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Numerical methods for solving minimum-time problem for linear systems

M E Buzikov, A M Mayer

Abstract

This paper offers a contemporary and comprehensive perspective on the classical algorithms utilized for the solution of minimum-time problem for linear systems (MTPLS). The use of unified notations supported by visual geometric representations serves to highlight the differences between the Neustadt-Eaton and Barr-Gilbert algorithms. Furthermore, these notations assist in the interpretation of the distance-finding algorithms utilized in the Barr-Gilbert algorithm. Additionally, we present a novel algorithm for solving MTPLS and provide a constructive proof of its convergence. Similar to the Barr-Gilbert algorithm, the novel algorithm employs distance search algorithms. The design of the novel algorithm is oriented towards solving such MTPLS for which the analytic description of the reachable set is available. To illustrate the advantages of the novel algorithm, we utilize the isotropic rocket benchmark. Numerical experiments demonstrate that, for high-precision computations, the novel algorithm outperforms others by factors of tens or hundreds and exhibits the lowest failure rate.

Numerical methods for solving minimum-time problem for linear systems

Abstract

This paper offers a contemporary and comprehensive perspective on the classical algorithms utilized for the solution of minimum-time problem for linear systems (MTPLS). The use of unified notations supported by visual geometric representations serves to highlight the differences between the Neustadt-Eaton and Barr-Gilbert algorithms. Furthermore, these notations assist in the interpretation of the distance-finding algorithms utilized in the Barr-Gilbert algorithm. Additionally, we present a novel algorithm for solving MTPLS and provide a constructive proof of its convergence. Similar to the Barr-Gilbert algorithm, the novel algorithm employs distance search algorithms. The design of the novel algorithm is oriented towards solving such MTPLS for which the analytic description of the reachable set is available. To illustrate the advantages of the novel algorithm, we utilize the isotropic rocket benchmark. Numerical experiments demonstrate that, for high-precision computations, the novel algorithm outperforms others by factors of tens or hundreds and exhibits the lowest failure rate.

Paper Structure

This paper contains 27 sections, 135 equations, 10 figures, 10 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Problem formulation
  4. General problem
  5. Assumptions
  6. Geometric view
  7. Maximum principle
  8. Boundary of the reachable set
  9. Distance estimates
  10. Inclination of hyperplanes
  11. Time boosting
  12. Algorithms of computing the distance
  13. Gilbert-Johnson-Keerthi distance algorithm
  14. Gilbert distance algorithm
  15. Steepest ascent distance algorithm
  16. ...and 12 more sections

Figures (10)

  • Figure 1: The support vector $\mathbf{p}$, the contact point $\boldsymbol{s}_{\mathcal{M}}(\mathbf{p})$, and the tangent hyperplane $\Gamma_{\mathcal{M}}(\mathbf{p})$
  • Figure 2: Connection of extremal trajectory $\boldsymbol{s}_E(\cdot; T, \mathbf{p})$ and contact function $\boldsymbol{s}_{\mathcal{R}(\cdot)}(\boldsymbol{p}(\cdot; T, \mathbf{p}))$. Here, $t < T$.
  • Figure 3: Distance estimates $\rho_{\mathrm{lower}}(t, \mathbf{p}) \leq \rho(t) \leq \rho_{\mathrm{upper}}(t, \mathbf{p})$
  • Figure 4: Inclination of hyperplanes $\Gamma_{\mathcal{R}(t)}(\mathbf{p})$, $\Gamma_{\mathcal{G}(t)}(-\mathbf{p})$ by changing $\mathbf{p}$ to $\tilde{\mathbf{p}} = \mathbf{p} + \gamma\boldsymbol{q}(t, \mathbf{p})$. Here, $\gamma$ is such that $\rho_{\mathrm{lower}}(t, \tilde{\mathbf{p}}) > \rho_{\mathrm{lower}}(t, \mathbf{p})$
  • Figure 5: Evaluating of the boosting-time function $t = F(T, \mathbf{p})$. Here, $\boldsymbol{p} = \boldsymbol{p}(t; T, \mathbf{p})$
  • ...and 5 more figures