## 3 Control Algorithm Synthesis
### Suspension Performance  
In the above analysis, we have proved that the tracking error of vertical displacement \(z_{s}\) is retained within the corresponding prescribed bound and the vertical velocity \(\dot{z}_{s}\) is bounded in a certain range as well. In this subsection, we will conduct the stability analysis of the zero dynamics to ensure that the CL system meets the safety constraints given at the end of Section 3. Redefining the state variable vector as follows:  
\[\overline{\mathbf{x}}=[\overline{x}_{1},\overline{x}_{2}]^{T}=\big{[}z_{s}-z_{v} \dot{z}_{s}\big{]}^{T}, \tag{43}\]  
Assuming that all the estimated values are the same as the real values, the zero dynamics equation can be obtained in the matrix form:  
\[\begin{split}\dot{\overline{\mathbf{x}}}=\mathbf{A}\overline{\mathbf{x}}+ \mathbf{\omega}\\ \mathbf{\omega}=\mathbf{B}\dot{z}_{v}+\mathbf{C}\Big{(}F_{a}+F_{f}+f\Big{)}, \end{split} \tag{44}\]  
with the coefficient matrixes \(\mathbf{A},\mathbf{B}\), and \(\mathbf{C}\) given as follows:  
\[\mathbf{A}=\begin{bmatrix}0&1\\ -\frac{k_{s}}{M}&-\frac{c_{s}}{M}\end{bmatrix},\;\mathbf{B}=\begin{bmatrix} -1\\ \frac{c_{s}}{M}\end{bmatrix},\;\mathbf{C}=\begin{bmatrix}0\\ \frac{1}{M}\end{bmatrix}. \tag{45}\]  
It can be verified that all the eigenvalues of matrix \(\mathbf{A}\) have a negative real part, which means there always exists a pair of positive matrixes \(\mathbf{P}>0\) and \(\mathbf{Q}>0\) satisfying the equity \(\mathbf{A}^{T}\mathbf{P}+\mathbf{P}\mathbf{A}=-\mathbf{Q}\). Therefore, we can select a Lyapunov function as follows:  
\[V_{z}=\overline{\mathbf{x}}^{T}\mathbf{P}\overline{\mathbf{x}}. \tag{46}\]  
Taking the time derivative of Equation (46) gives  
\[\dot{V_{z}}=\overline{\mathbf{x}}^{T}\Big{(}\mathbf{A}^{T}\mathbf{P}+\mathbf{P}\mathbf{A}\Big{)} \overline{\mathbf{x}}+2\overline{\mathbf{x}}^{T}\mathbf{P}\mathbf{\omega}. \tag{47}\]  
Note that, because \(\dot{z}_{v}\), \(F_{a}\), \(F_{f}\), and \(f\) are all bounded, \(\mathbf{\omega}\) is bounded for a positive constant \(\overline{\omega}\), i.e., \(\|\mathbf{\omega}\|\leq\overline{\omega}\), we can further obtain  
\[\begin{split}\dot{V_{z}}\leq-\lambda_{min}(\mathbf{Q})\|\overline{\mathbf{ x}}\|^{2}+\lambda_{max}(\mathbf{P})\|\overline{\mathbf{x}}\|\|\mathbf{\omega}\|\\ \leq-[\lambda_{min}(\mathbf{Q})-\lambda_{max}(\mathbf{P})/\eta]\|\overline{\mathbf{x}} \|^{2}+\eta\lambda_{max}(\mathbf{P})\overline{\omega}^{2},\end{split} \tag{48}\]where \(\eta\) is a positive design parameter. By appropriately choosing \(\eta>\lambda_{max}(P)/\lambda_{min}(Q)\), the above inequity can be rewritten as  
\[\dot{V_{z}}\leq-\alpha V_{z}+\beta, \tag{49}\]  
where \(\alpha=(\lambda_{min}(Q)-\lambda_{max}(P)/\eta)/\lambda_{min}(P)\) and \(\beta=\eta\lambda_{max}(P)\overline{\omega}^{2}\). The above inequity indicates that \(V_{z}(t)\) is bounded with the upper limit \(\overline{V_{z}}=V_{z}(0)+\beta/\alpha\), which means that \(\overline{\mathrm{x}}_{\mathrm{i}}\) is bounded as follows:  
\[|\overline{x}_{i}|\leq\sqrt{\left[V_{z}(0)+\beta/\alpha\right]/\lambda_{min}( P)},\ \ (i=1,2). \tag{50}\]  
Therefore, the bounds of the suspension deformation can be guaranteed as  
\[|\Delta z|=|\overline{x}_{1}|\leq\sqrt{\left[V_{z}(0)+\beta/\alpha\right]/ \lambda_{min}(P)}. \tag{51}\]  
Furthermore, the maximum actuating force can be regulated by appropriately tuning the design parameters \(\eta\) and \(P\) such that  
\[|F_{a}|\leq(\mathbf{k_{s}}+\mathbf{\epsilon_{s}})\sqrt{\left[V_{z}(0)+\beta/\alpha \right]/\lambda_{min}(P)}+\mathbf{c_{s}}\big{\|}\dot{\mathbf{z}}_{\oplus}\big{\|}_{ \infty}+\mathbf{F_{c}}<\mathbf{F_{a-max}}, \tag{52}\]  
where \(\mathbf{F_{c}}\) is the Coulomb friction. By substituting the above inequity into Equation (1), the second safety constraint can be guaranteed.