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Moment LMI approach to LTV impulsive control

Mathieu Claeys, Denis Arzelier, Didier Henrion, Jean-Bernard Lasserre

Abstract

In the 1960s, a moment approach to linear time varying (LTV) minimal norm impulsive optimal control was developed, as an alternative to direct approaches (based on discretization of the equations of motion and linear programming) or indirect approaches (based on Pontryagin's maximum principle). This paper revisits these classical results in the light of recent advances in convex optimization, in particular the use of measures jointly with hierarchy of linear matrix inequality (LMI) relaxations. Linearity of the dynamics allows us to integrate system trajectories and to come up with a simplified LMI hierarchy where the only unknowns are moments of a vector of control measures of time. In particular, occupation measures of state and control variables do not appear in this formulation. This is in stark contrast with LMI relaxations arising usually in polynomial optimal control, where size grows quickly as a function of the relaxation order. Jointly with the use of Chebyshev polynomials (as a numerically more stable polynomial basis), this allows LMI relaxations of high order (up to a few hundreds) to be solved numerically.

Moment LMI approach to LTV impulsive control

Abstract

In the 1960s, a moment approach to linear time varying (LTV) minimal norm impulsive optimal control was developed, as an alternative to direct approaches (based on discretization of the equations of motion and linear programming) or indirect approaches (based on Pontryagin's maximum principle). This paper revisits these classical results in the light of recent advances in convex optimization, in particular the use of measures jointly with hierarchy of linear matrix inequality (LMI) relaxations. Linearity of the dynamics allows us to integrate system trajectories and to come up with a simplified LMI hierarchy where the only unknowns are moments of a vector of control measures of time. In particular, occupation measures of state and control variables do not appear in this formulation. This is in stark contrast with LMI relaxations arising usually in polynomial optimal control, where size grows quickly as a function of the relaxation order. Jointly with the use of Chebyshev polynomials (as a numerically more stable polynomial basis), this allows LMI relaxations of high order (up to a few hundreds) to be solved numerically.

Paper Structure

This paper contains 11 sections, 3 theorems, 31 equations, 3 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Relaxed linear optimal control
  3. Problem of moments
  4. Primal and dual conic LP
  5. Integration of the LTV ODE
  6. Solving the LP on measures
  7. Examples
  8. Scalar polynomial example
  9. Nonsmooth trajectories
  10. A fuel-optimal linear impulsive guidance rendezvous problem for an elliptic reference orbit
  11. Conclusion

Key Result

Lemma 1

The infimum is attained in problem (p) and it is equal to the infimum of problem (ocp), i.e. $q^*=p^*$.

Figures (3)

  • Figure 1: Trajectories in the orbital plane: 2-impulse solution (pink), optimal solution (blue), Direct 20-impulses solution (cyan).
  • Figure 2: Trajectories in the orbital plane: optimal solution (blue), Direct 20-impulses solution (cyan) .
  • Figure 3: Impulses for 2-impulse solution (red and blue), direct 20-impulse solution (cyan), optimal solution (black).

Theorems & Definitions (4)

  • Lemma 1
  • Lemma 2
  • Remark 1
  • Lemma 3