Double-layer nonsingular terminal sliding-mode active-disturbance-rejection control method of permanent magnet synchronous motor based on super-spiral sliding-mode observer

Abstract

The invention discloses a double-layer nonsingular terminal sliding-mode active-disturbance-rejection control method for a permanent magnet synchronous motor based on a superspiral sliding-mode observer, and the method specifically comprises the following steps: 1, building a voltage equation under an alpha-beta static coordinate system, and then building a mathematical model under a d-q rotating coordinate system through coordinate transformation; 2, designing a fractional order linear super-spiral sliding mode observer; step 3, designing a double-layer nonsingular terminal sliding mode active-disturbance-rejection controller according to the mathematical model under the d-q rotating coordinate system obtained in the step 1; and step 4, making a difference between the mechanical angular velocity given value omega ref and the mechanical angular velocity observed value, and obtaining a control signal of a PMSM current loop through a double-layer nonsingular terminal sliding mode active-disturbance-rejection controller according to the obtained error. According to the method, global rapid convergence of the speed tracking error can be ensured, the anti-interference capability of the system is improved to a certain extent, and buffeting is reduced.

Description

Technical Field

[0001] This invention belongs to the field of electronic power technology and relates to a two-layer non-singular terminal sliding mode active disturbance rejection control method for permanent magnet synchronous motors based on a super-helical sliding mode observer. Background Technology

[0002] Permanent magnet synchronous motors (PMSMs) have become core drive units in industrial automation, electric vehicles, and other fields due to their superior performance. The key to achieving high-performance vector control lies in the precise acquisition and regulation of rotor speed. Sensorless technology estimates speed from electrical signals through algorithms, effectively overcoming the inherent drawbacks of high cost and low reliability of mechanical sensors, and represents an important direction for current technological development. As the final executor of system performance, the dynamic and steady-state quality of the speed loop controller directly impacts the user experience. However, existing sensorless observers face severe challenges when dealing with complex operating conditions: changes in motor parameters with operating conditions lead to model mismatch; weak or even absent back EMF signals in the low-speed region cause a sharp drop in observation accuracy and stability risks. These observation errors directly contaminate the input signal of the speed control loop. Meanwhile, widely used speed loop PI controllers struggle to balance rapid response and overshoot suppression when dealing with dynamic processes such as sudden load changes and wide-range speed regulation, resulting in limited disturbance rejection capabilities. Especially when observer errors or delays exist, fluctuations in speed loop adjustments further interfere with observer convergence, forming negative coupling and significantly limiting the system's performance limits in complex scenarios.

[0003] As applications increasingly demand higher levels of dynamic response, control precision, and operational robustness from drive systems, there is an urgent need to overcome the bottlenecks of existing observers in parameter robustness and low-speed performance, and to design advanced speed control strategies that can be optimized in tandem with observers, so as to fully unleash the potential of sensorless PMSM drives in high-dynamic, wide-range speed regulation, and complex disturbance environments. Summary of the Invention

[0004] The purpose of this invention is to provide a two-layer non-singular terminal sliding mode active disturbance rejection control method for permanent magnet synchronous motors based on a super-helical sliding mode observer. This method not only improves the estimation accuracy of rotor position and speed, but also achieves a synergistic improvement in anti-interference capability and dynamic response.

[0005] The technical solution adopted in this invention is a two-layer non-singular terminal sliding mode active disturbance rejection control method for permanent magnet synchronous motors based on a superspiral sliding mode observer, which specifically includes the following steps: Step 1, establish on α-β The voltage equation in the stationary coordinate system is then used to establish a mathematical model in the dq rotating coordinate system through coordinate transformation. Step 2: Design a fractional-order linear superhelical sliding mode observer; Step 3: The mathematical model in the dq rotating coordinate system obtained in Step 1 is used to design a two-layer non-singular terminal sliding mode active disturbance rejection controller. Step 4, set the mechanical angular velocity value. oh ref Compared with mechanical angular velocity observations The difference is calculated, and the resulting error is used to obtain the control signal for the PMSM current loop through a two-layer non-singular terminal sliding mode active disturbance rejection controller.

[0006] The invention is further characterized by: The specific process of step 1 is as follows: For surface-mounted permanent magnet synchronous motors, based on α-β The voltage equation in the stationary coordinate system is as follows: (1) In the formula, R Stator resistance; L s For stator direct-axis and quadrature-axis inductors; This is the stator inductor voltage; Stator current; To extend the back electromotive force, we have: (2) The voltage equation in equation (1) can be changed to the current state equation as follows: (3) In the formula, , , , oh e Electric angular velocity; Through coordinate transformation α-β The mathematical model of the PMSM in the stationary coordinate system is transformed into the dq coordinate system, and the voltage equation is as follows: (4) In the formula, u d and u q These are the stator voltages in the dq coordinate system, respectively. i d and i q Let be the stator currents in the dq coordinate system; the mechanical motion equations are: (5) In the formula, ; T e Electromagnetic torque; T L This is the load torque;J It is the moment of inertia; B The damping coefficient; oh ω is the rotor mechanical angular velocity.

[0007] Step 2 is as follows: Fractional calculus operators The definition is shown in equation (6), where α、β These are the upper and lower limits; r R is the fractional order; r )for r The real part: (6) The fractional calculus is defined as Caputo as: (7) In the formula, n -1< i ≤ n ; Let Gamma be Euler's function. Simplified to D λ Formal expression; If the fractional order and the integer order satisfy the following conditions: (8) In the formula, 0 < l <1, A ∈ R - . The solution to equation (8) is: (9) In the formula, E λ,β ( t The definition of ) is as follows: (10) In the formula, 0 < l <1 is the transfer function, Γ( y ) is defined as: (11) In the formula, the integer-order transfer function expands to: (12) Using the Matignon conclusion to analyze fractional-order systems, we have (13) The superhelical algorithm is as follows: (14) In the formula, xFor state variables; For the estimated value of the state variable and actual value x The difference between them; l 1. l 2 represents the gain coefficient; It is a symbolic function; r 1. r 2 is the disturbance term, when r 1. r 2. Satisfying equation (15): (15) and l 1. l 2. When equation (16) is satisfied, the system can converge in a finite time. (16) In the formula, a For any constant that satisfies equation (15); In equation (3) Treating it as a disturbance term, and combining it with equation (14), the super-helical sliding mode observer is designed as follows: (17) In the formula, for α , β Estimated value of shaft current, , The disturbance term is: (18) Since the disturbance term must satisfy equation (15), substituting the disturbance term equation (18) into equation (15) yields: (19) Subtracting equation (3) from equation (17) yields the current error equation: (20) In the formula, , ; Define the sliding surface When the observer's state variable reaches the sliding surface Afterward, the observer state will remain on the sliding mode surface. According to the equivalent principle of sliding mode control, the control quantity at this time is regarded as the equivalent control quantity, and the back electromotive force is obtained: (twenty one) The transfer function of the low-pass filter is: (twenty two) In the formula, oh cLet be the cutoff frequency of the low-pass filter; then: (twenty three) In the formula, éα and éβ are the back electromotive forces along the α-axis and β-axis in the α-β coordinate system after being filtered by a low-pass filter, respectively; to obtain rotor position and electrical angular velocity information, the arctangent function needs to be used, i.e.: (twenty four) The mathematical expression for the linear superhelical sliding mode observer is: (25) In the formula, r 1( x , t ) ≤ a | x | 0.5 ; r 2( v , t ) = 0; The back electromotive force estimated by the linear superhelical sliding mode observer is: (26) The mathematical expression for the fractional-order linear superspiral algorithm is as follows: (27) In the formula, ; ; ; The back electromotive force estimated by the fractional-order linear superhelical sliding mode observer is: (28) The rotor position and electrical angular velocity information can be obtained directly using the arctangent function method, that is: (29) In the formula, These are the observed values ​​of the rotor position. The observed value of electric angular velocity. It is a magnetic flux.

[0008] Step 3 specifically involves: According to equation (5), the PMSM speed loop state equation is as follows: (30) Equation (30) can be written in the form of the active disturbance rejection paradigm as follows: (31) In the formula, ; ; It is a control quantity, and ; Equation (31) can be rewritten as: (32) In the formula, , This is due to the time-varying nature of motor parameters, load uncertainty, and interference caused by environmental changes. Define state variables , Then the state equation of the system is: (33) According to equation (33), the traditional LESO can be designed as follows: (34) In the formula, ; , for , Observed values; The tracking error and its derivative are used to replace the traditional ESO. The speed tracking error is defined as: (35) In the formula, Given a mechanical angular velocity, design to replace : (36) In the formula, , ; Therefore, PD-LESO is designed as follows: (37) Based on a single PD-LESO, the observations of the disturbance are imported into a second LESO. The second LESO is used to reconstruct the remaining disturbances that were not estimated by the first PD-LESO, and finally a two-layer non-singular terminal sliding mode active disturbance rejection controller is designed.

[0009] In step 3, the second-level LESO design is as follows: (38) Combining equations (37) and (38), we can obtain the CLESO pair. More accurate observations for: (39) In the formula, They are respectively A more accurate estimate; for The residual observations; , These are the gain coefficients, and all are greater than 0.

[0010] In step 3, the inner sliding surface of the two-layer non-singular terminal sliding mode controller is defined as... l , l The expression is: (40) In the formula, ; , g1 All are parameters to be designed, and satisfy... , , ; Lemma 1: Suppose a first-order nonlinear inequality satisfies the following condition: (41) In the formula, V ( x ) represents a state variable x ∈ R Lyapunov function, h ∈ R + 0< v <1; then for any initial value V ( x ( t 0)), V ( x It can converge to the origin in a finite amount of time, and the convergence time satisfies the formula: (42) Therefore, when l = 0 is: (43) Constructing Lyapunov functions V l for: (44) right V l Differentiating, we get: (45) According to Lemma 1, On the sliding surface l The motion on the surface takes a finite amount of time t x2 Converging to 0, assuming The initial value is The convergence time is...t x2 Satisfying the inequality: (46) According to equation (43), since It can converge in a finite amount of time, therefore guaranteeing It can also converge in a finite amount of time, with the same convergence time. t x2 . In step 3, to ensure that the sliding surface converges in a finite time and to reduce chattering, the outer sliding surface, i.e., the double-layer integral terminal sliding surface s, is defined as: (47) If variable The initial values ​​satisfy the following relation: (48) Substituting equation (47) into equation (47), we can obtain s ( t 0) = 0, when satisfying s ( t When the condition is 0, we have l = - it This describes the sliding mode variable. l and t There is a proportional relationship between them to ensure the sliding mode variables l and t Simultaneous convergence; according to equation (47): (49) According to equation (49), l , t Convergence time t l , t τ for: (50) right Differentiating, we get: (51) make Combining equations (35), (32), and (51), the equivalent control term can be obtained. u eq for: (52) In the formula, , All of these are parameters to be designed.

[0011] In step 3, to accelerate the convergence of state variables to the sliding surface, an improved terminal sliding mode reaching law is adopted: (53) Combining the equivalent control terms (52) and (53), the output term of the designed feedback controller is: (54) In the formula, , , .

[0012] The beneficial effects of this invention are as follows: 1. The fractional-order linear superspiral sliding mode observer used in this invention significantly optimizes the core performance of sensorless control of permanent magnet synchronous motors through structural innovation. First, a linear correction term is introduced based on the traditional superspiral sliding mode observer: on the one hand, the linear term effectively suppresses high-frequency noise interference in the back EMF estimation process, avoiding the use of a low-pass filter, thereby eliminating the phase delay problem caused by the filtering stage, simplifying the system structure while improving the estimation accuracy of rotor position and electrical angular velocity; on the other hand, the linear correction enhances the algorithm's ability to suppress the linear growth of disturbances, especially in dynamic scenarios, enabling more accurate responses to changes in motor parameters or external disturbances, improving system robustness. Next, the fractional-order theory is combined with the linear superspiral sliding mode observer to produce a fractional-order linear superspiral sliding mode observer. Compared to the linear superspiral sliding mode observer, the fractional-order linear superspiral sliding mode observer shows significant improvements in its ability to suppress chattering, accelerate convergence, improve accuracy, and resist noise.

[0013] 2. The sliding mode active disturbance rejection control method adopted in this invention firstly designs a cascaded LESO. The front-stage PD-LESO uses the tracking error and its derivative to replace the observation error in the traditional ESO, improving observation performance and suppressing noise. To address the issues of residual observations and lag in the observation speed of the first-stage LESO, a LESO is cascaded after the original PD-LESO. The cascaded LESO improves observation accuracy and accelerates the disturbance estimation speed of the observer. Secondly, the feedback controller of LADRC is replaced with a two-layer non-singular terminal sliding mode controller. This sliding mode controller has two recursive sliding surfaces. When the outer sliding surface converges to the origin, the inner sliding surface can satisfy the finite-time convergence condition. Simultaneously, when the inner sliding surface converges to the origin, the state variables can also converge within a finite time. When both sliding surfaces converge sequentially, the convergence of the state variables within a finite time is guaranteed. This effectively reduces system chattering while improving the convergence speed, thus enhancing the response speed and anti-interference performance of the permanent magnet synchronous motor speed control system. Furthermore, the improved terminal sliding mode convergence law accelerates the convergence of state variables to the sliding surface and reduces chattering. The organic combination of these two aspects achieves a synergistic improvement in anti-interference capability and dynamic response, providing a new solution for high-performance PMSM drive systems. Attached Figure Description

[0014] Figure 1 This is a block diagram showing the results of the invention's method for dual-layer non-singular terminal sliding mode active disturbance rejection control of permanent magnet synchronous motors based on a super-helical sliding mode observer; Figure 2 This is a block diagram of the cascaded LESO control strategy in the two-layer non-singular terminal sliding mode active disturbance rejection control method for permanent magnet synchronous motors based on super-helical sliding mode observers in this invention; Figure 3 This is a block diagram of the bilayer non-singular terminal sliding mode controller (HNTSMC) in the bilayer non-singular terminal sliding mode active disturbance rejection control method for permanent magnet synchronous motors based on superhelical sliding mode observers of the present invention. Figure 4(a) is a waveform diagram of motor acceleration under PI control; Figure 4(b) is a diagram of the motor acceleration waveform under the control of the dual-layer non-singular terminal sliding mode active disturbance rejection control method of the permanent magnet synchronous motor based on the super-helical sliding mode observer of the present invention. Figure 4(c) shows the motor deceleration waveform under PI control; Figure 4(d) is a waveform diagram of motor deceleration under the control of the double-layer non-singular terminal sliding mode active disturbance rejection control method of permanent magnet synchronous motor based on super-helical sliding mode observer of the present invention. Figure 5(a) is a waveform diagram of motor loading under PI control; Figure 5(b) is a waveform diagram of motor loading under the control of the double-layer non-singular terminal sliding mode active disturbance rejection control method of permanent magnet synchronous motor based on super-helical sliding mode observer of the present invention; Figure 5(c) is a motor reducer diagram under PI control; Figure 5(d) is a motor reduction curve diagram under the control of the dual-layer non-singular terminal sliding mode active disturbance rejection control method of permanent magnet synchronous motor based on the super-helical sliding mode observer of the present invention. Detailed Implementation

[0015] The following detailed description is provided in conjunction with specific implementation methods.

[0016] Example 1 This invention presents a two-layer non-singular terminal sliding mode active disturbance rejection control method for permanent magnet synchronous motors (PMSMs) based on a superspiral sliding mode observer. The fractional-order linear superspiral sliding mode observer used is simple and reliable, accurately observing the electrical angular velocity of the PMSM while virtually eliminating high-frequency signal interference. This improves the system's steady-state performance and effectively suppresses chattering. The employed two-layer non-singular terminal sliding mode active disturbance rejection control not only ensures rapid global convergence of speed tracking errors but also enhances the system's anti-interference capability and reduces chattering to a certain extent.

[0017] Example 2 This invention relates to a two-layer non-singular terminal sliding mode active disturbance rejection control method for permanent magnet synchronous motors based on a superspiral sliding mode observer, specifically implemented according to the following steps: Step 1, for a surface-mounted PMSM, first establish its... α-β The voltage equation in the stationary coordinate system is obtained, and then its mathematical model in the dq rotating coordinate system is established through coordinate transformation; Step 2, based on the equation obtained in Step 1... α-β A mathematical model in a stationary coordinate system was used to design a fractional-order linear superhelical sliding mode observer, and to observe the electrical angular velocity of the PMSM. Step 3: Based on the mathematical model obtained in Step 1 in the dq rotating coordinate system, design a two-layer non-singular terminal sliding mode active disturbance rejection controller; Step 4: Set the mechanical angular velocity to... oh ref Compared with mechanical angular velocity observations The difference is calculated, and the resulting error is used to obtain the control signal for the PMSM current loop via a two-layer non-singular terminal sliding mode active disturbance rejection controller. The PMSM current loop still uses PI control.

[0018] Example 3 The specific process of step 1 is as follows: For surface-mounted permanent magnet synchronous motors, based on α-β The voltage equation in the stationary coordinate system is as follows: (1) In the formula, R Stator resistance;L s For stator direct-axis and quadrature-axis inductors; This is the stator inductor voltage; Stator current; To extend the back electromotive force (induced electromotive force), we have: (2) The voltage equation in equation (1) can be changed to the current state equation as follows: (3) In the formula, , , , oh e It represents the electric angular velocity.

[0019] Through coordinate transformation α-β The mathematical model of the PMSM in the stationary coordinate system is transformed into the dq coordinate system, and its voltage equation is as follows: (4) In the formula, u d and u q These are the stator voltages in the dq coordinate system, respectively. i d and i q These are the stator currents in the dq coordinate system, respectively. L s For stator inductance; R Stator resistance; p n This represents the number of pole pairs of the motor. ψ f For permanent magnet flux linkage; oh Let be the mechanical angular velocity of the rotor. The mechanical equation of motion is: (5) In the formula, , T e Electromagnetic torque; T L This is the load torque; J It is the moment of inertia; B The damping coefficient; oh ω is the rotor mechanical angular velocity.

[0020] Example 4 Step 2 specifically involves the following fractional calculus theory: fractional calculus operators. The definition is shown in equation (6), where α、β These are the upper and lower limits; r R is the fractional order; r )for r The real part. (6) Since its inception in the 1980s, the theory of fractional calculus has evolved to the present day. The definitions of fractional calculus can be broadly categorized into three types, among which Caputo's definition is: (7) In the formula, n -1< i ≤ n ; Let Gamma be Euler's function. Simplified to D λ Formal expression. Next, let's analyze the advantages of fractional-order controllers. If fractional and integer orders satisfy the following conditions: (8) In the formula, 0 < l <1, A ∈ R - The solution to equation (8) is: (9) In the formula, E λ,β ( t The definition of ) is as follows: (10) In the formula, 0 < l <1 is the transfer function, Γ( y ) is defined as: (11) In the formula, the integer-order transfer function expands to: (12) Using the Matignon conclusion to analyze fractional-order systems, we have (13) From (12) and (13), we can see that the system state is... e ( t The degree of decay varies greatly, with fractional orders and above. t - l The rate of decay is integer order. eAt The speed decays rapidly. This indicates that fractional-order signals have smaller fluctuations compared to integer-order sliding surfaces. Furthermore, fractional-order chattering layers are smaller. Therefore, in principle, fractional-order systems can more effectively suppress chattering.

[0021] By adding a linear term to the traditional superhelical sliding mode observer, a linear superhelical sliding mode observer is formed. The optimized observer contains almost no high-frequency signal when estimating the back electromotive force, so a low-pass filter is no longer needed, which improves the accuracy of rotor position and electric angular velocity estimation.

[0022] Super-twisting sliding mode control (STSMC) is an advanced nonlinear control strategy that effectively improves system performance and ensures the stability and accuracy of motor operation. The super-twisting algorithm is given below: (14) In the formula, x For state variables; For the estimated value of the state variable and actual value x The difference between them; l 1. l 2 represents the gain coefficient; It is a symbolic function; r 1. r 2 represents the disturbance term. When r 1. r 2. Satisfying equation (15): (15) and l 1. l 2. When equation (16) is satisfied, the system can converge in a finite time.

[0023] (16) In the formula, a For any constant that satisfies equation (15).

[0024] In equation (3) Treating it as a disturbance term, and combining it with equation (14), the super-helical sliding mode observer can be designed as follows: (17) In the formula, for α , β Estimated value of shaft current, , Its disturbance term is: (18) Since the disturbance term must satisfy equation (15), substituting the disturbance term equation (18) into equation (15) yields the following: (19) From equation (19), it can be seen that when the value of a is large, equation (17) holds true. Subtracting equation (3) from equation (17) yields the current error equation: (20) In the formula, , Define the sliding surface. When the observer's state variable reaches the sliding surface Afterward, the observer state will remain on the sliding mode surface. According to the equivalent principle of sliding mode control, the control quantity at this time can be regarded as the equivalent control quantity, and the back electromotive force can be obtained: (twenty one) The back electromotive force estimate obtained by the above method often contains some high-frequency components. Therefore, it is usually necessary to set a low-pass filter. The transfer function of the low-pass filter is: (twenty two) In the formula, ωc is the cutoff frequency of the low-pass filter. Therefore: (twenty three) In the formula, éα and éβ are the back electromotive forces along the α-axis and β-axis in the α-β coordinate system after being filtered by a low-pass filter, respectively. To obtain rotor position and electrical angular velocity information, the arctangent function is required, i.e.: (twenty four) In the formula, These are the observed values ​​of the rotor position. The observed value of electric angular velocity. ψ f It is a magnetic flux.

[0025] The linear superhelical sliding mode observer improves the traditional superhelical sliding mode observer architecture by introducing a linear correction term. This not only enhances the system's robustness against strong disturbances in the equilibrium neighborhood but also effectively suppresses the tendency for disturbances to grow linearly with the system state. The addition of this linear correction term significantly optimizes the back EMF estimation process: firstly, by constraining the generation amplitude of the high-frequency oscillation signal, it greatly reduces the dependence on subsequent filtering stages; secondly, it eliminates the phase lag effect introduced by traditional filters, resulting in a significant improvement in the observer's dynamic characteristics. This dual improvement mechanism ultimately leads to a step-wise improvement in the overall anti-interference performance of the system, especially demonstrating stronger robustness in complex disturbance environments. The mathematical expression of the linear superhelical sliding mode observer is: (25) In the formula, ρ1(x,t) ≤ a|x|0.5; ρ2(v,t) = 0. Similarly, referring to the design method of the superspiral sliding mode observer, the back electromotive force estimated by the linear superspiral sliding mode observer is: (26) By integrating fractional-order theory with the linear superspiral observer, we obtain the fractional-order linear superspiral sliding mode observer. This observer, by combining the flexibility of fractional-order calculus with the robustness of the superspiral algorithm, significantly improves upon the linear superspiral sliding mode observer in terms of chatter suppression, accelerated convergence, enhanced accuracy, and noise resistance. It is particularly suitable for complex systems with high precision, strong nonlinearity, or fractional-order characteristics. The mathematical expression of the fractional-order linear superspiral algorithm is as follows: (27) In the formula, ; ; Similarly, the back electromotive force estimated by the fractional-order linear superspiral sliding mode observer is: (28) Since the back electromotive force obtained by the fractional-order linear superspiral sliding mode observer contains almost no high-frequency signals, the use of a low-pass filter can be omitted, and the rotor position and electric angular velocity information can be obtained directly through the arctangent function method, that is: (29) Example 5 The specific process of step 3 is as follows: According to equation (5), the PMSM speed loop state equation is: (30) Equation (30) can be written in the form of the active disturbance rejection paradigm as follows: (31) In the formula, ; ; It is a control quantity, and .

[0026] Considering the actual engineering calculations It is unknown and generally cannot be obtained accurately; therefore, it is defined... For control quantity gain The estimated value. In addition, under complex and variable operating conditions, permanent magnet synchronous motors (PMSMs) are difficult to maintain an ideal working state for a long time. Their internal mechanical and electromagnetic characteristic parameters are easily affected by nonlinear factors such as magnetic circuit saturation effect and temperature rise fluctuation, exhibiting time-varying fluctuations, which leads to a significant deviation between the actual electromagnetic torque equation, i.e., the motor parameters, and the ideal model. Therefore, equation (31) is rewritten as: (32) In the formula, , This addresses the time-varying nature of motor parameters, load uncertainties, and disturbances caused by environmental changes. State variables are defined. , Then the state equation of the system is: (33) According to equation (33), the traditional LESO can be designed as follows: (34) In the formula, ; , for , The observed values. To improve observation performance and suppress noise, this invention uses the sum of the tracking error and its derivative to replace the traditional ESO. The speed tracking error is defined as: (35) In the formula, This is a given value for the mechanical angular velocity. (Design) to replace : (36) In the formula, , Therefore, PD-LESO can be designed as follows: (37) The accuracy of the total disturbance estimation directly affects the controller's disturbance suppression performance. To improve the reconstruction accuracy and estimation speed of the disturbance, this paper uses two LESOs to build a cascaded linear extended state observer (CLESO) to enhance the disturbance estimation and reconstruction capability. The design idea is to import the disturbance observations of a traditional single (proportional-derivative-type linear extended state observer, PD-LESO) into a second LESO, and use the second LESO to reconstruct the remaining disturbances that were not estimated by the first PD-LESO. The design of the second-level LESO is as follows: (38) In the formula, They are respectively A more accurate estimate; for The residual observations; , These are the gain coefficients, and all are greater than 0. Combining equations (37) and (38), we can obtain the CLESO gain coefficient. More accurate observations for: (39) The block diagram of the cascaded LESO control strategy is as follows: Figure 2 As shown, to improve the dynamic response characteristics and anti-interference capability of a permanent magnet synchronous motor (PMSM) speed control system, this invention employs a speed loop controller based on dual-layer non-singular terminal sliding mode control (HNTSMC). An innovative dual-layer terminal sliding surface with a dual-layer structure is designed, which not only inherits the strong robustness of traditional sliding mode control but also achieves layer-by-layer convergence of dynamic errors through a dual-layer mechanism, effectively suppressing the high-frequency chattering phenomenon inherent in traditional sliding mode control, thereby improving the dynamic response quality of the system. The inner sliding surface of the dual-layer non-singular terminal sliding mode controller is defined as... l , l The expression is: (40) In the formula, ; , g1 All are parameters to be designed, and satisfy... , , .

[0027] Lemma 1: Suppose a first-order nonlinear inequality satisfies the following condition: (41) In the formula, V ( x ) represents a state variable x ∈ R Lyapunov function, h ∈ R + 0< v <1. Then for any initial value V ( x ( t 0)), V ( x It can converge to the origin in a finite amount of time, and the convergence time satisfies the formula: (42) Therefore, when l = 0 is: (43) Constructing Lyapunov functions V l for: (44) right V l Differentiation yields: (45) According to Lemma 1, On the sliding surface l The motion on the surface takes a finite amount of time t x2 Converging to 0, assuming The initial value is The convergence time is... t x2 Satisfying the inequality: (46) According to equation (43), since It can converge in a finite amount of time, therefore guaranteeing It can also converge in a finite amount of time, with the same convergence time. t x2 . To ensure that the sliding surface converges in a finite time and to reduce chattering, an outer sliding surface is defined, namely the double-layer integral terminal sliding surface. s for: (47) In the formula, , All are parameters to be designed. If variables... The initial values ​​satisfy the following relation: (48) Substituting equation (47) into equation (47), we can obtain s ( t 0) = 0. When satisfied s ( t When the condition is 0, we have l = - it This describes the sliding mode variable. l and t There is a proportional relationship between them, which can guarantee the sliding mode variable. l and t They converge simultaneously. According to equation (47), we can obtain: (49) According to equation (49), l , t Convergence time t l , tτ for: (50) right Differentiation yields: (51) make Combining equations (35), (32), and (51), the equivalent control term can be obtained. u eq for: (52) To accelerate the convergence of state variables to the sliding surface, an improved terminal sliding mode convergence law is adopted: (53) In the formula, , , Therefore, combining the equivalent control terms (52) and (53), the output term of the designed feedback controller is: (54) A schematic diagram of the dual-layer non-singular terminal sliding mode controller (HNTSMC) proposed in this invention is shown below. Figure 3 As shown.

[0028] Example 6 To better reflect real-world production scenarios and further verify the feasibility and effectiveness of the proposed control strategy, a comparative analysis of the PI control strategy and the strategy of this invention will be conducted on a Hardware-in-the-Loop (HIL) experimental platform to comprehensively evaluate the advantages and disadvantages of each strategy. The sampling frequency of the HIL experimental platform is set to 20kHz to ensure data accuracy. First, a simulation circuit is built based on Simulink to simulate the actual system control logic; then, the simulation circuit is loaded into the MT6020 and RCP1050 via a host computer to simulate the actual hardware behavior and response; finally, the platform's real-time signal is connected to an oscilloscope via an adapter board to monitor and record key experimental data in real time for subsequent analysis and verification. Motor parameters are shown in Table 1.

[0029] Table 1 PMSM Parameters

[0030] Operating Condition 1: The motor is started under no-load at a given speed of 1200 r / min. After the motor stabilizes, the speed is increased to 1500 r / min, and after stabilization, the speed is reduced to 1200 r / min. The experimental results are shown in Figures 4(a) to 4(d). Figures 4(a) to 4(d) show the speed curves of the motor during speed changes. By observing Figures 4(a) to 4(d), it can be found that the proposed HNTSMADRC control strategy achieves good control performance while ensuring minimal chattering. Specifically, when the motor speed suddenly increases from 1200 r / min to 1500 r / min, the speed overshoot of the traditional PI control in Figure 4(a) reaches 1608 r / min with a settling time of 59.4 ms, while the speed overshoot of the RNFTSMADRC control in Figure 4(b) is not observed with a settling time of 11.4 ms. When the speed suddenly drops from 1500 r / min to 1200 r / min, the traditional PI controller in Figure 4(c) has a settling time of 28.3 ms and a speed ripple of 214 r / min; while the controller proposed in this invention in Figure 4(d) has a settling time of 13.9 ms and a speed ripple of 104 r / min. The comparison shows that the HNTSMADRC controller has better dynamic performance.

[0031] Operating Condition 2: The motor is started under no-load at a given speed of 1200 r / min. After the motor stabilizes, the acceleration and deceleration patterns are shown in Figures 5(a) to 5(d). When a sudden load is applied, the speed drop and recovery time after the load torque is increased are selected as performance indicators. In Figure 5(a), the performance indicators of the traditional PI controller are 70 r / min and 37.6 ms; in contrast, the speed drop of the HNTSMADRC control in Figure 5(b) is only 40 r / min, and the settling time is only 6.6 ms. When the load is suddenly reduced, it can be seen from Figure 5(c) that the PI controller settling time is 33.4 ms, and the speed overshoots to 1284 r / min; Figure 5(d) shows the controller in this paper, with a settling time of 6.1 ms and a speed change of 16 r / min. In summary, the HNTSMADRC control method based on FORSTSM exhibits excellent robustness and tracking accuracy, and can effectively improve the performance of the speed control system.

[0032] This invention presents a sliding mode active disturbance rejection control (ADRC) method for permanent magnet synchronous motors (PMSMs) based on a fractional-order linear superspiral sliding mode observer. Compared to traditional superspiral sliding mode observers, the fractional-order linear superspiral sliding mode observer used in this invention significantly improves the estimation accuracy, reduces velocity estimation errors, and enhances the system's anti-interference capability. This invention designs a sliding mode ADRC controller for the PMSM's speed loop. The cascaded LESO improves observation accuracy and accelerates the observer's disturbance estimation speed. The double-layer terminal sliding surface with a double-layer structure not only inherits the strong robustness of traditional sliding mode control but also achieves layer-by-layer convergence of dynamic errors through a double-layer mechanism, effectively suppressing the inherent high-frequency chattering phenomenon of traditional sliding mode control, thereby improving the system's dynamic response quality. The improved terminal sliding mode reaching law accelerates the convergence of state variables to the sliding surface and reduces chattering. The fractional-order linear superspiral sliding mode observer used is simple and reliable, accurately observing the PMSM's electrical angular velocity, almost avoiding high-frequency signal interference, improving the system's steady-state performance while effectively suppressing chattering. The dual-layer non-singular terminal sliding mode active disturbance rejection control not only ensures rapid global convergence of speed tracking error, but also improves the system's anti-interference capability and reduces chattering to a certain extent.

[0033] This invention provides a two-layer hierarchical non-singular terminal sliding mode active disturbance rejection control (HNTSMADRC) method for permanent magnet synchronous motors (PMSMs) based on a fractional-order linear super twisting sliding mode observer (FOLSTSMO). First, a method for surface-mounted permanent magnet synchronous motors is established... α-β Mathematical models in the stationary coordinate system and the dq rotating coordinate system. Secondly, based on the permanent magnet synchronous motor in... α-β A fractional-order linear superhelical sliding mode observer is designed based on a mathematical model in a stationary coordinate system. Then, a two-layer non-singular terminal sliding mode active disturbance rejection controller is designed based on the mathematical model of the permanent magnet synchronous motor in the dq rotating coordinate system. Finally, simulation and experimental verification show that the fractional-order linear superhelical sliding mode observer exhibits significant advantages in improving speed estimation accuracy and suppressing chattering. The two-layer non-singular terminal sliding mode active disturbance rejection controller can improve the system's response speed, reduce steady-state error, and effectively enhance the system's anti-interference capability.

Claims

1. A two-layer non-singular terminal sliding mode active disturbance rejection control method for permanent magnet synchronous motors based on a superspiral sliding mode observer, characterized in that: Specifically, the steps include the following: Step 1, establish on α-β The voltage equation in the stationary coordinate system is then used to establish a mathematical model in the dq rotating coordinate system through coordinate transformation. Step 2: Design a fractional-order linear superhelical sliding mode observer; Step 3: The mathematical model in the dq rotating coordinate system obtained in Step 1 is used to design a two-layer non-singular terminal sliding mode active disturbance rejection controller. Step 4, set the mechanical angular velocity value. ω ref Compared with mechanical angular velocity observations The difference is calculated, and the resulting error is used to obtain the control signal for the PMSM current loop through a two-layer non-singular terminal sliding mode active disturbance rejection controller.

2. The method for dual-layer non-singular terminal sliding mode active disturbance rejection control of a permanent magnet synchronous motor based on a superspiral sliding mode observer according to claim 1, characterized in that: The specific process of step 1 is as follows: For surface-mounted permanent magnet synchronous motors, based on α-β The voltage equation in the stationary coordinate system is as follows: (1) In the formula, R Stator resistance; L s For stator direct-axis and quadrature-axis inductors; This is the stator inductor voltage; Stator current; To extend the back electromotive force, we have: (2) The voltage equation in equation (1) can be changed to the current state equation as follows: (3) In the formula, , , , ω e It is the electric angular velocity; through coordinate transformation, it becomes... α-β The mathematical model of the PMSM in the stationary coordinate system is transformed into the dq coordinate system, and the voltage equation is as follows: (4) In the formula, u d and u q These are the stator voltages in the dq coordinate system, respectively. i d and i q These are the stator currents in the dq coordinate system, respectively. L s For stator inductance; R Stator resistance; p n This represents the number of pole pairs of the motor. ψ f For permanent magnet flux linkage; ω Let be the mechanical angular velocity of the rotor; the mechanical motion equation is: (5) In the formula, ; T e Electromagnetic torque; T L This is the load torque; J It is the moment of inertia; B The damping coefficient; ω ω is the rotor mechanical angular velocity.

3. The method for dual-layer non-singular terminal sliding mode active disturbance rejection control of a permanent magnet synchronous motor based on a superspiral sliding mode observer according to claim 2, characterized in that: Step 2 specifically involves fractional-order calculus operators. The definition is shown in equation (6), where α and β are the upper and lower limits; ρ is the fractional order; and R(ρ) is the real part of ρ. (6) The fractional calculus is defined as Caputo as: (7) In the formula, n -1< θ ≤ n ; Let Gamma be Euler's function. Simplified to D λ Formal expression; if fractional and integer orders satisfy the following conditions: (8) In the formula, 0 < λ <1, A ∈ R - ; The solution to equation (8) is: (9) In the formula, E λ,β ( t The definition of ) is as follows: (10) In the formula, 0 < λ <1 is the transfer function, Γ( y ) is defined as: (11) In the formula, the integer-order transfer function expands to: (12) Using the Matignon conclusion to analyze fractional-order systems, we have (13) The superhelical algorithm is as follows: (14) In the formula, x For state variables; For the estimated value of the state variable and actual value x The difference between them; λ 1. λ 2 represents the gain coefficient; It is a symbolic function; ρ 1. ρ 2 is the disturbance term, when ρ 1. ρ 2. Satisfying equation (15): (15) and λ 1. λ 2. When equation (16) is satisfied, the system can converge in a finite time. (16) In the formula, a For any constant that satisfies equation (15); In equation (3) Treating it as a disturbance term, and combining it with equation (14), the super-helical sliding mode observer is designed as follows: (17) In the formula, for α , β Estimated value of shaft current, , The disturbance term is: (18) Since the disturbance term must satisfy equation (15), substituting the disturbance term equation (18) into equation (15) yields: (19) Subtracting equation (3) from equation (17) yields the current error equation: (20) In the formula, , ; Define the sliding surface When the observer's state variable reaches the sliding surface Afterward, the observer state will remain on the sliding mode surface. According to the equivalent principle of sliding mode control, the control quantity at this time is regarded as the equivalent control quantity, and the back electromotive force is obtained: (21) The transfer function of the low-pass filter is: (22) In the formula, ω c This is the cutoff frequency of the low-pass filter; So: (23) In the formula, éα and éβ are the back electromotive forces along the α-axis and β-axis in the α-β coordinate system after being filtered by a low-pass filter, respectively; to obtain rotor position and electrical angular velocity information, the arctangent function needs to be used, i.e.: (24) The mathematical expression for the linear superhelical sliding mode observer is: (25) In the formula, ρ1(x,t) ≤ a|x|0.5; ρ2(v,t) = 0; the back electromotive force estimated by the linear superhelical sliding mode observer is: (26) The mathematical expression for the fractional-order linear superspiral algorithm is as follows: (27) In the formula, ; ; The back electromotive force estimated by the fractional-order linear superspiral sliding mode observer is: (28) The rotor position and electrical angular velocity information can be obtained directly using the arctangent function method, that is: (29) In the formula, These are the observed values ​​of the rotor position. The observed value of electric angular velocity. It is a magnetic flux.

4. The method for dual-layer non-singular terminal sliding mode active disturbance rejection control of a permanent magnet synchronous motor based on a superspiral sliding mode observer according to claim 3, characterized in that: Step 3 specifically involves: According to equation (5), the PMSM speed loop state equation is as follows: (30) Equation (30) can be written in the form of the active disturbance rejection paradigm as follows: (31) In the formula, ; ; It is a control quantity, and Rewrite equation (31) as follows: (32) In the formula, , This is due to the time-varying nature of motor parameters, load uncertainty, and interference caused by environmental changes. Define state variables , Then the state equation of the system is: (33) According to equation (33), the traditional LESO can be designed as follows: (34) In the formula, ; , for , The observed values; the tracking error and its differential sum are used to replace the traditional ESO. The speed tracking error is defined as: (35) In the formula, Given a mechanical angular velocity, design to replace : (36) In the formula, , Therefore, PD-LESO is designed as follows: （37） Based on a single PD-LESO, the observations of the disturbance are imported into a second LESO. The second LESO is used to reconstruct the remaining disturbances that were not estimated by the first PD-LESO, and finally a two-layer non-singular terminal sliding mode active disturbance rejection controller is designed.

5. The method for dual-layer non-singular terminal sliding mode active disturbance rejection control of a permanent magnet synchronous motor based on a superspiral sliding mode observer according to claim 4, characterized in that: In step 3, the second-level LESO design is as follows: (38) Combining equations (37) and (38), we obtain CLESO's pair. More accurate observations for: (39) In the formula, They are respectively A more accurate estimate; for The residual observations; , These are the gain coefficients, and all are greater than 0.

6. The method for dual-layer non-singular terminal sliding mode active disturbance rejection control of a permanent magnet synchronous motor based on a superspiral sliding mode observer according to claim 5, characterized in that: In step 3, the inner sliding surface of the dual-layer non-singular terminal sliding mode controller is defined as... l , l The expression is: (40) In the formula, ; , g1 All are parameters to be designed, and satisfy... , , ; Lemma 1: Suppose a first-order nonlinear inequality satisfies the following condition: (41) In the formula, V ( x ) represents a state variable x ∈ R Lyapunov function, h ∈ R + 0 < v < 1; then for any initial value V ( x ( t 0)), V ( x It can converge to the origin in a finite amount of time, and the convergence time satisfies the formula: (42) Therefore, when l = 0 is: (43) Constructing Lyapunov functions V l for: (44) right V l Differentiating, we get: (45) According to Lemma 1, On the sliding surface l The motion on the surface takes a finite amount of time t x2 Converging to 0, assuming The initial value is The convergence time is... t x2 Satisfying the inequality: (46) According to equation (43), since It can converge within a finite time.

7. The method for dual-layer non-singular terminal sliding mode active disturbance rejection control of a permanent magnet synchronous motor based on a superspiral sliding mode observer according to claim 6, characterized in that: In step 3, to ensure that the sliding surface converges in a finite time and to reduce chattering, the outer sliding surface, i.e., the double-layer integral terminal sliding surface s, is defined as: (47) If variable The initial values ​​satisfy the following relation: (48) Substituting equation (47) into equation (47), we can obtain s ( t 0) = 0, when satisfying s ( t When the condition is 0, we have l = - ητ This describes the sliding mode variable. l and τ There is a proportional relationship between them to ensure the sliding mode variables l and τ Simultaneous convergence; according to equation (47): (49) According to equation (49), l , τ Convergence time t l , t τ for: (50) right Differentiating, we get: (51) make Combining equations (35), (32), and (51), the equivalent control term can be obtained. u eq for: (52) In the formula, , All of these are parameters to be designed.

8. The method for dual-layer non-singular terminal sliding mode active disturbance rejection control of a permanent magnet synchronous motor based on a superspiral sliding mode observer according to claim 7, characterized in that: In step 3, to accelerate the convergence of state variables to the sliding surface, an improved terminal sliding mode reaching law is adopted: (53) Combining the equivalent control terms (52) and (53), the output term of the designed feedback controller is: (54) In the formula, , , .

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