Control of nonlinear systems with partial state constraints using a barrier Lyapunov function

This article addresses the problem of control design for strict-feedback systems with constraints on the states. To prevent the states from violating the constraints, we employ a barrier Lyapunov function (BLF), which grows to infinity whenever its arguments approaches some finite limits. Based on BLF-based backstepping, we show that asymptotic output tracking is achieved without violation of any constraint, provided that the initial states and control parameters are feasible. We also establish sufficient conditions to ensure feasibility, which can be checked offline without precise knowledge of the initial states. The feasibility conditions are relaxed when handling the partial state constraint problem as compared to the full state constraint problem. In the presence of parametric uncertainties, BLF-based adaptive backstepping is useful in preventing the states from transgressing the constrained region during the transient stages of online parameter adaptation. To relax the feasibility conditions, asymmetric error bounds are considered and asymmetric barrier functions are used for control design. The performance of the BLF-based control is illustrated with two simulated examples.