# System Identification Using Energy-Bounded Noise Models:
A Full Characterization of Chebyshev Centers and Radii

Amir Shakouri, Henk van Waarde, Kanat Camlibel The authors are with the
Bernoulli Institute for Mathematics, Computer Science and Artificial
Intelligence, University of Groningen, The Netherlands. Henk van Waarde
acknowledges financial support by the Dutch Research Council under the NWO
Talent Programme Veni Agreement (VI.Veni.22.335). Email<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

This paper studies the identification of linear time-invariant systems using
noisy input-output data in a set-membership framework. It is assumed that the
noise sequence belongs to a bounded set, induced by a quadratic matrix
inequality, that allows for capturing various energy-bounded noise models. In
this framework, the identification problem is to find an estimation for the
true system so that the worst-case estimation error with respect to a given
norm is minimized. For such a problem, we present closed-form solutions with
respect to arbitrary _unitarily invariant_ matrix norms. Examples of these
norms include the Frobenius norm, spectral norm, nuclear norm, Schatten
$p$-norms, and Ky Fan $k$-norms.

##  I INTRODUCTION

Identification of linear time-invariant systems from noisy input-output data
has been approached in various ways depending on the nature of the noise and
how it is modeled. The majority of the system identification literature is
devoted to stochastic noise models with assumptions on the distribution,
correlation, ergodicity, and expected value of the noise (see, e.g., [1]). In
the absence of a reliable statistical hypothesis on the noise, a common
modeling approach assumes that the noise is unknown but belongs to a known
bounded set [2, 3, 4, 5, 6, 7]. In this setting, the input-output data and
knowledge about the noise give rise to the _set of explaining systems_ , which
consists of all systems that could have generated the data. In the case that
the set of explaining systems is bounded, the identification problem seeks a
solution that minimizes the worst-case estimation error with respect to a
given norm. This approach is known as set-membership system identification
[8].

Given a set of explaining systems, the center and radius of a smallest ball
containing the set are referred to as the _Chebyshev center_ and _Chebyshev
radius_ , respectively. A Chebyshev center has the minimum distance to the
farthest element within the set, making it a best worst-case estimation for
the true system. The error of such an estimation is the Chebyshev radius. In
addition, the farthest distance between two elements within the set is another
measure of the identification error, which is referred to as the _diameter_ of
the set. Hence, for a given norm, the system identification problem can be
cast as the problem of finding a Chebyshev center, the Chebyshev radius, and
the diameter for the set of explaining systems.

The complexity of finding a Chebyshev center and the Chebyshev radius depends
on the characteristics of the set and the given norm. For instance, if the set
is an ellipsoid and the given norm is the Euclidean norm, then the Chebyshev
center is unique and is simply the center of the ellipsoid with the Chebyshev
radius being the length of the largest semi-axis. However, for the
intersection of ellipsoids, there are no closed-form solutions and the
existing algorithms only converge to a proximity of the Chebyshev center [9,
10]. The effect of the given norm on the computational complexity of finding
the Chebyshev radius is evident as a norm can itself be NP-hard to compute
(such as matrix $p$-norms when $p\neq 1,2,\infty$ [11]). The reader can refer
to [12] and the references therein for more details on Chebyshev centers and
their applications.

In system identification, the characteristics of the set of explaining systems
are affected by the noise model. The state-of-the-art methods for computing a
Chebyshev center and the Chebyshev radius for the set of explaining systems
are mainly algorithmic solutions offered for specific choices of norms that
either converge to a true Chebyshev center [13] or a proximity of that [10,
14]. In many practical cases, algorithmic solutions are inevitable as the
noise model does not allow for the optimal estimation error to have a simple
analytic expression. An example of such noise models is the case where the
noise admits sample bounds [3, 4, 5, 6, 7]. Except for some well-known cases
such as minimizing the Frobenius norm of the data error via least-squares
problem, system identification seldom offers closed-form solutions.
Nevertheless, exploring cases for which a closed-form solution exists may lead
to valuable information on how the input-output data affect the identification
error with respect to different norms.

In this paper, we consider multiple-input multiple-output (MIMO)
autoregressive models with exogenous input (ARX models) with the noise
sequence satisfying a quadratic matrix inequality (QMI). The considered
assumption on the noise captures a range of energy-bounded models. We use the
recent results on QMI-induced sets [15] to parameterize the set of explaining
systems. Our main result is a full characterization of a Chebyshev center, the
Chebyshev radius, and the diameter of the set of explaining systems with
respect to arbitrary unitarily invariant matrix norms (e.g., Frobenius norm,
spectral norm, nuclear norm, Schatten $p$-norms, and Ky Fan $k$-norms). The
freedom over the choice of norm makes it possible to express the worst-case
estimation error with respect to the norm that is most relevant to the target
application. For instance, the results of this paper make it possible to
directly implement tools from robust control, where the system uncertainty is
assumed to be bounded by different norms such as the spectral norm studied in
[16], or the Frobenius norm studied in [17, 18]. The main result of this
letter is followed by several discussions on cases in which the Chebyshev
center is unique, conditions under which the solution to a generalized least-
squares problem is a Chebyshev center for all norms, and how to incorporate
prior knowledge to enhance the estimation error.

The remainder of this paper is organized as follows. Section II is devoted to
preliminaries. Section III formulates the to-be-studied system identification
problem. The main results are presented in Section IV and their proofs are
given in Section V. Section VI contains numerical examples. Finally, Section
VII concludes the paper.

##  II PRELIMINARIES

###  II-A Notation

Let $\mathbb{R}_{+}^{n}$ denote the set of $n$-dimensional real vectors with
nonnegative entries. We define
$\mathbb{R}^{n}_{\downarrow}\coloneqq\\{x\in\mathbb{R}_{+}^{n}:x_{i}\geq
x_{i+1},i=1,\ldots,n\\}$. For $x\in\mathbb{R}^{n}$ and $k\leq n$, we define
the vector containing the first $k$ entries of $x$ as
$x_{[k]}\coloneqq[x_{1},\ldots,x_{k}]^{\top}$. For two vectors
$x,y\in\mathbb{R}^{n}$, we say $y$ _weakly majorizes_ $x$, and we denote it by
$x\prec_{w}y$, if $\sum_{i=1}^{k}x_{i}\leq\sum_{i=1}^{k}y_{i}$ for all
$k\in[1,n]$.

The vector of singular values of matrix $M\in\mathbb{C}^{n\times m}$, arranged
in a nondecreasing order, is denoted by
$\sigma(M)\in\mathbb{R}^{\min\\{n,m\\}}_{\downarrow}$. We denote the kernel of
matrix $M\in\mathbb{R}^{n\times m}$ by $\ker
M\coloneqq\\{x\in\mathbb{R}^{m}:Mx=0\\}$. The Moore-Penrose pseudo-inverse of
a real matrix $M$ is denoted by $M^{\dagger}$. Moreover, we denote the
Hadamard (or Schur) product of two matrices with the same size
$M,N\in\mathbb{R}^{n\times m}$ by $M\circ N$.

Let $\mathbb{S}^{n}$ denote the set of $n\times n$ symmetric matrices. A
matrix $M\in\mathbb{S}^{n}$ is called _positive definite_ (resp., _positive
semi-definite_) and denoted by $M>0$ (resp., $M\geq 0$) if all its eigenvalues
are positive (resp., nonnegative). By $M<0$ (resp., $M\leq 0$) we imply that
$-M>0$ (resp., $-M\geq 0$).

A norm $|\cdot|$ on $\mathbb{R}^{p\times q}$ is said to be _essentially
strictly convex_ if the unit ball in $\mathbb{R}^{p\times q}$ with respect the
norm is a strictly convex set, or equivalently,
$X_{1},X_{2}\in\mathbb{R}^{p\times q}$ and $|X_{1}+X_{2}|=|X_{1}|+|X_{2}|$
imply $X_{1}=\alpha X_{2}$ for some $\alpha\geq 0$ (cf. [19, p. 106], [20, eq.
2.8]).

For a set of matrices $\mathcal{X}$ and matrices $X_{0},M,N$ of appropriate
sizes, we define $X_{0}+\mathcal{X}\coloneqq\\{X_{0}+X:X\in\mathcal{X}\\}$,
$\mathcal{X}-\mathcal{X}\coloneqq\\{X_{1}-X_{2}:X_{1},X_{2}\in\mathcal{X}\\}$,
and $M\mathcal{X}N\coloneqq\\{MXN:X\in\mathcal{X}\\}$.

###  II-B Center, radius and diameter of sets

Let $\mathcal{X}\subset\mathbb{R}^{p\times q}$ be a compact set and let
$|\cdot|$ be a norm on $\mathbb{R}^{p\times q}$. We define

$\operatorname{diam}\mathcal{X}\coloneqq\max_{\Delta\in\mathcal{X}-\mathcal{X}}|\Delta|$
(1)

as the _diameter_ of $\mathcal{X}$,

$\operatorname{rad}\mathcal{X}\coloneqq\min_{C\in\mathbb{R}^{p\times
q}}\max_{X\in\mathcal{X}}|C-X|$ (2)

as the _Chebyshev radius_ of $\mathcal{X}$, and

$\operatorname{cent}\mathcal{X}\coloneqq\\{C\in\mathbb{R}^{p\times
q}:|C-X|\leq\operatorname{rad}\mathcal{X}\ \ \forall X\in\mathcal{X}\\}$ (3)

as the _set of Chebyshev centers_ 111Note that Chebyshev centers and radii are
defined differently in some references such as [21, Chapter 8]. of
$\mathcal{X}$. Due to compactness of $\mathcal{X}$,
$\operatorname{diam}\mathcal{X}$ and $\operatorname{rad}\mathcal{X}$ are well-
defined, and $\operatorname{cent}\mathcal{X}$ is nonempty.

The unique Chebyshev center with Euclidean normThe set of Chebyshev centers
with infinity norm$\mathcal{X}$ Figure 1: The Chebyshev radius and the set of
Chebyshev centers for the closed area defined by an equilateral triangle in
terms of Euclidean and infinity norms.

Based on the above definitions we have the following results222The proof of
Proposition 1 is available in the Appendix of the extended version of this
letter at “https://arxiv.org/abs/2403.05315.”.

###### Proposition 1

Let $\mathcal{X}\subset\mathbb{R}^{p\times q}$ be a compact set. Then:

  1. (a)

$\tfrac{1}{2}\operatorname{diam}\mathcal{X}\leq\operatorname{rad}\mathcal{X}\leq\operatorname{diam}\mathcal{X}$.

  2. (b)

$\operatorname{cent}\mathcal{X}$ is compact and convex with no interior
points.

  3. (c)

If the norm $|\cdot|$ is essentially strictly convex, then the set
$\mathcal{X}$ has a unique Chebyshev center.

As an example, consider the set $\mathcal{X}\subset\mathbb{R}^{2}$ to be the
closed shaded area defined by an equilateral triangle as depicted in Fig. 1.
With respect to the Euclidean norm, which is an essentially strictly convex
norm, the set of Chebyshev centers is a singleton. However, with respect to
the infinity norm, the set of Chebyshev centers is not a singleton, but a line
segment. The diameter is the length between the farthest elements within the
set, equal to the length of the triangle’s side for both Euclidean and
infinity norms.

###  II-C Sets induced by quadratic matrix inequalities

We define

$\begin{split}\boldsymbol{\Pi}_{q,p}\coloneqq\Bigg{\\{}\begin{bmatrix}\Pi_{11}&\Pi_{12}\\\
\Pi_{21}&\Pi_{22}\end{bmatrix}\in\mathbb{S}^{q+p}:\Pi_{11}\in\mathbb{S}^{q},\Pi_{22}\in\mathbb{S}^{p},\\\
\Pi_{22}\leq 0,\Pi|\Pi_{22}\geq
0,\ker\Pi_{22}\subseteq\ker\Pi_{12}\Bigg{\\}}.\end{split}$ (4)

where $\Pi|\Pi_{22}\coloneqq\Pi_{11}-\Pi_{12}\Pi_{22}^{\dagger}\Pi_{21}$ is
the Schur complement of $\Pi$ with respect to $\Pi_{22}$.

For a given $\Pi\in\boldsymbol{\Pi}_{q,p}$, the matrices
$\Pi_{11}\in\mathbb{S}^{q}$, $\Pi_{12}\in\mathbb{R}^{q\times p}$,
$\Pi_{21}=\Pi_{12}^{\top}$, and $\Pi_{22}\in\mathbb{S}^{p}$ are defined by

$\begin{bmatrix}\Pi_{11}&\Pi_{12}\\\ \Pi_{21}&\Pi_{22}\end{bmatrix}=\Pi.$ (5)

We also define the QMI-induced set $\mathcal{Z}_{p}(\Pi)$ associated with a
matrix $\Pi\in\boldsymbol{\Pi}_{q,p}$ to be

$\mathcal{Z}_{p}(\Pi)\coloneqq\left\\{Z\in\mathbb{R}^{p\times
q}:\begin{bmatrix}I\\\ Z\end{bmatrix}^{\top}\Pi\begin{bmatrix}I\\\
Z\end{bmatrix}\geq 0\right\\}.$ (6)

The basic properties of $\mathcal{Z}_{p}(\Pi)$ are recalled in the following
proposition.

######  Proposition 2 ([15, Thms. 3.2 & 3.3])

Let $\Pi\in\boldsymbol{\Pi}_{q,p}$. Then:

  1. (a)

$\mathcal{Z}_{p}(\Pi)$ is nonempty and convex.

  2. (b)

$\mathcal{Z}_{p}(\Pi)$ is bounded if and only if $\Pi_{22}<0$.

  3. (c)

Assuming $\Pi_{22}<0$, $Z\in\mathcal{Z}_{p}(\Pi)$ if and only if

$Z=-\Pi_{22}^{-1}\Pi_{21}+(-\Pi_{22})^{-\frac{1}{2}}S(\Pi|\Pi_{22})^{\frac{1}{2}}$
(7)

for some $S\in\mathbb{R}^{p\times q}$ satisfying $SS^{\top}\leq I$.

As an example, consider the three-dimensional ellipsoid
$\tfrac{z_{1}^{2}}{a^{2}}+\tfrac{z_{2}^{2}}{b^{2}}+\tfrac{z_{3}^{2}}{c^{2}}=1$,
depicted in Fig. 2. The region enclosed by this ellipsoid,
$\tfrac{z_{1}^{2}}{a^{2}}+\tfrac{z_{2}^{2}}{b^{2}}+\tfrac{z_{3}^{2}}{c^{2}}\leq
1$, is a QMI-induced set with
$Z^{\top}=\begin{bmatrix}z_{1}&z_{2}&z_{3}\end{bmatrix}$ and $\Pi$ being a
diagonal matrix with diagonal entries of $1$, $-\tfrac{1}{a^{2}}$,
$-\tfrac{1}{b^{2}}$, and $-\tfrac{1}{c^{2}}$. In this example, (7) is
equivalent to the parametrization of the ellipsoid in the spherical
coordinates.

$z_{3}$$z_{2}$$z_{1}$ Figure 2: The closed shaded area defined by an ellipsoid
is an example of a QMI-induced set.

###  II-D Unitarily invariant norms

In this paper, we focus on unitarily invariant matrix norms.

######  Definition 1 ([22, p. 835])

A matrix norm $|\cdot|$ on $\mathbb{C}^{n\times m}$ is called unitarily
invariant if $|UMV|=|M|$ for all $M\in\mathbb{C}^{n\times m}$ and all unitary
$U\in\mathbb{C}^{n\times n}$ and $V\in\mathbb{C}^{m\times m}$.

For instance, for a matrix $M\in\mathbb{C}^{n\times m}$, its

  1. -

Frobenius norm
$\sqrt{\operatorname{tr}(MM^{*})}=\sqrt{\sum_{i=1}^{\min\\{n,m\\}}\sigma_{i}^{2}(M)}$,

  2. -

Schatten $p$-norms $(\sum_{i=1}^{\min\\{n,m\\}}\sigma_{i}^{p}(M))^{1/p}$,
$p\geq 1$, and

  3. -

Ky Fan $k$-norms $\sum_{i=1}^{k}\sigma_{i}(M)$, $k\in[1,\min\\{n,m\\}]$,

are all unitarily invariant. This class of norms have an interesting
association with so-called symmetric gauge functions.

###### Definition 2

A function $g:\mathbb{R}^{n}\rightarrow\mathbb{R}_{+}$ is called a _symmetric
gauge function_ if it satisfies the following properties:

  1. (a)

$g$ is a norm,

  2. (b)

$g(Px)=g(x)$ for all permutation matrices $P\in\mathbb{R}^{n\times n}$,

  3. (c)

$g(x_{1},\ldots,x_{n})=g(|x_{1}|,\ldots,|x_{n}|)$.

Having the above definition, it is not surprising that a symmetric gauge
function of the vector of singular values of a matrix defines a unitarily
invariant norm. However, strikingly, von Neumann proved in [23] that the
converse is also true.

######  Proposition 3 ([22, Fact 11.9.59])

A norm $|\cdot|$ on $\mathbb{C}^{n\times m}$ is unitarily invariant if and
only if there exists a symmetric gauge function
$g(\cdot):\mathbb{R}^{\min\\{n,m\\}}\rightarrow\mathbb{R}_{+}$ such that
$|M|=g(\sigma(M))$ for all $M\in\mathbb{C}^{n,m}$.

We know from Proposition 1(c) that for essentially strictly convex norms, the
Chebyshev center of a compact set of matrices is unique. The following result
shows that the essential strict convexity of a unitarily invariant norm is
equivalent to that of its symmetric gauge function.

######  Proposition 4 ([20, Thm. 3.1])

A unitarily invariant norm is essentially strictly convex if and only if its
associated symmetric gauge function is essentially strictly convex.

For instance, based on Proposition 4, the Frobenius norm is essentially
strictly convex, and thus from Proposition 1(c), the Chebyshev center of a
compact set with respect to the Frobenius norm is unique. However, this is not
necessarily the case for the spectral norm.

##  III PROBLEM FORMULATION

We consider ARX models given by

$y(t+L)=\sum_{i=0}^{L-1}P_{i}y(t+i)+\sum_{j=0}^{M}Q_{j}u(t+j)+w(t),$ (8)

where $y(t)\in\mathbb{R}^{p}$ is the output, $u(t)\in\mathbb{R}^{m}$ is the
input, $w(t)\in\mathbb{R}^{p}$ is the process noise, the integers $L$ and $M$
are known, and matrices $P_{i}\in\mathbb{R}^{p\times p}$, $i=0,\ldots,L-1$,
and $Q_{j}\in\mathbb{R}^{p\times m}$, $j=0,\ldots,M$, are unknowns. We assume
that the system is causal, i.e., $M\leq L$. Note that input-state models can
be captured by choosing $L=1$ and $M=0$. We refer to (8) as the true system
and we define

$P_{\mathrm{true}}\coloneqq\begin{bmatrix}P_{0}&\cdots&P_{L-1}\end{bmatrix}\
\text{and}\
Q_{\mathrm{true}}\coloneqq\begin{bmatrix}Q_{0}&\cdots&Q_{M}\end{bmatrix}.$ (9)

We collect the input-output data

$u(0),\ldots,u(T+M-L),\hskip 14.22636pty(0),\ldots,y(T),$ (10)

from (8), where $T\geq L$. The process noise $w(t)$ that affects the dynamics
during this experiment is unknown, but the collection of them,

$W_{-}\coloneqq\begin{bmatrix}w(0)&\cdots&w(T-L)\end{bmatrix},$

is assumed to satisfy an energy bound.

###### Assumption 1

The noise matrix $W_{-}$ satisfies

$\begin{bmatrix}I\\\
W_{-}^{\top}\end{bmatrix}^{\top}\begin{bmatrix}\Phi_{11}&\Phi_{12}\\\
\Phi_{12}^{\top}&\Phi_{22}\end{bmatrix}\begin{bmatrix}I\\\
W_{-}^{\top}\end{bmatrix}\geq 0,$ (11)

i.e., $W_{-}^{\top}\in\mathcal{Z}_{T-L+1}(\Phi)$, for a known matrix
$\Phi\in\boldsymbol{\Pi}_{p,T-L+1}$ with $\Phi_{22}<0$.

The reader can refer to [15, p. 4] for the special cases that can be captured
by this noise model.

Define the matrices

$\begin{split}U_{-}&\coloneqq\begin{bmatrix}u(0)&u(1)&\cdots&u(T-L)\\\
\vdots&\vdots&\ddots&\vdots\\\ u(M)&u(M+1)&\cdots&u(T+M-L)\end{bmatrix},\\\
Y_{-}&\coloneqq\begin{bmatrix}y(0)&y(1)&\cdots&y(T-L)\\\
\vdots&\vdots&\ddots&\vdots\\\ y(L-1)&y(L)&\cdots&y(T-1)\end{bmatrix},\
\text{and}\\\
Y_{+}&\coloneqq\begin{bmatrix}y(L)&y(L+1)&\cdots&y(T)\end{bmatrix}.\end{split}$

A pair of real matrices $(P,Q)$ satisfying

$Y_{+}=PY_{-}+QU_{-}+W_{-}$ (12)

for some $W_{-}^{\top}\in\mathcal{Z}_{T-L+1}(\Phi)$ is called an explaining
system. We define the set of all explaining systems as

$\Sigma\coloneqq\\{(P,Q):(Y_{+}-PY_{-}-QU_{-})^{\top}\in\mathcal{Z}_{T-L+1}(\Phi)\\},$
(13)

which is the set of all $(P,Q)$ that could have generated the available data
for some noise sequence agreeing with Assumption 1.

One can validate that the set of explaining systems is equivalent to a QMI-
induced set as (cf. [24, Lem. 4] and [25, Sec. IV])

$(P,Q)\in\Sigma\Longleftrightarrow\begin{bmatrix}P&Q\end{bmatrix}^{\top}\in\mathcal{Z}_{s}(N)$
(14)

where $s\coloneqq Lp+(M+1)m$ and

$N\coloneqq\begin{bmatrix}I&Y_{+}\\\ 0&-Y_{-}\\\
0&-U_{-}\end{bmatrix}\begin{bmatrix}\Phi_{11}&\Phi_{12}\\\
\Phi_{12}^{\top}&\Phi_{22}\end{bmatrix}\begin{bmatrix}I&Y_{+}\\\ 0&-Y_{-}\\\
0&-U_{-}\end{bmatrix}^{\top}.$ (15)

Furthermore, it can be observed that matrix $N$ in (15) satisfies
$N\in\boldsymbol{\Pi}_{p,s}$. Hence, based on Proposition 2, we have the
following facts.

###### Proposition 5

The following statements hold:

  1. (a)

$\Sigma$ is nonempty.

  2. (b)

$\Sigma$ is bounded if and only if
$\operatorname{rank}\begin{bmatrix}Y_{-}^{\top}&U_{-}^{\top}\end{bmatrix}=s$.

  3. (c)

Assume that $\Sigma$ is bounded. Then, $\Sigma$ is a singleton if and only if
$N|N_{22}=0$. In this case,
$\Sigma=\\{(P_{\mathrm{true}},Q_{\mathrm{true}})\\}$ and
$\begin{bmatrix}P_{\mathrm{true}}&Q_{\mathrm{true}}\end{bmatrix}=-N_{12}N_{22}^{-1}$.

In the noiseless case, $\Phi_{11}=0$, $\Phi_{12}=\Phi_{21}^{\top}=0$, and
$\Phi_{22}=-I$, the condition of Proposition 5(c), $N|N_{22}=0$, is always
satisfied. In the presence of noise, a necessary condition for unique
identification is that the noise sequence is such that the left-hand side of
(11) is equal to zero, i.e., the noise is of bound-exploring type (cf. [26,
Ass. 5]). However, in general, $\mathcal{Z}_{s}(N)$ is not a singleton and
$(P_{\mathrm{true}},Q_{\mathrm{true}})$ cannot be uniquely recovered from the
data. Nevertheless, in this scenario, we can still formalize system
identification as the problem of finding a Chebyshev center, the Chebyshev
radius, and the diameter of $\mathcal{Z}_{s}(N)$.

###### Problem 1

Given a unitarily invariant norm $|\cdot|$ and the input-output data with
$\begin{bmatrix}Y_{-}^{\top}&U_{-}^{\top}\end{bmatrix}$ of full column rank,
find a Chebyshev center, the Chebyshev radius, and the diameter of
$\mathcal{Z}_{s}(N)$.

##  IV MAIN RESULTS

As discussed in Section III, the set of explaining systems $\Sigma$ can be
expressed in terms of the solution set $\mathcal{Z}_{s}(N)$ of a QMI for a
matrix $N\in\boldsymbol{\Pi}_{p,s}$ constructed from the data and the noise
model. The following theorem is the main result of this paper that presents
the solution of Problem 1.

###### Theorem 1

Let $|\cdot|$ be a unitarily invariant matrix norm associated with a symmetric
gauge function $g$. Then, the following statements hold for the set of
explaining systems:

  1. (a)

$\operatorname{diam}\mathcal{Z}_{s}(N)=2g(\sigma_{[p]}((-N_{22})^{-\frac{1}{2}})\circ\sigma((N|N_{22})^{\frac{1}{2}}))$.

  2. (b)

$\operatorname{rad}\mathcal{Z}_{s}(N)=\tfrac{1}{2}\operatorname{diam}\mathcal{Z}_{s}(N)$.

  3. (c)

$-N_{22}^{-1}N_{21}\in\operatorname{cent}\mathcal{Z}_{s}(N)$.

In the following, we discuss some aspects of Theorem 1.

###  IV-A The common Chebyshev center

Recall from the example in Fig. 1 that the sets of Chebyshev centers for
different norms do not necessarily intersect. However, Theorem 1 reveals that
for QMI-induced sets,

$\begin{split}\begin{bmatrix}\hat{P}&\hat{Q}\end{bmatrix}&\coloneqq-
N_{12}N_{22}^{-1}\\\ &=\begin{bmatrix}I\\\
Y_{+}^{\top}\end{bmatrix}^{\top}\begin{bmatrix}\Phi_{12}\\\
\Phi_{22}\end{bmatrix}\begin{bmatrix}Y_{-}\\\
U_{-}\end{bmatrix}^{\top}\left(\begin{bmatrix}Y_{-}\\\
U_{-}\end{bmatrix}\Phi_{22}\begin{bmatrix}Y_{-}\\\
U_{-}\end{bmatrix}^{\top}\right)^{-1}\end{split}$ (16)

is a Chebyshev center for every unitarily invariant norm. Moreover, it can be
proved using Lemma 3(a) that (16) is a common Chebyshev center for all matrix
norms. Recall from Proposition 1(c) that for essentially strictly convex norms
(e.g., Frobenius norm), the set of Chebyshev centers is a singleton. This
implies that (16) is the only common center among all (unitarily invariant)
norms. In addition, we note that the common Chebyshev center (16) is an
explaining system, i.e., $N_{12}N_{22}^{-1}\in\mathcal{Z}_{s}(N)$.

###  IV-B Least-squares solution as a Chebyshev center

One can validate that the Chebyshev center (16) is also the solution to the
following problem:

$\arg\\!\min_{P,Q}|(Y_{+}-PY_{-}-QU_{-}-\Phi_{12}\Phi_{22}^{-1})(-\Phi_{22})^{\frac{1}{2}}|_{\mathrm{F}},$
(17)

where $|\cdot|_{\mathrm{F}}$ denotes the Frobenius norm. Note that (17)
corresponds to the generalized least-squares problem with the bias term
$\Phi_{12}\Phi_{22}^{-1}$ taken into account and $\Phi_{22}$ acting as the
precision matrix. If $\Phi_{22}=-I$ and $\Phi_{12}=0$, then (17) is the
ordinary least-squares problem (cf. [2, 27, 28]).

###  IV-C Prior knowledge

The set of explaining systems discussed in Theorem 1 is merely constructed
based on the collected data and the known model class. However, incorporating
further prior knowledge may result in a smaller estimation error. Let us
introduce the set $\Sigma_{\mathrm{pk}}$ representing any prior knowledge of
the true system coefficients. Given $\Sigma$ and $\Sigma_{\mathrm{pk}}$, the
set of explaining systems complying with the prior knowledge is
$\Sigma\cap\Sigma_{\mathrm{pk}}$. In the case that
$\Sigma\cap\Sigma_{\mathrm{pk}}$ is bounded and is the solution to a QMI, the
result of Theorem 1 can be extended accordingly. Note that prior knowledge
could also reduce the rank condition on the data matrix
$\begin{bmatrix}Y_{-}^{\top}&U_{-}^{\top}\end{bmatrix}$ as
$\Sigma\cap\Sigma_{\mathrm{pk}}$ may be bounded even if $\Sigma$ is unbounded
(see Proposition 5(b)).

As an example, suppose that for known matrices
$\Omega_{2}\in\mathbb{R}^{h\times s}$ and $\Omega_{1}\in\mathbb{R}^{p\times
s}$ we have

$\Sigma_{\mathrm{pk}}\\!\coloneqq\\{(P,Q)\\!:\\!\begin{bmatrix}P&Q\end{bmatrix}\\!=\\!H\Omega_{2}+\Omega_{1}\
\text{for some}\ H\in\mathbb{R}^{p\times h}\\}.$ (18)

This type of constraint can be used when some columns of
$\begin{bmatrix}P_{\mathrm{true}}&Q_{\mathrm{true}}\end{bmatrix}$ are known a
priori so that the unknown terms are included in $H$. Now, one can validate
that $(P,Q)\in\Sigma\cap\Sigma_{\mathrm{pk}}$ if and only if

$H^{\top}\in\mathcal{Z}_{h}(N_{\mathrm{pk}})$ (19)

with $N_{\mathrm{pk}}$ being

$N_{\mathrm{pk}}\coloneqq\begin{bmatrix}I&\Omega_{1}\\\
0&\Omega_{2}\end{bmatrix}N\begin{bmatrix}I&\Omega_{1}\\\
0&\Omega_{2}\end{bmatrix}^{\top}.$ (20)

In this case, one can observe that for $\Sigma\cap\Sigma_{\mathrm{pk}}$ to be
bounded, we need
$\begin{bmatrix}Y_{-}^{\top}&U_{-}^{\top}\end{bmatrix}\Omega_{2}^{\top}$ to be
full column rank (cf. Proposition 5(b)). Having a bounded
$\Sigma\cap\Sigma_{\mathrm{pk}}$, the results of Theorem 1 hold for the set of
explaining systems incorporating prior knowledge (18) by replacing $N$ with
$N_{\mathrm{pk}}$.

###  IV-D The choice of norm

Theorem 1 is applicable to general unitarily invariant norms. Depending on the
application, certain norms could be more relevant than others. For instance,
suppose that the goal is to design a robust controller for the set of
explaining systems. In this case, considering a Chebyshev center
$(\hat{P},\hat{Q})$ to be the nominal system with the bound on the uncertainty
$(\Delta_{P},\Delta_{Q})$ given by the Chebyshev radius with respect to the
spectral norm, one may check the feasibility to design a robust controller for
all systems in the form of $(\hat{P}+\Delta_{P},\hat{Q}+\Delta_{Q})$ (e.g.,
see [16]). To this end, having $\begin{bmatrix}\hat{P}&\hat{Q}\end{bmatrix}$
as the Chebyshev center (16), from Theorem 1 it can be verified that

$\sigma_{1}(\begin{bmatrix}\Delta_{P}&\Delta_{Q}\end{bmatrix})\leq\rho,$ (21)

for all $(\hat{P}+\Delta_{P},\hat{Q}+\Delta_{Q})\in\Sigma$ if and only if

$\lambda_{\mathrm{max}}(N|N_{22})\leq\rho^{2}\lambda_{\mathrm{min}}(-N_{22}).$
(22)

For the special case $L=1$, $M=0$, $\Phi_{22}=-I$, and
$\Phi_{12}=\Phi_{21}^{\top}=0$, it was proved in [29, Lem. IV.5] that the set
of explaining systems $\mathcal{Z}_{s}(N)$ is contained in a ball with radius
$\sqrt{\lambda_{\mathrm{max}}(N|N_{22})/\lambda_{\mathrm{min}}(-N_{22})}$
centered at $-N_{22}^{-1}N_{21}$. Interestingly, our result shows that this
ball is the one with _smallest_ radius containing $\mathcal{Z}_{s}(N)$.

The result of Theorem 1 can be extended to arbitrary submatrices of
$\mathcal{Z}_{s}(N)$. In particular, given $\Theta_{L}\in\mathbb{R}^{l\times
s}$, $\Theta_{R}\in\mathbb{R}^{p\times r}$, and $k\coloneqq\min\\{l,r,p\\}$,
we have
$\operatorname{rad}\Theta_{L}\mathcal{Z}_{s}(N)\Theta_{R}=\tfrac{1}{2}\operatorname{diam}\Theta_{L}\mathcal{Z}_{s}(N)\Theta_{R}$
with

$\begin{split}&\operatorname{diam}\Theta_{L}\mathcal{Z}_{s}(N)\Theta_{R}\\\
&=2g(\sigma_{[k]}(\Theta_{L}(-N_{22})^{-\frac{1}{2}})\circ\sigma_{[k]}((N|N_{22})^{\frac{1}{2}})\Theta_{R}),\end{split}$
(23)

and
$-\Theta_{L}N_{22}^{-1}N_{21}\Theta_{R}\in\operatorname{cent}\Theta_{L}\mathcal{Z}_{s}(N)\Theta_{R}$.

##  V PROOF OF THEOREM 1

To prove Theorem 1, we need three auxiliary results. The first one deals with
an upper bound on the unitarily invariant norm of a product of matrices.

###### Lemma 1

Let $L\in\mathbb{R}^{l\times p}$, $S\in\mathbb{R}^{p\times q}$,
$R\in\mathbb{R}^{q\times r}$, and $k\coloneqq\min\\{l,p,q,r\\}$. Also, let
$g:\mathbb{R}^{k}\rightarrow\mathbb{R}_{+}$ be a symmetric gauge function.
Then,

$g(\sigma_{[k]}(LSR))\leq
g(\sigma_{[k]}(L)\circ\sigma_{[k]}(S)\circ\sigma_{[k]}(R)).$ (24)

###### Proof:

The following majorization inequality holds for the singular values of the
product of two matrices $X\in\mathbb{R}^{n_{x}\times m}$ and
$Y\in\mathbb{R}^{m\times n_{y}}$:

$\sigma_{[s]}(XY)\prec_{w}\sigma_{[s]}(X)\circ\sigma_{[s]}(Y),$ (25)

for any $s\leq\min\\{n_{x},m,n_{y}\\}$. A proof of inequality (25) can be
found in [30, p. 342] for square matrices, which can be straightforwardly
extended to rectangular matrices by taking care of the dimension of the
singular value vectors (see [30, p. 299] for further explanations). Next, to
extend inequality (25) to the product of three matrices, we consider the fact
that if two vectors $x,y\in\mathbb{R}^{n}_{+}$ satisfy $x\prec_{w}y$, then for
any $z\in\mathbb{R}_{\downarrow}^{n}$ we have $x\circ z\prec_{w}y\circ z$ (see
[31, Prob. 11.5.16]). Hence, as
$\sigma_{[k]}(R)\in\mathbb{R}_{\downarrow}^{k}$, inequality (25) implies
$\sigma_{[k]}(LS)\circ\sigma_{[k]}(R)\prec_{w}\sigma_{[k]}(L)\circ\sigma_{[k]}(S)\circ\sigma_{[k]}(R)$
by choosing $X=L$ and $Y=S$. Thus, by choosing $X=LS$ and $Y=R$ in (25), we
have $\sigma_{[k]}(LSR)\prec_{w}\sigma_{[k]}(LS)\circ\sigma_{[k]}(R)$. Hence,
we obtain

$\sigma_{[k]}(LSR)\prec_{w}\sigma_{[k]}(L)\circ\sigma_{[k]}(S)\circ\sigma_{[k]}(R).$
(26)

In addition, from Ky Fan’s lemma, for vectors $x,y\in\mathbb{R}^{n}_{+}$, we
have $g(x)\leq g(y)$ for all symmetric gauge functions if and only if
$x\prec_{w}y$ (see [30, 4.B.6]). Therefore, inequality (26) implies (24) for
all symmetric gauge functions. ∎

The second auxiliary result deals with nonexpansive matrices.

###### Lemma 2

Let $\mathcal{S}\coloneqq\\{X\in\mathbb{R}^{n\times m}:XX^{\top}\leq I\\}$.
Then, $\mathcal{S}-\mathcal{S}=2\mathcal{S}$.

###### Proof:

Let $Z\in 2\mathcal{S}$. Then, $Z=2X$ for some $X\in\mathcal{S}$. Since
$-X\in\mathcal{S}$ and $Z=X-(-X)$, we see that $Z\in\mathcal{S}-\mathcal{S}$.
This proves that $2\mathcal{S}\subseteq\mathcal{S}-\mathcal{S}$. For the
reverse inclusion, let $Z\in\mathcal{S}-\mathcal{S}$. Then, $Z=X-Y$ where
$X,Y\in\mathcal{S}$. Note that

$\begin{bmatrix}I&X\\\ X^{\top}&I\end{bmatrix}\geq 0,\hskip
7.11317pt\text{and}\hskip 7.11317pt\begin{bmatrix}I&-Y\\\
-Y^{\top}&I\end{bmatrix}\geq 0.$ (27)

Adding the matrices in (27) yields

$\begin{bmatrix}2I&Z\\\ Z^{\top}&2I\end{bmatrix}\geq 0,$ (28)

equivalently, $ZZ^{\top}\leq 4I$. Then, $\tfrac{1}{2}Z\in\mathcal{S}$ and thus
$Z\in 2\mathcal{S}$. Therefore, $\mathcal{S}-\mathcal{S}\subseteq
2\mathcal{S}$, which completes the proof. ∎

The third auxiliary result characterizes the diameter and Chebyshev radius and
center for sets obtained by scaling nonexpansive matrices.

###### Lemma 3

Consider the set $\mathcal{X}\coloneqq\\{LSR:SS^{\top}\leq I\\}$ where
$L\in\mathbb{R}^{l\times p}$ and $R\in\mathbb{R}^{q\times r}$. Let
$k\coloneqq\min\\{p,q,r,l\\}$. The following statements hold:

  1. (a)

$0\in\operatorname{cent}\mathcal{X}$.

  2. (b)

If $|\cdot|$ is a unitarily invariant matrix norm associated with a symmetric
gauge function $g$, then
$\operatorname{rad}\mathcal{X}=\tfrac{1}{2}\operatorname{diam}\mathcal{X}=g(\sigma_{[k]}(L)\circ\sigma_{[k]}(R))$.

###### Proof:

Let $\rho=\max\\{|X|:X\in\mathcal{X}\\}$. It follows from Lemma 2 that
$\mathcal{X}-\mathcal{X}=2\mathcal{X}$ and from the definition of the diameter
(1) that $\operatorname{diam}\mathcal{X}=\max\\{|\Delta|:\Delta\in
2\mathcal{X}\\}=2\rho$. On the one hand, Proposition 1(a) implies that
$\operatorname{rad}\mathcal{X}\geq\rho$. On the other hand, we have

$\operatorname{rad}\mathcal{X}=\min_{C_{0}\in\mathbb{R}^{p\times
q}}\max_{X\in\mathcal{X}}|C_{0}-X|\leq\max_{X\in\mathcal{X}}|X|=\rho.$ (29)

Therefore,
$\operatorname{rad}\mathcal{X}=\tfrac{1}{2}\operatorname{diam}\mathcal{X}=\rho$.
Hence, it follows from the definition of the set of Chebyshev centers (3) that
$0\in\operatorname{cent}\mathcal{X}$. As such, what remains to be proven is
that $\rho=g(\sigma_{[k]}(L)\circ\sigma_{[k]}(R))$. To do so, observe that

$|LSR|\leq g(\sigma_{[k]}(L)\circ\sigma_{[k]}(S)\circ\sigma_{[k]}(R)),$ (30)

due to Lemma 1. Since $\sigma_{1}(S)\leq 1$ whenever $SS^{\top}\leq I$,
inequality (30) yields

$\rho\leq g(\sigma_{[k]}(L)\circ\sigma_{[k]}(R)).$ (31)

For $M\in\\{L,R\\}$, let $M=U_{M}\Sigma_{M}V_{M}^{\top}$ be a singular value
decomposition where $U_{M}$ and $V_{M}$ are orthogonal matrices, and
$\Sigma_{M}$ is the matrix containing zero entries apart from its main
diagonal which equals $\sigma(M)$. Moreover, let $\Sigma_{S}$ be a $p\times q$
matrix of the form $\begin{bmatrix}I_{k}&0\\\ 0&0\end{bmatrix}$. Note that
$S=V_{L}\Sigma_{S}U_{R}^{\top}$ satisfies $SS^{\top}\leq I$ and
$|LSR|=|\Sigma_{L}\Sigma_{S}\Sigma_{R}|=g(\sigma_{[k]}(L)\circ\sigma_{[k]}(R))$.
Together with (31) this proves that
$\rho=g(\sigma_{[k]}(L)\circ\sigma_{[k]}(R))$. ∎

Once more consider the example of the ellipsoid
$\tfrac{z_{1}^{2}}{a^{2}}+\tfrac{z_{2}^{2}}{b^{2}}+\tfrac{z_{3}^{2}}{c^{2}}\leq
1$ shown in Fig. 2. For this example, the Chebyshev radius presented in Lemma
3(b) with respect to the maximum singular value is equal to the largest semi-
axis $\max\\{a,b,c\\}$.

It is now straightforward to prove Theorem 1 as follows.

_Proof of Theorem 1:_ Recall that since $N\in\boldsymbol{\Pi}_{p,s}$, from
Propositions 2(b) and 5(b) we know that
$\begin{bmatrix}Y_{-}^{\top}&U_{-}^{\top}\end{bmatrix}$ being full column rank
is equivalent to $N_{22}<0$. Therefore, from Proposition 2(c) we have

$\mathcal{Z}_{s}(N)=X_{0}+\mathcal{X}$ (32)

where $\mathcal{X}=\\{LSR:SS^{\top}\leq I\\}$, $X_{0}=-N_{22}^{-1}N_{21}$,
$L=(-N_{22})^{-\frac{1}{2}}$, and $R=(N|N_{22})^{\frac{1}{2}}$. Note that
$\operatorname{diam}\mathcal{Z}_{s}(N)=\operatorname{diam}\mathcal{X}$,
$\operatorname{rad}\mathcal{Z}_{s}(N)=\operatorname{rad}\mathcal{X}$, and
$\operatorname{cent}\mathcal{Z}_{s}(N)=X_{0}+\operatorname{cent}\mathcal{X}$.
Thus, Theorem 1 now follows from Lemma 3. ∎

##  VI EXAMPLES

###  VI-A Example 1: mass-spring-damper

Consider a mass-spring-damper system with parameters $m=2\ \mathrm{kg}$, $k=3\
\mathrm{N/m}$, and $c=1\ \mathrm{Ns/m}$, discretized with a sample time of $1\
\mathrm{s}$. The mass position $y(t)\in\mathbb{R}$ is measured through a noisy
experiment during a time horizon of $5$ seconds. For such a system, we
consider the following model:

$y(t+2)=P_{0}y(t)+P_{1}y(t+1)+Q_{0}u(t)+Q_{1}u(t+1)+w(t).$ (33)

The true system coefficients are $P_{0}=-0.6065$, $P_{1}=0.5659$,
$Q_{0}=0.1583$, and $Q_{1}=0.1886$. The process noise $w(t)$ is assumed to
satisfy the energy bound $\sum_{t=0}^{3}(w(t)-0.005)^{2}\leq 10^{-4}$ that can
be captured by Assumption 1, $W_{-}^{\top}\in\mathcal{Z}_{4}(\Phi)$ and
$\Phi\in\boldsymbol{\Pi}_{1,4}$, with $\Phi_{11}=0$, $\Phi_{22}=-I_{4}$, and
$\Phi_{12}=\Phi_{21}^{\top}=0.005\begin{bmatrix}1&1&1&1\end{bmatrix}$.

During the experiment, starting from $y(0)=0$, $y(1)=1$, we apply the input
sequence $u(0)=1$, $u(2)=0$, $u(3)=-1$, $u(4)=0$, and $u(5)=1$, which makes
$U_{-}$ to be

$U_{-}=\begin{bmatrix}1&0&-1&0\\\ 0&-1&0&1\\\ \end{bmatrix}.$ (34)

Now, consider a simulation with the noise sequence

$W_{-}=\begin{bmatrix}0.0105&-0.0013&0.0092&0.0084\end{bmatrix},$ (35)

for which the collected output data is

$\begin{split}Y_{-}&=\begin{bmatrix}0&1&0.7347&-0.3807\\\
1&0.7347&-0.3807&-0.8101\end{bmatrix},\\\
Y_{+}&=\begin{bmatrix}0.7347&-0.3807&-0.8101&-0.0305\end{bmatrix}.\end{split}$
(36)

For this case, the Chebyshev center (16) is

$\hat{P}=\begin{bmatrix}-0.6065&0.5659\end{bmatrix},\hskip
7.11317pt\hat{Q}=\begin{bmatrix}0.1583&0.1886\end{bmatrix},$ (37)

and the Chebyshev radius in terms of the Euclidean norm is
$\operatorname{rad}\mathcal{Z}_{4}(N)=0.0458$. For the described simulation,
the set of explaining systems, the true system, and the Chebyshev center (16)
are shown in Fig. 3.

To visualize the results beyond a single simulation, we obtain 1000
simulations for uniformly distributed random sequences of noise normalized to
comply with $W_{-}^{\top}\in\mathcal{Z}_{4}(\Phi)$. The Chebyshev centers (16)
are obtained for each simulation and are shown by the gray points in Fig. 4.

Figure 3: The set of explaining systems is shown by the gray area. The red and
black squares show the Chebyshev center and the true system, respectively.
Figure 4: The gray points show the Chebyshev centers obtained for different
sequences of the noise. The black square shows the true system.

###  VI-B Example 2: Resistor-capacitor circuit

Consider the model of the resistor-capacitor (RC) circuit in Fig. 5,
discretized with a sample time of $1\ \mathrm{s}$, where the output variables
$y_{1}$, $y_{2}$, and $y_{3}$ are the voltages across the capacitors $C_{1}$,
$C_{2}$, and $C_{3}$, respectively, and the input variables $u_{1}$ and
$u_{2}$ are the voltage sources. The true parameters of the system are
$R_{1}=7\ \mathrm{\Omega}$, $R_{2}=5\ \mathrm{\Omega}$, $R_{3}=10\
\mathrm{\Omega}$, $R_{4}=15\ \mathrm{\Omega}$, $C_{1}=0.5\ \mathrm{F}$,
$C_{2}=0.4\ \mathrm{F}$, and $C_{3}=0.6\ \mathrm{F}$. We collect the input-
output data for a duration of $10$ seconds, starting from $y(0)=0$, where the
noise sequence is assumed to satisfy $W_{-}^{\top}\in\mathcal{Z}_{10}(\Phi)$
and $\Phi\in\boldsymbol{\Pi}_{3,10}$ with $\Phi_{11}=10^{-6}I_{3}$,
$\Phi_{22}=-I_{10}$, and $\Phi_{12}=\Phi_{21}^{\top}=0$. Let the signal-to-
noise ratio (SNR) between the input and the noise be defined as
$\sigma_{m}(U_{-})/\sigma_{1}(W_{-})$. We run $1000$ simulations per SNR for a
range of SNR from $1$ to $10^{3}$. For each simulation, uniformly distributed
random sequences of noise and input are generated and then normalized,
respectively, to attain the required noise bound $\sigma_{1}(W_{-})\leq
10^{-3}$ and the SNR value. Fig. 6 shows the Chebyshev radii with respect to
the nuclear, spectral, and Frobenius norms. The solid curves are the mean
values and the shaded areas indicate the standard deviation computed through
1000 simulations per SNR333MATLAB codes for the examples of this section can
be found on “https://github.com/a-shakouri/set-membership-system-
identification”..

$+$$-$$u_{1}$$R_{1}$$R_{2}$$R_{3}$$R_{4}$$+$$-$$u_{2}$$C_{1}$$C_{2}$$C_{3}$
Figure 5: RC circuit for Example 2. Figure 6: The Chebyshev radii with respect
to different norms. The curves indicate the mean values with the shaded area
indicating the standard deviation over 1000 simulations per SNR.

##  VII CONCLUSIONS

In this work, system identification using energy-bounded noise models has been
studied. We have formulated analytic expressions for a Chebyshev center, the
Chebyshev radius, and the diameter of the set of explaining systems with
respect to arbitrary unitarily invariant matrix norms. It has been shown that
the presented Chebyshev center is the only common center among all unitarily
invariant norms. Also, it has been shown that the solution to a particular
form of the generalized least-squares problem coincides with the common
Chebyshev center.

Extending the presented results to systems with unknown order is an
interesting topic for future work. A notion of the identification error that
has not been studied in this work is the volume of the set of explaining
systems, which can also be a subject for future studies. Moreover, future work
could focus on applications of the obtained results in experiment design for
system identification with prescribed accuracy. The results of this letter can
also be extended to stochastic noise models using the approach explained in
[15, Sec. 5.4].

## APPENDIX: PROOF OF PROPOSITION 1

(a) Observe that from the triangle inequality
$\tfrac{1}{2}|X_{1}-X_{2}|\leq\tfrac{1}{2}|C-X_{1}|+\tfrac{1}{2}|C-X_{2}|$.
Therefore, taking $C$ to be a Chebyshev center, we have
$\tfrac{1}{2}|X_{1}-X_{2}|\leq\operatorname{rad}\mathcal{X}$ for all
$X_{1},X_{2}\in\mathcal{X}$, which implies
$\tfrac{1}{2}\operatorname{diam}\mathcal{X}\leq\operatorname{rad}\mathcal{X}$.
The second inequality is obvious from the definitions (1) and (2).

(b) To prove the boundedness of $\operatorname{cent}\mathcal{X}$, we consider
the fact that for any $C\in\operatorname{cent}\mathcal{X}$ we have
$|C|\leq|C-X|+|X|\leq\operatorname{rad}\mathcal{X}+|X|$. Since $\mathcal{X}$
is bounded, we see that $\operatorname{cent}\mathcal{X}$ is bounded. To prove
closedness, consider a sequence $C_{i}\in\operatorname{cent}\mathcal{X}$,
$i\in\mathbb{N}$, with $\lim_{i\rightarrow\infty}C_{i}=C$. Let
$X\in\mathcal{X}$. Note that
$|C-X|\leq|C-C_{i}|+|C_{i}-X|\leq|C-C_{i}|+\operatorname{rad}\mathcal{X}$.
Take the limit as $i\rightarrow\infty$ to conclude that
$|C-X|\leq\operatorname{rad}\mathcal{X}$. Thus,
$C\in\operatorname{cent}\mathcal{X}$ and therefore
$\operatorname{cent}\mathcal{X}$ is closed. To prove convexity, suppose that
$C_{1},C_{2}\in\operatorname{cent}\mathcal{X}$. Then, by the triangle
inequality, we have the following for any $\alpha\in[0,1]$:

$\begin{split}&|\alpha C_{1}+(1-\alpha)C_{2}-X|\\\
&\leq\alpha|C_{1}-X|+(1-\alpha)|C_{2}-X|\leq\operatorname{rad}\mathcal{X}.\end{split}$
(38)

Therefore, for any $C_{1},C_{2}\in\operatorname{cent}\mathcal{X}$ and any
$\alpha\in[0,1]$ we have $\alpha
C_{1}+(1-\alpha)C_{2}\in\operatorname{cent}\mathcal{X}$, which implies
$\operatorname{cent}\mathcal{X}$ is convex.

(c) Suppose that $C_{1},C_{2}\in\operatorname{cent}\mathcal{X}$ and $|\cdot|$
is essentially strictly convex. Then, there exists an $X$ such that

$|\tfrac{1}{2}(C_{1}+C_{2})-X|=\operatorname{rad}\mathcal{X}.$ (39)

Moreover, from the triangle inequality

$|\tfrac{1}{2}(C_{1}+C_{2})-X|\leq\tfrac{1}{2}|C_{1}-X|+\tfrac{1}{2}|C_{2}-X|\leq\operatorname{rad}\mathcal{X}.$
(40)

Since $|C_{i}-X|\leq\operatorname{rad}\mathcal{X}$ for $i=1,2$, we conclude
that

$|C_{1}-X|=|C_{2}-X|=\operatorname{rad}\mathcal{X}.$ (41)

As the norm is essentially strictly convex, (41) implies that
$C_{1}-X=\alpha(C_{2}-X)$ for some $\alpha$. Since $|C_{1}-X|=|C_{2}-X|$, we
have $\alpha=1$, which implies $C_{1}=C_{2}$. ∎

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