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

TL;DR

This paper provides 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 and designs an algorithm on the actuator side such that it can restore the control input from the lower bits under a finite modulus.

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.

Figures (4)

• Figure 1: Encrypted control architecture. Dashed lines represent networks. The sensor and reference provider transmit $\textbf{Enc}( Q(\bar{y}(k) \,\,\text{mod}\,\,q )$ and $\textbf{Enc}\left( Q\left( \frac{S}{s} \tilde{v}_p(k) - \frac{S}{s} \tilde{v}(k)\right)\text{mod}\,q \right)$ to the encrypted controller over networks, respectively. The encrypted controller sends $\mathbf{\bar{u}}(k)$ to the actuator over networks. The actuator computes $\bar{u}_a(k)$ based on $\bar{u}_a(k-j)$ ($j=1, \cdots, v$), and further computes $u_a(k)$. It feeds $u_a(k)$ to the process and stores $\bar{u}_a(k)$ in the memory for being utilized at $k+1$.
• Figure 2: Encrypted control architecture. Dashed lines represent networks. The sensor and reference provider send the same messages as in Fig. \ref{['Control archi 1']}. The encrypted controller generates $\mathbf{\bar{u}}(k)$ and $\mathbf{m}(k)$. It transmits $\mathbf{m}(k)$ to the actuator over networks and stores $\mathbf{\bar{u}}(k)$ in the memory for being utilized at $k+1$. The actuator computes $m_a(k)$, and $u_a(k)$ based on $u_a(k-j)$ ($j= 1, \cdots, v$). $u_a(k)$ is then fed to the process and stored in the memory for being utilized at $k+1$.
• Figure 3: Time responses of tracking errors $y_p(k)-v_p(k)$
• Figure 4: Time responses of $\| \bar{u}(k) +C_v \bar{U}_a(k-1)\|_\infty$.

Theorems & Definitions (9)

• Lemma 1
• Remark 1
• Remark 2
• Lemma 2
• Theorem 1
• Remark 3
• Remark 4
• Proposition 1
• Remark 5