## 3 Implementation in a Two-Wheel Vehicle System
### Numerical Simulations
#### 3.3.2 Balance Control Simulation Results  
Simulation results are presented in Figures 10-12. For the first simulation, Figure 10 shows the system responses of the TWVS balancing with no external disturbances and the initial condition is \((\hat{\varepsilon}_{d}(0),\hat{\varepsilon}_{d}(0))=(1,0)\). We note that the settling time to reach \(\hat{\varepsilon}_{d}\) and the error of the PISMC are smaller and converge quicker than that of the conventional SMC. It is evident that the PISMC presented in this paper is effective for the TWVS to reach the desired attitudes around the equilibrium point in short a period. The stability and accuracy are both maintained in the balance simulation.  
Figure 10: Balance response to PISMC without disturbances: (**top**) angular response; (**middle**) response of angular rate; (**bottom**) applied control signals.  
Figure 9: The experiment focused on the tilt angle control (\(\tilde{x}=\theta\)) only, due to the inherit complexity and difficulty in the two-wheel vehicle system (TWVS).  
For the second simulation, Figure 12 shows the system responses of the TWVS with external disturbances. To examine the disturbance rejection property afforded by the proposed control, we consider to simulate the system with initial condition of \((\tilde{x}_{d}(0),\tilde{x}_{d}(0))=(-0.5,0)\), and the system external disturbance is \(\sin(2\pi t)\). Figure 11 shows that \(\tilde{x}_{d}\) settles down to the steady state and the  
Figure 11: Switching and sliding signal for PISMC without disturbances: (**top**) history of sliding surface values; (**bottom**) history of saturation conditions.  
Figure 12: Balance response to PISMC with disturbances of \(\sin(2\pi t)\): (**top**) angular response; (**middle**) response of angular rate; (**bottom**) applied control signals.  
amplitude of the errors are smaller. The results also verify that the PISMC is capable of tracking the desired states and the errors converge steadily to the neighborhood of zero.  
The performance of the system with the proposed PISMC is better than the use of the SMC controller such that no overshoot, shorter rise time, and shorter settling time were obtained from the proposed controller. The variations of the control efforts are shown in Figure 10 which indicates that the SMC signal is smaller in magnitude and the system settles down in 1.2 s. Several quantitative performance indicators of the attitude tracking qualities for SMC and PISMC controllers are summarized in Table 3. Variations of the sliding signal for SMC and PISMC are illustrated in Figure 10. It shows that magnitudes of the variations are smaller in PISMC. Variations of the sliding surface during the control are illustrated in Figure 10 as well. One notes that the sliding function is \(s(t)\
eq 0\) when the error signal is not zero. This means that the sliding mode is in the reaching surface up to about 0.92 s, and then arrives the sliding surface. When it reaches to the sliding surface, theoretically it is expected the sliding function to be zero (i.e., \(s(t)=0\)). In practical applications, there are always some small deviations and fluctuations at the output measured variable because of uncertainties and disturbances. Here, the average value of the sliding function is zero \(s(t)=0\). As shown in Figures 10-12, we note that these remarkable results are obtained without involving too much control effort.