Robust control for temperature variation of the special suction cup based on feedforward decoupling algorithm

Abstract

In order to improve the stability and consistency for the surface temperature rise of heating suction cup about silicon carbide wafer, the study adopts the special suction cup based on metal oxide conductive coating heating. It puts forward the feedforward decoupling of Smith fuzzy multi-level integral separation PID based on the four-region temperature control system, which improves the problem about the non-uniformity for the surface temperature rise of heating the special suction cup due to the coupling interference between each region. First, a mathematical model of the temperature control system is established based on the heating structure and heat transfer mechanism of the special sucker. Then, Smith fuzzy multi-level integral separation PID simulation models with and without the feedforward decoupling algorithm are built, and comparative experiments are carried out, with stability analysis conducted using Bode plots. Afterwards, experimental tests are performed on the field of the special sucker temperature control system built on the basis of a PLC controller. Both simulation and test results show that the proposed control algorithm effectively solves the problem of uneven surface temperature of the special sucker and achieves robust temperature control based on the feedforward decoupling algorithm.

Keywords

robust control temperature variation special suction cup PID feedforward decoupling algorithm

Introduction

With the continuous advancement of industrial automation,¹ the special suction cup capable of realizing precise temperature adjustment are widely used for temperature control in semiconductor manufacturing, chemical synthesis,² biopharmaceuticals,^3,4 and other fields. With the excellent control system,⁵ the special suction cup to achieve fast and accurate temperature regulation, to meet the needs of large-plane uniform temperature control. Such as in the semiconductor manufacturing process, the special suction cup can regulate and control the surface temperature and its distribution, which plays an important role in photolithography equipment.⁶ So the special suction cup for temperature control has become a kind of special equipment widely used in semiconductor manufacturing.

The temperature of the special suction cup is mainly controlled by heaters, with temperature control coating as the heating medium of it although the time of its development is late,^7,8 because it is not subject to the limitations of the shape and volume, it has a higher heating rate and temperature uniformity.⁹ How to design the temperature control system of the special suction cup is an important problem in current research, and the difficulty lies in the fact that the temperature control of the special suction cup has time-varying, large inertia, hysteresis, and multivariate characteristics.¹⁰ It usually needs to maintain high accuracy and uniformity within ±0.1 °C, which puts forward a higher requirement for the temperature control algorithm.

In the history of temperature control technology, PID controllers have been widely adopted due to the advantages of the simple structure, easy implementation, strong robustness, and easy operation of PID algorithm.¹¹ In contemporary industrial temperature control, more than 90% of temperature control loops are still regulated by PID control structures. However, as the complexity of temperature control tasks increases, traditional PID controllers often suffer from overshooting, decreased stability, and slower response to perturbations, making it difficult to achieve optimal control.

In order to solve these problems, researchers continue to optimize the temperature control algorithms. Entering the modern control stage, the temperature control technology has transitioned from the traditional PID control to the improved PID control as well as the adaptive control technology. And the application of these methods significantly improves the performance and efficiency of the control system for the time-varying, large inertia, hysteresis, and other problems in the temperature control system.^12–15 Dawei et al. take the temperature control system of electric heating furnace as the background, design the fuzzy PID system by combining the fuzzy algorithm, build the fuzzy inference system, and the fuzzy rule table by using the MATLAB fuzzy control box, and use the Simulink platform to build the temperature control system simulation model of the ordinary PID and fuzzy PID, and the comparative simulation results show that the fuzzy PID control system effectively alleviates the traditional PID control method of overshoot or static error. problems such as overshoot or static error.¹⁶ Kaliappan et al. addressed the problems in the design of temperature control of continuously stirred tank reactor (CSTR) plant in chemical industry by controlling the liquid temperature by controlling the coolant flow rate with the help of modified model reference adaptive controller (MRAC) and improved the temperature of the liquid by using PID controller, PID based on differential improvement algorithm and fuzzy based DE controller. DE controller to improve the transient response of the temperature process.¹⁴ Wang et al. introduced a PID feedback control strategy at the end of the finishing zone in order to achieve accurate control of the final rolling temperature of high-performance steels, a Smith compensator was used to predict the compensation of the large hysteresis problem of the final rolling temperature, and the nonlinearity was eliminated by the Smith GA-PID optimization control strategy system so that problem and better dynamic performance was obtained.¹⁷ Liu et al. show that the heating furnace and other narrow closed space temperature control system has hysteresis, inertia, nonlinearity, and other characteristics, first introduced Tent chaos mapping, nonlinear feedback update factor to improve the standard mud mold optimization algorithm (SMA), and then optimized quantization factor and proportionality factor on the basis of the improved SMA, to achieve a shorter rise time, smaller tracking error, and superior control effect.¹⁸ Zhang et al. used nonlinear segmented logistic chaos mapping to start the particle swarm during the performance testing of precision devices in aerospace field, optimized the ACPSO-Fuzzy-PID parameters and introduced the contraction factor to ensure the convergence of the algorithm, so that the overall and local search ability of the particle swarm is more efficient, and invoked the ACPSO-Fuzzy-PID control system, with faster response time, fewer overshoots, and superior control effect of the tuning. faster, less overshoot, shorter adjustment time, more stable temperature control fluctuations, and stronger anti-interference ability, which significantly improves the temperature control accuracy of the incubator in the aerospace field.¹⁵ He et al. used Smith fuzzy control combined with a PID controller in the temperature control system, allowing the PID controller to adaptively change the controller parameters according to the changes in the temperature control system and to compensate for the time lag in advance, thus reducing the control error caused by the time-varying nature of the system and the hysteresis; Cao et al. used a multi-level integral separation PID algorithm for the design of high-precision temperature control system, by doing multilevel control of the output of the integral link of the PID controller, effectively reducing the amount of system overshooting and improving the stability of the system¹⁹; Zhang et al. for the temperature control system with more than one control variable, by decoupling control to do the calculation of the compensation of the advance, the system will be turned into a number of independent control loops to reduce the interference caused by coupling.²⁰

As a result, the Smith fuzzy multilevel integral separation PID control algorithm²¹ is proposed to solve the impact of time-varying, large inertia and response lag on the control accuracy in the system, and then the temperature coupling model of each partition of the system is established in the simulation software, and the temperature control model of each region is decoupled through the feedforward compensator, so as to put forward a feedforward decoupling Smith fuzzy multilevel integral separation PID-based control algorithm to improve the stability of the heating surface temperature of the special suction cup.

Heating structure and heat transfer mechanism of the special suction cup

Heating structure of the special suction cup

The main part of the special suction cup for the silicon carbide material, silicon carbide has a very high hardness, excellent thermal conductivity, fire resistance to high temperatures and continuous heating in the case of strong oxidation resistance, and has a lower conduction and switching losses,^22,23 temperature control coating part of the conductive layer of metal oxides and silicon dioxide,⁷ polyethylene insulating layer²⁴ composed of, silver has a good electrical conductivity, in the temperature control coating surface plated with a number of groups of silver electrode contacts used to connect the heating power supply. Model structure of a special suction cup based on temperature-controlled coating is shown in Figure 1.

Figure 1.

Model structure of a special suction cup based on temperature-controlled coating.

Studies related to heat transfer simulation have shown that the temperature distribution of the special sucker exhibits radial non-uniformity from the center to the outside. To reduce the temperature difference between different areas of the sucker surface, the temperature control structure is divided into multiple annular zones, and a decentralized independent control method is adopted to individually regulate the heating power of each zone, thereby improving the overall temperature uniformity. In the temperature control of special suction cup for semiconductor, temperature control coating is the physical basis for the realization of temperature control uniformity, in order to improve the uniformity of temperature distribution from the center to the edge of the temperature control coating is designed to be divided into four regions from the center outward as shown in Figure 2. First, second, third, and fourth regions were lined up with 1, 5, 6, 4 heating channels, each channel has a uniform width, length is close to the same length, the resistance value of the resistance value of the resistance value of the resistance value of the resistance value of the resistance value of the resistance value of the resistance value of the resistance value of the resistance value of the resistance value of the resistance value of the heating channels, each circuit is distributed a head to tail Connect the silver electrode point of the heating power supply, the heating power of each channel tends to be consistent, and four adsorption through-holes are evenly distributed in the middle of the sucker at 90° intervals, and the inner ring adopts PEEK material as the heat insulation ring, which constitutes the structural composition of the special sucker model.

Figure 2.

Temperature control coating circuit structure.

In order to obtain the data of temperature control, the temperature control of the special suction cup from the center to the outside of the radial uniform distribution of 11 temperature measurement holes, each temperature measurement holes connected to a high-precision PT100 temperature sensors, the special suction cup in kind as shown in Figure 3.

Figure 3.

Physical drawing of the special suction cup.

Heat transfer mechanism of the special suction cup

In the special suction cup based on temperature-controlled coating, the temperature-controlled coating with high thermal conductivity is used as the heating medium, and the circuit structure of the temperature-controlled coating is designed to make its resistance distribution more uniform, so as to ensure the temperature distribution uniformity of the temperature-controlled coating from the physical structure. In the special suction cup based on temperature-controlled coating, there are three heat transfer modes, respectively, thermal conduction, thermal convection, and thermal radiation. Figure 4 shows a schematic diagram of heat transfer in the special suction cup. The wafer is adsorbed on the surface of the special suction cup and fully attached to it. Thermal conduction refers to the transfer of heat from a region with a higher temperature to a region with a lower temperature through direct contact between objects, which mainly occurs between the upper and lower surfaces of the temperature-controlled coating and the special suction cup. Thermal convection refers to the transfer of heat from one place to another in space through a flowing medium, which mainly occurs between the upper surface of the wafer and the air. Thermal radiation refers to the phenomenon of heat transfer in the form of electromagnetic waves, which mainly occurs between the temperature-controlled coating and the surrounding environment.

Figure 4.

Schematic diagram of thermal transfer of temperature-controlled special suction cup.

The mathematical model of temperature control system for the special suction cup

The methods for establishing mathematical models include experimental modeling, mechanism modeling, and system identification. This paper adopts the system identification method, which is highly applicable and flexible for model establishment when internal mechanism knowledge of the system is insufficient or experimental data is inadequate. Its principle lies in combining experimental modeling with mechanism modeling: first, the model structure of the temperature control system is determined based on theoretical principles and practical experience; then, the model is identified by analyzing the input and output data of the system.

Step response is suitable for evaluating the transient behavior of a system, such as start-up time, overshoot, and settling time. In contrast, frequency response is more effective for analyzing the frequency characteristics of a system, including bandwidth, resonance peak, as well as the sensitivity of system to noise and interference.

The input of this temperature control system is a step signal. Therefore, using the step response method for system identification can fully stimulate the dynamic response of the system and obtain abundant dynamic information. By applying a step change to the heater, the response process of how the system transitions from one steady state to another is studied. The model structure of the temperature control system is established based on experience, and the system response is solved to obtain information such as the steady-state gain, time constant, and lag time of the system, thereby establishing an accurate mathematical model of the temperature control system.

Since the temperature control system of the special suction cup is powered by an external DC power supply, causing the temperature-controlled coating to generate heat due to the Joule effect, and the temperature-controlled coating is directly applied as a film on the suction cup surface, the generated heat mainly enters the suction cup through heat conduction. The temperature control system model can be approximated using a first-order inertial lag transfer function, with its input-output relationship described by equation (1) and the transfer function model structure given by equation (2):

Y (s) = G (s) \cdot U (s)

(1)

G (s) = \frac{K}{Ts + 1} e^{- τ s}

(2)

where Y(s) is the temperature of the suction cup, G(s) is the system transfer function, U(s) is the heating power, K is the system open-loop gain, T is the system time constant, and τ is the system lag time.

Using the system identification method of step response, set the heating voltage as 80 V adjustable DC voltage, current peak 2.5 A, the initial for the temperature 25 °C, the target temperature of 30 °C, the average temperature of all temperature sensors as the output of the system, Figure 5 shows the temperature response curves of the special suction cup.

Figure 5.

Temperature response curve of the special suction cup.

After obtaining the step response data, it is also necessary to solve the specific open-loop gain K, time constant T, and hysteresis time τ. For the temperature control system of the special suction cup, the System identification Tool module of MATLAB software is used, and the step signal is used as the system input signal, and the average temperature of all temperature sensors is used as the system output signal, so that the temperature control system transfer function is obtained as shown in equation (3). The temperature control system transfer function shown. The model obtained by identification is compared with the excitation signal obtained from the experiment, and Figure 6 shows the degree of agreement of the identification model, and the degree of agreement of the model is as high as 85.76%, which indicates that the model obtained by identification can respond to the system characteristics to a certain extent.

G (s) = \frac{1.0024}{23.6792 s + 1} e^{- 1.275 s}

(3)

Figure 6.

Identifying the degree of model fit.

The algorithm of smith fuzzy multi-level integral separation PID based on feedforward decoupling

PID controller

The system uses the difference between the current temperature measured by the temperature sensor and the preset target temperature as the input of the controller according to the set sampling interval, and the output of the controller is temperature control by PWM signal,²⁵ therefore, a digital PID controller is used in this system. The specific control algorithm expression is shown in equation (4).

P (k) = k_{p} E (k) + k_{i} \sum_{j = 0}^{k} E (j) + k_{d} [E (k) - E (K - 1)]

(4)

where k is the number of samples; P(k) is the output at the kth sample; E(k) is the deviation between the set value and the controlled quantity at the “k”th sample; and k_p, k_i, and k_d are the proportional, integral, and differential parameters of the PID controller, respectively.

Smith predictor

Due to the lag characteristic of the system that it increases the inertia of the system and may cause oscillations, thus increasing the steady state error of the system. In order to solve this problem, a Smith predictor is introduced to compensate the time lag of the system.²⁶ The basic principle of the Smith predictor is to predict the system response in advance in the feedback loop of the main controller.²⁷

Where the Smith predictor is expressed as in equation (5):

H (s) = G_{p} (s) \cdot (1 - e^{- τ s})

(5)

where H(s) is the transfer function of Smith’s predictive compensator and G_P(s) is the transfer function of the temperature control system without time-lag link. Combined with the system transfer function equation (3) obtained from the system identification above, the transfer function of the temperature control system without time lag link is equation (6):

G_{p} (s) = \frac{1.0024}{23.6792 s + 1}

(6)

Substituting the above equation (6) into equation (5), the Smith predictive compensator is solved as equation (7):

H (s) = \frac{1.0024}{23.6792 s + 1} (1 - e^{- 1.275 s})

(7)

Fuzzy adaptive controller

In order to reduce the overshoots and oscillations caused by the time-varying nature of the temperature control system of the special suction cups, fuzzy control is introduced to adaptively compensate the proportional and differential parameters of the PID controller,²⁸ and the steps of fuzzy control implementation in the temperature control system of the special suction cup²⁹ are as follows:

1. Fuzzification: First, it is necessary to determine the input-output range, the fuzzy universe. The universe of discourse is similar to the range of values, generally referring to the variation range of variables. The fuzzy control in this temperature control system adopts a two-input and two-output structure. The two input variables are temperature deviation $E$ and temperature deviation rate of change $Ec$ , respectively; the two output variables $Δ k_{p}$ , $Δ k_{d}$ are the compensation values of the proportional parameter and the differential parameter of the PD controller, respectively.

Their actual variation ranges are selected as the basic universes of discourse. For the convenience of calculation, both the basic universe of discourse and the fuzzy universe of discourse of temperature deviation are set as (6, 6), where 6 and 6 represent the minimum and maximum values of the set temperature deviation for heating and temperature rise in each zone. The basic universe of discourse of the temperature deviation rate of change is (0.1, 0.1), and its fuzzy universe of discourse is (10, 10); these values are derived from the variation rate range of temperatures obtained in experimental tests. The basic universes of discourse for the compensation values of the proportional parameter and the differential parameter $Δ k_{p}$ , $Δ k_{d}$ are (0.5, 0.5) and (1, 1), respectively, while their corresponding fuzzy universes of discourse are (5, 5) and (10, 10), respectively. The fuzzy subsets of the input and output variables of the fuzzy controller are all divided into seven levels: Negative Large, Negative Medium, Negative Small, Zero, Positive Small, Positive Medium, Positive Large, which are denoted by the symbols (NL, NM, NS, ZO, PS, PM, PL).

Common types of membership functions for fuzzy sets include triangular membership functions, trapezoidal membership functions, Gaussian membership functions, sigmoid membership functions, and so forth. In practical applications, these membership functions can be selected and adjusted according to the characteristics and requirements of specific problems.

The triangular membership function is a commonly used type of membership function. Its membership degree exhibits a triangular profile as the independent variable increases or decreases, and it is symmetric in nature. It is typically applied to describe the variation of membership degree when the variable takes values within a certain range. On the basis of the triangular membership function, the trapezoidal membership function incorporates a flat segment, which allows the membership degree to maintain a relatively high level over a specific interval. It is often used to characterize scenarios where the membership degree remains stable within a defined range. The Gaussian membership function is constructed based on the Gaussian distribution; its curve assumes a shape similar to that of the normal distribution, and it can be used to describe membership scenarios that conform to the law of normal distribution. The sigmoid membership function is a common type of membership function with an S-shaped curve, and it is suitable for describing cases where the membership degree increases or decreases rapidly as the independent variable changes.

Given that the temperature control system in this study demonstrates distinct symmetry, and the triangular membership function is generally defined by three parameters—the left boundary a, the midpoint b, and the right boundary c—the membership degree reaches its maximum value near the midpoint b, and then gradually decreases to zero toward both sides. The shape of this function resembles a triangle, hence the name triangular membership function. It is often used to describe fuzzy sets with strong symmetry, such as the “cold, moderate, hot” classification of temperature. Therefore, it is more appropriate to select the triangular membership function.

Since the temperature control system in this study exhibits obvious symmetry, a triangular membership function is selected. Figure 7 shows a schematic diagram of the triangular membership function.

Figure 7.

Triangular affiliation function.

The triangular affiliation function in Figure 7 is usually determined by three parameters: the left boundary “a,” the middle point “b” and the right boundary “c.” The affiliation reaches its maximum value near the middle point b, and then decreases gradually to both sides until it falls to 0. The shape of this function is similar to a triangle, so it is called triangular affiliation function, which is often used to describe the fuzzy set of strong symmetry, such as the temperature of the “cold, moderate, hot” and so on, and its expression formula is shown in equation (8).

f (x, a, b, c) = {\begin{matrix} 0 & x \leq a \\ \frac{x - a}{b - a} & a \leq x \leq b \\ \frac{c - x}{c - b} & b \leq x \leq c \\ 0 & x \geq c \end{matrix}

(8)

2. Establishment of fuzzy rules: In order to find out the optimal fuzzy rules, the fuzzy control rules are designed based on the temperature deviation E and the temperature deviation change rate Ec from experience and experiments.

At the beginning of heating, the rise time period, k_p should be increased appropriately, and a larger Δk_p should be output to accelerate the response speed; at the same time, k_d needs to be increased, and a larger Δk_d needs to be output to prevent overshooting.

Before entering the steady-state tolerance b and, k_p should be appropriately reduced, the output is smaller Δk_p reduce the overshoot; at this time, k_d is moderately adjusted, Δk_d should be moderately valued, to prevent the differential inhibition is too large to enter the steady-state tolerance band.

When entering the steady state tolerance band, in order to maintain the system stability k_p moderately adjusted, Δk_p should be taken as a moderate value; in order to reduce the impact of the system oscillations and disturbances on the differentiation, k_d should be appropriately reduced, Δk_d is taken as a smaller value. Based on the above laws, the final table of fuzzy control rules is shown in Table 1.

Table 1.

Δk_p, Δk_d fuzzy control rules.

E	Ec
E	NL	NM	NS	ZO	PS	PM	PL
NL	PL/NL	PS/PM	PM/PS	PL/ZO	PS/NL	ZO/NM	ZO/PS
NM	PM/NL	PL/NS	PM/NL	PS/NM	PS/NM	ZO/NS	ZO/ZO
NS	PS/NM	PM/NS	PS/NM	PS/NM	ZO/NS	NS/NS	NS/ZO
ZO	ZO/NS	PM/NS	PS/NS	ZO/NS	NS/NS	NM/NS	NM/ZO
PS	NS/ZO	PS/ZO	ZO/ZO	NS/ZO	NS/ZO	NM/ZO	NM/ZO
PM	NM/PM	ZO/PS	NS/PS	NM/PS	NM/PS	NM/PS	NL/PS
PL	NL/PL	ZO/PM	NM/PS	NM/PS	NM/PS	NL/PS	PL/PM

3. Defuzzification: Converting the membership degrees of fuzzy outputs into specific control output values is to determine the final control action. Compared with other methods, the centroid method has a very significant regulating effect on the output—even if there is a slight change in the input, the output after defuzzification using the centroid method will change accordingly. When the fuzzy domain has “n” output quantization levels, the formula of the center of gravity method is shown in equation (9).

W = \frac{\sum_{j = 1}^{n} u_{j} A (u_{j})}{\sum_{j = 1}^{n} A (u_{j})}

(9)

Where “W” is the horizontal coordinate value of the center of gravity point, and this value is used as the representative value of defuzzification in the fuzzy theory domain, and A(u_j) is the affiliation function of u_j. The fuzzy quantities are clarified using the center of gravity method to obtain the compensation values of the proportional and differential parameters of the PD control Δk_p,Δk_d. The initial values of the proportional and differential parameters of the PID controller are set to be k_p0, k_d0, then the adaptive correction equations for the proportional ad differential parameters of the PID controller, k_p, k_d, are shown in equation (10).

{\begin{matrix} k_{p} = k_{p 0} + Δ k_{p} \\ k_{d} = k_{d 0} + Δ k_{d} \end{matrix}

(10)

Smith fuzzy multi-level integral separation PID algorithm

In modern temperature control systems, PID controllers are widely used for temperature regulation. PID control is a classical closed-loop control algorithm that adjusts the control inputs by comparing the error between the actual and desired outputs of the system, using proportional, integral, and differential control links to minimize the error.

Which in the integral link of PID controller is to eliminate static error, but due to the temperature control system in the initial stage of heating within a short period of time the output of the system and the set value has a large deviation, often lead to the accumulation of integral link is too large, so that the output control volume is easy to exceed the target value, and the existence of the special suction cup temperature control system of time-varying, large inertia, and hysteresis, all of these problems will result in the system to appear overshooting and prolong the system stability time. Adjustment and extend the system stabilization time, due to the actual working process of the suction cup natural heat dissipation rate is very slow, once the overshoot is difficult to eliminate within a short period of time, which will seriously affect the accuracy of temperature control, and a partition error will also have an impact on the neighboring partition, error coupling, resulting in a greater error.

To address the above problems based on the conventional PID controller, a multistage integral separation PID algorithm³⁰ is introduced, and a Smith fuzzy multistage integral separation PID algorithm is proposed by combining fuzzy adaptive control and Smith predictor. For different temperature difference thresholds, the integral term is multiplied by different integral switching parameters to eliminate the static error step by step, and the proportional and differential parameters are adaptively compensated by fuzzy control, and the Smith predictor compensates for the time lag to ensure the response speed while minimizing the overshooting amount of the system, the specific steps are as follows:

First, set the multilevel threshold $ε_{1}, ε_{2}, ε_{3}, \dots, ε_{n}$ and make $ε = ε_{1} > ε_{2} > ε_{3} > \dots > ε_{n} = 0$ . When $| E | > ε$ , the Smith fuzzy PD control is used to make the output of the integral link zero, avoiding overshooting and oscillation caused by the integral link, and also improving the response speed of the system; when $| E | \leq ε$ , the Smith fuzzy multilevel integral-separated PID control is used to eliminate the static error step by step, reducing the amount of overshooting of the system and improving the response speed at the same time. Then, a multilevel integral switch parameter $α$ is introduced into the formula to get the multilevel integral separation PID expression as shown in equation (11).

P (k) = k_{p} E (k) + α k_{i} \sum_{j = 0}^{k} E (j) + k_{d} [E (k) - E (k - 1)]

(11)

α = {\begin{matrix} α_{1} & ε_{2} < | E (k) | \leq ε_{1} \\ α_{2} & ε_{3} < | E (k) | \leq ε_{2} \\ \dots \\ α_{n} & ε_{n} < | E (k) | \leq ε_{n - 1} \end{matrix}

The core of the multilevel integral separation PID control strategy is to switch different control modes by comparing the temperature deviation of the suction cup with a preset integral separation threshold.³¹ When the temperature deviation exceeds the threshold, the controller sets the integral output to 0, essentially using PD control to improve the system response speed; when the deviation is below the threshold, the controller adjusts the integral link hierarchically according to the magnitude of the deviation to reduce the overshoot and maintain the response speed.

Feedforward decoupler

Through the heat transfer simulation related research found,³² the temperature distribution of the special suction cup presents a radial inhomogeneity from the center outward, in order to reduce the temperature difference between different areas of the disk surface, the temperature control structure is divided into a ring-shaped multiple regions, using a decentralized independent control method, and individually controlling the heating power of each region, in order to improve the overall temperature uniformity. Figure 8 shows the regional division of the designed special suction cup, divided into four temperature control region, the temperature of each region to do independent control, but this also makes the temperature control system of the special suction cup into a coupled system with four input variables and four output variables,³³ the temperature control of each region will inevitably interfere with the temperature of the other regions, and therefore need to use appropriate methods to reduce the impact of this coupled interference.

Figure 8.

Division of temperature control area for the special suction cup.

In order to reduce the coupling interference between different temperature control regions of the special suction cup, firstly, the decentralized control method³⁴ is used to do independent control for each region, and since the coupling is brought by different temperature control loops within the same system, the coupling interference of other regions can be regarded as a total perturbation, and then the influence brought by the perturbation is compensated through the feedforward decoupling method,³⁵ and a single-input single-output feedforward decoupled temperature control system can be obtained, combined with the feedback controller to realize the dynamic tracking of the target temperature, through this control strategy can be the temperature control system of each partition of the special suction cup can be equated to the feedforward decoupled temperature control system shown in Figure 9.

Figure 9.

Feedforward decoupled temperature control system.

In the figure, R(s) and Y(s) are the input and output signals of independent temperature control partition, and D(s) is the coupling interference caused by the inputs of other partitions, according to the feedforward compensation decoupling temperature control system can be listed as the transfer function of the coupling interference to the output as equation (12):

\frac{Y (s)}{D (s)} = \frac{G_{d} (s) + G_{f} (s) \cdot G_{p} (s)}{1 + G_{c} (s) \cdot G_{p} (s)}

(12)

To make the output Y(s) only related to R(s), even if the effect of coupled disturbances on the system is 0, there is equation (13):

G_{d} (s) + G_{f} (s) \cdot G_{p} (s) = 0

(13)

Simplifying the equation yields a decoupled controller for an independent temperature control loop as equation (14):

G_{f} (s) = - \frac{G_{d} (s)}{G_{p} (s)}

(14)

Similarly, the control system of special suction cups with four inputs and outputs can be turned into four independent feedforward decoupled temperature control systems. This method focuses on the temperature control of each partition to ensure the temperature uniformity of the special suction cup and effectively reduce the degree of coupling between systems.

Framework for the algorithm of smith fuzzy multi-level integral separation PID based on feedforward decoupling

In Figure 10, r(k) is the temperature value set at the “k”th sampling; y(k) is the actual temperature value; E(k) is the temperature deviation value; Ec(k) is the temperature deviation rate; Δk_p and Δk_d are the compensation values of the fuzzy controller for the proportional parameter and differential parameter of the PD controller, respectively.

Figure 10.

Feedforward decoupled Smith fuzzy multilevel integral separation PID algorithm framework.

Smith fuzzy multilevel integral separation PID as a feedback control algorithm in the temperature control system to realize the dynamic tracking of the target temperature of a single partition, combined with feedforward compensation decoupling, to obtain the feedforward decoupling Smith fuzzy multilevel integral separation PID algorithm framework shown in Figure 10. Take the control of the center temperature-controlled partition of the special suction cup as an example, through the system identification method, while turning off the power of the center temperature-controlled partition, make a step response to the other temperature-controlled partition, and take the temperature change of the center temperature-controlled partition at this time as a coupling interference. The target temperature as a system input, the temperature change of the center temperature-controlled partition as a system output in the simulation software to select the model structure to solve the parameters of the coupling model, to get the center temperature-controlled partition of the coupled channel transfer function as equation (15):

G_{d} (s) = \frac{1.0375}{10.5472 s + 1} e^{- 2.129 s}

(15)

Comparative analysis of temperature control system modeling simulation

Since the temperature control of all temperature control partitions of the special suction cup is based on the same temperature control coating, so the dynamic performance of the temperature control process transfer function of each partition is similar, equation (3) will be used as the transfer function of the center temperature control partition G_p(s) substituting equation (14) solves the transfer function of the decoupled controller of the center temperature control partition as equation (16):

G_{f} (s) = - \frac{G_{d} (s)}{G (s)} = - \frac{24.5672 s + 1.0375}{10.5725 s + 1.0024} e^{- 0.854 s}

(16)

Figure 11 shows the simulation model of Smith fuzzy multi-level integral separation PID algorithm based on feedforward decoupling. In the simulation model, a step signal with a set amplitude of 5 is input, while a random signal with a mean value of 2 is introduced at the output of the system to simulate the coupling interference brought about by other temperature control partitions for temperature control. In order to evaluate the decoupling effect, simulation experiments with and without decoupling are carried out, respectively. Figure 12 shows the simulation results in these two cases. From the results, it can be seen that the coupling interference leads to a 4.3% overshoot and continuous oscillation in the system response, and the system reaches a steady state at 26.78 s. However, by implementing the feedforward decoupling strategy, the effect of the coupling interference on the system is effectively eliminated, and the system is free of overshooting and oscillations, and the system reaches a steady state in 14.53 s. This result shows that the feedforward decoupling algorithm has a significant effect on coping with coupled disturbances in the control system.

Figure 11.

Simulation model of feedforward decoupled Smith fuzzy multi-level integral separation PID algorithm.

Figure 12.

Response curves with and without decoupling.

Figure 13 illustrates the amplitude variation characteristics of the frequency response of this temperature control system. Analysis of the curve shows that the system has a relatively high gain of ∼25–30 dB in the low-frequency band (0.001–0.01 rad/s), which ensures good steady-state accuracy and anti-interference capability of the system. In the mid-frequency band (0.01–0.1 rad/s), the curve crosses the 0 dB line at an ideal slope close to −20 dB/dec, with a crossover frequency of about 0.095 rad/s. This characteristic determines the response speed and relative stability of the system. In the high-frequency band (>0.1 rad/s), the curve attenuates rapidly at a slope of −40 dB/dec, indicating that the system has excellent suppression capability against high-frequency disturbances. The entire amplitude-frequency curve exhibits the characteristics of a typical high-performance control system: high gain in the low-frequency band, ideal slope in the mid-frequency band, and sufficient attenuation in the high-frequency band. Combined with a gain margin of 15.8 dB, it shows that the temperature control system has sufficient stability and robustness while ensuring accuracy.

Figure 13.

Amplitude-frequency characteristic.

Figure 14 reflects the phase variation law of the frequency response of this temperature control system. As can be seen from the curve, the phase transitions smoothly from −90° in the low-frequency band to around −270° in the high-frequency band, with an overall smooth change and no sharp drop, indicating that the system has favorable phase characteristics. At the gain crossover frequency $ω_{c} = 0.095 rad / s$ , the phase value is −114.7°, and the corresponding phase margin reaches 65.3°. This means the system has stability margin and can effectively resist parameter variations and external disturbances. The curve is always far away from the −180° stability boundary, which further verifies the strong robustness and reliability of the system. Such ideal phase-frequency characteristics benefit from the time-delay compensation of the Smith predictor and the adaptive adjustment of fuzzy control, which jointly ensure the stable operation of the system under complex working conditions.

Figure 14.

Phase-frequency characteristic.

Figure 15 shows that the temperature control system exhibits excellent frequency response characteristics. The amplitude-frequency characteristic curve indicates that the system has a high gain of 57.8 dB in the low-frequency band, which ensures outstanding steady-state accuracy and anti-interference capability. In the mid-frequency band, the curve crosses the 0 dB line at an ideal slope close to −20 dB/dec, demonstrating the system’s superior dynamic performance and stability margin. In the high-frequency band, the curve attenuates rapidly at a slope of −40 dB/dec, exhibiting strong suppression capability against high-frequency noise. The phase-frequency characteristic curve transitions smoothly from −90° in the low-frequency band to −270° in the high-frequency band, with a continuous variation process and no abrupt phase drop. At the gain crossover frequency, the system has a sufficient phase margin and stays well away from the −180° stability boundary. A comprehensive evaluation shows that the system’s phase margin exceeds 60° and its amplitude margin is >10 dB, achieving ideal stability. The system has strong robustness, can withstand large parameter variations and external disturbances, which fully verifies the effectiveness of the control algorithm in the large time-delay temperature control system.

Figure 15.

Analysis of amplitude-frequency characteristic slope and variation of phase characteristic.

Simulation and testing

Simulation

To verify the robustness of the temperature control system for the special sucker, simulations were conducted under conditions of external temperature disturbance variations. Dynamic temperature simulations were performed on the designed heating film layer with additional devices applied under external voltage, so as to observe the temperature uniformity and check whether the temperature difference between the front and back surfaces of the heating film layer could consistently meet the accuracy requirements, as shown in Table 2.

Table 2.

Robustness analysis.

Environmental factors	External environmental disturbance of 0.2 °C	External environmental disturbance of 0.5 °C
Simulation results
At the 20th s	Front surface
	Temperature difference: 0.534 °C	Temperature difference: 0.529 °C
	Back surface	Back surface
	Temperature difference: 0.671 °C	Temperature difference: 0.671 °C
Analysis	Under the action of different critical voltages (the voltages of the four zones from outside to inside are 133, 136, 138, and 170 V, respectively), heating is conducted through silver terminals with only room temperature taken into consideration. After installing the additional device and under the condition of environmental disturbance, the temperature difference on the front surface reaches 0.534 °C and that on the back surface reaches 0.671 °C when the temperature rises to 30 °C, which is roughly the same as the temperature difference without the additional device	Under the action of different critical voltages, the voltages of the four zones from outside to inside are 133, 136, 138, and 170 V, respectively. With only room temperature taken into consideration, heating is conducted via silver terminals. After installing the additional device and under the condition of temperature disturbance, the temperature difference on the front surface reaches 0.529 °C and that on the back surface reaches 0.671 °C when the temperature rises to 30 °C. Compared with the scenario without temperature disturbance, the temperature difference on the front surface increases by 0.02 °C
Conclusion	According to the obtained simulation results, the temperature difference under the condition of temperature disturbance is not significantly different from the original temperature difference, indicating that the control system has good anti-interference capability and strong robustness

Testing

To address the issue of uneven temperature distribution across the disk when the special sucker is heated up, this paper adopts a four-zone temperature control strategy. The experiment is conducted in a closed test chamber, which is equipped with the special sucker, temperature sensors, and a dedicated cooling device. Fifteen temperature sensors are evenly distributed radially on the surface to monitor the temperature changes of the sucker in real time. After doing each experiment, the cooling device is used to rapidly reduce the temperature of the sucker, thereby shortening the waiting time between experiments. The presence of the temperature sensors and cooling device ensures the constancy and safety of the experimental environment, which in turn guarantees the accuracy and reliability of the experimental results.

The experimental environment of the special suction cup control system is shown in Figure 16. The experimental environment was set up in a stable constant temperature air-conditioned room to ensure the stability of the ambient temperature during the experiment. The control system consists of three main components, namely, the control panel, the experiment box, and the control cabinet. Among them, the experiment box contains an infrared thermometer, a cooling device, the special suction cup, and temperature sensor.

Figure 16.

The temperature test site of the special suction cup.

The special suction cup using decentralized control, that is, four different partitions for independent temperature control, the special suction cup temperature control system using feedforward compensation based decoupling Smith fuzzy multilevel integral separation PID algorithm for validation, after trial and error method to obtain the algorithm parameters, the control of special suction cups in a relatively stable constant temperature room for experiments, the PLC controller on the special suction cups of the four partitions were output The PLC controller outputs different PWM signals to each of the four partitions of the special suction cup to control the heating power to realize the temperature control of the four partitions.

Temperature distribution consistency and stability is an important index to measure the robustness of the special suction cup control system, in order to intuitively grasp the consistency and stability of the temperature distribution, the standard variance of temperature along the radial direction of the suction cup is used to reflect its temperature consistency, and the temperature stability is reflected by the difference between the average temperature and the set temperature per second.

The temperature consistency function of the specialty suction cups is shown in equation (17).

σ = \sqrt{\frac{1}{n} \sum_{i = 1}^{n} {(T_{i} - \bar{T})}^{2}}

(17)

The temperature stability function is shown in equation (18).

Δ t = \bar{t} - t

(18)

The table $σ$ indicates the temperature consistency of the special suction cup; n indicates the number of collected temperatures; $T_{i}$ indicates the temperature of the ith point in the radial direction; $\bar{T}$ indicates the average temperature in the radial direction; $Δ t$ indicates the temperature stability of the special suction cup; $\bar{t}$ indicates the average temperature of all the temperature measurement points every second; t indicates the set temperature.

Figure 17 shows that the special suction cup four sections of their respective average temperature and the difference between the set temperature change, from top to bottom for the overall average temperature stability, consistency, sections 1–4 stability. As shown in Figure 17 can be seen in the constant temperature box under the temperature rise of 0.5 °C, the stability of the overall temperature of the suction cup in the beginning of the temperature rise when there are large fluctuations, but with the adjustment of the system quickly converge to a smooth, while the consistency of the overall temperature of the special suction cup although there is a small fluctuation, but has always maintained a smooth state, due to the change in the overall stability of the sucker by the change in the stability of each small sub-partition, so the stability change of each sub-partition with the Because the overall stability change of the special suction cup is affected by the stability change of each small partition, the stability change of each partition is basically consistent with the overall stability change of the suction cup, which shows that the control algorithm is more accurate after the application of decoupling, reduces the temperature coupling interference between the partitions, and improves the robustness of the control system of the special suction cup.

Figure 17.

Average temperature difference between the four partitions of the special suction cup.

In order to further verify the stability and consistency of the special suction cup temperature control system, ten consecutive temperature control tests were carried out in a constant temperature environment, starting from room temperature 25 °C, each time the temperature was increased by 0.5 °C, and the test data obtained were shown in Table 3, from which it can be seen that the average temperature steady-state error in the ten temperature control tests was maintained at 0.04 °C or less, and the maximum temperature difference between all the temperature measurement points in the whole temperature increase process has been <0.1 °C, the adjustment time is <14 s, the whole temperature control system has good stability.

Table 3.

Ten consecutive temperature control test data.

Target temperature ( °C)	Average temperature steady state error ( °C)	Maximum temperature difference ( °C)	Adjustment time (s)
25.5	0.039	0.075	14
26	0.038	0.079	14
26.5	0.034	0.096	10
27	0.035	0.088	10
27.5	0.029	0.094	11
28	0.036	0.084	11
28.5	0.034	0.089	10
29	0.033	0.078	10
29.5	0.028	0.092	12
30	0.029	0.079	12

Conclusion

The paper firstly establishes the mathematical model of temperature control system through the heating structure and heat transfer mechanism of special suction cups, and then establishes the temperature control system model by using the system identification technology in the simulation software, and proposes the multilevel integral separation PID control algorithm combined with Smith fuzzy to guarantee the control accuracy for the time-varying, large inertia, and hysteresis of the special suction cup’ temperature control, and proposes the coupling interference based on feedforward compensation decoupling method to improve the consistency and stability of the special suction cup heating and warming, and modeling simulation to initially verify the simulation results in both cases. Considering the coupling interference between different temperature control regions of the special suction cup, a feedforward compensation decoupling method is proposed to improve the consistency and stability of the heating of the special suction cup, and the modeling simulation is used to initially verify the simulation results in these two cases. From the results, it can be seen that the coupling interference leads to 4.3% overshooting and continuous oscillation of the system response, and the system reaches a steady state in 26.78 s. However, by implementing the feedforward decoupling strategy, the effect of the coupling interference on the system is effectively eliminated, and the system is free of overshooting and oscillations, and the system reaches a steady state in 14.53 s. This result shows that the feedforward decoupling algorithm has a significant effect on coping with coupled disturbances in the control system. The proposed feedforward decoupled Smith fuzzy multilevel integral separation PID algorithm is applied in a practical experimental environment to carry out on-site temperature tests to verify the temperature control effect of the designed control system. The test results show that in 10 consecutive temperature control experiments, the average steady state error is maintained at about 0.03 °C, the maximum temperature difference between all the temperature measurement points in the whole heating process is <0.1 °C, and the adjustment time is <14 s. The Smith fuzzy multilevel integral separation PID algorithm based on feedforward decoupling algorithm in the temperature control system of the special suction cup shows strong robustness, which shows the high consistency of the surface heating and the high accuracy of the surface heating and the temperature control effect of the special suction cup. High consistency of surface heating and temperature rise as well as high stability.

The stability of this temperature control system is fully verified by frequency domain analysis. The amplitude-frequency characteristic curve exhibits desirable features including high gain of 57.8 dB in the low-frequency band, an ideal crossover slope of 20 dB/dec in the mid-frequency band, and sufficient attenuation in the high-frequency band. The phase-frequency characteristic curve is smooth, with a phase margin of 65.3 and an amplitude margin of 15.8 dB, both of which far exceed the engineering safety standards. Comprehensive analysis of the Bode plot indicates that the system is far from the stability boundary, has strong robustness, can reliably withstand internal parameter changes and external disturbances, and fully meets the stringent requirements of stability and reliability for large time-delay temperature control systems. Moreover, the simulation results of the temperature control system under environmental disturbances demonstrate its excellent anti-interference capability and robustness. When external temperature disturbances of 0.2 °C and 0.5 °C are applied, respectively, the maximum temperature differences between the front and back surfaces of the system are 0.534 °C and 0.529 °C, respectively, showing minimal changes compared with the undisturbed condition and being lower than the accuracy requirement threshold.

Due to the high temperature control accuracy requirements of the special suction cup, and the time-varying, large inertia, and hysteresis characteristics of the temperature control system will have a greater impact on the temperature control accuracy of the problem of PID temperature control algorithms combined with fuzzy control, Smith control, multilevel integral separation of PID algorithm, to reduce the overshooting amount of temperature control system, improve the accuracy of the temperature control system. Considering the problem of uneven temperature on the surface of the suction cup due to the coupling phenomenon between the various divisions of the special suction cup during heating and warming, the feedforward decoupling algorithm is used to decouple the independent temperature control loops, improve the consistency and stability of the heating and warming of the special suction cup, and arrive at the control effect of the strong robustness of the temperature control system of the special suction cup.

In the semiconductor manufacturing process, temperature control accuracy and uniformity directly affect the performance of the chip, the special suction cup for temperature control has become a widely used in semiconductor manufacturing special equipment. With excellent control system, the special suction cup can realize fast and accurate temperature adjustment to meet the demand of large plane uniform temperature control, so in chemical synthesis, biopharmaceuticals, and other fields of temperature control, the special suction cup that can realize accurate temperature adjustment have also been widely used, which is of great significance to enhance the manufacturing capacity of high-end scientific and technological products.

Footnotes

Handling Editor: Aarthy Esakkiappan

ORCID iDs

Xiaolei Shen

Xintian Liu

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: The authors received financial support for the Shanghai Top-List Leader Project (YDZX20213100003013).

Declaration of conflicting interests

The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

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