## 2 Operational Principle and Mathematical Model
### Mathematical Model  
According to [PERSON]'s second law, the dynamical model of typical EPS systems is described below. The column dynamics are described as follows:  
\[J_{c}\overset{\bullet\bullet}{\theta_{c}}=T_{d}-B_{c}\overset{\bullet}{\theta _{c}}-K_{c}(\theta_{c}-\frac{p_{r}}{r_{p}}) \tag{1}\]  
where \(\theta_{c}\), \(r_{p}\), \(B_{c}\), \(K_{c}\), \(J_{c}\), \(p_{r}\), \(T_{d}\) are the steering wheel angular position, pinion radius, steering wheel damping, steering column torsional stiffness, steering wheel moment of inertia, steering rack displacement and driver torque, respectively. The motor mechanical dynamics are described by the following equation:  
\[J_{m}\overset{\bullet\bullet}{\theta_{m}}=k_{e}I_{a}-B_{m}\overset{\bullet}{ \theta_{m}}-K_{m}(\theta_{m}-\frac{p_{r}}{r_{p}}N) \tag{2}\]  
where \(J_{m}\), \(\theta_{m}\), \(k_{e}\), \(B_{m}\), \(N\) are the motor moment of inertia motor torque of inertia, motor column angular position, motor torque and voltage constant, motor and gearbox damping, motor and gearbox torsional stiffness and motor gear ratio, respectively. The rack dynamics are described by the following equation:  
\[M_{r}\overset{\bullet\bullet}{p_{r}}=\frac{K_{c}}{r_{p}}(\theta_{c}-\frac{p_{r }}{r_{p}})-B_{r}\overset{\bullet}{p_{r}}+\frac{K_{m}N}{r_{p}}(\theta_{m}-\frac {p_{r}}{r_{p}}N)-K_{r}p_{r} \tag{3}\]  
where \(M_{r}\), \(K_{r}\), \(B_{r}\) are the rack and wheel assembly mass, the tire or rack centering spring rate and the rack damping, respectively. Figure 1: Electric power steering system model. Based on a fractional differential equation, the fractional mathematical model is written as follows:  
\[y(t)+a_{1}H^{\alpha_{1}}y(t)+\ldots+a_{m_{A}}H^{\alpha_{m_{A}}}y(t)=b_{0}H^{ \beta_{0}}u(t)+b_{1}H^{\beta_{1}}u(t)+\ldots+b_{m_{B}}H^{\beta_{m_{B}}}u(t) \tag{11}\]  
where \(\alpha_{1}<\alpha_{2}<\ldots<\alpha_{m_{A}},\ \beta_{0}<\beta_{1}<\ldots<\beta_{m_{B}}\) are the differentiation orders, \(u(t)\) and \(y(t)\) are the input signal and output signal, respectively, and \(H^{\gamma}=\left(\frac{d}{dt}\right)^{\gamma}\) is the differentiation to an arbitrary order. The measured noise \(p(t)\) is added to the measured output signal as below:  
\[y^{*}(t)=y(t)+p(t) \tag{12}\]  
The error can be written as follows:  
\[e(t)=y^{*}(t)-\psi^{*}(t)^{T}\varphi \tag{13}\]  
where  
\[\psi^{*}(t)^{T}=\left[-H^{\alpha_{1}}y^{*}(t),\ldots,-H^{\alpha_{m_{A}}}y^{*} (t),H^{\beta_{0}}u(t),\ldots,H^{\beta_{m_{B}}}u(t)\right] \tag{14}\]  
Hence, the estimated parameters of the model are defined as follows:  
\[\varphi^{T}=\left[a_{1},\ldots,a_{m_{A}},\,b_{0},\,b_{1},\ldots,b_{m_{B}}\right] \tag{15}\]  
Achieving model parameters that are close to the real plant parameters requires that the sum of the error squares is as small as possible. Hence, the least squares algorithm can be written as follows:  
\[\overset{\wedge}{\varphi}_{LS}=\left[\int\limits_{0}^{T}\psi^{*}(t)^{T}\psi^ {*}(t)dt\right]^{-1}\int\limits_{0}^{T}\psi^{*}(t)^{T}y^{*}(t)dt \tag{16}\]  
The input signal and output signal observed at regular sample \(T_{s},\,2T_{s},\ldots,KT_{s}\). Hence, the parameters estimation can be an approximated as follows:  
\[J_{m}\overset{\bullet\bullet}{\theta_{m}}=k_{e}I_{a}-B_{m}\overset{\bullet}{ \theta_{m}}-K_{m}(\theta_{m}-\frac{p_{r}}{r_{p}}N) \tag{2}\]0  
where  
\[J_{m}\overset{\bullet\bullet}{\theta_{m}}=k_{e}I_{a}-B_{m}\overset{\bullet}{ \theta_{m}}-K_{m}(\theta_{m}-\frac{p_{r}}{r_{p}}N) \tag{2}\]1  
The real system always includes noise, so estimating the parameters considering noise is very important. The noise output is used by direct fractional differentiations lead inaccurate results. Hence, [PERSON]'s filters can be adopted to be applied to the input and the output signals [29]. The output signal of the filter can be written as follows:  
\[\left\{ \begin{array}{l}H^{\beta_{i}}u_{f}=u(t)\otimes L^{-1}\{F_{\beta_{i} }(s)\}\\ H^{\alpha j}y^{*}_{f}=y(t)\otimes L^{-1}\{F_{\alpha j}(s)\}\end{array} \right. \tag{19}\]  
where \(L^{-1}\) denotes for the inverse Laplace transform.