# Control for Autonomous Intersection Management Based on Adaptive Control Barrier Function
## 3 Controller Design
### A Safe Adaptive Controller Design  
Regarding the power transmission loss, setting \(\theta_{i}=\theta_{i}^{*}+\frac{1+\alpha_{i}}{2}\) where \(\alpha_{i}>0\) is a small constant, we can see that \(\theta_{i}^{*}\in[\frac{\alpha_{i}-1}{2},\frac{1-\alpha_{i}}{2}]\). By selecting \(\theta_{i}=\frac{1-\alpha_{i}}{2}\), we have \(\theta_{i}^{*}\in[-\tilde{\theta}_{i},\tilde{\theta}_{i}]\). Then, system (9) becomes  
\[\dot{\mathbf{p}}_{i}=\theta_{i}^{*}\mathbf{u}_{i}+\frac{1+\alpha_{i}}{2}\mathbf{u}_{i}. \tag{12}\]  
To compensate for the effects of the unknown parameter \(\theta_{i}^{*}\) to the control system (12), corresponding update laws \(\hat{\theta}_{i}\) need to be designed for approximating these parameters. **Theorem 1**. : _By constructing the update law \(\hat{\theta}_{i}\) for the parameter estimation as_  
\[\hat{\theta}_{i}=\mathrm{Proj}\left(\hat{\theta}_{i},-\frac{1}{2K_{i}}\left( \frac{\partial h_{i}}{\partial\mathbf{p}_{i}}\right)^{\top}\mathbf{u}_{i}-\frac{\mu} {2}\hat{\theta}_{i},l(\hat{\theta}_{i})\right), \tag{13}\]  
_where \(\mu>0\), \(l(\hat{\theta}_{i})=\frac{\tilde{\theta}_{i}^{2}-\tilde{\theta}_{i}^{2}}{ \theta_{i}\alpha_{i}+(\alpha_{i}/2)^{2}}\), \(|\hat{\theta}_{i}(0)|\leq\frac{1}{2}\), and \(K_{i}\leq\frac{h_{i}(0)}{(|\hat{\theta}_{i}(0)|+\tilde{\theta}_{i})^{2}}\), any Lipschitz continuous control \(\mathbf{u}_{i}(\mathbf{p}_{i})\in K_{bf}(\mathbf{p}_{i},\hat{\theta}_{i})\) where_  
\[K_{bf}(\mathbf{p}_{i},\hat{\theta}_{i})\triangleq\bigg{\{}\mathbf{u}_{i}\in\mathbb{R} ^{m}\,|\,\left(\hat{\theta}_{i}+\frac{1+\alpha_{i}}{2}\right)\left(\frac{ \partial h_{i}}{\partial\mathbf{p}_{i}}\right)^{\top}\mathbf{u}_{i}-\left\|\frac{ \partial h_{i}}{\partial\mathbf{z}_{i}}\right\|V_{z}+\frac{\mu}{2}\left(h_{i}-K_{ i}\tilde{\theta}_{i}^{2}\right)\geq 0\bigg{\}}, \tag{14}\]  
_will guarantee the safety of \(\mathcal{C}_{i}\) in regard to system (9)._  
Proof. : Define \(\tilde{h}_{i}\) as  
\[\tilde{h}_{i}=h_{i}-K_{i}\tilde{\theta}_{i}^{2}, \tag{15}\]  
where \(\tilde{\theta}_{i}=\theta_{i}^{*}-\hat{\theta}_{i}\). To prove Theorem 1, one needs to show that \(\tilde{h}_{i}(t)\geq 0\) for \(\forall t\), such that \(h_{i}(t)\geq 0\) for \(\forall t\) as required by (10). This property holds if \(\tilde{h}_{i}\) can be expressed in the form of (or larger than) \(-\lambda\tilde{h}_{i}\), where \(\lambda>0\) with \(\tilde{h}_{i}(0)\geq 0\). Figure 2: Schematic diagram of CBF in application to the multi-agent system. Following the reconstruction of \(\hat{h}_{i}\) developed in [51], one obtains  
\[\hat{h}_{i}\geq\bigg{(}\hat{\theta}_{i}+\frac{1+\alpha_{i}}{2}\bigg{)}\bigg{(} \frac{\partial h_{i}}{\partial\mathbf{p}_{i}}\bigg{)}^{\top}\mathbf{u}_{i}-\frac{\mu}{2 }\bigg{(}K_{i}\bar{\theta}_{i}^{2}\bigg{)}-\bigg{\|}\frac{\partial h_{i}}{\partial \mathbf{z}_{i}}\bigg{\|}V_{z}+\frac{\mu}{2}K_{i}\bar{\theta}_{i}^{2}. \tag{16}\]  
If \(\mathbf{u}_{i}\) in (16) is selected from (14), the following condition is satisfied  
\[\bigg{(}\hat{\theta}_{i}+\frac{1+\alpha_{i}}{2}\bigg{)}\bigg{(}\frac{\partial h _{i}}{\partial\mathbf{p}_{i}}\bigg{)}^{\top}\mathbf{u}_{i}-\frac{\mu}{2}\bigg{(}K_{i} \bar{\theta}_{i}^{2}\bigg{)}-\bigg{\|}\frac{\partial h_{i}}{\partial\mathbf{z}_{i }}\bigg{\|}V_{z}\geq-\frac{\mu}{2}h_{i}, \tag{17}\]  
and thus, in virtue of (15), (16) can be reexpressed as  
\[\hat{h}_{i}\geq-\frac{\mu}{2}\bigg{(}h_{i}-K_{i}\bar{\theta}_{i}^{2}\bigg{)}= -\frac{\mu}{2}\bar{h}_{i}. \tag{18}\]  
In addition, as \(\theta_{i}^{*}\) is bounded by \(\bar{\theta}_{i}\) respectively, \(\bar{h}_{i}(0)\) satisfies  
\[\bar{h}_{i}(0)=h_{i}(0)-K_{i}\Big{(}\theta_{i}^{*}-\hat{\theta}_{i}(0)\Big{)} ^{2}\geq h_{i}(0)-K_{i}\Big{(}\bar{\theta}_{i}+|\hat{\theta}_{i}(0)|\Big{)}^{ 2}\geq 0. \tag{19}\]  
According to the comparison lemma [57], we know \(\bar{h}_{i}(t)\geq 0\) for \(\forall t\), such that \(h_{i}(t)\geq 0\) for \(\forall t\).