# Inducing Optimality in Prescribed Performance Control for Uncertain Euler-Lagrange Systems
## 3 Problem Formulation and Preliminaries
### Prescribed Performance Control (PPC)  
The PPC strategy [18] enables the tracking of a reference trajectory with the system's response fulfilling predefined performance criteria during the transient and the steady state, without requiring any knowledge of the system dynamics. For the case of a generic scalar error \(\sigma(t)\), the prescribed performance is achieved if the error remains bounded within a predefined region that is formed by decaying functions of time, as illustrated in Figure 1:  
\[-\rho(t)<\sigma(t)<\rho(t),\ \forall t\geqslant 0 \tag{3}\]  
The function \(\rho(t)\) is a smooth, bounded, strictly positive and decreasing function of time, called performance function, and is chosen as \(\rho(t)=(\rho_{0}-\rho_{\infty})e^{-lt}+\rho_{\infty}\), with \(\rho_{0},\rho_{\infty}\) and \(l\) being positive gains that are chosen to satisfy the designer's specifications. Specifically, \(\rho_{\infty}=\lim_{t\to\infty}\rho(t)\) is selected according to the maximum allowable tracking error at the steady state; \(l\) determines a lower bound on the speed of convergence; and \(\rho_{0}\) affects the maximum overshoot and is selected such that \(\rho_{0}>|\sigma(0)|\). In our case, we define the scalar tracking error \(\sigma_{i}(e_{i}(t))=\bar{\lambda}^{T}e_{i}(t)\) where \(\bar{\lambda}^{T}=[\lambda,1]\) with \(\lambda\) being a positive constant and \(e_{i}(t)=[q_{i}-q_{d_{i}},q_{i}-q_{d_{i}}]^{T}\) with \(q_{d_{i}}(t)\), \(i=1,\ldots,n\) being the reference trajectories. Notice that the performance specifications imposed on the errors \(\sigma_{i}\) are easily translated into equivalent performance specifications of the errors \(e_{i}\) as mentioned in Lemma 1 in [23]. The intrinsic property behind PPC lies on a mapping of the tracking error that transforms the constrained behavior as defined in (3) into a significantly relaxed unconstrained problem. More specifically, we define:  
\[\epsilon_{i}(t)=T(\xi_{i}(t)),\ \ i=1,\ldots,n \tag{4}\]  
where \(\epsilon_{i}(t)\) is the transformed error, \(T:(-1,1)\to(-\infty,\infty)\) is a strictly increasing, symmetric and bijective mapping, e.g., \(T(\star)=\frac{1}{2}\mathrm{ln}\big{(}\frac{1+\star}{1-\star}\big{)}\), and \(\xi_{i}(t)=\frac{\sigma_{i}(e_{i}(t))}{\rho_{i}(t)}\). To achieve the prescribed performance the following control signal [23] is used:  
\[\tau_{i}=-kT^{\prime}(\xi_{i})\rho_{i}^{-1}(t)T(\xi_{i}),\ k>0,\ \forall i=1, \ldots,n \tag{5}\]  
where \(k\) is a positive gain and \(T^{\prime}(\xi)\) the derivative of \(T(\xi)\). The following proposition guarantees that no matter how large the upper bound of the transformed error \(\epsilon_{i}(t)\) is (which is affected by the model uncertainty), the performance specifications encapsulated in the corresponding performance function \(\rho_{i}(t)\) are met. Proposition 1 ([25]): _The prescribed-performance control problem, as defined by (3), admits a solution if and only if the transformed error signals (4) can be kept bounded._  
Proof: Given any finite initial condition \([q_{i}(0),q_{i}(0)]^{T}\), we can always select \(\rho_{i}(0)\) to satisfy (3) at \(t=0\), hence ensuring that \(\xi_{i}(0)\in(-1,1)\), and thus, the transformed system is initially well defined. Furthermore, if the transformed system is robustly stabilizable, there  
Figure 1: Graphical representation of (3). exists a continuous control law that guarantees that \([q_{i}(t),q_{i}(t)]^{T},e_{i}(t)\in\mathcal{L}_{\infty}\) and as a consequence, there exist unknown constants \(\epsilon_{i_{tomor}}e_{i_{tapp}}\), such that \(\epsilon_{i_{tomor}}\leq e_{i}(t)\leq\epsilon_{i_{tapp}}\), \(i=1\ldots,N\;\forall t\geq 0\). Since T is a smooth, strictly increasing function, its inverse exists, and thus, \(T^{-1}(e_{i_{tomor}})\leq\xi_{i}(t)\leq T^{-1}(e_{i_{tapp}})\Rightarrow T^{-1 }(e_{i_{tomor}})\rho_{i}(t)\leq\sigma_{i}(t)\leq T^{-1}(e_{i_{tapp}})\;\forall t\geq 0\).