Permanent magnet synchronous motor control method of adaptive fractional order fast terminal sliding mode

Abstract

The invention discloses a permanent magnet synchronous motor control method of an adaptive fractional order fast terminal sliding mode. The method comprises the following steps: firstly, constructing a mathematical model of the permanent magnet synchronous motor in a d-q axis rotor rotating coordinate system; then designing a fractional order fast terminal sliding mode surface, and solving an equivalent control law of sliding mode control based on the designed fractional order fast terminal sliding mode surface; performing adaptive design on the switching control gain and the boundary layer thickness to obtain an adaptive switching control law; and integrating the equivalent control law and the adaptive switching control law to obtain a total control law. Compared with existing surface-mounted permanent magnet synchronous motor sliding mode control, the surface-mounted permanent magnet synchronous motor sliding mode control method introduces a fractional order fast terminal sliding mode surface on the basis of a traditional integral sliding mode so as to reduce system buffeting and ensure that tracking errors converge in finite time, and meanwhile designs a self-adaptive control law of switching control rate gain and boundary layer thickness. The uncertainty influence of external disturbance is counteracted, and the robustness of the system is improved.

Description

Technical Field

[0001] The present invention belongs to the technical field of motor control, and in particular relates to a permanent magnet synchronous motor control method capable of suppressing system chattering and ensuring that tracking errors converge within a limited time. Background Art

[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, usually exhibit slow dynamic response speed and low control accuracy when facing motor parameter changes and external disturbances. These problems not only limit the performance of the motor, but may 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 anti-disturbance control are used to improve the control performance of PMSMs. As a theory based on nonlinear control, sliding mode control is particularly suitable for dealing with system uncertainty, disturbances and nonlinear problems. However, the application of sliding mode control to PMSM control systems faces the problems of large system jitter and tracking error.

[0003] In recent years, domestic and foreign scholars have conducted a lot of research on the sliding mode control of permanent magnet synchronous motors. The system jitter and slow convergence of tracking errors faced by sliding mode control in permanent magnet synchronous motor vector control systems are the main concerns of most studies. Therefore, there is an urgent need for a sliding mode control method for permanent magnet synchronous motors that can suppress system jitter and ensure that the tracking error converges within a limited time. Summary of the invention

[0004] In view of the deficiencies in the prior art, the present invention provides a permanent magnet synchronous motor control method with an adaptive fractional-order fast terminal sliding mode to suppress system chattering and ensure that the tracking error converges within a limited time.

[0005] The specific technical solutions are as follows:

[0006] A permanent magnet synchronous motor control method with adaptive fractional-order fast terminal sliding mode comprises the following steps:

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

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

[0009] S3: Obtain the equivalent control law of sliding mode control based on the designed fractional-order fast terminal sliding surface.

[0010] S4: Adaptively design the switching control gain and the boundary layer thickness to obtain an adaptive switching control law.

[0011] 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.

[0012] S6: Stability analysis.

[0013] Furthermore, the mathematical model includes a stator voltage equation, an electromagnetic torque equation, and a mechanical motion equation; wherein the stator voltage equation is as follows:

[0014]

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

[0016] The electromagnetic torque equation is as follows:

[0017]

[0018] Where T e is the electromagnetic torque, P n is the pole pair number;

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

[0020] L d =L q =L s

[0021] The mechanical equation of motion is as follows:

[0022]

[0023] Where J is the moment of inertia, ω m is the rotor mechanical angular velocity, T L is the load torque, and B is the viscous friction coefficient.

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

[0025]

[0026] T imposed by the outside world LThe damping effect between the motor winding and the permanent magnet is usually regarded as a disturbance. d =L q , Therefore, the above mathematical model can be rewritten as:

[0027]

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

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

[0030] Among them, the expression of RL fractional calculus is as follows:

[0031]

[0032] in, is the differential operator, t0 is the lower limit of the operator, indicating the initial time of the time variable, t is the upper limit of the operator, indicating the current time point in the integration operation, r is the fractional integration order, dτ represents the increment of the integral variable τ, Γ(n) is the gamma function, and its expression is as follows:

[0033]

[0034] The state equation of the system is as follows:

[0035]

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

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

[0038]

[0039] Where 0 <a1<1,μ1> 0, tanh(·) is the hyperbolic tangent function, c is the gain for adjusting 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 the fractional sliding surface is:

[0041]

[0042] Furthermore, the S3 specific method is as follows:

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

[0044]

[0045] In order to ensure the control effect and reduce the high-frequency chattering of the system, the hyperbolic tangent function is selected as the switching control law, and the expression is as follows:

[0046]

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

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

[0049] In the actual permanent magnet synchronous motor vector control system, 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 is the improved switching control rate, is the adaptive control gain, φ(s) is the adaptive boundary layer thickness function, ε is the set convergence boundary, φ min is the expected accuracy near the sliding surface, φ max is the maximum tolerable control range when the sliding surface is far away, α is the adjustment factor, η is the proportional coefficient, and η>0. It is stipulated that when t=0, if |S(0)|>=ε / 2, the switch control law gain selects K1, and the time when it first converges to the [-ε / 2, ε / 2] region is recorded as If S(0)<ε / 2, then when When K bar For switching control gain.

[0053] Furthermore, the specific method of S5 is as follows:

[0054] Overall control law The formula is as follows:

[0055]

[0056] The q-axis reference current is obtained as follows:

[0057]

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

[0059] In order to verify the stability of the designed sliding mode controller, the Lyapunov function is used to prove the stability of the sliding mode controller:

[0060]

[0061] Among them, ξ is a constant used to assist in the proof. V1 is the Lyapunov function when K1 is the switching control gain, Auxiliary proof function.

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

[0063]

[0064] Taking the derivative of V1, we get:

[0065]

[0066] in, for The derivative of .

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

[0068] when When K bar is the switching control law gain:

[0069]

[0070] V2 is in K bar is the Lyapunov function when switching the control gain.

[0071] From the perspective of the potential barrier function range, for the uncertain disturbance F M , K bar is always greater than the disturbance F M So K bar -|F M |>0, Therefore, in At this moment, the sliding mode can converge to [-ε, ε].

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

[0073] Compared with the existing surface-mounted permanent magnet synchronous motor sliding mode control, the present invention introduces a fractional-order fast terminal sliding mode surface on the basis of traditional integral sliding mode to reduce system jitter and ensure that the tracking error converges within a limited time. At the same time, an adaptive control law of the switching control rate gain and the boundary layer thickness is designed to offset the uncertainty of external disturbances and improve the robustness of the system. BRIEF DESCRIPTION OF THE DRAWINGS

[0074] Figure 1 The figure is a flow chart of a method according to an embodiment of the present invention.

[0075] Figure 2 This is a block diagram of an application of the method according to an embodiment of the present invention in speed control of a permanent magnet synchronous motor. DETAILED DESCRIPTION

[0076] In order to make the purpose, technical solution and advantages of the embodiments of the present invention clearer, the following is a clear and complete description of the implementation scheme in the embodiments of the present invention in conjunction with the drawings in the embodiments of the present invention. Obviously, the described embodiments are part of the embodiments of the present invention, not all embodiments. Based on the embodiments of the present invention, all other examples obtained by ordinary technicians in this field without making creative work are within the scope of protection of the present invention.

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

[0078] S1: Construct the mathematical model of permanent magnet synchronous motor in dq axis rotor rotating 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 is the d-axis stator voltage, u q is the q-axis stator voltage, i dis the d-axis stator current, i q is the q-axis stator current, L d is the d-axis inductance, L q is the q-axis inductance, ω e is the rotor electrical angular velocity, R is the stator resistance, ψ f is the rotor flux.

[0082] The electromagnetic torque equation is as follows:

[0083]

[0084] Where T e is the electromagnetic torque, P n is the pole pair number;

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

[0086] L d =L q =L s

[0087] The mechanical equation of motion is as follows:

[0088]

[0089] Where J is the moment of inertia, ω m is the rotor mechanical angular velocity, T L is the load torque, and B is the viscous friction coefficient.

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

[0091]

[0092] T imposed by the outside world L The damping effect between the motor winding and the permanent magnet is usually regarded as a disturbance. d =L q , Therefore, the above mathematical model can be rewritten as:

[0093]

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

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

[0096] Among them, the expression of RL fractional calculus is as follows:

[0097]

[0098] in, is the differential operator, t0 is the lower limit of the operator, indicating the initial time of the time variable, t is the upper limit of the operator, indicating the current time point in the integration operation, r is the fractional integration order, dτ represents the increment of the integral variable τ, Γ(n) is the gamma function, and its expression is as follows:

[0099]

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

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

[0102] When 0 <n<1:t n-1 It is larger near t→0 and contributes significantly to the integral.

[0103] The state equation of the system is as follows:

[0104]

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

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

[0107]

[0108] Where 0 <a1<1,μ1> 0, tanh(·) is the hyperbolic tangent function, c is the gain for adjusting 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 the fractional sliding surface is:

[0110]

[0111] S3: Obtain the equivalent control law of sliding mode control based on the designed fractional-order fast terminal sliding surface;

[0112] Ignoring the total uncertainty of the system, let the derivative of the fractional sliding surface definition Then the equivalent control law is:

[0113]

[0114] In order to ensure the control effect and reduce the high-frequency chattering of the system, the hyperbolic tangent function is selected as the switching control law, and the expression is as follows:

[0115]

[0116] Among them, u s is the switching control law, φ is the boundary layer thickness, and k is the switching control gain.

[0117] S4: To improve the performance of the sliding mode controller, the switching control gain k and the boundary layer thickness φ are adaptively designed:

[0118] In the actual permanent magnet synchronous motor vector control system, 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:

[0119]

[0120] φ(s)=φ min +(φ max -φ min )·e -α|s|

[0121] Among them, u as is the improved switching control rate, is the adaptive control gain, φ(s) is the adaptive boundary layer thickness function, ε is the set convergence boundary, φ min is the expected accuracy near the sliding surface, φ max is the maximum tolerable control range when the sliding surface is far away, α is the adjustment factor, η is the proportional coefficient, and η>0. It is stipulated that when t=0, if |S(0)|>=ε / 2, the switch control law gain selects K1, and the time when it first converges to the [-ε / 2, ε / 2] region is recorded as If S(0)<ε / 2, then when When K bar For switching control gain.

[0122] When t = 0, S(0)> = ε / 2, then the switch control law gain selects K1. When the system error converges to the sliding surface, the exponential term in K1 can make the sliding mode s approach [-ε / 2, ε / 2] in a short time, and the farther away from the origin, the closer the boundary layer thickness φ(s) is to φ min , the closer the switching control rate is to the sign function, the faster s converges. After first converging 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 bar It will increase significantly with the occurrence of uncertain disturbances until the uncertain disturbances are offset. At the same time, when the sliding mode s decreases to 0, the adaptive gain K bar will be reduced accordingly to avoid oversaturation of the system control input, and the boundary layer thickness tends to φ max , the system enters a larger boundary layer, the switching control is smoother, and the jitter is smaller.

[0123] 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.

[0124] Overall control law The formula is as follows:

[0125]

[0126] The q-axis reference current is obtained as follows:

[0127]

[0128] S6: Stability analysis: After S1-S5, the input is ω re f、ω m , the output is i q In order to verify the stability of the designed sliding mode controller, the Lyapunov function is used to prove the stability of the sliding mode controller:

[0129]

[0130] Among them, ξ is a constant used to assist in the proof. V1 is the Lyapunov function when K1 is the switching control gain, Auxiliary proof function.

[0131] When the initial value S(0)>=ε / 2, the switch control law gain selects K1. Under the action of the integral term, K1 will increase to K1>|F M |, record this time as △t, after this time, K1 continues to increase until S(t)<ε / 2, at which point K1 reaches its maximum value At the same time, exist Similarly, the same conclusion can be obtained when S(0)<ε / 2.

[0132]

[0133] Taking the derivative of V1, we get:

[0134]

[0135] in, for The derivative of .

[0136] because and ξ is small enough to make ξ≤ηe s ,therefore According to Lyapunov stability theory, when there is uncertainty in the system, the system state error can converge to the [ε, ε] region.

[0137] when When K bar is the switching control law gain:

[0138]

[0139] V2 is in K bar is the Lyapunov function when switching the control gain.

[0140] From the perspective of the potential barrier function range, for the uncertain disturbance F M , K bar is always greater than the disturbance F M So K bar -|F M |>0, Therefore, in At this moment, the sliding mode can converge to [-ε, ε].

[0141] Figure 2 The application of an adaptive fractional-order fast terminal sliding mode permanent magnet synchronous motor control method provided in an embodiment of the present invention in the speed control of a permanent magnet synchronous motor is shown. Figure 2 It can be seen that the next control quantity iqref is calculated by the given motor speed and the current motor speed through the fractional-order sliding mode controller.

[0142] The above contents are further detailed descriptions of the present invention in combination with specific / preferred embodiments, and it cannot be determined that the specific implementation of the present invention is limited to these descriptions. For ordinary technicians in the technical field to which the present invention belongs, they can also make several substitutions or modifications to these described embodiments without departing from the concept of the present invention, and these substitutions or modifications should be regarded as belonging to the protection scope of the present invention.

[0143] Parts of the present invention that are not described in detail belong to the well-known technologies of those skilled in the art.

Claims

1. A permanent magnet synchronous motor control method with adaptive fractional-order fast terminal sliding mode, characterized in that: The steps include: S1: Construct the mathematical model of permanent magnet synchronous motor in dq axis rotor rotating coordinate system; S2: Design fractional-order fast terminal sliding surface; S3: Obtain the equivalent control law of sliding mode control based on the designed fractional-order fast terminal sliding surface; S4: Adaptively design the switching control gain and the boundary layer thickness to obtain an adaptive switching control law; 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 an 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 is the d-axis stator voltage, u q is the q-axis stator voltage, i d is the d-axis stator current, i q is the q-axis stator current, L d is the d-axis inductance, L q is the q-axis inductance, ω e is the rotor electrical angular velocity, R is the stator resistance, ψ f is the rotor flux; The electromagnetic torque equation is as follows: Where T e is the electromagnetic torque, P n is the pole pair number; For a surface-mounted three-phase permanent magnet synchronous motor, the stator inductance satisfies: L d =L q =L s The mechanical equation of motion is as follows: Where J is the moment of inertia, ω m is the rotor mechanical angular velocity, T L is the load torque, B is the viscous friction coefficient; When the field-oriented control method is used to control the surface-mounted three-phase permanent magnet synchronous motor, id = 0, and the following mathematical model is obtained: T imposed by the outside world L The damping effect between the motor winding and the permanent magnet is usually regarded as 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 is the total uncertainty factor of the system, assuming that F M is bounded, |F M |<=k, k is a constant.

3. The method for controlling a permanent magnet synchronous motor with an adaptive fractional-order fast terminal sliding mode according to claim 1 or 2, characterized in that: The specific method of S2 is as follows: Among them, the expression of RL fractional calculus is as follows: in, is the differential operator, t0 is the lower limit of the operator, indicating the initial time of the time variable, t is the upper limit of the operator, indicating the current time point in the integration operation, r is the fractional integration order, dτ represents the increment of the integral variable τ, Γ(n) is the gamma function, and its expression is as follows: The state equation of the system is as follows: where ω ref is the motor reference speed, ω m is the current speed of the motor, e1 is the speed tracking error, and e2 is the derivative of e1; The designed fractional-order terminal sliding surface is: Where 0 <a1<1,μ1> 0, tanh(·) is the hyperbolic tangent function, c is the gain for adjusting 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; The derivative of the fractional sliding surface is:

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

5. The method for controlling a permanent magnet synchronous motor with an adaptive fractional-order fast terminal sliding mode according to claim 4 is characterized in that: The specific method of S4 is as follows: In the actual permanent magnet synchronous motor vector control system, 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 is the improved switching control rate, is the adaptive control gain, φ(s) is the adaptive boundary layer thickness function, ε is the set convergence boundary, φ min is the expected accuracy near the sliding surface, φ max is the maximum tolerable control range when the sliding surface is far away, α is the adjustment factor, η is the proportional coefficient, η>0; it is stipulated that when t=0, if |S(0)|>=ε / 2, the switching control law gain selects K1, and the time for the first convergence to the [-ε / 2, ε / 2] region is recorded as If S(0)<ε / 2, then when When K bar For switching control gain.

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

7. The method for controlling a permanent magnet synchronous motor with an adaptive fractional-order fast terminal sliding mode according to claim 6, characterized in that: S6 specific methods are as follows: In order to verify the stability of the designed sliding mode controller, the Lyapunov function is used to prove the stability of the sliding mode controller: Among them, ξ is a constant used to assist in the proof; V1 is the Lyapunov function when K1 is the switching control gain, is the auxiliary proof function; When the initial value S(0)>=ε / 2, the switch control law gain selects K1. Under the action of the integral term, K1 will increase to K1>|F M |, record this time as △t, after this time, K1 continues to increase until S(t)<ε / 2, at which point K1 reaches its maximum value At the same time, exist Similarly, the same conclusion can be obtained when S(0)<ε / 2; Taking the derivative of V1, we get: in, for The derivative of because and ξ is small enough to make ξ≤ηe s ,therefore According to Lyapunov stability theory, when there is uncertainty in the system, the system state error can converge to the [ε, ε] region; when When K bar is the switching control law gain: V2 is in K bar is the Lyapunov function when switching the control gain; From the perspective of the potential barrier function range, for the uncertain disturbance F M , K bar is always greater than the disturbance F M The upper bound of K bar -|F M |>0, Therefore, in At this moment, the sliding mode can converge to [-ε, ε].

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