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Peak Estimation of Rational Systems using Convex Optimization

Jared Miller, Roy S. Smith

TL;DR

Algorithms that upper-bound the peak value of a state function along trajectories of a continuous-time system with rational dynamics, using a sum-of-rational method based on absolute continuity of measures are presented.

Abstract

This paper presents algorithms that upper-bound the peak value of a state function along trajectories of a continuous-time system with rational dynamics. The finite-dimensional but nonconvex peak estimation problem is cast as a convex infinite-dimensional linear program in occupation measures. This infinite-dimensional program is then truncated into finite-dimensions using the moment-Sum-of-Squares (SOS) hierarchy of semidefinite programs. Prior work on treating rational dynamics using the moment-SOS approach involves clearing dynamics to common denominators or adding lifting variables to handle reciprocal terms under new equality constraints. Our solution method uses a sum-of-rational method based on absolute continuity of measures. The Moment-SOS truncations of our program possess lower computational complexity and (empirically demonstrated) higher accuracy of upper bounds on example systems as compared to prior approaches.

Peak Estimation of Rational Systems using Convex Optimization

TL;DR

Algorithms that upper-bound the peak value of a state function along trajectories of a continuous-time system with rational dynamics, using a sum-of-rational method based on absolute continuity of measures are presented.

Abstract

This paper presents algorithms that upper-bound the peak value of a state function along trajectories of a continuous-time system with rational dynamics. The finite-dimensional but nonconvex peak estimation problem is cast as a convex infinite-dimensional linear program in occupation measures. This infinite-dimensional program is then truncated into finite-dimensions using the moment-Sum-of-Squares (SOS) hierarchy of semidefinite programs. Prior work on treating rational dynamics using the moment-SOS approach involves clearing dynamics to common denominators or adding lifting variables to handle reciprocal terms under new equality constraints. Our solution method uses a sum-of-rational method based on absolute continuity of measures. The Moment-SOS truncations of our program possess lower computational complexity and (empirically demonstrated) higher accuracy of upper bounds on example systems as compared to prior approaches.
Paper Structure (24 sections, 7 theorems, 36 equations, 2 figures, 4 tables)

This paper contains 24 sections, 7 theorems, 36 equations, 2 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notation
  4. Occupation Measures
  5. Rational Linear Program
  6. Assumptions
  7. Measure Program
  8. Absolute Continuity Formulation
  9. Function Program
  10. Finite-Dimensional Truncation
  11. Sum of Squares Background
  12. SOS Program
  13. Computational Complexity
  14. Numerical Examples
  15. Two-Species Chemical Reaction Network
  16. ...and 9 more sections

Key Result

Theorem 4

Under Assumption A1, Problem prob:peak_meas will upper-bound prob:peak_traj with $p^* \geq P^*$.

Figures (2)

  • Figure 1: Trajectories and $k=6$ bounds for \ref{['eq:ex_mm']}, along with position of the unique equilibrium point $x_{eq}$
  • Figure 2: Trajectories and $k=6$ bound for \ref{['eq:twist_rat_dynamics']}

Theorems & Definitions (18)

  • Remark 2
  • Theorem 4
  • proof
  • Theorem 5
  • proof
  • Definition 6
  • Definition 7
  • Remark 8
  • Proposition 9
  • Corollary 11
  • ...and 8 more