# Integrated Adaptive Steering Stability Control for Ground Vehicle with Actuator Saturations
## 3 Integrated Controller Design
Choosing a Lyapunov candidate \(V_{10}=\frac{k_{\zeta}}{2}\varepsilon_{10}^{2}+\frac{1}{2}\zeta^{2}\), we then have  
\[\dot{V}_{10}=e_{10}(f_{\gamma}(x_{\gamma},\delta_{f})+M_{\Delta}+u_{y}-I_{z} \dot{x}_{10\gamma})-k_{\zeta}\zeta^{2}+\zeta M_{\Delta} \tag{15}\]  
We design the control law of \(u_{y}\) as  
\[u_{y}=-f_{\gamma}(x_{\gamma},\delta_{f})+I_{z}\dot{x}_{10\gamma}-k_{10}e_{10}- k_{11}\zeta,\quad k_{10}>0. \tag{16}\]  
Then, substituting (16) into (15) yields  
\[\dot{V}_{10} = -k_{10}e_{10}^{2}-k_{\zeta}\zeta^{2}+e_{10}M_{\Delta}-k_{11} \zeta e_{10}+\zeta M_{\Delta}\] \[= -\frac{k_{10}}{3}e_{10}^{2}+(e_{10}M_{\Delta}-\frac{k_{10}}{3}e_ {10}^{2})+(-\frac{k_{10}}{3}e_{10}^{2}-\frac{k_{\zeta}}{3}\zeta^{2}-k_{11} \zeta e_{10})\] \[-\frac{k_{\zeta}}{3}\zeta^{2}+(\zeta M_{\Delta}-\frac{k_{\zeta}} {3}\zeta^{2})\] \[= -\frac{k_{10}}{3}e_{10}^{2}-(\sqrt{\frac{k_{10}}{3}}e_{10}-\sqrt{ \frac{3}{4k_{10}}}M_{\Delta})^{2}+\frac{3}{4k_{10}}M_{\Delta}^{2}\] \[u_{I}=\frac{du_{z}+u_{\theta}}{2d},\quad u_{r}=\frac{du_{z}-u_{\theta}}{2d}. \tag{13}\]0 \[u_{I}=\frac{du_{z}+u_{\theta}}{2d},\quad u_{r}=\frac{du_{z}-u_{\theta}}{2d}. \tag{13}\]1 \[u_{I}=\frac{du_{z}+u_{\theta}}{2d},\quad u_{r}=\frac{du_{z}-u_{\theta}}{2d}. \tag{13}\]2  
where \(\bar{M}\) is the upper bound of \(M_{\Delta}\). If we take \(\rho=\min\{\frac{4k_{10}-3k_{11}}{3l_{z}},\frac{4k_{\zeta}}{3}-k_{11}\}\), \(\frac{2k_{10}}{3}>\frac{k_{11}}{2}\), and \(\frac{2k_{\zeta}}{3}>\frac{k_{11}}{2}\), then  
\[u_{I}=\frac{du_{z}+u_{\theta}}{2d},\quad u_{r}=\frac{du_{z}-u_{\theta}}{2d}. \tag{13}\]3  
where \(\varepsilon=(\frac{3}{4k_{10}}+\frac{3}{4k_{\zeta}})\bar{M}^{2}\). Integrating both sides of (17) generates  
\[u_{I}=\frac{du_{z}+u_{\theta}}{2d},\quad u_{r}=\frac{du_{z}-u_{\theta}}{2d}. \tag{13}\]4  
When \(t\to\infty\), we have \(V(\infty)\leq\frac{e_{0}}{\rho}\), meaning that the signals \(e_{10}\) and \(\zeta\) are ultimately bounded and the attraction domain of \(V_{10}\) can be adjusted to be smaller, with a smaller \(\sigma\) and a larger \(k_{10}\). The integrated controller design yields a fifth-order error dynamics system, while the original system is a tenth-order system. Hence, the zero dynamics contains five states. Setting \(e_{1}=e_{3}=e_{10}=0\) generates the following zero dynamics:  
\[u_{I}=\frac{du_{z}+u_{\theta}}{2d},\quad u_{r}=\frac{du_{z}-u_{\theta}}{2d}. \tag{13}\]5  
where  
\[x_{0}=\left[ \begin{array}{c}x_{5}\\ x_{6}\\ x_{7}\\ x_{8}\\ x_{9}\end{array} \right],\quad A=\left[ \begin{array}{cccc}0&1&0&0&0\\ -\frac{k_{x}}{m_{w}}&-\frac{cz_{w}}{m_{w}}&0&0&0\\ 0&0&0&1&0\\ 0&0&-\frac{k_{w}}{m_{w}}&-\frac{cz_{w}}{m_{w}}&0\\ 0&0&0&0&-\frac{2(cz+cz_{r})}{m_{w}}\end{array} \right],\]\[B=\left[ \begin{array}{ccccc}0&0&0&0&0\\ \frac{k_{x}}{m_{w}}&\frac{c_{x}}{m_{w}}&0&0&0\\ 0&0&0&0&0\\ 0&0&\frac{k_{w}}{m_{w}}&\frac{c_{w}}{m_{w}}&0\\ 0&0&0&0&\frac{2c_{f}}{m_{w}}\end{array} \right],\quad z_{0}=\left[ \begin{array}[ ]{c}z_{rl}\\ z_{rl}\\ z_{rr}\\ \delta_{f}\end{array} \right],\] \[C=\left[ \begin{array}{ccccc}0&0&0\\ -\frac{m_{a}}{2}&-\frac{l_{x}}{2d}&0\\ 0&0&0\\ -\frac{m_{a}}{2}&\frac{l_{x}}{2d}&0\\ 0&0&\frac{2(bc_{r}-ac_{f})}{m_{w}2}-1\end{array} \right],\quad d_{0}=\left[ \begin{array}{c}0\\ \frac{r_{x}h_{x}}{2d}\\ 0\\ -\frac{r_{y}h_{x}}{2d}\\ 0\end{array} \right],\quad x_{r}=\left[ \begin{array}{c}\dot{x}_{2r}\\ \dot{x}_{4r}\\ x_{10r}\end{array} \right].\]  
It is easy to see that \(A\) is Hurwitz; accordingly the zero dynamics is stable. The control scheme proposed in this paper is based on full-state feedback; thus, the full-state variable acquisition becomes a requirement. However, as certain states, especially for the vehicle sideslip angle \(\beta\)[32] and body roll angle \(\theta\), are not easy to measure directly by sensors, this requirement is somewhat demanding. In addition, the tire lateral force model is involved in the controller design; tire state measurement has been a difficult problem for many decades, and may lead to the limitation of the controller implementation in practice.