Permanent magnet synchronous motor sliding mode control method based on preset time

Abstract

The invention discloses a permanent magnet synchronous motor sliding mode control method based on preset time, and the method specifically comprises the steps: carrying out the modeling of a system mathematical model, and building a rotating speed ring super-local model according to a model-free theory; based on a predetermined time stability theory, designing a predetermined time sliding mode controller, and carrying out stability proving on the predetermined time sliding mode controller; regarding an unknown part and external disturbance of the model as lumped disturbance, defining an intermediate variable, designing an extended state observer to estimate the lumped disturbance, and proving the stability of the lumped disturbance. Modeling is carried out based on the model-free theory, the sliding mode controller and the expansion state observer in the preset time are designed based on the built model, convergence stability of the system in the preset time is finally achieved, the convergence time is irrelevant to the initial state of the system, buffeting of a traditional sliding mode control method can be effectively restrained, and the control accuracy is improved. And the control precision, the anti-interference capability and the robustness of the permanent magnet synchronous motor are improved.

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] At present, permanent magnet synchronous motors have been widely used in aerospace, electric vehicles and other fields due to their advantages such as high efficiency and high dynamic performance. However, since the permanent magnet synchronous motor servo system is a strongly coupled nonlinear system, it has characteristics such as parameter perturbation, friction torque, mismatch disturbance, and unmodeled dynamics. The PID control method commonly used in industrial control has a simple structure and good control performance. However, when facing a strongly coupled nonlinear system such as a permanent magnet synchronous motor, it is unable to accurately and quickly adjust the control parameters to adapt to the changes in the system, thereby affecting the response speed and control quality of the entire system. However, the fields of aerospace, electric vehicles and other fields have extremely high requirements for the stability and response speed of motor control performance. Therefore, traditional PID control algorithms are difficult to meet their requirements. In recent years, sliding mode control algorithms have been widely used in actual engineering control because they have low requirements on the mathematical model of the controlled object and have strong anti-interference performance. However, traditional sliding mode control algorithms are prone to produce small-amplitude high-frequency jitter, which will cause the system to vibrate, reduce the system's working efficiency, and seriously affect the control performance of the controlled system. At present, there are many relevant literatures on the study of permanent magnet synchronous motor control. Reference ["Adaptive position tracking control of permanent magnet synchronous motor based on RBF fast terminal sliding mode control." Neurocomputing 115(2013):23-30] proposed a neural network adaptive sliding mode control algorithm. By combining the fast terminal sliding mode method with the radial basis function, it can not only compensate for the network approximation error, but also solve the problem that the fast terminal sliding mode is highly dependent on the parameters of the permanent magnet synchronous motor, and finally achieve the stability of the closed-loop system. Reference ["Finite-time adaptive neural network event-triggered output feedback control for PMSMs." Neurocomputing(2023).] proposed a finite-time adaptive neural network event-triggered output feedback control method, which can make the system closed-loop stable within a finite time. In recent years, the scheduled time control method has been proposed. The convergence time is not affected by the initial state of the system and most of the control parameters. The system can be stable within the scheduled time, which can meet the requirements of motor control performance in aerospace, electric vehicles and other fields. However, there are still few related studies on the application of the scheduled time control method in the field of motor control. However, the permanent magnet synchronous motor is a complex nonlinear system, which makes it difficult to accurately describe it with the constructed 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 hyperlocal model based on model-free theory and combine the scheduled time method with the sliding mode control method, how to effectively suppress the jitter 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 purpose of the present invention is to overcome the defects of the prior art, provide a sliding mode control method for a permanent magnet synchronous motor based on a predetermined time, establish a super-local mathematical model based on the model-free theory, reduce the dependence on the accuracy of the mathematical model of the motor system, design an extended state observer based on the established model, and estimate the system lumped disturbance, design a sliding mode controller with a predetermined time, realize the system convergence and stability within the predetermined time, and the convergence time is independent of the initial state of the system, and can effectively suppress the chattering of the traditional sliding mode control algorithm. Ultimately, the response speed, anti-interference, robustness and stability of the motor control system can be improved, thereby overcoming the deficiencies mentioned in the background technology.

[0004] To achieve the above-mentioned object of the invention, the present invention provides a sliding mode control method of a permanent magnet synchronous motor based on a predetermined time, characterized in that it comprises 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 scheduled time stability theory, design a scheduled time 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 them, and prove their stability.

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

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

[0010] Figure 1 It is a schematic diagram of 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 of a permanent magnet synchronous motor based on predetermined time of the present invention. DETAILED DESCRIPTION

[0012] In order to make the technical solutions and advantages of the present invention clearer, the specific implementation of the present invention is described below in conjunction with the accompanying drawings so that those skilled in the art can better understand the present invention. It should be noted that in the following description, when the detailed description of known functions and designs may dilute the main content of the present invention, these descriptions will be omitted here.

[0013] Combination Figure 1 The present invention discloses 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 a speed loop according to the model-free theory; designing a predetermined time sliding mode controller based on the predetermined time stability theory, and proving its stability; regarding 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] Combination 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 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-mounted permanent magnet synchronous motor is:

[0018]

[0019] In the formula, 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] In the formula, 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] In the formula, 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] In the formula, r is

[0029] Step 2: Based on the scheduled time stability theory, design a scheduled time 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] In the formula, e is the given reference mechanical angular velocity of the system The actual mechanical angular velocity ω m The error, n 1 >0,n 2 >0, c>0, d>0, d<c.

[0033] For the sliding surface function sω The derivative is:

[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] In the formula, 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] In the formula, 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] In the formula, q 1 >0,q 2 >0,q 3 >0,q 4 >0, T cω >0, T cω is the system scheduled time, indicating 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 preset 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(t 0 )=x 0 , There exists an initial state x 0 The system convergence time function T(x 0 ), so 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(x 0 ) is the convergence time of the system and 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(x 0 )≤T max , then the system (13) is said to be fixed-time stable.

[0055] Definition of predetermined time stability: Consider a nonlinear system (13), and let the system convergence time be T(x 0 ), T(x 0 ) 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(x 0 )≤T c , Then the system (13) is said to be stable within a predetermined time.

[0056] To prove the stability of the sliding mode controller, a Lyapunov function is selected, and the expression is:

[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, x 0 is the initial state of the system, and Because for the Lyapunov function, V 0 >0, V will be at the scheduled time T cω If 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 scheduled time stability, the designed speed loop scheduled 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 them, and prove their stability.

[0069] First, define

[0070]

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

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

[0073]

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

[0075]

[0076] In the formula, f 1 >0,f 2 >0, define α 2 =2α 1 -1,

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

[0078]

[0079] In the formula, 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] In the formula,

[0087] Let Z = [z 1 z 2 ] T ,but

[0088] Therefore, by choosing a suitable parameter f 1 、f 2 , α 1 , so that B 1 and B 2 is a Hurwitz matrix. Therefore, for any positive definite symmetric matrix There exists a positive definite symmetric matrix Make B 1 and B 2 They satisfy equation (26) respectively.

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

[0090] Choose a Lyapunov function, expressed as:

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

[0092] V cTaking the derivative we get:

[0093]

[0094] Scaling formula (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≤λ max (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 Q 2 Z≤-λ min (Q 2 )||Z‖ 2 .

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

[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, 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 κ 1 >0,κ 2 >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∈D 0 \{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<τ 1 <1,λ 1 >0,λ 2 > 0, the system will be in a limited time Stablize.

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

[0110]

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

[0112] The above description is only a preferred specific implementation manner of the present invention, but the protection scope of the present invention is not limited thereto. Any technician familiar with the technical field can make equivalent replacements or changes according to the technical scheme and inventive concept of the present invention within the technical scope disclosed by the present invention, which should be covered by the protection scope of the present invention.

Claims

1. A sliding mode control method for a permanent magnet synchronous motor based on a 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 scheduled time stability theory, design a scheduled time sliding mode controller and prove its stability; 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 them, and prove their stability; The sliding mode control method of a permanent magnet synchronous motor based on a predetermined time according to claim 1 is characterized in that: The step 2 is based on the predetermined time stability theory to design a predetermined time sliding mode controller, which refers to the speed loop super local model established and based on a 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; The sliding mode control method of a permanent magnet synchronous motor based on a predetermined time according to claim 1 is characterized in that: The step 2 designs a predetermined time sliding mode controller based on the predetermined time stability theory, and designs the sliding mode controller as follows: In the formula, 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 is the system scheduled time, indicating that the designed controller will cω Internal stability; You can get: According to the sliding surface function You can get: In the formula, α m for r is P n is the number of motor poles, 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; that is, the expression of the speed loop preset time sliding mode controller is: The sliding mode control method of a permanent magnet synchronous motor based on a predetermined time according to claim 1 is characterized in that: The stability of the scheduled time sliding mode controller in step 2 is proved by choosing a Lyapunov function: Taking the derivative we get: Can get It is proved that the designed scheduled time sliding mode controller will cω Internal stability, assuming the convergence time is T ω , then T ω ≤T cω ; The sliding mode control method of a permanent magnet synchronous motor based on a predetermined time according to claim 1 is 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 local model of the speed loop and the estimated value of the motor mechanical angular velocity are used to calculate the value of the unknown part of the model and the external disturbance. The actual mechanical angular velocity ω m The error e ω ,definition Construct the extended state observer: In the formula, f1>0, f2>0, define α2=2α1-1, And define an intermediate variable Z = [z1 z2] T : In the formula, e p is the lumped disturbance prediction error, is the estimated value of the lumped disturbance p; The sliding mode control method of a permanent magnet synchronous motor based on a predetermined time according to claim 1 is 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 , 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, and finally it is proved that the designed extended state observer is stable.

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