Nominal and Robust Feedforward Design With Time Domain Constraints Applied to a Wafer Stage

In this paper, we studied the Position&Force servo control of the voice coil motor commonly used in semiconductor packaging equipment and analyzed its Position&Force servo control switching strategy. System identification is the basis for high-speed, high-precision motion control. We use the frequency domain identification method to identify the voice coil motor system. The frequency domain identification method uses sine sweep frequency and PRBS method to compare and verify. Feedback and feedforward servo control structures are used to ensure quick response and precise control of voice coil motors while ensuring system stability. In the feedforward control, we use an iterative learning control method. In semiconductor packaging equipment, the movement of the voice coil motor has always been a similar process cycle. Iterative learning control (ILC) has prominent control effects in situations where control objectives and external disturbances are periodic or repetitive. The iterative learning control algorithm is applied to the feedforward plus feedback servo control structure of the voice coil motor. From the actual effect point of view, the trajectory tracking ability of the voice coil motor during high-speed motion is significantly increased. The voice coil motor servo control algorithm described above is written on the DSP and verified on the motion control card. The experimental results show that the above servo control algorithm realizes the high speed and high precision motion control of the voice coil motor.

In this paper, the feedforward controller design problem for high-precision electromechanical servo systems that execute finite time tasks is addressed. The presented procedure combines the selection of the fixed structure of the feedforward controller and the optimization of the controller parameters by iterative trials. A linear parametrization of the feedforward controller in a two-degree-of-freedom control architecture is chosen, which results in a feedforward controller that is applicable to a class of motion profiles as well as in a convex optimization problem, with the objective function being a quadratic function of the tracking error. Optimization by iterative trials avoids the need for detailed knowledge of the plant, achieves the controller parameter values that are optimal with respect to the actual plant, and allows for the adaptation to possible variations that occur in the plant dynamics. Experimental results on a high-precision wafer stage and a desktop printer illustrate the procedure.

This paper presents a method for designing a controller that achieves the best positioning performance while maximizing the energy efficiency of a redundantly actuated hybrid feed drive. A two-degree-of-freedom controller, consisting of a feedforward (FF) controller for tracking and a feedback (FB) controller for regulation, is assumed. It is shown that the ideal FF controller, which achieves perfect tracking with maximum efficiency, is not always stable. Therefore, a method for designing a stable FF controller that achieves perfect tracking with near optimal efficiency is proposed. Furthermore, the optimal relationship between FB control inputs, which guarantees maximum efficiency for any specified regulation performance, is derived. Two approaches for using the derived optimal relationship to synthesize a FB controller that achieves the best positioning performance while maximizing efficiency are proposed. Simulations and machining experiments are conducted to demonstrate the effectiveness of the proposed energy-efficient FF and FB controller design methods. Significant improvements in energy efficiency (without sacrificing positioning performance) are reported.

Many manufacturing machines must execute motions as quickly as possible to achieve profitable high-volume production. Most of them exhibit some flexibility, which makes the settling time longer and controller design difficult. This paper develops a control strategy that combines feedforward and feedback control with command shaping for systems with collocated actuator and sensor. First, a feedback controller is designed to increase damping and eliminate steady-state error. Next, an appropriate reference profile is generated using command-shaping techniques to ensure fast point-to-point motions with minimum residual vibration. Finally, a feedforward controller is designed to speed up the transient response. The proposed proportional-integral-derivative (PID) controller design ensures that two important resonant frequencies nearly match, making the design of the input commands much simpler. The resulting control strategy is successfully demonstrated for a generic dimensionless system that incorporates some modeling errors to assess robustness.

The various robust linear programming models investigated so far in the literature essentially appear to be based either on what is referred to as rowwise uncertainty models or on columnwise uncertainty models (these basically assume that the rows, or the columns, of the constraint matrix are subject to changes within a well-specified uncertainty set). In this chapter, we discuss a special case of columnwise uncertainty, namely the subclass of robust linear programming (LP) models with uncertainty limited to the right hand side (RHS) only (this subclass does not appear to have been significantly investigated so far). In this context we introduce the concept of a ‘two-stage robust LP model’ as opposed to the standard case (which might be referred to as a ‘single-stage robust LP model’) and we address the question of whether LP duality can be used to convert a LP problem with RHS uncertainty into a robust LP problem with uncertainty on the objective function. We show how to derive both statements of (a) the dual to the robust model and (b) the robust version of the dual. The resulting expressions of the objective function to be optimized in both cases, appear to be clearly distinct. Moreover, from a complexity point of view, one appears to be efficiently solvable (it reduces to a convex optimization problem), whereas the other, as a nonconvex optimization problem, is expected to be computationally difficult in the general case. As an application of the two-stage robust LP model introduced here, we next investigate the PERT (program evaluation and review technique) scheduling problem, considering two possible natural ways of specifying the uncertainty set for the task durations: the case where the uncertainty set is a scaled ball with respect to the L ∞ norm; the case where the uncertainty set is a scaled Hamming ball of bounded radius (which, though leading to a quite different model, bears some resemblance to the well-known Bertsimas–Sim approach to robustness). We show that in both cases the resulting robust optimization problem can be efficiently solved in polynomial time.

A mechanically simple, high performance wafer stage is discussed in this paper, including detailed descriptions of the mechanical construction, laser gauging system, electrical configuration and servo control. This stage provides for simultaneous X, Y, and theta motion with an inherently rigid, monolithic platform propelled by Sawyer linear motors. The measurement of position, for closed loop control, is provided by four axes of interferometric laser gauging. Position control is accomplished with a power-efficient, robust, digital servo system. The enhanced wafer positioning precision capability of this stage offers accurate alignment in enhanced global alignment mode (EGA). Due to its low mass (weighing less than 18 lb.) the stage delivers optimal acceleration, velocity, and settling. The stage has been incorporated in a 1:1 photolithographic wafer stepper. Its simplistic design incorporates a minimum number of moving parts and provides high reliability and low maintenance. This stage offers significant productivity advantages as demonstrated by the throughput specifications and the stage settling data reported in this paper. Overlay data using this system operating in EGA mode is also included.

This article presents a data-driven algorithm that combines the advantages of iterative feedforward tuning and disturbance rejection control to satisfy the precision requirements and ensure extrapolation capability of wafer scanning. The proposed algorithm differs from pre-existing algorithms in terms of its low requirement of system model, high extrapolation capability for non repetitive trajectory tracking tasks, and high tracking precision. The feedforward controller is tuned based on instrumental variables. It utilizes tracking errors from past iterations to eliminate reference-induced errors without requiring a system model. Meanwhile, the system inverse is approximated during iterative process, and then a disturbance rejection control based on iterative tuning is constructed to compensate for disturbance-induced errors. The proposed algorithm is applied to a wafer stage. The experimental results validate the effectiveness and superiority of the proposed algorithm.

We consider the problem of designing piecewise affine policies for two-stage adjustable robust linear optimization problems under right-hand side uncertainty. It is well known that a piecewise affine policy is optimal although the number of pieces can be exponentially large. A significant challenge in designing a practical piecewise affine policy is constructing good pieces of the uncertainty set. Here we address this challenge by introducing a new framework in which the uncertainty set is “approximated” by a “dominating” simplex. The corresponding policy is then based on a mapping from the uncertainty set to the simplex. Although our piecewise affine policy has exponentially many pieces, it can be computed efficiently by solving a compact linear program given the dominating simplex. Furthermore, we can find the dominating simplex in a closed form if the uncertainty set satisfies some symmetries and can be computed using a MIP in general. We would like to remark that our policy is an approximate piecewise-affine policy and is not necessarily a generalization of the class of affine policies. Nevertheless, the performance of our policy is significantly better than the affine policy for many important uncertainty sets, such as ellipsoids and norm-balls, both theoretically and numerically. For instance, for hypersphere uncertainty set, our piecewise affine policy can be computed by an LP and gives a $$O(m^{1/4})$$ -approximation whereas the affine policy requires us to solve a second order cone program and has a worst-case performance bound of $$O(\sqrt{m})$$ .

Permanent magnet synchronous motors have the advantages of high power density and high efficiency. The development of permanent magnet synchronous motor position servo control strategy research has an important technical support for improving my country’s basic industry. Taking the surface-mounted permanent magnet synchronous motor as the object, the id=0 control strategy is selected, and the three closed-loop control structure of position, speed and current is adopted to control the system. Among them, the speed loop and current loop adopt proportional integral controller. For the position controller, a comparative analysis of two different controllers, proportional control and feedforward control, is creatively adopted. The two control systems of the position controller are simulated and verified in the MATLAB/Simulink environment. Given a control signal, the simulation shows that the system using the feedforward controller is optimized in terms of steady-state position tracking accuracy, position overshoot, and tracking speed compared to the proportional controller.

The robust counterpart of a given conic linear program with uncertain data in the constraints is defined as the robust conic linear program that arises from replacing the nominal feasible set by the robust feasible set of points that remain feasible for any possible perturbation of the data within an uncertainty set. Any minor changes in the size of the uncertainty set can result in significant changes, for instance, in the robust feasible set, robust optimal value and the robust optimal set. The concept of quantifying the extent of these deviations is referred to as the stability of robustness. This paper establishes conditions for the stability of robustness under which minor changes in the size of the uncertainty sets lead to only minor changes in the robust feasible set of a given linear program with cone constraints and ball uncertainty sets.

In many situations, decision-makers need to exceed a random target or make decisions using expected utilities. These two situations are equivalent when a decision-maker’s utility function is increasing and bounded. This article focuses on the problem where the random target has a concave cumulative distribution function (cdf) or a risk-averse decision-maker’s utility is concave (alternatively, the probability density function (pdf) of the random target or the decision-maker’ marginal utility is decreasing) and the concave cdf or utility can only be specified by an uncertainty set. Specifically, a robust (maximin) framework is studied to facilitate decision making in such situations. Functional bounds on the random target’s cdf and pdf are used. Additional general auxiliary requirements may also be used to describe the uncertainty set. It is shown that a discretized version of the problem may be formulated as a linear program. A result showing the convergence of discretized models for uncertainty sets specified using continuous functions is also proved. A portfolio investment decision problem is used to illustrate the construction and usefulness of the proposed decision-making framework.

Iterative feedforward tuning (IFFT) enables high performance for motion systems that perform varying tasks without the need for system models. In this paper, IFFT is employed for a wafer stage to achieve good trajectory tracking performance and excellent disturbance compensation ability. Recently, the instrumental variable (IV) approach has been introduced into IFFT algorithms (IV-IFFT), enabling unbiased estimates for the parameters of a feedforward controller in the presence of stochastic noise. However, the estimation variances achievable with IV-IFFT are larger than zero. The aim of this paper is to develop an IFFT algorithm that enables unbiased estimates with zero asymptotic variances, which can be achieved by the simultaneous use of the Kalman filtering (KF) approach and the IV approach in IFFT, yielding the KF-IV-IFFT algorithm. The different roles of KF and IV approaches to improve the noise-tolerant capability of IFFT are also revealed. Experimental results obtained on a wafer stage confirm the practical relevance of the proposed KF-IV-IFFT algorithm.

Mechanical resonance is one of the most pervasive problems in servo control. Closed-loop simulations are requisite when the servo control system with high accuracy is designed. The mathematical model of resonance mode must be considered when the closed-loop simulations of servo systems are done. There will be a big difference between the simulation results and the real actualities of servo systems when the resonance mode is not considered in simulations. Firstly, the mathematical model of resonance mode is introduced in this paper. This model can be perceived as a product of a differentiation element and an oscillating element. Secondly, the second-order differentiation element is proposed to simulate the resonant part and the oscillating element is proposed to simulate the antiresonant part. Thirdly, the simulation approach for two resonance modes in servo systems is proposed. Similarly, this approach can be extended to the simulation of three or even more resonances in servo systems. Finally, two numerical simulation examples are given.

Electro-hydraulic servo systems have been frequently used in the fin position servo system of a missile because of their high power and good positioning capability. The objective of this paper is to realize a self-tuning fuzzy controller for the position control of an electro-hydraulic fin servo system. In this paper, we propose a new self-tuning fuzzy controller, which can improve the overall performance of the electro-hydraulic fin position servo system. The uniqueness of this self-tuning fuzzy controller is that the performance evaluation of the fin position servo system is carried out in a phase plane mapped to a decision rule table. The effectiveness of this control scheme is verified by comparison with a fixed fuzzy control with initial rules through a series of simulation studies.

Servo system modeling and reduction of mechatronic system through finite element analysis for control design