# Model Reference Adaptive Control of Vehicle Slip Ratio Based on Speed Tracking
## 3 Model Reference Adaptive Control for Slip Ratio Control
### Road Adhesion Coefficient Observer  
Among the uncertainties, road condition is one of the most difficult variables, which affects the control performance directly. For this reason, it is necessary to design the road adhesion coefficient observer. The road adhesion coefficient observer is designed as follows.  
From (5), the road adhesion coefficient can be obtained  
\[b_{3}\mu(\lambda)=v_{w}-b_{1}v_{w}-b_{2}T_{b} \tag{17}\]  
Subsequently, the road adhesion coefficient observer is proposed as follows:  
\[\hat{\mu}(\lambda)=L(v_{w}-b_{1}v_{w}-b_{2}T_{b}-b_{3}\hat{\mu}(\lambda)) \tag{18}\]  
Let \(z=\hat{\mu}(\lambda)-h(v_{w})\). The observer gain is decided  
\[L=\frac{dh}{dv_{w}}\]  
where, \(h(v_{w})=cv_{w}\), \(c\) is a constant, and \(L=c\).  
Through the derivation of auxiliary variables \(z\), we have  
\[\begin{array}{ll}z&=\hat{\mu}(\lambda)-h(v_{w})=\hat{\mu}(\lambda)-Lv_{w}\\ &=L(-b_{1}v_{w}-b_{2}T_{b}-b_{3}\hat{\mu}(\lambda))\\ &=L(-b_{1}v_{w}-b_{2}T_{b}-b_{3}(z+h(v_{w})))\end{array} \tag{19}\]  
The observation error is  
\[\overline{\mu}(\lambda)=\mu(\lambda)-\hat{\mu}(\lambda)\]  
And  
\[\overline{\mu}(\lambda)=\mu(\lambda)-\hat{\mu}(\lambda) \tag{20}\]  
Substituting (20),  
\[\overline{\mu}(\lambda)+L(v_{w}-b_{1}v_{w}-b_{2}T_{b}-b_{3}\hat{\mu}(\lambda ))-\mu(\lambda)=0\] (21a) Considering (19), we have  
\[\overline{\mu}(\lambda)+L(v_{w}-b_{1}v_{w}-b_{2}T_{b}-b_{3}\hat{\mu}(\lambda )-\mu(\lambda)+L(-v_{w}+b_{1}v_{w}+b_{2}T_{b}+b_{3}\mu(\lambda))=0 \tag{21b}\] \[\overline{\mu}(\lambda)+Lb_{3}\overline{\mu}(\lambda)-\mu(\lambda )=0\]Based on auxiliary variable, the disturbance observer can be designed as:  
\[\hat{\mu}(\lambda)=z+h(v_{w})=z+cv_{w} \tag{22}\] \[z=\hat{\mu}(\lambda)-h(v_{w})\] \[=L(-b_{1}v_{w}-b_{2}T_{b}-b_{3}(z+h(v_{w})))\] \[=c(-b_{3}z-(b_{1}+b_{3}c)v_{w}-b_{2}T_{b})\]  
**Lemma 1**.: ([17; 31])_: For a nonlinear system (7), and a positive definite function \(V\) satisfies the following differential inequality:  
\[V\leq-\varsigma V+C \tag{23}\]  
where, \(\varsigma>0\) and \(C\geq 0\), and for the given \(t_{0}\), \(V\) satisfies:  
\[0\leq V\leq\left[C-(C-\varsigma V(t_{0}))\exp(-\varsigma(t-t_{0}))\right]/ \varsigma,\ \ \ \ \ \forall t\geq t_{0} \tag{24}\]  
Then \(V\) is eventually bounded by \(C/\varsigma\), and the exponential convergence rate of \(V\) to the bound is \(\varsigma\). The system (7) is uniformly and ultimately bounded.  
The stability of the system (7) is analyzed according to Lemma 1. Choose the Lyapunov candidate function as  
\[V=\frac{1}{2}\overline{\mu}^{2} \tag{25}\]  
According to (22),  
\[V=\overline{\mu\mu}=\overline{\mu}\mu-cb_{3}\overline{\mu}^{2} \tag{26}\]  
It is assumed \(\left|\mu\right|<D\), \(D>0\), and \(\overline{\mu}^{2}\leq 2q,q>0\) then  
\[V\leq\left|D\overline{\mu}\right|-cb_{3}\overline{\mu}^{2} \tag{27}\]  
According to [PERSON]'s inequality,  
\[\left|D\overline{\mu}\right|\leq\frac{1}{4}D^{2}+\overline{\mu}^{2} \tag{28}\]  
Then,  
\[V\leq\frac{1}{4}D^{2}+\overline{\mu}^{2}-cb_{3}\overline{\mu}^{2}=-(cb_{3}-1 )\overline{\mu}^{2}+\frac{1}{4}D^{2} \tag{29}\]  
Thus, according to Lemma 1, if \(\varsigma=cb_{3}-1>0\), \(C=\frac{1}{4}D^{2}\geq 0\), we can make \(\varsigma>\frac{C}{2q}\), and \(V\leq 0\) on \(V=q\), the road adhesion coefficient observer is semi-globally, uniformly and ultimately bounded.  
**Remark 3**.: The observer gain needs relatively large to satisfy \(\varsigma>\frac{C}{2q}\), such that the observer quickly converges to the actual value. In addition, compared with the assumption that \(\mu=0\) in [17], \(\left|\mu\right|<D\) in this paper is more in accordance with the actual condition and the method is simpler because it does not need to construct a virtual control law and filter. The stability is only related to the observer itself based on the [PERSON] stability theory. Furthermore, because the model reference adaptive control system has strong robustness, even if there is some observation error, it will not destroy the stability and control performance of the system.