## 3 The Design of Robust Cruise Control  
An intervention in the longitudinal dynamics of the autonomous vehicle at an intersection has great importance for achieving the required motion profile for time step \(k+1\). The control input of the vehicle is composed by two elements, \(a_{K}(k)\) and \(\Delta(k)\); see (3). The computation of \(\Delta(k)\) in Section 2 has already been proposed. In this section the design of the control intervention \(a_{K}(k)\) is proposed. The most important requirements for the controller are formed as follows. * The control system must guarantee safe longitudinal motion for the vehicle, even if \(a_{L}\) has degradation or a fault scenario. The safe motion is guaranteed through \(a_{K}\) and \(\Delta\). * The longitudinal controller must have robust characteristics, because \(a(k)\
eq a_{K}(k)\), if \(\Delta(k)\
eq 0\). Robustness must also be guaranteed in extreme vehicle dynamic scenarios: for example, if \(a_{K}(k)\) suggests full throttle, but it has been overridden by \(\Delta(k)\) to be full braking. Similarly, robustness must be guaranteed if \(a_{K}(k)\) suggests full braking, but it has been overridden by \(\Delta(k)\) to be full throttle. * The control signal \(a_{K}(k)\) must be computed to avoid the saturation of the control actuation \(a(k)\). Since in the computation process the computation of \(a_{K}(k)\) is prior to \(\Delta(k)\), the result of (13) is not considered in the computation of \(a_{K}(k)\). Thus, the role of the cruise controller is to guarantee \(a_{min}\leq a_{K}(k)\leq a_{max}\). The design of the longitudinal controller is based on the simplified vehicle model (1) with one state, such as the longitudinal velocity \(v_{1}(k)\). The controller is formed as a gain \(P\) for velocity tracking, whose input is the velocity error; see Figure 3. The reference signal for the tracking is \(v_{ref}\). The idea behind the formulation of the control design as a tracking problem is as follows. The vehicle must adapt to the environment, which determines the achievable velocity of the vehicle, e.g., through maximum speed limits. The aim of the cruise control is to consider the regulations and the motion of the other vehicles to achieve safe motion. The maximum speed limit information is provided by \(v_{ref}(k)\), and information on the motion of the vehicles in the environment in the computation of \(\Delta(k)\) is included. Thus, through the selections of \(v_{ref}(k)\) and \(\Delta(k)\), safe motion is guaranteed. The selection of \(P\) is determined by the requirement of robustness by the controlled system. In the presented structure, \(\Delta(k)\) is handled as a disturbance and the performance of the system is the minimization of the tracking error, such as \(|v_{ref}(k)-v_{1}(k)|\). Thus, for the robustness issue, it is necessary to limit the impact of \(\Delta(k)\) disturbance on the performance. The robustness criterion is based on the small-gain theorem [49]. As a consequence, the  
Figure 3: Closed-loop structure of the longitudinal control. norm transfer function from \(\Delta(k)\) on the velocity error must be smaller than 1, which guarantees disturbance attenuation. In the computation of the transfer function the maximum value of the \(\bar{\Delta}=|\Delta(k)|\) is also considered. The maximum is determined by the vehicle dynamic scenario, if \(\Delta(k)\) implies full throttle instead of full braking or full braking is implied instead of full throttle. Thus, the value of \(\bar{\Delta}\) is \(|a_{min}|+|a_{max}|\). The transfer function from \(\Delta_{i}(k)\) to the velocity error is  
\[G(z)=\frac{\bar{\Delta}\frac{T}{z-1}}{1+\frac{P-T}{z-1}}. \tag{14}\]  
The criterion on the selection of \(P\) is to guarantee \(||G(z)||_{\infty}<1\). The constraint on the control input \(a_{1}(k)\) is influenced by the limitation of the reference signal. The reference velocity in Figure 3 represents the maximum speed limit \(v_{max}(k)\) for the vehicle on the given road section.