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This work presents an unconstrained model predictive control (MPC) scheme for nonlinear time-delay systems with guaranteed closed-loop stability using neither terminal constraints nor terminal weighting terms. Therefore, we do not require the calculation of control Lyapunov-Krasovskii functionals for the nonlinear time-delay system and obtain a computationally more attractive online optimization problem. Based on similar previous results for discrete-time systems and finite-dimensional continuous-time systems, an extended asymptotic controllability assumption suitable for nonlinear time-delay systems is introduced. Since the stage cost is not positive definite in the full state, but only penalizes the instantaneous state of the system, additional arguments are required in order to guarantee closed-loop stability. It is particularly interesting to note that in contrast to essentially all other MPC schemes with guaranteed stability, the optimal cost is not used as Lyapunov function(al) of the closed-loop, and indeed the optimal cost can increase along trajectories of the closed loop due to the influence of the delayed states.