A speed control method for permanent magnet synchronous motor based on fractional-order sliding mode

Abstract

This invention discloses a speed control method for a permanent magnet synchronous motor based on a fractional-order sliding mode. This method exploits the slow decay of fractional-order calculus over time. First, a fractional-order sliding mode surface is introduced on top of the traditional sliding mode surface. Second, a novel fast reaching law is proposed that adaptively adjusts parameters based on the distance of the system state from the sliding mode surface. Finally, to avoid repeated trial and error in adjusting the reaching law parameters, a particle swarm optimization algorithm is introduced to tune the parameters. Simulation results show that, compared with the exponential reaching law and the novel reaching law, the speed loop sliding mode control designed using the proposed method not only improves the system's speed regulation performance but also reduces system chattering and enhances system robustness.

Description

Technical Field

[0001] The present invention relates to the technical field of permanent magnet synchronous motor speed regulation, and in particular to a permanent magnet synchronous motor speed regulation method based on fractional-order sliding mode. Background Art

[0002] Permanent magnet synchronous motors (PMSMs) are widely used in industrial control, aerospace, electric vehicles, and medical devices due to their high power density, compact size, wide speed regulation range, and high reliability. Currently, PI control is often used for the speed loop of PMSM vector control. PI control is favored by many engineers due to its simplicity, ease of implementation, and simple parameter adjustment. However, because PMSMs are multivariable, strongly coupled, and nonlinear control targets, PI control is only suitable for simple operating conditions. External disturbances and sudden changes in operating conditions significantly impact the robustness of the system. For high-precision control applications, simple PI control can no longer meet the demands of society. With the advancement of control theory, many scholars are constantly pursuing better control methods. Currently, fuzzy control, neural network control, predictive control, active disturbance rejection control, and sliding mode control are commonly used for PMSM speed regulation.

[0003] Sliding mode control, a nonlinear variable structure control method, offers excellent interference immunity and robustness when system parameters and operating conditions change, making it widely used in permanent magnet synchronous motor speed control systems. However, when in sliding mode, sliding mode control can cause chattering as the sliding surface traverses back and forth. If applied directly to permanent magnet synchronous motor control without prior processing, this can lead to system instability and poor sliding mode quality.

[0004] Therefore, many scholars have adopted different methods to eliminate the chattering caused by the control process. In order to improve the quality of sliding mode control, Chinese scholar Gao Weibing proposed the concept of reaching law and designed an exponential reaching law as shown below:

[0005]

[0006] Where ε,k are constants greater than zero, s is the sliding surface, sign() is the switching function of the sliding mode, ks is the exponential term, and -εsign(s) is the constant speed approach term.

[0007] When the initial state s(0) of the system is far away from the sliding surface, the approach mode is completed by adjusting the exponential term and the constant speed term parameters at the same time. When the system state point reaches the sliding surface, the exponential term is zero, and the sliding mode is completed by the constant speed term. Since the constant speed term contains a sign function, ε cannot be zero, so the system will always be accompanied by jitter of the size of ε. From the above, we can see that the dynamic quality of the sliding mode control can be guaranteed by adjusting the ε and k parameters. When the ε and k parameters take larger values, the system will converge to the sliding surface at a very fast speed, but will produce larger jitter. When the ε and k parameters take smaller values, the system converges slower and the jitter is smaller. At this time, by calculating the convergence time of the exponential approach law, we can clearly see the shortcomings of the approach law. The calculation is as follows:

[0008] In formula (1), when s>0

[0009]

[0010] Assuming that the initial state of the system is at s(0), the time when the system reaches the sliding surface is as follows:

[0011]

[0012]

[0013] From formula (4), we can clearly see that increasing the k value can speed up the system's arrival speed to improve rapidity, but it will cause the speed of reaching the sliding surface to be too high, and the system state will traverse the sliding surface back and forth, resulting in large chattering, and vice versa.

[0014] In summary, the problem with the reaching law used in existing permanent magnet synchronous motor speed regulation is that improving speed and reducing jitter are technically contradictory and cannot be achieved simultaneously. Summary of the Invention

[0015] In order to solve the above shortcomings and deficiencies of the prior art, an object of the present invention is to provide a speed control method of a permanent magnet synchronous motor based on a fractional-order sliding mode.

[0016] The technical solution of the present invention is: a speed control method of a permanent magnet synchronous motor based on a fractional-order sliding mode, comprising:

[0017] Let the reference value of the direct-axis current component be

[0018] The rotor position θ m Provided for Park and Anti-Park use;

[0019] The actual rotor speed ω m and reference speed ω refThe error is sent to the sliding mode controller, and the reference value of the quadrature axis current component is obtained after adjustment by the sliding mode controller.

[0020] The three-phase stator winding current i a ,i b ,i c The quadrature axis current component i in the rotating coordinate system is obtained by Clark and Park transformation q and the direct axis current component i d ;

[0021] The actual quadrature-axis current component i q and the direct axis current component i d The reference value of the quadrature axis component after sliding mode control adjustment and the reference value of the direct-axis current component Subtract and get the error;

[0022] The error is sent to PI control and then output u q ,u d ;

[0023] u q ,u d Obtain u through inverse Park transform α ,u β ;

[0024] will u α ,u β The input is sent to the space vector pulse width modulation module to generate a pulse signal to the three-phase inverter to drive the motor;

[0025] The sliding mode controller outputs the reference value of the quadrature-axis current component through the following reaching law

[0026]

[0027] The control expression of the sliding mode speed loop of the permanent magnet synchronous motor is as follows:

[0028]

[0029]

[0030] Among them, 0<γ<1, 0<σ<0.1, λ represents the order of the fractional order, x1 represents the velocity error, T L represents the load torque, J represents the motor moment of inertia, c represents the sliding surface coefficient, s represents the sliding surface, k1 is a constant greater than zero, k2 is a constant greater than zero, β is the integral gain, D represents the calculus operator, P n is the number of motor pole pairs, ψ f is the permanent magnet flux linkage.

[0031] Furthermore, an adaptive swarm particle algorithm is used to find the optimal sliding mode parameters. The method is as follows:

[0032] In the adaptive swarm particle algorithm, the fitness function is:

[0033] f=∫t*|x1|dt;

[0034] Here, t represents time.

[0035] Preferably, the objective function of the adaptive swarm particle algorithm is 6-dimensional, the initialization population is 30, and the maximum number of iterations is 30.

[0036] The beneficial effects of the present invention are as follows: compared with the prior art, the present invention introduces a new reaching law for the reference value of the quadrature axis current component output by the sliding mode controller, and introduces an adaptive factor γe for the constant speed term. -|s| +σ |s| , to solve the contradiction between the rapidity and chatter reduction of sliding mode control, the hyperbolic tangent function is introduced to further reduce the chattering caused by the sign function switching process, and the controller is designed by adopting the fractional-order sliding surface and the new reaching law. The simulation results show that the sliding mode speed loop designed with the fractional-order sliding surface and the new reaching law can also speed up the response speed of the system without overshoot of the speed, reduce the steady-state error of the system, and at the same time, the electromagnetic torque fluctuation is smaller; when the motor parameters and load change, the speed and electromagnetic torque can respond quickly to reach a steady state, thereby improving the robustness of the system. BRIEF DESCRIPTION OF THE DRAWINGS

[0037] Figure 1 This is a structural block diagram of the sliding mode speed loop of the present invention;

[0038] Figure 2 is the speed comparison curve of the present invention;

[0039] Figure 3 is the electromagnetic torque comparison curve of the present invention;

[0040] Figure 4 This is a speed comparison diagram when the moment of inertia changes according to the present invention;

[0041] Figure 5 This is a torque comparison diagram of the present invention. DETAILED DESCRIPTION

[0042] Mathematical model of permanent magnet synchronous motor:

[0043] In order to simplify the analysis, an ideal permanent magnet synchronous motor mathematical model is established, and the following assumptions are made for the PMSM: the saturation of the permanent magnet synchronous motor core is ignored, the influence of hysteresis and eddy current is ignored, the magnetomotive force is sinusoidally distributed, and the influence caused by the irregularity of the stator slot is ignored.

[0044] The mathematical model of the permanent magnet synchronous motor under the dq axis is as follows

[0045]

[0046] The magnetic flux equation is shown below

[0047]

[0048] u d ,u q is the stator voltage under dq, i d ,i q is the stator current under dq, L d ,L q is the stator inductance under the dq axis, R is the stator resistance, ψ d ,ψ q is the magnetic flux under dq, ψ f is the permanent magnet flux, ω e is the mechanical angular velocity, ω is the electrical angular velocity, P n is the number of motor pole pairs.

[0049] The electromagnetic torque equation is shown below

[0050]

[0051] This article takes the surface-mounted permanent magnet synchronous motor as an example, so L d＝ L q =L, so the new electromagnetic torque equation is as follows

[0052]

[0053] The equation of motion is

[0054]

[0055] The invention will be further described below with reference to the accompanying drawings and specific embodiments:

[0056] Implementation Example 1:

[0057] In order to solve the problem that improving speed and reducing chattering in the existing technology are technical contradictions and cannot be achieved simultaneously, the present invention adopts a speed control method of a permanent magnet synchronous motor based on a fractional-order sliding mode, including:

[0058] Let the reference value of the direct-axis current component be

[0059] The rotor position θ m Provided for Park and Anti-Park use;

[0060] The actual rotor speed ω m and reference speed ω ref The error is sent to the sliding mode controller, and the reference value of the quadrature axis current component is obtained after adjustment by the sliding mode controller.

[0061] The three-phase stator winding current i a ,i b ,i c The quadrature axis current component i in the rotating coordinate system is obtained by Clark and Park transformation q and the direct axis current component i d ;

[0062] The actual quadrature-axis current component i q and the direct axis current component i d The reference value of the quadrature axis component after sliding mode control adjustment and the reference value of the direct-axis current component Subtract and get the error;

[0063] The error is sent to PI control and then output u q ,u d ;

[0064] u q ,u d Obtain u through inverse Park transform α ,u β ;

[0065] will u α ,u β The input is sent to the space vector pulse width modulation module to generate a pulse signal to the three-phase inverter to drive the motor;

[0066] The sliding mode controller outputs the reference value of the quadrature-axis current component through the following reaching law

[0067]

[0068] The control expression of the sliding mode speed loop of the permanent magnet synchronous motor is as follows:

[0069]

[0070]

[0071] Among them, 0<γ<1, 0<σ<0.1, λ represents the order of the fractional order, x1 represents the velocity error, T L represents the load torque, J represents the motor moment of inertia, c represents the sliding surface coefficient, s represents the sliding surface, k1 is a constant greater than zero, k2 is a constant greater than zero, β is the integral gain, D represents the calculus operator, Pn is the number of motor pole pairs, ψ f is the permanent magnet flux linkage.

[0072] The reaching law proposed in the present invention is analyzed in terms of reaching mode and sliding mode as follows:

[0073] Approaching mode: When the system state is far away from the sliding surface, the exponential term and the constant velocity term act simultaneously. At this time, the constant velocity term is k1e |s| / γ, which makes the system approach speed very large, increases the speed at which the system approaches the sliding surface, and enables sliding mode control.

[0074] Sliding mode: When the system state is close to the sliding surface, that is, when s→0, the exponential term k2s is zero, and the constant speed term takes effect. At this time, e -|s| →1,σ |s| →1, the constant velocity term becomes k1 / (γ+1), and the system will slide on the sliding surface at a very small number, effectively reducing chattering.

[0075] Therefore, the convergence law of this patent can solve the technical contradiction between the rapidity and the reduction of chattering in sliding mode control, while achieving both rapidity and reduced chattering.

[0076] This patent selects the hyperbolic sine function as the switching function of the system sliding mode, making its switching process smoother and further reducing the chattering. Its tanh(as) image is as follows: Figure 1 shown.

[0077] In order to analyze the stability of the reaching law proposed in this paper, the following Lyapunov function is constructed:

[0078] By taking the derivative of the above formula, we can get the following:

[0079]

[0080] From the above formula (10), we can see that γe -|s| +σ |s| >0, according to the Lyapunov principle, when Therefore, when k1, k2>0, the designed new reaching law meets the reaching condition of sliding mode control, and the system can converge within a finite time.

[0081] The derivation process of the control expression of the sliding mode speed loop is as follows:

[0082]

[0083] Among them, x1 represents the velocity error, x2 represents the derivative of x1, ωref Reference speed, ω m is the actual speed.

[0084] Combining formulas (9) and (11), we can obtain:

[0085]

[0086] The selected fractional sliding surface is:

[0087] S=x1+βD -λ-1 x1 (13)

[0088] By taking the derivative of (13), we can get:

[0089]

[0090] Combining Equation (12) and Equation (14), the control expression of the sliding mode speed loop of the permanent magnet synchronous motor can be obtained as follows:

[0091]

[0092] Furthermore, an adaptive swarm particle algorithm is used to find the optimal sliding mode parameters. The method is as follows:

[0093] In the adaptive swarm particle algorithm, the fitness function is:

[0094] f=∫t*|x1|dt;

[0095] Here, t represents time.

[0096] The specific steps of the algorithm optimization are:

[0097] 1) First, assign values ​​to the particles. The space of the objective function here is 6-dimensional, the initial population is 30, the position of the i-th particle in space is expressed as Xi = [xi1,xi2,xi3,xi4,xi5,xi6], and the velocity is expressed as Vi = [vi1,vi2,vi3,vi4,vi5,vi6], where the individual optimal position of the i-th particle is pid = [pi1,pi2,pi3,pi4,pi5,pi6], and the global optimal position is pgd = [pg1,pg2,pg3,pg4,pg5,pg6].

[0098] 2) Select the fitness function as follows:

[0099] f=∫t*|x1|dt; (16)

[0100] 3) Set the maximum number of iterations to 30, and find the particle with the minimum fitness function through continuous iteration. The position of this particle in the search space is the optimal sliding mode parameter.

[0101] 4) Determine whether the algorithm meets the constraints. If so, end the optimization and output the optimal parameters. If not, continue the optimization.

[0102] Simulation experiment results:

[0103] Figure 1 The block diagram of the sliding mode speed loop of the permanent magnet synchronous motor based on the proposed new reaching law is shown in Figure 1. d =0 control strategy, using the error between the reference speed and the actual speed, a fractional-order sliding mode controller is designed to control the speed, and a particle swarm optimization algorithm is introduced to tune the sliding mode parameters, thereby improving the PMSM control performance.

[0104] The PMSM model parameters used in this paper in MATLAB / simulink simulation are as follows: stator resistance R = 2.875Ω, inductance Ld = Lq = 0.0085mH, magnetic flux The torque coefficient is 1.05Nm / A, and the moment of inertia J is 0.003Kg.m 2 , pole pairs = 4. In order to verify the superiority of the reaching law proposed in this paper, the proposed reaching law is compared with the exponential reaching law.

[0105] To ensure the accuracy of the simulation comparison, the current loop parameters were kept consistent, and the speed loop was controlled using a new reaching law and an exponential reaching law. The exponential reaching law parameters were selected as ε = 200, k2 = 300, and c = 60. The new reaching law was optimized online using a particle swarm, and its optimized parameters were:

[0106] k1=1.12, k2=125.25, γ=0.45, σ=0.01, c=2.51.

[0107] The fractional-order sliding mode control parameters are:

[0108] k1=0.12, k2=260, γ=0.55, σ=0.013, β=2, λ=0.089.

[0109] In order to verify the dynamic performance of the permanent magnet synchronous motor using the new sliding mode speed loop control, the no-load speed is set to 1000r / min at the initial moment of system simulation. In order to compare the anti-interference performance of the system, a 5N load is suddenly added to the system at 0.2s. The speed comparison and electromagnetic torque comparison are as follows Figure 2 and Figure 3 shown.

[0110] from Figure 2The speed comparison curves show that the sliding-mode speed loop designed with an exponential reaching law exhibits a faster response, but suffers from a large initial overshoot (31%) and a long tracking time (0.06s). The sliding-mode speed loop with the new reaching law exhibits a faster response (0.05s) and no speed overshoot. The fractional-order sliding-mode control approach also exhibits a faster response (0.02s) and no speed overshoot. When a sudden load is applied to the system within 0.2s, the exponential reaching law approach results in large speed fluctuations (50r / min) and a long recovery time. The new reaching law approach exhibits smaller speed fluctuations (10r / min) and a faster recovery time. The fractional-order sliding-mode approach also exhibits smaller speed fluctuations (5r / min) and a stronger load-carrying capability.

[0111] Figure 3 The electromagnetic torque comparison results show that the torque under exponential reaching law control fluctuates greatly at the initial moment. When the load is suddenly added within 0.2 seconds, the electromagnetic torque takes a long time to stabilize. Under the new reaching law control method, the electromagnetic torque fluctuations are small at the initial time. At the same time, when the load is suddenly added within 0.2 seconds, the electromagnetic torque can track and stabilize in a very short time. Under the fractional-order sliding mode control method, the electromagnetic torque fluctuations at the initial time are more stable. When the load is suddenly added within 0.2 seconds, the electromagnetic torque can quickly track and stabilize. From the above results comparison analysis chart, it can be concluded that the fractional-order sliding mode speed loop of the permanent magnet synchronous motor using the proposed reaching law design has an exponential reaching law. The new reaching law control method has good dynamic characteristics and strong robustness.

[0112] To verify the robustness of the fractional-order sliding-mode velocity loop control system under parameter variations and load disturbances, the speed was first set to 1000 rpm. The load was suddenly increased to 5 N in 0.2 seconds, and the speed was increased by 1200 rpm in 0.3 seconds. For parameter variations, this paper used the moment of inertia as the validation, comparing J = 0.5 J and J = 2 J.

[0113] from Figure 4 and Figure 5 It can be seen that when the moment of inertia changes, the speed can respond quickly and reach a steady state. The torque and current fluctuate slightly when the load suddenly changes in 0.2s. When the speed rises to 1200r / min, the torque fluctuates to a certain extent, but can reach a steady state in a short time.

[0114] in conclusion:

[0115] In order to solve the problems of poor dynamic characteristics and poor robustness of the speed loop control of permanent magnet synchronous motors using traditional sliding mode control, a fractional-order sliding mode speed loop control is proposed. It uses the slow decay characteristics of fractional-order calculus over time to effectively reduce system chattering. At the same time, a new reaching law is proposed. This reaching law improves the traditional exponential reaching law and introduces an adaptive factor γe to the constant speed term. -|s| +σ|s| To address the conflict between rapidity and chattering in sliding mode control, a hyperbolic tangent function is introduced to further reduce chattering caused by the sign function switching process. A controller is designed using a fractional-order sliding surface and a novel reaching law. Simulation results show that the sliding mode velocity loop designed with this approach accelerates system response without speed overshoot, reduces steady-state error, and minimizes electromagnetic torque fluctuations. When motor parameters and load change, the speed and electromagnetic torque quickly respond to steady-state conditions, improving system robustness and verifying the effectiveness of the proposed method.

[0116] The above is a further detailed description of the present invention in conjunction with specific preferred embodiments, and the specific implementation of the present invention should not be considered to be limited to these descriptions. For those skilled in the art of the present invention, without departing from the concept of the present invention, several simple deductions or substitutions can be made, which should be considered to fall within the scope of protection of the present invention.

Claims

1. A speed control method for a permanent magnet synchronous motor based on a fractional-order sliding mode, comprising: Let the reference value of the direct-axis current component be The rotor position θ m Provided for Park and Anti-Park use; The actual rotor speed ω m and reference speed ω ref The error is sent to the sliding mode controller, and the reference value of the quadrature axis current component is obtained after adjustment by the sliding mode controller. The three-phase stator winding current i a ,i b ,i c The quadrature axis current component i in the rotating coordinate system is obtained by Clark and Park transformation q and the direct axis current component i d ; The actual quadrature-axis current component i q and the direct axis current component i d The reference value of the quadrature axis component after sliding mode control adjustment and the reference value of the direct-axis current component Subtract and get the error; The error is sent to PI control and then output u q ,u d ; u q ,u d Obtain u through inverse Park transform α ,u β ; will u α ,u β The input is sent to the space vector pulse width modulation module to generate a pulse signal to the three-phase inverter to drive the motor; It is characterized in that the sliding mode controller outputs the reference value of the quadrature axis current component through the following reaching law The control expression of the sliding mode speed loop of the permanent magnet synchronous motor is as follows: Among them, 0<γ<1, 0<σ<0.1, λ represents the order of the fractional order, x1 represents the velocity error, T L represents the load torque, J represents the motor moment of inertia, s represents the sliding surface, k1 is a constant greater than zero, k2 is a constant greater than zero, β is the integral gain, D represents the calculus operator, P n is the number of motor pole pairs, ψ f is the permanent magnet flux linkage.

2. The speed control method of a fractional-order sliding mode permanent magnet synchronous motor according to claim 1, characterized in that: The adaptive swarm particle algorithm is used to find the optimal sliding mode parameters. The method is as follows: In the adaptive swarm particle algorithm, the fitness function is: f=∫t*|x1|dt; Here, t represents time.

3. The fractional-order sliding mode permanent magnet synchronous motor speed control method according to claim 2, characterized in that: The objective function of the adaptive swarm particle algorithm is 6-dimensional, the initial population is 30, and the maximum number of iterations is 30.

Images