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Design of Differentially Private Dynamic Controllers

As a quantitative criterion for privacy of "mechanisms" in the form of data-generating processes, the concept of differential privacy was first proposed in computer science and has later been applied to linear dynamical systems. However, differential privacy has not been studied in depth together with other properties of dynamical systems, and it has not been fully utilized for controller design. In this paper, first we clarify that a classical concept in systems and control, input observability (sometimes referred to as left invertibility) has a strong connection with differential privacy. In particular, we extend the concept of the Gramian to input observability and then show that differential privacy evaluates the maximum eigenvalue of the input observability Gramian. Next, enabled by our new insight into differential privacy, we develop a method to design dynamic controllers for a tracking problem while making the output information of the controlled plant differential private. We call the designed controller as such the \emph{differentially private controller}. The usage of such controllers is further illustrated by solving a diabetes control problem where privacy issues are of concern.