The development of the high-tech industry has pushed the requirements of motion applications to extremes regarding precision, speed and robustness. A clear example is given by the wafer and reticle stages that require rigorous demands like robust nanometer precision and high-speed motion profiles to ensure product quality and production efficiency. Industrial workhorse Proportional Integral Derivative (PID) has been widely used for its simple implementation and good performance. However, PID is insufficient to meet the ever-increasing demands in the high-tech industry due to its inherent constraints of linear controllers such as the waterbed effect. To overcome these fundamental limitations, researchers have turned to nonlinear controllers. Nevertheless, most of the nonlinear controllers are difficult to design and implement and thus are not widely accepted in the industry. Reset control is a
nonlinear controller that is easy to implement and design since it maintains compatibility with the PID loop shaping technique using a pseudo-linear analysis tool named describing function method. However, the reset control as a nonlinear controller also introduces high order harmonics to the system that can negatively affect system performance by causing unwanted dynamics. Hence, describing function analysis as a linear approximation approach that only considers first harmonics is not accurate enough. Recently, a theory to analyze high order harmonics of nonlinear system in frequency domain termed higher order sinusoidal describing function has been developed, which enables the possibility to perform more precise analysis on reset systems.
The majority of research on reset control has focused on the phase lag reduction but a novel reset element proposed in literature termed ”Constant in gain, Lead in phase” (CgLp) is used to provide broadband phase compensation and has been shown to improve system performance. However, there is no systematic designing and tuning approach in literature such that the full advantage of CgLp elements is extracted. This work focuses on the tuning of the CgLp elements in order to obtain optimal performance. High order harmonics are also considered in the tuning analysis since they are critical to system performance due to the effect of unwanted dynamics. When a group of CgLp elements are designed to provide pre-determined phase compensation at the crossover frequency, it is seen that the optimal tracking precision performance is always obtained with the case that has the highest frequency of third order harmonic peak and has almost the smallest magnitude of high order harmonics at low frequencies. Moreover,
the second order CgLp controllers are observed to outperform the first order CgLp controller regarding tracking precision. On the other hand, configurations that have the lowest magnitude of third order harmonic at high frequency are found to have the best noise attenuation performance. ... Read more

The PID controller is one of the most used controllers in the industry. However, fundamental limitations due to linearity restrict its performances when higher bandwidth, stability, and precision are required simultaneously in today’s high-tech industry. Reset control is a promising nonlinear control strategy which can overcome these limitations. But it also brings new problems.
High order harmonics are introduced into the system because of non-linearity which lead to unwanted dynamics and deterioration of performances. So it is necessary to reduce them as much as possible. It is found that the sequence of different parts of a reset controller has effects on the magnitude of high order harmonics. Through high order sinusoidal input describing functions (HOSIDOFs) tool, the optimal sequence of the open loop in which the magnitude of high order harmonics is minimum is achieved for a general reset controller. The superiority of the suggested sequence in the closed-loop system is validated through both simulation and experiments at a Lorentz-actuated precision positioning stage. ... Read more