A revised Terminal Sliding Mode Controller Design for Servo Implementation Khalid Abidi * , and Jian-Xin Xu * * National University of Singapore Faculty of Engineering, Department of Electrical and Computer Engineering 10 Kent Ridge Crescent, Singapore, 119260 Abstract—Terminal Sliding Mode (TSM) control is known for its high gain property nearby the vicinity of the equilibrium while retaining reasonably low gain elsewhere. This is desirable in digital implementation where the limited sampling frequency may incur chattering if the controller gain is overly high. In this work we integrate a linear switching surface with a terminal switching surface. The mixed switching surface can be designed according to the precision requirement. The analysis, simulations and experimental investigation show that the mixed SMC design outperforms the linear SMC as well as the pure TSMC. I. I NTRODUCTION SLIDING MODE control is a powerful technique that has been successfully used for the control of the linear and nonlinear systems. In order to design sliding mode control systems, a switching surface or a sliding mode is defined first, and a sliding mode controller is then designed to drive the system state variables to the sliding mode so that the desired convergence property can be obtained in the sliding mode, which is not affected by any modeling uncertainties and/or disturbances [1]. Reviewing the history of the development of the sliding mode control systems [2]-[4], it can be found that linear sliding mode has been widely used to describe the desired performance of closed loop systems, that is, the system state variables reach the system origin asymptotically in the linear sliding mode. Although the parameters of the linear slid- ing mode can be adjusted such that the convergence rate may be arbitrarily fast, the system states in the sliding mode cannot converge to zero in finite time. Recently, a new technique called terminal sliding mode control has been developed in [1] to achieve finite time convergence of the system dynamics in the terminal sliding mode. In [5]-[7], the first-order terminal sliding mode control technique is developed for the control of a simple second-order nonlinear system and an 4th-order nonlinear rigid robotic manipulator system with the result that the output tracking error can converge to zero in finite time. In this paper, a revised terminal sliding mode control law is developed. It is shown that the new method can achieve better performance than with the linear SM or pure TSM. To validate the proposed method simulation and experiments are conducted on a piezo-motor system. The paper is organized as follows. The problem formulation is presented in §2. Appropriate terminal sliding surface and SMC are designed in §3. In §4, numerical and experimental results will be presented. Conclusions are given in §5. II. PROBLEM FORMULATION A. System Properties Consider the following continuous-time model of piezo- motor driven linear stage ˙ x 1 (t) = x 2 (t) ˙ x 2 (t) = - k fv m x 2 (t)+ k f m u(t) - 1 m f (x,t) y(t) = x 1 (t) (1) where x 1 is the position, x 2 is the velocity, u is the voltage input, and f (x,t) is the friction disturbance and is assumed bounded such that |f (x,t)|≤ f max . The constants m, k fv , and k f are the nominal mass, damping, and force constants respectively. The objective is to design a TSM such that the output, y(t), of system (1) will converge to a desired reference trajectory r(t) in finite time. III. TSM CONTROLLER DESIGN In this section we will discuss the TSM controller design. The controller will be designed based upon an appropriate selection of a Lyapunov function. Further, the closed-loop system will be analyzed to derive the stability conditions. This section will conclude with a discsussion on the tracking error- bound. A. Controller Design Consider the terminal sliding surface defined below, σ = c 1 e + c 2 ˙ e + c 3 e p (2) where e = r - y is the tracking error, σ is the sliding function, and c 1 , c 2 , c 3 , p are design constants. Before we proceed with the controller design, we need to rewrite the system (1) in terms of the tracking error e. Consider the new state z =˙ e, it can be shown that the system can be rewritten as ˙ e(t) = ˙ r - x 2 (t)= z(t) ˙ z(t) = k fv m z(t) - k f m u(t)+¨ r - k fv m ˙ r + 1 m f (x,t) (3) 159 978-1-4244-2200-5/08/$25.00 ©2008 IEEE