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Exact Computation of LTI Reach Set from Integrator Reach Set with Bounded Input

Shadi Haddad, Pansie Khodary, Abhishek Halder

TL;DR

The paper addresses exact reach‑set computation for a controllable single‑input LTI system with bounded input. It develops a semi‑analytical approach that leverages a Brunovsky transformation to convert the LTI into an $n$th‑order integrator with time‑varying input bounds, then computes the integrator boundary and volume before mapping back to the LTI coordinates. A key contribution is the explicit σ‑parametric boundary formula for the integrator under time‑varying bounds, together with a method to derive the induced bounds $[u_{\min}(s),u_{\max}(s)]$ from the original $[v_{\min},v_{\max}]$ and to recover the LTI reach set as $\mathcal{Z}_t = M^{-1}\mathcal{X}_t$. The work also provides an exact volume computation via a Jacobian formulation and demonstrates the approach on a 2D example, showing that constant input bounds reduce to previous results. Overall, it yields precise geometric characterizations (boundary and volume) of LTI reach sets, with implications for safety verification and potential extensions to multi‑input systems.

Abstract

We present a semi-analytical method for exact computation of the boundary of the reach set of a single-input controllable linear time invariant (LTI) system with given bounds on its input range. In doing so, we deduce a parametric formula for the boundary of the reach set of an integrator linear system with time-varying bounded input. This formula generalizes recent results on the geometry of an integrator reach set with time-invariant bounded input. We show that the same ideas allow for computing the volume of the LTI reach set.

Exact Computation of LTI Reach Set from Integrator Reach Set with Bounded Input

TL;DR

The paper addresses exact reach‑set computation for a controllable single‑input LTI system with bounded input. It develops a semi‑analytical approach that leverages a Brunovsky transformation to convert the LTI into an th‑order integrator with time‑varying input bounds, then computes the integrator boundary and volume before mapping back to the LTI coordinates. A key contribution is the explicit σ‑parametric boundary formula for the integrator under time‑varying bounds, together with a method to derive the induced bounds from the original and to recover the LTI reach set as . The work also provides an exact volume computation via a Jacobian formulation and demonstrates the approach on a 2D example, showing that constant input bounds reduce to previous results. Overall, it yields precise geometric characterizations (boundary and volume) of LTI reach sets, with implications for safety verification and potential extensions to multi‑input systems.

Abstract

We present a semi-analytical method for exact computation of the boundary of the reach set of a single-input controllable linear time invariant (LTI) system with given bounds on its input range. In doing so, we deduce a parametric formula for the boundary of the reach set of an integrator linear system with time-varying bounded input. This formula generalizes recent results on the geometry of an integrator reach set with time-invariant bounded input. We show that the same ideas allow for computing the volume of the LTI reach set.
Paper Structure (7 sections, 6 theorems, 47 equations, 2 figures)

This paper contains 7 sections, 6 theorems, 47 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Brunovsky Normal Form and LTI Reach Set
  3. Main Idea
  4. Input Set in Brunovsky Coordinates
  5. Parametric boundary
  6. Volume
  7. Concluding Remarks

Key Result

Lemma 1

Suppose that the state matrix ${\bm{A}}\in\mathbb{R}^{n\times n}$ has $n$ distinct eigenvalues $\lambda_1,\hdots,\lambda_n\in\mathbb{C}$. Then the function $f$ in defIv can be expressed as

Figures (2)

  • Figure 1: The $f(\tau)$ (solid line) in \ref{['fAsFiniteSum']} for $\bm{A}=\tiny672-4-21-532$, and its four zeros (circular markers) for $\tau\in[0, 2]$. Here $\bm{A}$ has one real and two complex conjugate eigenvalues.
  • Figure 2: The reach sets $\mathcal{Z}_t\left(\{\bm{0}_{2\times 1}\}\right)$ at $t=1, 1.5, 2,2.5, 3$ (grey filled) for Example \ref{['ExampleZt']} shown in the right plot. These sets were computed via the proposed two step method in Sec. \ref{['subsecMainIdea']}. The 8 sample state trajectories shown here correspond to 8 randomly sampled truncated Gaussian process input paths in $\{v(\cdot)\in C([0,t])\mid v(s)\in[-0.2,0.2]\forall s\in[0,t]\}$ shown in the top left inset plot. The bottom left inset plot shows the time-varying range $[u_{\min}(s),u_{\max}(s)]$.

Theorems & Definitions (21)

  • Remark 1
  • Lemma 1
  • proof
  • Example 1
  • Example 2
  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Remark 2
  • ...and 11 more