Asymptotic tracking control of dynamic reference over homomorphically encrypted data with finite modulus

This paper considers a tracking control problem, in which the dynamic controller is encrypted with an additively homomorphic encryption scheme and the output of a process tracks a dynamic reference asymptotically. Our paper is motivated by the following problem: When dealing with both asymptotic tracking and dynamic reference, we find that the control input is generally subject to overflow issues under a finite modulus, though the dynamic controller consists of only integer coefficients. First, we provide a new controller design method such that the coefficients of the tracking controller can be transformed into integers leveraging the zooming-in factor of dynamic quantization. By the internal model principle on the actuator side, we present the control input as a linear combination of the previous control inputs, in which the information of the reference model is utilized. Leveraging the property above, we design an algorithm on the actuator side such that it can restore the control input from the lower bits under a finite modulus. A lower bound of the modulus is also provided. In the second part of the paper, we utilize a finite-range quantizer to design the encrypted controller. A lower bound of quantization range is provided, which can ensure that the quantizer is free of saturation. At last, we propose an innovation based encrypted control architecture, which requires the same modulus.