## 1 Introduction
More works related to sliding mode control can be found in [22; 23]. In [22] a SMC was designed to control a washout filter with constant impedance and nonlinear constant power loads. In [23] a PDSMC was designed to collaborate with a sliding perturbation observer to control a robot manipulator. One thing to note is the work, cited in [15], done by [PERSON] et al. in 2006. In this reference, [PERSON] suggested PISMC for the two-wheel inverted pendulum mobile robot to avoid high chattering drawbacks. Even though [15] had a similar topic and methodology to this paper, the details were significantly different. In [15], a linear model was investigated; the proportional and integral sliding modes shared the same coefficient, and the integration applied to the derivative of the states. In this paper, nevertheless, there are three major differences. First, we do not restrict our system to a linear one. Actually, the examples in this paper were presented and controlled using the original nonlinear form. Second, we introduce two parameters, instead of one coefficient as in [15], in the investigation. This setting provides us more degrees of freedom to tune parameters and improve the system performance. Third, in the integral we include not only the derivative of the states, but also the states themselves through the error term. Consequently, the work described here and our contributions in the PISMC apparently differ from those made by other researchers. More specifically, we cite the following list the differences from [15]:  
1. Design of sliding surface for the controlled system: Our paper presents \(s=2\dot{e}+(\lambda+2\alpha)e+\alpha^{2}\int ed\tau\). In [15], the sliding surface was designed as \(s=k_{p}e+k_{i}\int ed\tau\). 2. The system of control in this paper is nonlinear, whereas the system in [15] was a linearized system. 3. Because the system to control in [15] was linear, their proposed controller was merely a function of the system, given by \[u = Kx-(CB)^{-1}Cf-(CB)^{-1}\eta^{2}\text{sgn}(s),\] (1)where \(f\) is the original form representing a nonlinear function describing the deviation from linearity in term of disturbances and un-modeled dynamics. Our controller, however, has two major differences from theirs: First, we do not employ the remainder of the nonlinearity terms directly. Instead, we employ the bound of the remainder nonlinearity terms. Secondly, the saturation function in our controller is more than a \(\text{sgn}(s)\) function; the selection of parameters is different. 4. The tuning procedures of parameters are discussed in this paper, but not in [15]. The work presented in this paper extends the work presented in [19] to a wider range. The following novelties or contributions are presented in this paper. First, differently from the commonly known ISMC, PISMC, and PID sliding surface, which usually each have the sliding surface of \(s=k_{p}e+k_{l}\int ed\tau+k_{d}\dot{e}+k_{l}e(0)+k_{2}\dot{e}(0)\) and five parameters or more to tune, our algorithm in [19] extends the sliding surface to the \(n\)th order derivative of the output error with only two parameters to tune. In [19], however, a system of unweighted control was considered. Based on the work in [19], this paper proposes the sufficient condition to design a control for a system of weighted control. Secondly, our algorithm was realized and verified in a two wheel vehicle system. In the literature, researchers either applied their work to nonlinear systems using numerically simulated results [21; 24], or realized their algorithms in a linear system [20; 22; 23]. Although nonlinearities were considered in [22; 23], they were treated as disturbances instead of dynamics themselves. This paper not only verifies our capability of controlling a nonlinear system by numerical simulations, but also realizes the algorithms in a TWVS using its original nonlinear model. Instead of treating the nonlinearities as disturbances like works in the literature, we dealt with the nonlinearity as the dynamics itself.