## 4 Controller Design of AEBS
### Execution Controller Design at Lower Level  
Throttle valve control is a logic control process whose main aim is to judge whether the motor is working; this process does not include any tracking precision or speed problems, and hence no separate controller is required. On the contrary, the brake controller should be robust enough to adapt to many uncertainties, e.g., variations in the road surface parameters and sensor noises. Therefore, in this section, a robust EBS is employed in the execution system. For its robustness to model error and external disturbance, the sliding mode control has been widely applied in nonlinear systems. However, the sliding mode control is a progressive convergence system. Terminal Sliding Control has the advantage of convergence in a limited time. However, when the system is far from the equilibrium point, TSMC tends to converge quite slowly. Non-singularity Fast Terminal Sliding Mode Control can increase the absolute value of the state derivative and effectively boost the convergence speed. Meanwhile, the introduction of a terminal attractor can solve the chattering problem of Sliding Mode Control considering system non-singularity [53; 54; 55]. Hence NFTSM is adapted as the control algorithm of the electronic brake system (EBS)First, define the Sliding Mode Plane as  
\[s=\dot{e}+\alpha e+\beta e^{\frac{p}{q}}, \tag{30}\]  
where \(e\in R\), \(\alpha\) and \(\beta\) are constants with \(\alpha>0\). \(q\) is a given positive odd number with \(p<q<2p\). Define the following error as  
\[e=\lambda-\lambda_{\text{Ref.}}. \tag{31}\]From Equation (1), we get  
\[\dot{e}=\frac{-r_{b}^{2}m_{ij}g\mu\big{(}B_{ij},C_{ij},D_{ij},E_{ij},\lambda\big{)} /J+(1-\lambda)\dot{u}}{u}+\frac{T_{b}r_{b}}{uJ}-\dot{\lambda}_{\text{Ref}}, \tag{31}\]  
where \(m_{ij}g\) is the load allocation value and \(\mu\big{(}B_{ij},C_{ij},D_{ij},E_{ij},\lambda\big{)}\) is the coefficient equation of the Magic Formula, with \(B_{ij},C_{ij},D_{ij}\) and \(E_{ij}\) being coefficients. To avoid the system chattering problem of the FTSM, the key step is to eliminate the nonlinear switch item, so an attractor approach is adapted to design the sliding mode control law. The terminal attractor can drive the system state to a sliding mode plane in limited time and is robust against model error and external disturbance. In this paper, an item of \(e^{p/q-1}\) is added to the traditional terminal attractor according to the characteristic of NFTSM, expressed as follows:  
\[\dot{s}=(-\phi s-\gamma s^{\frac{m}{n}})e^{\frac{p}{q}-1}, \tag{32}\]  
where \(\phi\in R^{+}\); \(\gamma\in R^{+}\); m and n are positive odd numbers with \(0<m/n<1\). Solving simultaneously Equations (29), (30), (32), and (33) will give us the control rate of NFTSM:  
\[ \begin{array}{ll}T_{b}&=\frac{uJ}{P_{b}}\left\{\frac{q}{\dot{P}P}\Big{(}- \ddot{e}\cdot e^{1-\frac{p}{q}}-\alpha\cdot\dot{e}\cdot e^{1-\frac{p}{q}}-(- \phi s-\gamma s^{\frac{m}{n}})\Big{)}\right.\\ &\left.-\frac{-r_{b}^{2}m_{ij}g\mu\big{(}B_{ij}C_{ij},D_{ij},E_{ij}\lambda\big{)} /J+(1-\lambda)\dot{u}}{u}-\dot{\lambda}_{\text{Ref}}\right\}\end{array} . \tag{33}\]  
**Proof 1. ** In Equation (33), no singularity will occur to cause \(0<1-p/q\). \(\square\)  
**Proof 2. ** Define Lyapunov function as  
\[V=\frac{1}{2}s^{2}. \tag{34}\]  
So  
\[ \begin{array}{l}\dot{V}=s\dot{s}=s\Big{(}\ddot{e}+\alpha\dot{e}+\frac{\beta p }{q}e^{\frac{p}{q}-1}\dot{e}\Big{)}\\ =s\Big{[}\ddot{e}+\alpha\dot{e}+\frac{\beta p}{q}e^{\frac{p}{q}-1}\cdot\left( \frac{-r_{b}^{2}m_{ij}g\mu\big{(}B_{ij},C_{ij},D_{ij},E_{ij}\lambda\big{)}/J+( 1-\lambda)\dot{u}}{u}+\frac{T_{b}r_{b}}{uJ}-\dot{\lambda}_{\text{Ref}}\right) \Big{]}\end{array} . \tag{35}\]  
Substituting Equation (32) into Equation (34) will give  
\[\dot{V}=s\dot{s}=s\Big{(}-\phi s-\gamma s^{\frac{m}{n}}\Big{)}=-\phi s^{2}- \gamma s^{\frac{m+n}{n}}. \tag{36}\]  
Because \(m\) and \(n\) are positive odd numbers, \(s^{\frac{m+n}{n}}\geq 0\).