# A Speed Tracking Method for Autonomous Driving via ADRC with Extended State Observer
## 2 Control-Oriented Vehicle Longitudinal Model
### Mean Value Engine Model  
The control input of the longitudinal motion of an autonomous driving vehicle is the accelerator pedal depth, which is converted to the engine throttle angle. Thus, we need to find the model between engine torque and throttle. In general, the modeling of the engine system is intricate because of its \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Symbol** & **Variable** & **Symbol** & **Variable** \\ \hline \(p_{m}\) [bar] & Intake manifold pressure & \(m_{m}\) [kg/s] & Air mass flow rate into manifold \\ \(m_{m}\) & Air mass flow rate into cylinder & \(K_{a}\) [K] & Environment temperature \\ \(K_{m}\) [K] & Manifold temperature & \(\theta_{thr}\) [deg] & Engine throttle angle \\ \(N\) [RPM] & Engine speed & \(m_{fund}\) [kg/s] & Injected fuel mass flow rate \\ \(\lambda\) & Air–fuel ratio & \(T_{i}\) [N-m] & Indicated engine torque \\ \(T_{fric,c}\) [N-m] & Friction engine torque & \(r\) [rev/s] & engine (crankshaft) rotation speed \\ \(i_{g}\) & Transmission gear ratio & \(i_{0}\) & Final drive ratio \\ \(a_{pdl}\) & Accelerator pedal depth & \(F_{t}\) [N] & Vehicle traction force \\ \(\eta_{T}\) & Total transmission efficiency & \(R_{w}\) [m] & Tire radius \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Notation used in models. Figure 1: Simplified vehicle powertrain systems. nonlinear and time-varying characteristics. A physical-based model called the mean value engine model (MVEM), which predicts the mean values of major external engine variables like crankshaft speed and manifold pressure dynamically in time, has been widely used [26; 27]. This model is relatively simple but can describe major time-varying characteristics. Figure 2. shows the MVEM modules. For the modeling purpose, the intake manifold is regarded as an adiabatic system. Thus, the ideal gas law can be used to describe the pressure variation [26]:  
\[\dot{p}_{m}=\frac{\kappa R}{V_{m}}(\dot{m}_{ai}K_{a}-\dot{m}_{ao}K_{m}) \tag{1}\]  
where \(V_{m}\) is the manifold volume, \(\kappa\) is the ratio of specific heats (\(\kappa\) = 1.4 for air), and \(R\) is the gas constant. The air mass flow is a function of the throttle angle [26]:  
\[\dot{m}_{ai}=MAX\cdot TC(\theta_{thr})\cdot PRI \tag{2}\]  
where \(MAX\) is a constant dependent on the size of the throttle body and \(TC(\theta_{thr})\) is the throttle characteristic which is the projected area the flow sees as a function of the throttle angle:  
\[TC(\theta_{thr})=1-\cos(\theta_{thr}-\theta_{0}) \tag{3}\]  
where \(\theta_{0}\) is the minimum throttle angle of the engine and \(PRI\) is the pressure ratio influence function which describes the choked/sonic flow that occurs through the throttle valve. The air mass flow out of the manifold can be governed by:  
\[\dot{m}_{ao}=\sqrt{\frac{K_{m}}{K_{a}}}\frac{V_{d}\eta_{vol}}{120 RK_{m}}Np_{m} \tag{4}\]  
where \(V_{d}\) is the engine displacement and \(\eta_{vol}\) is the charging efficiency. Hence, the fuel mass flow that desired injected into the cylinder can be calculated through the air-fuel equivalence ratio:  
\[\dot{m}_{fuel}=\frac{\dot{m}_{ao}}{\lambda L_{th}} \tag{5}\]  
where \(L_{th}=14.7\) is the stoichiometric air-fuel mass ratio for SI engine. Then, the combustion of air-fuel mixtures produced engine indicated torque:  
\[T_{i}=\frac{H_{i}\eta_{i}\dot{m}_{fuel}}{2\pi N/60} \tag{6}\]  
where \(H_{i}\) is the fuel energy constant and \(\eta_{i}\) is the thermal efficiency multiplier. To obtain the effective engine torque, we need to calculate engine friction torque. A simple model for the total friction torque is given by [29]:  
\[T_{fric,e}=\frac{9.55(c_{1}+c_{2}r^{2}+c_{3}r^{2})V_{d}r}{n_{R}\cdot N} \tag{7}\]  
Figure 1:0  
where \(n_{R}\) denotes the ratio between the number of strokes of the pistons to the number of rotations of the crankshaft (\(n_{R}=2\) for a four-stroke engine). The coefficients \(\left[ \begin{array}{ccc}c_{1}&c_{2}&c_{3}\\ \end{array} \right]\) can be identified as  
Figure 2: The model blocks of the spark ignition (SI) engine. \(\left[