LMI Framework for Set Reachability Inclusion in Discrete-Time LTI Systems

In this article, we present a convex optimization framework to verify the reachability of a desired set for discrete-time linear time-invariant systems. Given elliptically bounded inputs, the set of reachable states in
$N$
time steps is the Minkowski sum of a finite number of ellipsoids. We formulate the inclusion verification problem as a chain of constraints in the form of linear matrix inequalities. As the time horizon grows, the number of constraints becomes unwieldy, and we present a technique to achieve a similar level of accuracy with far fewer terms, significantly reducing the computational cost of the method. Numerical examples presented in this article show that the method is highly adaptable.