Clegg Integrator Research Articles

A Nonlinear Integrator Based on the First-Order Reset Element

This work proposes a novel nonlinear Proportional-Integral (PI) controller, which utilizes a generalized first-order reset element. The proposed element can achieve similar magnitude-characteristics as its linear counterpart but with less phase lag at the open-loop crossover frequency (i.e. the control bandwidth), according to a sinusoidal-input describing function (SIDF) analysis. The same can be achieved with an existing reset-based integrator, the Clegg integrator (CI). However, it is known that a Proportional-CI (PCI) element can excessively generate higher-order harmonics of its input, which are neglected in the SIDF-analysis. Furthermore, a PCI can cause a limit cycle when placed in closed-loop with certain types of plants. The novel PI controller proposed in this work can prevent the limit cycle and can reduce the generation of higher-order harmonics, while retaining the beneficial phase advantage that is associated with existing reset-based PI controllers.

Reset control performance improvement using fractional derivatives

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.

Generalization of Reset Controllers to Fractional Orders

Reset control is a simple non-linear control technique that can help overcome the structural limitations of linear control. Fractional control uses the concept of fractional derivatives to expand the range of possibilities when modeling a controller, making it more robust. Fractional reset control merges the advantages of both areas and is the object of this paper. Fractional-order versions of different reset controllers were implemented, namely a fractional Clegg integrator, a fractional generalized first-order reset element, a fractional generalized second-order reset element, and fractional “constant in gain lead in phase” controllers with first- and second-order reset elements. These were computed directly from a numerical implementation of the Grünwald–Letnikov definition of fractional derivatives, and their performances were analyzed.

An optimal robust design method for fractional‐order reset controller

AbstractIn this paper, a novel design method for the reset controller structure (i.e., fractional‐order proportional and integral plus Clegg integrator (PI + CI )), is proposed for a second‐order plus time delay plant. To this end, the designer can get an optimal fractional reset controller that gives the control system more phase margin over the base linear PI controller and robust to loop gain variation. The describing function method is used to investigate the capability of phase lead and the frequency domain properties of PI + CI . The gain crossover frequency and phase margin specifications ensure the stability of the control system, and the flat phase constraint makes the control system robust to loop gain variations. Meanwhile, the integral of time and absolute error (ITAE) value is applied to achieve the optimal dynamic performance as the cost function. PI + CI is compared with its integer‐order counterpart (i.e., proportional and integral plus Clegg integrator (PI + CI) controller) and their base controllers (i.e., integer‐order PI and fractional‐order PI controllers) in terms of the step response and robustness to loop gain variations. The simulation results illustrate that the PI + CI control system obtains lower overshoot and oscillation and better robustness to loop gain variations than others. The experiments are performed on the speed control of an air bearing stage. Experimental results show that the designed PI + CI control system behaves better than others. The proposed PI + CI design method can be applied to other general control plants easily.

Reset-and-Hold Control of Systems With Time Delay

Reset control is a kind of hybrid control which is capable of overcoming fundamental limitations on the performance intrinsic to linear time invariant (LTI) systems. In this work, we develop a novel control strategy for first order LTI plants with time delay, based on the proportional-integral plus Clegg integrator (PI+CI) controller, a hybrid extension of the proportional-integral (PI) controller, augmented with a new mechanism for keeping its state constant during a given interval of time (reset-and-hold strategy). This strategy is capable of producing an approximation of a flat response which greatly improves the performance in comparison with non-hybrid linear strategies, extending previous results developed for first order systems without delay. Well-posedness and closed-loop stability of the resulting control system are analyzed under the Hybrid Inclusions (HI) framework, and a set of sufficient stability conditions are provided. Furthermore, a case study is developed that showcases the possibilities of this new approach.

Projection-based integrators for improved motion control: Formalization, well-posedness and stability of hybrid integrator-gain systems

In this paper we formally describe the hybrid integrator-gain system (HIGS), which is a nonlinear integrator designed to avoid the limitations typically associated with linear integrators. The HIGS keeps the sign of its input and output equal, thereby inducing less phase lag than a linear integrator, much like the famous Clegg integrator. The HIGS achieves the reduced phase lag by projection of the controller dynamics instead of using resets of the integrator state, which forms a potential benefit of this control element. To formally analyze HIGS-controlled systems, we present an appropriate mathematical framework for describing these novel systems. Based on this framework, HIGS-controlled systems are proven to be well-posed in the sense of existence and forward completeness of solutions. Moreover, we propose two approaches for analyzing (input-to-state) stability of the resulting nonlinear closed-loop systems: (i) circle-criterion-like conditions based on (measured) frequency response data, and (ii) LMI-based conditions exploiting a new construction of piecewise quadratic Lyapunov functions. A motion control example is used to illustrate the results.

Reset Control of Parallel MISO Systems

The proportional–integral plus Clegg integrator (PI + CI) controller is a hybrid extension of the proportional–integral (PI) controller that is able to overcome fundamental limitations of the linear and time-invariant control systems, potentially obtaining faster responses without increasing overshooting. This work focused on the analysis and design of PI + CI controllers and reset controllers in general, for the case of parallel multiple-input single-output (MISO) systems, extending previous design methods developed for the single-input single-output (SISO) case. Several design strategies were developed: one for first-order MISO plants achieving a flat response with a finite settling time, and for second-order MISO plants obtaining a fast response with a reduced overshoot and settling time in comparison with non-hybrid strategies. Several case studies were also developed to illustrate the potential of the proposed methods.

Reset control for synchronization of multi-agent systems

Standard approaches to synchronization problems in multi-agent systems apply local smooth control signals to achieve ultimate coherence. Most existing algorithms emphasize asymptotic behavior rather than transient performance. An alternative design framework based on reset control is presented to guarantee not only the asymptotic convergence but also a desired transient response. Specifically, the control protocol consists of a proportional integral (PI) compensator and a Clegg integrator (CI), where the integral term is reset to zero whenever the proportional term crosses zero. Numerical examples are included to illustrate the main result, and to compare the proposed reset control approach with the traditional static/dynamic state feedback control and the quasi-reset control methods.

Robust control and data‐driven tuning of a hybrid integrator‐gain system with applications to wafer scanners

SummaryA hybrid integrator‐gain system is discussed that aims for improved low‐frequency disturbance rejection, while, at the same time, does not deteriorate overshoot and settling times when compared with a linear integrator. The hybrid integrator has similar phase advantages as the well‐known Clegg integrator but without inducing the discontinuous behavior resulting from resetting system state values. Optimal tuning of the controller parameters of the hybrid integrator is strongly influenced by machine‐specific properties and therefore favors a data‐driven optimization approach. However, as a time‐domain optimization algorithm can easily lead to nonrobust solutions in the sense of large peaking of the closed‐loop frequency response functions, frequency‐domain robustness constraints will be imposed. By means of an adaptive weighting filter design, the parameter updates are penalized upon violation of said robustness constraints. Posed in an unconstrained problem formulation, this is subsequently solved by applying a Gauss‐Newton–based parameter update scheme. Closed‐loop stability of the linear time‐invariant plant and controller in feedback connection with a hybrid integrator‐gain system element follows from a circle‐criterion‐like analysis, which is based on evaluating (measured) frequency response data. Measurement results obtained from an industrial wafer scanner demonstrate the effectiveness of the approach.

Pseudo-linear correction as a means of improving the quality of control systems

The work shows the possibility to improve the quality of automatic control systems by applying the controller which includes a pseudo-linear correcting device with phase advance, one proportional and one differential element and an integrating link based on the Clegg integrator. The conducted research shows that such approach to constructing automatic control systems enables to improve significantly the performance indices for transient processes when controlling objects whose parameters change their values over time. The work concludes that the use of such systems is possible and appropriate for non-stationary objects when changing their parameters in quite a wide range.

Reset control of integrating plus dead time processes

This work develops a tuning method for a reset compensator and an integrating plus dead time system. The reset compensator consists of a proportional integral compensator plus a Clegg integrator (PI+CI), and is equipped with a variable band resetting law, which allows to reduce the harmful effect of time-delay, and a switching reset ratio that obtains a superior performance in comparison with a well-tuned PI compensator. This work exposes a simple tuning rule for setting all the parameters of the PI+CI compensator with the aim of improving both setpoint tracking and load disturbance rejection. A case study is analyzed by simulation, discussing both performance and robustness of several PI+CI and PI compensators. Finally, an experiment of liquid level control shows the advantages of the proposed reset compensator.

Stability of time-delay reset control systems with time-dependent resetting law

This work presents results on the stability of time-delay reset control systems under time-dependent resetting conditions. The stability of a reset control system composed by a time-delay process and a proportional and integrative plus Clegg integrator (PI+CI) compensator is tackled by using the framework of sampled-data systems. It leads to sufficient stability conditions expressed in terms of LMIs (Linear Matrix Inequality), that depend explicitly on the reset times. In contrast to previous results, the proposed conditions allow to guarantee the stability of reset systems with unstable base system.

Tuning of reset proportional integral compensators with a variable reset ratio and reset band

The PI+CI compensator, given by the parallel connection of a proportional integral (PI) compensator and a Clegg integrator (CI), has been shown to be effective in a number of practical applications, overcoming fundamental limitations of PI compensation. Two key design aspects of PI+CI compensation is the use of the resetting law (zero crossing/reset band) and a (constant/variable) reset ratio; these two design factors have been previously used in control practice, but using heuristic design methods. This work develops a systematic design method for the tuning of reset PI compensators, specifically for first- and second-order plants, obtaining in both cases simple tuning rules. In addition, several detailed examples are developed; therefore, a detailed analysis of cost of feedback and robustness again parametric uncertainty is also given. Finally, an experiment of pH In-Line control is made to show the advantages of the proposed tuning rule.

Generalized Clegg integrator for integral feedback control systems

This article presents a generalized Clegg integrator for integral feedback control systems. The integral feedback compensation with a linear integrator suffers from the constraint imposed by Bode’s gain–phase relationship. The nonlinear Clegg integrator releases this constraint. However, one element missing in the literature is a concrete formulation connecting the Clegg integrator with the linear integrator. The generalized Clegg integrator proposed in this article bridges the gap between the Clegg integrator and the linear integrator. Moreover, compared with the Clegg integrator, the generalized Clegg integrator can be employed to further improve system performance in both the transient and steady states. Experimental studies are conducted to practically demonstrate the effectiveness of the proposed scheme.

Sistemas de Control basados en Reset

A reset control system is formed by a linear controller endowed with certain resetting mechanism, that sets to zero the state. This zeroing of some of the state coordinates is applied when certain condition holds. The condition that triggers resetting is usually the zero crossing of the tracking error. The idea of re-set control dates back to Clegg Integrator (CI) in 1958, and was adopted in the 70's by I. Horowitz, who emphasized the ability of reset systems to overcome fundamental limitations of linear systems with delays or with poles or zeros in the right half-plane. Left aside for some time, the idea was recovered in the last decade and was given a new impulse by several research teams that contribute with advances in analysis, design and application of reset control systems. This interest appears in the context of the present boom of hybrid systems: a reset system is a particular case of hybrid system that, in spite of its simplicity, enables the possibility of overcoming linear limitations. The objetive of this work is to present, in the form of a tutorial, the current panorama in the field. First, a historical review is introduced, on the three main approaches to reset control: the classical approach (based on zero-crossing of the tracking error), the impulsive approach (where reset is applied on a temporal basis) and the hybrid approach (where some sector reset condition is defined). Next, a summary of results is presented on analysis topics (relation to fundamental limitations, stability) and finally several design proposals are discussed, focused to the tuning of PI+CI compensators.

Design of Reset Control Systems: The PI + CI Compensator

Reset compensation has been used to overcome limitations of linear and time invariant (LTI) compensation. In this work, a new reset compensator, referred to as proportional and integral (PI) + CI (Clegg integrator), is introduced. It basically consists of adding a Clegg integrator to a PI compensator, with the goal of improving the closed loop response by using the nonlinear characteristic of this element. It turns out that by resetting a percentage of the integral term in a PI compensator, a significant improvement can be obtained over a well-tuned PI compensator in some relevant practical cases, such as systems with dominant lag and integrating systems. The work is devoted to the development of PI + CI tuning rules for basic dynamic systems in a wide range of applications, including first and higher order plus dead time systems.

Reset passivation of nonlinear controllers via a suitable time-regular reset map

For a class of square continuous-time nonlinear controllers we design a suitable resetting rule inspired by the resetting rule for Clegg integrators and First Order Reset Elements (FORE). With this rule, we prove that the arising hybrid system with temporal regularization is passive in the conventional continuous-time sense with a small shortage of input passivity decreasing with the temporal regularization constant. Based on the passivity property, we then investigate the finite gain stability of the interconnection between this passive controller and a passive nonlinear plant.

RESULTS ON PASSIFICATION OF NONLINEAR CONTROLLERS VIA A SUITABLE TIME-REGULAR RESET MAP

For a class of square continuous time nonlinear controllers we design a suitable resetting rule inspired by the resetting rule for Clegg integrators and First Order Reset Elements (FORE). With this rule, we prove that the arising hybrid system with temporal regularization is passive in the conventional continuous time sense with a small shortage of input passivity decreasing with the temporal regularization constant. Based on the passivity property, we then investigate the finite gain stability of the interconnection between this passive controller and a passive nonlinear plant.

Stability properties of reset systems

Stability properties for a class of reset systems, such as systems containing a Clegg integrator, are investigated. We present Lyapunov based results for verifying L 2 and exponential stability of reset systems. Our results generalize the available results in the literature and can be easily modified to cover L p stability for arbitrary p ∈ [ 1 , ∞ ] . Several examples illustrate that introducing resets in a linear system may reduce the L 2 gain if the reset controller parameters are carefully tuned.

STABILITY PROPERTIES OF RESET SYSTEMS

Stability properties for a class of reset systems, such as systems containing a Clegg integrator, are investigated. We present Lyapunov based results for verifying L2 and exponential stability of reset systems. Our results generalize the available results in the literature and can be easily modified to cover Lp stability for arbitrary p ∈ [1, ∞]. Several examples illustrate that introducing resets in a linear system may reduce the L2 gain if the reset controller parameters are carefully tuned.