On the Stability of Nonlinear Receding Horizon Control: A Geometric Perspective

TL;DR

This paper investigates the stability of nonlinear Receding Horizon Control when the inner optimization is solved to first-order stationary points, revealing conditions under which stability can still be guaranteed despite non-global solutions.

Contribution

It provides the first analysis of stability guarantees for nonlinear RHC with local solutions, especially for feedback linearizable systems, highlighting the importance of system geometry.

Findings

First-order solutions can exponentially stabilize linearizable systems under certain conditions.

Stability guarantees can hold even with spurious local minima.

The compatibility of state costs with system geometry is crucial for stability.

Abstract

%!TEX root = LCSS_main_max.tex The widespread adoption of nonlinear Receding Horizon Control (RHC) strategies by industry has led to more than 30 years of intense research efforts to provide stability guarantees for these methods. However, current theoretical guarantees require that each (generally nonconvex) planning problem can be solved to (approximate) global optimality, which is an unrealistic requirement for the derivative-based local optimization methods generally used in practical implementations of RHC. This paper takes the first step towards understanding stability guarantees for nonlinear RHC when the inner planning problem is solved to first-order stationary points, but not necessarily global optima. Special attention is given to feedback linearizable systems, and a mixture of positive and negative results are provided. We establish that, under certain strong conditions, first-order solutions to RHC exponentially stabilize linearizable systems. Surprisingly, these conditions can hold even in situations where there may be \textit{spurious local minima.} Crucially, this guarantee requires that state costs applied to the planning problems are in a certain sense `compatible' with the global geometry of the system, and a simple counter-example demonstrates the necessity of this condition. These results highlight the need to rethink the role of global geometry in the context of optimization-based control.