Pre-defined time ZNN model design method for solving pseudo-inverse of time-varying matrix

Abstract

A predefined time ZNN model design method for solving pseudo-inverse of a time-varying matrix comprises the steps that a pseudo-inverse matrix is solved based on the time-varying matrix, an error function is established, the time-varying matrix is known, and the pseudo-inverse matrix to be solved is unknown; designing a predefined time function based on a high-gain method, and introducing the predefined time function into the ZNN model; aiming at bounded noise interference in a ZNN model solving process, introducing a disturbance estimator comprising a predefined time function, and constructing a subsystem of a neural network model; a Lyapunov function is constructed for stability analysis, and a ZNN model obtained through analysis can effectively suppress noise and solve a pseudo-inverse matrix of a time-varying matrix at predefined time. And applying the predefined time ZNN model to control of the permanent magnet synchronous motor chaotic system. According to the designed predefined time ZNN model, a complex parameter adjustment process is avoided in parameter adjustment. The method has the advantages of high robustness, high calculation precision and high solving speed.

Description

Technical Field

[0001] This invention relates to the field of ZNN model research technology, specifically to a predefined time ZNN model design method for solving the pseudo-inverse of a time-varying matrix. Background Technology

[0002] The problem of pseudoinverses of time-varying matrices is common in control theory, mathematics, robot motion tracing, and many other engineering science fields. However, with the increasing complexity of engineering computations, the accurate calculation of pseudoinverses of time-varying matrices has become increasingly important. This complexity stems from the fact that each element in the time-varying matrix is ​​a distinct time-varying function, and this complexity is further exacerbated by increasing matrix dimension.

[0003] Current research on nullable neural network (ZNN) models for solving time-varying matrix problems is still in its early stages, and methods such as changing the activation function are commonly used to improve the solution accuracy of neural network models. In the design of general ZNN models, ideal noise-free conditions are usually assumed. However, when solving time-varying matrix inverse problems, noise often arises due to system operation or external disturbances, such as residual errors in hardware operation and truncation errors during the solution process, which affects the ZNN model's solution of time-varying matrix pseudo-inverse problems, thus making computation in practical engineering problems difficult.

[0004] Research on ZNN models for solving time-varying matrix equations is still insufficient both domestically and internationally. This limitation significantly restricts the development of ZNN model design in practical engineering. Therefore, designing more robust ZNN models and improving their robustness and solution accuracy is of great significance to the development of solving time-varying matrix equations. Simultaneously, this will also provide theoretical and practical application value for applications such as image processing and robot control. Summary of the Invention

[0005] This invention provides a predefined-time ZNN model design method for solving the pseudoinverse of a time-varying matrix. The predefined-time ZNN model designed in this invention can achieve stability under pre-set parameters. Furthermore, compared with ZNN models designed in other inventions, the predefined-time ZNN model of this invention avoids a complex parameter tuning process. It has the advantages of strong robustness, high computational accuracy, and fast solution speed.

[0006] The technical solution adopted in this invention is as follows: A predefined time-varying ZNN model design method for solving the pseudo-inverse of a time-varying matrix includes the following steps: Step 1: Solve for the pseudo-inverse matrix based on the time-varying matrix and establish the error function. Here, the time-varying matrix is ​​known, while the pseudo-inverse matrix to be solved is unknown. Step 2: Design a predefined time function based on the high-gain method and introduce it into the ZNN model; Step 3: To address the bounded noise interference in the ZNN model solution process, a perturbation estimator containing a predefined time function is introduced to construct a subsystem of the neural network model; Step 4: Construct the Lyapunov function for stability analysis. The analysis shows that the ZNN model can effectively suppress noise and solve the pseudo-inverse matrix of the time-varying matrix in a predefined time.

[0007] Step 5: Apply the predefined time ZNN model to the control of the chaotic system of permanent magnet synchronous motor.

[0008] In step 1, the process of finding the pseudo-inverse matrix based on the time-varying matrix specifically includes: (1); In equation (1), The transpose symbol of a matrix; for any time-varying matrix in equation (1) ; Represents the set of real numbers; Indicates the number of rows in the matrix; Indicates the number of columns in the matrix.

[0009] If the unknown matrix If one or more of the equations in (1) are satisfied, then the unknown matrix is... yes The pseudo-inverse matrix; In addition, for time-varying matrices If either row or column is full rank, then they exist respectively. and This holds true for non-singular matrices; Therefore, the pseudo-inverse matrix can be obtained. The following equation must be satisfied: (2); In equation (2), express The transpose of the matrix; Then, based on equation (2), the following error matrix is ​​obtained. : (3); In step 2, the following predefined time function is proposed; (4); In equation (4), Represents a predefined time function; Indicates the preset time; Represents a time variable; It is an adjustable parameter that is greater than zero.

[0010] Introducing a predefined time function (4), based on the error matrix of equation (3) Construct a predefined time ZNN model: (5); In equation (5), Representing the error function The first derivative matrix; Represents a predefined time function The first derivative; Based on equations (3) and (5), the neural network state matrix for solving the pseudo-inverse matrix of the time-varying matrix is ​​obtained. Design formula: (6); In equation (6), Representing the state of a neural network The first derivative matrix; express The first derivative matrix; express The transpose of the matrix; Based on the noise interference in solving the time-varying matrix using the ZNN model, the following predefined time-varying ZNN model is designed: (7); In equation (7), This is a time-varying noise matrix; Each element in satisfy as well as ,at the same time, and These represent bounded constants that are greater than zero; express The first derivative matrix.

[0011] In step 3, estimators are designed respectively. and Used for estimation and : (8); In equation (8), express The first derivative matrix; express The first derivative matrix; definition This yields a predefined time ZNN model containing a perturbation estimator: (9); In equation (9), Representing the error function and estimator The difference matrix; express The first derivative matrix; express The first derivative matrix; and They represent and exist The matrix limit value.

[0012] Therefore, the definition The following error system is obtained: (10); In equation (10), express The square of the first derivative; Represents the time-varying noise matrix The derivative matrix; Represents the time-varying noise matrix and estimator The difference matrix; and They represent and exist The limit value of the matrix; Consider the following transformation: (11); In equation (11), Representing the error function and estimator The transformation matrix of the difference; Represents the time-varying noise matrix and estimator The transformation matrix of the difference.

[0013] because and Each element and They all have the same dynamic equations, and based on equations (10) and (11), the following error subsystem is obtained: (12); In equation (12), express child elements; express child elements; express The first derivative matrix of the subelements; express The second derivative; express The reciprocal of the first derivative; Indicates the number of rows in the matrix; Indicates the number of columns in the matrix; and They represent and exist The matrix limit value.

[0014] By simplifying the system formula (12), we can obtain: (13); In equation (13), This represents the simplified error subsystem. This represents the first derivative matrix of the simplified error subsystem. and Representing different coefficient matrices, express The second derivative, express The reciprocal of the first derivative, This represents a matrix containing noise in the subsystem. The simplified error subsystem is represented in The limit value of the matrix; in: , , , and .

[0015] For matrix sum matrix ,have Makes the following two inequalities true:

[0016] in: Represents a constant greater than zero. Represents the identity matrix.

[0017] Step 4 includes: Construct the following Lyapunov function: (14); In equation (14), Representing the error subsystem Lyapunov functions; Represents the simplified error subsystem The transpose of the matrix; Through derivation, we obtain the following: , (15); In equation (15), Representing the error matrix The square of; Represents the transformation matrix The square of; And when , (16); in: and Representing matrices respectively The maximum and minimum eigenvalues; Representing the noise matrix The maximum bounded value, It is an adjustable parameter that is greater than zero.

[0018] In step 5, the predefined time ZNN model is applied to the control of the chaotic system of permanent magnet synchronous motor to verify the performance of the predefined time ZNN model. The chaotic system of a controllable permanent magnet synchronous motor (PMSM) is described as follows: (17); in, , and These represent the three different state elements of the PMSM model; , and The first derivatives of the three different state elements of the PMSM model are represented. and Indicates the uncertainty term; and This indicates external interference to the system; and Indicates controller; definition It is a state variable , and The column vector formed; All of these parameters can be adjusted.

[0019] To better describe the simulation process of a predefined time ZNN model with a perturbation estimator applied to a chaotic system of a controllable permanent magnet synchronous motor (PMSM), an error vector is defined. as follows: (18); in: Represents the error vector. These are the error vectors. The three child elements; Therefore, the error dynamic system of the chaotic system equations of a controllable permanent magnet synchronous motor (PMSM) can be expressed as: (19); in: , and Representing error sub-elements respectively and The first derivative; In a noisy environment, the aforementioned controllable permanent magnet synchronous motor (PMSM) chaotic system can be simplified as follows: (20); in: Represents the error vector The first derivative vector; This represents the noise interference vector present in a chaotic system of a controllable permanent magnet synchronous motor (PMSM).

[0020] in, , In the above formula, and Indicates the uncertainty term; and This indicates external interference to the system; definition: , and .

[0021] Similarly, regarding the noise problem in the aforementioned chaotic system of a controllable permanent magnet synchronous motor (PMSM), the following estimator exists. and The designs are as follows: (twenty one); in: Represents the error vector The first derivative; This represents the composite noise interference vector present in a chaotic system of a controllable permanent magnet synchronous motor (PMSM); and They represent and The estimator; One of the estimators has the following specific form: (twenty two); in: , and They represent estimators respectively. child elements; , and They represent , and The first derivative; , and These represent the estimated values ​​of three different state elements of the PMSM model; Furthermore, the noise estimator is designed as follows: (twenty three); in: Representation estimator The first derivative vector.

[0022] Based on the above design, define and The following formula can be obtained: (twenty four); in: Representing the error function and estimator The difference vector; express The derivative vector; Represents the noise interference vector and estimator The difference vector; therefore, (25); in: , and They represent child elements; , and Represent the difference vectors respectively The first derivative of the sub-elements; , and These represent the three different state elements of the PMSM model; , and These represent the estimated terms for the three different state elements of the PMSM model; and These represent controllers with three different state variables; definition , and .

[0023] Controller can be obtained and The design formula is as follows: (26); in: Indicates composite noise and The difference vector; Indicates composite noise and The difference vector; Indicates composite noise and The difference vector; Parameters are designed as follows and .

[0024] Uncertainty Term and Set to respectively , and ; And the external interferences are respectively , and .

[0025] initial value .

[0026] This invention provides a predefined time-varying ZNN model design method for solving the pseudo-inverse of a time-varying matrix, which has the following advantages: 1) The predefined-time ZNN model designed in this invention introduces a predefined time function, which can obtain the actual matrix inverse at a preset time. After this preset time, the error between the approximate inverse and the actual inverse can still maintain high accuracy.

[0027] 2) Compared with other inventions, the estimator introduced in this invention can effectively suppress the influence of noise in the presence of unknown noise.

[0028] 3) The superior performance of the predefined time ZNN model of the present invention was verified by the control of the chaotic system of the controllable permanent magnet synchronous motor. Attached Figure Description

[0029] The present invention will be further described below with reference to the accompanying drawings and embodiments: Figure 1 This is a flowchart of the method of the present invention.

[0030] Figure 2 State curve 1 for a chaotic system of a controllable permanent magnet synchronous motor; Figure 3 State curve 2 for a chaotic system of a controllable permanent magnet synchronous motor; Figure 4 State curve 3 for a chaotic system of a controllable permanent magnet synchronous motor; Figure 5 State curve four for a chaotic system of a controllable permanent magnet synchronous motor. Detailed Implementation

[0031] A predefined time-varying ZNN model design method for solving the pseudoinverse of a time-varying matrix. It includes the following steps: Step 1: Establish an error function based on the basic knowledge of finding the pseudo-inverse of a time-varying matrix, where the time-varying matrix is ​​known, and the pseudo-inverse matrix to be solved is unknown; Step 2: Design a predefined time function based on the high-gain method and introduce it into the ZNN model. The ZNN model solution process contains bounded noise interference. Step 3: For bounded noise interference, introduce a perturbation estimator containing a predefined time function and construct a subsystem of a neural network model; Step 4: Construct the Lyapunov function and perform stability analysis. The stability analysis shows that ZNN can effectively suppress noise and solve the pseudo-inverse of the time-varying matrix in a predefined time. Step 5: Apply the predefined time ZNN model to the control of the chaotic system of permanent magnet synchronous motor to verify the performance of the predefined time ZNN model.

[0032] In step 1, the process of finding the pseudo-inverse matrix based on the time-varying matrix specifically includes: (1); In equation (1), The transpose symbol of a matrix; for any time-varying matrix in equation (1) ; Represents the set of real numbers; Indicates the number of rows in the matrix; Indicates the number of columns in the matrix.

[0033] If the unknown matrix If one or more of the equations in (1) are satisfied, then the unknown matrix is... yes The pseudo-inverse matrix; In addition, for time-varying matrices If either row or column is full rank, then they exist respectively. and This holds true for non-singular matrices; Therefore, the pseudo-inverse matrix can be obtained. The following equation must be satisfied: (2); In equation (2), express The transpose of the matrix; Then, based on equation (2), the following error matrix is ​​obtained. : (3); In step 2, the following predefined time function is proposed; (4); In equation (4), Represents a predefined time function; Indicates the preset time; Represents a time variable; It is an adjustable parameter that is greater than zero.

[0034] Introducing a predefined time function (4), based on the error matrix of equation (3) Construct a predefined time ZNN model: (5); In equation (5), Representing the error function The first derivative matrix; Represents a predefined time function The first derivative.

[0035] Based on equations (3) and (5), the neural network state matrix for solving the pseudo-inverse matrix of the time-varying matrix is ​​obtained. Design formula: (6); In equation (6), Representing the state of a neural network The first derivative matrix; express The first derivative matrix; express The transpose of the matrix; Based on the noise interference in solving the time-varying matrix using the ZNN model, the following predefined time-varying ZNN model is designed: (7); In equation (7), This is a time-varying noise matrix; Each element in satisfy as well as ,at the same time, and These represent bounded constants that are greater than zero; express The first derivative matrix.

[0036] In step 3, estimators are designed respectively. and Used for estimation and : (8); In equation (8), express The first derivative matrix; express The first derivative matrix; definition This yields a predefined time ZNN model containing a perturbation estimator: (9); In equation (9), Representing the error function and estimator The difference matrix; express The first derivative matrix; express The first derivative matrix; and They represent and exist The matrix limit value.

[0037] Therefore, the definition The following error system is obtained: (10); In equation (10), express The square of the first derivative; Represents the time-varying noise matrix The derivative matrix; Represents the time-varying noise matrix and estimator The difference matrix; and They represent and exist The limit value of the matrix; Consider the following transformation: (11); In equation (11), Representing the error function and estimator The transformation matrix of the difference; Represents the time-varying noise matrix and estimator The transformation matrix of the difference.

[0038] because and Each element and They all have the same dynamic equations, and based on equations (10) and (11), the following error subsystem is obtained: (12); In equation (12), express child elements; express child elements; express The first derivative matrix of the subelements; express The second derivative; express The reciprocal of the first derivative; Indicates the number of rows in the matrix; Indicates the number of columns in the matrix; and They represent and exist The matrix limit value.

[0039] By simplifying the system formula (12), we can obtain: (13); In equation (13), This represents the simplified error subsystem. This represents the first derivative matrix of the simplified error subsystem. and Representing different coefficient matrices, express The second derivative, express The reciprocal of the first derivative, This represents a matrix containing noise in the subsystem. The simplified error subsystem is represented in The limit value of the matrix; in: , , , and .

[0040] For matrix sum matrix ,have Makes the following two inequalities true:

[0041] in: Represents a constant greater than zero. Represents the identity matrix.

[0042] Step 4 includes: Construct the following Lyapunov function: (14); In equation (14), Representing the error subsystem Lyapunov functions; Represents the simplified error subsystem The transpose of the matrix; Through derivation, we obtain the following: , (15); In equation (15), Representing the error matrix The square of; Represents the transformation matrix The square of; And when , (16); in: and Representing matrices respectively The maximum and minimum eigenvalues; Representing the noise matrix The maximum bounded value, It is an adjustable parameter that is greater than zero.

[0043] In step 5, the predefined time ZNN model is applied to the control of the chaotic system of permanent magnet synchronous motor to verify the performance of the predefined time ZNN model. The chaotic system of a controllable permanent magnet synchronous motor (PMSM) is described as follows: (17); in, , and These represent the three different state elements of the PMSM model; , and The first derivatives of the three different state elements of the PMSM model are represented. and Indicates the uncertainty term; and This indicates external interference to the system; and Indicates controller; definition It is a state variable , and The column vector formed; All of these parameters can be adjusted.

[0044] To better describe the simulation process of a predefined time ZNN model with a perturbation estimator applied to a chaotic system of a controllable permanent magnet synchronous motor (PMSM), an error vector is defined. as follows: (18); in: Represents the error vector. These are the error vectors. The three child elements; Therefore, the error dynamic system of the chaotic system equations of a controllable permanent magnet synchronous motor (PMSM) can be expressed as: (19); in: , and Representing error sub-elements respectively and The first derivative; In a noisy environment, the aforementioned controllable permanent magnet synchronous motor (PMSM) chaotic system can be simplified as follows: (20); in: Represents the error vector The first derivative vector; This represents the noise interference vector present in a chaotic system of a controllable permanent magnet synchronous motor (PMSM).

[0045] in, , In the above formula, and Indicates the uncertainty term; and This indicates external interference to the system; definition: , and .

[0046] Similarly, regarding the noise problem in the aforementioned chaotic system of a controllable permanent magnet synchronous motor (PMSM), the following estimator exists. and The designs are as follows: (twenty one); in: Represents the error vector The first derivative; This represents the composite noise interference vector present in a chaotic system of a controllable permanent magnet synchronous motor (PMSM); and They represent and The estimator; One of the estimators has the following specific form: (twenty two); in: , and They represent estimators respectively. child elements; , and They represent , and The first derivative; , and These represent the estimated values ​​of three different state elements of the PMSM model; Furthermore, the noise estimator is designed as follows: (twenty three); in: Representation estimator The first derivative vector.

[0047] Based on the above design, define and The following formula can be obtained: (twenty four); in: Representing the error function and estimator The difference vector; express The derivative vector; Represents the noise interference vector and estimator The difference vector; therefore, (25); in: , and They represent child elements; , and Represent the difference vectors respectively The first derivative of the sub-elements; , and These represent the three different state elements of the PMSM model; , and These represent the estimated terms for the three different state elements of the PMSM model; and These represent controllers with three different state variables; definition , and .

[0048] Controller can be obtained and The design formula is as follows: (26); in: Indicates composite noise and The difference vector; Indicates composite noise and The difference vector; Indicates composite noise and The difference vector; Parameters are designed as follows and .

[0049] Uncertainty Term and Set to respectively , and ; And the external interferences are respectively , and .

[0050] initial value .

[0051] Based on the above parameter settings, apply the high-gain function, according to Figures 2-5 As shown, Figures 2-4 Three different state variables of the controllable permanent magnet synchronous motor chaotic system are shown respectively. , and The change curve shows that the controller designed using the predefined time ZNN in this invention... , and Enabling three different state variables of a controllable permanent magnet synchronous motor chaotic system , and At the preset time Below, completely suppress the disturbance. A stable state has been reached.

[0052] Figure 5 It can be seen that the spatial trajectories of the three different state variables of the controllable permanent magnet synchronous motor chaotic system, under noise disturbance, utilize the controller designed in this invention. , and Able to set a time It restores to the initial state, achieving stable operation at any time.

[0053] The chaotic system of the permanent magnet synchronous motor reached stability in about 0.2 seconds.

[0054] In summary, in a chaotic system of a permanent magnet synchronous motor with unknown noise, the predefined time ZNN model of this invention can achieve stability under pre-set parameters. Furthermore, compared with ZNN models designed by other inventions, the predefined time ZNN model of this invention avoids a complex parameter tuning process.

Claims

1. A predefined time-varying ZNN model design method for solving the pseudo-inverse of a time-varying matrix, characterized in that... Includes the following steps: Step 1: Solve for the pseudo-inverse matrix based on the time-varying matrix and establish the error function. Here, the time-varying matrix is ​​known, while the pseudo-inverse matrix to be solved is unknown. Step 2: Design a predefined time function based on the high-gain method and introduce it into the ZNN model; Step 3: To address the bounded noise interference in the ZNN model solution process, a perturbation estimator containing a predefined time function is introduced to construct a subsystem of the neural network model; Step 4: Construct the Lyapunov function to perform stability analysis and obtain the pseudo-inverse matrix of the time-varying matrix that the ZNN model can solve in a predefined time.

2. The predefined time ZNN model design method for solving the pseudo-inverse of a time-varying matrix according to claim 1, characterized in that: It also includes step 5: applying the predefined time ZNN model designed by the design method described in claim 1 to the control of the chaotic system of permanent magnet synchronous motor.

3. The predefined time ZNN model design method for solving the pseudo-inverse of a time-varying matrix according to claim 2, characterized in that: In step 1, the process of finding the pseudo-inverse matrix based on the time-varying matrix specifically includes: （1）； In equation (1), The transpose symbol of a matrix; for any time-varying matrix in equation (1) , Represents the set of real numbers; Indicates the number of rows in the matrix; Indicates the number of columns in the matrix; If the unknown matrix If one or more of the equations in (1) are satisfied, then the unknown matrix is... yes The pseudo-inverse matrix; In addition, for time-varying matrices If either row or column is full rank, then they exist respectively. and This holds true for non-singular matrices; Therefore, the pseudo-inverse matrix can be obtained. The following equation must be satisfied: （2）； In equation (2), express The transpose of the matrix; Based on equation (2), the following error matrix is ​​obtained. : （3）。 4. The predefined time ZNN model design method for solving the pseudo-inverse of a time-varying matrix according to claim 3, characterized in that: In step 2, the following predefined time function is proposed; （4）； In equation (4), Represents a predefined time function; Indicates the preset time; Represents a time variable; An adjustable parameter that is greater than zero; Introducing a predefined time function (4), based on the error matrix of equation (3) Construct a predefined time ZNN model: （5）； In equation (5), Representing the error function The first derivative matrix; Represents a predefined time function The first derivative.

5. The predefined time ZNN model design method for solving the pseudo-inverse of a time-varying matrix according to claim 4, characterized in that: Based on equations (3) and (5), the neural network state matrix for solving the pseudo-inverse matrix of the time-varying matrix is ​​obtained. Design formula: （6）； In equation (6), Representing the state of a neural network The first derivative matrix; express The first derivative matrix; express The transpose of the matrix; Based on the noise interference in solving the time-varying matrix using the ZNN model, the following predefined time-varying ZNN model is designed: （7）； In equation (7), This is a time-varying noise matrix; Each element in satisfy as well as ,at the same time, and These represent bounded constants that are greater than zero; express The first derivative matrix.

6. The predefined time-varying ZNN model design method for solving the pseudo-inverse of a time-varying matrix according to claim 5, characterized in that: In step 3, estimators are designed respectively. and Used for estimation and : （8）； In equation (8), express The first derivative matrix; express The first derivative matrix; definition This yields a predefined time ZNN model containing a perturbation estimator: （9）； In equation (9), Representing the error function and estimator The difference matrix; express The first derivative matrix; express The first derivative matrix; and They represent and exist The limit value of the matrix; Therefore, the definition The following error system is obtained: （10）； In equation (10), express The square of the first derivative; Represents the time-varying noise matrix The derivative matrix; Represents the time-varying noise matrix and estimator The difference matrix; and They represent and exist The limit value of the matrix; Consider the following transformation: （11）； In equation (11), Representing the error function and estimator The transformation matrix of the difference; Represents the time-varying noise matrix and estimator The transformation matrix of the difference; because and Each element and They all have the same dynamic equations, and based on equations (10) and (11), the following error subsystem is obtained: （12）； In equation (12), express child elements; express child elements; express The first derivative matrix of the subelements; express The second derivative; express The reciprocal of the first derivative; Indicates the number of rows in the matrix; Indicates the number of columns in the matrix; and They represent and exist The limit value of the matrix; By simplifying the system formula (12), we can obtain: （13）； In equation (13), This represents the simplified error subsystem. This represents the first derivative matrix of the simplified error subsystem. and Representing different coefficient matrices, express The second derivative, express The reciprocal of the first derivative, This represents a matrix containing noise in the subsystem. The simplified error subsystem is represented in The limit value of the matrix; in: , , , and ; For matrix sum matrix ,have Makes the following two inequalities true: in: Represents a constant greater than zero. Represents the identity matrix.

7. The predefined time ZNN model design method for solving the pseudo-inverse of a time-varying matrix according to claim 6, characterized in that: Step 4 includes: Construct the following Lyapunov function: （14）； In equation (14), Representing the error subsystem Lyapunov functions; Represents the simplified error subsystem The transpose of the matrix; Through derivation, we obtain the following: , （15）； In equation (15), Representing the error matrix The square of; Represents the transformation matrix The square of; And when , （16）； in: and Representing matrices respectively The maximum and minimum eigenvalues; Representing the noise matrix The maximum bounded value, It is an adjustable parameter that is greater than zero.

8. The predefined time ZNN model design method for solving the pseudo-inverse of a time-varying matrix according to claim 7, characterized in that: In step 5, the predefined time ZNN model is applied to the control of the chaotic system of permanent magnet synchronous motor to verify the performance of the predefined time ZNN model. The chaotic system of a controllable permanent magnet synchronous motor (PMSM) is described as follows: （17）； in, , and These represent the three different state elements of the PMSM model; , and The first derivatives of the three different state elements of the PMSM model are represented. and Indicates the uncertainty term; and This indicates external interference to the system; and Indicates controller; definition It is a state variable , and The column vector formed by these.

9. The predefined time ZNN model design method for solving the pseudo-inverse of a time-varying matrix according to claim 8, characterized in that: To better describe the simulation process of a predefined time ZNN model with a perturbation estimator applied to a chaotic system of a controllable permanent magnet synchronous motor (PMSM), an error vector is defined. as follows: （18）； in: Represents the error vector. These are the error vectors. The three child elements; Therefore, the error dynamic system of the chaotic system equations of the controllable permanent magnet synchronous motor (PMSM) can be expressed as: （19）； in: , and Representing error sub-elements respectively and The first derivative; In a noisy environment, the aforementioned controllable permanent magnet synchronous motor (PMSM) chaotic system can be simplified as follows: （20）； in: Represents the error vector The first derivative vector; This represents the noise interference vector present in a chaotic system of a controllable permanent magnet synchronous motor (PMSM); in, , In the above formula, and Indicates the uncertainty term; and This indicates external interference to the system; definition: , and .

10. The predefined time ZNN model design method for solving the pseudo-inverse of a time-varying matrix according to claim 9, characterized in that: To address the noise problem in a chaotic system of a controllable permanent magnet synchronous motor (PMSM), the following estimator is proposed. and The designs are as follows: （21）； in: Represents the error vector The first derivative; This represents the composite noise interference vector present in a chaotic system of a controllable permanent magnet synchronous motor (PMSM); and They represent and The estimator; One of the estimators has the following specific form: （22）； in: , and They represent estimators respectively. child elements; , and They represent , and The first derivative; , and These represent the estimated values ​​of three different state elements of the PMSM model; Furthermore, the noise estimator is designed as follows: （23）； in: Representation estimator The first derivative vector; Based on the above design, define and The following formula can be obtained: （24）； in: Representing the error function and estimator The difference vector; express The derivative vector; Represents the noise interference vector and estimator The difference vector; therefore, （25）； in: , and They represent child elements; , and Represent the difference vectors respectively The first derivative of the sub-elements; , and These represent the three different state elements of the PMSM model; , and These represent the estimated terms for the three different state elements of the PMSM model; and These represent controllers with three different state variables; definition , and ; Get controller and The design formula is as follows: （26）； in: Indicates composite noise and The difference vector; Indicates composite noise and The difference vector; Indicates composite noise and The difference vector; Uncertainty Term and Set to respectively , and ; And the external interferences are respectively , and .

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