Adaptive speed and torque control method for flywheel energy storage motor PMSM

Abstract

The invention discloses a flywheel energy storage motor PMSM adaptive speed torque control method, which relates to motor control. The method comprises: obtaining the actual current i output by the flywheel energy storage motor body; d 、i q , that is, the actual current value y output by the reference model; the stator current equation in the synchronous rotating coordinate system obtained from the reference model is transformed into an adjustable model, and the motor orthogonal axis current #imgabs0# estimated by the adjustable model is the current estimation value #imgabs1# output by the adjustable model. The state error e is obtained by subtracting the current estimation value #imgabs2# output by the adjustable model and the actual current value y output by the reference model, thereby obtaining the error state equation; the speed identification adaptive law is designed, so that the error state equation corrects the motor parameters of the adjustable model according to the speed identification adaptive law, so that the error e gradually decreases until the error is zero, that is, the estimated current i#imgabs3# quickly converges to the current true value i d 、i q The control method of the present invention uses the identified motor parameters to update the control model parameters in real time, identifies the state changes of the controlled motor online, and adjusts the system model parameters in time, thereby accurately positioning and controlling the motor in real time, greatly improving the accuracy and robustness of the flywheel motor system.

Description

Technical Field

[0001] The present invention relates to the technical field of motor control, and in particular to a flywheel energy storage motor (PMSM) adaptive speed and torque control method. Background Art

[0002] At present, there are many studies on the control strategies of permanent magnet synchronous motors, but the solutions for flywheel energy storage are relatively limited. Traditional vector control, direct torque control, constant voltage-frequency ratio control, sliding mode variable structure control, model predictive control, etc., all of these control strategies have achieved the control of permanent magnet synchronous motors, but there are certain shortcomings when used for flywheel motors. The actual torque controlled by direct torque control is not completely constant, but pulsating, the low-speed performance is poor, and the speed regulation range is limited. Because there is no feedback circuit in constant voltage-frequency ratio control, the control system cannot grasp the operating status of the motor in real time, and it cannot achieve real-time and accurate control, so the dynamic performance of the system is poor. Vector control requires precise positioning and control of the motor, and has high requirements for the accuracy of motor parameters. In the actual control process, the system may be affected by different degrees of disturbances from the outside or inside, and if the controller only uses a set of fixed parameters, it cannot cope with system changes, which leads to reduced control accuracy and poor stability. Summary of the invention

[0003] In view of one or more of the above-mentioned existing problems, the present invention proposes a flywheel energy storage motor (PMSM) adaptive speed and torque control method.

[0004] According to a first aspect of the present application, a flywheel energy storage motor PMSM adaptive speed and torque control method comprises the following steps:

[0005] S100. Obtain the actual current i output by the flywheel energy storage motor d 、i q , that is, the actual value y of the current output of the reference module;

[0006] S200. The stator current equation in the synchronous rotating coordinate system obtained from the reference model is transformed into an adjustable model, and the orthogonal axis current of the motor estimated by the adjustable model is That is, the current estimate of the adjustable model output

[0007] S300. The current estimate value output by the adjustable model The state error e is obtained by subtracting the actual current value y output by the reference model, thereby obtaining the error state equation;

[0008] S400. Design a speed identification adaptive law so that the error state equation corrects the motor parameters of the adjustable model according to the speed identification adaptive law, so that the error e gradually decreases until the error is zero, that is, the estimated current Quickly converge to the true value of current i d 、i q .

[0009] As an achievable preferred method, step S100 specifically includes:

[0010] S110. Obtain the voltage equation of the flywheel energy storage motor body in the synchronous rotating coordinate system:

[0011]

[0012] Where: u d is the d-axis component of the stator voltage; u q is the q-axis component of the stator voltage; R s is the stator resistance; i d is the d-axis component of the stator current; i q is the q-axis component of the stator inductance; L d is the d-axis component of the stator inductance; L q is the q-axis component of the stator inductance; ω is the rotor angular velocity; ψ f is the permanent magnet flux;

[0013] S120. Convert the obtained voltage equation into a current equation:

[0014]

[0015] As an achievable preferred method, step S200 specifically includes:

[0016] S210. Introducing four variables And set the preset values ​​of the above variables respectively, where: is the preset value of the stator voltage d-axis component, is the preset value of the q-axis component of the stator voltage, Preset value for the d-axis component of the stator current; The preset value of the q-axis component of the stator inductance is as follows:

[0017]

[0018] S220. Obtain the current equation of the adjustable model according to the preset value and the current equation of the reference model:

[0019]

[0020] S230. Convert the AC and DC axis current components and electrical angular velocity in formula (5) into estimated corresponding quantities, and obtain the adjustable model as follows:

[0021]

[0022] S240. The adjustable model obtains the estimated current value of the motor orthogonal axis when the motor starts by integral operation, and the formula is as follows:

[0023]

[0024] As a feasible preferred method, S300. The current estimation value output by the adjustable model is The state error e is obtained by subtracting the actual current value y output by the reference model, thereby obtaining the error state equation; wherein, the error state equation is as follows:

[0025]

[0026] in:

[0027]

[0028] As an achievable preferred method, step S400. Design a speed identification adaptive law so that the error state equation corrects the motor parameters of the adjustable model according to the speed identification adaptive law, so that the error e gradually decreases until the error is zero, that is, the estimated current Quickly converge to the true value of current i d 、i q , specifically including:

[0029] S410. According to the expression of the error state equation, let Output equation v=Ce, and construct corresponding feedback system according to the output equation;

[0030] S420. The feedback system includes a linear forward channel and a nonlinear feedback channel. The state space of the linear forward channel can be described as:

[0031] S430. According to Popov's hyperstability theory, in order to ensure the stability of the constructed feedback system, the feedback system must meet the following conditions:

[0032] The transfer function matrix G(s)=D+C(sI-A)B of the linear forward channel is a strictly positive definite matrix;

[0033] The nonlinear feedback path satisfies Popov's integral inequality.

[0034] As an achievable preferred method, the feedback system must satisfy the following conditions: The feedback system must satisfy the following conditions: The transfer function matrix G(s)=D+C(sI-A)B of the linear forward channel is a strictly positive definite matrix; The specific verification steps are: Because B is the unit matrix and D is the zero matrix; The matrix G(s)=D+C(sI-A)B can be simplified to G(s)=C(sI-A), G(s) is equivalent to the existence of a symmetric positive definite matrix P and a symmetric positive definite matrix Q, satisfying:

[0035]

[0036] Therefore, both the P matrix and the Q matrix are symmetric positive definite matrices, C is a linear compensator and is a symmetric positive definite matrix, so the P matrix is ​​also a symmetric positive definite matrix. After calculation, it is found that the Q matrix is ​​a symmetric positive definite matrix, that is, the transfer function matrix of the linear forward channel is a strictly positive real matrix, which meets the conditions of Popov's superstability theory.

[0037] As a feasible preferred method, S431. The feedback system needs to comply with Popov integral inequality, that is, satisfy:

[0038]

[0039] Where: γ0 is a positive constant that is independent of time and has a finite size;

[0040] S432. Substitute the expressions of W and v into the above formula to obtain:

[0041]

[0042] S433. For the adaptive law design in the MRAS system, the proportional integral is selected and the speed identification adaptive law is:

[0043]

[0044] Where: is the initial value of the estimated electrical angular velocity;

[0045] S434. Substituting the adaptive law of speed identification in formula (11) into formula (10) yields:

[0046]

[0047] make:

[0048]

[0049] Because η(0,t1)=η1(0,t1)+η2(0,t1), to satisfy Popov integral inequality, we need to prove and It can be proved γ0,γ1,γ2 are all time-independent positive constants of finite size;

[0050] S435. Constructor:

[0051]

[0052] Where k is a positive number, then:

[0053]

[0054] S436. Order but:

[0055] S437. Take the derivatives of both sides of the constructor and get:

[0056] For η2(0,t1), F2(v,t) can be:

[0057] Substituting into the above formula, we can get:

[0058]

[0059] therefore, It is established, so it is concluded that the nonlinear time-varying feedback link satisfies Popov's superstability theory;

[0060] S438. Substitute the expressions F1(v,t,τ) and F2(v,t) solved above into the speed identification adaptive law,

[0061]

[0062] From the above formula, the motor speed can be estimated:

[0063]

[0064] S439. The rotor position angular velocity can be obtained by integrating the rotation speed as follows:

[0065]

[0066] The beneficial effects of the present invention are:

[0067] The control method of the present invention uses the identified motor parameters to update the control model parameters in real time. By online identifying the state changes of the controlled motor and timely adjusting the system model parameters, the motor can be accurately positioned and controlled in real time, which greatly improves the accuracy and robustness of the flywheel motor system. BRIEF DESCRIPTION OF THE DRAWINGS

[0068] Figure 1 It is the speed waveform diagram of the flywheel energy storage motor PMSM adaptive speed torque control method;

[0069] Figure 2 The stator three-phase voltage waveform diagram of the flywheel energy storage motor PMSM adaptive speed torque control method of the present application;

[0070] Figure 3 Output torque waveform diagram of the flywheel energy storage motor PMSM adaptive speed torque control method of the present application;

[0071] Figure 4 It is a control schematic diagram of an application system of a flywheel energy storage motor PMSM adaptive speed torque control method;

[0072] Figure 5 It is the schematic diagram of the MRAS system in the flywheel energy storage motor PMSM adaptive speed and torque control method;

[0073] Figure 6 Schematic diagram of the feedback system designed for the flywheel energy storage motor PMSM adaptive speed and torque control method;

[0074] The above drawings show clear embodiments of the present disclosure, which will be described in more detail below. These drawings and text descriptions are not intended to limit the scope of the present disclosure in any way, but to illustrate the concepts of the present disclosure to those skilled in the art by referring to specific embodiments. DETAILED DESCRIPTION

[0075] The technical solution of the application is further described in detail below with reference to the accompanying drawings.

[0076] In order to make the purpose, technical solution and advantages of the embodiments of the present invention clearer, the technical solution in the embodiments of the present invention will be clearly and completely described below in conjunction with the drawings in the embodiments of the present invention. Obviously, the described embodiments are part of the embodiments of the present invention, not all of the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by ordinary technicians in this field without creative work are within the scope of protection of the present invention.

[0077] Unless otherwise defined, all technical and scientific terms used herein have the same meaning as those commonly understood by those skilled in the art of the present invention. The terms used herein in the specification of the present invention are only for the purpose of describing specific embodiments and are not intended to limit the present invention. The term "and / or" used herein includes any and all combinations of one or more related listed items.

[0078] The following is combined with Figure 1-6 , some embodiments of the present invention are described in detail. In the absence of conflict, the following embodiments and features in the embodiments may be combined with each other, and the same or similar concepts or processes may not be described in detail in some embodiments.

[0079] The present invention provides a flywheel energy storage motor PMSM adaptive speed torque control method, comprising the following steps:

[0080] S100. Obtain the actual current i output by the flywheel energy storage motor d 、i q , that is, the actual value y of the current output of the reference module;

[0081] S200. The stator current equation in the synchronous rotating coordinate system obtained from the reference model is transformed into an adjustable model, and the orthogonal axis current of the motor estimated by the adjustable model is That is, the current estimate of the adjustable model output

[0082] S300. The current estimate value output by the adjustable model The state error e is obtained by subtracting the actual current value y output by the reference model, thereby obtaining the error state equation;

[0083] S400. Design a speed identification adaptive law so that the error state equation corrects the motor parameters of the adjustable model according to the speed identification adaptive law, so that the error e gradually decreases until the error is zero, that is, the estimated current Quickly converge to the true value of current i d 、i q .

[0084] Step S100 specifically includes:

[0085] S110. Obtain the voltage equation of the flywheel energy storage motor body in the synchronous rotating coordinate system:

[0086]

[0087] Where: u d is the d-axis component of the stator voltage; u q is the q-axis component of the stator voltage; R s is the stator resistance; i d is the d-axis component of the stator current; i q is the q-axis component of the stator inductance; L d is the d-axis component of the stator inductance; L q is the q-axis component of the stator inductance; ω is the rotor angular velocity; ψ f is the permanent magnet flux;

[0088] S120. Convert the obtained voltage equation into a current equation:

[0089]

[0090] Step S200 specifically includes:

[0091] S210. Introducing four variables And set the preset values ​​of the above variables respectively, where: is the preset value of the stator voltage d-axis component, is the preset value of the q-axis component of the stator voltage, Preset value for the d-axis component of the stator current; The preset value of the q-axis component of the stator inductance is as follows:

[0092]

[0093] S220. Obtain the current equation of the adjustable model according to the preset value and the current equation of the reference model:

[0094]

[0095] S230. Convert the AC and DC axis current components and electrical angular velocity in formula (5) into estimated corresponding quantities, and obtain the adjustable model as follows:

[0096]

[0097] S240. The adjustable model obtains the estimated current value of the motor orthogonal axis when the motor starts by integral operation, and the formula is as follows:

[0098]

[0099] S300. The current estimate value output by the adjustable model The state error e is obtained by subtracting the actual current value y output by the reference model, thereby obtaining the error state equation; wherein, the error state equation is as follows:

[0100]

[0101] in:

[0102]

[0103] Step S400. Design a speed identification adaptive law so that the error state equation corrects the motor parameters of the adjustable model according to the speed identification adaptive law, so that the error e gradually decreases until the error is zero, that is, the estimated current Quickly converge to the true value of current i d 、iq , specifically including:

[0104] S410. According to the expression of the error state equation, let Output equation v=Ce, and construct corresponding feedback system according to the output equation;

[0105] S420. The feedback system includes a linear forward channel and a nonlinear feedback channel. The state space of the linear forward channel can be described as:

[0106] S430. According to Popov's hyperstability theory, in order to ensure the stability of the constructed feedback system, the feedback system must meet the following conditions:

[0107] The transfer function matrix G(s)=D+C(sI-A)B of the linear forward channel is a strictly positive definite matrix;

[0108] The nonlinear feedback path satisfies Popov's integral inequality.

[0109] It should be noted that the feedback system must satisfy the following conditions: the transfer function matrix G(s)=D+C(sI-A)B of the linear forward channel is a strictly positive definite matrix; the specific verification steps are: because B is the unit matrix and D is the zero matrix; the matrix G(s)=D+C(sI-A)B can be simplified to G(s)=C(sI-A), G(s) is equivalent to the existence of a symmetric positive definite matrix P and a symmetric positive definite matrix Q, satisfying:

[0110] Therefore, the matrix Q is a symmetric positive real matrix, that is, the transfer function matrix of the linear forward channel is a strictly positive real matrix, which satisfies the conditions of Popov's superstability theory.

[0111] This feedback system is the MRAS equivalent nonlinear feedback system, and the feedback system must meet the following conditions: the nonlinear feedback path satisfies the Popov integral inequality;

[0112] The specific verification steps include:

[0113] S431. The feedback system needs to comply with Popov integral inequality, that is, it satisfies:

[0114]

[0115] Where: γ0 is a positive constant that is independent of time and has a finite size;

[0116] S432. Substitute the expressions of W and v into the above formula to obtain:

[0117]

[0118] S433. For the adaptive law design in the MRAS system, the proportional integral is selected and the speed identification adaptive law is:

[0119]

[0120] Where: is the initial value of the estimated electrical angular velocity;

[0121] S434. Substituting the adaptive law of speed identification in formula (11) into formula (10) yields:

[0122]

[0123] make:

[0124]

[0125] Because η(0,t1)=η1(0,t1)+η2(0,t1), to satisfy Popov integral inequality, we need to prove and It can be proved γ0,γ1,γ2 are all time-independent positive constants of finite size;

[0126] S435. Constructor:

[0127]

[0128] Where k is a positive number, then:

[0129]

[0130] S436. Order but:

[0131] This section is and Proof process, through the constructor to find the expression of F1(v,t,τ) and F2(v,t) that satisfy Popov integral inequality, thus proving

[0132] S437. Take the derivatives of both sides of the constructor and get:

[0133] For η2(0,t1), F2(v,t) can be:

[0134] Substituting into the above formula, we can get:

[0135]

[0136] therefore, It is established, so it is concluded that the nonlinear time-varying feedback link satisfies Popov's superstability theory;

[0137] S438. Substitute the expressions F1(v,t,τ) and F2(v,t) solved above into the speed identification adaptive law,

[0138]

[0139] From the above formula, the motor speed can be estimated:

[0140]

[0141] S439. The rotor position angular velocity can be obtained by integrating the rotation speed as follows:

[0142]

[0143] From this we can see that Figure 3 The control schematic diagram of the application system of the flywheel energy storage motor PMSM adaptive speed torque control method is a block diagram of the permanent magnet synchronous motor vector control system based on MRAS; Figure 3 It can be seen that the three-phase stator current and rotor position information are first sampled. Since the vector sum of the three-phase current is zero, it is usually only necessary to measure the two-phase current. Then, the three-phase stator current is converted into an orthogonal alternating current component i under the two-phase stationary coordinate through Clarke transformation. α 、i β . The alternating current component i α 、i β Through Park transformation, the two-phase rotating coordinate system is converted to obtain two mutually orthogonal DC current components i d 、i q The given speed is compared with the actual speed, and the given torque current component i is output through the speed PI controller. qref , given value i dref 、i qref Respectively with the feedback amount i d 、i q The difference passes through the current PI controller and outputs the voltage component u on the dq axis d 、u q . Given a voltage component u d 、u q Input Ipark module, output given voltage u in stationary two-phase coordinate system α 、u β Finally, u α 、u βThe input is sent to the SVPWM module, which calculates the new PWM duty cycle and drives the inverter circuit's switch tube to turn off to control the motor's rotation. The MRAS system schematic is as follows: Figure 5 As shown in the figure, the MRAS system mainly consists of three parts, namely the reference model, the adjustable model and the adaptive law. The research object in this invention is the flywheel motor body, and the actual output current value y. The adjustable model is converted from the stator current equation in the rotating coordinate system, which contains the rotor information to be identified, and the output is the current estimation value The difference of the current output of the two models is input into the designed adaptive law, and the speed and position are estimated through the output of the adaptive law. At the same time, the adaptive law will adjust the adjustable model and the estimated speed, and finally make the output error between the adjustable model and the reference model zero.

[0144] The flywheel energy storage motor PMSM adaptive speed torque control method of the present application is based on a DC bus flywheel energy storage system, the bus voltage is 600-1000V, the rated power is 1000kW, the rotor pole pair number of the permanent magnet synchronous motor is 3, the moment of inertia J is 1000kg·m2, the bearing capacity is 50kN, the inner diameter is 500mm, the outer diameter is 800mm, the rated speed is 10000r / min, such as Figure 1 As shown in the figure, the initial motor speed is 3000r / min. When the input torque remains unchanged, the motor speed is changed to 8000r / min. The speed increases from 3000 to 8000 in 50 seconds. Figure 2-3 As shown, the whole process is completed in 50 seconds. When the DC bus voltage increases, the flywheel excitation current increases rapidly. After the DC bus voltage is controlled in a reasonable range, the excitation current is stable. The speed can respond quickly during charging and discharging switching and there is no abnormal change in the stator current. The output torque increases and then decreases rapidly to stabilize, thereby improving the stability of the control system.

[0145] This control method can identify online the changes in motor parameters caused by the temperature change of the flywheel motor, the increase in the service life, etc., so as to update the control algorithm parameters in real time according to the identified motor parameters, which greatly improves the accuracy and robustness of the flywheel motor system. In addition, the sensorless control method is used instead of the mechanical position sensor, which reduces the cost of the system and improves the reliability of the system. The experimental results show that the flywheel motor speed control system has good dynamic performance, fast response speed and ideal control accuracy, which verifies the effectiveness of the control strategy.

[0146] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention, rather than to limit it. Although the present invention has been described in detail with reference to the aforementioned embodiments, those skilled in the art should understand that they can still modify the technical solutions described in the aforementioned embodiments, or replace some or all of the technical features therein with equivalents. However, these modifications or replacements do not cause the essence of the corresponding technical solutions to deviate from the scope of the technical solutions of the embodiments of the present invention.

[0147] The above are only some embodiments of the present invention. For those skilled in the art, several modifications and improvements can be made without departing from the creative concept of the present invention, which all belong to the protection scope of the present invention.

Claims

1. A flywheel energy storage motor PMSM adaptive speed and torque control method, characterized in that: The estimation method comprises the following steps: S100. Obtain the actual current i output by the flywheel energy storage motor d 、i q , that is, the actual value y of the current output of the reference module; S200. The stator current equation in the synchronous rotating coordinate system obtained from the reference model is transformed into an adjustable model, and the orthogonal axis current of the motor estimated by the adjustable model is That is, the current estimate of the adjustable model output S300. The current estimate value output by the adjustable model The state error e is obtained by subtracting the actual current value y output by the reference model, thereby obtaining the error state equation; S400. Design the speed identification adaptive law so that the error state equation corrects the motor parameters of the adjustable model according to the speed identification adaptive law, so that the state error e gradually decreases until the error is zero, that is, the estimated current Quickly converge to the true value of current i d 、i q .

2. The flywheel energy storage motor PMSM adaptive speed and torque control method according to claim 1 is characterized in that: Step S100 specifically includes: S110. Obtain the voltage equation of the flywheel energy storage motor body in the synchronous rotating coordinate system: Where: u d is the d-axis component of the stator voltage; u q is the q-axis component of the stator voltage; R s is the stator resistance; i d is the d-axis component of the stator current; i q is the q-axis component of the stator inductance; L d is the d-axis component of the stator inductance; L q is the q-axis component of the stator inductance; ω is the rotor angular velocity; ψ f is the permanent magnet flux; S120. Convert the obtained voltage equation into a current equation:

3. The flywheel energy storage motor PMSM adaptive speed and torque control method according to claim 2 is characterized in that: Step S200 specifically includes: S210. Introducing four variables And set the preset values ​​of the above variables respectively, where: is the preset value of the stator voltage d-axis component, is the preset value of the q-axis component of the stator voltage, Preset value for the d-axis component of the stator current; The preset value of the q-axis component of the stator inductance is as follows: S220. Obtain the current equation of the adjustable model according to the preset value and the current equation of the reference model: S230. Convert the AC and DC axis current components and electrical angular velocity in formula (5) into estimated corresponding quantities, and obtain the adjustable model as follows: S240. The adjustable model obtains the estimated current value of the motor orthogonal axis when the motor starts by integral operation, and the formula is as follows:

4. The flywheel energy storage motor PMSM adaptive speed and torque control method according to claim 3 is characterized in that: S300. The current estimate value output by the adjustable model The state error e is obtained by subtracting the actual current value y output by the reference model, thereby obtaining the error state equation; wherein, the error state equation is as follows: in:

5. The flywheel energy storage motor PMSM adaptive speed and torque control method according to claim 4 is characterized in that: Step S400. Design a speed identification adaptive law so that the error state equation corrects the motor parameters of the adjustable model according to the speed identification adaptive law, so that the error e gradually decreases until the error is zero, that is, the estimated current Quickly converge to the true value of current i d 、i q , specifically including: S410. According to the expression of the error state equation, let Output equation v=Ce, and construct corresponding feedback system according to the output equation; S420. The feedback system includes a linear forward channel and a nonlinear feedback channel. The state space of the linear forward channel can be described as: S430. According to Popov's hyperstability theory, in order to ensure the stability of the constructed feedback system, the feedback system must meet the following conditions: The transfer function matrix G(s)=D+C(sI-A)B of the linear forward channel is a strictly positive definite matrix; The nonlinear feedback path satisfies Popov's integral inequality.

6. The flywheel energy storage motor PMSM adaptive speed and torque control method according to claim 5 is characterized in that: The feedback system must satisfy the following conditions: the transfer function matrix G(s)=D+C(sI-A)B of the linear forward channel is a strictly positive definite matrix; the specific verification steps are: because B is the unit matrix and D is the zero matrix; the matrix G(s)=D+C(sI-A)B can be simplified to G(s)=C(sI-A), G(s) is equivalent to the existence of a symmetric positive definite matrix P and a symmetric positive definite matrix Q, satisfying: Therefore, both the P matrix and the Q matrix are symmetric positive definite matrices, C is a linear compensator and is a symmetric positive definite matrix, so the P matrix is ​​also a symmetric positive definite matrix. After calculation, it is found that the Q matrix is ​​a symmetric positive definite matrix, that is, the transfer function matrix of the linear forward channel is a strictly positive real matrix, which meets the conditions of Popov's superstability theory.

7. The flywheel energy storage motor PMSM adaptive speed and torque control method according to claim 6 is characterized in that: The feedback system must meet the following conditions: the nonlinear feedback path satisfies the Popov integral inequality; the specific verification steps include: S431. The feedback system needs to comply with Popov integral inequality, that is, it satisfies: Where: γ0 is a positive constant that is independent of time and has a finite size; S432. Substitute the expressions of W and v into the above formula to obtain: S433. For the adaptive law design in the MRAS system, the proportional integral is selected and the speed identification adaptive law is: Where: is the initial value of the estimated electrical angular velocity; S434. Substituting the adaptive law of speed identification in formula (11) into formula (10) yields: make: Because η(0,t1)=η1(0,t1)+η2(0,t1), to satisfy Popov integral inequality, we need to prove and It can be proved γ0,γ1,γ2 are all time-independent positive constants of finite size; S435. Constructor: Where k is a positive number, then: S436. Order but: S437. Take the derivatives of both sides of the constructor and get: For η2(0,t1), F2(v,t) can be: Substituting into the above formula, we can get: therefore, It is established, so it is concluded that the nonlinear time-varying feedback link satisfies Popov's superstability theory; S438. Substitute the expressions F1(v,t,τ) and F2(v,t) solved above into the speed identification adaptive law, From the above formula, the motor speed can be estimated: S439. The rotor position angular velocity can be obtained by integrating the rotation speed as follows:

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