Moment LMI approach to LTV impulsive control

TL;DR

This paper revisits classical moment-based LTV impulsive control methods, enhancing them with modern convex optimization techniques and LMI relaxations to achieve high-order solutions efficiently.

Contribution

It introduces a simplified LMI hierarchy using moments of control measures, avoiding occupation measures, and employs Chebyshev polynomials for numerical stability, enabling high-order relaxations.

Findings

High-order LMI relaxations are computationally feasible.

The approach simplifies the control measure representation.

Numerical stability is improved with Chebyshev polynomials.

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.