Prandtl Ishlinskii Research Articles

Asymmetric Prandtl-Ishlinskii Hysteresis Model for Giant Magnetostrictive Actuator

An asymmetric Prandtl–Ishlinskii (API) hysteresis model for a giant magnetostrictive actuator (GMA) is proposed in this paper. The classical Prandtl–Ishlinskii (PI) model is analyzed and divided into two parts: linear function and operator summation. To enhance model asymmetry, a polynomial function is used in the API model as the center curve of the hysteresis instead of the linear function. The remaining curve of the hysteresis is modeled by a new operator that provides some basic asymmetric hysteresis. In this manner, the proposed API model requires relatively less operators and fewer parameters to describe the asymmetric hysteresis behavior of the GMA. All parameters of the API model are identified by a standard least square method. Simulation results show that the API model is very successful in formulating an asymmetric hysteresis of the GMA. In addition, it provides better identification results compared with the classical PI model.

Further Results on Hysteresis Compensation of Smart Micropositioning Systems With the Inverse Prandtl–Ishlinskii Compensator

A formula that characterizes the output of the inverse compensation is derived when the inverse Prandtl–Ishlinskii hysteresis model is applied as a feedforward compensator. For that the composition property as well as the initial loading curve of the Prandtl–Ishlinskii model is used to obtain this formula. We demonstrate, therefore, that the output of the feedforward controlled system is linear versus the input reference with an additional nonlinear and bounded term. To illustrate the interest of this theoretical result, we propose an experimental application with a piezoelectric actuator (PEA). First, we apply the Prandtl–Ishlinskii feedforward technique to control the PEA. Then, the formula of the previous theoretical result is used to construct an $H_\infty $ feedback control from the feedforward controlled actuator. The experimental results demonstrate the efficiency of the calculated controller and, therefore, the benefits of the formula concerning the output of the inverse compensation.

A generalized Prandtl-Ishlinskii model for characterizing the rate-independent and rate-dependent hysteresis of piezoelectric actuators.

In this paper, a generalized hysteresis model is developed to describe both rate-independent and rate-dependent hysteresis in piezoelectric actuators. Based on the classical Prandtl-Ishlinskii (P-I) model, the developed model adds a quadratic polynomial and makes other small changes. When it is used to describe rate-independent hysteresis, the parameters of the model are constants, which can be identified by self-adaptive particle swarm optimization. The effectiveness of this rate-independent modified P-I model is demonstrated by comparing simulation results of the developed model and the classic Prandtl-Ishlinskii model. Simulation results suggest that the rate-independent modified P-I model can describe hysteresis more precisely. Compared with the classical P-I model, the rate-independent modified P-I model reduces modeling error by more than 50%. When it is used to describe rate-independent hysteresis, a one-side operator is adopted and the parameters are functions with input frequency. The results of the experiments and simulations have shown that the proposed models can accurately describe both rate-independent and rate-dependent hysteresis in piezoelectric actuators.

Experimental comparison of rate-dependent hysteresis models in characterizing and compensating hysteresis of piezoelectric tube actuators

An experimental study has been carried out to characterize rate-dependent hysteresis of a piezoelectric tube actuator at different excitation frequencies. The experimental measurements were followed by modeling and compensation of the hysteresis nonlinearities of the piezoelectric tube actuator using both the inverse rate-dependent Prandtl–Ishlinskii model (RDPI) and inverse rate-independent Prandtl–Ishlinskii model (RIPI) coupled with a controller. The comparison of hysteresis modeling and compensation of the actuator with both models is presented.

Modeling and compensating the dynamic hysteresis of piezoelectric actuators via a modified rate-dependent Prandtl–Ishlinskii model

This paper presents a modified rate-dependent Prandtl–Ishlinskii (MRPI) model for the description and compensation of the rate-dependent asymmetric hysteresis in piezoelectric actuators. Different from the commonly used approach with dynamic weights or dynamic thresholds, the MRPI model is formulated by employing dynamic envelope functions into the play operators, while the weights and thresholds of the play operators are still static. By this way, the developed MRPI model has a relatively simple mathematic format with fewer parameters and easier parameter identification process. The benefit for the developed MRPI model also lies in the fact that the existing control approaches can be directly adopted with the MRPI model for hysteresis compensation in real-time applications. To validate the proposed model, an open-loop tracking controller and a closed-loop tracking controller are developed based on a dynamic hysteresis compensator, which is directly constructed with the MRPI model. Comparative experiments are carried out on a piezo-actuated nanopositioning stage. The experimental results demonstrate the effectiveness and superiority of the controllers based on the developed MRPI model compared to the controllers based on the rate-independent P–I model and the rate-dependent P–I model with dynamic weighting functions.

Improved direct inverse tracking control of a piezoelectric tube scanner for high-speed AFM imaging

For a piezoelectric tube scanner (PTS), this paper proposes an improved direct inverse tracking control algorithm and apply it to an atomic force microscope (AFM) to accomplish high-speed scanning tasks. That is, to enhance the high-speed tracking control performance of a PTS, an improved direct inverse rate-dependent Prandtl–Ishlinskii (P–I) model is firstly constructed, which includes a polynomial module to eliminate the structure nonlinearity. Based on the model, a practical feedforward control law is then designed to implement high-speed tracking control for a high-frequency trajectory with strong robustness, which presents the advantages of high-speed response, simple structure and convenient implementation. Subsequently, the designed feedforward law is combined with a feedback component, and the combined control strategy is employed in an AFM to accomplish fast imaging tasks. Numerous experimental results are then collected, which convincingly demonstrate the superior performance of the proposed practical model/control scheme.

Modeling of the interleaved hysteresis loop in the measurements of rotational core losses

The measurement of core losses in machine laminations reveals a fundamental difference between rotational and pulsating types. Rotational core losses under rotating fields decrease at high flux density, while pulsating losses keep increasing steadily. Experimental analyses of loss components Px and Py in x and y directions with rotating fields show that the loss decreases in one loss component and sometimes attains negative values. Tracking the evolution of hysteresis loops along this loss component discloses a peculiar behavior of magnetic hysteresis, where the loop changes its path from counterclockwise to clockwise within a cycle of magnetization process, the so called interleaved hysteresis loop. This paper highlights a successful procedure for modeling the interleaved hysteresis loop in the measurement of rotational core losses in electrical machine laminations using the generalized Prandtl–Ishlinskii (PI) model. The efficiency of the proposed model is compared to Preisach model. Results and conclusion of this work are of importance toward building an accurate model of rotational core losses.

Adaptive failure compensation of hysteretic actuators for a class of nonlinear systems via output feedback

SummaryIn this paper, an adaptive output feedback control scheme is proposed for a class of nonlinear systems with possible actuator failures. The system not only involves unknown parameters but also takes nonlinear terms linear in the unmeasured states into account and is preceded by hysteretic actuators whose nonlinearities are characterized by the saturated Prandtl–Ishlinskii model. By developing a high‐gain observer with one dynamic gain, the closed‐loop stability and arbitrarily small tracking error can be guaranteed. Furthermore, by estimating the upper bound of the unknown time‐varying (piecewise constant) parameters caused by the actuator failures, together with an initialization technique, it is proved that the tracking performance is achievable, which is fundamentally different from the commonly accepted concept that with actuator failures, it is impossible to improve the tracking performance by using initialization technique. Copyright © 2015 John Wiley & Sons, Ltd.

Output feedback adaptive control of a class of nonlinear discrete-time systems with unknown control directions and preceded by hysteresis

This paper considers the output feedback adaptive controller design problem for a class of discrete-time nonlinear systems in output feedback form with unknown control directions and preceded by unknown hysteresis. The problem of lacking in a-priori knowledge on the control directions and unknown hysteresis are solved by using the discrete Nussbaum gain and Prandtl–Ishlinskii model, respectively. The system is transformed into the form of a nonlinear auto regressive moving average (NARMA) model to construct an output feedback control. To overcome the noncausal problem in the control design, future output prediction laws and parameter update laws with the dead-zone technique are constructed on the basis of the NARMA model. The proposed control algorithm guarantees that all the signals in the controlled system are bound and the output tracking error is made to be neighbourhood around zero, ultimately. Finally, simulations are performed on a nonlinear system to show the effectiveness of the proposed method

Optimal compression of generalized Prandtl–Ishlinskii hysteresis models

Prandtl–Ishlinskii (PI) hysteresis models have been used widely in magnetic and smart material-based systems. A generalized PI model, consisting of a weighted superposition of generalized play operators, is capable of characterizing saturated and asymmetric hysteresis. The fidelity of the model hinges on accurate representation of the envelope functions, play operator radii, and corresponding weights. Existing work has typically adopted some predefined play radii, the performance of which could be far from optimal. In this paper, novel schemes are proposed for optimally compressing generalized PI models, subject to a complexity constraint characterized by the number of play operators. An information-theoretic tool, entropy, is adopted to capture the information loss in replacing a group of weighted play operators with a single play operator. The optimal compression algorithm is reformulated as an optimal control problem and solved with dynamic programming, the computational complexity of which is shown to be much lower than that of exhaustive search. Simulation results are first presented to examine the performance of the proposed approach in approximating a PI model consisting of a large number of play operators, where cases with different types of weight distributions are explored. The approach is further applied to compress an experimentally identified generalized PI model for the complex hysteresis behavior between the resistance and temperature of vanadium dioxide, a promising multifunctional material. Both simulation and experimental results demonstrate that the proposed algorithms in general yield far more superior performance than a typically adopted scheme where the play radii are assigned uniformly.

An inverse Prandtl–Ishlinskii model based decoupling control methodology for a 3-DOF flexure-based mechanism

A modified Prandtl–Ishlinskii (P–I) hysteresis model is developed to form the feedforward controller for a 3-DOF flexure-based mechanism. To improve the control accuracy of the P–I hysteresis model, a hybrid structure that includes backlash operators, dead-zone operators and a cubic polynomial function is proposed. Both the rate-dependent hysteresis modeling and adaptive dead-zone thresholds selection method are investigated. System identification was used to obtain the parameters of the newly-developed hysteresis model. Closed-loop control was added to reduce the influence from external disturbances such as vibration and noise, leading to a combined feedforward/feedback control strategy. The cross-axis coupling motion of the 3-DOF flexure-based mechanism has been explored using the established controller. Accordingly, a decoupling feedforward/feedback controller is proposed and implemented to compensate the coupled motion of the moving platform. Experimental tests are reported to examine the tracking capability of the whole system and features of the controller. It is demonstrated that the proposed decoupling control methodology can distinctly reduce the coupling motion of the moving platform and thus improve the positioning accuracy and trajectory tracking capability.

Output feedback prescribed performance control for interconnected time-delay systems with unknown Prandtl–Ishlinskii hysteresis

In this paper, output feedback prescribed performance control problem is investigated for interconnected time-delay systems with unknown Prandtl–Ishlinskii hysteresis. Decentralized reduced-order observer independent of time delay is first designed. Then, by using a novel state transformation, the prescribed performance control problem is transformed to stabilization problem. An adaptive variable structure control approach is proposed to address the Prandtl–Ishlinskii hysteresis model without necessarily constructing a hysteresis inverse. Subsequently, via backstepping method, the output feedback controller is designed. Based on Lyapunov stability theory, the constructed controllers are proved to guarantee the stability of the closed-loop system with the prescribed performance. Finally, a simulation example is given and the results verify the effectiveness of the proposed design technique.

Neural Controller Design-Based Adaptive Control for Nonlinear MIMO Systems With Unknown Hysteresis Inputs.

This paper studies an adaptive neural control for nonlinear multiple-input multiple-output systems in interconnected form. The studied systems are composed of N subsystems in pure feedback structure and the interconnection terms are contained in every equation of each subsystem. Moreover, the studied systems consider the effects of Prandtl-Ishlinskii (PI) hysteresis model. It is for the first time to study the control problem for such a class of systems. In addition, the proposed scheme removes an important assumption imposed on the previous works that the bounds of the parameters in PI hysteresis are known. The radial basis functions neural networks are employed to approximate unknown functions. The adaptation laws and the controllers are designed by employing the backstepping technique. The closed-loop system can be proven to be stable by using Lyapunov theorem. A simulation example is studied to validate the effectiveness of the scheme.

Hysteresis modeling and tracking control for a dual pneumatic artificial muscle system using Prandtl–Ishlinskii model

This study investigated pressure/length hysteresis characteristics. Instead of the conventional force/length hysteresis model, a Prandtl–Ishlinskii (P–I) model of a dual pneumatic artificial muscle (PAM) system is presented. For the comparison, an alternative hysteresis model such as Bouc–Wen (B–W) model is also considered. All model parameters are identified by real code genetic algorithm (RCGA). Different feedback control strategies are combined with a feed-forward controller based on a P–I model for hysteresis compensation to reduce the tracking error of the dual PAM system. The experimental results validated the use of the proposed controller for trajectory tracking of PAM systems.

Decentralised adaptive output feedback control for interconnected nonlinear systems preceded by unknown hysteresis

In this paper, a global decentralised adaptive output feedback control is proposed for a class of interconnected nonlinear systems with each subsystem involving not only unknown parameters but also nonlinear terms linear in the unmeasured states, and being preceded with hysteresis nonlinearity described by the saturated PI (Prandtl–Ishlinskii) model. By introducing a modified high-gain observer, together with an initialisation technique, the global stability of the overall closed-loop system can be guaranteed and the tracking performance for each subsystem can be achieved. Furthermore, with a new adaptive law, the restrictive assumption in the existing literature about the upper bound of the strength of the interactions between the subsystems is removed.

Adaptive Neural Network Dynamic Surface Control for a Class of Time-Delay Nonlinear Systems With Hysteresis Inputs and Dynamic Uncertainties.

In this paper, an adaptive neural network (NN) dynamic surface control is proposed for a class of time-delay nonlinear systems with dynamic uncertainties and unknown hysteresis. The main advantages of the developed scheme are: 1) NNs are utilized to approximately describe nonlinearities and unknown dynamics of the nonlinear time-delay systems, making it possible to deal with unknown nonlinear uncertain systems and pursue the L∞ performance of the tracking error; 2) using the finite covering lemma together with the NNs approximators, the Krasovskii function is abandoned, which paves the way for obtaining the L∞ performance of the tracking error; 3) by introducing an initializing technique, the L∞ performance of the tracking error can be achieved; 4) using a generalized Prandtl-Ishlinskii (PI) model, the limitation of the traditional PI hysteresis model is overcome; and 5) by applying the Young's inequalities to deal with the weight vector of the NNs, the updated laws are needed only at the last controller design step with only two parameters being estimated, which reduces the computational burden. It is proved that the proposed scheme can guarantee semiglobal stability of the closed-loop system and achieves the L∞ performance of the tracking error. Simulation results for general second-order time-delay nonlinear systems and the tuning metal cutting system are presented to demonstrate the efficiency of the proposed method.

A rate-dependent Prandtl-Ishlinskii model for piezoelectric actuators using the dynamic envelope function based play operator

In this paper, a novel rate-dependent Prandtl-Ishlinskii (P-I) model is proposed to characterize the rate-dependent hysteresis nonlinearity of piezoelectric actuators. The new model is based on a modified rate-dependent play operator, in which a dynamic envelope function is introduced to replace the input function of the classical play operator. Moreover, a dynamic density function is utilized in the proposed P-I model. The parameters of the proposed model are identified by a modified particle swarm optimization algorithm. Finally, experiments are conducted on a piezo-actuated nanopositioning stage to validate the proposed P-I model under the sinusoidal inputs. The experimental results show that the developed rate-dependent P-I model precisely characterize the rate-dependent hysteresis loops up to 1000 Hz.

Improving Atomic Force Microscopy Imaging by a Direct Inverse Asymmetric PI Hysteresis Model

A modified Prandtl–Ishlinskii (PI) model, referred to as a direct inverse asymmetric PI (DIAPI) model in this paper, was implemented to reduce the displacement error between a predicted model and the actual trajectory of a piezoelectric actuator which is commonly found in AFM systems. Due to the nonlinearity of the piezoelectric actuator, the standard symmetric PI model cannot precisely describe the asymmetric motion of the actuator. In order to improve the accuracy of AFM scans, two series of slope parameters were introduced in the PI model to describe both the voltage-increase-loop (trace) and voltage-decrease-loop (retrace). A feedforward controller based on the DIAPI model was implemented to compensate hysteresis. Performance of the DIAPI model and the feedforward controller were validated by scanning micro-lenses and standard silicon grating using a custom-built AFM.

Robust hydraulic actuator force control through relief discharge

Actuator force control in servo-hydraulic systems has numerous applications in machinery and industrial settings. The prevalent hydraulic configuration for force control utilizes a proportional directional control valve to properly route the supplied hydraulic fluid toward the chambers of the actuator. It is well known that this configuration imposes certain physical limitations on force-tracking performance. In this article, a different configuration is utilized to improve the hydraulic force control performance. This is done by replacing the proportional directional control valve with a proportional double-stage relief valve. It is shown that this new setting is superior in its achievable force-tracking performance. This comes at the cost of a more challenging control problem compared to the proportional directional control valve case, due to the presence of un-modeled dynamics and unknown parameters. A robust [Formula: see text] control approach has been followed as the main solution to address uncertainty problems. However, conservative robust design imposes its own limitations on force-tracking performance. To address this issue, a combination of additional switched-proportional–integral–derivative and Prandtl–Ishlinskii hysteresis compensation control loops is proposed. Conditions are derived under which the additional loops improve the performance of the robust [Formula: see text] controller. The effectiveness of the proposed configuration and control method is demonstrated by the experimental results.

Analysis and compensation of hysteresis of PZT micro-actuator used in high precision dual-stage servo system

Piezoelectric micro–actuator made from (PZT) has been a suitable choice as the secondary actuator of a dual–stage actuator system. However, like any other piezoelectric actuator, this secondary PZT micro–actuator also exhibits hysteresis which is the inherent nonlinearity characteristic of piezoelectric material. Therefore, the advantage gained by the precision of secondary actuator is somewhat lost by this hysteresis, if not compensated. This paper proposes a rigorous technique for analysing, identifying and compensating the hysteresis of PZT actuator used in dual–stage actuator system. Identification and compensation are the two parts of the paper. In the first part of the paper, the parameters of generalised Prandtl–Ishlinskii (GPI) model are identified offline by particle swarm optimisation (PSO) technique. In compensation part, inverse GPI as the hysteresis compensator is obtained from the identified hysteresis model. Then in case of parameter uncertainty, PSO–based online tuning method is employed to adaptively adjust the parameters of inverse GPI model. For the linear controller of dual–stage, a simple design approach is followed. Simulation results show that the proposed technique can be efficiently used for the identification and compensation of hysteresis of PZT micro–actuator.