# From Control to Mathematics–Part I: Controllability-Based Design for
Iterative Methods in Solving Linear Equations

Deyuan Meng, Senior Member, IEEE, and Yuxin Wu This work was supported by the
National Natural Science Foundation of China under Grant 61873013 and Grant
61922007.The authors are with the Seventh Research Division, Beihang
University (BUAA), Beijing 100191, P. R. China, and also with the School of
Automation Science and Electrical Engineering, Beihang University (BUAA),
Beijing 100191, P. R. China (e-mail: dymeng@buaa.edu.cn).

###### Abstract

In the interaction between control and mathematics, mathematical tools are
fundamental for all the control methods, but it is unclear how control impacts
mathematics. This is the first part of our paper that attempts to give an
answer with focus on solving linear algebraic equations (LAEs) from the
perspective of systems and control, where it mainly introduces the
controllability-based design results. By proposing an iterative method that
integrates a learning control mechanism, a class of tracking problems for
iterative learning control (ILC) is explored for the problem solving of LAEs.
A trackability property of ILC is newly developed, by which analysis and
synthesis results are established to disclose the equivalence between the
solvability of LAEs and the controllability of discrete control systems.
Hence, LAEs can be solved by equivalently achieving the perfect tracking tasks
of resulting ILC systems via the classic state feedback-based design and
analysis methods. It is shown that the solutions for any solvable LAE can all
be calculated with different selections of the initial input. Moreover, the
presented ILC method is applicable to determining all the least squares
solutions of any unsolvable LAE. In particular, a deadbeat design is
incorporated to ILC such that the solving of LAEs can be completed within
finite iteration steps. The trackability property is also generalized to
conventional two-dimensional ILC systems, which creates feedback-based
methods, instead of the common used contraction mapping-based methods, for the
design and convergence analysis of ILC.

###### Index Terms:

Controllability, iterative method, learning control, linear algebraic
equation, solvability, trackability.

##  I Introduction

Solving linear algebraic equations (LAEs) has been one of the most
fundamentally significant problems in science and engineering [1, 2]. It has
caught attention in lots of areas since many practical and complex problems
can ultimately boil down to the solving of LAEs, such as forecast, estimation,
and approximation of nonlinear systems and modelling of physical systems. In
general, there are two basic categories of methods for solving LAEs, that is,
direct methods and iterative methods. The direct methods are traditionally
adopted for solving LAEs, which leverage the Gaussian elimination and its
enhancements to directly calculate the exact solutions of LAEs with a process
of finite steps. Though the direct methods can lead to the exact solutions of
LAEs, it is lack of robustness and requires a large amount of computation and
storage, especially for LAEs with higher dimensions. The iterative methods
mainly make use of the successive iterations to update the approximate
solutions of LAEs such that the exact solutions of them can be determined when
the convergence of iterative process is achieved. Thanks to that the iterative
methods only need to perform the addition and multiplication operations over a
few matrices and vectors, they are generally preferred in practical
applications [4, 3].

The classic iterative methods for solving LAEs make certain use of the
feedback mechanism. However, they ignore the use of “control design” in
feedback, due to which the convergence rate, or even the convergence, for them
may not be guaranteed. What will emerge if we can incorporate the idea of
control into the iterative methods, and can this make them more effective in
solving LAEs? If the answers to them are affirmative, how can we reasonably
integrate control into the iterative methods? For this key problem, a
promising control-theoretic design method is explored in [5, 6], which not
only presents continuous-time and discrete-time algorithms but also is
effective, regardless of linear or nonlinear LAEs. In [7, 8], the design
methods from optimal control and robust control are successfully introduced to
solve LAEs, which yield solution algorithms with the global convergence or
tunable convergence properties. Recently, there have been reported insightful
consensus-based network control methods for solving LAEs, together with
providing distributed algorithms based on local information exchange (see,
e.g., [9, 10, 11, 12]). It has been revealed in [5, 6, 7, 8, 9, 10, 11, 12]
that the introduction of control design into iterative methods can make them
to possess better performances in the crucial aspects of, e.g., complexity,
convergence, and robustness, and can also provide a systematic way to design
solution algorithms to LAEs.

However, the use of control methods to better performances of iterative
methods is less developed especially in comparison with the popular
application of iterative methods in the control area (see also [13] for
similar discussions made on the relation between control design and numerical
analysis). Though some attempts are devoted to concerning this issue in, e.g.,
[5, 6, 7, 8, 9, 10, 11, 12], these existing results are mainly aimed at
solvable LAEs (with a unique or at least one solution). It is required to
further study whether and how the control design ideas can be leveraged to
determine all solutions (or least squares solutions) for solvable (or
unsolvable) LAEs. More fundamentally, are there inherent relationships between
the solvability or unsolvability problems for LAEs and the basic properties
for control systems, such as controllability, reachability, or
stabilizability? Furthermore, in case of a positive answer to this question,
how can the inherent relationships be reasonably disclosed? To our knowledge,
these problems still remain open in the interaction between iterative methods
for solving LAEs and control design methods, which will be addressed in this
paper. Our approach is to incorporate the strategies of iterative learning
control (ILC) in establishing effective solving methods for LAEs, in which a
salient idea of learning from experience can be leveraged to connect iterative
methods with control systems.

As a class of intelligent control methods, ILC contributes to realizing the
“perfect output tracking” of any desired reference, which is accomplished by
learning information from previous iterations and iteratively updating the
control input signal (see, e.g., [14, 15, 16]). Once the convergence for the
iterative process is ensured, together with decreasing the tracking error to
zero, the output tracking objective for any specified ILC system can be
achieved perfectly over a fixed interval from the beginning to the end. It has
been reported in the literature (see, e.g., [17, 18]) that ILC can be enabled
to improve the transient response performances of the controlled systems, and
be implemented in an off-line manner based on the input and output data,
together with using quite limited model knowledge. Simultaneously, the update
of the control input for ILC arrives at the approximation to the desired input
for the desired reference, which is actually consistent with the exploration
of iterative methods in realizing the approximation to the solution for an
LAE. This consistency result brings the possibility to solve LAEs from the
perspective of control analysis and synthesis based on constructing certain
ILC systems, which however has not been investigated in the literature to the
best of our knowledge.

With appropriate design of iterative learning controllers, the convergence
performance of iterative processes may be greatly enhanced. The design and
analysis of ILC mainly resorts to the contraction mapping method, rendering
ILC distinct especially from the popular feedback-based control methods and
separate from them simultaneously, e.g., see [19, 20, 21, 22, 23, 24, 25, 26,
27]. It is known that the controllability is considered as a fundamentally
significant property in most popular control methods, whereas it is seldom
used in ILC, and instead, the system relative degree is regarded as one of the
essential conditions in ILC. Are basic properties, such as reachability,
controllability, or stabilizability, of control systems really not significant
for ILC? The answer is negative. For the design and analysis of ILC, a
realizability hypothesis is usually imposed, which embeds the controllability
information of ILC systems (see, e.g., [28]). However, where and how do these
properties take effect in ILC? Furthermore, how can we benefit from them to
proceed the studies of ILC and establish a close connection of ILC to the
popular state feedback-based control methods? These questions still remain
unanswered.

In this paper, we provide a viewpoint of systems and control to solve LAEs by
constructing a framework of ILC, for which a fundamental trackability property
of ILC is newly introduced by requiring only the existence of some desired
inputs for ILC systems in generating the desired reference. It helps to
develop the equivalent relation between the solvability of LAEs and the
controllability of control systems, as shown in Fig. 1. Further, we can arrive
at an ILC algorithm to calculate the solutions of LAEs by leveraging the state
feedback-based control methods. In addition, we explore the trackability
property for traditional two-dimensional (2-D) ILC systems, and thus can
benefit from the controllability and state feedback-based design methods to
implement the perfect tracking tasks of ILC. Specifically, three main
contributions of this paper are summarized as follows.

  1. 1.

We can incorporate the state feedback-based control into the solving of LAEs.
It is thanks to the transformation of this problem to a tracking problem of
ILC, which can be further disclosed with a connection to the controllability
property. Thus, we can give an algorithm to determine all solutions
(respectively, all least squares solutions) of any solvable (respectively,
unsolvable) LAE by the selections of different initial inputs. By comparison,
we may extend the existing control-theoretic iterative methods given in, e.g.,
[5, 6, 7, 8, 9, 10, 11, 12] by incorporating the idea of ILC to derive all
(least squares) solutions for LAEs.

  2. 2.

We propose a design method to improve the convergence rate of the iterative
processes that result from the solving of LAEs. By integrating the idea of
deadbeat control into the design of ILC, we can enable the convergence of the
iterative processes to be accomplished in finite iteration steps. This not
only overcomes the shortcoming of slow convergence in the classic iterative
methods for solving LAEs, but also preserves their advantages of robustness
and simple calculation. It thus renders iterative methods more practical for
determining the solutions to LAEs.

  3. 3.

We disclose that the realizability hypothesis, required in many ILC literature
(see, e.g., [29, 30, 31, 32]), is unnecessary for the perfect tracking of ILC.
For any trackable desired reference, there generally allow multiple desired
inputs that generate this desired reference. We also extend the trackability
property to the traditional 2-D ILC systems, and hence the state feedback-
based design and analysis methods are effective in implementing the perfect
output tracking tasks for them, which narrows the gap between ILC and the
classic feedback-based control methods.

As a dual result of this paper, observability-based design results for solving
LAEs will be introduced in [33].

Figure 1: The equivalent relationship between the solvability of LAEs and the
controllability of control systems built via the trackability of ILC.

The rest of this paper is organized as follows. We establish a close
connection between the solving problem of LAEs and the tracking problem of ILC
in Section II. In Sections III and IV, the trackability and the
controllability of ILC are introduced, respectively, and their relation is
further disclosed and used to realize the perfect tracking of ILC based on the
state feedback-based design and analysis methods in Section V. In Section VI,
we incorporate the idea of deadbeat control into the design for ILC such that
we can develop the convergence for ILC within finite iterations. In Section
VII, we present two implementation algorithms of ILC for solving LAEs,
regardless of whether the LAEs are solvable. As an application, we apply the
trackability property to traditional 2-D ILC systems and achieve the perfect
tracking tasks for them with the state feedback-based methods in Section VIII.
We make concluding remarks in Section IX.

Notations: Let $\mathbb{Z}_{+}\triangleq\\{0,1,2,\cdots\\}$,
$\mathbb{Z}_{N}\triangleq\\{0,1,\cdots,N\\}$, and $I$ be an identity matrix
with required dimensions. For any vector $x\in\mathbb{R}^{n}$,
$\left\|x\right\|$ is its any norm, and $\left\|x\right\|_{2}$ and
$\left\|x\right\|_{\infty}$ are particularly the Euclidean and infinite norms
of $x$, respectively. Let $\rho(A)$ be the spectral radius of any matrix
$A\in\mathbb{R}^{n\times n}$. For any linear space $X\subseteq\mathbb{R}^{n}$,
$\dim(X)$ denotes its dimension. Let $X^{\bot}$ be the orthogonal complement
subspace of any linear subspace $X$ in $\mathbb{R}^{n}$, and thus the direct
sum of them is the entire space, which is denoted by $\mathbb{R}^{n}=X\oplus
X^{\bot}$. For any vector $z_{k}\in\mathbb{R}^{n}$ changing with respect to an
iteration axis that is denoted by $k\in\mathbb{Z}_{+}$, let
$\Delta:z_{k}\to\Delta z_{k}\triangleq z_{k+1}-z_{k}$ represent a forward
iteration operator.

##  II Problem-Solving of LAEs Via ILC Tracking

Let any desired reference $\bm{Y}_{d}\in\mathbb{R}^{p}$ be specified. The
problem of interest in this paper is to solve an LAE expressed by

$\bm{Y}_{d}=\bm{G}\bm{U}_{d}$ (1)

to obtain the solution $\bm{U}_{d}\in\mathbb{R}^{q}$, where
$\bm{G}\in\mathbb{R}^{p\times q}$ is the transfer or mapping matrix. If the
LAE (1) has solutions, then it is said to be solvable. Generally, there exist
multiple solutions for the LAE (1) when it is solvable. We thus need to not
only calculate a certain solution but also determine an analytical formulation
of the multiple solutions for the LAE (1).

We target at addressing the solving problem of the LAE (1) from the
perspective of ILC. Toward this end, we duplicate the LAE (1) in an iterative
manner to arrive at a process, described with respect to the iteration index
$k\in\mathbb{Z}_{+}$, as

$\bm{Y}_{k}=\bm{G}\bm{U}_{k},\quad\forall k\in\mathbb{Z}_{+}$ (2)

where we call $\bm{Y}_{k}\in\mathbb{R}^{p}$ and $\bm{U}_{k}\in\mathbb{R}^{q}$
the output and the input of (2), respectively. Since (2) can be employed for
the description of an ILC system implicitly involving an independent discrete-
time dynamics (see, e.g., [14, 15, 34] for more discussions), it is also
directly called a system for clarity, in accordance with which $\bm{U}_{d}$ is
called the desired input for generating the desired reference $\bm{Y}_{d}$.

With the construction of the system (2), the solving problem of the LAE (1)
can be interpreted as an ILC tracking problem.

Problem Statement: For the problem-solving of the LAE (1), our objective is to
design an input sequence $\left\\{\bm{U}_{k}:k\in\mathbb{Z}_{+}\right\\}$ for
the system (2), with any initial input $\bm{U}_{0}$ that can be arbitrarily
selected, such that the resulting output sequence
$\left\\{\bm{Y}_{k}:k\in\mathbb{Z}_{+}\right\\}$ can approach the desired
reference $\bm{Y}_{d}$ as $k\to\infty$, namely,

$\lim_{k\to\infty}\bm{Y}_{k}=\bm{Y}_{d}$ (3)

and we can also learn a desired input $\bm{U}_{d}$ through the design of
$\bm{U}_{k}$, $\forall k\in\mathbb{Z}_{+}$, and identify the unified
properties of all possible desired inputs simultaneously after the ILC
process.

To address the abovementioned tracking problem in ILC, we denote the tracking
error as $\bm{E}_{k}=\bm{Y}_{d}-\bm{Y}_{k}$, $\forall k\in\mathbb{Z}_{+}$.
Obviously, (3) holds if and only if $\lim_{k\to\infty}\bm{E}_{k}=0$. Let the
output deviation and input deviation between two sequential iterations for (2)
be denoted as $\Delta\bm{Y}_{k}=\bm{Y}_{k+1}-\bm{Y}_{k}$, $\forall
k\in\mathbb{Z}_{+}$ and $\Delta\bm{U}_{k}=\bm{U}_{k+1}-\bm{U}_{k}$, $\forall
k\in\mathbb{Z}_{+}$, respectively. Then the use of (2) yields

$\bm{E}_{k+1}=\bm{E}_{k}-\Delta\bm{Y}_{k}=\bm{E}_{k}+\bm{G}\bm{\Psi}_{k},\quad\forall
k\in\mathbb{Z}_{+}$ (4)

which describes a dynamic system evolving along the iteration axis $k$ and
having a control input given by

$\bm{\Psi}_{k}=-\Delta\bm{U}_{k},\quad\forall k\in\mathbb{Z}_{+}.$ (5)

For distinction from the notations adopted in the system (2), we present the
following definition to introduce some notations of the linear discrete
control system (4).

###### Definition 1

For the system (4), $\bm{E}_{k}$ and $\bm{\Psi}_{k}$ are called the $k$-state
and the $k$-input, respectively. This system, controlled with some certain
$k$-state feedbacks, is said to be $k$-stable if, for any initial $k$-state
condition, $\lim_{k\to\infty}\bm{E}_{k}=0$ can be accomplished.

Based on Definition 1, the tracking objective (3) is achieved for the system
(2) if and only if the $k$-stability is accomplished for the system (4). As a
consequence of this equivalence result, the design of the input $\bm{U}_{k}$
for the system (2) to get the tracking objective (3) can be transformed into
designing the $k$-input $\bm{\Psi}_{k}$ for the system (4) to realize the
$k$-stability. It can be easily seen that the $k$-state is accessible for the
feedback controller design of the system (4). This renders the classic state
feedback-based design tools in the Kalman state-space framework available for
the ILC design of the system (2) in the presence of the output tracking task
(3), and consequently available for the problem-solving of the LAE (1).
Motivated by such an observation, we will further develop a trackability
property for the system (2), which matches exactly with the solvability of the
LAE (1), and disclose the close relation between them and the controllability
of the system (4) (see also Fig. 1) such that the solutions of the LAE (1) can
be obtained by a $k$-state feedback-based method.

##  III Trackability in ILC

In this section, we propose a fundamental class of trackability problems for
the desired references in realizing the tracking tasks of ILC. We introduce
the basic trackability properties and criteria, and disclose tight relations
of them with the controlled systems of ILC at the same time.

###  III-A Trackability and Trackability Subspace

Typically, existing ILC results aim at how to design updating laws of inputs
to achieve the output tracking objectives and at how to analyze the
convergence performances of their resulting ILC processes. However, it is lack
of investigations for ILC to explore whether the desired references are
trackable with some certain inputs. To address this crucial problem, the
realizability assumption is often made directly for the desired references in
the ILC literature, but its reasonability has not been developed. For the sake
of our discussions on these fundamental problems of ILC, we introduce the
following concepts of trackability and realizability for the desired
references in ILC, and also present the related properties of the
corresponding ILC systems.

###### Definition 2

A desired reference $\bm{Y}_{d}$ is said to be trackable (respectively,
realizable) for the system (2) if there exists some (respectively, a unique)
desired input $\bm{U}_{d}$ fulfilling the LAE (1).

###### Definition 3

The system (2) is said to have the trackability property (respectively, the
realizability property) if any desired reference $\bm{Y}_{d}\in\mathbb{R}^{p}$
is trackable (respectively, realizable) for the system (2).

With Definitions 2 and 3, we connect the solvability problem of the LAE (1)
tightly to two basic problems of the trackability and the realizability in ILC
for the system (2).

###### Remark 1

From Definition 2, it is clear that the trackability and realizability
represent different concepts for characterizing the output tracking tasks of
ILC systems. The realizability of a desired reference requires the uniqueness
of the corresponding desired input for the controlled system, but by
comparison, the trackability does not. In the literature of ILC, the
realizability is usually adopted as a basic assumption, especially for under-
actuated systems with less number of the inputs than that of the outputs (see,
e.g., [30, 31, 32]). Despite this fact, the trackability provides a more
general property than the realizability for ILC by removing the uniqueness
requirement, through which more fundamentally important problems in ILC may be
explored.

###### Remark 2

From Definition 3, we know that an ILC system has the trackability
(respectively, realizability) property if any desired reference is trackable
(respectively, realizable). We can clearly see that these two properties for
an ILC system are two inherent characteristics independent from the selections
of the desired references. When the ILC system dose not possess the
trackability property, only a portion of the desired references may be
trackable, for which the corresponding desired inputs are available; and
otherwise, it is impossible to determine any desired input by iterations.
Thus, it is a crucial issue to identify whether the given desired reference is
trackable before seeking control inputs for the system (2).

It is worth highlighting, however, that few results of ILC are devoted to
exploring whether any desired reference of interest is trackable for the
system (2). If not, what properties do the trackable desired references have,
and how do they connect to the characteristics of the system (2)? Furthermore,
whether and how could some easy-to-check criteria be developed to validate the
trackability of the desired references of interest? To address these
fundamental problems, we focus on exploiting the output trackability of ILC
for the system (2) during accomplishing the tracking objective (3) in the
presence of any desired reference, and on characterizing the roles of the
trackability of the desired references in performing the tracking tasks of
ILC.

In Definition 2, the satisfaction of the LAE (1) can guarantee that the linear
combination of two trackable desired references for the system (2) is also
trackable. This evidently hints that all trackable desired references for the
system (2) span a subspace of $\mathbb{R}^{p}$. Let us denote this subspace as
$\mathcal{Y}_{T}\subseteq\mathbb{R}^{p}$ that is directly called the
trackability subspace of the system (2) for simplicity and adopt
$\bm{Y}_{d}\in\mathcal{Y}_{T}$ to denote any trackable $\bm{Y}_{d}$ in
$\mathbb{R}^{p}$. It clearly follows from Definition 3 that
$\mathcal{Y}_{T}=\mathbb{R}^{p}$ holds if and only if the system (2) has the
trackability property. However, by contrast, similar properties can not be
analogously shown for the system (2) when it is concerned with the realizable
desired references owing to the requirement of uniqueness. This problem
depends heavily upon the column independence of the matrix $\bm{G}$, which
will be addressed with details in the following subsection.

###  III-B Trackability Criteria

Let $\mathrm{span}\bm{G}$ be the space that is spanned by the columns of
$\bm{G}$. Let us also denote $\dim\left(\mathrm{span}\bm{G}\right)=m$. Then we
represent a set of basis vectors for $\mathrm{span}\bm{G}$, defined by
$\left\\{h_{i}\in\mathbb{R}^{p}:i=1,2,\cdots,m\right\\}$, with which we denote
$\bm{H}_{1}=\left[h_{1},h_{2},\cdots,h_{m}\right]\in\mathbb{R}^{p\times m}$.
Moreover, we choose some matrix
$\bm{H}_{2}=\left[h_{m+1},h_{m+2},\cdots,h_{p}\right]\in\mathbb{R}^{p\times(p-m)}$
such that $\bm{H}=\left[\bm{H}_{1}~{}\bm{H}_{2}\right]\in\mathbb{R}^{p\times
p}$ is nonsingular. Correspondingly, we denote $\bm{H}^{-1}=\bm{F}$, and let
$\bm{F}$ be in a structured form of

$\bm{F}=\begin{bmatrix}\bm{F}_{1}^{\mathrm{T}}\\\
\bm{F}_{2}^{\mathrm{T}}\end{bmatrix}~{}\hbox{with}~{}\bm{F}_{1}\in\mathbb{R}^{p\times
m},\bm{F}_{2}\in\mathbb{R}^{p\times(p-m)}.$

Clearly, $\bm{H}_{1}$, $\bm{H}_{2}$, $\bm{F}_{1}$, and $\bm{F}_{2}$ are full-
column rank matrices. These denotations also help develop a useful lemma as
follows.

###### Lemma 1

For any matrix $\bm{G}$, the following statements hold:

  1. 1.

$\left(\mathrm{span}\bm{G}\right)^{\bot}=\mathrm{span}\bm{F}_{2}$ is such that
$\mathrm{span}\bm{G}\oplus\mathrm{span}\bm{F}_{2}=\mathbb{R}^{p}$;

  2. 2.

$\left(\bm{H}_{1}\bm{F}_{1}^{\mathrm{T}}\right)^{2}=\bm{H}_{1}\bm{F}_{1}^{\mathrm{T}}$
and
$\left(\bm{H}_{2}\bm{F}_{2}^{\mathrm{T}}\right)^{2}=\bm{H}_{2}\bm{F}_{2}^{\mathrm{T}}$
are idempotent matrices such that
$\bm{H}_{1}\bm{F}_{1}^{\mathrm{T}}+\bm{H}_{2}\bm{F}_{2}^{\mathrm{T}}=I$.

###### Proof:

“1):” From $\bm{F}\bm{H}=I$, $\bm{F}_{2}^{\mathrm{T}}\bm{H}_{1}=0$ follows.
Since the columns of $\bm{H}_{1}$ form a set of basis vectors for
$\mathrm{span}\bm{G}$, we know $\mathrm{span}\bm{G}=\mathrm{span}\bm{H}_{1}$.
From these two facts, we can obtain

$\bm{\vartheta}^{\mathrm{T}}\bm{\theta}=0,\quad\forall\bm{\theta}\in\mathrm{span}\bm{G},\forall\bm{\vartheta}\in\mathrm{span}\bm{F}_{2}.$
(6)

As a consequence of (6), we can arrive at
$\left(\mathrm{span}\bm{G}\right)^{\bot}=\mathrm{span}\bm{F}_{2}$, and then
$\mathrm{span}\bm{G}\oplus\mathrm{span}\bm{F}_{2}=\mathbb{R}^{p}$ by
considering $\dim\left(\mathrm{span}\bm{G}\right)=m$ and
$\dim\left(\mathrm{span}\bm{F}_{2}\right)=p-m$.

“2):” From $\bm{F}\bm{H}=I$, it is direct to get
$\bm{F}_{1}^{\mathrm{T}}\bm{H}_{1}=I$ and
$\bm{F}_{2}^{\mathrm{T}}\bm{H}_{2}=I$. This leads to that
$\left(\bm{H}_{1}\bm{F}_{1}^{\mathrm{T}}\right)^{2}=\bm{H}_{1}\bm{F}_{1}^{\mathrm{T}}$
and
$\left(\bm{H}_{2}\bm{F}_{2}^{\mathrm{T}}\right)^{2}=\bm{H}_{2}\bm{F}_{2}^{\mathrm{T}}$
are idempotent. Moreover,
$\bm{H}_{1}\bm{F}_{1}^{\mathrm{T}}+\bm{H}_{2}\bm{F}_{2}^{\mathrm{T}}=I$ is a
straightforward consequence of $\bm{H}\bm{F}=I$. ∎

To proceed with Lemma 1, we establish a trackability result of any desired
reference in ILC.

###### Lemma 2

For any desired reference $\bm{Y}_{d}\in\mathbb{R}^{p}$ of the system (2), the
following statements are equivalent:

  1. 1.

$\bm{Y}_{d}\in\mathcal{Y}_{T}$;

  2. 2.

$\bm{Y}_{d}\in\mathrm{span}\bm{G}$;

  3. 3.

$\bm{F}_{2}^{\mathrm{T}}\bm{Y}_{d}=0$;

  4. 4.

$\bm{H}_{1}\bm{F}_{1}^{\mathrm{T}}\bm{Y}_{d}=\bm{Y}_{d}$.

###### Proof:

We prove the equivalences among four statements in a circular manner.

“1)$\Rightarrow$2):” If $\bm{Y}_{d}\in\mathcal{Y}_{T}$, then the LAE (1) holds
for some $\bm{U}_{d}$ from Definition 2. Clearly, (1) implies
$\bm{Y}_{d}\in\mathrm{span}\bm{G}$.

“2)$\Rightarrow$3):” With
$\left(\mathrm{span}\bm{G}\right)^{\bot}=\mathrm{span}\bm{F}_{2}$ in Lemma 1,
$\bm{F}_{2}^{\mathrm{T}}\bm{Y}_{d}=0$ is an immediate result of
$\bm{Y}_{d}\in\mathrm{span}\bm{G}$.

“3)$\Rightarrow$4):” By
$\bm{H}_{1}\bm{F}_{1}^{\mathrm{T}}+\bm{H}_{2}\bm{F}_{2}^{\mathrm{T}}=I$ in
Lemma 1 together with $\bm{F}_{2}^{\mathrm{T}}\bm{Y}_{d}=0$, we can deduce

$\bm{H}_{1}\bm{F}_{1}^{\mathrm{T}}\bm{Y}_{d}=\left(I-\bm{H}_{2}\bm{F}_{2}^{\mathrm{T}}\right)\bm{Y}_{d}=\bm{Y}_{d}.$

“4)$\Rightarrow$1):” Owing to $\mathrm{span}\bm{G}=\mathrm{span}\bm{H}_{1}$,
$\bm{H}_{1}=\bm{G}\bm{H}_{3}$ holds for some $\bm{H}_{3}\in\mathbb{R}^{q\times
m}$. We thus define $\bm{U}_{d}=\bm{H}_{3}\bm{F}_{1}^{\mathrm{T}}\bm{Y}_{d}$
and it holds

$\bm{Y}_{d}=\bm{H}_{1}\bm{F}_{1}^{\mathrm{T}}\bm{Y}_{d}=\bm{G}\left(\bm{H}_{3}\bm{F}_{1}^{\mathrm{T}}\bm{Y}_{d}\right)=\bm{G}\bm{U}_{d}$

namely, the LAE (1) holds. According to Definition 2,
$\bm{Y}_{d}\in\mathcal{Y}_{T}$ follows immediately. ∎

By moving Lemma 2 a bit further, we present a realizability result of any
desired reference in ILC.

###### Lemma 3

Consider the system (2) with any $\bm{Y}_{d}\in\mathcal{Y}_{T}$. Then
$\bm{Y}_{d}$ is realizable if and only if
$\mathrm{rank}\left(\bm{G}\right)=q$.

###### Proof:

For $\bm{Y}_{d}\in\mathcal{Y}_{T}$, let $\bm{U}_{d}^{1}$ and $\bm{U}_{d}^{2}$
be two desired inputs that both fulfill the LAE (1), namely,
$\bm{Y}_{d}=\bm{G}\bm{U}_{d}^{1}=\bm{G}\bm{U}_{d}^{2}$. This obviously leads
to $\bm{G}\left(\bm{U}_{d}^{1}-\bm{U}_{d}^{2}\right)=0$. Since
$\bm{U}_{d}^{1}$ and $\bm{U}_{d}^{2}$ denote any two desired inputs satisfying
(1), we can validate

$\bm{U}_{d}^{1}-\bm{U}_{d}^{2}=0~{}\Leftrightarrow~{}\mathrm{rank}\left(\bm{G}\right)=q$

which, together with Definition 2, implies that $\bm{Y}_{d}$ is realizable if
and only if $\mathrm{rank}\left(\bm{G}\right)=q$. ∎

With Lemma 3, we reveal that the realizability of the desired references in
ILC of the system (2) requires the matrix $\bm{G}$ being of full-column rank.
Under the satisfaction of this requirement, we can obtain that the linear
combination of any two realizable desired references for the system (2) is
also realizable. Thus, it follows that all realizable desired references for
the system (2) span a subspace of $\mathbb{R}^{p}$, denoted by
$\mathcal{Y}_{R}\subseteq\mathbb{R}^{p}$. We directly call $\mathcal{Y}_{R}$
the realizability subspace of the system (2) for convenience and use
$\bm{Y}_{d}\in\mathcal{Y}_{R}$ to denote any realizable $\bm{Y}_{d}$ in
$\mathbb{R}^{p}$. By Lemma 3, we particularly have
$\mathcal{Y}_{R}\subseteq\mathcal{Y}_{T}\subseteq\mathbb{R}^{p}$.

Based on Lemmas 2 and 3, the following theorem introduces basic trackability
and realizability properties in ILC.

###### Theorem 1

For the system (2), the following two properties can be developed.

  1. 1.

Trackability: $\mathcal{Y}_{T}=\mathrm{span}\bm{G}$ holds, and in particular,
$\mathcal{Y}_{T}=\mathbb{R}^{p}$ if and only if
$\mathrm{rank}\left(\bm{G}\right)=p$.

  2. 2.

Realizability: Either $\mathcal{Y}_{R}=\mathcal{Y}_{T}$ or
$\mathcal{Y}_{R}=\\{0\\}$ holds, where $\mathcal{Y}_{R}=\mathcal{Y}_{T}$ if
and only if $\mathrm{rank}\left(\bm{G}\right)=q$.

###### Proof:

“1):” From the equivalent results between 1) and 2) in Lemma 2,
$\mathcal{Y}_{T}=\mathrm{span}\bm{G}$ follows directly. We, in particular, can
arrive at

$\mathcal{Y}_{T}=\mathbb{R}^{p}\Leftrightarrow~{}\mathrm{span}\bm{G}=\mathbb{R}^{p}\Leftrightarrow~{}\dim\left(\mathrm{span}\bm{G}\right)=p\Leftrightarrow~{}\mathrm{rank}\left(\bm{G}\right)=p.$

Thus, the trackability result 1) is obtained.

“2):” With Lemma 3, we can verify that $\mathcal{Y}_{R}=\mathcal{Y}_{T}$ if
and only if $\mathrm{rank}\left(\bm{G}\right)=q$. Otherwise, let
$\mathrm{rank}\left(\bm{G}\right)\neq q$. Then
$\mathrm{rank}\left(\bm{G}\right)=m<q$ holds, and for the null space
$\mathrm{null}\bm{G}$ of $\bm{G}$, i.e.,

$\mathrm{null}\bm{G}=\left\\{\bm{\phi}\in\mathbb{R}^{q}\big{|}\bm{G}\bm{\phi}=0\right\\}$

we have
$\mathrm{null}\bm{G}=\left(\mathrm{span}\bm{G}^{\mathrm{T}}\right)^{\bot}$,
and as a consequence,

$\dim\left(\mathrm{null}\bm{G}\right)=q-\dim\left(\mathrm{span}\bm{G}^{\mathrm{T}}\right)=q-m>0.$
(7)

With (7), there exist more than one desired inputs that generate every nonzero
trackable desired reference based on Theorem 2 of [35, Subchapter 3.10].
Namely, for every nonzero $\bm{Y}_{d}\in\mathcal{Y}_{T}$, we have
$\bm{Y}_{d}\not\in\mathcal{Y}_{R}$, and thus $\mathcal{Y}_{R}=\\{0\\}$ holds
owing to $\mathcal{Y}_{R}\subseteq\mathcal{Y}_{T}$. The realizability result
2) is developed. ∎

###### Remark 3

By $\mathcal{Y}_{T}=\mathrm{span}\bm{G}$, it discloses that the trackability
subspace of the system (2) is exactly the spanning space for the columns of
$\bm{G}$. This, however, is generally no longer applicable for the
realizability subspace $\mathcal{Y}_{R}$ of the system (2), which only works
(namely, $\mathcal{Y}_{R}=\mathrm{span}\bm{G}$) if and only if $\bm{G}$ is of
full-column rank. Otherwise, $\mathcal{Y}_{R}=\\{0\\}$ emerges, namely, there
do not exist any nonzero realizable desired references. Therefore, we reveal
with Theorem 1 that the trackability for ILC systems is a much more available
property than the realizability. Simultaneously, for the system (2), the
trackability and realizability properties can also be determined according to
Theorem 1. To be specific, the system (2) has the trackability (respectively,
realizability) property if and only if $\mathrm{rank}\left(\bm{G}\right)=p$
holds (respectively, $\bm{G}$ is an invertible square matrix).

##  IV Controllability in ILC

In this section, we introduce the fundamental controllability property and its
related problems into the tracking tasks of ILC, which helps to bring a
viewpoint of addressing the tracking problems of ILC, and thus of dealing with
the problem-solving of LAEs, from the classic theories and methods of systems
and control. We develop the ideas for controllability, controllability
subspace, controllability criteria, and controllability decomposition of the
linear discrete systems (see, e.g., [36, Chapter 3]) to ILC systems.

###  IV-A Controllability and Controllability Subspace

Next, we turn to explore the tracking error system (4) that is formulated in
the state-space form of a linear discrete system. To this end, let us
introduce two concepts for the controllability of the linear discrete systems
(see, e.g., [36, Chapter 3]).

###### Definition 4

For the system (4), a nonzero $k$-state $\bm{\beta}$ in $\mathbb{R}^{p}$ is
said to be controllable if, for some finite $l\in\mathbb{Z}_{+}$, there exists
some $k$-input sequence $\bm{\Psi}_{k}$, $\forall k\in\mathbb{Z}_{l-1}$ that
transfers the $k$-state from $\bm{\beta}$ to the origin at the $l$th
iteration.

###### Definition 5

For the system (4), if all, some but not all, and none of the nonzero
$k$-states in $\mathbb{R}^{p}$ are controllable, respectively, then (4) is
said to be completely controllable, incompletely controllable, and completely
uncontrollable, respectively.

With Definition 4, we can validate that all the controllable $k$-states of the
system (4), together with the null vector, constitute a subspace of
$\mathbb{R}^{p}$. We call it the controllability subspace of the system (4),
which is denoted by $\mathcal{C}_{C}$. It follows from Definition 5 that
$\mathcal{C}_{C}=\mathbb{R}^{p}$ holds if and only if the system (4) is
completely controllable. For clarity, let
$\mathcal{C}_{NC}=\mathcal{C}_{C}^{\bot}$ be the uncontrollability subspace of
the system (4). We thus have
$\mathcal{C}_{C}\oplus\mathcal{C}_{NC}=\mathbb{R}^{p}$, and can develop
specific properties of both $\mathcal{C}_{C}$ and $\mathcal{C}_{NC}$ as
follows.

###### Theorem 2

For the system (4), the following controllability properties hold:

  1. 1.

$\mathcal{C}_{NC}=\mathrm{null}\bm{G}^{\mathrm{T}}$;

  2. 2.

$\mathcal{C}_{C}=\mathrm{span}\bm{G}$.

###### Proof:

“1):” For any controllable $k$-state $\bm{\beta}\in\mathcal{C}_{C}$, there
exist some $l\in\mathbb{Z}_{+}$ and the corresponding input
$\left\\{\bm{\Psi}_{k}:\forall k\in\mathbb{Z}_{l-1}\right\\}$ such that the
solution of the system (4) starting with $\bm{\beta}$ arrives at the origin
after $l$ iterations, namely,

$0=\bm{E}_{l}=\bm{E}_{0}+\sum_{k=0}^{l-1}\bm{G}\bm{\Psi}_{k}=\bm{\beta}+\sum_{k=0}^{l-1}\bm{G}\bm{\Psi}_{k}.$

This implies that $\bm{\beta}\in\mathcal{C}_{C}$ is a controllable $k$-state
for the system (4) if and only if it can be described in the form of

$\bm{\beta}=-\sum_{k=0}^{l-1}\bm{G}\bm{\Psi}_{k},\quad\forall\bm{\beta}\in\mathcal{C}_{C}.$
(8)

“$\mathcal{C}_{NC}\subseteq\mathrm{null}\bm{G}^{\mathrm{T}}$:” For any
$\bm{\alpha}\in\mathcal{C}_{NC}$, we can obtain
$\bm{\alpha}^{\mathrm{T}}\bm{\beta}=0$, $\forall\bm{\beta}\in\mathcal{C}_{C}$.
By specifically considering the controllable $k$-state generated by
$\bm{\Psi}_{k}=-l^{-1}\bm{G}^{\mathrm{T}}\bm{\alpha}$, we can resort to (8) to
derive

$0=\bm{\alpha}^{\mathrm{T}}\bm{\beta}=-\bm{\alpha}^{\mathrm{T}}\sum_{k=0}^{l-1}\bm{G}\bm{\Psi}_{k}=\bm{\alpha}^{\mathrm{T}}\bm{G}\bm{G}^{\mathrm{T}}\bm{\alpha}$

which leads to $\bm{G}^{\mathrm{T}}\bm{\alpha}=0$, i.e.,
$\bm{\alpha}\in\mathrm{null}\bm{G}^{\mathrm{T}}$. This clearly implies
$\mathcal{C}_{NC}\subseteq\mathrm{null}\bm{G}^{\mathrm{T}}$.

“$\mathcal{C}_{NC}\supseteq\mathrm{null}\bm{G}^{\mathrm{T}}$:” For any
$\bm{\alpha}\in\mathrm{null}\bm{G}^{\mathrm{T}}$,
$\bm{\alpha}^{\mathrm{T}}\bm{G}=0$ holds, and thus we can leverage (8) to
arrive at

$\bm{\alpha}^{\mathrm{T}}\bm{\beta}=-\sum_{k=0}^{l-1}\left(\bm{\alpha}^{\mathrm{T}}\bm{G}\right)\bm{\Psi}_{k}=0,\quad\forall\bm{\beta}\in\mathcal{C}_{C}$

which ensures $\bm{\alpha}\in\mathcal{C}_{NC}$. Consequently,
$\mathcal{C}_{NC}\supseteq\mathrm{null}\bm{G}^{\mathrm{T}}$ holds.

By combining $\mathcal{C}_{NC}\subseteq\mathrm{null}\bm{G}^{\mathrm{T}}$ with
$\mathcal{C}_{NC}\supseteq\mathrm{null}\bm{G}^{\mathrm{T}}$, the result 1) can
be immediately obtained.

“2):” For any $\bm{\alpha}\in\mathcal{C}_{NC}$, the use of
$\mathcal{C}_{NC}=\mathrm{null}\bm{G}^{\mathrm{T}}$ in 1) yields
$\bm{\alpha}^{\mathrm{T}}\bm{\beta}=0$,
$\forall\bm{\beta}\in\mathrm{span}\bm{G}$. This implies
$\mathrm{span}\bm{G}\subseteq\mathcal{C}_{C}$. Again using
$\mathcal{C}_{NC}=\mathrm{null}\bm{G}^{\mathrm{T}}$, we can derive

$\dim\left(\mathcal{C}_{NC}\right)+\dim\left(\mathrm{span}\bm{G}\right)=p.$
(9)

From $\mathcal{C}_{C}\oplus\mathcal{C}_{NC}=\mathbb{R}^{p}$, it is clear that
$\dim\left(\mathcal{C}_{NC}\right)+\dim\left(\mathcal{C}_{C}\right)=p$, which,
together with (9), yields

$\dim\left(\mathrm{span}\bm{G}\right)=\dim\left(\mathcal{C}_{C}\right)=p-\dim\left(\mathcal{C}_{NC}\right).$

Thanks to $\mathrm{span}\bm{G}\subseteq\mathcal{C}_{C}$, we actually have
$\mathcal{C}_{C}=\mathrm{span}\bm{G}$. Namely, the result 2) holds. ∎

We can see from Theorem 2 that the controllability of any $k$-state depends
only on the matrix $\bm{G}$. The good property benefits from the specific
structure of the system (4) that is established from the tracking task (3) of
ILC for the system (2). Obviously, we can use Theorem 2 and Definition 5 to,
respectively, obtain $\mathrm{span}\bm{G}=\mathbb{R}^{p}$,
$\\{0\\}\subset\mathrm{span}\bm{G}\subset\mathbb{R}^{p}$, and $\bm{G}=0$ when
(4) is completely controllable, incompletely controllable, and completely
uncontrollable, respectively. Next, the complete controllability of the system
(4) is directly called controllability for simplicity.

###  IV-B Controllability Criteria

With the property of the controllable $k$-states in Theorem 2, we can present
the controllability criteria for the system (4).

###### Theorem 3

For the system (4), it is controllable if and only if any of the following
conditions holds:

  1. 1.

(4) is $k$-stabilizable;

  2. 2.

$\mathrm{rank}\left(\bm{G}\right)=p$.

###### Proof:

If the system (4) is controllable, then it naturally is $k$-stabilizable, and
as a consequence, there exists some $k$-state feedback with a gain matrix
$\bm{K}\in\mathbb{R}^{q\times p}$ in the form of

$\bm{\Psi}_{k}=-\bm{K}\bm{E}_{k},\quad\forall k\in\mathbb{Z}_{+}$ (10)

to stabilize the resulting closed-loop system, given by

$\bm{E}_{k+1}=\left(I-\bm{G}\bm{K}\right)\bm{E}_{k},\quad\forall
k\in\mathbb{Z}_{+}.$ (11)

Clearly, the $k$-stability of the system (11) holds if and only if

$\rho\left(I-\bm{G}\bm{K}\right)<1$ (12)

which implies the nonsingularity of $\bm{G}\bm{K}$. Hence,
$\mathrm{rank}\left(\bm{G}\right)=p$ is immediate.

On the contrary, if $\mathrm{rank}\left(\bm{G}\right)=p$, then (12) holds for
some $\bm{K}\in\mathbb{R}^{q\times p}$, under which (11) is a $k$-stable
system. That is, the system (4) is $k$-stabilizable under some $k$-state
feedback controller (10). Moreover, we incorporate the result of Theorem 2 and
can deduce

$\dim\left(\mathcal{C}_{C}\right)=\dim\left(\mathrm{span}\bm{G}\right)=\mathrm{rank}\left(\bm{G}\right)=p$

and hence, $\mathcal{C}_{C}=\mathbb{R}^{p}$. That is, the controllability of
the system (4) is obtained. ∎

As a counterpart result of Theorem 3, the following theorem is introduced in
the presence of the incomplete controllability of the system (4).

###### Theorem 4

If the system (4) is incompletely controllable, then there exists some
nonsingular linear transformation $\widehat{\bm{E}}_{k}=\bm{F}\bm{E}_{k}$ such
that
$\widehat{\bm{E}}_{k}^{C}=\bm{F}_{1}^{\mathrm{T}}\bm{E}_{k}\in\mathbb{R}^{m}$
satisfies

$\widehat{\bm{E}}_{k+1}^{C}=\widehat{\bm{E}}_{k}^{C}+\bm{F}_{1}^{\mathrm{T}}\bm{G}\bm{\Psi}_{k},\quad\forall
k\in\mathbb{Z}_{+}$ (13)

and
$\widehat{\bm{E}}_{k}^{NC}=\bm{F}_{2}^{\mathrm{T}}\bm{E}_{k}\in\mathbb{R}^{p-m}$
satisfies

$\widehat{\bm{E}}_{k}^{NC}=\widehat{\bm{E}}_{0}^{NC},\quad\forall
k\in\mathbb{Z}_{+}.$ (14)

Moreover, the subsystem (13) is controllable.

###### Proof:

From (6), we have $\bm{F}_{2}^{\mathrm{T}}\bm{G}=0$. We thus apply
$\widehat{\bm{E}}_{k}=\bm{F}\bm{E}_{k}=\left[\left(\widehat{\bm{E}}_{k}^{C}\right)^{\mathrm{T}}~{}\left(\widehat{\bm{E}}_{k}^{NC}\right)^{\mathrm{T}}\right]^{\mathrm{T}}$
to the system (4) and can derive

$\displaystyle\begin{bmatrix}\widehat{\bm{E}}_{k+1}^{C}\\\
\widehat{\bm{E}}_{k+1}^{NC}\end{bmatrix}$
$\displaystyle=\begin{bmatrix}\widehat{\bm{E}}_{k}^{C}\\\
\widehat{\bm{E}}_{k}^{NC}\end{bmatrix}+\begin{bmatrix}\bm{F}_{1}^{\mathrm{T}}\\\
\bm{F}_{2}^{\mathrm{T}}\end{bmatrix}\bm{G}\bm{\Psi}_{k}$ (15)
$\displaystyle=\begin{bmatrix}\widehat{\bm{E}}_{k}^{C}\\\
\widehat{\bm{E}}_{k}^{NC}\end{bmatrix}+\begin{bmatrix}\bm{F}_{1}^{\mathrm{T}}\bm{G}\\\
0\end{bmatrix}\bm{\Psi}_{k},\quad\forall k\in\mathbb{Z}_{+}$

from which (13) is straightforward. Also, (15) implies

$\widehat{\bm{E}}_{k+1}^{NC}=\widehat{\bm{E}}_{k}^{NC},\quad\forall
k\in\mathbb{Z}_{+}$

which is equivalent to (14).

For the subsystem (13), we can further obtain
$\mathrm{rank}\left(\bm{F}_{1}^{\mathrm{T}}\bm{G}\right)=m$ because the use of
the nonsingularity of $\bm{F}$ and $\mathrm{rank}\left(\bm{G}\right)=m$ leads
to

$\mathrm{rank}\left(\bm{G}\right)=\mathrm{rank}\left(\bm{F}\bm{G}\right)=\mathrm{rank}\left(\begin{bmatrix}\bm{F}_{1}^{\mathrm{T}}\bm{G}\\\
0\end{bmatrix}\right)\\\
=\mathrm{rank}\left(\bm{F}_{1}^{\mathrm{T}}\bm{G}\right).$ (16)

Hence, (13) is a controllable system by considering Theorem 3 for this
subsystem. ∎

###### Remark 4

By Theorems 2, 3 and 4, we obtain fundamental controllability properties for
the tracking error systems of ILC. They actually provide criteria on the
design of $k$-state feedback controllers to achieve the $k$-stability. Of
particular note is the close relation between these results and rank
conditions of the matrix $\bm{G}$. The full-row rank of $\bm{G}$ is a
necessary and sufficient condition for the controllability of the system (4).
Otherwise, there exist uncontrollable $k$-states for the system (4), of which
a standard controllability decomposition is provided in (15). In particular,
it is interesting to derive (14) by this decomposition, which reveals that all
the uncontrollable components of the $k$-states are not dynamic but fixed
along the iteration axis.

##  V Controllability-Based Design of ILC

In this section, we connect the trackability problems for ILC systems to the
fundamental controllability problems, based on which we explore the design and
analysis of ILC in performing output tracking tasks. For this purpose, next we
present a result to reveal how to verify the trackability of the desired
references in ILC from the perspective of the controllability.

###### Theorem 5

For the system (2) and the associated tracking error system (4), it always
holds:

$\mathcal{Y}_{T}=\mathcal{C}_{C}.$ (17)

Further, for any $\bm{Y}_{d}\in\mathbb{R}^{p}$, $\bm{Y}_{d}\in\mathcal{Y}_{T}$
if and only if

$\widehat{\bm{E}}_{k}^{NC}=\bm{F}_{2}^{\mathrm{T}}\bm{E}_{0}=0,\quad\forall
k\in\mathbb{Z}_{+}$ (18)

and otherwise, $\bm{Y}_{d}\not\in\mathcal{Y}_{T}$ if and only if

$\widehat{\bm{E}}_{k}^{NC}=\bm{F}_{2}^{\mathrm{T}}\bm{E}_{0}\neq
0,\quad\forall k\in\mathbb{Z}_{+}.$ (19)

###### Proof:

From Theorems 1 and 2, we can directly gain (17) because two subspaces are
both identical with $\mathrm{span}\bm{G}$. Further, we leverage (2) and
$\bm{F}_{2}^{\mathrm{T}}\bm{G}=0$ by Lemma 1 to validate that for any
$\bm{Y}_{d}\in\mathbb{R}^{p}$,

$\displaystyle\widehat{\bm{E}}_{k}^{NC}=0,\quad\forall k\in\mathbb{Z}_{+}{}$
$\displaystyle\Leftrightarrow~{}\bm{F}_{2}^{\mathrm{T}}\left(\bm{Y}_{d}-\bm{G}\bm{U}_{k}\right)=0,\quad\forall
k\in\mathbb{Z}_{+}$
$\displaystyle\Leftrightarrow~{}\bm{F}_{2}^{\mathrm{T}}\bm{Y}_{d}=0$
$\displaystyle\Leftrightarrow~{}\bm{Y}_{d}\in\mathcal{Y}_{T}$

where we also incorporate the equivalent results of Lemma 2. For the same
reason, we can deduce

$\displaystyle\widehat{\bm{E}}_{k}^{NC}\neq 0,\quad\forall
k\in\mathbb{Z}_{+}{}$
$\displaystyle\Leftrightarrow~{}\bm{F}_{2}^{\mathrm{T}}\left(\bm{Y}_{d}-\bm{G}\bm{U}_{k}\right)\neq
0,\quad\forall k\in\mathbb{Z}_{+}$
$\displaystyle\Leftrightarrow~{}\bm{F}_{2}^{\mathrm{T}}\bm{Y}_{d}\neq 0$
$\displaystyle\Leftrightarrow~{}\bm{Y}_{d}\not\in\mathcal{Y}_{T}.$

Then all results of this theorem can be derived with (14). ∎

Based on Theorem 5, we can study the trackability problems of ILC systems by
instead treating the controllability problems of their resulting tracking
error systems. This makes it feasible to address the trackability analysis and
the updating law design of ILC by resorting only to its tracking error system.
Further, it is worth noting that all trackable desired references correspond
to $\widehat{\bm{E}}_{k}^{NC}=0$, $\forall k\in\mathbb{Z}_{+}$, namely, the
uncontrollable components of their relevant $k$-states are always equal to
zero.

To proceed with the development of Theorem 5, we develop a fundamental
tracking result of ILC in the following theorem, regardless of the system (2)
in the presence of any plant model.

###### Theorem 6

Let any desired reference $\bm{Y}_{d}\in\mathbb{R}^{p}$ be specified. Then the
system (2) can realize the tracking objective (3) under some updating law of
ILC in the form of

$\bm{U}_{k+1}=\bm{U}_{k}+\bm{K}\bm{E}_{k},\quad\forall k\in\mathbb{Z}_{+}$
(20)

if and only if $\bm{Y}_{d}\in\mathcal{Y}_{T}$. Otherwise, the system (2) can
no longer achieve the tracking objective (3), regardless of the application of
any input sequence $\left\\{\bm{U}_{k}:k\in\mathbb{Z}_{+}\right\\}$.

###### Proof:

From (5), we can validate the equivalence between the updating law (20) and
the $k$-state feedback (10). Note that
$\mathrm{rank}\left(\bm{F}_{1}^{\mathrm{T}}\bm{G}\right)=m$ and
$\mathrm{rank}\left(\bm{H}_{1}\right)=m$, namely,
$\bm{F}_{1}^{\mathrm{T}}\bm{G}$ and $\bm{H}_{1}$ are of full-row rank and of
full-column rank, respectively. Hence, there always exists some gain matrix
$\bm{K}\in\mathbb{R}^{q\times p}$ such that

$\rho\left(I-\bm{F}_{1}^{\mathrm{T}}\bm{G}\bm{K}\bm{H}_{1}\right)<1$ (21)

and some gain matrix $\widehat{\bm{K}}\in\mathbb{R}^{q\times m}$ such that

$\rho\left(I-\bm{F}_{1}^{\mathrm{T}}\bm{G}\widehat{\bm{K}}\right)<1.$ (22)

By these preparations, next we show that the tracking objective (3) holds for
the system (2) under the updating law (20) if and only if
$\bm{Y}_{d}\in\mathcal{Y}_{T}$.

Necessity: If (3) holds for (2) under (20), then
$\lim_{k\to\infty}\bm{E}_{k}=0$ holds for (11) (namely, the closed-loop system
resulting from (4) and (10)). With Lemma 1 and from (11), we can leverage
$\widehat{\bm{E}}_{k}=\bm{F}\bm{E}_{k}$,
$\widehat{\bm{E}}_{k}^{C}=\bm{F}_{1}^{\mathrm{T}}\bm{E}_{k}$, and
$\widehat{\bm{E}}_{k}^{NC}=\bm{F}_{2}^{\mathrm{T}}\bm{E}_{k}$ to obtain (14)
and

$\widehat{\bm{E}}_{k+1}^{C}=\left(I-\bm{F}_{1}^{\mathrm{T}}\bm{G}\bm{K}\bm{H}_{1}\right)\widehat{\bm{E}}_{k}^{C}-\bm{F}_{1}^{\mathrm{T}}\bm{G}\bm{K}\bm{H}_{2}\widehat{\bm{E}}_{k}^{NC},\quad\forall
k\in\mathbb{Z}_{+}.$ (23)

Based on $\lim_{k\to\infty}\bm{E}_{k}=0$ and with
$\widehat{\bm{E}}_{k}^{NC}=\bm{F}_{2}^{\mathrm{T}}\bm{E}_{k}$, the use of (14)
leads to (18). It hence follows from Theorem 5 that
$\bm{Y}_{d}\in\mathcal{Y}_{T}$. In addition, the substitution of (18) into
(23) yields

$\widehat{\bm{E}}_{k+1}^{C}=\left(I-\bm{F}_{1}^{\mathrm{T}}\bm{G}\bm{K}\bm{H}_{1}\right)\widehat{\bm{E}}_{k}^{C},\quad\forall
k\in\mathbb{Z}_{+}$ (24)

which is $k$-stable, and hence $\lim_{k\to\infty}\widehat{\bm{E}}_{k}^{C}=0$,
under the spectral radius condition (21). Clearly, this coincides with the two
facts of $\lim_{k\to\infty}\bm{E}_{k}=0$ and of
$\widehat{\bm{E}}_{k}^{C}=\bm{F}_{1}^{\mathrm{T}}\bm{E}_{k}$.

Sufficiency: Let us consider the controllable subsystem (13), and there always
exists some $k$-state feedback $\bm{\Psi}_{k}=-\widehat{\bm{K}}\bm{E}_{k}^{C}$
to generate the closed-loop system given by

$\widehat{\bm{E}}_{k+1}^{C}=\left(I-\bm{F}_{1}^{\mathrm{T}}\bm{G}\widehat{\bm{K}}\right)\widehat{\bm{E}}_{k}^{C},\quad\forall
k\in\mathbb{Z}_{+}$ (25)

of which the $k$-stability can be ensured with the spectral radius condition
(22). This implies $\lim_{k\to\infty}\widehat{\bm{E}}_{k}^{C}=0$. If
$\bm{Y}_{d}\in\mathcal{Y}_{T}$, then (18) holds based on Theorem 5. Hence, we
have $\lim_{k\to\infty}\widehat{\bm{E}}_{k}=0$ which, together with
$\bm{E}_{k}=\bm{H}\widehat{\bm{E}}_{k}$, yields
$\lim_{k\to\infty}\bm{E}_{k}=0$. Owing to
$\widehat{\bm{E}}_{k}^{C}=\bm{F}_{1}^{\mathrm{T}}\bm{E}_{k}$, we denote
$\bm{K}=\widehat{\bm{K}}\bm{F}_{1}^{\mathrm{T}}$, and as a consequence of
$\bm{\Psi}_{k}=-\widehat{\bm{K}}\bm{E}_{k}^{C}$, we can deduce that (10) is
equivalent to (20). By incorporating these facts, we can see that the tracking
objective (3) holds for the system (2) under the updating law (20) when
designing its gain matrices of the form
$\bm{K}=\widehat{\bm{K}}\bm{F}_{1}^{\mathrm{T}}$ and choosing
$\widehat{\bm{K}}$ according to (22), regardless of any
$\bm{Y}_{d}\in\mathcal{Y}_{T}$.

Otherwise, we can see from Theorem 5 that $\bm{Y}_{d}\not\in\mathcal{Y}_{T}$
if and only if (19) holds. It is immediate from (19) that
$\lim_{k\to\infty}\bm{E}_{k}=0$ does not hold any longer because of
$\widehat{\bm{E}}_{k}^{NC}=\bm{F}_{2}^{\mathrm{T}}\bm{E}_{k}$. Namely, (3) can
not be achieved for (2), despite any $\bm{U}_{k}$, $\forall
k\in\mathbb{Z}_{+}$. ∎

###### Remark 5

In Theorem 6, we identify a property of ILC that the trackability of the
desired references actually can provide a necessary and sufficient condition
to realize the output tracking tasks. The property makes us able to provide
ILC of the system (2) with an output tracking result for a general case,
regardless of any rank condition of $\bm{G}$. This is different from
conventional ILC results that generally require a full (row or column) rank
condition. In addition, Theorem 6 introduces a way to leverage the state
feedback-based design and analysis to address output tracking problems of ILC.
We can thus achieve the design and analysis of ILC from the perspective of
system stability, rather than resorting to traditional approaches for ILC
based on, e.g., contraction mapping analysis.

###### Remark 6

With Theorem 6, it reveals that for any $\bm{Y}_{d}\in\mathcal{Y}_{T}$, we can
design the updating law (20) of ILC for the system (2) to realize the tracking
objective (3). We can particularly design the gain matrix $\bm{K}$ of (20)
according to $\bm{K}=\widehat{\bm{K}}\bm{F}_{1}^{\mathrm{T}}$, for which
$\widehat{\bm{K}}$ is selected to satisfy (22). Since
$\bm{F}_{1}^{\mathrm{T}}\bm{G}$ is of full-row rank, there always exists
$\widehat{\bm{K}}$ that fulfills (22). This spectral radius condition in fact
ensures the $k$-stability of the system (25) corresponding to the controllable
components of the $k$-states. Because (20) is actually a generalized form of
the P-type updating law of ILC, Theorem 6 also provides the explanation about
why the P-type updating law is powerful and most applied in the ILC tracking
tasks from the viewpoints of state feedback and controllability.

In addition to the fundamental tracking problem of concern in Theorem 6,
another fundamental problem emerging is: what is the performance of the
correspondingly used input sequence, or what does ILC learn? To answer this
question, we separately consider two cases depending on whether the desired
reference is trackable for the system (2) or not.

We first contribute to addressing the case for the system (2) in the presence
of the trackable desired references, and denote the set of the desired inputs
that generate any $\bm{Y}_{d}\in\mathcal{Y}_{T}$ as

$\mathcal{U}_{d}\left(\bm{Y}_{d}\right)=\left\\{\bm{U}_{d}\in\mathbb{R}^{q}\big{|}\bm{Y}_{d}=\bm{G}\bm{U}_{d}\right\\},\quad\forall\bm{Y}_{d}\in\mathcal{Y}_{T}.$
(26)

It is clear that $\mathcal{U}_{d}\left(\bm{Y}_{d}\right)$ is exactly the set
of the solutions to the LAE (1) for any $\bm{Y}_{d}\in\mathcal{Y}_{T}$. We can
further reveal that ILC can be designed to learn all the desired inputs
collected in $\mathcal{U}_{d}\left(\bm{Y}_{d}\right)$ for any
$\bm{Y}_{d}\in\mathcal{Y}_{T}$ through the selection of the initial inputs.

###### Theorem 7

Consider the system (2) with any $\bm{Y}_{d}\in\mathcal{Y}_{T}$, and let the
updating law (20) be applied, and the condition (21) be satisfied. Then for
any initial input $\bm{U}_{0}$, the sequence of inputs $\bm{U}_{k}$, $\forall
k\in\mathbb{Z}_{+}$, generated by (20), converges such that the input learned
with ILC forms a convex set as

$\displaystyle\mathcal{U}_{\mathrm{ILC}}(\bm{Y}_{d})=\bigg{\\{}\bm{U}_{\infty}$
$\displaystyle=\left[I-\bm{K}\bm{H}_{1}\left(\bm{F}_{1}^{\mathrm{T}}\bm{G}\bm{K}\bm{H}_{1}\right)^{-1}\bm{F}_{1}^{\mathrm{T}}\bm{G}\right]\bm{U}_{0}$
(27)
$\displaystyle~{}~{}~{}+\bm{K}\bm{H}_{1}\left(\bm{F}_{1}^{\mathrm{T}}\bm{G}\bm{K}\bm{H}_{1}\right)^{-1}\bm{F}_{1}^{\mathrm{T}}\bm{Y}_{d}\Big{|}\bm{U}_{0}\in\mathbb{R}^{q}\bigg{\\}},$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\forall\bm{Y}_{d}\in\mathcal{Y}_{T}$

where $\bm{U}_{\infty}\triangleq\lim_{k\to\infty}\bm{U}_{k}$. Moreover, it
always holds

$\mathcal{U}_{\mathrm{ILC}}(\bm{Y}_{d})=\mathcal{U}_{d}(\bm{Y}_{d}),\quad\forall\bm{Y}_{d}\in\mathcal{Y}_{T}$
(28)

where it particularly follows

$\mathcal{U}_{\mathrm{ILC}}(\bm{Y}_{d})=\mathcal{U}_{d}(\bm{Y}_{d})=\left\\{\left(\bm{G}^{\mathrm{T}}\bm{G}\right)^{-1}\bm{G}^{\mathrm{T}}\bm{Y}_{d}\right\\},\quad\forall\bm{Y}_{d}\in\mathcal{Y}_{T}$
(29)

if and only if $\mathcal{Y}_{T}=\mathcal{Y}_{R}$ holds.

###### Proof:

See Appendix A. ∎

###### Remark 7

We reveal by Theorem 7, together with Theorem 6, that for any trackable
desired reference, we can leverage the design with a P-type ILC algorithm to
not only accomplish the tracking objective but also determine a corresponding
input to generate the desired reference. Furthermore, the inputs learned
through ILC are dependent linearly upon the desired reference and the initial
input, based on which we can develop all desired inputs associated with the
desired reference with the selections of different initial inputs. It is worth
noting also for Theorems 6 and 7 that under the spectral radius condition
(21), we ensure their convergence results with an exponentially fast speed.

When the full-row rank condition of $\bm{G}$ is ensured, Theorem 6 can be
developed to work for the tracking tasks of the system (2) in the presence of
any desired references.

###### Corollary 1

Let $\mathrm{rank}\left(\bm{G}\right)=p$ hold for the system (2). Then for any
desired reference $\bm{Y}_{d}\in\mathbb{R}^{p}$, there exists some updating
law (20) of ILC to achieve the tracking objective (3), together with the
resulting sequence of inputs converging into a convex set given by

$\displaystyle\mathcal{U}_{\mathrm{ILC}}(\bm{Y}_{d})=\bigg{\\{}\bm{U}_{\infty}$
$\displaystyle=\left[I-\bm{K}\left(\bm{G}\bm{K}\right)^{-1}\bm{G}\right]\bm{U}_{0}$
(30)
$\displaystyle~{}~{}~{}+\bm{K}\left(\bm{G}\bm{K}\right)^{-1}\bm{Y}_{d}\Big{|}\bm{U}_{0}\in\mathbb{R}^{q}\bigg{\\}},\quad\forall\bm{Y}_{d}\in\mathbb{R}^{p}$

for which the design condition of the gain matrix $\bm{K}$ is provided by
(12).

###### Proof:

Owing to $\mathrm{rank}\left(\bm{G}\right)=p$ and from Theorem 3,
$\bm{H}_{1}=I$ and $\bm{F}_{1}=I$ can be directly taken. Then this corollary
follows as an immediate consequence of Theorems 6 and 7. ∎

As a counterpart of Corollary 1, the following tracking result of ILC under
the full-column rank condition of $\bm{G}$ is presented.

###### Corollary 2

Let $\mathrm{rank}\left(\bm{G}\right)=q$ hold for the system (2). Then for any
realizable desired reference $\bm{Y}_{d}\in\mathcal{Y}_{R}$, there exists some
updating law (20) of ILC to both achieve the tracking objective (3) and ensure

$\lim_{k\to\infty}\bm{U}_{k}=\left(\bm{G}^{\mathrm{T}}\bm{G}\right)^{-1}\bm{G}^{\mathrm{T}}\bm{Y}_{d},\quad\forall\bm{U}_{0}\in\mathbb{R}^{q}$
(31)

for which the design condition is to guarantee the gain matrix $\bm{K}$ to
satisfy

$\rho\left(I-\bm{K}\bm{G}\right)<1.$ (32)

###### Proof:

For the system (2) under the updating law (20), the tracking result of (3)
follows straightforwardly from Theorem 6, regardless of any
$\bm{Y}_{d}\in\mathcal{Y}_{R}$. Simultaneously, we use Theorem 7 for the case
$\mathrm{rank}\left(\bm{G}\right)=q$, and can obtain (31) thanks to (29),
where
$\bm{U}_{d}=\left(\bm{G}^{\mathrm{T}}\bm{G}\right)^{-1}\bm{G}^{\mathrm{T}}\bm{Y}_{d}$
is exactly the unique solution for the LAE (1). Furthermore, if we denote
$\delta\bm{U}_{k}=\bm{U}_{d}-\bm{U}_{k}$, then we can equivalently deduce from
(1) and (53) that

$\delta\bm{U}_{k+1}=\left(I-\bm{K}\bm{G}\right)\delta\bm{U}_{k},\quad\forall
k\in\mathbb{Z}_{+}.$ (33)

Note that we can ensure $\lim_{k\to\infty}\delta\bm{U}_{k}=0$ by (31) for any
initial condition $\delta\bm{U}_{0}$. By considering this stability result for
(33), the design condition (32) of $\bm{K}$ can be developed
straightforwardly. The proof of this corollary is completed. ∎

###### Remark 8

In Corollaries 1 and 2, we specifically contribute to ILC for the controlled
systems with the full (row or column) rank conditions. This class of ILC
problems is one of the most considered classes in the ILC literature,
especially by applying the simple P-type updating law (20) with the design
conditions (12) and (32). Based on Corollaries 1 and 2, we reveal that this
specific class of ILC problems can be particularly addressed in our framework
developed by benefiting from the controllability and trackability criteria
related to ILC systems.

From Theorem 6, if the desired reference $\bm{Y}_{d}$ is untrackable, that is,
$\bm{Y}_{d}\notin\mathrm{span}\bm{G}$, then it is impossible to determine an
input sequence $\left\\{\bm{U}_{k}:k\in\mathbb{Z}_{+}\right\\}$ to realize the
tracking objective (3) for the system (2). In this circumstance, we contribute
to exploring the updating law (20) for the system (2) such that the sequence
of output $\left\\{\bm{Y}_{k}:k\in\mathbb{Z}_{+}\right\\}$ converges (namely,
$\bm{Y}_{\infty}\triangleq\lim_{k\to\infty}\bm{Y}_{k}$ exists), and
simultaneously produces a converged sequence of input
$\left\\{\bm{U}_{k}:k\in\mathbb{Z}_{+}\right\\}$ able to minimize
$\left\|\bm{Y}_{d}-\bm{G}\bm{U}_{\infty}\right\|_{2}$, namely,

$\left\|\bm{Y}_{d}-\bm{Y}_{\infty}\right\|_{2}=\left\|\bm{Y}_{d}-\bm{G}\bm{U}_{\infty}\right\|_{2}=\min_{\widetilde{\bm{U}}\in\mathbb{R}^{q}}\left\|\bm{Y}_{d}-\bm{G}\widetilde{\bm{U}}\right\|_{2},~{}\forall\bm{Y}_{d}\notin\mathcal{Y}_{T}.$

It obviously corresponds to the calculation of the least squares solutions for
the LAE (1). Towards this end, let us denote the set of all least squares
solutions for the LAE (1) in the presence of any untrackable desired reference
$\bm{Y}_{d}\notin\mathcal{Y}_{T}$ as

$\displaystyle\overline{\mathcal{U}}_{d}\left(\bm{Y}_{d}\right)=\left\\{\overline{\bm{U}}_{d}\in\mathbb{R}^{q}\Big{|}\left\|\bm{Y}_{d}-\bm{G}\overline{\bm{U}}_{d}\right\|_{2}=\min_{\widetilde{\bm{U}}\in\mathbb{R}^{q}}\left\|\bm{Y}_{d}-\bm{G}\widetilde{\bm{U}}\right\|_{2}\right\\}$
, (34) $\displaystyle\forall\bm{Y}_{d}\notin\mathcal{Y}_{T}$ .

For the tracking problem of the system (2) in the presence of any
$\bm{Y}_{d}\notin\mathcal{Y}_{T}$, we can present the following theorem to
develop learning results under the updating law (20).

###### Theorem 8

Consider the system (2) with any $\bm{Y}_{d}\notin\mathcal{Y}_{T}$, and let
the updating law (20) be applied with
$\bm{K}=\bm{\widehat{K}}\bm{F}^{\mathrm{T}}_{1}$, and the condition (22) be
satisfied. Then for any initial input $\bm{U}_{0}$, the sequences of the
output $\bm{Y}_{k}$, $\forall k\in\mathbb{Z}_{+}$ and the input $\bm{U}_{k}$,
$\forall k\in\mathbb{Z}_{+}$ converge, respectively, to

$\bm{Y}_{\infty}=\bm{H}_{1}\bm{F}_{1}^{\mathrm{T}}\bm{Y}_{d}$ (35)

and to $\bm{U}_{\infty}$ collected in the convex set as

$\displaystyle\overline{\mathcal{U}}_{\mathrm{ILC}}(\bm{Y}_{d})=\Bigg{\\{}\bm{U}_{\infty}$
$\displaystyle=\left[I-\widehat{\bm{K}}\left(\bm{F}_{1}^{\mathrm{T}}\bm{G}\widehat{\bm{K}}\right)^{-1}\bm{F}_{1}^{\mathrm{T}}\bm{G}\right]\bm{U}_{0}$
(36)
$\displaystyle~{}~{}~{}+\widehat{\bm{K}}\left(\bm{F}_{1}^{\mathrm{T}}\bm{G}\widehat{\bm{K}}\right)^{-1}\bm{F}_{1}^{\mathrm{T}}\bm{Y}_{d}\Big{|}\bm{U}_{0}\in\mathbb{R}^{q}\Bigg{\\}},$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\forall\bm{Y}_{d}\notin\mathcal{Y}_{T}.$

Moreover, it holds

$\overline{\mathcal{U}}_{\mathrm{ILC}}(\bm{Y}_{d})=\overline{\mathcal{U}}_{d}(\bm{Y}_{d}),\quad\forall\bm{Y}_{d}\notin\mathcal{Y}_{T}$
(37)

if and only if the selection of $\bm{H}$ is such that
$\bm{H}_{1}^{\mathrm{T}}\bm{H}_{2}=0$.

###### Proof:

See Appendix B. ∎

###### Remark 9

In Theorem 8, we reveal that we can still obtain convergent updating laws of
ILC in the presence of untrackable desired references. Since
$\bm{H}_{1}\bm{F}_{1}^{\mathrm{T}}$ is idempotent, (35) shows that for any
$\bm{Y}_{d}\notin\mathcal{Y}_{T}$, the output of the system (2) converges not
to $\bm{Y}_{d}$ but to its projection onto
$\mathrm{span}\left(\bm{H}_{1}\bm{F}_{1}^{\mathrm{T}}\right)$ along
$\mathrm{span}\left(\bm{H}_{2}\bm{F}_{2}^{\mathrm{T}}\right)$. In particular,
we can minimize the residual tracking error in the sense of the least squares
error by setting $\bm{H}$ such that $\bm{H}_{1}^{\mathrm{T}}\bm{H}_{2}=0$, and
correspondingly determine all least squares solutions for the LAE (1) by
resorting to the converged inputs of the system (2) under different selections
of the initial inputs. Since $\bm{H}_{2}$ is determined to arrive at a
nonsingular matrix together with $\bm{H}_{1}$, this property and the condition
$\bm{H}_{1}^{\mathrm{T}}\bm{H}_{2}=0$ can be accomplished simultaneously for
some $\bm{H}_{2}$. This actually renders it possible to always develop Theorem
8 for any selected matrix $\bm{H}_{1}$.

##  VI Finite-Iteration Convergence of ILC

In Theorems 6, 7, and 8, the convergence of ILC only resorts to the spectral
radius conditions (21) and (22) that can achieve the exponential convergence
for the sequences of both $\bm{Y}_{k}$ and $\bm{U}_{k}$. A further question
arising for the improvement of the ILC convergence rate is: whether and how
can $\bm{Y}_{k}$ and $\bm{U}_{k}$ be driven to converge within finite
iterations? If so, we can obtain control design tools to realize the
convergence of iterative methods for solving LAEs within finite iterations,
which works despite the calculation of solutions or least squares solutions
for LAEs.

To provide the abovementioned question with an affirmative answer, we
introduce the idea of deadbeat control to the design of ILC, and propose a
finite iteration convergence result below.

###### Theorem 9

Consider the system (2) with any $\bm{Y}_{d}\in\mathbb{R}^{p}$, and let the
updating law (20) be applied. For $\bm{Y}_{d}\in\mathcal{Y}_{T}$, there exists
some $\nu_{1}\in\mathbb{Z}_{m}$ ($m=\mathrm{rank}\left(\bm{G}\right)$) such
that

$\bm{U}_{k}\left\\{\begin{aligned} &\neq\bm{U}_{d},&\forall k&\leq\nu_{1}-1\\\
&=\bm{U}_{d},&\forall
k&\geq\nu_{1}\end{aligned}\right.,~{}~{}\bm{Y}_{k}\left\\{\begin{aligned}
&\neq\bm{Y}_{d},&\forall k&\leq\nu_{1}-1\\\ &=\bm{Y}_{d},&\forall
k&\geq\nu_{1}\end{aligned}\right.$ (38)

holds for some $\bm{U}_{d}\in\mathcal{U}_{\mathrm{ILC}}(\bm{Y}_{d})$ given by
(27) if and only if

$\rho\left(I-\bm{F}_{1}^{\mathrm{T}}\bm{G}\bm{K}\bm{H}_{1}\right)=0$ (39)

where $\nu_{1}$ is the minimal integer such that

$\left(I-\bm{F}_{1}^{\mathrm{T}}\bm{G}\bm{K}\bm{H}_{1}\right)^{\nu_{1}-1}\neq
0~{}\hbox{and}~{}\left(I-\bm{F}_{1}^{\mathrm{T}}\bm{G}\bm{K}\bm{H}_{1}\right)^{\nu_{1}}=0.$
(40)

For $\bm{Y}_{d}\notin\mathcal{Y}_{T}$, there exists some
$\nu_{2}\in\mathbb{Z}_{m}$ such that

$\bm{U}_{k}\left\\{\begin{aligned} &\neq\overline{\bm{U}}_{d},&\forall
k&\leq\nu_{2}-1\\\ &=\overline{\bm{U}}_{d},&\forall
k&\geq\nu_{2}\end{aligned}\right.,~{}~{}\bm{Y}_{k}\left\\{\begin{aligned}
&\neq\bm{H}_{1}\bm{F}_{1}^{\mathrm{T}}\bm{Y}_{d},&\forall k&\leq\nu_{2}-1\\\
&=\bm{H}_{1}\bm{F}_{1}^{\mathrm{T}}\bm{Y}_{d},&\forall
k&\geq\nu_{2}\end{aligned}\right.$ (41)

holds for some
$\overline{\bm{U}}_{d}\in\overline{\mathcal{U}}_{\mathrm{ILC}}(\bm{Y}_{d})$
given by (36) if and only if

$\rho\left(I-\bm{F}_{1}^{\mathrm{T}}\bm{G}\widehat{\bm{K}}\right)=0.$ (42)

Further, $\bm{K}=\bm{\widehat{K}}\bm{F}^{\mathrm{T}}_{1}$ holds for (20), and
$\nu_{2}$ is the minimal integer such that

$\left(I-\bm{F}_{1}^{\mathrm{T}}\bm{G}\widehat{\bm{K}}\right)^{\nu_{2}-1}\neq
0~{}\hbox{and}~{}\left(I-\bm{F}_{1}^{\mathrm{T}}\bm{G}\widehat{\bm{K}}\right)^{\nu_{2}}=0.$
(43)

###### Proof:

See Appendix C. ∎

###### Remark 10

With Theorem 9, a finite iteration convergence result of ILC is obtained,
regardless of trackable or untrackable desired references. This is realized by
appropriate selections of the ILC gain matrix. In addition, the number
$\nu_{1}$ (respectively, $\nu_{2}$) of iteration steps for ILC convergence is
exactly the degree of the minimal polynomial for
$I-\bm{F}_{1}^{\mathrm{T}}\bm{G}\bm{K}\bm{H}_{1}$ (respectively,
$I-\bm{F}_{1}^{\mathrm{T}}\bm{G}\widehat{\bm{K}}$) versus the trackable
(respectively, untrackable) desired reference. It actually incorporates the
idea of deadbeat control into the improvement of the convergence rate for ILC,
which is thus applicable for improving the convergence rate of iterative
methods for solving LAEs from the control design perspective.

##  VII Problem-Solving of LAEs Via ILC Methods

In this section, we consider how to leverage the ILC method to solve LAEs.
Before proceeding further, we present a lemma for the gain matrix design of
ILC.

###### Lemma 4

For any $\bm{G}\in\mathbb{R}^{p\times q}$, there exists some
$\bm{K}\in\mathbb{R}^{q\times p}$ to satisfy (21) (respectively, (39)) if and
only if there exists some $\widehat{\bm{K}}\in\mathbb{R}^{q\times m}$ to
satisfy (22) (respectively, (42)).

###### Proof:

Sufficiency: Due to $\bm{F}_{1}^{\mathrm{T}}\bm{H}_{1}=I$, we take
$\bm{K}=\widehat{\bm{K}}\bm{F}_{1}^{\mathrm{T}}$ and can arrive at
$\bm{F}_{1}^{\mathrm{T}}\bm{G}\bm{K}\bm{H}_{1}=\bm{F}_{1}^{\mathrm{T}}\bm{G}\widehat{\bm{K}}\left(\bm{F}_{1}^{\mathrm{T}}\bm{H}_{1}\right)=\bm{F}_{1}^{\mathrm{T}}\bm{G}\widehat{\bm{K}}$.
Thus, we can obtain (21) (respectively, (39)) from (22) (respectively, (42))
by taking $\bm{K}=\widehat{\bm{K}}\bm{F}_{1}^{\mathrm{T}}$.

Necessity: Let $\widehat{\bm{K}}=\bm{K}\bm{H}_{1}$, and then it is immediate
to deduce (22) (respectively, (42)) from (21) (respectively, (39)). ∎

By Lemma 4, we always design the updating law (20) with
$\bm{K}=\widehat{\bm{K}}\bm{F}_{1}^{\mathrm{T}}$ under the conditions (22) and
(42) without loss of generality. Since $\bm{F}_{1}^{\mathrm{T}}\bm{G}$ is of
the full-row rank based on (16), we can obtain the candidate selections of
$\widehat{\bm{K}}$, respectively, fulfilling (22) as

$\widehat{\bm{K}}=\bm{G}^{\mathrm{T}}\bm{F}_{1}\left(\bm{F}_{1}^{\mathrm{T}}\bm{G}\bm{G}^{\mathrm{T}}\bm{F}_{1}\right)^{-1}\left(I-\overline{\bm{K}}\right)$
(44)

and (42) as

$\widehat{\bm{K}}=\bm{G}^{\mathrm{T}}\bm{F}_{1}\left(\bm{F}_{1}^{\mathrm{T}}\bm{G}\bm{G}^{\mathrm{T}}\bm{F}_{1}\right)^{-1}\left(I-\widetilde{\bm{K}}\right)$
(45)

where $\overline{\bm{K}}\in\mathbb{R}^{m\times m}$ is any stable matrix (i.e.,
its eigenvalues are within the unit circle), and
$\widetilde{\bm{K}}\in\mathbb{R}^{m\times m}$ is any nilpotent matrix (i.e.,
$\widetilde{\bm{K}}^{n}=0$ holds for some positive integer $n$ [35]).

With Lemma 4, a fundamental solvability result is proposed for LAEs in the
following theorem.

###### Theorem 10

Given $\bm{Y}_{d}\in\mathbb{R}^{p}$, the LAE (1) is solvable if and only if
$\bm{Y}_{d}$ is trackable for the system (2), where the application of the
updating law (20) with $\bm{K}=\widehat{\bm{K}}\bm{F}_{1}^{\mathrm{T}}$ based
on the condition (22) determines the solution to the LAE (1) in the
convergence form of $\lim_{k\to\infty}\bm{U}_{k}$, which is collected with an
analytical form in the convex set (27). Further, if $\widehat{\bm{K}}$ is
designed according to (42), then the solution to the LAE (1) can be determined
within finite iterations.

###### Proof:

With Definition 2, this theorem can be developed based on Theorems 6, 7, and
9. ∎

From Lemma 2, we can resort to
$\bm{H}_{1}\bm{F}_{1}^{\mathrm{T}}\bm{Y}_{d}=\bm{Y}_{d}$ to establish the
solvability of the LAE (1) in Theorem 10 for any
$\bm{Y}_{d}\in\mathbb{R}^{p}$. We also note from (27) and (28) that the
solutions to the LAE (1) depend on the selection of the initial input
$\bm{U}_{0}$ of the system (2), and thus we may determine all solutions
through choosing all candidates of $\bm{U}_{0}$ in $\mathbb{R}^{q}$.

Next, we explore Theorem 10 to present an implementation algorithm of ILC to
determine solutions for solvable LAEs.

Algorithm 1: Solving the LAE (1) for any $\bm{Y}_{d}\in\mathcal{Y}_{T}$

  1. 1)

Set a tolerance $\varepsilon>0$, calculate
$\bm{K}=\widehat{\bm{K}}\bm{F}_{1}^{\mathrm{T}}$ by determining $\bm{F}_{1}$
and designing $\widehat{\bm{K}}$ according to (22) or (42), and select the
initial input $\bm{U}_{0}$.

  2. 2)

Let $k=0$, and go to the step 3) to start the iteration.

  3. 3)

Apply $\bm{U}_{k}$ to the system (2) to measure $\bm{Y}_{k}$.

  4. 4)

Calculate $\bm{E}_{k}=\bm{Y}_{d}-\bm{Y}_{k}$. If $\|\bm{E}_{k}\|<\epsilon$,
then go to the step 7) by returning the solution “$\bm{U}_{d}=\bm{U}_{k}$;”
and otherwise, go to the step 5).

  5. 5)

Apply the updating law (20) to calculate $\bm{U}_{k+1}$.

  6. 6)

Let $k=k+1$, and return to the step 3).

  7. 7)

Stop the iteration.

To determine $\bm{F}_{1}$ in the Algorithm 1, we only need to choose
$\bm{H}_{1}$ as a full-column rank matrix that satisfies
$\mathrm{span}\bm{H}_{1}=\mathrm{span}\bm{G}$, and can directly take
$\bm{F}_{1}=\bm{H}_{1}\left(\bm{H}_{1}^{\mathrm{T}}\bm{H}_{1}\right)^{-1}$.
This is thanks to the property that the selection of $\bm{H}_{2}$, together
with that of $\bm{F}_{2}$, does not have influence on both design condition
and convergence result of ILC though it plays a significant role in
implementing the convergence analysis of ILC.

When the LAE (1) is unsolvable, the system (2) is subject to the untrackable
desired references. However, the ILC updating law (20) is still applicable,
and helps calculate the least squares solutions to the LAE (1), as shown in
the following theorem.

###### Theorem 11

For the LAE (1) with any $\bm{Y}_{d}\notin\mathcal{Y}_{T}$, the system (2)
with the application of the updating law (20) for
$\bm{K}=\widehat{\bm{K}}\bm{F}_{1}^{\mathrm{T}}$ under the condition (22)
determines its least squares solution in the convergence form of
$\lim_{k\to\infty}\bm{U}_{k}$, which is collected with an analytical form in
the convex set (36) if and only if $\bm{H}_{1}^{\mathrm{T}}\bm{H}_{2}=0$ holds
for the selection of $\bm{H}$. Moreover, if $\widehat{\bm{K}}$ is designed
based on (42), then the least squares solution to the LAE (1) can be
determined within finite iterations.

###### Proof:

A consequence of applying Theorems 8 and 9. ∎

With Theorem 11, we show an implementation algorithm of ILC to determine least
squares solutions for unsolvable LAEs.

Algorithm 2: Solving the LAE (1) for any $\bm{Y}_{d}\notin\mathcal{Y}_{T}$

  1. 1)

Select a full-column rank matrix $\bm{H}_{1}$ such that
$\mathrm{span}\bm{H}_{1}=\mathrm{span}\bm{G}$, and calculate
$\bm{F}_{1}=\bm{H}_{1}\left(\bm{H}_{1}^{\mathrm{T}}\bm{H}_{1}\right)^{-1}$.

  2. 2)

Set a tolerance $\varepsilon>0$, calculate
$\bm{K}=\widehat{\bm{K}}\bm{F}_{1}^{\mathrm{T}}$ by determining
$\widehat{\bm{K}}$ based on (22) or (42), and select an initial input
$\bm{U}_{0}$.

  3. 3)

Let $k=0$, and go to the step 4) to start the iteration.

  4. 4)

Apply $\bm{U}_{k}$ to the system (2) to measure $\bm{Y}_{k}$.

  5. 5)

Calculate
$\overline{\bm{E}}_{k}=\bm{H}_{1}\bm{F}_{1}^{\mathrm{T}}\bm{Y}_{d}-\bm{Y}_{k}$.
If $\left\|\overline{\bm{E}}_{k}\right\|<\epsilon$, then go to the step 8) by
returning the least squares solution “$\overline{\bm{U}}_{d}=\bm{U}_{k}$;” and
otherwise, go to the step 6).

  6. 6)

Apply the updating law (20) to calculate $\bm{U}_{k+1}$.

  7. 7)

Let $k=k+1$, and return to the step 4).

  8. 8)

Stop the iteration.

For the Algorithm 2, the use of
$\bm{F}_{1}=\bm{H}_{1}\left(\bm{H}_{1}^{\mathrm{T}}\bm{H}_{1}\right)^{-1}$
ensures $\bm{H}_{1}^{\mathrm{T}}\bm{H}_{2}=0$ since the selection of
$\bm{H}_{2}$ needs to satisfy $\bm{F}=\bm{H}^{-1}$. This renders the
implementation of the Algorithm 2 capable of determining the least squares
solutions for LAEs. To determine whether the Algorithm 1 or 2 should be used
for any LAE (1), it depends on whether $\bm{Y}_{d}\in\mathcal{Y}_{T}$ or
$\bm{Y}_{d}\notin\mathcal{Y}_{T}$, which can instead be validated by whether
the criterion $\bm{H}_{1}\bm{F}_{1}^{\mathrm{T}}\bm{Y}_{d}=\bm{Y}_{d}$ is
satisfied or not. In particular, for the design of $\widehat{\bm{K}}$, we can
directly use (44) and (45) that render the Algorithms 1 and 2 exponentially
convergent and convergent within finite iterations, respectively.

To explain how to solve LAEs with our ILC algorithms, we provide illustrations
in the following example.

Example 1: Consider the LAE (1) with $\bm{G}$ given by

$\bm{G}=\left[\begin{matrix}1&0&0&2&2\\\ 0&3&4&0&0\\\
1&-3&-4&2&2\end{matrix}\right]^{\mathrm{T}}.$

Clearly, $\mathrm{rank}(\bm{G})=2$ holds. To select a full-column rank matrix
$\bm{H}_{1}$ such that $\mathrm{span}\bm{H}_{1}=\mathrm{span}\bm{G}$, we adopt

$\bm{H}_{1}=\left[\begin{matrix}1&0&0&2&2\\\
0&3&4&0&0\end{matrix}\right]^{\mathrm{T}}.$

As a consequence, to apply the Algorithms 1 and 2, we directly choose
$\bm{F}_{1}$ as

$\bm{F}_{1}=\bm{H}_{1}\left(\bm{H}_{1}^{\mathrm{T}}\bm{H}_{1}\right)^{-1}=\left[\begin{matrix}1/9&0&0&2/9&2/9\\\
0&3/25&4/25&0&0\end{matrix}\right]^{\mathrm{T}}$

and adopt $\bm{K}=\widehat{\bm{K}}\bm{F}_{1}^{\mathrm{T}}$, where we
particularly design $\widehat{\bm{K}}$ according to (45). Namely, by
specifying a nilpotent matrix as

$\widetilde{\bm{K}}=\begin{bmatrix}0&1\\\ 0&0\end{bmatrix}$

we can calculate $\widehat{\bm{K}}$ as

$\widehat{\bm{K}}=\bm{G}^{\mathrm{T}}\bm{F}_{1}\left(\bm{F}_{1}^{\mathrm{T}}\bm{G}\bm{G}^{\mathrm{T}}\bm{F}_{1}\right)^{-1}\left(I-\widetilde{\bm{K}}\right)=\left[\begin{matrix}2/3&1/3&1/3\\\
-1/3&1/3&-2/3\end{matrix}\right]^{\mathrm{T}}.$

Then we can leverage the Algorithms 1 and 2 to solve the LAE (1) for any
desired reference $\bm{Y}_{d}\in\mathbb{R}^{5}$. Due to
$I-\bm{F}_{1}^{\mathrm{T}}\bm{G}\widehat{\bm{K}}=\widetilde{\bm{K}}$ and
$\widetilde{\bm{K}}^{2}=0$, we can derive the calculation result after only
two iterations of the implementation of the Algorithms 1 and 2.

To proceed, we without any loss of generality use the initial input as
$\bm{U}_{0}=[1,0,0]^{\mathrm{T}}$, and consider two different cases of the
desired references as

  1. c1)

$\bm{Y}_{d}=\left[1,3,4,2,2\right]^{\mathrm{T}}$;

  2. c2)

$\bm{Y}_{d}=\left[1,2,1,1,2\right]^{\mathrm{T}}$.

We can easily verify that for the case c1),
$\bm{H}_{1}\bm{F}_{1}^{\mathrm{T}}\bm{Y}_{d}=\bm{Y}_{d}$ holds, whereas for
the case c2), it does not. This implies that the LAE (1) is solvable in the
case c1), but not in the case c2).

For the case c1), we apply the Algorithm 1, and can obtain the solution to the
LAE (1) after two iterations as

$\bm{U}_{d}=\left[4/3,2/3,-1/3\right]^{\mathrm{T}}.$

In fact, we can determine the solutions to the LAE (1) for any trackable
desired reference $\bm{Y}_{d}\in\mathcal{Y}_{T}$ in a general form of

$\bm{U}_{d}=\begin{bmatrix}1/3&-1/3&-1/3\\\ -1/3&1/3&1/3\\\
-1/3&1/3&1/3\end{bmatrix}\bm{U}_{0}+\begin{bmatrix}1\\\ 1\\\
0\end{bmatrix},\quad\forall\bm{U}_{0}\in\mathbb{R}^{q}.$

For the case c2), the application of the Algorithm 2 results in the least
squares solution to the LAE (1) after two iterations, which is given by

$\overline{\bm{U}}_{d}=\left[133/135,26/135,-28/135\right]^{\mathrm{T}}$

and as a result, the least squares error norm is
$\left\|\bm{Y}_{d}-\bm{G}\overline{\bm{U}}_{d}\right\|_{2}=\sqrt{14}/{3}$. In
addition, the least squares solutions to the LAE (1) for any untrackable
desired reference $\bm{Y}_{d}\notin\mathcal{Y}_{T}$ have a general form of

$\overline{\bm{U}}_{d}=\begin{bmatrix}1/3&-1/3&-1/3\\\ -1/3&1/3&1/3\\\
-1/3&1/3&1/3\end{bmatrix}\bm{U}_{0}+\begin{bmatrix}88/135\\\ 71/135\\\
17/135\end{bmatrix},\quad\forall\bm{U}_{0}\in\mathbb{R}^{q}.$

##  VIII Applications to Traditional 2-D ILC Systems

Traditionally, each ILC system involves dynamics along two independent axes.
In addition to an infinite iteration axis given by $k\in\mathbb{Z}_{+}$, it
generally has a fixed time axis denoted by $t\in\mathbb{Z}_{N}$. We consider
the ILC problem for a system described by

$\left\\{\begin{aligned} x_{k}(t+1)&=Ax_{k}(t)+Bu_{k}(t)\\\
y_{k}(t)&=Cx_{k}(t)\end{aligned}\right.,\quad\forall
t\in\mathbb{Z}_{N},\forall k\in\mathbb{Z}_{+}$ (46)

where $x_{k}(t)\in\mathbb{R}^{n_{s}}$, $u_{k}(t)\in\mathbb{R}^{n_{i}}$ and
$y_{k}(t)\in\mathbb{R}^{n_{o}}$ denote the state, input and output,
respectively; and $A$, $B$ and $C$ are three system matrices of appropriate
dimensions. The implementation of the system (46) requires two different
initial conditions, which are given by $x_{k}(0)=0$, $\forall
k\in\mathbb{Z}_{+}$ and $u_{0}(t)$, $\forall t\in\mathbb{Z}_{N-1}$. In
addition, let us denote
$C=\left[C_{1}^{\mathrm{T}},C_{2}^{\mathrm{T}},\cdots,C_{n_{o}}^{\mathrm{T}}\right]^{\mathrm{T}}$,
where $C_{i}\in\mathbb{R}^{1\times n_{s}}$, $\forall i=1$, $2$, $\cdots$,
$n_{o}$.

Let each of the $n_{o}$ input-output channels for the system (46) have a
relative degree $r$ ($r\geq 1$) without any loss of generality. Namely, it
holds

$\begin{aligned} C_{i}A^{j}B&=0,j=0,1,\cdots,r-2\\\ C_{i}A^{r-1}B&\neq
0\end{aligned},\quad\forall i=1,2,\cdots,n_{o}.$ (47)

Based on some standard algebraic manipulations, the condition (47) can be
generalized to describe different relative degrees $r_{i}$, $\forall i=1$,
$2$, $\cdots$, $n_{o}$ for the system (46) (see, e.g., [29, 30]). This
generalization, however, does not influence our investigations, which will not
be considered. Due to (47), we denote the series of the Markov parameter
matrices for the system (46) as

$G_{i}=CA^{i+r-1}B,\quad\forall i\in\mathbb{Z}_{+}.$

It is worth noting that $G_{0}=CA^{r-1}B$ is the first nonzero Markov
parameter matrix for the system (46), and also that the system relative degree
$r$ is exactly the delay of time steps in the output $y_{k}(t)$ when the input
$u_{k}(t)$ takes effect [19].

For the system (46), the evolution from iteration to iteration is to enable
its output to track the desired reference for all time steps of interest over
a fixed time interval. If we use $y_{d}(t)\in\mathbb{R}^{n_{o}}$ to represent
any desired reference, then in view of the relative degree $r$ for the system
(46), the tracking objective is to design appropriate updating laws of
$u_{k}(t)$, $\forall t\in\mathbb{Z}_{N-1}$ such that

$\lim_{k\to\infty}y_{k}(t)=y_{d}(t),\quad\forall t=r,r+1,\cdots,r+N-1.$ (48)

To cope with this tracking problem of ILC, we are motivated by Definition 2 to
give the trackability and realizability notions of the desired references for
the system (46).

###### Definition 6

For the system (46), a desired reference $y_{d}(t)$ is said to be trackable
(respectively, realizable) if there exists some (respectively, a unique)
desired input $u_{d}(t)$, together with the zero initial state condition
$x_{d}(0)=0$, such that

$\left\\{\begin{aligned} x_{d}(t+1)&=Ax_{d}(t)+Bu_{d}(t)\\\
y_{d}(t)&=Cx_{d}(t)\end{aligned}\right.,\quad\forall t\in\mathbb{Z}_{N}.$ (49)

In Definition 6, we may make extensions by considering any initial state
condition $x_{d}(0)\in\mathbb{R}^{n_{s}}$. Since $x_{d}(0)$ is closely tied to
$y_{d}(t)$, we employ $y_{d}(t)-CA^{t}x_{d}(0)$ instead of directly applying
$y_{d}(t)$ to bridge the connection with $u_{d}(t)$. By this replacement, we
can obtain the same trackability and realizability properties, thanks to which
we directly adopt $x_{d}(0)=0$ to keep consistent with the establishment of
Definition 2.

###  VIII-A ILC Reformulation Based on Super-Vectors

Let $p=Nn_{o}$ and $q=Nn_{i}$, and define $\bm{Y}_{k}$, $\bm{Y}_{d}$,
$\bm{U}_{k}$ and $\bm{U}_{d}$ as the super-vectors that result from
$y_{k}(t)$, $y_{d}(t)$, $u_{k}(t)$ and $u_{d}(t)$, respectively, based on the
lifting technique in the form of

$\displaystyle\bm{Y}_{k}$
$\displaystyle=\left[y_{k}^{\mathrm{T}}(r),y_{k}^{\mathrm{T}}(r+1),\cdots,y_{k}^{\mathrm{T}}(N+r-1)\right]^{\mathrm{T}},~{}\forall
k\in\mathbb{Z}_{+}$ (50) $\displaystyle\bm{Y}_{d}$
$\displaystyle=\left[y_{d}^{\mathrm{T}}(r),y_{d}^{\mathrm{T}}(r+1),\cdots,y_{d}^{\mathrm{T}}(N+r-1)\right]^{\mathrm{T}}$
$\displaystyle\bm{U}_{k}$
$\displaystyle=\left[u_{k}^{\mathrm{T}}(0),u_{k}^{\mathrm{T}}(1),\cdots,u_{k}^{\mathrm{T}}(N-1)\right]^{\mathrm{T}},~{}\forall
k\in\mathbb{Z}_{+}$ $\displaystyle\bm{U}_{d}$
$\displaystyle=\left[u_{d}^{\mathrm{T}}(0),u_{d}^{\mathrm{T}}(1),\cdots,u_{d}^{\mathrm{T}}(N-1)\right]^{\mathrm{T}}.$

We can reformulate (46) and (49) into (2) and (1), respectively, by defining
$\bm{G}$ in the structured form of a block Toeplitz matrix:

$\bm{G}=\begin{bmatrix}G_{0}&0&\cdots&0\\\ G_{1}&G_{0}&\ddots&\vdots\\\
\vdots&\vdots&\ddots&0\\\ G_{N-1}&G_{N-2}&\cdots&G_{0}\end{bmatrix}.$ (51)

Similarly to (50), $\bm{E}_{k}$ can be defined as

$\bm{E}_{k}=\left[e_{k}^{\mathrm{T}}(r),e_{k}^{\mathrm{T}}(r+1),\cdots,e_{k}^{\mathrm{T}}(N+r-1)\right]^{\mathrm{T}},\quad\forall
k\in\mathbb{Z}_{+}$

in which $e_{k}(t)=y_{d}(t)-y_{k}(t)$ denotes the output tracking error of the
system (46).

Since the matrix $\bm{G}$ has a special structure of (51), we benefit from
this property to get some rank conditions on $\bm{G}$, especially with the aid
of the rank conditions on $G_{0}$.

###### Lemma 5

For the matrix $\bm{G}$ defined by (51), it holds:

  1. 1.

$N\mathrm{rank}\left(G_{0}\right)\leq\mathrm{rank}\left(\bm{G}\right)\leq
N\min\left\\{n_{o},n_{i}\right\\}$;

  2. 2.

$\mathrm{rank}\left(\bm{G}\right)=N\min\left\\{n_{o},n_{i}\right\\}\Leftrightarrow\mathrm{rank}\left(G_{0}\right)=\min\left\\{n_{o},n_{i}\right\\}$.

###### Proof:

A direct consequence of the block Toeplitz matrix structure of $\bm{G}$ in
(51). ∎

By Lemma 5, we generally have $\mathrm{rank}\left(\bm{G}\right)\geq
N\mathrm{rank}\left(G_{0}\right)$, and
$\mathrm{rank}\left(\bm{G}\right)=N\mathrm{rank}\left(G_{0}\right)$ emerges
when $\mathrm{rank}\left(G_{0}\right)=\min\left\\{n_{o},n_{i}\right\\}$. We
also find that $\bm{G}$ is of full row (respectively, column) rank if and only
if $G_{0}$ is of full row (respectively, column) rank. These properties can
help achieve the conditions of $\bm{G}$ in establishing the trackability and
realizability properties for ILC systems. Of particular note is this benefit
for the system (2) that is induced from the system (46).

From Definitions 2 and 6 and with (50), we can establish the trackability and
realizability properties of $y_{d}(t)$ for the system (46) by exploring those
of $\bm{Y}_{d}$ for the system (2). Consequently, we also denote
$\mathcal{Y}_{T}$ and $\mathcal{Y}_{R}$ as the trackability and realizability
subspaces of the system (46), respectively. Besides, a trackable
(respectively, realizable) desired reference $y_{d}(t)$ is denoted by
$y_{d}(t)\in\mathcal{Y}_{T}$ (respectively, $y_{d}(t)\in\mathcal{Y}_{R}$).

With $\bm{E}_{k}$, we can still arrive at the system (4) to deal with the
tracking problem (48) for the system (46). Based on Definition 1, it follows
that we can achieve the tracking objective (48) for the system (46) by
transforming it into the $k$-stability problem of the system (4).

###  VIII-B Trackability-Based ILC Analysis and Design

For the trackability of the system (46), we present a helpful lemma to provide
it with some basic properties.

###### Lemma 6

For the system (46), $\mathcal{Y}_{T}=\mathrm{span}\bm{G}$ always holds, from
which two properties follow as:

  1. 1.

$N\mathrm{rank}\left(G_{0}\right)\leq\dim\left(\mathcal{Y}_{T}\right)\leq
N\min\left\\{n_{o},n_{i}\right\\}$;

  2. 2.

$\mathcal{Y}_{T}=\mathbb{R}^{Nn_{o}}$ if and only if
$\mathrm{rank}\left(G_{0}\right)=n_{o}$.

In particular, for any desired reference $y_{d}(t)$,
$y_{d}(t)\in\mathcal{Y}_{T}$ if and only if
$\bm{Y}_{d}\in\mathrm{span}\bm{G}$.

###### Proof:

With the applications of Theorem 1 and Lemmas 2 and 5, this lemma can be
proved by noting the lower triangular block form of (51) for the matrix
$\bm{G}$. ∎

Analogously to Lemma 6, the following lemma shows basic realizability
properties of the system (46).

###### Lemma 7

For the system (46), $\mathcal{Y}_{R}=\mathrm{span}\bm{G}$ follows if and only
if $\mathrm{rank}\left(G_{0}\right)=n_{i}$ holds; and
$\mathcal{Y}_{R}=\\{0\\}$ emerges, otherwise. In particular, for any desired
reference $y_{d}(t)$, $y_{d}(t)\in\mathcal{Y}_{R}$ if and only if
$y_{d}(t)\in\mathcal{Y}_{T}$ and $\mathrm{rank}\left(G_{0}\right)=n_{i}$.

###### Proof:

A consequence of Lemma 3 and Theorem 1 thanks to $\bm{G}$ given by (51). ∎

With Lemmas 6 and 7, we can notice by comparison that the proposed
trackability property may play a more fundamentally important role than the
usually employed realizability property in addressing the design and analysis
problems of ILC. To proceed, we explore the controllability properties of the
tracking error system (4) with $\bm{G}$ given by (51).

###### Lemma 8

The system (4) is controllable (or stabilizable) if and only if
$\mathrm{rank}\left(G_{0}\right)=n_{o}$, and admits a standard form for the
controllability decomposition in (15), otherwise. Furthermore,
$\mathcal{C}_{C}=\mathcal{Y}_{T}=\mathrm{span}\bm{G}$ always holds.

###### Proof:

With the specific structure of $\bm{G}$ in (51), this lemma can follow
directly from Theorems 2, 3 and 4. ∎

Based on the development of Lemmas 6 and 8, we establish a basic tracking
result of ILC for the system (46).

###### Theorem 12

For the system (46) with any desired reference $y_{d}(t)$, the tracking
objective (48) can be achieved under some updating law of ILC, with gain
matrices $K_{ij}\in\mathbb{R}^{n_{i}\times n_{o}}$, $\forall i$, $j=1$, $2$,
$\cdots$, $N$, given by

$u_{k+1}(t)=u_{k}(t)+\sum_{i=0}^{N-1}K_{t+1,i+1}e_{k}(i+r),\quad\forall
t\in\mathbb{Z}_{N-1},\forall k\in\mathbb{Z}_{+}$ (52)

if and only if $y_{d}(t)\in\mathcal{Y}_{T}$; and otherwise, the tracking
objective (48) can no longer be achieved for the system (46), regardless of
applying any input $u_{k}(t)$, $\forall t\in\mathbb{Z}_{N-1}$, $\forall
k\in\mathbb{Z}_{+}$. In particular, the design condition of the gain matrices
in (52) for any $y_{d}(t)\in\mathcal{Y}_{T}$ is such that
$\bm{K}=\left[K_{ij}\right]\in\mathbb{R}^{Nn_{i}\times Nn_{o}}$ fulfills the
spectral radius condition (21).

###### Proof:

By noting the equivalence between (20) and (52), we can develop this theorem
with Lemmas 6 and 8 and based on Theorem 6 as well as its proof. ∎

###### Remark 11

From Theorem 6, we know that the ILC results of Theorem 12 benefit from the
trackability result of Lemma 6, the controllability result of Lemma 8, and the
$k$-state feedback controller design in the Kalman state-space framework. It
hints that our trackability-based approach can provide a feasible way to
narrow the gap between ILC and the popular state feedback-based control
methods. As a result, more powerful design and analysis tools may be developed
for ILC, such as those gained based on the system stability (see also Remark
5).

Next, we use an example to demonstrate the effectiveness of ILC even in the
absence of full (row or column) rank condition of the controlled system.

Example 2: Consider the system (46) with $A$, $B$, and $C$ as

$A=\begin{bmatrix}1&0&0\\\ 0&1&0\\\ 0&0&1\end{bmatrix},\quad
B=\begin{bmatrix}1&-1\\\ 2&-2\\\ 0&0\end{bmatrix},\quad
C=\begin{bmatrix}1&0&1\\\ 0&1&-1\end{bmatrix}.$

Clearly, $G_{0}=CB\in\mathbb{R}^{2\times 2}$ is such that
$\mathrm{rank}\left(G_{0}\right)=1$. That is, $G_{0}$ is neither of full-row
rank nor of full-column rank. It generally renders the existing results for
ILC not applicable. By contrast, we reveal in Theorem 12 that we can still
perform the tracking tasks of ILC in the presence of trackable desired
references.

As an example, we consider the desired reference given by

$y_{d}(t)=\left[\sin(0.06t),2\sin(0.06t)\right]^{\mathrm{T}},\quad\forall
t\in\mathbb{Z}_{100}.$

It is not difficult to know $y_{d}(t)\in\mathcal{Y}_{T}$. To apply the
updating law (52), we directly take $\bm{K}=I\otimes
K_{0}\in\mathbb{R}^{200\times 200}$, and to determine
$K_{0}\in\mathbb{R}^{2\times 2}$, we select $\bm{H}_{1}=\bm{G}\bm{Z}$ with
$\bm{Z}=I\otimes[1,0]^{\mathrm{T}}\in\mathbb{R}^{200\times 100}$ and
$\bm{H}_{2}=I\otimes[0,1]^{\mathrm{T}}\in\mathbb{R}^{200\times 100}$. In this
case, if we choose $K_{0}=diag\\{1.5,0.73\\}$, then we can obtain
$\rho\left(I-\bm{F}_{1}^{\mathrm{T}}\bm{G}\bm{K}\bm{H}_{1}\right)=0.96$,
namely, the spectral radius condition (21) holds. In Fig. 2, we depict the
perfect tracking performance for ILC under the zero initial input. This
illustration demonstrates the effectiveness of our trackability-based ILC
results of Theorem 12.

Figure 2: The evolution of the tracking error for ILC versus iteration number.

##  IX Conclusions

In this paper, we have discussed and answered the questions about whether and
how we can achieve the objective of solving LAEs from the perspective of
systems and control, concerning which we have developed an iterative method
integrated with a learning control mechanism. By introducing a new
trackability problem and connecting it to typical tracking problems of ILC, we
have revealed an equivalent relation between the solvability of LAEs and the
controllability of discrete control systems. In view of this equivalence
relation, we have leveraged the classic state feedback-based methods to solve
LAEs, which is realized by equivalently accomplishing the perfect tracking
tasks of the resulting ILC systems. It has been shown that the solutions for
any solvable LAE are linearly dependent upon the initial inputs and thus all
its solutions can be derived by different selections of the initial inputs.
This property also applies to obtaining all the least squares solutions for
any unsolvable LAE. Moreover, we have incorporated the design idea of deadbeat
control into ILC such that the solving of LAEs can be realized within finite
iteration steps. In addition, we have generalized the trackability property to
conventional 2-D ILC systems, and have developed a state feedback-based method
for the analysis and synthesis of ILC, instead of resorting to the generally
employed contraction mapping-based methods for ILC. This brings new insights
into ILC and narrows the gap between it and the popular feedback-based control
methods. Our analysis and synthesis results may also provide a way to promote
the interaction between control and mathematics.

## Acknowledgement

We would like to express our thanks to Dr. Jingyao Zhang, Beihang University
(BUAA), for his helpful discussion on the development of Theorem 9.

##  Appendix A Proof of Theorem 6

###### Proof:

Two steps are included to develop this proof.

Step (i): Let us prove that for any $\bm{Y}_{d}\in\mathcal{Y}_{T}$,
$\bm{U}_{\infty}$ exists and satisfies
$\bm{U}_{\infty}\in\mathcal{U}_{\mathrm{ILC}}(\bm{Y}_{d})$.

From (2) and (20), we can arrive at

$\bm{U}_{k+1}=\left(I-\bm{K}\bm{G}\right)\bm{U}_{k}+\bm{K}\bm{Y}_{d},\quad\forall
k\in\mathbb{Z}_{+}.$ (53)

By combining two results of Lemma 1 with (53), we can derive

$\displaystyle\bm{U}_{k+1}$
$\displaystyle=\left[I-\bm{K}\left(\bm{H}_{1}\bm{F}_{1}^{\mathrm{T}}+\bm{H}_{2}\bm{F}_{2}^{\mathrm{T}}\right)\bm{G}\right]\bm{U}_{k}+\bm{K}\bm{Y}_{d}$
(54)
$\displaystyle=\left(I-\bm{K}\bm{H}_{1}\bm{F}_{1}^{\mathrm{T}}\bm{G}\right)\bm{U}_{k}+\bm{K}\bm{Y}_{d},\quad\forall
k\in\mathbb{Z}_{+}.$

Owing to $\mathrm{rank}\left(\bm{F}_{1}^{\mathrm{T}}\bm{G}\right)=m$, we
denote $\bm{G}=\left[\bm{G}_{1}~{}\bm{G}_{2}\right]$ with
$\bm{G}_{1}\in\mathbb{R}^{p\times m}$ and
$\bm{G}_{2}\in\mathbb{R}^{p\times(q-m)}$, where we consider
$\mathrm{rank}\left(\bm{F}_{1}^{\mathrm{T}}\bm{G}_{1}\right)=m$ without any
loss of generality. We correspondingly denote
$\bm{K}=\left[\bm{K}_{1}^{\mathrm{T}}~{}\bm{K}_{2}^{\mathrm{T}}\right]^{\mathrm{T}}$
with $\bm{K}_{1}\in\mathbb{R}^{m\times p}$ and
$\bm{K}_{2}\in\mathbb{R}^{(q-m)\times p}$. From the condition (21), we know
that $\bm{F}_{1}^{\mathrm{T}}\bm{G}\bm{K}\bm{H}_{1}$ is nonsingular, and thus
can develop a nonsingular matrix
$\bm{\Omega}=\left[\bm{\Omega}_{1}~{}\bm{\Omega}_{2}\right]\in\mathbb{R}^{q\times
q}$ with $\bm{\Omega}_{1}\in\mathbb{R}^{q\times m}$ and
$\bm{\Omega}_{2}\in\mathbb{R}^{q\times(q-m)}$ in the form of

$\displaystyle\bm{\Omega}_{1}$
$\displaystyle=\bm{K}\bm{H}_{1}\left(\bm{F}_{1}^{\mathrm{T}}\bm{G}\bm{K}\bm{H}_{1}\right)^{-1}=\begin{bmatrix}\bm{K}_{1}\bm{H}_{1}\left(\bm{F}_{1}^{\mathrm{T}}\bm{G}\bm{K}\bm{H}_{1}\right)^{-1}\\\
\bm{K}_{2}\bm{H}_{1}\left(\bm{F}_{1}^{\mathrm{T}}\bm{G}\bm{K}\bm{H}_{1}\right)^{-1}\end{bmatrix}$
$\displaystyle\bm{\Omega}_{2}$
$\displaystyle=\begin{bmatrix}-\left(\bm{F}_{1}^{\mathrm{T}}\bm{G}_{1}\right)^{-1}\bm{F}_{1}^{\mathrm{T}}\bm{G}_{2}\\\
I\end{bmatrix}$

of which the inverse matrix is given by
$\bm{\Omega}^{-1}\triangleq\bm{\Xi}=\left[\bm{\Xi}_{1}^{\mathrm{T}}~{}\bm{\Xi}_{2}^{\mathrm{T}}\right]^{\mathrm{T}}$
with $\bm{\Xi}_{1}\in\mathbb{R}^{m\times q}$ and
$\bm{\Xi}_{2}\in\mathbb{R}^{(q-m)\times q}$ satisfying

$\displaystyle\bm{\Xi}_{1}$
$\displaystyle=\bm{F}_{1}^{\mathrm{T}}\bm{G}=\begin{bmatrix}\bm{F}_{1}^{\mathrm{T}}\bm{G}_{1}&\bm{F}_{1}^{\mathrm{T}}\bm{G}_{2}\end{bmatrix}$
$\displaystyle\bm{\Xi}_{2}$
$\displaystyle=\begin{bmatrix}\left[-\bm{K}_{2}\bm{H}_{1}\left(\bm{F}_{1}^{\mathrm{T}}\bm{G}\bm{K}\bm{H}_{1}\right)^{-1}\bm{F}_{1}^{\mathrm{T}}\bm{G}_{1}\right]^{\mathrm{T}}\\\
\left[I-\bm{K}_{2}\bm{H}_{1}\left(\bm{F}_{1}^{\mathrm{T}}\bm{G}\bm{K}\bm{H}_{1}\right)^{-1}\bm{F}_{1}^{\mathrm{T}}\bm{G}_{2}\right]^{\mathrm{T}}\end{bmatrix}^{\mathrm{T}}.$

With $\bm{Y}_{d}\in\mathcal{Y}_{T}$ and
$\mathrm{span}\bm{G}=\mathrm{span}\bm{H}_{1}$,
$\bm{Y}_{d}=\bm{H}_{1}\bm{\eta}$ holds for some $\bm{\eta}\in\mathbb{R}^{m}$,
and thus we can leverage Lemmas 1 and 2 to derive

$\displaystyle\bm{\Xi}_{1}\bm{K}\bm{Y}_{d}$
$\displaystyle=\bm{F}_{1}^{\mathrm{T}}\bm{G}\bm{K}\bm{Y}_{d}$
$\displaystyle=\bm{F}_{1}^{\mathrm{T}}\bm{G}\bm{K}\left(\bm{H}_{1}\bm{F}_{1}^{\mathrm{T}}+\bm{H}_{2}\bm{F}_{2}^{\mathrm{T}}\right)\bm{Y}_{d}$
$\displaystyle=\bm{F}_{1}^{\mathrm{T}}\bm{G}\bm{K}\bm{H}_{1}\bm{F}_{1}^{\mathrm{T}}\bm{Y}_{d}$

and

$\displaystyle\bm{\Xi}_{2}\bm{K}\bm{Y}_{d}$
$\displaystyle=\bm{K}_{2}\bm{H}_{1}\bm{\eta}-\bm{K}_{2}\bm{H}_{1}\left(\bm{F}_{1}^{\mathrm{T}}\bm{G}\bm{K}\bm{H}_{1}\right)^{-1}\bm{F}_{1}^{\mathrm{T}}\bm{G}_{1}\bm{K}_{1}\bm{H}_{1}\bm{\eta}$
$\displaystyle~{}~{}~{}-\bm{K}_{2}\bm{H}_{1}\left(\bm{F}_{1}^{\mathrm{T}}\bm{G}\bm{K}\bm{H}_{1}\right)^{-1}\bm{F}_{1}^{\mathrm{T}}\bm{G}_{2}\bm{K}_{2}\bm{H}_{1}\bm{\eta}$
$\displaystyle=\bm{K}_{2}\bm{H}_{1}\bm{\eta}-\bm{K}_{2}\bm{H}_{1}\big{[}\left(\bm{F}_{1}^{\mathrm{T}}\bm{G}\bm{K}\bm{H}_{1}\right)^{-1}\bm{F}_{1}^{\mathrm{T}}\left(\bm{G}_{1}\bm{K}_{1}\right.$
$\displaystyle~{}~{}~{}\left.+\bm{G}_{2}\bm{K}_{2}\right)\bm{H}_{1}\big{]}\bm{\eta}$
$\displaystyle=0$

the combination of which yields

$\bm{\Omega}^{-1}\bm{K}\bm{Y}_{d}=\begin{bmatrix}\bm{\Xi}_{1}\bm{K}\bm{Y}_{d}\\\
\bm{\Xi}_{2}\bm{K}\bm{Y}_{d}\end{bmatrix}=\begin{bmatrix}\bm{F}_{1}^{\mathrm{T}}\bm{G}\bm{K}\bm{H}_{1}\bm{F}_{1}^{\mathrm{T}}\bm{Y}_{d}\\\
0\end{bmatrix}.$ (55)

Similarly, we can validate

$\bm{\Omega}^{-1}\bm{K}\bm{H}_{1}=\begin{bmatrix}\bm{F}_{1}^{\mathrm{T}}\bm{G}\bm{K}\bm{H}_{1}\\\
0\end{bmatrix},\quad\bm{F}_{1}^{\mathrm{T}}\bm{G}\bm{\Omega}=\begin{bmatrix}I&0\end{bmatrix}.$
(56)

By applying (55) and (56), we implement a nonsingular linear transformation of
(54) given by

$\bm{\Omega}^{-1}\bm{U}_{k}\triangleq\bm{U}_{k}^{\ast}=\begin{bmatrix}\bm{U}_{1,k}^{\ast}\\\
\bm{U}_{2,k}^{\ast}\end{bmatrix}~{}\hbox{with}~{}\left\\{\begin{aligned}
\bm{U}_{1,k}^{\ast}&=\Xi_{1}\bm{U}_{k}\in\mathbb{R}^{m}\\\
\bm{U}_{2,k}^{\ast}&=\Xi_{2}\bm{U}_{k}\in\mathbb{R}^{q-m}\end{aligned},\forall
k\in\mathbb{Z}_{+}\right.$

and can obtain

$\begin{aligned}
\bm{U}_{k+1}^{\ast}&=\bm{\Omega}^{-1}\left(I-\bm{K}\bm{H}_{1}\bm{F}_{1}^{\mathrm{T}}\bm{G}\right)\bm{\Omega}\bm{U}_{k}^{\ast}+\bm{\Omega}^{-1}\bm{K}\bm{Y}_{d}\\\
&=\begin{bmatrix}I-\bm{F}_{1}^{\mathrm{T}}\bm{G}\bm{K}\bm{H}_{1}&0\\\
0&I\end{bmatrix}\bm{U}_{k}^{\ast}+\begin{bmatrix}\bm{F}_{1}^{\mathrm{T}}\bm{G}\bm{K}\bm{H}_{1}\bm{F}_{1}^{\mathrm{T}}\bm{Y}_{d}\\\
0\end{bmatrix}\end{aligned},~{}~{}\forall k\in\mathbb{Z}_{+}$

which can be decomposed into two separate subsystems of

$\bm{U}_{1,k+1}^{\ast}=\left(I-\bm{F}_{1}^{\mathrm{T}}\bm{G}\bm{K}\bm{H}_{1}\right)\bm{U}_{1,k}^{\ast}+\bm{F}_{1}^{\mathrm{T}}\bm{G}\bm{K}\bm{H}_{1}\bm{F}_{1}^{\mathrm{T}}\bm{Y}_{d},\quad\forall
k\in\mathbb{Z}_{+}$ (57)

and of

$\bm{U}_{2,k+1}^{\ast}=\bm{U}_{2,k}^{\ast},\quad\forall k\in\mathbb{Z}_{+}.$
(58)

An immediate consequence of (58) is

$\bm{U}_{2,k}^{\ast}=\bm{U}_{2,0}^{\ast}=\bm{\Xi}_{2}\bm{U}_{0},\quad\forall
k\in\mathbb{Z}_{+}.$ (59)

By incorporating (21) into (57), we can verify

$\bm{U}_{1,\infty}^{\ast}\triangleq\lim_{k\to\infty}\bm{U}_{1,k}^{\ast}=\bm{F}_{1}^{\mathrm{T}}\bm{Y}_{d}.$
(60)

With $\bm{U}_{k}=\bm{\Omega}\bm{U}_{k}^{\ast}$, the use of (59) and (60)
ensures the existence of $\bm{U}_{\infty}$ such that

$\displaystyle\bm{U}_{\infty}$
$\displaystyle=\bm{\Omega}_{1}\bm{U}_{1,\infty}^{\ast}+\bm{\Omega}_{2}\bm{U}_{2,0}^{\ast}$
(61)
$\displaystyle=\bm{\Omega}_{1}\bm{F}_{1}^{\mathrm{T}}\bm{Y}_{d}+\bm{\Omega}_{2}\bm{\Xi}_{2}\bm{U}_{0}$
$\displaystyle=\left[I-\bm{K}\bm{H}_{1}\left(\bm{F}_{1}^{\mathrm{T}}\bm{G}\bm{K}\bm{H}_{1}\right)^{-1}\bm{F}_{1}^{\mathrm{T}}\bm{G}\right]\bm{U}_{0}$
$\displaystyle~{}~{}~{}+\bm{K}\bm{H}_{1}\left(\bm{F}_{1}^{\mathrm{T}}\bm{G}\bm{K}\bm{H}_{1}\right)^{-1}\bm{F}_{1}^{\mathrm{T}}\bm{Y}_{d}$

where we also insert the following fact:

$\bm{\Omega}_{2}\bm{\Xi}_{2}=I-\bm{\Omega}_{1}\bm{\Xi}_{1}=I-\bm{K}\bm{H}_{1}\left(\bm{F}_{1}^{\mathrm{T}}\bm{G}\bm{K}\bm{H}_{1}\right)^{-1}\bm{F}_{1}^{\mathrm{T}}\bm{G}.$

From (27) and (61), $\bm{U}_{\infty}\in\mathcal{U}_{\mathrm{ILC}}(\bm{Y}_{d})$
holds for any $\bm{Y}_{d}\in\mathcal{Y}_{T}$.

Step (ii): We prove (28), and the equivalence between (29) and
$\mathrm{rank}\left(\bm{G}\right)=q$.

“$\mathcal{U}_{\mathrm{ILC}}(\bm{Y}_{d})\subseteq\mathcal{U}_{d}(\bm{Y}_{d})$:”
For any $\bm{\chi}\in\mathcal{U}_{\mathrm{ILC}}(\bm{Y}_{d})$, there exists
some $\bm{U}_{0}\in\mathbb{R}^{q}$ such that

$\displaystyle\bm{\chi}$
$\displaystyle=\left[I-\bm{K}\bm{H}_{1}\left(\bm{F}_{1}^{\mathrm{T}}\bm{G}\bm{K}\bm{H}_{1}\right)^{-1}\bm{F}_{1}^{\mathrm{T}}\bm{G}\right]\bm{U}_{0}$
$\displaystyle~{}~{}~{}+\bm{K}\bm{H}_{1}\left(\bm{F}_{1}^{\mathrm{T}}\bm{G}\bm{K}\bm{H}_{1}\right)^{-1}\bm{F}_{1}^{\mathrm{T}}\bm{Y}_{d}$

which, together with the results of Lemmas 1 and 2, leads to

$\displaystyle\bm{G}\bm{\chi}$
$\displaystyle=\left(\bm{H}_{1}\bm{F}_{1}^{\mathrm{T}}+\bm{H}_{2}\bm{F}_{2}^{\mathrm{T}}\right)\bm{G}\bm{\chi}$
$\displaystyle=\bm{H}_{1}\bm{F}_{1}^{\mathrm{T}}\bm{G}\left[I-\bm{K}\bm{H}_{1}\left(\bm{F}_{1}^{\mathrm{T}}\bm{G}\bm{K}\bm{H}_{1}\right)^{-1}\bm{F}_{1}^{\mathrm{T}}\bm{G}\right]\bm{U}_{0}$
$\displaystyle~{}~{}~{}+\bm{H}_{1}\bm{F}_{1}^{\mathrm{T}}\bm{G}\bm{K}\bm{H}_{1}\left(\bm{F}_{1}^{\mathrm{T}}\bm{G}\bm{K}\bm{H}_{1}\right)^{-1}\bm{F}_{1}^{\mathrm{T}}\bm{Y}_{d}$
$\displaystyle=\bm{H}_{1}\bm{F}_{1}^{\mathrm{T}}\bm{Y}_{d}$
$\displaystyle=\bm{Y}_{d}$

namely, $\bm{\chi}\in\mathcal{U}_{d}(\bm{Y}_{d})$. This implies
$\mathcal{U}_{\mathrm{ILC}}(\bm{Y}_{d})\subseteq\mathcal{U}_{d}(\bm{Y}_{d})$.

“$\mathcal{U}_{\mathrm{ILC}}(\bm{Y}_{d})\supseteq\mathcal{U}_{d}(\bm{Y}_{d})$:”
For any $\bm{\chi}\in\mathcal{U}_{d}(\bm{Y}_{d})$, we have
$\bm{Y}_{d}=\bm{G}\bm{\chi}$ by (26). Let us define

$\bm{U}_{0}=\bm{\chi}-\bm{K}\bm{H}_{1}\bm{\eta},\quad\forall\bm{\eta}\in\mathbb{R}^{m}.$

Since $\bm{K}\bm{H}_{1}$ is a matrix of full-column rank under the condition
(21), there exists $\bm{U}_{0}\in\mathbb{R}^{q}$ as defined above such that

$\displaystyle\left[I-\bm{K}\bm{H}_{1}\left(\bm{F}_{1}^{\mathrm{T}}\bm{G}\bm{K}\bm{H}_{1}\right)^{-1}\bm{F}_{1}^{\mathrm{T}}\bm{G}\right]\left(\bm{\chi}-\bm{U}_{0}\right)$
$\displaystyle~{}~{}~{}=\left[I-\bm{K}\bm{H}_{1}\left(\bm{F}_{1}^{\mathrm{T}}\bm{G}\bm{K}\bm{H}_{1}\right)^{-1}\bm{F}_{1}^{\mathrm{T}}\bm{G}\right]\bm{K}\bm{H}_{1}\bm{\eta}$
$\displaystyle~{}~{}~{}=0$

which is equivalent to

$\displaystyle\bm{\chi}$
$\displaystyle=\left[I-\bm{K}\bm{H}_{1}\left(\bm{F}_{1}^{\mathrm{T}}\bm{G}\bm{K}\bm{H}_{1}\right)^{-1}\bm{F}_{1}^{\mathrm{T}}\bm{G}\right]\bm{U}_{0}$
$\displaystyle~{}~{}~{}+\bm{K}\bm{H}_{1}\left(\bm{F}_{1}^{\mathrm{T}}\bm{G}\bm{K}\bm{H}_{1}\right)^{-1}\bm{F}_{1}^{\mathrm{T}}\bm{G}\bm{\chi}$
$\displaystyle=\left[I-\bm{K}\bm{H}_{1}\left(\bm{F}_{1}^{\mathrm{T}}\bm{G}\bm{K}\bm{H}_{1}\right)^{-1}\bm{F}_{1}^{\mathrm{T}}\bm{G}\right]\bm{U}_{0}$
$\displaystyle~{}~{}~{}+\bm{K}\bm{H}_{1}\left(\bm{F}_{1}^{\mathrm{T}}\bm{G}\bm{K}\bm{H}_{1}\right)^{-1}\bm{F}_{1}^{\mathrm{T}}\bm{Y}_{d}$

namely, $\bm{\chi}\in\mathcal{U}_{\mathrm{ILC}}(\bm{Y}_{d})$. Thus, we can get
$\mathcal{U}_{\mathrm{ILC}}(\bm{Y}_{d})\supseteq\mathcal{U}_{d}(\bm{Y}_{d})$.

Thanks to
$\mathcal{U}_{\mathrm{ILC}}(\bm{Y}_{d})\subseteq\mathcal{U}_{d}(\bm{Y}_{d})$
and
$\mathcal{U}_{\mathrm{ILC}}(\bm{Y}_{d})\supseteq\mathcal{U}_{d}(\bm{Y}_{d})$,
we can develop (28). From Theorem 1, $\mathcal{Y}_{T}=\mathcal{Y}_{R}$ is
equivalent to $\mathrm{rank}\left(\bm{G}\right)=q$. Thus,
$\mathcal{Y}_{T}=\mathcal{Y}_{R}$ yields
$\mathrm{rank}\left(\bm{F}_{1}^{\mathrm{T}}\bm{G}\right)=\mathrm{rank}\left(\bm{G}\right)=q$,
and consequently $\bm{F}_{1}^{\mathrm{T}}\bm{G}$ and $\bm{K}\bm{H}_{1}$ are
nonsingular, owing to which (27) collapses into

$\mathcal{U}_{\mathrm{ILC}}(\bm{Y}_{d})=\left\\{\left(\bm{F}_{1}^{\mathrm{T}}\bm{G}\right)^{-1}\bm{F}_{1}^{\mathrm{T}}\bm{Y}_{d}\right\\},\quad\forall\bm{Y}_{d}\in\mathcal{Y}_{T}.$
(62)

Since we have
$\left(\bm{F}_{1}^{\mathrm{T}}\bm{G}\right)^{-1}\bm{F}_{1}^{\mathrm{T}}=\left(\bm{G}^{\mathrm{T}}\bm{G}\right)^{-1}\bm{G}^{\mathrm{T}}$
by
$\mathrm{rank}\left(\bm{F}_{1}^{\mathrm{T}}\bm{G}\right)=\mathrm{rank}\left(\bm{G}\right)=q$,
(29) follows from (28) and (62). On the contrary, if (29) holds, then for any
$\bm{Y}_{d}\in\mathcal{Y}_{T}$, there exists a unique $\bm{U}_{d}$ to
guarantee (1) (namely, $\bm{Y}_{d}\in\mathcal{Y}_{R}$), based on which
$\mathrm{rank}\left(\bm{G}\right)=q$ follows from Lemma 3, and equivalently
$\mathcal{Y}_{T}=\mathcal{Y}_{R}$ holds. That is, we have the equivalence
between (29) and $\mathcal{Y}_{T}=\mathcal{Y}_{R}$. ∎

##  Appendix B Proof of Theorem 8

We first present a useful lemma to provide the general form of the least
squares solutions to the LAE (1) for any $\bm{Y}_{d}\notin\mathcal{Y}_{T}$.

###### Lemma 9

For the LAE (1), the set $\overline{\mathcal{U}}_{d}\left(\bm{Y}_{d}\right)$
of all its least squares solutions for any $\bm{Y}_{d}\notin\mathcal{Y}_{T}$
can be described by

$\displaystyle\overline{\mathcal{U}}_{d}\left(\bm{Y}_{d}\right)=\bigg{\\{}\bm{G}^{\\{1,3\\}}\bm{Y}_{d}+\left(I-\bm{G}^{\\{1,3\\}}\bm{G}\right)\bm{Z}\Big{|}\bm{G}^{\\{1,3\\}}$
$\displaystyle\in\widetilde{\bm{G}},$ (63)
$\displaystyle\bm{Z}\in\mathbb{R}^{q}\bigg{\\}},\quad\forall\bm{Y}_{d}$
$\displaystyle\notin\mathcal{Y}_{T}$

where the matrix set $\widetilde{\bm{G}}$ is given by

$\displaystyle\widetilde{\bm{G}}=\bigg{\\{}\bm{G}^{\\{1,3\\}}\in\mathbb{R}^{q\times
p}\Big{|}\bm{G}\bm{G}^{\\{1,3\\}}\bm{G}$ $\displaystyle=\bm{G},$ (64)
$\displaystyle\left(\bm{G}\bm{G}^{\\{1,3\\}}\right)^{\mathrm{T}}$
$\displaystyle=\bm{G}\bm{G}^{\\{1,3\\}}\bigg{\\}}.$

###### Proof:

For any $\bm{G}^{\\{1,3\\}}\in\widetilde{\bm{G}}$, we notice (64) and can
derive

$\displaystyle\left(\bm{G}\bm{G}^{\\{1,3\\}}\bm{Y}_{d}-\bm{Y}_{d}\right)^{\mathrm{T}}\left(\bm{G}\widetilde{\bm{U}}-\bm{G}\bm{G}^{\\{1,3\\}}\bm{Y}_{d}\right)$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}=\left(\bm{Y}_{d}^{\mathrm{T}}\bm{G}\bm{G}^{\\{1,3\\}}-\bm{Y}_{d}^{\mathrm{T}}\right)\left(\bm{G}\widetilde{\bm{U}}-\bm{G}\bm{G}^{\\{1,3\\}}\bm{Y}_{d}\right)$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}=\bm{Y}_{d}^{\mathrm{T}}\bm{G}\bm{G}^{\\{1,3\\}}\bm{G}\widetilde{\bm{U}}-\bm{Y}_{d}^{\mathrm{T}}\left(\bm{G}\bm{G}^{\\{1,3\\}}\right)^{2}\bm{Y}_{d}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}-\bm{Y}_{d}^{\mathrm{T}}\bm{G}\widetilde{\bm{U}}+\bm{Y}_{d}^{\mathrm{T}}\bm{G}\bm{G}^{\\{1,3\\}}\bm{Y}_{d}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}=0,\quad\forall\widetilde{\bm{U}}\in\mathbb{R}^{q},\forall\bm{Y}_{d}\in\mathbb{R}^{p}.$

Namely, $\bm{G}\bm{G}^{\\{1,3\\}}\bm{Y}_{d}-\bm{Y}_{d}$ and
$\bm{G}\widetilde{\bm{U}}-\bm{G}\bm{G}^{\\{1,3\\}}\bm{Y}_{d}$ are orthogonal
for any $\widetilde{\bm{U}}\in\mathbb{R}^{q}$ and any
$\bm{Y}_{d}\in\mathbb{R}^{p}$, with which we can deduce

$\displaystyle\left\|\bm{G}\widetilde{\bm{U}}-\bm{Y}_{d}\right\|_{2}$
$\displaystyle=\left\|\left(\bm{G}\widetilde{\bm{U}}-\bm{G}\bm{G}^{\\{1,3\\}}\bm{Y}_{d}\right)+\left(\bm{G}\bm{G}^{\\{1,3\\}}\bm{Y}_{d}-\bm{Y}_{d}\right)\right\|_{2}$
$\displaystyle=\left\|\bm{G}\widetilde{\bm{U}}-\bm{G}\bm{G}^{\\{1,3\\}}\bm{Y}_{d}\right\|_{2}+\left\|\bm{G}\bm{G}^{\\{1,3\\}}\bm{Y}_{d}-\bm{Y}_{d}\right\|_{2}$
$\displaystyle\geq\left\|\bm{G}\bm{G}^{\\{1,3\\}}\bm{Y}_{d}-\bm{Y}_{d}\right\|_{2},\quad\forall\widetilde{\bm{U}}\in\mathbb{R}^{q},\forall\bm{Y}_{d}\in\mathbb{R}^{p}.$

This together with (34) ensures that
$\overline{\bm{U}}_{d}\in\overline{\mathcal{U}}_{d}\left(\bm{Y}_{d}\right)$
holds if and only if $\overline{\bm{U}}_{d}$ satisfies
$\left\|\bm{G}\overline{\bm{U}}_{d}-\bm{G}\bm{G}^{\\{1,3\\}}\bm{Y}_{d}\right\|_{2}=0$,
or equivalently,
$\overline{\bm{U}}_{d}\in\overline{\mathcal{U}}_{d}\left(\bm{Y}_{d}\right)$ is
exactly the solution to the resulting LAE as

$\bm{G}\left(\overline{\bm{U}}_{d}-\bm{G}^{\\{1,3\\}}\bm{Y}_{d}\right)=0.$
(65)

Thanks to
$\mathrm{null}\bm{G}=\mathrm{null}\left(\bm{G}^{\\{1,3\\}}\bm{G}\right)$ and
$\left(\bm{G}^{\\{1,3\\}}\bm{G}\right)^{2}=\bm{G}^{\\{1,3\\}}\bm{G}$, we can
leverage the properties of the idempotent matrices (see, e.g., Theorem 1 of
[35, Subchapter 5.8]) to obtain

$\mathrm{null}\bm{G}=\mathrm{span}\left(I-\bm{G}^{\\{1,3\\}}\bm{G}\right).$
(66)

For the LAE (65), since $\overline{\bm{U}}_{d}=\bm{G}^{\\{1,3\\}}\bm{Y}_{d}$
is a particular solution, we resort to (66) and can conclude that any solution
takes the general form of (see, e.g., Theorem 7 of [35, Subchapter 12.6])

$\overline{\bm{U}}_{d}=\bm{G}^{\\{1,3\\}}\bm{Y}_{d}+\left(I-\bm{G}^{\\{1,3\\}}\bm{G}\right)\bm{Z},\quad\forall\bm{Z}\in\mathbb{R}^{q}.$

Namely, the description of $\overline{\mathcal{U}}_{d}\left(\bm{Y}_{d}\right)$
in (63) is identical to that of (34). The proof of this lemma is complete. ∎

Based on Lemma 9, we are in position to present the proof of Theorem 8 as
follows.

###### Proof:

We next adopt two separate steps to develop this theorem.

Step (i): We prove that for any $\bm{Y}_{d}\notin\mathcal{Y}_{T}$,
$\bm{Y}_{\infty}$ and $\bm{U}_{\infty}$ exist and satisfy
$\bm{Y}_{\infty}=\bm{H}_{1}\bm{F}_{1}^{\mathrm{T}}\bm{Y}_{d}$ and
$\bm{U}_{\infty}\in\overline{\mathcal{U}}_{\mathrm{ILC}}(\bm{Y}_{d})$,
respectively.

With Theorem 4, we substitute (20) into (13) and can obtain

$\widehat{\bm{E}}_{k+1}^{C}=\widehat{\bm{E}}_{k}^{C}-\bm{F}_{1}^{\mathrm{T}}\bm{G}\bm{K}\bm{E}_{k}=\left(I-\bm{F}_{1}^{\mathrm{T}}\bm{G}\widehat{\bm{K}}\right)\widehat{\bm{E}}_{k}^{C},\quad\forall
k\in\mathbb{Z}_{+}$ (67)

where we also use $\bm{\Psi}_{k}=-\Delta\bm{U}_{k}$, $\forall
k\in\mathbb{Z}_{+}$, $\bm{K}=\bm{\widehat{K}}\bm{F}^{\mathrm{T}}_{1}$ and
$\widehat{\bm{E}}_{k}^{C}=\bm{F}_{1}^{\mathrm{T}}\bm{E}_{k}$. With the
condition (22), $\lim_{k\to\infty}\widehat{\bm{E}}_{k}^{C}=0$ follows directly
from (67). This, together with (19) and Lemma 1, yields

$\displaystyle\lim_{k\to\infty}\bm{E}_{k}$
$\displaystyle=\bm{H}\lim_{k\to\infty}\widehat{\bm{E}}_{k}=\left[\bm{H}_{1}~{}\bm{H}_{2}\right]\left[\begin{matrix}0\\\
\bm{F}_{2}^{\mathrm{T}}\bm{E}_{0}\end{matrix}\right]=\bm{H}_{2}\bm{F}_{2}^{\mathrm{T}}\left(\bm{Y}_{d}-\bm{G}\bm{U}_{0}\right)$
(68)
$\displaystyle=\left(I-\bm{H}_{1}\bm{F}_{1}^{\mathrm{T}}\right)\bm{Y}_{d}.$

As a consequence of (68),
$\bm{Y}_{\infty}=\lim_{k\to\infty}\left(\bm{Y}_{d}-\bm{E}_{k}\right)=\bm{H}_{1}\bm{F}_{1}^{\mathrm{T}}\bm{Y}_{d}$
is immediate, namely, (35) holds.

From (20), we can also deduce

$\displaystyle\bm{U}_{k+1}$
$\displaystyle=\bm{U}_{k}+\widehat{\bm{K}}\bm{F}_{1}^{\mathrm{T}}\bm{E}_{k}$
(69)
$\displaystyle=\bm{U}_{k}+\widehat{\bm{K}}\bm{F}_{1}^{\mathrm{T}}\left(\bm{Y}_{d}-\bm{G}\bm{U}_{k}\right)$
$\displaystyle=\left(I-\widehat{\bm{K}}\bm{F}_{1}^{\mathrm{T}}\bm{G}\right)\bm{U}_{k}+\widehat{\bm{K}}\bm{F}_{1}^{\mathrm{T}}\bm{Y}_{d},\quad\forall
k\in\mathbb{Z}_{+}.$

Since the condition (22) ensures the nonsingularity of
$\bm{F}_{1}^{\mathrm{T}}\bm{G}\widehat{\bm{K}}$, we can construct a
nonsingular matrix
$\overline{\bm{\Omega}}=\left[\overline{\bm{\Omega}}_{1}~{}\overline{\bm{\Omega}}_{2}\right]\in\mathbb{R}^{q\times
q}$ with $\overline{\bm{\Omega}}_{1}\in\mathbb{R}^{q\times m}$ and
$\overline{\bm{\Omega}}_{2}\in\mathbb{R}^{q\times(q-m)}$ given by

$\displaystyle\overline{\bm{\Omega}}_{1}$
$\displaystyle=\widehat{\bm{K}}\left(\bm{F}_{1}^{\mathrm{T}}\bm{G}\widehat{\bm{K}}\right)^{-1}=\begin{bmatrix}\widehat{\bm{K}}_{1}\left(\bm{F}_{1}^{\mathrm{T}}\bm{G}\widehat{\bm{K}}\right)^{-1}\\\
\widehat{\bm{K}}_{2}\left(\bm{F}_{1}^{\mathrm{T}}\bm{G}\widehat{\bm{K}}\right)^{-1}\end{bmatrix}$
$\displaystyle\overline{\bm{\Omega}}_{2}$
$\displaystyle=\begin{bmatrix}-\left(\bm{F}_{1}^{\mathrm{T}}\bm{G}_{1}\right)^{-1}\bm{F}_{1}^{\mathrm{T}}\bm{G}_{2}\\\
I\end{bmatrix}$

where $\mathrm{rank}\left(\bm{F}_{1}^{\mathrm{T}}\bm{G}_{1}\right)=m$ is
assumed as in the proof of Theorem 7. Then the inverse matrix of
$\overline{\bm{\Omega}}$ can be given by
$\overline{\bm{\Xi}}=\left[\overline{\bm{\Xi}}_{1}^{\mathrm{T}}~{}\overline{\bm{\Xi}}_{2}^{\mathrm{T}}\right]^{\mathrm{T}}$
with $\overline{\bm{\Xi}}_{1}\in\mathbb{R}^{m\times q}$ and
$\overline{\bm{\Xi}}_{2}\in\mathbb{R}^{(q-m)\times q}$ satisfying

$\displaystyle\overline{\bm{\Xi}}_{1}$
$\displaystyle=\bm{F}_{1}^{\mathrm{T}}\bm{G}=\begin{bmatrix}\bm{F}_{1}^{\mathrm{T}}\bm{G}_{1}&\bm{F}_{1}^{\mathrm{T}}\bm{G}_{2}\end{bmatrix}$
$\displaystyle\overline{\bm{\Xi}}_{2}$
$\displaystyle=\begin{bmatrix}\left[-\widehat{\bm{K}}_{2}\left(\bm{F}_{1}^{\mathrm{T}}\bm{G}\widehat{\bm{K}}\right)^{-1}\bm{F}_{1}^{\mathrm{T}}\bm{G}_{1}\right]^{\mathrm{T}}\\\
\left[I-\widehat{\bm{K}}_{2}\left(\bm{F}_{1}^{\mathrm{T}}\bm{G}\widehat{\bm{K}}\right)^{-1}\bm{F}_{1}^{\mathrm{T}}\bm{G}_{2}\right]^{\mathrm{T}}\end{bmatrix}^{\mathrm{T}}.$

By taking a nonsingular linear transformation of (69) with

$\overline{\bm{\Omega}}^{-1}\bm{U}_{k}\triangleq\overline{\bm{U}}_{k}^{\ast}=\begin{bmatrix}\overline{\bm{U}}_{1,k}^{\ast}\\\
\overline{\bm{U}}_{2,k}^{\ast}\end{bmatrix}~{}\hbox{with}~{}\left\\{\begin{aligned}
\overline{\bm{U}}_{1,k}^{\ast}&=\overline{\Xi}_{1}\bm{U}_{k}\in\mathbb{R}^{m}\\\
\overline{\bm{U}}_{2,k}^{\ast}&=\overline{\Xi}_{2}\bm{U}_{k}\in\mathbb{R}^{q-m}\end{aligned},\forall
k\in\mathbb{Z}_{+}\right.$

we can derive

$\begin{aligned}
\overline{\bm{U}}_{k+1}^{\ast}&=\overline{\bm{\Omega}}^{-1}\left(I-\widehat{\bm{K}}\bm{F}_{1}^{\mathrm{T}}\bm{G}\right)\overline{\bm{\Omega}}\overline{\bm{U}}_{k}^{\ast}+\overline{\bm{\Omega}}^{-1}\widehat{\bm{K}}\bm{F}_{1}^{\mathrm{T}}\bm{Y}_{d}\\\
&=\begin{bmatrix}I-\bm{F}_{1}^{\mathrm{T}}\bm{G}\widehat{\bm{K}}&0\\\
0&I\end{bmatrix}\overline{\bm{U}}_{k}^{\ast}+\begin{bmatrix}\bm{F}_{1}^{\mathrm{T}}\bm{G}\widehat{\bm{K}}\bm{F}_{1}^{\mathrm{T}}\bm{Y}_{d}\\\
0\end{bmatrix}\end{aligned},~{}~{}\forall k\in\mathbb{Z}_{+}$

which clearly consists of two separate subsystems as

$\overline{\bm{U}}_{1,k+1}^{\ast}=\left(I-\bm{F}_{1}^{\mathrm{T}}\bm{G}\widehat{\bm{K}}\right)\overline{\bm{U}}_{1,k}^{\ast}+\bm{F}_{1}^{\mathrm{T}}\bm{G}\widehat{\bm{K}}\bm{F}_{1}^{\mathrm{T}}\bm{Y}_{d},\quad\forall
k\in\mathbb{Z}_{+}$ (70)

and

$\overline{\bm{U}}_{2,k+1}^{\ast}=\overline{\bm{U}}_{2,k}^{\ast},\quad\forall
k\in\mathbb{Z}_{+}$

or equivalently,

$\overline{\bm{U}}_{2,k}^{\ast}=\overline{\bm{U}}_{2,0}^{\ast}=\overline{\bm{\Xi}}_{2}\bm{U}_{0},\quad\forall
k\in\mathbb{Z}_{+}.$ (71)

For the subsystem (70), the condition (22) ensures its solution to converge
and satisfy

$\lim_{k\to\infty}\overline{\bm{U}}_{1,k}^{\ast}=\bm{F}_{1}^{\mathrm{T}}\bm{Y}_{d}.$

With this fact and based on (71), we consider the nonsingular linear
transformation $\bm{U}_{k}=\overline{\bm{\Omega}}\overline{\bm{U}}_{k}^{\ast}$
and can arrive at the existence of $\bm{U}_{\infty}$. Further, we notice
$\overline{\bm{\Omega}}_{2}\overline{\bm{\Xi}}_{2}=I-\overline{\bm{\Omega}}_{1}\overline{\bm{\Xi}}_{1}$
and can deduce

$\displaystyle\bm{U}_{\infty}$
$\displaystyle=\overline{\bm{\Omega}}_{1}\overline{\bm{U}}_{1,\infty}^{\ast}+\overline{\bm{\Omega}}_{2}\overline{\bm{U}}_{2,0}^{\ast}$
$\displaystyle=\overline{\bm{\Omega}}_{1}\bm{F}_{1}^{\mathrm{T}}\bm{Y}_{d}+\overline{\bm{\Omega}}_{2}\overline{\bm{\Xi}}_{2}\bm{U}_{0}$
$\displaystyle=\left[I-\widehat{\bm{K}}\left(\bm{F}_{1}^{\mathrm{T}}\bm{G}\widehat{\bm{K}}\right)^{-1}\bm{F}_{1}^{\mathrm{T}}\bm{G}\right]\bm{U}_{0}+\widehat{\bm{K}}\left(\bm{F}_{1}^{\mathrm{T}}\bm{G}\widehat{\bm{K}}\right)^{-1}\bm{F}_{1}^{\mathrm{T}}\bm{Y}_{d}$

which yields
$\bm{U}_{\infty}\in\overline{\mathcal{U}}_{\mathrm{ILC}}(\bm{Y}_{d})$ for any
$\bm{Y}_{d}\notin\mathcal{Y}_{T}$ based on (36).

Step (ii): We prove that (37) holds if and only if
$\bm{H}_{1}^{\mathrm{T}}\bm{H}_{2}=0$. With Lemma 9, we only need to show that
$\widehat{\bm{K}}\left(\bm{F}_{1}^{\mathrm{T}}\bm{G}\widehat{\bm{K}}\right)^{-1}\bm{F}_{1}^{\mathrm{T}}\in\widetilde{\bm{G}}$
if and only if $\bm{H}_{1}^{\mathrm{T}}\bm{H}_{2}=0$.

Sufficiency: According to Lemma 1, we can arrive at

$\displaystyle\bm{G}\widehat{\bm{K}}\left(\bm{F}_{1}^{\mathrm{T}}\bm{G}\widehat{\bm{K}}\right)^{-1}\bm{F}_{1}^{\mathrm{T}}$
$\displaystyle=\left(\bm{H}_{1}\bm{F}_{1}^{\mathrm{T}}+\bm{H}_{2}\bm{F}_{2}^{\mathrm{T}}\right)\bm{G}\widehat{\bm{K}}\left(\bm{F}_{1}^{\mathrm{T}}\bm{G}\widehat{\bm{K}}\right)^{-1}\bm{F}_{1}^{\mathrm{T}}$
(72) $\displaystyle=\bm{H}_{1}\bm{F}_{1}^{\mathrm{T}}$

which can be employed to get

$\bm{G}\widehat{\bm{K}}\left(\bm{F}_{1}^{\mathrm{T}}\bm{G}\widehat{\bm{K}}\right)^{-1}\bm{F}_{1}^{\mathrm{T}}\bm{G}=\bm{H}_{1}\bm{F}_{1}^{\mathrm{T}}\bm{G}=\left(I-\bm{H}_{2}\bm{F}_{2}^{\mathrm{T}}\right)\bm{G}=\bm{G}.$

From (72), the use of $\bm{H}_{1}^{\mathrm{T}}\bm{H}_{2}=0$ and
$\bm{H}_{1}\bm{F}_{1}^{\mathrm{T}}+\bm{H}_{2}\bm{F}_{2}^{\mathrm{T}}=I$ yields

$\displaystyle\bm{G}\widehat{\bm{K}}\left(\bm{F}_{1}^{\mathrm{T}}\bm{G}\widehat{\bm{K}}\right)^{-1}\bm{F}_{1}^{\mathrm{T}}$
$\displaystyle=\left(\bm{H}_{1}\bm{F}_{1}^{\mathrm{T}}+\bm{H}_{2}\bm{F}_{2}^{\mathrm{T}}\right)^{\mathrm{T}}\bm{H}_{1}\bm{F}_{1}^{\mathrm{T}}$
$\displaystyle=\bm{F}_{1}\bm{H}_{1}^{\mathrm{T}}\bm{H}_{1}\bm{F}_{1}^{\mathrm{T}}$
$\displaystyle=\left(\bm{G}\widehat{\bm{K}}\left(\bm{F}_{1}^{\mathrm{T}}\bm{G}\widehat{\bm{K}}\right)^{-1}\bm{F}_{1}^{\mathrm{T}}\right)^{\mathrm{T}}.$

Thus, we have
$\widehat{\bm{K}}\left(\bm{F}_{1}^{\mathrm{T}}\bm{G}\widehat{\bm{K}}\right)^{-1}\bm{F}_{1}^{\mathrm{T}}\in\widetilde{\bm{G}}$
by (64) when $\bm{H}_{1}^{\mathrm{T}}\bm{H}_{2}=0$.

Necessity: If
$\widehat{\bm{K}}\left(\bm{F}_{1}^{\mathrm{T}}\bm{G}\widehat{\bm{K}}\right)^{-1}\bm{F}_{1}^{\mathrm{T}}\in\widetilde{\bm{G}}$,
then by (64), it follows

$\left(\bm{G}\widehat{\bm{K}}\left(\bm{F}_{1}^{\mathrm{T}}\bm{G}\widehat{\bm{K}}\right)^{-1}\bm{F}_{1}^{\mathrm{T}}\right)^{\mathrm{T}}=\bm{G}\widehat{\bm{K}}\left(\bm{F}_{1}^{\mathrm{T}}\bm{G}\widehat{\bm{K}}\right)^{-1}\bm{F}_{1}^{\mathrm{T}}$

and as a consequence of (72), it is equivalent to

$\bm{F}_{1}\bm{H}_{1}^{\mathrm{T}}=\bm{H}_{1}\bm{F}_{1}^{\mathrm{T}}.$ (73)

Due to $\bm{F}_{1}^{\mathrm{T}}\bm{H}_{2}=0$, we resort to (73) and can obtain

$\bm{F}_{1}\bm{H}_{1}^{\mathrm{T}}\bm{H}_{2}=\bm{H}_{1}\bm{F}_{1}^{\mathrm{T}}\bm{H}_{2}=0$

from which $\bm{H}_{1}^{\mathrm{T}}\bm{H}_{2}=0$ follows immediately, thanks
to the full-column rank property of $\bm{F}_{1}$. ∎

##  Appendix C Proof of Theorem 9

###### Proof:

For any $\bm{Y}_{d}\in\mathcal{Y}_{T}$, we follow the proof of Theorem 7 to
obtain that (38) is equivalent to

$\bm{U}_{1,k}^{\ast}\left\\{\begin{aligned}
&\neq\bm{F}_{1}^{\mathrm{T}}\bm{Y}_{d},&\forall k&\leq\nu_{1}-1\\\
&=\bm{F}_{1}^{\mathrm{T}}\bm{Y}_{d},&\forall
k&\geq\nu_{1}.\end{aligned}\right.$ (74)

Since we can rewrite (57) as

$\bm{U}_{1,k+1}^{\ast}-\bm{F}_{1}^{\mathrm{T}}\bm{Y}_{d}=\left(I-\bm{F}_{1}^{\mathrm{T}}\bm{G}\bm{K}\bm{H}_{1}\right)\left(\bm{U}_{1,k}^{\ast}-\bm{F}_{1}^{\mathrm{T}}\bm{Y}_{d}\right),\quad\forall
k\in\mathbb{Z}_{+}$

we consequently have

$\bm{U}_{1,k}^{\ast}-\bm{F}_{1}^{\mathrm{T}}\bm{Y}_{d}=\left(I-\bm{F}_{1}^{\mathrm{T}}\bm{G}\bm{K}\bm{H}_{1}\right)^{k}\left(\bm{U}_{1,0}^{\ast}-\bm{F}_{1}^{\mathrm{T}}\bm{Y}_{d}\right),\quad\forall
k\in\mathbb{Z}_{+}.$ (75)

Clearly, we can achieve (74) for (75) if and only if (40) holds, which is
equivalent to (39) by properties of nilpotent matrices.

In the same way, we can develop the equivalent relationships among (41), (42),
and (43) by applying Theorem 8 and further exploiting its proof, for which the
details are thus omitted. ∎

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