## 1 Introduction  
This paper presents a controller called the proportional-integral sliding mode controller (PISMC). The sliding mode control (SMC) has been widely accepted as an efficient method for the tracking or balance control of an uncertain nonlinear system. The capability of achieving perfect performance in principle is also demonstrated in the presence of parameter uncertainties and bounded external input disturbances. The proposed PISMC combines the proportional sliding mode control (PSMC) and the integral sliding mode control (ISMC). There are two main contributions in this paper. First, the PISMC provides more parameters with which to tune the controller. As a consequence, it reduces adverse effects due to the external disturbances effectively. Second, this paper sets up an experiment as an example to demonstrate the tracking accuracy in a nonlinear system. This example also shows how this algorithm can be implemented to a physical system. Though the name PISMC may sound similar to some algorithms regarding the PID sliding surface, the definition of sliding surface and the mathematical expression of PISMC are different. The main structure of SMC theory was brought out by [PERSON], a former USSR scholar, in 1977 [1]. After the original theory, various applications and extensions of the theory were proposed and presented. Some of the examples are given and reviewed in the following. In 1994, [PERSON] et al. [2] proposed a dynamic multivariable discontinuous feedback control strategy of the sliding mode type for the altitude stabilization of a nonlinear helicopter model in vertical flight. In 1996, [PERSON] and [PERSON] [3] proposed a new sliding mode design concept named integral sliding mode control (ISMC). ISMC is different from the conventional sliding mode design approaches, and the only difference is the integral form of the error term in place of the original error term. In 2000, [PERSON] et al. [4] designed an ISMC which has one separate boundary surface in order to decrease the chattering and improve the controllers' responses. In 2003, [PERSON] et. al [5] proposed a PID sliding surface, where the sliding mode motion occurs with global invariance. In the same year, [PERSON] and [PERSON] [6] employed the SMC for flight control design. In 2004, [PERSON] and [PERSON] [7] investigated the low chattering SMC for balance control design. In 2005, [PERSON] and [PERSON] et al. [8] applied the robust integral sliding mode control to uncertain stochastic systems. Besides that, [PERSON] et al. applied the design of ISMC to a multi-agent quadrotor test-bed. Moreover, [PERSON], et al. also applied the back-stepping and sliding-mode techniques to an indoor micro-quad-rotor. [PERSON] et al. [9] in 2005 proposed a dynamic SMC which increases accuracy under high-frequency switching of control. More studies regarding the applications of sliding mode technique and its extensions are discussed in [10; 11; 12; 13]. In 2009, [PERSON] et al. [14] studied the control of an underwater remote vehicle-manipulator system using sliding-mode control and \(H^{\infty}\) control. This paper showed that the sliding-mode and \(H^{\infty}\) controller combined approach provides superior dynamic coupling reduction performance. Speaking of PISMC, [PERSON], [PERSON], and [PERSON] have devoted themselves to the investigation of PISMC [15; 16; 17; 18] since 2002. In 2011, [PERSON] et al. applied the PISMC to stabilize the attitude and position of a hovering unmanned helicopter [19]. [PERSON] at al. applied the PISMC to a hydraulically-actuated active suspension system [17; 18]; [PERSON], et al. focused on the control and manipulation of of robotic systems. [15; 16]. In [20], robust integral sliding mode control (ISMC) was proposed for an underactuated rotary hook system in order to deal with parametric uncertainties presenting themselves in the system's parameters. [PERSON] et al. studied the modification of the integral sliding variable degrades, and suggested that the switching element is smoothed by using a low-pass filter [21].