# Control for Autonomous Intersection Management Based on Adaptive Control Barrier Function
## 3 Controller Design
### Control Problem Formalization  
The control strategy design employs the following single integrator model for the agents:  
\[\dot{\mathbf{p}}_{i}=\overline{\mathbf{u}}_{i},\quad i\in\{1,2,\ldots,n\}. \tag{7}\]  
where \(n\) is the number of agents, \(\mathbf{p}_{i}\in\mathbb{R}^{2}\) represents the state of an agent, \(\overline{\mathbf{u}}_{i}\in\mathbb{R}^{2}\) denotes the input signal. It is important to note that when actuators do not function as intended (e.g., power transmission loss), the expected performance and stability of the system can be severely compromised [55,56]. To enhance the reliability and performance of systems, control methods must be developed to counteract the impact of power transmission loss, which results in a fault-tolerant control problem. In order to investigate this issue, we introduce a power transmission loss model that considers partial loss of actuator effectiveness [55,56]. The power transmission loss model can be describe as follows:  
\[\overline{\mathbf{u}}_{i}=\theta_{i}\mathbf{u}_{i}, \tag{8}\]  
where \(\theta_{i}>0\) represents the power transmission efficiency of the \(i\)th vehicle, and it includes steering power transmission loss, brake power transmission loss, transmission power transmission loss, and so on. Then, (7) can be rearranged as  
\[\dot{\mathbf{p}}_{i}=\theta_{i}\mathbf{u}_{i}. \tag{9}\]  
where the power transmission efficiency \(\theta_{i}\) is assumed to satisfy the following assumptions.  
**Assumption 1**.: _To ensure the controllability of system (9), the power transmission efficiency \(\theta_{i}\) is assumed to satisfy \(\theta_{i}\in[\alpha_{i},1]\), where \(\alpha_{i}\) is a small positive constant \((0<\alpha_{i}<1)\). Without this condition, such as when \(\theta_{i}=0\), the system becomes uncontrollable, making the controller design infeasible._  
To deal with the AIM problem, in this paper we design a safe adaptive controller for the system (9) based on the CBF technique. The schematic diagram of the CBF in application to the multi-agent system can be seen in Figure 2. To do so, according to Section 2.1, we define a safe set \(\mathcal{C}_{i}\) for agent \(i\) as  
\[\mathcal{C}_{i}=\{\mathbf{p}_{i}\in\mathbb{R}^{n}\mid h_{i}(\mathbf{p}_{i})\geq 0\}, \tag{10}\]  
where \(h_{i}(\mathbf{p}_{i})\) is a Lipschitz continuous function with a relative degree 1. To avoid collisions, one can design \(h_{i}\) as  
\[h_{i}=\|\mathbf{p}_{i}-\mathbf{z}_{i}\|^{2}-(2r_{\text{safe}})^{2}, \tag{11}\]  
\(\mathbf{z}_{i}\) indicates the position of the nearest neighbor detected within agent \(i\)'s sensing range, \(\dot{\mathbf{z}}_{i}\) is the measurable velocity of this neighbor, and \(r_{\text{safe}}\) represents the safety radius that can fit the size of agent \(i\), as shown in Figure 2.  
**Assumption 2**.: _The velocity of the detected obstacle \(\mathbf{z}_{i}\) by the \(i\)th agent is bounded such that \(\|\mathbf{z}_{i}\|\leq V_{z}\), where \(V_{z}>0\) is a positive constant._  
The control problem can now be stated as follows. Construct the actual input \(\mathbf{u}_{i}\) such that the corresponding \(\mathbf{p}_{i}(t)\) resulting from system (9) will stay inside the safety set \(\mathcal{C}_{i}\) defined as (10), i.e., \(h(\mathbf{p}_{i}(t))\geq 0\) for \(\forall t>t_{0}\), in the presence of uncertainty \(\theta_{i}\in[\alpha_{i},1]\).  
To the best of our knowledge, CBF-based safe adaptive controller design for systems with the power transmission loss has not been proposed yet in the literature, which constitutes the novelty of this work.