1 research outputs found
Tight Decomposition Functions for Continuous-Time Mixed-Monotone Systems with Disturbances
The vector field of a mixed-monotone system is decomposable via a
decomposition function into increasing (cooperative) and decreasing
(competitive) components, and this decomposition allows for, e.g., efficient
computation of reachable sets and forward invariant sets. A main challenge in
this approach, however, is identifying an appropriate decomposition function.
In this work, we show that any continuous-time dynamical system with a
Lipschitz continuous vector field is mixed-monotone, and we provide a
construction for the decomposition function that yields the tightest
approximation of reachable sets when used with the standard tools for
mixed-monotone systems. Our construction is similar to that recently proposed
by Yang and Ozay for computing decomposition functions of discrete-time systems
[1] where we make appropriate modifications for the continuous-time setting and
also extend to the case with unknown disturbance inputs. As in [1], our
decomposition function construction requires solving an optimization problem
for each point in the state-space; however, we demonstrate through example how
tight decomposition functions can sometimes be calculated in closed form. As a
second contribution, we show how under-approximations of reachable sets can be
efficiently computed via the mixed-monotonicity property by considering the
backward-time dynamics.Comment: 6 pages. 3 figures. Submitted to Control Systems Letters (L-CSS