On maximum hands-off restricted hybrid control for discrete-time switched linear systems
Darsana U, Atreyee Kundu
TL;DR
This work addresses the problem of designing maximum hands-off hybrid control sequences for discrete-time switched linear systems, seeking to minimize the total number of active time instants for both discrete switches and continuous inputs while steering $x(0)=\xi$ to $x(T)=0_d$ in a fixed horizon $T$. The authors develop a state-space abstraction framework that partitions the state space into regions and builds a labeled transition graph whose edges encode switching and control actions. A xi-hands-off walk on this graph maps directly to a sparse hybrid control sequence, and the proposed algorithm guarantees a maximum hands-off solution under the abstraction's feasibility. This graph-theoretic approach, which does not rely on RIP-type conditions, yields practical maximum-sparsity controls for a broad class of discrete-time switched systems and includes sufficient conditions for when the state-space abstraction is admissible; the numerical examples illustrate effectiveness even when prior guarantees fail. Overall, the method enables energy- and resource-efficient control of switched systems by leveraging off-the-shelf graph algorithms on a principled state-space abstraction.
Abstract
This paper deals with design of maximum hands-off hybrid control sequences for discrete-time switched linear systems. It is a sparsest combination of a discrete control sequence (i.e. the switching sequence) and a continuous control sequence, both satisfying pre-specified restrictions on the admissible actions, that steers a given initial state of the switched system to the origin of the state-space in a pre-specified duration of time. Given the subsystems dynamics, the sets of admissible continuous and discrete control, the initial state and the time horizon, we present a new algorithm that, under certain conditions on the subsystems dynamics and the admissible control, designs maximum hands-off hybrid control sequences for the resulting switched system. The key apparatuses for our analysis are graph theory and linear algebra. Numerical examples are presented to demonstrate our results.