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.@en

Reset controllers have the potential to enhance the performance of high-precision industrial motion systems. However, similar to other non-linear controllers, the stability analysis for these controllers is complex and often requires parametric model of the system, which may hinder their applicability. In this paper a frequency-domain approach for assessing stability properties of control systems with first and second order reset elements is developed. The proposed approach is also able to determine uniformly bounded-input bounded-state (UBIBS) property for reset control systems in the case of resetting to non-zero values. An illustrative example to demonstrate the effectiveness of the proposed approach in using frequency response measurements to assess stability properties of reset control systems is presented.

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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.

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Currently, the demand for a better alternative to linear PID controllers is increasing due to the rising expectations of the high-tech industry. In literature, it has been shown that Constant in gain Lead in phase (CgLp) compensators, which are a type of reset element, have high potential to improve the performance of systems. Although there are few works which investigate tuning of these compensators, the high order harmonics and steady-state performances have not yet been considered in these methods. Recently, a frequency-domain framework has been developed to analyze closed-loop performances of reset control systems which includes high order harmonics. In this paper, this frequency-domain framework is combined with loop-shaping constraints to provide a reliable frequency-domain tuning method for CgLp compensators. Finally, different performance metrics of a CgLp compensator are compared with those of a PID controller on a precision positioning stage. The results show that the presented tuning method is effective, and the system with the CgLp compensator achieves superior dynamic performance to that of the PID controller.@en

Constant in gain Lead in phase (CgLp) compensators, which are a type of reset elements, have shown high potential to overcome limitations of linear control systems. There are few works which investigate the tuning of these compensators. However, there are some significant drawbacks which make those methods unreliable. First, their analyses are performed in the open-loop configuration which do not guarantee the existence of steady-state response of the closed-loop. If it is guaranteed, unlike linear control systems, open-loop analyses cannot precisely predict the closed-loop steady-state performance. In addition, the stability condition could not be assessed during the tuning process. These significant challenges have been separately solved in our recent works by proposing frequency-domain frameworks for analyzing the closed-loop performance and stability of reset control systems. However, they are not formulated and implemented for tuning CgLp compensators. In this paper, based on the loop-shaping approach, the recent frequency-domain framework and the frequency-domain stability method are utilized to provide a reliable frequency-domain tuning method for CgLp compensators. Finally, different performance metrics of a CgLp compensator, tuned by the proposed method, are compared with those of a PID controller on a precision positioning stage. The results show that this method is effective, and the tuned CgLp can achieve more favorable dynamic performance than the PID controller.

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Proportional Integral Derivative (PID) controllers dominate the industry and are used in more than 90 percent of machines in this era. One of the reason for the popularity of these controllers is the existence of easy to use frequency-domain analysis tools such as loop-shaping for this type of controller. Due to the advancement of technology in recent decades, industry needs machines with higher speed and precision. Thus, an advanced industry-compatible control capable of a simultaneous increase in precision and speed is needed. Unfortunately, linear controllers, including integer and fractional order controllers, cannot satisfy this requirement of industry because of fundamental limitations such as the “water-bed" effect. In other words, precision and speed are conflicting demands in linear controllers, and designers should consider a proper trade-off between them when they tune these controllers. The reset control strategy which is one of the well-known non-linear controllers, has shown a great capacity to overcome the limitation of linear controllers. In our group, a newtype of reset compensator, which is termed “Constant in gain Lead in phase (CgLp)”, has been proposed as a potential solution for this significant challenge. Considering the first harmonic of the steady-state response of the CgLp compensator, which is called the Describing Function (DF) analysis, this compensator has a constant gain while providing a phase lead. As a result, this novel compensator can improve the precision of the control system, while simultaneously maintaining the high quality level of transient response (throughput of the system). As mentioned before, industry favours designing controllers in the frequency-domain because it provides an easy to use tool for performance analysis of control systems. Therefore, in order to interface this compensatorwell with the current control design in industry and broaden its applicability, it is important to study this type of reset compensator in the frequency-domain. So far, CgLp compensators have been studied in the frequency-domain using the DF method. However, there are some major drawbacks which have to be solved in order to make these compensators ready for industry utilization. Essentially, there is a lack of knowledge about the closed-loop steady-state performance of reset control systems. In addition, since the high order harmonics generated by CgLp compensators are neglected in the DF method, this method by itself is not an appropriate method for predicting open-loop and closed-loop steady-state performance, particularly for precision motion applications. Second, it is necessary to develop an intuitive frequency-domain stability method to assess the stability of CgLp compensators, similar to the Nyquist plot for linear controllers. Finally, to achieve a favourable dynamic performance, it is highly needed to propose a systematic frequency-domain tuning method for this type of reset compensators. @en

Due to development of technology, linear controllers cannot satisfy requirements of high-tech industry. One solution is using nonlinear controllers such as reset elements to overcome this big barrier. In literature, the Constant in gain Lead in phase (CgLp) compensator is a novel reset element developed to overcome the inherent linear controller limitations. However, a tuning guideline for these controllers has not been proposed so far. In this paper, a recently developed method named higher-order sinusoidal input describing function (HOSIDF), which gives deeper insight into the frequency behaviour of non-linear controllers compared to sinusoidal input describing function (DF), is used to obtain a straight-forward tuning method for CgLp compensators. In this respect, comparative analyses on tracking performance of these compensators are carried out. Based on these analyses, tuning guidelines for CgLp compensators are developed and validated on a high-tech precision positioning stage. The results show the effectiveness of the developed tuning method.

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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.

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PID controllers cannot satisfy the high-performance requirements since they are restricted by the water-bed effect. Thus, the need for a better alternative to linear PID controllers increases due to the rising demands of the high-tech industry. This has led many researchers to explore nonlinear controllers like reset control. Although reset controllers have been widely used to overcome the limitations of linear controllers in literature, the performance of the system varies depending on the relative sequence of controller linear and nonlinear parts. In this paper, the optimal sequence is found using high order sinusoidal input describing functions (HOSIDF). By arranging controller parts according to this strategy, better performance in the sense of precision and control input is achieved. The performance of the proposed sequence is validated on a precision positioning setup. The experimental results demonstrate that the optimal sequence found in theory outperforms other sequences.

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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.@en

The potential of reset controllers to enhance the performance of control systems has been extensively demonstrated in the literature. However, similarly to other non-linear controllers, the stability analysis for these controllers is complex and relies on parametric models of the systems which may hinder the applicability of these controllers in industry. The well-known Hß method tries to solve this significant issue. However, assessing the Hß condition in the frequency-domain is complex, especially for high dimensional plants. In addition, it cannot be used to assess UBIBS stability of reset control systems in the case of reseting to non-zero values. These problems have been solved in this paper for the first order reset elements, and an easy-to-use frequency approach for assessing stability of reset control systems is proposed. The effectiveness of the proposed approach is demonstrated through a practical example.

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Today, there is a great tendency toward using fractional calculus to solve engineering problems. The control is one of the fields in which fractional calculus has attracted a lot of attention. On the one hand, fractional order dynamic models simulate characteristics of real dynamic systems better than integer order models. On the other hand, Fractional Order (FO) controllers outperform Integer Order (IO) controllers in many cases. FO-controllers have been studied in both time an frequency domain. The latter one is the fundamental tool for industry to design FO-controllers. The scope of this paper is to review research which has been carried out on FO-controllers in the frequency domain. In this review paper, the concept of fractional calculus and their applications in the control problems are introduced. In addition, basic definitions of the fractional order differentiation and integration are presented. Then, four common types of FO-controllers are briefly presented and after that their representative tuning methods are introduced. Furthermore, some useful continuous and discrete approximation methods of FO-controllers and their digital and analogue implementation methods are elaborated. Then, some Matlab toolboxes which facilitate utilizing FO calculus in the control field are presented. Finally, advantages and disadvantages of using FO calculus in the control area are discussed. To wrap up, this paper helps beginners to get started rapidly and learn how to select, tune, approximate, discretize, and implement FO-controllers in the frequency domain.

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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.

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This article attempts to increase the crashworthiness characteristics of energy absorbers. It is found that the effect of the bi-tubular arrangement on the energy absorption and peak force is nonlinear. This nonlinearity is somewhat related to friction but is mostly related to the changing of buckling modes. Therefore, it is possible to reach higher Specific Absorbed Energy (SAE) in the bi-tubular case than with two tubes since the weight is the same in both arrangements while the energy absorption is higher in the bi-tubular case. To exploit this, multi-objective optimization of bi-thin walled cylindrical aluminium tubes under axial impact loading is performed. The absorbed energy and the SAE are considered as the objective functions while the maximum crush load is regarded as a constraint. Finally, the optimal dimensions of tubes are found in order to maximize the SAE and energy absorption for a specified maximum crushing force.

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Today, fractional order proportional integration derivative (FO-PID) controllers have attracted much attention from academia and industry. Despite FO-PID controllers outperforming integer order (IO) ones in many cases, the latter continues to dominate industrial utilization. One of the big barriers which confine adoption of FO-PID controllers is the complexity of the current tuning methods for industrial application. In this paper, a practical tuning method for FO-PID controllers is introduced. In this line, classical loop-shaping tools are utilized to propose this new simple tuning rule. A rule of thumb and a guideline are given for non-expert and industrial users for controlling motion systems. Finally, the tuning method is validated in a high-tech precision positioning system.

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