## 1 Introduction  
Non-linear systems are almost ubiquitous in daily life and exist widely in various applications, such as motor [1], power [2], and electrical systems [3]. After more than half a century of development, the control of non-linear systems has made considerable progress, and various control methods and strategies have emerged, e.g., backstepping control strategies [4], neural networks [5], fuzzy-based control [6], and system identification [7]. For example, scenario-based model predictive control (MPC) approaches can mitigate the conservatism inherent in robust open-loop MPC. Reference [8] presents a method for evaluating the confidence intervals of RBNN predictions and determines the number of samples required to estimate the confidence interval for a given confidence level. The authors of [9] propose a security-model-based reinforcement learning approach to control nonlinear systems described by linear parameter variation models. In order to improve control accuracy, these methods require feedback. For example, a state feedback Smith predictive controller was proposed for the effective temperature control of a cement rotary kiln precalining furnace [10]. The authors of [11] present a fully distributed adaptive tracking control scheme for multi-agent systems with a strict feedback form. Generally, this type of state feedback control requires all systems' internal state information, which is challenging to achieve in reality. Therefore, control methods based on system output have also been proposed. These control algorithms require only the system's output information to complete the control process. For example, the authors of [12] studied a linear-quadratic (LQ) control problem with irregular output feedback in which a noisy linear system measured the state. In [13], the authors discuss the collaborative design of output-dependent switching functions and full-order affine filters for discrete-time switched affine systems. This control method has a certain degree of versatility and has achieved good application results. However, the controlled system does not have a strict feedback form. Strict feedback systems present a good lower triangle form, while the control systems for a flexible manipulator and some temperatures are in a strict feedback form. Commonly, the backstepping control method is used for such systems, e.g., in [14], a neural-network-based adaptive gain scheduling backward sliding mode control (NNAGS-BSMC) method is proposed for a class of non-linear systems with uncertain strict feedback. Reference [15] presents a novel tracking controller utilizing an event-triggering implementation for uncertain rigor feedback systems. Adaptive fuzzy decentralized optimal control problems for a class of large-scale non-linear systems with a strict feedback form have also been studied [16]. A backstepping method usually has specific prerequisite requirements for the system or control strategy and needs to calculate a higher differential, imposing a high calculation complexity. Such problems have been improved to a certain extent by combining some improved backstepping methods with adaptive control ideas. For example, [17] addresses the adaptive event-triggered control of non-linear continuous-time strict feedback systems. However, the overall calculation process of this method must meet trigger conditions before it is carried out, prohibiting it from meeting real-time performance requirements. With the development of neural networks, new methods have been proposed for non-linear control problems, exploiting the appealing approximation characteristics of neural networks. For example, [18] addresses the compound learning control of a perturbed uncertain strict feedback system. In [19], the authors studied the data-based compound neural control of an uncertain strict feedback system's online record using a backstepping framework.