Privacy-Preserving Cloud Computation of Algebraic Riccati Equations

We address the problem of securely outsourcing the solution of algebraic Riccati equations (ARE) to a cloud. Our proposed method explores a middle ground between privacy preserving algebraic transformations and perturbation techniques, aiming to achieve simplicity of the former and strong guarantees of the latter. Specifically, we modify the coefficients of the ARE in such a way that the cloud computation on the modified ARE returns the same solution as the original one, which can be then readily used for control purposes. Notably, the approach obviates the need for any algebraic decoding step. We present privacy-preserving algorithms with and without a realizability re-quirement, which asks for preserving sign-definiteness of certain ARE coefficients in the modified ARE. For the LQR problem, this amounts to ensuring that the modified ARE coefficients can be realized again as an LQR problem for a (dummy) linear system. The algorithm and its computational load is illustrated through a numerical example.