CN117518837B - Decoupling method based on parameterized model - Google Patents
Decoupling method based on parameterized model Download PDFInfo
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Abstract
The invention relates to the technical field of vibration control, in particular to a decoupling method based on a parameterized model, which is used for obtaining a system parameterized model through system identification; eliminating feed-through items of the state space through an integral operator to obtain a group of linear constraint equations; constructing a system optimization problem to obtain an optimization formula; constructing an optimization target to obtain an optimization target formula; obtaining a transformation matrix of the system by optimizing an optimization target; the method combines the optimization problem and the constraint equation to obtain the transformation matrix, can freely select a required decoupling mode, converts the problem of solving the decoupling matrix into the optimization problem, can assign an optimization target according to the engineering actual requirement, and utilizes the existing optimization algorithm to obtain the decoupling model suitable for the engineering actual, does not need to calculate the mass and rigidity matrix of the system, only needs the minimum state space of the system to realize, can be suitable for a general non-decoupling control system, and can achieve a good decoupling effect for a complex configuration parallel platform.
Description
Technical Field
The invention relates to the technical field of vibration control, and particularly provides a decoupling method based on a parameterized model.
Background
Along with the continuous deep research of the world front hot spot problems of dark hole, dark matter energy, universe origin, celestial origin and the like, the human beings put forward higher requirements on the performance of optical equipment, and the method has important application value in the future space exploration fields of space astronomical observation and the like. The effective load is the most important component of the spacecraft system, and the essence of ensuring the pointing precision and stability of the spacecraft is to ensure the pointing precision and stability of the effective load. The other parts of the spacecraft are essentially used for guaranteeing that the effective load of the spacecraft can finish tasks, such as a flywheel, a thruster, a refrigerator, a solar sailboard and the like, and vibration, such as dynamic-static unbalance and thermal tremble of the flywheel, can be additionally generated besides the respective corresponding tasks, and the vibration isolation means aiming at the common optical effective load mainly adopts a rigid structure, reduces vibration of a vibration source and actively controls the vibration by designing and applying a vibration isolation system to compensate the vibration, but the methods have great defects.
At present, the multidimensional vibration isolation of the spacecraft mainly adopts that the motion speed and displacement of a parallel mechanism due to micro vibration are generally very tiny, the nonlinear effect caused by centrifugal force, coriolis force and large-range angular motion is not obvious, and the parallel vibration isolator generally does not need to specially process nonlinear items like the general dynamics research. Because of the broadband characteristics of micro-vibration, the dynamics model of the parallel vibration isolator needs to take structural flexibility into wide consideration and is convenient to analyze in a frequency domain, the current most main modeling method of the dynamics problem is still a finite element method, but in some specific application occasions, a simplified theoretical modeling method also exists, the first six-order mode of the six-dimensional vibration isolation platform can be conveniently estimated by using the rigidity and quality matrix of the simplified model by neglecting the mass of the support legs and considering only the rigidity of the support legs, and the modeling method is the most commonly used modeling method for decoupling control of the six-axis vibration isolation platform.
In order to suppress the influence of load side disturbance on the optical system, only a feedback structure can be adopted. Loop shaping is a very practical and effective design method for a single-input single-output system, but is a design method based on an open-loop transfer function, and cannot be directly applied to a multiple-input multiple-output system (MIMO system). Although advanced control methods such as H-infinity loop forming, model matching, etc. have been proposed successively with the development of control theory, they are limited by the inherent complexity of MIMO control systems, which are far less flexible and convenient than the original loop forming method. If the model of the controlled object can be acquired more accurately and the decoupling condition is met, a designer prefers to perform model decoupling on the controlled object, so that the MIMO feedback control is disassembled into a plurality of independent single-input single-output system (SISO system) feedback control problems.
According to the mode control method for the spatial six-degree-of-freedom hydraulic motion platform, a mode matrix of the spatial six-degree-of-freedom hydraulic motion platform is adopted to convert a strong coupling physical space system into a decoupled mode space system, a mode control concept is introduced on the basis of the traditional spatial six-degree-of-freedom hydraulic motion platform control, the mode conversion matrix is utilized to decouple the strong dynamic coupling six-degree-of-freedom hydraulic motion platform, and an expected physical input signal and an actual output signal of the motion platform are converted into mode signals to perform independent mode control and adjustment, so that driving and control of the spatial six-degree-of-freedom hydraulic motion platform are realized, coupling influence among actuators and among degrees of freedom in the spatial six-degree-of-freedom hydraulic motion system is effectively weakened, indexes such as single-degree-of-freedom motion and multi-degree-of-freedom compound motion reproduction of the six-degree-of the hydraulic motion platform are improved, and the degree of freedom bandwidths other degrees of freedom of the degrees other than the degrees of freedom of the first-order mode approach are improved.
Because of the inherent properties of the parallel platforms, the respective degrees of freedom have serious coupling characteristics, so that the parallel platforms have good control performance and dynamic performance, the mutual decoupling among the degrees of freedom needs to be realized, and the control performance of the system is improved. The current mainstream method is to redesign the structural configuration of the decoupling parallel mechanism of the model through structural design, so that the influence of coupling can be reduced to a certain extent. For example, a symmetric Stewart platform with orthogonal properties, the structure can realize an analysis method for simplifying an inverse Jacobian matrix of the symmetric Stewart platform with decoupling capability into 2 simple matrices in a small working space, and the method can be used for producing the Stewart platform with decoupling properties; the structural design of the Stewart platform is optimized by utilizing isotropy of a frequency matrix, and decoupling conditions and dynamic isotropy conditions are discussed and proposed; and the expression of the coupling degree of the mechanism is deduced by combining an amplification coefficient matching method, and the decoupling of the structure is realized by optimizing a decoupling model, so that the coupling degree of the parallel mechanism is weakened. The method reduces the coupling performance of the parallel platform from the structural design angle, but has good decoupling effect only in a very small working space, and the mechanism still has strong coupling performance after exceeding the specified working range.
In addition, the parallel platform is controlled in physical space by PD decoupling control, neural network control, calculation torque control and other methods in the control scheme, but the influence of system coupling cannot be eliminated all the time, and the control strategy is specifically that each control channel cannot be independently regulated, and the physical space control method can only optimize the performance of the control channel with the lowest natural frequency and sacrifice the performance of other control channels.
Disclosure of Invention
The invention provides a model decoupling method based on a parameterized model to solve the problem of active damping vibration isolation of high-precision instruments and equipment.
The invention provides a decoupling method based on a parameterized model, which comprises the following steps:
s1: obtaining a system parameterized model through system identification;
s2: eliminating feed-through items of a state space of the parameterized model through an integral operator to obtain a linear constraint equation;
s3: constructing a system optimization problem according to the linear constraint equation;
s4: constructing an optimization target according to the optimization problem;
s5: obtaining a transformation matrix of the system by optimizing the optimization target;
s6: and combining the optimization problem with the linear constraint equation to obtain a transformation matrix.
Preferably, the S1 includes:
the system is used for identifying that any linear dynamic system always has the minimum state space, and the corresponding expression is as follows:
the number of input channels and the number of output channels of the preset system are bothAt the same time, the system matrix is also preset>The order of->；
Wherein A is a system matrix for describing the connection of the internal state of the system; b is a control matrix, C is an output matrix, D is a direct transfer matrix, s is a Laplacian operator, E is an order number 2k identity matrix,representing the transfer function of the main path of the system vibration transfer.
Preferably, if direct transfer of the system input vector is not considered, the term is fed throughCan be eliminated by an integral operator;
find a pair ofOrder matrix->、/>Decoupling the system, adding moreThe input and multiple output system is converted into a single input and single output system, and then the expression of the state control realization of the system after conversion is as follows:
wherein,、/>and->Is a block diagonal matrix having the form:
；
representing the transfer function of the decoupled system vibration transfer main path +.>The system matrix is used for describing the connection of the internal state of the system after decoupling; />For the decoupled control matrix +.>For decoupling the post output matrix,/a>Is a zero matrix after decoupling; />Representing the left decoupling matrix of the system,>representing the right decoupling matrix of the system,>representation->First value of inner kth diagonal element, < ->Representation->A second value of the inner kth diagonal element; />Representation->First value of inner kth diagonal element, < ->Representation->A second value of the inner kth diagonal element; />Representation->First value of inner kth diagonal element, < ->Representation->A second value of the inner kth diagonal element.
Preferably, the S2 includes:
the state space of the system before and after transformation is compared to obtain a transformation matrix、/>Is a set of linear constraint equations:
according to vectors、/>The number of elements is->More than vector->、/>The number of (1)>Namely, the number of the linear constraint equations is more than the total number of unknown quantities to be solved, and the linear constraint equations are converted into an optimization problem to be solved to obtain a decoupling matrix>Representing vectorization operator>The Kronecker product of the matrix is represented.
Preferably, the S3 includes:
preset vector、/>Vectors composed of non-zero elements in the formula are +.>、/>The vectors of the corresponding zero elements are +.>、/>Splitting each linear constraint equation into two vector linear equations:
wherein the matrix、/>、/>And->According to->、/>、/>And->Matrix for element sequence in medium、/>And combining the linear constraint equation to obtain the following relation:
。
preferably, the step S4 includes constructing the optimization objective of the optimization problem, and the corresponding objective function expression is:
the vectors corresponding to the complete decoupling matrix、/>Can be respectively enable the objective function->、Obtaining a minimum value of 0; if the decoupling is not completed, the smaller the objective function value is, the corresponding decoupling +.>、/>The better the decoupling effect of (c), the problem of solving the decoupling matrix is converted into an optimization problem, wherein +.>For the first optimization objective function, < >>For the second optimization objective function, < >>Dependent variable for the first optimization objective function, < +.>Dependent variable of the second optimization objective function, +.>Representing the transpose of the matrix.
Preferably, the step S5 includes:
solving an objective function by adopting the same method based on symmetry of two optimization problems、/>According to the matrix->、/>For a semi-positive symmetric matrix, a set of eigenvectors and non-negative eigenvalues can be found to satisfy:
according to the matrixRank of->The number of the characteristic values is +.>；
The feature values are marked as follows:
then there are properties according to Rayleigh quotient:
thenIs an optimization->Wherein->Representing the selected ith feature vector, +.>Characteristic value +.>Representing a set of real numbers.
Preferably, the vector is based on the nature of the Kronecker productAfter the matrix is restored, the rank is always 1, and reversible transformation cannot be performed;
selectingThe feature vectors of the minimum generalized feature values are linearly combined and then restored to +.>A dimension transformation matrix:
then the resulting matrix is guaranteedFull rank, optimal full rank transform in eigenvector linear combination, according to the Rayleigh quotient properties, where +.>Representing inverse quantization operator +.>Indicates presence of->Representing the reduction gain.
Preferably, the step S6 includes:
combining the optimization problem with the constraint equation to obtain a decoupling matrix of the system:
wherein,，/>representing the selected feature vector ∈>Moore-Penrose inverse of the matrix>And (5) after decoupling, controlling the target value of the matrix.
Preferably, the reduced order processing is adopted to simplify the excessive computational complexity of the system model in S1.
Compared with the prior art, the invention has the following beneficial effects:
(1) The decoupling method provided by the invention can freely select a required decoupling mode, convert the problem of solving the decoupling matrix into an optimization problem, assign an optimization target according to the actual engineering requirement, and obtain a decoupling model suitable for the actual engineering by using the existing optimization algorithm. Compared with modal decoupling and structural decoupling, the method effectively reduces the complexity of the algorithm and has better engineering applicability;
(2) The algorithm utilizes the parameterized model of the system, can be obtained according to the principle of system identification, does not need to calculate a mass matrix and a rigidity matrix of the system, only needs to be realized in the minimum state space of the system, and reduces the requirement on the identification precision of the system model compared with the mode decoupling which needs a more accurate mass matrix and rigidity matrix;
(3) The input transformation matrix and the output transformation matrix are independent in solution, so that the method can be suitable for a general non-decoupling control system, and a good decoupling effect can be achieved for a complex-configuration parallel platform.
Drawings
Fig. 1 is a schematic diagram of a multi-dimensional active vibration isolation provided by the present invention;
FIG. 2 is a flow chart of a parameterized model-based decoupling method provided in accordance with an embodiment of the present invention;
FIG. 3 is a schematic diagram of system parameterized decoupling provided in accordance with an embodiment of the present invention;
fig. 4 is a graph comparing amplitude-frequency characteristics of MIMO systems before and after order reduction according to an embodiment of the present invention;
fig. 5 is a comparison diagram of amplitude-frequency characteristics of a MIMO system before and after decoupling according to an embodiment of the present invention.
Detailed Description
Hereinafter, embodiments of the present invention will be described with reference to the accompanying drawings. In the following description, like modules are denoted by like reference numerals. In the case of the same reference numerals, their names and functions are also the same. Therefore, a detailed description thereof will not be repeated.
In order to make the objects, technical solutions and advantages of the present invention more apparent, the present invention will be further described in detail with reference to the accompanying drawings and specific embodiments. It should be understood that the specific embodiments described herein are for purposes of illustration only and are not to be construed as limiting the invention.
Referring to fig. 2 and 3, the decoupling method based on the parameterized model provided by the invention specifically includes the following steps:
s1: obtaining a system parameterized model through system identification;
s2: eliminating feed-through items of a state space of the parameterized model through an integral operator to obtain a linear constraint equation;
s3: constructing a system optimization problem according to the linear constraint equation;
s4: constructing an optimization target according to the optimization problem;
s5: obtaining a transformation matrix of the system by optimizing the optimization target;
s6: and combining the optimization problem with the linear constraint equation to obtain a transformation matrix.
In this embodiment, the system is a MIMO system, and if the system model is too high, the system is subjected to order reduction processing to simplify the computational complexity. The modal decoupling theory can convert the parallel platform from a multi-input multi-output system into a single-input single-output system, so as to realize decoupling among all control channels; the diagonalization method is applied to the multi-degree-of-freedom system, so that the interference between the position control system and the force control system is restrained, a bilateral control system is realized, the modal decoupling is carried out on the property of the space quality matrix of the Stewart platform joint, the limitation of the structural parameters and the load weight of the platform is avoided, and the control performance is effectively improved. After mode decoupling, each coupling channel element of the system is zero, and each diagonal channel is only influenced by one mode, so that the system is converted into a SISO control system design problem. The mass matrix and the rigidity matrix of the vibration isolation system are required to be obtained, and the modal matrix can be obtained through generalized eigenvalue decomposition. In practice, however, the multi-dimensional stiffness and multi-dimensional mass matrix are more difficult to measure, whereas the frequency response function and parameterized model of the system are more readily available.
It should be noted that the above decoupling process can be understood as follows: obtaining a system parameterized model through system identification; eliminating feed-through items of the state space through an integral operator to obtain a group of linear constraint equations; constructing a system optimization problem to obtain an optimization formula; constructing an optimization target to obtain an optimization target formula; obtaining a transformation matrix of the system by optimizing an optimization target; combining the optimization problem with the constraint equation to obtain a transformation matrix. The decoupling method provided by the invention can freely select a required decoupling mode, convert the problem of solving the decoupling matrix into an optimization problem, assign an optimization target according to the actual engineering requirement, and obtain a decoupling model suitable for the actual engineering by using the existing optimization algorithm. Compared with modal decoupling and structural decoupling, the method effectively reduces the complexity of the algorithm and has better engineering applicability; the digital model can be obtained according to the principle of system identification, a mass matrix and a rigidity matrix of the system are not required to be calculated, and only the minimum state space of the system is required to be realized, so that compared with the mode decoupling, the method has the advantages that the requirement on the identification precision of the system model is reduced; the input transformation matrix and the output transformation matrix are independent in solution, so that the method can be suitable for a general non-decoupling control system, and a good decoupling effect can be achieved for a complex-configuration parallel platform.
Further, the S1 includes:
the system is used for identifying that any linear dynamic system always has the minimum state space, and the corresponding expression is as follows:
the number of input channels and the number of output channels of the preset system are bothAt the same time, the system matrix is also preset>The order of->；
Wherein A is a system matrix for describing the connection of the internal state of the system; b is a control matrix, C is an output matrix, D is a direct transfer matrix, s is a Laplacian operator, E is an order number 2k identity matrix,representing the transfer function of the main path of the system vibration transfer.
In this embodiment, the novel vibration isolation directional optimization system provided by the invention uses a satellite star as a support module, and the laser communication antenna is an optical load. In order to simulate the influence of a flexible mode, a solar sailboard, a secondary mirror truss, a diaphragm spring and a supporting leg connecting rod are all arranged as flexible components in the model. The input is the output force of three fine legs, and the output is the three rotation direction structural model of the upper platform, and is specifically shown in (a) and (b) in fig. 1. Converting the model after the order reduction into a 6-order calculation transformation matrix, wherein a system amplitude-frequency phase-frequency curve before the order reduction and after the order reduction is shown as a figure 4, 4-1 in the figure 4 is a bird diagram of the comparison of the amplitude-frequency characteristics of the MIMO system before the order reduction and after the order reduction corresponding to 1-channel output, 4-2 in the figure 4 is a bird diagram of the comparison of the amplitude-frequency characteristics of the MIMO system before the order reduction and after the order reduction corresponding to 2-channel output, and 4-3 in the figure 4 is a bird diagram of the comparison of the amplitude-frequency characteristics of the MIMO system before the order reduction and after the order reduction corresponding to 3-channel input to 1-channel output; 4-4 in fig. 4 is a bode diagram of comparison of frequency characteristics of the MIMO system before and after the order reduction corresponding to the 1-channel input to the 2-channel output, 4-5 in fig. 4 is a bode diagram of comparison of frequency characteristics of the MIMO system before and after the order reduction corresponding to the 2-channel input to the 2-channel output, and 4-6 in fig. 4 is a bode diagram of comparison of frequency characteristics of the MIMO system before and after the order reduction corresponding to the 3-channel input to the 2-channel output; 4-7 in fig. 4 is a bode diagram of comparison of frequency characteristics of the MIMO system before and after the order reduction corresponding to the 1-channel input to the 3-channel output, 4-8 in fig. 4 is a bode diagram of comparison of frequency characteristics of the MIMO system before and after the order reduction corresponding to the 2-channel input to the 3-channel output, and 4-9 in fig. 4 is a bode diagram of comparison of frequency characteristics of the MIMO system before and after the order reduction corresponding to the 3-channel input to the 3-channel output; in other words, each plot in fig. 4 represents a bode plot of the input 123 channel to output 123 channel vibration transfer, respectively.
In the system equation, if direct transfer of the system input vector is not considered, the term is fed throughCan be eliminated by an integral operator;
find a pair ofOrder matrix->、/>Decoupling the system, inputting multiple inputsThe multi-output system is converted into a single-input single-output system, and then the expression of the state control realization of the system after conversion is as follows:
wherein,、/>and->Is a block diagonal matrix having the form:
；
representing the transfer function of the decoupled system vibration transfer main path +.>The system matrix is used for describing the connection of the internal state of the system after decoupling; />For the decoupled control matrix +.>For decoupling the post output matrix,/a>Is a zero matrix after decoupling; />Representing the left decoupling matrix of the system,>representing the right decoupling matrix of the system,>representation->First value of inner kth diagonal element, < ->Representation->A second value of the inner kth diagonal element; />Representation->First value of inner kth diagonal element, < ->Representation->A second value of the inner kth diagonal element; />Representation->First value of inner kth diagonal element, < ->Representation->A second value of the inner kth diagonal element.
Further, the S2 includes:
the state space of the system before and after transformation is compared to obtain a transformation matrix、/>Is a set of linear constraint equations:
according to vectors、/>The number of elements is->More than vector->、/>The number of (1)>Namely, the number of the linear constraint equations is more than the total number of unknown quantities to be solved, and the linear constraint equations are converted into an optimization problem to be solved to obtain a decoupling matrix>Representing vectorization operator>The Kronecker product of the matrix is represented.
It should be appreciated that the constraint equations are not solved unless the system is exactly fully decoupled. In actual control, complete decoupling is not required, constraint equations can not be strictly established for more general approximate decoupling, but two sides of the equations are similar, so that the equations can be converted into optimization problems to be solved, and a decoupling matrix is obtained.
Further, the step S3 includes:
preset vector、/>Vectors composed of non-zero elements in the formula are +.>、/>The vectors of the corresponding zero elements are +.>、/>Splitting each linear constraint equation into two vector linear equations:
wherein the matrix、/>、/>And->According to->、/>、/>And->Matrix for element sequence in medium、/>And combining the linear constraint equation to obtain the following relation:
。
further, the step S4 includes constructing the optimization objective of the optimization problem, and the corresponding objective function expression is:
the vectors corresponding to the complete decoupling matrix、/>Can be respectively enable the objective function->、Obtaining a minimum value of 0; if the decoupling is not completed, the smaller the objective function value is, the corresponding decoupling +.>、/>The better the decoupling effect of (c), the problem of solving the decoupling matrix is converted into an optimization problem, wherein +.>For the first optimization objective function, < >>For the second optimization objective function, < >>Dependent variable for the first optimization objective function, < +.>Dependent variable of the second optimization objective function, +.>Representing the transpose of the matrix.
Further, the S5 includes:
solving an objective function by adopting the same method based on symmetry of two optimization problems、/>According to the matrix->、/>For a semi-positive symmetric matrix, a set of eigenvectors and non-negative eigenvalues can be found to satisfy:
according to the matrixRank of->The number of the characteristic values is +.>；
The feature values are marked as follows:
then there are properties according to Rayleigh quotient:
thenIs an optimization->Wherein->Representing the selected ith feature vector, +.>Characteristic value +.>Representing a set of real numbers.
In this embodiment, the vector is based on the nature of the Kronecker productAfter the matrix is restored, the rank is always 1, and reversible transformation cannot be performed;
selectingThe feature vectors of the minimum generalized feature values are linearly combined and then restored to +.>A dimension transformation matrix:
then the resulting matrix is guaranteedFull rank, optimal full rank transform in eigenvector linear combination, according to the Rayleigh quotient properties, where +.>Representing inverse quantization operator +.>Indicates presence of->Representing the reduction gain.
It should be noted that due to the nature of Kronecker product, vectorsAfter the matrix is restored, the rank is always 1, and reversible transformation cannot be performed. To obtain a full rank transform matrix, one can choose +.>The feature vectors of the minimum generalized feature values are linearly combined and then restored to +.>Dimension transformation matrix, thus guaranteed matrix +.>Full rank, while according to the nature of the Rayleigh quotient, the optimal full rank transform is among these combinations. In fact, even if the system can be completely decoupled, the optimal decoupling matrix is not unique, and in general, a better decoupling effect can be obtained by directly using the above results. Wherein, rayleigh quotient is defined above Hermite quadratic form, which ensures the real nature of the result. Kronecker products are very common in tensor computation and are an important bridge for the calculation of the join matrix and tensor computation.
Further, the S6 includes:
combining the optimization problem with the constraint equation to obtain a decoupling matrix of the system:
wherein,，/>representing the selected feature vector ∈>Moore-Penrose inverse of the matrix>And (5) after decoupling, controlling the target value of the matrix.
In this embodiment, a difference in magnitude between the amplitude-frequency curve of the main diagonal and the original model pair is obtained, as shown in fig. 5, and the difference is compared with other directions, which indicates that vibration transmission mainly depends on the path of the main diagonal, and the system has good decoupling characteristics. Wherein, the Bode Diagram is Bode diagnostic, the Frequency is Frequency and unit is rad/s, the amplitude is magnitide and unit is dB, the Phase is Phase and unit is deg. The method comprises the steps of comparing the amplitude frequency characteristics of the MIMO system before decoupling and after decoupling, wherein 5-1 in the figure 5 is a bird chart of comparing the amplitude frequency characteristics of the MIMO system before decoupling and after decoupling, which corresponds to 1-channel output, 5-2 in the figure 5 is a bird chart of comparing the amplitude frequency characteristics of the MIMO system before decoupling and after decoupling, which corresponds to 1-channel output, and 5-3 in the figure 5 is a bird chart of comparing the amplitude frequency characteristics of the MIMO system before decoupling and after decoupling, which corresponds to 3-channel input to 1-channel output; 5-4 in fig. 5 is a bode diagram of comparison of amplitude frequency characteristics of the MIMO system before decoupling and after decoupling corresponding to 1-channel input to 2-channel output, 5-5 in fig. 5 is a bode diagram of comparison of amplitude frequency characteristics of the MIMO system before decoupling and after decoupling corresponding to 2-channel input to 2-channel output, and 5-6 in fig. 5 is a bode diagram of comparison of amplitude frequency characteristics of the MIMO system before decoupling and after decoupling corresponding to 3-channel input to 2-channel output; 5-7 in fig. 5 is a bode diagram of the comparison of the amplitude frequency characteristics of the MIMO system before and after decoupling corresponding to the 1-channel input to the 3-channel output, 5-8 in fig. 5 is a bode diagram of the comparison of the amplitude frequency characteristics of the MIMO system before and after decoupling corresponding to the 2-channel input to the 3-channel output, and 5-9 in fig. 5 is a bode diagram of the comparison of the amplitude frequency characteristics of the MIMO system before and after decoupling corresponding to the 3-channel input to the 3-channel output; in other words, each plot in fig. 5 represents a bode plot of the input 123 channel to output 123 channel vibration transfer, respectively.
While embodiments of the present invention have been illustrated and described above, it will be appreciated that the above described embodiments are illustrative and should not be construed as limiting the invention. Variations, modifications, alternatives and variations of the above-described embodiments may be made by those of ordinary skill in the art within the scope of the present invention.
The above embodiments of the present invention do not limit the scope of the present invention. Any other corresponding changes and modifications made in accordance with the technical idea of the present invention shall be included in the scope of the claims of the present invention.
Claims (6)
1. A decoupling method based on a parameterized model, comprising the steps of:
s1: obtaining a system parameterized model through system identification;
s2: eliminating feed-through items of a state space of the parameterized model through an integral operator to obtain a linear constraint equation;
s3: constructing a system optimization problem according to the linear constraint equation;
s4: constructing an optimization target according to the optimization problem;
s5: obtaining a transformation matrix of the system by optimizing the optimization target;
s6: combining the optimization problem with the linear constraint equation to obtain a transformation matrix;
if direct transfer of the system input vector is not considered, then the term is fed throughCan be eliminated by an integral operator;
find a pair ofOrder matrix->、/>Decoupling the system, converting the multi-input multi-output system into a single-input single-output system, and realizing the state control expression of the transformed system as follows:
wherein,、/>and->Is a block diagonal matrix having the form:
；
representing the transfer function of the decoupled system vibration transfer main path +.>The system matrix is used for describing the connection of the internal state of the system after decoupling; />After being decoupledControl matrix->For decoupling the post output matrix,/a>Is a zero matrix after decoupling;representing the left decoupling matrix of the system,>representing the right decoupling matrix of the system,>representation->First value of inner kth diagonal element, < ->Representation->A second value of the inner kth diagonal element; />Representation->First value of inner kth diagonal element, < ->Representation->A second value of the inner kth diagonal element; />Representation->First value of inner kth diagonal element, < ->Representation->A second value of the inner kth diagonal element;
the step S2 comprises the following steps:
the state space of the system before and after transformation is compared to obtain a transformation matrix、/>Is a set of linear constraint equations:
according to vectors、/>The number of elements is->More than vector->、/>The number of (1)>Namely, the number of the linear constraint equations is more than the total number of unknown quantities to be solved, and the linear constraint equations are converted into an optimization problem to be solved to obtain a decoupling matrix>Representing vectorization operator>Kronecker product representing a matrix;
the step S3 comprises the following steps:
preset vector、/>Vectors composed of non-zero elements in the formula are +.>、/>The vectors of the corresponding zero elements are +.>、/>Splitting each linear constraint equation into two vector linear equations:
wherein the matrix、/>And->According to->、/>、/>And->Matrix for element sequence of Chinese>、And combining the linear constraint equation to obtain the following relation:
；
and S4, constructing an optimization objective by constructing the optimization problem, wherein the corresponding objective function expression is as follows:
the vectors corresponding to the complete decoupling matrix、/>Can be respectively enable the objective function->、/>Obtaining a minimum value of 0; if the decoupling is not completed, the smaller the objective function value is, the corresponding decoupling +.>、/>The better the decoupling effect of (c), the problem of solving the decoupling matrix is converted into an optimization problem, wherein +.>For the first optimization objective function, < >>For the second optimization objective function, < >>Dependent variable for the first optimization objective function, < +.>Dependent variable of the second optimization objective function, +.>Representing the transpose of the matrix.
2. The method of parameterized model-based decoupling of claim 1, wherein S1 comprises:
the system is used for identifying that any linear dynamic system always has the minimum state space, and the corresponding expression is as follows:
the number of input channels and the number of output channels of the preset system are bothAt the same time, the system matrix is also preset>The order of->；
Wherein A is a system matrix for describing the connection of the internal state of the system; b is a control matrix, C is an output matrix, D is a direct transfer matrix, s is a Laplacian operator, E is an order number 2k identity matrix,representing the transfer function of the main path of the system vibration transfer.
3. The method of parameterized model-based decoupling of claim 2, wherein S5 comprises:
solving an objective function by adopting the same method based on symmetry of two optimization problems、/>According to the matrix、/>For a semi-positive symmetric matrix, a set of eigenvectors and non-negative eigenvalues can be found to satisfy:
according to the matrixRank of->The number of the characteristic values is +.>；
The feature values are marked as follows:
then there are properties according to Rayleigh quotient:
thenIs an optimization->Wherein->Representing the selected ith feature vector, +.>Characteristic value +.>Representing a set of real numbers.
4. A method of parameterized model-based decoupling as in claim 3, further comprising:
vector according to the nature of Kronecker productAfter the matrix is restored, the rank is always 1, and reversible transformation cannot be performed;
selectingThe feature vectors of the minimum generalized feature values are linearly combined and then restored to +.>A dimension transformation matrix:
then the resulting matrix is guaranteedFull rank, optimal full rank transform in eigenvector linear combination, according to the Rayleigh quotient properties, where +.>Representing inverse quantization operator +.>Indicates presence of->Representing the reduction gain.
5. The method of parameterized model-based decoupling of claim 4, wherein S6 comprises:
combining the optimization problem with the constraint equation to obtain a decoupling matrix of the system:
wherein,，/>representing the selected feature vector ∈>Representation ofThe Moore-Penrose inverse of the matrix,and (5) after decoupling, controlling the target value of the matrix.
6. The method of parameterized model-based decoupling of claim 1, wherein the system model in S1 is simplified with a reduced order process to account for excessive computational complexity.
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