Research on the control method of the structural stability of the precision tiled-grating

Abstract

In a chirped pulse amplification device, tiled grating is a key component for compressing ultrashort pulse laser. The control system of the grating tiling device is essential for maintaining the stability of the array grating. In this study, a proportional-integral-derivative algorithm based on the latch compensation method and four-point center difference method is proposed for a truss-type 2 $\times$ 1 grating tiling device. Numerical analyses show that this method avoids the saturation loss in the stability control process of the array grating, effectively suppresses noise interference, and ensures the stability of the array grating. The prototype experiment shows that the use of this control algorithm can significantly improve the convergence speed and stability of the grating tiling device. Specifically, compared with a mechanism without a control algorithm, the time to reach the reference position for the first time is reduced by 39.1%, and the stability of the grating device is improved by 58.5%.

Keywords

Chirped pulse amplification tiled grating stability control method PID

1. Introduction

Chirped pulse amplification (CPA) is an important technique for realizing the energy amplification of ultrashort pulse laser [1, 2, 3, 4]. However, the damage threshold and aperture of the grating in the CPA compression cell limit the energy of the output pulse [5, 6, 7]. Presently, the grating that offers the best performance is a multilayer dielectric (MLD) holographic grating [8, 9]; however, it is difficult to process it to meter size. Therefore, researchers mostly use the grating tiling method to obtain large aperture array gratings to improve the output pulse energy [10, 11].

Array grating involves the tiling of more than two small gratings as a single large grating to realize the function of a single large grating [12, 13]. However, it becomes a challenge when controlling the tiling error [14, 15]. This study aims to realize the similarity between array grating and single integral grating, as well as the spatio-temporal stability of array grating structure [16]. Maximum similarity includes two aspects: geometric similarity and physical similarity (Fig. 1). Geometric similarity refers to the similarity between an array grating composed of sub-gratings and a single overall grating in a geometric structure, which is the basis of physical similarity. The physical similarity requires that the effect of the beam passing through the array grating is equivalent to that passing through a single overall large grating to meet the requirements of the high-energy short-pulse laser system. Temporal and spatial stability also includes two aspects: 1) instability caused by environmental micro-vibration, transient temperature change, and other disturbances; 2) structural accuracy drift with time. The structural accuracy drift significantly impacts the micro–nano precision required structure and presents uncertainty.

Figure 1.

Relationship about the accuracy and spatial-temporal stability of tiled-grating.

To realize the stability control of the grating, the relative spatial position of the sub-grating must be detected in real time. To achieve this goal, some researchers have proposed a scheme for integrating a monitoring light into the compressor [17, 18]. However, Zuo [19] indicated that the method has defects, and the position of the array grating is easily disturbed by environmental factors. When a slight disturbance occurs, because the monitor light passes through the array grating twice, the monitoring light and the main laser are displaced, which makes it difficult to reflect the posture of the grating after the disturbance. Moreover, because the monitoring light and the main laser have different light sources and are far apart when disturbances occur, the environmental excitations of the two light sources are different. This explains the inconsistencies between the monitoring light and the main laser [20, 21]. A study by Wang Xiao revealed that the wavelength and incident angle of the light used for monitoring and adjustment must be consistent with the main laser to reflect the state of the main laser because the phase difference between the sub-gratings is directly affected by the incident light wavelength and incident angle [17]. Specifically, the monitoring adjustment light must occupy the optical path of the main laser; otherwise, a multi-wavelength and multi-level light must be adopted to separate various errors. When the main laser is running and shooting, the light guide mirror that monitors the optical path cannot be inserted into the main optical path [22]. Owing to the limitations of real-time monitoring of array gratings using optical methods, the tiling system must have sufficient stability. This self-stability requirement should be achieved using non-optical methods.

Regarding the requirements of stability, Zhu [23] explained the overall stability design of the shooting range and assigned the stability of the overall laser-shooting device.

The space-time stability of the array grating can be expressed by Eq. (1):

$\displaystyle S=S[u_{\textit{env}}(t),u_{\textit{str}}(t),u_{\textit{con}}(t)]$ (1)

where $S$ is the overall stability of the array grating, which is a function of the environmental incentives $u_{\textit{env}}$ , structural factors $u_{\textit{str}}$ , and control factors $u_{\textit{con}}$ . Environmental factors are referred to as exciting vibration factors, and structural factors and control factors as shock-absorbing factors because they are targeted to improve the stability of the tiling system during the design process.

For precision optomechanical systems, the environmental load is a problem that must be considered when studying the stability of optomechanical systems. Environmental factors $u_{env}$ include three parts: background vibration $u_{\textit{bac}}$ , temperature $u_{\textit{tem}}$ , and accidental factors $u_{\textit{ran}}$ , which is expressed as:

$\displaystyle u_{\textit{env}}=\begin{bmatrix}u_{\textit{bac}}\\ u_{\textit{tem}}\\ u_{\textit{ran}}\\ \end{bmatrix}$ (2)

The stability of the excitation source refers to the fluctuation of the load of the excitation source, which is characterized by its effect on the stability of the optical element. Because of the excitation of the external environment, the array grating produces an excited response, which causes position error. Consequently, the beam deviates from the predetermined position. By improving the structure of the tiling device and optimizing the control scheme, the stability of the tiling device can be improved in a targeted manner.

In the analysis and research on the stability of the optical components of large solid-state laser devices, the excitation factors that affect the optical components were divided [24].

Figure 2.

Classification of effect of environmental excitation.

As shown in Fig. 2, for the factors in Level 1, the influence of vibration, transient temperature, and contingency was conservatively included. Hence, for the total error, the stability can be expressed as follows [25]:

$\displaystyle\Delta_{\textit{Drift}}=\Delta_{\textit{Structural}}+\Delta_{% \textit{Thermal\_Transient}}+\Delta_{\textit{Contingency}}$ (3)

where:

$\Delta_{\textit{Drift}}$ : the total error caused by all drifts;

$\Delta_{\textit{Structural}}$ : the error caused by random vibration;

$\Delta_{\textit{Thermal\_Transient}}$ : the error caused by the thermal effect; and

$\Delta_{\textit{Contingency}}$ : the error caused by other accidental factors.

The two external loads, vibration and temperature, affect the tiling device in the form of displacement caused by the load, which is expressed as an absolute value from a numerical point of view.

The structural factor ( $\Delta_{\textit{Structural}l}$ ) is primarily the excitation generated by structural creep. Under constant temperature and stress, the creep process can be divided into three stages: 1. decelerated, 2. constant speed, and 3. accelerated creep stages according to the change of creep rate. To ensure that the structure is controllable, we make the grating tiling device in the constant speed creep stage when it works via aging treatment, structural optimization design and other methods. Thus, it can be suitably controlled.

The influence of temperature factors ( $\Delta_{\textit{Thermal\_Transient}}$ ) on the stability of optical elements, especially the influence of daily temperature change, is reflected in the low frequency, which is a slow changing process. Owing to the daily temperature change, the optical elements are affected by the thermal deformation of the building structure and the change period is one day. Such deformation can be adjusted according to the daily temperature change rule.

In this study, the control method addresses the stability problem in a short time, such as a few seconds, namely, the problem of controlling the vibration ( $\Delta_{\textit{Contingency}}$ ) generated by random factors.

Notably, the existence of uncertain factors such as accidental factors cannot be determined in advance, and the control system needs to be self-adjusted in real time to ensure the overall stability of the system.

2. Control system design of grating tiling device

The main purpose of the design of the grating tiling device control system is to ensure the stability, accuracy, and real-time requirements of the sub-grating adjustment. As shown in Fig. 2, the control system of the grating tiling system is mainly composed of a controller, an actuator, and a pose control algorithm. The controller is used to realize the calculation and judgment of the control signal of the respective degree adjustment amount between the sub-gratings during the entire tiling process and after the tiling adjustment is completed. The control signal is then outputted to the actuator to ensure that the movement of the actuator is consistent with the received control signal. The grating makes the corresponding displacement drive realize the adjustment of the corresponding degree of freedom. The detection device detects the pose of the moving grating in real time during the entire tiling adjustment and dynamic stabilization adjustment process. It also processes the detected data to perform the pose control algorithm processing, which is then transmitted to the controller to ensure that the controller can control the moving grating. The pose of the camera is then adjusted in real time. The pose control algorithm is fundamental to the whole system. It determines the stability, accuracy, and real-time performance of the grating pose control to a large extent.

Figure 3.

Process of the precision tiling device.

In this study, a closed-loop control system is adopted in the control scheme. As shown in the figure, before the grating tiling device is introduced into the laser compressor, the displacement value should be calibrated by measuring the light spot. This calibration value is used as the adjustment target of the control system. When the tiling grating works in the compressor, the closed-loop control system always acts on the grating tiling device, collects the sensor displacement signal in real time, compares it with the target value, and obtains the deviation signal. According to the deviation signal, it is performed on the system to ensure its stability.

This design uses the proportional-integral-derivative (PID) algorithm as the control algorithm for the grating tiling system. This algorithm has a simple structure and good robustness. It is a classic control theory in automatic control and is widely used in the control of production processes [26, 27, 28, 29]. In production practice, control objects are often complicated and uncertain [30]. Because the traditional PID algorithm structure is relatively simple, it is unable to ascertain the complexity and uncertainty of the control object in the production application process. Furthermore, the anti-jamming performance is poor and causes problems such as saturation and loss [31]. To improve the control accuracy and enhance the anti-interference ability, traditional PID is generally improved using methods such as incomplete differentiation and integral separation [32].

In engineering practice, the incremental PID algorithm and positional PID algorithm are two forms of digital PID. Compared with the positional PID algorithm, the incremental PID algorithm has the advantages of small cumulative calculation error and strong anti-interference ability, and it can easily realize impact-free manual/automatic conversion [33]. The digital form of the incremental PID algorithm is as follows [34]:

$\displaystyle\Delta u(n)=u(n)-u(n-1)=K_{p}[e(n)-e(n-1)]+K_{p}\frac{T}{T_{i}}e(% n)+K_{P}\frac{T_{d}}{T}[e(n)-2e(n-1)+e(n-2)]$ (4)

where,

$\Delta u(n)$ : the output value at the $n$ th sampling time;

$e(n)$ : the deviation of the $n$ th sampling;

$K_{p}$ : proportional coefficient;

$K_{P}\frac{T}{T_{i}}$ : integral coefficient; and

$K_{P}\frac{T_{d}}{T}$ : differential coefficient.

Although the incremental PID algorithm has several advantages, it presents challenges, such as integral saturation, and its anti-interference ability requires further improvement [35]. The grating tiling system requires the PID algorithm to achieve high control accuracy and stable performance. This study adopts the “latch compensation method” and the “four-point center difference method” to adjust the parameters of the incremental PID algorithm as a closed-loop control algorithm for grating tiling.

2.1 Latch compensation method

The incremental PID algorithm solves the integral saturation problem in the positional PID algorithm. However, in the control system, the operating range of the actuator is limited. When the control system fails or the external environment undergoes a sudden change, the output of the actuator exceeds the operating range of the actuator, resulting in a partial increase in the system saturation. Consequently, Information is lost [35].

The control precision of the grating tiling system is extremely high, and the stroke and resolution of the actuator are required to reach the sub-micron level. The fine adjustment of the structure, failure, and sudden change in environmental excitation can lead to the loss of saturation. In this regard, this study adopts the "latch compensation method" for parameter tuning of incremental PID to solve the saturation loss of the grating tiling system in the automatic control process.

Theoretically, the control signal output by PID can be infinite. However, in practice, the actuator has an objective limitation. Saturation occurs when the output signal reaches this limit. Above this limit, the excess is discarded, which is called saturation loss. The tiling grating involved in this study is prone to integral saturation owing to its high control accuracy and relatively small actuator limit. If the excess is discarded at this time, deviation in the error calculation of the tiling grating position will occur. Owing to this deviation, the control that should be performed in the next adjustment will be ignored, and the target value used in the control will deviate from its actual value. Thus, the expected goal will not be achieved. The latch compensation method described in this work is a targeted design to overcome saturation loss.

Figure 4.

Latch compensation method.

The control principle of the latch compensation method is shown in Fig. 4. The aim is to latch the $n$ time lost increment $\Delta y(n)$ by the latch, and subsequently sum it with the next output as the $n+1$ time control of the total output quantity. Regarding $u_{\min}$ and $u_{\max}$ as the control limit quantity of the actuator, the control quantity of the n time is $u(n)$ , and the calculated increment is $u(n)$ . The $n-1$ time lost increment for the $n-1$ time is $\Delta y(n-1)$ , and the control quantity of the first time can be calculated by Eq. (5):

$\displaystyle u(n+1)=u(n)+\Delta u(n)+\Delta y(n-1)$ (5)

If $u(n+1)>u_{\max}$ (or $u(n+1)<u_{\min}$ ), the $n+1$ time actual control quantity is $u_{\max}$ (or $u_{\min}$ ). Then, the system is saturated and lost, and the lost increment is

$\displaystyle\Delta y(n)=u(n+1)-u_{\max}\ \text{or}\ \Delta y(n)=u_{\min}-u(n+1)$ (6)

Here, the incremental information $\Delta y(n)$ is latched by the latch and added to the control quantity of $u(n+2)$ , etc.

2.2 Four-point central difference method

The digital PID algorithm is an approximation of the analog PID algorithm, which replaces the integral and differential terms in the analog PID control formula with sum and difference terms, respectively [36]. Because the differential term in the digital PID is very sensitive to noise and data errors, the appearance of interference may cause significant changes in the control amount [37]. As an indispensable part of the PID regulator, the differential component cannot be simply discarded. Although the working environment of the array grating has adopted measures, such as noise reduction and vibration isolation, there remain many uncertain factors, such as sudden changes in the vibration frequency of the foundation and the air conditioner, and the disturbance of the on-site staff. To suppress random interference and obtain better system stability, this study adopts the “four-point central difference method” to set the parameters of the differential term in the PID control formula to improve the stability of the array grating for a long time.

Figure 5.

Four-point center difference method.

As shown in Fig. 5, the four-point central difference method has to set the value of the differential coefficient $K_{p}\frac{T_{d}}{T}$ to be slightly smaller than the ideal state. When constructing the difference, the current deviation $e(n)$ is not used; however, $e(n)$ and the past four deviation values are averaged to obtain $\overline{e(k)}$ as the reference, namely:

$\displaystyle\overline{e(k)}=\frac{1}{4T}[e(k)+e(k-1)+e(k-2)+e(k-3)]$ (7)

The approximate differential term is obtained using the weighted summation:

$\displaystyle\frac{\Delta\overline{e(k)}}{T}=\frac{1}{4T}\left(\frac{e(k)-% \overline{e(k)}}{1.5}+\frac{e(k-1)-\overline{e(k)}}{0.5}+\frac{\overline{e(k)}% -e(k-2)}{0.5}+\frac{\overline{e(k)}-e(k-3)}{1.5}\right)=\frac{1}{6T}[e(k)+3e(k% -1)-3e(k-2)-3e(k-3)]$ (8)

Equation (7) is substituted for the different term $\frac{1}{T}[e(n)-2e(n-1)+e(n-2)]$ in Eq. (4), which is decomposed into two first-order differences: $\frac{1}{T}[e(n)-e(n-1)]$ and $-\frac{1}{T}[e(n-1)-e(n-2)]$ .

$k$ is replaced with $n$ and $n-1$ , and the improved incremental PID calculation formula can be obtained:

$\displaystyle\Delta u(n)=K_{P}[e(n)-e(n-1)]+K_{P}\frac{T}{T_{i}}e(n)+K_{P}% \frac{T_{d}}{6T}\begin{bmatrix}e(n)+2e(n-1)-6e(n-2)\\ +2e(n-3)+e(n-4)\\ \end{bmatrix}$ (9)

It can be observed from Eq. (9) that the incremental PID control algorithm is modified by the four-point central difference method, and the obtained control increment contains a total of five continuous deviations $e$ from the present and the past.

After adjustment, the deviation $e$ , that is regarded as the benchmark became $\bar{e}$ , and the adjusted benchmark was smoothed. In particular, its mean value remained unchanged, but its variance changed to one-fourth of the original, and the fluctuation range reduced by half, namely, $\textit{PV}_{\bar{e}}=\frac{1}{2}\textit{PV}_{e}$ .

The calculation process is described below:

According to the central limit theorem of independent identical distribution and the law of large numbers, the random variable obeys the normal distribution when the sampling value is large enough. Therefore, it can be assumed that the expected value of e before setting is $E(e)$ and the variance is $D(e)$ . Thus,

$\displaystyle e\sim N(E(e),D(e))$

According to Eq. (6), after adjustment

$\displaystyle\overline{e(k)}=\frac{1}{4T}[e(k)+e(k-1)+e(k-2)+e(k-3)],$

If $T=$ 1, $\bar{e}$ is the average of four continuous sampling quantities.

Now, $e\sim N(E(e),\frac{1}{4}D(e))$ .

As shown in Fig. 6, as the reference value, the vibration center did not change, but the vibration amplitude (PV value) reduced to half its original value.

Figure 6.

The deviation $e$ before and after adjusted.

To accurately compare the algorithms before and after the tuning, a simulation program is designed, and the controlled objects are as follows:

$\displaystyle G(s)=\frac{400}{s^{2}+50s}$ (10)

Because short-term fluctuations are reduced, the impact of environmental disturbances is also suppressed to a certain extent. Figure 7 shows the response of the algorithm to the step signal, and its response speed is also enhanced.

Figure 7.

Unit-step response of the improved and classical increment PID algorithm.

3. Experiment

3.1 Tiling-grating device

To tile the gratings successfully and make them equivalent to the function of a single large grating, the following prototype of the grating tiling system is designed (Fig. 8). Functionally, the system is composed of a grating tiling module, detection module, and control module. The structure of the system is composed of a grating tiling device, drive controller, industrial computer, and control system program.

Figure 8.

Prototype of grating tiling system.

In this system, the control module judges the relative position and attitude of the array grating according to the feedback value of the detection module. The control module is composed of an industrial computer, piezoelectric actuator, and a control algorithm. To realize the control and adaptive adjustment of grating tiling, a grating control program is designed. The control program can be adjusted manually and automatically, display the sensor indication and spot image simultaneously, and set and save the sampling time of the sensor and CCD. The automatic adjustment program adopts the improved incremental PID control program, calculates the current position and attitude of the grating through the detection value returned by the sensor, and adjusts the moving grating in real time to ensure the stability of the beam pointing.

Table 1

The key equipment

Equipment	Parameters or models
Light source	532-nm laser
Sensors	D-100 capacitive displacement transducer (Measuring range 0.1mm, resolution 0.0005%)
Actuators	PSt 150/4/100 VS20 piezoelectric actuators (manufactured by Harbin Core Tomorrow Science & Technology Co., Ltd.)
Controller	XE-500/501 PZT controller (manufactured by Harbin Core Tomorrow Science & Technology Co., Ltd.)

Figure 9.

Displacement of mirror when the control program is in work and not.

The detailed information is listed as Table 1.

3.2 Real-time stability test

Real-time stability refers to the ability of the grating to re-adjust itself to the specified position using external forces when it deviates from the specified position in the working process.

The grating tiling device requires the laser compressor (CPA) to pull the grating back to the reference position within 10 s, and its vibration range does not exceed 70 nm. However, we hope that the faster the convergence speed, the smaller the vibration range. Furthermore, if the control system fails, the device must still work. Therefore, this experiment verifies the adaptive control and adjustment performance of the grating tiling device, and compares the stability of the array grating with or without closed-loop control.

The test index for this experiment is the displacement value of the mirror surface. The experiment first adjusts the mirror to the specified position and subsequently locks the grating. Two sets of experiments are then performed. In the first step of the experiment, the grating is maintained at a working position for some time. After the position of the light spot is stabilized, the sensor value at that position is used as the stable state reference value of the mirror. Here, an impact is applied to the mirror surface, which forces it to produce jitter. The displacement of the mirror is collected at this point, and the other group turns on the actuator and sets it to the closed-loop automatic control state. An impact, which is similar to the previous experiment, is imposed on it, and the displacement of the mirror is collected in the same length of time.

As shown in Fig. 9, the test results under the control program on and off conditions are compared with the reference values. It can be seen that the speed of the mirror surface reaching a stable state (hereinafter referred to as convergence speed) is faster when control exists and is faster. After stabilization, the displacement of the mirror surface becomes smaller.

Table 2
Displacement of the mirror in the stable situation (nm)

	Max	Min	PV	Average	Positive offset (average)	Negative offset (average)	Standard deviation
Uncontrolled	65	12	53	36	25	20	25
Controlled	43	21	22	32	23	18	20

Figure 10.

Comparison of convergence speed when the control program is in work and not.

Figure 11.

Vibration displacement when the mirror is in a stable situation.

To accurately examine the convergence speed of the mirror and the displacement in the steady-state, the above figure is divided into two parts: the convergence period (0–10 s) and the stable period (30–60 s).

3.2.1 Convergence rate analysis

As shown in Fig. 10, the convergence of the displacement value can be clearly observed in the interval of 0–10 s. The stable reference value of the mirror is set at 36 nm. When the control program is turned on, the mirror surface reaches the reference position for the first and second time at 5.3 s and 6.4 s, respectively, after which the mirror position fluctuates in a stable interval. When the control program is off, the first time the mirror reaches the reference position is at 8.7s. This shows that the work of the control program has advanced the mirror position by 3.4s and converges to a stable position. The time for the mirror to reach the reference position for the first time is also reduced by 39.1%.

$\displaystyle\eta=\frac{3.4}{8.7}=39.1\%$ (11)

3.2.2 Steady-state analysis

As shown in Fig. 11, the mirror position stabilizes in the 30–60 s interval. The figure shows that, in a stable time, the position of the mirror reciprocates near the reference line, and because of the control program, the vibration range of the mirror is narrower.

The specific values are listed in Table 2. After the control program is turned on, the average displacement value of the device changes from 36 to 32 nm, and the standard deviation decreases from 25 to 20 nm. This is because under the influence of the control program, the device is affected by the piezoelectric actuator, and the device no longer performs free vibration, which is the result of the joint action of free vibration and the piezoelectric actuator. According to the program setting, when the device deviates from the reference value to a certain extent, the piezoelectric actuator pulls the mirror back to the reference position, thereby narrowing the vibration range of the mirror. The narrowing of its vibration range can also be reflected in the maximum, minimum, and peak-valley (PV) values of the amplitude under the two conditions.

The sensor indication value is then compared with the reference value. If the sensor indication value is greater than the reference value, the offset is called a positive offset; otherwise, it is called a negative offset. As shown in Table 1, without the control program, the average positive and negative offsets of the mirror are 25 nm and 20 nm, respectively. After the control program is turned on, the two values are reduced to 23 nm and 18 nm, respectively. This further illustrates that the control program improves the stability of the grating tiling device. Based on the PV value, the stability is improved by 58.5%.

4. Conclusion

This study proposed a PID control algorithm that combines the four-point central difference method and the latch compensation method for the stability control of the precision array grating. This algorithm effectively eliminated the loss of saturation during the stability process and reduced the impact of environmental disturbances.

The grating prototype experiment showed that the grating device using this control algorithm could recover to a stable position faster and had a superior stable state after being subjected to an external impact. Specifically, compared with a mechanism without a control algorithm, the time to reach the reference position for the first time is reduced by 39.1%, and the stability of the grating device is improved by 58.5%.

Footnotes

Acknowledgments

This paper was supported by the Science and Technology Research Program of Chongqing Municipal Education Commission. (Grant No. KJQN202203210).

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