A sliding mode control method for permanent magnet synchronous motor based on predetermined time

Abstract

The present invention discloses a method for sliding mode control of a permanent magnet synchronous motor based on predetermined time. The method specifically includes: modeling a mathematical model of the system and establishing a local model of the speed loop based on model-free theory; designing a predetermined time sliding mode controller based on predetermined time stability theory and proving its stability; treating the unknown parts of the model and external disturbances as lumped disturbances, defining an intermediate variable, designing an extended state observer to estimate the disturbances, and proving its stability. The present invention models the system based on model-free theory and designs a predetermined time sliding mode controller and an extended state observer based on the established model. Ultimately, the system converges and stabilizes within the predetermined time, and the convergence time is independent of the initial state of the system. The method can effectively suppress the chattering of traditional sliding mode control methods and improve the control accuracy, anti-interference capability, and robustness of the permanent magnet synchronous motor.

Description

Technical Field

[0001] The present invention relates to the field of permanent magnet synchronous motor control, and more specifically, to a predetermined time sliding mode control method based on an extended state observer for a nonlinear system of a permanent magnet synchronous motor. Background Art

[0002] Currently, permanent magnet synchronous motors (PMSMs) are widely used in aerospace, electric vehicles, and other fields due to their advantages such as high efficiency and high dynamic performance. However, because PMSM servo systems are strongly coupled nonlinear systems, they are subject to parameter perturbations, friction torque, mismatched disturbances, and unmodeled dynamics. While the PID control method commonly used in industrial control offers a simple structure and good control performance, it cannot accurately and quickly adjust control parameters to adapt to system changes in such strongly coupled nonlinear systems as PMSMs, thereby affecting the overall system response speed and control quality. Furthermore, fields such as aerospace and electric vehicles place extremely high demands on motor control stability and response speed, making traditional PID control algorithms difficult to meet. In recent years, sliding mode control algorithms have been widely used in practical engineering control due to their relaxed requirements for the mathematical model of the controlled object and their strong anti-interference performance. However, traditional sliding mode control algorithms are prone to small-amplitude, high-frequency chattering, which can cause system vibration, reduce system efficiency, and severely affect the control performance of the controlled system. Currently, a large number of relevant literature has been published on the control of PMSMs. The paper ["Adaptive position tracking control of permanent magnet synchronous motor based on RBF fast terminal sliding mode control." Neurocomputing 115(2013):23-30] proposes a neural network adaptive sliding mode control algorithm. By combining the fast terminal sliding mode method with the radial basis function, it not only compensates for the network approximation error but also solves the problem of the fast terminal sliding mode's high dependence on the parameters of the permanent magnet synchronous motor, ultimately achieving closed-loop system stability. The paper ["Finite-time adaptive neural network event-triggered output feedback control for PMSMs." Neurocomputing(2023).] proposes a finite-time adaptive neural network event-triggered output feedback control method, which can achieve closed-loop stability within a finite time. In recent years, the scheduled time control method has been proposed. The convergence time is not affected by the system's initial state and most control parameters. The system can be stabilized within the scheduled time, which can meet the motor control performance requirements of fields such as aerospace and electric vehicles. However, the application of the scheduled time control method to the motor control field is still relatively limited. However, the permanent magnet synchronous motor is a complex nonlinear system, which makes it difficult to accurately describe it with the established mathematical model. The model-free theory can overcome the challenges brought by the limitations of the motor mathematical model because it does not rely on the precise mathematical model of the motor.Therefore, it is urgent to study how to establish a super-local model based on model-free theory and combine the scheduled time method with the sliding mode control method, how to effectively suppress the chattering of the traditional sliding mode control algorithm, and apply the scheduled time sliding mode controller to the control of the motor system to improve the response speed, anti-interference, robustness and stability of the motor control system. Summary of the Invention

[0003] The present invention aims to overcome the shortcomings of the prior art by providing a time-scheduled sliding-mode control method for a permanent magnet synchronous motor. This method establishes a hyperlocal mathematical model based on model-free theory, reducing reliance on the accuracy of the motor system's mathematical model. Based on this model, an extended state observer is designed, and the system's lumped disturbance is estimated. A time-scheduled sliding-mode controller is then designed to achieve system convergence and stability within a predetermined time, independent of the system's initial state. This method also effectively suppresses chattering associated with traditional sliding-mode control algorithms. Ultimately, this method improves the motor control system's response speed, interference rejection, robustness, and stability, thereby overcoming the shortcomings mentioned in the background art.

[0004] To achieve the above object of the invention, the present invention provides a sliding mode control method for a permanent magnet synchronous motor based on a predetermined time, characterized in that it includes the following steps:

[0005] Step 1: Model the mathematical model of the system and establish a speed loop local model based on the model-free theory;

[0006] Step 2: Based on the time-scheduled stability theory, design a time-scheduled sliding mode controller and prove its stability;

[0007] Step 3: For the unknown parts of the model and external disturbances, treat them as lumped disturbances, define an intermediate variable, design an extended state observer to estimate it, and prove its stability.

[0008] The object of the present invention is achieved in this way.

[0009] The present invention provides a method for sliding mode control of a permanent magnet synchronous motor based on a predetermined time. The method specifically includes: establishing a super-local model of the permanent magnet synchronous motor speed loop based on model-free theory; designing a predetermined time sliding mode controller based on the established model and in accordance with predetermined time stability theory; and proving the stability of the controller; designing an extended state observer to estimate the lumped disturbance in the established speed loop mathematical model by defining an intermediate variable; compensating the predetermined time sliding mode controller; and proving the stability of the observer. The present invention can achieve system convergence and stability within the predetermined time, and the system convergence time is independent of the system's initial state. Ultimately, it can effectively suppress chattering in traditional sliding mode control algorithms and, to a certain extent, improve the response speed, anti-interference performance, robustness, and stability of the motor control system. BRIEF DESCRIPTION OF THE DRAWINGS

[0010] Figure 1 It is a schematic diagram of the implementation steps of a specific implementation method of a permanent magnet synchronous motor sliding mode control method based on predetermined time of the present invention.

[0011] Figure 2 It is a schematic diagram of the principle of a specific implementation of a sliding mode control method for a permanent magnet synchronous motor based on predetermined time of the present invention. DETAILED DESCRIPTION

[0012] To make the technical solutions and advantages of the present invention more clear, the following describes specific embodiments of the present invention in conjunction with the accompanying drawings to facilitate a better understanding of the present invention by those skilled in the art. It should be noted that in the following description, when detailed descriptions of known functions and designs may dilute the main content of the present invention, such descriptions will be omitted.

[0013] Combine Figure 1 The present invention provides a sliding mode control method for a permanent magnet synchronous motor based on predetermined time, comprising the following steps: modeling a mathematical model of the system, and establishing a local model of the speed loop based on the model-free theory; designing a predetermined time sliding mode controller based on the predetermined time stability theory, and proving its stability; for the unknown part of the model and the external disturbance, regard them as lumped disturbances, define an intermediate variable, design an extended state observer to estimate them, and prove their stability.

[0014] Combine Figure 2 The present invention provides a sliding mode control method for a permanent magnet synchronous motor based on a predetermined time, which involves establishing a local model of the permanent magnet synchronous motor speed loop, designing a predetermined time sliding mode control reaching law, and and scheduled time sliding mode controller The stability of the scheduled time sliding mode controller is proved, and an extended state observer is designed to estimate the lumped disturbance p(t), and its stability is proved.

[0015] The present invention discloses a sliding mode control method for a permanent magnet synchronous motor based on a predetermined time, and the specific implementation steps are as follows:

[0016] Step 1: Model the mathematical model of the system and establish a speed loop local model based on the model-free theory;

[0017] First, the mathematical equation for the surface-mount permanent magnet synchronous motor is:

[0018]

[0019] Where, ω m is the mechanical angular velocity of the rotor, i d,i q are the d-axis and q-axis components of the stator current respectively, J is the rotor moment of inertia, T L is the load torque, B m is the viscous friction coefficient, u d ,u q are the d-axis and q-axis components of the stator voltage, L is the stator inductance, P n is the number of pole pairs of the motor, R is the stator resistance of the motor, and Φ is the permanent magnet flux.

[0020] Based on the model-free theory, the hyperlocal mathematical model of a single-input single-output nonlinear system can be expressed as:

[0021]

[0022] Where, u is the input signal, y is the output signal, and dis is the sum of the known parameters of the system and the unknown disturbance part.

[0023] According to the motor speed equation (1) and the hyperlocal model equation (2), the motor speed loop hyperlocal model can be designed as:

[0024]

[0025] Where q is the sum of some known parameters and unknown disturbances in the system, α m for

[0026] Furthermore, according to formula (3), the speed loop local model can be further expressed as:

[0027]

[0028] Where r is

[0029] Step 2: Based on the time-scheduled stability theory, design a time-scheduled sliding mode controller and prove its stability;

[0030] First, the sliding surface function of the scheduled time sliding mode controller is defined as:

[0031]

[0032] Where, e is the given reference mechanical angular velocity of the system and the actual mechanical angular velocity ω m The error is n1>0, n2>0, c>0, d>0, d<c.

[0033] For the sliding surface function s ω Taking the derivative we get:

[0034]

[0035] According to formula (6), we can get

[0036]

[0037] According to the equivalent control principle, the speed loop preset time sliding mode controller is designed as follows:

[0038]

[0039] Where i qeq is the sliding surface function The input value, i qsw is the switching control law calculated by the reaching law of the sliding mode control with a predetermined time.

[0040] make The speed loop equivalent controller can be obtained, and the expression is:

[0041]

[0042] Where, is the estimated value of the lumped disturbance p obtained by the extended state observer.

[0043] Design the reaching law of the scheduled time sliding mode control, which is expressed as:

[0044]

[0045] Where, q1>0, q2>0, q3>0, q4>0, T cω >0, T cω The system scheduled time means that the designed controller will cω Internal stability.

[0046] Therefore, according to equations (7) and (10), the switching control law expression of the scheduled time sliding mode controller can be obtained as follows:

[0047]

[0048] Therefore, the expression of the speed loop scheduled time sliding mode controller is:

[0049]

[0050] The stability of the scheduled time sliding mode controller is proved.

[0051] Assume that there exists a class of nonlinear systems (13), is a class of open neighborhoods defined at the origin A continuous nonlinear function on , t is the time variable, is a state variable and satisfies h(0)=0.

[0052]

[0053] Definition of finite-time stability: Consider a nonlinear system (13) if the system is asymptotically stable at the origin and for any initial state x(t0) = x0, There exists a system convergence time function T(x0) determined by the initial state x0, such that the following conditions hold: and If x(t) = 0, the system (13) is said to be finite-time stable.

[0054] Definition of fixed-time stability: Consider a nonlinear system (13), let T(x0) be the convergence time of the system and it is bounded. If the system is stable at the origin for a finite time and there exists a positive constant T max And it has nothing to do with the initial state of the system, so that the following conditions hold: T(x0)≤T max , then the system (13) is said to be fixed-time stable.

[0055] Definition of fixed-time stability: Consider a nonlinear system (13), let the system convergence time be T(x0), T(x0) is bounded and positive, if the origin of the system is fixed-time stable and there exists a positive constant T c , so that T(x0)≤T c , Then the system (13) is said to be stable for a predetermined time.

[0056] To prove the stability of the sliding mode controller, a Lyapunov function is selected, which is expressed as:

[0057]

[0058] By taking the derivative of the Lyapunov function (14), we can obtain:

[0059]

[0060] Formula (15) can be rewritten as:

[0061]

[0062] Assume T ω is the controller convergence time, x0 is the initial state of the system, and Because for the Lyapunov function, V0>0, V will be at the predetermined time T cωConverges to 0, then V f =0.

[0063] Then further transformation of formula (16) can be obtained:

[0064]

[0065] So we can get:

[0066] T ω ≤T cω (18)

[0067] Therefore, according to the definition of predetermined time stability, the designed speed loop predetermined time sliding mode controller will be cω Internally stable and satisfies T ω ≤T cω .

[0068] Step 3: For the unknown parts of the model and external disturbances, treat them as lumped disturbances, define an intermediate variable, design an extended state observer to estimate it, and prove its stability.

[0069] First, define

[0070]

[0071] Where, e p is the lumped disturbance prediction error, is the estimated value of the lumped disturbance p.

[0072] The local model of the speed loop is written as:

[0073]

[0074] The extended state observer is constructed based on formula (20):

[0075]

[0076] Where, f1>0, f2>0, define α2=2α1-1,

[0077] Then, by subtracting formula (20) from formula (21), we can get:

[0078]

[0079] Where, is the estimated mechanical angular velocity of the speed ring motor, ω m is the actual mechanical angular velocity.

[0080] Based on formula (22), new variables are defined:

[0081]

[0082] Taking the derivative with respect to the new variable we get:

[0083]

[0084] By transforming formula (24), we can get:

[0085]

[0086] Where,

[0087] Let Z = [z1 z2] T ,but

[0088] Therefore, by selecting appropriate parameters f1, f2, α1, B1 and B2 are Hurwitz matrices. Therefore, for any positive definite symmetric matrix There exists a positive definite symmetric matrix Make B1 and B2 satisfy equation (26) respectively.

[0089] B i T P+PB i =-Q i ,i=1,2 (26)

[0090] Choose a Lyapunov function, the expression is:

[0091] V c =Z T PZ (27)

[0092] V c Taking the derivative we get:

[0093]

[0094] Scaling equation (28) yields:

[0095]

[0096] Lemma 1: If the matrix is a positive definite symmetric matrix, then for any vector And A≠0, all satisfy A T QA>0 holds, and A T QA meets:

[0097] λ min (Q)||A|| 2 ≤A T QA≤λmax (Q)||A|| 2 (30)

[0098] From Lemma 1, we can see that for a positive definite symmetric matrix Q i , there is λ min (Q i )||Z|| 2 ≤Z T Q i Z≤λ max (Q i )||Z|| 2 .so, -Z T Q2Z≤-λ min (Q2)||Z‖ 2 .

[0099] Therefore, formula (29) can be further obtained:

[0100]

[0101] Since P is a positive definite symmetric matrix, λ min (P)||Z|| 2 ≤Z T PZ≤λ max (P)||Z‖ 2 , that is, λ min (P)‖Z‖ 2 ≤V c ≤λ max (P)‖Z‖ 2 , so we can get

[0102] Also because so According to formula (23), we can get Established, therefore so Also because so,

[0103] So formula (31) can be obtained

[0104]

[0105] In the formula, let κ1>0,κ2>0.

[0106] Finite-time stability lemma: Consider the nonlinear system (13) and construct a Lyapunov function V defined on the set D o(x), the set D contains the origin, if for any x∈D0\{0}, and V o (x) all satisfy equation (33), then the nonlinear system can be said to be finite-time stable.

[0107]

[0108] Where, 0<τ1<1, λ1>0, λ2>0, the system will Stablize.

[0109] Therefore, according to the finite-time stability theorem, the designed extended state observer will be obs Internal stability:

[0110]

[0111] Therefore, the designed extended state observer is stable.

[0112] The above description is only a preferred specific embodiment of the present invention, but the scope of protection of the present invention is not limited thereto. Any technician familiar with the technical field, within the technical scope disclosed by the present invention, who makes equivalent replacements or changes based on the technical solution and inventive concept of the present invention, should be covered by the scope of protection of the present invention.

Claims

1. A sliding mode control method for a permanent magnet synchronous motor based on predetermined time, characterized in that: The following steps are involved: Step 1: Model the mathematical model of the system and establish a speed loop local model based on the model-free theory; Step 2: Based on the time-scheduled stability theory, design a time-scheduled sliding mode controller and prove its stability; The process of designing the predetermined time sliding mode controller is as follows: Based on the established speed loop local model and according to the given reference mechanical angular velocity and the actual mechanical angular velocity ω m Error Define the sliding surface function of the scheduled time sliding mode controller, the expression is: Wherein, n1>0, n2>0, c>0, d>0, d<c; Based on the predetermined time stability theory, a predetermined time sliding mode controller is designed, and the sliding mode controller is designed as follows: Where, is a sliding mode controller, i qeq for The input value at time i qsw is the switching control law obtained based on the sliding mode control reaching law; a predetermined time sliding mode control reaching law is designed, and the expression is: Where, q1>0, q2>0, q3>0, q4>0, T cω >0, T cω The system scheduled time means that the designed controller will cω internal stability; You can get: According to the sliding surface function You can get: The expression of the speed loop preset time sliding mode controller is: Where, α m for r is P n is the number of pole pairs of the motor, Φ is the permanent magnet flux, J is the rotor moment of inertia, B m is the viscous friction coefficient, is the estimated value of the lumped disturbance p obtained by the extended state observer; Step 3: For the unknown parts of the model and external disturbances, treat them as lumped disturbances, define an intermediate variable, design an extended state observer to estimate it, and prove its stability.

2. The sliding mode control method of a permanent magnet synchronous motor based on predetermined time according to claim 1, characterized in that: To prove the stability of the scheduled time sliding mode controller in step 2, a Lyapunov function is selected: Taking the derivative we can get: Can get It is proved that the designed scheduled time sliding mode controller will cω Internal stability, let the convergence time be T ω , then T ω ≤T cω .

3. The sliding mode control method of a permanent magnet synchronous motor based on predetermined time according to claim 1, characterized in that: In step 3, the unknown part of the model and the external disturbance are regarded as lumped disturbances, and an intermediate variable is defined. The extended state observer is designed to estimate it. The speed loop local model and the motor mechanical angular velocity estimation are used to calculate the value of the unknown part of the model and the external disturbance. and the actual mechanical angular velocity ω m The error e ω ,definition Construct the extended state observer: Where i q is the q-axis component of the stator current, f1>0, f2>0, define α2=2α1-1, And define an intermediate variable Z = [z1 z2] T : Where, e p is the lumped disturbance prediction error.

4. The sliding mode control method of a permanent magnet synchronous motor based on predetermined time according to claim 1, characterized in that: The step 3 proves the stability of the extended state observer and selects a Lyapunov function, which is expressed as: V c =Z T PZ, where Z = [z1 z2] T , e ω Estimated value of the motor mechanical angular velocity and the actual mechanical angular velocity ω m The error is defined as e p Estimate the total disturbance The error of the lumped disturbance estimate with the lumped disturbance value p is defined as is a positive definite symmetric matrix, for V c Taking the derivative we can get and The convergence time is In the formula, κ1>0, κ2>0, 0<σ1<1, x0 is the initial state of the system. Finally, it is proved that the designed extended state observer is stable.

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