Stage Motion Control: Nonlinear Integrators Revisited

Linear integrators are well known for their ability to counter static forces and improve low-frequency disturbance rejection properties in control systems. However, linear integrators introduce phase lag, which is a frequency-dependent time shift or delay. Since the early introduction of the Clegg integrator, nonlinear integrators have held the promise of providing phase advantages over linear integrators when evaluated from the perspective of a describing function. This could potentially reduce delay and therefore provide the means to surpass linear design limitations; for example, overshoot and settling times can be reduced or even avoided. In addition, loop gains can be increased, which improves the low-frequency disturbance rejection properties. For five nonlinear integrators—the Clegg integrator, a generalized first-order reset integrator called the constant-in-gain–lead-in-phase (CgLp) element, a hybrid integrator–gain system, a variable gain integrator, and a split-path integrator—this article provides a comparison and overview of recent developments in stage motion control. Benchmark examples are taken from the industrial practice of wafer scanners, which form the pivotal machines used in the manufacturing of computer chips.