MIMO FIR Feedforward Design for Zero Error Tracking Control
2014 American Control Conference (ACC) June 4-6, 2014. Portland, Oregon, USA
MIMO FIR Feedforward Design for Zero Error Tracking Control Marcel Heertjes1 and Dennis Bruijnen2
Abstract— This paper discusses a multi-input multi-output (MIMO) finite impulse response (FIR) feedforward design. The design is intended for systems that have (non-)minimum phase zeros in the plant description. The zeros of the plant (either minimum or non-minimum phase) are used in the shaping of the reference signals whereas the poles of the plant are used in constructing feedforward forces. The FIR coefficients themselves are obtained from data-based optimizations which are the result of iterative machine measurements on an industrial wafer scanner. The resulting experimental results demonstrate the ability to obtain zero error tracking.
I. INTRODUCTION Nanometer tracking performance in motion control systems like industrial wafer scanners is obtained with feedforward control using the principles of plant inversion. Though many researchers in the past have worked on the inversion problem, see for example [5], [11], [12] and the references therein, basically two directions evolved toward tracking error minimization. First, the design of feedforward controllers being approximations of the inverse of the (closed-loop) plant [6], [9], [13]. Second, the design of input shapers [8], [10] that modify the reference trajectory with the aim to provide less excitation of this plant. In this paper, similar to [4], both ideas are combined. But different from [4] we pose the problem in a discrete-time (measured) stage control context and consider an extension to MIMO FIR feedforward design. By shaping the reference trajectories in combination with a proper feedforward contribution, zero error tracking can be obtained in all the coupled directions of the MIMO system. This follows from individually delaying the reference contributions to these directions with respect to the feedforward contributions. It is important to realize that tracking the shaped reference rather than the original reference does not pose a performance problem within the context of wafer scanning. This is because scanning takes place under constant velocity. The shaped reference will therefore only be a time-shifted version of the original reference. The potential loss of throughput due to this time shift is more than compensated for by reduced settling times. The latter being the result of not exciting the plant dynamics. The MIMO FIR filter coefficients, which generally appear too sensitive to obtain from first principles modelling, are obtained from a data-based optimization approach, see also [1], [7]. The remainder of this paper is organized as follows. In Section 2, FIR feedforward design in the context of singleinput single-output (SISO) motion systems will be discussed. By means of simple examples of discretized plants, the FIR 1 Marcel
F. Heertjes is with Faculty of Mechanical Engineering, Eindhoven University of Technology, 5600 MB Eindhoven, The Netherlands
filter properties are illustrated. In Section 3, an extension toward MIMO FIR feedforward design will be given. In Section 4, the optimization approach used to obtain the FIR filter coefficients will be discussed. In Section 5, both numerical and experimental results will be presented. Section 6 summarizes the main conclusions of this work. II. SISO FIR FEEDFORWARD DESIGN Consider the simplified SISO control configuration of a motion system such as depicted in Fig.1. Herein r(k) is a
r(k)
F1
r˜(k)
e(k) Σ
C
u f b (k)
Σ
u(k)
P
y(k)
−
Fig. 1.
SISO control configuration.
data-sampled reference signal, i.e. the value of r at time tk = kT with k an integer and T the sampling time. This reference signal which is used to obtain point-to-point motion is applied to a set-point filter F1 with DC gain F1 (z = 1) = 1 (z indicating the z-domain) and to a feedforward filter F2 . The filtered reference r˜(k) along with the output y(k) of the double integrator-based plant P form the input to the stabilizing and linear time-invariant feedback controller C via e(k) = r˜(k) − y(k). The output of the controller u f b (k) along with the output of the feedforward filter u f f (k) form the input to the plant u(k) via u(k) = u f b (k) + u f f (k). Suppose P(z) stems from a continuous-time double integrator plant, which is representative for stage systems, or: P(s) :=
1 , ms2
(1)
with mass m and Laplace variable s, which is sampled using a zero-order hold circuit. In discrete-time, it follows that: P(s) P(z) = (1 − z−1)Z s (2) 1 −1 1 −2 z + z 2 2 = , c0 (1 − 2z−1 + z−2) with constant c0 = mT −2 , z−1 a unit time delay, and Z {·} denoting the z-transform. From Fig.1, it follows that if
[email protected] 2 Dennis
Bruijnen is with Mechatronics Technologies Group, Philips Innovation Services, High Tech Campus 7, 5656 AE Eindhoven, The Netherlands [email protected]
978-1-4799-3271-9/$31.00 ©2014 AACC
u f f (k)
F2
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1 1 F1 (z) = z−1 + z−2 2 2 F2 (z) = c0 (1 − 2z−1 + z−2 ),
(3)
i.e. two (3-taps) FIR filters, any bounded r(k) with r(k) = 0 for k ≤ 0 induces zero error tracking in the sense that e(k) = 0 for k > 0. From (2) and (3) it is clear that such error tracking is obtained by choosing F1 (z) equal to the numerator of P(z) in (2) whereas F2 (z) is chosen equal to the denominator of P(z), namely:
with constant c2 = mωn−2 T −2 m−1 1 . By choosing
e(z) = r˜(z) − y(z) = F1 (z)r(z) − P(z)F2 (z)r(z) − P(z)C (z)e(z) (4) F1 (z) F2 (z)r(z) = 0. = S (z)F1 (z)r(z) − S (z) F2 (z) Now consider an extended version of the double integrator plant given by the fourth-order system: k , (5) P(s) := m1 m2 s4 + k(m1 + m2 )s2 with mass m1 , m2 satisfying m = m1 + m2 and an interconnected spring with stiffness coefficient k, see Fig.2. The pur-
it follows from (9), (10), and Fig.1 that e(k) = 0 for k > 0. Since the denominator of P(z) in (9) equals the denominator of P(z) in (6), F2 (z) in (10) equals F2 (z) in (7). F1 (z) in (10), however, differs from F1 (z) in (7). Apart from the sampling zero at z = −1, which in Laplace domain implies a zero at the Nyquist frequency, an extra complex zero pair is added to F1 (z) in (10) to describe the anti-resonance. In Bode representation, this is shown in Fig.3 for F1 from (3), (7), and (10). From the figure, it can be concluded that
c2 −1 1 − c2 −2 1 − c2 −3 c2 −4 z + z + z + z 2 2 2 2 F2 (z) = c1 (1 − 2 cos(ωn T )z−1 + z−2 )(1 − 2z−1 + z−2 ), (10)
magnitude of F1 in dB
F1 (z) =
k
F1
m1 x1
m2 x2
40
0
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1
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1000
5000
100
1000
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Schematics of a 4th-order system.
pose of this non-collocated example (the actuator generates the force F1 whereas the sensor measures the displacement x2 ) is twofold. First, we want to demonstrate that the choice of FIR filters with F1 (z) equal to the numerator of P(z) and F2 (z) equal to the denominator induces zero error tracking. Second, we want to advocate that finding the FIR filter coefficients from first principles modeling is generally unfeasible. In discrete-time representation, the fourth-order system in (5) sampled using a zero-order-hold circuit is given by: 1 −2 1 −3 z + z 2 2 P(z) := , (6) c1 (1 − 2 cos(ωn T )z−1 + z−2 )(1 − 2z−1 + z−2 ) with constant c1 = m sin−1 (ωn T )ωn−1 T −3 and ωn = p k(m1 + m2 )/(m1 m2 ) the natural frequency. By choosing:
1 1 F1 (z) = z−2 + z−3 2 2 (7) F2 (z) = c1 (1 − 2 cos(ωn T )z−1 + z−2 )(1 − 2z−1 + z−2 ),
it is clear from (6), (7), and Fig.1 that e(k) = 0 for k > 0. However, obtaining the FIR filter coefficients in (7) is no easy task. This is because the physical parameters m1 , m2 , and k are often not exactly known. Finally, consider the collocated case in Fig.2 where the transfer between the force F1 and the displacement x1 reads: m2 s2 + k , (8) m1 m2 s4 + k(m1 + m2 )s2 Being sampled with a zero-order-hold circuit, (8) in discretetime representation is given by: c2 −1 1 − c2 −2 1 − c2 −3 c2 −4 z + z + z + z 2 2 2 2 , (9) P(z) := c1 (1 − 2 cos(ωn T )z−1 + z−2 )(1 − 2z−1 + z−2 )
phase of F1 in degrees
Fig. 2.
180
F1 from (3) F1 from (7) F1 from (10)
90
0
−90
−180 1
10
Fig. 3.
frequency in Hz
Bode diagrams of F1 .
tracking the reference r by the output y is only achieved at low frequencies, because r → r˜ for ω → 0. This also follows from the fact that the sum of the FIR filter coefficients of F1 (z) in the examples equals one, i.e. F1 (z = 1) = 1. At high frequencies, the plant output perfectly tracks r˜ but no longer r. For wafer scanning, however, this does not pose a performance problem. This is because during wafer scanning, point-to-point motions consist of two phases: a) an acceleration/deceleration phase, and b) a constant velocity phase in which exposure takes place. Performance is thus needed only during constant velocity where the shaped reference signal r˜ becomes a time-shifted version of the original reference signal r. In exposing a full wafer the effect of this time shift on wafer throughput is negligible. Besides, the elongation of the reference trajectory being the result of this time shift is much smaller than the effect of settling time reduction obtained from shaping r.
P(s) :=
III. MIMO FIR FEEDFORWARD CONTROL In the motion control industry, plants generally possess MIMO properties. However, the combination of a MIMO plant and a static decoupling structure often justifies the application of a diagonal feedback controller. Being weakly coupled, SISO stability properties derived from each of the
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diagonally controlled directions are not seriously compromised when being compared to the MIMO system. This, however, is not the case for the tracking performance which strongly benefits from MIMO feedforward design. For a 2×2 example, it will be demonstrated that zero error tracking is implied, similar as for the SISO case, by combining input shaping and feedforward control to the coupled directions. Consider the MIMO feedforward design depicted in Fig.4 with the 2 × 2 plant P(z) which can be represented in u f f ,1 (k)
F21
r(k)
r˜1 (k)
F11
Σ
e1 (k)
−
F12
r˜2 (k)
− Σ
e2 (k)
C1
C2
u f b,1 (k) u1 (k) Σ
u f b,2 (k) u2 (k) Σ
P11
y1 (k)
P21
y12 (k)
P22
The upper part of Fig.4 resembles the SISO case as considered in Fig.1, where r(k) is a data-sampled reference signal which is applied to an input shaper F11 and a feedforward filter F21 . Both F11 and F21 are given by FIR filters. The filtered reference r˜1 (k) along with the output y1 (k) of the plant P11 form the input to the feedback controller C1 via e1 (k) = r˜1 (k) − y1 (k). The output of the controller u f b,1 (k) along with the output of the feedforward filter u f f ,1 (k) form the input to the plant u1 (k) = u f b,1 (k) + u f f ,1 (k). The motion objective is to conduct a series of point-to-point movements, where the output y1 at both the start- and end-position should match with respectively the start- and end-position of the reference r. The lower part of Fig.4 describes the MIMO FIR extension. Given the tracking error e2 = r˜2 − y2 and the input u2 = u f b,2 + u f f ,2 to plant P22 (z), a proper choice of FIR filters F12 (z) and F22 (z) is required in view of the coupling induced by P21 (z). For this (non-scanning) direction, the control aim is essentially to keep the start- and end-position of the output y2 at zero. From Fig.4, it follows in z-domain (z is omitted for ease of notation) that:
y2 (k)
y1 = P11 u1 y2 = P21 u1 + P22u2 ,
(14)
u1 = u f b,1 + u f f ,1 = F21 r + C1 e1 u2 = u f b,2 + u f f ,2 = F22 r + C2 e2 .
(15)
Σ
and C
P u f f ,2 (k)
F22
For the error signal e1 (z) it therefore follows that: Fig. 4.
MIMO control configuration.
e1 = r˜1 − P11u1 = F11 r − P11 F21 r − P11 C1 e1
factorized form by:
P11 (z) P21 (z)
N11 (z) D (z) P12 (z) 11 = N21 (z) P22 (z) D11 (z)
0
(16)
= (1 + P11C1 )−1 (F11 − P11 F21 )r.
N22 (z) . (11) D22 (z)
From (16) it is clear that zero error tracking e1 (k) = 0 for k > 0 is obtained with:
Without loss of generality, we consider P12 = 0. The reason to do so is strictly performance driven. Namely, since the reference r(k) acts primarily in one direction, |u f f ,1 | ≫ |u f f ,2 |. As a result, the effect of u1 will generally crossover to y2 via P21 (z) despite the fact that the plant P is diagonally dominant; reversely, a cross-over effect from u2 to y1 will be negligible, hence the assumption P12 (z) = 0. The inverse of (11) is given by: D11 (z) 0 N11 (z) inv{P(z)} = (12) −N21 (z)D22 (z) D22 (z) . N11 (z)N22 (z) N22 (z)
(17)
P(z) =
In (11) and (12) the fact is used that P11 (z), P21 (z), and P22 (z) have common poles. The MIMO plant P(z) is controlled by the diagonal controller C (z): C (z) 0 . (13) C (z) = 1 0 C2 (z)
F11 (z) = N11 (z) F21 (z) = D11 (z),
i.e. with F11 (z) describing the zeros of plant P11 (z) and F21 (z) describing its poles. This is similar to the observations made in the previous section. −1 Remark 1: In terms of inverting the plant, see P11 (z) in (12), zero error tracking also follows from choosing: F11 (z) = 1 −1 (z) = F21 (z) = P11
D11 (z) , N11 (z)
(18)
i.e. a non-FIR solution in the case N11 (z) has zeros. If N11 (z) has zeros outside the unit circle (18) will render F21 (z) unstable, i.e. u1 becomes unbounded, which is obviously unacceptable; these problems do not occur with (17). For e2 (z) it follows with (14), (15), and (17) that:
Both C1 (z) and C2 (z) are linear time-invariant controllers that through C (z) stabilize P(z). 2168
e2 = r˜2 − P21 u1 − P22 u2 = F12 r − P21 (F21 r + C1 e1 ) − P22 (F22 r + C2 e2 ) (19) = (1 + P22C2 )−1 (F12 − P21F21 − P22F22 )r.
From (19), zero error tracking e2 (k) = 0 for k > 0 follows if: N22 (z) F22 (z) = 0. (20) F12 (z) − N21 (z) − D22 (z)
With the Gauss-Newton method, which is widely used for minimization of a sum of squared function values [3], the FIR coefficients pi ∈ R1+2n in (36) that minimize (24), or: p˜ i = arg min Ji ,
A particular FIR solution from (20) is given by: F12 (z) = N22 (z) + N21 (z) F22 (z) = D22 (z).
pi (κ )
(21)
The MIMO FIR feedforward design thus renders e1 (k) = e2 (k) = 0 for k > 0. Remark 2: For the physical interpretation of (21) con−1 sider P21 (z) in (12). With F12 (z) = 0, the feedforward term F22 (z) follows from: F22 (z) =
−N21 (z)D22 (z) , N11 (z)N22 (z)
(22)
that is an infinite impulse response (IIR) filter. For F22 (z) to become a FIR filter, the zeros in N11 (z) and N22 (z) need to be cancelled. Given (17), cancellation of the zeros in N11 (z) is done by F11 (z). This follows from the solution of F22 (z) in (20) with F12 (z) = 0, which reads: F22 (z) = −
N21 (z)D22 (z) , N22 (z)
are generally found with: pi (κ + 1) = pi (κ ) − ζ (∇T ei ∇ei )−1 (∇T ei )ei ,
In a data-driven context finding p˜ i in (27) using (28) essentially boils down to determining two gradient error signals: ∂ ei /∂ ai0 and ∂ ei /∂ bi0 . This is because for each iteration κ , ei in (28) can be obtained directly from measurement whereas the gradient error signals in (29) for example can be obtained using the fact that in the FIR filter structure, each FIR coefficient is delayed one sample with respect to its predecessor:
∂ ei ∂ ei −n ∂ ei ∂ ei −(n−2) z and = z . = ∂ ain ∂ ai0 ∂ bin−2 ∂ bi0
and which no longer contains N11 (z) in the denominator. The zeros in N22 (z) are being cancelled by F12 (z) in (21). For higher-order plants and more complex (possibly nonsquare) MIMO structures, the physical models underlying (17) and (21) generally have complex expressions in terms of FIR coefficients. It thus makes sense to attempt finding these coefficients by least squares optimization rather than by first principles modelling.
Ji = eTi ei ,
(24)
which represents the quadratic sum of the error signals ei = [ei (1) . . . ei (k)]T with i ∈ {1 . . . m}, k the number of data samples within a relevant performance window, i referring to a specific direction of the MIMO feedforward design, and m the number of considered directions. For each direction i let the FIR filter parameters to-beoptimized be given by: T pi (κ ) = ai0 (κ ) . . . ain (κ ) bi0 (κ ) . . . bin−2 (κ ) , (25)
with κ an iteration number. The set of FIR coefficients bi (κ ) has reduced dimensions because different from Fig.1 and Fig.4 not r but d 2 r/dt 2 is input to the FIR filters F2i (z), a logical step when considering double integratorbased systems. The FIR filters at each iteration κ are given by: F1i (z) = ai0 (κ ) + ai1(κ )z−1 · · · + ain(κ )z−n F2i (z) = bi0 (κ ) + bi1(κ )z−1 · · · + bin−2(κ )z−(n−2) ,
(26)
where F1i (z) refers to the FIR filter in the set-point path for the i-th direction and F2i (z) refers to the FIR filter in the feedforward path for the same direction, see also Fig.4.
(28)
with the initial set pi (0) = 0, damping coefficient 0 < ζ ≤ 1, and gradients: ∂ ei ∂ ei ∂ ei ∂ ei . . . . . . ∇ei = . (29) ∂ ain ∂ bi0 ∂ ai0 ∂ bin−2
(23)
IV. FIR FILTER OPTIMIZATION In finding the MIMO FIR filter coefficients through optimization, consider the H2 cost function:
(27)
(30)
From (16), (19), and (26) it follows that:
∂ ei (z) = (1 + Pii (z)Ci (z))−1 r(z) = Si (z)r(z) ∂ ai0 ∂ ei (z) = −Pii (z)(1 + Pii (z)Ci (z))−1 r(z) = S p,i (z)r(z), ∂ bi0 (31) with Si (z) the sensitivity function for direction i and S p,i (z) the corresponding process sensitivity function. By constructing for each direction i (either scanning or non-scanning) the Toeplitz matrices Si and S p,i using the impulse responses corresponding to the sensitivity and process sensitivity functions Si (z) and S p,i (z), see also [2], it can be obtained that:
∂ ei ∂ ei = Si r and = S p,i r, ∂ ai0 ∂ bi0
(32)
with r = [r(1) . . . r(k)]T . The impulse responses used in (32) are obtained from reconstruction of the controller characteristics, which are known, with the plant characteristics, which are unknown but which can be modelled fairly accurately using closed-loop frequency response data. Different from (30), we use the derivative basis from [4]:
∂ e∗i ∂ ei := i (1 − z−1)n . ∗i ∂ an ∂ a0
(33)
The reason to do so is that (33) gives a more natural basis to include constraints on the FIR coefficients a∗i (κ ). Constraints are needed to impose the required DC gains on F1 (z = 1), i.e. the scalings of the reference r˜i in comparison with r as to meet the motion objectives, recall earlier discussions. With (33), the update law in (28) becomes
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p∗ i (κ + 1) = p∗ i (κ ) − ζ (∇T e∗i ∇e∗i )−1 (∇T e∗i )ei ,
(34)
with:
∂ e∗i ∂ e∗i . . . ∇e∗i = ∂ a∗i ∂ a∗i n 1
∂ ei ∂ ei ... i , ∂ bi0 ∂ bn−2
(35)
T = 10−4 s. The result of closed-loop time-series simulations with and without the optimized SISO FIR filters F1 (z) and F2 (z) from Fig.1 are depicted in Fig.6. In these simulations,
and 30
acc.setp. FF with FIR FF without FIR
(36)
The FIR filters F1i (z) from (26) in view of the (derivative) basis in (33) are given by:
ey in nm
T ∗i i i p∗ i (κ ) = a∗i 1 (κ ) . . . an (κ ) b0 (κ ) . . . bn−2 (κ ) .
0
n
F1i (z) = ai0 + ∑ B j+1 (z)p∗ i [ j]
(37) −30
j=1
0
0.1
time in seconds
with B j+1 (z) = (1 − z−1 )B j , B1 = 1.
(38)
The sum of the FIR coefficients of F1i (z), the second term in the first equation of (37), is equal to zero regardless the values of p∗ i . In combination with the choice of respectively a10 = 1 or ai0 = 0 (the latter if i ≥ 2) this will either result in F1i (z = 1) = 1 in the scanning direction or F1i (z = 1) = 0 in the non-scanning directions. As a result, both the scanning output y1 and the non-scanning outputs yi with i ≥ 2 will have zero offset with respect to the original references. V. WAFER SCANNER EXAMPLES To demonstrate the effectiveness of the proposed MIMO FIR feedforward design consider the example of wafer scanning as depicted in Fig.5. (Extreme) ultraviolet light light path
reticle stage
the feedback controller C (z) is of the PID-type which is extended with extra loop shaping filters; this controller is also used in the experiments. The reference signal r, which is a point-to-point motion from y = 0 to y = 0.05 m, is obtained from a fourth-order setpoint generator with the following motion parameters: velocity vmax = 0.5 ms−1 , acceleration amax = 20 ms−2 , jerk jmax = 2 · 103 ms−3 , and snap smax = 1.5 · 106 ms−4 , see Fig.6 for the resulting (scaled) acceleration profile (thin black curves). In addition to the reference input, output disturbances in the sense of band-limited white noise are input to the closed-loop simulation model. In terms of ey (thick black curves) the result of the closed-loop system subject to these inputs is shown. The simulations include a nominal mass and snap feedforward controller, in which the acceleration feedforward gain is chosen equal to m1 + m2 and the snap feedforward gain is chosen at 75% of k1 , i.e. the error responses without optimized SISO FIR filters are the result of this gain mismatch. After FIR optimization with damping coefficient ζ = 1, the number of iterations κ = 2, and the order of the FIR filters n = 4, the error signal ey (red curves) is at the noise level. Fig.7 shows that the fourth-order plant characteristics (gray curves) from (6) match with the characteristics obmagnitude of P in dB
optical column
Fig. 6. Time-series simulation with (red) and without (black) SISO FIR filter optimization; the simulations include a nominal mass (m1 + m2 ) and snap (0.75k1 ) feedforward controller.
wafer stage
−40
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Schematics of wafer scanning.
containing an image of the integrated circuits to be processed travels via a light path through an optical column to expose the light sensitive layers on a wafer. The image is obtained from the reticle which is part of the reticle stage motion control system. Similarly the wafer is part of the wafer stage motion control system. In this paper, the MIMO FIR filter design will be demonstrated on the wafer stage system only. In simulations, consider the SISO FIR filter design based on the fourth-order plant P(z) from (6) with the parameter values: m1 = 1 kg, m2 = 17 kg, k1 = 4 107 Nm−1 , and
phase of P in degrees
Fig. 5.
180
90
0
−90
P(z) from (6) reconstructed P(z) from optimized F1 (z) and F2 (z) reconstructed P(z) from optimized F1 (z) and F2 (z) (no noise)
−180 1
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100
frequency in Hz
1000
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Fig. 7. Bode diagrams of the 4th order plant P(z) from (6) and the reconstructed plant obtained from the optimized filters F1 (z) and F2 (z).
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ex in nm
20
0
−20
0
0.1
ey in nm
30
acc.setp. FF without FIR FF with FIR
0
−30
0
0.1
erz in nrad
350
0
−350
0
0.1
time in seconds
magnitude of P in dB
Fig. 8. Time-series measurement with (red) and without (black) MIMO FIR filter optimization on a wafer stage system; the experiments include a nominal mass (m1 + m2 ) and snap (k1 ) feedforward controller. −40
−140
−240
phase of P in degrees
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VI. CONCLUSIONS A MIMO FIR feedforward design is demonstrated to be effective in obtaining zero error tracking. To this end, the SISO FIR filter structure with an input shaping filter containing the plant zeros, which are possibly non-minimum phase, and a feedforward filter containing the plant poles is largely preserved in the MIMO structure. In this structure the reference trajectories are delayed rather then solving for the inverse dynamics problem, i.e. zero error tracking is obtained with respect to the delayed trajectories. This has no consequences for the performancerelevant phase of constant velocity where scanning takes place and where the shaped reference trajectories are just time-shifted versions of the original reference trajectories. In this sense, throughput loss by extending the reference trajectories (the latter being the result of these time-shifts) is fully compensated by reduced settling times as a result of less excitation of the closed-loop dynamics. In obtaining the FIR filter coefficients, a gradient-based optimization scheme is used because of two reasons. First, the FIR coefficients which tend to become complex expressions in terms of the physical parameters are easily obtained within the convex optimization context. Second, to guarantee zero positioning offset errors, constraints imposed on the sum of the FIR filter coefficients in the set-point path are easily addressed in the optimization algorithm. R EFERENCES
1
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180
90
0
−90
P(z) from FRF measurements reconstructed P(z) from optimized F1 (z) and F2 (z)
−180 1
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frequency in Hz
Fig. 9. Bode diagrams of the plant P(z) from FRF measurements and the reconstructed plant obtained from the optimized filters F1 (z) and F2 (z).
tained from the optimized filters F1 (z) and F2 (z) either in the case without noise (thin curves) or approximately in the case with noise (dashed curves). In experiment, Fig.8 demonstrates the effectiveness of the MIMO FIR approach on the wafer stage systems of a real wafer scanner. For point-to-point motions similar to the motions used in simulation, it can be seen that the error signals in both x-, y-, and rz-direction after optimization (red curves) are close to the noise level. With nominal SISO feedforward control design (black curves) this is clearly not the case; the nominal SISO feedforward control design is based on an acceleration feedforward gain of m1 + m2 = 18 kg and a snap gain of k1 = 4 107 Nm−1 . In terms of frequency-domain characteristics Fig.9 shows that F1 (z) and F2 (z) with n = 9 approximate the measured plant with characteristics similar to the characteristics found in the simulation results of Fig.7.
[1] Baggen M, Heertjes MF, and Kamidi R. (2008) Data-based feedforward control in MIMO motion systems. In Proceedings of the American Control Conference, Seattle, Washington, USA: 3011-3016. [2] Bamieh B, Boyd PJ, Francis BA, and Tannenbaum A. (1991) A lifting technique for linear periodic systems with applications to sampled-data control. Systems and Control Letters, 17:79-88. [3] Bazanella AS, Campestrini L, and Eckhard D. (2012) Data-driven controller design; the H2 approach. Springer, New York. [4] Bruijnen D. and Van Dijk N. (2012) Combined input shaping and feedforward control for flexible motion systems, In Proceedings of the American Control Conference, Montr´eal, Canada: 2473-2478. [5] Butterworth J.A., Pao L.Y., and Abramovitch D.Y. (2008) The effect of nonminimum-phase zero locations on the performance of feedforward model-inverse control techniques in discrete-time systems. In Proceedings of the American Control Conference, Seattle, Washington, USA: 2696-2702. [6] Heertjes MF, Hennekens D, and Steinbuch M. (2010) MIMO feedforward design in wafer scanners using a gradient approximation-based algorithm. Control Engineering Practice, 15(5):495–506, 2010. [7] Heertjes M.F. and Nijmeijer H. (2012) Self-tuning of a switching controller for scanning motion systems, Mechatronics, 22:310-319. [8] Jouaneh M.K. and Anderson E. (2006) Input shaping using finite impulse response filters, In Proceedings of the Conference on Decision and Control, San Diego, California, USA:6525-6530. [9] Marconi L., Marro G., and Melchiorri C.. (2001) A solution for almost perfect tracking of non-minimum-phase, discrete-time linear systems. International Journal of Control, 74(5): 496-506. [10] Singh T., and Singhose W. (2002) Tutorial on input shaping/time delay control of maneuvering flexible structures. In Proceedings of the American Control Conference, Anchorage, AK, USA:1717-1731. [11] Rigney B.P., Pao L.Y., and Lawrence D.A. (2009) Nonminimum phase dynamic inversion for settle time applications. Transactions on Control Systems Technology, 17(5): 989-1005. [12] Tomizuka M. (1987) Zero phase error tracking algorithm for digital control. ASME Journal of Dynamic Systems, Measurement, and Control, 109(1): 65-68. [13] Van der Meulen S.H., Tousain R.L., and Bosgra O.H. (2008) Fixed structure feedforward controller design exploiting iterative trials: application to a wafer stage and a desktop printer, Journal of Dynamic Systems, Measurement, and Control, 130(5): 1-16.
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MIMO FIR Feedforward Design for Zero Error Tracking Control Marcel Heertjes1 and Dennis Bruijnen2
Abstract— This paper discusses a multi-input multi-output (MIMO) finite impulse response (FIR) feedforward design. The design is intended for systems that have (non-)minimum phase zeros in the plant description. The zeros of the plant (either minimum or non-minimum phase) are used in the shaping of the reference signals whereas the poles of the plant are used in constructing feedforward forces. The FIR coefficients themselves are obtained from data-based optimizations which are the result of iterative machine measurements on an industrial wafer scanner. The resulting experimental results demonstrate the ability to obtain zero error tracking.
I. INTRODUCTION Nanometer tracking performance in motion control systems like industrial wafer scanners is obtained with feedforward control using the principles of plant inversion. Though many researchers in the past have worked on the inversion problem, see for example [5], [11], [12] and the references therein, basically two directions evolved toward tracking error minimization. First, the design of feedforward controllers being approximations of the inverse of the (closed-loop) plant [6], [9], [13]. Second, the design of input shapers [8], [10] that modify the reference trajectory with the aim to provide less excitation of this plant. In this paper, similar to [4], both ideas are combined. But different from [4] we pose the problem in a discrete-time (measured) stage control context and consider an extension to MIMO FIR feedforward design. By shaping the reference trajectories in combination with a proper feedforward contribution, zero error tracking can be obtained in all the coupled directions of the MIMO system. This follows from individually delaying the reference contributions to these directions with respect to the feedforward contributions. It is important to realize that tracking the shaped reference rather than the original reference does not pose a performance problem within the context of wafer scanning. This is because scanning takes place under constant velocity. The shaped reference will therefore only be a time-shifted version of the original reference. The potential loss of throughput due to this time shift is more than compensated for by reduced settling times. The latter being the result of not exciting the plant dynamics. The MIMO FIR filter coefficients, which generally appear too sensitive to obtain from first principles modelling, are obtained from a data-based optimization approach, see also [1], [7]. The remainder of this paper is organized as follows. In Section 2, FIR feedforward design in the context of singleinput single-output (SISO) motion systems will be discussed. By means of simple examples of discretized plants, the FIR 1 Marcel
F. Heertjes is with Faculty of Mechanical Engineering, Eindhoven University of Technology, 5600 MB Eindhoven, The Netherlands
filter properties are illustrated. In Section 3, an extension toward MIMO FIR feedforward design will be given. In Section 4, the optimization approach used to obtain the FIR filter coefficients will be discussed. In Section 5, both numerical and experimental results will be presented. Section 6 summarizes the main conclusions of this work. II. SISO FIR FEEDFORWARD DESIGN Consider the simplified SISO control configuration of a motion system such as depicted in Fig.1. Herein r(k) is a
r(k)
F1
r˜(k)
e(k) Σ
C
u f b (k)
Σ
u(k)
P
y(k)
−
Fig. 1.
SISO control configuration.
data-sampled reference signal, i.e. the value of r at time tk = kT with k an integer and T the sampling time. This reference signal which is used to obtain point-to-point motion is applied to a set-point filter F1 with DC gain F1 (z = 1) = 1 (z indicating the z-domain) and to a feedforward filter F2 . The filtered reference r˜(k) along with the output y(k) of the double integrator-based plant P form the input to the stabilizing and linear time-invariant feedback controller C via e(k) = r˜(k) − y(k). The output of the controller u f b (k) along with the output of the feedforward filter u f f (k) form the input to the plant u(k) via u(k) = u f b (k) + u f f (k). Suppose P(z) stems from a continuous-time double integrator plant, which is representative for stage systems, or: P(s) :=
1 , ms2
(1)
with mass m and Laplace variable s, which is sampled using a zero-order hold circuit. In discrete-time, it follows that: P(s) P(z) = (1 − z−1)Z s (2) 1 −1 1 −2 z + z 2 2 = , c0 (1 − 2z−1 + z−2) with constant c0 = mT −2 , z−1 a unit time delay, and Z {·} denoting the z-transform. From Fig.1, it follows that if
[email protected] 2 Dennis
Bruijnen is with Mechatronics Technologies Group, Philips Innovation Services, High Tech Campus 7, 5656 AE Eindhoven, The Netherlands [email protected]
978-1-4799-3271-9/$31.00 ©2014 AACC
u f f (k)
F2
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1 1 F1 (z) = z−1 + z−2 2 2 F2 (z) = c0 (1 − 2z−1 + z−2 ),
(3)
i.e. two (3-taps) FIR filters, any bounded r(k) with r(k) = 0 for k ≤ 0 induces zero error tracking in the sense that e(k) = 0 for k > 0. From (2) and (3) it is clear that such error tracking is obtained by choosing F1 (z) equal to the numerator of P(z) in (2) whereas F2 (z) is chosen equal to the denominator of P(z), namely:
with constant c2 = mωn−2 T −2 m−1 1 . By choosing
e(z) = r˜(z) − y(z) = F1 (z)r(z) − P(z)F2 (z)r(z) − P(z)C (z)e(z) (4) F1 (z) F2 (z)r(z) = 0. = S (z)F1 (z)r(z) − S (z) F2 (z) Now consider an extended version of the double integrator plant given by the fourth-order system: k , (5) P(s) := m1 m2 s4 + k(m1 + m2 )s2 with mass m1 , m2 satisfying m = m1 + m2 and an interconnected spring with stiffness coefficient k, see Fig.2. The pur-
it follows from (9), (10), and Fig.1 that e(k) = 0 for k > 0. Since the denominator of P(z) in (9) equals the denominator of P(z) in (6), F2 (z) in (10) equals F2 (z) in (7). F1 (z) in (10), however, differs from F1 (z) in (7). Apart from the sampling zero at z = −1, which in Laplace domain implies a zero at the Nyquist frequency, an extra complex zero pair is added to F1 (z) in (10) to describe the anti-resonance. In Bode representation, this is shown in Fig.3 for F1 from (3), (7), and (10). From the figure, it can be concluded that
c2 −1 1 − c2 −2 1 − c2 −3 c2 −4 z + z + z + z 2 2 2 2 F2 (z) = c1 (1 − 2 cos(ωn T )z−1 + z−2 )(1 − 2z−1 + z−2 ), (10)
magnitude of F1 in dB
F1 (z) =
k
F1
m1 x1
m2 x2
40
0
−40
1
10
100
1000
5000
100
1000
5000
Schematics of a 4th-order system.
pose of this non-collocated example (the actuator generates the force F1 whereas the sensor measures the displacement x2 ) is twofold. First, we want to demonstrate that the choice of FIR filters with F1 (z) equal to the numerator of P(z) and F2 (z) equal to the denominator induces zero error tracking. Second, we want to advocate that finding the FIR filter coefficients from first principles modeling is generally unfeasible. In discrete-time representation, the fourth-order system in (5) sampled using a zero-order-hold circuit is given by: 1 −2 1 −3 z + z 2 2 P(z) := , (6) c1 (1 − 2 cos(ωn T )z−1 + z−2 )(1 − 2z−1 + z−2 ) with constant c1 = m sin−1 (ωn T )ωn−1 T −3 and ωn = p k(m1 + m2 )/(m1 m2 ) the natural frequency. By choosing:
1 1 F1 (z) = z−2 + z−3 2 2 (7) F2 (z) = c1 (1 − 2 cos(ωn T )z−1 + z−2 )(1 − 2z−1 + z−2 ),
it is clear from (6), (7), and Fig.1 that e(k) = 0 for k > 0. However, obtaining the FIR filter coefficients in (7) is no easy task. This is because the physical parameters m1 , m2 , and k are often not exactly known. Finally, consider the collocated case in Fig.2 where the transfer between the force F1 and the displacement x1 reads: m2 s2 + k , (8) m1 m2 s4 + k(m1 + m2 )s2 Being sampled with a zero-order-hold circuit, (8) in discretetime representation is given by: c2 −1 1 − c2 −2 1 − c2 −3 c2 −4 z + z + z + z 2 2 2 2 , (9) P(z) := c1 (1 − 2 cos(ωn T )z−1 + z−2 )(1 − 2z−1 + z−2 )
phase of F1 in degrees
Fig. 2.
180
F1 from (3) F1 from (7) F1 from (10)
90
0
−90
−180 1
10
Fig. 3.
frequency in Hz
Bode diagrams of F1 .
tracking the reference r by the output y is only achieved at low frequencies, because r → r˜ for ω → 0. This also follows from the fact that the sum of the FIR filter coefficients of F1 (z) in the examples equals one, i.e. F1 (z = 1) = 1. At high frequencies, the plant output perfectly tracks r˜ but no longer r. For wafer scanning, however, this does not pose a performance problem. This is because during wafer scanning, point-to-point motions consist of two phases: a) an acceleration/deceleration phase, and b) a constant velocity phase in which exposure takes place. Performance is thus needed only during constant velocity where the shaped reference signal r˜ becomes a time-shifted version of the original reference signal r. In exposing a full wafer the effect of this time shift on wafer throughput is negligible. Besides, the elongation of the reference trajectory being the result of this time shift is much smaller than the effect of settling time reduction obtained from shaping r.
P(s) :=
III. MIMO FIR FEEDFORWARD CONTROL In the motion control industry, plants generally possess MIMO properties. However, the combination of a MIMO plant and a static decoupling structure often justifies the application of a diagonal feedback controller. Being weakly coupled, SISO stability properties derived from each of the
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diagonally controlled directions are not seriously compromised when being compared to the MIMO system. This, however, is not the case for the tracking performance which strongly benefits from MIMO feedforward design. For a 2×2 example, it will be demonstrated that zero error tracking is implied, similar as for the SISO case, by combining input shaping and feedforward control to the coupled directions. Consider the MIMO feedforward design depicted in Fig.4 with the 2 × 2 plant P(z) which can be represented in u f f ,1 (k)
F21
r(k)
r˜1 (k)
F11
Σ
e1 (k)
−
F12
r˜2 (k)
− Σ
e2 (k)
C1
C2
u f b,1 (k) u1 (k) Σ
u f b,2 (k) u2 (k) Σ
P11
y1 (k)
P21
y12 (k)
P22
The upper part of Fig.4 resembles the SISO case as considered in Fig.1, where r(k) is a data-sampled reference signal which is applied to an input shaper F11 and a feedforward filter F21 . Both F11 and F21 are given by FIR filters. The filtered reference r˜1 (k) along with the output y1 (k) of the plant P11 form the input to the feedback controller C1 via e1 (k) = r˜1 (k) − y1 (k). The output of the controller u f b,1 (k) along with the output of the feedforward filter u f f ,1 (k) form the input to the plant u1 (k) = u f b,1 (k) + u f f ,1 (k). The motion objective is to conduct a series of point-to-point movements, where the output y1 at both the start- and end-position should match with respectively the start- and end-position of the reference r. The lower part of Fig.4 describes the MIMO FIR extension. Given the tracking error e2 = r˜2 − y2 and the input u2 = u f b,2 + u f f ,2 to plant P22 (z), a proper choice of FIR filters F12 (z) and F22 (z) is required in view of the coupling induced by P21 (z). For this (non-scanning) direction, the control aim is essentially to keep the start- and end-position of the output y2 at zero. From Fig.4, it follows in z-domain (z is omitted for ease of notation) that:
y2 (k)
y1 = P11 u1 y2 = P21 u1 + P22u2 ,
(14)
u1 = u f b,1 + u f f ,1 = F21 r + C1 e1 u2 = u f b,2 + u f f ,2 = F22 r + C2 e2 .
(15)
Σ
and C
P u f f ,2 (k)
F22
For the error signal e1 (z) it therefore follows that: Fig. 4.
MIMO control configuration.
e1 = r˜1 − P11u1 = F11 r − P11 F21 r − P11 C1 e1
factorized form by:
P11 (z) P21 (z)
N11 (z) D (z) P12 (z) 11 = N21 (z) P22 (z) D11 (z)
0
(16)
= (1 + P11C1 )−1 (F11 − P11 F21 )r.
N22 (z) . (11) D22 (z)
From (16) it is clear that zero error tracking e1 (k) = 0 for k > 0 is obtained with:
Without loss of generality, we consider P12 = 0. The reason to do so is strictly performance driven. Namely, since the reference r(k) acts primarily in one direction, |u f f ,1 | ≫ |u f f ,2 |. As a result, the effect of u1 will generally crossover to y2 via P21 (z) despite the fact that the plant P is diagonally dominant; reversely, a cross-over effect from u2 to y1 will be negligible, hence the assumption P12 (z) = 0. The inverse of (11) is given by: D11 (z) 0 N11 (z) inv{P(z)} = (12) −N21 (z)D22 (z) D22 (z) . N11 (z)N22 (z) N22 (z)
(17)
P(z) =
In (11) and (12) the fact is used that P11 (z), P21 (z), and P22 (z) have common poles. The MIMO plant P(z) is controlled by the diagonal controller C (z): C (z) 0 . (13) C (z) = 1 0 C2 (z)
F11 (z) = N11 (z) F21 (z) = D11 (z),
i.e. with F11 (z) describing the zeros of plant P11 (z) and F21 (z) describing its poles. This is similar to the observations made in the previous section. −1 Remark 1: In terms of inverting the plant, see P11 (z) in (12), zero error tracking also follows from choosing: F11 (z) = 1 −1 (z) = F21 (z) = P11
D11 (z) , N11 (z)
(18)
i.e. a non-FIR solution in the case N11 (z) has zeros. If N11 (z) has zeros outside the unit circle (18) will render F21 (z) unstable, i.e. u1 becomes unbounded, which is obviously unacceptable; these problems do not occur with (17). For e2 (z) it follows with (14), (15), and (17) that:
Both C1 (z) and C2 (z) are linear time-invariant controllers that through C (z) stabilize P(z). 2168
e2 = r˜2 − P21 u1 − P22 u2 = F12 r − P21 (F21 r + C1 e1 ) − P22 (F22 r + C2 e2 ) (19) = (1 + P22C2 )−1 (F12 − P21F21 − P22F22 )r.
From (19), zero error tracking e2 (k) = 0 for k > 0 follows if: N22 (z) F22 (z) = 0. (20) F12 (z) − N21 (z) − D22 (z)
With the Gauss-Newton method, which is widely used for minimization of a sum of squared function values [3], the FIR coefficients pi ∈ R1+2n in (36) that minimize (24), or: p˜ i = arg min Ji ,
A particular FIR solution from (20) is given by: F12 (z) = N22 (z) + N21 (z) F22 (z) = D22 (z).
pi (κ )
(21)
The MIMO FIR feedforward design thus renders e1 (k) = e2 (k) = 0 for k > 0. Remark 2: For the physical interpretation of (21) con−1 sider P21 (z) in (12). With F12 (z) = 0, the feedforward term F22 (z) follows from: F22 (z) =
−N21 (z)D22 (z) , N11 (z)N22 (z)
(22)
that is an infinite impulse response (IIR) filter. For F22 (z) to become a FIR filter, the zeros in N11 (z) and N22 (z) need to be cancelled. Given (17), cancellation of the zeros in N11 (z) is done by F11 (z). This follows from the solution of F22 (z) in (20) with F12 (z) = 0, which reads: F22 (z) = −
N21 (z)D22 (z) , N22 (z)
are generally found with: pi (κ + 1) = pi (κ ) − ζ (∇T ei ∇ei )−1 (∇T ei )ei ,
In a data-driven context finding p˜ i in (27) using (28) essentially boils down to determining two gradient error signals: ∂ ei /∂ ai0 and ∂ ei /∂ bi0 . This is because for each iteration κ , ei in (28) can be obtained directly from measurement whereas the gradient error signals in (29) for example can be obtained using the fact that in the FIR filter structure, each FIR coefficient is delayed one sample with respect to its predecessor:
∂ ei ∂ ei −n ∂ ei ∂ ei −(n−2) z and = z . = ∂ ain ∂ ai0 ∂ bin−2 ∂ bi0
and which no longer contains N11 (z) in the denominator. The zeros in N22 (z) are being cancelled by F12 (z) in (21). For higher-order plants and more complex (possibly nonsquare) MIMO structures, the physical models underlying (17) and (21) generally have complex expressions in terms of FIR coefficients. It thus makes sense to attempt finding these coefficients by least squares optimization rather than by first principles modelling.
Ji = eTi ei ,
(24)
which represents the quadratic sum of the error signals ei = [ei (1) . . . ei (k)]T with i ∈ {1 . . . m}, k the number of data samples within a relevant performance window, i referring to a specific direction of the MIMO feedforward design, and m the number of considered directions. For each direction i let the FIR filter parameters to-beoptimized be given by: T pi (κ ) = ai0 (κ ) . . . ain (κ ) bi0 (κ ) . . . bin−2 (κ ) , (25)
with κ an iteration number. The set of FIR coefficients bi (κ ) has reduced dimensions because different from Fig.1 and Fig.4 not r but d 2 r/dt 2 is input to the FIR filters F2i (z), a logical step when considering double integratorbased systems. The FIR filters at each iteration κ are given by: F1i (z) = ai0 (κ ) + ai1(κ )z−1 · · · + ain(κ )z−n F2i (z) = bi0 (κ ) + bi1(κ )z−1 · · · + bin−2(κ )z−(n−2) ,
(26)
where F1i (z) refers to the FIR filter in the set-point path for the i-th direction and F2i (z) refers to the FIR filter in the feedforward path for the same direction, see also Fig.4.
(28)
with the initial set pi (0) = 0, damping coefficient 0 < ζ ≤ 1, and gradients: ∂ ei ∂ ei ∂ ei ∂ ei . . . . . . ∇ei = . (29) ∂ ain ∂ bi0 ∂ ai0 ∂ bin−2
(23)
IV. FIR FILTER OPTIMIZATION In finding the MIMO FIR filter coefficients through optimization, consider the H2 cost function:
(27)
(30)
From (16), (19), and (26) it follows that:
∂ ei (z) = (1 + Pii (z)Ci (z))−1 r(z) = Si (z)r(z) ∂ ai0 ∂ ei (z) = −Pii (z)(1 + Pii (z)Ci (z))−1 r(z) = S p,i (z)r(z), ∂ bi0 (31) with Si (z) the sensitivity function for direction i and S p,i (z) the corresponding process sensitivity function. By constructing for each direction i (either scanning or non-scanning) the Toeplitz matrices Si and S p,i using the impulse responses corresponding to the sensitivity and process sensitivity functions Si (z) and S p,i (z), see also [2], it can be obtained that:
∂ ei ∂ ei = Si r and = S p,i r, ∂ ai0 ∂ bi0
(32)
with r = [r(1) . . . r(k)]T . The impulse responses used in (32) are obtained from reconstruction of the controller characteristics, which are known, with the plant characteristics, which are unknown but which can be modelled fairly accurately using closed-loop frequency response data. Different from (30), we use the derivative basis from [4]:
∂ e∗i ∂ ei := i (1 − z−1)n . ∗i ∂ an ∂ a0
(33)
The reason to do so is that (33) gives a more natural basis to include constraints on the FIR coefficients a∗i (κ ). Constraints are needed to impose the required DC gains on F1 (z = 1), i.e. the scalings of the reference r˜i in comparison with r as to meet the motion objectives, recall earlier discussions. With (33), the update law in (28) becomes
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p∗ i (κ + 1) = p∗ i (κ ) − ζ (∇T e∗i ∇e∗i )−1 (∇T e∗i )ei ,
(34)
with:
∂ e∗i ∂ e∗i . . . ∇e∗i = ∂ a∗i ∂ a∗i n 1
∂ ei ∂ ei ... i , ∂ bi0 ∂ bn−2
(35)
T = 10−4 s. The result of closed-loop time-series simulations with and without the optimized SISO FIR filters F1 (z) and F2 (z) from Fig.1 are depicted in Fig.6. In these simulations,
and 30
acc.setp. FF with FIR FF without FIR
(36)
The FIR filters F1i (z) from (26) in view of the (derivative) basis in (33) are given by:
ey in nm
T ∗i i i p∗ i (κ ) = a∗i 1 (κ ) . . . an (κ ) b0 (κ ) . . . bn−2 (κ ) .
0
n
F1i (z) = ai0 + ∑ B j+1 (z)p∗ i [ j]
(37) −30
j=1
0
0.1
time in seconds
with B j+1 (z) = (1 − z−1 )B j , B1 = 1.
(38)
The sum of the FIR coefficients of F1i (z), the second term in the first equation of (37), is equal to zero regardless the values of p∗ i . In combination with the choice of respectively a10 = 1 or ai0 = 0 (the latter if i ≥ 2) this will either result in F1i (z = 1) = 1 in the scanning direction or F1i (z = 1) = 0 in the non-scanning directions. As a result, both the scanning output y1 and the non-scanning outputs yi with i ≥ 2 will have zero offset with respect to the original references. V. WAFER SCANNER EXAMPLES To demonstrate the effectiveness of the proposed MIMO FIR feedforward design consider the example of wafer scanning as depicted in Fig.5. (Extreme) ultraviolet light light path
reticle stage
the feedback controller C (z) is of the PID-type which is extended with extra loop shaping filters; this controller is also used in the experiments. The reference signal r, which is a point-to-point motion from y = 0 to y = 0.05 m, is obtained from a fourth-order setpoint generator with the following motion parameters: velocity vmax = 0.5 ms−1 , acceleration amax = 20 ms−2 , jerk jmax = 2 · 103 ms−3 , and snap smax = 1.5 · 106 ms−4 , see Fig.6 for the resulting (scaled) acceleration profile (thin black curves). In addition to the reference input, output disturbances in the sense of band-limited white noise are input to the closed-loop simulation model. In terms of ey (thick black curves) the result of the closed-loop system subject to these inputs is shown. The simulations include a nominal mass and snap feedforward controller, in which the acceleration feedforward gain is chosen equal to m1 + m2 and the snap feedforward gain is chosen at 75% of k1 , i.e. the error responses without optimized SISO FIR filters are the result of this gain mismatch. After FIR optimization with damping coefficient ζ = 1, the number of iterations κ = 2, and the order of the FIR filters n = 4, the error signal ey (red curves) is at the noise level. Fig.7 shows that the fourth-order plant characteristics (gray curves) from (6) match with the characteristics obmagnitude of P in dB
optical column
Fig. 6. Time-series simulation with (red) and without (black) SISO FIR filter optimization; the simulations include a nominal mass (m1 + m2 ) and snap (0.75k1 ) feedforward controller.
wafer stage
−40
−140
−240
−340
1
10
100
1000
5000
Schematics of wafer scanning.
containing an image of the integrated circuits to be processed travels via a light path through an optical column to expose the light sensitive layers on a wafer. The image is obtained from the reticle which is part of the reticle stage motion control system. Similarly the wafer is part of the wafer stage motion control system. In this paper, the MIMO FIR filter design will be demonstrated on the wafer stage system only. In simulations, consider the SISO FIR filter design based on the fourth-order plant P(z) from (6) with the parameter values: m1 = 1 kg, m2 = 17 kg, k1 = 4 107 Nm−1 , and
phase of P in degrees
Fig. 5.
180
90
0
−90
P(z) from (6) reconstructed P(z) from optimized F1 (z) and F2 (z) reconstructed P(z) from optimized F1 (z) and F2 (z) (no noise)
−180 1
10
100
frequency in Hz
1000
5000
Fig. 7. Bode diagrams of the 4th order plant P(z) from (6) and the reconstructed plant obtained from the optimized filters F1 (z) and F2 (z).
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ex in nm
20
0
−20
0
0.1
ey in nm
30
acc.setp. FF without FIR FF with FIR
0
−30
0
0.1
erz in nrad
350
0
−350
0
0.1
time in seconds
magnitude of P in dB
Fig. 8. Time-series measurement with (red) and without (black) MIMO FIR filter optimization on a wafer stage system; the experiments include a nominal mass (m1 + m2 ) and snap (k1 ) feedforward controller. −40
−140
−240
phase of P in degrees
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VI. CONCLUSIONS A MIMO FIR feedforward design is demonstrated to be effective in obtaining zero error tracking. To this end, the SISO FIR filter structure with an input shaping filter containing the plant zeros, which are possibly non-minimum phase, and a feedforward filter containing the plant poles is largely preserved in the MIMO structure. In this structure the reference trajectories are delayed rather then solving for the inverse dynamics problem, i.e. zero error tracking is obtained with respect to the delayed trajectories. This has no consequences for the performancerelevant phase of constant velocity where scanning takes place and where the shaped reference trajectories are just time-shifted versions of the original reference trajectories. In this sense, throughput loss by extending the reference trajectories (the latter being the result of these time-shifts) is fully compensated by reduced settling times as a result of less excitation of the closed-loop dynamics. In obtaining the FIR filter coefficients, a gradient-based optimization scheme is used because of two reasons. First, the FIR coefficients which tend to become complex expressions in terms of the physical parameters are easily obtained within the convex optimization context. Second, to guarantee zero positioning offset errors, constraints imposed on the sum of the FIR filter coefficients in the set-point path are easily addressed in the optimization algorithm. R EFERENCES
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P(z) from FRF measurements reconstructed P(z) from optimized F1 (z) and F2 (z)
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Fig. 9. Bode diagrams of the plant P(z) from FRF measurements and the reconstructed plant obtained from the optimized filters F1 (z) and F2 (z).
tained from the optimized filters F1 (z) and F2 (z) either in the case without noise (thin curves) or approximately in the case with noise (dashed curves). In experiment, Fig.8 demonstrates the effectiveness of the MIMO FIR approach on the wafer stage systems of a real wafer scanner. For point-to-point motions similar to the motions used in simulation, it can be seen that the error signals in both x-, y-, and rz-direction after optimization (red curves) are close to the noise level. With nominal SISO feedforward control design (black curves) this is clearly not the case; the nominal SISO feedforward control design is based on an acceleration feedforward gain of m1 + m2 = 18 kg and a snap gain of k1 = 4 107 Nm−1 . In terms of frequency-domain characteristics Fig.9 shows that F1 (z) and F2 (z) with n = 9 approximate the measured plant with characteristics similar to the characteristics found in the simulation results of Fig.7.
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