On unified asymmetric barrier Lyapunov functions

Abstract

Barrier Lyapunov functions (BLFs) have been a popular choice when dealing with constrained control problems. In the current article, we present a unified asymmetric barrier Lyapunov function that generalizes the existing logarithmic symmetric Lyapunov function. We show that the proposed function is smooth and does not require the discontinuous switching function that is ubiquitous in the asymmetric barrier Lyapunov functions existing in the literature. Based on the proposed barrier Lyapunov function, a control law, guaranteeing exponentially fast output tracking, is designed for a class of single-input single-output nonlinear systems with output constraints. We show that the proposed control law unifies the control design and structure for a system with either symmetric or asymmetric output constraints. Furthermore, we constructively show that the proposed function can be bounded from above and below by symmetric class $\mathcal{K}$ functions which help in establishing local exponential convergence together with the proposed control law. Lastly, we provide numerical examples to illustrate the performance of the proposed control and compare the results with the existing logarithmic BLFs.