## 1 Introduction
Consequently, indirect adaptive control has garnered significant attention since it offers more advanced identification algorithms. A traditional approach in this domain is the identification algorithm based on the Kalman filter, which characterizes uncertainty through a noise signal with a predefined probability density function [24]. However, the limited capability of this algorithm in handling nonlinear systems restricts its utilization in specific application scenarios [25]. The set-membership identification approach ensures that the parameters belong to specific sets, offering increased flexibility in addressing intricate problems. Among the various set-membership identification methods, the OBE algorithm stands out as the most prominent. The OBE algorithm has significant importance in the realms of system identification and adaptive control, serving as an effective and precise tool for the estimation of unknown system parameters [26]. This iterative technique incrementally shrinks the volume of an ellipsoid to yield the optimal parameter estimate. Notably, ellipsoids are employed in this context to demarcate predefined sets [27; 28]. The learning gain matrix in the OBE algorithm exhibits dynamic characteristics, with an upper bound that prevents explosions in scenarios of inadequate excitation and a lower bound that guarantees robust identification performance for time-varying parameters. The OBE algorithm has been studied extensively in the literature. For example, [PERSON] et al. proposed a novel dynamic sliding mode control design scheme by combining sliding mode control with the attracting ellipsoid method, enabling the system state to rapidly enter and maintain the sliding surface during the dynamic process [29]. [PERSON] et al. introduced a directional forgetting algorithm that discards outdated data based on the informational content across various directions, above and below the bounded information matrix [30]. [PERSON] et al. developed an adaptive state estimator and a robust output control for a class of uncertain nonlinear systems, which enables the system to maintain stability and performance in the presence of uncertainty and disturbances, and the feedback gains \(K\), \(L\) can be updated online based on the attractive ellipsoid method [31]. [PERSON] et al. proposed an efficient OBE identification algorithm that ensures deterministic upper and lower bounds for the learning gain matrix [13; 32]. Applications of OBE-based algorithms for the identification of singularly perturbed systems have been detailed in [33; 34]. However, it is crucial to note that these papers rely solely on identification errors for updating NN weights. As highlighted in [22], incorporating both identification and tracking errors in weight updating laws enhances NN training outcomes. Therefore, it is advisable to incorporate the tracking error into a composite learning algorithm to further bolster identification performance. To tackle the aforementioned challenges, this paper introduces a novel identification and control algorithm tailored for the FJR. The key contributions of this work are outlined below:  
* An adaptive prescribed performance controller for the FJR is formulated, leveraging the SPT framework and continuous terminal sliding mode control technique. This simplifies the controller design process and ensures that the transient and steady-state behavior meets the pre-established specifications. * A composite-learning OBE-based identification algorithm is introduced, aiming to boost NN training performance by incorporating both identification and tracking errors. The subsequent sections of this paper are organized as follows: Section 2 presents preliminary information and demonstrates how the original high-order FJRs can be decoupled into two lower-order subsystems using SPT. Section 3 introduces an NN-based terminal sliding mode PPC for the simplified subsystems. The identification algorithm is presented in Section 4.