Modified Prandtl-Ishlinskii Research Articles

Dual-reference adaptive feedforward control of a magnetostrictive device for synchronous micropositioning and microvibration isolation

In this paper, we propose a novel dual-reference adaptive feedforward controller to realize synchronous micropositioning and microvibration isolation on a magnetostrictive device. The scheme of the proposed adaptive feedforward controller and its differences from the traditional single-reference controller are briefly introduced. The desired trajectory and available external disturbance as two input references are utilized to develop the proposed controller. The dynamics compensator is constructed based on the modified filtered-x normalized least mean square (MFxNLMS) algorithm with the discrete cosine transform (DCT) technique. The asymmetric hysteresis compensator is modeled via the arctangent-polynomial modified Prandtl–Ishlinskii (APMPI) model. The experimental setup is built, and the closed-loop control experiment is carried out according to the designed experimental process. Comparison of experimental results show that the proposed dual-reference DCT-MFxNLMS controller behaves better than the single-reference DCT-MFxNLMS and proportional-derivative-derivative (PID) integrated controller for the synchronous micropositioning and vibration isolation cases. Moreover, by the dual-reference DCT-MFxNLMS controller, the vibration isolation ratio enhances and the tracking bandwidth increases within the interest of frequency bandwidth, compared with those of open-loop system, respectively.

A feedforward-feedback controller based on modified Prandtl–Ishlinskii model and submodel for tracking control the trajectories of piezoelectric actuator

A feedforward-feedback controller based on modified Prandtl–Ishlinskii model and submodel for tracking control the trajectories of piezoelectric actuator

Dynamic hysteresis compensation and iterative learning control for underwater flexible structures actuated by macro fiber composites

Studies on the hydrodynamics of underwater flexible structures are gaining significant attention. To improve the performance and stability of underwater flexible structures in complex water environments, new control strategies must be proposed. This study proposes underwater rate-dependent hysteresis compensation and iterative learning control (ILC) of the flexible structure actuated by macro fiber composites (MFC). The underwater rate-dependent hysteresis model of the MFC partially-actuated structure consists of two components: a modal state space model characterizing the underwater rate-dependent behavior, and a modified Prandtl–Ishlinskii (MPI) model describing the bias bipolar hysteresis. The MPI model is combined with a PI and a polynomial model of fractional orders. To obtain a unique analytic inverse model, constraints for the inverse are proposed. Results show the varying bias ellipse-like hysteresis behaviors are well captured. Tracking performances are greatly improved with the feedforward compensator. Furthermore, an ILC is developed to address model uncertainties and unsteady hydrodynamics. The convergence and robustness of the ILC are discussed. Parameters of the zero-phase and learning filters are determined. Experimental results demonstrate the tracking errors of the underwater structure are significantly reduced. These findings are meaningful for the realization of flexible structures for marine engineering, underwater robots and other fields.

Disturbance observer-based robust optimal control of rate-dependent hysteretic systems: Application to a piezoelectric nano-manipulator

This article investigates the robust and optimal control problem of hysteretic systems with both parameter variations and unmodeled dynamics. By means of a hysteresis compensator based on a modified Prandtl–Ishlinskii model and a feedback controller, a novel robust optimal control architecture is proposed to address the asymmetric and rate-dependent hysteresis nonlinearities and various disturbances, where the compensation errors, parameter variations, and unmodeled dynamics are treated as a bounded disturbance. The proposed controller is composed of two components: a high order sliding mode observer and an extended linear quadratic controller. The high order sliding mode observer is introduced to guarantee that the disturbance observation error converges to the origin in a finite time such that the extended linear quadratic controller enables the predefined optimal performance to be achieved. Moreover, the exponential stability of the closed-loop system has been proven, and some sufficient conditions are given. Finally, the proposed control algorithm is applied to a piezoelectric nano-manipulator system, both the good hysteresis modeling accuracy and excellent tracking performance are demonstrated in the real-time experiments.

Simultaneous micropositioning and microvibration control of a magnetostrictive Stewart platform with synthesized strategy

In this paper, we present a hybrid adaptive feedforward and feedback controller with hysteresis compensator on a magnetostrictive-actuated multi degree of freedom (DOF) positioning and vibration isolation integrated system (PVIIS). The PVIIS is custom-developed and based on “cubic” Stewart configuration. It is challenging to develop an effective controller to systematically address the issues of cross-coupling, strut hysteresis and dynamics for the PVIIS. Magnetostrictive asymmetric hysteresis leads to the loss of closed-loop output performance with conventional linear controller. To this end, a novel comprehensive controller is proposed in this paper. With the inverse kinematics analysis, the spatial motion of the mobile plate of the PVIIS is decoupled into the independent motions of six actuator struts. As for controlling the single strut of the PVIIS, the hybrid adaptive feedforward (AFF) and proportional–integral–derivative (PID) feedback controller with asymmetric hysteresis compensator (HC) is constructed. Among them, the AFF sub-controller is characterized by an adaptive finite impulse response (FIR) filter whose coefficients are updated by normalized least-mean squares (NLMS) algorithm. The PID feedback sub-controller is introduced to enhance the ability of positioning and vibration isolation. The arctangent-polynomial modified Prandtl–Ishlinskii (APMPI) model is used to build the inversion-based asymmetric hysteresis compensator. The experimental set-up is built. The positioning and vibration isolation experiments are conducted. Experimental studies show that the output performances are best with the proposed controller compared against sole AFF, hybrid AFF and PID controllers. With the proposed controller, the relative positioning error can maintain around 2.83% when the PVIIS performs 2-DOF rotation in the presence of 0.5 Hz disturbance.

A Modified Prandtl-Ishlinskii Hysteresis Model for Modeling and Compensating Asymmetric Hysteresis of Piezo-Actuated Flexure-Based Systems.

Piezo-actuated flexure-based systems are widely used in applications with high accuracy requirements, but the intrinsic hysteresis has a detrimental effect on the performance which should be compensated. Conventional models were presented to model this undesired effect using additional dead-zone operators. This paper presents a new approach using two sets of operators with a distributed compensator to model and compensate for the asymmetric system hysteresis based on inversion calculation with a simplified digitized representation. The experimental results validate the effectiveness of the proposed model in modeling and compensating the asymmetric system hysteresis.

High-performance control of fast tool servos with robust disturbance observer and modified [formula omitted] control

Fast tool servo (FTS) is an effective technique for the generation of complicated surfaces in modern precision manufacture industry. In this paper, a novel three-degree-of-freedom (3-DOF) control scheme is proposed for the FTS to achieve good tracking and disturbance rejection performance. Specifically, the 3-DOF control scheme contains a hysteresis compensation term, a modified H∞ feedback control term, a robust disturbance observer (RDOB) term and a feedforward control term. Firstly, a hysteresis compensator is developed based on the modified Prandtl–Ishlinskii (MPI) model to eliminate the intrinsic hysteresis nonlinearity so as to identify the linear dynamic model of the FTS system. A modified mixed-sensitivity-based H∞ feedback control is then proposed to simultaneously provide an excellent damping and tracking performance with a deliberately designed input weighting function. A RDOB is designed for non-minimum phase (NMP) system based on all-pass factorization of the linear dynamics model and the standard H∞ control framework to compensate uncertain external disturbances as well as unmodeled dynamics while guaranteeing the robust stability of closed-loop system. To remove phase delay and enlarge control bandwidth, a closed-loop-inversion feedforward controller is synthesized by integrating a zero-phase error tracking control (ZPETC) with a finite impulse response (FIR) filter. A series of comparative experiments are conducted on a self-developed flux-biased normal-stressed electromagnetic actuated FTS. The experimental results consistently validate the effectiveness and superiority of the proposed scheme in terms of outstanding tracking performance and disturbance rejection.

High-Precision Displacement and Force Hybrid Modeling of Pneumatic Artificial Muscle Using 3D PI-NARMAX Model

Pneumatic artificial muscle (PAM) is attractive in rehabilitation and biomimetic robots due to its flexibility. However, there exists a strong hysteretic nonlinearity in PAMs and strong coupling between the output displacement and the output force. At present, most commonly used hysteresis models can be treated as two-dimensional models, which only consider the nonlinearity between the input and the output displacement of the PAM without considering the coupling of the output force. As a result, high-precision modeling and estimation of the PAM’s behavior is difficult, especially when the external load of the system varies significantly. In this paper, the influence of the output force on the displacement is experimentally investigated. A three-dimensional model based on the modified Prandtl–Ishlinskii (MPI) model and the Nonlinear AutoRegressive Moving Average with eXogenous inputs (NARMAX) model is proposed to describe the relationship and couplings among the input, the output displacement, and the output force of the PAM. Experiments are conducted to verify the modeling accuracy of the proposed model when the external load of the PAM varies across a wide range. The experimental results show that the proposed model captures well the hysteresis and couplings of the PAM and can precisely predict the PAM’s behavior.

Comprehensive Design Method of a High-Frequency-Response Fast Tool Servo System Based on a Full-Frequency Error Control Algorithm.

With the development of optoelectronic information technology, high-performance optical systems require an increasingly higher surface accuracy of optical mirrors. The fast tool servo (FTS) based on the piezoelectric actuator is widely used in the compensation machining of high-precision optical mirrors. However, with the low natural frequency of mechanical structures, hysteresis of the piezoelectric actuators, and phase delay of the control systems, conventional FTS systems face problems such as a low working frequency and a large tracking error. This study presents a method for the design of a high-performance FTS system. First, a flexure hinge servo turret with a high natural frequency was designed through multi-objective optimization and finite element simulations. Subsequently, a composite control algorithm was proposed, targeting the problems of hysteresis and phase delay. The modified Prandtl–Ishlinskii inverse hysteresis model was used to overcome the hysteresis effect and a zero-phase error tracker was designed to reduce the phase error. The experimental results reveal that the tracking error of the designed FTS system was <10% in the full frequency range (0–1000 Hz).

A Self-Sensing Method for Electromagnetic Actuators with Hysteresis Compensation

Self-sensing techniques are a commonly used approach for electromagnetic actuators since they allow the removal of position sensors. Thus, costs, space requirements, and system complexity of actuation systems can be reduced. A widely used parameter for self-sensing is the position-dependent incremental inductance. Nevertheless, this parameter is strongly affected by electromagnetic hysteresis, which reduces the performance of self-sensing. This work focuses on the design of a hysteresis-compensated self-sensing algorithm with low computational effort. In particular, the Integrator-Based Direct Inductance Measurement (IDIM) technique is used for the resource-efficient estimation of the incremental inductance. Since the incremental inductance exhibits a hysteresis with butterfly characteristics, it first needs to be transformed into a B-H curve-like hysteresis. Then, a modified Prandtl–Ishlinskii (MPI) approach is used for modeling this hysteretic behavior. By using a lumped magnetic circuit model, the hysteresis of the iron core can be separated from the air gap, thus allowing a hysteresis-compensated estimation of the position. Experimental studies performed on an industrial switching actuator show a significant decrease in the estimation error when the hysteresis model is considered. The chosen MPI model has a low model order and therefore allows a computationally lightweight implementation. Therefore, it is proven that the presented approach increases the accuracy of self-sensing on electromagnetic actuators with remarkable hysteresis while offering low computational effort which is an important aspect for the implementation of the technique in cost-critical applications.

Parameter identification and dynamic response analysis of a modified Prandtl–Ishlinskii asymmetric hysteresis model via least-mean square algorithm and particle swarm optimization

Hysteresis is a nonlinear phenomenon observed in the dynamic response behavior of numerous structural systems under high intensity cyclic or random loading, as well as in numerous mechanical and electromagnetic systems. For several decades, hysteretic response analysis of structural systems has been widely studied and numerous hysteresis models have been proposed and utilized in order to reproduce and better understand the complex hysteretically degrading behavior of structural systems. An important area of research in this regard has been the parameter identification of hysteretic systems. In this paper, we propose a modified Prandtl–Ishlinskii model to simulate the asymmetric hysteresis, which is the complex behavior in structural systems. In addition, a new approach based on particle swarm optimization and least-mean square algorithm is utilized for parameter identification of this hysteresis model. Finally, the model is applied in structural dynamic response analysis of a base isolated structural model under seismic load.

Inversion-Based Hysteresis Compensation Using Adaptive Conditional Servocompensator for Nanopositioning Systems

Abstract Nanopositioning stages are widely used in high-precision positioning applications. However, they suffer from an intrinsic hysteretic behavior, which deteriorates their tracking performance. This study proposes an adaptive conditional servocompensator (ACS) to compensate the effect of the hysteresis when tracking periodic references. The nanopositioning system is modeled as a linear system cascaded with hysteresis at the input side. The hysteresis is modeled with a modified Prandtl–Ishlinskii (MPI) operator. With an approximate inverse MPI operator placed before the system hysteresis operator, the resulting system takes a semi-affine form. The design of the ACS consists of two stages: first, we design a continuously implemented sliding mode control (SMC) law. The hysteresis inversion error is treated as a matched disturbance, and an analytical bound on the inversion error is used to minimize the conservativeness of the SMC design. The second part of the controller is the ACS. Under mild assumptions, we establish the well-posedness and periodic stability of the closed-loop system. In particular, the solution of the closed-loop error system will converge exponentially to a unique periodic solution in the neighborhood of zero. The efficacy of the proposed controller is verified experimentally on a commercial nanopositioning device under different types of periodic reference inputs, via comparison with multiple inversion-based and inversion-free approaches.

A Digitized Representation of the Modified Prandtl-Ishlinskii Hysteresis Model for Modeling and Compensating Piezoelectric Actuator Hysteresis.

Piezoelectric actuators are widely used in micromanipulation and miniature robots due to their rapid response and high repeatability. The piezoelectric actuators often have undesired hysteresis. The Prandtl–Ishlinskii (PI) hysteresis model is one of the most popular models for modeling and compensating the hysteresis behaviour. This paper presents an alternative digitized representation of the modified Prandtl–Ishlinskii with the dead-zone operators (MPI) hysteresis model to describe the asymmetric hysteresis behavior of piezoelectric actuators. Using a binary number with n digits to represent the classical Prandtl–Ishlinskii hysteresis model with n elementary operators, the inverse model can be easily constructed. A similar representation of the dead-zone operators is also described. With the proposed digitized representation, the model is more intuitive and the inversion calculation is avoided. An experiment with a piezoelectric stacked linear actuator is conducted to validate the proposed digitized MPI hysteresis model and it is shown that it has almost the same performance as compared to the classical representation.

Inversion-Free Hysteresis Compensation via Adaptive Conditional Servomechanism With Application to Nanopositioning Control

In this article, an adaptive conditional servocompensator is proposed to achieve precise tracking control of systems with hysteresis without requiring explicit inversion of the hysteresis. Motivated by a range of applications, such as the piezoelectric-actuated nanopositioning, the system considered in this work consists of a chain of integrators preceded by a hysteresis nonlinearity modeled by a modified Prandtl-Ishlinskii (MPI) operator with uncertainty. To facilitate the proposed control design, the MPI operator is rearranged into a form comprised of three parts: a linear term, a nominal hysteretic term represented by a classical Prandtl-Ishlinskii (PI) operator, and a hysteretic perturbation. The bound on the hysteretic perturbation is further derived based on the parameter uncertainty of the MPI operator. The controller consists of two major elements. The first is a continuously implemented sliding mode controller (SMC), which exploits the bound on hysteretic perturbation and drives the system states to a bounded set in finite time. To properly “cancel” the nominal hysteresis effect without inversion, a technique involving a low-pass filter is introduced. The second element of the proposed controller is an adaptive conditional servocompensator that aims to eliminate periodic components in the tracking error. We show that, with persistent excitation, the closed-loop variables are ultimately bounded and the tracking error approaches a neighborhood of zero, where the neighborhood can be made arbitrarily small via the choice of the SMC boundary-layer width parameter and the servocompensator order. The proposed approach is implemented experimentally on a commercial nanopositioner under different types of periodic references, and it shows superior tracking performance over the traditional proportional-integral control, as well as several hysteresis inversion-based approaches reported in the literature.

Asymmetric Hysteresis Modeling Approach Featuring "Inertial System + Shape Function" for Magnetostrictive Actuators.

Hysteresis of the actuators based on magnetostrictive materials influences the control performance of the application systems. It is of importance and significance to establish an effective hysteresis model for the magnetostrictive actuators for precision engineering. In this paper, based on the analysis of the Duhem model, a first-order inertial system with hysteresis characteristic under harmonic input is used to describe the hysteresis caused by the inertia of the magnetic domains of magnetostrictive materials. Shape function is employed to describe the pinning of domain walls, the interactions of different magnetic domains of magnetostrictive materials, and the saturation properties of the hysteresis. Specifically, under an architecture of “inertial system + shape function” (ISSF-Duhem model), firstly a new hysteresis model is proposed for magnetostrictive actuators. The formulation of the inertial system is constructed based on its general expression, which is capable of describing the hysteresis characteristics of magnetostrictive actuators. Then, the developed models with a Grompertz function-based shape function, a modified hyperbolic tangent function-based shape function employing an exponential function as an offset function, a one-sided dead-zone operator-based shape function are compared with each other, and further compared with the classic modified Prandtl–Ishlinskii model with a one-sided dead-zone operator. Sequentially, feasibility and capability of the proposed hysteresis model are verified and evaluated by describing and predicting the hysteresis characteristics of a commercial magnetostrictive actuator.

Modeling and compensation of hysteresis in piezoelectric actuators

Piezoelectric actuator has the advantages of high rigidity, wide bandwidth, fast response and high resolution. Therefore, they are widely used in many micro and nano positioning applications. However, the hysteresis characteristic in the piezoelectric actuator (PEA) seriously affects its positioning accuracy and even causes instability. In this paper, a modified Prandtl-Ishlinskii (MPI) model, which can describe the rate asymmetric hysteresis of piezoelectric actuator, is studied. The hysteresis compensation is realized by using the rate dependent Prandtl-Iishlinskii model based on the improved Prandtl-Iishlinskii hysteresis model and the hysteresis characteristics of the driver measured in the laboratory under the frequency input of up to 100 Hz. In order to further reduce the error of feedforward compensation, a sliding mode controller is designed. The stability of the control system is proved by Lyapunov theory. The experimental results show that the linear error of the system is reduced from 10% to less than 1%, and the tracking error can also be reduced by 90%.

Characterization and modeling of viscoelastic hysteresis in a dielectric elastomer actuator

This paper presents characterization and modeling for viscoelastic hysteresis exhibited by dielectric elastomer actuator (DEAs). Firstly, output–input characteristics in terms of hysteresis, asymmetry and output nonlinearity are quantitatively described by proposed indicators. Their variations with the excitation frequency are investigated, revealing significant asymmetric property and strong frequency-dependence. Next, the Hammerstein model with a cascaded structure is proposed for viscoelastic hysteresis description. A modified Prandtl–Ishlinskii model, representing the static nonlinear element, is employed to characterize the asymmetric property of static hysteresis, while a finite-impulse-response based adaptive filter as the linear part captures the dynamic effect. The hysteresis loops of DEAs under the input voltages with different frequencies, waveforms and amplitudes are experimentally measured and compared with those predicted results. The relative error in terms of root-mean-square value ranges from 1.92% to a maximum of 5.53% over a wide frequency range of 0.1–10 Hz, demonstrating the effectiveness of the proposed model of rate-dependent hysteresis in DEAs. This work can be useful to promote the applications of DEAs to soft robotics and haptics.

Modeling and Discrete-Time Terminal Sliding Mode Control of a DEAP Actuator with Rate-Dependent Hysteresis Nonlinearity

Dielectric electro-active polymer (DEAP) materials, also called artificial muscle, are a kind of EAP smart materials with extraordinary strains up to 30% at a high driving voltage. However, the asymmetric rate-dependent hysteresis is a barrier for trajectory tracking control of DEAP actuators. To overcome the barrier, in this paper, a Hammerstein model is established for the asymmetric rate-dependent hysteresis of a DEAP actuator first, in which a modified Prandtl-Ishlinskii (MPI) model is used to represent the static hysteresis nonlinear part, and an autoregressive with exogenous inputs (ARX) model is used to represent the linear dynamic part. Applying Levenberg-Marquardt (LM) algorithm identifies the parameters of the Hammerstein model. Then, based on the MPI model, an inverse hysteresis compensator is obtained to compensate the hysteresis behavior. Finally, a compound controller consisting of the hysteresis compensator and a novel discrete-time terminal sliding mode controller (DTSMC) without state observer is proposed to achieve the high-precision trajectory tracking control. Stability analysis of the closed-loop system is verified by using Lyapunov stability theorem. Experimental results based on a DEAP actuator show that the proposed controller has better tracking control performance compared with a conventional discrete-time sliding mode controller (DSMC).

Hammerstein model for hysteresis characteristics of pneumatic muscle actuators

As a kind of novel compliant actuators, pneumatic muscle actuators (PMAs) have been recently used in wearable devices for rehabilitation, industrial manufacturing and other fields due to their excellent actuation characteristics such as high power/weight ratio, safety and inherent compliance. However, the strong nonlinearity and asymmetrical hysteresis cause difficulties in the accurate control of robots actuated by PMAs. In this paper, a method for hysteresis modeling of PMA based on Hammerstein model is proposed, which introduces the BP neural network into the hysteretic system. In order to overcome the limitation of BP neural network’s single-valued mapping, an extended space input method is adapted while the Modified Prandtl–Ishlinskii model is applied to characterize the hysteretic phenomenon. A conventional PID control is implemented to track the trajectory of PMA with and without the feed-forward hysteresis compensation based on Hammerstein model, and experimental results validate the effectiveness of the designed model which has the advantages of high precision and easy identification.

Charge-based hysteresis compensation in low impedance piezoelectric actuators by a modified Prandtl–Ishlinskii model

Piezoelectric actuators are one of the most popular actuators in micro- and nano-applications. The main deficiency of these actuators is the hysteretic behavior. Hysteresis not only can destroy the positioning accuracy, but also may lead to instability. In previous researches, hysteresis in the mechanical domain (voltage–position) has been modeled and compensated by several approaches. The limiting condition has been position measurement by a high cost, fine resolution sensor. So, an alternative idea can be compensation in the electrical domain (voltage–charge). In fact, it can be demonstrated that hysteresis compensation in the electrical domain can simultaneously compensate the mechanical one. But, experimental results depict that voltage–charge relation may be time dependent due to low internal impedances. It would lead to “time-dependent hysteresis”. As a result, conventional models cannot be applied for hysteresis identification. In this paper, a modified time-dependent Prandtl–Ishlinskii model is proposed to identify the time-dependent hysteresis in low impedance actuators. Utilizing the proposed model, experimental results validate that the mechanical hysteresis would be appropriately compensated as a result of compensation in the electrical domain.