Nowadays, the demand for an alternative to linear PID controllers has increased because of the rising expectations of the high-precision industry. The potential of reset controllers to solve this important challenge 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 Hbeta method tries to solve this significant issue. However, assessing the H\b{eta} 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.
Today, linear PID controllers cannot satisfy requirements of high-precision industry due to the development of technology. According to the literature, reset controllers can overcome this important barrier. However, similar to other non-linear controllers, stability analysis for these controllers is complex and needs parametric models of the systems. Consequently, the applicability of these controllers may diminish in industry. The well-known Hb method is one of the solutions of this significant problem. Nevertheless, assessing the Hb condition in the frequency-domain is complex, particularly …
Exponential stability of a PI plus reset integrator controller by a sampled-data system approach
Gipsa-lab（吉普萨实验室）；CNRS, LAAS（法国国家科学研究中心，图卢兹航空航天实验室）；Univ de Toulouse, UPS, LAAS（图卢兹大学，国立高等邮电学校，图卢兹航空航天实验室）；Dpto. Informática y Sistemas, University of Murcia（穆尔西亚大学，计算机与系统系）；Univ de Toulouse, LAAS（图卢兹大学，图卢兹航空航天实验室）
A novel representation of reset control systems with a zero-crossing resetting law, in the framework of hybrid inclusions, is postulated. The problems of well-posedness and stability of the resulting hybrid dynamical system are investigated, with a strong motivation in how non-deterministic behavior is accomplished in control practice. Several stability conditions, based on the eigenstructure of matrices related with periods of the reset interval sequences, and on Lyapunov function-based conditions, are developed.
This paper 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, pseudo-sensitivi…
Soft-reset controllers are introduced as a way to approximate hard-reset controllers. The focus is on implementing reset controllers that are (strictly) passive and on analyzing their interconnection with passive plants. A passive hard-reset controller that has a strongly convex energy function can be approximated as a soft-reset controller. A hard-reset controller is a hybrid system whereas a soft-reset controller corresponds to a differential inclusion, living entirely in the continuous-time domain. This feature may make soft-reset controllers easier to understand and implement. A soft-reset…
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…
The precision motion industry has an ever-increasing demand for faster, more precise and more robust controllers. From the perspective of frequency domain and loopshaping technique, this demand has pushed the linear controllers to their inherent limits, namely, the waterbed effect and Bode's phase-gain relationship. Mathematically, complex-order transfer functions are not bound by Bode's phase-gain relationship. However, implementing them in practice is a challenge to be solved. This extended abstract will show review the previous work of the authors in shaping nonlinearities in reset controll…
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 ap…
The ever-growing demands on speed and precision from the precision motion industry have pushed control requirements to reach the limitations of linear control theory. Nonlinear controllers like reset provide a viable alternative since they can be easily integrated into the existing linear controller structure and designed using industry-preferred loop-shaping techniques. However, currently, loop-shaping is achieved using the describing function (DF) and performance analysed using linear control sensitivity functions not applicable for reset control systems, resulting in a significant deviation…
Beyond the Waterbed Effect: Development of Fractional Order CRONE Control with Non-Linear Reset
Delft Center for Systems and Control, Delft University of Technology（代尔夫特系统与控制中心，代尔夫特理工大学）；Department of Precision and Microsystems Engineering, Delft University of Technology（精密与微系统工程学院，代尔夫特理工大学）
The ever-increasing industry desire for improved performance makes linear controller design run into fundamental limitations. Nonlinear control methods such as Reset Control (RC) are needed to overcome these. RC is a promising candidate since, unlike other nonlinear methods, it easily integrates into the industry-preferred PID design framework. Thus far, RC has been analysed in the frequency domain either through describing function analysis or by direct closed-loop numerical computation. The former computes a simplified closed-loop RC response by assuming a sufficient low-pass behaviour. In d…
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 beha…
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…
University of California, Berkeley（加州大学伯克利分校）；University of Southern California（南加利福尼亚大学）
Contact-rich robotic systems, such as legged robots and manipulators, are often represented as hybrid systems. However, the stability analysis and region-of-attraction computation for these systems are often challenging because of the discontinuous state changes upon contact (also referred to as state resets). In this work, we cast the computation of region-ofattraction as a Hamilton-Jacobi (HJ) reachability problem. This enables us to leverage HJ reachability tools that are compatible with general nonlinear system dynamics, and can formally deal with state and input constraints as well as bou…
This paper presents a complex order filter developed and subsequently integrated into a PID-based controller design. The nonlinear filter is designed with reset elements to have describing function based frequency response similar to that of a linear (practically non-implementable) complex order filter. This allows for a design which has a negative gain slope and a corresponding positive phase slope as desired from a loop-shaping controller-design perspective. This approach enables improvement in precision tracking without compromising the bandwidth or stability requirements. The proposed desi…
High-order tuners are algorithms that show promise in achieving greater efficiency than classic gradient-based algorithms in identifying the parameters of parametric models and/or in facilitating the progress of a control or optimization algorithm whose adaptive behavior relies on such models. For high-order tuners, robust stability properties, namely uniform global asymptotic (and exponential) stability, currently rely on a persistent excitation (PE) condition. In this work, we establish such stability properties with a novel analysis based on a Matrosov theorem and then show that the PE requ…
Due to possible emergency faults and frequency regulation reserve shortage in the multi-infeed hybrid AC-DC (MIDC) system, the emergency frequency control (EFC) with LCC-HVDC systems participating is important for system frequency stability. Nevertheless, the existing decentralized EFC strategies cannot guarantee the transient stability constraints of lines and cannot meet the frequency restoration requirement. To fill this gap, this paper proposes a complementary distributed EFC strategy. By selecting the semi-distributed or fully-distributed control law according to the state of AC system co…
As renewable resources gradually replace conventional generation based synchronous machines, the dynamics of the modern grid changes significantly and the system synchronous inertia decreases substantially. This transformation poses severe challenges for power system stability; for instance, it may lead to larger initial rate of change of frequency and increase frequency excursions. However, new opportunities also arise as novelconverter control techniques, so-called grid-forming strategies, show higher efficiency and faster response than conventional synchronous generators. They mainly involv…
A Framework for Frequency Stability Assessment of Future Power Systems: An Australian Case Study
University of Sydney, Faculty of Electrical and Information Technology, School of Electrical and Information Engineering（悉尼大学电气与信息工程学院）；University of Engineering and Technology Lahore（拉合尔工程与技术大学）；The University of Hong Kong（香港大学）
The ongoing energy transition challenges the stability of the electrical power system. Stable operation of the electrical power grid requires both the voltage (amplitude) and the frequency to stay within operational bounds. While much research has focused on frequency dynamics and stability, the voltage dynamics has been neglected. Here, we study frequency and voltage stability in the case of simple networks via linear stability and bulk analysis. In particular, our linear stability analysis of the network shows that the frequency secondary control guarantees the stability of a particular elec…
This paper considers the phenomenon of distinct regional frequencies recently observed in some power systems. First, a reduced-order mathematical model describing this behaviour is developed. Then, techniques to solve the model are discussed, demonstrating that the post-fault frequency evolution in any given region is equal to the frequency evolution of the Centre Of Inertia plus certain inter-area oscillations. This finding leads to the deduction of conditions for guaranteeing frequency stability in all regions of a power system, a deduction performed using a mixed analytical-numerical approa…
