Rate-dependent Hysteresis Nonlinearity Research Articles

Precision positioning based on temperature dependence self-sensing magnetostrictive actuation mechanism

The self-sensing actuator that integrates both actuation and sensing functions is an effective way to simplify the structure of electromechanical systems. This paper proposes a temperature-dependent self-sensing actuation strategy based on a self-sensing giant magnetostrictive actuator (SSGMA) by sensing online stiffness. Utilizing the developed model considering temperature-dependent hysteresis nonlinearity, the self-sensing output characteristics of SSGMA are first evaluated. The self-sensing based control system is then designed in detail. The relationship between the self-sensing signal and the output displacement of SSGMA at different temperatures is constructed by General Regression Neural Network (GRNN) model for fast recognition by the controller. On this basis, a composite control scheme that takes into account rate-dependent asymmetric hysteresis nonlinearities and combines an adaptive feedforward controller with PID feedback controller is proposed for precision actuation of SSGMA. Finally, a temperature-controlled experimental platform is assembled for verification. Experimental results demonstrate that, based on the developed model, the SSGMA is capable of accurate self-sensing output at temperatures of 22 °C, 40 °C, and 70 °C, respectively. The SSGMA with the proposed control method enables the synchronous and precise self-sensing positioning of step and multiple-frequency sinusoidal expectations at different temperatures.

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.

Rate-Dependent Hysteresis Model of a Giant Magnetostrictive Actuator Based on an Improved Cuckoo Algorithm

A rate-dependent asymmetric Prandtl–Ishilinskii (RAPI) model was proposed to tackle the serious rate-dependent hysteresis nonlinearity of the giant magnetostrictive actuator(GMA) output. First, a polynomial function was introduced based on the PI model, and hysteresis factors were introduced to the Play operator, which accurately described the asymmetrical characteristic of the actuator output. On this basis, rate-dependent parameters were added to establish a rate-dependent RAPI model. Second, an improved cuckoo search (ICS) algorithm was proposed to solve the difficulty in the parameter identification of the RAPI model. For the ICS algorithm, the algorithm stability and optimization accuracy were improved using the adaptive step (AS) strategy and bird’s nest disturbance strategy. Then, the effectiveness of the ICS algorithm was tested by comparing it with other parameter identification algorithms. Finally, the rate-dependent RAPI model was verified by combining the output data of the giant magnetostrictive actuator under different frequencies. The results show that the rate-dependent RAPI model exhibits a higher accuracy than the PI model, thus verifying the effectiveness of the rate-dependent RAPI model.

Adaptive iterative learning control and inverse compensation of macro fiber composite system with hysteresis

Macro fiber composite (MFC) actuators, which exhibit excellent flexibility and strong deformability, are widely used in aerospace exploration, flexible robotics, and other applications. However, the rate-dependent hysteresis nonlinearity of the flexible beam driven by MFCs significantly affects positioning accuracy. This study utilizes a new Hammerstein model to describe the hysteresis of an MFC system and proposes a control strategy that combines adaptive iterative learning control and inverse compensation. The adaptive iterative learning control allows the learning gain parameters to be adjusted based on the tracking error. A high gain ensures the convergence speed of iterative learning, whereas a small learning gain provides high robustness and convergence accuracy. Trajectory tracking experiments are conducted to verify the effectiveness of the proposed control strategy. Compared with the case of direct inverse compensation, after the sixth iteration, the tracking maximum absolute error, relative error, and corresponding root mean square error of the proposed control strategy reduce by 73% on average. Moreover, compared with the case of P-type iterative learning control combined with inverse compensation, the metrics above reduce by 12% on average. The proposed control strategy offers higher tracking accuracy and greater practicality in the trajectory tracking of MFC systems.

Modelling and model-based tracking control of soft twisted and coiled actuators

Soft twisted and coiled actuators (STCAs) have promising prospects for application in the field of soft-bodied robots. However, the STCA has a complex energy conversion mechanism and a rate-dependent hysteresis nonlinearity, which brings great difficulties to its modelling and control. To solve these difficulties, this paper proposes modelling and model-based tracking control methods to realize the high-precision output-force tracking control of the STCA. First, a dynamics model framework is constructed, which consists of a thermo-electric model and a thermo-mechanical model, to describe the current-to-force process of the STCA. Among which, the thermo-mechanical model is established by combining the Bouc–Wen model and the linear dynamic system in series, which can accurately describe the rate-dependent hysteresis nonlinearity of the STCA. Then, a double closed-loop control architecture with a hysteresis inverse compensation controller is proposed, which can realize the timely control of the temperature and the compensation for the rate-dependent hysteresis nonlinearity of the STCA. Next, controller parameters are adjusted in the simulation environment to reduce the risk of the damage of the STCA and to obtain a satisfactory control effect. Finally, practical experiments are conducted, and root mean square errors of all experiment results are all less than 5.01%, which verifies the effectiveness of the proposed modelling and control methods.

Rate-Dependent Hysteresis Modeling and Displacement Tracking Control Based on Least-Squares SVM for Axially Pre-Compressed Macro-Fiber Composite Bimorph.

The new axially pre-compressed macro-fiber composite bimorph (MFC-PBP) can produce large displacement and output power. However, it has the property of strong rate-dependent hysteresis nonlinearity, which challenges the displacement tracking control of morphing structures. In this paper, the least-squares support vector machine (LS-SVM) is applied to model the rate-dependent hysteresis of MFC-PBP. Compared with the predicated results of the series model of the Bouc–Wen model and Hammerstein model (BW-H), the LS-SVM model achieved higher predication accuracy and better generalization ability. Based on the LS-SVM hysteresis compensation model, with the support vector pruning, the displacement tracking feedforward compensator is obtained. In order to improve the displacement tracking accuracy, the LS-SVM feedforward compensator combined with the proportional and integral (PI) controller and the feedforward plus feedback control experiment is carried out on the displacement tracking of MFC-PBP. The test results show that the feedforward plus feedback displacement tracking control loop based on the LS-SVM model also has a higher displacement tracking accuracy than that based on the inverse model of BW-H.

A fractional-order Duhem model of rate-dependent hysteresis for piezoelectric actuators

In this paper, a Fractional-Order Duhem Model (FODuhem) is proposed to describe the rate-dependent hysteresis nonlinearity of piezoelectric actuators (PEAs). A fractional-order operator is introduced on the basis of the traditional Duhem model, and the unique nonlocal memory property of the fractional-order operator makes it possible to describe the memory effect inherent in hysteresis. A differential evolutionary algorithm was used to identify the parameters of the FODuhem model. Finally, experimental results clearly show that the FODuhem model can better describe the rate-dependent hysteresis behavior of piezoelectric actuators compared with the conventional Duhem model.

A Fractional-Order Normalized Bouc–Wen Model for Piezoelectric Hysteresis Nonlinearity

This article presents a new fractional-order normalized Bouc–Wen (BW) (FONBW) model to describe the asymmetric and rate-dependent hysteresis nonlinearity of piezoelectric actuators (PEAs). In view of the fact that the classical BW (CBW) model is only efficient for the symmetric and rate-independent hysteresis description, the FONBW model is devoted to characterizing the asymmetric and rate-dependent behaviors of the hysteresis in PEAs by adopting an <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$N$</tex-math></inline-formula> th-order polynomial input function and two fractional operators, respectively. Different from the traditional modified BW models, the proposed FONBW model also eliminates the redundancy of parameters in the CBW model via the normalization processing. By this way, the developed FONBW model has a relatively simple mathematic expression with fewer parameters to simultaneously characterize the asymmetric and rate-dependent hysteresis behaviors of PEAs. Model parameters are identified by the self-adaptive differential evolution algorithm. To validate the effectiveness of the proposed model, a series of model verification and inverse-multiplicative-structure-based feedforward control experiments are carried out on a PEA system. Results show that the proposed model is superior to the CBW model and traditional modified BW model in modeling accuracy and hysteresis compensation.

Finite-time adaptive sliding mode control for high-precision tracking of piezo-actuated stages

Piezo-actuated stages are widely used in nanopositioning applications. However, they not only have inherent static hysteresis characteristics but also have dynamic rate-dependent hysteresis nonlinearity. Therefore, to address dynamic hysteresis nonlinearity and uncertainty in the model parameters, an adaptive switching-gain sliding mode controller with a proportional–integral–derivative surface is designed. In particular, the combination of Bouc–Wen model and second-order linear system is used to describe the dynamic hysteresis process. To improve the robustness and reduce chattering in the sliding mode control method, an adaptive switching-gain is added to the controller without knowing in advance the upper bound of uncertainties. Finite-time convergence conditions of the closed-loop system are also analyzed. Finally, the proposed control method is implemented in real time on an ARM experimental platform. Comparative experimental results demonstrate excellent tracking performance and robustness. The dynamic hysteresis characteristics are suppressed effectively, and this result provides a powerful reference for engineering applications.

Precision motion control of a piezoelectric cantilever positioning system with rate-dependent hysteresis nonlinearities

The study proposed a feedforward– feedback combined architecture to control a piezoelectric motion system with rate-dependent hysteresis nonlinearities. First, the feedforward control that considered the inverse rate-dependent Prandtl–Ishlinskii (RDPI) model was presented. Based on the inverse RDPI model, a formula that characterizes the output of the inverse compensation considering modeling errors and uncertainties was obtained. More precisely, the output of the inverse compensation of RDPI model consists of an invertible linear term and a bounded nonlinear term. Afterward, the feedforward control is extended with a feedback control architecture. In order to satisfy performance criteria in the presence of the uncertainties, the design of the feedback controller was based on the performances inclusion theorem from interval techniques. Experiments were conducted on a piezoelectric actuator to confirm the effectiveness of the combined feedforward–feedback control.

High-bandwidth modeling for rate-dependent hysteresis nonlinearity using a standard Prandtl–Ishlinskii model

Rate-dependent hysteresis of piezoelectric stack actuators (PSAs) is known to cause oscillations and instabilities. This paper introduces a novel electro-mechanical coupling model that can accurately characterize the high-bandwidth rate-dependent hysteresis of a PSA. The proposed model contains three components to fully represent the output response of the system, namely, an electric model that is used to describe the electric dynamic behaviors of the PSA and its power amplifier, a hysteresis model that is applied to represent the inverse piezoelectric effect, and a mechanical model that is able to characterize the mechanical dynamic behavior of the PSA. It should be noted that, in the mechanical model, the force generated by the inverse piezoelectric effect of piezoelectric wafers is considered a distributed force and the dynamic model of the PSA is regarded as a longitudinal vibration straight bar with a uniform cross section. Through this proposed model, high modeling accuracy is achieved well with high-bandwidth varying frequencies (1–1000 Hz) as well as varying amplitudes. This paper provides a simple and effective novel method for the high-bandwidth modeling of smart material-based actuators with rate-dependent hysteresis nonlinearity.

Output-feedback control of precision motion systems with uncertain nonlinearities

We propose output-feedback tracking control with an extended high-gain observer-based feedback control for a class of precision motion systems that include unknown rate-dependent hysteresis nonlinearities with linear dynamics. The proposed control system has a number of features; namely, (i) it can guarantee ultimate boundedness of the tracking error, where the ultimate bound can be made arbitrarily small, for any given initial conditions and for bounded unknown exogenous inputs and modeling parameters, (ii) it provides the possibility of shaping the transient response of the closed-loop system as desired, and (iii) the proposed technique is non-adaptive inversion free technique. To verify the effectiveness of these results, first simulation results are provided for an iron pendulum in a magnetic field system. Then, the output-feedback approach is also applied experimentally to a piezotube micropositioning actuator for a precision motion control.

Rate Dependent Krasnoselskii-Pokrovskii Modeling and Inverse Compensation Control of Piezoceramic Actuated Stages.

The piezoceramic actuated stages have rate-dependent hysteresis nonlinearity, which is not simply related to the current and historical input, but also related to the frequency of the input signal, seriously affects its positioning accuracy. Consider the influence of frequency on hysteresis modeling, a rate-dependent hysteresis nonlinearity model that is based on Krasnoselskii–Pokrovskii (KP) operator is proposed in this paper. A hybrid optimization algorithm of improved particle swarm optimization and cuckoo search is employed in order to identify the density function of rate-dependent KP model, avoiding the blind search process caused by the high randomness of Levy’s flight in the cuckoo search algorithm, and improving the parameter identification performance. For the sake of eliminating the hysteresis characteristics, an inverse feed-forward compensation control that is based on recursive method is proposed without any additional conditions, and a feed-forward compensation controller is designed accordingly. The experimental results show that, under different frequency input signals, as compared with the classic KP model, the proposed rate-dependent KP model can accurately describe the rate-dependent hysteresis characteristics of the piezoceramic actuated stages, and the recursive inverse feed-forward compensation control method can effectively mitigate the hysteresis behaviors.

Elman neural network–based identification of rate-dependent hysteresis in piezoelectric actuators

Rate-dependent hysteresis nonlinearity in piezoelectric actuators severely limits micro- and nanoscale system performance. It is necessary to establish a dynamic model to describe the full behavior of rate-dependent hysteresis. In this article, the Elman neural network–based hysteresis model is developed for piezoelectric actuators. An improved dynamic hysteretic operator is proposed to transform the multi-valued mapping of hysteresis into one-to-one mapping on a newly constructed expanded input space. Then, Elman neural network incorporated with the improved dynamic hysteretic operator is utilized to approximate the behavior of rate-dependent hysteresis. The combination of Elman neural network and the improved dynamic hysteretic operator can dually embody the dynamic property and is capable of fully extracting the characteristics of rate-dependent hysteresis. The experimental results are presented to illustrate the potential of the proposed modeling technique.

Feedforward Control of the Rate-Dependent Viscoelastic Hysteresis Nonlinearity in Dielectric Elastomer Actuators

Dielectric elastomer actuators have shown promising applications in the field of soft robotics. However, due to their rate-dependent viscoelastic hysteresis nonlinearity, it is still challenging to achieve precision tracking control of dielectric elastomer actuators. In this letter, we propose a feedforward control approach that can compensate for the rate-dependent viscoelastic hysteresis nonlinearity with maximum tracking errors of 6.18% and root-mean-square errors of 2.96% when the frequency of the input voltage is between 0.05 Hz and 1.5 Hz. Our control approach consists of two feedforward compensators: 1) for the ease of the hysteresis compensation, a creep compensator is firstly developed to remove the viscoelastic creep nonlinearity; 2) based on a phenomenological mathematical model, an inverse hysteresis compensator is then developed to compensate for the rate-dependent viscoelastic hysteresis nonlinearity. Experimental results of tracking various periodic trajectories demonstrate that: the maximum tracking errors are reduced by 87.17% and the root-mean-square errors are decreased by 89.53%, by comparing the results without the viscoelastic compensation. It is the first time to successfully compensate for both the viscoelastic creep nonlinearity and rate-dependent hysteresis nonlinearity of dielectric elastomer actuators by a feedforward control approach, which may pave the way for further applications in dielectric-elastomer-actuators based soft robotics.

Event-Triggered Neural Control of Nonlinear Systems With Rate-Dependent Hysteresis Input Based on a New Filter.

In controlling nonlinear uncertain systems, compensating for rate-dependent hysteresis nonlinearity is an important, yet challenging problem in adaptive control. In fact, it can be illustrated through simulation examples that instability is observed when existing control methods in canceling hysteresis nonlinearities are applied to the networked control systems (NCSs). One control difficulty that obstructs these methods is the design conflict between the quantized networked control signal and the rate-dependent hysteresis characteristics. So far, there is still no solution to this problem. In this paper, we consider the event-triggered control for NCSs subject to actuator rate-dependent hysteresis and failures. A new second-order filter is proposed to overcome the design conflict and used for control design. With the incorporation of the filter, a novel adaptive control strategy is developed from a neural network technique and a modified backstepping recursive design. It is proved that all the control signals are semiglobally uniformly ultimately bounded and the tracking error will converge to a tunable residual around zero.

Rate-dependent hysteresis modeling and compensation of piezoelectric actuators using Gaussian process

Rate-dependent hysteresis nonlinearity of the piezoelectric actuators (PEAs) deteriorates the positioning accuracy of the nano-positioning stage. To deal with it, the Gaussian Process (GP) is applied in this work to model the PEAs and also to compensate for it. The proposed GP-based model is capable of describing the nonlinear memorability as well as rate-dependence of hysteresis by introducing both the voltage value and its changing rate to the model input. The usage of the kernel function makes the model flexible and accurate without specifying a function form and the parameters. The kernel function contains only three hyperparameters, which can be determined by combining the differential evolution algorithm and Bayesian inference framework. An inverse hysteresis model is then obtained by interchanging the input and output variables of the GP-based hysteresis model to serve as a feedforward compensator. Based on this feedforward compensator, open-loop and closed-loop controllers are developed and tested. The comparative experimental studies are carried out on a PEA stage and the results demonstrate the effectiveness and superiority of the GP-based hysteresis model and compensator.

Prandtl-Ishlinskii Modeling for Giant Magnetostrictive Actuator Based on Internal Time-Delay Recurrent Neural Network

Micro and nano technology is an indispensable part of modern science and technology. Because of the excellent advantages of high energy density, rapid response and large mechanical force, GMA becomes more promising in precision positioning, microelectronic, and biomedicine field[1], [2]. However, the relationship between input current and output displacement of GMA is hysteresis nonlinearity, which shows that the output of GMA not only relates to the current input value, but also relates to previous output value. Besides, the hysteresis nonlinearity is rate-dependent, so that the output of GMA depends on the input frequency. The intrinsic rate-dependent hysteresis nonlinearity of GMA is the main sticking point preventing its application in the high precision positioning [3], [4]. Therefore, modeling of GMA has long been difficult to study and attracted the attention of researchers. In this paper, the Prandtl-Ishlinskii (PI) model with the parameters self-tuning ability is established by the internal time-delay recurrent neural network (RNN) to describe the hysteresis nonlinearity of GMA. PI model consists of play operator and density function. Play operator is a continuous hysteresis operator, whose output depends on not only the current input but also the previous input. Nevertheless, play operator is rate-independent. Identifying the applicable density function is an important part of modeling PI model of GMA. Neural network has the features of the nonlinear mapping property and high parallel process ability. Therefore it is suitable to be applied to identify the nonlinear model. In this paper, the internal time-delay RNN is used to replace the density function of PI model. The PI model structure identified by the internal time-delay RNN for GMA is shown in Fig. 1. The internal time-delay RNN is set by the input layer, output layer and hidden layer. Where output of play operator $v_{i}(t)$ is the $i$ th input sample of the network at time $t$, $y^{\ast }(t)$ is the output of the network. $^{1}w_{ji}$ and $^{2}w_{j}$ are the weights of input layer and output layer, respectively, $^{h}w_{jk}$ is the weight for the nodes of hidden layer. Compared with feedforward neural network, it has property of memory because there is time delay recurrent existed in the hidden layer. Due to the inherent feedback structure of internal time-delay RNN, it possesses the dynamic characteristics and can adjust the parameters of PI model adaptively. The simulation results of PI model identified by internal time-delay RNN at the different input frequency are shown in Fig. 2. The red solid lines are hysteresis loops measured in the experiments, the blue dotted lines are the hysteresis loops of PI model based on internal time-delay RNN, and the blue solid lines are the modeling error curves. To facilitate simulation, the normalized data is adopted. As shown in Fig.2, the maximum modeling error rate at 1Hz, 10Hz, 50Hz and 100Hz is 0.81%, 1.07%, 1.41% and 2.14%, respectively. The PI model identified in this paper can accurately describe the hysteresis nonlinearity of GMA with the increase of input frequency. Therefore, the ability of precise modeling by PI model based on internal time-delay RNN is certified. The proposed PI model can be used to eliminate the hysteresis nonlinearity at the compensation control of GMA and promote the application of GMA in precision positioning field in the future.

Further results on open-loop compensation of rate-dependent hysteresis in a magnetostrictive actuator with the Prandtl-Ishlinskii model

Abstract Apart from the output-input hysteresis loops, the magnetostrictive actuators also exhibit asymmetry and saturation, particularly under moderate to large magnitude inputs and at relatively higher frequencies. Such nonlinear input-output characteristics could be effectively characterized by a rate-dependent Prandtl-Ishlinskii model in conjunction with a function of deadband operators. In this study, an inverse model is formulated to seek real-time compensation of rate-dependent and asymmetric hysteresis nonlinearities of a Terfenol-D magnetostrictive actuator. The inverse model is formulated with the inverse of the rate-dependent Prandtl-Ishlinskii model, satisfying the threshold dilation condition, with the inverse of the deadband function. The inverse model was subsequently applied to the hysteresis model as a feedforward compensator. The proposed compensator is applied as a feedforward compensator to the actuator hardware to study its potential for rate-dependent and asymmetric hysteresis loops. The experimental results are obtained under harmonic and complex harmonic inputs further revealed that the inverse compensator can substantially suppress the hysteresis and output asymmetry nonlinearities in the entire frequency range considered in the study.

Modeling, identification and compensation of hysteresis nonlinearity for a piezoelectric nano-manipulator

Hysteresis nonlinearity widely exists in piezoelectric actuated nano-positioning applications, which degrades their tracking accuracy and limits their precision positioning applications. This paper presents a novel hysteresis modeling and compensation approach to alleviate the adverse effect of the asymmetric and rate-dependent hysteresis nonlinearity for a piezoelectric transducer actuated servo stage. By integrating a generalized input function with the play operator of the classical Prandtl–Ishlinskii model, a novel polynomial-based rate-dependent Prandtl–Ishlinskii (PRPI) model is proposed to capture the hysteresis behavior of the piezoelectric positioning stage, where a polynomial function of input and a time rate function of input are introduced to formulate the generalized input function. Meanwhile, a new adaptive differential evolution optimization algorithm is developed to identify the parameters of the proposed PRPI hysteresis model. Based on the PRPI hysteresis model with the identified parameters, an inverse feedforward controller is constructed to achieve the accurate tracking motion. Furthermore, the hysteresis compensation error of the proposed PRPI model is theoretically analyzed. Finally, comparative experiments are conducted, and the experimental results provided in this paper demonstrate the effectiveness and superiority of the proposed inverse PRPI model compensation approach.