Exponential Decay of Sensitivity in Nonlinear Model Predictive Control: A Graph-Theoretic Approach

Sungho Shin, Victor M. Zavala

Published 2021-01-16Version 1

Exponential decay of sensitivity (EDS) is a property of nonlinear model predictive control (NMPC) problems; the property indicates that the sensitivity of the solution at time $i$ against a data perturbation at time $j$ decays exponentially with $|i-j|$. We use a graph-theoretic analysis of the optimality conditions of NMPC problems to prove that EDS holds under uniform boundedness of the Lagrangian Hessian, a uniform second order sufficiency condition (uSOSC), and a uniform linear independence constraint qualification (uLICQ). Furthermore, we prove that uSOSC and uLICQ can be obtained from uniform controllability and observability (well-known system-theoretic properties). These results provide insights into how perturbations propagate along the horizon and enable the development of approximation and solution schemes. We illustrate the developments with numerical examples.