## 1 Introduction  
Since 1980s, scholars have paid more and more attention to nonlinear science, and nonlinear analysis in mathematics, nonlinear functional, and nonlinear dynamics in physics have developed rapidly [1]. At the same time, the theory of nonlinear systems has also developed vigorously [2, 3, 4, 5, 6], and more control theory experts have turned to research on nonlinear systems, and more engineers have tried to construct controllers with the theory of nonlinear systems, which has resulted in certain achievements. The feedback linearization method [8], backstepping method [9], sliding mode control method, and so on are proposed accordingly [10]. In order to ensure the security and improve the stability of the system, the main research result which is being focused on at present is the FSCs control method [11, 12, 13, 14, 15, 16, 17]. As a common state constraint method, FSC control has always been a research hotspot in the control field. At present, the algorithm combining barrier Lyapunov function (BLF), backstepping, and approximation control method is the main method used to solve the problem of FSCs control [11]. According to the definition, the value of the BLF tends to infinity as the function variable approaches the constraint boundary. When the BLF remains bounded under the action of the designed controller, the function variables will always remain within the constraint boundary; that is, the constraint is satisfied. This is also the key characteristic of the application of BLF to constraints control design. However, the BLF also has some shortcomings. This algorithm directly constrains the state of the CLS rather than the state of the controlled system; that is, the constraint interval includes not only the states of the system, but also the virtual controllers. In this way, this method can be referred to as indirect FSC control. Therefore, the BLF control method needs to obtain the constraint interval through complicated calculations. To solve the shortage of BLF, reference [18] proposed a mapping logarithmic constraint function to solve the problem of FSCs control. The constraint interval in this method only contains the system state. Therefore, reasonable constraint parameters or functions can be directly designed according to the actual needs of the state. This method can be referred to as direct FSC control. Both the BLF and the mapping logarithmic constraint function use constraint parameters or functions to play a security role in the system. For example, if the overshoot of the system is too large, we can reduce the overshoot of the system by designing constraint parameters or functions. However, due to the inherent characteristics of its method, the constraint parameters or functions cannot be designed to be zero, which means the overshoot of the system is unable to reach zero; that is, the NOTC cannot be realized. However, the NOTC is very useful in practical applications, and in some applications [19], it is also required that the control method can achieve the result with NOTC, such as the temperature control of sintering furnace [20], missile launch, attitude control of flight control system and welding operation, and so on. Therefore, it is necessary to study the NOTC. At present, the research results of NOTC are mainly aimed at linear systems [21; 22; 23; 24; 25], and there are very few studies which focus on nonlinear systems. For the nonlinear strict-feedback system, an improved controller was designed for the first time based on backstepping reference [19], and thus a linear CLS was obtained. Reference [26] pointed out that reference [19] mainly used backstepping technology to construct control law and obtain a chain CLS, in which the first CLS state was a tracking error. By selecting design parameters, the initial values of all CLS states were guaranteed to be negative, so that all CLS states remain negative in the convergence process. This method abandons other NOTC conditions that guarantee the tracking error will be negative.