# Inducing Optimality in Prescribed Performance Control for Uncertain Euler-Lagrange Systems
## 2 State of the Art and Contributions  
A common factor in the aforementioned works is that none of them take into consideration performance specifications, such as convergence speed and steady-state error. One way to achieve this is through the incorporation of the prescribed performance control (PPC) technique [18]. The PPC scheme has been utilized for Euler-Lagrange systems with unknown dynamics in several works [19; 20; 21]. In [19], a sliding-mode control technique is employed along with a novel prescribed performance function to guarantee finite-time stability. In [20], the saturated PPC problem is addressed by using a controller that utilizes an adaptive multilayer neural network of adjustable weights to compensate for state-dependent uncertainties of the system. Furthermore, ref. [21] presents a reinforcement learning-based control scheme that has the optimizing ability for a trade-off between performance and control cost. Nevertheless, note that optimality is only addressed in the latter. In this paper, we examine the optimal regulation problem and propose an algorithm that approximates the solution to the HJB equation without assuming any a priori knowledge of the system dynamics. In contrast to the vast majority of previous related works, the corresponding control policy is not only optimal with respect to the chosen cost function, but also satisfies prescribed performance criteria that have been set by the designer beforehand through the appropriate selection of certain control parameters. This is achieved by incorporating the PPC formulation within the definition of the optimal control problem. In addition, a significant part of our contribution lies in the provable guarantee of the system's trajectories evolution, strictly within the set for which the approximation capabilities of the adopted neural network structures are sufficient. This is necessary to ensure the stability of the derived optimal control policy, which is the primary requirement in control system design [22]. The proposed method follows a two-phase strategy. First, via a system identification phase, a neural network is utilized to learn the unknown open-loop system dynamics, even though it might be unstable (i.e., probing signals that are common in the related identification literature are not sufficient to excite the system dynamics as they may lead to instability). The novelty of our approach lies in the fact that it can be employed to learn the unknown dynamics even in the case of an open-loop unstable system. Moreover, the neural network is fitted over a series of trajectories that cover the desired subset of the state space in its entirety to ensure that the extracted knowledge is valid over the entire set and not only around the neighborhood of a single trajectory, as commonly met in related works [10]. It should be noted that knowledge of the system dynamics is not required to achieve prescribed performance [23]. However, to achieve optimality, the system dynamics are necessary since they relate the input-output mapping which is involved in the adopted cost function, as it will be made clear in the sequel. Despite the approximation error between the real and the acquired dynamics, the induction of optimality with prescribed performance in the derived control strategy provides robustness when dealing with such uncertainty. In the second phase, the optimal cost function and policy are approximated through an iterative technique that converges uniformly to their actual values. A constrained least-squares problem is solved at each iteration, and upon convergence, we obtain the optimal controller's parameters which also ensure that the system's trajectories evolve strictly within the set for which the approximation of the cost function is valid (an issue that is ignored in the related literature). Our key contributions in this work can be summarized as follows:  
* An identification framework that is able to retrieve the unknown system dynamics even in the case of open-loop instability.