Prandtl Ishlinskii Research Articles

Modeling and precise tracking control of spatial bending pneumatic soft actuators

In recent years, a variety of pneumatic soft actuators (PSAs) have been proposed due to the development of soft robots in biomimetic robots, medical devices, etc. At the same time, the modeling and control of PSAs remains an open question. In this paper, a spatial bending pneumatic soft actuator (SBPSA) modeling method based on the Prandtl–Ishlinskii (PI) model is proposed, and the inverse model is designed to compensate for hysteresis nonlinearity. Furthermore, an adaptive feedback controller combined with a hysteresis compensator is proposed for the precise control and tracking of SBPSAs. Finally, an experimental platform is built, and experimental results demonstrate the effectiveness of the proposed method for precise tracking.

Nonlinear Control System for Flat Plate Structures Considering Interference Based on Operator Theory and Optimization Method

In recent years, vibration control utilizing smart materials has garnered considerable attention. In this paper, we aim to achieve vibration suppression of a plate structure with a tail-fin shape by employing piezoelectric actuators—one of the smart materials. The plate structure is rigorously modeled based on the Kirchhoff–Love plate theory, while the piezoelectric actuators are formulated in accordance with the Prandtl–Ishlinskii model. This research proposed a control system that addresses the interference effects arising during vibration control by dividing multiple piezoelectric elements into two groups and implementing MIMO control. The efficacy of the proposed control method was validated through simulations and experiments.

Hysteresis Compensation and Trajectory Tracking Control Model for Pneumatic Artificial Muscles

The optimum performance position control of pneumatic artificial muscles (PAM) is restricted by their in-built hysteresis and nonlinearity. The hysteresis is usually depicted by a phenomenological model, while the model mentioned above always only describes the hysteresis phenomenon under certain conditions. Thus, the universality of the compensator is due to its weakness in handling disparate outside conditions. Our research employs the FN–QUPI (feedforward neural network–quadratic unparallel Prandtl–Ishlinskii) model to depict the phenomenon of pressure-displacement hysteresis in PAMs. This model has high-precision expression and generalization ability for the PAM hysteresis phenomenon. According to this, an inverse model of the QUPI operator is established as a feedforward control while combining with the feedback control of incremental PID-type iterative learning. The results show that due to the hysteresis of PAM, the compound control of feedforward control and iterative learning has better tracking performance than the ordinary PID compound control in terms of convergence rate and stability. According to the mean absolute error (MAE) and root mean square error (RMSE) of the tracking process, it can be seen that the control model can achieve accurate nonlinear compensation, and the control system shows excellent robustness to different input signals.

Research on active-passive integrated vibration isolator based on metal rubber and piezoelectric actuator

Abstract The paper introduces a novel active-passive integrated vibration isolator based on metal rubber and piezoelectric actuator, along with an adaptive active vibration control strategy. The active control strategy employs the adaptive dynamic step filtered-x normalized least mean squares algorithm, allowing the step size to adaptively adjust with the error. The secondary control path of the algorithm is modeled using the enhanced rate-dependent Prandtl–Ishlinskii model and the auto-regressive with extra inputs model. The active-passive integrated vibration isolator achieves broadband vibration isolation from 10 to 200 Hz. Compared to passive isolation, the transmissibility decreases from 0.99 to 0.056 at 10 Hz, from 3.02 to 0.068 at the resonance frequency, and from 0.057 to 0.046 at 200 Hz. This study provides a theoretical and experimental foundation for the design of a novel, broadband, and efficient active-passive integrated vibration isolator structure and active control method.

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.

Coupled hysteresis and resonance control of three-degree-of-freedom tip-tilt-piston piezoelectric stage

In order to solve the coupled hysteresis and resonance problems in three-degree-of-freedom tip-tilt-piston piezoelectric stage, a coupled hysteresis model and a coupled integral resonance compensation model are designed. The inverse coupled hysteresis feedforward control combined multi-axis coupled resonance control and high-gain integral tracking controller are used to improve the tracking speed and positioning accuracy of the stage. Firstly, a kinematic model of the stage is built to translate the three-degree-of-freedom motion of the end-effector into the outputs of three piezoelectric actuators. Then, a coupled hysteresis model based on the rate-dependent Prandtl–Ishlinskii model is established and parameterized to simultaneously compensate for the hysteresis of a single piezoelectric actuator and the coupled hysteresis between multiple piezoelectric actuators. The effectiveness of the model is verified by open-loop and closed-loop inverse model feedforward control experiments. Afterwards, a coupled integral resonance compensation model is also established and parameterized to suppresses the resonance and expand the system bandwidth. Finally, the proposed controller is used to conduct tracking experiments on a composite signal, and the results show that the accuracy and speed of trajectory tracking are further improved, and the mechanical coupling between axes are significantly reduced.

An improved dynamic Prandtl–Ishlinskii hysteresis model and a PID-type adaptive sliding mode controller for piezoelectric micro-motion platform

Piezoelectric actuators are widely used in the field of micro-nano driving due to their advantages of fast frequency response, high displacement resolution, large stiffness, small size, and low heat generation. However, the hysteresis non-linearity of piezoelectric actuators severely affects their positioning accuracy during operation. Therefore, the study of hysteresis model and compensation control for piezoelectric actuators has always been a hot topic in the field of micro-nano driving. However, existing hysteresis models and control methods cannot well describe the dynamic hysteresis curve of piezoelectric actuators and track the desired trajectory. To address this issue, this paper proposes an improved dynamic Prandtl–Ishlinskii (IDPI) model and a finite-time trajectory tracking adaptive sliding mode control method based on PID-type. Firstly, this paper introduces the construction and parameter identification method of the IDPI model. Secondly, the accuracy of the IDPI model in describing hysteresis curves under different input signals is verified through experiments, and the effectiveness of the feedforward controller based on the IDPI model is validated experimentally. Then, considering the modeling uncertainty, unmodeled internal dynamics, and external environmental disturbances of the piezoelectric micro-motion platform, a finite-time trajectory tracking adaptive sliding mode controller based on PID-type is designed to suppress the non-linear characteristics of the piezoelectric micro-motion platform. Finally, the performance of the compensator is verified through simulation experiments.

Application of adaptive inverse compensation feedforward—MPC feedback control to AFM piezoelectric micro-positioning platform

As the integral constituent of atomic force microscope (AFM), piezoelectric micro-positioning platform (PMP) plays an pivotal role in AFM working accuracy. However, the PMP platform has hysteretic nonlinear characteristics, which bring challenges to high-precision positioning applications, especially in large travel applications. In this paper, the nonlinear Prandtl–Ishlinskii (P–I) model and the linear auto-regressive with extra inputs dynamic model are cascaded to form the Hammerstein model to characterize the dynamic characteristics of PMP, and the mixed algorithm of the beetle antennae search-differential evolution is designed to identify the parameters of the established model. In order to eliminate the hysteresis effect, a compound controller based on adaptive inverse compensation is proposed, which is composed of feedforward controller of P–I inverse model and model prediction controller feedback controller. As the compound controller depends on modeling accuracy, the tracking error caused by model mismatch is improved by adaptive mechanism. The experimental tracking results of sinusoidal signals and triangular signals of different frequencies show that the proposed method can improve the tracking performance of PMP and verify its effectiveness.

Modeling and parameter identification of rate-dependent hysteresis behavior based on modified-generalized Prandtl–Ishlinskii model

The intrinsic characteristic of piezoelectric actuators (PEA), known as hysteresis, has been demonstrated to diminish the capability and stability of the system significantly. This paper proposes a modified-generalized Prandtl–Ishlinskii (MGPI) model to describe the rate-dependent hysteresis in piezoelectric actuators. The developed model incorporates a voltage change rate function to replace the first part of the generalized Prandtl–Ishlinskii (GPI) model. Additionally, the model integrates the cubic polynomial into the envelope function, along with the dynamic thresholds and weights. When describing the hysteresis of the piezoelectric actuator (PEA), the model parameters are identified using the Improved Grey Wolf Optimizer (IGWO) algorithm. To prevent the algorithm from getting trapped in local optima, the cubic chaotic mapping is utilized for population initialization, as well as a nonlinear convergence factor, and the Levy flight strategy factor is introduced to update the Wolf pack’s position. The rate-dependent hysteresis behavior of a PEA under excitation in the 1–200 Hz frequency range was experimentally measured. The measured data were used to demonstrate the validity of the proposed MGPI model. The relative root-mean-square error and the relative maximum error of the MGPI model are 1.41% and 6.00%, respectively, which are lower than those of the GPI model, which are 3.15% and 10.58%. Under the composite frequency driving, the outputs of the GPI model and MGPI model were compared with the measured data of the PEA, the results suggest that the MGPI model and the IGWO algorithm can more accurately describe the rate-dependent hysteresis of the piezoelectric actuators.

Comprehensive compensation of dynamic hysteresis and creep for piezoelectric actuator

This paper addresses the modeling of dynamic hysteresis and creep in piezoelectric actuators, and employs feedforward open-loop control based on inverse models to compensate for hysteresis and creep phenomena. The comprehensive model consists of quasi-dynamic and dynamic components. The quasi-dynamic model combines the quasi-dynamic Prandtl–Ishlinskii (PI) model with an PI-based linear time-invariant model, while the dynamic part utilizes the auto-regressive exogenous model. The model accurately describes creep and dynamic hysteresis with modeling errors of less than 0.01 μm and 0.14 μm, respectively. The inversion of the comprehensive model has been proven to exhibit unique convergence. Under inverse feedforward control, the improvement in dynamic hysteresis and hysteresis with creep can be achieved at 94% and 83%, respectively. The comprehensive model proposed in this paper accurately describes the dynamic hysteresis and creep phenomena in piezoelectric actuators and realizes open-loop compensation control, achieving precise actuation of piezoelectric actuators.

Development of a Prandtl-Ishlinskii hysteresis model for a large capacity magnetorheological fluid damper

Magnetorheological (MR) fluid (MRF) dampers, serving as fail-safe semi-active devices, exhibit nonlinear hysteresis characteristics, emphasizing the necessity for accurate modeling to formulate effective control strategies in smart systems. This paper introduces a novel stop operator-based Prandtl-Ishlinskii (PI) model, featuring a reduced parameter set (seven), designed to estimate the nonlinear hysteresis properties of a large-scale bypass MRF damper with variable stiffness capabilities under varying applied current. With only seven parameters, the model realizes current, displacement, and rate dependencies. The force-displacement and force-velocity responses of the designed MRF damper were experimentally characterized under broad ranges of applied current (0–2 A), excitation frequency (0.5–4 Hz), and displacement amplitude (1–2.5 mm). A training dataset was subsequently used to develop a novel field-dependent modified PI model, incorporating multiple hysteresis operators with and without a friction element. The proposed model accurately predicted the MRF damper behavior within the training dataset, and its validity was assessed against data from diverse experimental conditions. The PI model with friction element generally outperformed the model without friction when frequency exceeds 0.5 Hz, demonstrating its ability to characterize nonlinear hysteresis force-displacement and force-velocity properties of the MRF damper under the ranges of applied current and excitations considered with reasonable accuracy. Experimental data were also estimated by the Bouc–Wen model, and compared with those obtained via the formulated PI model, affirming the overall superiority of the proposed PI models considering the computational cost, and total number of parameters. Leveraging the simplicity, minimal parameter requirements, and analytic invertibility of PI models, the proposed PI model is considered a superior choice for modeling and subsequently controlling smart structures employing MRF dampers.

Model predictive control for a bending pneumatic muscle based on an online modified generalized Prandtl–Ishlinskii model

Model predictive control for a bending pneumatic muscle based on an online modified generalized Prandtl–Ishlinskii model

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

Model-free compensation learning control of asymmetric hysteretic systems with initial state learning

In this article, a model-free compensation learning control scheme is presented for asymmetric hysteretic systems to achieve high-precision output tracking, where the effects of asymmetric input hysteresis nonlinearities are denoted by the asymmetric Prandtl–Ishlinskii model (APIM). In the presented control scheme, the feedforward iterative learning technique is generalized to the control design of asymmetric hysteretic systems. To improve the robustness and dynamic performance of the existing feedforward learning control for hysteretic systems, both the feedback action and the differential term are utilized to design a discrete-time PD-P open-closed-loop learning control law for simultaneously compensating the asymmetric input hysteresis nonlinearities and the linear dynamics effects without constructing compensators based on their models. The initial state error of such a system is also considered, a modified initial state learning strategy is proposed to ensure the initial state error tends to a prescribed level with the increasing of iterations. By fully analyzing the properties of the APIM, the convergence conditions with respect to the input error, the state error, and the tracking error along the iteration domain are given. The simulation and experimental results are provided to demonstrate strong robustness and excellent tracking accuracy with the proposed model-free compensation learning control scheme.

Model-based linear control of nonlinear pneumatic soft bending actuators

Advanced model-based control techniques hold great promise for the precise control of pneumatic soft bending actuators (PSBAs) with strong nonlinearities. However, most previous controllers were designed in a cumbersome nonlinear form. Considering the simplicity of linear system theory, this paper presents a novel perspective on using model-based linear control to handle nonlinear PSBAs, and for the first time, summarizes two methodologies, global linearization and pseudo-linear construction. Derived from them, Koopman-based and hysteresis-based linear control approaches are proposed, respectively. For the former, a novel fusion prediction equation is developed to build a high-fidelity Koopman model, realizing global linearization, and then the linear model predictive control (MPC) is deployed. For the latter, the inverse of the generalized Prandtl–Ishlinskii (GPI) model cascades with the PSBA to construct a pseudo-linear system, eliminating the asymmetric hysteresis, which activates the linear proportional-integral-derivative (PID) control. It is worth noting that the above two are based on data-driven models adapted to various PSBAs with material and structural customization. Finally, the two model-based linear control approaches are verified and compared through a series of experiments. The results show that the proposed linear controls, with more concise designs, achieve comparable or even superior performance than nonlinear controls.

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.

On the hysteresis characteristics and compensation control strategy of a pneumatic hydrodynamic retarder

As the core component of the hydrodynamic retarder braking system, pneumatic proportional solenoid valve plays an important role in the process of braking torque control. However, the hysteresis characteristic caused by friction and air compressibility will lead to the deviation of braking torque. In order to solve this problem, the hysteresis characteristic of pneumatic proportional solenoid valve is experimentally studied, and the causes of hysteresis phenomenon are analyzed through mathematical modeling of the valve core. Then, a high-precision prediction model of braking torque is obtained by combining Computational Fluid Dynamics (CFD) and response surface method, which is used to construct the mapping relationship between control air pressure and braking torque. Furthermore, a feedforward controller based on Prandtl–Ishlinskii inverse model is designed to compensate the hysteresis characteristics. On this basis, a compound hysteresis characteristics compensation control strategy combined with Proportional Integral Derivative (PID) feedback control is proposed to realize the accurate control of braking torque. Finally, the test verification of braking torque control is carried out. The average error of compound control, feedforward control, and PID control under torque step working condition is 3.16%, 5.97%, and 6.23%, and the response time is 3.75, 8.75, and 3.2 s, respectively. The compound control strategy also has smaller torque error and shorter response time under both ramp torque tracking and constant torque conditions. To sum up, this compound control strategy can effectively compensate the hysteresis characteristics of the pneumatic hydrodynamic retarder and ensure the driving safety by improving the control accuracy and response speed of braking torque.

Reinforcement Learning-Based Adaptive Optimal Control for Nonlinear Systems With Asymmetric Hysteresis.

This article investigates the adaptive optimal tracking problem for a class of nonlinear affine systems with asymmetric Prandtl-Ishlinskii (PI) hysteresis nonlinearities based on actor-critic (A-C) learning mechanisms. Considering the huge obstacles arising from the uncertainty of hysteresis nonlinearity in actuators, we develop a scheme for the conflict between the construction of Hamilton functions and hysteresis nonlinearity. The actuator hysteresis forces the input into a hysteresis delay, thus preventing the Hamilton function from getting the current moment's input instantly and thus making optimization impossible. In the first step, an inverse model is constructed to compensate for the hysteresis model with a shift factor. In the second step, we compensate for the control input by designing a feedback controller and incorporating the estimation and approximation errors into the Hamilton error. Optimal control, the other part of the actual control input, is obtained by taking partial derivatives of the Hamiltonian function after the nonlinearities have been circumvented. At the end, a simulation is given to validate the developed solution.

Memory reduction of rate-dependent Prandtl–Ishlinskii compensators in applications on high-precision motion systems

Inverse compensation of rate-dependent Prandtl–Ishlinskii operators with finitely many thresholds can be carried out explicitly under some structural conditions. However, if the number of thresholds is very high, the formulas become complicated. A simple and elegant framework for the investigation of such problems encompasses a recently proposed extension of the class of rate-dependent Prandtl–Ishlinskii operators to the case of a whole continuum of play operators with time-dependent thresholds. Indeed, such a theory allows for reducing the number of necessary thresholds in the compensation procedure and estimating the errors of the memory-discrete compensation in terms of the distance between the thresholds and weights of the individual plays. Following these results, our goal in this work is the validation of these theoretical models via numerical simulations and experimental results. In particular, we show that high accuracy of the hysteresis compensation algorithm can be achieved even with a relatively small number of thresholds.

Robust Control of Dielectric Elastomer Smart Actuator for Tracking High-Frequency Trajectory

Dielectric elastomer smart actuator (DESA) has shown great potentials in soft robot applications. The control of the DESA is one of key issues for soft robots. However, the precise control of the DESA is a challenge work, especially when tracking high-frequency trajectories and guaranteeing a certain robustness. To this end, this paper presents a robust tracking control method for a DESA to track various high-frequency trajectories. Firstly, to characterize the complex nonlinear properties (including the quadratic input, asymmetric and rate-dependent hysteresis, as well as creep) of the DESA, its dynamic model is built by combining a square function module, a modified Prandtl-Ishlinskii (P-I) hysteresis model and a linear system. Subsequently, by combining the inverse of the modified P-I hysteresis model with a square root function module, the quadratic input and asymmetric hysteresis properties of the DESA are compensated to reduce tracking the control error. Since model uncertainties and external disturbances are unavoidable in practical applications, a robust controller is designed to achieve the tracking control of the DESA, the controller also guarantees the robustness of the control system. In the end, the effectiveness of the proposed control method is demonstrated through a series of tracking control experiments with different high-frequency trajectories, whose maximal frequency is 8 (Hz). The root- mean-square errors of all experimental results are lower than 1.5%, which proves that the presented method is remarkable from the respective of the practical application.