@ignorepar

# Sample Complexity of the Robust LQG Regulator
with Coprime Factors Uncertainty

Yifei Zhang♯, Sourav Kumar Ukil‡ and Serban Sabau♭ ♯Yifei Zhang is with the
Electrical and Computer Engineering Deptartment, Stevens Institute of
Technology, Hoboken, NJ 07030 USA. email<EMAIL_ADDRESS>Kumar
Ukil is with the Electrical and Computer Engineering Deptartment, Stevens
Institute of Technology, Hoboken, NJ 07030 USA. email:
<EMAIL_ADDRESS>Sabău is with the Electrical and Computer Engineering
Deptartment, Stevens Institute of Technology, Hoboken, NJ 07030 USA. email:
<EMAIL_ADDRESS>Yifei Zhang<EMAIL_ADDRESS>
Department of Electrical and Computer Engineering
Stevens Institute of Technology, NJ 07030, USA Sourav Kumar Ukil
<EMAIL_ADDRESS>
Department of Electrical and Computer Engineering
Stevens Institute of Technology, NJ 07030, USA Ephraim Neimand
<EMAIL_ADDRESS>
Department of Electrical and Computer Engineering
Stevens Institute of Technology, NJ 07030, USA Şerban Sabău<EMAIL_ADDRESS>
Department of Electrical and Computer Engineering
Stevens Institute of Technology, NJ 07030, USA Myron E. Hohil
<EMAIL_ADDRESS>
U.S. Army Combat Capabilities Development Command
(CCDC) Armaments Center, Picatinny Arsenal, NJ 07806, USA

###### Abstract

This paper addresses the end-to-end sample complexity bound for learning the
$\mathcal{H}_{2}$ optimal controller (the Linear Quadratic Gaussian (LQG)
problem) with unknown dynamics, for potentially unstable Linear Time Invariant
(LTI) systems. The robust LQG synthesis procedure is performed by considering
bounded additive model uncertainty on the coprime factors of the plant. The
closed-loop identification of the nominal model of the true plant is performed
by constructing a Hankel-like matrix from a single time-series of noisy finite
length input-output data, using the ordinary least squares algorithm from
Sarkar et al. (2020). Next, an $\mathcal{H}_{\infty}$ bound on the estimated
model error is provided and the robust controller is designed via convex
optimization, much in the spirit of Boczar et al. (2018) and Zheng et al.
(2020a), while allowing for bounded additive uncertainty on the coprime
factors of the model. Our conclusions are consistent with previous results on
learning the LQG and LQR controllers.

Keywords: Robust LQG Control, Coprime Factorization, LTI Systems, Sample
Complexity.

##  1 Introduction

Considerable research efforts have been spent within the last few years
towards approaching classical control problems with modern statistical and
optimization tools from the Machine Learning framework, envisaging practical
applications, see for example Dean et al. (2018), Mania et al. (2019), Dean et
al. (2020), Zheng et al. (2020a). The starting point of the aforementioned
research efforts has been the classical LQG control problem, which deals with
partially observed linear and time-invariant dynamical systems driven by
Gaussian noise and where the problem is finding the optimal output feedback
law that minimizes the expected value of a quadratic cost.

In this paper an end-to-end sample-complexity bound on learning LQG
controllers that stabilize the true system with high probability is
established by incorporating recent advances in finite time (non-asymptotic)
system identification (Sarkar et al. (2020)). The contribution resides in the
development of a tractable robust control synthesis procedure, that allows for
bounded additive model uncertainty on the coprime factors of the model of the
plant, thus circumventing both the need for state feedback and the restrictive
assumption on the plant’s open loop stability. The resulted sub-optimality gap
is bounded as a function of the level of the model uncertainty. The end-to-end
sample complexity bound for learning robust LQG controllers is
$\mathcal{O}(\sqrt{{logT}/{T}})$, where $T$ is the time horizon for learning.
For open-loop stable systems, Zheng et al. (2020a) recently proved that the
performance for LQG controllers deteriorates linearly with the model
estimation error, starting from the original analysis of Dean et al. (2018)
from the case of learning fully observed LQR controllers. The robust control
synthesis proposed here achieves the same scaling for the sub-optimality gap
as Dean et al. (2018), namely $\mathcal{O}(\gamma^{2})$, where $\gamma$ is the
model uncertainty level.

###  1.1 The Linear Quadratic Gaussian Problem

Within the last few years, modern statistical and algorithmic methods led to
new solutions for classical control problems, such as the Linear Quadratic
Gaussian problem. For a discrete-time LTI (Linear and Time Invariant) systems
driven by Gaussian process and sensor noise:

$\displaystyle x_{k+1}$ $\displaystyle={A}x_{k}+{B}u_{k}+{\delta x}_{k},$ (1)
$\displaystyle y_{k}$ $\displaystyle={C}x_{k}+{D}u_{k}+\nu_{k},$

where $x_{k}\in\mathbb{R}^{n}$ is the state of the system,
$u_{k}\in\mathbb{R}^{m}$ is the control input and $y_{k}\in\mathbb{R}^{p}$ is
the measurement output with ${\delta x}_{k}\in\mathbb{R}^{m}$,
$\nu_{k}\in\mathbb{R}^{p}$ are Gaussian noise with zero mean, covariance
$\sigma_{\delta x}^{2}I$ and $\sigma_{\nu}^{2}I$ respectively, the classical
LQG control problem is defined as:

$\displaystyle\min_{u_{0},u_{1},...}\quad\lim\limits_{T\rightarrow\infty}$
$\displaystyle\mathbb{E}\bigg{[}\dfrac{1}{T}\sum\limits_{t=0}^{T}(y^{T}_{t}P_{1}y_{t}+u^{T}_{t}P_{2}u_{t})\bigg{]}$
(2) subject to $\displaystyle\eqref{stateeq},$

where, $P_{1},P_{2}$ is positive definite. Without loss of generality, it is
assumed that $P_{1}=I_{p}$, $P_{2}=I_{m}$, $\sigma_{\delta x}=1$,
$\sigma_{\nu}=1$.

In a nutshell, the problem can be stated as learning with high probability and
in finite time the model of an unknown LTI system and subsequently designing
its optimal LQG controller, while accounting for the inherent model
uncertainty incurred at the learning stage.

###  1.2 The Main Technical Ingredient

Identifying LTI models from input-output data has been the focus of time-
domain identification. Using coprime factors instead of the state space
realization of the system has a great advantage as it ensures to work on
unstable system identification. The Transfer Function Matrix (TFM) of the
plant is written as:

$\textbf{G}(z)=\widetilde{\textbf{M}}^{-1}(z){\bf\widetilde{N}}(z)=\textbf{N}(z)\textbf{M}^{-1}(z)$
(3)

where $\widetilde{\textbf{M}}(z)$, $\widetilde{\textbf{N}}(z)$,
$\textbf{M}(z)$ and $\textbf{N}(z)$ are stable TFM. Doubly coprime
factorization of the TFM of a LTI system plays a key role in many sectors of
the factorization approach to filter synthesis and multi-variable control
systems analysis. A doubly coprime factorization of a given LTI plant is
closely related to the Parameterization of all stabilizing controllers for
this plant. With a realization of the TFM, various formulas to compute doubly
coprime factorizations over the ring of stable and proper Rational Matrix
Functions (RMF) have been proposed both for standard (proper) and for
generalized (improper, singular, or descriptor) systems. The formulas are
expressed either in terms of a stabilizable and detectable realization of the
underlying TFM or make additional use of a realization of a full or reduced
order observer based stabilizing controller.

###  1.3 Contributions

Recently, LQG control has been studied in a model based Reinforcement Learning
framework (Zheng et al. (2020a)) and the sub-optimality performance
degradation of the robust LQG controller was proved to scale as a function of
the modeling error. However, the results in Zheng et al. (2020a) are valid
only for open-loop stable systems, thus excluding many situations of practical
interest. This paper shows how to remove the stability assumption on the
unknown system, while at the same time streamlining the equivalent
optimization problem, by reducing the size of the subsequent linear
constraints. The proposed algorithm is consistent with previous results, while
allowing for a much stronger description of the modeling error as bounded
additive uncertainty on the coprime factors of the model of the plant (without
restriction on the McMillan degree of the true plant or on its number of
unstable poles). As expected, the presence of additive, norm-bounded factors
on the coprimes of the plant renders the cost functional non-convex, therefore
the derivation of an upper-bound on the cost functional is needed. This is
subsequently exploited to derive a quasi-convex approximation of the robust
LQG problem. An inner approximation of the quasi convex problems via FIR
truncation is employed. Previous results (Mania et al. (2019), Zheng et al.
(2020a)) show that indeed the certainty equivalent controller may achieve
superior sub-optimality scaling than our result, but only for the fully
observed LQR settings (Mania et al. (2019)), in the setup of a stricter
requirement on admissible uncertainty. Given the lack of prior gain margin for
the optimal LQG controller, which is known to be notoriously fragile, even
under small model uncertainty the stabilizability of the resulted controller
may be lost, thus the availability of a more general framework for modelling
of uncertainty is important.

Existing non-asymptotic identification methods (Sarkar et al. (2020)) have
been adapted in order to yield a comparable end-to-end sample complexity. The
identification of the unstable plant is performed in closed loop, directly on
its the coprime factors via the dual Youla Parameterization (Anderson (1998)).
The algorithm employed for system identification doesn’t require the knowledge
of the model’s order (Oymak and Ozay (2019)), which is the common scenario in
many applications. Pursuing the identification of the plant a
$\mathcal{H}_{2}$ bound for Hankel matrix estimation with high probability is
derived, followed by a $\mathcal{H}_{\infty}$ bound on the uncertainty on the
coprime factors, which quantifies the modeling error. The robust controller
design is recast as convex optimization for estimated nominal model within a
worst case scenario on the uncertainty. For the output feedback of potentially
unstable plants, the resulted sample complexity result is matched to the same
level as that obtained in recent papers (Boczar et al. (2018), Dean et al.
(2020), Zheng et al. (2020a)), where the robust so-called SLP or IOP
procedures (Zheng et al. (2020b)) are used for design.

The paper is organized as follows: the general setup is given in Section II.
The robust controller synthesis with uncertainty on the coprime factors is
included in Section III. The sub-optimality guarantees are discussed in
Section IV. A brief discussion on the closed-loop system identification is
provided in Section V with end-to-end sample complexity results. Conclusions
and future possible directions are given in Section V. All the proofs are
postponed to the Appendices, where literature review, mathematical
preliminaries and few remarks also have been discussed briefly.

##  2 General Setup and Technical Preliminaries

The notation used in this paper is fairly common in control systems. Upper and
lower case boldface letters (e.g. ${\bf z}$ and ${\bf G}$) are used to denote
signals and transfer function matrices, and lower and upper case letters (e.g.
$z$ and $A$) are used to denote vectors and matrices. The enclosed results are
valid for discrete-time linear systems, therefore $z$ denotes the complex
variable associated with the $\mathbf{Z}$-transform for discrete-time systems.
A LTI system is stable if all the poles of its TFM are situated inside the
unit circle for discrete time systems. The TFM of a LTI system is called
unimodular if it is square, stable and has a stable inverse. For the sake of
brevity the $z$ argument after a transfer function may be omitted. Some
frequently used notation is listed in the next page.

|  Nomenclature of Basic Notation
---|---
LTI |  Linear and Time Invariant
TFM |  Transfer Function Matrix
DCF |  Doubly Coprime Factorization
LCF |  Left Coprime Factorization
RCF |  Right Coprime Factorization
$x\overset{def}{=}y$ |  $x$ is by definition equal to $y$
$\mathbb{R}(z)$ |  Set of all real–rational transfer functions
$\mathbb{R}(z)^{p\times m}$ |  Set of $p\times m$ matrices having all entries in $\mathbb{R}(z)$
${\bf T}^{\ell\varepsilon}$ |  The TFM of the (closed-loop) map having $\varepsilon$ as input and $\ell$ as output
${\bf T}^{\ell\varepsilon}_{\bf Q}$ |  The TFM of the (closed-loop) map from the exogenous signal $\varepsilon$ to the signal $\ell$ inside the feedback loop, as a function of the Youla parameter ${\bf Q}$
$\|{\bf G}\|_{{F}}$ |  Frobenius norm, Schur norm or $l_{2}$ norm of ${\bf G}\in\mathbb{R}(z)$, defined as $|tr(\bf GG^{*})|^{1/2}$
$\|{\bf G}\|_{\infty}$ |  $\mathcal{H}_{\infty}$-norm of ${\bf G}\in\mathbb{R}(z)$, defined as $\sup_{\omega}\sigma_{\max}\big{(}{\bf G}(e^{j\omega})\big{)}$
$\|{\bf G}\|_{\mathcal{H}_{2}}$ |  $\mathcal{H}_{2}$-norm of ${\bf G}\in\mathbb{R}(z)$, defined as ${{{\dfrac{1}{2\pi}\Big{(}\bigintsss\limits_{-\pi}^{\pi}tr\big{(}{\bf G}^{*}(e^{j\omega}){\bf G}(e^{j\omega})\big{)}d\omega}}\Big{)}}^{1/2}$
pt |  Notations for true plant (e.g. ${\bf G}^{\mathtt{pt}}$, ${\bf K}^{\mathtt{pt}}$)
md |  Notations for nominal/estimated model (e.g. ${\bf G}^{\mathtt{md}}$, ${\bf K}^{\mathtt{md}}$)
${\bf G}^{\mathtt{pt}}$, ${\bf K}^{\mathtt{opt}}$ |  True Plant, Optimal Controller
${\bf G}^{\mathtt{md}}$, ${\bf K}^{\mathtt{md}}_{\bf Q}$ |  Estimated Model, Nominal stabilizing controller for any stable Youla parameter ${\bf Q}$

###  2.1 Standard Unity Feedback

A standard unity feedback configuration is depicted in Figure 1, where ${\bf
G}\in\mathbb{R}(z)^{p\times m}$ is a multi-variable LTI plant and ${\bf
K}\in\mathbb{R}(z)^{m\times p}$ is an LTI controller. Here $w$, $\nu$ and $r$
are the input disturbance, sensor noise and reference signal respectively
while $u$, $z$ and $y$ are the controls, regulated signals and measurements
vectors, respectively. If all the closed–loop maps from the exogenous signals
$\displaystyle[r^{T}\;\>w^{T}\;\>\nu^{T}\;]^{T}$ to any point inside the
feedback loop of Figure 1 are stable, then ${\bf K}$ is said to be an
(internally) stabilizing controller of ${\bf G}$ or equivalently that ${\bf
K}$ stabilizes ${\bf G}$.

+$r$$z$${\bf K}$$u$$v$+$w$+${\bf G}$$\nu$++$y$– Figure 1: Standard unity
feedback loop of the plant $\bf G$ with the controller $\bf K$

The notation ${\bf T}^{\ell\varepsilon}$ is used to indicate the mapping from
signal $\varepsilon$ to signal $\ell$ after combining all the ways in which
$\ell$ is a function of $\varepsilon$ and solving any feedback loops that may
exist. For example, ${\bf T}^{zw}$ in Figure 1 is the mapping from the
disturbances $w$ to the regulated measurements $z$.

###  2.2 The Youla-Kuçera Parameterization

###### Proposition 2.1

Given a TFM ${\bf K}\in\mathbb{R}(z)^{m\times p}$, a fractional representation
of the form ${\bf K}={\bf R}^{-1}{\bf S}$ with ${\bf
R}\in\mathbb{R}(z)^{m\times m}$, ${\bf S}\in\mathbb{R}(z)^{m\times p}$ is
called a left factorization of K. If ${\bf K}={\bf Y}^{-1}{\bf X}$ is a left
factorization of ${\bf K}$ then any other left factorization of ${\bf K}$ such
as ${\bf K}={\bf R}^{-1}{\bf S}$ is of the form ${\bf R}={\bf UY}$, ${\bf
S}={\bf UX}$, for some invertible TFM ${\bf U}$.

Given a plant ${\bf G}\in\mathbb{R}(z)^{p\times m}$, a left coprime
factorization of ${\bf G}$ is defined by ${\bf G}=\widetilde{\bf
M}^{-1}\widetilde{\bf N}$, with $\widetilde{\bf N}\in\mathbb{R}(z)^{p\times
m}$, $\widetilde{\bf M}\in\mathbb{R}(z)^{p\times p}$ both stable and
satisfying $\widetilde{\bf M}\widetilde{\bf Y}+\widetilde{\bf N}\widetilde{\bf
X}=I_{p}$, for certain stable TFMs $\widetilde{\bf X}\in\mathbb{R}(z)^{m\times
p}$, $\widetilde{\bf Y}\in\mathbb{R}(z)^{p\times p}$.

Analogously, a right coprime factorization of ${\bf G}$ is defined by ${\bf
G}={\bf NM}^{-1}$ with both factors ${\bf N}\in\mathbb{R}(z)^{p\times m}$,
${\bf M}\in\mathbb{R}(z)^{m\times m}$ being stable and for which there exist
${\bf X}\in\mathbb{R}(z)^{m\times p}$, ${\bf Y}\in\mathbb{R}(z)^{m\times m}$
also stable, satisfying ${\bf YM}+{\bf XN}=I_{m}$ (Vidyasagar, 1985, Ch. 4,
Corollary 17), with $I_{m}$ being the identity matrix.

###### Definition 2.2

(Vidyasagar, 1985, Ch.4, Remark pp. 79) A collection of eight stable TFMs
$\big{(}{\bf M},{\bf N}$, $\widetilde{\bf M},\widetilde{\bf N}$, ${\bf X},{\bf
Y}$, $\widetilde{\bf X},\widetilde{\bf Y}\big{)}$ is called a doubly coprime
factorization of ${\bf G}$ if $\widetilde{\bf M}$ and ${\bf M}$ are
invertible, yield the factorizations ${\bf G}=\widetilde{\bf
M}^{-1}\widetilde{\bf N}={\bf NM}^{-1}$, and satisfy the following equality
(Bézout’s identity):

$\left[\begin{array}[]{cc}\widetilde{\bf M}&\widetilde{\bf N}\\\ -{\bf X}&{\bf
Y}\end{array}\right]\left[\begin{array}[]{cc}\widetilde{\bf Y}&-{\bf N}\\\
\widetilde{\bf X}&{\bf
M}\end{array}\right]=I_{p+m},\left[\begin{array}[]{cc}\widetilde{\bf Y}&-{\bf
N}\\\ \widetilde{\bf X}&{\bf
M}\end{array}\right]\left[\begin{array}[]{cc}\widetilde{\bf M}&\widetilde{\bf
N}\\\ -{\bf X}&{\bf Y}\end{array}\right]=I_{p+m}.$ (4)

###### Theorem 2.3

(Youla-Kuc̆era) (Vidyasagar, 1985, Ch.5, theorem 1) Let $\big{(}{\bf M},{\bf
N}$, $\widetilde{\bf M},\widetilde{\bf N}$, ${\bf X},{\bf Y}$, $\widetilde{\bf
X},\widetilde{\bf Y}\big{)}$ be a doubly coprime factorization of ${\bf G}$.
Any controller ${\bf K}_{\bf Q}$ stabilizing the plant ${\bf G}$, in the
feedback interconnection of Figure 1, can be written as

${\bf K_{Q}}={\bf Y}_{\bf Q}^{-1}{\bf X_{Q}}={\bf\widetilde{X}}_{\bf
Q}{\bf\widetilde{Y}}_{\bf Q}^{-1},$ (5)

where ${\bf X_{Q}}$, ${\bf\widetilde{X}_{Q}}$, ${\bf Y_{Q}}$ and
${\bf\widetilde{Y}_{Q}}$ are defined as:

$\begin{split}{\bf X_{Q}}\overset{def}{=}{\bf X}+{\bf Q}\widetilde{\bf
M},\quad&{\bf\widetilde{X}_{Q}}\overset{def}{=}\widetilde{\bf X}+{\bf
MQ},\quad\\\ {\bf Y_{Q}}\overset{def}{=}{\bf Y}-{\bf Q}\widetilde{\bf
N},\quad&{\bf\widetilde{Y}_{Q}}\overset{def}{=}\widetilde{\bf Y}-{\bf
NQ}\end{split}$ (6)

for some stable ${\bf Q}$ in $\mathbb{R}(z)^{m\times p}$. It also holds that
${\bf K_{Q}}$ from (5) stabilizes ${\bf G}$, for any stable ${\bf Q}$ in
$\mathbb{R}(z)^{m\times p}$.

###### Proposition 2.4

Starting from any doubly coprime factorization (4), the following identity

$\\!\left[\begin{array}[]{cl}\\!{\bf U}_{1}\widetilde{\bf M}&{\bf
U}_{1}\widetilde{\bf N}\\\ -{\bf U}_{2}{\bf X_{Q}}&{\bf U}_{2}{\bf
Y_{Q}}\\!\end{array}\right]\\!\left[\begin{array}[]{lr}\\!\widetilde{\bf
Y}_{\bf Q}{\bf U}^{-1}_{1}&-{\bf N}{\bf U}^{-1}_{2}\\\ \widetilde{\bf X}_{\bf
Q}{\bf U}^{-1}_{1}&{\bf M}{\bf U}^{-1}_{2}\\!\end{array}\right]=I_{p+m}.$ (7)

provides the class of all doubly coprime factorizations of ${\bf G}$, where
${\bf Q}$ is stable in $\mathbb{R}(z)^{m\times p}$ and ${\bf
U}_{1}\in\mathbb{R}(z)^{p\times p}$, ${\bf U}_{2}\in\mathbb{R}(z)^{m\times m}$
are both unimodular.

###### Definition 2.5

$\mathbb{H}({\bf G},{\bf K}_{{\bf Q}})$ denotes the TFM whose entries are the
closed-loop maps from $\displaystyle[r^{T}\;\>w^{T}\;\>\nu^{T}\;]^{T}$ to
$\displaystyle[y^{T}\;\;u^{T}\;\;z^{T}\;\;v^{T}]^{T}$ achievable via the
stabilizing controllers (5), namely equation (8) here

$\left[\begin{array}[]{c}y\\\ u\\\ z\\\ v\\\
\end{array}\right]=\mathbb{H}({\bf G},{\bf K}_{{\bf
Q}})\left[\begin{array}[]{c}r\\\ w\\\ \nu\\\ \end{array}\right],$
$\mathbb{H}({\bf G},{\bf K}_{{\bf
Q}})\overset{def}{=}\left[\begin{array}[]{ccc}(I_{p}+{\bf GK_{Q}})^{-1}{\bf
GK_{Q}}&(I_{p}+{\bf GK_{Q}})^{-1}{\bf G}&(I_{p}+{\bf GK_{Q}})^{-1}\\\
(I_{m}+{\bf K_{Q}}{\bf G})^{-1}{\bf K_{Q}}&-(I_{m}+{\bf K_{Q}G})^{-1}{\bf
K_{Q}G}&-(I_{m}+{\bf K_{Q}}{\bf G})^{-1}{\bf K_{Q}}\\\ (I_{p}+{\bf
GK_{Q}})^{-1}&-(I_{p}+{\bf GK_{Q}})^{-1}{\bf G}&-(I_{p}+{\bf GK_{Q}})^{-1}\\\
(I_{m}+{\bf K_{Q}G})^{-1}{\bf K_{Q}}&(I_{m}+{\bf K_{Q}G})^{-1}&-(I_{m}+{\bf
K_{Q}G})^{-1}{\bf K_{Q}}\\\ \end{array}\right]$ (8)

###### Proposition 2.6

(Vidyasagar, 1985, (7)/ pp.101) ${\bf T}^{\ell\varepsilon}_{\bf Q}$ denotes
the dependency on the Youla parameter ${\bf Q}$ of the closed loop map from
the exogenous signal $\varepsilon$ to the signal $\ell$ inside the feedback
loop. The set of all closed loop maps (8) achievable via stabilizing
controllers (5) depends on the plant ${\bf G}$ alone and not on the particular
doubly coprime factorization (4) in which the Youla parameterization is
formulated. Furthermore, the parameterization of the closed loop maps (8) is
affine in the Youla parameter $\bf{Q}$.

$\small\begin{tabular}[]{|c | c| c| c|}\hline\cr${}$&${r}$&${w}$&${v}$\\\ \hline\cr&&&\\\\[-9.10623pt] ${y}$&$I_{p}-\widetilde{\bf Y}_{\bf Q}\widetilde{\bf M}$&$\widetilde{\bf Y}_{\bf Q}\widetilde{\bf N}$&$\widetilde{\bf Y}_{\bf Q}\widetilde{\bf M}$\\\\[2.90623pt] \hline\cr&&&\\\\[-9.10623pt] ${u}$&$\widetilde{\bf X}_{\bf Q}\widetilde{\bf M}$&$-\widetilde{\bf X}_{\bf Q}\widetilde{\bf N}$&$-\widetilde{\bf X}_{\bf Q}\widetilde{\bf M}$\\\\[2.90623pt] \hline\cr&&&\\\\[-9.10623pt] ${z}$&$\widetilde{\bf Y}_{\bf Q}\widetilde{\bf M}$&$-\widetilde{\bf Y}_{\bf Q}\widetilde{\bf N}$&$-\widetilde{\bf Y}_{\bf Q}\widetilde{\bf M}$\\\\[2.90623pt] \hline\cr&&&\\\\[-9.10623pt] ${v}$&$\widetilde{\bf X}_{\bf Q}\widetilde{\bf M}$&$I_{m}-\widetilde{\bf X}_{\bf Q}\widetilde{\bf N}$&${-\widetilde{\bf X}_{\bf Q}\widetilde{\bf M}}$\\\\[2.90623pt] \hline\cr\end{tabular}{\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color@gray@fill{1},}\begin{tabular}[]{|c | c| c| c|}\hline\cr${}$&${r}$&${w}$&${v}$\\\ \hline\cr&&&\\\\[-7.74997pt] ${y}$&${{\bf N}{\bf X}_{\bf Q}}$&${{\bf N}{\bf Y}_{\bf Q}}$&$I_{p}-{\bf N}{\bf X}_{\bf Q}$\\\\[2.90623pt] \hline\cr&&&\\\\[-7.74997pt] ${u}$&${{\bf M}{\bf X}_{\bf Q}}$&${-(I_{m}-{\bf M}{\bf Y}_{\bf Q})}$&$-{\bf M}{\bf X}_{\bf Q}$\\\\[2.90623pt] \hline\cr&&&\\\\[-7.74997pt] ${z}$&${I_{p}-{\bf N}{\bf X}_{\bf Q}}$&$-{\bf N}{\bf Y}_{\bf Q}$&$-(I_{p}-{\bf N}{\bf X}_{\bf Q})$\\\\[2.90623pt] \hline\cr&&&\\\\[-7.74997pt] ${v}$&${\bf M}{\bf X}_{\bf Q}$&${{\bf M}{\bf Y}_{\bf Q}}$&${-{\bf M}{\bf X}_{\bf Q}}$\\\\[2.90623pt] \hline\cr\end{tabular}$ (9)

###  2.3 Dual Youla-Kuçera Parameterization

###### Theorem 2.7

(Dual Youla-Kuçera)(Hof and Schrama (1992)) Let $\big{(}{\bf M},{\bf N}$,
$\widetilde{\bf M},\widetilde{\bf N}$, ${\bf X},{\bf Y}$, $\widetilde{\bf
X},\widetilde{\bf Y}\big{)}$ be a doubly coprime factorization of ${\bf G}$.
Any plant ${\bf G}_{R}$ stabilized by a fixed controller ${\bf K}$, can be
written as

${\bf G_{R}}={\bf\widetilde{M}_{R}}^{-1}{\bf\widetilde{N}_{R}}={\bf N_{R}}{\bf
M_{R}}^{-1},$ (10)

where ${\bf M_{R}}$, ${\bf\widetilde{M}_{R}}$, ${\bf N_{R}}$ and
${\bf\widetilde{N}_{R}}$ are defined as:

$\begin{split}{\bf M_{R}}\overset{def}{=}{\bf M}-\widetilde{\bf X}{\bf
R},\quad&{\bf\widetilde{M}_{R}}\overset{def}{=}\widetilde{\bf M}-{\bf
RX},\quad\\\ {\bf N_{R}}\overset{def}{=}{\bf N}+\widetilde{\bf Y}{\bf
R},\quad&{\bf\widetilde{N}_{R}}\overset{def}{=}\widetilde{\bf N}+{\bf
RY}\end{split}$ (11)

for some stable ${\bf R}$ in $\mathbb{R}(z)^{p\times m}$.

###### Proposition 2.8

(Hof and Schrama (1992)) Let $\bf{G}$ with LCF
${\bf\widetilde{M}}^{-1}{\bf\widetilde{N}}$ be any system that is stabilized
by a controller $\bf{K}$ with LCF ${\bf Y}^{-1}{\bf X}$. Then, the plant
$\bf{G_{R}}$ is stabilized by controller $\bf{K}$ iff there exists a stable
${\bf R}\in\mathbb{R}(z)^{p\times m}$, such that $\bf{G_{R}}$ =
$(\widetilde{\bf M}-{\bf RX})^{-1}(\widetilde{\bf N}+{\bf RY})$. Similarly,
for RCF of controller ${\bf K}$ = ${\bf\widetilde{X}}{\bf\widetilde{Y}}^{-1}$
and plant $\bf{G}$ = ${\bf N}{\bf M}^{-1}$, the plant $\bf{G}$ is stabilized
by controller $\bf{K}$ iff there exists a stable ${\bf
R}\in\mathbb{R}(z)^{p\times m}$, such that $\bf{G_{R}}$ = $({\bf
N}+\widetilde{\bf Y}{\bf R})({\bf M}-\widetilde{\bf X}{\bf R})^{-1}$.

##  3 Robust Controller Synthesis

Given a DCF of the nominal model of the plant ${\bf
G}^{\mathtt{md}}=(\widetilde{\bf M}^{\mathtt{md}})^{-1}\widetilde{\bf
N}^{\mathtt{md}}={\bf N}^{\mathtt{md}}({\bf M}^{\mathtt{md}})^{-1}$, we can
write the Bezout identity that incorporates the corresponding Youla
parameterization of all stabilizing controller for the nominal model ${{\bf
K}^{\mathtt{md}}_{\bf Q}}=({\bf Y}^{\mathtt{md}}_{\bf Q})^{-1}{\bf
X}^{\mathtt{md}}_{\bf Q}={\bf{\widetilde{X}}}^{\mathtt{md}}_{\bf
Q}({\bf\widetilde{Y}}^{\mathtt{md}}_{\bf Q})^{-1}$ as:

$\left[\begin{array}[]{cc}{\widetilde{\bf M}}^{\mathtt{md}}&{\widetilde{\bf
N}}^{\mathtt{md}}\\\ -{{\bf X}}^{\mathtt{md}}_{\bf Q}&{{\bf
Y}}^{\mathtt{md}}_{\bf
Q}\end{array}\right]\left[\begin{array}[]{cc}{\widetilde{\bf
Y}}^{\mathtt{md}}_{\bf Q}&-{{\bf N}}^{\mathtt{md}}\\\ {\widetilde{\bf
X}}^{\mathtt{md}}_{\bf Q}&{{\bf
M}}^{\mathtt{md}}\end{array}\right]=\left[\begin{array}[]{cc}{I_{p}}&{0}\\\
{0}&{I_{m}}\end{array}\right],$ (12)

where Q denotes as usually the Youla parameter.

######  Definition 3.1 (Model Uncertainty Set)

The $\gamma$-radius model uncertainty set (for the nominal plant ${\bf
G}^{\mathtt{md}}$ with $\Delta_{\bf\widetilde{M}}$,
$\Delta_{\bf\widetilde{N}}$ both stable) is defined as:

$\mathcal{G}_{\gamma}\overset{\mathrm{def}}{=}\\{{\bf G}={\widetilde{\bf
M}}^{-1}{\widetilde{\bf N}}\hskip 2.0pt\big{|}\hskip 2.0pt{\widetilde{\bf
M}}=({\widetilde{\bf
M}^{\mathtt{md}}}+\Delta_{\bf{\widetilde{M}}}),{\widetilde{\bf
N}}=({\widetilde{\bf
N}}^{\mathtt{md}}+\Delta_{\bf\widetilde{N}});\;\;\;\Big{\|}\left[\begin{array}[]{cc}\Delta_{\bf\widetilde{M}}&\Delta_{\bf\widetilde{N}}\end{array}\right]\Big{\|}_{\infty}<\gamma\\}$
(13)

######  Definition 3.2 ($\gamma$-Robustly Stabilizing)

A stabilizing controller ${\bf K}^{\mathtt{md}}$ of the nominal plant is said
to be $\gamma$-robustly stabilizing iff ${\bf K}^{\mathtt{md}}$ stabilizes not
only ${\bf G}^{\mathtt{md}}$ but also all plants ${\bf
G}\in\mathcal{G}_{\gamma}$.

###### Assumption 1

It is assumed that the true plant, denoted by ${\bf G}^{\mathtt{pt}}$, belongs
to the model uncertainty set introduced in Definition 3.1, i.e. that there
exist stable $\Delta_{\bf\widetilde{M}}$, $\Delta_{\bf\widetilde{N}}$ with
$\Big{\|}\left[\begin{array}[]{cc}\Delta_{\bf\widetilde{M}}&\Delta_{\bf\widetilde{N}}\end{array}\right]\Big{\|}_{\infty}<\gamma$
for which ${\bf G}^{\mathtt{pt}}=({\widetilde{\bf
M}}^{\mathtt{md}}+\Delta_{\bf\widetilde{M}})^{-1}({\widetilde{\bf
N}}^{\mathtt{md}}+\Delta_{\bf\widetilde{N}})$.

In the presence of additive uncertainty on the coprime factors the Bezout
identity in (12) no longer holds, however, the following holds for certain
stable ${\Delta_{\bf M}}$, ${\Delta_{\bf N}}$ factors:

$\left[\begin{array}[]{cc}({\widetilde{\bf
M}}^{\mathtt{md}}+\Delta_{\bf\widetilde{M}})&({\widetilde{\bf
N}}^{\mathtt{md}}+\Delta_{\bf\widetilde{N}})\\\ -{{\bf X}}^{\mathtt{md}}_{\bf
Q}&{{\bf Y}}^{\mathtt{md}}_{\bf
Q}\end{array}\right]\left[\begin{array}[]{cc}{\widetilde{\bf
Y}}^{\mathtt{md}}_{\bf Q}&-({{\bf N}}^{\mathtt{md}}+\Delta_{\bf{N}})\\\
{\widetilde{\bf X}}^{\mathtt{md}}_{\bf Q}&({{\bf
M}}^{\mathtt{md}}+\Delta_{\bf{M}})\end{array}\right]=\left[\begin{array}[]{cc}{\bf\Phi}_{11}&O\\\
O&{\bf\Phi}_{22}\end{array}\right].$ (14)

The block diagonal structure of the right hand side term in (14) is due to the
fact that ${\bf G}^{\mathtt{pt}}=({\widetilde{\bf
M}}^{\mathtt{md}}+\Delta_{\bf\widetilde{M}})^{-1}({\widetilde{\bf
N}}^{\mathtt{md}}+\Delta_{\bf\widetilde{N}})=({{\bf
N}}^{\mathtt{md}}+\Delta_{\bf{N}})({{\bf
M}}^{\mathtt{md}}+\Delta_{\bf{M}})^{-1}$ for the aforementioned certain stable
${\Delta_{\bf M}}$, ${\Delta_{\bf N}}$ factors.

###### Lemma 3.3

A stabilizing controller of the nominal plant ${{\bf K}^{\mathtt{md}}_{\bf
Q}}=({\bf Y}^{\mathtt{md}}_{\bf Q})^{-1}{\bf X}^{\mathtt{md}}_{\bf
Q}={\bf{\widetilde{X}}}^{\mathtt{md}}_{\bf
Q}({\bf\widetilde{Y}}^{\mathtt{md}}_{\bf Q})^{-1}$ is $\gamma$-robustly
stabilizing iff for any stable model perturbations
$\Delta_{\bf\widetilde{M}},\Delta_{\bf\widetilde{N}}$ with
$\Big{\|}\left[\begin{array}[]{@{}cc@{}}\Delta_{\bf\widetilde{M}}&\Delta_{\bf\widetilde{N}}\end{array}\right]\Big{\|}_{\infty}<\gamma$
the TFM

${\bf\Phi}_{11}={I_{p}}+\left[\begin{array}[]{cc}\Delta_{\bf\widetilde{M}}&\Delta_{\bf\widetilde{N}}\end{array}\right]\left[\begin{array}[]{c}{\widetilde{\bf
Y}}^{\mathtt{md}}_{\bf Q}\\\ {\widetilde{\bf X}}^{\mathtt{md}}_{\bf
Q}\end{array}\right],$ (15)

is unimodular (i.e. it is square, stable and has an inverse
${\bf\Phi}_{11}^{-1}$ which is also stable) from (14). A similar condition for
$\gamma$-robust stabilizability can be formulated in terms of ${\bf\Phi}_{22}$
TFM, whereas

${\bf\Phi}_{22}={I_{m}}+\left[\begin{array}[]{cc}{\bf X}^{\mathtt{md}}_{\bf
Q}&{\bf Y}^{\mathtt{md}}_{\bf
Q}\end{array}\right]\left[\begin{array}[]{c}\Delta_{\bf{N}}\\\
\Delta_{\bf{M}}\end{array}\right].$ (16)

Since ${\bf\Phi}_{11}$ in (15) clearly depends on the Youla parameter (via the
right coprime factors of the controller), the condition for the
$\gamma$-robust stabilizability of the controller can be recast in the
following particular form, which will be instrumental in the sequel:

###### Theorem 3.4

The Youla parameterization yields a $\gamma$-robustly stabilizing controller
iff its corresponding Youla parameter satisfies
$\Bigg{\|}\left[\begin{array}[]{c}{\widetilde{\bf Y}}^{\mathtt{md}}_{\bf Q}\\\
{\widetilde{\bf X}}^{\mathtt{md}}_{\bf
Q}\end{array}\right]\Bigg{\|}_{\infty}\leq\dfrac{1}{\gamma}$.

The proofs for Lemma 3.3 and Theorem 3.4 are given on Appendix C. As an
intermediary result, by employing Theorem 3.4 and the standard inequality (35)
from Appendix B it is concluded that:

$\Bigg{\|}\left[\begin{array}[]{cc}-\Delta_{\bf\widetilde{M}}&-\Delta_{\bf\widetilde{N}}\end{array}\right]\left[\begin{array}[]{c}{\widetilde{\bf
Y}}^{\mathtt{md}}_{\bf Q}\\\ {\widetilde{\bf X}}^{\mathtt{md}}_{\bf
Q}\end{array}\right]\Bigg{\|}_{\infty}\leq\Bigg{\|}\left[\begin{array}[]{cc}\Delta_{\bf\widetilde{M}}&\Delta_{\bf\widetilde{N}}\end{array}\right]\Bigg{\|}_{\infty}\Bigg{\|}\left[\begin{array}[]{c}{\widetilde{\bf
Y}}^{\mathtt{md}}_{\bf Q}\\\ {\widetilde{\bf X}}^{\mathtt{md}}_{\bf
Q}\end{array}\right]\Bigg{\|}_{\infty}<\gamma\times\dfrac{1}{\gamma}=1$.

###### Proposition 3.5

The square root of the LQG cost function from (2) has the form:

$\begin{split}\mathcal{H}({\bf G}^{\mathtt{pt}},{{\bf K}^{\mathtt{md}}_{\bf
Q}})\overset{def}{=}&\Bigg{\|}\left[\begin{array}[]{c}{\widetilde{\bf
Y}}^{\mathtt{md}}_{\bf Q}\\\ {\widetilde{\bf X}}^{\mathtt{md}}_{\bf
Q}\end{array}\right]{\bf\Phi}_{11}^{-1}\left[\begin{array}[]{cc}({\widetilde{\bf
M}^{\mathtt{md}}}+\Delta_{\bf\widetilde{M}})&({\widetilde{\bf
N}^{\mathtt{md}}}+\Delta_{\bf\widetilde{N}})\end{array}\right]\Bigg{\|}_{\mathcal{H}_{2}}\\\
\end{split}$ (17)

representing the following closed loop responses:

$\left[\begin{array}[]{cc}{{\bf T}^{y\nu}_{\bf Q}}&{{\bf T}^{yw}_{\bf Q}}\\\
{{\bf T}^{u\nu}_{\bf Q}}&{{\bf T}^{uw}_{\bf
Q}}\end{array}\right]=\left[\begin{array}[]{cc}{(I_{p}+{\bf
G}^{\mathtt{pt}}{{\bf K}^{\mathtt{md}}_{\bf Q}})^{-1}}&{{(I_{p}+{\bf
G}^{\mathtt{pt}}{{\bf K}^{\mathtt{md}}_{\bf Q}})^{-1}}{{\bf
G}^{\mathtt{pt}}}}\\\ {{{\bf K}^{\mathtt{md}}_{\bf Q}}{(I_{p}+{\bf
G}^{\mathtt{pt}}{{\bf K}^{\mathtt{md}}_{\bf Q}})^{-1}}}&{{{\bf
K}^{\mathtt{md}}_{\bf Q}}{(I_{p}+{\bf G}^{\mathtt{pt}}{{\bf
K}^{\mathtt{md}}_{\bf Q}})^{-1}}}{{\bf G}^{\mathtt{pt}}}\end{array}\right]$
(18)

The proofs for the Proposition 3.5 and Theorem 3.6 are given on Appendix C.

###### Theorem 3.6

The Robust LQG Control Problem is defined as :

$\displaystyle\min_{\bf Q\>\text{stable}}$
$\displaystyle\max_{\Big{\|}\left[\begin{array}[]{cc}\Delta_{\bf\widetilde{M}}&\Delta_{\bf\widetilde{N}}\end{array}\right]\Big{\|}_{\infty}<\gamma}\mathcal{H}({\bf
G}^{\mathtt{pt}},{{\bf K}^{\mathtt{md}}_{\bf Q}})$ (19) s.t.
$\displaystyle\Bigg{\|}\left[\begin{array}[]{c}{\widetilde{\bf
Y}}^{\mathtt{md}}_{\bf Q}\\\ {\widetilde{\bf X}}^{\mathtt{md}}_{\bf
Q}\end{array}\right]\Bigg{\|}_{\infty}\leq\dfrac{1}{\gamma}.$

whose solution, obtained for the optimal Youla parameter ${\bf Q_{*}}$ in (19)
will be denoted by ${\widetilde{\bf Y}}^{\mathtt{md}}_{\bf Q_{*}}$,
${\widetilde{\bf X}}^{\mathtt{md}}_{\bf Q_{*}}$ such that the optimal, robust
controller reads ${{\bf K}^{\mathtt{md}}_{\bf
Q_{*}}}={\bf{\widetilde{X}}}^{\mathtt{md}}_{\bf
Q_{*}}({\bf\widetilde{Y}}^{\mathtt{md}}_{\bf Q_{*}})^{-1}$.

Canonical min-max formulation caused by the additive uncertainty on the
coprime factors of the robust LQG controller synthesis renders the problem
non-convex. In order to circumvent this, an upper bound on the
$\mathcal{H}({\bf G}^{\mathtt{pt}},{{\bf K}^{\mathtt{md}}_{\bf Q}})$ cost
functional will be derived, much in the spirit of Dean et al. (2018) and Zheng
et al. (2020a). This bound will further be exploited to derive a quasi-convex
approximation for the robust LQG control problem in the next subsection.

###  3.1 Quasi-convex formulation

###### Proposition 3.7

Given any $\gamma$-robustly stabilizing controller satisfying
$\Bigg{\|}\left[\begin{array}[]{c}{\widetilde{\bf Y}}^{\mathtt{md}}_{\bf Q}\\\
{\widetilde{\bf X}}^{\mathtt{md}}_{\bf
Q}\end{array}\right]\Bigg{\|}_{\infty}<\dfrac{1}{\gamma}$ then for any
additive model perturbations
$\Big{\|}\left[\begin{array}[]{cc}\Delta_{\bf\widetilde{M}}&\Delta_{\bf\widetilde{N}}\end{array}\right]\Big{\|}_{\infty}<\gamma$,
the cost functional of the robust LQG problem from (19) admits the upper
bound:

$\mathcal{H}({\bf G}^{\mathtt{pt}},{{\bf K}^{\mathtt{md}}_{\bf
Q}})\leq\dfrac{1}{1-\gamma\bigg{\|}\left[\begin{array}[]{c}{\widetilde{\bf
Y}}^{\mathtt{md}}_{\bf Q}\\\ {\widetilde{\bf X}}^{\mathtt{md}}_{\bf
Q}\end{array}\right]\bigg{\|}_{\infty}}{\Bigg{[}{h\Big{(}\gamma,\bigg{\|}\left[\begin{array}[]{c}{\widetilde{\bf
Y}}^{\mathtt{md}}_{\bf Q}\\\ {\widetilde{\bf X}}^{\mathtt{md}}_{\bf
Q}\end{array}\right]\bigg{\|}_{\infty}\Big{)}}\Bigg{\|}\left[\begin{array}[]{c}{\widetilde{\bf
Y}}^{\mathtt{md}}_{\bf Q}\\\ {\widetilde{\bf X}}^{\mathtt{md}}_{\bf
Q}\end{array}\right]\bigg{\|}_{\mathcal{H}_{2}}\Bigg{]}}$ (20)

where ${h\bigg{(}\gamma,\bigg{\|}\left[\begin{array}[]{c}{\widetilde{\bf
Y}}^{\mathtt{md}}_{\bf Q}\\\ {\widetilde{\bf X}}^{\mathtt{md}}_{\bf
Q}\end{array}\right]\bigg{\|}_{\infty}\bigg{)}}\overset{def}{=}\bigg{(}1+{\gamma\bigg{\|}\left[\begin{array}[]{c}{\widetilde{\bf
Y}}^{\mathtt{md}}_{\bf Q}\\\ {\widetilde{\bf X}}^{\mathtt{md}}_{\bf
Q}\end{array}\right]\bigg{\|}_{\infty}}\bigg{)}\bigg{(}\bigg{\|}\left[\begin{array}[]{cc}{\widetilde{\bf
M}^{\mathtt{md}}}&{\widetilde{\bf
N}^{\mathtt{md}}}\end{array}\right]\bigg{\|}_{\infty}+\gamma\bigg{)}$.

Next one of the main results is stated that provides an aptly designed
approximation of the robust LQG control problem from Definition 3.6 by means
of the LQG cost upper bound from (20). The proofs for Proposition 3.7 and
Theorem 3.8 are given in Appendix E.

###### Theorem 3.8

For the true plant, ${\bf G}^{\mathtt{pt}}\in\mathcal{G}_{\gamma}$ and
$(\forall)\alpha>0$, the robust LQG control problem in (19) admits the
following upper bound:

$\displaystyle\min_{\delta\in[0,1/\gamma)}$
$\displaystyle\dfrac{1}{1-\gamma\delta}\min_{\bf
Q\>\text{stable}}{\bigg{(}{h\big{(}\gamma,\alpha\big{)}}\Bigg{\|}\left[\begin{array}[]{c}{\widetilde{\bf
Y}}^{\mathtt{md}}_{\bf Q}\\\ {\widetilde{\bf X}}^{\mathtt{md}}_{\bf
Q}\end{array}\right]\bigg{\|}_{\mathcal{H}_{2}}\bigg{)}}$ (21) s.t.
$\displaystyle\Bigg{\|}\left[\begin{array}[]{c}{\widetilde{\bf
Y}}^{\mathtt{md}}_{\bf Q}\\\ {\widetilde{\bf X}}^{\mathtt{md}}_{\bf
Q}\end{array}\right]\Bigg{\|}_{\infty}\leq\min\\{\delta,\alpha\\},$

where as before, ${h\big{(}\gamma,\alpha\big{)}}$ =
$(1+\gamma\alpha)\bigg{(}\bigg{\|}\left[\begin{array}[]{cc}{\widetilde{\bf
M}^{\mathtt{md}}}&{\widetilde{\bf
N}^{\mathtt{md}}}\end{array}\right]\bigg{\|}_{\infty}+\gamma\bigg{)}$.

###### Remark 1

(Numerical Computation) For each fixed $\delta$, the inner optimization is
convex but the dimension of ${\bf Q}(z)$ remains infinite.For numerical
computation, a (FIR) truncation is considered on ${\bf Q}(z)$, such that an
equivalent Semi-Definite Program (SDP) can be formulated for the inner
optimization problem. Details of this is discussed in Appendix E.

###### Remark 2

(Feasibility) Since the subsequent quasi-convex optimization problem in
Theorem-3.8 must include an additional constraint
$\bigg{\|}\left[\begin{array}[]{c}{\widetilde{\bf Y}}^{\mathtt{md}}_{\bf Q}\\\
{\widetilde{\bf X}}^{\mathtt{md}}_{\bf
Q}\end{array}\right]\bigg{\|}_{\infty}<\alpha$ proportional to $\alpha$, this
$\alpha$ should be chosen as small as possible. However, $\alpha$ can’t be
made arbitrarily small and therefore the feasibility of the quasi-convex
program from Theorem-3.8 cannot be guaranteed. This is caused by the fact that
$\alpha$ must be assimilated to the norm of a LCF of a stabilizing controller
for the true plant, whose $H_{\infty}$ attenuation is simultaneously greater
or equal to the one of the optimal $H_{\infty}$ controller for the nominal
model (see Theorem 3.4). This reflects the fact that the feasibility of the
closed loop "learning" problem depends inherently on the performance (with
respect to the model uncertainty of the plant) of the initially chosen
stabilizing controller (the one with which the closed loop "learning" is being
performed).

##  4 Analysis of End-to-End Performance

###  4.1 Sub-optimality guarantee

If we denote by ${{\bf K}^{\mathtt{opt}}}$ the optimal $\mathcal{H}_{2}$
controller for the true plant, then by Assumption 1 there exist stable
additive factors such that ${\bf G}^{\mathtt{pt}}$ = $(\widetilde{\bf
M}^{\mathtt{md}}+\Delta_{\bf\widetilde{M}})^{-1}(\widetilde{\bf
N}^{\mathtt{md}}+\Delta_{\bf\widetilde{N}})$ = $({\bf
N}^{\mathtt{md}}+\Delta_{\bf{N}})({\bf M}^{\mathtt{md}}+\Delta_{\bf{M}})^{-1}$
and furthermore, there always exists a Bezout identity of the true plant that
features the optimal controller ${{\bf K}^{\mathtt{opt}}}=({\bf
Y}^{\mathtt{opt}})^{-1}{\bf
X}^{\mathtt{opt}}={\bf{\widetilde{X}}}^{\mathtt{opt}}({\bf\widetilde{Y}}^{\mathtt{opt}})^{-1}$
as its “central controller”, thus reading:

$\left[\begin{array}[]{cc}({\widetilde{\bf
M}}^{\mathtt{md}}+\Delta_{\bf\widetilde{M}})&({\widetilde{\bf
N}}^{\mathtt{md}}+\Delta_{\bf\widetilde{N}})\\\ -{{\bf
X}}^{\mathtt{opt}}&{{\bf
Y}}^{\mathtt{opt}}\end{array}\right]\left[\begin{array}[]{cc}{\widetilde{\bf
Y}}^{\mathtt{opt}}&-({{\bf N}}^{\mathtt{pt}}+\Delta_{\bf{N}})\\\
{\widetilde{\bf X}}^{\mathtt{opt}}&({{\bf
M}}^{\mathtt{pt}}+\Delta_{\bf{M}})\end{array}\right]=\left[\begin{array}[]{cc}{I_{p}}&{{0}}\\\
{0}&{I_{m}}\end{array}\right]$ (22)

Consequently, the square root of the LQG cost functional for optimal
controller is given by:

$\mathcal{H}({\bf G}^{\mathtt{pt}},{\bf
K}^{\mathtt{opt}})\overset{def}{=}\Bigg{\|}\left[\begin{array}[]{c}{\widetilde{\bf
Y}}^{\mathtt{opt}}\\\ {\widetilde{\bf
X}}^{\mathtt{opt}}\end{array}\right]\left[\begin{array}[]{cc}({\widetilde{\bf
M}}^{\mathtt{md}}+\Delta_{\bf\widetilde{M}})&({\widetilde{\bf
N}}^{\mathtt{md}}+\Delta_{\bf\widetilde{N}})\end{array}\right]\Bigg{\|}_{\mathcal{H}_{2}}$
(23)

Next the main result on the sub-optimality guarantee for the performance of
the robust controller with model uncertainty of radius $\gamma$ is stated. The
proof for Theorem 4.1 is given in Appendix F.

###### Theorem 4.1

Let ${{\bf K}^{\mathtt{opt}}}$ be the optimal LQG controller and ${{\bf
G}^{\mathtt{pt}}}$ be the model of the true plant, with modeling error
uncertainty satisfying
${\Big{\|}\left[\begin{array}[]{cc}\Delta_{\bf\widetilde{M}}&\Delta_{\bf\widetilde{N}}\end{array}\right]\Big{\|}_{\infty}<\gamma}$.
Furthermore, let ${\bf Q_{*}}$ and $\delta_{*}$ denote the solution to (21).
Then, when applying the resulting controller ${{\bf K}^{\mathtt{md}}_{\bf
Q_{*}}}$ in feedback interconnection with the true plant ${{\bf
G}^{\mathtt{pt}}}$, the relative error in the LQG cost is upper bounded by:

$\dfrac{\mathcal{H}({\bf G}^{\mathtt{pt}},{{\bf K}^{\mathtt{md}}_{\bf
Q_{*}}})^{2}-\mathcal{H}({\bf G}^{\mathtt{pt}},{\bf
K}^{\mathtt{opt}})^{2}}{\mathcal{H}({\bf G}^{\mathtt{pt}},{\bf
K}^{\mathtt{opt}})^{2}}\leq\dfrac{1}{\Big{(}1-\gamma\bigg{\|}\left[\begin{array}[]{c}{\bf\widetilde{Y}}^{\mathtt{md}}_{\bf
Q_{*}}\\\ {\bf\widetilde{X}}^{\mathtt{md}}_{\bf
Q_{*}}\end{array}\right]\bigg{\|}_{\infty}\Big{)}^{2}}\times{g\bigg{(}\gamma,\bigg{\|}\left[\begin{array}[]{c}{\widetilde{\bf
Y}}^{\mathtt{md}}_{\bf Q_{*}}\\\ {\widetilde{\bf X}}^{\mathtt{md}}_{\bf
Q_{*}}\end{array}\right]\bigg{\|}_{\infty}\bigg{)}}^{2}-1,$ (24)
$\textrm{where}\hskip
4.0pt\small{g\bigg{(}\gamma,\bigg{\|}\left[\begin{array}[]{c}{\widetilde{\bf
Y}}^{\mathtt{md}}_{\bf Q_{*}}\\\ {\widetilde{\bf X}}^{\mathtt{md}}_{\bf
Q_{*}}\end{array}\right]\bigg{\|}_{\infty}\bigg{)}}\overset{def}{=}\bigg{\|}\left[\begin{array}[]{c}{\widetilde{\bf
Y}}^{\mathtt{md}}_{\bf Q_{*}}\\\ {\widetilde{\bf X}}^{\mathtt{md}}_{\bf
Q_{*}}\end{array}\right]\bigg{\|}_{\infty}\bigg{(}1+{\gamma\bigg{\|}\left[\begin{array}[]{c}{\widetilde{\bf
Y}}^{\mathtt{md}}_{\bf Q_{*}}\\\ {\widetilde{\bf X}}^{\mathtt{md}}_{\bf
Q_{*}}\end{array}\right]\bigg{\|}_{\infty}}\bigg{)}\bigg{(}\bigg{\|}\left[\begin{array}[]{cc}{\widetilde{\bf
M}^{\mathtt{md}}}&{\widetilde{\bf
N}^{\mathtt{md}}}\end{array}\right]\bigg{\|}_{\infty}+\gamma\bigg{)}.$

###### Remark 3

(Optimality vs Robustness) If
$\gamma\bigg{\|}\left[\begin{array}[]{c}{\bf\widetilde{Y}}^{\mathtt{md}}_{\bf
Q_{*}}\\\ {\bf\widetilde{X}}^{\mathtt{md}}_{\bf
Q_{*}}\end{array}\right]\bigg{\|}_{\infty}=\eta$, it’s easy to observe that
$\eta\in(0,1)$. Then it’s immediate to see that the upper bound of the
relative error in the LQG cost increases as a function of $\eta$. The price of
obtaining a faster rate is that the controller becomes less robust to model
uncertainty as pointed out in Mania et al. (2019), Zheng et al. (2020a). It
holds for this case too as shown in Theorem 4.1. In practice, using a
relatively large value for $\eta$ forces a trade-off of optimality for
robustness in the controller design procedure. In general, optimality stands
i.e. better controller performance is guaranteed as $\eta$ goes closer to $0$
and better robustness performance is guaranteed as $\eta$ goes closer to $1$
with the upper bound (24) of relative error in LQG cost might be large. This
is shown with an example below.

Let’s set $\eta=\dfrac{1}{5}$. Then by Theorem 4.1 relative error in the LQG
cost is

$\dfrac{\mathcal{H}({\bf G}^{\mathtt{pt}},{{\bf K}^{\mathtt{md}}_{\bf
Q_{*}}})^{2}-\mathcal{H}({\bf G}^{\mathtt{pt}},{\bf
K}^{\mathtt{opt}})^{2}}{\mathcal{H}({\bf G}^{\mathtt{pt}},{\bf
K}^{\mathtt{opt}})^{2}}\leq
2\times{g\bigg{(}\gamma,\bigg{\|}\left[\begin{array}[]{c}{\widetilde{\bf
Y}}^{\mathtt{md}}_{\bf Q_{*}}\\\ {\widetilde{\bf X}}^{\mathtt{md}}_{\bf
Q_{*}}\end{array}\right]\bigg{\|}_{\infty}\bigg{)}}\bigg{]}^{2}-1.$ (25)

Hence, the the relative error in the LQG cost grows as
$\mathcal{O}(\gamma^{2})$ as long as
$\gamma\bigg{\|}\left[\begin{array}[]{c}{\bf\widetilde{Y}}^{\mathtt{md}}_{\bf
Q_{*}}\\\ {\bf\widetilde{X}}^{\mathtt{md}}_{\bf
Q_{*}}\end{array}\right]\bigg{\|}_{\infty}<\dfrac{1}{5}$.

##  5 Closed Loop Identification Scheme

Figure 2 at the top of next page depicts the closed-loop identification setup
of a potentially unstable noise contaminated plant $\textbf{G}^{\mathtt{md}}$
with control input $u$, noise $\nu$ (taken $w$ = $0$) and output measurement
$y$ (where $u$ and $\nu$ are assumed independent and stationary), provided
that some initial stabilizing controller ${{\bf K}^{\mathtt{md}}}$ is
available beforehand.

Figure 2: Closed loop Identification for Noise Contaminated Plant

If any plant $\mathbf{G}^{\mathtt{pt}}$ is LTI and stable, for open-loop
identification it can be written that:

$y=\textbf{G}^{\mathtt{pt}}u+\nu$ (26)

Here $u$ and $y$ are available through measurements, then the model of the
plant can be estimated as $\mathbf{G}^{\mathtt{md}}$ (Anderson (1998)).
Suppose that the controller ${{\bf K}^{\mathtt{md}}}=({\bf
Y}^{\mathtt{md}})^{-1}{\bf
X}^{\mathtt{md}}={\bf{\widetilde{X}}}^{\mathtt{md}}({\bf\widetilde{Y}}^{\mathtt{md}})^{-1}$,
where ${\bf X}^{\mathtt{md}}$, ${\bf Y}^{\mathtt{md}}$ are left coprimes and
${\bf\widetilde{X}}^{\mathtt{md}}$, ${\bf\widetilde{Y}}^{\mathtt{md}}$ are
right coprimes of ${{\bf K}^{\mathtt{md}}}$. Then by the coprimeness there
exists ${{\bf M}}^{\mathtt{md}}$, ${{\bf N}}^{\mathtt{md}}$, ${\widetilde{\bf
M}}^{\mathtt{md}}$, ${\widetilde{\bf N}}^{\mathtt{md}}$ with

${\bf{X}}^{\mathtt{md}}{\bf{N}}^{\mathtt{md}}+{\bf{Y}}^{\mathtt{md}}{\bf{M}}^{\mathtt{md}}={I}_{m},\hskip
2.0pt{\bf\widetilde{M}}^{\mathtt{md}}{\bf\widetilde{Y}}^{\mathtt{md}}+{\bf\widetilde{N}}^{\mathtt{md}}{\bf\widetilde{X}}^{\mathtt{md}}={I}_{p}.$
(27)

This means that ${\bf G}^{\mathtt{md}}$ =
${\bf{N}}^{\mathtt{md}}{({\bf{M}}^{\mathtt{md}})}^{-1}$ =
${({\bf\widetilde{M}}^{\mathtt{md}})}^{-1}{\bf\widetilde{N}}^{\mathtt{md}}$ is
the nominal model stabilized by ${\bf K}^{\mathtt{md}}$. Alternatively, it can
be started with a RCF representation
${\bf{N}}^{\mathtt{md}}({\bf{M}}^{\mathtt{md}})^{-1}$ of the nominal model
${\bf G}^{\mathtt{md}}$ and then choose that particular representation
${\bf{\widetilde{X}}}^{\mathtt{md}}({\bf\widetilde{Y}}^{\mathtt{md}})^{-1}$ of
the known stabilizing ${\bf K}^{\mathtt{md}}$ so that equation (27) holds. The
set of all plants ${\bf G}_{\bf R}^{\mathtt{md}}$ stabilized by ${\bf
K}^{\mathtt{md}}$ is given by Dual Youla-Kuçera Parameterization in
Proposition 2.8:

${\bf G}_{\bf
R}^{\mathtt{md}}={({\bf{N}}^{\mathtt{md}}+{\bf{\widetilde{Y}}}^{\mathtt{md}}\mathbf{R}^{\mathtt{md}})({\bf{M}}^{\mathtt{md}}-{\bf{\widetilde{X}}}^{\mathtt{md}}\mathbf{R}^{\mathtt{md}})^{-1}}$
(28)

for some stable $\mathbf{R}^{\mathtt{md}}\in\mathbb{R}^{p\times m}$ and
${\bf{\widetilde{X}}}^{\mathtt{md}}({\bf\widetilde{Y}}^{\mathtt{md}})^{-1}$ is
the right coprime factorization of ${\bf K}^{\mathtt{md}}$.

Suppose that
${({\bf\widetilde{M}}^{\mathtt{md}})}^{-1}{\bf\widetilde{N}}^{\mathtt{md}}$ is
a left coprime fractional description of the nominal model ${\bf
G}^{\mathtt{md}}$ such that

$\begin{bmatrix}{\mathbf{Y}}^{\mathtt{md}}&{\mathbf{X}}^{\mathtt{md}}\\\
-{\mathbf{\widetilde{N}}}^{\mathtt{md}}&{\mathbf{\widetilde{M}}}^{\mathtt{md}}\end{bmatrix}\begin{bmatrix}{\mathbf{M}}^{\mathtt{md}}&-{\mathbf{\widetilde{X}}}^{\mathtt{md}}\\\
{\mathbf{N}}^{\mathtt{md}}&{\mathbf{\widetilde{Y}}}^{\mathtt{md}}\end{bmatrix}=\begin{bmatrix}{I}_{m}&0\\\
0&{I}_{p}\end{bmatrix}$ (29)

###### Lemma 5.1

The noise contaminated plant set-up of Figure 2 is identical to Figure 1 with
input $u$, noise $\nu$ and output $y$, it implies

$y={\bf G}_{\bf R}^{\mathtt{md}}u+\nu$ (30)

where ${\bf G}_{\bf R}^{\mathtt{md}}$ is given by (28).

The key idea dating back to Anderson (1998) is to identify the stable dual-
Youla parameter ${\bf R}^{\mathtt{md}}$ from Theorem 2.7 rather than ${\bf
G}^{\mathtt{md}}$, thus recasting the problem in a standard, open-loop
identification form. More specifically, direct inspection of Figure 2 shows
that

$e_{2}={\bf R}^{\mathtt{md}}e_{1}+({\bf\widetilde{M}}^{\mathtt{md}}-{\bf
R}^{\mathtt{md}}{\bf X}^{\mathtt{md}})\nu={\bf R}^{\mathtt{md}}(e_{1}-{\bf
X}^{\mathtt{md}}\nu)+{\bf\widetilde{M}}^{\mathtt{md}}\nu.$ (31)

In (31) we have the knowledge of ${\bf X}^{\mathtt{md}}$; $e_{1}$ and $e_{2}$
are available from measurements with noise $\nu$. The recent algorithm from
Sarkar et al. (2020) given below is employed toward identifying the dual-Youla
parameter.
Details of this idea and proof for Lemma 5.1 is given in Appendix G.1, also
choice of noise is discussed in Appendix G.2.

###  5.1 Identification Algorithms

Identification algorithms from Sarkar et al. (2020) has been employed here.
Preliminaries of system identification is given in Appendix G.3. The state
space representation of ${\bf R}^{\mathtt{md}}$ in $e_{2}={\bf
R}^{\mathtt{md}}u+{\bf\widetilde{M}}^{\mathtt{md}}\nu$ with $u$ = $e_{1}-{\bf
X}^{\mathtt{md}}\nu$ is:

$\displaystyle l_{t+1}$ $\displaystyle=A_{R}l_{t}+B_{R}u_{t}+\eta_{t+1}$ (32)
$\displaystyle z_{t}$
$\displaystyle=C_{R}l_{t}+{\bf\widetilde{M}}^{\mathtt{md}}\nu_{t}$

Here two important assumptions is stated as below:

###### Assumption 2

The noise process $\\{\eta_{t}\\}_{t=1}^{\infty}$ in the dynamics of ${\bf
R}^{\mathtt{md}}$ are i.i.d., and isotropic with sub-gaussian parameter 1. The
noise process $\\{r_{t}\\}_{t=1}^{\infty}$, $\\{w_{t}\\}_{t=1}^{\infty}$ and
$\\{\nu_{t}\\}_{t=1}^{\infty}$ are Gaussian processes with mean function
$m_{r}(t)=m_{w}(t)=m_{\nu}(t)$ $0$, and their spectral density
${\bf\phi}_{r}(\omega)$, ${\bf\phi}_{w}(\omega)$ and ${\bf\phi}_{\nu}(\omega)$
satisfy the constraint in Appendix G.

###### Assumption 3

There exists constants $\beta,R\geq 1$ s.t.
$\|\mathcal{T}_{0,\infty}\|_{2}<\beta$ and
$\dfrac{\|\mathcal{TO}_{k,d}\|_{2}}{\|\mathcal{T}_{0,\infty}\|_{2}}\leq\mathcal{R}$.
$\beta$ exists since ${\bf R}^{\mathtt{md}}$ is stable.

The Algorithms for system identification is illustrated in Appendix G.

####  5.1.1 Probabilistic guarantees

Let’s define, $T_{*}(\delta)$ =
$\inf\\{T|d_{*}(T,\delta)\in\mathcal{D}(T),d_{*}(T,\delta)\leq
2d_{*}(\frac{T}{256},\delta)\\}$ where, $d_{*}(T,\delta)$ =
$\inf\\{d|16\beta\mathcal{R}\alpha(d)$
$\geq\Big{\|}\mathcal{\widehat{H}}_{0,\widehat{d},\widehat{d}}-\mathcal{\widehat{H}}_{0,\infty,\infty}\Big{\|}_{2}\\}$,
with $\mathcal{\widehat{H}}_{p,q,r}$ is the $(p,q,r)$ \- dimensional estimated
Hankel matrix. Whenever $T\geq T_{*}(\delta)$ for the failure probability
$\delta$, then it follows with the probability at least $(1-\delta)$ that

$\Big{\|}\mathcal{\widehat{H}}_{0,\widehat{d},\widehat{d}}-\mathcal{\widehat{H}}_{0,\infty,\infty}\Big{\|}_{2}\leq
12c\beta\mathcal{R}\bigg{(}\sqrt{\dfrac{m\widehat{d}+p\widehat{d}^{2}+\widehat{d}log(T/\delta)}{T}}\bigg{)}.$
(33)

###  5.2 Sample Complexity

###### Lemma 5.2

The norm of the identification error incurred by the proposed scheme after
Algorithm-1 and 2 is bounded by:

$\Big{\|}{\bf{R}}^{\mathtt{md}}-{\bf
R}^{\mathtt{pt}}\Big{\|}_{\infty}\leq\Big{\|}\mathcal{\widehat{H}}_{0,\widehat{d},\widehat{d}}-\mathcal{\widehat{H}}_{0,\infty,\infty}\Big{\|}_{2}\leq
12c\beta\mathcal{R}\bigg{(}\sqrt{\dfrac{m\widehat{d}+p\widehat{d}^{2}+\widehat{d}log(T/\delta)}{T}}\bigg{)}$

Finally, the error on the model uncertainty can be directly checked using
Lemma 5.2 as:

$\displaystyle\Big{\|}\left[\begin{array}[]{cc}\Delta_{\bf\tilde{M}}&\Delta_{\bf\tilde{N}}\end{array}\right]\Big{\|}_{\infty}$
$\displaystyle=\Big{\|}\left[\begin{array}[]{cc}(\tilde{\bf
M}^{\mathtt{md}}+{\bf X}^{\mathtt{md}}{\bf R}^{\mathtt{md}})&({\tilde{\bf
N}}^{\mathtt{md}}-{\bf Y}^{\mathtt{md}}{\bf
R}^{\mathtt{md}})\end{array}\right]-\left[\begin{array}[]{cc}(\tilde{\bf
M}^{\mathtt{md}}+{\bf X}^{\mathtt{md}}{\bf R}^{\mathtt{pt}})&({\tilde{\bf
N}}^{\mathtt{md}}-{\bf Y}^{\mathtt{md}}{\bf
R}^{\mathtt{pt}})\end{array}\right]\Big{\|}_{\infty}$
$\displaystyle=\Big{\|}\left[\begin{array}[]{cc}{\bf X}^{\mathtt{md}}&{\bf
Y}^{\mathtt{md}}\end{array}\right]({\bf R}^{\mathtt{md}}-{\bf
R}^{\mathtt{pt}})\Big{\|}_{\infty}\leq\Big{\|}\left[\begin{array}[]{cc}{\bf
X}^{\mathtt{md}}&{\bf
Y}^{\mathtt{md}}\end{array}\right]\Big{\|}_{\infty}\Big{\|}{\bf
R}^{\mathtt{md}}-{\bf{R}}^{\mathtt{pt}}\Big{\|}_{\infty}$

Consequently, the uncertainty level on the LCF of the model satisfies
$\Big{\|}\left[\begin{array}[]{cc}\Delta_{\bf\tilde{M}}&\Delta_{\bf\tilde{N}}\end{array}\right]\Big{\|}_{\infty}<\gamma$.

###### Theorem 5.3

Define ${s}$ = $144\Big{\|}\left[\begin{array}[]{cc}{\bf X}^{\mathtt{md}}&{\bf
Y}^{\mathtt{md}}\end{array}\right]\Big{\|}_{\infty}^{2}c^{2}\beta^{2}\mathcal{R}^{2}$.
Then, the robust controller will achieve the relative cost within the bound
with probability $(1-\delta)$ provided $T\geq\max\\{T_{s},T_{*}({\delta})\\}$.
Here, $T_{s}$ is the right most zero of $g(T)$ =
$\gamma^{2}T-s\widehat{d}log(T/\delta)-s(m\widehat{d}+p\widehat{d}^{2})$. If
$g(T)$ doesn’t have any zero for $T>0$, then define $T_{s}=0$ and
$T_{*}(\delta)$ =
$\inf\\{T|d_{*}(T,\delta)\in\mathcal{D}(T),d_{*}(T,\delta)\leq
2d_{*}(\frac{T}{256},\delta)\\}$,

where, $d_{*}(T,\delta)$ =
$\inf\\{d|16\beta\mathcal{R}\alpha(d)\geq\Big{\|}\mathcal{\widehat{H}}_{0,d,d}-\mathcal{\widehat{H}}_{0,\infty,\infty}\Big{\|}_{2}\\}$,

$\mathcal{D}(T)=\\{d\in\mathbb{N}|d\leq\dfrac{T}{cm^{2}log^{3}(Tm/\delta)}\\}$
and $\alpha(h)=\sqrt{h}.\big{(}\sqrt{\dfrac{m+hp+log(T/\delta)}{T}}\big{)}$.

Combining Theorem 5.3 with Theorem 4.1, it follows that with high probability
the suboptimality gap behaves as

$\dfrac{\mathcal{H}({\bf G}^{\mathtt{pt}},{{\bf K}^{\mathtt{md}}_{\bf
Q_{*}}})^{2}-\mathcal{H}({\bf G}^{\mathtt{pt}},{\bf
K}^{\mathtt{opt}})^{2}}{\mathcal{H}({\bf G}^{\mathtt{pt}},{\bf
K}^{\mathtt{opt}})^{2}}\sim\mathcal{O}\Big{(}\sqrt{\dfrac{logT}{T}}\Big{)}$

Finally, we note here that the resulted sample complexity is on par with the
existing methods from Zheng et al. (2020a) and Dean et al. (2018).

##  6 Conclusion and Future work

In this paper, we have provided the sample complexity bounds for a robust
controller synthesis procedure for LQG problems with unknown dynamics, able to
cope with unstable plants. We combined finite-time, non-parametric LTI system
identification with the Youla parameterization for robust stabilization under
uncertainty on the coprime factors of the plant. One exciting avenue for
future research is the online learning LQG control problem under the same type
of model uncertainty. Another direction is to work out the sample complexity
of learning the optimal state feedback (LQR) controller in tandem with the
optimal state-observer (Kalman Filter (Tsiamis et al. (2020))) for a
potentially unstable system. Combining these two results, should yield
precisely the optimal LQG controller discussed above and reveal the separation
principle within this framework.

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###### Contents

  1. 1 Introduction
    1. 1.1 The Linear Quadratic Gaussian Problem
    2. 1.2 The Main Technical Ingredient
    3. 1.3 Contributions
  2. 2 General Setup and Technical Preliminaries
    1. 2.1 Standard Unity Feedback
    2. 2.2 The Youla-Kuçera Parameterization
    3. 2.3 Dual Youla-Kuçera Parameterization
  3. 3 Robust Controller Synthesis
    1. 3.1 Quasi-convex formulation
  4. 4 Analysis of End-to-End Performance
    1. 4.1 Sub-optimality guarantee
  5. 5 Closed Loop Identification Scheme
    1. 5.1 Identification Algorithms
      1. 5.1.1 Probabilistic guarantees
    2. 5.2 Sample Complexity
  6. 6 Conclusion and Future work
  7. A Related Works
  8. B Norm/Inequality preliminaries
  9. C Closed Loop Maps
  10. D Controller Parameterization
  11. E Quasi Convex Optimization
  12. F Suboptimality guarantee proof
  13. G Non-asymptotic System Identification
    1. G.1 Noise Contaminated Plant
    2. G.2 Choice of Noise
    3. G.3 Identification Preliminaries
    4. G.4 Identification Algorithms
    5. G.5 End to End Complexity Proof

## Appendix

This appendix is divided into eight parts. Appendix A presents a brief review
about the related works on LTI systems with coprime Factorization, controller
parameterization, non-asymptotic system identification and robust control in
reinforcement learning. Appendix B provides a handful mathematical
preliminaries on norm identities and inequalities (Zhou et al. (1996)). An
overview of the classical Youla Parameterization framework is given in
Appendix C. Also the closed loop maps of the system, proof for Lemma 3.3,
Theorem 3.4 and Proposition 3.5 is stated here. Appendix D presents the proof
for Theorem 3.6, which includes the equivalent $\mathcal{H}_{2}$ norm LQG cost
derivation and proof for Proposition 3.7. Appendix E discusses about the
quasi-convex approximation with proof for Theorem 3.8. Also numerical
computation has been discussed for Finite Impulse Response (FIR) truncation
which leads Semi-Definite Program (SDP) formulation for quasi-convex problem.
Appendix F completes the suboptimality guarantee proof in Theorem 4.1.
Appendix G presents the non-asymptotic system identification (Sarkar et al.
(2020)) for identifying Dual-Youla parameter by using Ordinary Least Square
algorithm. $\mathcal{H}_{\infty}$ bound Lemma 5.2 is also proved here.

##  A Related Works

Recently, considerable research efforts have been spent towards the finite
time (non-asymptotic) learning of the optimal Linear Quadratic Gaussian (LQG)
regulator for an unknown plant by employing the modern statistical and
optimization tools from the machine learning framework.

Identification of Unknown Dynamical Systems: Model identification of systems
has a long history in electrical engineering. The work of Ljung (1999) covers
the classical asymptotic results on identification for Linear and Time
Invariant (LTI) models. Classical methods are based on subspaces
identification in the absence of disturbances or noise, whereas state-space
parameters of a system are identified with the Hankel matrix obtained from
input-output measurements, followed by balanced realization of the Grammians,
via Hankel singular value decomposition (Ljung (1999), Overschee and Moor
(2012)). More recent approaches use rank minimization methods for system
identification by relaxing rank constraint as in Fazel et al. (2013), Grussler
et al. (2018). Very recently, LTI systems identification from a single
trajectory, with statistical guarantees (of high probability) has been
proposed based on a standard ordinary least squares criterion (OLS) in
Simchowitz et al. (2018), Sarkar and Rakhlin (2019), using measure
concentration type results. Time series online prediction is presented in this
framework in Hazan et al. (2018), Agarwal et al. (2018). More recent non-
asymptotic system identification algorithm inspired by the Kalman-Ho algorithm
from a single input-output trajectory is included in Oymak and Ozay (2019),
Sarkar et al. (2020). Multiple independent trajectories based system
estimation are presented in Tu et al. (2017), Zheng and Li (2020). The
algorithm for the system estimation in Oymak and Ozay (2019) needs the prior
knowledge of the model order which seriously limits its applicability in
practice. Other related approaches such as Campi and Weyer (2002), Shah et al.
(2012), Hardt et al. (2018) suffer from the same limitation. However, recent
results such as the algorithm for system estimation in Sarkar et al. (2020)
are able to surpass this problem. Their identification approach is to estimate
the system Hankel-like matrix from noisy data using OLS with high probability.
However, the method is restricted to open-loop stable systems. To overcome
this hypothesis, the results from Hof and Schrama (1992) have been used in
this paper for parameterizing all plants/models which are stabilized by a
prescribed controller, known as the Dual-Youla parameterization (Vidyasagar
(1985)), which re-casts the identification of the unstable, noise contaminated
plant into the problem of identifying the stable dual Youla parameter.
Therefore, the key identification idea is to identify the dual Youla parameter
from data rather than identifying the model itself, which is shown as an open
loop identification problem. As the dual Youla parameter is stable, the
algorithm from Sarkar et al. (2020) can be used to identify it with high
probability. For more details about dual Youla parameterizations and
converting closed loop approximation problems to open loop we refer to Schrama
(1991), Anderson (1998).

Controller Design: There is a rich history of convex parameterizations of
stabilizing controllers, under some restricting hypotheses guaranteeing their
existence (Boyd and Barratt (1991b)). In general, iterative schemes allowing
the design of a controller in tandem with the system identification of the
plant render (highly) non-convex problems and the direct search for the
controller inefficient, even intractable. Despite the non-convexity problem,
convergence guarantees have been given for gradient-based algorithms over
controller parameterization in few recent works such as Fazel et al. (2018),
Malik et al. (2019), Furieri et al. (2021). Another approach is based on
Lyapunov-based parameterizations in the state-space domain to obtain linear
matrix inequalities (LMIs) as in Boyd et al. (1994), Scherer et al. (1997).
Changing variables in different coordinates is also a popular approach in
controller parameterization (Gahinet and Apkarian (1994), Zhou et al. (1996)).
A parameterization over state-feedback policies has been proposed in Goulart
et al. (2006). The classical Youla-Kucera parameterization from Youla et al.
(1976), Kucera (1975) requires only approximate knowledge of the system and
gives a unique way in the frequency domain to get a convex formulation.
Recently, another frequency based parameterization has been proposed and
dubbed System Level Synthesis (SLS) in Wang et al. (2019), Xue and Matni
(2021). More recently, Input-Output Parameterization (IOP) has been introduced
in Furieri et al. (2019). Ultimately, some sort of equivalence exists between
IOP, SLS and Youla, under the notion of closed loop convexity (Boyd and
Barratt (1991a)), which is basically entails designing the closed loop maps
directly rather than designing a controller. These equivalences between IOP,
SLS and Youla has been discussed extensively in Zheng et al. (2020b), Tseng
(2021). However, and this is important, there are still essential distinctions
between these paramterizations in terms of their ability to allow the design
and the subsequent distributed implementation of optimal controllers. Here
preliminary results show that certain crippling working hypotheses, such as
the stability of the plant (assumed in the SLS, IOP frameworks), can be
elegantly overcome via the Youla parameterization.

Robust Control: Non-asymptotic estimation provides a family of plants
described by a nominal model with a set of bounded uncertainty/modeling error
for a fixed amount of data. Hence it is necessary to ensure that the designed
controller is robustly stable with performance guarantee for the family of
plants, known as robust controller design (Zhou et al. (1996)). When the
modeling errors is unstructured and arbitrary norm-bounded LTI operators,
classical “small-gain” type theorems can be used to solve the robust
stabilization problem with no conservatism. If the modeling error is
structured, other sophisticated techniques similar to $\mu$-synthesis
techniques in Doyle (1982), integral quadratic constraints (IQC) in Megretski
and Rantzer (1997), and sum-of-squares (SOS) optimization in Scherer (2006)
are less traditional than small-gain approaches as in Zheng et al. (2020a).
The strongest method available in the theoretical control machinery for coping
with systems with uncertainty are additive or multiplicative perturbations on
the coprime factors of the nominal model (Ionescu et al. (1999)). The modeling
errors are allowed to be unstructured and arbitrary LTI operators, with a mere
upper bound on their $\mathcal{H}_{\infty}$ norm. This describes
simultaneously uncertainty on the locations of both the poles and the Smith
zeros of the plant and on its McMillan degree. Modeling uncertainty as
additive perturbations on the coprimes also has the advantage over additive
uncertainty on the plant, in the fact that it enables to deal with unstable
plants.

Online Control: Applying online learning techniques for the learning and
optimal control of LTI systems, via time-varying cost functions has received a
lot of attention within the last year. These results bring significant
algorithmic benefits over the more classical online and adaptive control
methodologies, which have an abundant history of research (Sastry and Bodson
(2011), Ioannou and Sun (2012)). Here we restrict the literature review to
online control with unknown dynamics in the regret minimization framework
similar to Simchowitz et al. (2020). The notion of “regret” arises from the
time varying cost function. Essentially, the regret measures how well a system
adapts to the time varying costs. Most of the previous results in regret
minimization framework assume either Gaussian perturbations (or no
perturbations at all) for the fully-observed linear dynamic system and manage
to obtaining relatively very low regret margins in the online LQR setting as
depicted in Abbasi-Yadkori and Szepesvari (2011), Dean et al. (2018), Mania et
al. (2019), Cohen et al. (2019). A regret bound for fully observable systems
with adversarial disturbances able to cope with partially observable systems
(under convexity assumption for adversarial disturbances and strong-convexity
assumptions for semi-adversarial noises) in the time-varying case has been
very recently proposed in Simchowitz et al. (2020). Another regret bound for
the case of stochastic perturbations, time-varying strongly-convex functions,
and partially observed states has been provided in Lale et al. (2020). Regret
minimization with Kalman filtering has been discussed in Sabag and Hassibi
(2021). Distributed online LQ problem has been studied for identical LTI
systems in Chang and Shahrampour (2020) and for model-free systems in Furieri
et al. (2021).

##  B Norm/Inequality preliminaries

Some useful norm identities and inequalities for $\mathcal{H}_{2}$ and
$\mathcal{H}_{\infty}$ is presented here from Zhou et al. (1996) that are key
to the proofs.

First, the following norm triangular inequalities is defined:

$\begin{split}\|{\bf G_{1}+G_{2}}\|_{\mathcal{H}_{2}}\leq\|{\bf
G_{1}}\|_{\mathcal{H}_{2}}+\|{\bf G_{2}}\|_{\mathcal{H}_{2}},(\forall){\bf
G_{1},G_{2}}\in\mathcal{RH}_{2}\\\ \|{\bf G_{1}+G_{2}}\|_{\infty}\leq\|{\bf
G_{1}}\|_{\infty}+\|{\bf G_{2}}\|_{\infty},(\forall){\bf
G_{1},G_{2}}\in\mathcal{RH}_{\infty}\end{split}$ (34)

Since $\|.\|_{\infty}$ is an induced norm, we have the following sub-
multiplicative property

$\|{\bf G_{1}G_{2}}\|_{\infty}\leq\|{\bf G_{1}}\|_{\infty}\|{\bf
G_{2}}\|_{\infty},(\forall){\bf G_{1},G_{2}}\in\mathcal{RH}_{\infty}$ (35)

The sub multiplicative property in (35) does not hold for the
$\mathcal{H}_{2}$ norm since it is not an induced norm. Instead we have the
result below.

###### Lemma B.1

For TFM ${\bf G_{1}}$, ${\bf G_{2}}$ $\in\mathcal{RH}_{\infty}$, it can be
stated that

$\begin{split}\|{\bf G_{1}G_{2}}\|_{\mathcal{H}_{2}}\leq\|{\bf
G_{1}}\|_{\mathcal{H}_{2}}\|{\bf G_{2}}\|_{\infty},(\forall){\bf
G_{1}}\in\mathcal{RH}_{2},{\bf G_{2}}\in\mathcal{RH}_{\infty}\\\ \|{\bf
G_{1}G_{2}}\|_{\mathcal{H}_{2}}\leq\|{\bf G_{1}}\|_{\infty}\|{\bf
G_{2}}\|_{\mathcal{H}_{2}},(\forall){\bf G_{1}}\in\mathcal{RH}_{\infty},{\bf
G_{2}}\in\mathcal{RH}_{2}\end{split}$ (36)

Finally, from small-gain theorem in Theorem C.1, we have the following useful
inequalities by taking power expansion of the matrix inverse.

###### Lemma B.2

For TFM ${\bf G_{1}}$, ${\bf G_{2}}$ $\in\mathcal{RH}_{\infty}$ with $\|{\bf
G_{1}}\|_{\infty}\times\|{\bf G_{2}}\|_{\infty}<1$, we have

$\begin{split}\|(I-{\bf G_{1}G_{2}})^{-1}\|_{\infty}\leq\dfrac{1}{1-\|{\bf
G_{1}}\|_{\infty}\|{\bf G_{2}}\|_{\infty}}\\\ \|(I+{\bf
G_{1}G_{2}})^{-1}\|_{\infty}\leq\dfrac{1}{1-\|{\bf G_{1}}\|_{\infty}\|{\bf
G_{2}}\|_{\infty}}\end{split}$ (37)

##  C Closed Loop Maps

The Bezout identity is retrieved before finding the closed loop maps
associated with it by using (14) as below:

$\left[\begin{array}[]{cc}{\bf\Phi}_{11}^{-1}&0\\\
0&I_{m}\end{array}\right]\left[\begin{array}[]{cc}{\bf\Phi}_{11}&0\\\
0&{\bf\Phi}_{22}\end{array}\right]\left[\begin{array}[]{cc}I_{p}&0\\\
0&{\bf\Phi}_{22}^{-1}\end{array}\right]=\left[\begin{array}[]{cc}{I_{p}}&{0}\\\
{0}&{I_{m}}\end{array}\right],$

$\left[\begin{array}[]{cc}{\bf\Phi}_{11}^{-1}&0\\\
0&I_{m}\end{array}\right]\left[\begin{array}[]{cc}({\widetilde{\bf
M}}^{\mathtt{md}}+\Delta_{\bf\widetilde{M}})&({\widetilde{\bf
N}}^{\mathtt{md}}+\Delta_{\bf\widetilde{N}})\\\ -{{\bf X}}^{\mathtt{md}}_{\bf
Q}&{{\bf Y}}^{\mathtt{md}}_{\bf
Q}\end{array}\right]\left[\begin{array}[]{cc}{\widetilde{\bf
Y}}^{\mathtt{md}}_{\bf Q}&-({{\bf N}}^{\mathtt{md}}+\Delta_{\bf{N}})\\\
{\widetilde{\bf X}}^{\mathtt{md}}_{\bf Q}&({{\bf
M}}^{\mathtt{md}}+\Delta_{\bf{M}})\end{array}\right]\left[\begin{array}[]{cc}I_{p}&0\\\
0&{{\bf\Phi}}_{22}^{-1}\end{array}\right]=\left[\begin{array}[]{cc}{I_{p}}&{0}\\\
{0}&{I_{m}}\end{array}\right],$

$\small\left[\begin{array}[]{cc}{\bf\Phi}_{11}^{-1}({\widetilde{\bf
M}}^{\mathtt{md}}+\Delta_{\bf\widetilde{M}})&{\bf\Phi}_{11}^{-1}({\widetilde{\bf
N}}^{\mathtt{md}}+\Delta_{\bf\widetilde{N}})\\\ -{{\bf X}}^{\mathtt{md}}_{\bf
Q}&{{\bf Y}}^{\mathtt{md}}_{\bf
Q}\end{array}\right]\left[\begin{array}[]{cc}{\widetilde{\bf
Y}}^{\mathtt{md}}_{\bf Q}&-({{\bf
N}}^{\mathtt{md}}+\Delta_{\bf{N}}){\bf\Phi}_{22}^{-1}\\\ {\widetilde{\bf
X}}^{\mathtt{md}}_{\bf Q}&({{\bf
M}}^{\mathtt{md}}+\Delta_{\bf{M}}){\bf\Phi}_{22}^{-1}\end{array}\right]=\left[\begin{array}[]{cc}{I_{p}}&{0}\\\
{0}&{I_{m}}\end{array}\right].$ (38)

Proof for Lemma 3.3: From DCF matrix ${\bf\Phi}$ in (14), ${\bf\Phi}_{11}$ =
$({\widetilde{\bf
M}^{\mathtt{md}}}+\Delta_{\bf{\widetilde{M}}}){\widetilde{\bf
Y}}^{\mathtt{md}}_{\bf Q}+({\widetilde{\bf
N}^{\mathtt{md}}}+\Delta_{\bf{\widetilde{N}}}){\widetilde{\bf
X}}^{\mathtt{md}}_{\bf Q}$ and ${\bf{\Phi}}_{22}$ = ${\bf
X}^{\mathtt{md}}_{\bf Q}({\bf N}^{\mathtt{md}}+\Delta_{\bf N})+{\bf
Y}^{\mathtt{md}}_{\bf Q}({\bf M}^{\mathtt{md}}+\Delta_{\bf M})$. Next using
Bezout identity for nominal model in (12) it follows that

$\displaystyle{\bf\Phi}_{11}$
$\displaystyle=\left[\begin{array}[]{cc}({\widetilde{\bf
M}^{\mathtt{md}}}+\Delta_{\bf{\widetilde{M}}})&({\widetilde{\bf
N}^{\mathtt{md}}}+\Delta_{\bf{\widetilde{N}}})\end{array}\right]\left[\begin{array}[]{c}{\widetilde{\bf
Y}}^{\mathtt{md}}_{\bf Q}\\\ {\widetilde{\bf X}}^{\mathtt{md}}_{\bf
Q}\end{array}\right]$ $\displaystyle=\left[\begin{array}[]{cc}{\widetilde{\bf
M}^{\mathtt{md}}}&{\widetilde{\bf
N}^{\mathtt{md}}}\end{array}\right]\left[\begin{array}[]{c}{\widetilde{\bf
Y}}^{\mathtt{md}}_{\bf Q}\\\ {\widetilde{\bf X}}^{\mathtt{md}}_{\bf
Q}\end{array}\right]+\left[\begin{array}[]{cc}\Delta_{\bf\widetilde{M}}&\Delta_{\bf\widetilde{N}}\end{array}\right]\left[\begin{array}[]{c}{\widetilde{\bf
Y}}^{\mathtt{md}}_{\bf Q}\\\ {\widetilde{\bf X}}^{\mathtt{md}}_{\bf
Q}\end{array}\right]=I_{p}+\left[\begin{array}[]{cc}\Delta_{\bf\widetilde{M}}&\Delta_{\bf\widetilde{N}}\end{array}\right]\left[\begin{array}[]{c}{\widetilde{\bf
Y}}^{\mathtt{md}}_{\bf Q}\\\ {\widetilde{\bf X}}^{\mathtt{md}}_{\bf
Q}\end{array}\right];$

$\displaystyle{\bf\Phi}_{22}$ $\displaystyle=\left[\begin{array}[]{cc}{\bf
X}^{\mathtt{md}}_{\bf Q}&{\bf Y}^{\mathtt{md}}_{\bf
Q}\end{array}\right]\left[\begin{array}[]{c}({\bf N}^{\mathtt{md}}+\Delta_{\bf
N})\\\ ({\bf M}^{\mathtt{md}}+\Delta_{\bf M})\end{array}\right]$
$\displaystyle=\left[\begin{array}[]{cc}{\bf X}^{\mathtt{md}}_{\bf Q}&{\bf
Y}^{\mathtt{md}}_{\bf Q}\end{array}\right]\left[\begin{array}[]{c}{\bf
N}^{\mathtt{md}}\\\ {\bf
M}^{\mathtt{md}}\end{array}\right]+\left[\begin{array}[]{cc}{\bf
X}^{\mathtt{md}}_{\bf Q}&{\bf Y}^{\mathtt{md}}_{\bf
Q}\end{array}\right]\left[\begin{array}[]{c}\Delta_{\bf N}\\\ \Delta_{\bf
M}\end{array}\right]=I_{m}+\left[\begin{array}[]{cc}{\bf X}^{\mathtt{md}}_{\bf
Q}&{\bf Y}^{\mathtt{md}}_{\bf
Q}\end{array}\right]\left[\begin{array}[]{c}\Delta_{\bf N}\\\ \Delta_{\bf
M}\end{array}\right].$

Before giving the proof for Theorem 3.4, the small gain theorem is stated
here.

######  Theorem C.1 (Small Gain Theorem)

(Ionescu et al., 1999, Theorem 7.4.1/ page 225 ) Let ${\bf
G}_{1}\in\mathbb{R}(z)^{p\times m}$ and ${\bf G}_{2}\in\mathbb{R}(z)^{m\times
p}$ be two TFM’s respectively. If $\|{\bf
G}_{1}\|_{\infty}\leq\dfrac{1}{\gamma}$ and $\|{\bf
G}_{2}\|_{\infty}\leq\gamma$, for some $\gamma>0$, then the closed loop
feedback system of ${\bf G}_{1}$ and ${\bf G}_{2}$ is internally stable.

Proof for Theorem 3.4: For any stable ${\bf Q}$ satisfying
$\small\Bigg{\|}\left[\begin{array}[]{c}{\widetilde{\bf Y}}^{\mathtt{md}}_{\bf
Q}\\\ {\widetilde{\bf X}}^{\mathtt{md}}_{\bf
Q}\end{array}\right]\Bigg{\|}_{\infty}\leq\dfrac{1}{\gamma}$ it follows that
${\bf\Phi}_{11}=\bigg{(}{I_{p}}+$
$\small\left[\begin{array}[]{cc}\Delta_{\bf\widetilde{M}}&\Delta_{\bf\widetilde{N}}\end{array}\right]\left[\begin{array}[]{c}{\widetilde{\bf
Y}}^{\mathtt{md}}_{\bf Q}\\\ {\widetilde{\bf X}}^{\mathtt{md}}_{\bf
Q}\end{array}\right]\bigg{)}$ is unimodular (square and stable with a stable
inverse) due to the fact that: (a) The term
$\bigg{(}I_{p}+\small\left[\begin{array}[]{cc}\Delta_{\bf\widetilde{M}}&\Delta_{\bf\widetilde{N}}\end{array}\right]\left[\begin{array}[]{c}{\widetilde{\bf
Y}}^{\mathtt{md}}_{\bf Q}\\\ {\widetilde{\bf X}}^{\mathtt{md}}_{\bf
Q}\end{array}\right]\bigg{)}$ is stable since all factors are stable and (b)
We know that
$\small\Big{\|}\left[\begin{array}[]{cc}\Delta_{\bf\widetilde{M}}&\Delta_{\bf\widetilde{N}}\end{array}\right]\Big{\|}_{\infty}<\gamma$
from the definition of the Model Uncertainty Set. At the same time
$\Bigg{(}I_{p}+\small\left[\begin{array}[]{cc}\Delta_{\bf\widetilde{M}}&\Delta_{\bf\widetilde{N}}\end{array}\right]\left[\begin{array}[]{c}{\widetilde{\bf
Y}}^{\mathtt{md}}_{\bf Q}\\\ {\widetilde{\bf X}}^{\mathtt{md}}_{\bf
Q}\end{array}\right]\Bigg{)}^{-1}$ is guaranteed to be stable via the Small
Gain Theorem.

Conversely, if a Youla parameter ${\bf Q}$ yields a $\gamma$-robustly
stabilizable controller of the nominal model then necessarily
$\small\Bigg{\|}\left[\begin{array}[]{c}{\widetilde{\bf Y}}^{\mathtt{md}}_{\bf
Q}\\\ {\widetilde{\bf X}}^{\mathtt{md}}_{\bf
Q}\end{array}\right]\Bigg{\|}_{\infty}\leq\dfrac{1}{\gamma}$. The proof of
this claim is done by contradiction. Assume that
$\small\Bigg{\|}\left[\begin{array}[]{c}{\widetilde{\bf Y}}^{\mathtt{md}}_{\bf
Q}\\\ {\widetilde{\bf X}}^{\mathtt{md}}_{\bf
Q}\end{array}\right]\Bigg{\|}_{\infty}>\dfrac{1}{\gamma}$. Then by the
Spectral Mapping Theorem (Douglas, 1972, page 41-42) there must exist
$\small\Big{\|}\left[\begin{array}[]{cc}\Delta_{\bf\widetilde{M}}&\Delta_{\bf\widetilde{N}}\end{array}\right]\Big{\|}_{\infty}<\gamma$
such that ${\bf\Phi}_{11}=\bigg{(}{I_{p}}+$
$\small\left[\begin{array}[]{cc}\Delta_{\bf\widetilde{M}}&\Delta_{\bf\widetilde{N}}\end{array}\right]\left[\begin{array}[]{c}{\widetilde{\bf
Y}}^{\mathtt{md}}_{\bf Q}\\\ {\widetilde{\bf X}}^{\mathtt{md}}_{\bf
Q}\end{array}\right]\bigg{)}$ is not unimodular and consequently the Youla
parameter ${\bf Q}$ does not produce an $\gamma$-robustly stabilizable
controller, which is a contradiction. The proof ends.

Proof of Proposition 3.5: By using (17) the cost function can be calculated as

$\begin{split}\mathcal{H}({\bf G}^{\mathtt{pt}},{{\bf K}^{\mathtt{md}}_{\bf
Q}})=&\Bigg{\|}\left[\begin{array}[]{c}{\widetilde{\bf Y}}^{\mathtt{md}}_{\bf
Q}\\\ {\widetilde{\bf X}}^{\mathtt{md}}_{\bf
Q}\end{array}\right]{\bf\Phi}_{11}^{-1}\left[\begin{array}[]{cc}({\widetilde{\bf
M}}^{\mathtt{md}}+\Delta_{\bf\widetilde{M}})&({\widetilde{\bf
N}}^{\mathtt{md}}+\Delta_{\bf\widetilde{N}})\end{array}\right]\Bigg{\|}_{\mathcal{H}_{2}}\\\
=&\Bigg{\|}\left[\begin{array}[]{cc}{\widetilde{\bf Y}}^{\mathtt{md}}_{\bf
Q}{\bf\Phi}_{11}^{-1}({\widetilde{\bf
M}}^{\mathtt{md}}+\Delta_{\bf\widetilde{M}})&{\widetilde{\bf
Y}}^{\mathtt{md}}_{\bf Q}{\bf\Phi}_{11}^{-1}({\widetilde{\bf
N}}^{\mathtt{md}}+\Delta_{\bf\widetilde{N}})\\\ {\widetilde{\bf
X}}^{\mathtt{md}}_{\bf Q}{\bf\Phi}_{11}^{-1}({\widetilde{\bf
M}}^{\mathtt{md}}+\Delta_{\bf\widetilde{M}})&{\widetilde{\bf
X}}^{\mathtt{md}}_{\bf Q}{\bf\Phi}_{11}^{-1}({\widetilde{\bf
N}}^{\mathtt{md}}+\Delta_{\bf\widetilde{N}})\end{array}\right]\Bigg{\|}_{\mathcal{H}_{2}}\\\
=&\Bigg{\|}\left[\begin{array}[]{cc}{\widetilde{\bf Y}}^{\mathtt{md}}_{\bf
Q}{\bf\Phi}_{11}^{-1}({\widetilde{\bf
M}}^{\mathtt{md}}+\Delta_{\bf\widetilde{M}})&{\widetilde{\bf
Y}}^{\mathtt{md}}_{\bf Q}{\bf\Phi}_{11}^{-1}({\widetilde{\bf
N}}^{\mathtt{md}}+\Delta_{\bf\widetilde{N}})\\\ {\widetilde{\bf
X}}^{\mathtt{md}}_{\bf Q}{\bf\Phi}_{11}^{-1}({\widetilde{\bf
M}}^{\mathtt{md}}+\Delta_{\bf\widetilde{M}})&{\widetilde{\bf
X}}^{\mathtt{md}}_{\bf Q}{\bf\Phi}_{11}^{-1}({\widetilde{\bf
N}}^{\mathtt{md}}+\Delta_{\bf\widetilde{N}})\end{array}\right]\Bigg{\|}_{\mathcal{H}_{2}}\end{split}$
(39)

Next the closed loop maps is calculated by using (12) which leads to:

${\widetilde{\bf Y}}^{\mathtt{md}}_{\bf Q}{\bf\Phi}_{11}^{-1}({\widetilde{\bf
M}}^{\mathtt{md}}+\Delta_{\bf\widetilde{M}})={\widetilde{\bf
Y}}^{\mathtt{md}}_{\bf
Q}{\Bigg{(}{I_{p}}+\left[\begin{array}[]{cc}\Delta_{\bf\widetilde{M}}&\Delta_{\bf\widetilde{N}}\end{array}\right]\left[\begin{array}[]{c}{\widetilde{\bf
Y}}^{\mathtt{md}}_{\bf Q}\\\ {\widetilde{\bf X}}^{\mathtt{md}}_{\bf
Q}\end{array}\right]\Bigg{)}}^{-1}({\widetilde{\bf
M}}^{\mathtt{md}}+\Delta_{\bf\widetilde{M}})$

$\begin{split}{\widetilde{\bf Y}}^{\mathtt{md}}_{\bf
Q}{\bf\Phi}_{11}^{-1}({\widetilde{\bf
M}}^{\mathtt{md}}+\Delta_{\bf\widetilde{M}})&=\Bigg{[}({\widetilde{\bf
M}}^{\mathtt{md}}+\Delta_{\bf\widetilde{M}})^{-1}{\bigg{(}{I_{p}}+\left[\begin{array}[]{cc}\Delta_{\bf\widetilde{M}}&\Delta_{\bf\widetilde{N}}\end{array}\right]\left[\begin{array}[]{c}{\widetilde{\bf
Y}}^{\mathtt{md}}_{\bf Q}\\\ {\widetilde{\bf X}}^{\mathtt{md}}_{\bf
Q}\end{array}\right]\bigg{)}}({\widetilde{\bf Y}}^{\mathtt{md}}_{\bf
Q})^{-1}\Bigg{]}^{-1}\\\ &=\Bigg{[}({\widetilde{\bf
M}}^{\mathtt{md}}+\Delta_{\bf\widetilde{M}})^{-1}{\bigg{(}{({\widetilde{\bf
M}}^{\mathtt{md}}{\widetilde{\bf Y}}^{\mathtt{md}}_{\bf Q}+{\widetilde{\bf
N}}^{\mathtt{md}}{\widetilde{\bf X}}^{\mathtt{md}}_{\bf
Q})}+(\Delta_{\bf\widetilde{M}}{\widetilde{\bf Y}}^{\mathtt{md}}_{\bf
Q}+\Delta_{\bf\widetilde{N}}{\widetilde{\bf X}}^{\mathtt{md}}_{\bf
Q})\bigg{)}}({\widetilde{\bf Y}}^{\mathtt{md}}_{\bf Q})^{-1}\Bigg{]}^{-1}\\\
&=\Bigg{[}({\widetilde{\bf
M}}^{\mathtt{md}}+\Delta_{\bf\widetilde{M}})^{-1}{\bigg{(}{({\widetilde{\bf
M}}^{\mathtt{md}}+\Delta_{\bf\widetilde{M}}){\widetilde{\bf
Y}}^{\mathtt{md}}_{\bf Q}+({\widetilde{\bf
N}}^{\mathtt{md}}+\Delta_{\bf\widetilde{N}}){\widetilde{\bf
X}}^{\mathtt{md}}_{\bf Q}}\bigg{)}}({\widetilde{\bf Y}}^{\mathtt{md}}_{\bf
Q})^{-1}\Bigg{]}^{-1}\\\ &=\Bigg{[}{{I_{p}}+({\widetilde{\bf
M}}^{\mathtt{md}}+\Delta_{\bf\widetilde{M}})^{-1}({\widetilde{\bf
N}}^{\mathtt{md}}+\Delta_{\bf\widetilde{N}}){\widetilde{\bf
X}}^{\mathtt{md}}_{\bf Q}({\widetilde{\bf Y}}^{\mathtt{md}}_{\bf
Q})^{-1}}\Bigg{]}^{-1}={(I_{p}+{\bf G}^{\mathtt{pt}}{{\bf
K}^{\mathtt{md}}_{\bf Q}})^{-1}}.\\\ \end{split}$

Next the remaining three terms is calculated as below:

${\widetilde{\bf Y}}^{\mathtt{md}}_{\bf Q}{\bf\Phi}_{11}^{-1}({\widetilde{\bf
N}}^{\mathtt{md}}+\Delta_{\bf\widetilde{N}})={\widetilde{\bf
Y}}^{\mathtt{md}}_{\bf Q}{\bf\Phi}_{11}^{-1}({\widetilde{\bf
M}}^{\mathtt{md}}+\Delta_{\bf\widetilde{M}})({\widetilde{\bf
M}}^{\mathtt{md}}+\Delta_{\bf\widetilde{M}})^{-1}({\widetilde{\bf
N}}^{\mathtt{md}}+\Delta_{\bf\widetilde{N}})={{(I_{p}+{\bf
G}^{\mathtt{pt}}{{\bf K}^{\mathtt{md}}_{\bf Q}})^{-1}}{\bf G}^{\mathtt{pt}}};$

${\widetilde{\bf X}}^{\mathtt{md}}_{\bf Q}{\bf\Phi}_{11}^{-1}({\widetilde{\bf
M}}^{\mathtt{md}}+\Delta_{\bf\widetilde{M}})={\widetilde{\bf
X}}^{\mathtt{md}}_{\bf Q}({\widetilde{\bf Y}}^{\mathtt{md}}_{\bf
Q})^{-1}{\widetilde{\bf Y}}^{\mathtt{md}}_{\bf
Q}{\bf\Phi}_{11}^{-1}({\widetilde{\bf
M}}^{\mathtt{md}}+\Delta_{\bf\widetilde{M}})={{{\bf K}^{\mathtt{md}}_{\bf
Q}}{(I_{p}+{\bf G}^{\mathtt{pt}}{{\bf K}^{\mathtt{md}}_{\bf Q}})^{-1}}};\\\ $

${\widetilde{\bf X}}^{\mathtt{md}}_{\bf Q}{\bf\Phi}_{11}^{-1}({\widetilde{\bf
N}}^{\mathtt{md}}+\Delta_{\bf\widetilde{N}})={\widetilde{\bf
X}}^{\mathtt{md}}_{\bf Q}{\bf\Phi}_{11}^{-1}({\widetilde{\bf
M}}^{\mathtt{md}}+\Delta_{\bf\widetilde{M}})({\widetilde{\bf
M}}^{\mathtt{md}}+\Delta_{\bf\widetilde{M}})^{-1}({\widetilde{\bf
N}}^{\mathtt{md}}+\Delta_{\bf\widetilde{N}})={{{\bf K}^{\mathtt{md}}_{\bf
Q}}{(I_{p}+{\bf G}^{\mathtt{pt}}{{\bf K}^{\mathtt{md}}_{\bf Q}})^{-1}}{\bf
G}^{\mathtt{pt}}}.$

##  D Controller Parameterization

Proof for Theorem 3.6: First $\mathcal{H}_{2}$ norm of the LQG cost in (2) is
derived. Youla Parameterization focuses on the responses of a closed loop
system (Youla et al. (1976), Furieri et al. (2019)). By Proposition 3.5 the
closed Loop responses with the LQG cost (17) are:

$\useshortskip\left[\begin{array}[]{cc}{{\bf T}^{y\nu}_{\bf Q}}&{{\bf
T}^{yw}_{\bf Q}}\\\ {{\bf T}^{u\nu}_{\bf Q}}&{{\bf T}^{uw}_{\bf
Q}}\end{array}\right]=\left[\begin{array}[]{cc}{(I_{p}+{\bf
G}^{\mathtt{pt}}{{\bf K}^{\mathtt{md}}_{\bf Q}})^{-1}}&{{(I_{p}+{\bf
G}^{\mathtt{pt}}{{\bf K}^{\mathtt{md}}_{\bf Q}})^{-1}}{{\bf
G}^{\mathtt{pt}}}}\\\ {{{\bf K}^{\mathtt{md}}_{\bf Q}}{(I_{p}+{\bf
G}^{\mathtt{pt}}{{\bf K}^{\mathtt{md}}_{\bf Q}})^{-1}}}&{{{\bf
K}^{\mathtt{md}}_{\bf Q}}{(I_{p}+{\bf G}^{\mathtt{pt}}{{\bf
K}^{\mathtt{md}}_{\bf Q}})^{-1}}}{{\bf G}^{\mathtt{pt}}}\end{array}\right]$
(40)

The LQG cost in (2) is written by using Zheng et al. (2020a) as below:

$\displaystyle\lim\limits_{T\rightarrow\infty}$
$\displaystyle\mathbb{E}\bigg{[}\dfrac{1}{T}\sum\limits_{t=0}^{T}(y^{T}_{t}P_{1}y_{t}+u^{T}_{t}P_{2}u_{t})\bigg{]}=\mathbb{E}[y^{T}_{\infty}P_{1}y_{\infty}+u^{T}_{\infty}P_{2}u_{\infty}]$
$\displaystyle=$
$\displaystyle\sum\limits_{k=0}^{\infty}\mathbb{E}\bigg{[}\sigma^{2}_{\nu}tr\Big{(}({{\bf
T}^{y\nu}_{\bf Q}})^{T}P_{1}{{\bf T}^{y\nu}_{\bf
Q}}\Big{)}+\sigma^{2}_{w}tr\Big{(}({{\bf T}^{yw}_{\bf Q}})^{2}P_{1}{{\bf
T}^{yw}_{\bf Q}}\Big{)}+$ $\displaystyle+\sigma^{2}_{\nu}tr\Big{(}({{\bf
T}^{u\nu}_{\bf Q}})^{2}P_{2}{{\bf T}^{u\nu}_{\bf
Q}}\Big{)}+\sigma^{2}_{w}tr\Big{(}({{\bf T}^{uw}_{\bf Q}})^{2}P_{2}{{\bf
T}^{uw}_{\bf Q}}\Big{)}\bigg{]}$ $\displaystyle=$
$\displaystyle\sum\limits_{k=0}^{\infty}\Bigg{\|}\left[\begin{array}[]{cc}P_{1}^{1/2}&\\\
&P_{2}^{1/2}\end{array}\right]\left[\begin{array}[]{cc}{{\bf T}^{y\nu}_{\bf
Q}}&{{\bf T}^{yw}_{\bf Q}}\\\ {{\bf T}^{u\nu}_{\bf Q}}&{{\bf T}^{uw}_{\bf
Q}}\end{array}\right]\left[\begin{array}[]{cc}\sigma_{\nu}I_{p}&\\\
&\sigma_{w}I_{m}\end{array}\right]\Bigg{\|}^{2}_{F}$ $\displaystyle=$
$\displaystyle\dfrac{1}{2\pi}\int\limits_{-\pi}^{\pi}\Bigg{\|}\left[\begin{array}[]{cc}P_{1}^{1/2}&\\\
&P_{2}^{1/2}\end{array}\right]\left[\begin{array}[]{cc}{{\bf T}^{y\nu}_{\bf
Q}}&{{\bf T}^{yw}_{\bf Q}}\\\ {{\bf T}^{u\nu}_{\bf Q}}&{{\bf T}^{uw}_{\bf
Q}}\end{array}\right]\left[\begin{array}[]{cc}\sigma_{\nu}I_{p}&\\\
&\sigma_{w}I_{m}\end{array}\right]\Bigg{\|}^{2}_{F}d\theta$ $\displaystyle=$
$\displaystyle\Bigg{\|}\left[\begin{array}[]{cc}P_{1}^{1/2}&\\\
&P_{2}^{1/2}\end{array}\right]\left[\begin{array}[]{cc}{{\bf T}^{y\nu}_{\bf
Q}}&{{\bf T}^{yw}_{\bf Q}}\\\ {{\bf T}^{u\nu}_{\bf Q}}&{{\bf T}^{uw}_{\bf
Q}}\end{array}\right]\left[\begin{array}[]{cc}\sigma_{\nu}I_{p}&\\\
&\sigma_{w}I_{m}\end{array}\right]\Bigg{\|}^{2}_{\mathcal{H}_{2}},$

where the second last quality is due to the Parseval’s theorem with Frobenius
norm and last quality is the definition of the ${\mathcal{H}_{2}}$ norm.
Without loss of generality it is assumed that $P_{1}=I_{p}$, $P_{2}=I_{m}$,
$\sigma_{\nu}=1$, $\sigma_{w}=1$. Then the square root of the LQG cost can be
written as the closed loop responses defined in (39), (40):

$\small\mathcal{H}({\bf G}^{\mathtt{pt}},{{\bf K}^{\mathtt{md}}_{\bf
Q}})=\Bigg{\|}\left[\begin{array}[]{cc}{{\bf T}^{y\nu}_{\bf Q}}&{{\bf
T}^{yw}_{\bf Q}}\\\ {{\bf T}^{u\nu}_{\bf Q}}&{{\bf T}^{uw}_{\bf
Q}}\end{array}\right]\Bigg{\|}_{\mathcal{H}_{2}}=\Bigg{\|}\left[\begin{array}[]{cc}{\widetilde{\bf
Y}}^{\mathtt{md}}_{\bf Q}{\bf\Phi}_{11}^{-1}({\widetilde{\bf
M}}^{\mathtt{md}}+\Delta_{\bf\widetilde{M}})&{\widetilde{\bf
Y}}^{\mathtt{md}}_{\bf Q}{\bf\Phi}_{11}^{-1}({\widetilde{\bf
N}}^{\mathtt{md}}+\Delta_{\bf\widetilde{N}})\\\ {\widetilde{\bf
X}}^{\mathtt{md}}_{\bf Q}{\bf\Phi}_{11}^{-1}({\widetilde{\bf
M}}^{\mathtt{md}}+\Delta_{\bf\widetilde{M}})&{\widetilde{\bf
X}}^{\mathtt{md}}_{\bf Q}{\bf\Phi}_{11}^{-1}({\widetilde{\bf
N}}^{\mathtt{md}}+\Delta_{\bf\widetilde{N}})\end{array}\right]\Bigg{\|}_{\mathcal{H}_{2}}.$

Proof for Proposition 3.7: Given
$\Big{\|}\left[\begin{array}[]{cc}\Delta_{\bf\widetilde{M}}&\Delta_{\bf\widetilde{N}}\end{array}\right]\Big{\|}_{\infty}<\gamma,\Bigg{\|}\left[\begin{array}[]{c}{\widetilde{\bf
Y}}^{\mathtt{md}}_{\bf Q}\\\ {\widetilde{\bf X}}^{\mathtt{md}}_{\bf
Q}\end{array}\right]\Bigg{\|}_{\infty}<\dfrac{1}{\gamma}$ and ${\widetilde{\bf
M}}^{\mathtt{md}},{\widetilde{\bf N}}^{\mathtt{md}}\in\mathcal{RH}_{\infty}$,
by using (34), (35) and Lemma B.1 the LQG cost in (19) is obtained as

$\displaystyle\mathcal{H}$ $\displaystyle({\bf G}^{\mathtt{pt}},{{\bf
K}^{\mathtt{md}}_{\bf Q}})=\Bigg{\|}\left[\begin{array}[]{c}{\widetilde{\bf
Y}}^{\mathtt{md}}_{\bf Q}\\\ {\widetilde{\bf X}}^{\mathtt{md}}_{\bf
Q}\end{array}\right]{\bf\Phi}_{11}^{-1}\left[\begin{array}[]{cc}{\widetilde{\bf
M}^{\mathtt{md}}}+\Delta_{\bf{\widetilde{M}}}&{\widetilde{\bf
N}^{\mathtt{md}}}+\Delta_{\bf{\widetilde{N}}}\end{array}\right]\Bigg{\|}_{\mathcal{H}_{2}}$
$\displaystyle\leq$
$\displaystyle\Bigg{\|}\left[\begin{array}[]{c}{\widetilde{\bf
Y}}^{\mathtt{md}}_{\bf Q}\\\ {\widetilde{\bf X}}^{\mathtt{md}}_{\bf
Q}\end{array}\right]\Bigg{\|}_{\mathcal{H}_{2}}\Bigg{\|}{\bf\Phi}_{11}^{-1}\left[\begin{array}[]{cc}{\widetilde{\bf
M}^{\mathtt{md}}}+\Delta_{\bf{\widetilde{M}}}&{\widetilde{\bf
N}^{\mathtt{md}}}+\Delta_{\bf{\widetilde{N}}}\end{array}\right]\Bigg{\|}_{\infty}$
$\displaystyle\leq$
$\displaystyle\Bigg{\|}\left[\begin{array}[]{c}{\widetilde{\bf
Y}}^{\mathtt{md}}_{\bf Q}\\\ {\widetilde{\bf X}}^{\mathtt{md}}_{\bf
Q}\end{array}\right]\Bigg{\|}_{\mathcal{H}_{2}}\Bigg{\|}{\Bigg{(}{I_{p}}-\left[\begin{array}[]{cc}-\Delta_{\bf\widetilde{M}}&-\Delta_{\bf\widetilde{N}}\end{array}\right]\left[\begin{array}[]{c}{\widetilde{\bf
Y}}^{\mathtt{md}}_{\bf Q}\\\ {\widetilde{\bf X}}^{\mathtt{md}}_{\bf
Q}\end{array}\right]\Bigg{)}}^{-1}\Bigg{\|}_{\infty}\Bigg{\|}\bigg{(}\left[\begin{array}[]{cc}{\widetilde{\bf
M}^{\mathtt{md}}}&{\widetilde{\bf
N}^{\mathtt{md}}}\end{array}\right]+\left[\begin{array}[]{cc}\Delta_{\bf\widetilde{M}}&\Delta_{\bf\widetilde{N}}\end{array}\right]\bigg{)}\Bigg{\|}_{\infty}$
$\displaystyle=$
$\displaystyle\Bigg{\|}\left[\begin{array}[]{c}{\widetilde{\bf
Y}}^{\mathtt{md}}_{\bf Q}\\\ {\widetilde{\bf X}}^{\mathtt{md}}_{\bf
Q}\end{array}\right]\Bigg{\|}_{\mathcal{H}_{2}}{\Bigg{(}\Bigg{\|}{I_{p}}+\sum\limits_{j=1}^{\infty}\bigg{(}\left[\begin{array}[]{cc}-\Delta_{\bf\widetilde{M}}&-\Delta_{\bf\widetilde{N}}\end{array}\right]\left[\begin{array}[]{c}{\widetilde{\bf
Y}}^{\mathtt{md}}_{\bf Q}\\\ {\widetilde{\bf X}}^{\mathtt{md}}_{\bf
Q}\end{array}\right]\bigg{)}^{j}\Bigg{\|}_{\infty}\Bigg{)}}\bigg{(}\Bigg{\|}\left[\begin{array}[]{cc}{\widetilde{\bf
M}^{\mathtt{md}}}&{\widetilde{\bf
N}^{\mathtt{md}}}\end{array}\right]+\left[\begin{array}[]{cc}\Delta_{\bf\widetilde{M}}&\Delta_{\bf\widetilde{N}}\end{array}\right]\Bigg{\|}_{\infty}\bigg{)}$
$\displaystyle\leq$
$\displaystyle\Bigg{\|}\left[\begin{array}[]{c}{\widetilde{\bf
Y}}^{\mathtt{md}}_{\bf Q}\\\ {\widetilde{\bf X}}^{\mathtt{md}}_{\bf
Q}\end{array}\right]\Bigg{\|}_{\mathcal{H}_{2}}{\Bigg{(}1+\Bigg{\|}\sum\limits_{j=1}^{\infty}\bigg{(}\left[\begin{array}[]{cc}-\Delta_{\bf\widetilde{M}}&-\Delta_{\bf\widetilde{N}}\end{array}\right]\left[\begin{array}[]{c}{\widetilde{\bf
Y}}^{\mathtt{md}}_{\bf Q}\\\ {\widetilde{\bf X}}^{\mathtt{md}}_{\bf
Q}\end{array}\right]\bigg{)}^{j}\Bigg{\|}_{\infty}\Bigg{)}}\bigg{(}\Bigg{\|}\left[\begin{array}[]{cc}{\widetilde{\bf
M}^{\mathtt{md}}}&{\widetilde{\bf
N}^{\mathtt{md}}}\end{array}\right]\Bigg{\|}_{\infty}+\gamma\bigg{)}$
$\displaystyle=$
$\displaystyle\Bigg{\|}\left[\begin{array}[]{c}{\widetilde{\bf
Y}}^{\mathtt{md}}_{\bf Q}\\\ {\widetilde{\bf X}}^{\mathtt{md}}_{\bf
Q}\end{array}\right]\Bigg{\|}_{\mathcal{H}_{2}}{\Bigg{[}\Bigg{\|}\left[\begin{array}[]{cc}{\widetilde{\bf
M}^{\mathtt{md}}}&{\widetilde{\bf
N}^{\mathtt{md}}}\end{array}\right]\Bigg{\|}_{\infty}+\gamma}+{\bigg{(}\sum\limits_{j=1}^{\infty}\gamma^{j}\Bigg{\|}\left[\begin{array}[]{c}{\widetilde{\bf
Y}}^{\mathtt{md}}_{\bf Q}\\\ {\widetilde{\bf X}}^{\mathtt{md}}_{\bf
Q}\end{array}\right]\Bigg{\|}_{\infty}^{j}\Bigg{)}}\bigg{(}\Bigg{\|}\left[\begin{array}[]{cc}{\widetilde{\bf
M}^{\mathtt{md}}}&{\widetilde{\bf
N}^{\mathtt{md}}}\end{array}\right]\Bigg{\|}_{\infty}+\gamma\bigg{)}\Bigg{]}$
$\displaystyle=$
$\displaystyle\Bigg{\|}\left[\begin{array}[]{c}{\widetilde{\bf
Y}}^{\mathtt{md}}_{\bf Q}\\\ {\widetilde{\bf X}}^{\mathtt{md}}_{\bf
Q}\end{array}\right]\Bigg{\|}_{\mathcal{H}_{2}}{\Bigg{[}\Bigg{\|}\left[\begin{array}[]{cc}{\widetilde{\bf
M}^{\mathtt{md}}}&{\widetilde{\bf
N}^{\mathtt{md}}}\end{array}\right]\Bigg{\|}_{\infty}+\gamma}+{\dfrac{\gamma\Bigg{\|}\left[\begin{array}[]{c}{\widetilde{\bf
Y}}^{\mathtt{md}}_{\bf Q}\\\ {\widetilde{\bf X}}^{\mathtt{md}}_{\bf
Q}\end{array}\right]\Bigg{\|}_{\infty}}{1-\gamma\Bigg{\|}\left[\begin{array}[]{c}{\widetilde{\bf
Y}}^{\mathtt{md}}_{\bf Q}\\\ {\widetilde{\bf X}}^{\mathtt{md}}_{\bf
Q}\end{array}\right]\Bigg{\|}_{\infty}}}\bigg{(}\Bigg{\|}\left[\begin{array}[]{cc}{\widetilde{\bf
M}^{\mathtt{md}}}&{\widetilde{\bf
N}^{\mathtt{md}}}\end{array}\right]\Bigg{\|}_{\infty}+\gamma\bigg{)}\Bigg{]}$
$\displaystyle\leq$
$\displaystyle\dfrac{\Bigg{\|}\left[\begin{array}[]{c}{\widetilde{\bf
Y}}^{\mathtt{md}}_{\bf Q}\\\ {\widetilde{\bf X}}^{\mathtt{md}}_{\bf
Q}\end{array}\right]\Bigg{\|}_{\mathcal{H}_{2}}}{1-\gamma\Bigg{\|}\left[\begin{array}[]{c}{\widetilde{\bf
Y}}^{\mathtt{md}}_{\bf Q}\\\ {\widetilde{\bf X}}^{\mathtt{md}}_{\bf
Q}\end{array}\right]\Bigg{\|}_{\infty}}{\Bigg{[}\Bigg{\|}\left[\begin{array}[]{cc}{\widetilde{\bf
M}^{\mathtt{md}}}&{\widetilde{\bf
N}^{\mathtt{md}}}\end{array}\right]\Bigg{\|}_{\infty}+\gamma}+{\gamma\Bigg{\|}\left[\begin{array}[]{c}{\widetilde{\bf
Y}}^{\mathtt{md}}_{\bf Q}\\\ {\widetilde{\bf X}}^{\mathtt{md}}_{\bf
Q}\end{array}\right]\Bigg{\|}_{\infty}}\bigg{(}\Bigg{\|}\left[\begin{array}[]{cc}{\widetilde{\bf
M}^{\mathtt{md}}}&{\widetilde{\bf
N}^{\mathtt{md}}}\end{array}\right]\Bigg{\|}_{\infty}+\gamma\bigg{)}\Bigg{]}$
$\displaystyle=$
$\displaystyle\dfrac{\Bigg{\|}\left[\begin{array}[]{c}{\widetilde{\bf
Y}}^{\mathtt{md}}_{\bf Q}\\\ {\widetilde{\bf X}}^{\mathtt{md}}_{\bf
Q}\end{array}\right]\Bigg{\|}_{\mathcal{H}_{2}}}{1-\gamma\Bigg{\|}\left[\begin{array}[]{c}{\widetilde{\bf
Y}}^{\mathtt{md}}_{\bf Q}\\\ {\widetilde{\bf X}}^{\mathtt{md}}_{\bf
Q}\end{array}\right]\Bigg{\|}_{\infty}}{\Bigg{[}\bigg{(}1+{\gamma\Bigg{\|}\left[\begin{array}[]{c}{\widetilde{\bf
Y}}^{\mathtt{md}}_{\bf Q}\\\ {\widetilde{\bf X}}^{\mathtt{md}}_{\bf
Q}\end{array}\right]\Bigg{\|}_{\infty}}\bigg{)}\bigg{(}\Bigg{\|}\left[\begin{array}[]{cc}{\widetilde{\bf
M}^{\mathtt{md}}}&{\widetilde{\bf
N}^{\mathtt{md}}}\end{array}\right]\Bigg{\|}_{\infty}+\gamma\bigg{)}\Bigg{]}}$

Hence the LQG cost in (19) is upper bounded by:

$\displaystyle\mathcal{H}$ $\displaystyle({\bf G}^{\mathtt{pt}},{{\bf
K}^{\mathtt{md}}_{\bf
Q}})\leq\dfrac{1}{1-\gamma\Bigg{\|}\left[\begin{array}[]{c}{\widetilde{\bf
Y}}^{\mathtt{md}}_{\bf Q}\\\ {\widetilde{\bf X}}^{\mathtt{md}}_{\bf
Q}\end{array}\right]\Bigg{\|}_{\infty}}{\Bigg{[}{h\Big{(}\gamma,\Bigg{\|}\left[\begin{array}[]{c}{\widetilde{\bf
Y}}^{\mathtt{md}}_{\bf Q}\\\ {\widetilde{\bf X}}^{\mathtt{md}}_{\bf
Q}\end{array}\right]\Bigg{\|}_{\infty}\Big{)}}\Bigg{\|}\left[\begin{array}[]{c}{\widetilde{\bf
Y}}^{\mathtt{md}}_{\bf Q}\\\ {\widetilde{\bf X}}^{\mathtt{md}}_{\bf
Q}\end{array}\right]\Bigg{\|}_{\mathcal{H}_{2}}\Bigg{]}}$

with ${h\bigg{(}\gamma,\Bigg{\|}\left[\begin{array}[]{c}{\widetilde{\bf
Y}}^{\mathtt{md}}_{\bf Q}\\\ {\widetilde{\bf X}}^{\mathtt{md}}_{\bf
Q}\end{array}\right]\Bigg{\|}_{\infty}\bigg{)}}$ =
$\bigg{(}1+{\gamma\Bigg{\|}\left[\begin{array}[]{c}{\widetilde{\bf
Y}}^{\mathtt{md}}_{\bf Q}\\\ {\widetilde{\bf X}}^{\mathtt{md}}_{\bf
Q}\end{array}\right]\Bigg{\|}_{\infty}}\bigg{)}\bigg{(}\Bigg{\|}\left[\begin{array}[]{cc}{\widetilde{\bf
M}^{\mathtt{md}}}&{\widetilde{\bf
N}^{\mathtt{md}}}\end{array}\right]\Bigg{\|}_{\infty}+\gamma\bigg{)}$.

##  E Quasi Convex Optimization

To proof Theorem 3.8 an important lemma is stated below which is a standard
result optimization.

###### Lemma E.1

(Dean et al. (2020)) For functions $f:X\mapsto R$ and $g:X\mapsto R$ and
constraint set $C\subseteq X$, consider

$\min_{x\in C}\dfrac{f(x)}{1-g(x)}$ (41)

Assuming that $f(x)\geq 0$ and $0\leq g(x)<1$, $(\forall)x\in C$, this
optimization problem can be reformulated as an outer single-variable problem
and an inner-constrained optimization problem (the objective value of an
optimization over the empty set is defined to be infinity):

$\min_{x\in
C}\dfrac{f(x)}{1-g(x)}=\min_{\delta\in[0,1)}\dfrac{1}{1-\delta}\min_{x\in
C}\\{f(x)|g(x)\leq\delta\\}.$ (42)

Proof of Theorem 3.8: The LQG cost in (19) is upper bounded by:

$\mathcal{H}({\bf G}^{\mathtt{pt}},{{\bf K}^{\mathtt{md}}_{\bf
Q}})\leq\dfrac{1}{1-\gamma\Bigg{\|}\left[\begin{array}[]{c}{\widetilde{\bf
Y}}^{\mathtt{md}}_{\bf Q}\\\ {\widetilde{\bf X}}^{\mathtt{md}}_{\bf
Q}\end{array}\right]\Bigg{\|}_{\infty}}{\Bigg{[}{h\Big{(}\gamma,\Bigg{\|}\left[\begin{array}[]{c}{\widetilde{\bf
Y}}^{\mathtt{md}}_{\bf Q}\\\ {\widetilde{\bf X}}^{\mathtt{md}}_{\bf
Q}\end{array}\right]\Bigg{\|}_{\infty}\Big{)}}\Bigg{\|}\left[\begin{array}[]{c}{\widetilde{\bf
Y}}^{\mathtt{md}}_{\bf Q}\\\ {\widetilde{\bf X}}^{\mathtt{md}}_{\bf
Q}\end{array}\right]\Bigg{\|}_{\mathcal{H}_{2}}\Bigg{]}},$

with ${h\bigg{(}\gamma,\Bigg{\|}\left[\begin{array}[]{c}{\widetilde{\bf
Y}}^{\mathtt{md}}_{\bf Q}\\\ {\widetilde{\bf X}}^{\mathtt{md}}_{\bf
Q}\end{array}\right]\Bigg{\|}_{\infty}\bigg{)}}$ =
$\bigg{(}1+{\gamma\Bigg{\|}\left[\begin{array}[]{c}{\widetilde{\bf
Y}}^{\mathtt{md}}_{\bf Q}\\\ {\widetilde{\bf X}}^{\mathtt{md}}_{\bf
Q}\end{array}\right]\Bigg{\|}_{\infty}}\bigg{)}\bigg{(}\Bigg{\|}\left[\begin{array}[]{cc}{\widetilde{\bf
M}^{\mathtt{md}}}&{\widetilde{\bf
N}^{\mathtt{md}}}\end{array}\right]\Bigg{\|}_{\infty}+\gamma\bigg{)}$.

Let $f(x)={h\Big{(}\gamma,\Bigg{\|}\left[\begin{array}[]{c}{\widetilde{\bf
Y}}^{\mathtt{md}}_{\bf Q}\\\ {\widetilde{\bf X}}^{\mathtt{md}}_{\bf
Q}\end{array}\right]\Bigg{\|}_{\infty}\Big{)}}\Bigg{\|}\left[\begin{array}[]{c}{\widetilde{\bf
Y}}^{\mathtt{md}}_{\bf Q}\\\ {\widetilde{\bf X}}^{\mathtt{md}}_{\bf
Q}\end{array}\right]\Bigg{\|}_{\mathcal{H}_{2}}$ and
$g(x)=\gamma\Bigg{\|}\left[\begin{array}[]{c}{\widetilde{\bf
Y}}^{\mathtt{md}}_{\bf Q}\\\ {\widetilde{\bf X}}^{\mathtt{md}}_{\bf
Q}\end{array}\right]\Bigg{\|}_{\infty}$. Here for any stable ${\bf Q}$,
$f(x)\geq 0$ and $0\leq g(x)<1$. Then using Lemma E.1 by re scaling $\delta$
with $\delta/\gamma$ and introducing one additional constraint
$\Bigg{\|}\left[\begin{array}[]{c}{\widetilde{\bf Y}}^{\mathtt{md}}_{\bf Q}\\\
{\widetilde{\bf X}}^{\mathtt{md}}_{\bf
Q}\end{array}\right]\Bigg{\|}_{\infty}\leq\alpha$, we get the upper bound of
Theorem 3.8.

To check Remark 1, we state the following lemma using Dumitrescu (2007).

###### Lemma E.2

We overload the notation and denote $\left[\begin{array}[]{c}{\widetilde{\bf
Y}}^{\mathtt{md}}_{\bf Q}(z)\\\ {\widetilde{\bf X}}^{\mathtt{md}}_{\bf
Q}(z)\end{array}\right]$ to be the following truncated system:

$\left[\begin{array}[]{c}{\widetilde{\bf Y}}^{\mathtt{md}}_{\bf Q}(z)\\\
{\widetilde{\bf X}}^{\mathtt{md}}_{\bf
Q}(z)\end{array}\right]=\sum\limits_{t=0}^{N}F_{t}z^{-t},\quad
F_{t}\in\mathbb{R}^{(p+m)\times p},(\forall)t.$ (43)

Denote
$\mathcal{F}=\left[\begin{array}[]{cccc}F_{0}&F_{1}&\dots&F_{N}\end{array}\right]^{T}\in\mathbb{R}^{[(p+m)\times
N]\times p}$, then the constraint
$\Bigg{\|}\left[\begin{array}[]{c}{\widetilde{\bf Y}}^{\mathtt{md}}_{\bf Q}\\\
{\widetilde{\bf X}}^{\mathtt{md}}_{\bf
Q}\end{array}\right]\Bigg{\|}_{\infty}\leq\gamma$ is equivalent with the
existence of a positive semidefinite $S\in\mathcal{S}_{+}^{(p+m)(N+1)}$, such
that

$\left[\begin{array}[]{cc}S&\mathcal{F}\\\ \mathcal{F}^{T}&\delta
I_{p}\end{array}\right]\succeq
0,\quad\sum\limits_{i=1}^{N-k}S_{i+k,i}=\delta\xi(k)I_{p+m},\quad
k=0,1,...,N.$

where, $S_{i,k}\in\mathbb{R}^{(p+m)\times(p+m)}$ is the $(i,k)^{\mathrm{th}}$
block of $S$. $\xi(k)$ is the impulse function defined as

$\xi(k)=\begin{cases}1,\quad k=0.\\\ 0,\quad otherwise.\end{cases}$

###### Proposition E.3

(SDP formulation for Theorem 3.8) We denote the following:

$\left[\begin{array}[]{c}{\bf\widetilde{Y}}^{\mathtt{md}}_{\bf Q}(z)\\\
{\bf\widetilde{X}}^{\mathtt{md}}_{\bf
Q}(z)\end{array}\right]=\sum\limits_{j}F_{j}z^{-j},\quad\left[\begin{array}[]{c}{\bf\widetilde{Y}}^{\mathtt{md}}(z)\\\
{\bf\widetilde{X}}^{\mathtt{md}}(z)\end{array}\right]=\sum\limits_{j}C_{j}z^{-j},\quad\left[\begin{array}[]{c}{\bf{M}}^{\mathtt{md}}(z)\\\
{\bf{N}}^{\mathtt{md}}(z)\end{array}\right]=\sum\limits_{j}P_{j}z^{-j},$
with $F_{j},C_{j},P_{j}\in\mathbb{R}^{(m+p)\times p}$ and ${\bf
Q}(z)=\sum\limits_{j}Q_{j}z^{-j},$ with $Q_{j}\in\mathbb{R}^{p\times p}$. Let
$\mathtt{vec(.)}$ be the column vectorization of a matrix. Next we define
$\widehat{P}_{k}=\left[\begin{array}[]{c:c:c:c}I_{p}\otimes
P_{k}^{T}&I_{p}\otimes P_{k-1}^{T}&\dots&I_{p}\otimes
P_{k-n}^{T}\end{array}\right]$, where $\otimes$ denote the Kronecker product
and by convention, $P_{-1}=P_{-2}=\dots=0$. $\overline{C}$ and $\overline{q}$
are vectors containing the coefficients of the FIR system
$\left[\begin{array}[]{c}{\bf\widetilde{Y}}^{\mathtt{md}}(z)\\\
{\bf\widetilde{X}}^{\mathtt{md}}(z)\end{array}\right]$ and ${\bf Q}(z)$
respectively. Therefore

$\overline{q}=\left[\begin{array}[]{cccc}\mathtt{vec}(Q_{0})&\mathtt{vec}(Q_{1})&\dots&\mathtt{vec}(Q_{n})\end{array}\right]^{T},\quad\widehat{P}=\left[\begin{array}[]{c}\widehat{P}_{0}\\\
\hdashline\\\ \widehat{P}_{1}\\\ \hdashline\vdots\\\ \hdashline\\\
\widehat{P}_{n}\end{array}\right].$

The inner optimization problem in Theorem 3.8 after FIR truncation can be
expressed as:

$\displaystyle\min_{\epsilon,\overline{q},S}$ $\displaystyle
h(\gamma,\alpha)\epsilon$ (44) s.t. $\displaystyle
S\in\mathcal{S}_{+}^{(p+m)(n+1)},\quad\sum\limits_{i=1}^{n-k}S_{i+k,i}$
$\displaystyle=\min\\{\delta,\alpha\\}\xi(k)I_{p+m},\quad k=0:n,$
$\displaystyle\left[\begin{array}[]{cc}S&\mathcal{F}\\\
\mathcal{F}^{T}&\min\\{\delta,\alpha\\}I_{p}\end{array}\right]\succeq 0,$
$\displaystyle\|\overline{C}+\widehat{P}\overline{q}\|\leq\epsilon.$

##  F Suboptimality guarantee proof

Proof of Theorem 4.1:

Let ${\bf Q_{*}}$ is the optimal solution for Theorem 3.8. By using (22), it
is observed that

$\displaystyle\mathcal{H}({\bf G}^{\mathtt{pt}},{{\bf K}^{\mathtt{md}}_{\bf
Q_{*}}})=\Bigg{\|}\left[\begin{array}[]{c}{\bf\widetilde{Y}}^{\mathtt{md}}_{\bf
Q_{*}}\\\ {\bf\widetilde{X}}^{\mathtt{md}}_{\bf
Q_{*}}\end{array}\right]{\bf\Phi}_{11}^{-1}{I_{p}}\left[\begin{array}[]{cc}{\widetilde{\bf
M}^{\mathtt{md}}}+\Delta_{\bf\widetilde{M}}&{\widetilde{\bf
N}^{\mathtt{md}}}+\Delta_{\bf\widetilde{N}}\end{array}\right]\Bigg{\|}_{\mathcal{H}_{2}}$
$\displaystyle=\Bigg{\|}\left[\begin{array}[]{c}{\bf\widetilde{Y}}^{\mathtt{md}}_{\bf
Q_{*}}\\\ {\bf\widetilde{X}}^{\mathtt{md}}_{\bf
Q_{*}}\end{array}\right]{\bf\Phi}_{11}^{-1}\left[\begin{array}[]{cc}{\widetilde{\bf
M}^{\mathtt{md}}}+\Delta_{\bf\widetilde{M}}&{\widetilde{\bf
N}^{\mathtt{md}}}+\Delta_{\bf\widetilde{N}}\end{array}\right]\left[\begin{array}[]{c}{\widetilde{\bf
Y}}^{\mathtt{opt}}\\\ {\widetilde{\bf
X}}^{\mathtt{opt}}\end{array}\right]\left[\begin{array}[]{cc}{\widetilde{\bf
M}^{\mathtt{md}}}+\Delta_{\bf\widetilde{M}}&{\widetilde{\bf
N}^{\mathtt{md}}}+\Delta_{\bf\widetilde{N}}\end{array}\right]\Bigg{\|}_{\mathcal{H}_{2}}$
$\displaystyle\leq\Bigg{\|}\left[\begin{array}[]{c}{\bf\widetilde{Y}}^{\mathtt{md}}_{\bf
Q_{*}}\\\ {\bf\widetilde{X}}^{\mathtt{md}}_{\bf
Q_{*}}\end{array}\right]{\bf\Phi}_{11}^{-1}\left[\begin{array}[]{cc}{\widetilde{\bf
M}^{\mathtt{md}}}+\Delta_{\bf\widetilde{M}}&{\widetilde{\bf
N}^{\mathtt{md}}}+\Delta_{\bf\widetilde{N}}\end{array}\right]\Bigg{\|}_{\infty}\Bigg{\|}\left[\begin{array}[]{c}{\widetilde{\bf
Y}}^{\mathtt{opt}}\\\ {\widetilde{\bf
X}}^{\mathtt{opt}}\end{array}\right]\left[\begin{array}[]{cc}{\widetilde{\bf
M}^{\mathtt{md}}}+\Delta_{\bf\widetilde{M}}&{\widetilde{\bf
N}^{\mathtt{md}}}+\Delta_{\bf\widetilde{N}}\end{array}\right]\Bigg{\|}_{\mathcal{H}_{2}}$
$\displaystyle\leq\Bigg{\|}\left[\begin{array}[]{c}{\bf\widetilde{Y}}^{\mathtt{md}}_{\bf
Q_{*}}\\\ {\bf\widetilde{X}}^{\mathtt{md}}_{\bf
Q_{*}}\end{array}\right]{\bf\Phi}_{11}^{-1}\left[\begin{array}[]{cc}{\widetilde{\bf
M}^{\mathtt{md}}}+\Delta_{\bf\widetilde{M}}&{\widetilde{\bf
N}^{\mathtt{md}}}+\Delta_{\bf\widetilde{N}}\end{array}\right]\Bigg{\|}_{\infty}\times\mathcal{H}({\bf
G}^{\mathtt{pt}},{\bf K}^{\mathtt{opt}})$

Next using same procedure of Proposition 3.7 for $\|.\|_{\infty}$ it follows
that

$\Bigg{\|}\left[\begin{array}[]{c}{\bf\widetilde{Y}}^{\mathtt{md}}_{\bf
Q_{*}}\\\ {\bf\widetilde{X}}^{\mathtt{md}}_{\bf
Q_{*}}\end{array}\right]{\bf\Phi}_{11}^{-1}\left[\begin{array}[]{cc}{\widetilde{\bf
M}^{\mathtt{md}}}+\Delta_{\bf\widetilde{M}}&{\widetilde{\bf
N}^{\mathtt{md}}}+\Delta_{\bf\widetilde{N}}\end{array}\right]\Bigg{\|}_{\infty}\leq\dfrac{1}{1-\gamma\bigg{\|}\left[\begin{array}[]{c}{\bf\widetilde{Y}}^{\mathtt{md}}_{\bf
Q_{*}}\\\ {\bf\widetilde{X}}^{\mathtt{md}}_{\bf
Q_{*}}\end{array}\right]\Bigg{\|}_{\infty}}\times{g\bigg{(}\gamma,\Bigg{\|}\left[\begin{array}[]{c}{\widetilde{\bf
Y}}^{\mathtt{md}}_{\bf Q_{*}}\\\ {\widetilde{\bf X}}^{\mathtt{md}}_{\bf
Q_{*}}\end{array}\right]\Bigg{\|}_{\infty}\bigg{)}}$

with $\small{g\bigg{(}\gamma,\Bigg{\|}\left[\begin{array}[]{c}{\widetilde{\bf
Y}}^{\mathtt{md}}_{\bf Q_{*}}\\\ {\widetilde{\bf X}}^{\mathtt{md}}_{\bf
Q_{*}}\end{array}\right]\Bigg{\|}_{\infty}\bigg{)}}=\Bigg{\|}\left[\begin{array}[]{c}{\widetilde{\bf
Y}}^{\mathtt{md}}_{\bf Q_{*}}\\\ {\widetilde{\bf X}}^{\mathtt{md}}_{\bf
Q_{*}}\end{array}\right]\Bigg{\|}_{\infty}\bigg{(}1+{\gamma\Bigg{\|}\left[\begin{array}[]{c}{\widetilde{\bf
Y}}^{\mathtt{md}}_{\bf Q_{*}}\\\ {\widetilde{\bf X}}^{\mathtt{md}}_{\bf
Q_{*}}\end{array}\right]\Bigg{\|}_{\infty}}\bigg{)}\bigg{(}\Bigg{\|}\left[\begin{array}[]{cc}{\widetilde{\bf
M}^{\mathtt{md}}}&{\widetilde{\bf
N}^{\mathtt{md}}}\end{array}\right]\Bigg{\|}_{\infty}+\gamma\bigg{)}$.

$\scriptsize\mathcal{H}({\bf G}^{\mathtt{pt}},{{\bf K}^{\mathtt{md}}_{\bf
Q_{*}}})^{2}-\mathcal{H}({\bf G}^{\mathtt{pt}},{\bf
K}^{\mathtt{opt}})^{2}\leq\Bigg{[}\Bigg{(}\dfrac{1}{1-\gamma\Bigg{\|}\left[\begin{array}[]{c}{\bf\widetilde{Y}}^{\mathtt{md}}_{\bf
Q_{*}}\\\ {\bf\widetilde{X}}^{\mathtt{md}}_{\bf
Q_{*}}\end{array}\right]\Bigg{\|}_{\infty}}\times{g\bigg{(}\gamma,\Bigg{\|}\left[\begin{array}[]{c}{\widetilde{\bf
Y}}^{\mathtt{md}}_{\bf Q_{*}}\\\ {\widetilde{\bf X}}^{\mathtt{md}}_{\bf
Q_{*}}\end{array}\right]\Bigg{\|}_{\infty}\bigg{)}}\Bigg{)}^{2}-1\Bigg{]}\mathcal{H}({\bf
G}^{\mathtt{pt}},{\bf K}^{\mathtt{opt}})^{2}.$

Therefore the relative error on the LQG cost becomes

$\displaystyle\dfrac{\mathcal{H}({\bf G}^{\mathtt{pt}},{{\bf
K}^{\mathtt{md}}_{\bf Q_{*}}})^{2}-\mathcal{H}({\bf G}^{\mathtt{pt}},{\bf
K}^{\mathtt{opt}})^{2}}{\mathcal{H}({\bf G}^{\mathtt{pt}},{\bf
K}^{\mathtt{opt}})^{2}}$
$\displaystyle\leq\Bigg{[}\dfrac{1}{1-\gamma\Bigg{\|}\left[\begin{array}[]{c}{\bf\widetilde{Y}}^{\mathtt{md}}_{\bf
Q_{*}}\\\ {\bf\widetilde{X}}^{\mathtt{md}}_{\bf
Q_{*}}\end{array}\right]\Bigg{\|}_{\infty}}\times{g\bigg{(}\gamma,\Bigg{\|}\left[\begin{array}[]{c}{\widetilde{\bf
Y}}^{\mathtt{md}}_{\bf Q_{*}}\\\ {\widetilde{\bf X}}^{\mathtt{md}}_{\bf
Q_{*}}\end{array}\right]\Bigg{\|}_{\infty}\bigg{)}}\Bigg{]}^{2}-1,$
$\displaystyle=\dfrac{1}{\Big{(}1-\gamma\Bigg{\|}\left[\begin{array}[]{c}{\bf\widetilde{Y}}^{\mathtt{md}}_{\bf
Q_{*}}\\\ {\bf\widetilde{X}}^{\mathtt{md}}_{\bf
Q_{*}}\end{array}\right]\Bigg{\|}_{\infty}\Big{)}^{2}}\times{g\bigg{(}\gamma,\Bigg{\|}\left[\begin{array}[]{c}{\widetilde{\bf
Y}}^{\mathtt{md}}_{\bf Q_{*}}\\\ {\widetilde{\bf X}}^{\mathtt{md}}_{\bf
Q_{*}}\end{array}\right]\Bigg{\|}_{\infty}\bigg{)}}^{2}-1.$

$\textrm{with}\hskip
2.0pt\small{g\bigg{(}\gamma,\Bigg{\|}\left[\begin{array}[]{c}{\widetilde{\bf
Y}}^{\mathtt{md}}_{\bf Q_{*}}\\\ {\widetilde{\bf X}}^{\mathtt{md}}_{\bf
Q_{*}}\end{array}\right]\Bigg{\|}_{\infty}\bigg{)}}=\Bigg{\|}\left[\begin{array}[]{c}{\widetilde{\bf
Y}}^{\mathtt{md}}_{\bf Q_{*}}\\\ {\widetilde{\bf X}}^{\mathtt{md}}_{\bf
Q_{*}}\end{array}\right]\Bigg{\|}_{\infty}\bigg{(}1+{\gamma\Bigg{\|}\left[\begin{array}[]{c}{\widetilde{\bf
Y}}^{\mathtt{md}}_{\bf Q_{*}}\\\ {\widetilde{\bf X}}^{\mathtt{md}}_{\bf
Q_{*}}\end{array}\right]\Bigg{\|}_{\infty}}\bigg{)}\bigg{(}\bigg{\|}\left[\begin{array}[]{cc}{\widetilde{\bf
M}^{\mathtt{md}}}&{\widetilde{\bf
N}^{\mathtt{md}}}\end{array}\right]\bigg{\|}_{\infty}+\gamma\bigg{)}.$

##  G Non-asymptotic System Identification

###  G.1 Noise Contaminated Plant

Proof for Lemma 5.1: The proof is started by writing the Bezout identity for
Dual Youla Parametrization using Theorem 2.7,

$\begin{split}\left[\begin{array}[]{cc}\widetilde{\bf M}^{\mathtt{md}}-{\bf
R}^{\mathtt{md}}{\bf X}^{\mathtt{md}}&\widetilde{\bf N}^{\mathtt{md}}+{\bf
R}^{\mathtt{md}}{\bf Y}^{\mathtt{md}}\\\ -{\bf X}^{\mathtt{md}}&{\bf
Y}^{\mathtt{md}}\end{array}\right]\left[\begin{array}[]{cc}\widetilde{\bf
Y}^{\mathtt{md}}&-({\bf N}^{\mathtt{md}}+\widetilde{\bf Y}^{\mathtt{md}}{\bf
R}^{\mathtt{md}})\\\ \widetilde{\bf X}^{\mathtt{md}}&{\bf
M}^{\mathtt{md}}-\widetilde{\bf X}^{\mathtt{md}}{\bf
R}^{\mathtt{md}}\end{array}\right]=I_{p+m},\\\
\left[\begin{array}[]{cc}\widetilde{\bf Y}^{\mathtt{md}}&-({\bf
N}^{\mathtt{md}}+\widetilde{\bf Y}^{\mathtt{md}}{\bf R}^{\mathtt{md}})\\\
\widetilde{\bf X}^{\mathtt{md}}&{\bf M}^{\mathtt{md}}-\widetilde{\bf
X}^{\mathtt{md}}{\bf
R}^{\mathtt{md}}\end{array}\right]\left[\begin{array}[]{cc}\widetilde{\bf
M}^{\mathtt{md}}-{\bf R}^{\mathtt{md}}{\bf X}^{\mathtt{md}}&\widetilde{\bf
N}^{\mathtt{md}}+{\bf R}^{\mathtt{md}}{\bf Y}^{\mathtt{md}}\\\ -{\bf
X}^{\mathtt{md}}&{\bf Y}^{\mathtt{md}}\end{array}\right]=I_{p+m}.\end{split}$
(45)

Next the closed loop maps with $e_{1}$ is checked in Figure 2. First, for the
left part $e_{1}=({\bf
M}^{\mathtt{md}})^{-1}\Big{[}u+{\bf\widetilde{X}}^{\mathtt{md}}\Big{(}{\bf
R}^{\mathtt{md}}e_{1}+({\bf\widetilde{M}}^{\mathtt{md}}-{\bf
R}^{\mathtt{md}}{\bf X}^{\mathtt{md}})\nu\Big{)}\Big{]}$, such that $({\bf
M}^{\mathtt{md}}-{\bf\widetilde{X}}^{\mathtt{md}}{\bf
R}^{\mathtt{md}})e_{1}=u+{\bf\widetilde{X}}^{\mathtt{md}}({\bf\widetilde{M}}^{\mathtt{md}}-{\bf
R}^{\mathtt{md}}{\bf X}^{\mathtt{md}})\nu$, therefore we get

$e_{1}=({\bf M}^{\mathtt{md}}-{\bf\widetilde{X}}^{\mathtt{md}}{\bf
R}^{\mathtt{md}})^{-1}\Big{[}u+{\bf\widetilde{X}}^{\mathtt{md}}({\bf\widetilde{M}}^{\mathtt{md}}-{\bf
R}^{\mathtt{md}}{\bf X}^{\mathtt{md}})\nu\Big{]}.$ (46)

Then for the right part $y={\bf
N}^{\mathtt{md}}e_{1}+{\bf\widetilde{Y}}^{\mathtt{md}}\Big{(}{\bf
R}^{\mathtt{md}}e_{1}+({\bf\widetilde{M}}^{\mathtt{md}}-{\bf
R}^{\mathtt{md}}{\bf X}^{\mathtt{md}})\nu\big{)}$, such that $y=({\bf
N}^{\mathtt{md}}+{\bf\widetilde{Y}}^{\mathtt{md}}{\bf
R}^{\mathtt{md}})e_{1}+{\bf\widetilde{Y}}^{\mathtt{md}}({\bf\widetilde{M}}^{\mathtt{md}}-{\bf
R}^{\mathtt{md}}{\bf X}^{\mathtt{md}})\nu$.
Next using (45) and (46) it follows that

$\displaystyle y$ $\displaystyle=({\bf
N}^{\mathtt{md}}+{\bf\widetilde{Y}}^{\mathtt{md}}{\bf R}^{\mathtt{md}})({\bf
M}^{\mathtt{md}}-{\bf\widetilde{X}}^{\mathtt{md}}{\bf
R}^{\mathtt{md}})^{-1}\Big{[}u+{\bf\widetilde{X}}^{\mathtt{md}}({\bf\widetilde{M}}^{\mathtt{md}}-{\bf
R}^{\mathtt{md}}{\bf
X}^{\mathtt{md}})\nu\Big{]}+{\bf\widetilde{Y}}^{\mathtt{md}}({\bf\widetilde{M}}^{\mathtt{md}}-{\bf
R}^{\mathtt{md}}{\bf X}^{\mathtt{md}})\nu$ $\displaystyle=({\bf
N}^{\mathtt{md}}+{\bf\widetilde{Y}}^{\mathtt{md}}{\bf R}^{\mathtt{md}})({\bf
M}^{\mathtt{md}}-{\bf\widetilde{X}}^{\mathtt{md}}{\bf
R}^{\mathtt{md}})^{-1}u+$ $\displaystyle+({\bf
N}^{\mathtt{md}}+{\bf\widetilde{Y}}^{\mathtt{md}}{\bf R}^{\mathtt{md}})({\bf
M}^{\mathtt{md}}-{\bf\widetilde{X}}^{\mathtt{md}}{\bf
R}^{\mathtt{md}})^{-1}{\bf\widetilde{X}}^{\mathtt{md}}({\bf\widetilde{M}}^{\mathtt{md}}-{\bf
R}^{\mathtt{md}}{\bf
X}^{\mathtt{md}})\nu+{\bf\widetilde{Y}}^{\mathtt{md}}({\bf\widetilde{M}}^{\mathtt{md}}-{\bf
R}^{\mathtt{md}}{\bf X}^{\mathtt{md}})\nu$ $\displaystyle={\bf G}_{\bf
R}^{\mathtt{md}}u+({\bf N}^{\mathtt{md}}+{\bf\widetilde{Y}}^{\mathtt{md}}{\bf
R}^{\mathtt{md}})({\bf M}^{\mathtt{md}}-{\bf\widetilde{X}}^{\mathtt{md}}{\bf
R}^{\mathtt{md}})^{-1}(\widetilde{\bf M}^{\mathtt{md}}-{\bf
R}^{\mathtt{md}}{\bf
X}^{\mathtt{md}}){\bf{X}}^{\mathtt{md}}\nu+{\bf\widetilde{Y}}^{\mathtt{md}}({\bf\widetilde{M}}^{\mathtt{md}}-{\bf
R}^{\mathtt{md}}{\bf X}^{\mathtt{md}})\nu$ $\displaystyle={\bf G}_{\bf
R}^{\mathtt{md}}u+({\bf N}^{\mathtt{md}}+{\bf\widetilde{Y}}^{\mathtt{md}}{\bf
R}^{\mathtt{md}}){\bf{X}}^{\mathtt{md}}\nu+{\bf\widetilde{Y}}^{\mathtt{md}}({\bf\widetilde{M}}^{\mathtt{md}}-{\bf
R}^{\mathtt{md}}{\bf X}^{\mathtt{md}})\nu$ $\displaystyle={\bf G}_{\bf
R}^{\mathtt{md}}u+\Big{(}{\bf\widetilde{Y}}^{\mathtt{md}}({\bf\widetilde{M}}^{\mathtt{md}}-{\bf
R}^{\mathtt{md}}{\bf X}^{\mathtt{md}})+({\bf
N}^{\mathtt{md}}+{\bf\widetilde{Y}}^{\mathtt{md}}{\bf
R}^{\mathtt{md}}){\bf{X}}^{\mathtt{md}}\Big{)}\nu={\bf G}_{\bf
R}^{\mathtt{md}}u+\nu.$

It can observed from Figure 2 and from proof of Lemma 5.1 that

$\begin{split}({\bf M}^{\mathtt{md}}-{\bf\widetilde{X}}^{\mathtt{md}}{\bf
R}^{\mathtt{md}})e_{1}&=u+{\bf\widetilde{X}}^{\mathtt{md}}({\bf\widetilde{M}}^{\mathtt{md}}-{\bf
R}^{\mathtt{md}}{\bf X}^{\mathtt{md}})\nu\\\ &=w+({\bf
Y}^{\mathtt{md}})^{-1}{\bf X}^{\mathtt{md}}r-({\bf Y}^{\mathtt{md}})^{-1}{\bf
X}^{\mathtt{md}}y+{\bf\widetilde{X}}^{\mathtt{md}}({\bf\widetilde{M}}^{\mathtt{md}}-\mathbf{R}{\bf\widetilde{X}}^{\mathtt{md}})\nu;\\\
\vspace{4pt}({\bf N}^{\mathtt{md}}+{\bf\widetilde{Y}}^{\mathtt{md}}{\bf
R}^{\mathtt{md}})e_{1}&=y-{\bf\widetilde{Y}}^{\mathtt{md}}({\bf\widetilde{M}}^{\mathtt{md}}-{\bf
R}^{\mathtt{md}}{\bf X}^{\mathtt{md}})\nu.\end{split}$ (47)

Multiplying first part of (47) by ${\bf Y}^{\mathtt{md}}$ and the second part
of (47) by ${\bf X}^{\mathtt{md}}$ it follows that

$e_{1}={\bf X}^{\mathtt{md}}r+{\bf Y}^{\mathtt{md}}w$ (48)

Furthermore

${\bf M}^{\mathtt{md}}e_{1}=u+{\bf\widetilde{X}}^{\mathtt{md}}e_{2};\hskip
5.0pt{\bf N}^{\mathtt{md}}e_{1}=y-{\bf\widetilde{Y}}^{\mathtt{md}}e_{2}$ (49)

Multiplying first part of (49) by ${\bf\widetilde{N}}^{\mathtt{md}}$ and
second part of (49) by ${\bf\widetilde{M}}^{\mathtt{md}}$ and subtracting it
is obtained that

$e_{2}={\bf\widetilde{M}}^{\mathtt{md}}y-{\bf\widetilde{N}}^{\mathtt{md}}u$
(50)

Hence $e_{1}$ and $e_{2}$ can be measured directly from (48) and (50)
respectively.

###  G.2 Choice of Noise

###### Lemma G.1

Let ${\bf H}$ be a discrete time dynamic system with input ${\bf X}$ and
output ${\bf Y}$ s.t. ${\bf Y}(z)={\bf H}(z){\bf X}(z)$. Let ${x}[n]$ be a
stationary stochastic process with mean function $m_{x}[n]$ and spectral
density $\phi_{\bf X}(z)$. Similarly, ${y}[n]$ be a stationary stochastic
process with mean function $m_{y}[n]$ and spectral density $\phi_{\bf Y}(z)$
Then $m_{y}[n]={\bf H}(z)\Big{|}_{z=1}m_{x}[n]$ and $\phi_{\bf Y}(z)={\bf
H}(z)\phi_{\bf X}(z){\bf H}^{T}(z^{-1})$.

######  Lemma G.2 (Spectral Factorization)

Ionescu et al. (1999) Given any arbitrary rational matrix ${\bf G}(z)$, one
can always always find a rational matrix ${\bf V}(z)$, such that

${\bf G}(z){\bf G}^{T}(z^{-1})={\bf V}(z){\bf V}^{T}(z^{-1}),$ (51)

where ${\bf V}$ is analytic outside the closed unit circle including at
infinity.

For the identification algorithm, it is needed that
$\phi_{{\bf\widetilde{M}}^{\mathtt{md}}\nu}(z)=\dfrac{1}{2\pi}I$. Therefore
${\bf\widetilde{M}}^{\mathtt{md}}(z)\phi_{\nu}(z)\times$
$({{\bf\widetilde{M}}^{\mathtt{md}}})^{T}(z^{-1})=\dfrac{1}{2\pi}I$ such that
$\phi_{\nu}(z)=\dfrac{1}{2\pi}({{\bf\widetilde{M}}^{\mathtt{md}}})^{-1}(z)({{\bf\widetilde{M}}^{\mathtt{md}}})^{-T}(z^{-1})$.
However, $\phi_{\nu}(z)$ may not necessarily be a density function for a
stochastic process since the stability of
$({\bf\widetilde{M}}^{\mathtt{md}})^{-1}$ cannot be guaranteed. Therefore we
consider the spectral factorization of $\phi_{\nu}(z)$ as:

$({\bf\widetilde{M}}^{\mathtt{md}})^{-1}(z)({\bf\widetilde{M}}^{\mathtt{md}})^{-T}(z^{-1})=I_{\nu}(z)I_{\nu}^{T}(z^{-1}),$
(52)

where $I_{\nu}$ has no poles outside the closed unit circle including at
infinity. Next we choose $\\{\nu[n]\\}_{n=1}^{T}$ to be a Gaussian process
with zero mean and spectral density
$\dfrac{1}{2\pi}I_{\nu}(z)I_{\nu}^{T}(z^{-1})$. Furthermore, as the input $u$
= ${\bf X}^{\mathtt{md}}r+{\bf Y}^{\mathtt{md}}w-{\bf X}^{\mathtt{md}}\nu$,
and we want to make $\\{u[n]\\}$ isotropic, consider the following: generate
$\\{r[n]\\}_{n=1}^{T}=\\{\nu[n]\\}_{n=1}^{T}$, then $u={\bf Y}^{\mathtt{md}}w$
such that ${\bf Y}^{\mathtt{md}}(z)\phi_{w}(z)({\bf
Y}^{\mathtt{md}})^{T}(z^{-1})=\dfrac{1}{2\pi}I$ with
$\phi_{w}(z)=\dfrac{1}{2\pi}({\bf Y}^{\mathtt{md}})^{-1}(z)({\bf
Y}^{\mathtt{md}})^{-T}(z^{-1})$. Next compute $I_{w}(z)$ by spectral
factorization such that

$({\bf Y}^{\mathtt{md}})^{-1}(z)({\bf
Y}^{\mathtt{md}})^{-T}(z^{-1})=I_{w}(z)I_{w}^{T}(z^{-1}),$

and $I_{w}(z)$ has no poles outside the closed unit circle including at
infinity. Finally, we choose $\\{r[n]\\}_{n=1}^{T}=\\{\nu[n]\\}_{n=1}^{T}$ and
$\\{w[n]\\}_{n=1}^{T}$ as Gaussian processes with zero mean and spectral
density $\dfrac{1}{2\pi}I_{w}(z)I_{w}^{T}(z^{-1})$.

###  G.3 Identification Preliminaries

Let’s define an LTI system with dynamics as (1) by its Markov parameters
$\mathcal{M}=(C,A,B)$. For any matrix $A$, $\sigma_{i}(A)$ is defined as the
$i^{th}$ singular value of $A$ with $\sigma_{i}(A)\geq\sigma_{i+1}(A)$.
Similarly let $\rho_{i}(A)$ = $|\lambda_{i}(A)|$, where $\lambda_{i}(A)$ is
the eigenvalue of $A$ with $\rho_{i}(A)\geq\rho_{i+1}(A)$.

Definition: A matrix $A\in\mathbb{R}^{n\times n}$ is stable if $\rho(A)<1$.

Definition: The transfer function of $\mathcal{M}=(C,A,B)$ is given by
$G(z)=C(zI-A)^{-1}B$, where $z\in\mathbb{C}$.

Definition: The $(k,p,q)$ \- dimensional Hankel matrix for
$\mathcal{M}=(C,A,B)$ is defined as,

$\small\mathcal{H}_{k,p,q}(R)=\begin{bmatrix}C_{R}A_{R}^{k}B_{R}&C_{R}A_{R}^{k+1}B_{R}&\dots&C_{R}A_{R}^{q+k-1}B_{R}\\\
C_{R}A_{R}^{k+1}B_{R}&C_{R}A_{R}^{k+2}B_{R}&\dots&C_{R}A_{R}^{q+k}B_{R}\\\
\vdots&\vdots&\ddots&\vdots\\\
C_{R}A_{R}^{p+k-1}B_{R}&\dots&\dots&C_{R}A_{R}^{p+q+k-2}B_{R}\\\
\end{bmatrix}=\mathcal{H}_{k,p,q}\in\mathbb{R}^{p\times q}$ (53)

and its associated Toeplitz matrix is defined as $\mathcal{M}$ = $(C,A,B)$ is
defined as,

$\small\mathcal{T}_{k,d}(R)=\begin{bmatrix}0&0&\dots&0&0\\\
C_{R}A_{R}^{k}B_{R}&0&\dots&0&0\\\ \vdots&\vdots&\ddots&\vdots&\vdots\\\
C_{R}A_{R}^{d+k-3}B_{R}&\dots&C_{R}A_{R}^{k}B_{R}&0&0\\\
C_{R}A_{R}^{d+k-2}B_{R}&C_{R}A_{R}^{d+k-3}B_{R}&\dots&C_{R}A_{R}^{k}B_{R}&0\\\
\end{bmatrix}=\mathcal{T}_{k,d}\in\mathbb{R}^{d\times d}$ (54)

Also we define the Toeplitz matrix corresponding to the process noise as,

$\small\mathcal{TO}_{k,d}(R)=\begin{bmatrix}0&0&\dots&0&0\\\
C_{R}A_{R}^{k}&0&\dots&0&0\\\ \vdots&\vdots&\ddots&\vdots&\vdots\\\
C_{R}A_{R}^{d+k-3}&\dots&C_{R}A_{R}^{k}&0&0\\\
C_{R}A_{R}^{d+k-2}&C_{R}A_{R}^{d+k-3}&\dots&C_{R}A_{R}^{k}&0\\\
\end{bmatrix}=\mathcal{TO}_{k,d}\in\mathbb{R}^{p\times q}$ (55)

Let’s also define, $\bar{z}_{l,d}^{+}$ = $\left[\begin{array}[]{cccc}z_{l}\\\
z_{l+1}\\\ \vdots\\\ z_{l+d-1}\end{array}\right]$ and $\bar{z}_{l,d}^{-}$ =
$\left[\begin{array}[]{cccc}z_{l}\\\ z_{l-1}\\\ \vdots\\\
z_{l-d+1}\end{array}\right]$

###  G.4 Identification Algorithms

The algorithms for System Identification using the result from Sarkar et al.
(2020) are stated below:

Input : $T=$ Time horizon for learning, $d=$ Hankel matrix size, $m=$ Input
dimension, $p=$ Output dimension

Data: signal $\\{r_{j}\\}_{j=1}^{2T},\\{w_{j}\\}_{j=1}^{2T}$, signal
$\\{y_{j}\\}_{j=1}^{2T},\\{u_{j}\\}_{j=1}^{2T}$

Result: Estimated Hankel Matrix for ${\bf R}^{\mathtt{md}}$,
$\mathcal{\widehat{H}}_{0,d,d}$

1 Generate $2T$ noises, $\\{r_{t}\\}_{t=1}^{2T},\\{w_{t}\\}_{t=1}^{2T}$,
$\\{\nu_{t}\\}_{t=1}^{2T}$ ;

2 Select input,
$\\{u_{t}\\}_{t=1}^{2T}=X\\{r_{t}\\}_{t=1}^{2T}+Y\\{w_{t}\\}_{t=1}^{2T}-X\\{\nu_{t}\\}_{t=1}^{2T}\in\mathbb{R}^{m\times
2T}$;

3 Collect $2T$ input-output pairs, $\\{u_{t},z_{t}\\}_{t=1}^{2T}$;

4 Do
$\mathcal{\widehat{H}}_{0,d,d}=\arg\min\limits_{\mathcal{H}}\sum\limits_{l=0}^{T-1}\Big{\|}\bar{z}_{l+d+1,d}^{+}-\mathcal{H}\bar{u}_{l+d,d}^{-}\Big{\|}_{2}^{2}$;

Return $\mathcal{\widehat{H}}_{0,d,d}$.

Algorithm 1 Derive Hankel Matrix for ${\bf R}^{\mathtt{md}}$

By running Algorithm-1 for $T$ times the desired
$\mathcal{\widehat{H}}_{0,d,d}$ is obtained. It can be shown that,

$\mathcal{\widehat{H}}_{0,d,d}=\bigg{(}\sum\limits_{l=0}^{T-1}\bar{z}_{l+d+1,d}^{+}(\bar{U}_{l+d,d}^{-})^{T}\bigg{)}\bigg{(}\sum\limits_{l=0}^{T-1}\bar{U}_{l+d,d}^{-}(\bar{U}_{l+d,d}^{-})^{T}\bigg{)}$

Parameters needed before Algorithm-2: $m$: Input Dimension, $p$: Output
Dimension, $\delta$: Error probability, $\beta$: Upper bound on