Contraction Estimates for Abstract Real-Time Algorithms for NMPC

In this paper, we derive a general contraction estimate for a class of algorithms that can be used in the context of nonlinear model predictive control (NMPC) to compute approximate and computationally cheap feedback policies. The main idea lies in performing a small number or, in the limit, a single iteration of a Q-linearly convergent algorithm in order to trade control performance for computational speed. In this way, we aim at generalizing existing results based on specific settings. We derive a contraction estimate that provides a bound on the numerical error incurred when using such schemes under the assumption of Robinson's strong regularity of the underlying parametric nonlinear optimization problem. Moreover, we show that several algorithms fall in the class of methods to which our results are applicable. Finally, we present a numerical example where, using one of the proposed real-time methods, a speedup of more than an order of magnitude can be obtained, at the cost of moderate closed-loop suboptimality.