Progressive Barrier Lyapunov Functions for Nonlinear Constrained Control Systems

This paper introduces the Progressive Barrier Lyapunov Function (p-BLF) for output- and full-state-constrained nonlinear control systems. Unlike traditional BLF methods, where control effort continuously increases as the state approaches the constraint boundaries, the p-BLF maintains minimal control effort in unconstrained regions and increases it progressively toward the boundaries. In contrast to previous methods with predefined safe zones and abrupt control activation, the p-BLF provides a smooth transition, improving continuity in the control response and enhancing stability by reducing chattering. Two forms of the p-BLF, logarithmic-based and rational-based, are developed to handle systems with either output constraints or full-state constraints. Theoretical analysis guarantees that all system states remain within the defined constraints, ensuring boundedness and stability of the closed-loop system. Simulation results validate the effectiveness of the proposed method in constrained nonlinear control.