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High precision and stability temperature control system for the immersion liquid in immersion lithography
5G Model
pp.1–9
JFMI 1249 Flow Measurement and Instrumentation ∎ (∎∎∎∎) ∎∎∎–∎∎∎
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High precision and stability temperature control system for the immersion liquid in immersion lithography Xiaoping Li, Yiwen Zhao, Min Lei n School of Mechanical Science & Engineering, Huazhong University of Science & Technology, Wuhan 430074, China
art ic l e i nf o
a b s t r a c t
Article history: Received 3 March 2016 Received in revised form 30 July 2016 Accepted 22 August 2016
The temperature stability of immersion liquid is one of the main factors that affect the performance of the immersion lithography tool. Since the temperature control system of immersion liquid has the characteristics of time delays, full of disturbance and non-linear, the system and control algorithm should be carefully designed to control the temperature of the immersion liquid within the specification. In this paper, a control system of cascade structure with feed forward and lag compensation is proposed to reduce the time delays and the disturbance caused by the temperature fluctuation of ambient environment. Then, the mathematical model of the temperature control system is built, and the parameters of the model are obtained by ‘gray’ identification method. Based on the model, an algorithm which combines hierarchical control algorithm, integral separation PI algorithm, feed forward algorithm and lag compensation algorithm is designed. Last, experiments are conducted to evaluate the algorithm. The results show that the algorithm improves the robustness, compensates the time delays and reduces the overshoot. The system achieves a temperature stability of the immersion liquid within 227 0.01 °C/30 mins, and it also has a good characteristic of anti-interference. & 2016 Elsevier Ltd. All rights reserved.
Keywords: Temperature control High stability Cascade structure Composite control algorithm Immersion liquid
1. Introduction Immersion lithography has been developed as an important approach to drive the resolution of optical lithography into 50 nm and below [1]. Due to the large refractive index of the immersion liquid, the NA (Numerical Aperture) of the lithographer, NA = nsinθ , can be larger. And according to Rayleigh’s rule, r = kλ /NA , the resolution will be smaller with large NA. In practice, the resolution of ArF immersion lithography can be smaller than 65 nm [2,3], and with MPL (Multiple Patterning Lithography) technology, the resolution can even reach sub 20 nm [4]. The high resolution of immersion lithography requires the high stability of the immersion liquid, especially the temperature stability, which is closely related to the stability of the refractive index, density, surface tension, and gas solubility [5]. According to Nikon’s research, the temperature coefficient of the refractive index of the immersion liquid, dn/dT , is about 2.0 10 4 K 1, and the formation of bubble has a close relationship with the temperature variation of water [6]. And currently, the temperature stability requirement of the immersion liquid is at least within the range of 22 70.01 °C/30 mins [7]. The temperature control or the thermal management is a hot n
Corresponding author. E-mail address: [email protected] (M. Lei).
topic in lithography. Nie et al. studied the temperature control system for the projection lens, he proposed a two inputs and two outputs nonlinear PI algorithm to improve the convergence speed and the steady state precision, and it achieved a high convergence speed and 70.006 °C temperature stability [8]. He et al. proposed a cascade-connected fuzzy PID feedback algorithm which controlled the temperature of immersion liquid by adjusting the flow rate of the PCW (Process Cooling Water) through heat exchangers [9]. Entegris proposed a re-circulating water bath method for the water purification system and a second stage “on-tool” temperature control system which served as a “polisher” that controlled the final temperature of immersion liquid close to the wafer in immersion lithography [5]. About the high precision temperature control of the liquid, Chinglain Chou from Stanford University proposed a steady state optimal control law which was composed of state feedback and time-dependent disturbance feedforward, and it controlled the shower oil temperature variance within 2.2 m °C [10]. However, there is little work about the temperature control system of the immersion liquid, and here we will study the system. First, the structure of the temperature system of cascade structure is proposed. Then the mathematical model of the system is built. To suppress the time-dependent disturbance, a smith predictor and a feedforward controller have also been added. Besides, a hierarchical control algorithm is adopted. Last, experiments are conducted to verify the system.
http://dx.doi.org/10.1016/j.flowmeasinst.2016.08.014 0955-5986/& 2016 Elsevier Ltd. All rights reserved.
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2
2. The temperature control system of the immersion liquid
different.
2.1. The immersion system in immersion lithography
The temperature system can be simplified as shown in Fig. 2-3. The random disturbance from the first flow control valve and heat exchanger can be merged as the secondary disturbance to the pilot area. The pump, the second flow control valve and the long pipe to the immersion hood can be treated as the inert zone, and the load disturbance from the pump and environment temperature variation of this area can be viewed as the primary disturbance. The corresponding block diagram is shown in Fig. 2-4. The first flow control valve and heat exchanger in the fast adjusting loop can be simplified as G2(s ) and the disturbance to these components can be viewed as N2(s ). The pump, the second flow control valve and the long pipe can be simplified as G1(s ), and the disturbance to these components is simplified as primary disturbance N1(s ).
The temperature control system is a subsystem of the immersion system which supplies the requirements satisfied liquid to the immersion hood. As shown in Fig. 2-1, the immersion system is composed of the degassing module which decreases the soluted gas in the liquid, UV module which sterilizes the liquid, ion/silica removal module which removes all the unwanted ions, filtration module which filters all the tiny particles, and the temperature control module (or system) which ensures the temperature stability of the liquid is within specification. Since the temperature of the liquid can be easily affected by other thermal factors, for example, environment changes and the self-heating element, so it is placed at the end of immersion system.
2.3. Improved cascade control structure
2.2. Temperature control system of cascade structure
Cascade control structure can improve the system’s respond speed and has the ability to suppress the secondary disturbance to the inner loop. However, the primary disturbance to the system is from the temperature fluctuation of the ambient environment which is in the outer loop, and the cascade control can do nothing to suppress it. As shown in Fig. 2-5, when the environment temperature varies about 1.5 °C, the fluctuation of the output temperature exceeds 0.05 °C, which cannot be ignored when compared with the stability specification. To suppress the primary disturbance, which is mainly the environment disturbance, in the outer loop, an environment feedforward controller is added to the cascade control system. The
The schematic of the temperature control system is shown in Fig. 2-2. The heat exchangers are used as the thermal control elements, and the outlet temperature of the heat exchanger can be controlled by adjusting the flow rate of PCW. Since there is a long pipe between the temperature measurement point T4 and the last servo valve, there will be serious time delays and the environment disturbance will be easily introduced. So a single loop controller is not enough and a cascade control structure, as shown in Fig. 2-3, is used. To achieve the high precision temperature control, FLUKE’s NTC reference thermistor probe is used, its short-term repeatability is 70.006 °C and long term accuracy is better than 70.01 °C. The cascade control system consists of a fast-adjustment loop (inner loop) and main control loop (outer loop). Compared with the single feedback loop control configuration, the cascade control has following advantages in temperature control system [11]: 1) The sub controller is utilized to correct the disturbance arising within the inner loop before they can affect the controlled variable. 2) The effect of phase lag existing in the inner loop may be reduced by the sub controller, thus allowing the speed of response of the main loop to be improved. 3) The effect of parameter variation within the inner loop is corrected by the sub controller. 4) The effect of nonlinearity of the inner loop can be suppressed, especially when the time constant of two controllers is very P 1
Fig. 2-2. Schematic of the temperature control system.
F 1
T 1
P 2
T 4
IH
IFC1 UPW Flow Controller
Booster Pump
T 3
Degassing Module
Temperature Control System
Heat Exchanger
UPW
T 5
UV Moudle
PCW Flow Controller T 2
IFC2
IFC3 Recirculation Pump
Iron/sillica Removal
PCW inlet
Filtration
Fig. 2-1. The schematic of the immersion system of immersion lithography.
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Fig. 2-3. Diagram of the temperature control system of cascade structure.
Td(s ). Td(s ) is the temperature of environment and serves as the input of Gd( s ) and Gf (s ). Besides the disturbance caused by the environment variation, the heat exchange process in the pilot zone and the liquid transportation in the inert zone will cause significant time delays which can deteriorate the dynamic performance and the stability of the system. To overcome these drawbacks, a smith predictor GB(s ) is added as shown in Fig. 2-7. Fig. 2-4. Block diagram of the control system of cascade structure.
3. The modeling of the temperature control system
temperature /Celsius
23
Ty Environment
22.5
3.1. Modeling of the system The flow controller’s input has a linear relationship with the output as shown in Fig. 3-1. The adjustment time is less than 2 s, and the time delays can almost be ignored. So a proportional model, Gt ( s ) = Kt , can be used to describe it. The heat exchanger’s dynamic model can be obtained by mechanism modeling. Through the heat balance equation and the heat conduction equation, the transfer function Gh( s ) of the heat exchanger can be got, and it is
22 21.5 21 100
200
300 400 500 sample point
600
700
Fig. 2-5. The fluctuation of target temperature caused by the environment temperature variation.
Gh( s )=
∆Tho( s) ∆mci( s)
=
(
)
Kh 1+ταs 2 2 τb s +2ψτbs +1
(3-1)
where ∆mci(s) is the flow rate increment of the cooling water, ∆Tho( s) is the temperature increment of immersion liquid, Kh is the gain of the heat exchanger, τα is the first order lead time constant, τb is the second order delay time constant, and ψ is the damp coefficient. Considering the thermal lag, the mathematical model of pilot area can be expressed as
G2( s ) =
K2(1+ταs ) τb2s 2+2ψτbs +1
e−τ 2s
(3-2)
Fig. 2-6. Block diagram of feedforward-cascade control structure.
where K2 = KtKh , and τ2 is the time constant. The inert area is generally regarded as a second order inertia element in practice, so it can be expressed in the following form:
main principle of feedforward controller is to introduce a feedforward adjustment in the inner loop by measuring the environment temperature variation, it can offset the temperature fluctuation in the long pipe caused by environment disturbance. The block diagram of the feedforward-cascade control structure is shown in Fig. 2-6, the model of disturbance Gd(s ) is introduced at the place of the primary disturbance. The feedforward controller Gf (s ) is introduced in the inner loop to suppress the disturbance
Fig. 2-7. Block diagram of the feedforward-cascade control structure with smith predicator.
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G1( s ) =
K1(1+τms ) τn2s 2+2ωτns +1
e−τ1s
(3-3)
Since a zero is very close to a pole and far to another pole, it can be further simplified as
G1′( s ) = 3.2. Parameter identification For model parameter identification of the pilot zone and inert zone, the off-line and open-loop gray identification method is used. The input is the flow percentage Wc of the flow controller decided by the M-sequence with time interval of τ . The output of pilot model is the temperature at the outlet of the heat exchanger. The identification experiment results are shown in Fig. 3-2 and in Fig. 3-3. Based on the identification data, the unknown constants in Eqs. (3-2) and (3-3) can be determined. The mathematical model of the pilot area can be considered as first order or second order, and the corresponding identification result is shown in Table 3-1. Table 3-1 shows that both the mean square errors of the second order model and the first order model are less than 2%. The second order model is better than first order model when considering the time delays, but considering the complexity of the model, the first order model is preferable. Here we chose the first order model, and the pilot area can be expressed as:
G2′( s ) =
−1. 3361 −11.63s e 1 + 11. 834s
1. 0092 e−35.75s 1 + 138. 06s
(3-6)
As to the influence of the environment disturbance to the outlet temperature, the mathematical model is unknown, so we just get it by identification using black box experiment. Fig. 3-4 is the result of the disturbance identification experiment. The input is the environmental temperature Td , which varies with a period of 90–100 mins, and the output is the temperature Ty at the outlet of the system. In order to reduce the complexity, the model of first order with time delays has been selected to describe the influence of the disturbance. With parameter identification, the transfer
(3-4)
The inert area is considered as a second order model, and the identification result is shown in Table 3-2. The mathematical model of the inert area can be expressed as
G1( s ) =
1. 0092(1 + 7. 11s ) e−35.75s (1 + 14. 25s )(1 + 116. 71s )
Fig. 3-3. The temperature at the outlet of the heat exchanger and the outlet of the system with manipulated temperature setting Tt.
(3-5) Table 3-1 The identification result of the pilot area. Orders
Gain
Zero
Pole
Second First
1.2898 0.01367 1.3361 0
Mean square error
0.021, 0.032 1.53% 0.0845 1.58%
Time delay 6.7 11.63
Table 3-2 The identification result of inert area. Orders
Gain
Zero
Pole
Mean square error
Time delay
Second
1.0092
0.141
0.00857, 0.07
0.18%
35.75
24 Fig. 3-1. The mA output vs flow rate of the flow controller.
Ty
Temperature /Celsius
23.5 Td 23 22.5 22 21.5
0
500
1000
1500
2000
sample count Fig. 3-2. The temperature at the outlet of the heat exchanger and the outlet of the system with manipulated flow rate of the cooling water.
Fig. 3-4. The outlet temperature fluctuation under the manipulated environment temperature change.
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function can be expressed as:
Gd( s ) =
0. 90078 106 + s
e−294s
(3-7)
Unlike the manipulated environment temperature change, the actual random disturbance from ambient is somewhat like white noise, and the fluctuation of the ambient temperature is shown in Fig. 3-5.
4. The controllers and control algorithm 4.1. The sub controller In cascade control system, the sub controller Gs(s ) is generally a PI or P controller. However, as the sub controller locates in the fast regulating loop, it will be easy to be integral windup when regulating frequently. So an IS-PI (Integral Separated PI) algorithm is applied in the sub-controller here. It can decrease the regulating time by offset the integral action in the initial stage which will introduce a considerable error. The key of IS-PI algorithm is setting the threshold E0 of the separation properly. When e( kT ) ≤ E0 , which means the error is very small, it is time for PI control. And when e( kT ) >| E0 |, it is time for P control. The algorithm can be expressed as:
⎧ 100% Wc ( k )≥100% ⎪ ⎪ Wc ( k ) = ⎨ Wc ( k−1)+∆Wc ( k ) 20% ≤ Wc ( k )<100% ⎪ ⎪ Wc ( k )<20% ⎩ 20%
(4-1)
∆Wc ( k ) = KPs⋅∆es ( k ) + KF KIs⋅es ( k )
(4-2)
where k is the sampling number, Wc( k ) is percentage of immersion liquid’s flow, ∆Wc( k ) is the variation value of flow output, KPs is the proportional coefficient, KIs is the integral coefficient, es( k ) = Tt ( k ) − Tx(k ) is the difference between the target temperature and the output of heat exchanger, ∆es( k ) = es( k ) − es( k −1) is ⎧ 1 | e k | ≤ 0. 2 ⎪ s( ) is the integral the change rate of error, and KF = ⎨ ⎪ 0 | es( k )| > 0. 2 ⎩ separation control coefficient. 4.2. The main controller The functionality of the main controller is to produce a set-
5
point for the sub controller. In order to improve the system dynamics and shorten the adjustment time, a hierarchical control algorithm is adopted. According to the algorithm, the temperature control process can be defined as follows: 1) To make immersion liquid’s temperature converge as fast as possible, the system is divided into five modes or five stages (D1 D5) from booting the system to reaching the stable state. 2) The algorithm can also be divided into two layers, the upper layer and lower layer. The upper layer is focused on the switch among different modes, and the selection of the mode is based on the decision-making information and inference rules. 3) The lower layer fulfills the precise control using the algorithm of PID, the control in different mode will utilize similar PID algorithm, but different coefficients will be used. The detailed description of the five modes (D1 to D5) is as follows. Here Tcw is the temperature of the cooling water, and Tsc is set point of cooling water. Tu is the outlet temperature of the system, TH is output temperature of heat exchanger, Tsm is the setpoint of the main controller, and Tss is the set-point of the sub controller. The threshold values of different stage are Ta=0.2 °C , Tb=0.05 °C and Tc =0.01 °C . D1 is the system initialization stage. In this stage, the set-point of sub-controller Tss(0) and the initial value of the cooling water Tcw (0) are set first. If the temperature control system is stopped as D5 last time, TH (0) can be got from the historical stable data. Otherwise, the Tsm(0) can be set according to the experience. In addition, the feedforward controller and other controllers will also be initialized. D2 is the fast adjustment stage of the target temperature. The system adjusts the deviation from a big one to a comparable small one with large step size. D3 is the fine adjustment stage. At this stage, the deviation is very small, and a slow PI regulation with smaller step size and longer adjustment period will be enough. Besides, the temperature precision is in the range of Tb . D4 is the stage of optimization. The system will optimize the output with smaller step size. When the output temperature of immersion liquid keeps in the range of Tb in the period of h2, it can be regarded as the optimal value and recorded as the optimal temperature of D5 stage. D5 is the ultra-stable stage. The system can be viewed as stable and all need to do is adjust the deviation between the current value and the optimal value at the outlet and make it smaller than Tc . As to the switching among the modes, it depends on the historical data, process data and reasoning rules. There are two reasoning rules in the system: Ψ1 and Ψ2. Ψ1 is mode selection based on the normal running of the whole immersion system, and Ψ2 is the mode selection for the booting of the temperature control system. Ψ1 is used for the process change from D1 to D5 step by step when the condition is OK. And the switching rules are as follows: TR1: If TR2: If TR3: If TR4: If to D5.
Tcw( k ) − Tsc < Ta , then from D1 to D2; TH ( k ) − Tss < Tb , then from D2 to D3; Tsm − Tu(k ) h1, then from D3 to D4; Tsm − Tu(k ) h2, then from D4
Here Ch1 is the period to keep value on the condition of TR3 and h1 is the threshold value of period for TR3. Similar, Ch2 is the period to keep value on the condition of TR4 and h2 is the threshold value of period for TR4. Fig. 3-5. The fluctuation of the ambient temperature.
Ψ1 = {TR1, TR2, TR3, TR4}
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(4-3)
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Each time the system boots, the initial condition will be different, so the time cost is different to reach the stable state. If we still switch from D1 to D5 as done in the normal operation mode, it will be time consuming. So Ψ2 is prepared, and the switching rules for booting are as follows: SR1: then SR2: then SR3: then
If Tsm − Tu( 0) > Tb , or Tcw( 0) − Tsc > Ta , or TH ( k ) − Tss > Tb , from start to D1; If Tsm − Tu( 0) < Tb , Tcw( 0) − Tsc < Ta , and Tss − TH ( 0) < Tb , from start to D4; If Tsm − Tu( 0) < Tc , Tcw( 0) − Tsc < Tb , and Tss(0)−TH ( 0) < Tc , from start to D5.
Ψ2 = {SR1, SR2, SR3}
Fig. 4-2. The equivalent transformation of feedforward-cascade diagram.
(4-4)
In practice, the immersion system sometimes needs to be changed from injecting liquid mode to exhausting liquid mode which will cause flow rate shock and induce some variation of the temperature. In order to improve the robustness and stability of system, the switching rule TR5 from D5 to D4 is added. TR5: If Tsm − Tu( k ) > Tc , then from D5 to D4. Here the new switching rules of Ψ1 is:
Ψ1′ = {TR1, TR2, TR3,TR4,TR5}
(4-5)
And the mode selection and transformation can be expressed by Fig. 4-1. The main controller’s algorithm is a multi-stage PID algorithm which can be formulated as:
⎧ 22. 5 Tt ( k )≥22. 5 ⎪ ⎪ Tt ( k ) = ⎨ Tt ( k−1)+∆Tt ( k ) 20. 5
⎧ K ⋅∆e ( k ) + K ⋅e ( k ) T ( k ) ≥ e Pi n Ii n t n ⎪ ⎪ ∆Tt ( k ) = ⎨ KPi⋅∆e( k ) + KIi⋅e( k ) em <| e( k )|
Fig. 4-3. Simplified block diagram of the feedforward controller.
Gf *( s ) =
(4-8)
Gs( s )G1( s )G2( s ) 1+Hs( s )Gs( s )G2( s )
(4-9)
As shown by Fig. 4-3, according to principle of full compensation, the disturbance produced by Td(s ) will be compensated by G*f ( s ). So we can get the following equation:
Td( s )Gf *( s )Gc ( s ) + Td( s )Gd( s )=0
(4-7)
where i = 1, 2, 3,4, 5 and it indicates the five modes in the hierarchical control algorithm, KPi is the proportional coefficient of mode i , KIi is the integral coefficient, Tt ( k ) is the setpoint of subcontroller, ∆Tt ( k ) is the change in the set-point of the sub-controller, e( k ) = Tr ( k ) − Ty( k ) is the deviation of main loop’s temperature, ∆e( k ) = e( k ) − e( k −1), ∆en( k ) = en( k ) − e( k −1) is the change rate of the endpoint’s temperature, em is the dead zone of deviation and the main controller does nothing on the zone, and en is the maximum of the deviation.
4.3. The feedforward controller The improved cascade control structure with feedforward controller shown in Fig. 2-6 can be simplified as shown in Fig. 4-2, and the equivalent feedforward controller is
(4-10)
According to Eqs. (4–8) and (4–10), we can get
Gf *( s ) =
Gf ( s ) Gs( s )
=−
Gd( s ) Gc ( s )
(4-11)
And substitute Gc ( s ) with Eq. (4-9), then
Gf ( s ) = −
Gd( s )( 1+Gs( s )G2( s )Hm(s )) G1( s )G2( s )
(4-12)
where Gs( s ) = (0. 3238s+ 0. 0312) /s is the PI sub-controller, and Hm(s ) is a unit gain. Substitute all the variables in Eq. (4-12), we can get
Gf ( s ) = −9. 01×10−4
1633. 25s +1
103s +1 ( 1 + 10. 36s)( 1 + 138. 06s) e−11.63s −2. 36×10−7 s 1+10−3s
(
)
(4-13)
We can see that the feedforward controller is composed of two parts: the major part which is a first order element and the minor part which is a high-order element with time delays. Since Eq. (413) is too complex to be applied in industry, the high order part is ignored here, and
Gf ( s ) = Fig. 4-1. Diagram of the mode selection and transformation.
Gs( s )
Then, it can be further simplified as shown in Fig. 4-3, and the equivalent transfer function in main channel Gc (s ) is
Gc ( s ) = (4-6)
Gf ( s )
yd ( s
)=K
Td( s )
d
T2s +1 T1s +1
(4-14)
With Laplace transformation, Eq. (4-14) can also be expressed
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in time domain as
T1
dyd (t ) dt
(
GB( s ) = G1′( s ) 1 − e−(τ1+ τ 2)s
dT (t ) + yd ( t ) = KdT2 d + KdTd(t ) dt
(4-15)
(4-16)
where T0 is the sampling period. Here we assume that α =
β=
T2 , T0
T1 T0
and
then Eq. (4-16) can be expressed as
α∆yd ( t ) + yd ( t ) = Kd⎡⎣ β∆Td( t ) + Td( t )⎤⎦
(4-17)
The input Td( s ) and the output Ty( s ) has been described in the Fig. 3-4, and the mathematical model of the inert zone has been got in the Section 3.2, so the estimated fully compensated value y^d (s ) under the disturbance Td(s ) can be got. Substituting the parameters in Eq. (4-17) with the values in Table 4-1, we can get
⎧ ⎪ −0. 00346α = K ( 0. 14484β + 22. 15) d ⎨ ⎪ ⎩ 23. 28Kd = − 0. 027
780s +1 74. 9925s +1
(4-19)
4.4. The controller based on Smith predictor As described in Section 2.3, a controller based on Smith predictor is proposed to solve the problem of significant time delays in temperature control system. In order to obtain the Smith predictor, the controlled object is simplified as an element of first order with time delays, G1′(s )e−(τ1+ τ 2)s (Fig. 4-4). The mathematical model of the controlled object is
G1′( s )e−(τ1+ τ 2)s =
(4-21)
GB( s ) can also be expressed in z-domain as
⎡ 1−e−T0s ⎤ 1. 0092 GB( z ) = z⎢ 1−e−(τ1+ τ 2)s ⎥ s 1 + 138. 06 s ⎣ ⎦
(
= 1−z−4
)
−1
z ) 1 0.− 071 0. 93z
−1
(4-22)
where T0 is the minimum sampling period. In order to get the difference equation, the following transformation is made
GB( z ) =
TB(z ) T (z ) TA(z ) = B Tt (z ) TA(z ) Tt (z )
(4-23)
Here we assume that
⎧ TB(z ) = 1−z−4 ⎪ ⎪ TA(z ) ⎨ ⎪ TA(z ) 0. 071z−1 ⎪ T (z ) = ⎩ t 1 − 0. 93z−1
(4-24)
So the final difference equation is
(4-18)
From Fig. 3-4 we can see that the peak time of the disturbance is 156T0 . Since ∆Td( t ) plays a dominant role in β∆Td( t ) + Td( t ), so ^ |β∆Td( t )| > | T¯d( t )|, and we take β=156. With Eq. (4-18), we can also get α = 14. 9985, β = 156, Kd = −0. 0011598, and the final result of feedforward controller is
Gf ( s ) = −0. 0011598
)
(
Replace the differential part with difference, we can get
⎡T ⎤ T1 ∆yd ( t ) + yd ( t ) = Kd⎢ 2 ∆Td( t ) + Td( t )⎥ T0 ⎣ T0 ⎦
7
1−e−Ts 1. 0092 e−47.4s s 1 + 138. 06s
(4-20)
And the corresponding Smith predictor is
⎧ ⎪ TA( k )=0. 93TA( k−1)+0. 071Tt (k−1) ⎨ ⎪ ⎩ TB( k ) = TA( k ) − TA( k−4)
5. The experiment results 5.1. The results of different controllers In order to compare the performance of the temperature control system with different controllers, a series of experiments are conducted. There are three configurations: only with feedforward controller, only with Smith predictor, and with both feedforward controller and Smith predictor. The result is shown in Fig. 5-1, and the ISE (Integral Error Square) criterion is used here to evaluate the controller’s performance. We can see that the system with both feedforward controller and Smith predictor has the smallest ISE, so it has better performance than the other configurations. Then the performance of the feedforward controller is also evaluated. A manipulated disturbance, the temperature variance of the environment, which is shown in Fig. 5-2 is introduced, and the feedforward controller is not added until the sampling point of 1981. The outlet temperature is shown in Fig. 5-3, we can see that the temperature stability at the system outlet has improved a lot
Table 4-1 Parameter estimation of full compensation. Index
^ Td( t )
^ ∆T d ( t )
y^d ( t )
∆y^d ( t )
Initial state Peak state
22.15 23.28
0.14480 0
0 0.027
0.00346 0
Fig. 4-4. Simplified block diagram of the system with smith predictor.
(4-25)
Fig. 5-1. The performance of the system with different controllers.
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8
with the feedforward controller. 5.2. The experiment results under different conditions Firstly, the system is tested without manipulated disturbance, and the system is operated just as it is. The result is shown in Fig. 5-4, the sampling interval is 10 s, and the total sampling time
24.4
24
24
23.8
23.5
Td
23
Ty
23.6 23.4 23.2 0
500
1000
1500 2000 sample count
2500
3000
Temperature /Celsius
T d /Celsius
24.2
is about eight hours. We can see that the temperature fluctuation is almost within 227 0.005 °C. Then, the manipulated environment disturbance is introduced. Fig. 5-5 shows the temperature at the outlet when the environment temperature Td varies from 21 to 24 °C which also has a fluctuation value of about 2 °C@10 mins, and Fig. 5-6 is the corresponding temperature fluctuation. We can see that the temperature Ty at the outlet of the system can maintain a high-precision and the stability which is within 22 70.008 °C@30 mins. Finally, the disturbance from the system is introduced, and the result is shown in Fig. 5-7. The flow rate of immersion liquid changes from 1 lpm to 0.8 lpm at the sampling point of 2016 and
Fig. 5-2. The manipulated temperature fluctuation of the environment.
22.5 22 21.5 21 20.5
0
22.08
2000
3000 4000 sample count
5000
6000
Fig. 5-5. The outlet temperature with manipulated environment temperature fluctuation.
22.06 22.04 1981 22.02 22 21.98 0
500
1000
1500 2000 sample count
2500
3000
Fig. 5-3. The outlet temperature of the system with feedforward and without feedforward controller.
Tem p eratu re v arian ce /Celsiu s
T y /Celsius
1000
0.04
0.03
0.02
0.01
0
0
1000
2000
3000 4000 sample count
5000
6000
Fig. 5-6. The outlet temperature fluctuation with the environment disturbance.
22.1
T y /Celsius
22.05
3733
2016 22
21.95
0
Fig. 5-4. The outlet temperature during a long period without manipulated disturbance.
1000
2000
3000
4000 5000 sample count
6000
7000
8000
Fig. 5-7. The outlet temperature when changing the inlet condition of the system.
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the corresponding thermal load increases sharply which causes a 0.2 °C fluctuation of the temperature at the outlet of the system, and it takes about 47 mins for the system to recover to the stable state of 227 0.01 °C@30 mins. At the sampling point of 3733, the temperature of the cooling water is reduced by 0.5 °C, it causes a 0.2 °C fluctuation of the outlet temperature, and the system returns to normal state at the sampling point of 3928, which costs about 32 mins.
[3]
[4]
[5] [6]
6. Conclusions [7]
A temperature control system of high precision and stability was built, which adopted a cascade structure with feedforward controller and Smith predictor. Then, the mathematical model of the system was built where the gray and black box experiments were used. And to reach the temperature requirements, four controllers were used: the main controller using a hierarchical control algorithm and multi-stage PID algorithm, the sub controller using the integral separated algorithm, the feedforward controller, and the Smith predictor. Last, the experiments were conducted, and the results showed that the robustness of the system was excellent, and the outlet temperature of the system would maintain a stability of 22 70.008 °C@30 mins with a 2 °C@10 mins fluctuation of the environment temperature.
[8]
[9] [10]
[11]
9
Jong, et al. Immersion lithography exposure systems: today’s capabilities and tomorrow's expectations (2005), pp. 710–724 〈http://dx.doi.org/10.1117/12. 606799〉. C. Robinson, D. Corliss, J. Barns, K. Cummings, H. Jansen, B. Lee, et al., Immersion Lithography Water Quality at Albany Nanotech, in: International Symposium on Immersion Immersion Lithography, Brugges, Belgium, 2005. S.H. Park, D.O. Shin, B.H. Kim, D.K. Yoon, K. Kim, S.Y. Lee, et al., Block copolymer multiple patterning integrated with conventional ArF lithography, Soft Matter 6 (2010) 120–125, http://dx.doi.org/10.1039/B913853F. U.S. Entegris, Contamination control for ultra-pure water for the emerging applications in immersion lithography, (n.d.). S. Owa, H. Nagasaka, Immersion lithography; its potential performance and issues, Proc. SPIE. Opt. Microlithogr. XVI 5040 (2003) 724–733, http://dx.doi. org/10.1117/12.504599. B. Budhlall, G. Parris, P. Zhang, X. Gao, Z. Zarkov, B. Ross, et al., High refractive index immersion fluids for 193 nm immersion lithography (in:), Proc. SPIE Opt. Microlithogr. XVIII (2005) 622–629, http://dx.doi.org/10.1117/12.600025. X. Li, H. Nie, B. Yu, Non-linear PI-control temperature control algorithm for projection lens, in: 3rd International Symposium on Advanced Optical Manufacturing and Testing Technologies: Design, Manufacturing, and Testing of Micro- and Nano-Optical Devices and Systems, 2007, pp. 67241Q–67241Q. J. He, X. Li, M. Lei, B. Chen, J. Wang, A temperature control algorithm of immersion liquid for immersion lithography, SPIE Adv. Lithogr. (2014), p. 90522F–90522F. C. Chou, S.U.D. of Aeronautics, Astronautics, S.U. Guidance, C. Laboratory, A precise fluid temperature control for precision machines, Stanford University, 1988 〈https://books.google.com.hk/books?id ¼csgDAAAAIAAJ〉. F.G. Shinskey, Process Control Systems: Application, Design, and Tuning, McGraw-Hill, New York, USA, 1996 〈https://books.google.com.hk/books? id ¼EqdTAAAAMAAJ〉.
References [1] W. Chen, Y. Chen, J. Zou, X. Fu, H. Yang, X. Ruan, et al., Effect of liquid dispensing on flow field for immersion lithography, J. Vac. Sci. Technol. B 27 (2009) 2192–2199, http://dx.doi.org/10.1116/1.3212916. [2] J. Mulkens, B. Streefkerk, M. Hoogendorp, R. Moerman, M. Leenders, F. de
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JFMI 1249 Flow Measurement and Instrumentation ∎ (∎∎∎∎) ∎∎∎–∎∎∎
Contents lists available at ScienceDirect
Flow Measurement and Instrumentation journal homepage: www.elsevier.com/locate/flowmeasinst
High precision and stability temperature control system for the immersion liquid in immersion lithography Xiaoping Li, Yiwen Zhao, Min Lei n School of Mechanical Science & Engineering, Huazhong University of Science & Technology, Wuhan 430074, China
art ic l e i nf o
a b s t r a c t
Article history: Received 3 March 2016 Received in revised form 30 July 2016 Accepted 22 August 2016
The temperature stability of immersion liquid is one of the main factors that affect the performance of the immersion lithography tool. Since the temperature control system of immersion liquid has the characteristics of time delays, full of disturbance and non-linear, the system and control algorithm should be carefully designed to control the temperature of the immersion liquid within the specification. In this paper, a control system of cascade structure with feed forward and lag compensation is proposed to reduce the time delays and the disturbance caused by the temperature fluctuation of ambient environment. Then, the mathematical model of the temperature control system is built, and the parameters of the model are obtained by ‘gray’ identification method. Based on the model, an algorithm which combines hierarchical control algorithm, integral separation PI algorithm, feed forward algorithm and lag compensation algorithm is designed. Last, experiments are conducted to evaluate the algorithm. The results show that the algorithm improves the robustness, compensates the time delays and reduces the overshoot. The system achieves a temperature stability of the immersion liquid within 227 0.01 °C/30 mins, and it also has a good characteristic of anti-interference. & 2016 Elsevier Ltd. All rights reserved.
Keywords: Temperature control High stability Cascade structure Composite control algorithm Immersion liquid
1. Introduction Immersion lithography has been developed as an important approach to drive the resolution of optical lithography into 50 nm and below [1]. Due to the large refractive index of the immersion liquid, the NA (Numerical Aperture) of the lithographer, NA = nsinθ , can be larger. And according to Rayleigh’s rule, r = kλ /NA , the resolution will be smaller with large NA. In practice, the resolution of ArF immersion lithography can be smaller than 65 nm [2,3], and with MPL (Multiple Patterning Lithography) technology, the resolution can even reach sub 20 nm [4]. The high resolution of immersion lithography requires the high stability of the immersion liquid, especially the temperature stability, which is closely related to the stability of the refractive index, density, surface tension, and gas solubility [5]. According to Nikon’s research, the temperature coefficient of the refractive index of the immersion liquid, dn/dT , is about 2.0 10 4 K 1, and the formation of bubble has a close relationship with the temperature variation of water [6]. And currently, the temperature stability requirement of the immersion liquid is at least within the range of 22 70.01 °C/30 mins [7]. The temperature control or the thermal management is a hot n
Corresponding author. E-mail address: [email protected] (M. Lei).
topic in lithography. Nie et al. studied the temperature control system for the projection lens, he proposed a two inputs and two outputs nonlinear PI algorithm to improve the convergence speed and the steady state precision, and it achieved a high convergence speed and 70.006 °C temperature stability [8]. He et al. proposed a cascade-connected fuzzy PID feedback algorithm which controlled the temperature of immersion liquid by adjusting the flow rate of the PCW (Process Cooling Water) through heat exchangers [9]. Entegris proposed a re-circulating water bath method for the water purification system and a second stage “on-tool” temperature control system which served as a “polisher” that controlled the final temperature of immersion liquid close to the wafer in immersion lithography [5]. About the high precision temperature control of the liquid, Chinglain Chou from Stanford University proposed a steady state optimal control law which was composed of state feedback and time-dependent disturbance feedforward, and it controlled the shower oil temperature variance within 2.2 m °C [10]. However, there is little work about the temperature control system of the immersion liquid, and here we will study the system. First, the structure of the temperature system of cascade structure is proposed. Then the mathematical model of the system is built. To suppress the time-dependent disturbance, a smith predictor and a feedforward controller have also been added. Besides, a hierarchical control algorithm is adopted. Last, experiments are conducted to verify the system.
http://dx.doi.org/10.1016/j.flowmeasinst.2016.08.014 0955-5986/& 2016 Elsevier Ltd. All rights reserved.
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2. The temperature control system of the immersion liquid
different.
2.1. The immersion system in immersion lithography
The temperature system can be simplified as shown in Fig. 2-3. The random disturbance from the first flow control valve and heat exchanger can be merged as the secondary disturbance to the pilot area. The pump, the second flow control valve and the long pipe to the immersion hood can be treated as the inert zone, and the load disturbance from the pump and environment temperature variation of this area can be viewed as the primary disturbance. The corresponding block diagram is shown in Fig. 2-4. The first flow control valve and heat exchanger in the fast adjusting loop can be simplified as G2(s ) and the disturbance to these components can be viewed as N2(s ). The pump, the second flow control valve and the long pipe can be simplified as G1(s ), and the disturbance to these components is simplified as primary disturbance N1(s ).
The temperature control system is a subsystem of the immersion system which supplies the requirements satisfied liquid to the immersion hood. As shown in Fig. 2-1, the immersion system is composed of the degassing module which decreases the soluted gas in the liquid, UV module which sterilizes the liquid, ion/silica removal module which removes all the unwanted ions, filtration module which filters all the tiny particles, and the temperature control module (or system) which ensures the temperature stability of the liquid is within specification. Since the temperature of the liquid can be easily affected by other thermal factors, for example, environment changes and the self-heating element, so it is placed at the end of immersion system.
2.3. Improved cascade control structure
2.2. Temperature control system of cascade structure
Cascade control structure can improve the system’s respond speed and has the ability to suppress the secondary disturbance to the inner loop. However, the primary disturbance to the system is from the temperature fluctuation of the ambient environment which is in the outer loop, and the cascade control can do nothing to suppress it. As shown in Fig. 2-5, when the environment temperature varies about 1.5 °C, the fluctuation of the output temperature exceeds 0.05 °C, which cannot be ignored when compared with the stability specification. To suppress the primary disturbance, which is mainly the environment disturbance, in the outer loop, an environment feedforward controller is added to the cascade control system. The
The schematic of the temperature control system is shown in Fig. 2-2. The heat exchangers are used as the thermal control elements, and the outlet temperature of the heat exchanger can be controlled by adjusting the flow rate of PCW. Since there is a long pipe between the temperature measurement point T4 and the last servo valve, there will be serious time delays and the environment disturbance will be easily introduced. So a single loop controller is not enough and a cascade control structure, as shown in Fig. 2-3, is used. To achieve the high precision temperature control, FLUKE’s NTC reference thermistor probe is used, its short-term repeatability is 70.006 °C and long term accuracy is better than 70.01 °C. The cascade control system consists of a fast-adjustment loop (inner loop) and main control loop (outer loop). Compared with the single feedback loop control configuration, the cascade control has following advantages in temperature control system [11]: 1) The sub controller is utilized to correct the disturbance arising within the inner loop before they can affect the controlled variable. 2) The effect of phase lag existing in the inner loop may be reduced by the sub controller, thus allowing the speed of response of the main loop to be improved. 3) The effect of parameter variation within the inner loop is corrected by the sub controller. 4) The effect of nonlinearity of the inner loop can be suppressed, especially when the time constant of two controllers is very P 1
Fig. 2-2. Schematic of the temperature control system.
F 1
T 1
P 2
T 4
IH
IFC1 UPW Flow Controller
Booster Pump
T 3
Degassing Module
Temperature Control System
Heat Exchanger
UPW
T 5
UV Moudle
PCW Flow Controller T 2
IFC2
IFC3 Recirculation Pump
Iron/sillica Removal
PCW inlet
Filtration
Fig. 2-1. The schematic of the immersion system of immersion lithography.
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Fig. 2-3. Diagram of the temperature control system of cascade structure.
Td(s ). Td(s ) is the temperature of environment and serves as the input of Gd( s ) and Gf (s ). Besides the disturbance caused by the environment variation, the heat exchange process in the pilot zone and the liquid transportation in the inert zone will cause significant time delays which can deteriorate the dynamic performance and the stability of the system. To overcome these drawbacks, a smith predictor GB(s ) is added as shown in Fig. 2-7. Fig. 2-4. Block diagram of the control system of cascade structure.
3. The modeling of the temperature control system
temperature /Celsius
23
Ty Environment
22.5
3.1. Modeling of the system The flow controller’s input has a linear relationship with the output as shown in Fig. 3-1. The adjustment time is less than 2 s, and the time delays can almost be ignored. So a proportional model, Gt ( s ) = Kt , can be used to describe it. The heat exchanger’s dynamic model can be obtained by mechanism modeling. Through the heat balance equation and the heat conduction equation, the transfer function Gh( s ) of the heat exchanger can be got, and it is
22 21.5 21 100
200
300 400 500 sample point
600
700
Fig. 2-5. The fluctuation of target temperature caused by the environment temperature variation.
Gh( s )=
∆Tho( s) ∆mci( s)
=
(
)
Kh 1+ταs 2 2 τb s +2ψτbs +1
(3-1)
where ∆mci(s) is the flow rate increment of the cooling water, ∆Tho( s) is the temperature increment of immersion liquid, Kh is the gain of the heat exchanger, τα is the first order lead time constant, τb is the second order delay time constant, and ψ is the damp coefficient. Considering the thermal lag, the mathematical model of pilot area can be expressed as
G2( s ) =
K2(1+ταs ) τb2s 2+2ψτbs +1
e−τ 2s
(3-2)
Fig. 2-6. Block diagram of feedforward-cascade control structure.
where K2 = KtKh , and τ2 is the time constant. The inert area is generally regarded as a second order inertia element in practice, so it can be expressed in the following form:
main principle of feedforward controller is to introduce a feedforward adjustment in the inner loop by measuring the environment temperature variation, it can offset the temperature fluctuation in the long pipe caused by environment disturbance. The block diagram of the feedforward-cascade control structure is shown in Fig. 2-6, the model of disturbance Gd(s ) is introduced at the place of the primary disturbance. The feedforward controller Gf (s ) is introduced in the inner loop to suppress the disturbance
Fig. 2-7. Block diagram of the feedforward-cascade control structure with smith predicator.
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G1( s ) =
K1(1+τms ) τn2s 2+2ωτns +1
e−τ1s
(3-3)
Since a zero is very close to a pole and far to another pole, it can be further simplified as
G1′( s ) = 3.2. Parameter identification For model parameter identification of the pilot zone and inert zone, the off-line and open-loop gray identification method is used. The input is the flow percentage Wc of the flow controller decided by the M-sequence with time interval of τ . The output of pilot model is the temperature at the outlet of the heat exchanger. The identification experiment results are shown in Fig. 3-2 and in Fig. 3-3. Based on the identification data, the unknown constants in Eqs. (3-2) and (3-3) can be determined. The mathematical model of the pilot area can be considered as first order or second order, and the corresponding identification result is shown in Table 3-1. Table 3-1 shows that both the mean square errors of the second order model and the first order model are less than 2%. The second order model is better than first order model when considering the time delays, but considering the complexity of the model, the first order model is preferable. Here we chose the first order model, and the pilot area can be expressed as:
G2′( s ) =
−1. 3361 −11.63s e 1 + 11. 834s
1. 0092 e−35.75s 1 + 138. 06s
(3-6)
As to the influence of the environment disturbance to the outlet temperature, the mathematical model is unknown, so we just get it by identification using black box experiment. Fig. 3-4 is the result of the disturbance identification experiment. The input is the environmental temperature Td , which varies with a period of 90–100 mins, and the output is the temperature Ty at the outlet of the system. In order to reduce the complexity, the model of first order with time delays has been selected to describe the influence of the disturbance. With parameter identification, the transfer
(3-4)
The inert area is considered as a second order model, and the identification result is shown in Table 3-2. The mathematical model of the inert area can be expressed as
G1( s ) =
1. 0092(1 + 7. 11s ) e−35.75s (1 + 14. 25s )(1 + 116. 71s )
Fig. 3-3. The temperature at the outlet of the heat exchanger and the outlet of the system with manipulated temperature setting Tt.
(3-5) Table 3-1 The identification result of the pilot area. Orders
Gain
Zero
Pole
Second First
1.2898 0.01367 1.3361 0
Mean square error
0.021, 0.032 1.53% 0.0845 1.58%
Time delay 6.7 11.63
Table 3-2 The identification result of inert area. Orders
Gain
Zero
Pole
Mean square error
Time delay
Second
1.0092
0.141
0.00857, 0.07
0.18%
35.75
24 Fig. 3-1. The mA output vs flow rate of the flow controller.
Ty
Temperature /Celsius
23.5 Td 23 22.5 22 21.5
0
500
1000
1500
2000
sample count Fig. 3-2. The temperature at the outlet of the heat exchanger and the outlet of the system with manipulated flow rate of the cooling water.
Fig. 3-4. The outlet temperature fluctuation under the manipulated environment temperature change.
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function can be expressed as:
Gd( s ) =
0. 90078 106 + s
e−294s
(3-7)
Unlike the manipulated environment temperature change, the actual random disturbance from ambient is somewhat like white noise, and the fluctuation of the ambient temperature is shown in Fig. 3-5.
4. The controllers and control algorithm 4.1. The sub controller In cascade control system, the sub controller Gs(s ) is generally a PI or P controller. However, as the sub controller locates in the fast regulating loop, it will be easy to be integral windup when regulating frequently. So an IS-PI (Integral Separated PI) algorithm is applied in the sub-controller here. It can decrease the regulating time by offset the integral action in the initial stage which will introduce a considerable error. The key of IS-PI algorithm is setting the threshold E0 of the separation properly. When e( kT ) ≤ E0 , which means the error is very small, it is time for PI control. And when e( kT ) >| E0 |, it is time for P control. The algorithm can be expressed as:
⎧ 100% Wc ( k )≥100% ⎪ ⎪ Wc ( k ) = ⎨ Wc ( k−1)+∆Wc ( k ) 20% ≤ Wc ( k )<100% ⎪ ⎪ Wc ( k )<20% ⎩ 20%
(4-1)
∆Wc ( k ) = KPs⋅∆es ( k ) + KF KIs⋅es ( k )
(4-2)
where k is the sampling number, Wc( k ) is percentage of immersion liquid’s flow, ∆Wc( k ) is the variation value of flow output, KPs is the proportional coefficient, KIs is the integral coefficient, es( k ) = Tt ( k ) − Tx(k ) is the difference between the target temperature and the output of heat exchanger, ∆es( k ) = es( k ) − es( k −1) is ⎧ 1 | e k | ≤ 0. 2 ⎪ s( ) is the integral the change rate of error, and KF = ⎨ ⎪ 0 | es( k )| > 0. 2 ⎩ separation control coefficient. 4.2. The main controller The functionality of the main controller is to produce a set-
5
point for the sub controller. In order to improve the system dynamics and shorten the adjustment time, a hierarchical control algorithm is adopted. According to the algorithm, the temperature control process can be defined as follows: 1) To make immersion liquid’s temperature converge as fast as possible, the system is divided into five modes or five stages (D1 D5) from booting the system to reaching the stable state. 2) The algorithm can also be divided into two layers, the upper layer and lower layer. The upper layer is focused on the switch among different modes, and the selection of the mode is based on the decision-making information and inference rules. 3) The lower layer fulfills the precise control using the algorithm of PID, the control in different mode will utilize similar PID algorithm, but different coefficients will be used. The detailed description of the five modes (D1 to D5) is as follows. Here Tcw is the temperature of the cooling water, and Tsc is set point of cooling water. Tu is the outlet temperature of the system, TH is output temperature of heat exchanger, Tsm is the setpoint of the main controller, and Tss is the set-point of the sub controller. The threshold values of different stage are Ta=0.2 °C , Tb=0.05 °C and Tc =0.01 °C . D1 is the system initialization stage. In this stage, the set-point of sub-controller Tss(0) and the initial value of the cooling water Tcw (0) are set first. If the temperature control system is stopped as D5 last time, TH (0) can be got from the historical stable data. Otherwise, the Tsm(0) can be set according to the experience. In addition, the feedforward controller and other controllers will also be initialized. D2 is the fast adjustment stage of the target temperature. The system adjusts the deviation from a big one to a comparable small one with large step size. D3 is the fine adjustment stage. At this stage, the deviation is very small, and a slow PI regulation with smaller step size and longer adjustment period will be enough. Besides, the temperature precision is in the range of Tb . D4 is the stage of optimization. The system will optimize the output with smaller step size. When the output temperature of immersion liquid keeps in the range of Tb in the period of h2, it can be regarded as the optimal value and recorded as the optimal temperature of D5 stage. D5 is the ultra-stable stage. The system can be viewed as stable and all need to do is adjust the deviation between the current value and the optimal value at the outlet and make it smaller than Tc . As to the switching among the modes, it depends on the historical data, process data and reasoning rules. There are two reasoning rules in the system: Ψ1 and Ψ2. Ψ1 is mode selection based on the normal running of the whole immersion system, and Ψ2 is the mode selection for the booting of the temperature control system. Ψ1 is used for the process change from D1 to D5 step by step when the condition is OK. And the switching rules are as follows: TR1: If TR2: If TR3: If TR4: If to D5.
Tcw( k ) − Tsc < Ta , then from D1 to D2; TH ( k ) − Tss < Tb , then from D2 to D3; Tsm − Tu(k )
Here Ch1 is the period to keep value on the condition of TR3 and h1 is the threshold value of period for TR3. Similar, Ch2 is the period to keep value on the condition of TR4 and h2 is the threshold value of period for TR4. Fig. 3-5. The fluctuation of the ambient temperature.
Ψ1 = {TR1, TR2, TR3, TR4}
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(4-3)
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Each time the system boots, the initial condition will be different, so the time cost is different to reach the stable state. If we still switch from D1 to D5 as done in the normal operation mode, it will be time consuming. So Ψ2 is prepared, and the switching rules for booting are as follows: SR1: then SR2: then SR3: then
If Tsm − Tu( 0) > Tb , or Tcw( 0) − Tsc > Ta , or TH ( k ) − Tss > Tb , from start to D1; If Tsm − Tu( 0) < Tb , Tcw( 0) − Tsc < Ta , and Tss − TH ( 0) < Tb , from start to D4; If Tsm − Tu( 0) < Tc , Tcw( 0) − Tsc < Tb , and Tss(0)−TH ( 0) < Tc , from start to D5.
Ψ2 = {SR1, SR2, SR3}
Fig. 4-2. The equivalent transformation of feedforward-cascade diagram.
(4-4)
In practice, the immersion system sometimes needs to be changed from injecting liquid mode to exhausting liquid mode which will cause flow rate shock and induce some variation of the temperature. In order to improve the robustness and stability of system, the switching rule TR5 from D5 to D4 is added. TR5: If Tsm − Tu( k ) > Tc , then from D5 to D4. Here the new switching rules of Ψ1 is:
Ψ1′ = {TR1, TR2, TR3,TR4,TR5}
(4-5)
And the mode selection and transformation can be expressed by Fig. 4-1. The main controller’s algorithm is a multi-stage PID algorithm which can be formulated as:
⎧ 22. 5 Tt ( k )≥22. 5 ⎪ ⎪ Tt ( k ) = ⎨ Tt ( k−1)+∆Tt ( k ) 20. 5
⎧ K ⋅∆e ( k ) + K ⋅e ( k ) T ( k ) ≥ e Pi n Ii n t n ⎪ ⎪ ∆Tt ( k ) = ⎨ KPi⋅∆e( k ) + KIi⋅e( k ) em <| e( k )|
Fig. 4-3. Simplified block diagram of the feedforward controller.
Gf *( s ) =
(4-8)
Gs( s )G1( s )G2( s ) 1+Hs( s )Gs( s )G2( s )
(4-9)
As shown by Fig. 4-3, according to principle of full compensation, the disturbance produced by Td(s ) will be compensated by G*f ( s ). So we can get the following equation:
Td( s )Gf *( s )Gc ( s ) + Td( s )Gd( s )=0
(4-7)
where i = 1, 2, 3,4, 5 and it indicates the five modes in the hierarchical control algorithm, KPi is the proportional coefficient of mode i , KIi is the integral coefficient, Tt ( k ) is the setpoint of subcontroller, ∆Tt ( k ) is the change in the set-point of the sub-controller, e( k ) = Tr ( k ) − Ty( k ) is the deviation of main loop’s temperature, ∆e( k ) = e( k ) − e( k −1), ∆en( k ) = en( k ) − e( k −1) is the change rate of the endpoint’s temperature, em is the dead zone of deviation and the main controller does nothing on the zone, and en is the maximum of the deviation.
4.3. The feedforward controller The improved cascade control structure with feedforward controller shown in Fig. 2-6 can be simplified as shown in Fig. 4-2, and the equivalent feedforward controller is
(4-10)
According to Eqs. (4–8) and (4–10), we can get
Gf *( s ) =
Gf ( s ) Gs( s )
=−
Gd( s ) Gc ( s )
(4-11)
And substitute Gc ( s ) with Eq. (4-9), then
Gf ( s ) = −
Gd( s )( 1+Gs( s )G2( s )Hm(s )) G1( s )G2( s )
(4-12)
where Gs( s ) = (0. 3238s+ 0. 0312) /s is the PI sub-controller, and Hm(s ) is a unit gain. Substitute all the variables in Eq. (4-12), we can get
Gf ( s ) = −9. 01×10−4
1633. 25s +1
103s +1 ( 1 + 10. 36s)( 1 + 138. 06s) e−11.63s −2. 36×10−7 s 1+10−3s
(
)
(4-13)
We can see that the feedforward controller is composed of two parts: the major part which is a first order element and the minor part which is a high-order element with time delays. Since Eq. (413) is too complex to be applied in industry, the high order part is ignored here, and
Gf ( s ) = Fig. 4-1. Diagram of the mode selection and transformation.
Gs( s )
Then, it can be further simplified as shown in Fig. 4-3, and the equivalent transfer function in main channel Gc (s ) is
Gc ( s ) = (4-6)
Gf ( s )
yd ( s
)=K
Td( s )
d
T2s +1 T1s +1
(4-14)
With Laplace transformation, Eq. (4-14) can also be expressed
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in time domain as
T1
dyd (t ) dt
(
GB( s ) = G1′( s ) 1 − e−(τ1+ τ 2)s
dT (t ) + yd ( t ) = KdT2 d + KdTd(t ) dt
(4-15)
(4-16)
where T0 is the sampling period. Here we assume that α =
β=
T2 , T0
T1 T0
and
then Eq. (4-16) can be expressed as
α∆yd ( t ) + yd ( t ) = Kd⎡⎣ β∆Td( t ) + Td( t )⎤⎦
(4-17)
The input Td( s ) and the output Ty( s ) has been described in the Fig. 3-4, and the mathematical model of the inert zone has been got in the Section 3.2, so the estimated fully compensated value y^d (s ) under the disturbance Td(s ) can be got. Substituting the parameters in Eq. (4-17) with the values in Table 4-1, we can get
⎧ ⎪ −0. 00346α = K ( 0. 14484β + 22. 15) d ⎨ ⎪ ⎩ 23. 28Kd = − 0. 027
780s +1 74. 9925s +1
(4-19)
4.4. The controller based on Smith predictor As described in Section 2.3, a controller based on Smith predictor is proposed to solve the problem of significant time delays in temperature control system. In order to obtain the Smith predictor, the controlled object is simplified as an element of first order with time delays, G1′(s )e−(τ1+ τ 2)s (Fig. 4-4). The mathematical model of the controlled object is
G1′( s )e−(τ1+ τ 2)s =
(4-21)
GB( s ) can also be expressed in z-domain as
⎡ 1−e−T0s ⎤ 1. 0092 GB( z ) = z⎢ 1−e−(τ1+ τ 2)s ⎥ s 1 + 138. 06 s ⎣ ⎦
(
= 1−z−4
)
−1
z ) 1 0.− 071 0. 93z
−1
(4-22)
where T0 is the minimum sampling period. In order to get the difference equation, the following transformation is made
GB( z ) =
TB(z ) T (z ) TA(z ) = B Tt (z ) TA(z ) Tt (z )
(4-23)
Here we assume that
⎧ TB(z ) = 1−z−4 ⎪ ⎪ TA(z ) ⎨ ⎪ TA(z ) 0. 071z−1 ⎪ T (z ) = ⎩ t 1 − 0. 93z−1
(4-24)
So the final difference equation is
(4-18)
From Fig. 3-4 we can see that the peak time of the disturbance is 156T0 . Since ∆Td( t ) plays a dominant role in β∆Td( t ) + Td( t ), so ^ |β∆Td( t )| > | T¯d( t )|, and we take β=156. With Eq. (4-18), we can also get α = 14. 9985, β = 156, Kd = −0. 0011598, and the final result of feedforward controller is
Gf ( s ) = −0. 0011598
)
(
Replace the differential part with difference, we can get
⎡T ⎤ T1 ∆yd ( t ) + yd ( t ) = Kd⎢ 2 ∆Td( t ) + Td( t )⎥ T0 ⎣ T0 ⎦
7
1−e−Ts 1. 0092 e−47.4s s 1 + 138. 06s
(4-20)
And the corresponding Smith predictor is
⎧ ⎪ TA( k )=0. 93TA( k−1)+0. 071Tt (k−1) ⎨ ⎪ ⎩ TB( k ) = TA( k ) − TA( k−4)
5. The experiment results 5.1. The results of different controllers In order to compare the performance of the temperature control system with different controllers, a series of experiments are conducted. There are three configurations: only with feedforward controller, only with Smith predictor, and with both feedforward controller and Smith predictor. The result is shown in Fig. 5-1, and the ISE (Integral Error Square) criterion is used here to evaluate the controller’s performance. We can see that the system with both feedforward controller and Smith predictor has the smallest ISE, so it has better performance than the other configurations. Then the performance of the feedforward controller is also evaluated. A manipulated disturbance, the temperature variance of the environment, which is shown in Fig. 5-2 is introduced, and the feedforward controller is not added until the sampling point of 1981. The outlet temperature is shown in Fig. 5-3, we can see that the temperature stability at the system outlet has improved a lot
Table 4-1 Parameter estimation of full compensation. Index
^ Td( t )
^ ∆T d ( t )
y^d ( t )
∆y^d ( t )
Initial state Peak state
22.15 23.28
0.14480 0
0 0.027
0.00346 0
Fig. 4-4. Simplified block diagram of the system with smith predictor.
(4-25)
Fig. 5-1. The performance of the system with different controllers.
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8
with the feedforward controller. 5.2. The experiment results under different conditions Firstly, the system is tested without manipulated disturbance, and the system is operated just as it is. The result is shown in Fig. 5-4, the sampling interval is 10 s, and the total sampling time
24.4
24
24
23.8
23.5
Td
23
Ty
23.6 23.4 23.2 0
500
1000
1500 2000 sample count
2500
3000
Temperature /Celsius
T d /Celsius
24.2
is about eight hours. We can see that the temperature fluctuation is almost within 227 0.005 °C. Then, the manipulated environment disturbance is introduced. Fig. 5-5 shows the temperature at the outlet when the environment temperature Td varies from 21 to 24 °C which also has a fluctuation value of about 2 °C@10 mins, and Fig. 5-6 is the corresponding temperature fluctuation. We can see that the temperature Ty at the outlet of the system can maintain a high-precision and the stability which is within 22 70.008 °C@30 mins. Finally, the disturbance from the system is introduced, and the result is shown in Fig. 5-7. The flow rate of immersion liquid changes from 1 lpm to 0.8 lpm at the sampling point of 2016 and
Fig. 5-2. The manipulated temperature fluctuation of the environment.
22.5 22 21.5 21 20.5
0
22.08
2000
3000 4000 sample count
5000
6000
Fig. 5-5. The outlet temperature with manipulated environment temperature fluctuation.
22.06 22.04 1981 22.02 22 21.98 0
500
1000
1500 2000 sample count
2500
3000
Fig. 5-3. The outlet temperature of the system with feedforward and without feedforward controller.
Tem p eratu re v arian ce /Celsiu s
T y /Celsius
1000
0.04
0.03
0.02
0.01
0
0
1000
2000
3000 4000 sample count
5000
6000
Fig. 5-6. The outlet temperature fluctuation with the environment disturbance.
22.1
T y /Celsius
22.05
3733
2016 22
21.95
0
Fig. 5-4. The outlet temperature during a long period without manipulated disturbance.
1000
2000
3000
4000 5000 sample count
6000
7000
8000
Fig. 5-7. The outlet temperature when changing the inlet condition of the system.
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the corresponding thermal load increases sharply which causes a 0.2 °C fluctuation of the temperature at the outlet of the system, and it takes about 47 mins for the system to recover to the stable state of 227 0.01 °C@30 mins. At the sampling point of 3733, the temperature of the cooling water is reduced by 0.5 °C, it causes a 0.2 °C fluctuation of the outlet temperature, and the system returns to normal state at the sampling point of 3928, which costs about 32 mins.
[3]
[4]
[5] [6]
6. Conclusions [7]
A temperature control system of high precision and stability was built, which adopted a cascade structure with feedforward controller and Smith predictor. Then, the mathematical model of the system was built where the gray and black box experiments were used. And to reach the temperature requirements, four controllers were used: the main controller using a hierarchical control algorithm and multi-stage PID algorithm, the sub controller using the integral separated algorithm, the feedforward controller, and the Smith predictor. Last, the experiments were conducted, and the results showed that the robustness of the system was excellent, and the outlet temperature of the system would maintain a stability of 22 70.008 °C@30 mins with a 2 °C@10 mins fluctuation of the environment temperature.
[8]
[9] [10]
[11]
9
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Please cite this article as: X. Li, et al., (2016), http://dx.doi.org/10.1016/j.flowmeasinst.2016.08.014i