Adaptive Tracking Control of High-Order Nonlinear Systems With Time-Varying Delays Under Asymmetric Output Constraints

This paper addresses the problem of adaptive tracking control of high-order nonlinear systems with time-varying delays under asymmetric output constraints. Unlike the existing researches, the studied system exhibits constantly changing unknown powers, which makes the existing Lyapunov Functionals invalid. Therefore, the Lyapunov Functionals are carefully designed to be suitable for the changing powers, where the changing powers include low power ( $0 < p_{i}\le 1$ ) and high power ( $p_{i} > 1$ ). Besides, we also incorporate the well-constructed dynamic gain signals into the Lyapunov Functionals to enhance the robustness, accuracy, and response speed of the system. Further, we cleverly deal with the time-varying delays by combining the novel dynamic gain signals and Lyapunov-Krasovskii (L-K) Functionals in the controller design process. For the system performance requirement, the asymmetric output constraints are considered by introducing a nonlinear transformation for the output signal $x_{1}$ and the reference signal $y_{r}$ . While the derivative of this nonlinear transformation generates a gain term in the form of function, which makes the controller design more difficult. We utilize the adding one power integrator technique to design the desired controller and solve the design difficulty caused by the gain term. Under the proposed controller, all the signals in the closed-loop system remain bounded, the asymmetric output constraints are not violated as well as the tracking error stays in a small neighborhood of the origin. Finally, a simulation example is given to demonstrate the effectiveness of the present strategy. Note to Practitioners—In the article, we consider a class of high-order nonlinear systems with time-varying delays under asymmetric output constraints, and their models are widely utilized in the engineering field, such as transportation, manufacturing sector and so on. Further, the tracking problem is regarded as a popular topic in the control field, so designing a desired tracking controller for high-order nonlinear systems is a significant research topic. Since the considered system has the changing powers and time-varying delays, the existing methods cannot be applied to the studied problem, so we provide some new strategies by the introduction of the well-constructed dynamic gain signals, the improved L-K functionals and the nonlinear transformation. From what has been mentioned above, an effective adaptive tracking control strategy is developed for the studied system, where the proposed strategy has great application prospect.

I. Introduction

In recent years, high-order nonlinear systems have been receiving increasing attention in the control field. High-order nonlinear systems often exhibit complicated dynamic characteristics and are typically more difficult to design the controller [1], [2], [3], [4], [5], [6], [7]. The traditional design tools such as feedback linearisation [8] and backstepping [9], [10], [11] are hardly applicable to high-order nonlinear systems, these methods are more suitable for the lower order nonlinear systems, where system power is only one. The adaptive stabilization of high-order nonlinear systems is a fundamental and challenging problem. Fortunately, [12] proposed a genius method called the adding one power integrator technique, which has been seen as a great victory to solve the problem, while some satisfactory results have been achieved. The stabilization problems of different classes of high-order nonlinear systems, such as high-order stochastic nonlinear systems, high-order interconnected nonlinear systems, and high-order switched nonlinear systems have been successfully solved [13], [14], [15]. However, it is worth noting that many existing articles focused on the system power in the case of constants or separately. Therefore, combining the above two cases to form a more general system power(, , and are the changing powers) and design the relevant controller is a challenging task. As shown in [16], the authors have proposed that the system has the changing system powers and multiple uncertainties, so the existing methods cannot be applied to high-order nonlinear system. In brief, it is a challenging and meaningful topic in developing an effective controller for high-order nonlinear system, which is still open for research.