Sliding mode control method based on improved extended state observer

Abstract

The invention relates to the technical field of motor control, in particular to a sliding mode control method based on an improved extended state observer, which comprises the following steps: S1, analyzing the influence of factors such as parameter mismatch, flux harmonic and cogging torque in motor control, and establishing a motor motion equation according to the structural characteristics of a motor; s2, designing an improved extended state observer according to the motor motion equation, performing discretization processing on the improved extended state observer, and performing real-time observation on total disturbance by using the observer; s3, designing a discretized sliding mode controller, and combining the discretized sliding mode controller with the discretized improved extended state observer to perform disturbance observation and feed-forward compensation; the improved extended state observer is designed to enlarge the observation range of disturbance, the improved extended state observer and sliding mode control are combined, and disturbance suppression, dynamic response and steady-state precision of sliding mode control are enhanced.

Description

Technical Field

[0001] The present invention relates to the technical field of motor control, and specifically to a sliding mode control method based on an improved extended state observer. Background Art

[0002] As a new type of special motor, the permanent magnet arc motor has the advantages of simple structure, fast response speed, high torque density, and direct drive. With these characteristics, it is widely used in many fields. During the operation of the arc motor, it is affected by various complex disturbances, mainly including the cogging torque caused by the motor's own structure, the torque ripple caused by magnetic flux harmonics, and the torque change caused by parameter mismatch. These disturbances significantly inhibit the accuracy of speed control and the dynamic response ability.

[0003] The extended state observer is usually used to observe disturbances and perform feedforward compensation in the speed loop. As the bandwidth increases, the extended state observer can detect a larger range of disturbances, but the expansion of the bandwidth will also introduce noise, thereby reducing the stability of the system. When the motor operates at low speed, the noise introduced by the bandwidth of the extended state observer has a more significant impact on speed fluctuations, thus reducing the stability of the system. Summary of the Invention

[0004] Technical Problem: To solve the deficiencies mentioned in the above background art, the purpose of the present invention is to provide a sliding mode control method with an improved extended state observer. The present invention improves the extended state observer, adds a speed error integral term to the disturbance observation equation, and the observed value of speed is used for feedforward compensation in the sliding mode control to achieve speed control of the arc motor.

[0005] Technical Solution: A sliding mode control method based on an improved extended state observer of the present invention is realized through the following method, which specifically includes the following steps:

[0006] S1. Analyze the influence of factors such as parameter mismatch, magnetic flux harmonics, and cogging torque in motor control, and establish a motor motion equation according to the structural characteristics of the motor;

[0007] S2. Design an improved extended state observer according to the motor motion equation, perform discretization processing on it, and use the observer to perform real-time observation of the total disturbance;

[0008] S3. Design a discretized sliding mode controller, combine it with the discretized improved extended state observer, and perform disturbance observation and feedforward compensation.

[0009] Wherein,

[0010] The specific content of S1 is as follows:

[0011] The expression of the motor motion equation is:

[0012]

[0013] Where ω m is the rotational speed of the motor, J is the moment of inertia of the motor, p is the number of pole pairs of the motor, ψ f is the permanent magnet flux linkage of the motor, i q is the q-axis current, let T L be the load torque of the motor, ΔT cog be the cogging torque of the rotor, ΔT f be the torque ripple caused by flux linkage harmonics, ΔT p be the torque caused by parameter mismatch, and they can be expressed as

[0014]

[0015] Where N c is determined by the least common multiple of the number of stator slots and the number of rotor pole pairs, T n is the amplitude of the nth torque harmonic, T0 is the DC component of the torque, T 6n is the amplitude of the 6nth torque harmonic of the torque, Δψ f is the change in flux linkage, ΔJ is the change in moment of inertia, θ e is the electrical angle.

[0016] The specific implementation manner of the said S2 is as follows:

[0017] According to the motion equation, an improved extended state observer is established as

[0018]

[0019] Where β1, β2, β3 are the gains of the improved extended state observer, e is the rotational speed error, u is the control quantity of the speed loop of the motor drive system, z1 is the estimated value of ω m and z2 is the estimated value of f, and are the differential values of z1 and z2;

[0020] According to the bandwidth method, their gains are respectively: β1 = 3ω0, where ω0 is the observation bandwidth;

[0021] Taking the Laplace transform of Equation (3), the closed-loop transfer functions of the observed values z1 and z2 can be obtained as

[0022]

[0023] Where s is the Laplace operator. Discretizing Equation (3) using the Euler method, the discrete equation is obtained as

[0024]

[0025] Wherein, T is the sampling period, k is the sampling time, ω m (k) is the rotational speed at time k, e(k) is the rotational speed error at time k, u(k) is the control quantity of the speed loop of the motor drive system at time k, and z1(k) is the estimate of ω m (k) at time k, z2(k) is the estimate of f(k) at time k, z3(k) is the integral value of e(k) at time k, and z1(k + 1) is the estimate of ω m (k + 1) at time k + 1, z2(k + 1) is the estimate of f(k + 1) at time k + 1, and z3(k + 1) is the integral value of e(k + 1) at time k + 1.

[0026] The specific implementation manner of the said S3 is as follows:

[0027] The integral sliding mode surface function s(k) is

[0028] s(k) = e(k) + c∑e(k) (6)

[0029] Wherein, the integral coefficient c > 0, and e(k) is the rotational speed ω m error value at time k. According to formula (6), the form of the sliding mode surface function at time k - 1 is

[0030] s(k - 1) = e(k - 1) + c∑e(k - 1) (7)

[0031] Subtracting formula (7) from formula (6) gives

[0032] s(k) - s(k - 1) = e(k) - e(k - 1) + ce(k) (8)

[0033] Then the sliding mode surface function s(k) is expressed as

[0034] s(k) = (1 + c)e(k) - e(k - 1) + s(k - 1) (9)

[0035] According to formula (9), the sliding mode surface function s(k + 1) at the next moment is

[0036] s(k + 1) = s(k) + (1 + c)e(k + 1) - e(k) (10)

[0037] Wherein, e(k + 1) is the rotational speed error at time k + 1.

[0038] Adopt the form of discrete exponential reaching law

[0039] s(k + 1) = (1 - Tq)s(k) - Tεsign(s(k)) (11)

[0040] Wherein, the proportionality coefficients ε > 0, q > 0, and satisfy 0 < Tq < 1, sign is the sign function; from Equation (11) and Equation (11) we get

[0041] (1 + c)e(k + 1) - e(k) = -Tεsign(s(k)) - Tqs(k) (12)

[0042] Equation (12) is transformed to obtain

[0043]

[0044] Substituting Equation (13) into Equation (1) gives

[0045]

[0046] Wherein, f(k) is the disturbance at time k.

[0047] The control variable u(k) is expressed as

[0048] u(k) = (u0(k) - z2(k)) / b (15).

[0049] Beneficial effects: The present invention improves the extended state observer, adds an integral term of the rotational speed error to the disturbance observation equation, and the observed value of the rotational speed is used for feedforward compensation in the sliding mode control to achieve the rotational speed control of the arc motor; compared with the traditional extended state observer, it can improve the suppression ability of low-frequency disturbances and the estimation accuracy of the total system disturbance. Therefore, the present invention can greatly improve the rotational speed stability performance and anti-disturbance performance of the arc motor control system at low speeds. Description of the Drawings

[0050] In order to more clearly illustrate the technical solutions in the embodiments of the present invention or the prior art, the following will briefly introduce the drawings required for use in the description of the embodiments or the prior art. Obviously, for those of ordinary skill in the art, without creative efforts, other drawings can also be obtained based on these drawings;

[0051] Figure 1 is a schematic diagram of the sliding mode control process based on the improved extended state observer of the present invention;

[0052] Figure 2 is a schematic diagram of the structure of the improved extended state observer of the present invention;

[0053] Figure 3 is a Bode diagram of the improved extended state observer of the present invention;

[0054] Figure 4It is a schematic diagram of the sliding mode control structure based on an improved extended state observer;

[0055] Figure 5 It is a schematic diagram for comparing the rotational speed waveforms of the motor under three control methods. Figure 5 In (a) is a schematic diagram of traditional PI control. Figure 5 In (b) is a schematic diagram of sliding mode control based on an extended state observer. Figure 5 In (c) is a schematic diagram of sliding mode control based on an improved extended state observer. Specific implementation manners

[0056] Next, the technical solutions in the embodiments of the present invention will be clearly and completely described in conjunction with the accompanying drawings in the embodiments of the present invention. Obviously, the described embodiments are only a part of the embodiments of the present invention, rather than all the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those of ordinary skill in the art without making creative efforts belong to the scope of protection of the present invention.

[0057] The sliding mode control method based on an improved extended state observer of the present invention includes the following steps:

[0058] S1. Analyze the influences of factors such as parameter mismatch, flux harmonic, and cogging torque in motor control, and establish a motor motion equation according to the structural characteristics of the motor;

[0059] S2. Design an improved extended state observer according to the motor motion equation, and perform discretization processing on it. Use the observer to perform real-time observation of the total disturbance;

[0060] S3. Design a discretized sliding mode controller, and combine it with the discretized improved extended state observer to perform disturbance observation and feedforward compensation.

[0061] During the operation of the motor, the motor is affected by various complex disturbances, mainly including cogging torque caused by the motor's own structure, torque fluctuations caused by flux harmonics, and torque changes caused by parameter mismatch. Extended state observers are usually used to observe disturbances and perform feedforward compensation in sliding mode control. To effectively suppress the influence of sensor noise and periodic disturbances on the observer, the present invention adopts an improved extended state observer. This improved scheme can enhance the ability to suppress low-frequency disturbances and improve the estimation accuracy of the total system disturbance, thereby improving the overall performance of the motor control system.

[0062] The present invention further improves the extended state observer to obtain stronger disturbance suppression ability and estimation accuracy. Next, in conjunction with Figure 2 、 3 、4, the sliding mode control method based on an improved extended state observer is elaborated and analyzed:

[0063] Specific implementation manner of S1:

[0064] The expression of the motion equation of the motor is:

[0065]

[0066] In the formula, ω m is the rotational speed of the motor, J is the moment of inertia of the motor, p is the number of pole pairs of the motor, ψ f is the magnetic flux of the permanent magnet of the motor, i q is the q-axis current. Let T L be the load torque of the motor, ΔT cog is the cogging torque of the rotor, ΔT f is the torque ripple caused by magnetic flux harmonics, ΔT p is the torque caused by parameter mismatch. They can be expressed as

[0067]

[0068] In the formula, N c is determined by the least common multiple of the number of stator slots and the number of rotor pole pairs, T n is the amplitude of the nth torque harmonic. T0 is the DC component of the torque, T 6n is the amplitude of the 6nth torque harmonic of the torque. Δψ f is the change in magnetic flux, ΔJ is the change in moment of inertia, θ e is the electrical angle.

[0069] The specific implementation manner of S2 is as follows:

[0070] According to the motion equation, an improved extended state observer is established as

[0071]

[0072] In the formula, β1, β2, β3 are the gains of the improved extended state observer, e is the rotational speed error, u is the control quantity of the speed loop of the motor drive system, z1 is the estimated value of ω m estimated value, z2 is the estimated value of f, and are the differential values of z1 and z2.

[0073] The extended state observer observes the rotational speed z1 and the actual rotational speed ω mThe difference between them is calculated to obtain the rotational speed error \(e\). Subsequently, the integral term of the rotational speed error is added to the disturbance observation equation, thereby obtaining the total disturbance \(z_2\). The observed total disturbance \(z_2\) is used for feedforward compensation in the sliding mode control, thereby enhancing the ability to suppress low-frequency disturbances and improving the estimation accuracy of the total system disturbance. By reasonably selecting parameters such as \(\beta_3\), it can be ensured that the state observer can quickly and accurately track the dynamic uncertainty of the system.

[0074] According to the bandwidth method, their gains are determined as: \(\beta_1 = 3\omega_0\), where \(\omega_0\) is the observation bandwidth.

[0075] Taking the Laplace transform of Equation (4), the closed-loop transfer functions of the observed values \(z_1\) and \(z_2\) are

[0076]

[0077] where \(s\) is the Laplace operator. The Bode plot of the transfer function is as Figure 3 shown. Adding the integral term of the rotational speed error to the observation disturbance equation can expand the observation range of the disturbance.

[0078] Discretizing Equation (4) using the Euler method, the discrete equation is

[0079]

[0080] where \(T\) is the sampling period, \(k\) is the sampling time, \(\omega\) m (k) is the rotational speed at time \(k\), \(e(k)\) is the rotational speed error at time \(k\), \(u(k)\) is the control quantity of the speed loop of the motor drive system at time \(k\), \(z_1(k)\) is the estimated value of \(\omega\) m (k) at time \(k\), \(z_2(k)\) is the estimated value of \(f(k)\) at time \(k\), \(z_3(k)\) is the integral value of \(e(k)\) at time \(k\), \(z_1(k + 1)\) is the estimated value of \(\omega\) m (k + 1) at time \(k + 1\), \(z_2(k + 1)\) is the estimated value of \(f(k + 1)\) at time \(k + 1\), and \(z_3(k + 1)\) is the integral value of \(e(k + 1)\) at time \(k + 1\).

[0081] The specific process in S3 is as follows:

[0082] The integral sliding mode surface function is

[0083] \(s(k)=e(k)+c\sum e(k)\ (6)\)

[0084] where the integral coefficient \(c\gt0\). According to Equation (7), the form of the sliding mode surface function at time \(k - 1\) is

[0085] s(k - 1) = e(k - 1) + c∑e(k - 1) (7)

[0086] Subtracting Equation (6) from Equation (7) gives

[0087] s(k) - s(k - 1) = e(k) - e(k - 1) + ce(k) (8)

[0088] Then the sliding mode surface function s(k) can be expressed as

[0089] s(k) = (1 + c)e(k) - e(k - 1) + s(k - 1) (9)

[0090] According to Equation (9), the sliding mode surface function s(k + 1) at the next moment is

[0091] s(k + 1) = s(k) + (1 + c)e(k + 1) - e(k) (10)

[0092] Where e(k + 1) is the rotational speed error at time k + 1.

[0093] Adopting the discrete exponential reaching law form

[0094] s(k + 1) = (1 - Tq)s(k) - Tεsign(s(k)) (11)

[0095] Where the proportionality coefficients ε > 0, q > 0, and 0 < Tq < 1 are satisfied, and sign is the sign function. From Equation (12) and Equation (10), we can get

[0096] (1 + c)e(k + 1) - e(k) = -Tεsign(s(k)) - Tqs(k) (12)

[0097] Equation (12) can be transformed to get

[0098]

[0099] Substituting Equation (13) into Equation (1) gives

[0100]

[0101] Where f(k) is the disturbance at time k.

[0102] The control variable u(k) can be expressed as

[0103] u(k) = (u0(k) - z2(k)) / b (15)

[0104] The structure of the sliding mode controller based on the improved extended state observer is as Figure 4As shown, the controller observes the total disturbance through the proposed observer and performs feedforward compensation in the sliding mode control to obtain the final control variable u(k).

[0105] The following conclusions can be drawn:

[0106] During the operation of the motor, there are a large number of periodic disturbances in both the motor body and the control system. The traditional extended state observer expands the disturbance observation range by increasing the bandwidth, but this method may introduce noise. To overcome this problem, an improved extended state observer introduces an integral term in the disturbance observation equation, thereby expanding the disturbance observation range and effectively suppressing the influence of noise and periodic disturbances on the observer.

[0107] Experimental results:

[0108] Figure 5 The rotational speed waveforms under three control methods are shown when the given rotational speed is 50 r / min and a disturbance of 0.5 N·m is added at t = 1 s. Specifically, after the disturbance is added, the time required for the rotational speed to rise under the traditional PI control method is 0.409 s, and the rotational speed drops by 12.61 r / min; under the sliding mode control method based on the extended state observer, the time required for the rotational speed to rise is 0.314 s, and the rotational speed drops by 6.62 r / min; under the sliding mode control method based on the improved extended state observer, the time required for the rotational speed to rise is 0.178 s, and the rotational speed drops by 2.42 r / min.

[0109] From Figure 5 it can be seen that the method proposed in the present invention can significantly improve the rotational speed response speed and at the same time enhance the anti-disturbance ability of the rotational speed.

[0110] The above shows and describes 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. What is described in the above embodiments and the specification only illustrates 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.

Claims

1. A sliding mode control method based on an improved extended state observer, characterized in that, The sliding mode control method includes the following steps: S1. Analyze the influences of factors such as parameter mismatch, flux linkage harmonics, and cogging torque in motor control, and establish a motor motion equation according to the structural characteristics of the motor; S2. Design an improved extended state observer according to the motor motion equation, perform discretization processing on it, and use the observer to perform real-time observation of the total disturbance; S3. Design a discretized sliding mode controller, combine it with the discretized improved extended state observer, and perform disturbance observation and feedforward compensation.

2. The sliding mode control method based on an improved extended state observer according to claim 1, characterized in that, The specific content of S1 is as follows: The expression of the motor motion equation is: where ω m is the rotational speed of the motor, J is the moment of inertia of the motor, p is the number of pole pairs of the motor, ψ f is the permanent magnet flux linkage of the motor, i q is the q-axis current, let T L be the load torque of the motor, ΔT cog be the cogging torque of the rotor, ΔT f be the torque ripple caused by flux linkage harmonics, ΔT p be the torque caused by parameter mismatch, and they can be expressed as follows respectively where N c is determined by the least common multiple of the number of stator slots and the number of rotor pole pairs, T n is the amplitude of the nth torque harmonic, T0 is the DC component of the torque, T 6n is the amplitude of the 6nth torque harmonic of the torque, Δψ f is the change in magnetic flux linkage, ΔJ is the change in moment of inertia, θ e is the electrical angle.

3. The sliding mode control method based on an improved extended state observer according to claim 2, wherein, The specific implementation manner of S2 is as follows: According to the motion equation, an improved extended state observer is established as where β1, β2, and β3 are the gains of the improved extended state observer, e is the rotational speed error, u is the control quantity of the speed loop of the motor drive system, z1 is the estimated value of ω m and z2 is the estimated value of f, and are the differential values of z1 and z2. Determine its gains according to the bandwidth method as: β1 = 3ω0, where ω0 is the observation bandwidth. Performing Laplace transform on Equation (3), the closed-loop transfer functions of the observed values z1 and z2 can be obtained as where s is the Laplace operator. Using the Euler method to discretize Equation (3), the discrete equation is obtained as where T is the sampling period and k is the sampling time, ω m (k) is the rotational speed at time k, e(k) is the rotational speed error at time k, u(k) is the control quantity of the speed loop of the motor drive system at time k, z1(k) is the estimated value of ω m (k) at time k, z2(k) is the estimated value of f(k) at time k, z3(k) is the integral value of e(k) at time k, z1(k + 1) is the estimated value of ω m (k + 1) at time k + 1, z2(k + 1) is the estimated value of f(k + 1) at time k + 1, z3(k + 1) is the integral value of e(k + 1) at time k + 1.

4. A sliding mode control method based on an improved extended state observer according to claim 2, characterized in that The specific implementation manner of S3 is as follows: The integral sliding mode surface function s(k) is s(k) = e(k) + c∑e(k) (6) where the integral coefficient c > 0. According to Equation (7), the form of the sliding mode surface function at the (k - 1)th moment is s(k - 1) = e(k - 1) + c∑e(k - 1) (7) Subtracting Equation (7) from Equation (6) gives s(k) - s(k - 1) = e(k) - e(k - 1) + ce(k) (8) Then the sliding mode surface function s(k) is expressed as s(k) = (1 + c)e(k) - e(k - 1) + s(k - 1) (9) According to Equation (9), the sliding mode surface function s(k + 1) at the next moment is s(k + 1) = s(k) + (1 + c)e(k + 1) - e(k) (10) where e(k + 1) is the rotational speed error at the (k + 1)th moment. Adopt the discrete exponential reaching law form s(k + 1) = (1 - Tq)s(k) - Tεsign(s(k)) (11) where the proportionality coefficients ε > 0, q > 0, and 0 < Tq < 1 are satisfied, and sign is the sign function; from Equation (10) and Equation (11), we get (1 + c)e(k + 1) - e(k) = -Tεsign(s(k)) - Tqs(k) (12) Equation (12) is transformed to get Substituting Equation (13) into Equation (1) gives where f(k) is the disturbance at the kth moment. The control variable u(k) is expressed as u(k) = (u0(k) - z2(k)) / b (15).

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