Realization of Reachability for the Control of a Class of Nonlinear Systems

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.