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Input Matrix Optimization for Desired Reachable Set Warping of Linear Systems

Hrishav Das, Melkior Ornik

Abstract

Shaping the reachable set of a dynamical system is a fundamental challenge in control design, with direct implications for both performance and safety. This paper considers the problem of selecting the optimal input matrix for a linear system that maximizes warping of the reachable set along a direction of interest. The main result establishes that under certain assumptions on the dynamics, the problem reduces to a finite number of linear optimization problems. When these assumptions are relaxed, we show heuristically that the same approach yields good results. The results are validated on two systems: a linearized ADMIRE fighter jet model and a damped oscillator with complex eigenvalues. The paper concludes with a discussion of future directions for reachable set warping research.

Input Matrix Optimization for Desired Reachable Set Warping of Linear Systems

Abstract

Shaping the reachable set of a dynamical system is a fundamental challenge in control design, with direct implications for both performance and safety. This paper considers the problem of selecting the optimal input matrix for a linear system that maximizes warping of the reachable set along a direction of interest. The main result establishes that under certain assumptions on the dynamics, the problem reduces to a finite number of linear optimization problems. When these assumptions are relaxed, we show heuristically that the same approach yields good results. The results are validated on two systems: a linearized ADMIRE fighter jet model and a damped oscillator with complex eigenvalues. The paper concludes with a discussion of future directions for reachable set warping research.

Paper Structure

This paper contains 14 sections, 1 theorem, 32 equations, 5 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Pontryagin's Maximum Principle
  4. Reachable Set and its Boundary
  5. Computing the Reachable Set Boundary
  6. Problem Statement
  7. Technical Results
  8. Relaxing Assumptions of Theorem \ref{['thm:directional_optimal_B']}
  9. Complex eigenvalues of $A^\top$ (both Assumptions relaxed)
  10. Real eigenvalues of $A^\top$ but $d$ not an eigenvector (Assumption 2 relaxed)
  11. Examples
  12. ADMIRE Linearized Fighter Jet Model
  13. System with Imaginary Eigenvalues of $A$
  14. Summary and Conclusion

Key Result

Theorem 1

Consider the family of systems lineardynamics parameterized by $B \in \mathcal{B}$, where $\mathcal{U} \subset \mathbb{R}^d$ is a compact convex polytope (i.e., the convex hull of finitely many vertices $\{u_i\}_{i=1}^N$) and $T > 0$. Under Assumptions 1--2 of Problem prob:main, define $P_0 \triangl Then $B^\ast \triangleq B_{i^\ast}$ solves Problem prob:main, i.e., $B^\ast \in \arg\max_{B\in\math

Figures (5)

  • Figure 1: Parameters of the Directional Growth Metric $G_d(B) = d^\top(X_{d,B} - c_0)$. The blue region is the reachable set $\mathcal{R}_B(T; X_0)$, $c_0$ is the zero-input trajectory endpoint (interior point), $X_{d,B}$ is the boundary point reached by the trajectory with terminal co-state $P(T) = d$, and the red arrow indicates the direction $d$.
  • Figure 2: Growth along $p$ direction ($d = [1,0,0]^\top$). Red: nominal $\mathcal{R}_B$. Green: optimized $\mathcal{R}_{B^\ast}$.
  • Figure 3: Shrinkage along $p$ direction ($d = [1,0,0]^\top$, $\arg\min$ variant). Red: nominal $\mathcal{R}_B$. Green: optimized $\mathcal{R}_{B^\ast}$.
  • Figure 4: Growth along $d = [0.3536,\,0.6124,\,0.7071]^\top$ (approximately equal pitch and yaw weighting). Red: nominal $\mathcal{R}_B$. Green: optimized $\mathcal{R}_{B^\ast}$.
  • Figure 5: Reachable set growth for the damped oscillator with complex eigenvalues of $A$, $d = [1,0]^\top$. Red: nominal $\mathcal{R}_{B_{\mathrm{nom}}}$. Green: optimized $\mathcal{R}_{B^\ast}$. Horizontal axis: $x_1$. Vertical axis: $x_2$. Growth occurs in all directions, consistent with Section \ref{['complexeigdiscussion']}.

Theorems & Definitions (2)

  • Theorem 1
  • proof