A command correction method considering frequency domain characteristics of servo control system

Abstract

The invention discloses an instruction correction method considering the frequency domain characteristics of a servo control system. The method comprises the following steps: firstly, the instruction is decomposed into a combination of a Fourier fitting term and a remainder term; then, the remainder term is decomposed into a basic mode component term and a new remainder term by using empirical mode decomposition; then, the Fourier fitting term is corrected, and a frequency response curve of the servo control system from the instruction position to the motion output is established; the amplitude and phase of the sine and cosine signals of each frequency in the correction are based on the frequency characteristics of the servo control system, and the attenuation and lag of each frequency component in the servo control system are offset; then, the instantaneous amplitude and instantaneous phase of each basic mode component are solved by using Hilbert transform, and the attenuation and lag of the servo control system are offset; finally, the corrected Fourier series fitting term, the basic mode component term and the remainder term are used as a new instruction as input; the invention not only meets the higher tracking error control requirements, but also ensures the processing efficiency, and realizes the improvement of the tracking performance of the servo control system.

Description

Technical Field

[0001] The invention belongs to the technical field of servo control systems, and in particular relates to an instruction correction method taking into account frequency domain characteristics of a servo control system. Technical Background

[0002] Servo control systems are widely used in high-precision motion control in CNC machine tools, 3D printing, robotics and other fields. The main goal of servo control systems is fast and accurate tracking. Following error is the most fundamental indicator for measuring the performance of servo control systems. There are many advanced control methods for servo tracking. Since the 1980s, researchers have been committed to continuously improving the bandwidth of control systems and reducing phase lag, hoping to solve the following error once and for all. Various control strategies have emerged one after another, including representative ones such as zero phase difference feedforward control, H ∞ Control, sliding film control, etc., their emergence has greatly improved the servo bandwidth; in addition, notch filters, modal filters, etc. are combined with traditional PID three-loop control to suppress the mechanical modes that limit the increase of servo bandwidth, further improve the servo bandwidth, and reduce following errors.

[0003] In addition to starting from the servo itself, instruction optimization is also an important research direction. Professor Weck of Aachen University of Technology in Germany first wrote an article to explain the impact of acceleration and deceleration on following error. Since then, researchers have carried out a lot of research work, such as tool path smoothing, acceleration and deceleration strategies based on jerk control, and feed rate planning. Their common point is to smooth the instructions or reduce the instruction acceleration and jerk to reduce the instruction bandwidth, make the instructions easier to respond, and thus reduce the following error.

[0004] As the requirements for servo control system performance in fields such as precision industrial instruments, robots, and military industry become increasingly higher, the improvement of servo control bandwidth tends to be saturated due to limitations in mechanical characteristics, and command smoothing is always accompanied by a reduction in speed at the expense of processing efficiency.

[0005] In summary, the ceilings of servo bandwidth improvement and instruction optimization have already appeared. In order to meet the higher tracking error control requirements in the future while ensuring processing efficiency, it is necessary to explore new control strategies. Summary of the invention

[0006] In order to overcome the shortcomings of the above-mentioned prior art, the purpose of the present invention is to provide a command correction method that takes into account the frequency domain characteristics of the servo control system, which can meet higher tracking error control requirements while ensuring processing efficiency and improving the tracking performance of the servo control system.

[0007] In order to achieve the above object, the technical solution adopted by the present invention is:

[0008] A command correction method considering the frequency domain characteristics of a servo control system comprises the following steps:

[0009] 1) Decompose the command into simple frequency components: Let x(t) be the input command of the servo control system, perform Fourier series fitting on the input command of the servo control system, and decompose the command into a combination of Fourier fitting term F(t) and a remainder term R(t), where F(t) is a combination of multiple sine and cosine waves with different frequencies, and R(t) is the error between the fitting term and the command. The command is represented as shown in formula (1);

[0010] x(t)=F(t)+R(t) (1)

[0011] 2) Decompose the remainder R(t): Use empirical mode decomposition to decompose the remainder R(t) into a basic mode component term H(t) and a new remainder r(t), where H(t) is the linear sum of multiple basic mode components, and r(t) is a monotonic component or a component with a very small value and does not require additional processing. The instruction representation is shown in formula (2);

[0012] x(t)=F(t)+H(t)+r(t) (2)

[0013] 3) Correction of the Fourier fitting term F(t): Establish the frequency response curve of the servo control system from the command position to the motion output. In order to make the amplitude and phase of each frequency of the input and output of F(t) in the servo control system the same, correct the amplitude and phase of the sine and cosine signals of each frequency in F(t) according to the frequency characteristics of the servo control system, offset the attenuation and lag of each frequency component in F(t) by the servo control system, and record the corrected F(t) as F′(t);

[0014] 4) Correction of the basic mode component term H(t): Using Hilbert transform, solve the instantaneous amplitude and instantaneous phase of each basic mode component; similarly, in order to make the instantaneous amplitude and phase of the input and output of H(t) in the servo control system the same, use the amplitude attenuation and phase lag corresponding to the instantaneous frequency to correct the instantaneous amplitude and instantaneous phase of each basic mode component, offset the attenuation and lag of H(t) by the servo control system, and record the corrected H(t) as H′(t);

[0015] 5) Reconstruct the command: The revised F′(t), H′(t) and r(t) are combined into a new command as the input of the servo control system. The revised command is recorded as x′(t), as shown in formula (3);

[0016] x'(t)=F'(t)+H'(t)+r(t) (3).

[0017] In the step 1), the input command of the servo control system is assumed to be x(t), and its n-th order Fourier series fitting expression is shown in formula (4), where a0 is a constant term, a m and b m is the expansion coefficient of the mth harmonic, ω is the fundamental frequency;

[0018]

[0019] Formula (4) is a data approximation processing method, in which the constant term and the expansion coefficient are calculated by the Fourier series coefficient formula, or obtained by a fitting algorithm such as the least squares method or the gradient descent method.

[0020] The two conditions that the basic mode component of step 2) needs to meet are:

[0021] a) The number of extreme points N in the entire data column, including maximum and minimum points e and the number of zero crossings N z must be equal, or at most differ by one, that is:

[0022] (N z -1)≤N e ≤(N z +1) (5)

[0023] b) At any time point t i The upper envelope f determined by the local maximum value of the signal max (t) and the lower envelope f determined by the local minimum min The mean of (t) is zero, that is:

[0024] [f max (t i )+f min (t i )] / 2=0 (6)

[0025] Before solving the instantaneous frequency of the remainder R(t), empirical mode decomposition is used to decompose R(t) into multiple basic mode components. The decomposition steps are as follows:

[0026] 2.1) Take R(t) as the sequence to be processed, determine all local extreme points of R(t), connect all the maximum and minimum points with cubic spline curves, obtain the upper and lower envelopes of R(t), and record the mean of the upper and lower envelopes as the sequence m(t);

[0027] 2.2) Subtract the mean m(t) of the upper and lower envelopes from R(t), and the resulting sequence is denoted as c1(t);

[0028] c1(t)=R(t)-m(t) (7)

[0029] Check whether c1 meets the conditions of the basic mode component. If not, take c1(t) as the sequence to be processed and repeat steps 2.1)-2.2) until c1 is a basic mode component, recorded as h1(t);

[0030] 2.3) Subtract the first basic mode component h1(t) from R(t) to obtain the remaining sequence denoted as r1(t);

[0031] r1(t)=R(t)-h1(t) (8)

[0032] 2.4) Take r1(t) as the new signal to be processed and repeat steps 2.1)-2.3) to obtain the second, third, and nth basic mode components. When the remaining sequence r n When (t) satisfies the set conditions, the decomposition stops and the remaining sequence is recorded as r(t); thus R(t) is decomposed into the sum of several basic mode components and a new remainder:

[0033]

[0034] Set the remaining sequence r n (t) When it becomes a monotonic function and can no longer filter out basic mode components, stop decomposing and record the sum of all basic mode components is H(t), and the instruction is expressed as shown in formula (2).

[0035] The amplitude-frequency characteristic and phase-frequency characteristic of the servo control system in step 3) are expressed as:

[0036]

[0037] The corrected F(t) is denoted as F′(t), as shown in formula (11), and each sine and cosine component in F(t) is corrected separately;

[0038]

[0039] Step 4) for each basic mode component h i (t) performing Hilbert transform to obtain the instantaneous frequency, instantaneous amplitude and instantaneous phase of each component;

[0040] Signal h i Hilbert transform of (t) for:

[0041]

[0042] Construct the analytical function z(t):

[0043]

[0044] In the formula, Yes i Hilbert transform of (t), j is the sign of the imaginary number;

[0045] The amplitude function is:

[0046]

[0047] The phase function is:

[0048]

[0049] The derivative of the phase function is the instantaneous frequency:

[0050]

[0051] The calculation process shown in formulas (12)-(16) is performed on all basic mode components to obtain n h i (t) instantaneous amplitude and instantaneous phase; similarly, in order to make the instantaneous amplitude and phase of H(t) input and output in the servo control system the same, the amplitude attenuation and phase lag corresponding to the instantaneous frequency are used to correct each basic mode component h i The instantaneous amplitude and instantaneous phase of H(t) offset the attenuation and lag of H(t) caused by the servo system.

[0052] The corrected instantaneous amplitude function is:

[0053] a' i (t) = A[ω i (t)]a i (t) (17)

[0054] The corrected instantaneous phase function is:

[0055]

[0056] Reconstruct the real part of the signal:

[0057] h' i (t) = a' i (t)·cos[φ' i (t)] (19)

[0058] The above correction is performed on each basic mode component, and the corrected H(t) is recorded as H′(t). The corrected instruction remainder is expressed as:

[0059]

[0060] The command correction method considering the frequency domain characteristics of the servo control system is applicable to any steady-state linear servo control system and can be used simultaneously with any additional linear control method.

[0061] The command correction method considering the frequency domain characteristics of the servo control system is suitable for reducing the single-axis following error and thus indirectly reducing the contour error of multi-axis linkage.

[0062] The present invention has the following beneficial effects:

[0063] The method of the present invention realizes the reasonable decomposition of the instruction by combining the two frequency analysis methods of Fourier series and Hilbert-Huang, and uses the frequency domain characteristics of the servo system to correct the instruction in the frequency domain to offset the attenuation and lag of the servo system to the instruction, thereby greatly improving the tracking performance of the servo system. Compared with the traditional control method for improving the servo bandwidth, the method of the present invention does not need to add a complex control structure, avoids the problem that the servo bandwidth is restricted by the mechanical mode, and the modified instruction changes are very small, which does not affect the processing efficiency. BRIEF DESCRIPTION OF THE DRAWINGS

[0064] Figure 1 It is a flow chart of the present invention.

[0065] Figure 2 Schematic diagram of the control structure of the servo control system according to an embodiment of the present invention.

[0066] Figure 3 This is a closed-loop frequency response curve of the position loop of the servo control system according to an embodiment of the present invention.

[0067] Figure 4 Schematic diagram of comparison before and after instruction correction in an embodiment of the present invention.

[0068] Figure 5 It is a schematic diagram comparing the following errors before and after instruction correction according to an embodiment of the present invention. DETAILED DESCRIPTION

[0069] The present invention is described in detail below with reference to the accompanying drawings and embodiments.

[0070] Reference Figure 1 , a command correction method considering the frequency domain characteristics of a servo control system, comprising the following steps:

[0071] 1) Decompose the command into simple frequency components: This embodiment adopts the ABB MicroFlexe190 series servo drive system. The servo control system adopts PID three-loop semi-closed loop control, in which the current loop is equivalent to the first-order inertia link, the speed loop adopts PI control, and the position loop adopts P control. On this basis, a speed feedforward controller is added. The control structure is as follows: Figure 2As shown; the input command of the servo control system is fitted by Fourier series, and the command is decomposed into a combination of multiple positive and residual signals and a residual term. Assuming that the input command of the servo control system is x(t), its n-order Fourier series fitting expression is shown in formula (4), where a0 is a constant term, a m and b m is the expansion coefficient of the mth harmonic, ω is the fundamental frequency;

[0072]

[0073] Formula (4) is a data approximation processing method. The constant term and expansion coefficient can be calculated by the Fourier series coefficient formula, or obtained by fitting algorithms such as the least squares method and the gradient descent method. In actual calculations, due to the limitation of the fitting order n, there is a certain error between the fitting instruction and the original instruction.

[0074] The error between the fitting instruction F(t) and the original instruction x(t) is recorded as R(t). At this time, the original instruction can be expressed as a combination of the Fourier fitting term F(t) and a remainder term R(t), where F(t) is a combination of multiple sine and cosine waves with different frequencies, and R(t) is the error between the fitting term and the instruction. The instruction representation is shown in formula (1);

[0075] x(t)=F(t)+R(t) (1)

[0076] 2) Based on step 1), the remainder R(t) is further decomposed: using empirical mode decomposition, the remainder R(t) is decomposed into a basic mode component term H(t) and a new remainder r(t), where H(t) is the linear sum of multiple basic mode components, and r(t) is a monotonic component or a component with a very small value and does not require additional processing. The instruction representation is shown in formula (2);

[0077] x(t)=F(t)+H(t)+r(t) (2)

[0078] The remainder R(t) usually contains components with drastic frequency changes, and its instantaneous frequency information needs to be solved. The instantaneous frequency of a signal is defined by the Hilbert transform, but the instantaneous frequency is only meaningful when the signal satisfies the basic mode component conditions. The two conditions that the basic mode components need to meet are:

[0079] a) The number of extreme points N in the entire data series e (including maximum and minimum points) and the number of zero-crossing points N z must be equal, or at most differ by one, that is:

[0080] (N z -1)≤N e ≤(Nz +1) (5)

[0081] b) At any time point t i The upper envelope f determined by the local maximum value of the signal max (t) and the lower envelope f determined by the local minimum min The mean of (t) is zero, that is,

[0082] [f max (t i )+f min (t i )] / 2=0 (6)

[0083] Most actual signals cannot meet this condition. Before solving the instantaneous frequency of the remainder R(t), empirical mode decomposition is used to decompose R(t) into multiple basic mode components. The decomposition steps are as follows:

[0084] 2.1) Take R(t) as the sequence to be processed, determine all local extreme points of R(t), connect all the maximum and minimum points with cubic spline curves, obtain the upper and lower envelopes of R(t), and record the mean of the upper and lower envelopes as the sequence m(t);

[0085] 2.2) Subtract the mean m(t) of the upper and lower envelopes from R(t), and the resulting sequence is denoted as c1(t);

[0086] c1(t)=R(t)-m(t) (7)

[0087] Check whether c1 meets the conditions of the basic mode component. If not, take c1(t) as the sequence to be processed and repeat steps 2.1)-2.2) until c1 is a basic mode component, recorded as h1(t);

[0088] 2.3) Subtract the first basic mode component h1(t) from R(t) to obtain the remaining sequence denoted as r1(t);

[0089] r1(t)=R(t)-h1(t) (8)

[0090] 2.4) Take r1(t) as the new signal to be processed and repeat steps 2.1)-2.3) to obtain the second, third, and nth basic mode components. When the remaining sequence r n When (t) satisfies the set conditions, the decomposition stops and the remaining sequence is recorded as r(t); thus R(t) is decomposed into the sum of several basic mode components and a new remainder:

[0091]

[0092] Set the remaining sequence r n(t) When it becomes a monotonic function and can no longer filter out basic mode components, stop decomposing and record the sum of all basic mode components is H(t), and the instruction is expressed as shown in formula (2);

[0093] 3) F(t) item correction: refer to Figure 3 , establish the frequency response curve of the servo control system from the command position to the motion output, the amplitude-frequency characteristic and phase-frequency characteristic of the servo control system can be expressed as:

[0094]

[0095] In order to make the amplitude and phase of each frequency of the input and output of F(t) in the servo control system the same, the amplitude and phase of each frequency sine and cosine signal in F(t) are corrected according to the frequency characteristics of the servo control system to offset the attenuation and lag of each frequency component in F(t) by the servo control system. The corrected F(t) is recorded as F′(t), as shown in formula (11). Each sine and cosine component in F(t) is corrected separately.

[0096]

[0097] 4) H(t) correction: Use Hilbert transform to solve the instantaneous amplitude and instantaneous phase of each basic mode component; for each basic mode component h i (t) Performing Hilbert transform, the instantaneous frequency, instantaneous amplitude and instantaneous phase of each component can be obtained;

[0098] Signal h i Hilbert transform of (t) for:

[0099]

[0100] Construct the analytical function z(t):

[0101]

[0102] In the formula, Yes i Hilbert transform of (t), j is the sign of the imaginary number;

[0103] The amplitude function is:

[0104]

[0105] The phase function is:

[0106]

[0107] The derivative of the phase function is the instantaneous frequency:

[0108]

[0109] The calculation process shown in formulas (12)-(16) is performed on all basic mode components to obtain n h i (t) instantaneous amplitude and instantaneous phase; similarly, in order to make the instantaneous amplitude and phase of H(t) input and output in the servo control system the same, the amplitude attenuation and phase lag corresponding to the instantaneous frequency are used to correct each basic mode component h i The instantaneous amplitude and instantaneous phase of H(t) offset the attenuation and lag of H(t) caused by the servo system.

[0110] The corrected instantaneous amplitude function is:

[0111] a' i (t) = A[ω i (t)]a i (t) (17)

[0112] The corrected instantaneous phase function is:

[0113]

[0114] Reconstruct the real part of the signal:

[0115] h' i (t) = a' i (t)·cos[φ' i (t)] (19)

[0116] The above correction is performed on each basic mode component, and the corrected H(t) is recorded as H′(t). The corrected instruction remainder can be expressed as:

[0117]

[0118] 5) Reconstruction command: The revised F′(t), H′(t) and r(t) are combined into a new command as the input of the servo system. The revised command is recorded as x′(t), as shown in formula (3).

[0119] x'(t)=F'(t)+H'(t)+r(t) (3)

[0120] This embodiment uses a section of the Z-axis machining trajectory of the EMO211 exhibition specimen, decomposes the selected instruction according to step 1) and step 2), and corrects the instruction according to step 3) and step 4) based on the measured frequency response curve; finally, the corrected components are combined to form a new input instruction; the instruction comparison before and after correction is as follows: Figure 4As shown in the figure, the modified instruction is very small compared to the original instruction, so the instruction correction will not destroy the processing efficiency of the original trajectory.

[0121] The original command and the corrected command are used as the input of the servo system respectively, and the motor encoder position is collected as the output. The following error comparison between the previous and the next commands is corrected. Figure 5 As shown in the figure, after the instruction was corrected, the fluctuation of the following error was significantly improved, and the maximum following error dropped from 261μm to 28.7μm, a decrease of 81.08%. The overall average following error dropped from 38.6μm to 7.3μm, a decrease of 89.02%.

Claims

1. A command correction method considering the frequency domain characteristics of a servo control system, characterized in that: The following steps are involved: 1) Decompose the command into simple frequency components: Let x(t) be the input command of the servo control system, perform Fourier series fitting on the input command of the servo control system, and decompose the command into a combination of Fourier fitting term F(t) and a remainder term R(t), where F(t) is a combination of multiple sine and cosine waves with different frequencies, and R(t) is the error between the fitting term and the command. The command is represented as shown in formula (1); x(t)=F(t)+R(t) (1) 2) Decompose the remainder R(t): Use empirical mode decomposition to decompose the remainder R(t) into a basic mode component term H(t) and a new remainder r(t), where H(t) is the linear sum of multiple basic mode components, and r(t) is a monotonic component or a component with a very small value and does not require additional processing. The instruction representation is shown in formula (2); x(t)=F(t)+H(t)+r(t) (2) 3) Correction of the Fourier fitting term F(t): Establish the frequency response curve of the servo control system from the command position to the motion output. In order to make the amplitude and phase of each frequency of the input and output of F(t) in the servo control system the same, correct the amplitude and phase of the sine and cosine signals of each frequency in F(t) according to the frequency characteristics of the servo control system, offset the attenuation and lag of each frequency component in F(t) by the servo control system, and record the corrected F(t) as F′(t); 4) Correction of the basic mode component term H(t): Using Hilbert transform, solve the instantaneous amplitude and instantaneous phase of each basic mode component; similarly, in order to make the instantaneous amplitude and phase of the input and output of H(t) in the servo control system the same, use the amplitude attenuation and phase lag corresponding to the instantaneous frequency to correct the instantaneous amplitude and instantaneous phase of each basic mode component, offset the attenuation and lag of H(t) by the servo control system, and record the corrected H(t) as H′(t); 5) Reconstruct the command: The revised F′(t), H′(t) and r(t) are combined into a new command as the input of the servo control system. The revised command is recorded as x′(t), as shown in formula (3); x'(t)=F'(t)+H'(t)+r(t) (3).

2. The method according to claim 1, characterized in that: In the step 1), the input command of the servo control system is assumed to be x(t), and its n-th order Fourier series fitting expression is shown in formula (4), where a0 is a constant term, a m and b m is the expansion coefficient of the mth harmonic, ω is the fundamental frequency; Formula (4) is a data approximation processing method, in which the constant term and the expansion coefficient are calculated by the Fourier series coefficient formula, or obtained by a fitting algorithm such as the least squares method or the gradient descent method.

3. The method according to claim 2, characterized in that The two conditions that the basic mode component of step 2) needs to meet are: a) The number of extreme points N in the entire data column, including maximum and minimum points e and the number of zero crossings N z must be equal, or at most differ by one, that is: (N z -1)≤N e ≤(N z +1) (5) b) At any time point t i The upper envelope f determined by the local maximum value of the signal max (t) and the lower envelope f determined by the local minimum min The mean of (t) is zero, that is: [f max (t i )+f min (t i )] / 2=0 (6) Before solving the instantaneous frequency of the remainder R(t), empirical mode decomposition is used to decompose R(t) into multiple basic mode components. The decomposition steps are as follows: 2.1) Take R(t) as the sequence to be processed, determine all local extreme points of R(t), connect all the maximum and minimum points with cubic spline curves, obtain the upper and lower envelopes of R(t), and record the mean of the upper and lower envelopes as the sequence m(t); 2.2) Subtract the mean m(t) of the upper and lower envelopes from R(t), and the resulting sequence is denoted as c1(t); c1(t)=R(t)-m(t) (7) Check whether c1 meets the conditions of the basic mode component. If not, take c1(t) as the sequence to be processed and repeat steps 2.1)-2.2) until c1 is a basic mode component, recorded as h1(t); 2.3) Subtract the first basic mode component h1(t) from R(t) to obtain the remaining sequence denoted as r1(t); r1(t)=R(t)-h1(t) (8) 2.4) Take r1(t) as the new signal to be processed and repeat steps 2.1)-2.3) to obtain the second, third, and nth basic mode components. When the remaining sequence r n When (t) satisfies the set conditions, the decomposition stops and the remaining sequence is recorded as r(t); thus R(t) is decomposed into the sum of several basic mode components and a new remainder: Set the remaining sequence r n (t) When it becomes a monotonic function and can no longer filter out basic mode components, stop decomposing and record the sum of all basic mode components is H(t), and the instruction is expressed as shown in formula (2).

4. The method according to claim 3, characterized in that The amplitude-frequency characteristic and phase-frequency characteristic of the servo control system in step 3) are expressed as: The corrected F(t) is denoted as F′(t), as shown in formula (11), and each sine and cosine component in F(t) is corrected separately; 5. The method according to claim 4, characterized in that: Step 4) for each basic mode component h i (t) performing Hilbert transform to obtain the instantaneous frequency, instantaneous amplitude and instantaneous phase of each component; Signal h i Hilbert transform of (t) for: Construct the analytical function z(t): In the formula, Yes i Hilbert transform of (t), j is the sign of the imaginary number; The amplitude function is: The phase function is: The derivative of the phase function is the instantaneous frequency: The calculation process shown in formulas (12)-(16) is performed on all basic mode components to obtain n h i (t) instantaneous amplitude and instantaneous phase; similarly, in order to make the instantaneous amplitude and phase of H(t) input and output in the servo control system the same, the amplitude attenuation and phase lag corresponding to the instantaneous frequency are used to correct each basic mode component h i The instantaneous amplitude and instantaneous phase of H(t) offset the attenuation and lag of H(t) caused by the servo system. The corrected instantaneous amplitude function is: a′ i (t)=A[ω i (t)]a i (t) (17) The corrected instantaneous phase function is: Reconstruct the real part of the signal: h′ i (t)=a′ i (t)·cos[φ′ i (t)] (19) The above correction is performed on each basic mode component, and the corrected H(t) is recorded as H′(t). The corrected instruction remainder is expressed as:

6. The method according to claim 1, characterized in that: It is applicable to any steady-state linear servo control system and can be used simultaneously with any additional linear control method.

7. The method according to claim 1, characterized in that: It is suitable for reducing single-axis following error and thus indirectly reducing contour error of multi-axis linkage.

Images