# Comparative Analysis of Different Model-Based Controllers Using Active Vehicle Suspension System
## 3 Control Methods
### First-Order Sliding Mode Control  
FOSMC strategy is acknowledged as one of the efficient solution for vibrant workplace with higher-order systems as it offers robust feedback for nonlinear dynamic systems. In FOSMC design process, performance specifications and closed-loop stability are two primary objectives. Initially, it is essential to find sliding mode surface that enables the system to reach its destined final value. It helps the system to remain unaware of the internal and insignificant disturbances. Final phase is to design feedback control law which directs the controlled variable to its target value. By using sign function, sliding surface is given as follows:  
\[s=\left(\frac{d}{dt}+\lambda\right)e \tag{17}\]  
where \(e=y_{ref}-y\) and \(y=X_{c}-X_{wh}\). Now, for the stability condition, consider Lyapunov function as follows:  
\[n=\frac{1}{2}s^{2} \tag{18}\]  
\[\dot{n}=s\dot{s} \tag{19}\]  
The time derivative of (17) is derived as follows:  
\[\dot{s}=\ddot{e}+\lambda\dot{e} \tag{20}\]  
By using (7) in (20) yields:  
\[\dot{s}= \ddot{y}_{ref}-X_{2}+X_{4}+\lambda\dot{y}_{ref}-\lambda(X_{2}-X_{4}) \tag{21}\] \[= \ddot{y}_{ref}\bigg{(}-\frac{C_{c}}{M_{c}}X_{1}-\frac{d_{c}}{M_{c }}X_{2}+\frac{d_{c}}{M_{c}}X_{4}+d\bigg{)}+\bigg{(}\frac{C_{c}}{M_{wh}}X_{1}+ \frac{d_{c}}{M_{wh}}X_{2}-\frac{C_{wh}}{M_{wh}}X_{3}-\frac{d_{c}}{M_{wh}}X_{4}\] \[+\frac{d_{wh}}{M_{wh}}X_{4}+\frac{1}{M_{wh}}u_{d}-d\bigg{)}+ \lambda\dot{y}_{ref}-\lambda(X_{2}-X_{4})-\bigg{(}\frac{1}{M_{c}}+\frac{1}{M_ {wh}}\bigg{)}u(t)\]Using (21) and (17) in (19) we get:  
\[ \begin{split}\dot{n}=& s[\dot{y}_{ref}-\dot{X}_{2}+\dot{X }_{4}+\lambda\dot{y}_{ref}-\lambda(X_{2}-X_{4})]\\ =&[\ddot{y}_{ref}-\bigg{(}-\frac{C_{c}}{M_{c}}X_{1}- \frac{d_{c}}{M_{c}}X_{2}+\frac{d_{c}}{M_{c}}X_{4}+d\bigg{)}+\bigg{(}\frac{C_{c} }{M_{wh}}X_{1}+\frac{d_{c}}{M_{wh}}X_{2}-\frac{C_{wh}}{M_{wh}}X_{3}-\frac{d_{c }}{M_{wh}}X_{4}\\ &+\frac{d_{wh}}{M_{wh}}X_{4}+\frac{d_{wh}}{M_{wh}}u_{d}-d\bigg{)} +\lambda\dot{y}_{ref}-\lambda(X_{2}-X_{4})-\bigg{(}\frac{1}{M_{c}}+\frac{1}{M _{wh}}\bigg{)}u(t)]s\end{split}  \tag{22}\]  
Thus, the stability condition can be written as follows:  
\[\dot{n}\leq-\xi\left|s\right| \tag{23}\]  
The Lyapunov function should be \(V>0\) and \(\dot{V}\leq 0\). Initially, the system states will be directed to the sliding mode surface and then hold them on the sliding mode surface as far as the source approaches asymptotically; where \(\xi\) is the positive constant. The above expression is known as reaching or sliding condition of the system. For FOSMC, the standard form of discontinuous control law is given as follows:  
\[U_{D}=ksgn(s) \tag{24}\]  
where \(k\) is constant parameter for manual configuration and is culpable for the reaching part. Chattering phenomena in FOSMC drops the performance of the system which occurs close to the desired equilibrium position. This issue could be fixed more comprehensively if the discontinuous control \(U_{D}\) is defined as in [31]. \[U_{D}=k\frac{s}{s+\delta} \tag{25}\]  
Chattering can be minimized up to significant level through the manual adjustment of the parameter \(\delta\), which is the chattering control component. If \(e(t)\) is zero, then system can remain on the sliding mode surface all the time. Thus,  
\[\frac{ds}{dt}=0 \tag{26}\]  
Feedback control design for SMC consists of two parts, i.e., equivalent control and switching control. Equivalent control \(U_{eq}\) is accountable to hold the system trajectories on the sliding surface, while switching/discontinuous control \(U_{D}\) is responsible for moving the system trajectories in the direction of sliding surface. \(U_{eq}\) can easily be solved when \(\dot{s}=0\).