Iterative Learning Control for Waferstage Positioning based on Orthogonal Projection

Transcription

1 Proceedings of the 46th IEEE Conference on Decision and Control New Orleans, LA, USA, Dec , 27 WeA9.5 Iterative Learning Control for Waferstage Positioning based on Orthogonal Projection Sandipan Mishra and Masayoshi Tomizuka Abstract In this paper, design and analysis of Iterative Learning Control (ILC) based on partial information from previous cycles is developed. Typically, in a discrete-time repetitive process, ILC schemes use error from the entire previous cycle for updating the control input in the current cycle. Partial information of error from the previous cycle can be modeled as a projection onto a lower dimensional subspace. By appropriate choice of the subspace, noise and other nonrepetitive disturbances in the error can be suppressed, leading to cleaner learning signals. In particular, two aspects of the orthogonal projection-based ILC schemes are investigated. First, conditions on stabilityof theprojection-based ILC scheme are developed. Second, given a stable ILC scheme which uses the full error vector, the possible choices of projection subspaces is discussed. Implementation of the projection-based ILC scheme in precision positioning of a waferstage is presented. The major sources of repetitive error in precision tracking are phasemismatch and force ripple. These effects are mathematically modeled and the subspace spanned by them is obtained from initial experimentation. A standard P-type ILC scheme based on the proposed projection method is then used in control of a prototype one DOF waferstage to effectively reject the error caused by these disturbances, thereby verifying the effectiveness of the proposed projection ILC scheme. I. INTRODUCTION Iterative Learning Control (ILC) has been widely used in control of repetitive processes because of its simplicity of design, analysis and ease of implementation. ILC is loosely derived from the idea of learning, i.e. on improving performance by incorporating information from a previous run of the process. The first rigorous formulation of ILC was developed by Arimoto [1] and Uchiyama [2]. Arimoto [1] used ILC for control of robotic manipulators. Since then, ILC has found several applications in control of repetitive processes. In particular, it has been successfully implemented in industrial robotics [3], computer-numerical control tools [4], injection molding systems [5], and micro-scale robotic deposition [6]. Hand in hand with industrial implementation, many ILC design techniques have been developed that guarantee better performance. The P-type, PD-type ILC schemes developed by Arimoto were designed primarily based on stability constraints and were tuned for linear systems. Similar schemes This work was supported in part by Nikon Research Center of America, Belmont, and UC Discovery Grant ele S.Mishra is a PhD Candidate in Mechanical Engineering, University of California, Berkeley, CA 9472 USA. sandipan@me.berkeley.edu M.Tomizuka is with the Faculty of the Department of Mechanical Engineering, University of California, Berkeley, CA 9472 USA. tomizuka@me.berkeley.edu for non linear systems were also designed based on stability of the ILC loop [7]. For guaranteeing robustness and performance, design based on stability was found to be insufficient. Lee and Bien evaluated the robustness of the PD-type controller [8]. Filtering in the frequency domain has been used in many applications and design of Q-filters for learning has also been the focus of much research [9]. While a lot of research effort has been directed at enhancing performance based on improved learning algorithm design, there remain some open questions about utilization of data for most effective learning. In other words, it is important to consider not only how to learn efficiently, but also what information to learn. In many learning systems, although the learning signal may have very high complexity, most of the information that is needed for learning may be concentrated in only a few directions. Some ILC schemes based on projection of the control signal onto a subspace were developed by Hamamoto and Sugie [1]. The use of sine and cosine basis functions, legendre basis functions in learning control has been investigated already [11]. We intend to extend this to design and analysis of ILC systems with arbitrary basis functions, with application to precision positioning of wafer stages. The choice of basis functions is then decided by understanding the physical sources of the repetitive disturbances. II. ITERATIVE LEARNING CONTROL FOR WAFERSTAGE POSITIONING Linear permanent magnet motors (LPMM) are especially suited for positioning of wafer and reticle stages because of their high speed and accuracy. Since no gears are needed in linear motors, backlash effects are also avoided. However, one of the major issues in linear permanent magnet motors is force ripple. Force ripple is attributed to three major causes: (a) cogging, (b) self inductance variations and (c) commutation error due to coil misalignment. Cogging can be eliminated by using iron-less LPMMs. Self inductance effects are also fairly small in LPMMs. Commutation error caused by misalignment of coils results in a position dependant variation of the force constant of the motor. This variation is periodic with respect to position, the period being equal to the electrical pitch of the motor. Force ripple can be modeled simply as a sum of sine and cosine functions of position and input current [12]. Eq. 1 shows this model. x is the position, p is the electrical pitch of the motor, u is the current/voltage input and F r is the force /7/$ IEEE. 27

2 46th IEEE CDC, New Orleans, USA, Dec , 27 WeA9.5 Position (m) This paper is organized as follows. Section III introduces notation. Section IV introduces the standard and projection ILC problems. Section V proposes stability conditions for the projection ILC scheme, and addresses three key design issues associated with projection ILC. Section VI describes the implementation of projection ILC for precision positioning of wafer stages. Finally section VII outlines the conclusions and open problems that need to be tackled III. NOTATION ripple. F r = u l n= Fig. 1. Reference trajectory ( a n sin( 2πx p n)+b n cos( 2πx ) p n) In repetitive processes, ILC can be used to compensate for errors caused by force ripple [13]. Considering the repetitive nature of scanning, ILC has been used extensively for improving trajectory tracking and repetitive disturbance rejection in wafer scanners [14]. While ILC is extremely effective in rejection of disturbances that repeat in every cycle of the operation, its performance is significantly degraded with introduction of disturbances that can vary from iteration to iteration. Therefore, it is important to separate out the non-repetitive disturbances from the repetitive disturbances. In a process with both repetitive and non repetitive disturbances, one way to address this issue of efficient learning is by filtering out non repetitive disturbances. This approach involves filtering out those frequency components from the learning signal at which non repetitive disturbances dominate. However, frequency domain distinction between repetitive and non repetitive disturbances is not always effective, especially when there is significant temporal variation in the disturbance signal. For example, non repetitive disturbances may dominate over repetitive disturbances in a certain time segment of the repetitive process, as opposed to being dominant in a certain frequency region. In such scenarios, it is intuitive to incorporate this temporal variation of the disturbance into the ILC scheme, i.e., use a time-domain segmented ILC algorithm [15]. A typical (single) scanning operation requires a trajectory shown in Fig 1. This trajectory consists of two major phases (a) acceleration to a fixed velocity ν, (b) scanning at constant velocity ν. During the acceleration phase, we expect high frequency components in the reference trajectory, which may result in tracking error due to phase mismatch, especially close to the bandwidth. In the constant velocity scan phase force ripple and cable force effects can be expected to cause most of the error. In both the phases, we also expect to have structural vibrations and measurement noise to cause non repeating disturbances beyond the bandwidth. Keeping in mind these constraints, we will develop the set of suitable basis functions for a projection ILC scheme. (1) N = Period of the Repetitive Process F k ( j) = F(kN + j) r k = Lifted Reference Input Vector at the kth cycle r = y k = e k = d = n k = ρ(f) = σ(f) = I l = R(F) = Lifted Desired Trajectory Vector Lifted Plant Output Vector at the kth cycle Lifted Output Error Vector at the kth cycle Lifted Input Disturbance Vector Lifted Output Noise Vector at the kth cycle Spectral Radius of the Matrix F Largest Singular Value of the Matrix F Identity Matrix in R l l Range Space of the matrix F IV. PROBLEM FORMULATION Let us consider a stable discrete time linear time invariant single input single output system, denoted by G(z ), with a known delay n d = 1. This system executes a repetitive process with period of N samples. We want the output of the system to track a trajectory r( j), where j ranges from to N. This process is repeated, with the system coming back to rest condition at the end of each iteration of the cycle, and starting at rest condition at the beginning of each iteration. The output of the plant for each iteration is denoted by y k ( j), where j ranges from to N 1, and k denotes the iteration number. Therefore, we have the following relationship y k ( j) =G(z )(r k ( j)) (2) e k ( j) =r( j) y k ( j) (3) where r k ( j) is the input to the plant, and e k ( j) is the error from the desired trajectory r( j). Standard ILC Problem Formulation The standard ILC design problem can be formulated in lifted form by stacking all the signals into N 1 vectors [14]. Assuming zero initial conditions, we get the following lifted formulation of the ILC system. y k = T (r k + d+ n k ) (4) e k = r y k (5) r k+1 = r k + Le k (6) e k+1 = (I N TL)e k (7) 271

3 46th IEEE CDC, New Orleans, USA, Dec , 27 WeA9.5 where T is a toeplitz matrix composed of the markov parameters of the linear system G(z ) g()... g(1) g()... T =..... (8) g(n 1) g(n 2)... g() where g(i) is the i th term of the impulse response. The ILC scheme is stable if ρ (I N TL) < 1. Further, the ILC scheme is monotonically stable in the sense of the 2- norm in R N if σ (I N TL) < 1. This is a desirable property to have if we wish to avoid poor learning transients. Projection-based ILC Problem Formulation Consider a set of m orthonormal basis vectors b 1,b 2,...,b m in R N. Construct B = [ b 1 b 2... b m ] R N m V B =R(B) Note that, since the basis vectors are orthonormal, B B = I m. In projection ILC, the projection of the error e k onto the subspace V B (denoted by ē k ) is used in the learning update law. The learning system resulting from this is shown below (9) r k+1 = r k + Lē k (1) ē k = BB e k = Πe k (11) e k+1 = (I N TLΠ)e k (12) In the following section, we will develop stability conditions and performance bounds for the projection ILC problem. A. Stability Analysis V. PROJECTION-BASED ILC For stability of the projection ILC system, we propose the following lemma. Lemma 1: The ILC system described by Eqs.1, 11, 12 is (marginally) stable if ρ (I m B TLB) < 1. Proof: The ILC system described by Eqs.1, 11, 12 is marginally stable iff ρ (I N TLBB ) 1 (Considering the error evolution equation). First, we construct an orthonormal set of vectors b m+1,b m+2,...,b N which, together with the vectors in B, constitute an orthonormal basis for R N. Let B = [ b m+1 b m+2... b N ] R N (N m) U = [ B B ] R N N We see that B B =, U is a unitary matrix in R N ( UU = I N ), and spec{f} = spec{u FU}. U (I N TLBB )U =I N U T LBB U [ ] B =I N B T LBB [ ] B B [ ] B =I N B T LB [ I m ] [ Im B = ] TLB B TLB I N m So, spec{i N T LBB } = spec{u (I N TLBB )U} = spec{i m B TLB} 1. Hence, The ILC system described by Eqs.1, 11, 12 is (marginally) stable if ρ (I m B T LB) < 1. q.e.d. Corollary 1 : The projection ILC system is (marginally) stable if ρ (B (I N TL)B) < 1. This result relates the closed loop matrix of the original standard ILC scheme (I N TL) with the closed loop matrix of the projection ILC scheme. B. Projection ILC design Let us consider two specific design problems related to the projection ILC. Problem 1: How should the learning matrix (L R N N ) be designed so that, for any selection of basis vectors B, the projection ILC scheme remains stable, and converges to a steady state error e ss? This design problem may arise when the set of vectors B is not known apriori during the design process, or may vary from run to run. Sufficiency Condition : If L is designed such that σ (I N TL) < 1, then, for any set of orthonormal basis vectors B, the projection ILC system remains (marginally) stable. Further, the projection of e ss onto the subspace V B converges monotonically to zero. This is clear from the fact that σ (I N TL) < 1 σ (B (I N TL)B) < 1 ρ (B (I N TL)B) < 1. The above condition ensures that if we design our original ILC system such that it is monotonically stable, we can guarantee that the components of the error along the basis vectors b i, will converge monotonically to zero by using the projected error ē k in the learning scheme. Further, we are also guaranteed that the overall error remains bounded in the 2-norm sense, and e k+1 e k. Problem 2: Given an existing (stable) learning scheme/matrix L, what possible choices of sets of basis vectors B result in a stable projection ILC system? This design problem may arise when we wish to remove the nonrepetitive components of the learning signal by projection without changing the original learning scheme. Sufficiency Condition: If the basis vector set B is chosen such that R(B)(= V B ) is invariant with respect to the matrix (I N TL), then the stability of the original learning scheme is preserved under projection. As an example, let us pick m eigenvectors of the matrix (I N TL), and use a gram-schmidt decomposition to obtain the set B of orthonormal basis functions. If we pick any vector v R(B) = V B, we get (I N TL)v V B. Hence the set V B is invariant with respect to (I N TL). We claim that under this projection, ρ (B (I N TL)B) < 1. Further, it can be easily shown that the eigenvalues of (B (I N TL)B) are the eigenvalues of (I N TL) corresponding to the m eigenvectors that were chosen. It is interesting to note that if L and T are LTI systems, then lifted sine and cosine vectors form invariant subspaces. Filtering of the error in frequency domain therefore preserves stability of the ILC system. This idea is commonly used in many ILC design techniques. 272

4 46th IEEE CDC, New Orleans, USA, Dec , 27 WeA9.5 VI. PROJECTION ILC IMPLEMENTATION FOR WAFERSTAGE CONTROL 1.5 x 1 5 Iteration Fig. 4. Following Error for Iterations 3,6 and 9 with full cycle ILC using a 12Hz Filter Fig. 2. A. Experimental Setup Schematic of stage/countermass system. The experimental setup, shown in Figure 2, includes a stage and countermass system, both driven by linear motors. The stage can be modeled as a simple mass-damper system as below P(s) = 5.3s 2 (13) + 7.2s The reference trajectory is designed as shown in Figure 1. This trajectory replicates the movement of one of the axes of a waferstage during scanning. The goal is to achieve constant velocity as soon as possible to within a certain accuracy so that scanning can be performed. A feedback PID controller C(z ) was designed keeping in mind the bandwidth and sensitivity constraints. The sampling time of the controller was set to 4µs. Inertial feedforward was also used to improve the transient response of the overall system. B. Results: Comparison of ILC, Switching Filter ILC and Projection ILC In order to improve performance of the repetitive tracking process, an ILC loop was designed, based on a simple P-type learning algorithm. The learning gain α was picked so that x 1 5 Constant Velocity Constant Velocity Fig. 3. First Cycle Following Error the overall ILC loop was monotonically stable (α =.8). The error obtained from each iteration was filtered by a filter with cut off frequency at the bandwidth 12Hz, and then used in the learning scheme. The trajectory following error after 3, 6 and 9 cycles is shown in figure 4. We observed that following error convergence was rapid in the acceleration phase. However, in the constant velocity scan phase, tracking was significantly degraded. Figure 5 shows a section of the constant velocity scan phase illustrating this This performance degradation can be explained by the fact that the learning system picked up significant nonrepetitive disturbances like vibration during the constant velocity phase. In another experiment, the entire learning signal (e) was filtered by a a filter with bandwidth 4Hz in order to filter out non-repetitive disturbances. The results of following error from iterations 3, 6 and 9 are plotted in figure 6. During the constant velocity scan phase, shown in figure 7, the following error converged to under 1nm. However, during the acceleration phase, following error still remained big. This is because all the components of the learning signal beyond 4Hz were filtered out. Hence tracking could not be improved in frequencies larger than 4Hz. To strike a compromise between these two situations, a switching filter was used in a third set of experiments. During the initial acceleration phase, where higher frequencies are required in the learning signal for better tracking, the 12Hz filter was used. In the constant velocity scan phase, only low frequency disturbances like the force ripple need to be removed to reduce following error. So, a 4Hz filter was used during the constant velocity scan period. Figure 8 shows the plot of following error versus time for iterations 3, 6 and 9. A detail of the same set of experiments in figure 9 shows that the following error can be reduced significantly in both the acceleration and constant velocity phases by choosing different filters based on the structure of the repetitive and nonrepetitive disturbances. During the experiments above, it was noted that the complexity of the following error during the constant velocity scan was restricted to the frequencies of the force ripple, which were harmonics of a fundamental frequency at 2.71Hz. Therefore, during the constant velocity scan, a very 273

5 46th IEEE CDC, New Orleans, USA, Dec , 27 WeA x 1 7 Iteration Fig. 5. Following Error for Iterations 3,6 and 9 during Constant Velocity Scan Phase with 12Hz Filter. Fig x 1 5 Iteration Following Error for Iterations 3,6 and 9 with 4Hz Filter x 1 7 Iteration Fig. 9. Detail of Following Error for Iterations 3,6 and 9 with Switched Filtering. We can see that the peak error is under 3nm during the acceleration phase, and under 1nm during the constant velocity scan phase for the 9 th iteration x True Error Signal Approximated Error Signal x 1 6 Constant Velocity Scan Iteration Fig. 1. Detail of True and Approximated Following Error for the Constant Velocity Scan Phase. In the acceleration phase, the following error is not approximated. Fig Detail of Following Error for Iterations 3,6 and 9 with 4Hz Filter. 2 x 1 5 Iteration Fig. 8. Following Error for Iterations 3,6 and 9 with Switched Filtering. In the acceleration phase, a 12Hz filter is used to remove noise from the learning signal, while in the constant velocity scan phase, a 4Hz filter is used. good approximation of the error could be obtained by just computing the coefficients associated with each harmonic. By using only six harmonics, a DC offset and a linear term, an approximation of the following error signal was obtained. This is shown in figure 1. The advantage of this method is that the coefficients can be computed and stored much more conveniently. We need to only store 14 coefficients, instead of 3 points. Since monotonic convergence was guaranteed by design of the P-type ILC, we can use any set of basis functions for projection ILC. The simple P-type ILC scheme was then implemented using such a projected approximation of the following error as the learning signal ē k (instead of the error e k ). Figure 11 shows the detail of the following error for the scanning phase for iterations 3,6 and 9. We can observe that the following error is comparable to the case when the entire segment was used in learning (Figure 9). However, there is some degradation of performance, especially at the beginning and end of the scan phase. This can be attributed to the fact that we can no longer learn all frequencies during the constant veocity scan phase. 274

6 46th IEEE CDC, New Orleans, USA, Dec , 27 WeA x 1 7 Iteration Fig. 11. Following Error for the Constant Velocity Scan Phase, using a Projection Approximation of the Learning Signal. At the beginning and the end of the constant scan phase, there is degradation of performance. VII. CONCLUSION Wafer scanning requires ultra-high precision positioning capabilities. In addition to smart design techniques, advanced control schemes are important for achieving the stringent performance standards. Iterative Learning Control improves performance of the waferstage from scan to scan, since the process is repetitive. Typically, in a discrete-time repetitive process, ILC schemes use error from the entire previous cycle for updating the control input in the current cycle. In this paper, we proposed a projection based ILC scheme to reduce implementation complexity and memory requirements for ILC. The projection based ILC scheme uses the projection of the full cycle lifted error vector onto a smaller dimensional subspace as the learning signal. The stability and convergence properties of the proposed scheme were investigated. Conditions on the possible choice of basis vectors for the lower dimensional approximation space were obtained. In this context, connections with filtering, time varying filters and segmentation in existing ILC schemes were also made. The ideas developed for projection ILC were implemented on a prototype single degree of freedom waferstage. First, a standard P-type ILC scheme was used to improve tracking performance of the waferstage. Next, a time-varying filter with different cut off frequencies during acceleration and constant velocity phase was then used to filter the learning signal. Significant tracking error reduction was obtained by using this learning signal instead of the single frequency filtered error. It was further noted that the complexity of the learning signal was limited to only a few harmonics in the constant velocity phase. These harmonics were caused by the force ripple. This motivated the idea of using a projection based ILC scheme. The structure for the basis functions of the projection subspace was obtained by understanding the sources of repetitive error. Two major sources of error were identified, phase mismatch and force ripples. Phase mismatch was the major source of tracking error in the acceleration phase and force ripples caused tracking error in the constant velocity phase. Based on this, the corresponding basis functions were constructed. An approximation of the error signal in the subspace spanned by these basis functions was obtained and used as the learning signal in the P- type learning algorithm. The projection ILC scheme s performance was then compared to the switched filter P-type ILC scheme, and it was observed that performance degradation was minimal. There remain several open questions about orthogonal projection based ILC. Robustness and steady state analysis must be done to understand the effects of model uncertainty and basis selection on performance and convergence. Design of ILC schemes in lower dimensional spaces, specifically tuned towards projection ILC schemes is another interesting direction for future research effort. REFERENCES [1] S. Arimoto, S. Kawamura, and F. Miyazaki, Bettering operation of robots by learning, J. of Robotic Systems, vol. 1, no. 2, pp , [2] M. Uchiyama, Formulation of high-speed motion pattern of a mechanical arm by trial, Trans. SICE (Soc. Instrum. Contr. Eng.), vol. 14, no. 6, pp (in Japanese), [3] K. Moore, M. Dahleh, and S. Bhattacharyya, Learning control for robotics, in Proceedings of 1988 International Conference on Communications and Control, Baton Rouge, Louisiana, October 1988, pp [4] D.-I. Kim and S. 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