A control method of permanent magnet synchronous motor based on adaptive fractional order fast terminal sliding mode

Abstract

The application discloses a permanent magnet synchronous motor control method of self-adaptive fractional order fast terminal sliding mode. First, a mathematical model of a permanent magnet synchronous motor in a d-q axis rotor rotating coordinate system is constructed; then, a fractional order fast terminal sliding surface is designed, and an equivalent control law of sliding mode control is obtained based on the designed fractional order fast terminal sliding surface; adaptive design is performed on a switching control gain and a boundary layer thickness to obtain an adaptive switching control law; and a total control law is obtained by comprehensively integrating the equivalent control law and the adaptive switching control law. Compared with existing surface-mounted permanent magnet synchronous motor sliding mode control, the application introduces a fractional order fast terminal sliding surface on the basis of a traditional integral sliding mode to reduce system chattering and ensure that tracking error converges within a limited time, and simultaneously designs an adaptive control law of the switching control rate gain and the boundary layer thickness to offset the influence of external disturbance uncertainty and improve the robustness of the system.

Description

Technical Field

[0001] This invention belongs to the field of motor control technology, specifically relating to a permanent magnet synchronous motor control method that can suppress system chattering and ensure that tracking errors converge within a finite time. Background Technology

[0002] In modern industrial automation and electric drive systems, permanent magnet synchronous motors (PMSMs) are widely used due to their high efficiency, high power density, and excellent control performance. However, traditional PMSM control methods, such as PID-based control strategies, typically exhibit slow dynamic response and low control accuracy when faced with changes in motor parameters and external disturbances. These problems not only limit the performance of the motor but can also affect the stability and reliability of the entire system. With the development of modern control theory, new control methods such as model predictive control, sliding mode control, fuzzy control, and active disturbance rejection control have been used to improve the control performance of PMSMs. Sliding mode control, as a theory based on nonlinear control, is particularly suitable for handling uncertainties, disturbances, and nonlinear problems in systems. However, applying sliding mode control to PMSM control systems faces problems such as system chattering and large tracking errors.

[0003] In recent years, scholars both domestically and internationally have conducted extensive research on sliding mode control of permanent magnet synchronous motors (PMSMs). However, the main concerns in most studies regarding the application of sliding mode control to PMSM vector control systems are system chattering and slow convergence of tracking errors. Therefore, there is an urgent need for a sliding mode control method for PMSMs that can suppress system chattering and ensure that the tracking error converges within a finite time. Summary of the Invention

[0004] To address the shortcomings of existing technologies, this invention provides a permanent magnet synchronous motor control method with adaptive fractional-order fast terminal sliding mode, which suppresses system chattering and ensures that tracking errors converge within a finite time.

[0005] The specific technical solution is as follows:

[0006] A method for controlling a permanent magnet synchronous motor with adaptive fractional-order fast terminal sliding mode includes the following steps:

[0007] S1: Construct a mathematical model of the permanent magnet synchronous motor in the dq-axis rotor rotation coordinate system.

[0008] S2: Design a fractional-order fast terminal sliding surface.

[0009] S3: Based on the designed fractional-order fast terminal sliding surface, find the equivalent control law for sliding mode control.

[0010] S4: Adaptive design of switching control gain and boundary layer thickness is performed to obtain adaptive switching control law.

[0011] S5: The total control law is obtained by combining the equivalent control law obtained in S3 with the adaptive switching control law designed in S4.

[0012] S6: Stability analysis.

[0013] Furthermore, the mathematical model includes stator voltage equations, electromagnetic torque equations, and mechanical motion equations; among which, the stator voltage equations are as follows:

[0014]

[0015] Among them, u d U is the d-axis stator voltage. q i is the q-axis stator voltage. d For the d-axis stator current, i q L is the q-axis stator current. d For the d-axis inductance, L q For q-axis inductance, ω e R is the rotor electric angular velocity, R is the stator resistance, and ψ is the rotor electric angular velocity. f For rotor flux linkage.

[0016] The electromagnetic torque equation is as follows:

[0017]

[0018] Where T e For electromagnetic torque, P n It is the extreme logarithm;

[0019] For surface-mounted three-phase permanent magnet synchronous motors, the stator inductance satisfies:

[0020] L d =L q =L s

[0021] The equations of motion for the machine are as follows:

[0022]

[0023] Where J is the moment of inertia, ω m T is the rotor's mechanical angular velocity. L Where is the load torque, and B is the coefficient of viscous friction.

[0024] When using the field-oriented control method to control a surface-mounted three-phase permanent magnet synchronous motor with id=0, the following mathematical model is obtained:

[0025]

[0026] T applied from the outside LThe damping effect between the motor windings and the permanent magnets is generally considered a disturbance. Furthermore, the surface-mounted three-phase permanent magnet synchronous motor L... d =L q , Therefore, the above mathematical model can be rewritten as:

[0027]

[0028] F M It is the total uncertainty factor of the system, assuming F M It is bounded, |F M |<=k, where k is a constant.

[0029] Furthermore, the specific method of S2 is as follows:

[0030] The expression for the fractional calculus of RL is as follows:

[0031]

[0032] in, Let be the differential operator, t0 be the lower bound of the operator representing the initial time of the time variable, t be the upper bound of the operator representing the current time point in the integration operation, r be the fractional order of integration, dv be the increment of the integration variable τ, and Γ(n) be the gamma function, the expression of which is as follows:

[0033]

[0034] The system's state equations are as follows:

[0035]

[0036] Where ω ref ω is the reference speed of the motor. m e1 is the current motor speed, e2 is the speed tracking error, and e2 is the derivative of e1.

[0037] The fractional-order terminal sliding surface is designed as follows:

[0038]

[0039] Among them 0 <a1<1,μ1> 0, tanh(·) is the hyperbolic tangent function, c is the gain that adjusts the smoothness. a is the feedback gain proportional to the error e1, a1 is the power exponent of the nonlinear feedback term, and μ1 is the gain of the fractional calculus feedback term.

[0040] The derivative of a fractional-order sliding surface is:

[0041]

[0042] Furthermore, the specific methods for S3 are as follows:

[0043] Ignoring the total uncertainty of the system, let the derivative of the fractional sliding surface... definition The equivalent control law is:

[0044]

[0045] To ensure effective control while reducing high-frequency chattering, the hyperbolic tangent function is chosen as the switching control law, with the following expression:

[0046]

[0047] Among them, u s The switching control law is φ, where φ is the boundary layer thickness, and k is the switching control gain.

[0048] Furthermore, the specific method for S4 is as follows:

[0049] In practical vector control systems for permanent magnet synchronous motors, since the upper limit of system uncertainty is difficult to obtain accurately, the following adaptive switching control law is designed when the upper limit of system uncertainty is unknown:

[0050]

[0051] φ(s)=φ min +(φ max -φ min )·e -α|s|

[0052] Among them, u as For the improved switching control law, Here, φ(s) is the adaptive control gain, φ(s) is the adaptive boundary layer thickness function, and ε is the set convergence boundary. min For the desired accuracy near the sliding surface, φ max Let α be the maximum tolerable control range when the sliding surface is far away, η be the adjustment factor, and η be the proportional coefficient, where η > 0. It is stipulated that at t = 0, if |S(0)| >= ε / 2, switch the control law gain to K1, and record the time for the first convergence to the [-ε / 2, ε / 2] region. If S(0) < ε / 2, then when When choosing K bar To switch control gain.

[0053] Furthermore, the specific methods of S5 are as follows:

[0054] Total control law The formula is expressed as follows:

[0055]

[0056] Therefore, the q-axis reference current is obtained as follows:

[0057]

[0058] Furthermore, the specific method for S6 is as follows:

[0059] To verify the stability of the designed sliding mode controller, the stability of the sliding mode controller is proved by applying Lyapunov functions:

[0060]

[0061] Where ξ is a constant used to aid the proof. V1 is the Lyapunov function with K1 as the switching control gain. This is a function used to assist in the proof.

[0062] When the initial value S(0) >= ε / 2, switching the control law gain to K1 will increase K1 to K1 > |F under the action of the integral term. M |, record this moment as △t, after this moment K1 continues to increase until S(t) < ε / 2, at which point K1 reaches its maximum value. At the same time, this moment is exist Similarly, the same conclusion can be reached when S(0) < ε / 2.

[0063]

[0064] Differentiating with respect to V1, we get:

[0065]

[0066] in, for The derivative of .

[0067] because And ξ is small enough that ξ≤ηe s ,therefore According to Lyapunov's stability theory, when there is uncertainty in a system, the system state error can converge to the region [ε, ε].

[0068] when When, use K bar To switch the control law gain:

[0069]

[0070] V2 is based on K bar This is the Lyapunov function for switching control gain.

[0071] From the perspective of the potential barrier function range, for uncertain disturbances F M K bar Always greater than the disturbance F M The upper bound of K. Therefore, K bar -|F M |>0, Therefore, in At time t, the sliding mode can converge to [-ε, ε].

[0072] The beneficial effects of this invention are as follows:

[0073] Compared with existing surface-mounted permanent magnet synchronous motor sliding mode control, this invention introduces a fractional-order fast terminal sliding mode surface on the basis of traditional integral sliding mode to reduce system chattering and ensure that the tracking error converges within a finite time. At the same time, an adaptive control law is designed to switch the control law gain and boundary layer thickness to offset the uncertainty of external disturbances and improve the robustness of the system. Attached Figure Description

[0074] Figure 1 This is a flowchart of a method according to an embodiment of the present invention.

[0075] Figure 2 This is a block diagram illustrating the application of the method of this invention in the speed control of a permanent magnet synchronous motor. Detailed Implementation

[0076] To make the objectives, technical solutions, and advantages of the embodiments of the present invention clearer, the implementation schemes of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, not all embodiments. Based on the embodiments of the present invention, all other instances obtained by those skilled in the art without creative effort are within the protection scope of the present invention.

[0077] like Figure 1 As shown, this embodiment discloses a control method for a permanent magnet synchronous motor with adaptive fractional-order fast terminal sliding mode, including the following steps:

[0078] S1: Construct a mathematical model of the permanent magnet synchronous motor in the dq-axis rotor rotation coordinate system;

[0079] The mathematical model includes the stator voltage equation, the electromagnetic torque equation, and the mechanical motion equation; among them, the stator voltage equation is as follows:

[0080]

[0081] Among them, u d U is the d-axis stator voltage. q i is the q-axis stator voltage. dFor the d-axis stator current, i q L is the q-axis stator current. d For the d-axis inductance, L q For q-axis inductance, ω e R is the rotor electric angular velocity, R is the stator resistance, and ψ is the rotor electric angular velocity. f For rotor flux linkage.

[0082] The electromagnetic torque equation is as follows:

[0083]

[0084] Where T e For electromagnetic torque, P n It is the extreme logarithm;

[0085] For surface-mounted three-phase permanent magnet synchronous motors, the stator inductance satisfies:

[0086] L d =L q =L s

[0087] The equations of motion for the machine are as follows:

[0088]

[0089] Where J is the moment of inertia, ω m T is the rotor's mechanical angular velocity. L Where is the load torque, and B is the coefficient of viscous friction.

[0090] When using the field-oriented control method to control a surface-mounted three-phase permanent magnet synchronous motor with id=0, the following mathematical model is obtained:

[0091]

[0092] T applied from the outside L The damping effect between the motor windings and the permanent magnets is generally considered a disturbance. Furthermore, the surface-mounted three-phase permanent magnet synchronous motor L... d =L q , Therefore, the above mathematical model can be rewritten as:

[0093]

[0094] F M It is the total uncertainty factor of the system, assuming F M It is bounded, |F M |<=k, where k is a constant.

[0095] S2: Design a fractional-order fast terminal sliding surface;

[0096] The expression for the fractional calculus of RL is as follows:

[0097]

[0098] in, Let be the differential operator, t0 be the lower bound of the operator representing the initial time of the time variable, t be the upper bound of the operator representing the current time point in the integration operation, r be the fractional order of integration, dτ be the increment of the integration variable τ, and Γ(n) be the gamma function, the expression of which is as follows:

[0099]

[0100] Among them, t n-1 It is a weighting factor.

[0101] When n>1: t n-1 It gradually increases with increasing t, and contributes more to the integral when t is large.

[0102] When 0 <n<1:t n-1 The value is relatively large near t→0, and its contribution to the integral is significant.

[0103] The system's state equations are as follows:

[0104]

[0105] Where ω ref ω is the reference speed of the motor. m e1 is the current motor speed, e2 is the speed tracking error, and e2 is the derivative of e1.

[0106] The fractional-order terminal sliding surface is designed as follows:

[0107]

[0108] Among them 0 <a1<1,μ1> 0, tanh(·) is the hyperbolic tangent function, c is the gain that adjusts the smoothness. a is the feedback gain proportional to the error e1, a1 is the power exponent of the nonlinear feedback term, and μ1 is the gain of the fractional calculus feedback term.

[0109] The derivative of a fractional-order sliding surface is:

[0110]

[0111] S3: Calculate the equivalent control law for sliding mode control based on the designed fractional-order fast terminal sliding surface; neglecting the total uncertainty of the system, let the derivative of the fractional-order sliding surface... definition The equivalent control law is:

[0112]

[0113] To ensure effective control while reducing high-frequency chattering, the hyperbolic tangent function is chosen as the switching control law, with the following expression:

[0114]

[0115] Among them, u s The switching control law is φ, where φ is the boundary layer thickness, and k is the switching control gain.

[0116] S4: To improve the performance of the sliding mode controller, an adaptive design is made for the switching control gain k and the boundary layer thickness φ:

[0117] In practical vector control systems for permanent magnet synchronous motors, since the upper limit of system uncertainty is difficult to obtain accurately, the following adaptive switching control law is designed when the upper limit of system uncertainty is unknown:

[0118]

[0119] φ(s)=φ min +(φ max -φ min )·e -α|s|

[0120] Among them, u as For the improved switching control law, Here, φ(s) is the adaptive control gain, φ(s) is the adaptive boundary layer thickness function, and ε is the set convergence boundary. min For the desired accuracy near the sliding surface, φ max Let α be the maximum tolerable control range when the sliding surface is far away, η be the adjustment factor, and η be the proportional coefficient, where η > 0. It is stipulated that at t = 0, if |S(0)| >= ε / 2, switch the control law gain to K1, and record the time for the first convergence to the [-ε / 2, ε / 2] region. If S(0) < ε / 2, then when When choosing K bar To switch control gain.

[0121] When t = 0, S(0) >= ε / 2, then switch the control law gain to K1. During the process of the system error converging to the sliding surface, the exponential term in K1 can make the sliding mode s tend to [-ε / 2, ε / 2] in a short time, and the farther away from the origin, the more the boundary layer thickness φ(s) tends to φ. min The closer the switching control law is to the sign function, the faster the convergence speed of s. After the first convergence to the [-ε / 2, ε / 2] region, If the sliding mode s converges to the boundary of the [-ε, ε] region, the adaptive gain K barThe adaptive gain K will increase significantly with the occurrence of uncertain disturbances until the uncertain disturbances are canceled out. Meanwhile, when the sliding mode s decreases to 0, the adaptive gain K... bar This will be reduced accordingly to avoid oversaturation of the system control input, while the boundary layer thickness will tend towards φ. max As the system enters a larger boundary layer, the switching control becomes smoother and the chattering is reduced.

[0122] S5: The total control law is obtained by combining the equivalent control law obtained in S3 with the adaptive switching control law designed in S4.

[0123] Total control law The formula is expressed as follows:

[0124]

[0125] Therefore, the q-axis reference current is obtained as follows:

[0126]

[0127] S6: Stability Analysis: From S1-S5, the input is ω ref ω m The output is i q A sliding mode controller is proposed. To verify the stability of the designed sliding mode controller, Lyapunov functions are used to prove its stability:

[0128]

[0129] Where ξ is a constant used to aid the proof. V1 is the Lyapunov function with K1 as the switching control gain. This is a function used to assist in the proof.

[0130] When the initial value S(0) >= ε / 2, switching the control law gain to K1 will increase K1 to K1 > |F under the action of the integral term. M |, record this moment as △t, after this moment K1 continues to increase until S(t) < ε / 2, at which point K1 reaches its maximum value. At the same time, this moment is exist Similarly, the same conclusion can be reached when S(0) < ε / 2.

[0131]

[0132] Differentiating with respect to V1, we get:

[0133]

[0134] in, for The derivative of .

[0135] because And ξ is small enough that ξ≤ηe s ,therefore According to Lyapunov's stability theory, when there is uncertainty in a system, the system state error can converge to the region [ε, ε].

[0136] when When, use K bar To switch the control law gain:

[0137]

[0138] V2 is based on K bar This is the Lyapunov function for switching control gain.

[0139] From the perspective of the potential barrier function range, for uncertain disturbances F M K bar Always greater than the disturbance F M The upper bound of K. Therefore, K bar -|F M |>0, Therefore, in At time t, the sliding mode can converge to [-ε, ε].

[0140] Figure 2 This paper illustrates the application of an adaptive fractional-order fast terminal sliding mode control method for permanent magnet synchronous motors (PMSMs) according to an embodiment of the present invention in the speed control of PMSMs. Figure 2 It can be seen that the control quantity iqref for the next iteration is calculated by the fractional-order sliding mode controller based on the given motor speed and the current motor speed.

[0141] The above description, in conjunction with specific / preferred embodiments, provides a further detailed explanation of the present invention. It should not be construed that the specific implementation of the present invention is limited to these descriptions. Those skilled in the art can make various substitutions or modifications to these described embodiments without departing from the inventive concept, and all such substitutions or modifications should be considered within the scope of protection of the present invention.

[0142] The parts of this invention not described in detail are well-known to those skilled in the art.

Claims

1. A control method for a permanent magnet synchronous motor with adaptive fractional-order fast terminal sliding mode, characterized in that, The steps include the following: S1: Construct a mathematical model of the permanent magnet synchronous motor in the dq-axis rotor rotation coordinate system; S2: Design a fractional-order fast terminal sliding surface; S3: Determine the equivalent control law for sliding mode control based on the designed fractional-order fast terminal sliding surface; S4: Adaptive design of the switching control gain and boundary layer thickness yields an adaptive switching control law; the specific method is as follows: In practical vector control systems for permanent magnet synchronous motors, since the upper limit of system uncertainty is difficult to obtain accurately, the following adaptive switching control law is designed when the upper limit of system uncertainty is unknown: φ(s)=φ min +(φ max -f min )·e -α||s| Among them, u as For the improved switching control law, Here, φ(s) is the adaptive control gain, φ(s) is the adaptive boundary layer thickness function, and ε is the set convergence boundary. min For the desired accuracy near the sliding surface, φ max The maximum tolerable control range when the sliding surface is far away is defined as follows: α is the adjustment factor, η is the proportional coefficient, and η>0. It is stipulated that at t=0, if |S(0)|>=ε / 2, the control law gain is switched to K1, and the time for the first convergence to the [-ε / 2, ε / 2] region is recorded. If S(0) < ε / 2, then when When choosing K bar To switch control gain; S5: The total control law is obtained based on the equivalent control law obtained in S3 and the adaptive switching control law designed in S4; S6: Stability analysis.

2. The method for controlling a permanent magnet synchronous motor with adaptive fractional-order fast terminal sliding mode according to claim 1, characterized in that, The mathematical model includes the stator voltage equation, the electromagnetic torque equation, and the mechanical motion equation; among them, the stator voltage equation is as follows: Among them, u d U is the d-axis stator voltage. q i is the q-axis stator voltage. d For the d-axis stator current, i q L is the q-axis stator current. d For the d-axis inductance, L q For q-axis inductance, ω e R is the rotor electric angular velocity, R is the stator resistance, and ψ is the rotor electric angular velocity. f For rotor flux linkage; The electromagnetic torque equation is as follows: Where T e For electromagnetic torque, P n It is the extreme logarithm; For surface-mounted three-phase permanent magnet synchronous motors, the stator inductance satisfies: L d L q L s The equations of motion for the machine are as follows: Where J is the moment of inertia, ω m T is the rotor's mechanical angular velocity. L Where is the load torque, and B is the coefficient of viscous friction; When using the field-oriented control method to control a surface-mounted three-phase permanent magnet synchronous motor with id=0, the following mathematical model is obtained: T applied from the outside L The damping effect between the motor windings and the permanent magnet is considered a disturbance; and because the surface-mounted three-phase permanent magnet synchronous motor L... d =L q , Therefore, the above mathematical model can be rewritten as: F M It is the total uncertainty factor of the system, assuming F M It is bounded, |F M |<=k, where k is a constant.

3. The method for controlling a permanent magnet synchronous motor with adaptive fractional-order fast terminal sliding mode according to claim 2, characterized in that, The specific method for S2 is as follows: The expression for the fractional calculus of RL is as follows: in, Let be the differential operator, t0 be the lower bound of the operator representing the initial time of the time variable, t be the upper bound of the operator representing the current time point in the integration operation, r be the fractional order of integration, dτ be the increment of the integration variable τ, and Γ(n) be the gamma function, the expression of which is as follows: The system's state equations are as follows: Where ω ref ω is the reference speed of the motor. m e1 is the current motor speed, e2 is the speed tracking error, and e2 is the derivative of e1. The fractional-order terminal sliding surface is designed as follows: Among them 0 <a1<1,μ1> 0, tanh(·) is the hyperbolic tangent function, c is the gain that adjusts the smoothness; a is the feedback gain that is proportional to the error e1, a1 is the power exponent of the nonlinear feedback term, and μ1 is the gain of the fractional calculus feedback term. The derivative of a fractional-order sliding surface is:

4. The method for controlling a permanent magnet synchronous motor with adaptive fractional-order fast terminal sliding mode according to claim 3, characterized in that, The specific method for S3 is as follows: Ignoring the total uncertainty of the system, let the derivative of the fractional sliding surface... definition The equivalent control law is: To ensure effective control while reducing high-frequency chattering, the hyperbolic tangent function is chosen as the switching control law, with the following expression: Among them, u s For the switching control law, φ represents the boundary layer thickness; k represents the switching control gain.

5. The method for controlling a permanent magnet synchronous motor with adaptive fractional-order fast terminal sliding mode according to claim 4, characterized in that, The specific method for S5 is as follows: Total control law The formula is expressed as follows: Therefore, the q-axis reference current is obtained as follows:

6. The method for controlling a permanent magnet synchronous motor with adaptive fractional-order fast terminal sliding mode according to claim 5, characterized in that, The specific method for S6 is as follows: To verify the stability of the designed sliding mode controller, the stability of the sliding mode controller is proved by applying Lyapunov functions: Where ξ is a constant used to aid the proof; V1 is the Lyapunov function with K1 as the switching control gain. This is an auxiliary proof function; When the initial value S(0) >= ε / 2, switching the control law gain to K1 will increase K1 to K1 > |F under the action of the integral term. M |, record this moment as △t, after this moment K1 continues to increase until S(t) < ε / 2, at which point K1 reaches its maximum value. At the same time, this moment is exist Similarly, the same conclusion can be reached when S(0) < ε / 2; Differentiating with respect to V1, we get: in, for The derivative; because And ξ is small enough that ξ≤ηe s ,therefore According to Lyapunov's stability theory, when there is uncertainty in the system, the system state error can converge to the region [ε, ε]. when When, use K bar To switch the control law gain: V2 is based on K bar This is the Lyapunov function for switching control gain; From the perspective of the potential barrier function range, for uncertain disturbances F M K bar Always greater than the disturbance F M The upper bound of K; therefore K bar -|F M |>0, Therefore, in At time t, the sliding mode can converge to [-ε, ε].

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