A Speed Control Method for Permanent Magnet Synchronous Motor Based on Fractional-Order Active Disturbance Rejection

Abstract

The present invention discloses a speed control method for a permanent magnet synchronous motor based on fractional order active disturbance rejection, which includes the following steps. First, a new nonlinear function newfal(x,a) is designed. Then, the fastest function is optimized and a fractional order tracking differentiator PO-TD is designed in combination with the fractional order theory. Next, a fractional order state observer PO-ESO is designed based on the new nonlinear function. Then, a nonlinear state error feedback law is constructed and a nonlinear feedback control law NLSEF is designed. Subsequently, a fractional order active disturbance rejection controller PO-ADRC is designed based on the fractional order tracking differentiator PO-TD, the fractional order state observer PO-ESO, and the nonlinear feedback control law NLSEF. Finally, the speed of the permanent magnet synchronous motor is controlled by using the control method of the fractional order active disturbance rejection controller PO-ADRC in the permanent magnet synchronous motor. The present invention realizes the use of the fractional order ADRC control method in the permanent magnet synchronous motor, and stably controls the permanent magnet synchronous motor, having strong anti-interference performance and fast response ability, and is suitable for being widely promoted and used.

Description

Technical Field

[0001] The present invention relates to the technical field of permanent magnet synchronous motor control, and particularly relates to a speed control method for a permanent magnet synchronous motor based on fractional order active disturbance rejection control (ADRC). Background Art

[0002] During the normal operation of a permanent magnet synchronous motor, disturbances are inevitable. Under normal circumstances, it is vulnerable to disturbance factors such as parameter uncertainties, load mutations, and friction, making it difficult for the motor's response speed to maintain a stable state during stable operation. Traditional control methods such as PID control and sliding mode control are difficult to eliminate the influence of external disturbances on the motor. Researchers have introduced the ADRC control method to control the motor, which has greatly reduced the influence of external disturbances and parameter uncertainties on the motor.

[0003] When the traditional ADRC control method is used to control the speed of a permanent magnet synchronous motor, although it greatly reduces the influence of factors such as load mutations and parameter uncertainties, traditional control methods are difficult to meet the actual anti-disturbance requirements when the control object is a non-integer order target; traditional control methods have weak resistance to large disturbances and long recovery times, affecting the normal operation of the permanent magnet synchronous motor. Therefore, it is necessary to design a speed control method for a permanent magnet synchronous motor based on fractional order ADRC. Summary of the Invention

[0004] The purpose of the present invention is to overcome the deficiencies of the prior art. In order to better and effectively solve the problems that when the traditional ADRC control method is used to control the speed of a permanent magnet synchronous motor, although it greatly reduces the influence of factors such as load mutations and parameter uncertainties, traditional control methods are difficult to meet the actual anti-disturbance requirements when the control object is a non-integer order target, and have weak resistance to large disturbances and long recovery times, affecting the normal operation of the permanent magnet synchronous motor, a speed control method for a permanent magnet synchronous motor based on fractional order ADRC is provided. It realizes the stable control of the permanent magnet synchronous motor and has the advantages of strong anti-disturbance performance and fast response ability.

[0005] In order to achieve the above purpose, the technical solution adopted by the present invention is as follows:

[0006] A speed control method for a permanent magnet synchronous motor based on fractional order ADRC includes the following steps:

[0007] Step (A), designing a new nonlinear function newfal(x,a);

[0008] Step (B), optimizing the fastest function and designing a fractional order tracking differentiator PO-TD in combination with the fractional order theory;

[0009] Step (C), designing a fractional order state observer PO-ESO based on the new nonlinear function;

[0010] Step (D): Construct a non - linear state error feedback law and design a non - linear feedback control law NLSEF;

[0011] Step (E): Based on the fractional - order tracking differentiator PO - TD, the fractional - order state observer PO - ESO, and the non - linear feedback control law NLSEF, design a fractional - order active disturbance rejection controller PO - ADRC;

[0012] Step (F): Use the control method of the fractional - order active disturbance rejection controller PO - ADRC in the permanent magnet synchronous motor to control its speed.

[0013] Preferably, in step (A), the expression of the new non - linear function newfal(x,a) is shown in formula (1),

[0014]

[0015] where a represents the non - linear factor and x represents the error signal.

[0016] Preferably, the design of the fractional - order tracking differentiator PO - TD in step (B) specifically combines the principle of fractional - order calculus, and the expression of the fastest function is shown in formula (2),

[0017]

[0018] where r is the set - point, v 1 and v 2 are the transition process and its differential signal, and θ is the derivative symbol; is the Caputo fractional - order calculus, and its specific expression is shown in formula (3),

[0019]

[0020] where m is a positive integer satisfying m - 1 < a E < m, f(x) is a smooth function, Due to the fractional - order calculus 0 < a E < 1, and then v 1 (t)≈v(t), r 1 and r 2 are constants, W = 1 - Γ 2 (a E +1) / Γ(2a E +1), and n is an odd number.

[0021] Preferably, the specific algorithm expression of the fractional - order state observer PO - ESO in step (C) is shown in formula (4),

[0022]

[0023] where ω 1 is the observation error; z 1 、z 2 and z 3 are the observation outputs of the states x 1 、x 2 and x 3 of the system respectively; the parameters β 1 、β 2 and β 3 are the gain coefficients of the extended state observer; b is the system gain parameter, and a represents the Caputo fractional order of t0 D t a .

[0024] Preferably, the nonlinear state error feedback law in step (D) is used to perform nonlinear calculations on the proportional, differential, and integral of the error signal, and the expression is as shown in formula (5),

[0025]

[0026] where e 0 、e 1 and e 2 are the differential of the error, the error of the input signal, and the error of the differential of the input signal respectively; a 1 、a 2 and a 3 are the nonlinear factors; β 21 、β 22 and β 23 are the gain of the differential of the error of the input signal, the gain of the error of the input signal, and the gain of the error of the differential of the input signal respectively.

[0027] The beneficial effects of the present invention are as follows: A speed control method for a permanent magnet synchronous motor based on fractional order active disturbance rejection control of the present invention first designs a new non-linear function newfal(x,a), which improves the system response speed while reducing jitter, and also improves the continuity and smoothness of the function at the origin; then optimizes the fastest function and designs a fractional order tracking differentiator PO-TD in combination with the fractional order theory. By arranging the transition process, a smooth model is obtained, thereby establishing a fast response without overshoot; subsequently, by designing a fractional order state observer PO-ESO, the observation effect of the ESO is enhanced, and the control performance of the entire controller for non-integer order objects is strengthened; then, using the non-linear state error feedback control law NLSEF, the overall anti-interference ability of the ADRC is improved, the response speed is accelerated, the system control accuracy is also improved, and the robustness of the system is enhanced; immediately afterwards, the PO-ADRC control method is used to control the motor, improving the anti-interference performance and response speed of the permanent magnet synchronous motor; finally, by using the fractional order ADRC control method in the permanent magnet synchronous motor, stable control of the permanent magnet synchronous motor is achieved, and the anti-interference performance and fast response ability of the present invention are verified through simulation experiments. Description of the Drawings

[0028] Figure 1 is the overall flow chart of the present invention;

[0029] Figure 2 is a comparison diagram of the newfal(x,α) function and the fal(x,α,c) function of the present invention;

[0030] Figure 3 is a schematic diagram of the error gain of the non-linear function of the present invention;

[0031] Figure 4 is a schematic diagram of the PO-TD response curve of the present invention;

[0032] Figure 5 is a schematic diagram of the framework of the fractional order state observer of the present invention;

[0033] Figure 6 is a schematic diagram of the mathematical model of the permanent magnet synchronous motor of the present invention;

[0034] Figure 7 is a schematic diagram of the simulation experiment results of the present invention;

[0035] Figure 8 is a schematic diagram of the verification experiment results of the present invention. Detailed Embodiment

[0036] The present invention will be further described below in conjunction with the accompanying drawings of the specification.

[0037] Such as Figure 1As shown, a speed control method for a permanent magnet synchronous motor based on fractional order active disturbance rejection of the present invention includes the following steps:

[0038] Step (A), design a new non-linear function newfal(x,a), and the expression of the new non-linear function newfal(x,a) is as shown in formula (1):

[0039]

[0040] where a represents the non-linear factor and x represents the error signal; the non-linear function is the core of the active disturbance rejection control mechanism, and the performance of the non-linear function largely determines the control performance of the active disturbance rejection; in order to reduce the problem of active disturbance rejection parameter tuning, the present invention designs for the non-linear function. On the basis of satisfying continuity, smoothness and convergence at the origin, a new non-linear function is designed, which can achieve interval stable switching while reducing the number of input parameters.

[0041] Perform a simulation comparison between the newfal(x,α) function and the fal(x,α,c) function. When c = 0.01 and α = 0.5, the function curves are as Figure 2 shown. It can be seen that the new non-linear function has better convergence and smoothness at the origin and the break point; effectively overcomes the problem that the traditional fal function has a large slope due to the inflection point near the origin. Through the function error curve Figure 3 it can be known that compared with the traditional function, the improved function satisfies the principle of "small error, large gain, large error, small gain", and avoids problems such as excessive increment of the fal function at the origin, effectively reducing problems such as system jitter.

[0042] Step (B), optimize the fastest function and design a fractional order tracking differentiator PO-TD in combination with the fractional order theory. The design of the fractional order tracking differentiator PO-TD specifically combines the principle of fractional order calculus, and the expression of the fastest function is as shown in formula (2):

[0043]

[0044] where r is the set point, v 1 and v 2 are the transition process and its differential signal, and θ is the derivative symbol; is the Caputo fractional order calculus, and its specific expression is as shown in formula (3):

[0045]

[0046] where m is a positive integer satisfying m - 1 < a E < m, and f(x) is a smooth function. Since the fractional order calculus 0 < aE <1, and then v 1 (t) ≈ v(t), r 1 and r 2 are constants, W = 1 - Γ 2 (a E +1) / Γ(2a E +1), n is odd.

[0047] The tracking differentiator, as an important part of the active disturbance rejection, mainly solves problems such as the reasonable extraction of the tracking signal and the differentiating signal; while reducing the response jitter, it also improves the response speed, as Figure 4 shown.

[0048] Step (C), based on the new nonlinear function, design the fractional-order state observer PO-ESO, and the specific algorithm expression of the fractional-order state observer PO-ESO is shown in formula (4),

[0049]

[0050] where ω 1 is the observation error; z 1 , z 2 and z 3 are the observation outputs of the system states x 1 , x 2 and x 3 respectively; the parameters β 1 , β 2 and β 3 are the gain coefficients of the extended state observer; b is the system gain parameter, a represents the Caputo fractional order of t0 D t a and the fractional-order ESO structure is as Figure 5 shown.

[0051] Step (D), construct the nonlinear state error feedback law and design the nonlinear feedback control law NLSEF, and the nonlinear state error feedback law is used for non-linear calculations of the proportional, differentiating, and integrating of the error signal, and the expression is shown in formula (5),

[0052]

[0053] where e 0 , e 1 and e 2 are the differentiating of the error, the error of the input signal, and the error of the differentiating of the input signal respectively; a 1 , a 2 and a 3 are the non-linear factors; β 21 , β22 and β 23 are the gains of the differential of the input signal error, the gain of the input signal error, and the gain of the error of the input signal differential, respectively.

[0054] Although the traditional non - linear feedback control law can meet the requirements of the entire controller, its control efficiency and effect are not optimal. The NLSEF in the present invention uses the non - linear state error feedback law to perform non - linear calculations on the proportion, differential, and integral of the error signal, improving the control accuracy of the control system and enhancing the system robustness.

[0055] Step (E), design a fractional - order active disturbance rejection controller PO - ADRC based on the fractional - order tracking differentiator PO - TD, the fractional - order state observer PO - ESO, and the non - linear feedback control law NLSEF;

[0056] Step (F), use the control method of the fractional - order active disturbance rejection controller PO - ADRC in the permanent - magnet synchronous motor to control its speed.

[0057] To further elaborate on the usage effect of the present invention, the following introduces the simulation experiment of the present invention: The present invention conducts simulations for various interferences under different input signals and conducts simulation comparisons with the parameters shown in Table 1.

[0058] Table 1 Parameters of the input signal

[0059]

[0060] By establishing the mathematical model of the permanent - magnet synchronous motor, assuming: 1. Do not consider the eddy current and hysteresis loss of the iron core; 2. Neglect the damping on the rotor winding; 3. The rotor magnetic field is symmetrically distributed about the magnetic pole center line; then the PMSM mathematical model can be as Figure 6 shown, use the proposed method to control the permanent - magnet synchronous motor for simulation, where the permanent - magnet synchronous motor mathematical model, J = 2.2×10 -5 kg·m 2 , B = 3×10 -3 N·m·s, T L = 0.2N·m·s.

[0061] The simulation experiment is as Figure 7As shown in the figure, where Figure a shows the response curves of two types of ADRCs when the input is a step input of 1 rad and the disturbance value is a step signal of 1000 rad; Figure b shows the error curves of the two; Figure c shows the response curves of two types of ADRCs when the input is a step input of 1 rad and the disturbance value is a pulse signal of 500 rad; Figure d shows the error curves of the two; Figure e shows the response curves of two types of ADRCs when the input is a step input of 1 rad and the input power is a noise signal of e-1; Figure f shows the error curves of the two; Figure g shows the response curves of two controllers when the input is a step input of 1 rad; Figure h shows the response curves of two observers when a pulse input signal is selected. And the system response data results are shown in Table 2.

[0062] Table 2 Experimental Results

[0063] Input signal External interference / value Feature Traditional ADRC Improved ADRC Step signal Step signal interference / 1000 Abrupt change quantity 0.25 0.03 Step signal Pulse signal interference / 500 Abrupt change quantity 0.10 0.01 Step signal <![CDATA[Random signal interference / e -1 > Abrupt change quantity 0.004 0.0001 Step signal None Response speed 0.580 0.443 Pulse signal None Response speed 0.648 0.543

[0064] From Figure 7 and Table 2, it can be seen that when the input is a step input of 1 rad and the disturbance amount is a step signal of 1000 rad, the disturbance change amount of the original ADRC is 0.25, and the disturbance change amount of the optimized ADRC is 0.03; when the input is a step input of 1 rad and the disturbance amount is a pulse signal of 500 rad, the disturbance change amount of the original ADRC is 0.10, and the disturbance change amount of the optimized ADRC is 0.01; when the input is a step input of 1 rad and the input power is a noise signal of e -1 noise signal, the average disturbance change amount of the original ADRC is 0.004, and the average disturbance change amount of the optimized ADRC is 0.0001; when the input value is a step signal of 1 rad, the response speed of the original ADRC is 0.580, and the response speed of the optimized ADRC is 0.443, an increase of 23.4%; when the input is a pulse signal, the response speed of the original ADRC is 0.648, and the response speed of the optimized ADRC is 0.543, an increase of 16.2%. Through comparison, it can be concluded that the optimized ADRC has better anti-interference ability and faster response speed than the traditional ADRC.

[0065] To verify the feasibility of the method of the present invention, a servo motor is used as the experimental object, where the input voltage is 200 V, the rated current is 2.5 A, the rated power is 400 W, the working frequency is 250 Hz, the torque is 1.27 N*m, the rated speed is 3000 r / min, and the maximum torque is 6000 r / min. The verification experimental results are as Figure 8 shown. From Figure 8 it can be seen that the proposed improved fractional-order control method is feasible and has good use effects, and also has outstanding use value.

[0066] In summary, for the speed control method of a permanent magnet synchronous motor based on fractional-order active disturbance rejection, a new nonlinear function newfal(x,a) is first designed, which improves the system response speed, reduces jitter, and also enhances the continuity and smoothness of the function at the origin. Then, the fastest function is optimized and combined with the fractional-order theory to design a fractional-order tracking differentiator PO-TD. By arranging the transition process, a smooth signal is obtained, thereby establishing a fast response without overshoot. Subsequently, a fractional-order state observer PO-ESO is designed to enhance the observation effect of the ESO and strengthen the control performance of the entire controller for non-integer-order objects. Then, the nonlinear state error feedback control law NLSEF is used to improve the overall anti-interference ability of the ADRC, accelerate the response speed, improve the system control accuracy, and enhance the robustness of the system. Immediately afterwards, the PO-ADRC control method is used to control the motor, improving the anti-interference performance and response speed of the permanent magnet synchronous motor. Finally, by using the fractional-order ADRC control method in the permanent magnet synchronous motor, stable control of the permanent magnet synchronous motor is achieved.

[0067] The foregoing has shown and described the basic principles, main features, and advantages of the present invention. Those skilled in the art should understand that the present invention is not limited by the above embodiments. The above embodiments and the descriptions in the specification are only for explaining the principles of the present invention. Without departing from the spirit and scope of the present invention, the present invention will have various changes and improvements, and these changes and improvements all fall within the scope of the present invention claimed. The scope of the present invention claimed is defined by the appended claims and their equivalents.

Claims

1. A speed control method for permanent magnet synchronous motors based on fractional order active disturbance rejection, characterized in that: It includes the following steps, Step (A), design a new nonlinear function newfal(x,a), where the expression of the new nonlinear function newfal(x,a) is shown in formula (1), where a represents the nonlinear factor and x represents the error signal; Step (B), optimize the fastest function and design a fractional order tracking differentiator PO-TD in combination with the fractional order theory. The design of the fractional order tracking differentiator PO-TD specifically combines the principle of fractional order calculus, and the expression of the fastest function is shown in formula (2), where r is the set point, v 1 and v 2 are the transient process and its differential signal, and θ is the derivative symbol; is the Caputo fractional calculus, and its specific expression is shown in formula (3). where m is a positive integer satisfying m - 1 < a E < m, and f(x) is a smooth function, Due to the fractional calculus 0 < a E < 1, and thus v 1 (t) ≈ v(t), r 1 and r 2 are constants, W = 1 - Γ 2 (a E + 1) / Γ(2a E + 1), and n is odd; Step (C), design a fractional order state observer PO-ESO based on the new nonlinear function. The specific algorithm expression of the fractional order state observer PO-ESO is shown in formula (4), where, ω 1 is the observation error; z 1 , z 2 and z 3 are the observation outputs of the states x 1 , x 2 and x 3 of the system respectively; the parameters β 1 , β 2 and β 3 are the gain coefficients of the extended state observer; b is the system gain parameter, and a represents the Caputo fractional order of t0 D t a ; Step (D), construct a nonlinear state error feedback law and design a nonlinear feedback control law NLSEF. The nonlinear state error feedback law is used for nonlinear calculation of the proportional, differential, and integral of the error signal, and the expression is shown in formula (5), where e 0 , e 1 and e 2 are the differential of the error, the error of the input signal, and the error of the differential of the input signal, respectively; a 1 , a 2 and a 3 are non-linear factors; β 21 , β 22 and β 23 are the gains of the differential of the error of the input signal, the gain of the error of the input signal, and the gain of the error of the differential of the input signal, respectively; Step (E), design a fractional order active disturbance rejection controller PO-ADRC based on the fractional order tracking differentiator PO-TD, the fractional order state observer PO-ESO, and the nonlinear feedback control law NLSEF; Step (F), use the control method of the fractional order active disturbance rejection controller PO-ADRC in the permanent magnet synchronous motor to control its speed.

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