Practical Explicit-time Stabilization of a Proportional Control System

TL;DR

The paper addresses the challenge of presetting settling time in proportional control by introducing a practical explicit-time stabilization framework. It develops a state-constrained proportional control law based on a Lyapunov structure and a logarithmic gain function, yielding a control input $u_{ep} = -\frac{\ln(x_c/x_s)}{T_c}x$ that drives the state to $|x|\le x_s$ within an explicit time bound $T_c$. Theoretical analysis shows conditional fixed-time stability and an explicit reaching time, while comparisons with predefined-time control demonstrate reduced initial input requirements. MATLAB simulations validate the method, confirming fast convergence to the desired accuracy under bounded initial conditions. The approach promises practical benefits for engineering systems requiring guaranteed settling-time performance with simple proportional control laws.

Abstract

Proportional control can be realized directly through the amplification of analog signals, and it also has the advantage of easy tuning parameters in digital signal control. However, it is difficult for the proportional control to preset the upper bound of settling time. To address this problem, a novel practical explicit-time control method is proposed. In bounded initial condition, this method makes this system error converge to a predefined neighborhood of zero within an explicit time. More specifically, the initial condition set and conditionally stable set are solved by practical explicit-time stabilization theorem. Based on that, a proportional feedback control is founded to achieve practical conditional fixed-time stability.

Figures (3)

• Figure 1: Block diagram of four stability theories on the relationship between settling time and initial state. $T_c$ represents the upper bound of settling time.
• Figure 2: The diagram of proportional control.
• Figure 3: The state and control input curves with different initial condition.