Learning in the synthesis of data-driven variable-gain controllers Marcel Heertjes, Bram Hunnekens, Nathan van de Wouw, and Henk Nijmeijer Abstract— To deal with performance trade-offs in the control of motion systems, a method is developed for designing variable- gain feedback controllers. The idea is to select a piecewise affine controller structure and, subsequently, to find the nonlinear controller parameter values of this structure by data-driven per- formance optimization. Herein an H2 performance objective is minimized. As a result, variable-gain controllers are synthesized using techniques from the field of learning and optimization. The method is applied to a wafer stage simulation model. Index Terms— data-driven optimization, gradient methods, Lur’e systems, motion control, self-tuning, wafer scanners I. INTRODUCTION In the wafer scanning industry [4] variable-gain control is used to deal with performance trade-offs otherwise occurring under linear feedback; see also [9], [12], [13], [20]. The rationale is that low-frequency vibrations induced by the motion set-points of wafer scanners are better suppressed under high-gain feedback. Contrarily, high-frequency noise encountered during constant (scanning) velocity, thus in the absence of said set-points, becomes less amplified under low-gain feedback. Exploiting the property of continuously varying the controller gain, for example by using a deadzone nonlinearity in the controller structure [6], variable-gain control provides the means to suppress set-point induced vibrations under high-gain feedback in one part of the scan while keeping a low-gain noise response in another part. Tuning of the variable gain controller generally refers to frequency-domain loop shaping of the underlying linear sys- tem [6]. This is done by splitting up the nonlinear system into a linear system in feedback connection with a nonlinearity, i.e. adopt a Lur’e system formulation [21]. The linear system represents a closed-loop motion system, whose characteris- tics are the result of frequency-domain tunings [18]. On the one hand, these tunings should render the closed-loop system robustly stable, i.e. being able to deal with the effect of plant resonances (and the uncertainty thereabout) on the closed- loop stability properties. On the other hand, the tunings aim at improved closed-loop performance properties in view of the trade-off between low-frequency disturbance suppression and high-frequency noise amplification. Stability of the nonlinear closed-loop system can be guar- anteed by frequency-domain evaluation through the circle criterion [6]. Performance, however, very much depends on the choice of the variable gains, the plant characteristics, and the (unknown) disturbances acting on the system, which vary from machine to machine but also from field (wafer area during exposure) to field. It therefore makes sense to All authors are with the Department of Mechanical Engineering, Eind- hoven University of Technology, 5600 MB Eindhoven, The Nether- lands. M.F. Heertjes is also with ASML, Veldhoven, The Netherlands. marcel.heertjes@asml.com This work is financially supported by the Dutch Technology Foundation STW. assess (servo) performance per machine or field and in time domain, the latter in view of the nonlinearity in the loop. This paper explicitly deals with the performance-based design of the nonlinear part of the Lur’e system. Observing that an accurate model description of the plant and the disturbances (suited for performance evaluation) is often lacking, an early selection of a fixed nonlinear structure, e.g. deadzone or saturation [1], is not likely to induce best time-domain performances on individual machines during later machine qualifications. In [7], this problem was (only partly) tackled by self-tuning of the parameters of a deadzone nonlinearity, which was done per machine. With an itera- tive data-driven approach, see [2], [3], [5], [14] for other approaches, sampled data provided the information to find an updated set of parameters on the basis of least-squares optimization; for self-tuning in the nonlinear context, see also [8], [11], [16]. In this paper, the nonlinearity is given by piecewise affine functions having neither pre-defined gains nor switching lengths. Adopting the iterative approach from [7] gives a method for data-driven variable-gain controller synthesis in which the gains and switching lengths can be tuned per machine or even (in the learning sense) from field to field. The latter has strong similarities with learning approaches having varying learning gains and/or Q-filters [19]. In contrast with these approaches, however, the method presented here considers (nonlinear) filter design rather than signal design. Also, it classifies under feedback instead of under feedforward control. In a companion paper [10], a model-based approach is considered which is more suited for (large-scale) parameter studies conducted in the design phase when no machine measurements are yet available. De- pending on the specific disturbances and plant characteristics, the nonlinearity can become deadzone, saturation, or any other structure the piecewise affine function supports; this is different from [7] where a fixed deadzone structure is used. Dedicated optimization of machine performance is obtained with guaranteed (robust) stability properties. This is favorable for the motion industry in dealing with performance variation among machines, and for the wafer scanning industry in particular. The remainder of the paper is organized as follows. In Section 2, the Lur’e-type system description of the variable- gain motion control system is discussed. This includes the introduction of a three-parameter piecewise affine (variable- gain) function, a stability analysis using the circle criterion, and a lifted system description describing the closed-loop dynamics of a sampled-data implementation of the nonlinear controller. In Section 3, a data-driven H 2 optimization ap- proach is presented that is used to find the optimal parameters of the piecewise affine function. Section 4 addresses a gen- eralization of the optimization method toward more arbitrary 2013 American Control Conference (ACC) Washington, DC, USA, June 17-19, 2013 978-1-4799-0176-0/$31.00 ©2013 AACC 6700