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Periodic solutions of nonlinear control systems with switching: a Lie-algebraic and contraction approach

Alexander Zuyev, Peter Benner

Abstract

This paper is devoted to the analysis of periodic solutions of nonlinear control-affine systems with bang-bang controls. Such problems naturally arise in periodic optimal control with constrained inputs, which have, in particular, important applications in the performance optimization of chemical reactions. We reduce the problem of constructing a periodic solution to that of finding a fixed point of a composition of exponential maps. The latter problem is then addressed using the Baker-Campbell-Hausdorff-Dynkin (BCHD) formula. We establish the equivalence between periodic solutions of the original control system and those of an associated autonomous system involving iterated Lie brackets. In the planar case, applying the Bendixson-Dulac theorem allows us to further simplify the problem to finding the equilibria of this autonomous system. The developed theory is then applied to nonlinear chemical reaction models with constrained controls.

Periodic solutions of nonlinear control systems with switching: a Lie-algebraic and contraction approach

Abstract

This paper is devoted to the analysis of periodic solutions of nonlinear control-affine systems with bang-bang controls. Such problems naturally arise in periodic optimal control with constrained inputs, which have, in particular, important applications in the performance optimization of chemical reactions. We reduce the problem of constructing a periodic solution to that of finding a fixed point of a composition of exponential maps. The latter problem is then addressed using the Baker-Campbell-Hausdorff-Dynkin (BCHD) formula. We establish the equivalence between periodic solutions of the original control system and those of an associated autonomous system involving iterated Lie brackets. In the planar case, applying the Bendixson-Dulac theorem allows us to further simplify the problem to finding the equilibria of this autonomous system. The developed theory is then applied to nonlinear chemical reaction models with constrained controls.
Paper Structure (7 sections, 3 theorems, 33 equations, 4 figures, 1 table)

This paper contains 7 sections, 3 theorems, 33 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction
  2. Problem Formulation and Preliminaries
  3. Main Results
  4. Case Studies
  5. Reaction of the type "$A\to$ Product"
  6. Reaction of the type "$A\to B\to Product$"
  7. Conclusions

Key Result

Theorem 3.1

Assume that the vector fields $f_1,f_2,...,f_N:D\to \mathbb R^n$ of system cs_driftless are analytic in a domain $D$, and let $x^0\in D$. Then, for any sufficiently small $\tau>0$, the following two conditions are equivalent: Here,

Figures (4)

  • Figure 1: Phase portrait of system \ref{['FM_system']} with $M=4$.
  • Figure 2: Periodic solutions of \ref{['CSTR_affine']} with controls \ref{['u_bangbang']} for $N=2$.
  • Figure 3: Periodic solution of \ref{['CSTR3_affine']}.
  • Figure 4: Trajectory of \ref{['CSTR3_affine']} attracted to the periodic regime.

Theorems & Definitions (10)

  • Theorem 3.1
  • proof
  • Remark 3.1
  • Theorem 3.2
  • proof
  • Remark 3.2
  • Definition 3.1
  • Theorem 3.3
  • proof
  • Remark 3.3