Prescribed-time convergence with input constraints: A control Lyapunov function based approach
This work addresses the problem of prescribed-time convergence under input constraints for control-affine nonlinear systems, offering a theoretical guarantee for a specific class of systems.
The paper presents a control framework for nonlinear systems to achieve prescribed-time convergence with bounded control inputs, using a time transformation and a parameter μ to handle input constraints. The method guarantees convergence within a user-defined time from a characterized set of initial conditions.
In this paper, we present a control framework for a general class of control-affine nonlinear systems under spatiotemporal and input constraints. Specifically, the proposed control architecture addresses the problem of reaching a given final set $S$ in a prescribed (user-defined) time with bounded control inputs. To this end, a time transformation technique is utilized to transform the system subject to temporal constraints into an equivalent form without temporal constraints. The transformation is defined so that asymptotic convergence in the transformed time scale results into prescribed-time convergence in the original time scale. To incorporate input constraints, we characterize a set of initial conditions $D_M$ such that starting from this set, the closed-loop trajectories reach the set $S$ within the prescribed time. We further show that starting from outside the set $D_M$, the system trajectories reach the set $D_M$ in a finite time that depends upon the initial conditions and the control input bounds. We use a novel parameter $μ$ in the controller, that controls the convergence-rate of the closed-loop trajectories and dictates the size of the set $D_M$. Finally, we present a numerical example to showcase the efficacy of our proposed method.