#  System Design Approach for Control of Differentially Private
Dynamical Systems

Raman Goyal, Dhrubajit Chowdhury, and Shantanu Rane R. Goyal, D. Chowdhury,
and S. Rane are with Palo Alto Research Center - Part of SRI International,
Palo Alto, CA, USA. {raman.goyal, dhruba.chowdhury<EMAIL_ADDRESS>
###### Abstract

This paper introduces a novel approach to concurrently design dynamic
controllers and correlated differential privacy noise in dynamic control
systems. An increase in privacy noise increases the system’s privacy but
adversely affects the system’s performance. Our approach optimizes the noise
distribution while shaping closed-loop system dynamics such that the privacy
noise has the least impact on system performance and the most effect on system
privacy. We further add privacy noise to both control input and system output
to privatize the system’s state for an adversary with access to both
communication channels and direct output measurements. The study also suggests
tailored privacy bounds for different states, providing a comprehensive
framework for jointly optimizing system performance and privacy in the context
of differential privacy.

##  I Introduction

In today’s increasingly interconnected and data-driven world, it has become
important for connected entities to share information with each other to work
efficiently. This applies not just to individuals, but also to Cyber-Physical
Systems (CPS) in various sectors, including industrial control systems, power
grids, financial markets, and commercial and military communication networks.
This pervasive data sharing has also brought forth heightened concerns
regarding system security, safety and privacy. With private entities,
government organizations, and adversaries engaging in extensive data
collection and analysis, the risk of exposing sensitive information has
significantly increased, posing potential harm to both individuals and
critical systems. This has led to the development of mechanisms providing
different kinds of privacy guarantees. Examples include differential privacy,
information-theoretic privacy, and privacy based on secure multiparty
computation.

Differential privacy was originally designed to protect the data of
individuals in static databases, but its application has expanded to address
the privacy challenges posed by dynamic and interconnected data ecosystems,
such as Cyber-Physical Systems (CPS) and Internet of Things (IoT) devices [1,
2]. At a basic level, a differentially private mechanism ensures that the
results of a query remain approximately unchanged if data belonging to any
single user, or a single row, in the database are modified [3]. Informally,
differential privacy makes similar data appear approximately indistinguishable
from one another [4]. The most important feature of differential privacy is
its protection from post-processing or its robustness in the presence of side
information [5]. However, there is a price associated with making the system
differentially private. Differential privacy works by adding noise to the
system which leads to a degradation in system performance both in static and
dynamic cases [5, 6].

In recent years, researchers have extended the work on differential privacy
for static databases to differential privacy for dynamic filters [6], control
and dynamical systems [3], differentially private LQ Control [4], multi-agent
formation control [7], and Differentially private distributed constrained
optimization [8]. In differentially private LQ control [4], the authors
consider a multi-agent system described using linear system dynamics and add
privacy noise such that every agent’s state trajectory is made approximately
indistinguishable from all other state trajectories. The paper provides lower
and upper bounds on mean square error (MSE) in state estimation for some
minimum and maximum privacy noise among agents, where the combined state is
estimated using a standard Kalman filter while designing an LQG control for
the overall system. The paper further provides guidelines for choosing the
privacy level $\epsilon_{i}$ to bound the MSE in the cloud’s state estimates
and further provides the cost of privacy in terms of the increase of the
quadratic cost. Kawan and Cao [9, 10] show that the Gaussian mechanism
evaluates the maximum eigenvalue of the input observability Gramian and thus
the addition of even small noise is enough to make the less input observable
Gaussian mechanism highly differentially private.

In this paper, we consider the joint design of dynamic controller and
differentially private noise (correlated noise with different variances across
channels) such that the system performance loss is minimized for a given
privacy metric or the privacy metric is maximized for a given system
performance. The idea is to find the optimal privacy noise distribution and
simultaneously design the closed-loop system dynamics such that the correlated
noises enter the system through channels that have the least impact on system
performance and maximize system privacy. It can be understood as the
following: the larger noise would only be added through the input/output
channels whose effect on the system state has been minimized by designing the
closed-loop poles. We design a dynamic controller that directly processes the
controller states to generate the control signal instead of first estimating
the states and then using it to obtain the control input. We assume a smart
adversary that will develop an optimal estimator to generate individual
signals for more accurate state estimation by leveraging the additional
information about the system dynamics. We consider two kinds of adversaries,
one that has access to the communication channel and another that has direct
access to the measurements. We express a privacy metric in the presence of
smart adversaries in terms of uncertainty in the estimation of individual
states. We will further design different privacy bounds in different states as
some states might need stricter privacy guarantees than other states. This
will also be considered for bounding the system performance across different
states based on the underlying system.

The organization of the paper can be laid out as follows: Section §II provides
the necessary background on differential privacy and the notation required for
the rest of the paper. Section §III gives the relationship between
differential privacy and error in state estimate and then formulates the final
design problem. Section §IV elaborates the system design approach for a
general dynamic controller and correlated differential privacy input and
output noises and provides the solution as a convex optimization problem.
Section §V gives simulation results for differential privacy of a networked
power distribution system with load frequency control under unknown power
demand and §VI provides the final concluding remarks along with the future
work.

##  II Review of Differential Privacy

In this section, we review the basic definitions and define the Gaussian
mechanism used to enforce differential privacy in dynamical systems. We define
the expectation operator by $\mathbb{E}[\cdot]$ and $\mathcal{N}({x},{Y})$
denotes the Gaussian distribution with mean ${x}$ and covariance ${Y}$. The
diagonal matrix generated from a vector ${x}$ is denoted as
$\operatorname{diag}({x})$ and the block diagonal matrix is denoted as
$\operatorname{blkdiag}({Y}_{1},{Y}_{2},\cdots,{Y}_{3})$. The symbol $O$
defines a zero matrix with suitable dimensions and $I$ defines the unit matrix
of appropriate dimensions. The notations ${X\succ 0}$ and ${Y\succeq 0}$
denote the symmetric positive definite and symmetric positive semidefinite
matrices, respectively.

Let us consider agent’s state trajectories of the form $x=(x(1),x(2),\ldots)$,
where $x(k)\in\mathbb{R}^{n_{x}}$ and $\|x(k)\|_{2}<\infty$ for all $k$, and
let us denote the set of all such sequences by $x\in{\ell}_{2}^{n_{x}}$. Let
us define our adjacency relation over ${\ell}_{2}^{n_{x}}$.

######  Definition 1.

(Adjacency for trajectories): Let us choose $\beta>0$ as the adjacency
parameter and $v,w\in{\ell}_{2}^{n_{x}}$ as two trajectories that are adjacent
if $\left\|v-w\right\|_{\ell_{2}}\leq\beta$. We write
$\operatorname{Adj}_{\beta}\left(v,w\right)=1$ if $v$ and $w$ are adjacent,
and $\operatorname{Adj}_{\beta}\left(v,w\right)=0$, otherwise.

This adjacency relation requires that an agent’s state trajectory be made
approximately indistinguishable within distance $\beta$ from all other state
trajectories. Let us consider that the agent’s output signal is of dimension
$n_{y}$ at each point in time and is in the set ${\ell}_{2}^{n_{y}}$. Next, we
define the sensitivity of a dynamical system.

######  Definition 2.

(Sensitivity): The p-norm sensitivity of a system $\mathcal{G}$ is the
greatest distance between two output trajectories that correspond to adjacent
state trajectories:

$\displaystyle\Delta_{p}\mathcal{G}:=\sup_{x,\tilde{x}|\operatorname{Adj}_{B}(x,\tilde{x})=1}\|\mathcal{G}(x)-\mathcal{G}(\tilde{x})\|_{p}.$

Now, we define differential privacy for dynamic systems (see [6] for a formal
construction).

######  Definition 3.

(Differential privacy for trajectories): Let $\epsilon>0$ and
$\delta\in(0,1/2)$ be given. A mechanism
$\mathcal{M}(\cdot)\in\ell_{2}^{n_{y}}$ is
$\left(\epsilon,\delta\right)$-differentially private if, for all adjacent
$x,x^{\prime}\in\ell_{2}^{n_{x}}$, we have:

$\displaystyle\mathbb{P}\left[\mathcal{M}\left(x\right)\in S\right]\leq
e^{\epsilon}\mathbb{P}\left[\mathcal{M}\left(x^{\prime}\right)\in
S\right]+\delta\text{ for all }S\in\Sigma_{2}^{n_{y}}.$

We now define the Gaussian mechanism.

######  Lemma 1.

(Gaussian mechanism; [6]): Let us use privacy parameters $\epsilon>0$ and
$\delta\in(0,1/2)$ and adjacency parameter $\beta>0$. Let $\mathcal{G}$ denote
a dynamical system and $\Delta_{2}\mathcal{G}$ denote its 2-norm sensitivity.
Then the Gaussian mechanism $\mathcal{M}=\mathcal{G}(x)+v^{p}$ makes the
system $\left(\epsilon,\delta\right)$-differentially private with respect to
$\operatorname{Adj}_{\beta}$, if
$v^{p}(k)\sim\mathcal{N}\left(0,\sigma^{2}I_{n_{y}}\right)$, and
$\sigma\geq\Delta_{2}\mathcal{G}~{}\beta~{}\kappa\left(\delta,\epsilon\right)$,
where
$\kappa\left(\delta,\epsilon\right)=\frac{1}{2\epsilon}\left(K_{\delta}+\sqrt{K_{\delta}^{2}+2\epsilon}\right)$,
with $K_{\delta}:=\mathcal{Q}^{-1}\left(\delta\right)$, $\mathcal{Q}$
representing the Gaussian tail integral.

##  III Problem Formulation

The main objective of the research is to make the state of the agents
differentially private by adding privacy noise while achieving the desired
system performance. The privacy noise can be added to (refer fig. 1):

  * •

System output: Add differential privacy noise to the outputs measured by the
sensors as $v^{p}_{k}$.

  * •

Control input: Add differential privacy noise directly to the control input as
$w^{p}_{k}$.

Figure 1: Design architecture for making agent’s state differentially private
by adding privacy noise to both system inputs and output.

Notice that the control input noise is a physically feasible way to insert
privacy noise into the system. Also, the actual privacy noise should be
calculated by accounting for the actuator noise present in the system. We
further consider two cases based on the capability of the adversary (refer
fig. 1):

  * •

In the first case, the adversary listens to the communication between the
agents and the centralized controller ((1) $y^{a}_{k}=\bar{y}_{k}$). We add
privacy noise to both the outputs measured by the sensor and the control
inputs to make the system state differentially private.

  * •

In the second case, the adversary has his own sensors and has direct access to
the output of the system ((2) $y^{a}_{k}=y_{k}$), and thus adding output
privacy noise alone would not make the system differentially private. Although
there is no benefit in adding privacy noise to the sensor side, we still add
it and expect the design to remove the privacy noise on the output side.

Let us consider a discrete-time linear time-invariant (LTI) system, along with
the addition of output privacy noise and control input privacy noise,
described by the following state-space representation:

$\displaystyle x_{k+1}$ $\displaystyle=Ax_{k}+B(u_{k}+w^{p}_{k})+Dw_{k},$ (1)
$\displaystyle y_{k}$ $\displaystyle=Cx_{k}+v_{k},$ (2)
$\displaystyle\bar{y}_{k}$ $\displaystyle=Cx_{k}+v_{k}+v^{p}_{k},$ (3)
$\displaystyle z_{k}$ $\displaystyle=C_{z}x_{k},$ (4)

where $x_{k}\in\mathbb{R}^{n_{x}}$ is the state of the system at time $k$,
$u_{k}\in\mathbb{R}^{n_{u}}$ is the control vector at time $k$. The initial
state vector $x_{0}$ and the process noise at time $k$, $w_{k}$, are assumed
to be independent random variables. In particular,
$w_{k}\sim\mathcal{N}(\mathbf{0},W)$, $\forall k$, with
$W\in\mathbb{R}^{n_{w}\times n_{w}}$ to be known and fixed covariance matrix.
The output of the system $y_{k}\in\mathbb{R}^{n_{y}}$, is measured by a sensor
network with sensor noise modeled as another independent Gaussian random
variable $v_{k}\sim\mathcal{N}(\mathbf{0},V)$, $\forall k$ with
$V\in\mathbb{R}^{n_{v}\times n_{v}}$ to be the known and fixed covariance
matrix. The performance variable $z_{k}\in\mathbb{R}^{n_{z}}$ defines the
variables of interest for the system.

The control input privacy noise, $w^{p}_{k}$, and output privacy noise,
$v^{p}_{k}$, are modeled as random variables,
$w^{p}_{k}\sim\mathcal{N}(\mathbf{0},W^{p})$ and
$v^{p}_{k}\sim\mathcal{N}(\mathbf{0},V^{p})$, $\forall k$, with $W^{p}$ and
$V^{p}$ being the covariance matrix, representing the strength of the added
noise. We further define the inverse of the respective noise covariance
matrices as:

$\Gamma_{w}=W^{p^{-1}},~{}~{}\Gamma_{v}=V^{p^{-1}}.$

We assume that the adversary is smart and has full information about the
system dynamics, i.e., knowledge of system matrices, $A,B,C,D$. Moreover, the
adversary will design an optimal estimator to estimate the system state
$\hat{x}_{k}$ using a general estimator of the form:

$\displaystyle\hat{x}_{k+1}$
$\displaystyle=\hat{A}\hat{x}_{k}+Bu_{k}+\hat{B}y^{a}_{k},$ (5) $\displaystyle
e_{k}$ $\displaystyle=x_{k}-\hat{x}_{k},$ (6)

such that the error in the state estimate $e_{k}$ is minimized.

###  III-A Relationship between differential privacy and error in state
estimates due to control input and output privacy noise

In this subsection, we show that the differential privacy of the system state
can be represented by the error in the estimation of the state while using the
optimal state estimator by the adversary of the form eqs. 5 and 6. In
particular, the covariance of the state error estimates
$E_{k}=\mathbb{E}[e_{k}e_{k}^{T}]$ can be used as a metric to quantify
differential privacy, and an increase in error covariance results in an
increase in $\left(\epsilon,\delta\right)$-differentially privacy.

######  Lemma 2.

(Gaussian mechanism for dynamical system; [6]): Let $\mathcal{G}$ denote an
LTI dynamical system and $\|\mathcal{G}\|_{\infty}<\infty$ and let us use
privacy parameters $\epsilon,\delta>0$. Then the Gaussian mechanism
$\mathcal{M}u=\mathcal{G}u+w^{p}$, where $w^{p}$ is a Gaussian noise with
$w^{p}\sim\mathcal{N}\left(0,\sigma^{2}I_{n_{y}}\right)$, and
$\sigma\geq\beta~{}\kappa\left(\delta,\epsilon\right)\|\mathcal{G}\|_{\infty}$,
makes the system $\left(\epsilon,\delta\right)$-differentially private with
respect to $\operatorname{Adj}_{\beta}$ in $u$ , i.e.,
$\|u-u^{\prime}\|_{2}\leq\beta$ with $\beta>0$.

######  Remark 1.

The above lemma allows us to make the control input differential private by
directly adding the noise to the control inputs when the output is queried and
obtained by passing through an LTI dynamical system. In this paper, we want to
make the system state differentially private by adding privacy noise to both
system inputs and outputs.

Yazdani et. al. [4] used the level of privacy to calculate the impact on
estimation error and showed the relationship between the privacy noise and the
trace of covariance of the state error estimates $\operatorname{tr}(E_{k})$
where the state estimates are calculated using a Kalman filter. However,
another way to look at the impact of differential privacy from the point of
view of an adversary is to hinder his capability to estimate the state
trajectories accurately. So if an adversary designs an optimal estimator, the
error in estimating state trajectories can be used as a metric of differential
privacy. Next, we expand on the results generated in [4] to quantify standard
($\epsilon,\delta$)-differential privacy as the error in adversary’s state
estimates due to both control input and output privacy noise.

######  Lemma 3.

For the given dynamical systems in (Eq. (1)-(3)) with both control input
privacy noise $w^{p}_{k}\sim\mathcal{N}(\mathbf{0},W^{p})$ and output privacy
noise $v^{p}_{k}\sim\mathcal{N}(\mathbf{0},\sigma^{2}I_{n_{y}}-V)$, with
$\sigma=\bar{S}(C)\beta k(\delta,\epsilon)$, and for a given adjacency
$\|x-x^{\prime}\|_{2}\leq\beta$ with $\beta>0$, if the states are
($\epsilon,\delta$)-differential private with $\delta\in[10^{-5},10^{-1}]$ and

$\displaystyle\epsilon$
$\displaystyle\leq\left(\frac{\bar{S}(C)^{2}\beta^{2}(n_{x}-\operatorname{tr}\left(\underline{E})\underline{\lambda}(\Psi)^{-1}\right)}{\operatorname{tr}(\underline{E})C_{u}^{2}}\right)^{1/2},$
$\displaystyle\Psi$ $\displaystyle=DWD^{T}+BW^{p}B^{T},$ $\displaystyle V^{p}$
$\displaystyle=\sigma^{2}I_{n_{y}}-V$

where $\bar{S}(\cdot)$ represents the maximum singular value,
$\underline{\lambda}(\cdot)$ represents the smallest eigenvalue of the matrix,
and $C_{u}$ is the value of $C$ corresponding to the index for which the
diagonal element of $C^{T}(V+V^{p})^{-1}C)$ is maximum, then the state error
estimate is lower bounded by $\operatorname{tr}(\underline{E})$ with
$\mathbb{E}[e_{k}e_{k}^{T}]>\underline{{E}}$.

###### Proof.

Here we consider both control input privacy noise and output privacy noise
along with the already present process and measurement noise. Thus the
equation for a priori state error covariance follows:

$\Sigma=A(\Sigma^{-1}+C^{T}(V+V^{p})^{-1}C)^{-1}A^{T}+DWD^{T}+BW^{p}W^{T},$

and for a posteriori state error covariance follows:

$\overline{\Sigma}=(\Sigma^{-1}+C^{T}(V+V^{p})^{-1}C)^{-1}.$

After that, it follows directly from (Theorem 2 of [4]) where we consider a
single agent with state dimension $n_{x}$ instead of the multi-agent case. ∎

The above result is used to show that enforcing differential privacy to the
systems’ state ensures a lower bound on state estimation error. Please note
that the above result provides a necessary condition for differential privacy
based on the lower bound on state estimation error. More work is needed to
find the bounds for sufficiency.

######  Remark 2.

Please note that both the performance norm $\mathbb{E}[z_{k}z_{k}^{T}]$ and
the estimator error covariance $\mathbb{E}[e_{k}e_{k}^{T}]$ increases with an
increase in output and control input privacy noises, and our objective is to
find the optimum noise level along with the controller and estimator to
minimize the performance norm for a given error covariance.

Main Design Problem Formulation: Design the strength of privacy noises,
$W^{p}(\Gamma_{w})$ and $V^{p}(\Gamma_{v})$, and an optimal state estimator of
the form eqs. 5 and 6, and a general linear dynamic controller of the form:

$\displaystyle x^{c}_{k+1}$
$\displaystyle=A_{c}\,x^{c}_{k}+B_{c}\,\bar{y}_{k},$ (7) $\displaystyle u_{k}$
$\displaystyle=C_{c}\,x^{c}_{k}+D_{c}\,\bar{y}_{k}.$ (8)

such that the state error covariance $\mathbb{E}[e_{k}e_{k}^{T}]$ is maximized
while closed-loop system performance is bounded
$\mathbb{E}[z_{k}z_{k}^{T}]\leq\bar{\mathbf{Z}}$.

$\displaystyle\begin{array}[]{lll}\max_{\\{A_{c},B_{c},C_{c},D_{c},\Gamma_{w},\Gamma_{v},\hat{A},\hat{B}\\}}&\operatorname{tr}(\mathbb{E}[e_{k}e_{k}^{T}])\\\
\textit{s.t.}&\mathbb{E}[z_{k}z_{k}^{T}]\leq\bar{\mathbf{Z}}.\end{array}$ (11)

Another problem of interest can be to minimize the closed-loop system
performance $\mathbb{E}[z_{k}z_{k}^{T}]$ while lower bounding the state error
covariance $\mathbb{E}[e_{k}e_{k}^{T}]>\underline{{E}}$ for some given
$\underline{{E}}$, i.e., to have higher differential privacy than some
specified limit.

$\displaystyle\begin{array}[]{lll}\min_{\\{A_{c},B_{c},C_{c},D_{c},\Gamma_{w},\Gamma_{v},\hat{A},\hat{B}\\}}&\operatorname{tr}(\mathbb{E}[z_{k}z_{k}^{T}])\\\
\textit{s.t.}&\mathbb{E}[e_{k}e_{k}^{T}]\geq\mathbf{\underline{E}}.\end{array}$
(14)

For the two cases that we discussed based on the capabilities of the
adversary, the information available to the estimator would change from (1)
$y^{a}_{k}=\bar{y}_{k}$ to (2) $y^{a}_{k}=y_{k}$.

######  Remark 3.

Notice that the estimator design from the point of view of the adversary is
general and can be used to simultaneously design the estimator with the
privacy noise for the case of open loop system dynamics also, i.e. with
$u_{k}=0$.

##  IV Final Design Solution Development

In this section, we develop frameworks for the co-design of input and output
privacy noise with a dynamic feedback controller; and the co-design of input
and output privacy noise with an optimal estimator. We further provide the
final design algorithm for the two cases of adversarial capabilities. In both
cases, we formulate the problem such that the controller gets the output
signal with added privacy noise $v^{p}$ and let the optimization solve the
optimal privacy noise.

###  IV-A Adversary with access to communication channels

For the case where the adversary listens to the noisy output passed through
the communication channel $y^{a}_{k}=\bar{y}_{k}=Cx_{k}+v_{k}+v^{p}_{k}$, the
final design problem can be solved using the following result.

######  Theorem 1.

For the dynamical system given in eqs. 1 and 3 with adversary listening
through the communication channel, and to maximize differential privacy for a
fixed performance bound, the optimal design solution with privacy noises,
$W^{p}(\Gamma_{w})$ and $V^{p}(\Gamma_{v})$, an optimal state estimator of the
form eqs. 5 and 6, and a general linear dynamic controller of the form eqs. 7
and 8, can be solved as a convex optimization problem using the following
LMIs:

$\displaystyle\textit{minimize}_{\\{A_{c},B_{c},C_{c},\Gamma_{w},\Gamma_{v},\hat{A},\hat{B}\\}}~{}\operatorname{trace}\left(\Gamma_{w}+\Gamma_{v}\right),$
$\begin{bmatrix}X&I&AX+BL&A&D&B&O&O\\\ (\cdot)^{T}&Y&Q&YA+FC&YD&YB&F&F\\\
(\cdot)^{T}&(\cdot)^{T}&X&I&O&O&O&O\\\
(\cdot)^{T}&(\cdot)^{T}&(\cdot)^{T}&Y&O&O&O&O\\\
(\cdot)^{T}&(\cdot)^{T}&(\cdot)^{T}&(\cdot)^{T}&W^{-1}&O&O&O\\\
(\cdot)^{T}&(\cdot)^{T}&(\cdot)^{T}&(\cdot)^{T}&(\cdot)^{T}&\Gamma_{w}&O&O\\\
(\cdot)^{T}&(\cdot)^{T}&(\cdot)^{T}&(\cdot)^{T}&(\cdot)^{T}&(\cdot)^{T}&V^{-1}&O\\\
(\cdot)^{T}&(\cdot)^{T}&(\cdot)^{T}&(\cdot)^{T}&(\cdot)^{T}&(\cdot)^{T}&(\cdot)^{T}&\Gamma_{v}\end{bmatrix}>O,$
(15) $\begin{bmatrix}\bar{\mathbf{Z}}&C_{z}X&C_{z}\\\ (\cdot)^{T}&X&I\\\
(\cdot)^{T}&(\cdot)^{T}&Y\end{bmatrix}>O,$ (16)
$\begin{bmatrix}\hat{X}&I&A\hat{X}&A&D&B&O&O\\\
(\cdot)^{T}&\hat{Y}&\hat{Q}&\hat{Y}A+\hat{F}C&\hat{Y}D&\hat{Y}B&\hat{F}&\hat{F}\\\
(\cdot)^{T}&(\cdot)^{T}&\hat{X}&I&O&O&O&O\\\
(\cdot)^{T}&(\cdot)^{T}&(\cdot)^{T}&\hat{Y}&O&O&O&O\\\
(\cdot)^{T}&(\cdot)^{T}&(\cdot)^{T}&(\cdot)^{T}&W^{-1}&O&O&O\\\
(\cdot)^{T}&(\cdot)^{T}&(\cdot)^{T}&(\cdot)^{T}&(\cdot)^{T}&\Gamma_{w}&O&O\\\
(\cdot)^{T}&(\cdot)^{T}&(\cdot)^{T}&(\cdot)^{T}&(\cdot)^{T}&(\cdot)^{T}&V^{-1}&O\\\
(\cdot)^{T}&(\cdot)^{T}&(\cdot)^{T}&(\cdot)^{T}&(\cdot)^{T}&(\cdot)^{T}&(\cdot)^{T}&\Gamma_{v}\end{bmatrix}>O,$
(17) $\begin{bmatrix}\bar{\mathbf{E}}&\hat{X}-\hat{U}&I\\\
(\cdot)^{T}&\hat{X}&I\\\ (\cdot)^{T}&(\cdot)^{T}&\hat{Y}\end{bmatrix}>O,$ (18)

and finally, the optimal estimator and dynamic controller can be calculated
as:

$\displaystyle\hat{A}$
$\displaystyle=\hat{S}^{-1}(\hat{Q}-\hat{Y}A\hat{X}-\hat{F}C\hat{X})\hat{U}^{-1},$
(19) $\displaystyle\hat{B}$ $\displaystyle=\hat{S}^{-1}\hat{F},$ (20)
$\begin{split}\begin{bmatrix}A_{c}&B_{c}\\\
C_{c}&D_{c}\end{bmatrix}=&\begin{bmatrix}S^{-1}&-S^{-1}YB\\\
O&I\end{bmatrix}\begin{bmatrix}Q-YAX&F\\\ L&O\end{bmatrix}\\\
&\cdot\begin{bmatrix}U^{-1}&O\\\ -CXU^{-1}&I\end{bmatrix}.\end{split}$ (21)

###### Proof.

We first design the dynamic controller of the form eqs. 7 and 8. We assume the
direct feedforward term in the dynamic controller to be zero $D_{c}=O$ for the
bounded control input covariance. Using the above compensator, the closed-loop
system dynamics can be written using the augmented state vector
$\mathbf{x}^{T}:=\begin{bmatrix}x^{T}&{x^{c}}^{T}\end{bmatrix}$ with augmented
process noise
$\mathbf{w}^{T}:=\begin{bmatrix}w^{T}&w^{p^{T}}&v^{T}&v^{p^{T}}\end{bmatrix}$
as:

$\displaystyle\mathbf{x}_{k+1}$
$\displaystyle=\mathbf{A}\,\mathbf{x}_{k}+\mathbf{B}\,\mathbf{w}_{k},$ (22)
$\displaystyle{z}_{k}$ $\displaystyle=\mathbf{C}\,\mathbf{x}_{k},$ (23)
$\displaystyle\text{where}~{}~{}\mathbf{A}=\begin{bmatrix}A&BC_{c}\\\
B_{c}C&A_{c}\end{bmatrix},~{}\mathbf{B}=\begin{bmatrix}D&B&O&O\\\
O&O&B_{c}&B_{c}\end{bmatrix},$

$\mathbf{C}=\begin{bmatrix}C_{z}&O\end{bmatrix}$ and
$\mathbf{w}_{k}\sim\mathcal{N}(\mathbf{0},\mathbf{W})$, where

$\displaystyle\mathbf{W}^{-1}=\begin{bmatrix}W^{-1}&O&O&O\\\
O&\Gamma_{w}&O&O\\\ O&O&V^{-1}&O\\\ O&O&O&\Gamma_{v}\end{bmatrix}.$ (24)

It is a standard result that the above closed loop system is stable and a
steady-state state covariance matrix ($\mathbf{X}>0$) exists, if:

$\mathbf{A}\mathbf{X}\mathbf{A}^{T}+\mathbf{B}\mathbf{W}\mathbf{B}^{T}<\mathbf{X},$
(25)

which using Schur’s complement gives:

$\begin{bmatrix}\mathbf{X}&\mathbf{A}\mathbf{X}&\mathbf{B}\\\
(\cdot)^{T}&\mathbf{X}&O\\\
(\cdot)^{T}&(\cdot)^{T}&\mathbf{W}^{-1}\end{bmatrix}>O,$ (26)

where $(\cdot)^{T}$ represents the corresponding transpose of the symmetric
block. It is straightforward to show that the performance covariance can be
bounded as: $\mathbf{C}\mathbf{X}\mathbf{C}^{T}<\bar{\mathbf{Z}},$ which can
be written as:

$\begin{bmatrix}\bar{\mathbf{Z}}&\mathbf{C}\mathbf{X}\\\
(\cdot)^{T}&\mathbf{X}\end{bmatrix}>O.$ (27)

Notice that the constraint in Eqn. (26) is not an LMI. We need to perform
congruence transformation and change of variables to convert them to LMIs [11,
12]. Let us define and partition the matrix as:

$\mathbf{X}\triangleq\begin{bmatrix}X&U^{T}\\\
U&\hat{X}\end{bmatrix},\quad\mathbf{X}^{-1}\triangleq\begin{bmatrix}Y&S\\\
S^{T}&\hat{Y}\end{bmatrix},$

and the transformation matrix

$\mathbf{T}\triangleq\begin{bmatrix}I&Y\\\ O&S^{T}\end{bmatrix}$

and associated congruence transformation matrices

$\mathbb{T}\triangleq\begin{bmatrix}\mathbf{T}&O&O\\\ O&\mathbf{T}&O\\\
O&O&I\end{bmatrix},\tilde{\mathbb{T}}\triangleq\begin{bmatrix}I&O\\\
O&\mathbf{T}\end{bmatrix}.$

Applying $\\{\mathbb{T},\tilde{\mathbb{T}}\\}$ to Eqn. (26) and Eqn. (27), we
obtain:

$\mathbb{T}^{T}\begin{bmatrix}\mathbf{X}&\mathbf{A}\mathbf{X}&\mathbf{B}\\\
(\cdot)^{T}&\mathbf{X}&O\\\
(\cdot)^{T}&(\cdot)^{T}&\mathbf{W}^{-1}\end{bmatrix}\mathbb{T}>O,$ (28)

$\tilde{\mathbb{T}}^{T}\begin{bmatrix}\bar{\mathbf{Z}}&\mathbf{C}\mathbf{X}\\\
(\cdot)^{T}&\mathbf{X}\end{bmatrix}\tilde{\mathbb{T}}>O.$ (29)

Expansion of eqs. 28 and 29 under an appropriate change of variables leads to
a set of LMIs eqs. 15 and 16 that do not depend on $S$ or $U$. Once the $X,Y$
are obtained, matrices $S$ and $U$ need to be constructed using:

$YX+SU=I,$ (30)

and, a handy choice of $U$ and $S$ satisfying eq. 30 is $S=Y$, then
$U=Y^{-1}-X$. Notice that when the controller has the same order as the plant,
$S$ and $U$ are square and non-singular matrices, in which case the controller
gain matrices can be calculated using eq. 21.

Although the original problem was to maximize the $\mathbb{E}[e_{k}e_{k}^{T}]$
to increase the differential privacy, but an increase in error covariance can
also result from suboptimal estimator gains. However, as we consider a smart
adversary, who would always design an optimal estimator, we update the design
problem to maximize the privacy noises while bounding the error covariance.

Let us design the estimator to bound the error covariance for different states
with the estimator dynamics given as:

$\displaystyle\hat{x}_{k+1}=\hat{A}\hat{x}_{k}+Bu_{k}+\hat{B}y^{a}_{k}.$ (31)

Combining the above estimator and the underlying dynamics given in eqs. 1 and
2, the combined dynamics can be written using the augmented state vector
$\mathbf{\hat{x}}^{T}:=\begin{bmatrix}x^{T}&{\hat{x}}^{T}\end{bmatrix}$ as:

$\displaystyle\mathbf{\hat{x}}_{k+1}$
$\displaystyle=\mathbf{\hat{A}}\,\mathbf{\hat{x}_{k}}+\mathbf{\hat{B}}\,\mathbf{w}_{k},$
(32)

$\displaystyle\text{where}~{}\mathbf{\hat{A}}=\begin{bmatrix}A&O\\\
\hat{B}C&\hat{A}\end{bmatrix},~{}~{}\mathbf{\hat{B}}=\begin{bmatrix}D&B&O&O\\\
O&O&\hat{B}&\hat{B}\end{bmatrix},$ (33)

and the error in estimation can be written as:

$\displaystyle\mathbf{e}_{k}=\mathbf{\hat{C}}\mathbf{\hat{x}}_{k},~{}~{}~{}\mathbf{\hat{C}}=\begin{bmatrix}I&-I\end{bmatrix},$
(34)

Similar to the previous development for the existence of the steady-state
state covariance matrix ($\mathbf{\hat{X}}>0$) and stability of the system, we
write:

$\begin{bmatrix}\mathbf{\hat{X}}&\mathbf{\hat{A}}\mathbf{\hat{X}}&\mathbf{\hat{B}}\\\
(\cdot)^{T}&\mathbf{\hat{X}}&O\\\
(\cdot)^{T}&(\cdot)^{T}&\mathbf{W}^{-1}\end{bmatrix}>O,$ (35)

and the error covariance can be bounded as:

$\begin{bmatrix}\bar{\mathbf{E}}&\mathbf{\hat{C}}\mathbf{\hat{X}}\\\
(\cdot)^{T}&\mathbf{\hat{X}}\end{bmatrix}>O.$ (36)

Again noticing that the constraint in Eqn. (35) is not an LMI, we follow a
similar procedure, by partitioning the state covariance matrix and performing
the congruence transformation to obtain the LMIs given in eqs. 17 and 18. Once
the $\hat{X},\hat{Y},\hat{U},\hat{F},\hat{Q}$ are obtained, matrix $\hat{S}$
can be constructed using:

$\hat{S}=(I-\hat{Y}\hat{X})\hat{U}^{-1},$ (37)

and the estimator matrices can be constructed using eqs. 19 and 20. ∎

###  IV-B Adversary with direct access to measurements

For the case where the adversary uses his own sensors to measure the system
output $y^{a}_{k}=y_{k}=Cx_{k}+v_{k}$, the output privacy noise will not help
in privatizing the system and the final design problem will change as follows.

######  Theorem 2.

For the dynamical system given in eqs. 1 and 3 with an adversary using his own
sensors to measure the system output, and to maximize differential privacy for
a fixed performance bound, the optimal design solution with privacy noises,
$W^{p}(\Gamma_{w})$ and $V^{p}(\Gamma_{v})$, an optimal state estimator (eqs.
5 and 6), and a general linear dynamic controller (eqs. 7 and 8), can be
solved as a convex optimization problem using the following LMIs:

$\displaystyle\textit{minimize}_{\\{A_{c},B_{c},C_{c},\Gamma_{w},\Gamma_{v},\hat{A},\hat{B}\\}}~{}\operatorname{trace}\left(\Gamma_{w}\right),$

$\mathbb{E}[z_{k}z_{k}^{T}]<\bar{\mathbf{Z}}\rightarrow$ (eqs. 15 and 16
(LMIs),

$\begin{bmatrix}\hat{X}&I&A\hat{X}&A&D&B&O\\\
(\cdot)^{T}&\hat{Y}&\hat{Q}&\hat{Y}A+\hat{F}C&\hat{Y}D&\hat{Y}B&\hat{F}\\\
(\cdot)^{T}&(\cdot)^{T}&\hat{X}&I&O&O&O\\\
(\cdot)^{T}&(\cdot)^{T}&(\cdot)^{T}&\hat{Y}&O&O&O\\\
(\cdot)^{T}&(\cdot)^{T}&(\cdot)^{T}&(\cdot)^{T}&W^{-1}&O&O\\\
(\cdot)^{T}&(\cdot)^{T}&(\cdot)^{T}&(\cdot)^{T}&(\cdot)^{T}&\Gamma_{w}&O\\\
(\cdot)^{T}&(\cdot)^{T}&(\cdot)^{T}&(\cdot)^{T}&(\cdot)^{T}&(\cdot)^{T}&V^{-1}\end{bmatrix}>O,$
(38) $\begin{bmatrix}\bar{\mathbf{E}}&\hat{X}-\hat{U}&I\\\
(\cdot)^{T}&\hat{X}&I\\\ (\cdot)^{T}&(\cdot)^{T}&\hat{Y}\end{bmatrix}>O,$ (39)

and finally, the optimal estimator and dynamic controller can be calculated as
eqs. 19 and 20 and eq. 21.

###### Proof.

The design solution for the dynamic controller is the same as the previous
solution. The proof follows similarly to the previous design solution with
$y^{a}_{k}=\bar{y}_{k}$ replaced with $y^{a}_{k}={y}_{k}$ and thus the
derivation for the equation for the estimator follows naturally from eqs. 17
and 18 to the above-mentioned results. ∎

######  Remark 4.

Notice that the design solution presented in Theorem 1 and Theorem 2 is
overparameterized, i.e., can help many different controllers/estimators to
obtain the same result with different realizations of $YX+SU=I$ and that can
further serve as an optimization domain for some other higher level objective
function.

###  IV-C Estimator for unstable systems

The discrete estimator design approach presented earlier is not valid for
unstable systems as eq. 35 requires the system matrix $\mathbf{\hat{A}}$ to be
stable for a valid positive definite solution for $\mathbf{\hat{X}}>O$. Thus,
we now present the results for unstable system dynamics which restricts the
estimator design to:

$\displaystyle\hat{x}_{k+1}=\hat{A}\hat{x}_{k}+Bu_{k}+\hat{B}y^{a}_{k},~{}\text{where}~{}\hat{A}=A-\hat{B}C,$

and thus $\hat{B}$ is the only design variable for the estimator. Notice that
the control input will cancel out in the estimator as the adversary also has
direct access to it. Combining the above estimator and the underlying dynamics
given in eqs. 1 and 2, the error dynamics can be written as:

$\displaystyle
e_{k+1}=(A-\hat{B}C)e_{k}+Dw_{k}+Bw^{p}_{k}-\hat{B}v_{k}-\hat{B}v^{p}_{k}.$

Now for the unstable dynamical system, the system design problem can be solved
using the following results. Notice that the approach can also be used to only
design an estimator and input/output privacy noises for the case of an open-
loop unstable dynamical process.

######  Corollary 1.

For the unstable dynamical system given in eqs. 1 and 3 with an adversary
using his own sensors to measure the output, and to maximize differential
privacy for a fixed performance bound, the optimal design solution with
privacy noises, $W^{p}(\Gamma_{w})$ and $V^{p}(\Gamma_{v})$, an optimal state
estimator of the form eqs. 5 and 6, and a general linear dynamic controller of
the form eqs. 7 and 8, can be solved as a convex optimization problem using
the following LMIs:

$\displaystyle\textit{minimize}_{\\{A_{c},B_{c},C_{c},\Gamma_{w},\Gamma_{v},\hat{B}\\}}~{}\operatorname{trace}\left(\Gamma_{w}+\Gamma_{v}\right),$

$\mathbb{E}[z_{k}z_{k}^{T}]<\bar{\mathbf{Z}}\rightarrow$ (eqs. 15 and 16
(LMIs),

$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\begin{bmatrix}\bar{\mathbf{E}}&I\\\
I&\hat{Y}\end{bmatrix}>O,$ (40)
$\displaystyle\begin{bmatrix}\hat{Y}&\hat{Y}A-\hat{Z}C&\hat{Y}D&\hat{Y}B&\hat{Z}&\hat{Z}\\\
(\cdot)^{T}&\hat{Y}&O&O&O&O\\\ (\cdot)^{T}&(\cdot)^{T}&W^{-1}&O&O&O\\\
(\cdot)^{T}&(\cdot)^{T}&(\cdot)^{T}&\Gamma_{w}&O&O\\\
(\cdot)^{T}&(\cdot)^{T}&(\cdot)^{T}&(\cdot)^{T}&V^{-1}&O\\\
(\cdot)^{T}&(\cdot)^{T}&(\cdot)^{T}&(\cdot)^{T}&(\cdot)^{T}&\Gamma_{v}\end{bmatrix}>O,$
(41)

and finally, the dynamic controller can be calculated as eq. 21 and optimal
estimator as: $\hat{B}=\hat{Y}^{-1}\hat{Z}.$

###### Proof.

We use the standard result that the error dynamics is stable and a steady-
state error covariance matrix (${E}>0$) is bounded, if: $E<\bar{\mathbf{E}}$

$\displaystyle(A-\hat{B}C)E(A-\hat{B}C)^{T}$
$\displaystyle+DWD^{T}+BW^{p}B^{T}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+\hat{B}V\hat{B}^{T}+$
$\displaystyle\hat{B}V^{p}\hat{B}^{T}<E,$

which can be written using Schur’s complement:

$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}E<\bar{\mathbf{E}},$
$\displaystyle\begin{bmatrix}E&A-\hat{B}C&D&B&\hat{B}&\hat{B}\\\
(\cdot)^{T}&E^{-1}&O&O&O&O\\\ (\cdot)^{T}&(\cdot)^{T}&W^{-1}&O&O&O\\\
(\cdot)^{T}&(\cdot)^{T}&(\cdot)^{T}&\Gamma_{w}&O&O\\\
(\cdot)^{T}&(\cdot)^{T}&(\cdot)^{T}&(\cdot)^{T}&V^{-1}&O\\\
(\cdot)^{T}&(\cdot)^{T}&(\cdot)^{T}&(\cdot)^{T}&(\cdot)^{T}&\Gamma_{v}\end{bmatrix}>O.$

Then, we defined $\hat{Y}=E^{-1}$ and multiplied both sides by matrix
$[\operatorname{blkdiag}(\hat{Y},I,I,I)]$ to apply congruence transformation
and finally defined $\hat{Z}=\hat{Y}\hat{B}$ to obtain the LMIs in
$\hat{Y},\hat{Z},\Gamma_{w}$ and $\Gamma_{v}$ as eqs. 40 and 41. ∎

######  Corollary 2.

Similar to Corollary 1, but with an adversary using his own sensors to measure
the system output, the optimal design problem can be solved as a convex
optimization problem using the following LMIs:

$\displaystyle\textit{minimize}_{\\{A_{c},B_{c},C_{c},\Gamma_{w},\Gamma_{v},\hat{B}\\}}~{}\operatorname{trace}\left(\Gamma_{w}\right),$

$\mathbb{E}[z_{k}z_{k}^{T}]<\bar{\mathbf{Z}}\rightarrow$ (eqs. 15 and 16
(LMIs),

$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\begin{bmatrix}\bar{\mathbf{E}}&I\\\
I&\hat{Y}\end{bmatrix}>O,$ (42)
$\displaystyle\begin{bmatrix}\hat{Y}&\hat{Y}A-\hat{Z}C&\hat{Y}D&\hat{Y}B&\hat{Z}\\\
(\cdot)^{T}&\hat{Y}&O&O&O\\\ (\cdot)^{T}&(\cdot)^{T}&W^{-1}&O&O\\\
(\cdot)^{T}&(\cdot)^{T}&(\cdot)^{T}&\Gamma_{w}&O\\\
(\cdot)^{T}&(\cdot)^{T}&(\cdot)^{T}&(\cdot)^{T}&V^{-1}\end{bmatrix}>O.$ (43)

and finally, the dynamic controller can be calculated as eq. 21 and optimal
estimator as: $\hat{B}=\hat{Y}^{-1}\hat{Z}.$

##  V Simulation Results

The Load Frequency Control (LFC) system maintains a balanced power
distribution across different regions by continuously aligning energy demand
with generation. LFC involves the transmission of data from remote areas to a
central control center, and back to the power production facilities. This
communication process in power grids has well-known privacy concerns and thus
becomes the motivation for our example [4, 13]. We illustrate the performance
of the proposed architecture on a connected four-area network which is
obtained from a network-reduced IEEE New England 39-bus system [14]. We
consider a lossless, connected, and network-reduced power system with each
generator modeled by the following equation [14]:

$\displaystyle\dot{\theta}_{i}(t)$ $\displaystyle=\omega_{i}(t),$
$\displaystyle M_{i}\dot{\omega}_{i}(t)$
$\displaystyle=-D_{i}\omega_{i}(t)-\sum_{j=1}^{n}B_{ij}V_{i}V_{j}\sin(\theta_{i}(t)-\theta_{j}(t))$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+P_{t_{i}}(t)+w_{p_{i}}(t),$
$\displaystyle\tau_{t_{i}}\dot{P}_{t_{i}}(t)$
$\displaystyle=-P_{t_{i}}(t)-R_{i}^{-1}\omega_{i}(t)+u_{i}(t),$

where $\theta_{i}(t)$ is the generator rotor angles w.r.t a synchronously
rotating reference axis, $\omega_{i}(t)$ is the frequency deviation w.r.t a
synchronous frequency which is $120\pi$ for a 60 Hz system, $M_{i}$ represents
the inertia, $D_{i}$ represents the damping matrix, $w_{p_{i}}$ represents the
unknown power demand modeled as disturbance, $R_{i}$ represents the frequency-
droop, and $P_{t_{i}}(t)$ and $\tau_{t_{i}}$ are the turbine power and time
constants, respectively [14]. We linearize the generator model and define the
state with four-area network system as:

$\dot{x}(t)=A_{c}x(t)+B_{c}u(t)+Dw_{p}(t),$
$x=[\theta_{1}~{}\omega_{1}~{}P_{t_{1}}~{}\theta_{2}~{}\omega_{2}~{}P_{t_{2}}~{}\theta_{3}~{}\omega_{3}~{}P_{t_{3}}~{}\theta_{4}~{}\omega_{4}~{}P_{t_{4}}]^{T},$
$u(t)=[u_{1}(t)~{}u_{2}(t)~{}u_{3}(t)~{}u_{4}(t)]^{T},$

with the parameters for the networked system given in table I. Finally, we
discretize the system dynamics with $A=e^{A_{c}\Delta t}$ and
$B=\int_{0}^{\Delta t}e^{A_{c}\tau}B_{c}d\tau$, where $\Delta t$ is the
sampling period.

TABLE I: Network Parameters Parameters | Area 1 | Area 2 | Area 3 | Area 4
---|---|---|---|---
$M_{i}$ | 0.1667 | 0.2222 | 0.16 | 0.1304
$D_{i}$ | 0.0083 | 0.0088 | 0.0080 | 0.0088
$R_{i}$ | $2.4$ | $2.7$ | $2.5$ | $2$
$\tau_{t}$ | 0.3 | 0.33 | 0.35 | 0.375
Figure 2: Interconnected four‐area power distribution system

The communication graph structure is the same as the physical connection graph
(fig. 2), with all the per unit line voltages chosen to be $V_{i}=V_{j}=1$ and
line coefficients of the power flow are taken as
$B_{12}=B_{21}=B_{13}=B_{31}=$ $B_{23}=B_{32}=B_{14}=B_{41}=0.545$ p.u. and
$B_{24}=B_{42}=B_{34}=$ $B_{43}=0$ [14]. We assume the measurement model to be
graph Laplacian:

$\displaystyle y_{1}$
$\displaystyle=(\theta_{1}-\theta_{2})+(\theta_{1}-\theta_{3})+(\theta_{1}-\theta_{4}),$
$\displaystyle y_{2}$
$\displaystyle=(\theta_{2}-\theta_{3})+(\theta_{2}-\theta_{1}),$
$\displaystyle y_{3}$
$\displaystyle=(\theta_{3}-\theta_{1})+(\theta_{3}-\theta_{2}),$
$\displaystyle y_{4}$
$\displaystyle=(\theta_{4}-\theta_{1})+(\theta_{4}-\theta_{ref}).$

The above measurement model implies that each individual area measures the sum
of the phase difference between itself and physically connected areas through
net tie-line active power flow measurement. We assume that we can measure the
absolute phase angle of area 4 by comparing it with known reference
$\theta_{ref}=0$. We bound the deviation in turbine power by choosing the
performance variable state as:
$z_{t}=[P_{t_{1}}~{}P_{t_{2}}~{}P_{t_{3}}~{}P_{t_{4}}]$ and want to obtain the
same level of privacy in frequency deviation for each area in the system
$\omega_{i}$.

Figure 3: Optimal input private noise for given values of differential privacy
for (L) an adversary with access to communication channels
($y^{a}_{k}=\bar{y}_{k}$), and (R) an adversary with direct access to
measurements ($y^{a}_{k}=y_{k}$).

The design problem is to find the optimal privacy noise in control input and
output channels and simultaneously design the controller to bound the
covariance of the deviation in the turbine power $Z_{P_{t_{i}}}$ while
preserving the privacy of the frequency deviation for each area
$E_{\omega_{i}}$. Figure 3 shows the optimal input privacy noises for each of
the control inputs to obtain the same level of privacy for each of
$E_{\omega_{i}}$. Notice that the intensity of the noise required is different
in each channel based on the open loop and finally the closed-loop dynamics of
each area. Also, the intensity of the noise increases with the level of
privacy but there is a change in respective ratios of the noise intensity
between different channels showing a non-scaled parameterization of the
privacy noise and thus the need for the co-design of the noises and the
controller. Also, notice that the amount of privacy noise required for the
second case with ($y^{a}_{k}={y}_{k}$) is more than the first case
($y^{a}_{k}=\bar{y}_{k}$) as there is no contribution from the output privacy
noise towards the privacy of the system.

Figure 4 shows similar plots for the optimal output privacy noise for two
cases that are based on adversarial capabilities. Notice that the amount of
privacy signal required to obtain the desired performance bound increases with
an increase in desired differential privacy. Moreover, the amount of privacy
noise added in the output channel is much higher than the input channels as
the output channel directly affects the state estimates and indirectly affects
the system performance after passing through the controller dynamics, but the
input channel directly affects the system performance and indirectly affects
state estimates after passing through the system dynamics. Notice that the
amount of noise added in the second case with ($y^{a}_{k}={y}_{k}$) is zero as
it does not help increase the differential privacy but adversely affects the
system performance.

Figure 4: Optimal output private noise for given values of differential
privacy for (L) an adversary with access to communication channels
($y^{a}_{k}=\bar{y}_{k}$), and (R) an adversary with direct access to
measurements ($y^{a}_{k}=y_{k}$). Figure 5: Optimal system performance norm
for given values of differential privacy for (L) an adversary with access to
communication channels ($y^{a}_{k}=\bar{y}_{k}$), and (R) an adversary with
direct access to measurements ($y^{a}_{k}=y_{k}$).

Finally, fig. 5 shows the plots for the system performance in terms of
variance of deviation in turbine power for different areas. Notice that higher
differential privacy results in a higher variance in deviation, i.e., an
adverse effect on system performance. Notice that we minimized the performance
loss for the given privacy level and thus different areas result in different
performance levels. Also, the variance in deviation is higher for the second
case with a stronger adversary with direct access to the measurement as only
input privacy noise is effective in providing privacy which has a worse effect
on system performance.

##  VI Conclusion

The paper showed that the joint design of differential privacy noise
distribution and a general dynamic controller can be posed as a convex
optimization problem using the Linear Matrix Inequalities framework. The
framework adds privacy noise to both control input and system output to
privatize the system’s state. The co-design problem also designs an optimal
estimator from the perspective of the adversary with access to both
communication channels and direct output measurements. The simulation results
show the interplay between the controller gains and the privacy noise to
obtain the desired level of privacy while minimizing the system performance as
a measure of the variance of deviation from reference. The results show the
effectiveness of input and output privacy noise based on the capabilities of
the adversary and show the need for the co-design of the privacy noises with
the controller.

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