H 0 ΦΦ T B u ，H 0 A position influence matrix representing acceleration measurement signals for system identification, the matrix D being a reversible matrix, the superscript "T" representing a transpose of the matrix or vector;
step 2, establishing a modal state transfer equation and an observation equation of a linear system time discretization containing process noise, wherein the modal state transfer equation and the observation equation are shown in the following formula:
z k+1 ＝f k (z k ,u k )+w k ；k＝1,2…N；
y k ＝h k (z k )+D k u k +v k ；k＝1,2…N；
wherein the subscript k represents the kth sampling time, N is a positive integer, z k An augmented state vector representing the kth sample time, u k Represents the external load excitation at the kth sampling instant, y k Representing acceleration measurement response, w, at the kth sample time k The mean and variance of the system noise representing the kth sampling time are set to 0 and G, respectively k The mean and variance of which are assumed to be 0 and G, respectively k ；v k The mean and variance of the observation noise representing the kth sampling time are set to 0 and R, respectively k ；f(z k ,u k ) Representing the vector z in the modal state transfer equation k 、u k Is a nonlinear function of h (z k ) Representing the vector z in the observation equation k Is a nonlinear function of (2);
step 3, establishing an extended GDF filter for fusing strain response and acceleration response: the method comprises the following four steps:
step S301: vector z k|k Is the true value z k In the observation vector (y) 0 ,y 1 ,y 2 ,…,y k ) The posterior estimate below;is the true value u k In the observation vector (y) 0 ,y 1 ,y 2 ,…,y k ) The posterior estimate below;
the state variance matrix is assumed to beInitial value z of given augmented state vector 1|0 Sum of variance value->
Step S302, load identification: the method comprises the following formula:
wherein,variance matrix for load at kth sampling instant, +.>For the sensitivity matrix at the kth sampling instant, D k For the load position influence matrix at the kth sampling instant, R εk Strain observation noise vector v for the kth sample time εk Is calculated as follows:
；
wherein the subscript l represents the amount of measured strain and the subscript udof represents the amount of unknown load;
step S303 measures the update step: the method comprises the following formula:
wherein,variance matrix for the augmentation state at the kth sampling instant, +.>For the covariance matrix of the load and the augmentation state at the kth sampling instant,H ε a strain-displacement transfer matrix for the kth sampling moment;
step S304, a time updating step: the method comprises the following formula:
wherein,for the sensitivity matrix, the following formula is calculated:
wherein,
B c ＝[[0] Φ T B u [0]] T 。
step 4, identifying a structure augmentation state vector { z ] by an extended GDF filter fusing the strain response and the acceleration response according to the structure dynamic acceleration response and the strain response measured in real time 1|1 ,…,z k|k ,…,z N|N Sum of unknown loadThe structure augmentation state vector contains modal state and uncertainty structural parameters;
step 5, transforming p according to the modal coordinate k|k ＝Φq k|k Identifying modal states as knots in physical spaceA structure state.
Comparative example 1:
comparative example 1 is prior art, and differs from the present invention in step 3, and the other steps are the same as in example 1.
Step 3 of comparative example 1: given the initial value and variance value of the augmented state vector, an extended GDF filter is built comprising the following four steps:
step S301 (initialization): definition vector z k|k 、Respectively the true value z k 、u k In the observation vector (y) 0 ,y 1 ,y 2 ,…,y k ) The state variance matrix is assumed to be +.>Initial value z of given augmented state vector 1|0 Sum of variance value->
Step S302 (load recognition step): the method comprises the following formula:
wherein,for variance matrix of load, +.>For the sensitivity matrix, the following formula is calculated:
step S303 (measurement update step): the method comprises the following formula:
wherein,variance matrix for augmented state, +.>Is the covariance matrix of the augmented state and load.
Step S304 (time update step): the method comprises the following formula:
wherein,for the sensitivity matrix, the following formula is calculated:
wherein,
B c ＝[[0] Φ T B u [0]] T 。
the following example analyses were performed for example 1 and comparative example 1:
as shown in FIG. 2, the object is a planar truss comprising 31 rod units, each rod having a uniform cross-sectional dimension, the length of the rod units disposed horizontally is 2m, and the length of the rod units disposed 45 ° to each other isThe common structural parameters for all bars are as follows: the cross-sectional area of the rod unit was 8.95×10 -5 m 2 The modulus of elasticity is 2X 10 7 Pa, density is 7.85X10 3 kg/m 3 . Each rod unit in this example adopts a centralized mass unit, and is composed of two nodes, each node contains 2 degrees of freedom in the transverse and longitudinal directions, and nodes 1 and 17 are fixed constraints. In addition, the structural system damping is assumed to be proportional c=γm+βk, and the damping coefficients are γ=0.1523, β= 4.6203 ×10, respectively -4 . The two external loads respectively act on the node 4 and the node 9 and are vertical force and load u 1 In the form of double sinusoidal excitation
u 1 (t)＝40sin(10πt)+30sin(20πt)；
And load u 2 (t) taking the form of random excitation.
The black squares in fig. 2 (b) represent the positions of the acceleration sensor arrangements, 6 acceleration sensor arrangements being arranged at the nodes of 2, 3, 5, 7, 8 and 10, respectively. The stiffness values of the 6 rod units 5, 7, 10, 14, 15 and 17 are indeterminate, requiring joint identification with external loads, assuming initial values of 759.5, 633.0, 1342.5, 1163.5, 759.5, 633.0N/m, respectively. The first 7 th order dominant modality is employed for load/state/parameter identification of the structure. Firstly, 7 acceleration measurement signals are selected to participate in the recognition calculation, namely a vertical acceleration response signal of the node 3, 4, 5, 7, 9, 10 and a horizontal acceleration response signal of the node 9. 5% of the ambient noise was added to all measured responses.
Two external loads u were obtained using the extended GDF filter of step 3 of comparative example 1 1 (t)、u 2 As shown in FIGS. 3-4, the identification results of (t) show that the load identification values have obvious low-frequency drift phenomenon, especially the load u 1 The relative error of (t) has reached 29.3%.
In addition, the extended GDF filter of step 3 in comparative example 1 is used to identify the state values of all nodes, as shown in fig. 5, which is a graph of theoretical values of vertical displacement and speed of node 8 versus identification values, the graph shows that the result of the speed identification value is very good, the relative error is small, but the displacement identification value obviously follows the load identification value, and false low-frequency drift phenomenon also occurs.
The strain responses of the individual units were measured using the extended GDF filter of example 1, which fuses the strain response and acceleration response of step 3, and the structural state unknown parameters and loads were identified in combination with the acceleration measurement signals described above, where the two newly added measurement signals were the vertical strain responses of units 6, 17, respectively. As shown in FIGS. 6-8, the actual value and the identification value of the external load, displacement and speed are respectively compared, the graph of the identification result is basically overlapped with the graph of the actual value, and the data fusion of the invention can be seenThe method can greatly relieve the false low-frequency drift problem of load and displacement identification, wherein the load u 1 (t) the relative error in the identification value has been reduced to 4.75%. Wherein the data in fig. 3, 4, 6, 7 are relatively dense, each of which is shown exaggerated in one section for clarity of illustration and ease of viewing.
In addition, as shown in table 1, the identification values of the 6 uncertain rigidities of the plane truss are very accurate, and the error is very small.
Table 1: identification value of 6 uncertain rigidities of plane truss
The acceleration sensor is small in size, easy to install and small in influence on the characteristics of a structural system, so that the acceleration sensor is widely applied to engineering practice to measure the vibration response of the structure, as described in step 3 of comparative example 1. However, using only acceleration measurement signals to identify the structural system GDF filters has intrinsic instability, and the identified displacement and load values can produce significant false low frequency drift phenomena. The reason for this is because the acceleration signal is not sensitive enough to the quasi-static component of the input load, resulting in a lack of partial low frequency information in the recognition result. Considering that the strain gauge is small in size, easy to install and low in price, the measurement response also comprises low-frequency displacement response information, and the invention adopts a fused strain response and acceleration response strategy to improve and expand the GDF filter. From the results of the above embodiments, it can be seen that the present invention can realize the joint identification of the state/parameter/load of the structure only by using individual strain and acceleration response measurement signals, which is simple and convenient to operate, and can effectively solve the problem of false low-frequency drift of displacement and load, thereby improving the identification precision; the application of the modal subtraction method improves the calculation efficiency, and is suitable for popularization in engineering application.
The present invention is not specifically described in the prior art or may be implemented by the prior art, and the specific embodiments described in the present invention are only preferred embodiments of the present invention, and are not intended to limit the scope of the present invention. Equivalent changes and modifications of the invention are intended to be within the scope of the present invention.
Claims (5)
1. The structure state/parameter/load joint identification method based on the extended GDF is characterized by comprising the following steps:
step 1, introducing modal coordinate transformation, constructing an augmented state vector containing structural states and uncertainty structural parameters, and establishing a state transfer equation and an observation equation of a linear system;
step 2: establishing a modal state transfer equation and an observation equation of a linear system time discretization containing process noise;
step 3, setting an initial value and a variance value of an augmentation state vector, and establishing an extended GDF filter for fusing strain response and acceleration response based on a first-order linearization idea of extended Kalman filtering;
the extended GDF filter that establishes the fusion strain response and acceleration response includes the following four steps:
step S301: definition vector z k|k 、Respectively the true value z k 、u k In the observation vector (y) 0 ,y 1 ,y 2 ,…,y k ) The state variance matrix is assumed to be +.>Initial value z of given augmented state vector 1|0 Sum of variance value->
Step S302, load identification: the method comprises the following formula:
wherein,variance matrix for load at kth sampling instant, +.>For the sensitivity matrix at the kth sampling instant, D k For the load position influence matrix at the kth sampling instant, R εk Strain observation noise vector v for the kth sample time εk Is calculated as follows:
；
wherein the subscript l represents the amount of measured strain and the subscript udof represents the amount of unknown load;
step S303 measures the update step: the method comprises the following formula:
wherein,variance matrix for the augmentation state at the kth sampling instant, +.>Covariance matrix of the amplified state and load at the kth sampling time, H ε A strain-displacement transfer matrix for the kth sampling moment;
step S304, a time updating step: the method comprises the following formula:
wherein, for the sensitivity matrix, the following formula is calculated:
wherein,
B c ＝[[0]Φ T B u [0]] T ；
step 4, identifying a structure augmentation state and an unknown load through an extended GDF filter integrating the strain response and the acceleration response according to the structure dynamic acceleration response and the strain response which are measured in real time;
and 5, recognizing the modal state as a structural state in the physical space.
2. The extended GDF-based joint identification method of structural states/parameters/loads according to claim 1, wherein,
in the step 1, a modal coordinate transformation p (t) =Φq (t), wherein Φ is a modal shape matrix, q (t) is a node displacement vector with respect to time t, and p (t) is a modal displacement vector with respect to time t;
building an augmented state vector containing structural states and uncertainty structural parametersWherein the structural state comprises displacement and velocity, alpha= [ alpha ] 1 α 2 … α α ] T An uncertainty parameter representing a structure;
and establishing a state transfer equation and an observation equation of the linear system, wherein the state transfer equation and the observation equation are shown in the following formula:
wherein,is a modal velocity vector with respect to time t, +.>For modal acceleration vector with respect to time t, Λ is the normalized modal frequency matrix, Γ is the modal damping matrix, u (t) is the external load excitation with respect to time t, B u Is the position influence matrix of the external load vector, y (t) represents the acceleration measurement response with respect to time t, h= [ -H 0 ΦΛ -H 0 ΦΓ]，D＝H 0 ΦΦ T B u ，H 0 The position impact matrix representing the acceleration measurement signal for system identification, matrix D being a reversible matrix, the superscript "T" representing the transpose of the matrix or vector.
3. The method for identifying the structural state/parameter/load combination based on the extended GDF according to claim 2, wherein in the step 2, a modal state transfer equation and an observation equation of a linear system time discretization containing process noise are established, and the following formula is shown:
z k+1 ＝f k (z k ,u k )+w k ；k＝1,2…N；
y k ＝h k (z k )+D k u k +v k ；k＝1,2…N；
wherein the subscript k represents the kth sampling time, N is a positive integer, z k An augmented state vector representing the kth sample time, u k Represents the external load excitation at the kth sampling instant, y k Representing acceleration measurement response, w, at the kth sample time k The mean and variance of the system noise representing the kth sampling time are set to 0 and G, respectively k The mean and variance of which are assumed to be 0 and G, respectively k ；v k The mean and variance of the observation noise representing the kth sampling time are set to 0 and R, respectively k ；f(z k ,u k ) Representing the vector z in the modal state transfer equation k 、u k Is not a line of (2)Sexual function, h (z k ) Representing the vector z in the observation equation k Is a non-linear function of (2).
4. The method for combined structural state/parameter/load identification based on extended GDF according to claim 3, wherein in step 4, the structural augmented state vector { z ] is identified by an extended GDF filter that fuses the strain response and the acceleration response based on the real-time measured structural dynamic acceleration response and the strain response 1|1 ,…,z k|k ,…,z N|N Sum of unknown loadThe structure augmentation state vector contains modal state and uncertainty structural parameters; vector z k|k Is the true value z k In the observation vector (y) 0 ,y 1 ,y 2 ,…,y k ) The posterior estimate below; />Is the true value u k In the observation vector (y) 0 ,y 1 ,y 2 ,…,y k ) The following posterior estimates.
5. The method for joint identification of structural states/parameters/loads based on extended GDF according to claim 4, wherein in step 5, the p is transformed according to modal coordinates k|k ＝Φq k|k The modal state is identified as a structural state in physical space.
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