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

TL;DR

This paper develops a novel encrypted control method for asymptotic tracking of dynamic references, addressing overflow issues under finite modulus by integer transformation and control input restoration techniques.

Contribution

It introduces a new controller design with integer coefficients and algorithms for control input recovery under finite modulus in encrypted control systems.

Findings

Successfully addresses overflow issues in encrypted control with finite modulus.

Provides a control input restoration algorithm for encrypted dynamic control.

Validates the approach through simulation examples.

Abstract

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 Cayley-Hamilton theorem, we represent the control input as linear combination of the previous control inputs. 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. As an extension of the first result, we further solve the problem of unbounded internal state taking place in the actuator. In particular, the actuator can restore the correct control input under the same modulus. A simulation example is provided to verify the control schemes proposed in our paper.