In classical model reference adaptive control (MRAC), the adaptive rates must be tuned to meet multiple competing objectives. Large adaptive rates guarantee rapid convergence of the trajectory tracking error to zero. However, large adaptive rates may also induce saturation of the actuators and excessive overshoots of the closed-loop system’s trajectory tracking error. Conversely, low adaptive rates may produce unsatisfactory trajectory tracking performances. To overcome these limitations, in the classical MRAC framework, the adaptive rates must be tuned through an iterative process. Alternative approaches require to modify the plant’s reference model or the reference command input. This paper presents the first MRAC laws for nonlinear dynamical systems affected by matched and parametric uncertainties that constrain both the closed-loop system’s trajectory tracking error and the control input at all times within user-defined bounds, and enforce a user-defined rate of convergence on the trajectory tracking error. By applying the proposed MRAC laws, the adaptive rates can be set arbitrarily large and both the plant’s reference model and the reference command input can be chosen arbitrarily. The user-defined rate of convergence of the closed-loop plant’s trajectory is enforced by introducing a user-defined auxiliary reference model, which converges to the trajectory tracking error obtained by applying the classical MRAC laws before its transient dynamics has decayed, and steering the trajectory tracking error to the auxiliary reference model at a rate of convergence that is higher than the rate of convergence of the plant’s reference model. The ability of the proposed MRAC laws to prescribe the performance of the closed-loop system’s trajectory tracking error and control input is guaranteed by barrier Lyapunov functions. Numerical simulations illustrate both the applicability of our theoretical results and their effectiveness compared to other techniques such as prescribed performance control, which allows to constrain both the rate of convergence and the maximum overshoot on the trajectory tracking error of uncertain systems.

在经典模型参考自适应控制 (MRAC) 中，必须调整自适应速率以满足多个竞争目标。大自适应率保证轨迹跟踪误差快速收敛到零。然而，大的自适应速率也可能导致执行器饱和和闭环系统轨迹跟踪误差的过度超调。相反，低自适应率可能会产生不令人满意的轨迹跟踪性能。为了克服这些限制，在经典 MRAC 框架中，必须通过迭代过程调整自适应速率。替代方法需要修改工厂的参考模型或参考命令输入。本文介绍了受匹配和参数不确定性影响的非线性动力系统的第一个 MRAC 定律，这些不确定性将闭环系统的轨迹跟踪误差和控制输入始终限制在用户定义的范围内，并强制执行用户定义的收敛速度关于轨迹跟踪误差。通过应用建议的 MRAC 定律，自适应率可以设置为任意大，并且可以任意选择工厂的参考模型和参考命令输入。闭环设备轨迹的用户定义收敛率是通过引入用户定义的辅助参考模型来强制执行的，该模型收敛到通过应用经典 MRAC 定律在其瞬态动态衰减之前获得的轨迹跟踪误差，并且以高于被控装置参考模型收敛速度的收敛速度将轨迹跟踪误差导向辅助参考模型。所提出的 MRAC 定律规定闭环系统轨迹跟踪误差和控制输入的性能的能力由障碍李雅普诺夫函数保证。数值模拟说明了我们的理论结果的适用性及其与其他技术（例如规定的性能控制）相比的有效性，这允许限制收敛速度和不确定系统轨迹跟踪误差的最大超调。所提出的 MRAC 定律规定闭环系统轨迹跟踪误差和控制输入的性能的能力由障碍李雅普诺夫函数保证。数值模拟说明了我们的理论结果的适用性及其与其他技术（例如规定的性能控制）相比的有效性，这允许限制收敛速度和不确定系统轨迹跟踪误差的最大超调。所提出的 MRAC 定律规定闭环系统轨迹跟踪误差和控制输入的性能的能力由障碍李雅普诺夫函数保证。数值模拟说明了我们的理论结果的适用性及其与其他技术（例如规定的性能控制）相比的有效性，这允许限制收敛速度和不确定系统轨迹跟踪误差的最大超调。