# Integrated Adaptive Steering Stability Control for Ground Vehicle with Actuator Saturations
## 3 Integrated Controller Design  
Let \(x_{1}=z_{s}\), \(x_{2}=\dot{z}_{s}\), \(x_{3}=\theta\), \(x_{4}=\dot{\theta}\), \(x_{5}=z_{wl}\), \(x_{6}=\dot{z}_{wl}\), \(x_{7}=z_{wr}\), \(x_{8}=\dot{z}_{wr}\), \(x_{9}=\beta\), and \(x_{10}=\gamma\); then, the vehicle dynamics can be reformulated as  
\[\dot{x}_{1} = x_{2},\quad\dot{x}_{2}=\frac{1}{m_{s}}(-F_{sdl}-F_{sdr}-F_{ssl}-F _{ssr}+u_{z})\] \[\dot{x}_{3} = x_{4},\quad\dot{x}_{4}=\frac{1}{I_{x}}(-d(F_{sdl}+F_{ssl})+d(F_{ sdr}+F_{ssr})+F_{y}h_{g}+u_{\theta})\] \[\dot{x}_{5} = x_{6},\quad\dot{x}_{6}=\frac{1}{m_{wl}}(F_{sdl}+F_{ssl}-F_{swl}- F_{usl}-u_{l})\] \[\dot{x}_{7} = x_{8},\quad\dot{x}_{8}=\frac{1}{m_{wr}}(F_{sdr}+F_{ssr}-F_{wdr}- F_{wsr}-u_{r})\] \[\dot{x}_{9} = -\frac{2(c_{f}+c_{r})}{mw}x_{9}+\left(\frac{2(bc_{r}-ac_{f})}{mw^ {2}}-1\right)x_{10}+\frac{2c_{f}}{mw}\delta_{f}\] \[\dot{x}_{10} = \frac{2(bc_{r}-ac_{f})}{I_{z}}x_{9}-\frac{2(a^{2}c_{f}+b^{2}c_{r} )}{I_{z}v}x_{10}+\frac{2 ac_{f}}{I_{z}}\delta_{f}+\frac{1}{I_{z}}M \tag{7}\]  
Accordingly, the integrated controller can be synthesised via the following three steps. Step 1: let \(e_{1}=x_{1}-x_{1r}\), and \(e_{2}=x_{2}-x_{2r}\), where \(x_{1r}\) is the reference value of \(x_{1}\) and \(x_{2r}\) is the virtual control input; then,  
\[\dot{e}_{1}=\dot{x}_{1}-\dot{x}_{1r}=e_{2}+x_{2r}-\dot{x}_{1r}.\]  
Choose \(V_{1}=\frac{1}{2}e_{1}^{2}\) as a Lyapunov candidate; its derivation along \(\dot{e}_{1}\) yields  
\[\dot{V}_{1}=e_{1}(e_{2}+x_{2r}-\dot{x}_{1r})\]  
Take \(x_{2r}=\dot{x}_{1r}-k_{1}e_{1}\), where \(k_{1}>0\) is the design parameter; then, we have  
\[\dot{V}_{1}=e_{1}e_{2}-k_{1}e_{1}^{2}.\]  
Choose \(V_{2}=V_{1}+\frac{m_{s}}{2}e_{2}^{2}+\frac{1}{2I_{z}}\dot{m}_{s}^{2}\) as a Lyapunov candidate, where \(\dot{m}_{s}=\dot{m}_{s}-m_{s}\) and \(\dot{m}_{s}\) is the estimation value of \(m_{s}\); then, taking its deviation yields  
\[\dot{V}_{2}=e_{1}e_{2}-k_{1}e_{1}^{2}+e_{2}(-F_{sdl}-F_{sdr}-F_{ssl}-F_{ssr}+u_ {z}-m_{s}\dot{x}_{2r})+\frac{1}{r_{1}}\dot{m}_{s}\dot{m}_{s}.\]  
Take  
\[\dot{x}_{3} = x_{4},\quad\dot{x}_{4}=\frac{1}{I_{x}}(-d(F_{sdl}+F_{ssl})+d(F_{ sdr}+F_{ssr})+F_{y}h_{g}+u_{\theta})\]0  
and the adaptation law  
\[\dot{x}_{3} = x_{4},\quad\dot{x}_{4}=\frac{1}{I_{x}}(-d(F_{sdl}+F_{ssl})+d(F_{ sdr}+F_{ssr})+F_{y}h_{g}+u_{\theta})\]1where \(k_{2}>0\) and \(r_{1}>0\) are design parameters; then, we have  
\[\dot{x}_{3} = x_{4},\quad\dot{x}_{4}=\frac{1}{I_{x}}(-d(F_{sdl}+F_{ssl})+d(F_{ sdr}+F_{ssr})+F_{y}h_{g}+u_{\theta})\]2  
which indicates that \(e_{1}\), \(e_{2}\) and \(\tilde{m}_{s}\) are bounded. Taking the derivative of \(\dot{V}_{2}\) leads to  
\[\dot{x}_{3} = x_{4},\quad\dot{x}_{4}=\frac{1}{I_{x}}(-d(F_{sdl}+F_{ssl})+d(F_{ sdr}+F_{ssr})+F_{y}h_{g}+u_{\theta})\]3  
thus, \(\dot{V}_{2}\) is uniformly continuous, conditional on \(x_{1r}\) and \(\dot{x}_{1r}\in L_{\infty}\). Consequently, the error dynamics of \(e_{1}\) and \(e_{2}\) are asymptotically convergent to \(0\) with Lyapunov-like lemma. Step 2: to facilitate the development of the control law, the following simplicities are used: \(\theta_{I}=\frac{1}{I_{x}}\) and \(f_{\theta}(x_{\theta},\delta_{f})=-d(F_{ssl}+F_{ssl})+d(F_{sdr}+F_{ssr})+F_{y} h_{g}\), where \(x_{\theta}=(x_{1},x_{2},x_{3},x_{4},x_{5},x_{6},x_{7},x_{8})^{T}\). Let \(e_{3}=x_{3}-x_{3r}\), \(e_{4}=x_{4}-x_{4r}\), \(\tilde{\theta}_{I}=\tilde{\theta}_{I}-\theta_{I}\), where \(x_{3r}\) is the reference value of \(x_{3}\), \(x_{4r}\) is the virtual control input, and \(\tilde{\theta}_{I}\) is the estimation of \(\tilde{\theta}_{I}\).