Efficient computation of Lipschitz constants for MPC with symmetries

TL;DR

This work tackles the problem of computing the minimum Lipschitz constant $L_p^*$ for linear MPC without constructing the explicit control law, which is often intractable for complex systems. It refines a prior MILP formulation by incorporating saturation-based reductions and, crucially, symmetry exploitation to dramatically reduce the number of binary variables and thus the solution time. The authors introduce primal LP-based norm reformulations, constraint-exclusion criteria, and symmetry-based domain reductions, then validate the approach on literature examples, showing significant runtime and scalability improvements over the baseline. The method enables robust Lipschitz-based certifications for more complex MPC implementations and suggests future extensions to the Lipschitz constants of the MPC value function.

Abstract

Lipschitz constants for linear MPC are useful for certifying inherent robustness against unmodeled disturbances or robustness for neural network-based approximations of the control law. In both cases, knowing the minimum Lipschitz constant leads to less conservative certifications. Computing this minimum Lipschitz constant is trivial given the explicit MPC. However, the computation of the explicit MPC may be intractable for complex systems. The paper discusses a method for efficiently computing the minimum Lipschitz constant without using the explicit control law. The proposed method simplifies a recently presented mixed-integer linear program (MILP) that computes the minimum Lipschitz constant. The simplification is obtained by exploiting saturation and symmetries of the control law and irrelevant constraints of the optimal control problem.

Figures (2)

• Figure 1: Sate space partition of the control law.
• Figure 2: Optimal control law for System $1$ with $\mathcal{T}=\mathcal{X}$.

Theorems & Definitions (9)

• Lemma 1
• proof
• Lemma 2
• proof
• Theorem 3
• proof
• Corollary 4
• Lemma 5
• proof