Solution method for time-frequency responds of servo control systems based on frequency domain characteristics

Abstract

In view of that the tracking error originates from the amplitude attenuation and phase lag of the setpoints frequency component caused by the servo control system at each moment, this paper puts forward the idea of obtaining the setpoints time-frequency response (STFR) by solving the amplitude attenuation and phase lag of each frequency component of the setpoints at each moment. First, the feasibility of the proposed idea is proved theoretically; Then, the setpoints time-frequency transform is carried out and the solving method of STFR and setpoints time-frequency response error (STFRE) is established; Finally, taking the servo control system as the object, the STFR and STFRE are solved, and the corresponding relationship with the output time-frequency transformation and tracking error is analyzed to verify the proposed method. The contributions of the proposed method are: the method achieves the solution of STFR based on the amplitude attenuation and phase lag of the servo system; it avoids the limitation that only harmonic wavelet bases must be used in the existing method and can be applied to the solution of the time-frequency response of actual setpoints; it can separate the setpoints loss or tracking error caused by amplitude attenuation and phase lag; it can determine the redundancy or lack of strategies such as filters for certain setpoints and servo control system.

Introduction

Continuous trajectory control is one of the basic research fields of motion control. In order to control the contour error of continuous trajectory, the response of each axis servo system to the input setpoints must be sufficiently precise or coordinated at each position (or at each time of movement). The analysis of the formation mechanism of tracking error at each moment and the response process of the servo system to the setpoint at each moment is important for the control of continuous trajectory profile error.

The amplitude-frequency and phase-frequency characteristics of the servo control system profoundly explain the mechanism of the tracking error from the perspective of frequency-domain, that is, the tracking error at each time originates from the amplitude attenuation and phase lag of the frequency components of the setpoints at each time by the servo control system. Accordingly, the research on control strategies for tracking errors are mainly from the frequency-domain perspective, focusing on the servo control characteristics of the servo control system itself and the setpoints inputted to the servo control system.

Focusing on the servo control characteristics, the feedforward controller, ¹ notch filter, ² modal filter, ³ and sliding mode controller ⁴ can greatly improve the amplitude-frequency characteristics and phase-frequency characteristics and hence significantly reduce the tracking error. Koren and Lo,^5,6 Altintas et al., ⁷ and Lyu et al. ⁸ systematically summarized the research progress in each period. Around the setpoints inputted to the servo control system, the frequency-domain analysis of the setpoints by FFT is the main method. Smith ⁹ and Altintas et al. ⁷ used fast Fourier transform (FFT) to analyze the frequency of different types of setpoints, and concluded that setpoints with acceleration second order continuous jerk have the least potential excitation frequencies and are most suitable for use as servo system inputs.

The magnitude of the tracking error is related not only to the servo control characteristics of the servo control system itself, but also to the setpoints inputted to the servo control system. The former can be considered not to change with time. However, the speed, acceleration and jerk of the setpoints change with time, resulting in that the tracking error presents a significant time-dependent characteristic. The frequency-domain characteristics of the servo control system and the FFT analysis of the setpoints do not have time information, which cannot focus on the specific time to analyze the time-dependent characteristic of the tracking error. Analyzing a servo control system from a time-domain perspective is a direct means to study the time-dependent characteristic of the tracking error. From a time-domain perspective, a servo control system needs to be described as a differential equation or transfer function. Servo system performance is measured using metrics such as overshoot, adjustment time, and rise time. The setpoints and system errors are then plotted as time-domain curves that allow visualizing the extremum values of velocity, acceleration and jerk to determine if the physical limits of the motor are exceeded.^10–16 However, the analysis of tracking error from the time-domain perspective can only provide an intuitive view of the control effect and does not allow exploring new control strategies.

To sum up, no matter the frequency-domain analysis or time-domain analysis of the servo control system, it is not possible to quantitatively express the response process of the servo control system to the setpoints at each moment in the frequency domain. In general, the time domain analysis method and frequency domain analysis method are independent of each other and need to introduce the time-frequency analysis method.

Time-frequency analysis is a modern branch of harmonic analysis. It analyzes signal in both the time- and frequency- domain simultaneously, which is broadly studied and used in image processing, signal analysis and communication theory, and so on. However, in motion control, there are only a few related literatures so far. Rotariu et al. ¹⁷ introduced time-frequency analysis into motion control. They combined the time-frequency analysis of the Wigner distribution with the field of iterative learning control (ILC) and designed a time-frequency adaptive filter to control the wafer in ILC.^18,19 The Wigner distribution was successfully used for tracking error control, demonstrating the feasibility and promise of time-frequency analysis for motion system analysis. Galleani ²⁰ used the Wegener distribution to transform the motion control system from the time domain to the time-frequency domain so that the time-frequency spectrum of the system output can be calculated. This method achieves the solution of the time-frequency response by time-frequency transformation of the system, but the calculation of the time-frequency transformation of the system is very complicated. In order to simplify the time-frequency response solving method, it becomes another choice to realize the time-frequency response solving by time-frequency transformation of the setpoints. Tratskas and Spanos ²¹ proposed the introduction of wavelet transform to describe the excitation-response relation for multiple-input multiple-output systems in the time-frequency domain, and they successfully described the response process of a second-order system to a random input under the premise of harmonic wavelets as basis functions. Subsequently Spanos et al. ²² further optimized the effectiveness of the method in lightly damped systems and successfully derived the excitation-response relation in a single-degree-of-freedom linear structure. However, the method can only be used in the case where the wavelet basis is a harmonic wavelet basis. Harmonic wavelet basis has good smoothness, and has excellent performance when dealing with smooth signals, such as sine functions and spline functions. However, in the servo control system, the input setpoint signal often has discontinuous features such as start-stop, corner, and spike, so the application range of the method based on harmonic wavelet basis only is narrow. In order to realize the response solution for arbitrary input, we need to study a time-frequency response solution method that is not limited by the wavelet basis type.

In essence, the output of the servo control system has changed in amplitude and phase relative to the input. From this point, this paper proposes a method for solving the STFR of servo systems. First, the frequency component of setpoints at each time is characterized by time-frequency analysis method. Then, using the frequency-domain feature of servo control system, the amplitude and phase changes of the output of the servo control system relative to the input at each time are given. Finally, the STFR of the servo control system is approximated by integrating the time-frequency characterization of the setpoints and the amount of amplitude and phase change of the servo system to the setpoints.

Time-frequency transformation of setpoint

This section uses time-frequency analysis to transform setpoints from time-domain to time-frequency domain, and solves the frequency components of setpoints at each time.

The choice of time-frequency analysis methods needs to consider the characteristics of setpoints and the needs of analysis:

• (1) The effective frequency component of the setpoints are mainly concentrated in the low frequency band, which requires high low frequency accuracy of the time-frequency analysis method. In addition, the frequency distribution range after time-frequency analysis needs to be able to cover the position loop bandwidth of the servo control system.

• (2) The time resolution of the setpoints after the time-frequency analysis needs to be far lower than the processing time of conventional part trajectory.

The time-frequency analysis methods such as Wigner-Ville distribution (WVD) and Hilbert-Huang transform (HHT) give the time-frequency information of the signal based on the characteristics of the signal itself. Its frequency band division and frequency resolution are only related to the signal itself and cannot be adjusted actively. Continuous wavelet transform (CWT) divides the signal at each time into a set of artificially set frequency bands through the similarity between wavelet basis and signal components. Although CWT is slightly inferior to WVD and HHT in accuracy, it has initiative in setting frequency scale and time scale ²³ which can be set according to the requirements of frequency resolution, frequency analysis range and time resolution.

The setpoints position sequence input to a servo control system is:

$$p(t)=p(k\mathrm{\Delta }T)k=1,2,\cdots ,(n-1),n$$

(1)

where, k is the number of interpolation cycles, $\mathrm{\Delta }T$ is the interpolation cycle.

The CWT of the setpoints position sequence $p(t)$ is defined as:

$$W_p(s,u)=\int p\frac{1}{\sqrt{s}}{\psi }^{*}\left(\frac{t-u}{s}\right)\mathrm{dt}$$

(2)

where, $\psi (t)$ is the wavelet basis function, $s$ is the scale factor, $u$ is the time shift factor, * is the conjugate operator, t is the time, and $W_p(s,u)$ is the wavelet coefficient.

Unless otherwise specified, $W_p(s,u)$ represents the CWT of $p(t)$ in the following sections. Through wavelet transform, the position sequence $p(t)$ is decomposed into a function of two variables $u$ and $s$ . In order to correspond the time shift factor u to time t, the arithmetic sequence of u is constructed by interpolation period:

$$\mathit{u}(k)=\left[\begin{array}{cc}\frac{1}{f_s}\hfill & \frac{2}{f_s}\cdots \hfill \end{array}\begin{array}{cc}\frac{n-1}{f_s}\hfill & \frac{n}{f_s}\hfill \end{array}\right]$$

(3)

Where, k represents the position of the element in the sequence, and the value is an integer from 1 to n; $f_s$ is the output frequency of the setpoints (the reciprocal of $\mathrm{\Delta }T$ ).

In order to correspond the scale factor $s$ to the frequency $f$ , the scale sequence s is constructed:

$$\mathit{s}(i)=\left[\begin{array}{cc}\frac{2f_{c}L}{1}\hfill & \begin{array}{ccc}\frac{2f_{c}L}{2}\hfill & \cdots \hfill & \begin{array}{cc}\frac{2f_{c}L}{L-1}\hfill & \frac{2f_{c}L}{L}\hfill \end{array}\hfill \end{array}\hfill \end{array}\right]$$

(4)

where, i represents the position of the element in the sequence, and the value is an integer from 1 to L, $f_c$ is the center frequency of wavelet basis function, L is the length of scale sequence.

The length L of the scale sequence determines the resolution of the frequency $f$ . The frequency $f$ corresponding to the scale factor $s$ is:

$$f=\frac{f_c{f}_s}{s}$$

(5)

Substituting equation (4) into equation (5), the frequency sequence can be obtained as:

$$\mathit{f}(i)=\left[\begin{array}{ccccc}\frac{f_s}{2}\frac{1}{L}\hfill & \frac{f_s}{2}\frac{2}{L}\hfill & \cdots \hfill & \frac{f_s}{2}\frac{L-1}{L}\hfill & \frac{f_s}{2}\frac{L}{L}\hfill \end{array}\right]$$

(6)

It can be seen from equation (6) that the frequency sequence $\mathit{f}$ is an isometric sequence with a tolerance of $\frac{f_s}{2}\frac{1}{L}$ .

In the remaining examples of this paper, the wavelet basis function $\psi (t)$ for plotting the time-frequency diagrams are all sym2. In equation (6), set the value of L is 500 and the setpoints output frequency $f_s$ is 1000 Hz, the resolution of frequency $\mathit{f}$ is 1 Hz and the frequency analysis range is 0–500 Hz, the time resolution is 1 ms, which is consistent with the interpolation cycle of setpoints.

Through equations (2)–(6), CWT can be carried out on the setpoints position sequence, as shown in equation (7).

$$W_p(s,u)={\mathit{W}}_{\mathit{p}}=\left[\begin{array}{cccc}{A}_{1,1}\hfill & A_{1,2}\hfill & \cdots \hfill & A_{1,n}\hfill \\ \multicolumn{1}{c}{A_{2,1}}& A_{2,2}\hfill & \cdots \hfill & A_{2,n}\hfill \\ \multicolumn{1}{c}{⋮}& ⋮\hfill & ⋮\hfill & ⋮\hfill \\ \multicolumn{1}{c}{A_{L-1,1}}& A_{L-1,2}\hfill & \cdots \hfill & A_{L-1,n}\hfill \\ \multicolumn{1}{c}{A_{L,1}}& A_{L,2}\hfill & \cdots \hfill & A_{L,n}\hfill \end{array}\right]$$

(7)

where, ${\mathit{W}}_{\mathit{p}}$ is a matrix of L rows and n columns, the row corresponds to the scale sequence s , representing the frequency points with an interval of $\frac{f_s}{2}\frac{1}{L}$ in the frequency range $\left[1\frac{f_s}{2}\right]$ , and the column corresponds to the time sequence u , representing the time points with an interval of $\frac{1}{f_s}$ .

Equation (7) transforms the setpoints from time-domain to frequency-domain, expressed as time-frequency matrix. The rows and columns in the matrix correspond to frequency and time, respectively. The values of the elements in the matrix are wavelet coefficients, which can be regarded as the setpoints amplitude. The smaller the wavelet coefficient, the smaller the setpoints amplitude in the corresponding frequency and time.

Solving idea and feasibility theory proof of STFR

The frequency-domain characteristic characterizes the response ability of the servo control system to the setpoints. Among them, the amplitude-frequency characteristic represents the amplitude attenuation of each frequency component of the setpoints by the servo control system; The phase-frequency characteristic represents the phase lag of the servo control system to each frequency component of the setpoints. Through CWT, the frequency components of the setpoints at each time are obtained. Combined with the connotation of frequency-domain characteristics of servo control system, this section puts forward the idea of solving the STFR by solving the amplitude attenuation and phase lag of setpoints frequency components at each time. Next, the feasibility of the proposed solution is proved theoretically.

CWT has the following properties:

• (1) CWT is a kind of linear transform, which can be superimposed. If the CWT of $x(t)$ is $W_x(s,u)$ and the CWT of $y(t)$ is $W_y(s,u)$ , the wavelet transform of $z(t)=k_{1}x(t)+k_{2}y(t)$ is given by equation (8):

$$W_z(s,u)=k_1{W}_x(s,u)+k_2{W}_y(s,u)$$

(8)

• (2) CWT has translation. If the CWT of $x(t)$ is $W_x(s,u)$ , then the CWT of $x(t-t_0)\mathrm{is}{W}_x(s,u-t_0).$

With a linear time-invariant servo control system $G(s)$ , set $s=j\omega$ , then the frequency characteristics of the system can be obtained. For an input signal whose frequency does not change with time, the amplitude ratio and phase difference between input and output can be expressed as:

$$\begin{array}{c}A=|G(j\omega )|\hfill \\ \phi =\angle G(j\omega )\hfill \end{array}$$

(9)

Rewrite the equation (7) of the position setpoints $p(t)$ into the form of the sum of multiple matrices:

$$\begin{array}{c}\multicolumn{1}{c}{{\mathit{W}}_{\mathit{p}}=\left[\begin{array}{c}{\mathit{r}}_{\mathit{1}}(u)\hfill \\ \multicolumn{1}{c}{{\mathit{r}}_{\mathit{2}}(u)}\\ \multicolumn{1}{c}{⋮}\\ \multicolumn{1}{c}{{\mathit{r}}_{\mathit{L}}(u)}\end{array}\right]=\left[\begin{array}{c}{\mathit{r}}_{\mathit{1}}(u)\hfill \\ \multicolumn{1}{c}{0}\\ \multicolumn{1}{c}{⋮}\\ \multicolumn{1}{c}{0}\end{array}\right]+\left[\begin{array}{c}0\hfill \\ \multicolumn{1}{c}{{\mathit{r}}_{\mathit{2}}(u)}\\ \multicolumn{1}{c}{⋮}\\ \multicolumn{1}{c}{0}\end{array}\right]+\cdots +\left[\begin{array}{c}0\hfill \\ \multicolumn{1}{c}{0}\\ \multicolumn{1}{c}{⋮}\\ \multicolumn{1}{c}{{\mathit{r}}_{\mathit{L}}(u)}\end{array}\right]}\\ ={\sum }_{i=1}^L{\mathit{R}}_{\mathit{i}}(u)\hfill \end{array}$$

(10)

where, ${\mathit{r}}_{\mathit{i}}\left(u\right)=[A_{i,1}{A}_{i,2}\cdots A_{i,n}]$ , ${\mathit{r}}_{\mathit{i}}\left(u\right)$ is the i-th row vector in equation (7) and is a function of time sequence u ; ${\mathit{R}}_{\mathit{i}}\left(u\right)=[0\cdots {\mathit{r}}_{\mathit{i}}\left(u\right)\cdots 0{]}^T$ , ${\mathit{R}}_{\mathit{i}}(u)$ is a matrix with the same size as ${\mathit{W}}_{\mathit{p}}$ , in the row corresponding to the scale sequence $\mathit{s}\left(i\right)$ of equation (4), there are non-zero elements and the remaining elements are 0.

For any CWT that satisfies the wavelet admissibility condition, there is its inverse transform which can be expressed as:

$$p(t)=\frac{1}{C_{\psi }}\underset{0}{\overset{\infty }{\int }}\frac{\mathrm{ds}}{s^2}\underset{-\infty }{\overset{\infty }{\int }}{W}_p(s,u)\frac{1}{\sqrt{s}}\psi \left(\frac{t-u}{s}\right)\mathrm{du}$$

(11)

Where, $C_{\psi }=\underset{0}{\overset{\infty }{\int }}\frac{{|\psi (\omega )|}^2}{\omega }d\omega <\infty$ is the admissible condition of wavelet $\psi (t)$ and $W_p(s,u)$ is the CWT of $p(t)$ .

Substituting the time sequence u of equation (3), the scale sequence s of equations (4) and (10) into equation (11) to obtain:

$$\begin{array}{c}\multicolumn{1}{c}{p(t)=\frac{1}{C_{\psi }}\underset{0}{\overset{\infty }{\int }}\frac{\mathrm{ds}}{s^2}\underset{-\infty }{\overset{\infty }{\int }}{\sum }_{i=1}^L{\mathit{R}}_{\mathit{i}}(u)\frac{1}{\sqrt{s}}\psi \left(\frac{t-u}{s}\right)\mathrm{du}}\\ ={\sum }_{i=1}^L\frac{1}{C_{\psi }}\underset{0}{\overset{\infty }{\int }}\frac{\mathrm{ds}}{s^2}\underset{-\infty }{\overset{\infty }{\int }}{\mathit{R}}_{\mathit{i}}(u)\frac{1}{\sqrt{s}}\psi \left(\frac{t-u}{s}\right)\mathrm{du}\hfill \\ ={\sum }_{i=1}^L{p}_i(t)\hfill \end{array}$$

(12)

Where, $p_i(t)$ is a single frequency signal in the time domain.

At this time, the original setpoints position $p(t)$ is represented as the superposition of multiple $p_i(t)$ . Since ${\mathit{R}}_{\mathit{i}}(u)$ has non-zero value only in the row corresponding to the scale sequence $\mathit{s}(i)$ , $p_i(t)$ is an ideal signal with frequency $\mathit{f}(i)$ which is the frequency corresponding to the scale sequence $\mathit{s}(i)$ .

Since $p_i(t)$ is a fixed frequency signal, its response $p{\prime }_i(t)$ in system $G(s)$ is determined by equation (9), and the output response $p{\prime }_i(t)$ is expressed as:

$$p{\prime }_i(t)=A_i·p_i(t+\frac{{\phi }_i}{360·f(i)})$$

(13)

where, $A_i$ is the ratio of the amplitude between the input and output of the system when the frequency is $\mathit{f}(i)$ ; ${\phi }_i$ is the phase difference between the input and output of the system at frequency $\mathit{f}(i)$ .

Because $G(s)$ is a linear time invariant system, the output response $p^{\prime }(t)$ of $p(t)$ can be expressed as the sum of the outputs generated when each $p_i(t)$ acts alone. Therefore, the output response $p^{\prime }(t)$ is expressed as:

$$p\prime (t)=\sum _{i=1}^{L}p{\prime }_i(t)$$

(14)

From equation (12), ${\mathit{R}}_{\mathit{i}}(u)$ is the CWT of $p_i(t)$ , which is recorded as $W_{p_i}(s_i,u)$ . According to the superposition characteristic of wavelets and the translation characteristic of wavelets as shown in equation (13), the CWT $W_{{p^{\prime }}_i}(s_i,u)$ of ${p^{\prime }}_i(t)$ can be indicate by using $W_{p_i}(s_i,u)$ :

$$\begin{array}{c}\multicolumn{1}{c}{W_{p{\prime }_i}(s_i,u)=A_i·W_{p_i}(s_i,u+\frac{{\phi }_i}{360·f(i)})}\\ =\left[\begin{array}{c}0\hfill \\ \multicolumn{1}{c}{⋮}\\ \multicolumn{1}{c}{A_i·{\mathit{r}}_{\mathit{i}}(u+\frac{{\phi }_i}{360·f(i)})}\\ \multicolumn{1}{c}{⋮}\\ \multicolumn{1}{c}{0}\end{array}\right]\hfill \end{array}$$

(15)

According to equation (14) and the superposition characteristics of wavelet, the wavelet transform $W_{p^{\prime }}(s,u)$ of output response $p^{\prime }(t)$ can be expressed by the CWT $W_{p{\prime }_i}(s_i,u)$ of each component ${p^{\prime }}_i(t)$ linear superposition:

$$\begin{array}{c}\multicolumn{1}{c}{W_{p\prime }(s,u)=\sum _{i=1}^L{W}_{p{\prime }_i}(s_i,u)=\sum _{i=1}^L{A}_i·W_{p_i}(s_i,u+\frac{{\phi }_i}{360·f(i)})}\\ =\left[\begin{array}{c}{A}_1·{\mathit{r}}_{\mathit{1}}(u+\frac{{\phi }_1}{360·f(1)})\hfill \\ \multicolumn{1}{c}{A_2·{\mathit{r}}_{\mathit{2}}(u+\frac{{\phi }_2}{360·f(2)})}\\ \multicolumn{1}{c}{⋮}\\ \multicolumn{1}{c}{A_L·{\mathit{r}}_{\mathit{L}}(u+\frac{{\phi }_L}{360·f(L)})}\end{array}\right]\hfill \end{array}$$

(16)

where, $W_{p^{\prime }}(s,u)$ is the object to be solved, that is, STFR; $W_{p_i}(s_i,u)$ is what has been solved in section 2, that is, CWT of the input setpoints; $A_i$ and ${\phi }_i$ represents the amplitude attenuation and phase lag of the servo system, respectively.

Equation (16) decomposes a sequence of setpoints into multiple ideal single frequency signals by using CWT, and solves the response of each single frequency signal according to the frequency-domain characteristics of the servo control system. The method is carried out in a linear premise, and obviously, the entire solution process wavelet basis function does not require a specific representation.

Solving method of STFR

The essence of equation (16) is to use the frequency-domain characteristics of the servo control system to carry out amplitude attenuation and time lag for different frequency components obtained by the CWT of the setpoints. This solution requires not only wavelet transform of setpoints, but also conversion of frequency-domain characteristics of servo control system into amplitude attenuation and time lag. Frequency-domain characteristics of servo control system are generally represented by Bode diagram. In order to express equation (16) in the form of a practically usable discrete matrix, in this section, the amplitude attenuation ratio expression is constructed based on the amplitude-frequency characteristics and the lag time expression is constructed based on the phase-frequency characteristics of the Bode diagram.

Establishment of discrete expression of amplitude attenuation ratio (AAR)

According to the definition of unit of amplitude (decibel) of amplitude-frequency characteristic, the amplitude-frequency characteristic can be converted into an expression of AAR, that is the ratio of output amplitude to input amplitude, as shown in equation (17).

$$H(f)={10}^{\frac{H{\left(f\right)}^{*}}{20}}$$

(17)

where, $H\left(f\right)$ represents the AAR of the servo control system to the setpoints; $H{\left(f\right)}^{*}$ is the amplitude-frequency characteristic.

In the amplitude-frequency characteristic, take the frequency range $\left[\begin{array}{cc}0,\hfill & \frac{f_s}{2}\hfill \end{array}\right]$ . The interval is discretized according to the frequency interval of $\frac{f_s}{2}\frac{1}{L}$ , and the interval is discretized into L frequency points. The AAR is expressed in the form of diagonal matrix as shown in equation (18).

$$\mathit{H}(i)=\left[\begin{array}{ccc}{H}_1\hfill & \cdots \hfill & 0\hfill \\ \multicolumn{1}{c}{⋮}& \ddots \hfill & ⋮\hfill \\ \multicolumn{1}{c}{0}& \cdots \hfill & H_L\hfill \end{array}\right]$$

(18)

where, $H_1,H_2\cdots H_{L-1},H_L$ represents the AAR corresponding to each frequency point $\left[\begin{array}{ccccc}0\hfill & \frac{f_s}{2}\frac{1}{L}\hfill & \frac{f_s}{2}\frac{2}{L}\hfill & \cdots \hfill & \frac{f_s}{2}\hfill \end{array}\right]$ on the amplitude-frequency characteristic.

Establishment of discrete expression of Lag Time (LT)

Convert the phase-frequency characteristic of Bode diagram into LT, as shown in equation (19).

$$D(f)=\left|\frac{\theta (f)}{360·f}\right|$$

(19)

where, $D\left(f\right)$ represents the LT, in seconds; θ represents the angle value on the phase-frequency curve; $f$ is the frequency point, in Hz.

In the phase-frequency curve, take the frequency range $\left[\begin{array}{cc}0,\hfill & \frac{f_s}{2}\hfill \end{array}\right]$ . The interval is discretized into L frequency points according to the frequency interval of $\frac{f_s}{2}\frac{1}{L}$ . The LT is discretized into a matrix form as shown in equation (20).

$$\mathit{D}(i)={\left[\begin{array}{ccccc}{D}_1\hfill & D_2\hfill & \cdots \hfill & D_{L-1}\hfill & D_L\hfill \end{array}\right]}^T$$

(20)

where, $\begin{array}{ccccc}{D}_1\hfill & D_2\hfill & \cdots \hfill & D_{L-1}\hfill & D_L\hfill \end{array}$ represents the LT corresponding to each frequency point $\left[\begin{array}{ccccc}0\hfill & \frac{f_s}{2}\frac{1}{L}\hfill & \frac{f_s}{2}\frac{2}{L}\hfill & \cdots \hfill & \frac{f_s}{2}\hfill \end{array}\right]$ on the phase-frequency characteristic.

Solving method flow of STFR

Equation (17) expresses the AAR at different frequencies. The attenuation effect of the servo control system is equivalent to scaling the setpoints time-frequency matrix according to the amplitude-frequency characteristics of the servo control system, that is, each row element of equation (7) is attenuated according to equation (18). Therefore, the attenuated STFR can be expressed as ${\mathit{W}}_{\mathit{p}}^{\mathit{H}}$ :

$$\begin{array}{c}{\mathit{W}}_{\mathit{p}}^{\mathit{H}}=\mathit{H}\times {\mathit{W}}_{\mathit{p}}=\hfill \\ \left[\begin{array}{cccc}{A}_{1,1}{H}_1\hfill & A_{1,2}{H}_1\hfill & \cdots \hfill & A_{1,n}{H}_1\hfill \\ \multicolumn{1}{c}{A_{2,1}{H}_2}& A_{2,2}{H}_2\hfill & \cdots \hfill & A_{2,n}{H}_2\hfill \\ \multicolumn{1}{c}{⋮}& ⋮\hfill & ⋮\hfill & ⋮\hfill \\ \multicolumn{1}{c}{A_{L-1,1}{H}_{L-1}}& A_{L-1,2}{H}_{L-1}\hfill & \cdots \hfill & A_{L-1,n}{H}_{L-1}\hfill \\ \multicolumn{1}{c}{A_{L,1}{H}_L}& A_{L,2}{H}_L\hfill & \cdots \hfill & A_{L,n}{H}_L\hfill \end{array}\right]\hfill \end{array}$$

(21)

where, $\mathit{H}(i)$ is the AAR of the servo control system, and ${\mathit{W}}_{\mathit{p}}$ is the time-frequency matrix of the setpoints.

According to the translation of wavelet transform, the wavelet coefficients of the delayed signal can be obtained by translating the wavelet coefficients of the original signal in the time sequence u . Under the action of the lag of the servo control system, each row element of the setpoints time-frequency matrix shown in equation (7) shifts to the right. Through dividing the LT $\mathit{D}(i)$ by the time resolution of the time-frequency matrix (the interval size of time sequence u ), the number of columns that all elements in each row should move right is obtained, as shown in equation (22). Fill the space with 0 after moving right.

$$\mathit{C}(i)=[D_1{f}_s{D}_2{f}_s\cdots D_{L-1}{f}_s{D}_L{f}_s]$$

(22)

After time lag as shown in equation (22), the STFR ${\mathit{W}}_{\mathit{p}}^{\mathit{D}}$ is shown in equation (23).

$${\mathit{W}}_{\mathit{p}}^{\mathit{D}}=\left[\begin{array}{ccccc}\stackrel{C(1)\mathrm{zeros}}{\overbrace{\begin{array}{cc}0\hfill & \cdots \hfill \end{array}}}\hfill & A_{1,1}\hfill & A_{1,2}\hfill & \cdots \hfill & A_{1,n}\hfill \\ \multicolumn{1}{c}{\stackrel{C(2)\mathrm{zeros}}{\overbrace{\begin{array}{cc}0\hfill & \cdots \hfill \end{array}}}}& A_{2,1}\hfill & A_{2,2}\hfill & \cdots \hfill & A_{2,n}\hfill \\ \multicolumn{1}{c}{\begin{array}{cc}⋮\hfill & ⋮\hfill \end{array}}& ⋮\hfill & ⋮\hfill & ⋮\hfill & ⋮\hfill \\ \multicolumn{1}{c}{\begin{array}{cc}0\hfill & A_{L-1,1}\hfill \end{array}}& A_{L-1,2}\hfill & \cdots \hfill & A_{L-1,n}\hfill & 0\hfill \\ \multicolumn{1}{c}{\begin{array}{cc}{A}_{L,1}\hfill & A_{L,2}\hfill \end{array}}& \cdots \hfill & A_{L,n}\hfill & \cdots \hfill & 0\hfill \end{array}\right]$$

(23)

where, $C(1),C(2)\cdots$ denotes the number of right-shift columns of elements with different rows, then ${\mathit{W}}_{\mathit{p}}^{\mathit{D}}$ should be a matrix of L row $\mathrm{Max}(\mathit{C})+n$ column, $\mathrm{Max}(\mathit{C})$ represents the maximum number in $\mathit{C}(i)$ .

Combining equations (21) and (23), the STFR attenuated and lagged by the servo control system can be obtained, which is expressed in matrix form ${\mathit{W}}_{\mathit{p}}^{\mathit{DH}}$ :

$$\begin{array}{c}{\mathit{W}}_{\mathit{p}}^{\mathit{DH}}=\hfill \\ \left[\begin{array}{ccccc}\stackrel{C(1)\mathrm{zeros}}{\overbrace{\begin{array}{cc}0\hfill & \cdots \hfill \end{array}}}\hfill & A_{1,1}{H}_1\hfill & A_{1,2}{H}_1\hfill & \cdots \hfill & A_{1,n}{H}_1\hfill \\ \multicolumn{1}{c}{\stackrel{C(2)\mathrm{zeros}}{\overbrace{\begin{array}{cc}0\hfill & \cdots \hfill \end{array}}}}& A_{2,1}{H}_2\hfill & A_{2,2}{H}_2\hfill & \cdots \hfill & A_{2,n}{H}_2\hfill \\ \multicolumn{1}{c}{\begin{array}{cc}⋮\hfill & ⋮\hfill \end{array}}& ⋮\hfill & ⋮\hfill & ⋮\hfill & ⋮\hfill \\ \multicolumn{1}{c}{\begin{array}{cc}0\hfill & A_{L-1,1}{H}_{L-1}\hfill \end{array}}& A_{L-1,2}{H}_{L-1}\hfill & \cdots \hfill & A_{L-1,n}{H}_{L-1}\hfill & 0\hfill \\ \multicolumn{1}{c}{\begin{array}{cc}{A}_{L,1}{H}_L\hfill & A_{L,2}\hfill \end{array}{H}_L}& \cdots \hfill & A_{L,n}{H}_L\hfill & \cdots \hfill & 0\hfill \end{array}\right]\hfill \end{array}$$

(24)

Definition and solving method of STFRE

Setpoints loss (SL)

In the research and application of servo control, attention should be paid not only to response, but also to error. The amplitude attenuation and phase lag of the servo control system to the setpoints results in a difference between the input and output, that is, SL. When expressing this loss in the time-frequency domain, it is referred to as STFRE.

By subtracting the time-frequency matrix of the original setpoints from the time-frequency matrix of the response, that is, equations (24) and (7), the difference matrix ${\mathit{W}}_{\mathit{p}}^{\mathit{E}}$ between the input and output is obtained, as shown in equation (25).

$${\mathit{W}}_{\mathit{p}}^{\mathit{E}}\mathbf{=}{\mathit{W}}_{\mathit{p}}\mathbf{-}{\mathit{W}}_{\mathit{p}}^{\mathit{DH}}$$

(25)

where, ${\mathit{W}}_{\mathit{p}}$ is the time-frequency matrix of the setpoints; ${\mathit{W}}_{\mathit{p}}^{\mathit{DH}}$ is the time-frequency response matrix of the servo control system to the setpoints.

Time lag leads to an increase in the number of ${\mathit{W}}_{\mathit{p}}^{\mathit{DH}}$ columns. To ensure the size of ${\mathit{W}}_{\mathit{p}}^{\mathit{DH}}$ and ${\mathit{W}}_{\mathit{p}}$ are consistent, ${\mathit{W}}_{\mathit{p}}$ is extend to the L row and $\mathrm{Max}(\mathit{C})+n$ column matrix, where $n+1$ to $\mathrm{Max}(\mathit{C})+n$ columns of the matrix are filled with 0 elements.

According to Moyal’s theorem, the integral of the wavelet coefficient square is proportional to the energy of the signal, and this proportional coefficient is related to the choice of wavelet basis. It means that although the wavelet coefficient is positively correlated with the amplitude of the signal, the value of the wavelet coefficient will change with the change of the wavelet basis function. Therefore, ${\mathit{W}}_{\mathit{p}}^{\mathit{E}}$ cannot be directly used as an indicator to measure the SL. A relative quantity, that is, the ratio of elements of the setpoints time-frequency difference matrix to element of original time-frequency matrix, should be constructed to define the SL generated by the servo control system. SL matrix ${\mathit{W}}_{\mathit{p}}^{\mathit{R}}$ can be expressed as:

$$\begin{array}{c}\multicolumn{1}{c}{{\mathit{W}}_{\mathit{p}}^{\mathit{R}}=\left[\frac{\alpha {W^E}_{\mathrm{ij}}}{W_{\mathrm{ij}}+\beta }\right]i=1,2,\cdots ,Lj=1,2,\cdots ,n+\mathrm{Max}(C)}\end{array}$$

(26)

where $W_{\mathrm{ij}}$ represents all elements in matrix ${\mathit{W}}_{\mathit{p}}$ of equation (7); ${W^E}_{\mathrm{ij}}$ represents all elements in matrix ${\mathit{W}}_{\mathit{p}}^{\mathit{E}}$ in equation (24); $\alpha$ and $\beta$ are the adjustment factors.

When establishing the SL matrix, normalization processing is carried out. However, since the AAR applied in calculating the setpoints response is already a ratio, simple normalization will make the SL only related to the amplitude frequency characteristics and independent of the setpoints itself, which will lead to the inconsistency between the loss of high-frequency part and its actual amplitude. Therefore, the adjustment coefficients $\alpha$ and $\beta$ are introduced into equation (26) to adjust the weight of two factors in the SL matrix: the attenuation caused by the amplitude-frequency characteristics of the system and the distribution of the frequency components of the setpoint itself.

In equation (26), if ${W^R}_{\mathrm{ij}}=0$ , it means that under frequency $i$ and time $j$ , the SL is 0 which means that the servo control system fully responds to the setpoints and the tracking error is 0; If $0<\left|{W^R}_{\mathrm{ij}}\right|<1$ , it means that there will be a certain SL through the servo control system. The closer ${W^R}_{\mathrm{ij}}$ is to 1, the greater the SL and the greater the tracking error.

Setpoints loss at each time (SLET)

The frequency integral of equation (26) is defined as the SLEL $P_t(t)$ :

$$P_t(t)=\int {\mathit{W}}_{\mathit{p}}^{\mathit{R}}\mathrm{df}$$

(27)

In equation (27), $P_t$ is a function of time, which characterizes the fluctuation of SL over time. If $P_t(t)=0$ , the servo control system fully responds to the setpoints at time t. If $P_t(t)=L$ , the setpoints are totally lost through the servo control system. In theory, the amount of SLEL should correspond to the tracking error in time-domain.

Equation (21) establishes the setpoints time-frequency matrix ${\mathit{W}}_{\mathit{p}}^{\mathit{H}}$ after amplitude attenuation, while equation (23) establishes the setpoints time-frequency matrix ${\mathit{W}}_{\mathit{p}}^{\mathit{D}}$ after time lag. These two matrices can be used to separate the losses caused by amplitude attenuation and phase lag. The SLET caused by the amplitude attenuation of the servo system is calculated by replacing ${\mathit{W}}_{\mathit{p}}^{\mathit{DH}}$ with ${\mathit{W}}_{\mathit{p}}^{\mathit{H}}$ in equations (25)–(27), which is recorded as $P_{\mathrm{tm}}(t)$ . In the same way, the SLET caused by the phase lag of the servo system is calculated by replacing ${\mathit{W}}_{\mathit{p}}^{\mathit{DH}}$ with ${\mathit{W}}_{\mathit{p}}^{\mathit{D}}$ , which is recorded as $P_{\mathrm{tp}}(t)$ .

Setpoints loss at each frequency (SLEF)

The time integral of equation (26) is defined as the SLEF $P_f(f)$ :

$$P_f(f)=\int {\mathit{W}}_{\mathit{p}}^{\mathit{R}}\mathrm{dt}$$

(28)

In equation (28), $P_f$ is a function of frequency. If $P_f(f)=0$ , it means that the servo control system fully responds to the setpoints at frequency f; If $P_f(f)=n$ , it means that the setpoints are completely lost through the servo control system. The SLEF reflects the contribution of the loss of each frequency to the global loss.

Verification

This section establishes the servo control model and designs the verification scheme to verify the solving methods for the STFR and the STFRE.

Servo control model and its AAR and LT

The servo dynamic model of x-axis of three-axis vertical milling machine established by Lyu et al. ²⁴ is adopted for verifying the solving method. In this model, the feed system is equivalent to a four rigid body dynamic model, as shown in Figure 1.

Figure 1.

The dynamic model of the x-axis of three-axis vertical milling machine.

The dynamic equation is shown in equation (29).

$$\mathit{M}\stackrel{··}{\mathit{X}}+\mathit{C}\stackrel{·}{\mathit{X}}+\mathit{KX}=F$$

(29)

where, M is the mass matrix; C is the damping matrix; K is the stiffness matrix; F is the external force; $\stackrel{··}{\mathit{X}},\stackrel{·}{\mathit{X}}$ and $\mathit{X}$ are acceleration, velocity and displacement vectors, respectively.

The control block diagram of the servo control system is shown in Figure 2. The current loop is equivalent to the first-order inertia element, the speed loop adopts PI control and the position loop adopts P control. On this basis, the speed feedforward controller is added.

Figure 2.

The control block diagram of the servo control system.

The speed loop transfer function is shown in equation (30):

$$G_v(s)=\frac{C_v(s)H_m(s)G(s)}{1+C_v(s)H_m(s)G(s)}$$

(30)

where, $H_m(s)=\frac{K_t}{T_{c}s+1}$ is the transfer function of current loop controller; $C_v(s)=K_{\mathrm{vp}}+\frac{K_{\mathrm{vi}}}{s}$ is the transfer function of the speed loop controller; $G(s)$ is the transfer function of the motor torque speed encoder.

The position loop transfer function of the servo control system is shown in equation (31).

$$G_p(s)=\frac{K_{\mathrm{vf}}{G}_v(s)s+K_p{G}_v(s)}{s+K_p{G}_v(s)}$$

(31)

where, $K_{\mathrm{vf}}$ is the speed feedforward controller coefficient; $G_v(s)$ is the transfer function of velocity loop; $K_p$ is the position loop gain.

The Bode diagram of the servo control system is drawn according to equation (31), as shown in Figure 3.

Figure 3.

The position loop Bode diagram of the servo control model.

According to equations (17) and (19), the AAR and LT of the servo control system are established, as shown in Figure 4.

Figure 4.

The AAR and LT at each frequency point.

Verification of the solving method for the STFR

Verification scheme

In this subsection, a basic signal is constructed, and the time-frequency response matrix of the signal is obtained by the proposed method and simulation experiments to verify STFR solution method, respectively.

Sinusoidal signal is a basic signal with the simplest frequency component. For the signal superimposed by sinusoidal signals of multiple frequencies, the components of frequencies can be well separated in the time-frequency diagram. Therefore, sinusoidal signals are used to verify the solving method of the STFR.

The signal expressed as equation (32) are designed as the setpoints inputted to the servo control system:

$$S(t)=\sin (16\pi t)+\sin (40\pi t)+\sin (80\pi t)$$

(32)

The signal includes three frequency components: 8, 20, and 40 Hz, and its amplitude are set to 1. The total duration of the signal is 2.5 s and the output frequency is 4000 Hz.

The designed verification scheme is as follows:

• (1) Calculate the time-frequency matrix of the signal described in equation (32), mark the sample points from it, and record their values and positions.

• (2) Input the signal described in equation (32) into the servo control model established in Section 6.1 to obtain the output signal. Calculate the time-frequency matrix of the output signal, and record the value and position of sample points in the output time-frequency matrix as the standard value.

• (3) Use the STFR solving method described in Section 4 to solve the STFR matrix. Record the value and position of sample points after attenuation and lag as the calculated value.

• (4) Compare the calculated values with the standard values to verify the solving method.

Verification results

Input $S(t)$ into the servo control model established in section 6.1 to obtain the output signal $S^{\prime }(t)$ . According to equations (1)–(7), the time-frequency matrixes of the input and output signals are established. Figure 5 shows the time-frequency diagram corresponding to the matrixes. In the figure, the abscissa is time whose resolution is 0.25 ms. The ordinate is frequency whose resolution is 1 Hz. And the color depth represents the value of wavelet coefficient.

Figure 5.

The time-frequency diagrams of input and output signals: (a) time frequency matrix of input signal and (b) time frequency matrix of output signal.

In order to accurately locate the sample points in both the input time-frequency diagram and the output time-frequency diagram, it is necessary to select some easily distinguishable points as sample points.

For sinusoidal signal, after CWT, when the frequency remains unchanged, the variation law of wavelet coefficient with time is still sinusoidal curve, and the interval between peak points on the curve is half of the cycle of sinusoidal curve. Therefore, in the time-frequency matrix of $S(t)$ , the minimum time interval between two peak points of the same frequency is 12.5 ms in the 40 Hz signal, which is much greater than the lag time. Taking these peak points as sample points, it is easy to find the corresponding points of input and output, which can accurately reflect the effect of servo system phase lag in the input time-frequency matrix. Table 1 shows the sample point information marked in the time-frequency matrix of the input signal $S(t)$ .

Table 1.

Sample point information.

Sample point	Frequency (Hz)/Number of rows	Time (s)/Number of columns	Wavelet coefficient value
1	8/8	0.34425/1377	25.1214
2	12/12	0.71875/2875	5.7614
3	20/20	1.288/5152	15.6949
4	25/25	1.83775/6252	9.3956
5	40/40	2.26925/9077	10.8301

The position and wavelet coefficient value of the corresponding sample point in the time-frequency matrix of the output signal $S^{\prime }(t)$ are taken as the standard values. Using the method described in Section 4, the time-frequency response matrix of the input signal $S\left(t\right)$ is calculated. The position of the sample points in the matrix and the value of the wavelet coefficient are taken as the calculated values. The standard values, calculated values and the errors are shown in Table 2.

Table 2.

Comparison of standard value and calculated value of each sample point.

		Calculated values		Standard values		errors
Sample point	Frequency (Hz)	Number of right shifted columns	Wavelet coefficient values	Number of right shifted columns	Wavelet coefficient values	Timeerror	Amplitude error
1	8	8	27.1796	8	27.1645	0	0.46%
2	12	7.8	6.7434	8	6.2069	2.5%	8.64%
3	20	15	20.7860	15	20.8408	0	0.26%
4	25	19.3	12.0860	18	12.3614	7.2%	2.22%
5	40	17.4	4.7434	18	4.4324	3%	7.01%

It can be seen from Table 2 that in sample points 1 and 3, the calculated values are very close to the standard values. However, there are some errors in sample points 2, 4, and 5. In the table, the number of columns shifted to the right represents the time lag. Because the sampling frequency of signal $S(t)$ is 4000 Hz, one column shifting corresponds to the time lag of 0.25 ms which is the time resolution in the time-frequency matrixes. For the sample points 2, 4, and 5, the number of right shifted columns are not the integers. The errors are caused by the non-integer number of right shifted columns. On the whole, the calculated values is in good agreement with the standard values, which verifies the solving method of STFR.

Verification of the solving method for SLET

The SLET should have a corresponding relationship with the tracking error at each time. In this section, two motion trajectories are designed to analyze the SLET. Inputting the two trajectory setpoints into the servo control model established in section 6.1, the tracking errors can be calculated. The corresponding relationship between SLET and tracking error at each time is analyzed to verify the proposed method.

Trajectory design and time-frequency transformation of its setpoints

Taking the to and fro motions as examples, two cases with different motion parameters are designed to carried out the time-frequency transformation. The main parameters of the motions are shown in Table 3

Table 3.

Parameters of the to and fro motions.

Cases	Stroke (mm)	The maximum velocity (mm/min)	The maximum acceleration (g)	Acceleration establishment time (ms)
1	−25 to 50	5000	0.3	100
2	0 to 50	5000/2500	0.3	100

The setpoints of the two motions are interpolated and collected by the KEDE GNC61 CNC system. The setpoints position, velocity and acceleration of the two motions are shown in Figure 6.

Figure 6.

Setpoints position, velocity and acceleration curves of the two motions.

According to the method described in Section 2, the time-frequency analysis of setpoints position sequences is carried out to construct the setpoints time-frequency distribution matrix. The matrix is L row and n column, corresponding to the time sequence and frequency sequence shown in equations (3) and (6). The corresponding time-frequency distribution diagrams are drawn, as shown in Figure 7.

Figure 7.

The time-frequency distribution diagrams of the setpoints of the two cases: (a) Case 1 and (b) Case 2.

Corresponding relationship between SLET and tracking error at each time

According to the method described in Section 5.2, the SLET $P_{\mathrm{tm}}(t)$ , $P_{\mathrm{tp}}(t)$ , and $P_t(t)$ are solved, as shown in Figure 8.

Figure 8.

SLET of the two cases: (a) $P_t$ , $P_{\mathrm{tm}}$ , and $P_{\mathrm{tp}}$ of Case 1 and (b) $P_t$ , $P_{\mathrm{tm}}$ , and $P_{\mathrm{tp}}$ of Case 2.

The value of $P_t(t)$ reflects the SLET. The ratio of its value to the length L of the setpoints analysis frequency sequence can approximately represent the percentage of the SL at time t.

In the inverse transformation equation (12), the derivative is derived for time on each side of the equal sign. The left side is the setpoints speed, while the right side is frequency integral of the product of wavelet coefficients and wavelet basis. Considering that the SL matrix in equation (26) is a relative quantity in the calculation of $P_t(t)$ , the influence of scale factor and wavelet basis is removed, and $P_t(t)$ represents the difference between input and output of servo system. Therefore, $P_t(t)$ should show a corresponding relationship with the speed tracking error of the servo control system.

The setpoints of the two cases are inputted into the servo control model established in Section 6.1 to obtain the speed tracking error. Figure 9 shows the speed tracking error and $P_t(t)$ for comparison. It can be seen from the figure that the change trends of the two curves are very consistent, which verifies the solving methods of the STFR and STFRE.

Figure 9.

Comparison between speed tracking error and $P_t$ : (a) Case 1 and (b) Case 2.

Combined with the results of Figure 9, given that the initial value of trajectory displacement is 0, the integral of $P_t(t)$ to time should show a linear correlation with the position tracking error. Figure 10 shows $\int P_t(t)\mathrm{dt}$ and position tracking error for comparison. The value of $\int P_t(t)\mathrm{dt}$ here has no physical significance and is only used as a reference for the change trend. Again, two curves show the same trend, which is in line with the expected results.

Figure 10.

Comparison of position tracking error and $\int p_t(t)\mathrm{dt}$ : (a) Case 1 and (b) Case 2.

$P_{\mathrm{tm}}\left(t\right)$ and $P_{\mathrm{tp}}(t)$ represent the SL caused by amplitude attenuation and phase lag, respectively. The tracking error caused by amplitude attenuation and phase lag can be separated through $\int P_{\mathrm{tm}}(t)\mathrm{dt}$ and $\int P_{\mathrm{tp}}(t)\mathrm{dt}$ . The calculation results are shown in Figure 11. By calculating the ratio of $\int P_{\mathrm{tm}}(t)\mathrm{dt}$ or $\int P_{\mathrm{tp}}(t)\mathrm{dt}$ to $\int p_t(t)\mathrm{dt}$ , the proportion of the tracking error caused by amplitude attenuation or phase lag can be separated.

Figure 11.

Tracking error under the action of amplitude attenuation or phase lag: (a) Case 1 and (b) Case 2.

Discussion

After the verification of the solving methods for STFR and STFRE, its theoretical significance and application prospect will be discussed in this section.

Theoretical significance of STFR and STFRE

Figure 12 shows the STFR diagram and STFRE diagram of the two cases. The diagrams of STFRE clearly show the SL at each time and the SL of each frequency. For a certain time, the loss of each frequency component is caused by the amplitude-frequency characteristics and phase-frequency characteristics of the servo control system. The SL at a certain time corresponds to the tracking error at that time. This analysis method realizes the unification of the time-domain phenomenon and frequency-domain mechanism of the tracking error.

Figure 12.

The STFR diagram and STFRE diagram of the two cases.

Theoretical significance and application prospects of SLET

The tracking error is the result of the simultaneous action of amplitude attenuation and phase lag of the servo system, which cannot be distinguished only by traditional methods such as time-domain comparison or frequency-domain FFT analysis. The solution method of SLET can separate the effects of amplitude attenuation and phase lag on the setpoints.

As shown in Figure 11, $P_{\mathrm{tm}}$ and $P_{\mathrm{tp}}$ correspond to the SL caused by amplitude attenuation and phase lag, respectively. Based on $P_{\mathrm{tm}}$ and $P_{\mathrm{tp}}$ , the proportion of the tracking error caused by amplitude attenuation and phase lag can be separated, which may play an important role in optimizing the setpoints, determining the performance defects of the servo control system and studying the mechanism of the tracking error.

Theoretical significance and application prospect of SLEF

Generally, the frequency component loss of an input signal passing through the servo control system can be characterized by the difference between the input and output Fourier transform. This method is actually equivalent to making the signal pass through a system with zero phase lag, only considering the effect of the amplitude frequency characteristics, but losing the action process of phase lag. It is obvious that the calculation method of frequency component loss based on Fourier transform is deficient. The SLEF proposed is helpful to explore the relationship between the SL in the frequency-domain and the tracking error.

According to the method described in Section5.3, the SLEF $P_f(f)$ of the two cases is calculated as shown in Figure 13. The value of $P_f(f)$ reflects the global contribution of the SL at the frequency f. The positive value indicates that there is a loss in the setpoints at this frequency; The negative value indicates that the amplitude of the setpoints is amplified. The ratio of its value to the setpoints time sequence length n can approximately represent the percentage of loss of the setpoints at that frequency. The peak and inverse peak in $P_f(f)$ curve not only represent the performance defects in the frequency-domain of the system at this frequency, but also indicate that this frequency band is the frequency band with concentrated setpoints energy.

Figure 13.

The SLEF of the two cases: (a) Case 1 and (b) Case 2.

It can be seen from Figure 13 that the dominant SL occurs near 15 and 40 Hz. It means the servo control system has insufficient performance near the frequency points, meanwhile the setpoints energy is concentrated around the frequency points.

Compared with the amplitude-frequency characteristics of Bode diagram, $P_f(f)$ additionally considers the distribution of the amplitude (energy) of the setpoints itself, as shown in Figure 14.

Figure 14.

Comparison between $P_f(f)$ of Cases 1 and amplitude-frequency curve of servo control system.

Adding filter controller is a common method to improve the servo bandwidth and to reduce the tracking error. Usually, the frequency band of the filter should be judged by amplitude-frequency characteristic of the servo control system. As shown in Figure 14, only from the perspective of the amplitude-frequency characteristics, the frequency bands near 16, 41, 318, and 428 Hz may be the dominant frequency band that should be filtered. However, if the $P_f(f)$ curve is used as a reference, for this trajectory setpoints, the main SL occurs near 16, 41, and 82 Hz. Measured by the amount of SL in frequency, dealing with the frequencies of 16, 41, and 82 Hz has much greater benefit than the frequencies of 318 and 428 Hz. In addition, the amplitude-frequency curve does not give the information that the frequency of 82 Hz should be considered.

A high-precision servo control system should produce the minimum distortion of the input. Considering the impossibility of the ideal filter, the spectrum of the real filter cannot be completely box shaped, and the unnecessary addition of filter may deteriorate the system performance. $P_f$ can clearly reflect which frequency band needs to be optimized priority for a specific setpoints, and which filters are redundant for the tracking error.

Solving process

In order to clearly present the computational flow of the proposed method, the solution flow of STFR and STFRE are plotted as shown in Figure 15.

Figure 15.

The solving process.

Conclusions

In this paper, a method for solving STFR and STFRE based on the frequency domain characteristics of servo control systems is proposed, which can be used for the analysis of the time-frequency response process of actual motion setpoints.

• (1) First, the setpoint is transformed from the time domain to the time-frequency domain by CWT; Then, the AAR expression and LT expression are constructed of the servo control system based on the amplitude-frequency characteristics and phase-frequency characteristics; Finally, the STFR is solved by applying amplitude attenuation and time lag to each frequency component of the setpoints at each moment.

• (2) The STFRE including SL, SLET and SLEF are constructed. The former is the ratio of the difference between the setpoints time-frequency matrix and the response time-frequency matrix to the setpoints time-frequency matrix, which represents the loss of each frequency component of the setpoints at each time; The latter two are the integration of SL to frequency and time respectively, representing the setpoints loss at each time and the contribution of each frequency loss to the global loss.

The theoretical significance and application prospects of the present method were found by solving the time-frequency response of two sets of real setpoints:

• (1) Compared with the existing analysis methods, the proposed method is not limited to the type of wavelet basis functions and can be applied to the solution of arbitrary STFR.

• (2) The time-frequency response solution method is based on the amplitude decay and phase lag of the servo system to the setpoints and is able to separate the SL or tracking error caused by both, and analyze the cause of the tracking error in depth.

• (3) The solving method of STFR can determine the redundancy and deficiency of filter and other strategies for a setpoints and servo control system.

The proposed method introduces the adjustment coefficients $\alpha$ and $\beta$ , which is to ensure the correct measurement of the SL while completing the normalization of the SL matrix. However, the current value standards of $\alpha$ and $\beta$ still rely on manual adjustment, and the tuning process needs to be optimized in future research.