Efficient computation of Robust Positively Invariant sets with linear state-feedback gain as a variable of optimization

In this paper, we develop an algorithm for the efficient computation of Robust Positively Invariant sets for linear discrete-time systems subject to bounded additive disturbances and polytopic input constraints. The proposed algorithm simultaneously computes both the optimal invariant set and the corresponding state-feedback control law in one step by solving a single semidefinite program. Ellipsoidal as well as a polytopic characterization of the invariant sets is derived. In addition to the input constraints, the proposed method also allows for the incorporation of state constraints in a non-conservative manner. Furthermore, it is shown that for the case with a fixed control law, the proposed algorithm computes the optimal polytopic invariant set by solving a single Linear Program. The viability of the proposed scheme is demonstrated through numerical examples.

I. Introduction

Robust Positively Invariant (RPI) set defines a bounded state-space region to which the system state can be confined, for all possible disturbances, through the application of a state-feedback control law . RPI sets are important in the robustness analysis and synthesis of controllers for uncertain systems [1]. These sets also play a fundamental role in various robust model predictive control schemes [2], [3]. Furthermore, the RPI set is also a suitable target set in robust time-optimal control schemes [4]–[6].