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Analysis and Control of Input-Affine Dynamical Systems using Infinite-Dimensional Robust Counterparts

Jared Miller, Mario Sznaier

TL;DR

This work develops a framework to analyze and control input-affine dynamical systems under uncertainty by decomposing Lie derivative constraints with infinite-dimensional robust counterparts over SDR (spectrahedral) input sets. By reframing the Lie constraint via parameterized robust counterparts, the method eliminates disturbance variables and reduces the dimensionality of the resulting semidefinite programs, enabling scalable moment-SOS relaxations. The approach is validated across peak, distance, reachable-set, and region-of-attraction problems, and extended to discrete-time dynamics and data-driven settings. The contributions include convergence guarantees under mild regularity, SDP-size reductions compared to traditional formulations, and practical demonstrations on several dynamical systems with SDR disturbances. The work highlights opportunities for improved online estimation and broader application to control problems where uncertainty is structured as SDRs, while also noting computational bottlenecks in SOS tightening and directions for future efficiency gains.

Abstract

Input-affine dynamical systems often arise in control and modeling scenarios, such as the data-driven case when state-derivative observations are recorded under bounded noise. Common tasks in system analysis and control include optimal control, peak estimation, reachable set estimation, and maximum control invariant set estimation. Existing work poses these types of problems as infinite-dimensional linear programs in auxiliary functions with sum-of-squares tightenings. The bottleneck in most of these programs is the Lie derivative nonnegativity constraint posed over the time-state-control set. Decomposition techniques to improve tractability by eliminating the control variables include vertex decompositions (switching), or facial decompositions in the case where the polytopic set is a scaled box. This work extends the box-facial decomposition technique to allow for a robust-counterpart decomposition of semidefinite representable sets (e.g. polytopes, ellipsoids, and projections of spectahedra). These robust counterparts are proven to be equivalent to the original Lie constraint under mild compactness and regularity constraints. Efficacy is demonstrated under peak/distance/reachable set data-driven analysis problems and Region of Attraction maximizing control.

Analysis and Control of Input-Affine Dynamical Systems using Infinite-Dimensional Robust Counterparts

TL;DR

This work develops a framework to analyze and control input-affine dynamical systems under uncertainty by decomposing Lie derivative constraints with infinite-dimensional robust counterparts over SDR (spectrahedral) input sets. By reframing the Lie constraint via parameterized robust counterparts, the method eliminates disturbance variables and reduces the dimensionality of the resulting semidefinite programs, enabling scalable moment-SOS relaxations. The approach is validated across peak, distance, reachable-set, and region-of-attraction problems, and extended to discrete-time dynamics and data-driven settings. The contributions include convergence guarantees under mild regularity, SDP-size reductions compared to traditional formulations, and practical demonstrations on several dynamical systems with SDR disturbances. The work highlights opportunities for improved online estimation and broader application to control problems where uncertainty is structured as SDRs, while also noting computational bottlenecks in SOS tightening and directions for future efficiency gains.

Abstract

Input-affine dynamical systems often arise in control and modeling scenarios, such as the data-driven case when state-derivative observations are recorded under bounded noise. Common tasks in system analysis and control include optimal control, peak estimation, reachable set estimation, and maximum control invariant set estimation. Existing work poses these types of problems as infinite-dimensional linear programs in auxiliary functions with sum-of-squares tightenings. The bottleneck in most of these programs is the Lie derivative nonnegativity constraint posed over the time-state-control set. Decomposition techniques to improve tractability by eliminating the control variables include vertex decompositions (switching), or facial decompositions in the case where the polytopic set is a scaled box. This work extends the box-facial decomposition technique to allow for a robust-counterpart decomposition of semidefinite representable sets (e.g. polytopes, ellipsoids, and projections of spectahedra). These robust counterparts are proven to be equivalent to the original Lie constraint under mild compactness and regularity constraints. Efficacy is demonstrated under peak/distance/reachable set data-driven analysis problems and Region of Attraction maximizing control.
Paper Structure (62 sections, 23 theorems, 118 equations, 10 figures)

This paper contains 62 sections, 23 theorems, 118 equations, 10 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Acronyms/Initialisms
  4. Notation
  5. Analysis
  6. Robust Counterparts
  7. Polynomial Matrix Inequalities
  8. Analysis and Control Problems
  9. Peak Estimation
  10. Distance Estimation
  11. Reachable Set Estimation
  12. Region of Attraction Maximization
  13. Decomposed Lie Constraint
  14. Peak Estimation Program
  15. Parameterized Robust Counterparts
  16. ...and 47 more sections

Key Result

Theorem 2.1

Assume that each $K_s$ is a convex and pointed cone with nonempty interior. Further assume that there exists a Slater point ($\exists \bar{w} \in \mathbb{R}^L, \forall s: \exists \bar{\lambda}_s \in \mathbb{R}^{q_s}\mid A_s \bar{w} + G_s \bar{\lambda}_s + e_s \in \text{int}(K_s)$) if $K$ is non-poly

Figures (10)

  • Figure 1: Order-5 bound on minimal $x_2$ for Flow \ref{['eq:flow_pillow']} under elliptope-constrained noise
  • Figure 2: Observed data of Flow system \ref{['eq:flow']} within a circle
  • Figure 3: Minimizing $x_2$ on the observed Flow system \ref{['eq:flow_data_driven']} at order-4 SOS tightening
  • Figure 4: Distance estimate of \ref{['eq:flow']} at order 5
  • Figure 5: 100 observations of Twist system \ref{['eq:twist_dynamics']}
  • ...and 5 more figures

Theorems & Definitions (71)

  • Definition 2.1: helton2010semidefinite
  • Definition 2.2: Equation (1.3.14) of ben2009robust
  • Theorem 2.1: Theorem 1.3.4 of ben2009robust
  • Remark 1
  • Remark 2
  • Lemma 2.2
  • Remark 3
  • Remark 4
  • Theorem 3.1
  • proof
  • ...and 61 more