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Optimization-based Constrained Funnel Synthesis for Systems with Lipschitz Nonlinearities via Numerical Optimal Control

Taewan Kim, Purnanand Elango, Taylor P. Reynolds, Behçet Açıkmeşe, Mehran Mesbahi

Abstract

This paper presents a funnel synthesis algorithm for computing controlled invariant sets and feedback control gains around a given nominal trajectory for dynamical systems with locally Lipschitz nonlinearities and bounded disturbances. The resulting funnel synthesis problem involves a differential linear matrix inequality (DLMI) whose solution satisfies a Lyapunov condition that implies invariance and attractivity properties. Due to these properties, the proposed method can balance maximization of initial invariant funnel size, i.e., size of the funnel entry, and minimization of the size of the attractive funnel for attenuating the effect of disturbance. To solve the resulting funnel synthesis problem with the DLMI as constraints, we employ a numerical optimal control approach that uses a multiple shooting method to convert the problem into a finite dimensional semidefinite programming problem. This framework does not require piecewise linear system matrices and funnel parameters, which is typically assumed in recent related work. We illustrate the proposed funnel synthesis method with a numerical example.

Optimization-based Constrained Funnel Synthesis for Systems with Lipschitz Nonlinearities via Numerical Optimal Control

Abstract

This paper presents a funnel synthesis algorithm for computing controlled invariant sets and feedback control gains around a given nominal trajectory for dynamical systems with locally Lipschitz nonlinearities and bounded disturbances. The resulting funnel synthesis problem involves a differential linear matrix inequality (DLMI) whose solution satisfies a Lyapunov condition that implies invariance and attractivity properties. Due to these properties, the proposed method can balance maximization of initial invariant funnel size, i.e., size of the funnel entry, and minimization of the size of the attractive funnel for attenuating the effect of disturbance. To solve the resulting funnel synthesis problem with the DLMI as constraints, we employ a numerical optimal control approach that uses a multiple shooting method to convert the problem into a finite dimensional semidefinite programming problem. This framework does not require piecewise linear system matrices and funnel parameters, which is typically assumed in recent related work. We illustrate the proposed funnel synthesis method with a numerical example.
Paper Structure (14 sections, 4 theorems, 33 equations, 3 figures)

This paper contains 14 sections, 4 theorems, 33 equations, 3 figures.

Table of Contents

  1. INTRODUCTION
  2. Constrained Funnel Synthesis
  3. Locally Lipschitz Nonlinear Systems
  4. Lyapunov Conditions
  5. Feasibility of State and Input Funnels
  6. Objectives
  7. Continuous-time funnel synthesis problem
  8. Optimizing Funnel via Optimal Control
  9. Changing a DLMI to a Differential Matrix Equality
  10. Multiple Shooting Numerical Optimal Control
  11. Discrete-time convex funnel synthesis problem
  12. Inter-sample constraint violation
  13. Numerical Simulation
  14. Conclusions

Key Result

Lemma 1

Suppose that the Lyapunov condition eq:Lyapunov_condition holds with a positive definite matrix-valued continuous function $Q(t)$ , then the time-varying ellipsoid defined as is invariant for the closed-loop system eq:diff_closedloop, that is, if $\eta(\cdot)$ is any solution with $\eta(t_{0})\in\mathcal{E}(t_{0})$, then $\eta(t)\in\mathcal{E}(t)$ for all $t\in[t_{0},t_{f}]$. Furthermore, the ell

Figures (3)

  • Figure 1: Illustration of the ellipsoids $\mathcal{E}(t)$ and $\mathcal{E}_{c}(t)$. An example of solution $\eta(t)$ is given as a dashed red line. Since $\mathcal{E}$ is attractive, any solution $\eta(\cdot)$ starting with $\eta(t_{0})\in\mathcal{E}_{c}(t_{0})\backslash\mathcal{\mathcal{E}}(t_{0})$ converges to $\mathcal{E}$ if $t_{f}$ is sufficiently large. The proposed funnel synthesis aims to maximize the size of $\mathcal{E}_c (t_0)$ and minimize that of $\mathcal{E}(t)$ for all $t$ in $[t_0,t_f]$.
  • Figure 2: The figure of the nominal trajectory and synthesized funnels projected on position coordinates. It shows the projection of $\mathcal{E}$ (brown ellipsoid) and $\mathcal{E}_c$ (blue ellipsoid) at each node point.
  • Figure 3: The figure of input funnel (top left and right) and support value $1/c(t)$ (bottom).

Theorems & Definitions (7)

  • Lemma 1
  • Lemma 2
  • proof
  • Theorem 1
  • proof
  • Proposition 1
  • proof