Set Invariance Conditions for Singular Linear Systems Subject to Actuator Saturation

In this note, we establish a set of conditions under which an ellipsoid is contractively invariant with respect to a singular linear system under a saturated linear feedback. These conditions can be expressed in terms of linear matrix inequalities, and the largest contractively invariant ellipsoid can be determined by solving an optimization problem with LMI constraints. With the feedback gain viewed as an additional variable, this optimization problem can be readily adapted for the design of feedback gain that results in the largest contractively invariant ellipsoid. Moreover, in the degenerate case where the singular linear system reduces to a regular system, our set invariance conditions reduce to the existing set invariance conditions for normal linear systems.