## 1 Introduction
In the literature, [PERSON]. [19] developed a TSMC scheme for precision tracking control and finite time convergence of a piezoelectric-driven micropositioning system. [PERSON]. [20] proposed an adaptive-gain fast nonsingular TSMC scheme for a piezo positing stage system which can quickly converge in finite time and reduce chattering. The aforementioned SMC and TSMC schemes are all applied in continuous-time domain; however, with the wide use of computers in control systems, the DSMC theory [21] has attracted considerable attention in recent years [22; 23; 24]. It must be pointed out that the continuous-time SMC cannot directly convert into its discrete counterpart through simple equivalence. DSMC and DTSMC schemes have been successfully applied to other smart actuators. For instance, [PERSON]. [25] proposed a novel DSMC scheme for precision tracking control of a piezoelectric actuator. In addition, for making the system state converge to the equilibrium point asymptotically within finite steps, [PERSON]. [26; 27] proposed DTSMC schemes for precision tracking control of piezoelectric actuators, and the stability of the control systems and  
Figure 1: The working principle of dielectric electro-active polymer (DEAP) materials. (**a**) Power cut: Initial state of DEAP materials. (**b**) Power supply: Actuation direction of DEAP materials. the effectiveness of the proposed control schemes were demonstrated through experimental studies. However, to the knowledge of the author, DTSMC scheme has not been applied for precision trajectory tracking of DEAP actuators in the literature. This paper focuses on the asymmetric rate-dependent hysteresis modeling and trajectory tracking control of a DEAP actuator under different input frequencies. A Hammerstein model with hysteresis nonlinearity is established to describe the asymmetric rate-dependent hysteresis of the DEAP actuator. To estimate the parameters of the model, LM algorithm [28; 29] is utilized. Based on the MPI model, an inverse hysteresis compensator is obtained to suppress the asymmetric hysteresis. In order to achieve precision tracking control and robustness of the closed-loop system, a DTSMC with a new discrete-time terminal sliding function is proposed. The main contributions of this paper are summarized as follows:  
(1) A Hammerstein model with hysteresis nonlinearity is established to account the asymmetric rate-dependent hysteresis of the DEAP actuator and the lumped nonlinear disturbance term is well formulated. It provides with a detailed model for the closed-loop system. (2) A DTSMC scheme based on a new discrete-time terminal sliding function is first proposed for the DEAP actuator. The proposed control scheme not only improves the trajectory tracking control accuracy and error convergence but also suppresses the chattering phenomenon. In addition, the proposed control scheme does not require a state observer. (3) Compared with other DTSMC schemes, a terminal-sliding-mode-type switching control law is defined in the proposed DTSMC scheme, which improves the robustness feature of the closed-loop system in the presence of unmodeled hysteresis nonlinearity, residual dynamics, parameter uncertainties and external disturbances. (4) Finite steps convergence and stability of the closed-loop system are theoretically verified and an experimental investigation is conducted to validate the effectiveness of the proposed control scheme to the DEAP actuator. Then, a comparison between the proposed control scheme and the DSMC scheme is undertaken. The remainder of the paper is organized as follows. In Section 2, the analytical formulation of the Hammerstein model and the inverse MPI model are established first, and then the parameters of the Hammerstein model are identified by LM algorithm. In Section 3, a compound controller is proposed, and the system stability is proved. Experimental results of the proposed controller to the DEAP actuator are presented in Section 4. Conclusions are given in Section 5.