Position-free control based on set time sliding mode observer and time-varying parameter extended state observer

Abstract

The invention discloses position-free control based on a set time sliding mode observer and a time-varying parameter expansion state observer, and belongs to the technical field of permanent magnet synchronous motor sensorless control. The invention provides a composite control method combining a set time sliding-mode observer and a time-varying parameter extended state observer, and aims to solve the problems that the convergence time of a traditional sliding-mode observer depends on an initial state, buffeting is serious, disturbance is sensitive and the like. The method specifically comprises the following steps: designing a sliding-mode observer with a set time convergence characteristic, and ensuring that an observation error is converged in a preset time and is irrelevant to an initial value by dynamically adjusting a sliding-mode surface index item; high-frequency buffeting is suppressed by combining an integral filtering effect; a time-varying parameter expansion state observer is further designed by utilizing the saturation characteristic of a bitangent function and the characteristics of small error, large gain, large error and small gain of a fal function, the initial differential peak value is effectively inhibited, and the total disturbance of the system is compensated in real time. And the advantages of controllable convergence time, small buffeting and high interference resistance are verified through simulation. The anti-interference capability and the dynamic response speed of the system are remarkably improved, and the method is suitable for permanent magnet synchronous motor sensorless control under medium and high speed working conditions.

Description

Technical Field

[0001] The present invention relates to sensorless control based on a set-time sliding mode observer and a time-varying parameter extended state observer, and particularly to the technical field related to permanent magnet synchronous motor control. Background Art

[0002] Permanent magnet synchronous motors have a large power density, high energy conversion efficiency, and a simple structure, so they are widely used in high-performance speed regulation systems. In engineering practice, in order to achieve vector control of permanent magnet synchronous motors, it is necessary to rely on mechanical sensors to instantaneously obtain the speed and position information of the motor rotor. However, the installation of mechanical sensors has multiple adverse effects, increasing the system hardware cost and overall volume, and being vulnerable to environmental influences and prone to failures, reducing system reliability. Therefore, sensorless control has extremely important research and application value in the current industrial field.

[0003] Sensorless control techniques for permanent magnet synchronous motors are mainly divided into two methods: high-frequency signal injection method and model-based method. The high-frequency signal injection method performs excellently in the stationary and low-speed stages where the back electromotive force can be neglected by utilizing the motor characteristics. However, its drawback is the introduction of large noise and complex demodulation processes. On the contrary, the model-based method relies on the back electromotive force of the motor and is widely applicable in the medium and high-speed regions because it can accurately capture the motor position and speed. However, the performance of such methods is limited by the accuracy of motor parameters.

[0004] Among various model-based methods, model reference adaptive systems, extended Kalman filters, and sliding mode observers are common representatives. Among them, the sliding mode observer is favored due to its strong robustness, simplified structure design, and fast convergence performance. However, the traditional sliding mode observer uses a discontinuous sign function as the control strategy, inevitably causing chattering phenomena during motor operation.

[0005] Chinese Patent No. 202210291733.X proposes a sliding mode observer method for the load torque of a permanent magnet synchronous motor. This method uses a sliding mode observer to observe the load torque of the permanent magnet synchronous motor and feed forward compensate its observed value to the current regulator, which can effectively reduce system chattering. However, the performance of the sliding mode observer depends on the accuracy of motor parameters, and its convergence time is affected by the initial load torque and speed of the motor, affecting the fast response performance of the system. And the convergence time of this patent can converge within a set time.

[0006] Chinese Patent No. 201811184495.2 proposes a sensorless control method for permanent magnet synchronous motors based on a smooth nonsingular terminal sliding mode observer. This method does not require an additional low-pass filter and can directly obtain the back electromotive force observation value, thus solving the phase lag problem. Through the new control method, the stator current error is reduced, and the estimation accuracy of the back electromotive force is improved, thereby enhancing the observation performance of the rotational speed and rotor position. However, there may be a problem of increased cost; it cannot converge within the set time, and the convergence time depends on the sliding mode observer with the initial value. In contrast, the convergence time of this patent is set and does not depend on the sliding mode observer with the initial value.

[0007] Chinese Patent No. 201711349302.X proposes a method for estimating the rotor position error of a sensorless permanent magnet synchronous motor. This method calculates the back electromotive force by collecting the stator current and voltage of the permanent magnet synchronous motor, obtains the rotor position estimation value through coordinate transformation, etc., and then calculates and compensates for the position error to improve the sensorless control accuracy. However, its convergence time is not fixed and is affected by multiple factors such as the algorithm, motor operating state, system parameters, signal processing, and measurement errors. In contrast, the innovative control strategy adopted in this patent can ensure that the system converges within the set time range.

[0008] Chinese Patent No. 201610303247.X proposes a control method and system for permanent magnet synchronous motors based on sliding mode observation. This method designs a load torque sliding mode observer, uses sliding mode control to compensate in the speed loop, rebuilds the speed controller, thereby obtaining a more stable q-axis reference current, and finally achieving relatively ideal rotational speed and torque effects. However, this method has relatively high requirements for parameter selection and design strategies, requires certain optimization and adjustment, and the design process is relatively complex; moreover, the convergence time does not converge within the set time. In contrast, with the unique design of the control mechanism in this patent, the convergence time can be accurately limited to complete the convergence within the preset time interval.

[0009] Chinese Patent No. 202410727686.8 proposes a control system and method for a controller of a sensorless permanent magnet synchronous motor. This method realizes the accurate calculation of the rotational speed of the permanent magnet synchronous motor without a position sensor through the synchronous motor motion equations, and efficiently and quickly iterates the duty ratio through an incremental PI model, calculates and outputs three-phase SPWM, adjusts the controller, and modulates the bus DC power. However, the convergence time of the incremental PI model may be related to the initial value, and different initial duty ratio and other parameters may lead to differences in the convergence time of the system to reach the stable state. In contrast, this patent uses a sophisticated control logic and algorithm architecture to successfully achieve that the convergence time is firmly within the established clear time threshold to complete the convergence process.

[0010] This patent has developed a sensorless control technology for permanent magnet synchronous motors that can converge within a set time and the convergence time is independent of the initial value, effectively solving the problems presented in the above patent. Summary of the Invention

[0011] The present invention proposes a positionless control based on a set-time sliding mode observer and a time-varying parameter extended state observer, with smaller system observation errors, and at the same time improving the system's rapidity and anti-disturbance ability.

[0012] The technical solution of the present invention is as follows:

[0013] Positionless control based on a set-time sliding mode observer and a time-varying parameter extended state observer, the method comprising the following steps:

[0014] Step 1: In the α-β axis coordinate system, use a set-time sliding mode observer to track the stator current of the permanent magnet synchronous motor, construct a current state observer equation, and obtain a stator current error state equation;

[0015] Step 2: In the continuous sliding mode motion state, design a set-time sliding mode surface with the stator current error as the state variable. After processing by the set-time sliding mode surface, then through the control rate switching and integral action designed based on the current error state equation and the set-time sliding mode surface, obtain an output vector v without high-frequency switching, and obtain the back electromotive force accordingly;

[0016] Step 3: Use the back electromotive force formula of the permanent magnet synchronous motor and the back electromotive force information obtained in Step 2 to calculate the speed and rotor position of the permanent magnet synchronous motor, and then realize sensorless motor control;

[0017] Step 4: According to the motor speed obtained in Step 3, use a time-varying parameter extended state observer to estimate the total system disturbance in real time and compensate its influence on the speed.

[0018] Step 5: Select a Lyapunov function to prove the stability of the system and obtain the upper bound of the convergence time.

[0019] Further, in Step 1, in Step 1, use a set-time sliding mode observer to track the stator current of the permanent magnet synchronous motor to obtain a stator current error;

[0020] In the α-β axis coordinate system, the mathematical model of the surface-mounted permanent magnet synchronous motor without a position sensor is as follows:

[0021]

[0022] In the formula, where Lm and R are the magnetization inductance and resistance of the stator phase winding; e sBack electromotive force in the axis coordinate system. The expression of the back electromotive force is:

[0023]

[0024] ω is the mechanical angular velocity of the rotor, θ is the electrical angle of the rotor position, φ r is the magnetic flux per pair of magnetic poles;

[0025] When using sliding mode observer control, more ripples will be generated in the sampled frequency of the observed motor back electromotive force. To overcome this problem, by inserting a first-order low-pass filter, we get

[0026]

[0027] The set-time sliding mode observer designed according to Formula 1 is:

[0028]

[0029] In the formula, are the estimated values of the stator current under the α-β axes respectively; v α , v β is the SMO control law function.

[0030] Subtracting Formula 3 from Formula 4, the stator current error state equation is

[0031]

[0032] In the formula, are the errors between the estimated values and the actual values of the stator current under the α-β axes respectively.

[0033] Discretizing Formula 5 gives

[0034]

[0035] In the formula, T s is the sampling period.

[0036] Writing Formula 3 in the following form

[0037]

[0038] Parameter ω c = 2πf c , f c is the cut-off frequency of the filter.

[0039] The discrete form of Formula 7 is

[0040]

[0041] In the continuous sliding mode motion state, the stator current deviation is first processed by the sliding surface and then by the smooth nonsingular terminal sliding mode control law, which contains two key links: switching and integration. Through such processing, an output vector v without high-frequency switching can be finally obtained:

[0042] Preliminary knowledge of finite-time stability. Consider the following system

[0043]

[0044] where: x is the system state, is a nonlinear function of x on the open region U containing the origin. f(x) may also be a discontinuous function. The trajectory of the system is defined in the sense of Filippov, and the origin is a globally finite-time convergent equilibrium point.

[0045] Definition 1 If the system of Equation (9) is globally finite-time stable and there exists a finite-time convergence upper bound T independent of the initial value max > 0 such that T(x0) ≤ T max , then the system of Equation (6) is finite-time stable.

[0046] Lemma 1 Consider a class of nonlinear systems

[0047]

[0048] where l1, l2 > 0 and m1 > 1, 0 < m2 < 1. Then the system of Equation (10) is globally finite-time stable, and the upper bound of the finite-time convergence is

[0049]

[0050] The estimated stable time T of the system max does not depend on the initial state of the system, but only on the design parameters l1, l2, m1, m2.

[0051] Assumption 1 Assume that is bounded. There exists a constant D > 0 such that

[0052] Furthermore, in Step 2, according to the finite-time stability theory, a sliding surface with finite-time convergence characteristics is designed. The sliding surface with finite-time convergence characteristics realizes that the observation error converges to zero in finite time and the upper bound of the convergence time is independent of the initial value of the system. The expression of the sliding surface with finite-time convergence is:

[0053]

[0054] where \(a,b\gt0\), \(0\lt p\lt1\), \(q\gt1\). Compared with the terminal sliding mode with constant exponential form, the sliding mode surface with set-time convergence characteristic designed in the present invention can change its own exponential value in real time according to the system state error value. When the state error is far from the sliding mode surface the exponential term of the sliding mode surface is greater than 1. When the state error approaches the sliding mode surface the exponential term of the sliding mode surface is less than 1, thereby ensuring the global convergence speed of the system. Finally, when the state error approaches zero, the exponential term switches to a real number not less than 1 to solve the singularity problem existing in the traditional terminal sliding mode control.

[0055] It can be seen from Lemma 1 that the system state can converge to \(s = 0\) within the set time \(T_1\), and the convergence time is:

[0056]

[0057] Furthermore, in the step 2 described above, the double power reaching law maintains fast convergence both when far from and close to the sliding mode surface. At the same time, the double power reaching law has the global fast set-time convergence characteristic, and the convergence time has an upper bound independent of the initial value of the sliding mode. The designed sliding mode surface control law with set time is:

[0058]

[0059] where \(a_1\gt0\), \(b_1\gt0\), \(D\) is the switching gain. The sliding mode control law usually consists of two parts: equivalent control and switching control. The switching control makes the system state approach the sliding mode surface, while the equivalent control is used to control the deterministic part of the system to keep it on the sliding mode surface. In addition, the state exponents in the control law are all greater than 0, and there are no singularities, so it is non-singular. It can be obtained from Equation 14 that although the sign function \(sign(\cdot)\) is incorporated into the control law, the introduction of the integral effectively filters the high-frequency signals generated by the sign function. Therefore, the estimated extended back electromotive force presents a smooth signal, weakening the sliding mode chattering.

[0060] Furthermore, the time-varying parameter extended state observer can estimate the uncertainty and total disturbance in real time. By estimating the disturbance in real time, the influence of the disturbance on the system can be compensated.

[0061] Consider the following first-order system:

[0062]

[0063] According to Equation 15, it can be obtained that

[0064]

[0065] where \(b\) is the system coefficient and \(D(t)\) is the total disturbance received by the system. Based on the system motion equation, \(D(t)\) is defined

[0066] For a new expanded state, an equivalent system is formed, expressed as:

[0067]

[0068] In the formula, η(t) is the derivative of the total disturbance D(t). Therefore, a time-varying parameter extended state observer can be designed as follows:

[0069]

[0070] In the formula, z1 is the observed value of the rotational speed ω, and z2 is the observed value of the disturbance term; e0 is the estimated error between z1 and ω, and k1 and k2 are the gains of the NTLESO, which are related to the bandwidth of the observer. Selecting an appropriate bandwidth can make the state variables observed by the observer track the corresponding state variables. The difference between the proposed observer and the extended state observer lies in the compensated estimated disturbance quantity z2. In the traditional ESO, z2 is the integral of the observation error e0. While in the time-varying parameter extended state observer, z2 is the sum of the proportion and integral of e0.

[0071] Subtracting Equation (17) from Equation (18), the error system equation is obtained as:

[0072]

[0073] When the difference between the initial state of the ESO and the true state of the system is large, there will be peaks in the initial differential and acceleration signal estimations, and the differential peak will be more obvious with a larger gain coefficient. To suppress the initial peak of the ESO, the saturation characteristic of the hyperbolic function and the "small error, large gain, large error, small gain" characteristic of the output of the nonlinear function with respect to the input are utilized. Replace the fixed parameters k1 and k2 with time-varying parameters as follows:

[0074]

[0075] In the formula, k 11 , k 21 are fixed value parameters greater than 0 selected according to the system, and r1, r2 are time variable coefficients, and r1 > 0, r2 > 0.

[0076] The specific process of the stability proof by Lyapunov analysis is as follows: Select the following Lyapunov function Taking the derivative with respect to time gives:

[0077]

[0078] According to the stator current error state equation and the sliding mode surface expression of the set time convergence characteristic, it can be obtained that:

[0079]

[0080] Substituting the sliding mode surface control law with a set time into this formula, we can obtain:

[0081]

[0082] Substituting into the original formula, we can obtain:

[0083]

[0084] According to the above Hypothesis 1, the following inequality can be obtained:

[0085]

[0086] Therefore, there is

[0087]

[0088] Based on the negative characteristic of the derivative of the Lyapunov function, it can be concluded that the time required for the system to reach the sliding mode surface is T1, and the time required for the system to reach the state origin within the set time after reaching the sliding mode surface is T2. The system will remain stable within the set time, and the convergence time t a is

[0089] t a ≤T max = T1 + T2 Formula 27,

[0090] where Therefore, the system will be stable within the set time t a and

[0091] therefore, continuous sliding mode motion state will be achieved within a finite time According to the equivalent principle of sliding mode control,

[0092] if the stator current error equation is zero, we can obtain:

[0093] e = v Formula 28,

[0094] The control law v is a smooth output without high-frequency switching and can directly estimate the back electromotive force.

[0095] Advantages of the present invention:

[0096] 1) Based on the set-time stability theory, a sliding mode surface with set-time convergence characteristics is designed, realizing that the observation error converges to zero within the set time and the upper bound of the convergence time is independent of the initial value of the system.

[0097] 2) The integral operation included in the control law has a filtering effect on discontinuous control quantities, improving the situation of large chattering amplitude.

[0098] 3) The time-varying parameter extended state observer effectively suppresses the initial peak and compensates for system disturbances in real time through the hyperbolic tangent function and the nonlinear error function. Brief Description of the Drawings

[0099] Figure 1 This is the structural framework diagram of the present invention.

[0100] Figure 2 This is the simulation result diagram of the rotor speed estimation value and its error of the traditional sliding mode observer.

[0101] Figure 3 This is the simulation result diagram of the rotor position estimation value and its error of the traditional sliding mode observer.

[0102] Figure 4 This is the simulation result diagram of the rotor speed estimation value and its error of the present invention.

[0103] Figure 5 This is the simulation result diagram of the rotor position estimation value and its error of the present invention. Detailed Embodiment

[0104] 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. All other embodiments obtained by those of ordinary skill in the art based on the embodiments of the present invention without creative efforts shall fall within the protection scope of the present invention.

[0105] The system structure diagram of the set-time sliding mode observer of the present invention is as Figure 1 shown, and the parameters of the permanent magnet synchronous motor in the simulation are shown in Table 1.

[0106] Table 1 shows the parameters of the permanent magnet synchronous motor used in the simulation

[0107]

[0108] For the positionless control based on the set-time sliding mode observer and the time-varying parameter extended state observer, it is characterized in that the implementation process of the method is as follows:

[0109] In the step 1, in step one, the set-time sliding mode observer is used to track the stator current of the permanent magnet synchronous motor to obtain the stator current error;

[0110] In the α-β axis coordinate system, the mathematical model of the surface-mounted permanent magnet synchronous motor without a position sensor is as follows:

[0111]

[0112] In the formula, where Lm and R are the magnetization inductance and resistance of the stator phase winding; e s is the back electromotive force in the axis coordinate system. The expression of the back electromotive force is:

[0113]

[0114] ω is the mechanical angular velocity of the rotor, θ is the electrical angle of the rotor position, φ r is the magnetic flux per pair of magnetic poles;

[0115] When the sliding mode observer control is adopted, more ripples will be generated in the observed motor back electromotive force at the sampling frequency. To overcome this problem, by inserting a first-order low-pass filter, we get

[0116]

[0117] The set-time sliding mode observer designed according to Formula 1 is:

[0118]

[0119] In the formula, are the estimated values of the stator current under the α-β axes respectively; v α 、v β is the SMO control law function.

[0120] Subtracting Formula 3 from Formula 4, the stator current error state equation is

[0121]

[0122] In the formula, are the errors between the estimated values and the actual values of the stator current under the α-β axes respectively.

[0123] Discretizing Formula 5 gives

[0124]

[0125] In the formula, T s is the sampling period.

[0126] Writing Formula 3 in the following form

[0127]

[0128] The parameter ω c = 2πf c ,f c is the cut-off frequency of the filter.

[0129] The discrete form of Formula 7 is

[0130]

[0131] In the second step, in the continuous sliding mode motion state, the stator current deviation is first processed by the sliding surface, and then through the action of the smooth nonsingular terminal sliding mode control law, which contains two key links: switching and integration. Through such processing, an output vector v without high-frequency switching can be finally obtained:

[0132] Preliminary knowledge of finite-time stability. Consider the following system

[0133]

[0134] where: x is the system state, is a nonlinear function of x on the open region U containing the origin, and f(x) may also be a discontinuous function. The trajectory of the system is defined in the sense of Filippov, and the origin is a globally finite-time convergent equilibrium point.

[0135] Definition 1 If the system of Equation (6) is globally finite-time stable and there exists a finite-time convergence upper bound T independent of the initial value max > 0 such that T(x0) ≤ T max , then the system of Equation (9) is finite-time stable.

[0136] Lemma 1 Consider a class of nonlinear systems

[0137]

[0138] where l1, l2 > 0 and m1 > 1, 0 < m2 < 1. Then the system of Equation (10) is globally finite-time stable, and the upper bound of the finite-time convergence is

[0139]

[0140] The estimated stable time T of the system max does not depend on the initial state of the system, but only on the design parameters l1, l2, m1, m2.

[0141] Assumption 1 Assume that is bounded, and there exists a constant D > 0 such that

[0142] Furthermore, in the second step, according to the finite-time stability theory, a sliding surface with the characteristics of finite-time convergence is designed. The sliding surface with the characteristics of finite-time convergence realizes that the observation error converges to zero in finite time and the upper bound of the convergence time is independent of the initial value of the system. The expression of the sliding surface with the characteristics of finite-time convergence is:

[0143]

[0144] where \(a,b\gt0\), \(0\lt p\lt1\), \(q\gt1\), and the exponential value of the sliding mode surface is dynamically adjusted with the error. When far from the sliding mode surface, the exponent is greater than 1, and when close, the exponent is less than 1, thus ensuring the global convergence rate of the system. Finally, when the state error approaches zero, the exponential term switches to a real number not less than 1, further ensuring the global convergence rate of the system. The dynamic exponential design realizes the optimization of the global convergence rate and solves the singularity problem of the traditional terminal sliding mode near the equilibrium point.

[0145] It can be seen from Lemma 1 that the system state can converge to \(s = 0\) within the set time \(T_1\), and the convergence time is:

[0146]

[0147] Furthermore, in step 2, the double-power reaching law maintains fast convergence both when far from and close to the sliding mode surface. At the same time, the double-power reaching law has the characteristic of global fast set-time convergence, and the convergence time has an upper bound independent of the initial value of the sliding mode. The designed sliding mode surface control law with set time is:

[0148]

[0149] where \(a_1\gt0\), \(b_1\gt0\), \(c\gt1\), \(0\lt d\lt1\), \(k = D\) is the switching gain. The sliding mode control law usually consists of two parts: equivalent control and switching control. The switching control makes the system state approach the sliding mode surface, while the equivalent control is used to control the deterministic part of the system to keep it on the sliding mode surface. In addition, the state exponents in the control law are all greater than 0, and there are no singularities, so it is non-singular. Although the sign function is incorporated into the control law, the integral term can filter out high-frequency chattering. Therefore, the estimated extended back electromotive force presents a smooth signal, weakening the sliding mode chattering. Furthermore, the time-varying parameter extended state observer can estimate uncertainties and total disturbances in real time. By estimating the disturbances in real time, the influence of the disturbances on the system can be compensated.

[0150] Consider the following first-order system:

[0151]

[0152] According to Equation 15, we can obtain

[0153]

[0154] where \(b\) is the system coefficient and \(D(t)\) is the total disturbance received by the system. Based on the system motion equation, \(D(t)\) is defined

[0155] as a new extended state to form an equivalent system, expressed as:

[0156]

[0157] Where η(t) is the derivative of the total disturbance D(t). Therefore, a time-varying parameter extended state observer can be designed as follows:

[0158]

[0159] Where z1 is the observed value of the rotational speed ω, and z2 is the observed value of the disturbance term; e0 is the estimation error between z1 and ω, and k1 and k2 are the gains of the NTLESO, which are related to the bandwidth of the observer. Selecting an appropriate bandwidth can make the observed state variables of the observer track the corresponding state variables. The difference between the proposed observer and the extended state observer lies in the compensated estimated disturbance quantity z2. In the traditional ESO, z2 is the integral of the observation error e0. While in the time-varying parameter extended state observer, z2 is the sum of the proportion and integral of e0.

[0160] Subtracting Equation (17) from Equation (18), the error system equation is obtained as:

[0161]

[0162] When the difference between the initial state of the ESO and the true state of the system is large, peaks will appear in the initial differential and acceleration signal estimations, and the differential peak will be more obvious with a large gain coefficient. To suppress the initial peak of the ESO, the saturation characteristic of the hyperbolic function and the characteristic of the nonlinear function that realizes "small error, large gain, large error, small gain" for its output with respect to the input are utilized. Replace the fixed parameters k1 and k2 with time-varying parameters as follows:

[0163]

[0164] Where k 11 and k 21 are fixed value parameters greater than 0 selected according to the system, and r1 and r2 are time variable coefficients, and r1 > 0, r2 > 0.

[0165] The specific process of the stability proof by Lyapunov analysis is as follows: Select the following Lyapunov function Taking the derivative with respect to time gives:

[0166]

[0167] According to the stator current error state equation and the sliding mode surface expression of the set time convergence characteristic, it can be obtained that:

[0168]

[0169] Substituting the sliding mode surface control law of the set time into this formula, it can be obtained that:

[0170]

[0171] Substituting into the original formula, we can get:

[0172]

[0173] According to the above assumption 1, the following inequality can be obtained:

[0174]

[0175] Therefore, there is

[0176]

[0177] Based on the negative characteristic of the derivative of the Lyapunov function, it can be concluded that the time required for the system to reach the sliding mode surface is T1, and the time required for the system to reach the state origin within the set time after reaching the sliding mode surface is T2. The system will remain stable within the set time, and the convergence time t a is

[0178] t a ≤T max = T1 + T2 Formula 27,

[0179] where, Therefore, the system will be stable within the set time t a and

[0180] Therefore, continuous sliding mode motion state will be achieved within a finite time According to the equivalent principle of sliding mode control,

[0181] if the stator current error equation is zero, we can get:

[0182] e = v Formula 28,

[0183] The control law v is a smooth output without high-frequency switching and can directly estimate the back electromotive force.

[0184] Specifically, at t = 0.2 s, a step load of 5 Nm is suddenly applied, and the parameters of the set-time sliding mode observer are selected as: a = 5, b = 10, p = 0.5, q = 1.5, c = 0.5, a1 = 10, b1 = 1.5, d = 1000.

[0185] When using the time-varying parameter extended state observer to estimate the disturbance, the selected parameters are: k1 = 700, k2 = 140000.

[0186] Design the set-time sliding mode observer, and verify the control effect of the present invention through simulation comparison. Specifically, set the desired speed of the permanent magnet synchronous motor to 1000 rpm and set the above-mentioned suddenly applied load. Figure 2and Figure 3 is the simulation result diagram of the traditional sliding mode observer, Figure 2 is the simulation result diagram of the estimated value of the rotor speed and its error, Figure 3 is the simulation result diagram of the estimated value of the rotor position and its error. Figure 4 and Figure 5 is the simulation result diagram of the set-time sliding mode observer, Figure 4 is the simulation result diagram of the estimated value of the rotor speed and its error, Figure 5 is the simulation result diagram of the estimated value of the rotor position and its error.

[0187] It can be seen from the simulation results that generally speaking, compared with the traditional sliding mode observer, the set-time sliding mode observer has less chattering and better stability performance.

[0188] The series of detailed descriptions listed above are only specific descriptions of the feasible implementation modes of the present invention, and they are not intended to limit the protection scope of the present invention. Any equivalent implementation modes or changes made without departing from the technical spirit of the present invention should be included within the protection scope of the present invention.

Claims

1. A positionless control based on a set-time sliding mode observer and a time-varying parameter extended state observer, characterized in that The method includes the following steps: Step S1: In the α-β axis coordinate system, a set-time sliding mode observer is used to track the stator current of the permanent magnet synchronous motor, construct the current state observer equation, and obtain the stator current error state equation; Step S2: In the continuous sliding mode motion state, a set-time sliding mode surface is designed with the stator current error as the state variable. After being processed by the set-time sliding mode surface, through the control rate switching and integral action designed based on the current error state equation and the set-time sliding mode surface, an output vector v without high-frequency switching is obtained, and based on this, the back electromotive force is obtained; Step S3: Using the back electromotive force formula of the permanent magnet synchronous motor and the back electromotive force information obtained in Step 2, calculate the speed and rotor position of the permanent magnet synchronous motor, and then realize sensorless motor control; Step S4: According to the motor speed obtained in Step 3, a time-varying parameter extended state observer is used to estimate the total system disturbance in real time and compensate its influence on the speed; Step S5: Select the Lyapunov function to prove the stability of the system and obtain the upper bound of the convergence time.

2. The positionless control method based on a set-time sliding mode observer and a time-varying parameter extended state observer according to claim 1, wherein In the above Step S1, the sensorless mathematical model of the surface-mounted permanent magnet synchronous motor is: where i α , i β , u α , u β are the stator currents and voltages under the α-axis and β-axis respectively; e α and e β are the back electromotive force components in the axis coordinate system. The back electromotive force expression is: ω is the mechanical angular velocity of the rotor, θ is the electrical angle of the rotor position, and φ r is the magnetic flux linkage per pair of magnetic poles; Derive the back electromotive force: In the formula, there is an upper bound, which is jointly determined by the rotational speed and acceleration of the motor. Considering the key parameters of the motor such as the rated speed, rated current, moment of inertia, etc., and combining with the load condition borne by the motor, the upper bound can be estimated.

3. The method according to claim 1, wherein In the above Step S2, in order to make the convergence time of the system state variables independent of the initial values, a set-time sliding mode observer is designed based on the set-time sliding mode surface. The convergence time of this set-time sliding mode observer is independent of the initial state of the system. The set-time sliding mode observer is: Wherein, are respectively the estimated values of the stator current under the α-β axis; v α , v β is the SMO control law function. The design of the set-time sliding mode observer is based on the set-time sliding mode surface. The expression of the sliding mode surface with set-time convergence characteristics is: Where a, b > 0, 0 < p < 1, q > 1, and the exponential value of the sliding mode surface is dynamically adjusted with the error, greater than 1 when far from the sliding mode surface and less than 1 when approaching, thus ensuring the global convergence speed of the system. Finally, when the state error is close to zero, the exponential term switches to a real number not less than 1, further ensuring the global convergence speed of the system. The dynamic exponential design realizes the optimization of the global convergence speed and solves the singularity problem of the traditional terminal sliding mode near the equilibrium point. The system state can converge to S = 0 at the set time T1, and the convergence time is: The control law of the set-time sliding mode surface is: Where a1 > 0, b1 > 0, c > 1, 0 < d < 1, k = D is the switching gain. The sliding mode control law usually consists of two parts: equivalent control and switching control. The switching control makes the system state approach the sliding mode surface, and the equivalent control is used to control the deterministic part of the system to keep it on the sliding mode surface. In addition, the state exponents in the control law are all greater than 0 and there are no singularities, so it is non-singular. Although the sign function sign(·) is included in the control law, the integral term can filter out high-frequency chattering. Therefore, the estimated extended back electromotive force presents a smooth signal, weakening the sliding mode chattering. This set-time sliding mode observer is used for sensorless control of permanent magnet synchronous motors. Through the set-time sliding convergence mode surface and the double power approach law, it effectively solves the problems of inaccurate estimation, large chattering, and slow response existing in traditional sliding mode observers.

4. The positionless control method based on a set-time sliding mode observer and a time-varying parameter extended state observer according to claim 1, characterized in that In step S4, to address the problem of the initial differential peak phenomenon in the traditional extended state observer, the saturation characteristic of the hyperbolic tangent function and the characteristics of the fal function of "small error, large gain; large error, small gain" are utilized. The fixed parameters of the extended state observer are replaced with time-varying parameters to suppress the initial peak and improve the stability and control accuracy of the system during the startup phase and under disturbances. The design of the time-varying parameter extended state observer is as follows: For a first-order system: Based on this first-order system, we can obtain: where b is the system coefficient and D(t) is the total disturbance received by the system. Based on the system motion equation, D(t) is defined as a new extended state to form an equivalent system, expressed as: The time-varying parameter extended state observer is: where z1 is the observed value of the rotational speed ω, z2 is the observed value of the disturbance term; e0 is the estimation error between z1 and ω, and k1 and k2 are the gains of the NTLESO, which are related to the bandwidth of the observer. Selecting an appropriate bandwidth can enable the state variables observed by the observer to track the corresponding state variables. The difference between the proposed observer and the extended state observer lies in the compensated estimated disturbance quantity z2. In the traditional ESO, z2 is the integral of the observation error e0. While in the time-varying parameter extended state observer, z2 is the sum of the proportion and integral of e0. When the initial state of the ESO differs greatly from the true state of the system, a peak will appear in the initial differential and acceleration signal estimation, and the larger the gain coefficient, the more obvious the peak. To suppress this initial peak, the saturation characteristic of the hyperbolic function and the characteristics of the nonlinear function of "small error, large gain; large error, small gain" are utilized. By replacing the fixed parameters k1 and k2 with time-varying parameters to suppress the initial peak of the ESO, the time-varying parameters are: where k 11 and k 21 are fixed value parameters greater than 0 selected according to the system, r1 and r2 are time variable coefficients, and r1 > 0, r2 > 0.

5. The positionless control method based on a set-time sliding mode observer and a time-varying parameter extended state observer according to claim 1, characterized in that In step S5, the specific process is as follows: Select the following Lyapunov function Taking the derivative with respect to time gives: Based on the stator current error state equation and the expression of the sliding mode surface with set-time convergence characteristics, we can obtain: Substituting the sliding mode surface control law with set time into this formula, we can obtain: Substituting it into the original formula, we can obtain: Based on the above assumption 1, the following inequality can be obtained: Therefore, we have Based on the negative characteristic of the derivative of the Lyapunov function, it can be concluded that the time required for the system to reach the sliding mode surface is T1, and the time required for the system to reach the state origin within the set time after reaching the sliding mode surface is T2. The system will remain stable within the set time, and the convergence time is t a is t a ≤T max =T1+T2, Among them, Therefore, the system will be stable a within the set time t Therefore, continuous sliding mode motion state will be achieved within a finite time According to the equivalent principle of sliding mode control, If the stator current error equation is zero, we can obtain: e = v The control law v is a smooth output without high-frequency switching and can directly estimate the back electromotive force.

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