# On Updating Static Output Feedback Controllers Under State-Space
Perturbation

MirSaleh Bahavarnia† and Ahmad F. Taha† †The authors are with the Department
of Civil and Environmental Engineering, Vanderbilt University, 2201 West End
Avenue, TN 37235, USA. Ahmad F. Taha is also affiliated with the Department of
Electrical and Computer Engineering. Emails:
<EMAIL_ADDRESS><EMAIL_ADDRESS>This work is
supported by the National Science Foundation under Grants 2152450 and 2151571.

###### Abstract

In this paper, we propose a novel update of a nominal stabilizing static
output feedback (SOF) controller for a perturbed linear system. In almost
every classical feedback controller design problem, a stabilizing feedback
controller is designed given a stabilizable unstable system. In realistic
scenarios, the system model is usually imperfect and subject to perturbations.
A typical approach to attenuate the impacts of such perturbations on the
system stability is repeating the whole controller design procedure to find an
updated stabilizing SOF controller. Such an approach can be inefficient and
occasionally infeasible. Using the notion of minimum destabilizing real
perturbation (MDRP), we construct a simple norm minimization problem (a least-
squares problem) to propose an efficient update of a nominal stabilizing SOF
controller that can be applied to various control engineering applications in
the case of perturbed scenarios like abrupt changes or inaccurate system
models. In particular, considering norm-bounded known or unknown
perturbations, this paper presents updated stabilizing SOF controllers and
derives sufficient stability conditions. Geometric metrics to quantitatively
measure the approach’s robustness are defined. Moreover, we characterize the
corresponding guaranteed stability regions, and specifically, for the case of
norm-bounded unknown perturbations, we propose non-fragility-based robust
updated stabilizing SOF controllers. Through extensive numerical simulations,
we assess the effectiveness of the theoretical results.

###### Index Terms:

Stability of linear systems, robust control, output feedback control,
uncertain linear systems.

##  I Introduction

Stability robustness is a significant classical notion in robust control
theory [1, 2, 3, 4, 5, 6, 7, 8]. Stability robustness simply means how
sensitive the stability of the control system is against the
perturbations/uncertainties. The varying nature of engineering systems’ models
necessitates the thorough analysis of stability robustness and its potential
applications to develop robustly stable engineering systems. Several studies
have quantitatively investigated the impacts of perturbations on the stability
robustness of the control systems. In [1, 3], a class of non-destabilizing
linear constant perturbations is characterized for the linear-quadratic state
feedback (LQSF) designs. The authors in [2], propose a guaranteed cost LQSF
for which the closed-loop system is stable for any variation of a vector-
valued parameter. In [4], for the LQSF designs, the stability robustness
bounds are derived based on the Algebraic Riccati equation and Lyapunov
stability theory. In [5], bounds on the non-destabilizing time-varying
nonlinear perturbations are obtained for asymptotically stable linear systems
to provide computationally efficient quantitative robustness measures. Various
stability robustness tests are investigated in [6] to highlight the trade-off
between the stability robustness conservatism and the information about the
perturbation. In [7], utilizing the Lyapunov stability theory, the author has
proposed an improved non-destabilizing perturbation bound over the bound
proposed by [5]. Taking advantage of appropriately chosen coordinate
transformations, the authors in [8] have reduced the conservatism of non-
destabilizing perturbation bounds proposed by [5, 7].

In this paper, in contrast to the aforementioned studies, we do not go through
the derivation of non-destabilizing perturbation bounds. Instead, we mainly
focus on attenuating the impacts of perturbations on the system stability via
updating a nominal stabilizing static output feedback (SOF) controller. With
that in mind, the control problem considered in this paper is an SOF
controller update problem. To put into perspective, it is noteworthy that our
considered problem slightly differs from the robust feedback controller design
problems for uncertain linear systems (with norm-bounded unknown perturbation)
[9, 10, 11, 12, 13, 14, 15, 16] in the sense that, the robust feedback
controller in those problems is robustly stabilizing for all perturbations
$\Delta$ satisfying $0<\|\Delta\|_{F}\leq\rho$ while in our case, the robust
updated stabilizing SOF controller is robustly stabilizing for a subset of
perturbations $\Delta$ satisfying $0<\|\Delta\|_{F}\leq\rho$ that will
mathematically be characterized. Specifically, the more accurate estimate
$\hat{\Delta}$ of a norm-bounded unknown perturbation we have, the more
robustly stabilizing updated stabilizing SOF controller we propose.

In general, the SOF controller stabilization problem is known to be an NP-hard
problem as it is intrinsically equivalent to solving a bi-linear matrix
inequality (BMI) [17]. Then, utilizing a typical approach by repeating the
whole controller design procedure can become computationally cumbersome. Also,
we avoid utilizing any Lyapunov-based approach as it enforces an extra
computational burden (mostly in the case of bi-linear matrix inequality (BMI)
or linear matrix inequality (LMI) formulations in semi-definite programs
(SDPs) [18]) which is not desired in terms of computational efficiency.
Remarkably, the Lyapunov-based SOF controller synthesis hinges on
approximately solving BMIs [19, 20] or incorporating sufficient LMI conditions
[21, 13] which induces a conservatism. The alternative non-Lyapunov approach
that we take is built upon the notion of minimum destabilizing real
perturbation (MDRP) [22] which has inspired [23, 24] to synthesize sparse
feedback controllers for the large-scale systems. Throughout the paper, we
utilize the fundamental linear algebraic results from [25] where needed.

Paper Contributions. The main contributions of this paper can be itemized as
follows:

  * •

Built upon the notion of minimum destabilizing real perturbation [22], we
construct a simple norm minimization problem (a least-squares problem) to
propose a novel update of a nominal stabilizing SOF controller that can be
applied to various control engineering applications in the case of perturbed
scenarios like abrupt changes or inaccurate system models.

  * •

Considering known perturbations and unknown perturbations with a known upper
bound on their norm, we propose novel updates of nominal stabilizing SOF
controllers and derive sufficient stability conditions.

  * •

We define geometric metrics to quantitatively measure the stability robustness
of the proposed updates of nominal stabilizing SOF controllers, characterize
the corresponding guaranteed stability regions, and specifically, for the case
of unknown perturbations with a known upper bound on their norm, we propose
non-fragility-based robust updated stabilizing SOF controllers.

  * •

Through extensive numerical simulations, we validate the effectiveness of the
theoretical results and present a thorough analysis of the empirical
visualizations.

Paper Structure. The remainder of the paper is structured as follows: Section
II states the main objective of the paper by arising a question to be answered
throughout the following sections. Section III presents a novel updated
stabilizing SOF controller via updating a nominal stabilizing SOF controller
built upon a simple norm minimization problem (a least-squares problem).
Section IV contains the main results of the paper detailing the stability
regions for the corresponding updated stabilizing SOF controllers. Through
various numerical simulations, Section V empirically verifies the
effectiveness of the theoretical results. Finally, the paper is concluded via
drawing a few concluding remarks in Section VI.

Paper Notation. We denote the vectors and matrices by lowercase and uppercase
letters, respectively. To represent the set of real numbers, $n$-dimensional
real-valued vectors, and $m\times n$-dimensional real-valued matrices, we
respectively use $\mathbb{R}$, $\mathbb{R}^{n}$, and $\mathbb{R}^{m\times n}$.
We show the set of positive real numbers with $\mathbb{R}_{++}$. We denote the
identity matrix of dimension $n$ with $I_{n}$. For a square matrix $M$,
$\alpha(M)$ represents the spectral abscissa (i.e., the maximum real part of
the eigenvalues) of $M$. We say a square matrix $M$ is stable (Hurwitz) if
$\alpha(M)<0$ holds. For a matrix $M$, symbols $M^{T}$, $\|M\|_{F}$,
$\mathbf{vec}(M)$, and $U_{M}\Sigma_{M}V_{M}^{T}$ denote its transpose,
Frobenius norm, vectorization, and singular value decomposition (SVD),
respectively. Given a full-column rank matrix $M$, $M^{+}:=(M^{T}M)^{-1}M^{T}$
denotes the Moore-Penrose inverse of $M$. We represent the Kronecker product
with the symbol $\otimes$. For a vector $v$, we respectively denote its
Euclidean norm and vectorization inverse with $\|v\|$ and
$\mathbf{vec}^{-1}(v)$ where $\mathbf{vec}^{-1}(v)$ is a matrix that satisfies
$\mathbf{vec}(\mathbf{vec}^{-1}(v))=v$. We represent the set union with
$\cup$. Given two real numbers $a<b$, we denote the open, closed, and half-
open intervals with $]a,b[$, $[a,b]$, $[a,b[$, and $]a,b]$, respectively. We
represent the logical or and the logical and with $\lor$ and $\land$,
respectively. We show the computation complexity with big O notation, i.e.,
$\mathcal{O}()$. We denote the Gamma function with $\Gamma(.)$. Symbols
$\mathcal{U}(0,1)$ and $\mathcal{N}(0,I)$ respectively represent the uniform
distribution on $[0,1]$ and the normal distribution with zero mean and unit
variance.

##  II Problem Statement

We consider the following linear state-space model:

$\displaystyle\dot{x}(t)$ $\displaystyle=(A+BFC)x(t),$ (1)

where $x(t)\in\mathbb{R}^{n}$, $A\in\mathbb{R}^{n\times n}$,
$B\in\mathbb{R}^{n\times m}$, $C\in\mathbb{R}^{p\times n}$, and
$F\in\mathbb{R}^{m\times p}$ denote the state vector, state matrix, input
matrix, output matrix, and a nominal stabilizing SOF controller matrix (i.e.,
$\alpha(A+BFC)<0$ holds), respectively.

Suppose that a norm-bounded perturbation $\Delta\in\mathbb{R}^{n\times n}$
with an upper bound $\rho>0$ on its Frobenius norm, (i.e.,
$0<\|\Delta\|_{F}\leq\rho$) hits the state-space model (1) as follows:

$\displaystyle\dot{x}(t)$ $\displaystyle=(A+BFC+\Delta)x(t).$ (2)

Similar to [4, 5, 6, 22, 8], we choose the Frobenius norm over the spectral
norm as it provides more analytic convenience. On the one hand, for non-
destabilizing perturbations (e.g., sufficiently small perturbations), although
$A+BFC+\Delta$ in (2) is still a stable matrix, the stability robustness can
be degraded. On the other hand, for destabilizing perturbations (e.g., more
severe perturbations), $A+BFC+\Delta$ in (2) can become unstable. To attenuate
the impacts of such perturbations on the stability robustness and the
stability, a typical approach can be repeating the whole controller design
procedure to find a new SOF controller, namely $F^{\mathrm{typical}}$, to
stabilize $A+\Delta$ and get a stable $A+\Delta+BF^{\mathrm{typical}}C$. Such
a typical approach can be inefficient in terms of scalability and even
infeasible in some cases. Motivated by such an issue and utilizing a simple
norm minimization problem (a least-squares problem) built upon the notion of
MDRP [22], we propose a novel update of a nominal stabilizing SOF controller
that can be applied to various control engineering applications in the case of
perturbed scenarios like abrupt changes or inaccurate system models. In a
nutshell, the main objective of this paper is to find an answer to the
following question:

Q1: Given the perturbed state-space model (2), how can we update a nominal
stabilizing SOF controller $F$ such that the closed-loop system remains
stable?

##  III A Novel Update of a Nominal Stabilizing SOF Controller

This section consists of twofold: (i) motivation and (ii) main idea. First, we
present what motivates us to propose a novel update of a nominal stabilizing
SOF controller. Second, we detail the main idea behind the proposed updated
stabilizing SOF controller.

###  III-A Motivation

To improve the stability robustness of the perturbed state-space (2), let us
consider the updated stabilizing SOF controller, as $F+G$, with the following
state-space model:

$\displaystyle\dot{x}(t)$ $\displaystyle=(A+\Delta+B(F+G)C)x(t).$ (3)

For instance, for the special case of the typical approach,
$G^{\mathrm{typical}}=F^{\mathrm{typical}}-F$ holds.

Defining the notion of minimum destabilizing real perturbation (MDRP) of a
given stable matrix $\mathcal{A}\in\mathbb{R}^{n\times n}$, namely
$\beta_{\mathbb{R}}(\mathcal{A})$, as follows ($(3.2)$ in [22]):

$\displaystyle\beta_{\mathbb{R}}(\mathcal{A})$
$\displaystyle:=\min\\{\|\mathcal{X}\|_{F}:\alpha(\mathcal{A}+\mathcal{X})=0,\mathcal{X}\in\mathbb{R}^{n\times
n}\\},$

and choosing $\mathcal{A}=A+BFC$ and $\mathcal{X}=BGC+\Delta$ based on the
updated perturbed state-space model (3), we see that if

$\displaystyle\|BGC+\Delta\|_{F}$ $\displaystyle<\beta_{\mathbb{R}}(A+BFC),$
(4)

holds, then $A+\Delta+B(F+G)C$ is stable, i.e., $F+G$ is an updated
stabilizing SOF controller for $A+\Delta$. Inequality (4) motivates us to
search for an efficient update $F+G$ via minimizing the $\|BGC+\Delta\|_{F}$.

In the sequel, we present the lower and upper bounds on MDRP of $A+BFC$
followed by a brief description of its exact value computation.

####  III-A1 Lower bound

Considering the fact that $\alpha(X)$ is a continuous function with respect to
$X$, we have by definition

$\displaystyle\forall\epsilon>0,\exists\delta(\epsilon)>0,\leavevmode\nobreak\
\mathrm{s.t.}\leavevmode\nobreak\ \mathrm{if}\leavevmode\nobreak\
\|\mathcal{X}\|_{F}<\delta(\epsilon)\leavevmode\nobreak\
\mathrm{holds},\leavevmode\nobreak\ \mathrm{then}$
$\displaystyle\alpha(\mathcal{A})-\epsilon<\alpha(\mathcal{A}+\mathcal{X})<\alpha(\mathcal{A})+\epsilon\leavevmode\nobreak\
\mathrm{holds},$

Then, choosing $\mathcal{A}=A+BFC$ and $\mathcal{X}=BGC+\Delta$, we realize
that for any $\epsilon$ satisfying $\epsilon<-\alpha(A+BFC)$, if
$\|BGC+\Delta\|_{F}<\delta(\epsilon)$ holds, then $A+\Delta+B(F+G)C$ is
stable. That suggests the following lower bound on MDRP of $A+BFC$:

$\displaystyle 0<\delta_{\sup}\leq\beta_{\mathbb{R}}(A+BFC),$ (5a)
$\displaystyle\delta_{\sup}:=\sup\\{\delta(\epsilon):\epsilon\in]0,-\alpha(A+BFC)[\\},$
(5b)

####  III-A2 Upper bound

On one hand, since $\alpha(\mathcal{A}+\mathcal{X})=0$ holds for the choice of
$\mathcal{X}=-\alpha(\mathcal{A})I_{n}$, then choosing $\mathcal{A}=A+BFC$ and
$\mathcal{X}=-\alpha(A+BFC)I_{n}$, we get the following upper bound on MDRP of
$A+BFC$ [22]:

$\displaystyle\beta_{\mathbb{R}}(A+BFC)\leq-\sqrt{n}\alpha(A+BFC).$ (6)

On the other hand, given
$\mathcal{A}=U_{\mathcal{A}}\Sigma_{\mathcal{A}}V_{\mathcal{A}}^{T}$ as the
singular value decomposition (SVD) of $\mathcal{A}$ and choosing
$\mathcal{X}=-\sigma_{\mathcal{A}}^{\min}u_{\mathcal{\mathcal{A}}}^{\min}v_{\mathcal{A}}^{\min
T}$ (superscript $\min$ denotes the corresponding minimum singular value and
vectors), it can be verified that $\alpha(\mathcal{A}+\mathcal{X})=0$ holds.
Then, choosing $\mathcal{A}=A+BFC$ and according to (6), we get the following
upper bound on MDRP of $A+BFC$ [22]:

$\displaystyle\beta_{\mathbb{R}}(A+BFC)\leq\beta_{\mathbb{R}}^{u},$ (7a)
$\displaystyle\beta_{\mathbb{R}}^{u}=\min\\{\sigma^{\min}(A+BFC),-\sqrt{n}\alpha(A+BFC)\\}.$
(7b)

For the special case of a symmetric matrix $A+BFC$, since
$\sigma^{\min}(A+BFC)=-\alpha(A+BFC)$ holds, (7) reduces to

$\displaystyle\beta_{\mathbb{R}}(A+BFC)\leq-\alpha(A+BFC),$ (8)

which is a tighter bound compared to the upper bound in (6). According to
Corollary $3.5.$ in [26] and noting that
$\|\mathcal{X}\|\leq\|\mathcal{X}\|_{F}$ holds for any $\mathcal{X}$ [25], it
can be verified that $\sigma^{\min}(A+BFC)\leq\beta_{\mathbb{R}}(A+BFC)$ holds
and the equality in (8) is consequently satisfied.

####  III-A3 Exact value

Unfortunately, computing the exact value of $\beta_{\mathbb{R}}(A+BFC)$ is not
theoretically possible [22]. Also, there is no systematic tractable way to
compute the exact value of the lower bound $\delta_{\sup}$ in (5) since we
only know about the existence of $\delta(\epsilon)$ and nothing more. However,
taking advantage of the upper bounds on $\beta_{\mathbb{R}}(A+BFC)$ (derived
in (7) and (8)), we may utilize heuristics to obtain an appropriate
approximate value of $\beta_{\mathbb{R}}(A+BFC)$ in a reasonable computational
time. Since (4) plays a significant role in the characterization of the
stability regions, the tightness of the upper bound on
$\beta_{\mathbb{R}}(A+BFC)$ in (7) becomes crucial. Remarkably, if the
equality in (7) becomes active (i.e., the case of a tight upper bound), then
the proposed updated stabilizing SOF controller in this paper becomes
efficient as it only requires the value of $\beta_{\mathbb{R}}^{u}$ which can
efficiently be computed (e.g., the case of a symmetric $A+BFC$ for which
$\beta_{\mathbb{R}}(A+BFC)=-\alpha(A+BFC)$ holds). For the special case of
structured perturbation, i.e., $\Delta=BMC$ for a matrix
$M\in\mathbb{R}^{m\times p}$, one may compute MDRP via (frequency
domain)-based algorithms detailed by [27].

###  III-B Main idea

Since (4) provides a sufficient condition on the stability of
$A+\Delta+B(F+G)C$, our main idea to propose an efficient updated stabilizing
SOF controller $F+G$ is to compute $G$ via minimizing $\|BGC+\Delta\|_{F}^{2}$
and verifying that under which conditions, the minimized value of
$\|BGC+\Delta\|_{F}^{2}$ would be less than $\beta_{\mathbb{R}}(A+BFC)^{2}$.
It is noteworthy that if the most optimistic scenario occurs, (i.e., the
scenario in which for a known $\Delta$, equation $\|BGC+\Delta\|_{F}=0$ has a
solution $G$), then one can completely cancel out the effect of the hitting
perturbation $\Delta$ and retrieve the primary unperturbed $A+BFC$ as detailed
later on. With that in mind and to find a reasonable answer to the question
stated in Section II (Q1), we consider the following optimization problem:

$\displaystyle\underset{G\in\mathbb{R}^{m\times p}}{\min}\leavevmode\nobreak\
\|BGC+\Delta\|_{F}^{2}.$ (9)

By vectorizing $BGC+\Delta$, defining
$g:=\mathbf{vec}(G),\delta:=\mathbf{vec}(\Delta),H:=C^{T}\otimes B$, and
noting that
$\mathbf{vec}(\mathcal{XYZ})=(\mathcal{Z}^{T}\otimes\mathcal{X})\mathbf{vec}(\mathcal{Y})$
holds for any triplet $(\mathcal{X},\mathcal{Y},\mathcal{Z})$ with consistent
dimensions and $\|\mathbf{vec}(X)\|=\|X\|_{F}$ holds for any $X$, optimization
problem (9) can equivalently be cast as the following least-squares problem
[28]:

$\displaystyle\underset{g\in\mathbb{R}^{mp}}{\min}\leavevmode\nobreak\
\|Hg+\delta\|^{2}.$ (10)

In this paper, we assume that the following standard assumption holds for $B$
and $C$.

######  Assumption 1.

We assume that $B$ and $C$ are full-column rank and full-row rank,
respectively.

According to Assumption 1 and noting that identity $(C^{T}\otimes
B)^{+}={C}^{T+}\otimes B^{+}$ holds, optimization problem (10) can
analytically be solved as

$\displaystyle g^{\ast}_{\delta}$ $\displaystyle=-({C}^{T+}\otimes
B^{+})\delta,$ (11)

and the analytic optimal solution of (9) can subsequently be presented as
follows:

$\displaystyle G^{\ast}_{\Delta}$
$\displaystyle=\mathbf{vec}^{-1}(g^{\ast}_{\delta})=-B^{+}\Delta({C}^{T+})^{T},$
(12)

for which the computation complexity is $\mathcal{O}(n^{2}\min\\{m,p\\})$
while the computation complexity of (11) is $\mathcal{O}(n^{2}m^{2}p^{2})$.
Substituting $g^{\ast}_{\delta}$ of (11) in (10), the optimal value of the
objective function in (10), namely $J^{\ast}(\delta)$, becomes

$\displaystyle
J^{\ast}(\delta):=\|Hg^{\ast}_{\delta}+\delta\|^{2}=\|(I_{n^{2}}-HH^{+})\delta\|^{2}.$
(13)

Defining $P:=I_{n^{2}}-HH^{+}$ and noting that $P^{T}P=P$ holds (since
$H^{+}H=I_{mp}$ holds), (13) reduces to

$\displaystyle J^{\ast}(\delta)$ $\displaystyle=\delta^{T}P\delta.$ (14)

For the sake of preciseness, with a bit of abuse of notation, we simply define
$J^{\ast}(\Delta):=J^{\ast}(\mathbf{vec}(\Delta))=J^{\ast}(\delta)$.

##  IV Main Results

This section consists of the main results of the paper. The main results are
twofold: (i) In Section IV-A, given a known norm-bounded perturbation $\Delta$
with $0<\|\Delta\|_{F}\leq\rho$, we investigate the dependency of
$J^{\ast}(\Delta)$ on $\Delta$ via inspecting the linear algebraic properties
of $P$ in (14). Proposition 1 analytically parameterizes the norm-bounded
perturbation and proposes a closed-form formula for $J^{\ast}(\Delta)$.
Proposition 2 elaborates on deriving sufficient conditions for the stability
of the proposed updated stabilizing SOF controllers while analytically
characterizing the guaranteed stability regions. Furthermore, we define a
geometric metric to quantify the stability robustness of the proposed updated
stabilizing SOF controllers, (ii) in Section IV-B, given an unknown norm-
bounded perturbation $\Delta$ with a known upper bound $\rho$ on its Frobenius
norm, we derive sufficient conditions on the stability of the proposed updated
stabilizing SOF controllers in Proposition 3. Proposition 4 mathematically
characterizes the guaranteed stability regions for which the proposed updated
SOF controllers are stabilizing. Similarly, we define a geometric metric to
quantify the stability quality of the proposed updated stabilizing SOF
controllers. Also, built upon a notion of non-fragility utilized in the
literature of robust non-fragile PID controller designs [29, 30, 31, 32], we
propose non-fragility-based robust updated stabilizing SOF controllers. In the
sequel, to save space, whenever needed, we refer to
$\beta_{\mathbb{R}}(A+BFC)$ as $\beta$.

###  IV-A Known norm-bounded perturbation

In the following lemma, we present an SVD-based parameterization of $P$ in
(14) that facilitates parameterizing the norm-bounded perturbation $\Delta$
and subsequently proposing a closed-form expression for $J^{\ast}(\Delta)$.

######  Lemma 1.

Suppose that $H=U_{H}\Sigma_{H}V_{H}^{T}$ is the SVD of $H$. Then, $P$ in (14)
can be parameterized as follows:

$\displaystyle P$ $\displaystyle=U_{H}\begin{bmatrix}0&0\\\
0&I_{n^{2}-mp}\end{bmatrix}U_{H}^{T},$ (15)

where $U_{H}=(V_{C}\otimes U_{B})U_{\Omega}$ holds provided that
$B=U_{B}\Sigma_{B}V_{B}^{T}$, $C=U_{C}\Sigma_{C}V_{C}^{T}$, and
$\Omega:=\Sigma_{C}^{T}\otimes\Sigma_{B}=U_{\Omega}\Sigma_{\Omega}V_{\Omega}^{T}$
denote the SVDs of $B$, $C$, and $\Omega$, respectively.

###### Proof:

See Appendix A. ∎

####  IV-A1 Norm-bounded perturbation analytic parameterization

Built upon Lemma 1, we present the following proposition that analytically
parameterizes the norm-bounded perturbation $\Delta$ while proposing a closed-
form expression for $J^{\ast}(\Delta)$.

######  Proposition 1.

Given the norm-bounded perturbation $\Delta$ with $\|\Delta\|_{F}=r$ and
$r\in]0,\rho]$, and considering $r=\rho\sin(\frac{\pi\tau}{2})$ with
$\tau\in]0,1]$, the norm-bounded perturbation $\Delta$ can be parameterized as
follows:

$\displaystyle\Delta$
$\displaystyle=\rho\sin\Big{(}\frac{\pi\tau}{2}\Big{)}U_{B}\mathbf{vec}^{-1}\bigg{(}U_{\Omega}\begin{bmatrix}\phi_{c}\cos(\frac{\pi\theta}{2})\\\
\phi_{s}\sin(\frac{\pi\theta}{2})\end{bmatrix}\bigg{)}V_{C}^{T},$ (16)

where $\phi_{c}\in\mathbb{R}^{mp}$ with $\|\phi_{c}\|=1$,
$\phi_{s}\in\mathbb{R}^{n^{2}-mp}$ with $\|\phi_{s}\|=1$, and
$\theta\in[0,1]$, and we can compute $J^{\ast}(\Delta)$ in (14) as follows:

$\displaystyle J^{\ast}(\Delta)$
$\displaystyle=\bigg{(}\rho\sin\Big{(}\frac{\pi\tau}{2}\Big{)}\sin\Big{(}\frac{\pi\theta}{2}\Big{)}\bigg{)}^{2}.$
(17)

###### Proof:

See Appendix B. ∎

The following corollary provides an alternative formula to compute
$G^{\ast}_{\Delta}$ in (12).

######  Corollary 1.

Considering the following identities:

$\displaystyle(U_{H},\Sigma_{H},V_{H})=((V_{C}\otimes
U_{B})U_{\Omega},\Sigma_{\Omega},(U_{C}\otimes V_{B})V_{\Omega}),$
$\displaystyle
B=U_{B}\Sigma_{B}V_{B}^{T},C=U_{C}\Sigma_{C}V_{C}^{T},\Sigma_{C}^{T}\otimes\Sigma_{B}=U_{\Omega}\Sigma_{\Omega}V_{\Omega}^{T},$

(12) can alternatively be computed as follows:

$\displaystyle
G^{\ast}_{\Delta}=-\mathbf{vec}^{-1}(V_{H}\begin{bmatrix}(\begin{bmatrix}I_{mp}&0\end{bmatrix}\Sigma_{H})^{-1}&0\end{bmatrix}U_{H}^{T}\mathbf{vec}(\Delta)).$

Fig. 1 depicts the dependency of $\frac{J^{\ast}(\Delta)}{\rho^{2}}$ on $\tau$
and $\theta$. As expected, since functions $\sin(\frac{\pi\tau}{2})$ and
$\sin(\frac{\pi\theta}{2})$ have monotonic behaviors versus $\tau$ (for
$\tau\in]0,1]$) and $\theta$ (for $\theta\in[0,1]$), respectively, the smaller
$\tau$ and/or $\theta$, the smaller $\frac{J^{\ast}(\Delta)}{\rho^{2}}$ we
get. Note that the smaller value of $\frac{J^{\ast}(\Delta)}{\rho^{2}}$ is
equivalent to the higher chance of satisfaction of the sufficient stability
condition (4). In other words, its intuitive interpretation is that handling a
less severe perturbation via an updated stabilizing SOF controller
$F+G^{\ast}_{\Delta}$ with $G^{\ast}_{\Delta}$ in (12) is easier.

Figure 1: The dependency of $\frac{J^{\ast}(\Delta)}{\rho^{2}}$ on $\tau$ and
$\theta$.

####  IV-A2 The guaranteed stability region analytic characterization

We state the following proposition that derives sufficient conditions on the
stability of the proposed updated stabilizing SOF controllers while
analytically characterizing the guaranteed stability regions.

######  Proposition 2.

Given the norm-bounded perturbation $\Delta$ parameterized by (16),
$F+G^{\ast}_{\Delta}$ with $G^{\ast}_{\Delta}$ in (12) is an updated
stabilizing SOF controller,

  1. i

. if $\rho<\beta_{\mathbb{R}}(A+BFC)$ holds.

  2. ii

. else if $\rho\geq\beta_{\mathbb{R}}(A+BFC)$ and
$(\tau_{\Delta},\theta_{\Delta})\in S_{\kappa}$ hold where the guaranteed
stability region $S_{\kappa}$ is defined as:

$\displaystyle S_{\kappa}$ $\displaystyle:=\check{S}\cup\tilde{S},$ (18a)
$\displaystyle\check{S}$
$\displaystyle:=\\{(\tau,\theta):\tau\in]0,\kappa[,\theta\in[0,1]\\},$ (18b)
$\displaystyle\kappa$
$\displaystyle:=\frac{2}{\pi}\arcsin\Big{(}\frac{\beta_{\mathbb{R}}(A+BFC)}{\rho}\Big{)},$
(18c) $\displaystyle\tilde{S}$
$\displaystyle:=\\{(\tau,\theta):\tau\in[\kappa,1],\theta\in[0,\zeta_{\tau,\kappa}[\\},$
(18d) $\displaystyle\zeta_{\tau,\kappa}$
$\displaystyle:=\frac{2}{\pi}\arcsin\Big{(}\frac{\sin(\frac{\pi\kappa}{2})}{\sin(\frac{\pi\tau}{2})}\Big{)}.$
(18e)

Moreover, the following geometric metric provides a percentage-based lower
bound on the stability of the updated perturbed state-space (3):

$\displaystyle\xi_{\kappa}\leavevmode\nobreak\ (\%)$
$\displaystyle:=100\times\bigg{(}\kappa+\int_{\kappa}^{1}\zeta_{\tau,\kappa}d\tau\bigg{)},$
(19)

and $\xi_{\kappa}$ is an increasing function of $\kappa$ (equivalently
$\xi_{\rho}$ is a decreasing function of $\rho$ for a fixed
$\beta_{\mathbb{R}}(A+BFC)$ and an increasing function of
$\beta_{\mathbb{R}}(A+BFC)$ for a fixed $\rho$).

###### Proof:

See Appendix C. ∎

For the case of $\rho<\beta_{\mathbb{R}}(A+BFC)$, the guaranteed stability
region would be $]0,1]\times[0,1]=S_{\kappa}|_{\kappa=1}\cup\\{(1,1)\\}$,
i.e., the unit square in the non-negative quadrant of $(\tau,\theta)$. For the
sake of notation simplicity, we define $\mathbb{S}=]0,1]\times[0,1]$ and
utilize the unified notation of $S$ to refer to both guaranteed stability
regions $S_{\kappa}$ and $\mathbb{S}$. The following corollary thoroughly
sheds light on the dependency and limiting behaviors of $\xi_{\rho}$ and
$\xi_{\beta}$ on $\rho$ and $\beta$, respectively.

######  Corollary 2.

For the case of $\rho\geq\beta_{\mathbb{R}}(A+BFC)$, considering the following
expression for $\xi_{\rho}$:

$\displaystyle\xi_{\rho}$
$\displaystyle=\frac{2}{\pi}\arcsin\Big{(}\frac{\beta}{\rho}\Big{)}+\frac{2}{\pi}\int_{\frac{2}{\pi}\arcsin\big{(}\frac{\beta}{\rho}\big{)}}^{1}\arcsin\Big{(}\frac{\beta}{\rho\sin(\frac{\pi\tau}{2})}\Big{)}d\tau,$

we compute the derivative of $\xi_{\rho}$ with respect to $\rho$ as follows:

$\displaystyle\frac{d\xi_{\rho}}{d\rho}$
$\displaystyle=-\frac{2}{\pi\rho}\int_{\frac{2}{\pi}\arcsin\big{(}\frac{\beta}{\rho}\big{)}}^{1}\frac{\beta}{\rho\sqrt{\sin(\frac{\pi\tau}{2})^{2}-(\frac{\beta}{\rho})^{2}}}d\tau.$
(20)

Moreover, as $\rho$ tends to $\beta$ and $\infty$ in (20), we get

$\displaystyle\lim_{\rho\to\beta^{+}}\frac{d\xi_{\rho}}{d\rho}=-\frac{2}{\pi\beta},\lim_{\rho\to\infty}\frac{d\xi_{\rho}}{d\rho}=0,\lim_{\rho\to\beta^{+}}\xi_{\rho}=1,\lim_{\rho\to\infty}\xi_{\rho}=0.$

Similarly, considering the following expression for $\xi_{\beta}$:

$\displaystyle\xi_{\beta}$
$\displaystyle=\frac{2}{\pi}\arcsin\Big{(}\frac{\beta}{\rho}\Big{)}+\frac{2}{\pi}\int_{\frac{2}{\pi}\arcsin\big{(}\frac{\beta}{\rho}\big{)}}^{1}\arcsin\Big{(}\frac{\beta}{\rho\sin(\frac{\pi\tau}{2})}\Big{)}d\tau,$

we compute the derivative of $\xi_{\beta}$ with respect to $\beta$ as follows:

$\displaystyle\frac{d\xi_{\beta}}{d\beta}$
$\displaystyle=\frac{2}{\pi\rho}\int_{\frac{2}{\pi}\arcsin\big{(}\frac{\beta}{\rho}\big{)}}^{1}\frac{1}{\sqrt{\sin(\frac{\pi\tau}{2})^{2}-(\frac{\beta}{\rho})^{2}}}d\tau.$
(21)

Moreover, by tending $\beta$ to $0$ and $\rho$ in (21), we get

$\displaystyle\lim_{\beta\to
0^{+}}\frac{d\xi_{\beta}}{d\beta}=\infty,\lim_{\beta\to\rho^{-}}\frac{d\xi_{\beta}}{d\beta}=\frac{2}{\pi\rho},\lim_{\beta\to
0^{+}}\xi_{\beta}=0,$ $\displaystyle\lim_{\beta\to\rho^{-}}\xi_{\beta}=1.$

Fig. 2 visualizes the guaranteed stability region $S_{\kappa}$ for
$\kappa=\frac{1}{3}$ and the percentage-based lower bounds on the stability of
the updated perturbed state-space (3) versus $\kappa$, $\rho$, and $\beta$. As
expected, the empirical observations of Fig. 2 are consistent with the
theoretical results of Proposition 2 and Corollary 2. Precisely, as $\kappa$
decreases, e.g., for an increased perturbation upper bound $\rho$ or a
decreased MDRP $\beta$, the percentage-based lower bound on the stability of
the updated perturbed state-space (3) $\xi\leavevmode\nobreak\ (\%)$ degrades
which is expected. As Fig. 2 (Top-Left) depicts, for the sufficiently large
values of $\tau$ and/or $\theta$, i.e., more severe perturbations,
$(\tau,\theta)$ lies outside the $S_{\kappa}$ and there is no stability
guarantee for the proposed updated SOF controller which is aligned with the
expectations around the negative impacts of perturbations on the stability.

(a) a

(b) b

(c) c

(d) d

Figure 2: (a) The guaranteed stability region $S_{\kappa}$ for
$\kappa=\frac{1}{3}$, (b) the percentage-based lower bound on the stability of
the updated perturbed state-space (3) $\xi_{\kappa}\leavevmode\nobreak\ (\%)$
versus $\kappa$, (c) the percentage-based lower bound on the stability of the
updated perturbed state-space (3) $\xi_{\rho}\leavevmode\nobreak\ (\%)$ versus
$\rho$ for $\beta=1$, and (d) the percentage-based lower bound on the
stability of the updated perturbed state-space (3)
$\xi_{\beta}\leavevmode\nobreak\ (\%)$ versus $\beta$ for $\rho=1$.

###  IV-B Unknown norm-bounded perturbation

Given an unknown norm-bounded perturbation $\Delta$ with
$0<\|\Delta\|_{F}\leq\rho$, let us denote a known norm-bounded perturbation
with an upper bound $\rho$ on its Frobenius norm as $\hat{\Delta}$. We refer
to $\hat{\Delta}$ as an estimate of unknown $\Delta$. Also, whenever needed,
for ease of representation, we will simply denote $\tau_{\hat{\Delta}}$ and
$\theta_{\hat{\Delta}}$ with $\hat{\tau}$ and $\hat{\theta}$, respectively.
Moreover, we represent the guaranteed stability regions associated with
$\hat{\Delta}$ by $\hat{S}_{\kappa}$, $\hat{\mathbb{S}}$, and $\hat{S}$ (the
unified notation for both $\hat{S}_{\kappa}$ and $\hat{\mathbb{S}}$). In the
following proposition, we derive sufficient stability conditions under which
the proposed updated SOF controllers are stabilizing.

######  Proposition 3.

Given an unknown norm-bounded perturbation $\Delta$ and its estimate
$\hat{\Delta}$ both with an upper bound $\rho$ on their Frobenius norms,
$F+G^{\ast}_{\hat{\Delta}}$ with $G^{\ast}_{\hat{\Delta}}$ in (12) is an
updated stabilizing SOF controller,

  1. i

. if $\rho<\beta_{\mathbb{R}}(A+BFC)$ and

$\displaystyle\|\Delta-\hat{\Delta}\|_{F}<\upsilon,$ (22a)
$\displaystyle\upsilon:=\beta_{\mathbb{R}}(A+BFC)-\rho\sin\Big{(}\frac{\pi\tau_{\hat{\Delta}}}{2}\Big{)}\sin\Big{(}\frac{\pi\theta_{\hat{\Delta}}}{2}\Big{)},$
(22b)

hold.

  2. ii

. if $\rho\geq\beta_{\mathbb{R}}(A+BFC)$, (22), and
$(\tau_{\hat{\Delta}},\theta_{\hat{\Delta}})\in\hat{S}_{\kappa}$ hold.

###### Proof:

See Appendix D. ∎

The following lemma enables us to have a more thorough quantitative
understanding of the estimation inaccuracy and its dependency on various
factors.

######  Lemma 2.

Given the $\Delta$ and $\hat{\Delta}$ as in Proposition 3, the following
identity holds:

$\displaystyle\|\Delta-\hat{\Delta}\|_{F}$
$\displaystyle=\rho\sqrt{s_{\tau}^{2}+s_{\hat{\tau}}^{2}-2s_{\tau}s_{\hat{\tau}}c_{2\eta}},$
(23a) $\displaystyle s_{\tau}$
$\displaystyle:=\sin\Big{(}\frac{\pi\tau_{\Delta}}{2}\Big{)},s_{\hat{\tau}}:=\sin\Big{(}\frac{\pi\tau_{\hat{\Delta}}}{2}\Big{)},$
(23b) $\displaystyle c_{2\eta}$
$\displaystyle:=\cos(\pi\eta),\eta:=\frac{1}{\pi}\arccos(\psi^{T}\hat{\psi}),$
(23c) $\displaystyle\psi$
$\displaystyle:=\psi_{\Delta},\hat{\psi}:=\psi_{\hat{\Delta}}.$ (23d)

###### Proof:

See Appendix E. ∎

Note that $\pi\eta$ denotes the phase difference between $\psi$ and
$\hat{\psi}$.

####  IV-B1 The guaranteed stability region mathematical characterization

Built upon Proposition 3 and Lemma 2, we present the following proposition
that lists all the possible parametric scenarios for mathematically
characterizing the guaranteed stability regions.

######  Proposition 4.

Given the $\Delta$ and $\hat{\Delta}$ as in Proposition 3 and defining

$\displaystyle
s_{\hat{\theta}}:=\sin\bigg{(}\frac{\pi\theta_{\hat{\Delta}}}{2}\bigg{)},\iota:=\frac{\beta_{\mathbb{R}}(A+BFC)}{\rho
s_{\hat{\tau}}}-s_{\hat{\theta}}=\frac{\upsilon}{\rho s_{\hat{\tau}}},$
$\displaystyle\bar{\eta}:=\frac{1}{\pi}\arcsin(\iota),\mathrm{for}\leavevmode\nobreak\
0<\iota\leq 1,\leavevmode\nobreak\
\hat{b}:=\frac{s_{\hat{\tau}}}{2}(1-\iota^{2})+\frac{1}{2s_{\hat{\tau}}},$
$\displaystyle\underline{\eta}:=\frac{1}{\pi}\arccos(\hat{b}),\mathrm{for}\leavevmode\nobreak\
|\hat{b}|\leq 1,\leavevmode\nobreak\ s_{2\eta}:=\sin(\pi\eta),$ $\displaystyle
b_{l}(\eta):=s_{\hat{\tau}}\Big{(}c_{2\eta}-\sqrt{\iota^{2}-s_{2\eta}^{2}}\Big{)},$
$\displaystyle\varphi_{l}(\eta):=\frac{2}{\pi}\arcsin(b_{l}(\eta)),\mathrm{for}\leavevmode\nobreak\
0\leq b_{l}(\eta)<1,$ $\displaystyle
b_{u}(\eta):=s_{\hat{\tau}}\Big{(}c_{2\eta}+\sqrt{\iota^{2}-s_{2\eta}^{2}}\Big{)},$
$\displaystyle\varphi_{u}(\eta):=\frac{2}{\pi}\arcsin(b_{u}(\eta)),\mathrm{for}\leavevmode\nobreak\
0<b_{u}(\eta)\leq 1,$

if $(\eta_{\Delta},\tau_{\Delta})\in\mathcal{S}$ holds, then
$F+G^{\ast}_{\hat{\Delta}}$ with $G^{\ast}_{\hat{\Delta}}$ in (12) is an
updated stabilizing SOF controller where the guaranteed stability region
$\mathcal{S}$ in $2$-dimensional parametric space of $(\eta,\tau)$ can be
characterized via the following itemized approach:

  1. i

. if $0<\iota\leq 1$ and $\hat{b}\leq 1$ hold, then $\mathcal{S}$ is defined
as

$\displaystyle\mathcal{S}$
$\displaystyle:=\check{\mathcal{S}}\cup\tilde{\mathcal{S}},$ (24a)
$\displaystyle\check{\mathcal{S}}$
$\displaystyle:=\\{(\eta,\tau):\eta\in[0,\underline{\eta}[,\tau\in]\varphi_{l}(\eta),1]\\},$
(24b) $\displaystyle\tilde{\mathcal{S}}$
$\displaystyle:=\\{(\eta,\tau):\eta\in[\underline{\eta},\bar{\eta}[,\tau\in]\varphi_{l}(\eta),\varphi_{u}(\eta)[\\},$
(24c)

  2. ii

. if $0<\iota\leq 1$ and $\hat{b}>1$ hold, then $\mathcal{S}$ is defined as

$\displaystyle\mathcal{S}$
$\displaystyle:=\\{(\eta,\tau):\eta\in[0,\bar{\eta}[,\tau\in]\varphi_{l}(\eta),\varphi_{u}(\eta)[\\},$
(25)

  3. iii

. if $\iota>1$ and $|\hat{b}|\leq 1$ hold, then $\mathcal{S}$ is defined as

$\displaystyle\mathcal{S}$
$\displaystyle:=\check{\mathcal{S}}\cup\tilde{\mathcal{S}},$ (26a)
$\displaystyle\check{\mathcal{S}}$
$\displaystyle:=\\{(\eta,\tau):\eta\in[0,\underline{\eta}[,\tau\in]0,1]\\},$
(26b) $\displaystyle\tilde{\mathcal{S}}$
$\displaystyle:=\\{(\eta,\tau):\eta\in[\underline{\eta},1],\tau\in]0,\varphi_{u}(\eta)[\\},$
(26c)

  4. iv

. if $\iota>1$ and $\hat{b}>1$ hold, then $\mathcal{S}$ is defined as

$\displaystyle\mathcal{S}$
$\displaystyle:=\\{(\eta,\tau):\eta\in[0,1],\tau\in]0,\varphi_{u}(\eta)[\\},$
(27)

  5. v

. if $\iota>1$ and $\hat{b}<-1$ hold, then $\mathcal{S}$ is defined as

$\displaystyle\mathcal{S}$
$\displaystyle:=\\{(\eta,\tau):\eta\in[0,1],\tau\in]0,1]\\},$ (28)

Moreover, if $\rho<\beta_{\mathbb{R}}(A+BFC)$ holds, then $0<\iota$
automatically holds. Also, in the case of $\rho\geq\beta_{\mathbb{R}}(A+BFC)$,
$0<\iota$ holds if and only if
$(\tau_{\hat{\Delta}},\theta_{\hat{\Delta}})\in\hat{S}_{\kappa}$ holds.

###### Proof:

See Appendix F. ∎

Utilizing the following equivalences:

$\displaystyle\iota>0\iff
s_{\hat{\tau}}s_{\hat{\theta}}\leq\frac{\beta}{\rho},\leavevmode\nobreak\
\iota\leq 1\iff\frac{\beta}{\rho}\leq s_{\hat{\tau}}(s_{\hat{\theta}}+1),$
$\displaystyle\hat{b}\leq 1\iff\iota\geq\frac{1}{s_{\hat{\tau}}}-1\iff
1+s_{\hat{\tau}}(s_{\hat{\theta}}-1)\leq\frac{\beta}{\rho},$
$\displaystyle\hat{b}\geq-1\iff\iota\leq\frac{1}{s_{\hat{\tau}}}+1\iff\frac{\beta}{\rho}\leq
1+s_{\hat{\tau}}(s_{\hat{\theta}}+1),$

the following corollary facilitates the itemized characterization proposed by
Proposition 4.

######  Corollary 3.

The if parts of the items presented by Proposition 4 can be simplified into
the following items:

  1. i

. if $1+s_{\hat{\tau}}(s_{\hat{\theta}}-1)\leq\frac{\beta}{\rho}\leq
s_{\hat{\tau}}(s_{\hat{\theta}}+1)$ and
$\frac{1}{3}\leq\tau_{\hat{\Delta}}\leq 1$ hold.

  2. ii

. 1) if $s_{\hat{\tau}}s_{\hat{\theta}}<\frac{\beta}{\rho}\leq
s_{\hat{\tau}}(s_{\hat{\theta}}+1)$ and $0<\tau_{\hat{\Delta}}<\frac{1}{3}$
hold,
or
2) if
$s_{\hat{\tau}}s_{\hat{\theta}}<\frac{\beta}{\rho}<1+s_{\hat{\tau}}(s_{\hat{\theta}}-1)$
and $\frac{1}{3}\leq\tau_{\hat{\Delta}}<1$ hold.

  3. iii

. 1) if $1+s_{\hat{\tau}}(s_{\hat{\theta}}-1)\leq\frac{\beta}{\rho}\leq
1+s_{\hat{\tau}}(s_{\hat{\theta}}+1)$ and $0<\tau_{\hat{\Delta}}<\frac{1}{3}$
hold,
or
2) if $s_{\hat{\tau}}(s_{\hat{\theta}}+1)<\frac{\beta}{\rho}\leq
1+s_{\hat{\tau}}(s_{\hat{\theta}}+1)$ and
$\frac{1}{3}\leq\tau_{\hat{\Delta}}\leq 1$ hold.

  4. iv

. if
$s_{\hat{\tau}}(s_{\hat{\theta}}+1)<\frac{\beta}{\rho}<1+s_{\hat{\tau}}(s_{\hat{\theta}}-1)$
and $0<\tau_{\hat{\Delta}}<\frac{1}{3}$ hold.

  5. v

. if $1+s_{\hat{\tau}}(s_{\hat{\theta}}+1)<\frac{\beta}{\rho}$ holds.

Note that the upper bounds of $\frac{\beta}{\rho}$ in item ii in Corollary 3
and the lower bounds of $\frac{\beta}{\rho}$ in item iii in Corollary 3 can
compactly be expressed as follows:

$\displaystyle\textit{ii}.\leavevmode\nobreak\
s_{\hat{\tau}}s_{\hat{\theta}}+\min\\{s_{\hat{\tau}},1-s_{\hat{\tau}}\\},$
$\displaystyle\textit{iii}.\leavevmode\nobreak\
s_{\hat{\tau}}s_{\hat{\theta}}+\max\\{s_{\hat{\tau}},1-s_{\hat{\tau}}\\},$

where highlights the appearance of the threshold
$\tau_{\hat{\Delta}}=\frac{1}{3}$, i.e.,
$\|\hat{\Delta}\|_{F}=\frac{\rho}{2}$.

Fig. 3 illustrates the guaranteed stability region $\mathcal{S}$ in
$2$-dimensional parametric space of $(\eta,\tau)$ for various cases itemized
by Corollary 3. Remarkably, item v in Corollary 3 can only occur for the case
of $\rho<\beta$ as $1<1+s_{\hat{\tau}}(s_{\hat{\theta}}+1)$ should be
satisfied. Also, the non-trivial boundary points with $\eta=0$
($(0,\tau_{l}^{0})$ and $(0,\tau_{u}^{0})$) or $\eta=1$ ($(0,\tau_{u}^{1})$)
can be computed via the following formulas:

$\displaystyle\tau_{l}^{0}$
$\displaystyle=\frac{2}{\pi}\arcsin\bigg{(}s_{\hat{\tau}}(1+s_{\hat{\theta}})-\frac{\beta}{\rho}\bigg{)},$
$\displaystyle\tau_{u}^{0}$
$\displaystyle=\frac{2}{\pi}\arcsin\bigg{(}s_{\hat{\tau}}(1-s_{\hat{\theta}})+\frac{\beta}{\rho}\bigg{)},$
$\displaystyle\tau_{u}^{1}$
$\displaystyle=\frac{2}{\pi}\arcsin\bigg{(}s_{\hat{\tau}}(-1-s_{\hat{\theta}})+\frac{\beta}{\rho}\bigg{)}.$

It is noteworthy that the extreme cases $\eta=0$ (no phase difference) and
$\eta=1$ (maximum phase difference) represent the special cases
$\hat{\psi}=\psi$ and $\hat{\psi}=-\psi$, respectively.

(a) a

(b) b

(c) c

(d) d

(e) e

(f) f

(g) g

Figure 3: The guaranteed stability region $\mathcal{S}$ in $2$-dimensional
parametric space of $(\eta,\tau)$ for various cases itemized by Corollary 3
(a) i, (b) ii-1, (c) ii-2, (d) iii-1, (e) iii-2, (f) iv, and (g) v.

Similar to the case with known perturbation, we define a geometric metric to
provide a percentage-based lower bound on the stability of the updated
perturbed state-space (3). Given the guaranteed stability region $\mathcal{S}$
in $2$-dimensional parametric space of $(\eta,\tau)$ (as presented by
Proposition 4 and Corollary 3), we define the following geometric metric:

$\displaystyle\Xi_{\tau_{\hat{\Delta}},\theta_{\hat{\Delta}};\frac{\beta}{\rho},n}\leavevmode\nobreak\
(\%)$
$\displaystyle:=100\times\frac{\mathcal{V}_{n^{2}}(\texttt{D}(\mathcal{S}))}{\mathcal{V}_{n^{2}}(\texttt{S}_{\rho}^{n^{2}})}.$
(29)

where $\mathcal{V}_{N}(.)$, $\texttt{S}_{r}^{N}$, and
$\texttt{D}(\mathcal{S})$ denotes the $N$-dimensional volume of an object,
$N$-dimensional hypersphere of radius $r$ centered at origin, and set of all
$\delta$ with $\|\mathbf{vec}^{-1}(\delta)\|_{F}\leq\rho$ corresponding to
$\mathcal{S}$.

To compute
$\Xi_{\tau_{\hat{\Delta}},\theta_{\hat{\Delta}};\frac{\beta}{\rho},n}\leavevmode\nobreak\
(\%)$ (defined by (29)), we need to compute
$\mathcal{V}_{n^{2}}(\texttt{D}(\mathcal{S}))$ and
$\mathcal{V}_{n^{2}}\big{(}\texttt{S}_{\rho}^{n^{2}}\big{)}$. We compute both
volumes via integral computation techniques similarly utilized by [33]. First,
$\mathcal{V}_{n^{2}}\big{(}\texttt{S}_{\rho}^{n^{2}}\big{)}$ can simply be
computed as follows:

$\displaystyle\mathcal{V}_{n^{2}}\big{(}\texttt{S}_{\rho}^{n^{2}}\big{)}$
$\displaystyle=\frac{\pi^{\frac{n^{2}}{2}}\rho^{n^{2}}}{\Gamma\big{(}\frac{n^{2}}{2}+1\big{)}}.$
(30)

Second, according to the spherical symmetry,
$\mathcal{V}_{n^{2}}(\texttt{D}(\mathcal{S}))$ can be computed as follows:

$\displaystyle\mathcal{V}_{n^{2}}(\texttt{D}(\mathcal{S}))$
$\displaystyle=\int_{\pi\eta_{u}}^{\pi\eta_{l}}f_{u}(\varphi)-f_{l}(\varphi)\leavevmode\nobreak\
d\varphi,$ (31a) $\displaystyle f_{u}(\varphi)$
$\displaystyle:=\mathcal{V}_{n^{2}-1}\Big{(}\texttt{S}_{r_{u}(\varphi)\sin(\varphi)}^{n^{2}-1}\Big{)}\frac{d(r_{u}(\varphi)\cos(\varphi))}{d\varphi},$
(31b) $\displaystyle f_{l}(\varphi)$
$\displaystyle:=\mathcal{V}_{n^{2}-1}\Big{(}\texttt{S}_{r_{l}(\varphi)\sin(\varphi)}^{n^{2}-1}\Big{)}\frac{d(r_{l}(\varphi)\cos(\varphi))}{d\varphi}.$
(31c)

where $\varphi:=\pi\eta$, $r:=\rho\sin(\frac{\pi\tau}{2})$, and
$r_{u}(\varphi)$/$\eta_{u}$ and $r_{l}(\varphi)$/$\eta_{l}$ represent the
upper and lower curves/bounds corresponding to $\mathcal{S}$, respectively.
Note that the following identities:

$\displaystyle\mathcal{V}_{n^{2}-1}\Big{(}\texttt{S}_{r_{u}(\varphi)\sin(\varphi)}^{n^{2}-1}\Big{)}=\frac{\pi^{\frac{n^{2}-1}{2}}(r_{u}(\varphi)\sin(\varphi))^{n^{2}-1}}{\Gamma\big{(}\frac{n^{2}-1}{2}+1\big{)}},$
(32a)
$\displaystyle\mathcal{V}_{n^{2}-1}\Big{(}\texttt{S}_{r_{l}(\varphi)\sin(\varphi)}^{n^{2}-1}\Big{)}=\frac{\pi^{\frac{n^{2}-1}{2}}(r_{l}(\varphi)\sin(\varphi))^{n^{2}-1}}{\Gamma\big{(}\frac{n^{2}-1}{2}+1\big{)}},$
(32b)
$\displaystyle\frac{d(r_{u}(\varphi)\cos(\varphi))}{d\varphi}=-r_{u}(\varphi)\sin(\varphi)+\frac{dr_{u}(\varphi)}{d\varphi}\cos(\varphi),$
(32c)
$\displaystyle\frac{d(r_{l}(\varphi)\cos(\varphi))}{d\varphi}=-r_{l}(\varphi)\sin(\varphi)+\frac{dr_{l}(\varphi)}{d\varphi}\cos(\varphi),$
(32d)

hold. Then, utilizing (30), (31), and (32) enables us to compute
$\Xi_{\tau_{\hat{\Delta}},\theta_{\hat{\Delta}};\frac{\beta}{\rho},n}\leavevmode\nobreak\
(\%)$.

Fig. 4 depicts the dependency of
$\Xi_{\tau_{\hat{\Delta}},\theta_{\hat{\Delta}};\frac{\beta}{\rho},n}\leavevmode\nobreak\
(\%)$ on $\tau_{\hat{\Delta}}$ and $\theta_{\hat{\Delta}}$ for
$\frac{\beta}{\rho}=\frac{1}{2}$ and $n=4$. As observed, approaching the
origin, the value of the geometric metric gets improved (a maximum value of
$1.5259\times 10^{-3}\%$), meaning that a larger amount of perturbations can
be handled provided that they are less severe. Similarly, approaching the
instability boundary, the value of the geometric metric gets degraded, that
is, a smaller amount of perturbations can be handled provided that they are
more severe. Then, there exists a fundamental trade-off between the potential
severeness of perturbations and the successfully handled amount of
perturbations.

Figure 4: The dependency of
$\Xi_{\tau_{\hat{\Delta}},\theta_{\hat{\Delta}};\frac{\beta}{\rho},n}\leavevmode\nobreak\
(\%)$ (defined by (29)) on $\tau_{\hat{\Delta}}$ and $\theta_{\hat{\Delta}}$
for $\frac{\beta}{\rho}=\frac{1}{2}$ and $n=4$.

####  IV-B2 Non-fragility-based robust update

Inspired by Propositions 1, 2, 3, and 4 and employing a notion of non-
fragility (NF) utilized by [29, 30, 31, 32], we propose a robust update for
the case of dealing with an unknown norm-bounded perturbation $\Delta$ with a
known upper bound $\rho$ on its Frobenius norm, based on the following
criterion:

C1: Choose the point deepest inside the guaranteed stability region $\hat{S}$
(i.e., farthest from the boundary) as a robust update.
To choose the point deepest inside the guaranteed stability region $\hat{S}$,
we utilize three well-known geometric notions: (i) Chebyshev center, (ii)
centroid, and (iii) weighted centroid.

Chebyshev center: A robust update based on the Chebyshev center can be
computed as follows:

$\displaystyle
G^{\ast}_{\hat{\Delta}_{\mathrm{NF}}}=-B^{+}\hat{\Delta}_{\mathrm{NF}}(C^{T+})^{T},$
(33a)
$\displaystyle\hat{\Delta}_{\mathrm{NF}}=\rho\sin\Big{(}\frac{\pi\hat{\tau}_{\mathrm{NF}}}{2}\Big{)}\mathbf{vec}^{-1}\Bigg{(}U_{H}\begin{bmatrix}\hat{\phi}_{c}\cos(\frac{\pi\hat{\theta}_{\mathrm{NF}}}{2})\\\
\hat{\phi}_{s}\sin(\frac{\pi\hat{\theta}_{\mathrm{NF}}}{2})\end{bmatrix}\Bigg{)},$
(33b)
$\displaystyle\hat{\tau}_{\mathrm{NF}}=\frac{4-2\sqrt{2}}{\pi}\arcsin\Bigg{(}\sqrt{\frac{\beta}{\rho}}\Bigg{)},\hat{\theta}_{\mathrm{NF}}=\hat{\tau}_{\mathrm{NF}}.$
(33c)

Centroid: A robust update based on centroid can be computed as follows:

$\displaystyle
G^{\ast}_{\hat{\Delta}_{\mathrm{NF}}}=-B^{+}\hat{\Delta}_{\mathrm{NF}}(C^{T+})^{T},$
(34a)
$\displaystyle\hat{\Delta}_{\mathrm{NF}}=\rho\sin\Big{(}\frac{\pi\hat{\tau}_{\mathrm{NF}}}{2}\Big{)}\mathbf{vec}^{-1}\Bigg{(}U_{H}\begin{bmatrix}\hat{\phi}_{c}\cos(\frac{\pi\hat{\theta}_{\mathrm{NF}}}{2})\\\
\hat{\phi}_{s}\sin(\frac{\pi\hat{\theta}_{\mathrm{NF}}}{2})\end{bmatrix}\Bigg{)},$
(34b)
$\displaystyle\hat{\tau}_{\mathrm{NF}}=\frac{\int_{\hat{S}}\hat{\tau}d\hat{\theta}d\hat{\tau}}{\int_{\hat{S}}d\hat{\theta}d\hat{\tau}},\hat{\theta}_{\mathrm{NF}}=\frac{\int_{\hat{S}}\hat{\theta}d\hat{\theta}d\hat{\tau}}{\int_{\hat{S}}d\hat{\theta}d\hat{\tau}}.$
(34c)

wherein $\hat{\theta}_{\mathrm{NF}}=\hat{\tau}_{\mathrm{NF}}$ holds due to the
symmetry of $\hat{S}$ with respect to $\hat{\theta}=\hat{\tau}$.
Specifically, for the case of $\rho<\beta_{\mathbb{R}}(A+BFC)$, both of the
robust updates in (33) and (34) reduce to the following form:

$\displaystyle G^{\ast}_{\hat{\Delta}_{\mathrm{NF}}}$
$\displaystyle=-B^{+}\hat{\Delta}_{\mathrm{NF}}(C^{T+})^{T},$ (35a)
$\displaystyle\hat{\Delta}_{\mathrm{NF}}$
$\displaystyle=\frac{\rho}{2}\mathbf{vec}^{-1}\bigg{(}U_{H}\begin{bmatrix}\hat{\phi}_{c}\\\
\hat{\phi}_{s}\end{bmatrix}\bigg{)}.$ (35b)

Note that in (35)
$(\hat{\tau}_{\mathrm{NF}},\hat{\theta}_{\mathrm{NF}})=(\frac{1}{2},\frac{1}{2})$
holds as $\hat{S}=\hat{\mathbb{S}}$ holds.

Weighted centroid: A robust update based on a weighted centroid can be
computed as follows:

$\displaystyle
G^{\ast}_{\hat{\Delta}_{\mathrm{NF}}}=-B^{+}\hat{\Delta}_{\mathrm{NF}}(C^{T+})^{T},$
(36a)
$\displaystyle\hat{\Delta}_{\mathrm{NF}}=\rho\sin\Big{(}\frac{\pi\hat{\tau}_{\mathrm{NF}}}{2}\Big{)}\mathbf{vec}^{-1}\Bigg{(}U_{H}\begin{bmatrix}\hat{\phi}_{c}\cos(\frac{\pi\hat{\theta}_{\mathrm{NF}}}{2})\\\
\hat{\phi}_{s}\sin(\frac{\pi\hat{\theta}_{\mathrm{NF}}}{2})\end{bmatrix}\Bigg{)},$
(36b)
$\displaystyle\hat{\tau}_{\mathrm{NF}}=\frac{\int_{\hat{S}}\Xi(\hat{\tau},\hat{\theta})\hat{\tau}d\hat{\theta}d\hat{\tau}}{\int_{\hat{S}}\Xi(\hat{\tau},\hat{\theta})d\hat{\theta}d\hat{\tau}},\hat{\theta}_{\mathrm{NF}}=\frac{\int_{\hat{S}}\Xi(\hat{\tau},\hat{\theta})\hat{\theta}d\hat{\theta}d\hat{\tau}}{\int_{\hat{S}}\Xi(\hat{\tau},\hat{\theta})d\hat{\theta}d\hat{\tau}}.$
(36c)

The following corollary highlights that since
$G^{\ast}_{\hat{\Delta}_{\mathrm{NF}}}$ in (33), (34), and (36) all lie inside
the guaranteed stability region $\hat{S}$, the corresponding
$F+G^{\ast}_{\hat{\Delta}_{\mathrm{NF}}}$ is a robust updated stabilizing SOF
controller by construction.

######  Corollary 4.

Given an unknown norm-bounded perturbation $\Delta$ with an upper bound $\rho$
on its Frobenius norm and considering the guaranteed stability region
$\hat{S}$, then $F+G^{\ast}_{\hat{\Delta}_{\mathrm{NF}}}$ with
$G^{\ast}_{\hat{\Delta}_{\mathrm{NF}}}$ in (33), (34), and (36) is a robust
updated stabilizing SOF controller.

We highlight that for an arbitrary choice of $(\hat{\tau},\hat{\theta})$, one
can similarly compute the corresponding $G^{\ast}_{\hat{\Delta}}$ via

$\displaystyle G^{\ast}_{\hat{\Delta}}=-B^{+}\hat{\Delta}(C^{T+})^{T},$ (37a)
$\displaystyle\hat{\Delta}=\rho\sin\Big{(}\frac{\pi\hat{\tau}}{2}\Big{)}\mathbf{vec}^{-1}\Bigg{(}U_{H}\begin{bmatrix}\hat{\phi}_{c}\cos(\frac{\pi\hat{\theta}}{2})\\\
\hat{\phi}_{s}\sin(\frac{\pi\hat{\theta}}{2})\end{bmatrix}\Bigg{)}.$ (37b)

Given $(\rho,n)$, computing
$(\beta_{\mathbb{R}}(A+BFC),\hat{\tau}_{\mathrm{NF}},\hat{\theta}_{\mathrm{NF}})$,
and having access to sufficiently accurate estimates
$(\hat{\phi}_{c},\hat{\phi}_{s})$ of $(\phi_{c},\phi_{s})$, we can utilize
(33), (34), and (36) to propose robust updated stabilizing SOF controllers. We
emphasize that by construction, the set of robust updated stabilizing SOF
controllers proposed by Corollary 4 is a proper subset of the exact (ideal)
set of solutions as the special form of the solutions in Corollary 4 is built
upon engineered sufficient conditions.

##  V Numerical Simulations

This section is naturally divided into two main parts: (i) known norm-bounded
perturbation and (ii) unknown norm-bounded perturbation. To assess the
effectiveness of the theoretical results, we employ two benchmarks of the SOF
controller benchmarks collected by [34]. To design a nominal stabilizing SOF
controller $F$, we utilize MATLAB built-in function $\texttt{hinfstruct}(.)$
[35] that has been developed built upon [36] to synthesize structured
$\mathcal{H}_{\infty}$ controllers.

As mentioned earlier in the paper, computing the exact value of MDRP $\beta$
is theoretically impossible. However, we utilize the following optimization
problem:

$\displaystyle\underset{v\in\mathbb{R}^{n^{2}},\leavevmode\nobreak\
\beta\in\mathbb{R}_{++}}{\max}\leavevmode\nobreak\
\alpha\bigg{(}A+BFC+\beta\mathbf{vec}^{-1}\bigg{(}\frac{v}{\|v\|}\bigg{)}\bigg{)}.$
(38)

along with a specialized bisection method (fixing the value of $\beta$ and
solving for a $v\in\mathbb{R}^{n^{2}}$), to obtain a near-optimal value of
$\beta$. We initialize $\beta$ with $\beta_{\mathbb{R}}^{u}$ and at each step,
we check if the maximum value, namely $\alpha^{\ast}$, is non-negative or not.
To solve the optimization problem, one could utilize MATLAB’s built-in
function fminunc(.). We emphasize that the efficiency of the proposed updated
stabilizing SOF controller mainly depends on the computational efficiency of
MDRP $\beta$ as the computation complexity of (12) is
$\mathcal{O}(n^{2}\min\\{m,p\\})$. Furthermore, it is noteworthy that solving
the optimization problem (38) along with a specialized bisection method to
obtain a near-optimal value of $\beta$ can become challenging for higher-order
systems as the dimension of $v$ is $n^{2}$ and quadratically increasing with
$n$.

###  V-A Known norm-bounded perturbation

Let us consider a lateral axis model of an $L-1011$ aircraft in cruise flight
conditions ($AC3$) [34]. For such a model, we have $(n,m,p)=(5,2,4)$. We
design the following nominal stabilizing SOF controller $F$ via
$\texttt{hinfstruct}(.)$:

$\displaystyle F$ $\displaystyle=\begin{bmatrix}0&0&0&-0.5057\\\
0.7521&0&-3.0713&1.1408\end{bmatrix}.$

for which $\alpha(A+BFC+\Delta)=0.0483$ (i.e., a destabilizing $\Delta$),
$\beta=0.1931$, and $\beta_{\mathbb{R}}^{u}=0.3230$ hold.

Fig. 5 (Left) visualizes the stability regions for $AC3$ benchmark with
$\frac{\beta}{\rho}=\frac{1}{2}$: the guaranteed (conservative) stability
region based on Proposition 2 and the exact one based on
$\alpha(A+BFC+\Delta+BG_{\Delta}^{\ast}C)<0$ with $G_{\Delta}^{\ast}$ in (12).
As expected, the guaranteed (conservative) stability region is a subset of the
exact one. For instance, the update $G_{\Delta}^{\ast}$ for
$(\tau_{\Delta},\theta_{\Delta})=(0.45,0.45)$ is as follows:

$\displaystyle G_{\Delta}^{\ast}$
$\displaystyle=\begin{bmatrix}0.0745&-0.2034&0.0214&-0.0939\\\
0.0115&-0.0302&0.0018&-0.0169\end{bmatrix},$

for which $\alpha(A+BFC+\Delta+BG_{\Delta}^{\ast}C)=-0.0637$ holds and the
updated stabilizing SOF controller $F+G_{\Delta}^{\ast}$ is as follows:

$\displaystyle F+G_{\Delta}^{\ast}$
$\displaystyle=\begin{bmatrix}0.0745&-0.2034&0.0214&-0.5996\\\
0.7636&-0.0302&-3.0695&1.1239\end{bmatrix}.$

(a) a

(b) b

Figure 5: The stability regions for $AC3$ benchmark with (a)
$\rho=2\beta^{\mathrm{accurate}}$ and $\beta=\beta^{\mathrm{accurate}}$ and
(b) $\rho=2\beta^{\mathrm{accurate}}$ and $\beta=\beta^{\mathrm{inaccurate}}$:
the guaranteed (conservative) stability regions based on Proposition 2 (filled
with blue circles) and the exact ones based on
$\alpha(A+BFC+\Delta+BG_{\Delta}^{\ast})<0$ with $G_{\Delta}^{\ast}$ in (12)
(filled with red asterisks).

Remarkably, the accurate computing of $\beta$ plays a significant role in
accurately identifying the stability regions. As Fig. 5 (Right) depicts,
choosing $\rho$ equal to $2\times 0.1931$ (as chosen for Fig. 5 (Left)) and
$\beta$ equal to $0.3230$ (an inaccurate value), leads to the misleading
stability regions. First, the guaranteed (conservative) stability region has
erroneously been enlarged. Second, the guaranteed (conservative) stability
region has erroneously become the superset of the exact one.

###  V-B Unknown norm-bounded perturbation

Let us consider the autopilot control problem for an air-to-air missile
($AC4$) [34]. For such a model, we have $(n,m,p)=(4,1,2)$. The number of
states for such a control problem is $n=4$. For
$\frac{\beta}{\rho}=\frac{1}{2}$ and $n=4$, we get the following NF-based
designs:

$\displaystyle(\hat{\tau}_{\mathrm{NF}}^{\mathrm{Cheb.\leavevmode\nobreak\
center}},\hat{\theta}_{\mathrm{NF}}^{\mathrm{Cheb.\leavevmode\nobreak\
center}})=\bigg{(}\frac{2-\sqrt{2}}{2},\frac{2-\sqrt{2}}{2}\bigg{)},$
$\displaystyle(\hat{\tau}_{\mathrm{NF}}^{\mathrm{Centroid}},\hat{\theta}_{\mathrm{NF}}^{\mathrm{Centroid}})=(0.3787,0.3787),$
$\displaystyle(\hat{\tau}_{\mathrm{NF}}^{\mathrm{W-centroid}},\hat{\theta}_{\mathrm{NF}}^{\mathrm{W-centroid}})=(0.1603,0.2278),$

where attain
$\Xi_{\frac{\beta}{\rho},\tau_{\hat{\Delta}},\theta_{\hat{\Delta}}}^{\mathrm{Cheb.\leavevmode\nobreak\
center}}=5.0077\times 10^{-7}\%$,
$\Xi_{\frac{\beta}{\rho},\tau_{\hat{\Delta}},\theta_{\hat{\Delta}}}^{\mathrm{Centroid}}=2.0450\times
10^{-10}\%$, and
$\Xi_{\frac{\beta}{\rho},\tau_{\hat{\Delta}},\theta_{\hat{\Delta}}}^{\mathrm{W-centroid}}=7.0944\times
10^{-5}\%$, respectively.

Fig. 6 visualizes the guaranteed stability region $\mathcal{S}$ in
$2$-dimensional parametric space of $(\eta,\tau)$ for various NF-based robust
updates. As observed, the weighted centroid update attains the best average
performance as it considers both being far from the boundary and obtaining a
large guaranteed stability region (i.e., a large value of the geometric
metric). Also, unlike the Chebyshev center update and the centroid update, for
the case of weighted centroid update the identity
$\hat{\theta}_{\mathrm{NF}}=\hat{\tau}_{\mathrm{NF}}$ does not necessarily
hold as $\Xi(c_{1},c_{2})=\Xi(c_{2},c_{1})$ does not necessarily hold for
$c_{1}\neq c_{2}$. Fig. 7 illustrates the weighted centroid updates for
$\frac{\beta}{\rho}=\frac{1}{2}$ and various values of $n$. As Fig. 7 depicts,
the higher the dimension $n$, the closer to the origin, the weighted centroid
update we get. Tab. I reflects the corresponding values of the geometric
metric
$\Xi_{\tau_{\hat{\Delta}},\theta_{\hat{\Delta}};\frac{\beta}{\rho},n}\leavevmode\nobreak\
(\%)$ for the weighted centroid updates with $\frac{\beta}{\rho}=\frac{1}{2}$
and various values of $n$. As Tab. I shows, the higher the dimension $n$, the
smaller geometric metric
$\Xi_{\tau_{\hat{\Delta}},\theta_{\hat{\Delta}};\frac{\beta}{\rho},n}\leavevmode\nobreak\
(\%)$ we get.

Figure 6: The guaranteed stability region $\mathcal{S}$ in $2$-dimensional
parametric space of $(\eta,\tau)$ for various NF-based robust updates
(Chebyshev center in red, Centroid in green, and Weighted centroid in blue)
for $\frac{\beta}{\rho}=\frac{1}{2}$ and $n=4$. Colored circles on the
vertical axis represent the corresponding perturbations in the ideal case,
i.e., $\hat{\Delta}_{\mathrm{NF}}=\Delta$.

(a) a

(b) b

Figure 7: The weighted centroid updates for $\frac{\beta}{\rho}=\frac{1}{2}$ and various values of $n$ (a) inside $\hat{S}$ and (b) zoomed version. TABLE I: the corresponding values of the geometric metric $\Xi_{\tau_{\hat{\Delta}},\theta_{\hat{\Delta}};\frac{\beta}{\rho},n}\leavevmode\nobreak\ (\%)$ for the weighted centroid updates with $\frac{\beta}{\rho}=\frac{1}{2}$ and various values of $n$. $n$ | $(\hat{\tau}_{\mathrm{NF}}^{\mathrm{W-centroid}},\hat{\theta}_{\mathrm{NF}}^{\mathrm{W-centroid}})$ | $\Xi_{\tau_{\hat{\Delta}},\theta_{\hat{\Delta}};\frac{\beta}{\rho},n}\leavevmode\nobreak\ (\%)$
---|---|---
$2$ | $(0.2210,0.2865)$ | $1.5360\times 10^{0}\%$
$3$ | $(0.1837,0.2505)$ | $2.1317\times 10^{-2}\%$
$4$ | $(0.1603,0.2278)$ | $7.0944\times 10^{-5}\%$
$5$ | $(0.1444,0.2120)$ | $5.5841\times 10^{-8}\%$
$6$ | $(0.1330,0.2003)$ | $1.0332\times 10^{-11}\%$
$7$ | $(0.1244,0.1912)$ | $4.4808\times 10^{-16}\%$
$8$ | $(0.1179,0.1838)$ | $4.5271\times 10^{-21}\%$
$9$ | $(0.1127,0.1777)$ | $1.0651\times 10^{-26}\%$
$10$ | $(0.1085,0.1725)$ | $5.8343\times 10^{-33}\%$

Given an arbitrary point $(\eta^{\mathrm{ap}},\tau^{\mathrm{ap}})$ in
$2$-dimensional parametric space of $(\eta,\tau)$ and utilizing the itemized
characterization proposed by Proposition 4, we visualize all the
$(\hat{\tau},\hat{\theta})$’s belonging to $\hat{S}$ for which the guaranteed
stability region $\mathcal{S}$ contains
$(\eta^{\mathrm{ap}},\tau^{\mathrm{ap}})$. For instance, Fig. 8 depicts such a
visualization for $(\eta^{\mathrm{ap}},\tau^{\mathrm{ap}})=(0.1,0.5)$ and
$\frac{\beta}{\rho}=\frac{1}{2}$. Fig. 9 visualizes all the
$G_{\hat{\Delta}}^{\ast}$-stabilizable points
$(\eta^{\mathrm{ap}},\tau^{\mathrm{ap}})$ in $2$-dimensional parametric space
of $(\eta,\tau)$ for $\frac{\beta}{\rho}=\frac{1}{2}$. As expected, from Fig.
9, we realize that the perturbations with both high gain ($\propto\tau$) and
high phase difference ($\propto\eta$) are not
$G_{\hat{\Delta}}^{\ast}$–stabilizable.

Figure 8: All the $(\hat{\tau},\hat{\theta})$’s belonging to $\hat{S}$ for
which the guaranteed stability region $\mathcal{S}$ contains
$(\eta^{\mathrm{ap}},\tau^{\mathrm{ap}})=(0.1,0.5)$ and
$\frac{\beta}{\rho}=\frac{1}{2}$. Colored circles identify the corresponding
items: item iv (filled with red circles), item ii (filled with green circles),
and item i (filled with blue circles). Figure 9: All the
$G_{\hat{\Delta}}^{\ast}$-stabilizable points
$(\eta^{\mathrm{ap}},\tau^{\mathrm{ap}})$ in $2$-dimensional parametric space
of $(\eta,\tau)$ for $\frac{\beta}{\rho}=\frac{1}{2}$.

Given a $G_{\hat{\Delta}}^{\ast}$-stabilizable point
$(\eta^{\mathrm{ap}},\tau^{\mathrm{ap}})$ in $2$-dimensional parametric space
of $(\eta,\tau)$, we define the following geometric metric:

$\displaystyle\mathcal{M}_{\eta,\theta;\frac{\beta}{\rho}}\leavevmode\nobreak\
(\%)$ $\displaystyle:=100\times
G_{\hat{\Delta}}^{\ast}\mathrm{-stabilizing\leavevmode\nobreak\
Region\leavevmode\nobreak\ Area},$ (39)

to quantify the $G_{\hat{\Delta}}^{\ast}$-stabilizability. Fig. 10 visualizes
the $G_{\hat{\Delta}}^{\ast}$-stabilizability geometric metric
$\mathcal{M}_{\eta,\theta;\frac{\beta}{\rho}}\leavevmode\nobreak\ (\%)$ for
$\frac{\beta}{\rho}=\frac{1}{2}$. The larger
$\mathcal{M}_{\eta,\theta;\frac{\beta}{\rho}}\leavevmode\nobreak\ (\%)$, the
easier to stabilize a $G_{\hat{\Delta}}^{\ast}$-stabilizable point
$(\eta^{\mathrm{ap}},\tau^{\mathrm{ap}})$ in $2$-dimensional parametric space
of $(\eta,\tau)$ we have. As Fig. 10 depicts, the largest value of
$\mathcal{M}_{\eta,\theta;\frac{\beta}{\rho}}\leavevmode\nobreak\ (\%)$, i.e.,
$39.1386\%$, is attained by
$(\eta^{\mathrm{ap}},\tau^{\mathrm{ap}})=(0,\frac{1}{3})$. A possible
justification for such an observation can be the fact that
$\eta^{\mathrm{ap}}=0$ corresponds to a zero phase difference and
$\tau^{\mathrm{ap}}=\frac{1}{3}$ corresponds to
$r=\frac{\rho}{2}=\frac{0+\rho}{2}$. Note that
$\mathcal{M}_{\eta,\theta;\frac{\beta}{\rho}}\leavevmode\nobreak\ (\%)=0\%$ in
Fig. 10 represents the points $(\eta^{\mathrm{ap}},\tau^{\mathrm{ap}})$ in
$2$-dimensional parametric space of $(\eta,\tau)$ that are not
$G_{\hat{\Delta}}^{\ast}$-stabilizable. Fig. 11 depicts the corresponding
$G_{\hat{\Delta}}^{\ast}$-stabilizing region for
$(\eta^{\mathrm{ap}},\tau^{\mathrm{ap}})=(0,\frac{1}{3})$.

Figure 10: The $G_{\hat{\Delta}}^{\ast}$-stabilizability geometric metric
$\mathcal{M}_{\eta,\theta;\frac{\beta}{\rho}}\leavevmode\nobreak\ (\%)$ for
$\frac{\beta}{\rho}=\frac{1}{2}$. Figure 11: All the
$(\hat{\tau},\hat{\theta})$’s belonging to $\hat{S}$ for which the guaranteed
stability region $\mathcal{S}$ contains
$(\eta^{\mathrm{ap}},\tau^{\mathrm{ap}})=(0,\frac{1}{3})$ and
$\frac{\beta}{\rho}=\frac{1}{2}$. Colored circles identify the corresponding
items: item iv (filled with red circles) and item ii (filled with green
circles).

To empirically verify the relative performance of the weighted centroid update
compared to: the centroid update (Case $1$), the Chebyshev center update (Case
$2$), and the update based on a point close to the origin
$(\hat{\tau},\hat{\theta})=(0.01,0)$ (Case $3$), we generate uniformly random
samples of an unknown norm-bounded perturbation $\Delta$ with
$0<\|\Delta\|_{F}\leq\rho$ [37] and check for how many percentages of the
samples if $\|\Delta-\hat{\Delta}\|_{F}<\upsilon$ holds inducing a percentage-
based performance associated with each update. Then, to compute the relative
performance values, the percentage-based performance associated with the
weighted centroid update is compared to the percentage-based performance
associated with the updates of Cases $1$–$3$. Such comparisons can simply be
classified as three categories for each case: (i) Better, (ii) Equal, and (i)
Worse. Precisely, we generate the uniformly random samples as follows [37]:

$\displaystyle\Delta=\mathbf{vec}^{-1}\bigg{(}r\frac{\vartheta}{\|\vartheta\|}\bigg{)},\leavevmode\nobreak\
r\in\rho\times\mathcal{U}(0,1)^{\frac{1}{n^{2}}},\vartheta\in\mathcal{N}(0,I_{n^{2}}).$

According to (23), considering

$\displaystyle\psi=\begin{bmatrix}\phi_{c}\cos(\frac{\pi\theta}{2})\\\
\phi_{s}\sin(\frac{\pi\theta}{2})\end{bmatrix},\hat{\psi}=\begin{bmatrix}\hat{\phi}_{c}\cos(\frac{\pi\hat{\theta}}{2})\\\
\hat{\phi}_{s}\sin(\frac{\pi\hat{\theta}}{2}),\end{bmatrix}$

and defining
$(\gamma_{c},\gamma_{s}):=(\phi_{c}^{T}\hat{\phi}_{c},\phi_{s}^{T}\hat{\phi}_{s})$,
$c_{\theta}:=\cos(\frac{\pi\theta}{2})$,
$s_{\theta}:=\sin(\frac{\pi\theta}{2})$, and
$c_{\hat{\theta}}:=\cos(\frac{\pi\hat{\theta}}{2})$, we get

$\displaystyle\eta=\frac{1}{\pi}\arccos(\psi^{T}\hat{\psi})=\frac{1}{\pi}\arccos(\gamma_{c}c_{\theta}c_{\hat{\theta}}+\gamma_{s}s_{\theta}s_{\hat{\theta}}).$
(40)

It is noteworthy that $\gamma_{c}\in[-1,1]$ and $\gamma_{s}\in[-1,1]$ hold.
For the ideal case of estimated $\hat{\Delta}$, i.e., $\hat{\Delta}=\Delta$,
on the one hand, we have $\hat{\theta}=\theta$ and subsequently
$(c_{\hat{\theta}},s_{\hat{\theta}})=(c_{\theta},s_{\theta})$. Also, we have
$(\hat{\phi}_{c},\hat{\phi}_{s})=(\phi_{c},\phi_{s})$ and subsequently
$(\gamma_{c},\gamma_{s})=(1,1)$. Consequently, according to (40), we observe
that $\eta=0$ holds. On the other hand, for the ideal case of estimated
$\hat{\Delta}$, $\hat{\tau}=\tau$ or equivalently $\hat{r}=r$ holds.

Since $U_{H}\psi=\frac{\vartheta}{\|\vartheta\|}$ and
$U_{H}^{T}U_{H}=I_{n^{2}}$ hold, we have
$\psi=U_{H}^{T}\frac{\vartheta}{\|\vartheta\|}$. Then, defining
$\mu:=\begin{bmatrix}I_{mp}&0\end{bmatrix}\psi$ and
$\nu:=\begin{bmatrix}0&I_{n^{2}-mp}\end{bmatrix}\psi$, we get

$\displaystyle\phi_{c}$
$\displaystyle=\frac{\mu}{\|\mu\|},\leavevmode\nobreak\
\mu=\begin{bmatrix}I_{mp}&0\end{bmatrix}U_{H}^{T}\frac{\vartheta}{\|\vartheta\|},$
(41a) $\displaystyle\phi_{s}$
$\displaystyle=\frac{\nu}{\|\nu\|},\leavevmode\nobreak\
\nu=\begin{bmatrix}0&I_{n^{2}-mp}\end{bmatrix}U_{H}^{T}\frac{\vartheta}{\|\vartheta\|},$
(41b) $\displaystyle\theta$ $\displaystyle=\frac{2}{\pi}\arcsin(\|\nu\|).$
(41c)

Note that given $\vartheta$, we can compute $\phi_{c}$, $\phi_{s}$, and
$\theta$ via (41). To compute $G^{\ast}_{\hat{\Delta}}$ in (37), we need to
choose $(\hat{\phi}_{c},\hat{\phi}_{s})$ given $(\gamma_{c},\gamma_{s})$. The
more accurate $(\gamma_{c},\gamma_{s})$ (i.e., the larger values of
$\gamma_{c}$ and/or $\gamma_{s}$) and/or $(\hat{\tau},\hat{\theta})$ (i.e.,
the smaller values of $\hat{\tau}-\tau$ and/or $\hat{\theta}-\theta$), the
more accurate estimate $\hat{\Delta}$ we have. Given $(\gamma_{c},\gamma_{s})$
and computing $(\phi_{c},\phi_{s})$, we solve the following equations:

$\displaystyle\phi_{c}^{T}\hat{\phi}_{c}-\gamma_{c}=0,\phi_{s}^{T}\hat{\phi}_{s}-\gamma_{s}=0,$
(42)

for $(\hat{\phi}_{c},\hat{\phi}_{s})$ via the MATLAB built-in function
$\texttt{fsolve}(.)$. We generate $N_{\Delta}=10^{6}$ uniformly random samples
inside the $n^{2}$-dimensional hypersphere of radius $\rho$ centered at origin
by the Cartesian product of $N_{r}=10^{4}$ samples of $r$ and
$N_{\vartheta}=10^{2}$ samples of $\vartheta$ [37].

Fig. 12 depicts the relative performance of the weighted centroid update
compared to: the centroid update (Case $1$), the Chebyshev center update (Case
$2$), and the update based on a point close to the origin
$(\hat{\tau},\hat{\theta})=(0.01,0)$ (Case $3$) for various choices of
$(\gamma_{c},\gamma_{s})$. As Fig. 12 shows, for the case of a more accurate
estimate $\hat{\Delta}$ (i.e., the larger values of $\gamma_{c}$ and/or
$\gamma_{s}$), the weighted centroid update outperforms all the other updates.
Interestingly, as the estimation quality degrades (i.e., the values of
$\gamma_{c}$ and/or $\gamma_{s}$ decrease as visualized by the trend from Fig.
12 (Top-Left) to Fig. 12 (Bottom-Right)), a point close to the origin attains
the best relative performance. Such an observation can be interpreted in this
way that when we have no accurate information about the perturbation, the best
strategy is choosing a point close to the origin (e.g.,
$(\hat{\tau},\hat{\theta})=(0.01,0)$) as it attains the largest value of the
geometric metric
$\Xi_{\tau_{\hat{\Delta}},\theta_{\hat{\Delta}};\frac{\beta}{\rho},n}\leavevmode\nobreak\
(\%)$. Also, as Fig. 12 (Top-Left) depicts, we observe that the Chebyshev
center update outperforms the centroid update.

(a) a

(b) b

(c) c

(d) d

(e) e

(f) f

Figure 12: The relative performance of the weighted centroid update compared
to the centroid update (Case $1$), the Chebyshev center update (Case $2$), and
the update based on a point close to the origin
$(\hat{\tau},\hat{\theta})=(0.01,0)$ (Case $3$). The Left (blue), Middle
(red), and Right (yellow) bars in each case respectively correspond to Better,
Equal, and Worse relative performances. Note that in each case the scenarios
in which $\|\Delta-\hat{\Delta}\|_{F}<\upsilon$ holds neither by the weighted
centroid update nor by the counterpart, are eliminated. (a)
$(\gamma_{c},\gamma_{s})=(1,1)$, (b) $(\gamma_{c},\gamma_{s})=(0.9,0.9)$, (c)
$(\gamma_{c},\gamma_{s})=(0.8,0.8)$, (d) $(\gamma_{c},\gamma_{s})=(0.7,0.7)$,
(e) $(\gamma_{c},\gamma_{s})=(0.6,0.6)$, and (f)
$(\gamma_{c},\gamma_{s})=(0.5,0.5)$.

Fig. 13 depicts the corresponding plots for the case of checking
$\|BG^{\ast}_{\hat{\Delta}}C+\Delta\|_{F}<\beta$ (the exact one) instead of
$\|\Delta-\hat{\Delta}\|_{F}<\upsilon$ (the guaranteed (conservative)
stability region). Similar observations/trends to the observations/trends
depicted in Fig. 12 are also observed in Fig. 13. One difference is that
fortunately, the relative performance of the NF-based robust updates in the
exact scenario can be better than the guaranteed (conservative) scenario.

(a) a

(b) b

(c) c

(d) d

(e) e

(f) f

Figure 13: The relative performance of the weighted centroid update compared
to the centroid update (Case $1$), the Chebyshev center update (Case $2$), and
the update based on a point close to the origin
$(\hat{\tau},\hat{\theta})=(0.01,0)$ (Case $3$). The Left (blue), Middle
(red), and Right (yellow) bars in each case respectively correspond to Better,
Equal, and Worse relative performances. Note that in each case the scenarios
in which $\|BG^{\ast}_{\hat{\Delta}}C+\Delta\|_{F}<\beta$ holds neither by the
weighted centroid update nor by the counterpart, are eliminated. (a)
$(\gamma_{c},\gamma_{s})=(1,1)$, (b) $(\gamma_{c},\gamma_{s})=(0.9,0.9)$, (c)
$(\gamma_{c},\gamma_{s})=(0.8,0.8)$, (d) $(\gamma_{c},\gamma_{s})=(0.7,0.7)$,
(e) $(\gamma_{c},\gamma_{s})=(0.6,0.6)$, and (f)
$(\gamma_{c},\gamma_{s})=(0.5,0.5)$.

Fig. 14 visualizes the relative performance (both guaranteed (conservative)
and exact scenarios) of the weighted centroid update compared to: the centroid
update (Case $1$), the Chebyshev center update (Case $2$), and the update
based on a randomly generated point
$(\hat{\tau},\hat{\theta})=(0.4081,0.3969)$ (Case $3$) for a randomly
generated choice of $(\gamma_{c},\gamma_{s})=(0.9212,0.8315)$. As Fig. 14
(Left) shows, the weighted centroid update is the only successful update among
all the updates. Fig. 14 (Right) similarly depicts the outperformance of the
weighted centroid update compared to the other updates. Also, it depicts that
in the exact scenario, the other updates have attained some positive results.
The descending order of the performance according to Fig. 14 (Right) is the
W-centroid update, the Cheb. center update, the centroid update, and the
random update. Interestingly, we observe that the corresponding values of the
geometric metric
$\Xi_{\tau_{\hat{\Delta}},\theta_{\hat{\Delta}};\frac{\beta}{\rho},n}\leavevmode\nobreak\
(\%)$ have the same order $(7.0944\times 10^{-5}\%,5.0077\times
10^{-7}\%,2.0450\times 10^{-10}\%,7.1872\times 10^{-12}\%)$.

(a) a

(b) b

Figure 14: The relative performance of the weighted centroid update compared
to the centroid update (Case $1$), the Chebyshev center update (Case $2$), and
the update based on a randomly generated point
$(\hat{\tau},\hat{\theta})=(0.4081,0.3969)$ (Case $3$) for a randomly
generated choice of $(\gamma_{c},\gamma_{s})=(0.9212,0.8315)$. The Left
(blue), Middle (red), and Right (yellow) bars in each case respectively
correspond to Better, Equal, and Worse relative performances. (a) guaranteed
(conservative) scenario and (b) exact scenario.

##  VI Concluding Remarks

In this paper, we propose a simple yet efficient update of a nominal
stabilizing SOF controller. According to the derived theoretical and empirical
results throughout the paper, we present the following answer to the question
stated in Section II (Q1):

A1: A least-squares problem built upon the notion of MDRP enables us to
propose an efficient updated stabilizing SOF controller. For both known and
unknown perturbations with a known upper bound on their norm, we derive
sufficient stability conditions followed by the characterized guaranteed
stability regions. Moreover, we define geometric metrics to quantify the
stability robustness of the proposed updated stabilizing SOF controllers.
Specifically, for unknown perturbations with a known upper bound on their
norm, we interestingly observe that the NF-based robust updates attain better
performance compared to the random update. Moreover, in the case of a
sufficiently accurate estimation of the unknown perturbation, the descending
order of the NF-based robust updates in terms of performance is the weighted
centroid update, the Cheb. center update, and the centroid design.

Limitations: Like any engineering solution, the proposed updated stabilizing
SOF controller has some limitations. The main limitations are three-fold: (i)
we propose a semi-dynamic solution to a dynamic problem. The static nature
comes from the utilized least-squares problem and the dynamic nature comes
from the information stored in the nominal stabilizing SOF controller $F$ for
the state-space triplet $(A,B,C)$ (i.e., $\beta_{\mathbb{R}}(A+BFC)$), (ii)
computing the exact value of MDRP $\beta_{\mathbb{R}}(A+BFC)$ is theoretically
impossible and the practical heuristics to estimate
$\beta_{\mathbb{R}}(A+BFC)$ may provide the less accurate values. The less
accurate $\beta_{\mathbb{R}}(A+BFC)$, the less accurate guaranteed stability
we get for the proposed update. Also, the more time-consuming practical
heuristics we utilize to estimate $\beta_{\mathbb{R}}(A+BFC)$, the less
efficient update we get, and (iii) Unlike the typical update, the proposed
update can be destabilizing for a subset of perturbations as illustrated by
the region outside the guaranteed stability region $S_{\kappa}$ for
$\kappa<1$, i.e., $\beta_{\mathbb{R}}(A+BFC)<\rho$. However, note that the
positive point about the proposed update is that, unlike the typical update,
it always provides a non-empty guaranteed stability region (the typical
approach can fail to propose an updated stabilizing SOF controller as it is a
sophisticated problem in general).

Future directions: As a pertinent future direction, a comprehensive comparison
can be conducted between the proposed robust control approach in this paper
and other specialized alternative robust control approaches including sliding
mode control (SMC) and $\mathcal{H}_{\infty}$ control. Also, for the scenario
of unknown perturbations with a known upper bound on their norm, exploring the
case of time-varying perturbations can be considered as another potential
future direction.

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##  Appendix A Proof of Lemma 1

Considering $\Sigma_{H}=\begin{bmatrix}\Gamma_{H}^{T}&0\end{bmatrix}^{T}$ and
noting that $U_{H}^{T}U_{H}=I_{n^{2}}$ and $V_{H}^{T}V_{H}=I_{mp}$ hold, we
have

$\displaystyle P:=I_{n^{2}}-HH^{+}=I_{n^{2}}-H(H^{T}H)^{-1}H^{T}=$
$\displaystyle
I_{n^{2}}-U_{H}\Sigma_{H}V_{H}^{T}(V_{H}\Sigma_{H}^{T}U_{H}^{T}U_{H}\Sigma_{H}V_{H}^{T})^{-1}V_{H}\Sigma_{H}^{T}U_{H}^{T}=$
$\displaystyle
I_{n^{2}}-U_{H}\Sigma_{H}V_{H}^{T}(V_{H}(\Gamma_{H}^{2})^{-1}V_{H}^{T})V_{H}\Sigma_{H}^{T}U_{H}^{T}=$
$\displaystyle
I_{n^{2}}-U_{H}\begin{bmatrix}\Gamma_{H}^{T}&0\end{bmatrix}^{T}(\Gamma_{H}^{2})^{-1}\begin{bmatrix}\Gamma_{H}^{T}&0\end{bmatrix}U_{H}^{T}=$
$\displaystyle U_{H}I_{n^{2}}U_{H}^{T}-U_{H}\begin{bmatrix}I_{mp}&0\\\
0&0\end{bmatrix}U_{H}^{T}=U_{H}\begin{bmatrix}0&0\\\
0&I_{n^{2}-mp}\end{bmatrix}U_{H}^{T}.$

Moreover, according to $H:=C^{T}\otimes B$ and the properties of Kronecker
product, we get

$\displaystyle H:=$ $\displaystyle\leavevmode\nobreak\ C^{T}\otimes
B=(V_{C}\otimes U_{B})(\Sigma_{C}^{T}\otimes\Sigma_{B})(U_{C}\otimes
V_{B})^{T}=$ $\displaystyle\leavevmode\nobreak\ (V_{C}\otimes
U_{B})(U_{\Omega}\Sigma_{\Omega}V_{\Omega}^{T})(U_{C}\otimes V_{B})^{T}=$
$\displaystyle\leavevmode\nobreak\ ((V_{C}\otimes
U_{B})U_{\Omega})\Sigma_{\Omega}((U_{C}\otimes V_{B})V_{\Omega})^{T}.$

Then, we have

$\displaystyle(U_{H},\Sigma_{H},V_{H})$ $\displaystyle=((V_{C}\otimes
U_{B})U_{\Omega},\Sigma_{\Omega},(U_{C}\otimes V_{B})V_{\Omega}),$

which completes the proof.

##  Appendix B Proof of Proposition 1

Substituting (15) of Lemma 1 in (14), we get

$\displaystyle J^{\ast}(\Delta)=$ $\displaystyle\leavevmode\nobreak\
\mathbf{vec}(\Delta)^{T}U_{H}\begin{bmatrix}0&0\\\
0&I_{n^{2}-mp}\end{bmatrix}U_{H}^{T}\mathbf{vec}(\Delta)=$
$\displaystyle\leavevmode\nobreak\ \delta^{T}U_{H}\begin{bmatrix}0&0\\\
0&I_{n^{2}-mp}\end{bmatrix}U_{H}^{T}\delta.$

Then, defining $\chi:=U_{H}^{T}\delta$ and noting that
$U_{H}U_{H}^{T}=I_{n^{2}}$ holds (because $U_{H}$ is a unitary matrix), we get
$\delta=U_{H}\chi$. Since $\delta^{T}\delta=\chi^{T}U_{H}^{T}U_{H}\chi$,
$U_{H}^{T}U_{H}=I_{n^{2}}$, and $\delta^{T}\delta=\|\Delta\|_{F}^{2}=r^{2}$
hold, we get $\chi^{T}\chi=r^{2}$ that inspires us to define
$\psi:=\frac{\chi}{\|\chi\|}=\frac{\chi}{r}$. Note that
$\psi\in\mathbb{R}^{n^{2}}$ and $\|\psi\|=1$ hold. Then, we have $\chi=r\psi$
and subsequently we get $\delta=U_{H}\chi=rU_{H}\psi$. Defining
$\mu\in\mathbb{R}^{mp}$ and $\nu\in\mathbb{R}^{n^{2}-mp}$ as follows:

$\displaystyle\mu$
$\displaystyle:=\begin{bmatrix}I_{mp}&0\end{bmatrix}\psi,\nu:=\begin{bmatrix}0&I_{n^{2}-mp}\end{bmatrix}\psi,$

we get $\psi=\begin{bmatrix}\mu^{T}&\nu^{T}\end{bmatrix}^{T}$. Since
$\|\psi\|^{2}=\|\mu\|^{2}+\|\nu\|^{2}=1$ holds, we can consider
$\|\mu\|=\cos(\frac{\pi\theta}{2})$ and $\|\nu\|=\sin(\frac{\pi\theta}{2})$
for a $\theta\in[0,1]$. Then, we have

$\displaystyle\mu$
$\displaystyle=\frac{\mu}{\|\mu\|}\cos\Big{(}\frac{\pi\theta}{2}\Big{)},\nu=\frac{\nu}{\|\nu\|}\sin\Big{(}\frac{\pi\theta}{2}\Big{)}.$

Defining $\phi_{c}:=\frac{\mu}{\|\mu\|}$ and $\phi_{s}:=\frac{\nu}{\|\nu\|}$,
we get $\mu=\phi_{c}\cos(\frac{\pi\theta}{2})$ and
$\nu=\phi_{s}\sin(\frac{\pi\theta}{2})$ (Note that $\|\phi_{c}\|=1$ and
$\|\phi_{s}\|=1$ hold). Then, considering the $r=\rho\sin(\frac{\pi\tau}{2})$
with $\tau\in]0,1]$, we have

$\displaystyle\Delta=$ $\displaystyle\leavevmode\nobreak\
\mathbf{vec}^{-1}(\delta)=\mathbf{vec}^{-1}(rU_{H}\psi)=$
$\displaystyle\leavevmode\nobreak\ r\mathbf{vec}^{-1}\bigg{(}(V_{C}\otimes
U_{B})U_{\Omega}\begin{bmatrix}\phi_{c}\cos(\frac{\pi\theta}{2})\\\
\phi_{s}\sin(\frac{\pi\theta}{2})\end{bmatrix}\bigg{)}=$
$\displaystyle\leavevmode\nobreak\
\rho\sin\Big{(}\frac{\pi\tau}{2}\Big{)}U_{B}\mathbf{vec}^{-1}\bigg{(}U_{\Omega}\begin{bmatrix}\phi_{c}\cos(\frac{\pi\theta}{2})\\\
\phi_{s}\sin(\frac{\pi\theta}{2})\end{bmatrix}\bigg{)}V_{C}^{T}.$

which completes the proof of (16).

Also, for $J^{\ast}(\Delta)$ in (14), we have

$\displaystyle
J^{\ast}(\Delta)=\mathbf{vec}(\Delta)^{T}P\mathbf{vec}(\Delta)=\delta^{T}P\delta=(rU_{H}\psi)^{T}P(rU_{H}\psi)$
$\displaystyle
r^{2}\psi^{T}U_{H}^{T}PU_{H}\psi=r^{2}\begin{bmatrix}\mu^{T}&\nu^{T}\end{bmatrix}\begin{bmatrix}0&0\\\
0&I_{n^{2}-mp}\end{bmatrix}\begin{bmatrix}\mu\\\ \nu\end{bmatrix}=$
$\displaystyle
r^{2}\nu^{T}\nu=r^{2}\Big{(}\phi_{s}\sin\Big{(}\frac{\pi\theta}{2}\Big{)}\Big{)}^{T}\phi_{s}\sin\Big{(}\frac{\pi\theta}{2}\Big{)}=$
$\displaystyle\Big{(}r\sin\Big{(}\frac{\pi\theta}{2}\Big{)}\Big{)}^{2}\|\phi_{s}\|^{2}=\Big{(}\rho\sin\Big{(}\frac{\pi\tau}{2}\Big{)}\sin\Big{(}\frac{\pi\theta}{2}\Big{)}\Big{)}^{2},$

which completes the proof of (17).

##  Appendix C Proof of Proposition 2

We use (4) as a sufficient condition on the stability of the updated perturbed
state-space (3). By substituting $G^{\ast}_{\Delta}$ in (4), we get

$\displaystyle\sin\Big{(}\frac{\pi\tau}{2}\Big{)}\sin\Big{(}\frac{\pi\theta}{2}\Big{)}<\frac{\beta_{\mathbb{R}}(A+BFC)}{\rho}.$
(43)

If $\rho<\beta_{\mathbb{R}}(A+BFC)$ holds, then $F+G^{\ast}_{\Delta}$ with
$G^{\ast}_{\Delta}$ in (12) is an updated stabilizing SOF controller because
the left-hand-side of (43) is at most $1$ and the right-hand-side of (43) is
greater than $1$. Then, (43) holds.

If $\rho\geq\beta_{\mathbb{R}}(A+BFC)$ holds, since
$\sin(\frac{\pi\theta}{2})$ attains its maximum value at $\theta=1$, (43)
reduces to

$\displaystyle\sin\Big{(}\frac{\pi\tau}{2}\Big{)}<\frac{\beta_{\mathbb{R}}(A+BFC)}{\rho},$

or equivalently

$\displaystyle\tau<\frac{2}{\pi}\arcsin\Big{(}\frac{\beta_{\mathbb{R}}(A+BFC)}{\rho}\Big{)},$

from which, we define $\kappa$ in (18). Similarly, we may extract the
definition of $\zeta_{\tau,\kappa}$ in (18). Thus, if
$(\tau_{\Delta},\theta_{\Delta})\in S_{\kappa}$ holds, then
$F+G^{\ast}_{\Delta}$ with $G^{\ast}_{\Delta}$ in (12) is an updated
stabilizing SOF controller.

The expression in (19) expresses the area of $S_{\kappa}$ divided by the area
of unit square $]0,1]\times[0,1]$ in $2$-dimensional parametric space of
$(\tau,\theta)$. Note that $\int_{0}^{\kappa}1d\tau=\kappa$ has simplified the
right-hand-side of (19). To show that $\xi_{\kappa}$ is an increasing function
of $\kappa$, we compute the derivative of $\xi_{\kappa}$ with respect to
$\kappa$ as follows (utilizing the Leibniz integral rule [38]):

$\displaystyle\frac{d\xi_{\kappa}}{d\kappa}$
$\displaystyle=\cos\Big{(}\frac{\pi\kappa}{2}\Big{)}\int_{\kappa}^{1}\frac{1}{\sqrt{\sin(\frac{\pi\tau}{2})^{2}-\sin(\frac{\pi\kappa}{2})^{2}}}d\tau.$
(44)

According to (44), $\frac{d\xi_{\kappa}}{d\kappa}\geq 0$ holds and noting that

$\displaystyle\frac{d\kappa}{d\rho}$
$\displaystyle=-\frac{2\beta}{\pi\rho\sqrt{\rho^{2}-\beta^{2}}}<0,\frac{d\kappa}{d\beta}=\frac{2}{\pi\sqrt{\rho^{2}-\beta^{2}}}>0,$

hold, the proof is complete.

##  Appendix D Proof of Proposition 3

On the one hand, using (4) as a sufficient condition on the stability of the
updated perturbed state-space (3) and substituting $G^{\ast}_{\hat{\Delta}}$
in (4), we get

$\displaystyle\|BG^{\ast}_{\hat{\Delta}}C+\Delta\|_{F}<\beta_{\mathbb{R}}(A+BFC).$
(45)

On the other hand, by applying the triangle inequality, we get

$\displaystyle\|BG^{\ast}_{\hat{\Delta}}C+\Delta\|_{F}\leq\|BG^{\ast}_{\hat{\Delta}}C+\hat{\Delta}\|_{F}+\|\Delta-\hat{\Delta}\|_{F}.$
(46)

Considering (45) and (46), we derive the following sufficient condition:

$\displaystyle\|\Delta-\hat{\Delta}\|_{F}<\beta_{\mathbb{R}}(A+BFC)-\|BG^{\ast}_{\hat{\Delta}}C+\hat{\Delta}\|_{F}.$
(47)

Substituting
$\|BG^{\ast}_{\hat{\Delta}}C+\hat{\Delta}\|_{F}=\rho\sin(\frac{\pi\tau_{\hat{\Delta}}}{2})\sin(\frac{\pi\theta_{\hat{\Delta}}}{2})$
in (47), we get (22). Since $\|\Delta-\hat{\Delta}\|_{F}\geq 0$ holds, the
satisfaction of (22) implies that $\upsilon>0$ must hold. In the case of
$\rho<\beta_{\mathbb{R}}(A+BFC)$, it automatically holds because
$(\tau_{\hat{\Delta}},\theta_{\hat{\Delta}})\in\hat{\mathbb{S}}$ holds and in
the case of $\rho\geq\beta_{\mathbb{R}}(A+BFC)$, it holds if and only if
$(\tau_{\hat{\Delta}},\theta_{\hat{\Delta}})\in\hat{S}_{\kappa}$ holds which
completes the proof.

##  Appendix E Proof of Lemma 2

Noting that $U_{H}^{T}U_{H}=I_{n^{2}}$ and $\|\psi\|=1$, $\|\hat{\psi}\|=1$,
and $\psi^{T}\hat{\psi}=\|\psi\|\|\hat{\psi}\|\cos(\pi\eta)$ hold, we have

$\displaystyle\|\Delta-\hat{\Delta}\|_{F}^{2}=\|\delta-\hat{\delta}\|^{2}=\|\rho
s_{\tau}U_{H}\psi-\rho s_{\hat{\tau}}U_{H}\hat{\psi}\|^{2}=$
$\displaystyle\rho^{2}(s_{\tau}U_{H}\psi-
s_{\hat{\tau}}U_{H}\hat{\psi})^{T}(s_{\tau}U_{H}\psi-
s_{\hat{\tau}}U_{H}\hat{\psi})=$
$\displaystyle\rho^{2}(s_{\tau}^{2}+s_{\hat{\tau}}^{2}-2s_{\tau}s_{\hat{\tau}}\psi^{T}\hat{\psi})=\rho^{2}(s_{\tau}^{2}+s_{\hat{\tau}}^{2}-2s_{\tau}s_{\hat{\tau}}c_{2\eta}).$

which ends the proof taking the square root of both sides.

##  Appendix F Proof of Proposition 4

Utilizing the results of Proposition 3 and Lemma 2, and dividing the both
sides of (22) by $\rho s_{\hat{\tau}}$, we get

$\displaystyle\sqrt{\frac{1}{s_{\hat{\tau}}^{2}}s_{\tau}^{2}+1-\frac{2c_{2\eta}}{s_{\hat{\tau}}}s_{\tau}}<\iota.$
(48)

Then, $0<\iota$ must hold. Assuming that $0<\iota$ holds, squaring the both
sides of (48) and multiplying the both sides by $s_{\hat{\tau}}^{2}$, we
obtain the following quadratic inequality (in terms of $s_{\tau}$):

$\displaystyle
s_{\tau}^{2}-2s_{\hat{\tau}}c_{2\eta}s_{\tau}+s_{\hat{\tau}}^{2}(1-\iota^{2})<0.$
(49)

Observe that (49) holds if and only if

$\displaystyle b_{l}(\eta)<s_{\tau}<b_{u}(\eta),$ (50)

holds for which $s_{2\eta}<\iota$ must hold. Since $0<s_{\tau}$ holds, then
$0<b_{u}(\eta)$ must hold based on (50). Moreover, it can be verified that the
following equivalence holds:

$\displaystyle 0<b_{u}(\eta)\iff\iota>1\lor\bigg{(}0<\iota\leq 1\land
0\leq\eta<\frac{1}{2}\bigg{)}.$ (51)

In the case of $0<\iota\leq 1$, $s_{2\eta}<\iota$ is equivalent to
$0\leq\eta<\bar{\eta}$ as $\arcsin(\pi\eta)$ is an increasing function of
$\eta$ for $0\leq\eta<\frac{1}{2}$ (according to (51)) and in the case of
$\iota>1$, observe that $s_{2\eta}<\iota$ automatically holds as
$s_{2\eta}\leq 1$ is satisfied. Then, these observations can compactly be
expressed as

$\displaystyle s_{2\eta}<\iota\iff\big{(}\iota>1\lor(0<\iota\leq 1\land
0\leq\eta<\bar{\eta})\big{)}.$ (52)

Under the $b_{u}(\eta)>1$, observe that $s_{\tau}<b_{u}(\eta)$ in (50)
automatically holds as $s_{\tau}\leq 1$ is satisfied. Moreover, it can be
verified that the following equivalence holds:

$\displaystyle b_{u}(\eta)\leq 1\iff c_{2\eta}\leq\hat{b}.$ (53)

Also, based on $|\hat{b}|$, (53) can equivalently be expressed as

$\displaystyle b_{u}(\eta)\leq 1\iff\Big{(}\hat{b}>1\lor\big{(}|\hat{b}|\leq
1\land\eta\geq\underline{\eta}\big{)}\Big{)}.$ (54)

Under the $b_{l}(\eta)<0$, observe that $b_{l}(\eta)<s_{\tau}$ in (50)
automatically holds as $0<s_{\tau}$ is satisfied. Moreover, it can be verified
that the following equivalence holds:

$\displaystyle 0\leq b_{l}(\eta)\iff\bigg{(}0<\iota\leq 1\land
0\leq\eta<\frac{1}{2}\bigg{)}.$ (55)

It is noteworthy that for the case of $0<\iota\leq 1$

$\displaystyle 0<\bar{\eta}\leq\frac{1}{2},$ (56)

holds. Also, for the case of $0<\iota\leq 1$ and $|\hat{b}|\leq 1$

$\displaystyle\underline{\eta}<\bar{\eta},$ (57)

holds. To prove that, we have

$\displaystyle\frac{1}{\pi}\arccos(\hat{b})<\frac{1}{\pi}\arcsin(\iota)\iff\sqrt{1-\iota^{2}}<\hat{b},$

where $\sqrt{1-\iota^{2}}<\hat{b}$ is satisfied according to the arithmetic-
geometric inequality as

$\displaystyle\sqrt{1-\iota^{2}}\leq\frac{s_{\hat{\tau}}(1-\iota^{2})+\frac{1}{s_{\hat{\tau}}}}{2},$

where the equality cannot occur as
$s_{\hat{\tau}}(1-\iota^{2})<1\leq\frac{1}{s_{\hat{\tau}}}$ holds.

Case i. According to (50)-(52) and (54)-(57), $\mathcal{S}$ in (24) can be
defined.

Case ii. According to (50)-(52) and (54)-(56), $\mathcal{S}$ in (25) can be
defined.

Case iii. According to (50)-(52), (54), and (55), $\mathcal{S}$ in (26) can be
defined.

Case iv. According to (50)-(52), (54), and (55), $\mathcal{S}$ in (27) can be
defined.

Case v. According to (50)-(52), (54), and (55), $\mathcal{S}$ in (28) can be
defined which ends the proof.