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

TL;DR

This work tackles stability guarantees for nonlinear Receding Horizon Control (RHC) when planning problems are solved only to first-order stationary points rather than to global optima, addressing a gap between theory and practice in derivative-based optimization loops.The authors introduce convex time-varying approximations based on Jacobian linearizations, derive sufficient conditions on cost convexity, stabilizability, drift matching, and input nonlinearity to ensure exponential decay of all approximate first-order stationary points, and connect these results to feedback linearization.Through a mix of positive results and counterexamples, the paper shows that stabilizing stationary points need not be global optima, highlighting the essential role of the global geometry to guide local optimization toward stabilizing trajectories and informing the design of RHC cost functionals.Extending to First-Order RHC (FO-RHC), the authors provide explicit exponential stability guarantees under a warm-start and planned descent, with bounds that improve as horizon grows and optimality tightens, offering a practical pathway for stability with computationally tractable planning.

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.

Figures (1)

• Figure 1: (a) Schematic for the simple inverted pendulum (b) schematic for the inverted pendulum with a flexible joint.

Theorems & Definitions (14)

• Definition 1
• Definition 2
• Lemma 1
• Lemma 2
• proof
• Theorem 1
• Lemma 3
• Lemma 4
• Proposition 1
• Remark 1