# Model Reference Adaptive Control of Vehicle Slip Ratio Based on Speed Tracking
## 3 Model Reference Adaptive Control for Slip Ratio Control
### Model Reference Adaptive Controller  
The plant model (5) is described in the form of a differential operator:  
\[\begin{array}{l}pv_{w}=b_{1}v_{w}+b_{2}T_{b}+b_{3}\mu(\lambda)\\ (p-b_{1})v_{w}=b_{2}T_{b}+b_{3}\mu(\lambda)\\ A_{p}(p)v_{w}(t)=b_{p}T_{b}+b_{3}\mu(\lambda)\end{array} \tag{7}\]  
where, \(A_{p}=p-b_{1}\), \(b_{p}=b_{2}\), and \(p=\frac{d}{dt}\) is the differential operator.  
The chosen reference model is defined as (8), in which the desired output is \(v_{m}(t)=\hat{v}(t)(1+\lambda^{*})\), and \(\hat{v}\) is estimated by vehicle velocity observer.  
\[A_{m}v_{m}(t)=b_{m}T_{m}(t) \tag{8}\]  
where \(b_{m}\) is a positive gain, \(A_{m}=p+a_{0}\), \(a_{0}\) is a given constant determined by a performance index requirement, \(T_{m}(t)\) is the control input of the reference model which represents the basic value of braking torque. The tracking error is defined as  
\[e(t)=v_{m}(t)-v_{w}(t) \tag{9}\]  
The generalized error equation and model reference adaptive adaptation law are derived as follows.  
The relationship of braking torque \(T_{b}(t)\), the control input of the reference model \(T_{m}(t)\) and the output of the adaptive controller \(u_{1}(t)\) is constructed as  
\[T_{b}(t)=T_{m}(t)-u_{1}(t) \tag{10}\]  
Subtracting (7) from (8), and adding \(A_{p}v_{m}(t)\) to both sides of the equation results in:  
\[\begin{array}{l}A_{m}v_{m}(t)-A_{p}v_{w}(t)+A_{p}v_{m}(t)=b_{m}T_{m}(t)-b_{p }T_{b}-b_{3}\mu(\lambda)+A_{p}v_{m}(t)\\ A_{p}(v_{m}(t)-v_{w}(t))=b_{m}T_{m}(t)-b_{p}T_{b}-b_{3}\mu(\lambda)+A_{p}v_{m}( t)-A_{m}v_{m}(t)\\ A_{p}e(t)=(A_{p}-A_{m})v_{m}(t)+b_{m}T_{m}(t)-b_{p}T_{b}-b_{3}\mu(\lambda)\end{array} \tag{11}\]  
The control variable \(u_{1}(t)\) is introduced to (11) via (10), so the generalized error is expressed  
\[e(t)=\frac{1}{A_{p}}\{(A_{p}-A_{m})v_{m}(t)+b_{m}u_{1}(t)+(b_{m}-b_{p})T_{b}( t)-b_{3}\mu(\lambda)\} \tag{12}\]  
The adaptive controller \(u_{1}(t)\) is obtained  
\[\begin{array}{l}u_{1}(t)=\frac{1}{b_{m}}[K_{p}v_{m}(t)+G_{p}T_{b}(t)+L_{p} \mu]\\ K_{p}(p)=k_{1}p+k_{0}\\ G_{p}(p)=g_{0}\\ L_{p}(p)=l_{0}\end{array} \tag{13}\]  
where, \(k_{1}\), \(k_{0}\), \(g_{0}\) and \(l_{0}\) are the adaptive gains. \(\mu\in[0,1]\) is bound and can be observed by the road adhesion coefficient observer \(\hat{\mu}\) in 3.2. By introducing the road adhesion coefficient observer \(\hat{\mu}\) into the model reference adaptive controller, the control system will have enhanced robustness and adaptability.  
The proportion-integration adaptive control law is described as follows:  
\[\begin{array}{l}k_{i}=-k_{1i}ep^{i}v_{m}-k_{2}\frac{d}{dt}(k_{i1}ep^{i}v_{m }),\ i=\ 0,\ \ \ 1\ \,\\ g_{0}=-g_{01}eT_{b}-g_{02}\frac{d}{dt}(g_{01}eT_{b})\\ l_{0}=-l_{01}e\hat{\mu}-l_{02}\frac{d}{dt}(l_{01}e\hat{\mu})\end{array} \tag{14}\]  
where \(k_{i1}\), \(k_{2}\), \(g_{01}\), \(g_{02}\), \(l_{01}\), \(l_{02}>0\).  
Finally, the general error equation of the system is given by:  
\[v(t)=\frac{1}{A_{p}}\{(A_{p}-A_{m}+K_{p})v_{m}(t)+(b_{m}-b_{p}+G_{p})T_{b}(t)-b_{ 3}\hat{\mu}(\lambda)\} \tag{15}\]  
For the wheel speed dynamic in (7) and the reference model in (8), if the system transfer function is strictly positive and real, and the adaptive control law is satisfied by (14) based on the Popov hyperstability theory [30], then the tracking error of the system \(e(t)\) can asymptotically converge to zero.  
\[\lim_{t\to\infty}e(t)=\lim_{t\to\infty}(v_{m}(t)-v_{w}(t))=0 \tag{16}\]  
**Remark 2**.: _The model reference adaptive control does not require higher model accuracy and its performance does not rely on the model accuracy although the system model (7) is simplified. The adaptive parameters in (14) can compensate and adapt the parameter uncertainty and the modeling error, such as the rolling radius change \(r_{w}\) and the load variation in terms in \(\hat{\mu}\)._