# Comparative Analysis of Different Model-Based Controllers Using Active Vehicle Suspension System
## 4 Numerical Simulation Results and Discussion  
This section presents numerical simulations results to show the effectiveness of the developed methods. It also demonstrates the impact of various controllers on the system performance. Before illustrating the numerical simulations discussion, a brief description about practical implementation of active vibration control is deliberated. A practical test rig (Quanser) active quarter-car suspension system consists of an amplifier, data acquisition card and a personal computer. The test rig is consisted of three plates led on four columns using linear bearings. The topmost plate is the vehicle body mass \(M_{c}\), the middle plate is the wheel axle assembly with mass \(M_{wh}\) and the lowest plate is for road inputs driven by a DC motor. There are springs between the plates that present \(C_{c}\) and \(C_{wh}\). Also, the linear bearings generate damping with \(d_{c}\) and \(d_{wh}\), respectively. To measure the acceleration of vehicle body an accelerometer is used. Three encoders are used to measure the vehicle body displacement \(\dot{X}_{c}\), suspension deflection travel \((X_{c}-X_{wh})\), and road input \(X_{g}\). A DC motor placed between sprung (vehicle body) and unsprung (tire) masses is used to generate desired forces \(u(t)\) based on active control laws to overwhelm vibrations of the sprung mass [27]. In this study, simulation based on mathematical model active control algorithms such as PID, LQR, SMC law, ISMC law and HOSMC law are implemented using MATLAB/SIMULINK. Performance of the suspension system in terms of ride performance and vehicle handling is taken into consideration under road disturbances. The main objective is to attain small amplitude value for vehicle body displacement \(\dot{X}_{c}\), suspension travel \((X_{c}-X_{wh})\) and wheel deflection \((X_{wh}-u_{d})\). Figure 4 shows the block control system of active suspension. The main goal of the controller is to decrease the acceleration and displacement with respect to the open-loop or passive suspension system, to give ride comfort to passengers. The problem is to find the suitable SMC signal \(u(t)\) that derives system states to the desired trajectory, i.e., \(X(t)=X_{d}\). The controller gain is chosen suitably by hit and trial method until the error \(X(t)-X_{d}\) approaches to zero. External disturbances \(d\) are added to the system as shown in Figure 5.  
Table 2 represents the control parameters for each control strategy. \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline
**Active Controllers** & \(k\) & \(\delta\) & \(\lambda\) & \(k_{1}\) & \(k_{2}\) & \(Q\) & \(R\) & \(k_{p}\) & \(k_{i}\) & \(k_{d}\) \\ \hline HOSMC & - & - & 100 & 31.5 & 2.2 & - & - & - & - & - \\ ISMC & 10 & 0.5 & 50 & - & - & - & - & - & - & - \\ FOSMC & - & - & - & - & - & diag(1000, 20, 80) & 1 & - & - & - \\ LQR & 20 & 0.04 & 3 & - & - & - & - & - & - & - \\ PID & - & - & - & - & - & - & - & 100 & 4000 & 50 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Active controller’s parameters. Figure 4: Schematic diagram of the closed-loop control system. Figure 5: Road disturbances. Moreover, Figure 9 shows the performance comparison analysis of all proposed controllers in terms of wheel deflection of the system. Figure 10 shows the control efforts of each controllers, i.e., PID, LQR, SMC, FOMSC, ISMC and HOSMC. It can be clearly seen that sliding mode control laws produce better performance for active vehicle control. Simulation results prove that PID and LQR give better road holding than passive suspension, while SMC and ISMC achieve better control ride performance. Of all, STA-based HOSMC provides better road holding against road disturbance and also provides superior performance to other active proposed controllers. It can be clearly noticed that the classical PID and FOSMC performance are superior than passive suspension system and it also reduces the undesired oscillations known as chattering.