CN112327805A - Control system stability analysis method and device and computer readable storage medium - Google Patents
Control system stability analysis method and device and computer readable storage medium Download PDFInfo
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Abstract
The invention discloses a method and a device for analyzing the stability of a control system and a computer readable storage medium, wherein the core idea is to construct a Kronecker-Lyapunov matrix, calculate the boundary value of an uncertain parameter when a control matrix is positioned at a stable and unstable critical position by using the Kronecker-Lyapunov matrix, and finally calculate the stable interval of the control system by using the boundary value; the method has the advantages that the automatic system with the feedback function is effectively controlled, the key is to ensure that the control system operates in a stable interval, the stable interval is obtained, the control system can be effectively controlled all the time, high-quality production is realized, and the production efficiency is improved. The calculation method is simple and clear, has low calculation complexity, and has a good help function on the judgment work of the stable interval of the uncertainty control system.
Description
Technical Field
The invention relates to stability analysis of a control system, in particular to a stability analysis method which can define an effective range of uncertainty factors when the control system contains the uncertainty factors and determine whether production equipment needs to be debugged again according to a steady-state range of the uncertainty factors.
Background
In the process of robot welding, due to the use aging of automation equipment, the effective distance between a welding gun and a welding point can deviate from a normal numerical range. By assuming a smart camera, the distance between the welding gun and the welding spot can be accurately measured, and in the analysis of the production system, the distance is taken as an indeterminate quantity which is exogenous to the system. The method effectively controls an automatic system with a feedback function, and has the key point of ensuring that a control matrix operates in a stable interval. According to the knowledge of modern control theory, a necessary condition for the stability of the control matrix is that all complex eigenroots of the control matrix are located in the left half plane (excluding the imaginary axis) of the complex plane. In a real production process, due to the existence of uncertain disturbance factors, a control matrix often contains uncertain parameters, and in order to ensure the effectiveness of a control system, the ranges of the parameters are limited. The traditional algorithm always needs to perform operation of an inverse matrix on a control matrix containing indefinite parameters, and for a high-dimensional control matrix, the calculation amount and the calculation time are very large, and the actual requirement of quick response cannot be met.
For example, in the welding production process, the solution of the uncertain parameters is very complicated, and the uncertain parameters cause the following problems to the stability control of the control matrix:
(1) is it judged within what range the distance between the welding torch and the spot is, the quality of arc welding is efficient and excellent?
(2) What is the effective distance between the corresponding torch and the weld spot for different weld plans?
(3) When the system has been in operation for a long period of time, the robot and the welding gun have degraded in function, and what is the effective distance between the welding gun and the welding spot at that time?
(4) How to better position the workpiece for welding, so that the welding quality can be improved for different welding requirements?
In past practice, hundreds of experiments are often needed to obtain an empirical interval of the effective distance between the welding gun and the welding point, and the interval is not high in adaptability when facing different welding plans and equipment changes along with the time.
Disclosure of Invention
In order to solve the technical problems, the invention aims to provide a method for judging the stability of a control matrix containing 1 unknown parameter by using a Kronecker-Lyapunov matrix, so that the formulation efficiency of an uncertain factor control strategy in the production process is improved, high-quality production is realized, and the production efficiency is improved.
The control system stability analysis method comprises the following steps:
the control of the automatic system is performed, and the most important is to keep the control matrix in a stable interval. In order to calculate the stability of the control matrix, it is usually necessary to calculate an ideal control matrix of the control system. However, due to long-term use of the control system, actual operation errors, and the like, there is an uncertainty error between the ideal matrix equation of the control system and the actual control system. In this case, it is very important to determine the stability of the control system.
A novel method is provided for calculating and deriving a new critical point of a stable interval by using a Kronecker-Lyapunov matrix. The object of the method is a control system containing 1 uncertainty parameter, which is expressed by a matrix of modern control theory; the method avoids massive operation of an inverse matrix through calculation, and simplifies the complexity of the operation to a great extent, and the control system stability analysis method comprises the following steps:
analyzing the condition dependence relationship of a production flow to obtain a deterministic factor and an uncertain factor in the production process, and constructing a control matrix with 1 unknown parameter according to a modern analysis theory;
step two, calculating a corresponding Kronecker-Lyapunov matrix according to the obtained production control matrix;
setting a determinant corresponding to the Kronecker-Lyapunov matrix, and enabling the value of the determinant to be 0;
step four, solving the boundary value of the unknown parameter by using the condition that the determinant corresponding to the Kronecker-Lyapunov matrix is 0 to obtain a group of ordered real number sequences;
step five, for the ordered real number sequence, dividing the ordered real number sequence into interval blocks consisting of 2 adjacent real numbers, randomly taking 1 value in 1 interval block, and testing the stability of the control matrix;
and step six, obtaining the value interval of the parameters of the control matrix in the stable state by using all the test results, and obtaining decision help for the control problem with uncertainty.
Obtaining 1 control system matrix with parameters, namely quantifying uncertainty factors in the production process by using theoretical knowledge of modern control, so as to prepare for analyzing the stability of the control system, wherein the first step specifically comprises the following steps:
step 1.1, analyzing the condition dependence relationship of the production flow to obtain an input variable, an output variable, an intermediate variable and an exogenous variable;
step 1.2, analyzing uncertainty factors in the production process, calculating the uncertainty factors and the correlation coefficients of the results by using a data formula of statistical calculation by adopting an expert analysis method, and finding out 1 most main factors;
step 1.3, arranging input variables, output variables, intermediate variables and exogenous variables according to corresponding periods by utilizing the modern analysis principle and according to the time sequence of a control system to form a control matrix;
and step 1.4, adding the extracted 1 most main uncertainty factors serving as parameters of the control matrix to a control matrix equation.
Suppose, in the control system, there is a control matrix of a ∈ R:
A＝A0+aA1
here, A0,A1∈Rn×nIs now a problem in the known a0,A1On the premise of (1), a range in which the control matrix a can be stabilized is calculated.
And calculating a corresponding Kronecker-Lyapunov matrix according to the obtained production control matrix. Since the matrix A is stable, all the eigenvalues λ of the matrix Ai(i ═ 1,2, L, n), must lie in the left half plane of the complex number plane (excluding the imaginary axis). The Kronecker-Lyapunov matrix corresponding to the construction matrix A is set as
The second step is specifically as follows:
step 2.1, setting matrix
The elements of the pq th row and the rs th column of (1) are:
epq,rs(p＝1,2,L,n；q＝1,2,L,p；r＝1,2,L,n；s＝1,2,L r)；
let the ith row and jth column element of matrix A be aij
Step 2.2, when p > q:
step 2.3, when p ═ q:
step 2.4, Kronecker-Lyapunov matrix
Using its row-by-column elements epq,rsExpressed as:
obtaining a Kronecker-Lyapunov matrix corresponding to the control matrix
Is/are as follows
The characteristic values are: lambda [ alpha ]i+λj(i-1, 2, …, n; j-1, 2, …, i), then the unknown parameter a may be included
Determinant value is 0, is satisfied
The set of all a includes all values of a that make the matrix a have characteristic values on the imaginary axis, and the step three is specifically:
step 3.1, setting a linear differential equation as:
where t is time and x (t) is all inputs to the control system.
Step 3.2, if x (t) is the solution of (2.1), xp(t) (p ═ 1,2, … n) is the p-th element of the n-dimensional column vector x (t), λi(i-1, 2, … n) are n different eigenvalues of a, x is thenp(t) is
1 linear combination of, i.e.
Wherein
Is the p-th element x of composition x (t)p(t) linear combination
In
The coefficient of (a);
step 3.3, set xp(t),xq(t) (p 1,2, … n; q 1,2, … p) are the p-th and q-th elements, respectively, of an n-dimensional column vector x (t),
wherein c ispq,ijIs composed of wpq(t) linear combination
In
Coefficient of (1), then wpq(t) is
1 linear combination of (1).
Step 3.4, set wpq(t) is 1 element of w (t), and a matrix B satisfying the solution w (t) is constructed, thereby obtaining
The eigenvalues of matrix a lie on the imaginary axis. Because all eigenvalues λ of the matrix Ai(i ═ 1,2, …, n), which must lie in the left half of the complex plane (excluding the imaginary axis), so at the marginal points, the value of a is such that the presence characteristic λ of a is presentiLocated on the imaginary axis.
Step 3.5, assuming it is located on the imaginary axis
By means of the calculation of characteristic values
Conjugated complex number of
Is also another characteristic value lambda of AjAccording to
Has a characteristic value ofi+λjThe traversal of (i ═ 1,2, …, n;. j ═ 1,2, …, i) yields λ ═ 0
1 characteristic value of (a).
Step 3.6, assume λ lying on the imaginary axisiIs equal to 0, according to
Has a characteristic value ofi+λj(i-1, 2, …, n; j-1, 2, …, i) and λ -0 is when i-j
1 characteristic value of (a).
Deriving an alternative for the interval of the parameter a that stabilizes the control system, said step four being in particular:
step 4.1, according to
In calculation, the set B of satisfied a is a set of a finite number of real numbers arranged in order of a from small to large1,a2,L,amThen set a0＝-∞,am＝1＝+∞；
Testing the effectiveness of the alternative interval of each parameter a, wherein the fifth step specifically comprises:
step 5.1, for selected i, arbitrarily choosing a e (a)i,ai+1) Calculating all characteristic values of A;
step 5.2, if all the real parts b of the eigenvalues are<0, then A is stable, thus the interval (a)i,ai+1) A is a value range which can make A stable;
step 5.3, if there is a real part b of the eigenvalue>When 0, a is unstable, and thus a is in the interval (a)i,ai+1) A is not a value interval which can make A stable;
step 5.4, because of the continuity of the real number a, yields a ∈ (a ∈) (a)i,ai+1) Within the range, the stability of a is consistent;
and six steps are specifically as follows: all the stable intervals are obtained by calculating from i to 0 to i to m. Then measuring the range of actual unknown parameters in the production equipment, and if the range of the unknown parameters is in a stable interval, the production system does not need to be debugged again; if the range of unknown parameters exceeds the stability interval, the production system needs to be debugged again.
Advantageous effects
The stability analysis method of the control system includes uncertain factors in the control system into system stability analysis, and the stability of the control matrix containing 1 unknown parameter is judged by using the Kronecker-Lyapunov matrix, so that the calculation intensity is greatly reduced, massive operation of an inverse matrix is avoided in calculation, the complexity of operation is simplified to a great extent, high-quality production is realized, and the production efficiency is improved; the calculation steps and the calculation method are based on the mathematical theory of mature matrix theory and have rigor; the calculation method takes the space complexity and the time complexity of computer processing into consideration, and has feasibility in operation.
Drawings
FIG. 1 is a diagram of a control matrix construction according to the present invention;
FIG. 2 is a schematic data flow diagram;
FIG. 3 is a computational logic diagram;
fig. 4 is a production feedback and tuning diagram.
Detailed Description
The invention will be further explained with reference to the drawings and the specific examples below:
the invention provides a new method for calculating and deriving a new critical point of a stable interval by using a Kronecker-Lyapunov matrix. The object of the method is a control system containing 1 uncertainty parameter, which is expressed by a matrix of modern control theory; the calculation of the method avoids mass operation of an inverse matrix, and simplifies the complexity of the operation to a great extent.
The application provides a control system stability analysis method, which is used for analyzing a control system containing uncertain parameters, and specifically comprises the following steps:
analyzing the condition dependence relationship of the production flow to obtain a deterministic factor and an uncertain factor in the production process, and constructing a control matrix with unknown parameters corresponding to the production system according to a modern analysis theory;
step two, calculating a corresponding Kronecker-Lyapunov matrix according to the obtained control matrix corresponding to the production process;
calculating a determinant corresponding to the corresponding Kronecker-Lyapunov matrix, and enabling the value of the determinant to be 0;
step four, solving a boundary value of an unknown parameter contained in the matrix by using a condition that the corresponding determinant of the Kronecker-Lyapunov matrix is 0 to obtain an ordered real number sequence;
step five, according to the ordered real number sequence, dividing a real number axis into interval blocks consisting of 2 adjacent real numbers, randomly taking one value in 1 interval block, testing the stability of a control matrix, and if the control matrix calculated by the selected value is stable, the matrix is stable on the whole real number interval block where the selected value is located;
step six, combining all test results to obtain a value interval of the parameters of the control matrix in a stable state, namely obtaining a stable state interval of the corresponding uncertain parameters in the production flow; in the production process, analyzing the actual production environment, calculating the range of uncertain parameters in real production, and if the uncertain parameters are in a steady-state interval, the system does not need to be adjusted; if the uncertain parameters exceed the steady state interval, the actual production system needs to be debugged again.
Further, the first step specifically comprises:
step 1.1, analyzing the condition dependence relationship of the production flow to obtain an input variable, an output variable, an intermediate variable and an exogenous variable;
step 1.2, analyzing uncertainty factors in the production process, and analyzing the correlation coefficients of the uncertainty factors and results by using domain expert knowledge, so as to find 1 most main uncertainty factors which can be numerically quantized and have large influence among the uncertainty factors;
step 1.3, arranging input variables, output variables, intermediate variables and exogenous variables according to corresponding periods by utilizing a modern control analysis theory according to a time sequence of a control system to form a control matrix;
step 1.4, adding the most main uncertainty factors extracted in the step 1.2 as parameters of a control matrix to an equation;
it is assumed that, in the control matrix, there is a matrix of parameters of a ∈ R:
A＝A0+aA1
here, A0,A1Are respectively A0,A1∈Rn×nThe constant matrix of (2) converts the problem of analyzing the stability of the control system into: known as A0,A1On the premise of (1), a range in which the control matrix a can be stabilized is calculated.
The Kronecker-Lyapunov matrix corresponding to the structural control matrix A is set as
The second step is specifically as follows:
step 2.1, setting
In the form of a matrix of a plurality of,
the elements of the pq th row and the rs th column of (c) are:
epq,rs(p＝1,2,…,n；q＝1,2,…,p；r＝1,2,…,n；s＝1,2,…r)；
let the element in the ith row and the jth column of the matrix A be aij，
Step 2.2, when p > q:
step 2.3, when p ═ q:
step 2.4, Kronecker-Lyapunov matrix
Using the elements e of its rows and columnspq,rsExpressed as:
further, the third step is specifically:
step 3.1, setting a linear differential equation as:
wherein t is a time variable, and x (t) is all input functions of the control system;
step 3.2, if x (t) is the solution of step 3.1, then x (t) is a column vector of n dimensions, let xp(t) (p ═ 1,2, … n) is the p-th element of x (t), λi(i-1, 2, … n) are n different eigenvalues of the control matrix a, x is thenp(t) is
1 linear combination of, i.e.
Wherein
Is the p-th element x constituting an n-bit column vector x (t)p(t) linear combination
In
The coefficient of (a);
step 3.3, set xp(t),xq(t) (p 1,2, … n; q 1,2, … p) are the p-th and q-th elements, respectively, of an n-dimensional column vector x (t),
wherein c ispq,ijIs composed of wpq(t) linear combination
In
Coefficient of (1), then wpq(t) is
A linear combination of (a);
step 3.4, construct the solution, let w (t) be the element wpq(t) a column vector of the composition, thereby obtaining
When the characteristic value of the control matrix A is positioned on the virtual axis;
step 3.5, assuming it is located on the imaginary axis
By means of the calculation of characteristic values
Conjugated complex number of
Is also another eigenvalue λ of the control matrix AjAccording to
Has a characteristic value ofi+λjThe traversal of (i ═ 1,2, …, n;. j ═ 1,2, …, i) yields λ ═ 0
A characteristic value of (d);
step 3.6, assume λ lying on the imaginary axisiIs equal to 0, according to
Characteristic value ofIs λi+λj(i-1, 2, …, n; j-1, 2, …, i) and λ -0 is when i-j
1 characteristic value of (a).
Further, the fourth step is specifically:
according to
The set B of a satisfied by the calculation is a set of a finite number of real numbers arranged in order from small to large as a1,a2,…,amThen set a0＝-∞,am＝1＝+∞。
Since the values of the determinant are 1-dimensional real variable functions with respect to a, and are therefore continuous, for i ∈ {0,1, …, m }, the production control matrix A is for any a ∈ (a) }i,ai+1) A sufficient condition for stabilization is the presence of a ∈ (a)i,ai+1) So that the production control matrix A is stable;
since the values of the determinant are 1-dimensional real variable functions with respect to a, and are therefore continuous, for i ∈ {0,1, …, m }, the production control matrix A is for any a ∈ (a) }i,ai+1) The requirement for sufficient instability is the presence of a ∈ (a)i,ai+1) Making the production control matrix a unstable.
Further, the fifth step is specifically:
step 5.1, for selected i, arbitrarily choosing a e (a)i,ai+1) Calculating all characteristic values of the control matrix A;
step 5.2, if all the real parts b of the eigenvalues are<0, then the control matrix A is stable, thus the interval (a)i,ai+1) A is a value range which can make A stable;
step 5.3, if there is a real part b of the eigenvalue>If 0, the control matrix a is unstable, and thus the interval (a)i,ai+1) A is not a value interval which can make A stable;
step 5.4, becauseThe continuity of the real number a is obtained in the interval a ∈ (a)i,ai+1) Within the range, the stability of the control matrix a is consistent.
And six steps are specifically as follows:
and 6.1, calculating from i to 0 to i to m one by one to obtain all stable intervals, and obtaining the value intervals of the parameters of the control matrix in the stable state, namely obtaining the steady-state intervals of the control matrix.
The invention is based on the implementation of an automatic production line and can be applied to various systems which use the automatic production line for control.
According to the stability analysis method of the control system, uncertain factors in the control system are brought into system stability analysis, the stability of the control matrix containing 1 unknown parameter is judged by using the Kronecker-Lyapunov matrix, the calculation intensity is greatly reduced, massive operation of an inverse matrix is avoided in calculation, the complexity of operation is simplified to a great extent, high-quality production is realized, and the production efficiency is improved.
The present application further provides a control system stability analysis device, the analysis device includes:
the control matrix construction module analyzes the condition dependence relationship of the production process to obtain the deterministic factor and the nondeterministic factor in the production process, and constructs a control matrix with unknown parameters corresponding to the production process according to the modern analysis theory;
the system comprises a Kronecker-Lyapunov matrix generation module, a control matrix generation module and a data processing module, wherein the Kronecker-Lyapunov matrix generation module is used for calculating to obtain a corresponding Kronecker-Lyapunov matrix based on the control matrix;
the boundary value solving module is used for solving a boundary value of an unknown parameter contained in the Kronecker-Lyapunov matrix by utilizing the condition that the corresponding determinant of the Kronecker-Lyapunov matrix is 0 to obtain an ordered real number sequence;
the control system stability interval solving module is used for carrying out the following processing on the ordered real number sequence: dividing a real number axis into interval blocks consisting of 2 adjacent real numbers, taking a value in 1 interval block, testing the stability of a control matrix, and if the control matrix calculated by the selected value is stable, the matrix is stable on the whole real number interval block where the selected value is located, so that a stable interval corresponding to uncertain parameters in the production flow is obtained;
and the test module is used for confirming and obtaining the value interval of the parameters of the control matrix in the stable state by combining all the test results.
The core of the analysis device is that the stability of a control matrix containing 1 unknown parameter is judged by using a Kronecker-Lyapunov matrix, so that the calculation intensity is greatly reduced, massive operation of an inversion matrix is avoided in calculation, the complexity of operation is simplified to a great extent, high-quality production is realized, and the production efficiency is improved.
Example 1
It is because of the technical difficulties in the background art that we propose a corresponding method. Firstly, the automation structure of the welding gun system should be analyzed, and a preliminary matrix of the control system is obtained aiming at the automation structure. Then, the distance between the welding gun and the welding point is set in the control matrix as an uncertain element, and the control matrix with 1 position parameter is obtained. Then, constructing a corresponding Kronecker-Lyapunov matrix of the original matrix, and judging whether the eigenvalue of the original matrix is positioned on the virtual axis by judging the range of the eigenvalue of the Kronecker-Lyapunov matrix, so that the stability of the control matrix can be analyzed; the following describes the implementation process of the technical scheme in detail by taking a welding process production system as an example:
in 1 system using current and voltage control robot welding, the magnitude of current and voltage values is required to identify the welding quality. In the robot system which completes teaching playback, if the robot is in a stable control interval, the welding current and voltage are automatically adjusted to achieve the optimal state; when the range of the robot's control system exceeds the stable current and voltage ranges, the control system loses linear stability, resulting in a degradation of the weld quality.
Step one, listing a corresponding differential equation set according to the relation between current and voltage of the historical robot under a normal condition;
secondly, setting the distance between the welding gun and the welding spot as an uncertain control quantity according to a modern control theory, and rewriting a high-order derivative differential equation set into a 1-order differential equation set comprising time t and t +1 to obtain a control matrix A and a parameter a;
step three, calculating a Kronecker-Lyapunov matrix corresponding to the matrix A
Step four, solving
The result of the solution is arranged as a according to the order of small to large1,a2,L,amThen set a0＝-∞,am＝1＝+∞；
Step five, for the selected i, randomly selecting a e (a)i,ai+1) Calculating all eigenvalues of A if the real parts b of all eigenvalues<0, then A is stable, thus the interval (a)i,ai+1) A is a value range which can make A stable;
step six, calculating i to m for i to 0 to obtain all stable intervals;
measuring the fluctuation range of the actual uncertainty parameter a of the system inductance, and calibrating the welding quality to be good when the fluctuation ranges of the welding gun and the welding spot are in a stable range; otherwise, judging that the welding quality is poor;
and step eight, continuously and repeatedly iterating and learning, and improving the accuracy of judging the distance between the welding gun and the welding spot.
Based on the method, the uncertain parameters in the welding process can be rapidly solved, the influence of the uncertain parameters on the stability control of the control matrix is reduced, the calculation intensity is greatly reduced, the massive operation of an inverse matrix is avoided in the calculation, the complexity of the operation is simplified to a great extent, and the welding quality and the welding precision are improved.
Example 2
In addition to welding production systems, other problems such as those encountered in industrial production systems are similar, and in 1 system using current and voltage to control engine operation, including resistive, inductive and capacitive devices, the inductive devices of the system will vary randomly within 1 amplitude range due to aging, which is an uncertain factor. It is now desirable to calculate that the inductance factor varies within those ranges without affecting the stability of the engine operation; under the action of the factors, the stability of the engine system is influenced; it can thus be decided whether this inductance is to be replaced or not. The solution method is as follows:
step one, listing a corresponding differential equation set according to a circuit relation of current and voltage of an engine;
step two, according to the modern control theory, rewriting a high-order derivative differential equation set into a 1-order differential equation set containing time t and t +1 to obtain a control matrix A and a parameter a;
step three, calculating a Kronecker-Lyapunov matrix corresponding to the matrix A
Step four, solving
The result of the solution is arranged as a according to the order of small to large1,a2,…,amThen set a0＝-∞,am＝1＝+∞；
Step five, for the selected i, randomly selecting a e (a)i,ai+1) Calculating all eigenvalues of A if the real parts b of all eigenvalues<0, then A is stable, thus the interval (a)i,ai+1) A is a value range which can make A stable;
step six, calculating i to m for i to 0 to obtain all stable intervals;
measuring the fluctuation range of the actual uncertainty parameter a of the system inductance, wherein if the actual fluctuation range is within the stable interval, the system does not need to replace parts; otherwise the inductive components need to be replaced.
Based on the method, the uncertain parameters of the engine caused by equipment aging can be rapidly solved, the influence of the uncertain parameters on the stability control of the control matrix is reduced, the calculation intensity is greatly reduced, massive calculation of the inverse matrix is avoided in calculation, the calculation complexity is simplified to a great extent, the use effect of an engine system is improved, and the service life of the engine system is prolonged.
The foregoing is a detailed description of the invention with reference to specific embodiments, and the practice of the invention is not to be construed as limited thereto. For those skilled in the art to which the invention pertains, several simple deductions or substitutions can be made without departing from the spirit of the invention, and all shall be considered as belonging to the protection scope of the invention.
Claims (10)
1. A control system stability analysis method is characterized in that the analysis method is used for analyzing a control system containing uncertainty parameters, and the analysis method specifically comprises the following steps:
analyzing the condition dependence relationship of the production process to obtain a deterministic factor and an uncertain factor in the production process, and constructing a control matrix with unknown parameters corresponding to the production process according to a modern analysis theory;
step two, calculating a corresponding Kronecker-Lyapunov matrix according to the obtained control matrix corresponding to the production process;
calculating a determinant corresponding to the Kronecker-Lyapunov matrix, and enabling the value of the determinant to be 0;
step four, solving a boundary value of an unknown parameter contained in the matrix by using a condition that the corresponding determinant of the Kronecker-Lyapunov matrix is 0 to obtain an ordered real number sequence;
step five, according to the ordered real number sequence, dividing a real number axis into interval blocks consisting of 2 adjacent real numbers, randomly taking one value in 1 interval block, testing the stability of a control matrix, and if the control matrix calculated by the selected value is stable, the matrix is stable on the whole real number interval block where the selected value is located;
and step six, combining all the test results to obtain a value interval of the parameters of the control matrix in the stable state, namely obtaining the stable state interval of the corresponding uncertain parameters in the production flow.
2. The method for analyzing the stability of the control system according to claim 1, wherein the first step is specifically:
step 1.1, analyzing the condition dependence relationship of the production flow to obtain an input variable, an output variable, an intermediate variable and an exogenous variable;
step 1.2, analyzing uncertainty factors in the production process, analyzing correlation coefficients of the uncertainty factors and results, and determining 1 most main uncertainty factors which can be numerically quantized and have large influence in the uncertainty factors;
step 1.3, arranging input variables, output variables, intermediate variables and exogenous variables according to corresponding periods by utilizing a modern control analysis theory according to a time sequence of a control system to form a control matrix;
step 1.4, adding the most main uncertainty factors extracted in the step 1.2 as parameters of a control matrix to an equation;
it is assumed that, in the control matrix, there is a matrix of parameters of a ∈ R:
A＝A0+aA1
here, A0,A1Are respectively A0,A1∈Rn×nThe constant matrix of (2) converts the problem of analyzing the stability of the control matrix into: known as A0,A1On the premise of (1), a range in which the control matrix a can be stabilized is calculated.
3. The method as claimed in claim 2, wherein the control matrix A is constructed corresponding to Kronecker-The Lyapunov matrix is set to
The second step is specifically as follows:
step 2.1, setting
In the form of a matrix of a plurality of,
the elements of the pq th row and the rs th column of (c) are:
epq,rs(p＝1,2,…,n；q＝1,2,…,p；r＝1,2,…,n；s＝1,2,…r)；
let the element in the ith row and the jth column of the matrix A be aij，
Step 2.2, when p > q:
step 2.3, when p ═ q:
step 2.4, Kronecker-Lyapunov matrix
Using the elements e of its rows and columnspq,rsExpressed as:
4. the method for analyzing the stability of the control system according to claim 3, wherein the third step is specifically:
step 3.1, setting a linear differential equation as:
wherein t is a time variable, and x (t) is all input functions of the control system;
step 3.2, if x (t) is the solution of step 3.1, then x (t) is a column vector of n dimensions, let xp(t) (p ═ 1,2, … n) is the p-th element of x (t), λi(i-1, 2, … n) are n different eigenvalues of the control matrix a, x is thenp(t) is
1 linear combination of, i.e.
Wherein
Is the p-th element x constituting an n-bit column vector x (t)p(t) linear combination
In
The coefficient of (a);
step 3.3, set xp(t),xq(t) (p 1,2, … n; q 1,2, … p) are the p-th and q-th elements, respectively, of an n-dimensional column vector x (t),
wherein c ispq,ijIs composed of wpq(t) linear combination
In
Coefficient of (1), then wpq(t) is
A linear combination of (a);
step 3.4, construct the solution, let w (t) be the element wpq(t) a column vector of the composition, thereby obtaining
When the characteristic value of the control matrix A is positioned on the virtual axis;
step 3.5, assuming it is located on the imaginary axis
By means of the calculation of characteristic values
Conjugated complex number of
Is also another eigenvalue λ of the control matrix AjAccording to
Has a characteristic value ofi+λjThe traversal of (i ═ 1,2, …, n;. j ═ 1,2, …, i) yields λ ═ 0
A characteristic value of (d);
step 3.6, assume λ lying on the imaginary axisiIs equal to 0, according to
Has a characteristic value ofi+λj(i＝1,2,…,n；j＝1,2,…,i) When i is j, λ is 0
1 characteristic value of (a).
5. The method for analyzing the stability of the control system according to claim 1, wherein the fourth step is specifically:
step 4.1, according to
The set B of a satisfied by the calculation is a set of a finite number of real numbers arranged in order from small to large as a1,a2,…,am，a0＝-∞,am＝1＝+∞。
6. The method for analyzing the stability of the control system according to claim 1, wherein the step five is specifically:
step 5.1, for selected i, arbitrarily choosing a e (a)i,ai+1) Calculating all characteristic values of the control matrix A;
step 5.2, if all the real parts b of the eigenvalues are<0, then the control matrix A is stable, thus the interval (a)i,ai+1) A is a value range which can make A stable;
step 5.3, if there is a real part b of the eigenvalue>If 0, the control matrix a is unstable, and thus the interval (a)i,ai+1) A is not a value interval which can make A stable;
step 5.4, because of the continuity of the real number a, yields a ∈ (a ∈) (a)i,ai+1) Within the range, the stability of the control matrix a is consistent.
7. The method for analyzing the stability of the control system according to claim 1, wherein the validity of the candidate interval of each parameter a is tested, and the sixth step is specifically:
and 6.1, calculating from i to 0 to i to m one by one to obtain all stable intervals, and obtaining the value intervals of the parameters of the control matrix in the stable state, namely obtaining the steady-state intervals of the control matrix.
8. A control system stability analysis device, the analysis device comprising:
the control matrix construction module analyzes the condition dependence relationship of the production process to obtain the deterministic factor and the nondeterministic factor in the production process, and constructs a control matrix with unknown parameters corresponding to the production process according to the modern analysis theory;
the system comprises a Kronecker-Lyapunov matrix generation module, a control matrix generation module and a data processing module, wherein the Kronecker-Lyapunov matrix generation module is used for calculating to obtain a corresponding Kronecker-Lyapunov matrix based on the control matrix;
the boundary value solving module is used for solving a boundary value of an unknown parameter contained in the Kronecker-Lyapunov matrix by utilizing the condition that the corresponding determinant of the Kronecker-Lyapunov matrix is 0 to obtain an ordered real number sequence;
the control system stability interval solving module is used for carrying out the following processing on the ordered real number sequence: dividing a real number axis into interval blocks consisting of 2 adjacent real numbers, taking a value in 1 interval block, testing the stability of a control matrix, and if the control matrix calculated by the selected value is stable, the matrix is stable on the whole real number interval block where the selected value is located, so that a stable interval corresponding to uncertain parameters in the production flow is obtained;
and the test module is used for confirming and obtaining the value interval of the parameters of the control matrix in the stable state by combining all the test results.
9. The control system stability analysis device of claim 8, wherein the kronecker-Lyapunov matrix is configured to be applied to a control system of a vehicle
Using the elements e of its rows and columnspq,rsExpressed as:
wherein, when p > q:
wherein, aps、apr、app、aqq、aqs、aqrElements of a p row, an s column, a p row, an r column, a p row, a p column, a q row, a q column, a q row, an s column and a q row, an r column of the matrix A are respectively arranged; a is a control matrix.
10. A computer-readable storage medium storing computer-executable instructions for performing the control system stability analysis method of any one of claims 1 to 7.
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