Discrete low-pass sliding mode control method based on extended state observer
Through the discrete low-pass slip mode control method combined with an expanded state observer and a low-pass filter, the torque fluctuation and vibration problems during low-speed operation of arc motors are solved, and higher speed stability and disturbance resistance are achieved.
Patent Information
- Application Number
- CN202510579738.6
- Authority / Receiving Office
- CN · China
- Patent Type
- Applications(China)
- Current Assignee / Owner
- Filing Date
- 2025-05-07
- Publication Date
- 2025-07-25
AI Technical Summary
When the arc motor is running at low speed, torque fluctuations caused by cogging torque, magnetic fluctuations and other factors affect the control accuracy. There is vibration phenomenon in sliding mode control, which reduces the control accuracy and causes additional burden on the system hardware.
A discrete low-pass sliding mode control method is constructed using an expanded state observer, combining the integrated sliding mode surface and the low-pass filter, feed-forward compensation is performed by observing the total disturbance, suppressing the vibration phenomenon and improving the speed response speed and anti-disturbance ability.
It effectively reduces the jitter amplitude, improves the speed stability and anti-interference performance of the arc motor drive system at low speeds, and enhances the stability of the system in complex environments.
Smart Images
Figure CN120377734A_ABST
Abstract
Description
Technical Field
[0001] The present invention relates to the technical field of motor control, and specifically to a discrete low-pass sliding mode control method based on an extended state observer. Background Art
[0002] The stator of the arc motor adopts a segmented splicing structure, which has the advantages of being easy to process, install, disassemble and repair, so it is widely used in fields such as robot joint drives and direct drive systems for large-aperture astronomical telescopes. When operating at low speeds, factors such as cogging torque and magnetic flux harmonics exist in the structure of the arc motor. At the same time, due to parameter mismatches and sampling errors, torque fluctuations may occur during the control process, thereby affecting the control accuracy of the motor. Therefore, when the permanent magnet synchronous motor operates at low speeds, speed stability and control accuracy become important indicators of the low-speed servo system.
[0003] Sliding mode control has been widely applied to various control systems due to its excellent robustness and strong adaptability to external disturbances. However, sliding mode control also faces a common problem - the "chattering" phenomenon. Frequent high-frequency switching will not only reduce the control accuracy but may also impose an additional burden on the system hardware. 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 discrete low-pass sliding mode control method based on an extended state observer, improve the sliding mode controller, combine the integral type sliding mode surface with a low-pass filter, and achieve the speed control of the arc motor.
[0005] Technical Solution: A discrete low-pass sliding mode control method based on an extended state observer of the present invention is realized in the following manner.
[0006] The sliding mode control method includes the following steps:
[0007] S1. Analyze the disturbances existing in motor control, and establish the motor motion equation of the reference disturbance.
[0008] S2. Construct an extended state observer from the motion equation, use the observer to observe the total disturbance, perform feedforward compensation using the total disturbance, and obtain its discrete form.
[0009] S3. Design a discrete low-pass sliding mode control model to suppress the chattering phenomenon in sliding mode control and improve the response speed and anti-disturbance ability of the speed.
[0010] Among them,
[0011] The specific content of step S1 is as follows:
[0012] During operation control, the motor is affected by the cogging torque and magnetic flux harmonics caused by the motor structure, and is also affected by torque ripple caused by motor parameter changes and current sampling errors during the actual operation of the motor control;
[0013] The motion equation expression of the motor is:
[0014]
[0015] In the formula, ω m is the rotational speed of the motor, d = T L + T cog + T flux + T p + T o + T c is the disturbance in the motor drive system, T L represents the load torque, T cog represents the cogging torque, T flux represents the torque ripple caused by magnetic flux harmonics, T p represents the torque disturbance caused by motor parameter errors, T o represents the torque caused by current calibration errors, T c represents the torque caused by current calibration errors, 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 of the motor, i q is the q-axis current.
[0016] The cogging torque is expressed as
[0017]
[0018] 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, θ e is the electrical angle.
[0019] The torque ripple caused by the magnetic flux harmonics is
[0020]
[0021] In the formula, T0 is the DC component of the torque, T 6n is the amplitude of the 6nth torque harmonic of the torque.
[0022] The torque disturbance caused by the motor parameter errors is expressed as
[0023]
[0024] In the formula, Δψ f is the change in magnetic flux, ΔJ is the change in moment of inertia.
[0025] The current sampling error includes current bias error and current calibration error, and the torque caused by the current bias error is expressed as
[0026]
[0027] In the formula, Δi a and Δi b are the bias error values of the phase current, α is the phase angle,
[0028] The torque caused by the current calibration error is expressed as
[0029]
[0030] In the formula, K a and K b are the calibration error coefficients of the phase current, I is the amplitude of the phase current, β is the phase angle; The specific content of S2 is as follows:
[0031] An extended state observer corresponding to the second-order motion equation is established as:
[0032]
[0033] In the formula, β1 and β2 are the gains of the extended state observer. Let x1 = ω m , x2 = f = -d / J, b is the control gain, e m is the speed error, u is the control quantity of the speed loop of the motor drive system, z1 is the estimated value of x1, z2 is the estimated value of x2, and are the differential values of z1 and z2;
[0034] According to the bandwidth method, their gains are determined respectively as: β1 = 2ω0, ω0 is the observation bandwidth,
[0035] The above formula is discretized by the Euler method to obtain the discrete equation as
[0036]
[0037] In the formula, T is the sampling period, k is the sampling time, ω m (k) is the speed at time k, e m (k) is the 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 x1 at time k, z2(k) is the estimated value of x2 at time k, z1(k + 1) is the estimated value of x1 at time k + 1, and z2(k + 1) is the estimated value of x2 at time k + 1.
[0038] The specific steps of step S3 are as follows:
[0039] The transfer function of the second-order low-pass filter is
[0040]
[0041] In the formula, ω c represents the cut-off frequency, K represents the gain, ζ represents the damping ratio, s is the Laplace operator,
[0042] Using the bilinear transformation method to transform the s-domain into the z-domain, let s = (2(1 - z -1 )) / (T(1 + z -1 ))), and the above formula is transformed into a discrete form
[0043]
[0044] In the formula, a1 = (2ω2cT 2 - 8) / (4 + 2ζω c T + ω2cT 2 ), a2 = (4 - 2ζω c T + ω2cT 2 ) / (4 + 2ζω c T + ω2cT 2 ),
[0045] b0 = Kω2cT 2 / (4 + 2ζω c T + ω2cT 2 ), b1 = 2b0, b2 = b0,
[0046] According to the above formula, the corresponding difference equation is
[0047] y(k) = b0e m (k) + b1e m (k - 1) + b2e m (k - 2) - a1y(k - 1) - a2y(k - 2)
[0048] In the formula, e m (k - 1) is the rotational speed error at the (k - 1)th moment, e m (k - 2) is the rotational speed error at the (k - 2)th moment, y(k) is the output value of the low-pass controller at the kth moment, y(k - 1) is the output value of the low-pass controller at the (k - 1)th moment, and y(k - 2) is the output value of the low-pass controller at the (k - 2)th moment. The sliding mode surface function is
[0049]
[0050] In the formula, c is the integral coefficient.
[0051] The discrete sliding mode surface functions s(k) and y(k) combined with low-pass control are expressed as
[0052]
[0053] The form of the sliding mode surface function at the (k - 1)th moment is
[0054] s(k - 1) = e m (k - 1) + c∑e m (k - 1) + y(k - 1)
[0055] Subtracting the above two equations gives
[0056]
[0057] The sliding mode surface function s(k) is expressed as
[0058]
[0059] From the above equation, the sliding mode surface function s(k + 1) is
[0060]
[0061] In the formula, e m (k + 1) is the rotational speed error at the (k + 1)th moment.
[0062] The discrete form of the exponential reaching law is
[0063] s(k + 1) = (1 - Tq)s(k) - Tεsign(s(k))
[0064] In the formula, the proportionality coefficients ε > 0, q > 0, and 0 < Tq < 1 are satisfied, and sign is the sign function. According to the above two equations, we can get
[0065]
[0066] Combining the above equation with the expression of the motor's motion equation gives
[0067]
[0068] In the formula, f(k) is the disturbance at the kth moment. The control variable u(k) is expressed as
[0069] u(k) = (u0(k) - z2(k)) / b.
[0070] Beneficial effects: A discrete low-pass sliding mode control method based on an extended state observer according to the present invention improves the sliding mode controller, combines an integral sliding mode surface with a low-pass filter to achieve speed control of an arc motor; compared with traditional sliding mode control, it reduces the chattering phenomenon caused by high-frequency switching, making the control smoother. Therefore, the present invention can greatly improve the speed stability performance and anti-disturbance performance of the arc motor drive system at low speeds. Description of the Drawings
[0071] 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 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;
[0072] Figure 1 is a schematic flowchart of the low-pass sliding mode control method of the present invention;
[0073] Figure 2 is a schematic structural diagram of the low-pass filter of the present invention;
[0074] Figure 3 is a schematic diagram for comparing the effects of the present invention with the sliding mode control method based on an extended state observer; Figure 3 In (a) is a schematic diagram of discrete sliding mode control based on an extended state observer, Figure 3 In (b) is a schematic diagram of discrete low-pass sliding mode control based on an extended state observer. Detailed Embodiments
[0075] The following will clearly and completely describe the technical solutions in the embodiments of the present invention with reference to the drawings in the embodiments of the present invention. Obviously, the described embodiments are only part of the embodiments of the present invention, rather than all of them. Based on the embodiments of the present invention, all other embodiments obtained by those of ordinary skill in the art without creative efforts belong to the scope of protection of the present invention.
[0076] A discrete low-pass sliding mode control method based on an extended state observer of the present invention, as Figure 1 shown, includes the following steps:
[0077] S1. Analyze the disturbances existing in motor control, and establish a motor motion equation for the reference disturbance;
[0078] S2. Construct an extended state observer from the motion equation, use the observer to observe the total disturbance, perform feedforward compensation using the total disturbance, and obtain its discrete form;
[0079] S3. Design a discrete low-pass sliding mode control model to suppress the chattering phenomenon in sliding mode control and improve the response speed of the rotational speed and the anti-disturbance ability.
[0080] During low-speed operation, there are factors such as cogging torque and flux harmonics in the structure of the arc-shaped motor. At the same time, due to parameter mismatch and sampling error, torque fluctuations may occur during the control process, thus affecting the control accuracy of the motor. In the present invention, a low-pass filter is added to the sliding mode control to filter out the high-frequency components in the control signal, smooth the output control quantity, effectively reduce the chattering amplitude, further enhance the anti-disturbance ability of the system to external disturbances, and thus improve the stability of the system in a complex environment.
[0081] The present invention further improves the sliding mode control strategy to obtain stronger anti-disturbance performance and chattering suppression ability. The discrete low-pass sliding mode control method based on the extended state observer is elaborated and analyzed as follows:
[0082] S1 Specific implementation method:
[0083] During the operation control of the motor, it is affected by the cogging torque and flux harmonics caused by the motor structure, and is also affected by torque pulsations caused by motor parameter changes and current sampling errors during the actual operation of the motor control.
[0084] The cogging torque can be expressed as
[0085]
[0086] where N c is determined by the least common multiple of the number of stator slots and the number of rotor pole pairs, θ e is the electrical angle, and T n is the amplitude of the nth torque harmonic.
[0087] The torque pulsation caused by flux harmonics is
[0088]
[0089] where T0 is the DC component of the torque, and T 6n is the amplitude of the 6nth torque harmonic of the torque.
[0090] The torque disturbance caused by motor parameter errors can be expressed as
[0091]
[0092] where 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 of the motor, i q is the q-axis current, T L is the load torque of the motor, and Δψ fΔΨ is the change in magnetic flux linkage, and ΔJ is the change in moment of inertia.
[0093] The current sampling error includes current offset error and current calibration error. The torque caused by the current offset error can be expressed as
[0094]
[0095] where Δi a and Δi b are the offset error values of the phase current, α is the phase angle,
[0096] The torque caused by the current calibration error can be expressed as
[0097]
[0098] where K a and K b are the calibration error coefficients of the phase current, I is the amplitude of the phase current, and β is the phase angle.
[0099] The expression of the motion equation of the motor is:
[0100]
[0101] where ω m is the rotational speed of the motor, d = T L + T cog + T flux + T p + T o + T c is the disturbance in the motor drive system, and T L is the load torque.
[0102] Specific implementation of S2:
[0103] An extended state observer corresponding to the second-order motion equation is established as:
[0104]
[0105] where β1 and β2 are the gains of the extended state observer. Let x1 = ω m , x2 = f = -d / J, e m is the rotational speed error, b is the control gain, u is the control quantity of the speed loop of the motor drive system, z1 is the estimated value of x1, z2 is the estimated value of x2, and are the differential values of z1 and z2;
[0106] According to the bandwidth method, their gains are determined as: β1 = 2ω0, where ω0 is the observation bandwidth.
[0107] The Euler method is used to discretize Equation (7), and the discrete equation obtained is
[0108]
[0109] where T is the sampling period and k is the sampling time. ω m (k) is the rotational speed at time k, e m (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 x1 at time k, z2(k) is the estimated value of x2 at time k, z1(k + 1) is the estimated value of x1 at time k + 1, and z2(k + 1) is the estimated value of x2 at time k + 1.
[0110] The specific implementation manner of S3 is as follows:
[0111] The transfer function of the second-order low-pass filter is
[0112]
[0113] where ω c represents the cut-off frequency, K represents the gain, ζ represents the damping ratio, and s is the Laplace operator.
[0114] The bilinear transformation method is used to transform the s-domain into the z-domain. Let s = (2(1 - z -1 )) / (T(1 + z -1 ))), and Equation (51) can be transformed into the discrete form
[0115]
[0116] where a1 = (2ω2cT 2 - 8) / (4 + 2ζω c T + ω2cT 2 ), a2 = (4 - 2ζω c T + ω2cT 2 ) / (4 + 2ζω c T + ω2cT 2 ), b0 = Kω2cT 2 / (4 + 2ζω c T + ω2cT 2 ), b1 = 2b0, and b2 = b0.
[0117] According to Equation (10), the corresponding difference equation can be obtained as
[0118] y(k) = b0e m (k) + b1e m(k - 1)+b2e m (k - 2)-a1y(k - 1)-a2y(k - 2) (11)
[0119] Where e m (k - 1) is the rotational speed error at time k - 1, e m (k - 2) is the rotational speed error at time k - 2, y(k) is the output value of the low - pass controller at time k, y(k - 1) is the output value of the low - pass controller at time k - 1, and y(k - 2) is the output value of the low - pass controller at time k - 2. Figure 2 is the structure diagram of the low - pass filter.
[0120] The sliding mode surface function is
[0121]
[0122] Where c is the integral coefficient.
[0123] The discrete sliding mode surface function s(k) combined with low - pass control can be expressed as
[0124]
[0125] The form of the sliding mode surface function at time k - 1 is
[0126] s(k - 1)=e m (k - 1)+c∑e m (k - 1)+y(k - 1) (14)
[0127] Subtracting Equation (13) from Equation (14) gives
[0128]
[0129] The sliding mode surface function s(k) is expressed as
[0130]
[0131] From Equation (16), the sliding mode surface function s(k + 1) is
[0132]
[0133] Where e m (k + 1) is the rotational speed error at time k + 1.
[0134] The discrete form of the exponential reaching law is
[0135] s(k + 1)=(1 - Tq)s(k)-Tεsign(s(k)) (18)
[0136] Wherein, the proportionality coefficients ε>0, q>0, and 0<Tq<1 is satisfied, and sign is the sign function.
[0137] According to equations (17) and (18), it can be obtained that
[0138]
[0139] Combining equations (19) and (6) gives
[0140]
[0141] Wherein, f(k) is the disturbance at time k.
[0142] The control variable u(k) can be expressed as
[0143] u(k) = (u0(k) - z2(k)) / b (21)
[0144] A low-pass filtering term is added to the sliding mode control to filter out the high-frequency components in the control signal, smooth the output control quantity, effectively reduce the chattering amplitude, and further enhance the anti-interference ability of the system to external disturbances, thereby improving the stability of the system in a complex environment.
[0145] The following conclusions can be obtained:
[0146] Due to its excellent robustness and strong adaptability to external disturbances, sliding mode control has been widely applied to various control systems. However, sliding mode control also faces a common problem - the "chattering" phenomenon. Frequent high-frequency switching not only reduces the control accuracy but may also impose an additional burden on the system hardware. To suppress the chattering phenomenon, the present invention introduces a low-pass filter into the sliding mode controller.
[0147] Experimental results:
[0148] Figure 3 (a) shows the rotational speed response graph of discrete sliding mode control based on an extended state observer when a disturbance of 0.5 N·m is added at t = 5 s under a given rotational speed of 50 r / min. Figure 3 (b) shows the rotational speed response graph of discrete low-pass sliding mode control based on an extended state observer under the same conditions. Specifically, under the discrete sliding mode control method based on an extended state observer, the rotational speed response time is 0.314 s, the rotational speed fluctuation range is 48.9 - 51.7 r / min, and the rotational speed drop amplitude after adding the disturbance is 6.62 r / min; while under the discrete low-pass sliding mode control method based on an extended state observer, the rotational speed response time is 0.151 s, the rotational speed fluctuation range is 49.4 - 51.2 r / min, and the rotational speed drop amplitude after adding the disturbance is 4.30 r / min.
[0149] From Figure 3It can be seen that the method of the present invention can not only significantly improve the rotational speed response speed, reduce the rotational speed fluctuation, but also make the decrease in rotational speed smaller after adding disturbances, indicating that the method of the present invention has remarkable effects in improving the rotational speed response speed and anti-disturbance ability.
[0150] 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 discrete low-pass sliding mode control method based on an extended state observer, characterized in that, The discrete low-pass sliding mode control method includes the following steps: S1. Analyze the disturbances existing in motor control, and establish a motor motion equation with reference to the disturbances; S2. Construct an extended state observer from the motion equation, use the observer to observe the total disturbance, perform feedforward compensation using the total disturbance, and obtain its discrete form; S3. Design a discrete low-pass sliding mode control model to suppress the chattering phenomenon in sliding mode control, and improve the response speed of the rotational speed and the anti-disturbance ability.
2. The discrete low-pass sliding mode control method based on an extended state observer according to claim 1, characterized in that The specific content of step S1 is as follows: During the operation control of the motor, it will be affected by the cogging torque and flux linkage harmonics caused by the motor structure, and will also be affected by the torque ripple caused by the changes in motor parameters and current sampling errors during the actual operation of motor control; The expression of the motor motion equation is: where ω m is the rotational speed of the motor, d = T L + T cog + T flux + T p + T o + T c is the disturbance in the motor drive system, T L represents the load torque, T cog represents the cogging torque, T flux represents the torque ripple caused by flux harmonics, T p represents the torque disturbance caused by motor parameter errors, T o represents the torque caused by current calibration errors, T c represents the torque caused by current calibration errors, 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 of the motor, i q is the q-axis current.
3. A discrete low-pass sliding mode control method based on an extended state observer according to claim 2, characterized in that, The cogging torque is expressed as 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 n-th torque harmonic, and θ e is the electrical angle.
4. A discrete low-pass sliding mode control method based on an extended state observer according to claim 2, characterized in that The torque ripple caused by the flux linkage harmonics is where T0 is the DC component of the torque, and T 6n is the amplitude of the 6n-th torque harmonic of the torque.
5. A discrete low-pass sliding mode control method based on an extended state observer according to claim 2, characterized in that, The torque disturbance caused by the motor parameter error is expressed as where Δψ f is the change in magnetic flux linkage, and ΔJ is the change in moment of inertia.
6. A discrete low-pass sliding mode control method based on an extended state observer according to claim 2, characterized in that The current sampling error includes current offset error and current calibration error, and the torque caused by the current offset error is expressed as where Δi a and Δi b are the offset error values of the phase currents, and α is the phase angle.
7. A discrete low-pass sliding mode control method based on an extended state observer according to claim 2, characterized in that, The torque caused by the current calibration error is expressed as where K a and K b are the calibration error coefficients of the phase current, I is the amplitude of the phase current, and β is the phase angle.
8. A discrete low-pass sliding mode control method based on an extended state observer according to claim 1, characterized in that The specific content of S2 is as follows: An extended state observer corresponding to the second-order motion equation is established as: where β1 and β2 are the gains of the extended state observer, and let x1 = ω m , x2 = f = -d / J, b is the control gain, e m 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 x1, z2 is the estimated value of x2, and are the differential values of z1 and z2; Determine its gains according to the bandwidth method as follows: β1 = 2ω0, where ω0 is the observation bandwidth, The above formula is discretized by the Euler method to obtain the discrete equation as where T is the sampling period, k is the sampling time, ω m (k) is the rotational speed at time k, e m (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 x1(k) at time k, z2(k) is the estimated value of x2(k) at time k, z1(k + 1) is the estimated value of x1(k + 1) at time k + 1, and z2(k + 1) is the estimated value of x2(k + 1) at time k + 1.
9. A discrete low-pass sliding mode control method based on an extended state observer according to claim 1, characterized in that The specific content of step S3 is as follows: The transfer function of the second-order low-pass filter is where ω c represents the cut-off frequency, K represents the gain, ζ represents the damping ratio, and s is the Laplace operator The bilinear transformation method is used to convert the s-domain to the z-domain. Let s = (2(1 - z -1 )) / (T(1 + z -1 )) and transform the above formula into a discrete form where a1 = (2ω2cT 2 - 8) / (4 + 2ζω c T + ω2cT 2 ), a2 = (4 - 2ζω c T + ω2cT 2 / (4 + 2ζω c T + ω2cT 2 ), b0 = Kω2cT 2 / (4 + 2ζω c T + ω2cT 2 ), b1 = 2b0, b2 = b0, The corresponding difference equation is obtained according to the above formula as y(k) = b0e m (k) + b1e m (k - 1) + b2e m (k - 2) - a1y(k - 1) - a2y(k - 2) where e m (k - 1) is the rotational speed error at time k - 1, e m (k - 2) is the rotational speed error at time k - 2, y(k) is the output value of the low-pass controller at time k, y(k - 1) is the output value of the low-pass controller at time k - 1, and y(k - 2) is the output value of the low-pass controller at time k - 2. The sliding mode surface function is In the formula, c is the integral coefficient. Combined with the discrete sliding mode surface function s(k) of the low-pass control, y(k) is expressed as The form of the sliding mode surface function at the (k - 1)th moment is s(k - 1)=e m (k - 1)+c∑e m (k - 1)+y(k - 1) Subtracting the above two formulas gives The sliding mode surface function s(k) is expressed as From the above formula, the sliding mode surface function s(k + 1) is obtained as where, e m (k + 1) is the rotational speed error at the (k + 1)-th moment. The discrete form of the exponential reaching law is s(k + 1) = (1 - Tq)s(k) - Tεsign(s(k)) In the formula, the proportionality coefficients ε > 0, q > 0, and 0 < Tq < 1 are satisfied, and sign is the sign function. According to the above two formulas, it can be obtained that Combining the above equation with the expression of the motor's motion equation gives In the formula, f(k) is the disturbance at the kth moment. The control variable u(k) is expressed as u(k) = (u0(k) - z2(k)) / b.