In this paper, a feedback controller designed to realize reachability is investigated in a class of nonlinear systems. Different from stability, reachability defines that there exists an input such that a closed-loop system can reach the desired states from any other initial states within finite time. In other words, the state errors can converge to zero in finite time if reachability can be realized. In order to realize reachability, it is proved that the state and all derivatives of the state must simultaneously satisfy a group of functions, called triple constraints. Then, it can be derived that the best approximation to realize the triple constraints is to guarantee the value of the highest order derivatives of the state. As a result, the controller can be obtained by solving a group of algebraic equations described by the highest order derivatives and the input. However, the solution will be difficult to obtain if the system is subjected to unknown parameters and disturbance. An online parameter estimation algorithm is proposed and the highest order derivatives are updated according to the current state to handle the uncertainties. With this newly developed online estimation algorithm and a scheme for updating the highest order derivatives online, reachability can be approximately realized for a class of nonlinear systems with uncertainties. To further illustrate the implementation and effectiveness of the proposed controller based on reachability, numerical simulations are conducted on a two-link robot arm. The results show the proposed controller outperforms conventional controllers such as computed-torque proportional-derivative (PD) control, sliding mode control, and finite-time controller.

中文翻译：

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一类非线性系统控制可达性的实现


本文研究了一类非线性系统中旨在实现可达性的反馈控制器。与稳定性不同，可达性定义了存在一个输入使得闭环系统可以在有限时间内从任何其他初始状态到达期望状态。换句话说，如果可以实现可达性，则状态误差可以在有限时间内收敛到零。为了实现可达性，需要证明状态和状态的所有导数必须同时满足一组函数，称为三重约束。由此可知，实现三重约束的最佳近似是保证状态的最高阶导数的值。这样，通过求解一组由最高阶导数和输入描述的代数方程就可以得到控制器。然而，如果系统受到未知参数和扰动，则求解将变得困难。提出了一种在线参数估计算法，并根据当前状态更新最高阶导数以处理不确定性。利用这种新开发的在线估计算法和在线更新最高阶导数的方案，可以近似实现一类具有不确定性的非线性系统的可达性。为了进一步说明所提出的基于可达性的控制器的实现和有效性，在两连杆机器人手臂上进行了数值模拟。结果表明，所提出的控制器优于传统控制器，例如计算扭矩比例微分 (PD) 控制、滑模控制和有限时间控制器。