Synthesis of Input-to-State Attractivity Controllers for Transition Systems with Disturbances
Résumé
In this paper, we introduce a notion of input-to-state
attractivity (ISA) controllers for a class of finite transition systems
with disturbances. The performances of an ISA controller are characterized
by a gain function that quantifies the deviation of closedloop
trajectories from the target set as a function of the amplitude
of past disturbances on a bounded time window. We prove the
existence of controllers that are gain-optimal (GO) in the sense that
their gain function is minimal (with respect to a given order on the
set of gain functions) over all possible ISA controllers. Then, we
consider the problem of synthesizing ISA controllers. We present
an approach based on successive refinements of controllers: starting
from a controller synthesized against worst-case disturbances,
the controller is iteratively refined in order to improve the closedloop
behavior under lower disturbances. We prove that our method
makes it possible to synthesize an ISA controller that is shown
to be a GO-ISA controller (for the colexicographic order) when a
condition, which can be easily checked a posteriori, is satisfied.
Finally, an application to adaptive cruise control demonstrates the
effectiveness of our approach.
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