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The goal of this paper is to present a systematic method to compute referencedependent positively invariant sets for systems subject to constraints. To thisend, we first characterize these sets as level sets of reference dependentLyapunov functions. Based on this characterization and using Sum of Squares(SOS) theory, we provide a polynomial certificate for the existence of suchsets. Subsequently, through some algebraic manipulations, we express thiscertificate in terms of a Semi-Definite Programming (SDP) problem whichmaximizes the size of the resulting reference dependent invariant sets. We thenpresent the results of implementing the proposed method to an example systemand propose some variations of the proposed method that may help in reducingthe numerical issues of the method. Finally, the proposed method is employed inthe Model Predictive Control (MPC) for Tracking scheme to compute the terminalset, and in the Explicit Reference Governor (ERG) scheme to compute theso-called Dynamic Safety Margin (DSM). The effectiveness of the proposed methodin each of the schemes is demonstrated through a simulation study.

Reference Dependent Invariant Sets: Sum of Squares Based Computation and Applications in Constrained Control