(untitled) (The International Journal for Scientific Research, 2026)
This paper addresses the challenges of implementing linear dynamic controllers on encrypted data using homomorphic encryption, focusing on the problem of scaling factor accumulation that leads to performance instability when operating systems with non-integer matrices over long time horizons. The paper demonstrates that integer matrices represent a fundamental solution to this problem, allowing signal scale to remain fixed throughout the operation without the need for re-encryption or resetting. The paper also presents two tests for analyzing the stability of nonlinear systems at equilibrium points: the first determines global stability based on the sign of determinants, while the second relies on Gershgorin criteria for local stability. The tests are applied to a dynamic system model of the pantograph-catenary system, identifying stability and instability regions in the parameter plane, and studying the influence of pantograph speed on system stability. Numerical results demonstrate the effectiveness of the proposed mathematical models in analyzing the behavior of complex physical systems.