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

Abstract

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.