A bilevel optimization approach for D-invariant set design*

This paper is dedicated to the set invariance characterization of dynamical systems affected by time-delays. Starting from a discrete-time dynamical system described by a delay-difference equation, the construction of a positive invariant set is sought. An optimization-based procedure is defined to include the properties related to the polyhedral structure, boundedness and positive invariance. This last property is related to the Minkowski sum and its translation into linear constraints or optimization-based subproblems is discussed together with the computational details. It will be shown that bilevel optimization problems can be used for D-invariance design problems. The difficulties related to the nonlinearity of the optimization and the complementarity constraints will be discussed as well as the objective functions which can translate additional features of the D-invariant sets.