Mechanical resonance suppression method of AC servo system based on improved BFGS adaptive notch filter

Abstract

The present invention discloses a method for suppressing mechanical resonance of an AC servo system based on an improved BFGS adaptive notch filter. By improving the BFGS iterative optimization algorithm, the notch filter parameters can be automatically adjusted to achieve online suppression of mechanical resonance, and the method comprises the following steps: establishing a mechanical resonance model, adding a notch filter behind a speed regulator of a servo system, minimizing the cost function of the speed error by using the improved BFGS iterative optimization algorithm, and achieving adaptive adjustment of the notch filter parameters, thereby completing online suppression of mechanical resonance. When the mechanical resonance frequency changes or is inconsistent with the center point frequency of the notch filter, the traditional adaptive notch filter will cause system instability. Compared with the traditional adaptive notch filter, the present invention effectively avoids the occurrence of such a situation, can quickly adjust the notch filter parameters, suppress mechanical resonance in real time, and improve the stability and robustness of the servo system.

Description

Technical Field

[0001] The invention relates to the technical field of AC servo control systems, and in particular to a mechanical resonance suppression method for an AC servo system based on an improved BFGS adaptive notch filter. Background Art

[0002] AC servo systems are used in many fields due to their good stability, fast speed, and high precision. However, mechanical resonance will limit the gain and bandwidth of the servo system, which will in turn affect the control accuracy and control performance of the system, damage the mechanical device, and even cause safety hazards. The traditional method of suppressing mechanical resonance is to improve the mechanical structure so that the resonant frequency is much larger than the system bandwidth, but the cost is too high and the implementation is difficult. In recent years, the main means of suppressing mechanical resonance include active suppression and passive suppression. At present, the most commonly used method is passive suppression, which adds a notch filter to the system speed loop to reduce the system gain at the resonant frequency.

[0003] In order to effectively suppress mechanical resonance, it is particularly important to properly set the center point frequency, depth and width of the notch filter in offline conditions. However, in actual engineering, due to the influence of complex working conditions such as changes in electrical parameters, load inertia and torque time variation, the parameters of the notch filter set offline cannot match its inherent characteristics in real time, and the suppression effect is not good, and the resonance may even become more serious. Therefore, in order to improve the servo control performance and intelligence level, it is necessary to self-tune the notch filter parameters. In the prior art, an adaptive frequency estimation algorithm based on interpolation fast Fourier transform and improved ANF is adopted to realize the rapid estimation of the resonant signal, but it depends on the speed of FFT frequency detection. In the prior art, an improved enhanced phase-locked loop is proposed to improve the application of adaptive notch filter in damped oscillation analysis, estimate the amplitude, phase angle, frequency and damping factor, and the simulation results verify the analytical derivation and performance, but the accuracy needs to be improved. The prior art adopts a dynamic behavior that is proportional to the difference between the estimated frequency and the actual frequency, which allows the selection of estimated parameters that are independent of the resonant frequency value. The improved algorithm is effective in suppressing low-frequency and high-frequency resonances, but the algorithm convergence speed needs to be improved.

[0004] In view of the shortcomings of the above suppression methods, the present invention proposes to adopt an AC servo system mechanical resonance suppression method based on an improved BFGS adaptive notch filter. Summary of the invention

[0005] The technical problem to be solved by the present invention is to provide a method for suppressing mechanical resonance of an AC servo system based on an improved BFGS adaptive notch filter in view of the defects in the prior art. By improving the BFGS iterative optimization algorithm, the notch filter parameters can be automatically adjusted to achieve online suppression of mechanical resonance with the purpose of minimizing the cost function of the speed error.

[0006] The technical solution adopted by the present invention to solve its technical problem is:

[0007] The present invention provides a method for suppressing mechanical resonance of an AC servo system based on an improved BFGS adaptive notch filter, the method comprising the following steps:

[0008] S1, obtaining the operating parameters of the servo system and establishing a mechanical resonance model for the servo system;

[0009] S2. Add a three-parameter notch filter after the speed regulator of the servo system and perform digital processing on it. The notch filter is designed using a bilinear variation method.

[0010] S3. Sample the input and output of the servo system to obtain a cost function consisting of speed error and notch filter parameters. An improved BFGS iterative optimization algorithm is proposed to minimize the cost function of the speed error, realize automatic adjustment of notch filter parameters, and complete online suppression of mechanical resonance.

[0011] Furthermore, the specific method in step S1 of the present invention is:

[0012] The mechanical part of the servo system with elastic load includes: servo motor, transmission device and load, in which the transmission device transmits the movement and power of the servo motor to the load, thereby controlling the movement of the load to meet the requirements of the system;

[0013] The mechanical equation of the servo motor is:

[0014] Among them, J m is the motor moment of inertia, ω m is the motor speed, T e is the electromagnetic torque, T l is the load torque, B m is the damping coefficient, t is the time;

[0015] The mechanical equations are simplified and the motor side transfer function is calculated:

[0016]

[0017] Among them, K s is the transmission shaft stiffness coefficient, J L is the load moment of inertia, s is the Laplace operator;

[0018] Mechanical resonance is related to the stiffness coefficient of the transmission shaft. When the system bandwidth continues to increase and reaches or even exceeds the natural frequency of the transmission shaft, the motor-driven load will produce mechanical resonance.

[0019] Furthermore, the specific method in step S2 of the present invention is:

[0020] In view of the inherent characteristics of the mechanical resonance equation, the most commonly used method is to introduce a notch filter after the speed regulator to compensate and correct the current loop input torque. In order to be able to control the notch depth and notch width separately, a three-parameter notch filter is used, and its transfer function is in the form of:

[0021]

[0022] Where d is the notch width, p is the notch depth, ω n Represents the angular frequency of the center point of the notch filter;

[0023] The digital notch filter is designed by using the bilinear variation method, and s = (2 / T)·(1-z -1 ) / (1+z -1 ), T is the sampling time, and the notch filter expression after discretization is:

[0024] H(z)=(x1+x2z -1 +x3z -2 ) / (1+x4z -1 +x5z -2 )(1)

[0025] in,

[0026]

[0027]

[0028]

[0029]

[0030] Where z represents the difference factor.

[0031] Furthermore, the specific method in step S3 of the present invention is:

[0032] The cost function composed of speed error and notch filter parameters in iterative tuning is:

[0033]

[0034] Where, k is the number of iterations; e(k) is the servo motor speed error;

[0035] The improved BFGS is used to iteratively optimize the cost function, and the parameter x update formula of the notch filter is:

[0036] x(k+1)=x(k)-α(k)·B -1 (k)·g(k) (3)

[0037] Wherein, x = [x1 x2 x3 x4 x5]; x(k+1) and x(k) are the parameters of the notch filter of the k+1th iteration and the kth iteration respectively; α(k) is the step size of the kth iteration; B(k) is the Hessian matrix approximation; g(k) is the gradient value of the cost function f(k) with respect to the notch filter parameter x(k);

[0038] New secant line equation:

[0039] In the formula,

[0040]

[0041] Wherein, s(k)=x(k+1)-x(k) is the increment of the notch filter parameter at each iteration; y(k)=g(k+1)-g(k) is the increment of the gradient value;

[0042] The improved BFGS algorithm is proposed in formula (4). It is known that the initial Hessian matrix B0 is symmetric and positive definite. In order to maintain the positive definiteness of the update, the update of the Hessian matrix is ​​as follows:

[0043]

[0044] Among them, δ is a positive real number;

[0045] According to the iteratively updated Hessian matrix B(k+1), the notch filter parameters are adjusted to obtain x(k+1), so as to minimize the cost function and achieve online suppression of mechanical resonance.

[0046] Furthermore, the specific steps of improving the secant equation of the BFGS algorithm in step S3 of the present invention are:

[0047] The new secant equation uses three iteration parameters x(k-1), x(k) and x(k+1), so the following chain rule exists:

[0048]

[0049] in, is the gradient value of the cost function f(k+1) with respect to x(k+1);

[0050] is the gradient value of the cost function f(k-1) with respect to x(k-1); X(t) is an interpolation curve, and X(-1) = x(k-1), X(0) = x(k) and X(1) = x(k+1); is the gradient of g(k) with respect to the notch filter parameter x(k);

[0051] For X(t) and Using the Lagrange interpolation method, we can get:

[0052]

[0053]

[0054] Among them, L j (t), j = 0, 1, 2 is the Lagrange polynomial:

[0055]

[0056] Let t0 = -1, t1 = 0 and t2 = 1 to obtain:

[0057] L1(t)=1-t 2 ,

[0058] Substitute equation (11) into equation (9), and then take the derivatives on both sides to obtain:

[0059]

[0060] Substituting formula (10) into formula (8), we get:

[0061]

[0062] Rearranging formula (13) yields:

[0063]

[0064] Among them, B(k-1), B(k) and B(k+1) are and Substituting equations (11) and (12) into equation (14) yields the new secant equation (4).

[0065] Furthermore, step S3 of the present invention also includes a method for simplifying and improving the calculation amount of the BFGS algorithm, and the specific steps are:

[0066] It is known that there are a lot of matrix-vector multiplication operations in formula (6). An effective method is proposed to reduce the computational cost and determine

[0067] Multiply both sides of equation (6) by s T (k-1), we get:

[0068]

[0069]

[0070] It is known that B(k)·s(k)=-α(k)·g(k), and B(k) satisfies B(k)·s(k-1)=y(k-1). Formula (15) can be rewritten as:

[0071]

[0072]

[0073] From formula (16), we can get The approximate value of can effectively reduce the amount of calculation, so formula (6) is modified as follows:

[0074]

[0075] Where u is an arbitrary real vector. To ensure s T (k)·u≠0.

[0076] Furthermore, step S3 of the present invention also includes a method for calculating the gradient of the cost function to the notch filter parameter, and the specific steps are:

[0077] In the servo system, the input signal is the given speed value ω r , after passing through the speed regulator, it becomes the notch filter input value T m , the speed regulator expression is:

[0078] T m (k) = T m (k-1)+k p ·(e(k)-e(k-1))+k i ·e(k) (19)

[0079] Among them, k p , k i They are the proportional and integral parameters of the PI speed regulator respectively;

[0080] According to equations (1) and (19), the notch filter output T is calculated e , as the input T of the speed loop controlled object e , recorded as:

[0081] T e (k) = x(k)·T n (k) (20)

[0082] Wherein, x(k) = [x1(k) x2(k) x3(k) x4(k) x5(k)] is the parameter of adjusting the notch filter for the kth iteration; T n (k) = [T e (k-1) T e (k-2) T m (k) Tm (k-1) T m (k-2)];

[0083] Calculate the gradient of the cost function with respect to the notch filter parameters:

[0084]

[0085] in, By symbolic function Approximate processing.

[0086] The beneficial effects produced by the present invention are:

[0087] 1. When the model structure of the controlled object is known, the improved BFGS algorithm is adopted. The three-iteration parameters x(k-1), x(k) and x(k+1) are used in the secant equation to iteratively update the Hessian matrix, and then the notch filter parameters are adaptively adjusted. The algorithm has strong real-time performance and high identification accuracy.

[0088] 2. Compared with the traditional adaptive notch filter frequency estimation algorithm, the present invention directly iteratively adjusts the notch filter parameters by minimizing the cost function of the speed error to achieve online suppression of mechanical resonance, and does not require real-time detection of the resonance frequency value;

[0089] 3. The present invention has a suppressive effect on both high-frequency and low-frequency mechanical resonances. When the frequency changes, the algorithm can quickly adjust the notch filter parameters so that it can suppress the resonance in time and ensure the stability of the servo system. BRIEF DESCRIPTION OF THE DRAWINGS

[0090] The present invention will be further described below with reference to the accompanying drawings and embodiments, in which:

[0091] Figure 1 It is a flow chart of an improved BFGS iterative optimization algorithm according to an embodiment of the present invention.

[0092] Figure 2 It is a schematic diagram of mechanical resonance suppression of an AC servo system based on an improved BFGS adaptive notch filter according to an embodiment of the present invention.

[0093] Figure 3 The invention is a flow chart of mechanical resonance suppression of an AC servo system based on an improved BFGS adaptive notch filter according to an embodiment of the present invention. DETAILED DESCRIPTION

[0094] In order to make the purpose, technical solution and advantages of the present invention more clearly understood, the present invention is further described in detail below in conjunction with the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are only used to explain the present invention and are not used to limit the present invention.

[0095] Figure 1This is a flowchart of the improved BFGS iterative optimization algorithm according to an embodiment of the present invention. After the algorithm is initialized, the input and output of the system are sampled to obtain the cost function and determine the algorithm termination condition || g k Whether ||<0.5 is satisfied, if not, the termination condition is satisfied, the search direction and step size are calculated, the Hessian matrix is ​​updated, and the iterative adjustment of the parameters is completed.

[0096] Figure 2 The present invention is a schematic diagram of mechanical resonance suppression of an AC servo system based on an improved BFGS adaptive notch filter. After initializing the notch filter parameters, the notch filter parameters are iteratively updated according to the input and output data of the servo system by minimizing the cost function of the speed error through the improved BFGS algorithm, thereby achieving online suppression of mechanical resonance and meeting the requirements of transient response and anti-interference capability of the AC servo system.

[0097] Figure 3 This is a flow chart of mechanical resonance suppression of an AC servo system based on an improved BFGS adaptive notch filter according to an embodiment of the present invention, which mainly includes the following steps:

[0098] first step,

[0099] The digital notch filter is designed by using the bilinear variation method, and s = (2 / T)·(1-z -1 ) / (1+z -1 ), T is the sampling time, and the notch filter expression after discretization is:

[0100]

[0101] Among them, the five parameters x1, x2, x3, x4, and x5 in the notch filter are the adjustment output values ​​of the improved BFGS iterative optimization algorithm. e is the notch filter output, T m is the notch filter input, and k is the number of iterations.

[0102] In the servo system, the input signal is a given speed value ω r , after passing through the speed regulator, it becomes the notch filter input value T m , the speed regulator expression is:

[0103] T m (k) = T m (k-1)+k p ·(e(k)-e(k-1))+k i ·e(k) (2)

[0104] Among them, k p , k i are the proportional and integral parameters of the PI speed regulator respectively, and e(k) is the speed error of the servo motor.

[0105] According to equations (1) and (2), the notch filter output T can be calculated: e , as the input of the speed loop controlled object, recorded as:

[0106] T e (k) = x(k)·T n (k) (3)

[0107] Wherein, x(k) = [x1(k) x2(k) x3(k) x4(k) x5(k)] is the parameter of adjusting the notch filter for the kth iteration; T n (k) = [T e (k-1) T e (k-2) T m (k) T m (k-1) T m (k-2)].

[0108] Thus, for each input ω r The corresponding actual output T e and ω m , and in the servo system, the output should change synchronously with the input value, that is, the expected output is ω m , so the speed error is e = ω r -ω m .

[0109] Step 2:

[0110] The input and output of the servo system are sampled to obtain a cost function consisting of velocity error and notch filter parameters:

[0111]

[0112] The improved BFGS is used to iteratively optimize the cost function, and the parameter x update formula of the notch filter is:

[0113] x(k+1)=x(k)-α(k)·B -1 (k)·g(k) (5)

[0114] Among them, x(k+1) and x(k) are the parameters of the notch filter of the k+1th iteration and the kth iteration respectively; α(k) is the step size of the kth iteration; B(k) is the Hessian matrix approximation; g(k) is the gradient value of the cost function with respect to the notch filter parameter x(k).

[0115] Calculate the gradient of the cost function with respect to the notch filter parameters:

[0116]

[0117] in, The symbolic function Approximate processing.

[0118] Step 3.

[0119] Use the new secant equation to complete the Hessian matrix B k The improved BFGS iterative optimization algorithm uses three iteration parameters x(k-1), x(k) and x(k+1) for the secant equation, so there is the following chain rule:

[0120]

[0121] in, is the gradient value of the cost function f(k+1) with respect to the parameter x(k+1); is the gradient value of the cost function f(k-1) with respect to the parameter x(k-1); X(t) is the interpolation curve, and X(-1) = x(k-1), X(0) = x(k), X(1) = x(k+1); is the gradient of g(k) with respect to the notch filter parameter x(k).

[0122] For X(t) and Using Lagrange interpolation method we get:

[0123]

[0124]

[0125] Among them, L j (t), j = 0, 1, 2 is the Lagrange polynomial:

[0126]

[0127] Let t0 = -1, t1 = 0 and t2 = 1 to obtain:

[0128] L1(t)=1-t 2 ,

[0129] Substitute equation (10) into equation (8), and then take the derivatives on both sides to obtain:

[0130]

[0131] Substituting formula (9) into formula (7), we get:

[0132]

[0133] Rearranging formula (12) yields:

[0134]

[0135] Among them, B(k-1), B(k) and B(k+1) are and

[0136] Approximate value of .

[0137] Substituting equations (10) and (11) into equation (13) and performing some algebraic calculations, we can obtain:

[0138] New secant line equation:

[0139] In the formula,

[0140]

[0141] Among them, s(k)=x(k+1)-x(k) is the increment of the notch filter parameter in each iteration; y(k)=g(k+1)-g(k) is the increment of the gradient value of the cost function to the notch filter parameter.

[0142] Formula (16) involves a large number of matrix-vector multiplication operations. Here we propose an effective method to reduce the computational cost and determine the new

[0143] Multiply both sides of equation (16) by s T (k-1), we can get:

[0144]

[0145]

[0146] It is known that B(k)·s(k)=-α(k)·g(k), B(k) satisfies B(k)·s(k-1)=y(k-1), formula (17)

[0147] can be rewritten as:

[0148]

[0149] From formula (18), we can get the new

[0150] in, Given by formula (19); u is an arbitrary real vector. To ensure s T (k)·u≠0.

[0151] Obviously, the computational cost of formula (20) is much lower than that of formula (16), especially for relatively large Hessian matrices, so the new secant equation can be obtained:

[0152]

[0153] in, Given by formula (15); It is given by formula (19).

[0154] It is known that the initial Hessian matrix B0 is symmetric and positive definite. In order to maintain the positive definiteness of the update, the update of the Hessian matrix is:

[0155]

[0156] Here, δ is a positive real number.

[0157] After updating the Hessian matrix by equation (22), equation (5) is used to iteratively adjust the parameters of the notch filter to achieve online suppression of mechanical resonance.

[0158] The method of the present invention utilizes the mechanical resonance suppression of the improved BFGS iterative optimization adaptive notch filter to replace the mechanical resonance suppression of the traditional adaptive notch filter frequency estimation method, so that the parameters of the notch filter can be automatically adjusted without detecting the frequency value of the resonance, thereby realizing online suppression of the mechanical resonance, and has the advantages of simple control structure, strong anti-interference ability and fast speed response.

[0159] It should be understood that those skilled in the art can make improvements or changes based on the above description, and all these improvements and changes should fall within the scope of protection of the appended claims of the present invention.

Claims

1. A method for suppressing mechanical resonance of an AC servo system based on an improved BFGS adaptive notch filter, characterized in that: The method comprises the following steps: S1, obtaining the operating parameters of the servo system and establishing a mechanical resonance model for the servo system; S2. Add a three-parameter notch filter after the speed regulator of the servo system and perform digital processing on it. The notch filter is designed using a bilinear variation method. S3, sampling the input and output of the servo system, deriving a cost function composed of speed error and notch filter parameters, proposing an improved BFGS iterative optimization algorithm to minimize the cost function of speed error, realizing automatic adjustment of notch filter parameters, and completing online suppression of mechanical resonance; The specific method in step S2 is: The digital notch filter is designed using the bilinear variation method. , is the sampling time, and the notch filter expression after discretization is: （1） Among them, z represents the difference factor, and the five parameters in the notch filter are , , , , That is, to improve the adjusted output value of the BFGS iterative optimization algorithm; in, is the notch filter output, is the notch filter input, is the number of iterations, is the notch width, is the notch depth, Represents the angular frequency of the center point of the notch filter; The specific method in step S3 is: The input and output of the servo system are sampled to obtain a cost function consisting of velocity error and notch filter parameters: （2） in, is the number of iterations; is the servo motor speed error; The improved BFGS is used to iteratively optimize the cost function and the parameters of the notch filter The update formula is: （3） in, ; and Respectively Iteration and Parameters of the iterative notch filter; For the The step size of the iteration; is the Hessian matrix approximation; The cost function Notch filter parameters The gradient value of Calculate the gradient of the cost function with respect to the notch filter parameters: （4） in, By symbolic function Approximate processing; Complete the Hessian matrix using the new secant equation The improved BFGS iterative optimization algorithm uses three iterative parameters for the secant equation. , and , so the following chain rule exists: （5） in, The cost function Parameters The gradient value of The cost function Parameters The gradient value of is the interpolation curve, and ; for Notch filter parameters The gradient value of .

2. The AC servo system mechanical resonance suppression method based on the improved BFGS adaptive notch filter according to claim 1 is characterized in that: The specific method in step S1 is: The mechanical part of the servo system with elastic load includes: servo motor, transmission device and load, in which the transmission device transmits the movement and power of the servo motor to the load, thereby controlling the movement of the load to meet the requirements of the system; The mechanical equation of the servo motor is: in, is the motor moment of inertia, is the motor speed, is the electromagnetic torque, is the load torque, is the damping coefficient, t is the time; The mechanical equations are simplified and the motor side transfer function is calculated: in, is the transmission shaft stiffness coefficient, is the load moment of inertia, is the Laplace operator; Mechanical resonance is related to the stiffness coefficient of the transmission shaft. When the system bandwidth continues to increase and reaches or even exceeds the natural frequency of the transmission shaft, the motor-driven load will produce mechanical resonance.

3. The AC servo system mechanical resonance suppression method based on the improved BFGS adaptive notch filter according to claim 1 is characterized in that: The specific steps of improving the secant equation of the BFGS algorithm in step S3 are: right and Using the Lagrange interpolation method, we can get: （6） （7） in, is a Lagrange polynomial: ， ， make , and have to: ， ， （8） Substitute equation (8) into equation (6), and then take the derivatives on both sides to obtain: （9） Substituting formula (7) into formula (5), we get: （10） Rearranging formula (10) yields: （11） in, , and for , and Substituting equations (8) and (9) into equation (11) yields a new secant equation.

4. The AC servo system mechanical resonance suppression method based on the improved BFGS adaptive notch filter according to claim 3 is characterized in that: The step S3 also includes a method for simplifying and improving the calculation amount of the BFGS algorithm, and the specific steps are: New secant line equation: (12) In the formula, (13) （14） in, is the increment of the notch filter parameter for each iteration; is the increment of the gradient value of the cost function to the notch filter parameter; It is known that there are a lot of matrix-vector multiplication operations in equation (14). An effective method is proposed to reduce the computational cost and determine ; Multiply both sides of equation (14) by ,get: （15） Known , satisfy , formula (15) can be rewritten as: （16） （17） From formula (16), we can get The approximate value of and effectively reduce the amount of calculation, so the formula (6) is modified as follows: （18） in, is any real vector, in order to ensure .

5. The AC servo system mechanical resonance suppression method based on the improved BFGS adaptive notch filter according to claim 1 is characterized in that: The step S3 also includes a method for calculating the gradient of the cost function to the notch filter parameters, and the specific steps are: In the servo system, the input signal is the given speed value. , after passing through the speed regulator, it becomes the notch filter input value , the speed regulator expression is: （19） in, , They are the proportional and integral parameters of the PI speed regulator respectively; According to equations (1) and (19), the notch filter output is calculated as , as the input of the speed loop controlled object , recorded as: （20） in, For the The parameters of the notch filter are adjusted in the iteration; ; Calculate the gradient of the cost function with respect to the notch filter parameters: （21） in, By symbolic function Approximate processing.

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