Modified Prandtl-Ishlinskii Model Research Articles

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

Adaptive Set-Membership Filter Based Discrete Sliding Mode Control for Pneumatic Artificial Muscle Systems With Hardware Experiments

Pneumatic artificial muscle (PAM), featuring good flexibility and safety, has been widely used in rehabilitation and bionic robots. However, the complex hysteretic nonlinearities and uncertainties of the PAM cause great difficulties and challenges to the accurate modeling and controller design, especially when confronted with unknown external disturbances in applications. This paper proposes a robust control strategy with disturbance compensation for the hysteresis compensation and trajectory tracking of PAMs. Considering the high hysteretic nonlinearity of the PAM, a modified Prandtl-Ishlinskii model is used as a feedforward hysteresis compensator. For the linearized system, adaptive set-membership filtering (ASMF) is used to estimate the nonlinear terms and external disturbances of the overall system. A sliding mode controller (SMC) with disturbance compensation is designed and cascaded to the feedforward hysteresis compensator in series. The stability of the closed-loop system is theoretically proved. The proposed method guarantees that the tracking error of the PAM system is bounded. Finally, the effectiveness and robustness of the proposed controller are verified via a series of experiments on an in-house built testbench for PAMs. <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Note to Practitioners</i> —With the increasing demand on human-robot interaction, the safety and compliance of robots have become one key requirement. PAM is a compliant actuator, exhibiting good flexibility, safety, and clean energy. PAM is widely used in rehabilitation robots, whereas its strong hysteresis nonlinearity and sensitivity to external disturbances affect its motion accuracy. This paper proposes an ASMF-based discrete SMC, which uses an inverse hysteresis model to compensate for the strong hysteresis of the PAM and uses ASMF to estimate the lumped disturbance of the system. Compared with the other filters, ASMF is unique in that its estimation error is bounded, which is very useful in the stability proof of the overall system. The effectiveness of the proposed controller is experimentally verified. Experimental results show that PAM’s hysteresis can be efficiently compensated, and the influence of external disturbances can be attenuated by the proposed controller, resulting in improved motion accuracy and robustness. In future work, efforts will be directed towards the modeling and control of PAMs in multi-DOF robots.

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.

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.

Hysteresis modeling and compensation of a pneumatic end-effector based on Gaussian process regression

This paper proposes a data-driven statistical learning method to identify the force-pressure hysteresis of a pneumatic end-effector based on Gaussian process regression (GPR). Given the actual complex characteristics of the hysteresis phenomena, GPR is used to establish the relationship between the input and output variables of the hysteresis as a first-order nonlinear differential equation ignoring the high-order dynamics without specifying any hyperparameters or considering the special features of hysteresis loops. The inverse hysteresis model can be derived directly. The parameters of the GPR are determined by choosing low-frequency triangle-wave pressure excitations as the training set, and then the prediction performance of the present model is tested under different types, amplitudes, and period conditions of pressure signals as the verification sets. Compared with two types of modified Prandtl-Ishlinskii model, the proposed model achieves better accuracy in both training and verification sets. Based on the inverse hysteresis feedforward compensator, comparative experiments of the force-tracking control are conducted in the form of the open-loop and closed-loop controllers, of which the results indicate the effectiveness and superiority of the proposed model.

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.

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.

Multi-DOF compensation of piezoelectric actuators with recursive databases

Abstract Piezoelectric actuators are subject to nonlinear effects when voltage-driven in open-loop control. In particular, hysteresis and creep effects are dominating nonlinearities that significantly deteriorate performance in tracking control scenarios. In this paper, we present an online compensator suitable for piezoelectric actuators that is based on the modified Prandtl-Ishlinskii model and utilizes recursive databases for the compensation of nonlinearities. The compensator scheme is furthermore extended to systems with more than one degree of freedom (DOF) such as Cartesian manipulators by employing a decoupling control design to mitigate inherent cross-coupling disturbances. In order to validate our theoretical derivations, experiments are conducted with coupled trajectories on a commercial 3-DOF micro-positioning unit driven by piezoelectric actuators.

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.

Hysteresis modeling and trajectory tracking control of the pneumatic muscle actuator using modified Prandtl–Ishlinskii model

The pneumatic muscle actuator (PMA) has attracted extensive attentions from both industrial and academic fields due to its high power/weight ratio and significant compliance. However, on the one hand the inherent hysteresis nonlinearity has high influence on the accuracy of trajectory tracking control of PMA, and on the other hand a system actuated by PMAs is at a high cost. This paper presents a modified Prandtl–Ishlinskii (MPI) model for the asymmetric hysteresis characterization and compensation of the PMA using fast switching valves. By using the Levenberg–Marquardt (L-M) method, the parameters in the MPI model are identified. To compensate the nonlinear length/pressure hysteresis and reduce the cost, a cascade control scheme using fast switching valves based on the inverse MPI model is developed to realize the trajectory tracking control. The experimental result shows the effectiveness of the proposed control scheme on compensating the asymmetric length/pressure hysteresis of the PMA.

Positive acceleration, velocity and position feedback based damping control approach for piezo-actuated nanopositioning stages

This paper proposes a new damping control approach with positive acceleration, velocity and position feedback (PAVPF) scheme for piezo-actuated nanopositioning stages to implement high-bandwidth operation. To achieve this objective, the intrinsic hysteresis nonlinearity of the piezoelectric actuator is firstly handled by a feedforward compensator with a modified Prandtl–Ishlinskii model. Afterwards, the PAVPF controller with the pole-placement method is implemented to suppress the lightly damped resonant mode of the hysteresis compensated system. With the PAVPF controller, the poles of the damped system in a third-model can be placed to arbitrary positions with an analytical method. Finally, for accurately tracking a predefined trajectory, a high-gain proportional-integral (PI) controller is designed, which could deal with the disturbance and the unmodeled dynamics. For verifying the proposed PAVPF-based control approach, comparative experiments with positive velocity and position feedback controller and with PI controller are conducted on a piezo-actuated nanopositioning stage. Experimental results demonstrate that the developed control approach with PAVPF controller is effective on damping control and improves the control bandwidth of the conventional PI controller from 111 Hz to 766 Hz, which leads to the significant increase of the tracking speed.

Recursive Online Compensation of Piezoelectric Nonlinearities via a Modified Prandtl-Ishlinskii Approach

Piezoelectric actuators are often employed in micro- and nanopositioning devices due to their extremely fine positioning resolution but exhibit strong nonlinear effects (predominantly hysteresis and creep) which pose a considerable challenge for the control community. For online compensation of these effects, the modified Prandtl-Ishlinskii model is particularly suitable since its inverse can be found analytically by a parameter transformation. However, this model-based approach has not yet made its way to devices that target positioning tasks. Therein, trajectories typically contain segments with varying final times, small ranges of motion, or stationary states such that the hysteresis and creep effects are not optimally excited and frequency-dependence is induced which ultimately leads to a deteriorated compensation performance. This paper proposes an online approach to overcome the problem by taking into account prior hysteresis/creep information in a recursive manner based on databases for better tracking performance and a more robust compensation of the aforementioned nonlinearities. In order to show the efficacy of the proposed approach, experimental results are provided by using trajectories with varying final times, stationary states and alternating small/large ranges of motion on a micro-positioning unit driven by piezoelectric actuators.

Position control of a single pneumatic artificial muscle with hysteresis compensation based on modified Prandtl-Ishlinskii model.

High-performance position control of pneumatic artificial muscles is limited by their inherent nonlinearity and hysteresis. This study aims to model the length/pressure hysteresis of a single pneumatic artificial muscle and to realize its accurate position tracking control with forward hysteresis compensation. The classical Prandtl-Ishlinskii model is widely used in hysteresis modelling and compensation. But it is only effective for symmetric hysteresis. Therefore, a modified Prandtl-Ishlinskii model is built to characterize the asymmetric length/pressure hysteresis of a single pneumatic artificial muscle, by replacing the classical play operators with two more flexible elementary operators to independently describe the ascending branch and descending branch of hysteresis loops. On the basis, a position tracking controller, which is composed of cascade forward hysteresis compensation and simple proportional pressure controller, is designed for the pneumatic artificial muscle. Experiment results show that the MPI model can reproduce the length/pressure hysteresis of the pneumatic artificial muscle, and the proposed controller for the pneumatic artificial muscle can track the reference position signals with high accuracy. By modelling the length/pressure hysteresis with the modified Prandtl-Ishlinskii model and using its inversion for compensation, precise position control of a single pneumatic artificial muscle is achieved.

Full closed-loop controls of micro/nano positioning system with nonlinear hysteresis using micro-vision system

The design and full closed-loop controls of micro/nano positioning system are presented in this paper. The micro/nano positioning system mainly consists of a 3-DOF compliant mechanism, a micro-vision system and three piezoelectric actuators. In order to compensate for nonlinear hysteresis in the piezoelectric actuators, an adaptive controller using direct inverse modeling approach based on a modified Prandtl–Ishlinskii model is presented. Instead of using traditional displacement sensors, a micro-vision system (MVS) is used to obtain feedback signals online. Based on the micro-vision system, a full closed-loop tracking control is developed to enhance the positioning accuracy of micro/nano positioning system. Taking into account the coupling motions, another full closed-loop tracking control with a decoupling feedforward controller is also proposed. Experimental results demonstrate that the full closed-loop controls can improve the positioning accuracy when compared with an open-loop control. Furthermore, the decoupling feedforward controller can improve the tracking performance of full closed-loop tracking control as well.

Modelling Length/Pressure Hysteresis of a Pneumatic Artificial Muscle using a Modified Prandtl-Ishlinskii Model

Pneumatic artificial muscles have been widely used in various fields owing to their inherent compliance and high power-to-weight ratio. However, the natural hysteresis nonlinearity including length/pressure hysteresis and force/pressure hysteresis degrades their performance in precise tracking control, making it necessary to build a mathematical hysteresis model for hysteresis compensation. This paper deals with the modelling of length/pressure hysteresis of pneumatic artificial muscles. The length/pressure hysteresis loops measured by the isotonic test are found to be asymmetric and independent of the external load when the load is small. Considering that the classical Prandtl-Ishlinskii model is only effective for symmetric hysteresis, a modified Prandtl-Ishlinskii model is proposed to describe the length/pressure hysteresis behaviour. The developed model utilizes two asymmetric operators with simple mathematical forms to independently model the ascending branch and descending branch of hysteresis loops. The model parameters are identified using the recursive least square algorithm. Comparisons between simulation results and experimental measurements demonstrate that the proposed model can characterize the asymmetric major hysteresis loop and minor hysteresis loops with high accuracy.

Position control of a bio-inspired semi-active joint with direct inverse hysteresis modeling and compensation

Accurate position control of pneumatic artificial muscle actuated systems has always been difficult due to their inherent nonlinear hysteresis characteristics. This article designs a bio-inspired semi-active robotic joint that is synergistically driven by a pneumatic artificial muscle and an extension spring by imitating the mechanism of energy storage and return in animals and then studies its position tracking control with hysteresis compensation. To simplify the derivation of the inverse hysteresis model which functions as hysteresis compensator, a novel approach termed as direct inverse hysteresis modeling is adopted. With this method, the inversion of the asymmetric angle–pressure hysteresis of the semi-active joint is modeled directly from experimental measurements by a modified Prandtl–Ishlinskii model. On the basis, a closed-loop position tracking controller composed of forward hysteresis compensation and conventional proportional–integral–derivative control is designed for the joint. Experimental results indicate that the proposed controller can track the reference position signals with high accuracy.