Prescribed-Time Optimal Control of Nonlinear Dynamical Systems With Application to a Coupled Tank System

This article presents a solution to the problem of achieving optimal prescribed-time stability and stabilization for nonlinear dynamical systems. In contrast to existing prescribed-time control methods, this article initiates by establishing sufficient conditions for prescribed-time stability through the use of continuous Lyapunov candidate functions. Building upon these conditions, we introduce an optimal prescribed-time stabilization method that incorporates specific differential inequalities. This method complies with the Hamilton-Jacobi-Bellman steady-state equation, ensuring both optimality and prescribed-time stability. Furthermore, we derive a set of optimal prescribed-time stabilizing control laws for a class of affine nonlinear dynamical systems. Finally, we demonstrate the effectiveness of the proposed approach through simulations and experiments involving the reference level tracking of a coupled tank system, thus ensuring that the tracking performance aligns with practical user specifications
Note to Practitioners
—This article was instigated by the challenge of devising optimal feedback control strategies for a specific class of nonlinear dynamical systems at predetermined time instances. In recent years, there has been a growing interest in prescribed time stability and stabilization approaches, driven by their potential applications across diverse fields, including control engineering, robotics, and aerospace engineering. These methods facilitate the regulation of nonlinear dynamical systems to reach a desired steady state within a predefined finite time, offering a valuable solution for situations demanding rapid stabilization. In this article, we introduce a novel optimal prescribed-time stabilization method that relies on specific differential inequalities. This method not only adheres to the Hamilton-Jacobi-Bellman steady-state equation but also guarantees both optimality and prescribed-time stability. Furthermore, we derive a family of optimal prescribed-time stabilizing control laws tailored to a particular class of affine nonlinear dynamical systems. To validate the effectiveness of our proposed stabilization approach, we conduct experiments focusing on tracking the desired water level within a coupled tank system. Ultimately, the presented prescribed-time optimal feedback control strategy marks a significant stride forward in the advancement of optimal and efficient control methods for nonlinear dynamical systems, offering solutions that hold immense promise in practical applications.