A data-driven method for optimal speed regulation of permanent magnet synchronous motor based on singular perturbation

Abstract

The present invention discloses a method for optimal speed regulation of a data-driven permanent magnet synchronous motor based on singular perturbation, comprising: based on the mathematical model of the permanent magnet synchronous motor, under the framework of vector control technology, proposing the quadratic stability problem of the d-axis current and the linear quadratic regulation problem of the motor speed; based on the singular perturbation theory, obtaining the speed loop reduced-order model by system decomposition, and constructing a new augmented error system, reinterpreting the solution of the standard linear quadratic regulation problem and deriving an equivalent convex optimization problem, thereby avoiding directly solving the algebraic Riccati equation; giving the data expression of the system, replacing the system model information in the quadratic stability problem and the linear quadratic regulation problem, and designing the corresponding PI controller based on the system I / O data. The present invention solves the problem of data-driven optimal speed regulation of a permanent magnet synchronous motor with an unknown system model.

Description

Technical Field

[0001] The invention relates to the technical field of speed tracking control of permanent magnet synchronous motors, and in particular to an optimal speed regulation method for data-driven permanent magnet synchronous motors based on singular perturbation. Background Art

[0002] It is difficult to establish a mathematical model of the system when the parameters of permanent magnet synchronous motors are affected by factors such as temperature and operating point changes. At present, the research on data-driven optimal tracking control has achieved rich results, providing a reference direction for model establishment, but these studies are mainly based on adaptive dynamic programming and neural network identification methods, which require adjusting a large number of parameters and consume a lot of computing resources. This prompted us to use other data-driven control methods to design the optimal speed regulator of permanent magnet synchronous motors. Studies have shown that the trajectory of a fully excited linear system can be used to parameterize all trajectories that the system can generate. Therefore, data-driven simulation methods based on finite-dimensional vector spaces containing all dynamic information of linear time-invariant systems have caused a new wave of research. Through this method, stability criteria in the form of inequalities that are easy to solve can be obtained based on system I / O data. Summary of the invention

[0003] In order to overcome the defects in the prior art, the present invention provides a data-driven permanent magnet synchronous motor optimal speed regulation method based on singular perturbation, designs a controller and speed regulator that are completely based on system I / O data, and avoids dependence on permanent magnet synchronous motor model knowledge.

[0004] The technical solution adopted by the present invention is: the present invention proposes a data-driven permanent magnet synchronous motor optimal speed regulation method based on singular perturbation, comprising the following steps:

[0005] Step S1, based on the mathematical model of the permanent magnet synchronous motor, under the framework of vector control technology, the quadratic stability problem of the d-axis current and the linear quadratic regulation problem of the motor speed are proposed;

[0006] Step S2, based on the singular perturbation theory, the velocity loop reduced-order model is obtained by system decomposition, and a new augmented error system is constructed, the solution of the standard linear quadratic regulation problem is reinterpreted and an equivalent convex optimization problem is derived; Step S3, the data expression of the augmented error system is given, replacing the system model information in the quadratic stability problem and the linear quadratic regulation problem; Step S4, the corresponding PI controller based on the system I / O data of the quadratic stability problem and the linear quadratic regulation problem is given.

[0007] As a further improvement of the present invention, the step S1 is specifically as follows:

[0008] Step S11, obtaining the dynamics of the permanent magnet synchronous motor under the dq coordinate axis:

[0009]

[0010] Among them, T e =1.5n p [ψ f i q +(L d -L q )i d i q ],i d (i q )、u d (u q ) are the current and voltage of the d(q) axis respectively, is the mechanical rotor speed, T L is the external load, stator inductance L d =L q =L = 9.8 × 10 -3 H, stator winding resistance R = 1.06Ω, permanent magnet flux ψ f =0.081Wb, rotor inertia moment Pole pair number n p =4, viscous friction coefficient B = 5.71×10 -3 N·m·s / rad;

[0011] Step S12: According to vector control, let i d =0, the linearization of the dynamics of the permanent magnet synchronous motor can be obtained:

[0012]

[0013] Step S13: Then the current i d The dynamics of is rewritten as:

[0014]

[0015] Let ε = L / R, A 21 =-n p ψ f / R,A 22 =-1, B 1 =0, B 2 =1 / R. Then formula (2) is rewritten as:

[0016]

[0017] Step S14: define a quadratic stability problem according to formula (3):

[0018]

[0019] Among them, find means to search, st means to obey, P d represents the weight matrix, K d represents the controller gain to be designed;

[0020] Step S15: According to the singular perturbation theory, let ε=0 in formula (4):

[0021]

[0022] Substituting formula (6) into formula (4), we get the reduced-order system:

[0023]

[0024] Among them, the reduced-order system parameter matrix

[0025] Assume that the fixed-point speed reference signal to be tracked is The output tracking error is then expressed as:

[0026]

[0027] Therefore, a standard linear quadratic regulation problem is defined according to formula (7) and formula (8):

[0028]

[0029] Among them, min means minimization, J means cost function, and st means subject to, represents the transpose of the matrix, the weight matrix

[0030] As a further improvement of the present invention, the step S2 is specifically as follows:

[0031] Step S21: define a new augmented vector Combining formula (7) we can get the augmented system:

[0032]

[0033] Among them, the augmented system parameter matrix Step S22: According to formula (10), the standard linear quadratic regulation problem is changed from formula (9) to:

[0034]

[0035] Among them, the weight matrix Step S23: Define K q =[K pq K iq ], then the optimal control law in, P is the algebraic Riccati equation The only positive solution of ; therefore, the optimal control strategy is:

[0036]

[0037] Step S24: Assume that for all t≥0, the formula If it holds, then formula (10) can be obtained:

[0038]

[0039] Integrate both sides of formula (13) from 0 to ∞:

[0040]

[0041] It shows that the lower bound of the cost function J is Step S25, combining formula (13) and formula (14), the standard linear quadratic adjustment problem, i.e., the equivalent convex optimization problem of formula (11), is:

[0042]

[0043] Among them, max means maximize;

[0044] Step S26: Substituting the formula (11) At this time, the cost function J is rewritten as:

[0045]

[0046] From standard results in control theory, we know that J can be evaluated as in, satisfy Therefore, the convex optimization problem, i.e. the dual problem of formula (15), is:

[0047]

[0048] Step S27: Introduce new variables Using Schur's lemma and due to the trace and are equivalent, decoupling problem, that is, formula (17) can be obtained:

[0049]

[0050] As a further improvement of the present invention, the step S3 is specifically as follows:

[0051] Step S31, design a PI controller for the d-axis current loop:

[0052]

[0053] Among them, K d =[K pd K id ];

[0054] Step S32: define augmented vector From formula (3), we can get:

[0055]

[0056] Among them, the augmented system parameter matrix Step S33: According to formula (5), the quadratic stability problem corresponding to formula (20) is obtained:

[0057]

[0058] Step S34: Set sampling time T s > 0, continuously sample formula (20) and give the data expression of formula (20):

[0059]

[0060] in, Continuously collect derivative data of N state variables, Indicates the continuous collection of real data of N state variables. Indicates continuous collection of N control input data, W 0,N :=[ω(0) ω(T s ) … ω((N-1)T s )],W 0,N Indicates continuous collection of N external disturbance data;

[0061] Step S35: From formula (22), we can get:

[0062]

[0063] in, represents the Moore-Penrose generalized inverse of a matrix;

[0064] Further, we can get:

[0065]

[0066] Among them, G satisfies Step S36: Replace the formula (21) with the formula (24): The quadratic stability problem, i.e., the data form of formula (21), can be obtained as:

[0067]

[0068] Step S37: define new variables Get the augmented system:

[0069]

[0070] Among them, the augmented system parameter matrix Step S38: Set sampling time T s > 0, continuously sample formula (26) and give the data expression of formula (26):

[0071] X 1,N =AX 0,N +BU 0,N +D 0,N (27)

[0072] in, X 1,N Continuously collect derivative data of N state variables, X 0,N :=[χ(0)χ(T s )…χ((N-1)T s )],X 0,N Indicates the continuous collection of real data of N state variables, U 0,N :=[u q (0) u q (T s ) … u q ((N-1)T s )],U 0,N Indicates continuous collection of N control input data, D 0,N :=[dd …d],D 0,N Indicates continuous collection of N external disturbance data;

[0073] Step S39, repeating steps S35-S36, the linear quadratic regulation problem, i.e., the data expression of formula (18), can be obtained:

[0074]

[0075] As a further improvement of the present invention, the step S4 is specifically as follows:

[0076] Step S41: Available Therefore, the controller gain of the d-axis current loop is expressed as:

[0077]

[0078] in, is the solution to the following quadratic stability problem:

[0079]

[0080] Step S42: Available Therefore, the data expression of the tracking controller gain of the speed loop is:

[0081]

[0082] where Y is the solution to the following convex optimization problem:

[0083]

[0084] The beneficial effects of the present invention are:

[0085] (1) According to the mathematical model of permanent magnet synchronous motor, based on vector control technology and singular perturbation theory, the quadratic stability problem of d-axis current and the linear quadratic regulation problem of motor speed are proposed, and the equivalent convex optimization problem of the standard linear quadratic regulation problem is derived, avoiding the difficulty of directly solving the algebraic Riccati equation and the potential numerical stiffness problem caused by the dynamic coupling between current and speed;

[0086] (2) By collecting the I / O data of the permanent magnet synchronous motor during operation, a fully data-driven PI controller and speed regulator were designed, avoiding the problem of difficulty in establishing the system mathematical model when the motor parameters are affected by factors such as temperature and operating point changes. BRIEF DESCRIPTION OF THE DRAWINGS

[0087] The present invention is further described in detail below with reference to the accompanying drawings and specific embodiments.

[0088] Figure 1 It is a flow chart of the method for optimal speed regulation of a data-driven permanent magnet synchronous motor based on singular perturbation of the present invention;

[0089] Figure 2 is a block diagram of data-driven optimal speed tracking control of a permanent magnet synchronous motor in an embodiment of the present invention;

[0090] Figure 3 is the current i of the d-axis current loop under closed-loop control in the embodiment of the present invention d and voltage u d Trajectory diagram of

[0091] Figure 4 is the current i of the q-axis speed loop under closed-loop control in the embodiment of the present invention q and voltage u q Trajectory diagram of

[0092] Figure 53 is a trajectory diagram of the motor speed and target speed under closed-loop control obtained experimentally in an embodiment of the present invention. DETAILED DESCRIPTION

[0093] Please refer to Figure 1 The present invention provides a method for adjusting the optimal speed of a data-driven permanent magnet synchronous motor based on singular perturbation, comprising the following steps:

[0094] Step 1, based on the mathematical model of the permanent magnet synchronous motor, under the framework of vector control technology, the quadratic stability problem of the d-axis current and the linear quadratic regulation problem of the motor speed are proposed, including the following steps;

[0095] Step 101, obtain the dynamics of the permanent magnet synchronous motor under the dq coordinate axis:

[0096]

[0097] Among them, T e =1.5n p [ψ f i q +(L d -L q )i d i q ],i d (i q )、u d (u q ) are the current and voltage of the d(q) axis respectively, is the mechanical rotor speed, T L is the external load, stator inductance L d =L q =L = 9.8 × 10 -3 H, stator winding resistance R = 1.06Ω, permanent magnet flux ψ f =0.081Wb, rotor inertia moment Pole pair number n p =4, viscous friction coefficient B = 5.71×10 -3 N·m·s / rad;

[0098] Step 102: According to vector control, let i d =0, the linearization of the dynamics of the permanent magnet synchronous motor can be obtained:

[0099]

[0100] Step 103: To facilitate writing, Then the current i d The dynamics of is rewritten as:

[0101]

[0102] Let ε = L / R, A 21 =-n p ψ f / R,A 22 =-1, B 1 =0, B 2 =1 / R. Then system (2) is rewritten as:

[0103]

[0104] Step 104: Define a quadratic stability problem based on system (3):

[0105]

[0106] Among them, find means to search, st means to obey, P d represents the weight matrix, K d represents the controller gain to be designed;

[0107] Step 105: According to the singular perturbation theory, let ε=0 in formula (4):

[0108]

[0109] Substituting system (6) into system (4), we obtain the reduced-order system:

[0110]

[0111] Among them, the reduced-order system parameter matrix

[0112] Assume that the fixed-point speed reference signal to be tracked is The output tracking error is then expressed as:

[0113]

[0114] Therefore, a standard linear quadratic regulation problem is defined according to system (7) and error system (8):

[0115]

[0116] Among them, min means minimization, J means cost function, and st means subject to, represents the transpose of the matrix, the weight matrix

[0117] Step 2, based on the singular perturbation theory, the velocity loop reduced-order model is obtained by system decomposition, and a new augmented error system is constructed to reinterpret the solution of the standard linear quadratic regulation problem and derive an equivalent convex optimization problem, including the following steps;

[0118] Step 201: Define a new augmented vector Combining formula (7) we can get the augmented system:

[0119]

[0120] Among them, the augmented system parameter matrix

[0121] Step 202: According to formula (10), the standard linear quadratic regulation problem is changed from formula (9) to:

[0122]

[0123] Among them, the weight matrix

[0124] Step 203: Define K q =[K pq K iq ], then the optimal control law in, P is the algebraic Riccati equation The only positive solution of ; therefore, the optimal control strategy is:

[0125]

[0126] Step 204: Assume that for all t≥0, the formula If it holds, then formula (10) can be obtained:

[0127]

[0128] Integrate both sides of formula (13) from 0 to ∞:

[0129]

[0130] It shows that the lower bound of the cost function J is

[0131] Step 205: Combining formula (13) and formula (14), the standard linear quadratic regulation problem, i.e., the equivalent convex optimization problem of formula (11), is:

[0132]

[0133] Among them, max means maximize;

[0134] Step 206: Use Substituting the formula (11) At this time, the cost function J is rewritten as:

[0135]

[0136] From standard results in control theory, we know that J can be evaluated as in, satisfy Therefore, the convex optimization problem, i.e. the dual problem of formula (15), is:

[0137]

[0138] Step 207: Introduce new variables Using Schur's lemma and due to the trace and are equivalent, decoupling problem, that is, formula (17) can be obtained:

[0139]

[0140] Step 3, giving a data expression of the system, replacing the system model information in the quadratic stability problem and the linear quadratic regulation problem, including the following steps;

[0141] Step 301, design a PI controller for the d-axis current loop:

[0142]

[0143] Among them, K d =[K pd K id ];

[0144] Step 302: Define augmented vector From formula (3), we can get:

[0145]

[0146] Among them, the augmented system parameter matrix

[0147] Step 303: According to formula (5), the quadratic stability problem of the corresponding system (20) is obtained:

[0148]

[0149] Step 304: Set sampling time T s > 0, the system (20) is continuously sampled and the data expression of formula (20) is given as:

[0150]

[0151] in, Continuously collect derivative data of N state variables, Indicates the continuous collection of real data of N state variables. Indicates continuous collection of N control input data, W 0,N :=[ω(0) ω(T s ) … ω((N-1)T s )],W 0,N Indicates continuous collection of N external disturbance data;

[0152] Step 305: From formula (22), we can get:

[0153]

[0154] in, represents the Moore-Penrose generalized inverse of a matrix;

[0155] Further, we can get:

[0156]

[0157] Among them, G satisfies

[0158] Step 306: Replace the formula (21) with the formula (24). The quadratic stability problem, i.e., the data form of formula (21), can be obtained as:

[0159]

[0160] Step 306: Define new variables Get the augmented system:

[0161]

[0162] Among them, the augmented system parameter matrix

[0163] Step 308: Set sampling time T s > 0, the system (26) is continuously sampled and the data expression of formula (26) is given as:

[0164] X 1,N =AX 0,N +BU 0,N +D 0,N (27)

[0165] in, X 1,N Continuously collect derivative data of N state variables, X 0,N :=[χ(0)χ(Ts )…χ((N-1)T s )],X 0,N Indicates the continuous collection of real data of N state variables, U 0,N :=[u q (0) u q (T s ) … u q ((N-1)T s )],U 0,N Indicates continuous collection of N control input data, D 0,N :=[dd …d],D 0,N Indicates continuous collection of N external disturbance data;

[0166] Step 309, repeating steps 305-306, the linear quadratic regulation problem, i.e., the data expression of formula (18), can be obtained:

[0167]

[0168] Step 4, providing a PI controller based on system I / O data corresponding to the quadratic stability problem and the linear quadratic regulation problem, including the following steps;

[0169] Step 401: Available Therefore, the controller gain of the d-axis current loop is expressed as:

[0170]

[0171] in, is the solution to the following quadratic stability problem:

[0172]

[0173] Step 402: Available Therefore, the data expression of the tracking controller gain of the speed loop is:

[0174]

[0175] where Y is the solution to the following convex optimization problem:

[0176]

[0177] [Example]

[0178] The speed tracking control process of the permanent magnet synchronous motor is combined to verify the effectiveness of the new data-driven permanent magnet synchronous motor optimal speed regulation method.

[0179] The block diagram of the data-driven optimal speed regulation control of the permanent magnet synchronous motor is as follows: Figure 2 shown.

[0180] Setting the reference speed Run the permanent magnet synchronous motor and collect data. Assuming Q = 10, R = 0.1, the closed-loop stable control gain K of the d-axis current loop can be obtained d =[39.3187 7.8909], closed-loop optimal control gain K of the speed loop q =[-0.0001 -0.0016]. The experimental results of the current and voltage of the d-axis are as follows Figure 3 As shown, the experimental results of current and voltage on the q axis are as follows Figure 4 As shown in the figure, they tend to be stable after a certain period of time. and target speed The experimental results are as follows Figure 5 As shown, under closed loop, the motor speed quickly tracks the target speed and fluctuates within a small range.

[0181] Experimental results show that the optimal speed regulation method for a data-driven permanent magnet synchronous motor based on singular perturbation proposed in the present invention can effectively regulate the speed of the permanent magnet synchronous motor without requiring any model information of the permanent magnet synchronous motor.

[0182] The embodiments of the present invention are described in detail above in conjunction with the accompanying drawings, but the present invention is not limited thereto. Various changes that can be made within the knowledge scope of technicians in the relevant technical field without departing from the purpose of the present invention are all within the protection scope of the claims of the present invention.

Claims

1. A method for optimal speed regulation of a data-driven permanent magnet synchronous motor based on singular perturbation, characterized in that: The following steps are involved: Step S1, based on the mathematical model of the permanent magnet synchronous motor, under the framework of vector control technology, the quadratic stability problem of the d-axis current and the linear quadratic regulation problem of the motor speed are proposed; Step S2, based on the singular perturbation theory, a velocity loop reduced-order model is obtained by system decomposition, and a new augmented error system about the reduced-order velocity loop and the reference velocity trajectory is constructed, the solution of the standard linear quadratic regulation problem is reinterpreted and an equivalent convex optimization problem is derived; Step S3, providing a data expression of the augmented error system to replace the system model information in the quadratic stability problem and the linear quadratic regulation problem; Step S4: providing a PI controller based on system I / O data corresponding to the quadratic stability problem and the linear quadratic regulation problem.

2. The optimal speed adjustment method of a data-driven permanent magnet synchronous motor according to claim 1, characterized in that: In step S1, the mathematical model based on the permanent magnet synchronous motor proposes the quadratic stability problem of the d-axis current and the linear quadratic regulation problem of the motor speed, which is specifically: Step S11, obtaining the dynamics of the permanent magnet synchronous motor under the dq coordinate axis: Among them, T e =1.5n p [ψ f i q +(L d -L q )i d i q ],i d (i q )、u d (u q ) are the current and voltage of the d(q) axis respectively, is the mechanical rotor speed, T L is the external load, stator inductance L d =L q =L = 9.8 × 10 -3 H, stator winding resistance R = 1.06Ω, permanent magnet flux ψ f =0.081Wb, rotor inertia moment Pole pair number n p =4, viscous friction coefficient B = 5.71×10 -3 N·m·s / rad; Step S12: According to vector control, let i d =0, the linearization of the dynamics of the permanent magnet synchronous motor can be obtained: Step S13: Then the current i d The dynamics of is rewritten as: Let ε = L / R, A 21 =-n p ψ f / R,A 22 =-1, B1=0, B2=1 / R; then formula (2) is rewritten as: Step S14: define a quadratic stability problem according to formula (3): Among them, find means to search, st means to obey, P d represents the weight matrix, K d represents the controller gain to be designed; Step S15: According to the singular perturbation theory, let ε=0 in formula (4): Substituting formula (6) into formula (4), we get the reduced-order system: Among them, the reduced-order system parameter matrix Assume that the fixed-point speed reference signal to be tracked is The output tracking error is expressed as: Therefore, a standard linear quadratic regulation problem is defined according to formula (7) and formula (8): Among them, min means minimization, J means cost function, and st means subject to, represents the transpose of the matrix, the weight matrix 3. The data driven permanent magnet synchronous motor optimal speed adjustment method according to claim 1, characterized in that: In step S2, the speed loop reduced-order model is obtained by system decomposition, and a new augmented system is constructed to reinterpret the solution of the standard linear quadratic regulation problem and derive an equivalent convex optimization problem, specifically: Step S21: define a new augmented vector Combining formula (7) we can get the augmented system: in, is the mechanical rotor speed, u q is the current of the q axis, e is the output speed tracking error, and the augmented system parameter matrix Step S22: According to formula (10), the standard linear quadratic regulation problem is changed from formula (9) to: Among them, min means minimization, st means obeying, the weight matrix Step S23: define K q =[K pq K iq ], then the optimal control law Among them, the speed loop controller gain P is the algebraic Riccati equation The only positive solution of ; therefore, the optimal control strategy is: Step S24: Assume that for all t≥0, the formula If it holds, then formula (10) can be obtained: Integrate both sides of formula (13) from 0 to ∞: It shows that the lower bound of the cost function J is in represents the value of the augmented system (10) at zero initial time; Step S25, combining formula (13) and formula (14), the standard linear quadratic adjustment problem, i.e., the equivalent convex optimization problem of formula (11), is: Among them, max means maximize, st means obey; Step S26: Substituting the formula (11) At this time, the cost function J is rewritten as: From standard results in control theory, we know that J can be evaluated as in, satisfy Therefore, the convex optimization problem, i.e. the dual problem of formula (15), is: Where min means minimization, st means subject to, and trace means the trace of the matrix. Step S27: Introduce new variables Using Schur's lemma and due to the trace and are equivalent, decoupling problem, that is, formula (17) can be obtained: Among them, min means minimize, and st means obey.

4. The data driven permanent magnet synchronous motor optimal speed adjustment method according to claim 1, characterized in that: In step S3, the data expression of the augmented error system is given to replace the system model information in the quadratic stability problem and the linear quadratic regulation problem, specifically: Step S31, design a PI controller for the d-axis current loop: Where, the d-axis current loop controller gain K d =[K pd K id ]; Step S32: define augmented vector From formula (3), we can get: Among them, i d represents the d-axis current, u d represents the d-axis voltage, ω represents the disturbance, and the augmented system parameter matrix Step S33: According to formula (5), the quadratic stability problem corresponding to formula (20) is obtained: Find means to search, and st means to obey. Step S34: Set sampling time T s > 0, continuously sample formula (20) and give the data expression: in, Indicates the continuous collection of derivative data of N state variables, Indicates the continuous collection of real data of N state variables. Indicates continuous collection of N control input data, W 0,N :=[ω(0) ω(T s ) … ω((N-1)T s )],W 0,N Indicates continuous collection of N external disturbance data; Step S35: From formula (22), we can get: in, represents the Moore-Penrose generalized inverse of a matrix; Further, we can get: Among them, G satisfies Step S36: Replace the formula (21) with the formula (24): The data form of the quadratic stability problem, i.e., formula (21), can be obtained as: Find means to search, and st means to obey. Step S37: define new variables Get the augmented system: Among them, the augmented system parameter matrix Step S38: Set sampling time T s > 0, continuously sample formula (26) and give the data expression: X 1,N =AX 0,N +BU 0,N +D 0,N (27) in, X 1,N Indicates the continuous collection of derivative data of N state variables, X 0,N :=[χ(0) χ(T s ) … χ((N-1)T s )],X 0,N Indicates the continuous collection of real data of N state variables, U 0,N :=[u q (0) u q (T s ) … u q ((N-1)T s )],U 0,N Indicates continuous collection of N control input data, D 0,N :=[dd …d],D 0,N Indicates continuous collection of N external disturbance data; Step S39, repeating steps S35-S36, the linear quadratic regulation problem, i.e., the data expression of formula (18), can be obtained: Min means minimize, and st means obey.

5. The data driven permanent magnet synchronous motor optimal speed adjustment method according to claim 1, characterized in that: In step S4, the PI controller based on system I / O data corresponding to the quadratic stability problem and the linear quadratic regulation problem is specifically: Step S41: Available Therefore, the controller gain of the d-axis current loop is expressed as: in, is the solution to the following quadratic stability problem: Find means to search, and st means to obey. Step S42: Available Therefore, the data expression of the tracking controller gain of the speed loop is: where Y is the solution to the following convex optimization problem: Min means minimize, and st means obey.

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