## 3 Extremum Seeking with Prescribed Performance Control  
### Prescribed Performance Control  
In this work, the leader-following architecture, for the design of the prescribed performance controller that tracks the desired distance, is adopted. The efficiency of this architecture is fully described and tested both in simulation and in an experimental environment in the author's previous work [6]. An important aspect of the PPC controller is that it maintains the inter-vehicular distance (\(p_{i-1}-p_{i}\)) between boundaries that ensure collision avoidance while limiting connectivity failures due to sensor constraints. Let, \(\Delta_{col}<p_{i-1}-p_{i}<\Delta_{con}\), \(i=1,\ldots,N\). Therefore, for the desired distance \(\Delta_{col}<\Delta_{i-1,i}<\Delta_{con}\), where \(\Delta_{col}\),\(\Delta_{con}\) are artificially selected positive constants for the collision avoidance and connectivity constraints, respectively. The initial state of the platoon should also meet this assumption. Hence, the control law \(u_{i}\) is designed as:  
\[u(\xi_{v},t)\equiv\left[ \begin{array}{c}u_{1}(\xi_{v_{1}},t)\\ \cdot\\ \cdot\\ u_{N}(\xi_{v_{N}},t)\end{array} \right]=-\ k_{v}\ (\rho_{v}(t))^{-1}\ r_{v}(\xi_{v})\ \epsilon_{v}(\xi_{v}) \tag{4}\]  
where \(k_{v}>0\) and \(r_{v}(\xi_{v})\), \(\epsilon_{v}(\xi_{v})\) are:  
\[r_{v}(\xi_{v})=\text{diag}(\left[ \begin{array}{c}\frac{2}{(1+\xi_{v_{1}})(1 -\xi_{v_{1}})}\end{array} \right]_{i=1,\ldots,N}) \tag{5}\]  
\[\epsilon_{v}(\xi_{v})=\left[ \begin{array}{cc}\ \ln(\frac{1+\xi_{v_{1}}}{1-\xi_{v_{1 }}}),&\ldots,&\ln(\frac{1+\xi_{v_{N}}}{1-\xi_{v_{N}}})\end{array} \right]^{T} \tag{6}\]  
To determine the normalized error vector \(\xi_{v}\), we must first define the velocity error vector as \(e_{v}\equiv[e_{v_{1}},\ldots,e_{v_{N}}]^{T}=vs.-\ v_{d}(\xi_{p},t)\), where \(vs.\equiv[v_{1},\ldots,v_{N}]^{T}\) and \(v_{d}(\xi_{p},t)\) is the vector that contains the reference velocities based on the desired distance. Subsequently, the velocity performance function \(\rho_{v_{i}}(t),\ i=1,\ldots,N\) must be properly chosen in order to satisfy \(\rho_{v_{i}}(0)>|e_{v_{i}}(0)|,\ i=1,\ldots,N\). Additionally,  
\[\xi_{v}(e_{v},t)\equiv[\xi_{v_{1}}(e_{v_{1}},t),\ldots,\xi_{v_{N}}(e_{v_{N}},t )]^{T}\equiv(\rho_{v}(t))^{-1}\,e_{u} \tag{7}\]  
with \(\rho_{v}(t)=\text{diag}([\rho_{v_{i}}(t)]_{i=1,\ldots,N})\)  
Figure 1: The closed-loop system after the control design for the i-th vehicle with the PPC and ESC controllers and the drag coefficient observer. The intermediate control law that formulates the reference velocity vector \(v_{d}(\xi_{p},t)\) at the kinematic level is selected as:  
\[v_{d}(\xi_{p},t)\equiv\left[ \begin{array}{c}v_{d_{1}}(\xi_{p_{1}},t)\\.\\.\\ u_{d_{N-1}}(\xi_{p_{N-1}},t)\\ u_{d_{N}}(\xi_{p_{N}},t)\end{array} \right]=k_{p}\ (\rho_{p}(t))^{-1}\ r_{p}(\xi_{p})\ \epsilon_{p}(\xi_{p}) \tag{8}\]  
Similarly, \(k_{p}>0\) and \(r_{p}(\xi_{p})\), \(\epsilon_{p}(\xi_{p})\) are:  
\[r_{p}(\xi_{p})=\text{diag}(\left[ \begin{array}{c}\frac{\frac{1}{M_{p_{1}}}+ \frac{1}{M_{p_{1}}}}{(1+\frac{5r_{1}}{M_{p_{1}}}\ (1-\frac{5r_{1}}{M_{p_{1}}})}\\ \end{array} \right]_{i=1,\ldots,N}) \tag{9}\]  
\[\epsilon_{p}(\xi_{p})=\left[ \begin{array}{cc}\ln(\frac{1+\frac{5r_{1}}{M_{p _{1}}}}{1-\frac{5r_{1}}{M_{p_{1}}}}),&\ldots,&\ln(\frac{1+\frac{5r_{1}}{M_{p _{N}}}}{1-\frac{5r_{1}}{M_{p_{N}}}})\\ \end{array} \right]^{T} \tag{10}\]  
where, \(\underline{M}_{p_{i}}\), \(\overline{M}_{p_{i}}\) are positive parameters that are related to collision and connectivity boundaries. For the normalized position error vector \(\xi_{p}\), we define the position error between two neighboured vehicles \(e_{p_{i}}(t)=p_{i-1}(t)-p_{i}(t)-\Delta_{i-1,i}\), \(i=1,\ldots,N\) and properly choose function \(\rho_{p_{i}}(t)\) to achieve the prescribed performance objectives.