A Complementary Sliding Mode Position Control Method for Permanent Magnet Synchronous Motor Based on Sliding Mode Extended State Observer

Abstract

The invention discloses a complementary sliding mode position control method for a permanent magnet synchronous motor based on a sliding mode extended state observer. First, the sliding mode surface of this strategy combines a generalized sliding mode surface and a complementary sliding mode phase surface. Then, an equivalent control law without integral action is designed, and an adaptive law is introduced in the switching control to dynamically adjust the gain of the boundary layer. Finally, a sliding mode extended state observer is designed to observe the uncertain disturbance, and feedforward compensation is combined to suppress the influence of the disturbance on the position control accuracy. This strategy not only realizes the fast, accurate and overshoot-free following of the position of the permanent magnet synchronous motor, but also effectively improves the robustness of the system to uncertain disturbances.

Description

Technical Field

[0001] The present invention relates to the technical field of permanent magnet synchronous motor (PMSM) position control, and particularly to a complementary sliding mode position control method for five-phase PMSM based on a sliding mode extended state observer. It is applicable to occasions with high requirements for the performance of the motor position servo system, such as aerospace, military, and industrial robots. Background Art

[0002] In recent years, with the rapid development of power electronics technology, modern control theory, etc., the position servo performance of motors has been further improved, which has enabled servo motors to be widely used in various fields. Due to its advantages such as high efficiency and large power density, PMSM is widely used in servo drives.

[0003] Traditional PMSM position servo systems usually adopt a three-closed-loop control of a position loop, a speed loop, and a current loop. The position loop adopts proportional derivative control, and the speed loop and current loop adopt proportional integral control. Although this control method has advantages such as simple design and easy implementation, for a high-order system like PMSM with multiple variables, high coupling, and nonlinearity, using this control method, its performance will be affected by uncertain disturbances and cannot meet the requirements of high precision and high response for the position servo system in specific occasions. In recent years, sliding mode control (SMC) has strong robustness and has been widely used in PMSM servo control. However, when the uncertain disturbance is bounded, SMC has a chattering phenomenon, and the chattering will reduce the dynamic performance and tracking error of the system, seriously affecting the stability of the system. Usually, using a saturation function instead of a sign function in the sliding mode switching control can effectively weaken the chattering, but the robustness of the system is reduced. Summary of the Invention

[0004] The present invention proposes a complementary sliding mode control (CSMC) strategy based on a sliding mode extended state observer to achieve fast, accurate, and overshoot-free following of the PMSM position and have strong robustness to the uncertain disturbances of the system.

[0005] A PMSM complementary sliding mode position control method based on a sliding mode extended state observer includes the following steps:

[0006] Step 1, establish the mathematical model of PMSM;

[0007] Step 2, define the mechanical position angle tracking error of PMSM as the state variable e and establish the state equation of the system;

[0008] Step 3, according to the state variable e, design the generalized sliding mode surface s1 and the complementary sliding mode surface s2, determine the relationship between the sliding mode surfaces s1 and s2, and further obtain Introduce an adaptive law into the complementary sliding mode control law Dynamically adjust the gain of the boundary layer, and then design a complementary sliding mode control law where

[0009]

[0010] In the formula: A n = 5P n ψ f / 2J; B n = B / J; P n is the number of pole pairs; ψ f is the rotor permanent magnet flux linkage (Wb); J is the moment of inertia (kg·m 2 ); B is the damping coefficient (N·m·s / rad); θ is the mechanical position angle (rad); e is the mechanical position angle tracking error (rad); λ is the sliding mode surface parameter, λ > 0; k1, k2 are the gains of the controller adaptation law, k1 > 0, k2 > λ 3 ; Ф is the boundary layer thickness value; sat(·) is the saturation function, specifically expressed as

[0011]

[0012] Step 4, for the uncertain disturbance d(t), construct an extended state observer to obtain the mechanical position angle observation error ε θ , the mechanical angular velocity observation error ε ω and the uncertain disturbance observation error ε dis , and obtain the relationship between ε θ , ε ω and ε dis as

[0013]

[0014] In the formula: k3 and k4 are the observer gains, k3 > 0, k4 > 0.

[0015] Step 5, according to the mechanical angular velocity observation error ε ω , design the sliding mode surface σ and the sliding mode reaching law Combined with the relationship between ε θ , ε ω and ε dis in Step 4, obtain the uncertain disturbance observation error ε dis as

[0016]

[0017] In the formula: k6 is the parameter of the sliding mode surface, k6 > 0; k7, k8 are the exponential term coefficient and the switching gain coefficient of the reaching law respectively, k7 > 0, k8 > 0;

[0018] Step 6, based on the obtained uncertainty disturbance observation error ε dis , design a sliding mode extended state observer to obtain the uncertainty disturbance observation value which is

[0019]

[0020] Step 7, substitute the disturbance observation value into the complementary sliding mode control law to obtain the complementary sliding mode control law based on the sliding mode extended state observer which is

[0021]

[0022] Step 8, the complementary sliding mode control CSMC based on the extended state observer is the position controller of the PMSM, and the output of this controller is the reference value of the q-axis current Adopt a PI controller as the current inner loop controller to control the current in the synchronous rotating coordinate system

[0023] Furthermore, the mathematical model of the PMSM in step 1 is

[0024]

[0025] where: A n = 5P n ψ f / 2J; B n = B / J; θ is the mechanical position angle (rad); ω is the mechanical angular velocity (rad / s); P n is the number of pole pairs; ψ f is the rotor permanent magnet flux linkage (Wb); J is the moment of inertia (kg·m 2 ); B is the damping coefficient (N·m·s / rad); T L is the load torque (N·m); r(t) is the change rate of the system uncertainty disturbance; d(t) is the system uncertainty disturbance, which can be expressed as

[0026] d(t) = ΔAi q -ΔBω - T L -ΔT L

[0027] where: ΔA, ΔB, ΔT L are the change amounts of A n , B n , external disturbance change and friction respectively

[0028] Furthermore, the state equation of the system in step 2 is

[0029]

[0030] Where: θ * is the given mechanical position angle (rad); e is the mechanical position angle tracking error (rad).

[0031] Further, the specific process of step 3 is as follows:

[0032] Step 3.1, design the generalized sliding surface s1 and the complementary sliding surface s2 as

[0033]

[0034] Where: λ is the sliding surface parameter, λ>0.

[0035] Step 3.2, the relationship between s1 and s2 can be expressed as

[0036]

[0037] Where: s is the sum of the generalized sliding surface s1 and the complementary sliding surface s2.

[0038] Step 3.3, It can be expressed as

[0039]

[0040] Therefore, the complementary sliding mode control law is designed

[0041] Further, the specific process of step 4 is as follows:

[0042] Step 4.1: Based on the PMSM mathematical model in step 1, the extended state observer is constructed as

[0043]

[0044] Where: k3, k4 and k5 are observer gains, k3>0, k4>0, k5>0; and are the observed values of mechanical position angle and mechanical angular velocity; is the uncertainty perturbation observation value.

[0045] Step 4.2, the observation errors of mechanical position angle, mechanical angular velocity and uncertainty disturbance can be expressed as

[0046]

[0047] Where: ε θ is the mechanical position angle observation error, ε ω is the mechanical angular velocity observation error, ε disis the observation error of the uncertainty disturbance, obtain ε θ 、ε ω and ε dis the relationship between.

[0048] Furthermore, the specific process of step 5 is

[0049] Step 5.1, according to the mechanical angular velocity observation error ε ω design the sliding mode surface σ as

[0050]

[0051] where: k6 is the parameter of the sliding mode surface, k6 > 0.

[0052] Step 5.2, design the sliding mode reaching law as the exponential reaching law

[0053]

[0054] where: k7 and k8 are the exponential term coefficient and the switching gain coefficient of the reaching law respectively, k7 > 0, k8 > 0.

[0055] Step 5.3, combining the relationship between ε θ 、ε ω and ε dis in step 4, the designed sliding mode surface σ can be further expressed as

[0056]

[0057] According to steps 5.2 and 5.3, obtain the observation error ε dis of the uncertainty disturbance.

[0058] Furthermore, the permanent magnet synchronous motor PMSM can be a three-phase PMSM, or a five-phase PMSM, or a six-phase PMSM; it can be a rotary PMSM or a linear PMSM.

[0059] Advantages of the present invention:

[0060] 1. Compared with the traditional SMC strategy, the CMSC strategy based on the sliding mode extended state observer for PMSM position control proposed by the present invention has better dynamic performance, and the position tracking error is at least half of that of the traditional SMC, and the steady-state performance is better.

[0061] 2. In the equivalent control of the CSMC strategy based on the sliding mode extended state observer proposed by the present invention, there is no integral action, and an adaptive law is introduced in the switching control to ensure that the system can achieve fast, accurate and non-overshooting following of the position under stable conditions.

[0062] 3. The present invention expands the uncertain disturbance into a state variable, constructs a combination of an extended state observer and a sliding mode observer, and designs a sliding mode extended state observer to obtain the observed value of the uncertain disturbance. It not only has excellent observation accuracy but also improves the dynamic performance of the observer.

[0063] 4. The present invention combines the CSMC strategy with the sliding mode extended state observer to observe and feedforward compensate for the uncertain disturbance, enhancing the robustness and anti-interference ability of the system and improving the position control accuracy.

[0064] 5. The CSMC method based on the sliding mode extended state observer of the present invention only adopts the position loop and the current loop. Compared with the traditional position control method, it removes the speed loop and improves the position response speed. BRIEF DESCRIPTION OF THE DRAWINGS

[0065] Figure 1 is the CSMC control block diagram based on the sliding mode extended state observer for PMSM position control in the embodiment of the present invention;

[0066] Figure 2 is the structural schematic diagram of the sliding mode extended state observer in the embodiment of the present invention;

[0067] Figure 3 is the structural schematic diagram of the CSMC control based on the sliding mode extended state observer in the embodiment of the present invention;

[0068] Figure 4 is the comparison waveform diagram of the position responses of the CSMC based on the sliding mode extended state observer and the traditional SMC in the present invention under a position step;

[0069] Figure 5 is the comparison waveform diagram of the position responses of the CSMC based on the sliding mode extended state observer and the traditional SMC in the present invention under a load step. DETAILED DESCRIPTION OF THE INVENTION

[0070] A CSMC method based on a sliding mode extended state observer for PMSM position control proposed by the present invention. In order to make the technical solutions, objectives, and effects of the present invention clearer and more distinct, the technical solutions implemented by the present invention will be further described clearly and completely below with reference to the accompanying drawings.

[0071] Step 1, design a CSMC position controller

[0072] The control block diagram of the system is as Figure 1 shown. The position control system of the present invention consists of an outer position control loop and an inner current control loop. Considering that the five-phase PMSM motor system is affected by uncertain factors such as parameter variations, load disturbances, and non-linear friction, the mathematical model of the PMSM can be expressed as

[0073]

[0074] Where: A n =5P n ψ f / 2J;B n =B / J; θ is the mechanical position angle (rad); ω is the mechanical angular velocity (rad / s); P n is the pole pair number; ψ f is the rotor permanent magnet flux (Wb); J is the moment of inertia (kg·m 2 ); B is the damping coefficient (N·m·s / rad); T L is the load torque (N·m); r(t) is the rate of change of the system uncertainty disturbance; d(t) is the system uncertainty disturbance, which can be expressed as

[0075] d(t)=ΔAi q -ΔBω-T L -ΔT L (2)

[0076] Where: ΔA, ΔB, ΔT L A n The change of B n changes in the amount of external disturbances and friction, etc.

[0077] In order to make the actual θ of the five-phase PMSM accurately track the given θ * , define the mechanical position angle tracking error as the state variable e, combined with the system shown in formula (1), the state equation of the system can be obtained as follows:

[0078]

[0079] The generalized sliding surface s1 and the complementary sliding surface s2 are designed as follows:

[0080]

[0081] Where: λ is the sliding surface parameter, λ>0. The relationship between s1 and s2 can be expressed as

[0082]

[0083] Where: s is the sum of the generalized sliding surface s1 and the complementary sliding surface s2.

[0084] According to (3) to (5), we can get

[0085]

[0086] Based on this, the complementary sliding mode control law is designed for

[0087]

[0088] Where: k1, k2 are the gains of the controller's adaptive law, k1>0, k2>λ 3 ; Ф is the boundary layer thickness; sat(·) is the saturation function, specifically expressed as

[0089]

[0090] The sliding surface adopts a method combining the generalized sliding surface s1 and the complementary sliding surface s2. When the system satisfies the existence and reachability of the sliding mode, that is, the designed complementary sliding mode controller is asymptotically stable, the mechanical position angle tracking error will reach the saturation function boundary layer within a limited time, s=s1+s2<Ф, then the mechanical position angle tracking error can be limited to

[0091]

[0092] Therefore, compared with the traditional SMC strategy, the mechanical position angle tracking error of the CSMC strategy is at least reduced to half of the original. In addition, in formula (7), there is no integral effect in the equivalent control, and the adaptive law is introduced in the switching control. By dynamically adjusting the gain of the boundary layer, the system can suppress the windup phenomenon under stable conditions and achieve fast, accurate and overshoot-free tracking of the motor position.

[0093] Step 2: Observe the uncertainty perturbation d(t)

[0094] According to formula (1), the extended state observer is constructed as

[0095]

[0096] Where: k3, k4 and k5 are observer gains, k3>0, k4>0, k5>0; and are the observed values of mechanical position angle and mechanical angular velocity; is the observed value of the uncertainty disturbance.

[0097] Combining equations (1) and (10), the observation error can be expressed as

[0098]

[0099] Where: ε θ is the mechanical position angle observation error, ε ω is the mechanical angular velocity observation error, ε dis is the uncertainty perturbation observation error, Get ε θ , ε ωand ε dis The relationship between

[0100]

[0101] In order to weaken the sliding mode chattering and improve the observation accuracy, the mechanical angular velocity observation error ε is selected ω The sliding surface and sliding reaching law are designed to adopt the exponential reaching law. The sliding surface is designed to be

[0102]

[0103] Where: k6 is the parameter of the sliding surface, k6>0.

[0104] Using the exponential reaching law

[0105]

[0106] Where: k7 and k8 are the exponential term coefficient of the reaching law and the switching gain coefficient respectively, k7>0, k8>0.

[0107] Combined with ε θ , ε ω and ε dis The relationship between the designed sliding surface σ can be further expressed as

[0108]

[0109] Substituting equation (14) into equation (15), we can obtain the system uncertainty disturbance observation error ε dis

[0110]

[0111] The sliding mode extended state observer is designed to obtain the uncertainty disturbance observation value for

[0112]

[0113] From formula (17), we can see that the mechanical angular velocity observation error ε ω Design the sliding surface and combine it with the exponential reaching law to obtain the uncertainty disturbance observation value It can effectively weaken the sliding mode chattering and improve the observation accuracy. In addition, compared with the linear extended state observer, the sliding mode extended state observer has better dynamic performance. Figure 2 Shown is the structural diagram of the sliding mode expansion state observer.

[0114] Perturb the observations Substituting the complementary sliding mode control law, we get the CSMC control law based on the sliding mode extended state observer for

[0115]

[0116] Figure 3 The figure shows the schematic diagram of the CSMC structure based on the sliding mode expansion state observer. The controller is used as the position controller of the PMSM, and the output of the controller is the reference value of the q-axis current. A PI controller is used as the current inner loop controller to control the current in a synchronous rotating coordinate system.

[0117] Through the above analysis, the CSMC strategy based on the sliding mode expansion state observer can achieve fast, accurate and non-overshoot position tracking, and has strong robustness to system uncertainty disturbances. In addition, the speed loop is omitted, and the position response speed is faster. In order to verify the effectiveness and feasibility of this method, Figure 4 and 5 The corresponding simulation waveform is given.

[0118] Figure 4 The following waveforms compare the position response of the CSMC position control method based on the sliding mode extended state observer (SMO) and the traditional SMC position control method under a position step. Traditional SMC position control has a long adjustment time of approximately 0.25 seconds and a large position tracking error within 0.02 degrees. The CSMC position control method based on the sliding mode extended state observer (SMO) offers a fast position response with no overshoot and a short adjustment time of approximately 0.15 seconds. Furthermore, the position tracking error is limited to within 0.005 degrees. Therefore, the CSMC method based on the sliding mode extended state observer achieves fast, accurate, and overshoot-free position tracking.

[0119] Figure 5 The following waveforms compare the position responses of the CSMC based on the sliding mode extended state observer and the traditional SMC under load steps in an embodiment of the present invention. Using the traditional SMC, when the load steps from 0 N·m to 10 N·m, the motor position drops by approximately 0.5 degrees, with a recovery time of 0.1s. When the load steps from 10 N·m to 5 N·m, the position drops by 0.2 degrees, with a recovery time of 0.08s. Furthermore, when the entire process is stable, the position tracking error remains within 0.5 degrees. Using the CSMC based on the sliding mode extended state observer, the position fluctuations are 0.04 degrees and 0.005 degrees, respectively, with recovery times of only 0.08s and 0.05s. Furthermore, when the entire process is stable, the position tracking error remains within 0.005 degrees. It can be seen that compared to the traditional SMC, the CSMC position control method based on the sliding mode extended state observer is highly robust to system uncertainty disturbances and has higher position control accuracy.

[0120] As can be seen from the above, a PMSM complementary sliding mode position control strategy based on a sliding mode extended state observer proposed by the present invention can not only obtain good tracking accuracy, but also has excellent dynamic response quality, realizing fast, accurate and non-overshooting following of the motor position. Considering the influence of uncertain disturbances on the system control accuracy, the present invention designs a sliding mode extended state observer to observe uncertain disturbances, and combines feedforward compensation to suppress the influence of disturbances on the position control accuracy, making the system have strong robustness.

[0121] Although the embodiments of the present invention have been shown and described, those of ordinary skill in the art can understand that various changes, modifications, substitutions and variations can be made to these embodiments without departing from the principles and spirit of the present invention. The scope of the present invention is defined by the claims and their equivalents.

Claims

1. A permanent magnet synchronous motor (PMSM) complementary sliding mode position control method based on a sliding mode extended state observer, characterized in that, It includes the following steps: Step 1: Establish the mathematical model of the PMSM; Step 2: Define the mechanical position angle tracking error of the PMSM as the state variable e, and establish the state equation of the system; Step 3: According to the state variable e, design the generalized sliding mode surface s1 and the complementary sliding mode surface s2, determine the relationship between the sliding mode surfaces s1 and s2, and then obtain Introduce an adaptive law into the complementary sliding mode control law to dynamically adjust the gain of the boundary layer, and then design the complementary sliding mode control law For Where: A n = 5P n ψ f / 2J; B n = B / J; P n is the number of pole pairs; ψ f is the rotor permanent magnet flux linkage (Wb); J is the moment of inertia (kg·m 2 ); B is the damping coefficient (N·m·s / rad); θ is the mechanical position angle (rad); e is the mechanical position angle tracking error (rad); λ is the sliding mode surface parameter, λ > 0; k1, k2 are the gains of the controller adaptation law, k1 > 0, k2 > λ 3 ; Ф is the boundary layer thickness value; sat(·) is the saturation function, specifically expressed as Step 4: For the uncertainty disturbance d(t), construct an extended state observer to obtain the mechanical position angle observation error ε θ , the mechanical angular velocity observation error ε ω and the uncertainty disturbance observation error ε dis , and obtain the relationship among ε θ , ε ω and ε dis as follows In the formula: k3 and k4 are observer gains, k3>0, k4>0; Step 5, according to the mechanical angular velocity observation error ε ω , design the sliding mode surface σ and the sliding mode reaching law Combined with ε in Step 4 θ , ε ω and ε dis The relationship between them, obtain the uncertainty disturbance observation error ε dis as In the formula: k6 is the parameter of the sliding mode surface, k6>0; k7 and k8 are respectively the exponential term coefficient and the switching gain coefficient of the reaching law, k7>0, k8>0; Step 6, based on obtaining the uncertainty disturbance observation error ε dis , design a sliding mode extended state observer to obtain the uncertainty disturbance observation value as Step 7, substitute the disturbance observation value into the complementary sliding mode control law to obtain the complementary sliding mode control law based on the sliding mode extended state observer as Step 8, the complementary sliding mode control CSMC based on the extended state observer is used as the position controller of the PMSM, and the output of this controller is the reference value of the q-axis current. A PI controller is used as the current inner-loop controller to control the current in the synchronous rotating coordinate system.

2. A PMSM complementary sliding mode position control method based on a sliding mode extended state observer according to claim 1, characterized in that, The mathematical model of the PMSM in the said Step 1 is Where: A n = 5P n ψ f / 2J; B n = B / J; θ is the mechanical position angle (rad); ω is the mechanical angular velocity (rad / s); P n is the number of pole pairs; ψ f is the rotor permanent magnet flux linkage (Wb); J is the moment of inertia (kg·m 2 ); B is the damping coefficient (N·m·s / rad); T L is the load torque (N·m); r(t) is the change rate of the system uncertainty disturbance; d(t) is the system uncertainty disturbance, which can be expressed as d(t) = ΔAi q -ΔBω - T L -ΔT L Where: ΔA, ΔB, ΔT L are respectively the change in A n , the change in B n , the external disturbance change, and the friction force.

3. A PMSM complementary sliding mode position control method based on a sliding mode extended state observer according to claim 1, characterized in that The state equation of the system in the said Step 2 is where: θ * is the given mechanical position angle (rad); e is the mechanical position angle tracking error (rad).

4. A PMSM complementary sliding mode position control method based on a sliding mode extended state observer according to claim 1, characterized in that The specific process of the said Step 3 is Step 3.1: Design the generalized sliding mode surface s1 and the complementary sliding mode surface s2 as In the formula: λ is the parameter of the sliding mode surface, λ>0; The relationship between s1 and s2 can be expressed as In the formula: s is the sum of the generalized sliding mode surface s1 and the complementary sliding mode surface s2; Step 3.3, can be expressed as Therefore, a complementary sliding mode control law is designed 5. A PMSM complementary sliding mode position control method based on a sliding mode extended state observer according to claim 1, characterized in that, The specific process of the said Step 4 is Step 4.1: Construct an extended state observer according to the PMSM mathematical model in Step 1 as where: k3, k4, and k5 are observer gains, k3>0, k4>0, k5>0; and are the observed values of the mechanical position angle and the mechanical angular velocity; is the observed value of the uncertainty disturbance; The observation errors of the mechanical position angle, the mechanical angular velocity and the uncertainty disturbance can be expressed as Where: ε θ is the mechanical position angle observation error, ε ω is the mechanical angular velocity observation error, ε dis is the uncertainty disturbance observation error, Obtain the relationship between ε θ , ε ω and ε dis .

6. A PMSM complementary sliding mode position control method based on a sliding mode extended state observer according to claim 1, characterized in that The specific process of the said Step 5 is Step 5.1, according to the mechanical angular velocity observation error ε ω , design the sliding surface σ as In the formula: k6 is the parameter of the sliding mode surface, k6>0; Step 5.2: Design the sliding mode reaching law as the exponential reaching law In the formula: k7 and k8 are respectively the exponential term coefficient and the switching gain coefficient of the reaching law, k7>0, k8>0; Step 5.3, combining the relationship between ε θ , ε ω and ε dis , the designed sliding mode surface σ can be further expressed as According to steps 5.2 and 5.3, the uncertainty perturbation observation error ε is obtained dis .

7. A PMSM complementary sliding mode position control method based on a sliding mode extended state observer according to claim 1, characterized in that The permanent magnet synchronous motor PMSM can be a three-phase PMSM, or a five-phase PMSM, or a six-phase PMSM; it can be a rotary PMSM or a linear PMSM.

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