Method for designing observer based on fast adaptive reaching law of exponential function

Abstract

The invention relates to a method for designing an observer based on a fast adaptive reaching law of an exponential function, belongs to the field of high-speed motor control, and provides an adaptive fast reaching law, AQRL is based on second-order sliding mode characteristics and a sign function, and the sign function is based on an exponential term and a system state. The robustness to disturbance and parameter uncertainty is obtained by using a sliding mode disturbance observer. In addition, a second-order fast terminal sliding mode observer is adopted to respond to interference more accurately, and the switching gain of the SMDO is prevented from being too large. The provided AQRL is verified through a Lyapunov second method, and it is mathematically proved that the arrival time is shorter than the traditional arrival time.

Description

Technical Field

[0001] The invention relates to the field of high-speed motor control, and specifically refers to a method for designing an observer based on a fast adaptive reaching law of an exponential function. Background Art

[0002] Currently, permanent magnet synchronous motor (PMSM) has shown unique advantages in electric vehicles, aerospace engineering and robotics applications. Compared with other motor systems (such as brushless motors or DC motor systems), PMSM has high air gap flux density and high efficiency. However, the nonlinear and time-varying parameters of permanent magnet synchronous motors are related to model uncertainties and internal and external disturbances. Since PI control cannot eliminate external and internal errors, the PI control method is limited in terms of high performance.

[0003] In order to overcome the limited performance of PI control systems, many researchers have studied nonlinear control systems for PMSM speed regulation, such as predictive control, neural network control, fuzzy control, stepping control, and sliding mode control (SMC). Among various nonlinear control methods, SMC is widely used due to its robustness to external disturbances and insensitivity to parameter uncertainty. However, its robustness is related to the switching gain. In order to ensure the robustness of the SMC system, a larger switching gain is often required. However, a larger switching gain will cause time delay and high-frequency jitter in the switching control law due to the existence of the sign function. Summary of the invention

[0004] The invention aims to solve the above technical problems and provides a method for designing an observer based on a fast adaptive reaching law of an exponential function.

[0005] In order to solve the above technical problems, the technical solution provided by the invention is:

[0006] A method for designing an observer based on a fast adaptive reaching law of an exponential function comprises the following steps:

[0007] S1. Establishment of adaptive reaching law:

[0008] A stable convergence law AQRL is proposed to overcome the fragility of SMC, as shown in formula (3):

[0009]

[0010] S2. Use adaptive reaching law to establish PMSM sliding mode controller:

[0011] Considering disturbances and parameter uncertainties, the equation of motor angular acceleration is obtained by combining the electromagnetic torque and the motion equation of the PMSM system (7), and the integral sliding surface is obtained based on the tracking error, and then the controller input based on AQRL is obtained:

[0012]

[0013]

[0014] S3. Establishment of the second-order fast terminal sliding mode observer of permanent magnet synchronous motor:

[0015] With the motor angular velocity and disturbance as state variables, the state equation formula (10) is established, and the SFTSMO is established based on the terminal sliding mode to obtain the sliding mode surface of the SFTSMO. The specific expressions of the system control input and the system disturbance derivative under SFTSMO are obtained as shown in formula (23) and formula (24):

[0016]

[0017]

[0018]

[0019] u t2 =h2sgn(s ft ) (twenty four);

[0020] Under the terminal sliding mode control law, by adjusting u t2 The estimated disturbance is obtained by integrating the values Because u t2 =h2sgn(s ft ), the proposed SFTSMO ensures smooth perturbation observations, obtained from Convert the controller input to:

[0021]

[0022] Preferably, the integral sliding surface is based on the tracking error, as shown in formula (8):

[0023]

[0024] Where c is a positive constant.

[0025] Preferably, in formula (3), x is the system state, the value of x(∞) is 0, the parameters a, b, k1, k2 represent positive adjustment parameters, and the boundary conditions are 0<a<1, 0<b<ln2, k1>0, k2>0 respectively.

[0026] Preferably, the expression of the electromagnetic torque and is simply expressed as shown in Formula 5:

[0027]

[0028] The motion equation of the PMSM system is shown in formula (6):

[0029]

[0030] Among them, i d 、i q They represent the d-axis and q-axis stator currents respectively, R is the resistance, is the flux of the permanent magnet, the measured speed of the PMSM is represented by ω, and the electromagnetic torque generated depends on the flux, the number of pole pairs p n and inductance, θ represents the rotor angle of the PMSM, J, B and T L represent the moment of inertia, viscous friction coefficient and load torque respectively.

[0031] Preferably, λ, γ, and ε in the formula (7) are Δλ, Δγ, Δε represent the uncertainty of the parameters. To represent the uncertainty of the parameters and the load torque, the disturbance is d = Δλi q -Δγω-(ε+Δε)T L .

[0032] Preferably, the SFTSMO established based on the terminal sliding mode is as shown in Formula 11:

[0033]

[0034] Where l represents the derivative of the lumped disturbance, and denote the estimated angular velocity and estimated disturbance of PMSM, u t1 and u t2 Represent the designed control law and lumped disturbance derivative respectively. The estimated angular velocity error can be obtained from formula (10) and formula (11): and the estimated disturbance error Sliding surface of SFTSMO ft Designed as follows:

[0035]

[0036] In the formula for Rational numbers under the condition that p and q are positive odd numbers.

[0037] Preferably, in formula (23) and formula (24), v represents k3, h1 and h2 represent designed positive constants.

[0038] After adopting the above method, the invention has the following advantages:

[0039] The present invention proposes an adaptive fast reaching law, AQRL, based on a second-order sliding mode characteristic and a sign function, the sign function is based on an exponential term and the system state. By using a sliding mode disturbance observer, robustness to disturbances and parameter uncertainties is obtained. The present invention also uses a second-order fast terminal sliding mode observer to respond to disturbances more accurately and avoid excessive switching gain of SMDO. The proposed AQRL is verified by Lyapunov's second method, and it is mathematically proved that the arrival time is faster than the traditional arrival time.

[0040] The above summary is for illustrative purposes only and is not intended to be limiting in any way. In addition to the illustrative aspects, embodiments, and features described above, further aspects, embodiments, and features of the invention will be readily apparent by reference to the drawings and the following detailed description. BRIEF DESCRIPTION OF THE DRAWINGS

[0041] In order to more clearly illustrate the embodiments of the present application or the technical solutions in the prior art, the drawings required for use in the embodiments or technical descriptions will be briefly introduced below. Obviously, the drawings described below are only some embodiments of the present application. For ordinary technicians in this field, other drawings can be obtained based on these drawings without paying creative work.

[0042] Figure 1 It is a system block diagram of the invention. DETAILED DESCRIPTION

[0043] The embodiments of the present application are described in detail below, and examples of the embodiments are shown in the accompanying drawings, wherein the same or similar reference numerals throughout represent the same or similar elements or elements having the same or similar functions. The embodiments described below with reference to the accompanying drawings are exemplary and are only used to explain the present application, and cannot be understood as limiting the present application.

[0044] In the description of this application, it should be noted that, unless otherwise clearly specified and limited, the terms "installed", "connected", and "connected" should be understood in a broad sense, for example, it can be a fixed connection, a detachable connection, or an integral connection; it can be a mechanical connection or an electrical connection; it can be a direct connection or an indirect connection through an intermediate medium, it can be the internal connection of two elements or the interaction relationship between two elements. For ordinary technicians in this field, the specific meanings of the above terms in this application can be understood according to specific circumstances.

[0045] The invention is further described in detail below in conjunction with the full text.

[0046] Combined with Figure 1 , a method for designing an observer based on a fast adaptive reaching law of an exponential function, comprising the following steps:

[0047] S1. Establishment of adaptive reaching law:

[0048] A stable convergence law AQRL is proposed to overcome the fragility of SMC, as shown in formula (3):

[0049]

[0050] S2. Use adaptive reaching law to establish PMSM sliding mode controller:

[0051] Considering disturbances and parameter uncertainties, the equation of motor angular acceleration is obtained by combining the electromagnetic torque and the motion equation of the PMSM system (7), and the integral sliding surface is obtained based on the tracking error, and then the controller input based on AQRL is obtained:

[0052]

[0053]

[0054] S3. Establishment of the second-order fast terminal sliding mode observer of permanent magnet synchronous motor:

[0055] With the motor angular velocity and disturbance as state variables, the state equation formula (10) is established, and the SFTSMO is established based on the terminal sliding mode to obtain the sliding mode surface of the SFTSMO. The specific expressions of the system control input and the system disturbance derivative under SFTSMO are obtained as shown in formula (23) and formula (24):

[0056]

[0057]

[0058]

[0059] u t2 =h2sgn(s ft ) (twenty four);

[0060] Under the terminal sliding mode control law, by adjusting u t2 The estimated disturbance is obtained by integrating the values Because u t2 =h2sgn(s ft ), the proposed SFTSMO ensures smooth perturbation observations, obtained from Convert the controller input to:

[0061]

[0062] The integral sliding surface is based on the tracking error, as shown in formula (8):

[0063]

[0064] Where c is a positive constant.

[0065] In formula (3), x is the system state, the value of x(∞) is 0, the parameters a, b, k1, and k2 represent positive adjustment parameters, and the boundary conditions are 0<a<1, 0<b<ln2, k1>0, and k2>0, respectively.

[0066] The expression of the electromagnetic torque and is simply expressed as shown in Formula 5:

[0067]

[0068] The motion equation of the PMSM system is shown in formula (6):

[0069]

[0070] Among them, i d 、i q They represent the d-axis and q-axis stator currents respectively, R is the resistance, is the flux of the permanent magnet, the measured speed of the PMSM is represented by ω, and the electromagnetic torque generated depends on the flux, the number of pole pairs p n and inductance, θ represents the rotor angle of the PMSM, J, B and T L represent the moment of inertia, viscous friction coefficient and load torque respectively.

[0071] In the formula (7), λ, γ, and ε are Δλ, Δγ, Δε represent the uncertainty of the parameters. To represent the uncertainty of the parameters and the load torque, the disturbance is d = Δλi q -Δγω-(ε+Δε)T L .

[0072] The SFTSMO established based on the terminal sliding mode is shown in Formula 11:

[0073]

[0074] Where l represents the derivative of the lumped disturbance, and denote the estimated angular velocity and estimated disturbance of PMSM, u t1 and u t2Represent the designed control law and lumped disturbance derivative respectively. The estimated angular velocity error can be obtained from formula (10) and formula (11): and the estimated disturbance error Sliding surface of SFTSMO ft Designed as follows:

[0075]

[0076] In the formula for Rational numbers under the condition that p and q are positive odd numbers.

[0077] In formula (23) and formula (24), v represents k3, h1 and h2 represent designed positive constants.

[0078] When the present invention is specifically implemented, Figure 1 As shown, given the reference angular velocity ω of the PMSM (permanent magnet synchronous motor) r , and the speed error is obtained by subtracting the angular velocity fed back by the motor; and denote the estimated angular velocity and estimated disturbance of the PMSM respectively, and the tracking error (e) is defined as ω r The speed controller obtains the system control input according to the speed error and the system estimated disturbance obtained by SFTSMO (second-order fast terminal sliding mode observer) and i d =0 The system u is obtained through the current controller d ,u q ;u d ,u q Get u through Park changes α ,u β ;u α ,u β After SVPWM and inverter, the three-phase voltage u of the motor is obtained. a ,u b ,u c ; The three-phase voltage passes through the motor and outputs the three-phase current i a ,i b ,i c and rotor angle θ;i a ,i b ,i c After Anti-Clark and Anti-Park transformations, we get the system feedback i q and i d Tracking Reference q and i d .

[0079] The present invention first gives an expression of AQRL (Formula 3), and based on the Lyapunov stability criterion (Formulas 1, 13), proves that the given approach law is stable; by dividing the arrival process of the new approach law into two intervals, the time for AQRL to reach the sliding surface is obtained, and compared with the time for the traditional sliding mode approach law to approach the sliding surface, it is proved that the approach law is faster (Formula 19); by discretizing the AQRL switch gain, the discretized mode boundary under AQRL is obtained (Formula 22), and by comparing with the traditional approach law, it is proved that the approach law has no jitter phenomenon after reaching the sliding surface and is more stable. Therefore, the nonlinear sliding mode approach law based on exponential function proposed by the present invention can approach the sliding surface in a shorter time, and has no jitter phenomenon after reaching the sliding surface, and is more stable.

[0080] Based on the FOC principle, a mathematical model of the permanent magnet synchronous motor is established, including voltage equations, torque equations, and motion equations. Taking into account disturbances and parameter uncertainties, the equation for the motor angular acceleration (Equation 7) is obtained, and based on the tracking error (reference angular velocity ω during speed regulation), the motor angular acceleration is calculated. r The difference of the angular velocity ω measured by the encoder is used to design an integral sliding surface (Eq. 8), and then the controller input based on AQRL can be obtained.

[0081] Since the controller input In order to improve the robustness of PMSM, the SFTSMO is proposed to estimate the lumped disturbance, which contains unknown lumped disturbance d. Specifically: the state equation (Equation 10) is designed with the motor angular velocity and disturbance as state variables, and the SFTSMO (Equation 11) is designed based on the terminal sliding mode, and then the sliding surface of the SFTSMO (Equation 12) is designed, and the specific expressions of the system control input and the system disturbance derivative under the SFTSMO are obtained (Equations 23, 24). Similarly, the Lyapunov criterion is used to prove that the designed sliding surface is stable (Equation 25), that is, the proposed SFTSMO is stable within a finite time. By comparing with the approach time of the high-order terminal sliding surface, it is proved that the proposed approach law reaches the sliding surface in a shorter time (Equation 27). Since the derivative of the disturbance has been obtained according to the terminal sliding mode control rate, the expression of the disturbance can be obtained by integrating it, and the input of the controller no longer contains unknown terms (Equation 28).

[0082] exist Figure 1 In the block diagram shown: using i d =0 control mode, so the control input of the current is To obtain the control input of the current The speed controller adopts sliding mode control. Since the traditional sliding mode reaching law has oscillation when reaching the sliding surface, and the switching gain cannot be too large, a new AQRL reaching law is proposed, which can reach the sliding surface faster and has no switching on the sliding surface, so it is more stable. When the errors caused by external disturbances and modeling uncertainties are taken into account, the controller input contains unknown parameters, and the estimation of unknown parameters is completed by SFTSMO.

[0083] Specifically: Given the reference angular velocity ω of the PMSM r , and the speed error is obtained by subtracting the angular velocity feedback from the motor; the speed controller obtains the control input of the system based on the speed error and the system estimated disturbance obtained by SFTSMO and i d =0 The system u is obtained through the current controller d ,u q ;u d ,u q Get u through Park changes α ,u β ;u α ,u β After SVPWM and inverter, the three-phase voltage u of the motor is obtained. a ,u b ,u c ; The three-phase voltage passes through the motor and outputs the three-phase current i a ,i b ,i c and rotor angle θ;i a ,i b ,i c After Anti-Clark and Anti-Park transformations, we get the system feedback i q and i d Tracking Reference q and i d .

[0084] Embodiment 1:

[0085] Sliding mode control design based on AQRL:

[0086] SMC is a nonlinear control method that is less sensitive to parameter fluctuations and disturbances when the system trajectory reaches the sliding surface (s(t)) established by the error between the expected value and the measured value. In order to achieve the stability of the sliding mode controller, the Lyapunov function is usually used to process the input of the controller. The Lyapunov stability criterion can be expressed as:

[0087]

[0088] In order to satisfy the condition that the derivative of the Lyapunov function is less than or equal to zero, the traditional sliding mode reaching law is Where the switching gain η is a positive constant. Under the conventional reaching law, the reaching (arrival) time t from the initial state (s(0) = 0) to the sliding surface (s(t) = 0) is expressed as formula (2):

[0089]

[0090] From formula (2), it can be seen that the arrival time is affected by the initial state and the switch gain η. The larger the η value, the faster the system response speed and the shorter the arrival time. However, when η is large, due to the existence of sgn(·), a chattering phenomenon that is detrimental to the system performance will occur. Therefore, it is necessary to select a suitable η value to suppress the chattering while maintaining a fast arrival time on the sliding surface. The present invention proposes an AQRL to overcome the fragility of SMC:

[0091]

[0092] Where x is the system state, and the value of x(∞) is 0. Parameters a, b, k1, and k2 represent positive adjustment parameters, and the boundary conditions are 0<a<1, 0<b<ln2, k1>0, and k2>0, respectively.

[0093] PMSM motor model:

[0094] PMSM uses field-oriented control (FOC) technology to achieve control, thus ensuring high precision and high efficiency. Based on FOC technology, the three-phase current of PMSMabc coordinates can be converted into two-phase current on dq rotating coordinates by using Clark / Park transformation. In the dq coordinate system, the stator voltage (u d ,u q ) equation is shown in equation (4). The model equation of the present invention is based on a surface-mount PMSM. The d-axis and q-axis of the surface-mount PMSM have the same inductance value (L d =L q =L s ).

[0095]

[0096] where i d 、i q They represent the d-axis and q-axis stator currents respectively. R is the resistance, is the flux linkage of the permanent magnet. The measured speed of the PMSM is represented by ω, and the electromagnetic torque (T e ) depends on the flux linkage, pole pair number (p n ) and inductance. PMSM’s T e The expression is simply expressed as:

[0097]

[0098] The motion equation of the PMSM system is shown in equation (6):

[0099]

[0100] Where θ represents the rotor angle of the PMSM. J, B and T L represent the moment of inertia, viscous friction coefficient and load torque respectively.

[0101] A general PMSM control system is established based on the FOC control principle, such as Figure 1 As shown. Control strategy to decouple speed and current.

[0102] Design of PMSM sliding mode controller using adaptive fast reaching law:

[0103] In the presence of disturbances and parameter uncertainties, the mechanical modeling of the permanent magnet synchronous motor can be converted into:

[0104]

[0105] Where λ, γ, and ε are Δλ, Δγ, Δε represent the uncertainty of the parameters. To represent the parameter uncertainty and load torque, the disturbance is d = Δλi q -Δγω-(ε+Δε)T L In order to track the reference angular velocity (ω r ), the tracking error (e) is defined as ω r The difference between the angular velocity (ω) measured by the encoder. Based on the tracking error, the integrated sliding surface (s i ) is designed as follows:

[0106]

[0107] Where c is a positive constant. Under the sliding mode surface designed by equation (8), the input of the sliding mode controller based on AQRL is set Reach ω r ;

[0108]

[0109] Due to the inclusion of unknown lumped disturbance d, the control input of equation (9) is not fully realized. Therefore, the controller is not a completely stable input. In order to overcome the system jitter caused by disturbances, the present invention proposes SFTSMO to increase the robustness of SMC based on AQRL.

[0110] Design of a second-order fast terminal sliding mode observer for permanent magnet synchronous motor:

[0111] In order to accurately predict and identify disturbances under parameter uncertainty and load torque, the SFTSMO is designed as the state equation given by Eq. (10):

[0112]

[0113] Where l represents the derivative of the lumped disturbance. Then, the SFTSMO is designed based on the terminal sliding mode:

[0114]

[0115] in, and They represent the estimated angular velocity and estimated disturbance of PMSM respectively. t1 and u t2 Represent the designed control law and lumped disturbance derivative respectively. From equations (10) and (11), the estimated angular velocity error can be obtained: and the estimated disturbance error In order to reduce the estimation time and achieve better tracking accuracy, the sliding mode surface (s ft ) is designed as follows:

[0116]

[0117] In the formula for Rational numbers under the condition. p and q are positive odd numbers.

[0118] Technical effect.

[0119] Verification of the stability of the sliding mode approach rate According to formula 1, formula 3 is verified to obtain the following:

[0120] For the sake of simplicity, the present invention only focuses on s>0, and s<0 can be studied in a similar way. In addition, the time from s(0)>1 to s(t)=1 is recorded as t1, and the time from s(t)=1 to t2 is recorded as t2.

[0121] Using the Lyapunov stability criterion, it can be verified that the approach rate designed by the present invention is stable:

[0122]

[0123] In formula (13), (e a|s| -1) and e -b|s| is always greater than 0. Therefore, the proposed method satisfies the Lyapunov stability criterion inequality, so the system derived from equation (3) can reach stability in a finite time.

[0124] It can be seen from formula (3) that the switching gain g(x,s) is designed based on the exponential function and the nonlinear sliding surface. In addition, the initial sliding surface is affected by the switching gain g(x,s). When the initial condition s(0)>1, the arrival process can be divided into two intervals: the first interval refers to the time taken to reach s(t)=1 from the initial condition s(0)>1, and the second interval refers to the time taken to reach the sliding surface s(t)=0 from s(t)=1. When in the first interval, the first term in formula (3) plays a major role, at this time:

[0125]

[0126] Integrating the above formula gives the arrival time t1 of the first interval:

[0127]

[0128] Right now:

[0129]

[0130] In the second interval, the second term in formula (3) plays a major role. At this time:

[0131]

[0132] Integrating the above formula, we can get the arrival time t2 of the second interval as:

[0133]

[0134] By choosing the optimal k1 and k2, we can obtain the time t required for the system to reach the sliding surface when the initial state s(0)>1 r1 :

[0135]

[0136] Then, the time t to reach the sliding surface using AQRL can be obtained: r1 The relationship between the time t to reach the sliding surface using the conventional approach rate is:

[0137]

[0138] Therefore, when k1 satisfies the condition k1 ≥ η, it can be proved that the proposed reaching law reduces the time to reach the sliding surface compared with the traditional sliding mode reaching law.

[0139] According to the proposed reaching law, when the system state approaches the sliding mode surface, due to the switching gain k2x of the AQRL 2 / (1+|x|) is based on the exponential function, and its discrete expression is:

[0140]

[0141] Where T is the sampling period. Assuming that the system trajectory is tracked by the sliding surface in a finite time, the sliding surface at the nth time step is s[n] = 0 + or s[n]=0 - When the system enters the n+1 time step, the sliding surface is expressed as follows:

[0142]

[0143] Therefore, the discrete sliding mode boundary Δ obtained by equation (22) r It can be expressed as Δ r ≈k2x 2 On the other hand, the traditional sliding mode boundary is expressed as Δ = kT. When the system approaches the equilibrium point (0, 0), the chattering phenomenon does not decrease because k is a constant. However, in the proposed reaching law, when the sliding mode surface is reached, Δ r Approaches zero, that is, when the system state approaches the equilibrium point, the system state x converges to zero in a finite time. r Converging to zero, the chattering phenomenon of the sliding mode control system driven by this reaching law is effectively suppressed, while overcoming the disadvantages of the constant switching gain of the traditional sliding mode control system. When the initial states are close to the sliding surface, they no longer stay on the sliding surface; instead, they oscillate continuously on the sliding surface, resulting in chattering.

[0144] Second-order terminal sliding mode verification:

[0145] Substituting formula (10) into s ft , we can get u t1 for:

[0146]

[0147] According to the terminal sliding mode control law, we can get u t2 for:

[0148]

[0149] In the formula, v represents k3, h1 and h2 represent the designed positive constants. In order to prove the stability of the proposed observer, the Lyapunov function shown in equation (1) is used:

[0150]

[0151] Among them, h1 and h2 satisfy h1>|k3v| and h2>|d max| condition. Therefore, the proposed observer is stable in finite time. Under the design sliding surface shown in Eq. (12), compared with the traditional observer based on high-order terminal sliding surface, from the initial state s ft (0)≠0 reaches s ft (0) = the time required for 0 ft Higher-order terminal sliding surface From h (0)≠0 to s h (0) = 0 arrival time t h , the derivation method is similar to formula (2). ft With t h The difference is expressed as follows:

[0152]

[0153] To prove that the designed terminal sliding surface arrival time is shortened, an exponential function is applied to t ft With t h , we can get:

[0154]

[0155] The reduction in arrival time is shown in equation (27). Under the terminal sliding mode control law, by adjusting u t2 The estimated disturbance is obtained by integrating the values Therefore, due to u t2 =h2sgn(s ft ), the proposed SFTSMO ensures smooth perturbation observation. Convert the controller input to:

[0156]

[0157] Therefore, the robustness of the controller input is obtained by estimating the disturbances of the PMSM control system.

[0158] In summary, in SMC, the switching gain will cause time delay and high-frequency jitter in the switching control law. Therefore, the purpose of the present invention is to solve the time delay and high-frequency jitter in the switching process of the sliding mode controller, specifically:

[0159] (1) In order to improve the time to reach the sliding surface, the present invention proposes a new reaching law, called adaptive quick reaching law (AQRL). AQRL is based on the second-order sliding property and the sign function. In addition, the sign function is based on the exponential term and the system state.

[0160] (2) By using a sliding mode disturbance observer (SMDO), robustness to disturbances and parameter uncertainties is achieved. The present invention also uses a second-order fast terminal sliding mode observer (SFTSMO) to respond to disturbances more accurately and avoid excessive switching gain of the SMDO.

[0161] (3) The proposed AQRL is verified by Lyapunov’s second method and mathematically proved to be faster than the traditional arrival time.

[0162] The invention and its implementation methods are described above, which is not restrictive. What is shown in the full text is only one of the implementation methods of the invention, and the actual structure is not limited thereto. In short, if ordinary technicians in this field are inspired by it and design structural methods and embodiments similar to the technical solution without creativity without departing from the purpose of the invention, they should all fall within the scope of protection of the invention.

Claims

1. A method for designing an observer based on a fast adaptive reaching law of an exponential function, characterized in that: The following steps are involved: S1. Establishment of adaptive reaching law: A stable convergence law AQRL is proposed to overcome the fragility of SMC, as shown in formula (3): S2. Use adaptive reaching law to establish PMSM sliding mode controller: Considering disturbances and parameter uncertainties, the equation of motor angular acceleration is obtained by combining the electromagnetic torque and the motion equation of the PMSM system (7), and the integral sliding surface is obtained based on the tracking error, and then the controller input based on AQRL is obtained: S3. Establishment of the second-order fast terminal sliding mode observer of permanent magnet synchronous motor: With the motor angular velocity and disturbance as state variables, the state equation formula (10) is established, and the SFTSMO is established based on the terminal sliding mode to obtain the sliding mode surface of the SFTSMO. The specific expressions of the system control input and the system disturbance derivative under SFTSMO are obtained as shown in formula (23) and formula (24): Under the terminal sliding mode control law, by adjusting u t2 The estimated disturbance is obtained by integrating the values Because u t2 =h2sgn(s ft ), the proposed SFTSMO ensures smooth perturbation observations, obtained from Convert the controller input to:

2. The method for designing an observer based on a fast adaptive reaching law of an exponential function according to claim 1, characterized in that: The integral sliding surface is based on the tracking error, as shown in formula (8): Where c is a positive constant.

3. The method for designing an observer based on a fast adaptive reaching law of an exponential function according to claim 1, characterized in that: In formula (3), x is the system state, the value of x(∞) is 0, the parameters a, b, k1, and k2 represent positive adjustment parameters, and the boundary conditions are 0<a<1, 0<b<ln2, k1>0, and k2>0, respectively.

4. The method for designing an observer based on a fast adaptive reaching law of an exponential function according to claim 1, characterized in that: The expression of the electromagnetic torque and is simply expressed as shown in Formula 5: The motion equation of the PMSM system is shown in formula (6): Among them, i d 、i q They represent the d-axis and q-axis stator currents respectively, R is the resistance, is the flux of the permanent magnet, the measured speed of the PMSM is represented by ω, and the electromagnetic torque generated depends on the flux, the number of pole pairs p n and inductance, θ represents the rotor angle of the PMSM, J, B and T L represent the moment of inertia, viscous friction coefficient and load torque respectively.

5. The method for designing an observer based on a fast adaptive reaching law of an exponential function according to claim 1, characterized in that: In the formula (7), λ, γ, and ε are Δλ, Δγ, Δε represent the uncertainty of the parameters. To represent the uncertainty of the parameters and the load torque, the disturbance is d = Δλi q -Δγω-(ε+Δε)T L .

6. The method for designing an observer based on a fast adaptive reaching law of an exponential function according to claim 1, characterized in that: The SFTSMO established based on the terminal sliding mode is shown in Formula 11: Where l represents the derivative of the lumped disturbance, and denote the estimated angular velocity and estimated disturbance of PMSM, u t1 and u t2 Represent the designed control law and lumped disturbance derivative respectively. The estimated angular velocity error can be obtained from formula (10) and formula (11): and the estimated disturbance error Sliding surface of SFTSMO ft Designed as follows: In the formula for Rational numbers under the condition that p and q are positive odd numbers.

7. The method for designing an observer based on a fast adaptive reaching law of an exponential function according to claim 1, characterized in that: In formula (23) and formula (24), v represents k3, h1 and h2 represent designed positive constants.

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