## 1 Introduction
Design for a real hydraulic system is more suitable than the adaptive updating law caused by simple dynamic modeling. The dynamic model of this hydraulic system is simplified as a 2 nd order system. The identified system parameters are obtained by the signal compression method [15], which generates an equivalent impulse signal as the trajectory reference. SPO is a combination of a sliding observer with a perturbation observer. Perturbation consists of disturbances, nonlinear terms, modeling errors, and system uncertainties. That is also effective at reducing the chattering in conventional SMC's caused by its robustness against the disturbance [16; 17], and improving the system's robustness by estimating and compensating for the perturbation [18; 19]. SPO has been proven to be an accurate estimation within an interest designed for a low-frequency range through simulation [20; 21]. Meanwhile, the estimated perturbation can be used to reduce chattering and tracking errors through SMC. SMCSPO (Sliding Mode Control with Sliding Perturbation Observer) is a combination of the controller SMC with SPO, which has an excellent performance in trajectory tracking. It only utilizes the partial state feedback to estimate the other states and perturbation including non-linear control terms. It has been proven to have a high tracking performance in some electrical motor applications (e.g., in [22; 23] for a surgical robot). However, the convergence time of SMCSPO is slower than only an SMC with full states feed-back, and some undershoot occurs. This phenomenon is caused by the existing time delay of the sliding perturbation observer. In order to reduce the disadvantage of time delay brought by SPO and have a robust tracking performance, TSMCSPO was designed and used in our current study. TSMCCSPO combines TSMC (Terminal Sliding Mode Control) and SPO. TSMC uses the terminal sliding surface instead of the traditional linear sliding surface [24; 25], resulting in the tracking error being lead to zero in a finite time fast. TSMC has been used and examined in control of robot and manipulators [26; 27]. This fast controller is useful to speed up the convergence velocity around an equilibrium point. Due to the introduction of a non-linear sliding surface, preliminary TSMC may have a singularity problem, which makes it difficult to act on the actual mechanical application. In order to avoid the singularity problem, some methods have been developed and applied [28; 29; 30]. We use the method mentioned in [30] to avoid the singularity point in the TSMCSPO design. In this research, the main point is the new TSMCSPO controller, which uses the terminal sliding mode to improve the slow convergence time and reduce the effect of time delay that occurs in the SPO. Application on the hydraulic system is novel using TMCSPO, though the SMC technical and perturbation observer is well known and not new for electric motors. The new designed controller is proposed for the first time in this paper. Then, the implementation using a hydraulic system is introduced with its dynamical analysis. Finally, the performance of this new controller TSMCSPO is demonstrated by MATLAB simulations compared with the conventional SMC, SMCSPO, and TSMC, as well as in real applications for the hydraulic servo system. The remaining part of this paper is organized as follows. The controller design for a general robot manipulator system is described in Section 2. Section 3 introduces the hydraulic system and the application with TSMCSPO. A simulation carried out in MATLAB and the experimental result from the real hydraulic system are present in Sections 4 and 5, respectively. Section 6 concludes this work.