Conversion of Controllers to have Integer State Matrix for Encrypted Control: Non-Minimal Order Approach

To implement an encrypted dynamic controller based on homomorphic encryption that operates for an infinite time horizon, it is essential for every component of the controller's state matrix to be an integer. In this paper, we tackle the challenge of converting a pre-designed controller into a new one with an integer state matrix, while preserving its control performance. This enables encrypted dynamic systems to be realized without re-encryption and approximation of control parameters. To achieve this, we allow the order of the controller to be increased so that the resulting closed-loop system becomes a non-minimal realization of the original closed-loop, without losing internal stability. Two approaches are proposed to design such controller with an integer state matrix. The first approach is to design the new controller as an estimator of the original closed-loop system, and the conditions on the estimator gain are derived. Our second approach is to formulate a problem of finding certain polynomials, whose solution leads to the design of the new controller. In a special case when the numerator of the plant transfer function is a constant, we provide a constructive method to obtain such solution.