## 2 Description of the Mathematical Model
### Mathematical Model of a Quarter Car  
Figure 3 shows an illustration of a quarter-car with a PM-ASS, in which the PM-ASS has two MacPherson struts and a PM, as shown in Figure 3a. For simplicity, the mathematical models of the two MacPherson struts are combined in one model, thus, they are expressed as one damper, one spring, and an internal frictional force. The dynamics of a tire are modeled as a spring if the tire fully contacts with the road surface. In this condition, the quarter-car is modeled as a system with 2-DOF (two-degrees-of-freedom). The frictional force of the piston in the shock absorber prevents the smooth travel of the MacPherson struts so the dynamic behavior of the PM-ASS must consider the frictional force. The dynamic equation for a quarter-car with a PM-ASS is:  
\[M_{s}\ddot{Z}_{s}=K_{s}(Z_{u}-Z_{s})+B_{s}(\dot{Z}_{u}-\dot{Z}_{s})+F_{a}-F_{ \mu}\text{sgn}(\dot{Z}_{u}-\dot{Z}_{s}) \tag{10}\]  
and:  
\[M_{u}\ddot{Z}_{u}=-K_{s}(Z_{u}-Z_{s})-B_{s}(\dot{Z}_{u}-\dot{Z}_{s})+K_{t}(Z_{ r}-Z_{u})-F_{a}+F_{\mu}\text{sgn}(\dot{Z}_{u}-\dot{Z}_{s}) \tag{11}\]  
where \(F_{\mu}\) is the frictional force, \(F_{a}\) is the force that is exerted by the PM, \(Z_{r}\) is the variation in the position of the road surface, \(Z_{s}\) is the displacement of the sprung mass, \(Z_{u}\) is the displacement of the un-sprung mass, \(M_{s}\) is the weight of the rigid body (sprung mass), \(M_{u}\) is the weight of the un-sprung mass, \(K_{t}\) is a spring constant, and \(B_{s}\) is a damping constant. If \(x_{1}=Z_{s}\), \(x_{2}=\dot{Z}_{s}\), \(x_{3}=Z_{u}\), \(x_{4}=\dot{Z}_{u}\), and \(x_{5}=P\), then manipulating Equations (8)-(11) with \(x_{1}\  
and:  
\[g(\mathbf{x})=\left[ \begin{array}{c}0\\ 0\\ 0\\ 0\\ \frac{K_{V}}{T_{s}}\end{array} \right] \tag{15}\]  
Since \(L_{g(\mathbf{x})}L_{f(\mathbf{x})}^{\kappa}h(\mathbf{x})=0\) for all \(\kappa<3\), and \(L_{g(\mathbf{x})}L_{f(\mathbf{x})}^{\kappa}h(\mathbf{x})\
eq 0\) for all \(\kappa\geq 3\left(\forall x\right)\), the relative degree of the system is 3. Using input-output Feedback linearization, Equation (13) is transformed into a canonical form after simple manipulation, which is:  
\[y^{(3)}=L_{f(\mathbf{x})}^{3}h(\mathbf{x})+L_{g(\mathbf{x})}L_{f(\mathbf{x})}^ {2}h(\mathbf{x})u=F(\mathbf{x})+G(\mathbf{x})u, \tag{16}\]  
where:  
\[ \begin{array}{l}F(\mathbf{x})=-\frac{1}{M_{s}}\{K_{s}(x_{2}-x_{4})-B_{s}( \frac{1}{M_{s}}+\frac{1}{M_{u}})\{K_{s}(x_{1}-x_{3})+B_{s}(x_{2}-x_{4})-\frac{ [3[L_{0}+(x_{1}-x_{3})]^{2}-l^{2}]}{4\pi n^{2}}x_{5}+\\ \frac{[3[L_{0}+(x_{1}-x_{3})]^{2}-l^{2}]}{4\pi n^{2}}P_{e}+F_{\mu}\text{sgn}(x _{2}-x_{4})]\}+\frac{B_{s}K_{t}}{M_{u}}x-\frac{B_{s}K_{t}}{M_{u}}Z_{r}+\frac{ [3[L_{0}+(x_{1}-x_{3})]^{2}-l^{2}]}{4\pi n^{2}T_{s}}x_{5}-\\ \frac{[6[L_{0}+(x_{1}-x_{3})](x_{2}-x_{4})]}{4\pi n^{2}}x_{5}+\frac{[6[L_{0}+ (x_{1}-x_{3})]^{2}(x_{2}-x_{4})]}{4\pi n^{2}}P_{e}+\dot{F}_{\mu}\text{sgn}(x_ {2}-x_{4})]\end{array}  \tag{17}\]  
and:  
\[G(\mathbf{x})=\frac{K_{p}}{M_{s}T_{s}}\frac{\left\{3[L_{0}+(x_{1}-x_{3})]^{2}- l^{2}\right\}}{4\pi n^{2}} \tag{18}\]  
The states are selected as \(\mathbf{z}(t)=\left[ \begin{array}{cccc}z_{1}(t)&z_{2}(t)&z_{3}(t)\end{array}  \right]^{T}=\left[ \begin{array}{cccc}y(t)&\dot{y}(t)&\ddot{y}(t)&\ddot{y}(t) \end{array} \right]^{T}\in R^{3}\), thus, Equation (16) can be expressed as:  
\[ \begin{array}{l}\dot{z}_{1}=z_{2}\\ \dot{z}_{2}=z_{3}\\ \dot{z}_{3}=\overline{F}(\mathbf{z})+\overline{G}(\mathbf{z})u\\ y=z_{1}\end{array}  \tag{19}\]  
where the output \(z_{1}\) is the displacement of the sprung mass, i.e., \(z_{1}=x_{1}=y\), \(\overline{F}(\mathbf{z})\) and \(\overline{G}(\mathbf{z})\) cannot be assumed to be given continuous functions because they might contain unknown parameters, nonlinearities and uncertainties.