Command Filtering Fuzzy Adaptive Control Method for Permanent Magnet Synchronous Motor Stochastic System

Abstract

The present invention belongs to the technical field of position tracking control of permanent magnet synchronous motors, and discloses a random system command filtering fuzzy adaptive control method for permanent magnet synchronous motors. Aiming at the control accuracy requirements of a random system of permanent magnet synchronous motors considering time-varying state constraints and input saturation, as well as the existing random disturbances and nonlinear problems, a fuzzy adaptive backstepping controller is designed to achieve tracking of the target position, and an obstacle Lyapunov function is constructed to ensure that state variables such as current and speed do not violate the time-varying constraint conditions. The fuzzy logic theory is used to handle the unknown nonlinear terms in the motor random system. A method combining command filtering technology and an error compensation mechanism is adopted, which not only solves the "computational explosion" problem in the traditional backstepping method, but also eliminates the influence of filtering errors. The method of the present invention can effectively suppress the influence of input saturation and random disturbances, improve the control performance of the system, and at the same time ensure that all states of the motor system are within the given constraint range.

Description

Technical Field

[0001] The present invention belongs to the technical field of permanent magnet synchronous motor position tracking control, and particularly relates to a stochastic system command filtering fuzzy adaptive control method for a permanent magnet synchronous motor considering time-varying state constraints and input saturation. Background Technique

[0002] In recent years, permanent magnet synchronous motors (PMSMs) have been widely used in industrial and agricultural fields due to their simple structure, high system efficiency, good control performance, etc., and have also become the focus of many domestic and foreign scholars. The permanent magnet synchronous motor is a highly nonlinear, strongly coupled, and multi-variable control system, and its performance is easily affected by unknown factors such as the motor itself and load disturbances. At present, in order to enable the system to obtain better control performance, many effective control strategies have been proposed by researchers, such as advanced control techniques like backstepping control, robust control, and adaptive control.

[0003] However, the above control methods do not take into account the influence caused by random disturbances. During the operation of a permanent magnet synchronous motor, parameters such as motor torque and winding resistance will be affected by disturbances such as damping torque and magnetic circuit saturation, resulting in random perturbation phenomena. These random perturbations often become the key factors restricting system performance. Previously, great progress has been made in the research on the control of stochastic nonlinear systems. However, few existing control methods consider the state constraint problem. In many practical engineering applications, the state variables of the system need to be restricted within a reasonable constraint space according to factors such as the working environment and actual requirements. If the input signal or state exceeds the specified constraint conditions, the safety and stability of the system cannot be guaranteed. For example, during the operation of a permanent magnet synchronous motor, due to excessive current, the motor windings will heat up severely, leading to accelerated aging of the insulation layer and ultimately shortening the motor life. Therefore, imposing time-varying constraints on the state variables in the motor system, such as angular velocity and stator current, can better meet the actual engineering requirements. At the same time, considering the physical constraints of the actuator and mechanical design in the actual system, the input voltage of the motor may exhibit an input saturation problem. It should be noted that too high a voltage will cause the motor to overheat, affecting the normal use of the motor and even damaging the motor severely. Therefore, it is of great significance to consider time-varying constraints and input saturation in the control of the permanent magnet synchronous motor stochastic system.

[0004] In another frontier area, the proposal of a large number of advanced control methods provides more effective solutions for dealing with nonlinear system problems. Among them, the adaptive backstepping method has been successfully applied to the permanent magnet synchronous motor system and achieved good control effects. However, the disadvantages of the backstepping method are mainly reflected in that some functions of certain systems must be linear and the "computational explosion" problem will occur due to repeated differentiation in the design process. Among them, the existing technology solves the problem that some functions of certain systems must be linear by using a fuzzy logic system (FLS) or a neural network (NN) to approximate the nonlinear terms of the system. For the "computational explosion" problem, the existing technology has proposed a dynamic surface control (DSC) method to deal with it and achieved remarkable results. However, there will be a filtering error when using the dynamic surface control method, and this error cannot be eliminated, which will affect the control effect. Summary of the Invention

[0005] The object of the present invention is to propose a command filtering fuzzy adaptive control method for a permanent magnet synchronous motor stochastic system considering time-varying state constraints and input saturation. Under the full consideration of time-varying state constraints and input saturation, this method can enable the permanent magnet synchronous motor stochastic system to quickly track the desired signal.

[0006] In order to achieve the above object, the present invention adopts the following technical solutions:

[0007] The present invention proposes a command filtering fuzzy adaptive control method for a permanent magnet synchronous motor stochastic system considering time-varying state constraints and input saturation. Aiming at the control accuracy requirements of the permanent magnet synchronous motor stochastic system considering time-varying state constraints and input saturation, as well as the existing stochastic disturbances and nonlinear problems, a fuzzy adaptive backstepping controller is designed to achieve the tracking of the target position. An obstacle Lyapunov function is constructed to ensure that state variables such as current and speed do not violate the time-varying constraint conditions. Moreover, a method combining command filtering technology and an error compensation mechanism is adopted, which not only solves the "computational explosion" problem in the traditional backstepping method but also eliminates the influence of the filtering error. The fuzzy logic system is used to handle the high-order nonlinear terms in the permanent magnet synchronous motor stochastic system, and the adaptive control method is combined to solve the problems of unknown parameters and input saturation in the system, thereby constructing a command filtering fuzzy adaptive backstepping controller for the permanent magnet synchronous motor considering time-varying state constraints and input saturation.

[0008] The present invention has the following advantages:

[0009] (1) The method of the present invention is directed to the PMSM stochastic system, incorporating time-varying state constraints and input saturation into the controller design considerations simultaneously, avoiding the damage to the motor caused by the input saturation problem. Meanwhile, a time-varying barrier Lyapunov function (TVBLF) is designed in each process of the backstepping derivation to ensure that the system state variables are constrained within the given time-varying interval, making the designed controller better meet the requirements of actual engineering.

[0010] (2) The present invention uses the method of fuzzy logic system approximation to handle the unknown nonlinear functions in the PMSM stochastic system, simplifies the structure of the fuzzy adaptive backstepping controller, and effectively solves the problem of position tracking control of the PMSM under the conditions of parameter uncertainty and load torque disturbance.

[0011] (3) The present invention combines the command filtering technology and the error compensation mechanism, not only fundamentally solves the "computational explosion" problem, but also reduces the influence of the filtering error on the system performance, thereby improving the control accuracy of the system.

[0012] (4) When designing the controller, the present invention considers the random disturbances that occur during the operation of the motor, improves the robustness and stability of the system, and the designed controller is more conducive to practical applications. Description of the Drawings

[0013] Figure 1 It is a schematic diagram of the composite controlled object composed of the stochastic command filtering fuzzy adaptive backstepping controller for the PMSM considering time-varying state constraints and input saturation, coordinate transformation, and SVPWM inverter in the present invention.

[0014] Figure 2 It is the simulation diagram of the rotor angle, the rotor angle set value, and the constraint condition tracking after adopting the control method of the present invention.

[0015] Figure 3 It is the simulation diagram of the tracking error between the rotor angle and the rotor angle set value after adopting the control method of the present invention.

[0016] Figure 4 It is the simulation diagram of the d-axis stator voltage of the PMSM after adopting the control method of the present invention.

[0017] Figure 5 It is the simulation diagram of the q-axis stator voltage of the PMSM after adopting the control method of the present invention.

[0018] Figure 6 It is the simulation diagram of the state variable x2 of the PMSM after adopting the control method of the present invention.

[0019] Figure 7 It is the simulation diagram of the state variable x3 of the PMSM after adopting the control method of the present invention.

[0020] Figure 8 It is the simulation diagram of the state variable x4 of the permanent magnet synchronous motor after adopting the control method of the present invention. Specific embodiments

[0021] The present invention will be further described in detail below with reference to the accompanying drawings and specific embodiments:

[0022] Figure 1 It shows a schematic diagram of the composite controlled object composed of the fuzzy adaptive backstepping controller for the random system of the permanent magnet synchronous motor based on instruction filtering, coordinate transformation, and SVPWM inverter in the present invention.

[0023] Figure 1 The components involved mainly include the fuzzy adaptive backstepping controller 1 for the random system of the permanent magnet synchronous motor based on instruction filtering, the coordinate transformation unit 2, the SVPWM inverter 3, the speed detection unit 4, and the current detection unit 5. In Figure 1 U, V, and W represent three-phase voltages, and u α and u β are voltages in the two-phase stationary coordinate system, and ω is the rotor angular velocity. The speed detection unit 4 and the current detection unit 5 are mainly used to detect the speed-related variables and current values of the permanent magnet synchronous motor. The actually measured current and speed variables are used as inputs to the fuzzy adaptive backstepping controller, and voltage control is performed through the fuzzy adaptive backstepping controller 1 for the random system of the permanent magnet synchronous motor based on instruction filtering, and finally converted into three-phase electricity to control the rotor position of the permanent magnet synchronous motor. In order to design an effective controller, it is very necessary to establish a random system model of the permanent magnet synchronous motor.

[0024] The instruction filtering fuzzy adaptive control method for the random system of the permanent magnet synchronous motor considering time-varying state constraints and input saturation includes the following steps:

[0025] Step 1. Establish the dynamic mathematical model of the d-q coordinate axis of the permanent magnet synchronous motor as shown in formula (1):

[0026]

[0027] Among them, θ represents the rotor angle of the motor, ω represents the rotor angular velocity, and both θ and ω are state variables of the system; u d and u q represent the voltages on the d-axis and q-axis respectively, and u d and u q are the input signals of the system; i d and i q are the exciting currents on the d-axis and q-axis respectively; L d and L qis the stator inductance in the d-q coordinate system; J represents the moment of inertia of the motor, B represents the friction coefficient of the motor, and T L represents the load torque of the motor, and n p represents the number of pole pairs of the motor, Φ represents the magnetic flux generated by the permanent magnet of the motor, and R s represents the stator resistance of the motor; the following variables are defined to simplify the d-q axis dynamic mathematical model of the permanent magnet synchronous motor.

[0028]

[0029] Considering random disturbances, the dynamic mathematical model of the permanent magnet synchronous motor is as follows:

[0030]

[0031] where ψ2, ψ3, and ψ4 are unknown smooth disturbance functions.

[0032] Step 2. According to the command filtering technology and the principle of adaptive backstepping, design a command filtering fuzzy adaptive control method for the random system of the permanent magnet synchronous motor considering time-varying state constraints and input saturation. The control objective is to design the input signals u d and u q such that x1 can track the desired signal x d well, and all states in the system need to satisfy the time-varying constraint condition Γx j ={x j ∈R||x j | < k cj (t)}, R represents the set of real numbers, and k cj (t) is a designed time-varying continuous function, j = 1, 2, 3, 4.

[0033] For Equation (2), considering that the input signals u d and u q of the system are affected by saturation nonlinearity, here u is used to represent u d and u q .

[0034]

[0035] where v is the true control input, u is the control input under actual application, u max > 0 and u min < 0 are unknown saturation constants. It can be seen from Equation (3) that when v = u max or v = u min , non-differentiable points will appear.

[0036] Define the following smooth piecewise function to approximately replace the saturation function, that is:

[0037]

[0038] Obtained from formula (3) and formula (4), \(u = \text{sat}(v)=s(v)+d(v)\), and:

[0039] \(\vert d(v)\vert=\vert\text{sat}(v)-s(v)\vert\leq\max\{u max (1 - \tanh(1)),u min (\tanh(1)-1)\}=D\), where \(D\) represents a positive constant.

[0040] Then there exists a constant \(\lambda\) such that:

[0041]

[0042] where \(v_0\) represents the initial value of the true control input voltage \(v\) of the stator.

[0043]

[0044] When \(v_0 = 0\), we get:

[0045] where, b 1i is a positive constant, \(i = 1,2\). When on the \(q\)-axis, use to represent \(d_1(v)\) to represent \(d(v)\); when on the \(d\)-axis, use to represent \(d_2(v)\) to represent \(d(v)\).

[0046] For the following stochastic system: \(dx = g(x)dt+\mu(x)dw\).

[0047] where \(g(x)\) and \(\mu(x)\) are locally Lipschitz functions, and satisfy \(g(0)=\mu(0) = 0\), where \(g(0)\) represents the initial value of \(g(x)\) and \(\mu(0)\) represents the initial value of \(\mu(x)\).

[0048] Arbitrarily given \(V(x)\in C 2 , C 2 represents the set of complex numbers, define the function differential From the differential rule, we know that:

[0049]

[0050] where \(x\in R n is the state variable of the system, \(R n represents the set of \(n\)-dimensional real vectors, \(w\in R r is a standard Brownian motion, \(Rr denotes the set of r-dimensional real vectors, denotes the correction term, and Tr represents the sum of diagonal elements.

[0051] If there is a function V(x) ∈ C 2 , satisfying:

[0052]

[0053] where β1(|x|) and β2(|x|) are k ∞ class functions, a0 and b0 are positive constants, then when t ≥ t0, V(x) satisfies:

[0054]

[0055] where E[V(t)] is the expectation of V(t), then the signal of the stochastic nonlinear system is bounded in probability, t0 represents the initial time; let f(x) be a continuous function defined on the compact set Ω, there exists a constant δ(z) > 0 and a fuzzy system W T S(x) such that f(x) = W T S(x) + δ(z), and for any ε > 0, there is δ(z) is the approximation error, W is the fuzzy weight vector, S(x) = [p1(x), …, p N (x)] T is the basis function vector, p m (x) is the Gaussian function, that is the center vector η m = [η m1 , η m2 , …, η mN T , ρ m is the width of the Gaussian function, m = 1, …, N.

[0056] Define the following command filter:

[0057]

[0058] where α i is the input signal of the command filter, i = 1, 2; l 11 , l 12 are both the output signals of the command filter, and the initial value of l 11 l 11 (0) = α i (0), α i (0) is the initial value of α i , and the initial value of l 12 l 12 ​(0) = 0; If there exist two constants θ1 > 0, θ2 > 0, for any time t ≥ 0, it can be satisfied simultaneously Then for any There always exists a suitable ω n > 0 and Such that Are all bounded.

[0059] For all |v j | < k bj (t), the following inequality holds:

[0060]

[0061] Where k bj (t) is a time-varying function, j = 1, 2, 3, 4.

[0062] Step 2.1. Based on the dynamic mathematical model of the permanent magnet synchronous motor, design the following fuzzy adaptive backstepping controller based on command filtering: According to the backstepping principle, define the tracking error variable and the compensation error variable as follows:

[0063]

[0064]

[0065] Where v j Represents the compensation error variable, z j Represents the tracking error variable, x d Is the given desired signal, x i,c Is the output signal of the filter, ξ j Is the filter error compensation signal, i = 1, 2, j = 1, 2, 3, 4.

[0066] Define the compact set Ω v = {|v j | < k bj (t)}.

[0067] Step 2.2. Select the barrier Lyapunov function:

[0068]

[0069] Where k b1 (t) = k c1 (t) - A1(t), A1(t) is a variable that satisfies certain conditions, and the conditions satisfied by A1(t) will be given in the stability analysis of step three. Then within the compact set Ω v It can be obtained that:

[0070]

[0071] From Young's inequality, we have:

[0072]

[0073] Design the virtual control function α1 and the filtering error compensation signal ξ1 as:

[0074]

[0075] where k1 > 0,

[0076] Therefore, the formula always holds. From formulas (13) - (15), we get:

[0077]

[0078] Step 2.3. Select the barrier Lyapunov function:

[0079]

[0080] where k b2 (t) = k c2 (t) - x 1,c - A2(t).

[0081] A2(t) is a variable satisfying certain conditions, and the conditions that A2(t) satisfies will be given in the stability analysis of step three. γ2 is a constant, ζ2 = ||W2|| 2 , ||W2|| is the norm of the fuzzy weight vector W2, is the estimated value of ζ2. Similarly, we get:

[0082]

[0083] In the actual system, the load torque T L has an upper limit. The upper limit of the load torque T L is a positive number d, satisfying 0 ≤ |T L | ≤ d.

[0084] From Young's inequality, we have:

[0085]

[0086] where I2 > 0. Through formulas (18) - (19), we obtain:

[0087]

[0088] where, We obtain:

[0089]

[0090] where h2 is a constant, m2 is a constant; ε2 represents an arbitrarily small positive number.

[0091] Construct the virtual control function α2, the filtering error compensation signal ξ2 and the adaptation law as follows:

[0092]

[0093] where k2 > 0, obtained through formulas (20) to (22):

[0094]

[0095] Step 2.4. Let

[0096] where and |d1(v)| ≤ D H , where D H and b 11 are both positive numbers.

[0097] Select the following barrier Lyapunov function:

[0098]

[0099] where k b3 (t) = k c3 (t) - x 2,c - A3(t).

[0100] A3(t) is a variable that satisfies certain conditions, and the conditions that A3(t) satisfies will be given in the stability analysis of Step Three. γ3 is a constant, ζ3 = ||W3|| 2 , ||W3|| is the norm of the vector W3, is an estimate of ζ3; similarly, we get:

[0101]

[0102] From the Young's inequality, we have:

[0103]

[0104] where I3 > 0, obtained through formulas (25) to (26):

[0105]

[0106] where We get:

[0107]

[0108] Among them, h3 is a constant, m3 is a constant, and ε3 represents an arbitrarily small positive number.

[0109] Construct the following actual control law v q and the adaptation law

[0110]

[0111] Among them, the design parameter Take Get:

[0112]

[0113] Step 2.5. Let

[0114] Among them, and |d2(v)| ≤ D d , D d and b 12 are both positive numbers.

[0115] Select the following barrier Lyapunov function:

[0116]

[0117] Among them, k b4 (t) = k c4 (t), ζ4 = ||W4|| 2 , ||W4|| is the norm of the vector W4, γ4 is a constant, is the estimated value of ζ4. Similarly, get:

[0118]

[0119] According to the Young's inequality:

[0120]

[0121] Among them, I4 > 0. Through formulas (32) - (33), get:

[0122]

[0123] Get:

[0124]

[0125] Among them, m4 is a constant, h4 is a constant, and ε4 represents an arbitrarily small positive number.

[0126] Construct the actual control law vd and the adaptive law are as follows:

[0127]

[0128] Among them, the design parameter takes to obtain:

[0129]

[0130] Step 3. Conduct a stability analysis on the command-filtered fuzzy adaptive control method for the permanent magnet synchronous motor stochastic system considering time-varying state constraints and input saturation.

[0131] Select the Lyapunov function of the permanent magnet synchronous motor stochastic system as:

[0132] V = V4 (38)

[0133] When |v j | < k bj , there is Then formula (37) is transformed into:

[0134]

[0135]

[0136]

[0137] From formula (38), for there exists:

[0138]

[0139] Among them, V(0) represents the initial value of V(t).

[0140] According to formulas (12), (17), (24), (31) and (38), it is obtained that:

[0141]

[0142] From equations (40) - (41), it is obtained that:

[0143] Therefore, there is

[0144] Then select the Lyapuno function shown below to prove the boundedness of the compensation signal.

[0145]

[0146] Differentiating formula (43) gives:

[0147]

[0148] where where By Young's inequality, we have:

[0149]

[0150] Then

[0151] where We thus obtain:

[0152]

[0153] From formula (39), we have v j , ζ n are all bounded; from formula (46), we know that ξ j is bounded, so z j is also bounded.

[0154] From v1 = z1 - ξ1, z1 = x1 - x d , we have

[0155] Taking A1(t) ≥ |x d (t)| + |ξ1(t)|, we have |x1| < k c1 (t) - (A1(t) - |x d (t) + ξ1(t)|) < k c1 (t).

[0156] From formula (15), we know that α1 is a function related to z1 and , so α1 has its least upper bound ι1. By we obtain τ1 represents the least upper bound of x 1,c .

[0157] Then, according to v2 = z2 - ξ2, z2 = x2 - x 1,c , we obtain Taking A2(t) > |ξ2|, we have |x2| < k c2 - (A2(t) - |ξ2|) < k c2 ; taking A3(t) > 0, we further obtain |x3| < k c3 ; similarly, |x4| < k c4 .

[0158] From formula (29) and formula (36), we know that vq is a function related to v3 and ζ3, v d is a function related to v4 and ζ4. Therefore, v q and v d are both bounded.

[0159] Therefore, considering the time-varying state constraints and input saturation, all signals in this system are bounded.

[0160] In the method of the present invention, the instruction filtering control technology eliminates the influence of filtering errors by introducing a compensation signal, thus solving the above problems. At present, many existing adaptive control methods are used to solve nonlinear deterministic systems, but the problem of time-varying state constraints in stochastic systems has not been involved. The present invention proposes a reasonable control method. On the other hand, considering the input saturation problem in the design makes the designed system more suitable for engineering applications.

[0161] Next, the fuzzy adaptive control method for the stochastic permanent magnet synchronous motor based on instruction filtering proposed herein is simulated in a virtual environment to verify the feasibility of the control method proposed by the present invention:

[0162] Motor parameters: J = 0.003798 Kg·m 2 , R S = 0.68 Ω, B = 0.001158 N·m / (rad / s), L d = 0.00285 H, L q = 0.00315 H, Φ = 0.1245 H, n p = 3.

[0163] The selected fuzzy sets are: N represents an integer, and l ∈ [-5, 5].

[0164] The initial state of the permanent magnet synchronous motor simulation is selected as [0, 0, 0, -0.1].

[0165] The parameters of the fuzzy adaptive backstepping controller are selected:

[0166] k1 = 110, k2 = 5, k3 = 5, k4 = 2,

[0167] m2 = m3 = m4 = 0.05, γ1 = γ2 = γ3 = 0.02, h2 = h3 = 0.02, h4 = 0.0005,

[0168] k c1 (t) = 0.9sin t + 2.3, k c2 (t) = 0.9cos t + 2.3, k c3 (t) = 6 + 0.3sin 5t, k c4\(\theta(t) = 0.1 + 0.03\cos t\).

[0169] The load torque is: \(T\) L \(= 1.5\ N\cdot m\), and the desired position signal is: \(x\) d \(= \sin t\).

[0170] The simulation results of the fuzzy adaptive control method for the stochastic system of permanent magnet synchronous motor based on instruction filtering are as Figures 2 - 8 shown. Among them, the rotor position signal \(x_1\) and the desired position signal \(x\) d are as Figure 2 shown. The rotor position tracking error \(z_1 = x_1 - x\) d , as Figure 3 shown. From Figure 2 and Figure 3 it can be seen that the output of the system can quickly track the desired signal. From Figure 3 it can be seen that the tracking error of the system converges to a small range near the origin, and the tracking effect of the system is good and the tracking accuracy is high. The \(d\)-axis stator voltage and the \(q\)-axis stator voltage are as Figure 4 and Figure 5 shown. From Figure 4 and Figure 5 it can be seen that after the control method of the present invention, the actual control laws \(u\) d and \(u\) q are restricted within a reasonable region, avoiding damage to the motor caused by excessive starting voltage, and playing a guarantee role in the safe and reliable operation of the system. The state variables \(x_2\), \(x_3\), \(x_4\) of the permanent magnet synchronous motor are respectively as Figures 6 - 8 shown, and it can be seen that the controller can keep the system state within its preset time-varying constraint interval.

[0171] The above simulation results show that the fuzzy adaptive control method for the stochastic system of permanent magnet synchronous motor based on instruction filtering in the present invention can efficiently track the reference signal. Therefore, it has practical implementation significance.

[0172] Of course, the above description is only a preferred embodiment of the present invention. The present invention is not limited to listing the above embodiments. It should be noted that all equivalent substitutions and obvious deformation forms made by any person skilled in the art under the teaching of this specification fall within the substantial scope of this specification and should be protected by the present invention.

Claims

1. A stochastic system instruction filtering fuzzy adaptive control method for a permanent magnet synchronous motor considering time-varying state constraints and input saturation, characterized in that It includes the following steps: Step 1. Establish the d-q axis dynamic mathematical model of the permanent magnet synchronous motor, as shown in formula (1): Among them, θ represents the rotor angle of the motor, ω represents the rotor angular velocity, and both θ and ω are state variables of the system; u d and u q represent the voltages on the d-axis and q-axis respectively, and u d and u q are the input signals of the system; i d and i q are the excitation currents on the d-axis and q-axis respectively; L d and L q are the stator inductances in the d-q coordinate system; J represents the moment of inertia of the motor, B represents the friction coefficient of the motor, T L represents the load torque of the motor, n p represents the number of pole pairs of the motor, Φ represents the magnetic flux generated by the permanent magnet of the motor, and R s represents the stator resistance of the motor; the following variables are defined to simplify the d-q axis dynamic mathematical model of the permanent magnet synchronous motor; Considering random disturbances, the d-q axis dynamic mathematical model of the permanent magnet synchronous motor is as follows: where, ψ2, ψ3, ψ4 are unknown smooth disturbance functions; Step 2. According to the instruction filtering technology and the principle of adaptive backstepping method, design a stochastic system instruction filtering fuzzy adaptive control method for permanent magnet synchronous motor considering time-varying state constraints and input saturation. The control objective is to design the input signals u d and u q such that x1 can track the desired signal x d well, and all states in the system need to satisfy the time-varying constraint condition Γx j ={x j ∈R||x j |<k cj (t)}, where R represents the set of real numbers, k cj (t) is the designed time-varying continuous function, and j = 1, 2, 3, 4; For Equation (2), consider the input signal u of the system d and u q affected by the saturation non-linearity, and use u to represent u d and u q ; where, v is the true control input, u is the control input under actual application, u max > 0 and u min < 0 are unknown saturation constants; the following smooth piecewise function is defined to approximately replace the saturation function, i.e.: Obtained from formula (3) and formula (4), u = sat(v) = s(v) + d(v), and: |d(v)| = |sat(v) - s(v)| ≤ max{u max (1 - tanh(1)), u min (tanh(1) - 1)} = D, where D represents a positive constant; Then there exists a constant λ such that: where, v0 represents the initial value of the true control input voltage v of the stator; v λ = λv+(1 - λ)v0, 0 < λ < 1; When v0 = 0, we get: Among them, b 1i is a positive constant, i = 1, 2; When on the q-axis, use to denote d1(v) to denote d(v); when on the d-axis, use to denote d2(v) to denote d(v); for the following stochastic system: dx = g(x)dt + μ(x)dw; where, g(x) and μ(x) are locally Lipschitz functions, and satisfy g(0) = μ(0) = 0, where, g(0) represents the initial value of g(x), and μ(0) represents the initial value of μ(x); Given any \(V(x)\in\mathbb{C}\) 2 , \(\mathbb{C}\) 2 denotes the set of complex numbers. Define the derivative of the function By the derivative rule, we know that: where \(x\in R\) n is the state variable of the system, and \(R\) n denotes the set of \(n -\)dimensional real vectors, \(w\in R\) r is a standard Brownian motion, and \(R\) r denotes the set of \(r -\)dimensional real vectors, denotes the correction term, and \(Tr\) represents the sum of diagonal elements; If there is a function \(V(x)\in C\) 2 , which satisfies: where, β1(|x|) and β2(|x|) are k ∞ -type functions, a0 and b0 are positive constants, then when t ≥ t0, V(x) satisfies: E[V(t)] is the expectation of V(t), then the signal of the stochastic nonlinear system is bounded in probability, and t0 represents the initial time; Let \(f(x)\) be a continuous function defined on a compact set \(\Omega\). There exists a constant \(\delta(z)>0\) and a logical system \(W\) T \(S(x)\) such that \(f(x)=W\) T \(S(x)+\delta(z)\), and for any \(\varepsilon > 0\), there is \(\delta(z)\) is the approximation error, \(W\) is the fuzzy weight vector, \(S(x)=[p_1(x),\ldots,p\) N (x)] T is the basis function vector; \(p\) m (x) is a Gaussian function, that is \(\eta\) m is the center vector, \(\rho\) m is the width of the Gaussian function, \(m = 1,\ldots,N\); Define the following command filter: Among them, α i is the input signal of the instruction filter, i = 1, 2; l 11 , l 12 are both the output signals of the instruction filter, and the initial value of l 11 l 11 (0) = α i (0), α i (0) is the initial value of α i , the initial value of l 12 l 12 (0) = 0; If there exist two constants θ1 > 0, θ2 > 0, for any time t ≥ 0, it can be satisfied simultaneously Then for any there always exists a suitable ω n > 0 and such that are all bounded; For all |v j | < k bj (t), the following inequality holds: where k bj (t) is a time-varying function, and j = 1, 2, 3, 4; Step 2.

1. Based on the dynamic mathematical model of the permanent magnet synchronous motor, design the following command-filter-based fuzzy adaptive backstepping controller: According to the backstepping principle, define the tracking error variable and the compensation error variable as follows: where, v j represents the compensation error variable, z j represents the tracking error variable, x d is the given desired signal, x i,c is the output signal of the filter, ξ j is the filter error compensation signal, i = 1, 2, j = 1, 2, 3, 4; Define the compact set Ω v = {|v j | < k bj (t)}; Step 2.

2. Select the barrier Lyapunov function: where k b1 (t) = k c1 (t) - A1(t), where A1(t) is a variable satisfying certain conditions, and the conditions satisfied by A1(t) will be given in the stability analysis of step three. Then, in the compact set Ω v we obtain: Obtained from the Young's inequality: Design the virtual control function α1 and the filtering error compensation signal ξ1 as: where, k1 > 0, ▽ > 0; Therefore, the formula always holds. From formulas (13) to (15), we get: Step 2.

3. Select the barrier Lyapunov function: where k b2 (t) = k c2 (t) - x 1,c - A2(t); A2(t) is a variable that satisfies certain conditions, and the conditions satisfied by A2(t) will be given in the stability analysis of Step 3. γ2 is a constant, and ζ2 = ||W2|| 2 , where ||W2|| is the norm of the fuzzy weight vector W2, is an estimated value of ζ2, and we obtain: In an actual system, the load torque T L has an upper limit, and the upper limit of the load torque T L is a positive number d, satisfying 0 ≤ |T L | ≤ d; Obtained from the Young's inequality: where, I2 > 0, obtained from formula (18) - formula (19): Among them, Obtained: where, h2 is a constant, m2 is a constant; ε2 represents an arbitrarily small positive number; Construct the virtual control function α2, the filtering error compensation signal ξ2, and the adaptation law as follows: where, k2 > 0, obtained from formula (20) - formula (22): Step 2.

4. Let Among them, and |d1(v)| ≤ D H , where D H and b 11 are both positive numbers; Select the following barrier Lyapunov function: where k b3 (t) = k c3 (t) - x 2,c - A3(t); A3(t) is a variable that satisfies certain conditions, and the conditions that A3(t) satisfies will be given in the stability analysis of Step 3. γ3 is a constant, and ζ3 = ||W3|| 2 , where ||W3|| is the norm of the vector W3, is the estimated value of ζ3; we obtain: Obtained from the Young's inequality: where, I3 > 0, obtained from formula (25) - formula (26): Among them, Obtained: where, h3 is a constant, m3 is a constant, ε3 represents an arbitrarily small positive number; Construct the following actual control law \(v\) q and the adaptation law Among them, the design parameters take to obtain: Step 2.

5. Let Among them, and |d2(v)| ≤ D d , D d and b 12 are both positive numbers; Select the following barrier Lyapunov function: where k b4 (t) = k c4 (t), ζ4 = ||W4|| 2 , ||W4|| is the norm of the vector W4, γ4 is a constant, is an estimated value of ζ4, and we obtain: Obtained from the Young's inequality: where, I4 > 0, obtained from formula (32) - formula (33): Obtained: where, m4 is a constant, h4 is a constant, ε4 represents an arbitrarily small positive number; Construct the true control law v d and the adaptation law as follows: Among them, the design parameters take to obtain: Step 3. Conduct a stability analysis on the command-filter-based fuzzy adaptive control method for the permanent magnet synchronous motor stochastic system considering time-varying state constraints and input saturation; Select the Lyapunov function of the permanent magnet synchronous motor stochastic system as: V = V4 (38) When |v j | < k bj (t), there is Then formula (37) is transformed into: From Equation (38), for there exists: where, V(0) represents the initial value of V(t); Obtained from formula (12), formula (17), formula (24), formula (31) and formula (38): Obtained from formula (40) - (41): Therefore, there is Then select the following Lyapunov function to prove the boundedness of the compensation signal; Take the derivative of formula (43): Among them, Among them, Obtained from the Young's inequality: Then Among them, Thus, it is obtained that: Obtained from formula (39) v j , ζ n are all bounded; As can be seen from formula (46), ξ j is bounded. Therefore, z j is also bounded; From v1 = z1 - ξ1, z1 = x1 - x d , we have Take \(A1(t)\geq|x d (t)| + |\xi1(t)|\), and \(|x1|\lt k c1 (t)-(A1(t)-|x d (t)+\xi1(t)|)\lt k c1 (t); As known from formula (15), α1 is a function related to z1 and , so α1 has its least upper bound ι1. By obtained τ1 represents the least upper bound of x 1,c ; Then, according to v2 = z2 - ξ2 and z2 = x2 - x 1,c , we obtain Taking A2(t) > |ξ2|, we have |x2| < k c2 -(A2(t) - |ξ2|) < k c2 ; taking A3(t) > 0, we further obtain |x3| < k c3 ; |x4| < k c4 ; From formulas (29) and (36), it can be seen that v q is a function related to v3 and ζ3, and v d is a function related to v4 and ζ4. Therefore, v q and v d are both bounded; Therefore, considering time-varying state constraints and input saturation, all signals in this system are bounded.

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