In this work, the proportional Clegg integrator (PCI), a resetting proportional-integrator (PI) element, is studied with the aim of improving the performance of an industrial motion stage currently controlled by a linear controller. A novel parallel continuous reset (CR) architecture, based on the PI, is presented, along with frequency-based tuning guidelines, similar to linear time-invariant (LTI) loopshaping techniques. Open-loop higher order sinusoidal input describing functions (HOSIDFs) and pseudo-sensitivities computed through analytically derived approximate closed-loop HOSIDFs were effectively applied to predict steady-state performance. The experimental results, obtained on a wire bonding machine, confirmed that resonance-induced vibrations of the machine's base frame can be suppressed more effectively by adopting a PCI-PID controller compared to the currently used linear controller. The novel structure does not only reduce unwanted excitation of higher order harmonics of the base frame resonance, such as the series CR architecture recently introduced in literature, but also avoids amplification of noise when implemented in practice. With the novel parallel structure, a significant (32%) decrease in the root mean square (rms) of the settling error could be achieved when compared to the linear controller currently used and the series CR reset structure.

When a linear controller is replaced by a reset controller, it is possible to keep the gain behaviour essentially the same, while improving the phase behaviour. However, because reset control is nonlinear, higher order harmonics appear, which may deteriorate the results. In this paper, reset controllers are combined with fractional derivatives, decreasing the magnitude of higher order harmonics. In this way, better results are found in simulations of a Clegg integrator, of a FORE (first order reset element), and of a CgLp (constant in gain lead in phase) controller. The reason why better results are found is explained.

This article addresses nonlinearity in reset elements and its effects. Reset elements are known for having less phase lag based on describing function (DF) analysis compared to their linear counterparts; however, they are nonlinear elements and produce higher-order harmonics. This article investigates the steady-state higher-order harmonics for reset elements with one resetting state and proposes an architecture and a method of design that allows for band-passing the nonlinearity and its effects, namely, higher-order harmonics and phase advantage. The nonlinearity of reset elements is not entirely useful for all frequencies, for example, they are useful for reducing phase lag at crossover frequency regions; however, higher-order harmonics can compromise tracking and disturbance rejection performance at lower frequencies. Using the proposed “phase shaping” method, one can selectively suppress the nonlinearity of a single-state reset element in a desired range of frequencies and allow the nonlinearity to provide its phase benefit in a different desired range of frequencies. This can be especially useful for the reset elements in the framework of the “constant in gain, lead in phase” (CgLp) filter, which is a newly introduced nonlinear filter, bound to circumvent the well-known linear control limitation—the waterbed effect.

This article introduces a closed-loop frequency analysis tool for reset control systems. To begin with sufficient conditions for the existence of the steady-state response for a closed-loop system with a reset element and driven by periodic references are provided. It is then shown that, under specific conditions, such a steady-state response for periodic inputs is periodic with the same period as the input. Furthermore, a framework to obtain the steady-state response and to define a notion of closed-loop frequency response, including high order harmonics, is presented. Finally, pseudosensitivities for reset control systems are defined. These simplify the analysis of this class of systems and allow a direct software implementation of the analysis tool. This methods gives deeper insight into the performance of the system than that achieved with the describing function method.

This dissertation addresses the demand for faster, more precise, and robust controllers in the precision motion industry. Traditional linear controllers have limitations due to the waterbed effect and Bode’s phase-gain relationship. To overcome these limitations, complex-order controllers are explored in this study. The dissertation focuses on shaping nonlinearities in reset controllers to realize complex-order behavior. Various methods and approaches are investigated, each contributing to the understanding and improvement of reset control systems for linear time-invariant systems. The dissertation demonstrates that reset controllers exhibit first-order harmonic behavior, which can be advantageous in achieving complex-order behavior and enhancing controller performance compared to linear controllers. However, higher-order harmonics resulting from nonlinearities play a significant role and should not be neglected. The study explores methods to shape and manipulate these higher-order harmonics for improved performance. Different approaches are categorized into two main categories: methods based on shaping the reset phase (ψ) and continuous reset (CR) methods. In the first category, ψ-shaping methods focus on manipulating ψ to achieve desirable non-linear behaviors. This includes the introduction of elements such as fractional-order lag elements and filters to shape ψ and suppress higher-order harmonics. The dissertation presents practical frameworks for analyzing and utilizing ψ-shaping concepts. The second category, continuous reset methods, addresses both transient and steadystate performance. By introducing lead and lag elements as pre- and post-filters for reset elements, improvements in transient response are achieved. Additionally, CR methods preserve the first-order harmonic behavior while reducing higher-order harmonics across the entire frequency range. The dissertation highlights the advantages and tradeoffs between ψ-shaping and CR methods, providing insights for selecting the appropriate approach based on application requirements. Practical implementation aspects are also considered throughout the dissertation. Challenges such as noise amplification caused by lead elements in CR architectures are addressed, offering solutions through increased filter orders or observer-based filtering techniques. The dissertation demonstrates the effectiveness of the proposed approaches through implementation in industrial precision motion stages, showcasing the superiority of complex-order reset controllers over their linear counterparts. Overall, this dissertation contributes to the understanding and practical implementation of reset controllers for realizing complex-order behavior in precision motion control. It provides insights into shaping nonlinearities, optimizing steady-state and transient performance, and selecting suitable architectures based on specific application needs. The findings and guidelines presented in this study offer valuable contributions to the precision motion industry and pave the way for further advancements in controller design and performance.

According to the well-known loop shaping method for the design of controllers, the performance of the controllers in terms of step response, steady-state disturbance rejection and noise attenuation and robustness can be improved by increasing the gain at lower frequencies and decreasing it at higher frequencies and increasing the phase margin as much as possible. However, the inherent properties of linear controllers, the Bode’s phase-gain relation, create a limitation. In theory, a complex-order transfer function can break the Bode’s gain-phase relation; however, such transfer function cannot be directly implemented and should be approximated. This paper proposes a reset element and a tuning method to approximate a Complex-Order Controller (CLOC) and, through a simulation example, shows the benefits of using such a controller.

According to the well-known loop-shaping control design approach, the steady-state precision of control systems can be improved by stacking integrators. However, due to the waterbed effect in linear control systems, such an action will worsen the transient response by increasing overshoot and creating wind-up problems. This paper presents a new architecture for rest control systems that can significantly decrease the overshoot and create a no-overshoot performance even in the presence of stacked integrators. The steady-state analysis of the proposed system will also show that improved precision expected due to stacked integrators can be achieved as well. A numerical simulation study is presented to verify the results, and the tuning guide is presented.

This paper addresses the main goal of using reset control in precision motion control systems, breaking of the well-known “Waterbed effect”. A new architecture for reset elements will be introduced which has a continuous output signal as opposed to conventional reset elements. A steady-state precision study is presented, showing the steady-state precision is preserved while the peak of sensitivity is reduced. The architecture is then used for a “Constant in Gain Lead in Phase” (CgLp) element and a numerical analysis on transient response shows a significant improvement in transient response. It is shown that by following the presented guideline for tuning, settling time can be reduced and at the same time a non-overshoot step response can be achieved. A practical example is presented to verify the results and also to show that the proposed element can achieve a complex-order behaviour.

This paper proposes a fractional-order reset element whose architecture allows for the suppression of nonlinear effects for a range of frequencies. Suppressing the nonlinear effects of a reset element for the desired frequency range while maintaining it for the rest is beneficial, especially when it is used in the framework of a “Constant in gain, Lead in phase” (CgLp) filter. CgLp is a newly introduced nonlinear filter, bound to circumvent the well-known linear control limitation—the waterbed effect. The ideal behaviour of such a filter in the frequency domain is unity gain while providing a phase lead for a broad range of frequencies. However, CgLp’s ideal behaviour is based on the describing function, which is a first-order approximation that neglects the effects of the higher-order harmonics in the output of the filter. Although CgLp is fundamentally a nonlinear filter, its nonlinearity is not required for all frequencies. Thus, it is shown in this paper that using the proposed reset element architecture, CgLp gets closer to its ideal behaviour for a range of frequencies, and its performance will be improved accordingly.

Linear control such as PID possesses fundamental limitations, seen through the Waterbed effect. Reset control has been found to be able to overcome these limitations, while still maintaining the simplicity and ease of use of PID control due to its compatibility with the loop shaping method. However, the resetting action also gives rise to higher order harmonics that hinders consistent realization of the aforementioned expected improvement. In this paper, a fractional-order augmented state analogue of the reset integrator is investigated. This analogue is composed of a series of augmented states that each possesses unique reset values, providing the same first order harmonic behavior but reduced higher order harmonics magnitude compared to the reset integrator. The optimal number of augmented states along with the corresponding tuning values are investigated. To validate the improvement, the reset integrator and the optimal fractional order analogue are tuned to equally improve disturbance rejection of a high precision 1 DOF positioning stage while maintaining the stability level, with both designed to overcome linear control. From simulation and experimental results, it was found that the novel fractional-order augmented state analogue gives rise to disturbance rejection performance that is closer to the desired and expected improvement, compared to using the traditional reset integrator.

This paper presents the tuning of a reset-based element called “Constant in gain and Lead in phase” (CgLp) in order to achieve desired precision performance in tracking and steady state. CgLp has been recently introduced to overcome the inherent linear control limitation - the waterbed effect. The analysis of reset controllers including ones based on CgLp is mainly carried out in the frequency domain using describing function with the assumption that the relatively large magnitude of the first harmonic provides a good approximation. While this is true for several cases, the existence of higher-order harmonics in the output of these elements complicates their analysis and tuning in the control design process for high precision motion applications, where they cannot be neglected. While some numerical observation-based approaches have been considered in literature for the tuning of CgLp elements, a systematic approach based on the analysis of higher-order harmonics is found to be lacking. This paper analyzes the CgLp behaviour from the perspective of first as well as higher-order harmonics and presents simple relations between the tuning parameters and the gain-phase behaviour of all the harmonics, which can be used for better tuning of these elements. The presented relations are used for tuning a controller for a high-precision positioning stage and results used for validation.

This paper addresses a phenomenon caused by resetting only one of the two states of a so-called second order 'Constant in gain Lead in phase' (CgLp) element. CgLp is a recently introduced reset-based nonlinear element, bound to circumvent the well-known linear control limitation - the waterbed effect. The ideal behaviour of such a filter in the frequency domain is unity gain while providing a phase lead for a broad range of frequencies, which clearly violates the linear Bode's gain phase relationship. However, CgLp's ideal behaviour is based on a describing function, which is a first order approximation that neglects the effects of higher order harmonics in the output of the filter. Consequently, achieving the ideal behaviour is challenging when higher order harmonics are relatively large. It is shown in this paper that by resetting only one of the two states of a second order CgLp, the nonlinear filter will act as a linear one at a certain frequency, provided that some conditions are met. This phenomenon can be used to the benefit of reducing higher order harmonics of CgLp's output and achieving the ideal behaviour and thus better performance in terms of precision.

A controller with the frequency response of a complex order derivative may have a gain that decreases with frequency, while the phase increases. This behaviour may be desirable to ensure simultaneous rejection of high-frequency noise and robustness to variations of the open-loop gain. Implementations of such complex order controllers found in the literature are unsatisfactory for several reasons: the desired behaviour of the gain may be difficult or impossible to obtain, or non-minimum phase zeros may appear, or even unstable open-loop poles. We propose an alternative nonlinear approximation, combining a CRONE approximation of a fractional derivative with reset control, which does not suffer from these problems. An experimental proof of concept confirms the good results of this approximation and shows that nonlinear effects do not preclude the desired performance.