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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A Frequency-Domain Tuning Method for a Class of Reset Control Systems

This paper addresses nonlinearity in reset elements and their effects. Reset elements are known for having less phase lag compared to their linear counterparts; however, they are nonlinear elements and produce higher-order harmonics. This paper investigates the higher-order harmonics for reset elements with one resetting state and proposes an architecture and a method of design which 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, e.g., they are useful for reducing phase lag at cross-over frequency region; however, higher-order harmonics can compromise tracking and disturbance rejection performance at lower frequencies. Using proposed "phase shaping" method, one can selectively suppress 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 "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.

Band-Passing Nonlinearity in Reset Elements.

This paper addresses nonlinearity in reset elements and their effects. Reset elements are known for having less phase lag compared to their linear counterparts; however, they are nonlinear elements and produce higher-order harmonics. This paper investigates the higher-order harmonics for reset elements with one resetting state and proposes an architecture and a method of design which 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, e.g., they are useful for reducing phase lag at cross-over frequency region; however, higher-order harmonics can compromise tracking and disturbance rejection performance at lower frequencies. Using proposed "phase shaping" method, one can selectively suppress 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 "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. 

Band-Passing Nonlinearity in Reset Elements

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 frequencydomain 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.

Tuning of a Class of Reset Elements Using Pseudo-Sensitivities

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. 

/pdf/complex-order-reset-control-system-35hcbo1p.pdf

Complex-order Reset Control System

Tuning of CgLp based reset controllers : Application in precision positioning systems

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.

Mahmoud Shirdast Bahnamiri

Papers

Tuning of CgLp based reset controllers : Application in precision positioning systems

Tuning of CgLp based reset controllers: Application in precision positioning systems