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Reachability Analysis for Design Optimization

Steven Nguyen, Jorge Cortés, Boris Kramer

Abstract

We present an approach to approximate reachable sets for linear systems with bounded L-infinity controls in finite time. Our first approach investigates the boundaries of these sets and reveals an exact characterization for single-input, planar systems with real, distinct eigenvalues. The second approach leverages convergence of the Lp-norms to L-infinity and uses Lp-norm reachable sets as an approximation of the L-infinity-norm reachable sets. Our optimal control results yield insights that make computational approximations of the Lp-norm reachable sets more tractable, and yield exact characterizations for L-infinity with the previous assumptions on the system. As an example, we incorporate our reachability analysis into the design optimization of a highly-maneuverable aircraft. Introducing constraints based on reachability allow us to factor physical limitations to desired flight maneuvers into the design process.

Reachability Analysis for Design Optimization

Abstract

We present an approach to approximate reachable sets for linear systems with bounded L-infinity controls in finite time. Our first approach investigates the boundaries of these sets and reveals an exact characterization for single-input, planar systems with real, distinct eigenvalues. The second approach leverages convergence of the Lp-norms to L-infinity and uses Lp-norm reachable sets as an approximation of the L-infinity-norm reachable sets. Our optimal control results yield insights that make computational approximations of the Lp-norm reachable sets more tractable, and yield exact characterizations for L-infinity with the previous assumptions on the system. As an example, we incorporate our reachability analysis into the design optimization of a highly-maneuverable aircraft. Introducing constraints based on reachability allow us to factor physical limitations to desired flight maneuvers into the design process.
Paper Structure (15 sections, 5 theorems, 22 equations, 3 figures)

This paper contains 15 sections, 5 theorems, 22 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Control concepts in aircraft design
  3. Background/literature review
  4. Contributions
  5. Preliminaries
  6. Notation
  7. Reachable sets of linear systems
  8. Reachable set boundaries for p=inf
  9. Optimal switching controllers
  10. Parameteric function representation of reachable sets
  11. Reachable sets for finite p
  12. p-norm optimal controls
  13. Sufficient condition for inclusion in p-norm reachable set
  14. Incorporating reachability into design optimization
  15. Summary and Conclusion

Key Result

Lemma 1

[lemma]lem:saturated_controls_on_boundary For linear system eq:basic_linear_system with $\mathcal{U}_{\ul{\mathbf{u}},\overline{\mathbf{u}}}=\left\{ \mathbf{u} \in \mathbb{R}^m \;\vert\; \ul{\mathbf{u}} \leq \mathbf{u} \leq \overline{\mathbf{u}} \right\}$, where $\ul{\mathbf{u}},\overline{\mathbf{ where $\mathbf{\psi}_i(t;\mathbf{c}) = \mathbf{c}^\top e^{\mathbf{A}(T-t)}\mathbf{b}_i$, $\mathbf{b

Figures (3)

  • Figure 1: Comparison of the boundary parameterization using the switching control scheme as outlined in \ref{['thm:parameterizing_singleinput_2dim_boundary']} with a zonotopic inner-approximation of the reachable set.
  • Figure 2: Numerical approximation of $\mathcal{R}(1;\mathbb{U}_{\mathcal{L}^{6}})$ for \ref{['eq:example_2d_system']}. Blue points denote states that can be reached using unit $\mathcal{L}^{6}$-norm controls, while red points denote states that cannot be reached with unit $\mathcal{L}^{6}$-norm controls. A zonotope inner-approximation of $\mathcal{R}(1;\mathcal{U}_{-1,1})$ is shown by the shaded region.
  • Figure 3: The magenta points show a subset of the interior of $\mathcal{R}(1;\mathbb{U}_{\mathcal{L}^{6}})$ using trajectories that satisfy a norm bound on $\bm{\lambda}(0)$. The blue and red points are the same as in \ref{['fig:2d_p6_vs_cora']}.

Theorems & Definitions (17)

  • Lemma 1
  • Remark 1
  • Proposition 1
  • proof
  • Remark 2
  • Theorem 2
  • proof
  • Example 1
  • Definition 1: athans2007optimal
  • Theorem 3
  • ...and 7 more