Data-Driven Synthesis of Robust Positively Invariant Sets: From State Feedback to Output Feedback
This paper develops a direct data-driven framework for robust positively invariant (RPI) set synthesis for unknown linear time-invariant systems under state-feedback and observer-based output-feedback scenarios. The feedback and observer gains, along with the RPI sets, are directly synthesized from noisy offline data by solving semidefinite programs (SDPs), avoiding intermediate model identification or explicit model-uncertainty set construction. In the state-feedback case, a linear-quadratic (LQ)-type feedback gain is first computed, and an ellipsoidal RPI set is then synthesized for the resulting closed-loop dynamics. In the observer-based output-feedback case, offline data are used to compute the observer gain and the corresponding RPI set for the system state through an augmented-state formulation. This design provides a unified method for invariant-set computation in both scenarios, and the direct data-driven formulation avoids the explicit construction and propagation of an intermediate model-uncertainty set. Numerical examples illustrate the effectiveness of the proposed method.