Stabilisation in an infinite time horizon by homomorphically encrypted first-order controllers with regular Pisot poles

Mohammad Saleh Tavazoei

International Journal of Systems Science, 2025, vol. 56, issue 10, 2221-2230

Abstract: It has been previously found that partially homomorphic encryption based controllers cannot operate in an infinite time horizon, unless all elements of their state matrices are integer numbers. Considering this fact and the results of a recent research revealing the role of Pisot numbers to act as the poles of integer state matrix based state space models involving no unstable pole-zero cancellation, the present paper aims to introduce some classes of realisations implementing first-order discrete-time controllers, which can be used in an infinite time horizon in encrypted control systems. These classes of controller realisations are obtained based on sequences approaching to limit points of positive Pisot numbers settled in the range $ (1,2) $ (1,2). An analytical procedure is also suggested to find the exact range of the controller gain for ensuring the closed-loop stability.

Date: 2025
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DOI: 10.1080/00207721.2024.2441454

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