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Abstract

This article studies the precision motion control of a long-stroke reticle stage driven by the permanent magnet linear motor in wafer scanner. A robust sliding-mode control method is proposed for tracking the reference trajectory in the presence of un-modeled dynamics, parametric uncertainty and external disturbances including force ripple, cogging and friction in the controlled system. A modified sliding-mode term based on the variable structure technique for eliminating the tracking error is employed in the proposed control law. The system stability and tracking convergence of the closed-loop control system are guaranteed by Lyapunov theory theoretically. The feasibility and effectiveness of the proposed method are demonstrated by comparative experiments on a linear motion testbed. The experimental results show that better tracking performance can be achieved by the proposed method compared with the conventional proportional–integral–derivative method and it can be considered as a possible alternative in the precision motion control system.

Keywords

Lithography wafer scanner sliding-mode control variable structure precision motion control

Introduction

Wafer scanner is highly complex equipment used in the semiconductor industry for the production of integrated circuits (ICs). Wafer stage and reticle stage are the key sub-systems in the wafer scanner, containing the patterns of IC and the silicon wafer to be exposed, respectively.¹ In the semiconductor manufacturing, a laser beam passes the desired image mask held by the reticle stage, through an optical lens system which can scales down the image, onto the surface of the wafer covered by a layer of light-sensitive material. Then the patterns are transferred from the mask to the wafer and an exposure is achieved.² During scanning, only a small area on the wafer is exposed and then the stage will be stepped to the next area to finish the next cycle. The quality of wafer scanner including accuracy and throughput is mainly determined by the tracking performance of the two stages. Only several nanometers of position tracking error are allowable during the scanning. This makes the control of wafer scanner challenging.

Currently, to achieve nanometer-scale tracking and positioning, dual-stroke strategy is employed in the wafer scanner, with long strokes for coarse positioning and short strokes for high-speed precise positioning.² For the reason of presentation, the discussion is mainly limited to the long-stroke reticle stage in this article. Permanent magnet linear motor (PMLM) is widely used in the field of ultra-precision motion control. For wafer scanner, PMLM is usually adopted as the long-stroke actuator. In addition to the un-modeled dynamics and parametric uncertainty, many external disturbances may degrade the performance of the closed-loop control system such as force ripple, cogging, friction, and vibration.³ This makes the control system of wafer scanner more complex.

Some drawbacks limit the applications of conventional proportional–integral–derivative (PID) methods in the precision motion control field: (1) they work best with systems that have only one input and output; (2) the plant to be controlled might not behave in a linear fashion; and (3) the changing of the plant dynamics due to changes on the loads, wear, and tear, and mechanical effectiveness in mechanical elements may degrade the performance of PID control systems. Therefore, more advanced control methods are essential.

In Heertjes and Van de Molengraft,⁴ Heertjes and Tso,⁵ Mishra and Tomizuka,⁶ and Mishra et al.,⁷ iterative learning controllers (ILCs) are proposed for wafer scanner. These studies show that no plant information is required for ILC and a good performance can be achieved in the presence of repetitive disturbances. However, ILC may result in a degraded performance under the effect of non-repetitive disturbances such as cable forces and structure vibrations.³ Some researches focus on disturbance observer-based (DOB) method which can compensate for the impacts of the uncertainties and external disturbances.^8–13 DOB is used to compensate for the uncertainties of parameter variations and current measurements, in the velocity and current loops, respectively.¹⁰ A composite method based on DOB is proposed in Li et al.¹¹ The modeling error of the nonlinear effects is regarded as lumped disturbances which can be estimated by DOB. However, the results presented in Yao and Xu^12,13 show that the wide range of discontinuous disturbances and parametric uncertainties cannot be handled well by DOB.

Recently, sliding-mode control approach based on variable structure technique develops rapidly and has been widely applied in the field of precision motion control.^14–17 A sliding-mode control and a PI-based equivalent disturbance observer are proposed for the control system of a tubular synchronous linear motor.¹⁶ The distinctive peculiarity of the proposed scheme is the use of a control law that guarantees the stability of the system regardless of the payload mass. An enhanced sliding-mode tracking control methodology is proposed for piezoelectric actuators to track desired motion trajectories. The convergence of the position tracking error to 0 can be ensured by the proposed methodology in the presence of the hysteresis effect and un-modeled disturbances.¹⁷

In this article, a robust sliding-mode (RSM) control method is proposed for the control system of wafer scanner. This method is used to accommodate the parametric uncertainties, un-modeled dynamics, as well as external disturbances. The proposed method is formulated without prior information of the plant, and the stability of the closed-loop control system is guaranteed by Lyapunov theory.^18,19 The contributions are briefly described as follows: (1) an RSM control method is proposed for the wafer scanner, (2) a modified RSM control law based on boundary layer technique is proposed for reducing the chatting effect, and (3) the feasibility and effectiveness of the proposed method are demonstrated by comparative experiments.

This article is organized as follows. In section “Dynamic modeling,” a mathematical model of PMLM for the long-stroke reticle stage is presented. In section “RSM control method for wafer scanner,” a modified robust sliding-mode (MRSM) control method is established and the stability analysis is conducted. The experimental study is detailed in section “Experimental study.” The experimental results are presented and discussed in section “Results and discussion.” Finally, some conclusions are drawn.

Dynamic modeling

The long-stroke reticle stage is driven by a current-controlled three-phase PMLM. Since the bandwidth of the current controller loop is large enough, its effect on the dynamics of outer loop will be neglected. The electro-mechanics model of the long-stroke reticle stage is given by

M \overset{\cdot\cdot}{x} = F_{m} - F_{d}, F_{m} = K_{f} K_{a} u_{A}, F_{d} = F_{c} + F_{r} + F_{ed}

(1)

where $u_{A}$ is the control input into the servo amplifier, $F_{m}$ is the thrust force of PMLM, $x$ is the position output, $M$ denotes the moving mass, $K_{a}$ is the gain constant of the amplifier, $K_{f}$ is the force constant of PMLM, and $F_{d}$ denotes the normalized lumped effect of the uncertain nonlinearities. In general, $F_{d}$ includes friction $F_{c}$ , force ripple $F_{r}$ , and external disturbances $F_{ed}$ . $F_{r}$ is always considered to be position dependent.²⁰ Since all moving parts are supported by the air guide in the reticle stage, the friction effect is limited.

Dividing both sides of equation (1) by $K_{f} K_{a}$ , the model becomes

K_{1} \overset{\cdot\cdot}{x} + K_{2} = u_{A}

(2)

where $K_{1} = M / (K_{f} K_{a})$ is the equivalent inertial term and $K_{2} = F_{d} / (K_{f} K_{a})$ is the equivalent disturbance term.

Without loss of generality, assume that $K_{1}$ and $K_{2}$ are bounded, respectively. There exist positive constant numbers $Δ_{1}$ and $Δ_{2}$ such that

| K_{1} | \leq Δ_{1}, | K_{2} | \leq Δ_{2}

(3)

RSM control method for wafer scanner

For mathematical model (2), an RSM control method is proposed to track the reference position $x_{d}$ , velocity ${\overset{\cdot}{x}}_{d}$ , and acceleration ${\overset{\cdot\cdot}{x}}_{d}$ precisely. Without loss of generality, assume that $x_{d}$ is at least twice continuously differentiable and $x_{d}$ , ${\overset{\cdot}{x}}_{d}$ , and ${\overset{\cdot\cdot}{x}}_{d}$ are all uniformly continuous and bounded. Therefore, there exists positive number $Δ_{a}$ such that

| {\overset{\cdot\cdot}{x}}_{d} | \leq Δ_{a}

(4)

The position tracking error $e$ is stated as

e = x - x_{d}

(5)

Define special positive-definite function $ρ (e)$ , saturation error function $s (e)$ , and tracking error function $σ$ related to the position error $e$ as²¹

ρ (e) = \sqrt{c^{2} + e^{2}} - | c |

(6)

s (e) = \frac{d ρ (e)}{de} = \frac{e}{\sqrt{c^{2} + e^{2}}}

(7)

σ = \overset{\cdot}{e} + α s (e)

(8)

where $c$ is a constant number and $α$ is a positive scalar.

The time derivatives of $s (e)$ and $σ$ are therefore expressed as

\overset{\cdot}{s} (e) = \frac{ds (e)}{de} \overset{\cdot}{e} = \frac{c^{2} \overset{\cdot}{e}}{\sqrt{{(c^{2} + e^{2})}^{3}}}, \overset{\cdot}{σ} = \overset{\cdot\cdot}{e} + α \overset{\cdot}{s} (e)

(9)

According to equations (5) and (9), we have

\overset{\cdot\cdot}{x} = \overset{\cdot}{σ} + {\overset{\cdot\cdot}{x}}_{d} - α \overset{\cdot}{s} (e)

(10)

Substituting equation (10) into equation (2), the model becomes

K_{1} \overset{\cdot}{σ} = u_{A} - K_{1} {\overset{\cdot\cdot}{x}}_{d} + K_{1} α \overset{\cdot}{s} (e) - K_{2}

(11)

Design the RSM control law $u_{A}$ as

u_{A} = - k_{p} e - k_{v} \overset{\cdot}{e} - k_{t} σ - d \frac{σ}{| σ |}

(12)

where $k_{p}$ , $k_{v}$ , and $k_{t}$ are the control gains and $d$ is a positive constant number.

Substituting control law (12) into equation (11), the model becomes

K_{1} \overset{\cdot}{σ} = - k_{p} e - k_{v} \overset{\cdot}{e} - k_{t} σ - d \frac{σ}{| σ |} + K_{1} α \overset{\cdot}{s} (e) - K_{1} {\overset{\cdot\cdot}{x}}_{d} - K_{2}

(13)

Multiplying both sides of equation (13) by $σ$ , we have

σ K_{1} \overset{\cdot}{σ} = - β - d | σ | - σ (K_{1} {\overset{\cdot\cdot}{x}}_{d} + K_{2})

(14)

where $β$ is given by

β = σ [k_{p} e + k_{v} \overset{\cdot}{e} + k_{t} σ - K_{1} α \overset{\cdot}{s} (e)]

(15)

Replacing $σ$ in equation (15) by equation (8), $β$ can be expressed as

β = {\overset{\cdot}{v}}_{1} + v_{2}

(16)

and

{\begin{matrix} v_{1} = \frac{1}{2} k_{p} e^{2} + α (k_{v} + 2 k_{t}) ρ (e) - \frac{1}{2} α^{2} K_{1} s^{2} (e) \\ v_{2} = (k_{v} + k_{t}) {\overset{\cdot}{e}}^{2} + α k_{p} es (e) - α K_{1} \overset{\cdot}{s} (e) \overset{\cdot}{e} + α^{2} k_{t} s^{2} (e) \end{matrix}

(17)

where $v_{1}$ and $v_{2}$ are positive-definite for the gains $k_{p}$ , $k_{v}$ , and $k_{t}$ set as

k_{p} > \frac{α^{2} K_{1}}{c^{2}}, k_{v} + k_{t} > \frac{α K_{1}}{| c |}

(18)

Theorem 1

For electro-mechanics model (2), the RSM control law (12) assures the tracking convergence of the closed-loop control system with $e \to 0$ and $\overset{\cdot}{e} \to 0$ as $t \to \infty$ under conditions (18) and

d > Δ_{1} Δ_{a} + Δ_{2}

(19)

Proof

It is to be noted that for plant (2), with the proposed control law (12), the functions $v_{1}$ and $v_{2}$ are always positive-definite for any non-zero values of $e$ and $\overset{\cdot}{e}$ under the conditions given by equations (18) and (19).

Choose a Lyapunov function $V$ related to $e$ and $\overset{\cdot}{e}$ as

V = \frac{1}{2} K_{1} σ^{2} + v_{1}

(20)

Differentiating $V$ with respect to time yields

\overset{\cdot}{V} = σ K_{1} \overset{\cdot}{σ} + {\overset{\cdot}{v}}_{1}

(21)

Substituting equation (14) into equation (21), we have

\overset{\cdot}{V} = - β - d | σ | - σ (K_{1} {\overset{\cdot\cdot}{x}}_{d} + K_{2}) + {\overset{\cdot}{v}}_{1}

(22)

Substitution from equation (16), $\overset{\cdot}{V}$ becomes

\overset{\cdot}{V} = - v_{2} - | σ | [d + \frac{σ}{| σ |} (K_{1} {\overset{\cdot\cdot}{x}}_{d} + K_{2})]

(23)

Applying the bounds given by equations (3) and (4), we have

\overset{\cdot}{V} \leq - v_{2} - | σ | [d - (Δ_{1} Δ_{a} + Δ_{2})]

(24)

With condition (19), the time derivative $\overset{\cdot}{V}$ becomes

\overset{\cdot}{V} \leq - v_{2}

(25)

It means $\overset{\cdot}{V}$ is negative-definite and the equilibrium at $e = 0$ and $\overset{\cdot}{e} = 0$ is asymptotically stable.²² Then $e \to 0$ and $\overset{\cdot}{e} \to 0$ as $t \to \infty$ can be achieved, and both the system stability and tracking convergence can be guaranteed by control law (12). To be noted, in the realization of equation (12), the discontinuous item $σ / | σ |$ may cause chattering and excite un-modeled high-frequency dynamics.²³ To reduce this effect, the boundary layer technique is employed to smooth the control input.²⁴ Then, the item $σ / | σ |$ can be replaced by the saturation function

sat (\frac{σ}{ψ}) = {\begin{matrix} - 1 & σ < - ψ \\ \frac{σ}{ψ} & - ψ \leq σ \leq ψ \\ 1 & σ > ψ \end{matrix}

(26)

where $ψ$ is the boundary layer thickness.

Replacing $σ / | σ |$ in equation (12) with equation (26), the modified control law is realized as

u_{A} = - k_{p} e - k_{v} \overset{\cdot}{e} - k_{t} σ - dsat (\frac{σ}{ψ})

(27)

Theorem 2

For electro-mechanics model (2), the MRSM control law (27) can assure the system stability and tracking convergence with $e \to 0$ and $\overset{\cdot}{e} \to 0$ as $t \to \infty$ under the conditions provided by equation (18).

Proof

Similarly, choose the Lyapunov function $V$ as equation (20).

When $| σ | > ψ$ , the proof can be accomplished as Theorem 1.

When $| σ | < ψ$ and $| σ | \neq 0$ , we have

\begin{matrix} \overset{\cdot}{V} = - v_{2} - d \frac{σ^{2}}{ψ} - σ (K_{1} {\overset{\cdot\cdot}{x}}_{d} + K_{2}) \\ \leq - v_{2} - | σ | {d \frac{| σ |}{ψ} - (Δ_{1} Δ_{a} + Δ_{2})} \end{matrix}

(28)

If $d$ is set as

d > \frac{1}{ε_{0}} (Δ_{1} Δ_{a} + Δ_{2}) \geq \frac{ψ}{| σ |} (Δ_{1} Δ_{a} + Δ_{2})

(29)

there exists a small positive constant $ε_{0}$ satisfying $ε_{0} \leq | σ | / ψ$ . Then $\overset{\cdot}{V} \leq - v_{2}$ can be obtained. It means that with the modified control law (27), the system stability and tracking convergence can still be guaranteed. The structure of the proposed MRSM control method is shown in Figure 1.

Figure 1.

Structure of the MRSM control method.

Experimental study

To verify the effectiveness of the proposed method, a precision linear motion testbed is set up, as shown partially in Figure 2. The testbed consists of an industrial personal computer (IPC) with encoder acquisition card, a PMLM together with inbuilt optical encoder sensor, a servo amplifier, and a digital-to-analog (D/A) control card. The block diagram of the architecture of the testbed is shown in Figure 3. The PMLM model BLMC-92 is manufactured by AEROTECH and the force constant $K_{f}$ is 7.12 N/A. The linear servo amplifier model Soloist-CL is a pulse-width-modulation (PWM) type with closed current control loop. The maximum control input signal is 10 V corresponding to the peak current output 25 A. Then, the amplifier gain constant $K_{a}$ is determined as 2.5 A/V. The moving mass $M$ is about 6.5 kg. All moving parts of the testbed are supported by the air guide and the air pressure is chosen as 0.5 MPa. The resolution of the optical encoder is 1 nm. IPC is used as the control computer running the Ardence RTX real-time operating system in which the control algorithms are achieved. The encoder acquisition card in IPC is used to acquire the position signals from the encoder sensor. The motion control card equipped with a 16-bit D/A AD5547 is connected to the IPC by serial bus. The digital control command from IPC is converted by the motion control card into analog signals which are then sent to the servo amplifier to drive the PMLM. In the experiments, the sampling frequency of the closed-loop control system is selected as 5 kHz.

Figure 2.

Experimental study facility.

Figure 3.

The architecture of the testbed.

Standard identification processes based on frequency response of the open-loop system are carried out and the measured transfer function from the control input $u_{A}$ to the position output $x$ is shown in Figure 4.

Figure 4.

Measured frequency response function in Bode representation.

In the experiments, the reference motion trajectory is chosen as a class of third-order set-points, as shown in Figure 5. The trajectory consists of three phases: (1) accelerating to the specified scanning velocity, (2) scanning at this velocity, and (3) decelerating to stationary state.

Figure 5.

Reference motion trajectory.

To evaluate the performance of the control system for the wafer scanner, the methods will be compared based on the following indexes:

Moving average (Ma) and moving standard deviation (Msd) are the measures for the low-frequency and high-frequency parts of the positioning errors with²⁵

M_{a} (i) = \frac{1}{n} \sum_{j = i - n / 2}^{i + n / 2 - 1} e (j), \forall i \in {1, \dots, N}

(30)

M_{sd} (i) = \sqrt{\frac{1}{n} \sum_{j = i - n / 2}^{i + n / 2 - 1} {[e (j) - M_{a} (i)]}^{2}}, \forall i \in {1, \dots, N}

(31)

where $n \in N^{+}$ is the specific exposure time frame, $e$ denotes the position error, and N is the sampling number.

Settling times $T_{ma}$ and $T_{msd}$ , the time consumptions for Ma and Msd to enter into the specified error ranges in the constant velocity scanning phase, are important measures of the throughput for the wafer scanner.⁵

Peak position error $e_{max}$ is used as a measure of the transient performance.

$H_{2}$ , 2-norm of the position error during the entire cycle, is used as a measure of the average performance of the tracking accuracy and is given by

H_{2} = \sqrt{\sum_{i = 1}^{N} {| e (i) |}^{2}}, \forall i \in {1, \dots, N}

(32)

$S^{2}$ , the variance of the position error during the entire cycle, is used as a measure of the average fluctuation of the position tracking error

S^{2} = \sum_{i = 1}^{N} {| e (i) - \bar{e} |}^{2}, \forall i \in {1, \dots, N}

(33)

\bar{e} = \frac{1}{N} \sum_{i = 1}^{N} e (i), \forall i \in {1, \dots, N}

(34)

where $\bar{e}$ is the arithmetic mean of the position tracking error.

In this article, to further evaluate the effectiveness of the proposed method in the physical system, three control methods are compared in the experiments.

PID

Conventional PID controller with low-pass filter is given by

PID = \frac{1}{s} (K_{d} s^{2} + K_{p} s + K_{i}), LPF = \frac{w_{p}^{2}}{s^{2} + 2 β_{p} w_{p} s + w_{p}^{2}}

(35)

where $LPF$ is a second-order low-pass filter, $w_{p} \approx 6000 rad / s$ is the cut-off frequency, and $β_{p} = 0.7$ denotes the damping coefficient. Choose the control gains of PID controller as $K_{p} = 9658$ , $K_{i} = 310, 320$ , and $K_{d} = 151$ and a closed-loop bandwidth can be obtained of about 50 Hz.

RSM

RSM control method is described in equation (12). By trial and error, $α$ and $c$ are selected as 0.005 m/s and 6 × 10⁻⁶ m, respectively. In terms of equations (18) and (19), $k_{p}$ , $k_{v}$ , $k_{t}$ , and $d$ are chosen as 100 V/m, 0.01 V s/m, 200 V s/m, and 0.5 V, respectively.

MRSM

MRSM control method is presented in equation (27). Same parameter values are chosen as aforementioned RSM. $ψ$ is chosen as 0.1 m/s. In terms of equation (29), $d$ is still chosen as 0.5 V.

Results and discussion

In the experiments, the closed-loop system is required to follow the reference trajectory shown in Figure 5. To be noted, the testbed working in an open environment is easily influenced by the external disturbances such as vibration, air flow, and temperature. Therefore, the experimental results may be affected, to some extent. Three experiments based on the three methods PID, RSM, and MRSM involved in section “Experimental study” are carried out.

Despite the open environment problem, the results show that the proposed method achieves a promising tracking performance. The resulting position tracking errors in experiments are compared in Figure 6. And the tracking accuracy indexes $e_{max}$ , $H_{2}$ , and $S^{2}$ are listed in Table 1, accordingly.

Table 1.

Tracking accuracy indexes.

	PID	RSM	MRSM
$e_{max}$	57.60 μm	51.70 μm	48.03 μm
$H_{2}$	1.10 × 10⁻³ m	8.14 × 10⁻⁴ m	7.64 × 10⁻⁴ m
$S^{2}$	2.71 × 10⁻¹⁰	1.47 × 10⁻¹⁰	1.30 × 10⁻¹⁰

PID: proportional–integral–derivative; RSM: robust sliding-mode; MRSM: modified robust sliding-mode.

Figure 6.

Position tracking errors in the experiments.

From the experimental results, it can be seen that the convergence speeds of the two methods based on RSM technique are obviously faster than the PID method in the settling phase. The position tracking error for the modified method is much smoother than the RSM method and the boundary layer technique has a significant effect on reducing the chatting effect of the sliding-mode. The proposed MRSM has a better performance than the other two methods on the tracking accuracy indexes $e_{max}$ , $H_{2}$ , and $S^{2}$ with values of 48.03 μm, 7.64 × 10⁻⁴ m, and 1.30 × 10⁻¹⁰, respectively.

Figure 7 compares the estimated velocity errors in three experiments. We can see that there is a significant velocity tracking error during the non-steady phases for PID method and the peak error is up to 8.8 mm/s. By comparison, the performance of velocity tracking is much better for the proposed MRSM method and the peak velocity tracking error is only 4.9 mm/s. Because of the chatting effect, during the steady state, violent velocity fluctuations can be observed for RSM method and the range is about ±2 mm/s.

Figure 7.

Estimated velocity tracking errors in the experiments.

Choose the exposure time as 70 ms, the resulting Ma and Msd position errors of the first acceleration and constant velocity scanning phases are compared in Figure 8. Accordingly, the settling times $T_{ma}$ and $T_{msd}$ for 100, 50, and 20 nm accuracy ranges are listed in Table 2. We can see that the proposed MRSM method has a better performance on the settling time indexes than the other two methods for the specified error ranges. For RSM method, the high-frequency chattering effect can be clearly observed in the Msd plot and the high-frequency fluctuations of about 0.89 μm make Msd error unable to enter the specified error ranges. Because of the slow convergence speed, the Msd position error of PID method cannot enter into the 20 nm error range.

Figure 8.

Resulting Ma and Msd position errors.

Table 2.

Settling time specification.

	PID	RSM	MRSM
T_ma (100 nm)	69.6 ms	138.0 ms	43.0 ms
T_ma (50 nm)	115.8 ms	164.2 ms	56.8 ms
T_ma (20 nm)	162.4 ms	–	98.0 ms
T_msd (100 nm)	99.8 ms	–	58.8 ms
T_msd (50 nm)	132.2 ms	–	68.6 ms
T_msd (20 nm)	–	–	112.6 ms

PID: proportional–integral–derivative; RSM: robust sliding-mode; MRSM: modified robust sliding-mode.

“–” Indicates that the position error does not enter into the specified error range.

The control inputs in the experiments are shown in Figure 9. In accordance with the analysis above, there are significant fluctuations of about 0.7 V during the steady state for the control input of RSM method. And because of the less precise velocity tracking, the peak control input for PID method is larger than the other two methods. By comparison, the proposed method can achieve better tracking performance with less and smoother control input signals.

Figure 9.

Control inputs in the experiments.

In summary, the proposed MRSM control method is demonstrated to be stable and effective. Compared with RSM and conventional PID methods, better tracking accuracy and settling time performance can be achieved. Especially, the chattering effect of sliding mode can be greatly reduced by the proposed method. Our future work will focus on establishing a 3-degree of freedom long-stroke wafer stage and designing a multiple-input-multiple-output (MIMO) controller for it.

Conclusion

In this article, an MRSM control method is proposed to improve the tracking performance of the long-stroke reticle stage for wafer scanner. The feasibility and effectiveness of the proposed method are verified by comparative experiments on a linear motion testbed. The experimental results show that better tracking performance can be achieved by the proposed method in an open environment and it can be considered as a possible alternative to the conventional PID control method for wafer scanner.

Footnotes

Acknowledgements

The authors thank the editor(s) and reviewer(s) for their work and helpful suggestions for this article.

Academic Editor: Hamid Reza Shaker

Declaration of conflicting interests

The authors declare that there is no conflict of interest.

Funding

This research was supported by the National Science and Technology Major Project of the Ministry of Science and Technology of China (Grant No. 2009ZX02207) and by the National Basic Research Program of China (Grant No. 973-10007.07-LB7).

References

Butler

. Adaptive feedforward for a wafer stage in a lithographic tool. IEEE Trans Contr Syst Technol 2013; 21(3): 875–881.

Heertjes

Hennekens

Steinbuch

. MIMO feed-forward design in wafer scanners using a gradient approximation-based algorithm. Contr Eng Pract 2010; 18(5): 495–506.

Mishra

Coaplen

Tomizuka

. Precision positioning of wafer scanners segmented iterative learning control for nonrepetitive disturbances [applications of control]. IEEE Contr Syst Mag 2007; 27(4): 20–25.

Heertjes

Van de Molengraft

RMJG

. Set-point variation in learning schemes with applications to wafer scanners. Contr Eng Pract 2009; 17(3): 345–356.

Heertjes

Tso

. Nonlinear iterative learning control with applications to lithographic machinery. Contr Eng Pract 2007; 15(12): 1545–1555.

Mishra

Tomizuka

. Projection-based iterative learning control for wafer scanner systems. IEEE/ASME Trans Mech 2009; 14(3): 388–393.

Mishra

Topcu

Tomizuka

. Optimization-based constrained iterative learning control. IEEE Trans Contr Syst Technol 2011; 19(6): 1613–1621.

Kobayashi

Katsura

Ohnishi

. An analysis of parameter variations of disturbance observer for motion control. IEEE Trans Ind Electron 2007; 54(6): 3413–3421.

Tesfaye

Lee

Tomizuka

. A sensitivity optimization approach to design of a disturbance observer in digital motion control systems. IEEE/ASME Trans Mech 2000; 5(1): 32–38.

10.

Huang

Liu

Hsu

. Precision control and compensation of servomotors and machine tools via the disturbance observer. IEEE Trans Ind Electron 2010; 57(1): 420–429.

11.

Qiu

. Piezoelectric vibration control for all-clamped panel using DOB-based optimal control. Mechatronics 2011; 21(7): 1213–1221.

12.

Yao

. Adaptive robust motion control of linear motors for precision manufacturing. Mechatronics 2002; 12(4): 595–616.

13.

Yao

. Output feedback adaptive robust precision motion control of linear motors. Automatica 2001; 37(7): 1029–1039.

14.

Yau

Yan

. Adaptive sliding mode control of a high-precision ball-screw-driven stage. Nonlinear Anal R world Appl 2009; 10(3): 1480–1489.

15.

. Micro/nanopositioning using model predictive output integral discrete sliding mode control. IEEE Trans Ind Electron 2012; 59(2): 1161–1170.

16.

Cupertino

Naso

Mininno

. Sliding-mode control with double boundary layer for robust compensation of payload mass and friction in linear motors. IEEE Trans Ind Appl 2009; 45(5): 1688–1696.

17.

Liaw

Shirinzadeh

Smith

. Enhanced sliding mode motion tracking control of piezoelectric actuators. Sensor Actuator A Phys 2007; 138(1): 194–202.

18.

Shaker

. Lyapunov stability for continuous-time multidimensional nonlinear systems. Nonlinear Dynam 2014; 75(4): 717–724.

19.

Shaker

Tahavori

. Stability analysis for a class of discrete-time two-dimensional nonlinear systems. Multidim Syst Sign Process 2010; 21(3): 293–299.

20.

Zhao

Tan

. Adaptive feedforward compensation of force ripples in linear motors. Contr Eng Pract 2005; 13(9): 1081–1092.

21.

Liaw

Oetomo

Alici

. Special class of positive definite functions for formulating adaptive micro/nano manipulator control. In: 9th IEEE international workshop on advanced motion control, Istanbul, Turkey, 27–29 March 2006, pp.517–522. New York: IEEE.

22.

Slotine

JJE

. Applied nonlinear control. Englewood Cliffs, NJ: Prentice Hall, 1991.

23.

Liu

Zhang

. Neural network-based robust finite-time control for robotic manipulators considering actuator dynamics. Robot Comput Integrated Manuf 2013; 29(2): 301–308.

24.

Liaw

Shirinzadeh

Smith

. Robust neural network motion tracking control of piezoelectric actuation systems for micro/nanomanipulation. IEEE Trans Neural Netw 2009; 20(2): 356–367.

25.

Heertjes

Nijmeijer

. Self-tuning of a switching controller for scanning motion systems. Mechatronics 2012; 22(3): 310–319.

Modified robust sliding-mode control method for wafer scanner

Abstract

Keywords

Introduction

Dynamic modeling

RSM control method for wafer scanner

Theorem 1

Proof

Theorem 2

Proof

Experimental study

PID

RSM

MRSM

Results and discussion

Conclusion

Footnotes

Acknowledgements

Declaration of conflicting interests

Funding

References