It is typically proven in adaptive control that asymptotic stabilization and tracking holds, and that at best a bounded-noise bounded-state property is proven.
Recently, it has been shown in both the pole-placement control and the $d$-step ahead control settings that
if, as part of the adaptive controller, a
parameter estimator based on the original projection algorithm is used
and the parameter estimates
are restricted
to a convex
the closed-loop system experiences linear-like behavior:
exponential stability, a bounded gain on the noise in every $p$-norm, and a convolution bound on the exogenous inputs;
this can be leveraged to provide tolerance to unmodelled dynamics and plant parameter time-variation.
In this paper, we extend
the approach to
the more general Model Reference Adaptive Control (MRAC)
problem and demonstrate that
we achieve the same desirable linear-like closed-loop properties.

§ INTRODUCTION

Adaptive control is an approach used to deal with systems
with uncertain and/or time-varying
parameters. In the classical approach to adaptive control, one combines a
linear time-invariant (LTI) compensator together with a tuning mechanism to adjust
the compensator parameters to match the plant.
The first general proofs came around 1980, e.g. see [2], [3], [17], [19] and [20].
However, the original controllers
are typically not robust to unmodelled dynamics, do not tolerate time-variations well,
have poor transient behavior and do not handle noise/disturbances well, e.g.
see [21].
During the following two decades, a good deal of research was carried out
to alleviate these shortcomings; a number of small controller design changes were proposed, such as the use of signal normalization,
deadzones and $\sigma$-modification, e.g.
see [5], [6], [10], [9],
and [25];
also, simply using projection onto a convex set
of admissible parameters turned out to be powerful,
e.g. see [8], [18], [26], [27] and [28].
However, in general these redesigned
controllers provide asymptotic stability
and not exponential stability, with no bounded
gain on the noise[An exception is the work
of Ydstie [28] where a bounded gain is proven.];
that being said, some of them, especially those using projection,
provide a bounded-noise bounded-state property,
as well as tolerance of some degree
of unmodelled dynamics and/or time-variations.

LTI plants,
in both the $d$-step ahead control setting [12], [15], [16],
and the pole-placement control setting [13], [14],
a new approach has been proposed
which not only provides exponential
stability and a bounded gain on the noise, but also a
convolution bound on the exogenous inputs;
the resulting
convolution bound is leveraged to prove
to a degree of time-variations and to a degree of unmodelled
dynamics [23].
As far as the authors are aware, such
linear-like convolution bounds have never before been
proven in the adaptive setting.
The key idea is to use the
original projection algorithm in conjunction with a restriction
of the parameter estimates to a convex set,
this convexity requirement was relaxed in
[22], [14] and [24].
The goal of the present paper is
to extend these linear-like results
in the $d$-step ahead control setting to
the more general
Model Reference Adaptive Control (MRAC) problem.

Model reference adaptive control
is an important approach to adaptive control
where a pre-designed stable reference model is
used to model the desired closed-loop behavior.
Here we build on the results
proven for the $d $-step-ahead adaptive control problem
in [15] and [16],
which is a special case of the more general MRAC problem considered here;
because we are seeking stronger closed-loop properties than what
is normally proven in the literature, more detailed analysis is needed in dealing with the MRAC setup,
the introduction of
the reference model into the analysis brings
extra complexity.
We prove that the
desirable linear-like closed-loop properties of
exponential stability, a bounded gain on the noise in every $p$-norm and
a convolution bound on the exogenous inputs,
are achieved using a model reference adaptive controller;
we also prove a stronger tracking result than what is usually found in the literature.

Notation. We use standard notation throughout the paper. We denote $\R$, $\Z$, and $\C$ as the set of real numbers, integers and complex numbers, respectively.
We will denote the Euclidean-norm of a vector and the induced norm of a matrix by
the subscript-less default notation $\|\cdot\|$.
${\mathbb S}(\R^{p\times q} ) $ denote
the set of $\R^{p\times q} $-valued sequences.
Also, $\ellb_{\infty} $ denotes the set of bounded sequences.
For a signal $f\in\ellb_\infty $, define the $\infty $-norm by
$\|f \|_\infty:= \sup_{t\in \Z} | f(t) | $.
For a closed and convex set $\Omega\subset\R^p$, let the function
\left\{ \cdot \right\}:\R^{p}
\mapsto \Omega $
denote the projection
onto the set $\Omega  $;
because the set $\Omega $ is closed and convex,
the function $\mathrm{Proj}_{\Omega} $ is well-defined.
If $\Omega\subset\R^p$ is a compact (closed and bounded) set,
we define $\|\Omega\|:=\max_{x\in\Omega}\|x\|$.
Let $I_{p} $ denote the
identity matrix of size $p$.
Define the normal vector $\e_j \in\R^{p}$
of appropriate length $p$ as

\bigl[
\underbrace{\begin{matrix}0 & \cdots & 0 \end{matrix}}_{j-1\text{ elements} } \;\; \begin{matrix}1 & 0 & \cdots & 0 \end{matrix}
\bigr]^\top.$
Last of all, for a signal
$f\in {\mathbb S}(\R ) $ which is
sufficiently well-behaved to have a $z$-transform,
we let $F(z) $ denote this quantity.

§ THE SETUP

In this paper we consider the following linear time-invariant (LTI)
discrete-time plant:

\begin{equation} \label{plant1}
\sum_{i=0}^{n} a_i y(t-i) = \sum_{i=0}^{m} b_i u(t-d-i) +w(t), \quad t\in\Z,
\end{equation}

$y(t)\in\R $ as the measured output,
$u(t)\in\R $ as the control input,
and $w(t)\in\R $ as the noise/disturbance.
The plant parameters are regularized such that $a_0=1 $,
and the system delay is exactly $d$, i.e. $b_0\neq0 $.
Associated with this plant are the polynomials
${\mathbf A}(z^{-1}):= \sum_{i=0}^{n} a_i z^{-i}
$ and  $
\quad {\mathbf B}(z^{-1}):= \sum_{i=0}^{m} b_i z^{-i}$,
the transfer function $z^{-d} \frac{{\mathbf B}(z^{-1})}{{\mathbf A}(z^{-1})} $,
and the plant parameter vector:

$$ \theta:= \begin{bmatrix}
a_1 & a_2 & \cdots & a_n & b_0 & b_1& \cdots & b_m
\end{bmatrix};

we assume that $\theta $ belongs to a known set $ {\cal S}_{ab} \subset\R^{n+m+1}$.
Observe that such a plant can be expressed in the $z$-transform domain as

\begin{equation} \label{plant2}
{\mathbf A}(z^{-1}) Y(z) = z^{-d} {\mathbf B}(z^{-1}) U(z) + W(z).
\end{equation}

The control objective is closed-loop stability and asymptotic tracking
of a given reference signal $y^*(t)\in\R $ generated as the output of a stable reference model;
more specifically,
given pre-designed polynomials
${\mathbf H}(z^{-1}):= \sum_{i=0}^{n'-d} h_i z^{-i}$ and $
{\mathbf L}(z^{-1}):= 1+\sum_{i=1}^{n'} l_i z^{-i}$
(with $ n'\leq n $),
and given
a bounded exogenous signal $r(t)\in\R $,
we utilize the following reference model expressed in the $z $-transform form:

𝐋(z^-1)Y^*(z)=z^-d𝐇(z^-1) R(z).

We assume that the roots of ${\mathbf L}(z^{-1})$ belongs to the open unit desk,
i.e. the reference model is stable.
If we define the tracking error $\varepsilon$ by

then the goal
is to drive $\varepsilon $ to zero asymptotically.

Notice that for the $d$-step-ahead control problem
the reference model is simply $Y^*(z)=z^{-d} R(z) $.

We impose the following assumptions on the set of admissible parameters.

${\cal S}_{ab}$ is closed and bounded (compact),
for each $\theta\in {\cal S}_{ab}$,
the corresponding ${\mathbf B}(z^{-1})$ has roots in the open unit disk and
the sign of $b_{0}$ is always the same.

The boundedness requirement on ${\cal S}_{ab}$ is reasonable in practical situations;
it is used here to prove uniform bounds and decay rates on the closed-loop behavior.
The constraint on the roots of ${\mathbf B}(z^{-1})$ is a requirement that the plant
be minimum phase;
this is necessary to ensure tracking of
bounded reference signals [11].
Knowledge of the sign of the high-frequency gain $b_0 $ is common
in adaptive control [4].

It is implicit in the assumptions that we know
the system delay $d$
as well as the upper bounds on the orders
of ${\mathbf A}(z^{-1})$ and ${\mathbf B}(z^{-1})$.

To proceed, we use a parameter estimator
together with an adaptive control law based on the Certainty Equivalence Principle.
It is convenient to put the plant into the so-called predictor form.
To this end,
by long division
we can find ${\mathbf F}(z^{-1})=\sum_{i=0}^{d-1} f_i z^{-i} $ and ${\boldsymbol\alpha}(z^{-1})=\sum_{i=0}^{n-1} \alpha_i z^{-i}$ that satisfy the following:

\begin{equation*}
\frac{{\mathbf L}(z^{-1})}{{\mathbf A}(z^{-1})} = {\mathbf F}(z^{-1})
+ z^{-d} \frac{{\boldsymbol\alpha}(z^{-1})}{ {\mathbf A}(z^{-1})};
\end{equation*}

if we now define
{\boldsymbol\beta}(z^{-1})= \sum_{i=0}^{m+d-1} \beta_i z^{-i}:={\mathbf F}(z^{-1}){\mathbf B}(z^{-1}),
then it is easy to verify that the following is true:

\begin{equation} \label{alpha1}
z^{-d} \frac{{\mathbf B}(z^{-1})}{{\mathbf A}(z^{-1})} = \frac{ {\boldsymbol\beta}(z^{-1})}{z^{d}
{\mathbf L}(z^{-1})-{\boldsymbol\alpha}(z^{-1})} .
\end{equation}

So comparing (<ref>) with the plant equation in (<ref>),
we are able to re-write the plant equation as

\begin{equation} \label{plantM1}
{\mathbf L}(z^{-1})[z^d Y(z)] = {\boldsymbol\alpha}(z^{-1})Y(z) + {\boldsymbol\beta}(z^{-1}) U(z)
+ \overline W(z),
\end{equation}

with $ \overline W(z):= z^d {\mathbf F}(z^{-1}) W(z) $.

Now define
a weighted sum of the system output $\overline y $ by

y(t):= y(t) + ∑_j=1^n' l_j y(t-j);

clearly the $z$-transform
of $\overline y(t) $ is
{\mathbf L}(z^{-1})Y(z) $,
so the time-domain counterpart of (<ref>)
in predictor form is

\begin{equation} \label{plantM2}
\overline y(t+d) = \phi(t)^\top \theta^* + \overline w(t),
\end{equation}

\[
\phi(t):=
\begin{bmatrix}
y(t) \\ y(t-1) \\ \vdots \\ y(t-n+1) \\ u(t) \\ u(t-1) \\ \vdots \\ u(t-m-d+1)
\end{bmatrix},
\qquad
\theta^*:=
\begin{bmatrix}
\alpha_0 \\ \alpha_1 \\ \vdots \\ \alpha_{n-1} \\
\beta_0 \\ \beta_1 \\ \vdots \\ \beta_{m+d-1}
\end{bmatrix}.
% \qquad
\]

Let ${\cal S}_{\alpha\beta} \subset \R^{n+m+d} $ denote the set of admissible $\theta^* $
that arise from the original plant parameters which lie in ${\cal S}_{ab} $;
it is clear that the associated mapping from ${\cal S}_{ab} $ to ${\cal S}_{\alpha\beta}$
is analytic, so the compactness of ${\cal S}_{ab} $ means that ${\cal S}_{\alpha\beta}$ is compact as well.
Furthermore, it is easy to see that $\beta_0=b_0 $.
It is convenient that the set of admissible parameters in the new parameter space be convex and closed;
so at this point let ${\cal S} \subset \R^{n+m+d} $ be any compact and convex set
containing ${\cal S}_{\alpha\beta}$ for which the $(n+1)$th element is never zero; the convex hull of ${\cal S}_{\alpha\beta}$ will do,
although it may be more convenient to use a
hyperrectangle (for projection purposes).
We will show an example on obtaining such a set in the simulation section.

Now define
$\overline Y^*(z):= {\mathbf L}(z^{-1})Y^*(z); $
then the model reference control law is
given by

\begin{equation*}
\overline y^*(t+d) = \phi(t)^\top \theta^*.
\end{equation*}

In the absence of noise, and assuming the controller is applied for all
$t\in\Z $,
we can show that
we have $y(t)=y^*(t) $ for all $t\in\Z $.
In our case of unknown parameters, we seek an adaptive version of the control law
which is applied after some initial time,
i.e. $t\geq t_0 $.

§.§ Initialization

In most adaptive control results, the goal is to prove asymptotic behavior, so the
details of the initial condition are unimportant.
On the other hand, here we wish to obtain a bound
on the transient behavior so we must proceed carefully.
Here we adopt the approach used in
the $d$-step ahead control setting of [15] and [16].
With the definition (<ref>) in mind (and $n'\leq n $),
observe that if we wish to solve (<ref>)
for $y(\cdot)$ starting at time $t_0$, then it
is clear that we need an initial condition of

\begin{multline}
\bigl[
y(t_0) \;\;
y(t_0-1) \;\;
\cdots \;\; y(t_0-n-d+2)
\nonumber
\\
\qquad
u(t_0) \;\;
u(t_0-1) \;\;
\cdots \;\; u(t_0-m-2d+2)
\bigr]^\top
\end{multline}

observe that this is sufficient information to obtain
$\phi(t_0),\phi(t_0-1),\ldots,\phi(t_0-d+1) $.

§.§ Parameter Estimation

We can re-write the plant equation (<ref>) as

\begin{equation} \label{plantM3}
\overline y(t+1) = \phi(t-d+1)^\top \theta^* + \overline w(t-d+1),
\;\; t\geq t_0.
\end{equation}

Given an estimate $\hat\theta(t) $ of $\theta^* $ at time $t$,
we define the prediction error by

\begin{equation} \label{predict1}
e(t+1):=\overline y(t+1) - \phi(t-d+1)^\top \hat\theta(t);
\end{equation}

this is a measure of the error in $\hat\theta(t) $.
A common way to obtain a new estimate is from the solution of the optimization problem

\[
\underset{\theta}{\mathrm{argmin}} \left\{
\|\theta-\hat\theta(t) \| :
\overline y(t+1)=\phi(t-d+1)^\top \theta
\right\},
\]

yielding the (ideal) Projection Algorithm:

\begin{equation}
\hat\theta(t+1)
\left\{
\begin{matrix*}[l]
\hat\theta(t)
& \phi(t-d+1)=0
\\
\hat\theta(t)
\frac{\phi(t-d+1)}{\|\phi(t-d+1)\|^2}
& \text{otherwise};
\end{matrix*}
\right.
% \nonumber
\label{orig1}
\end{equation}

at this point, we can also constrain it to ${\cal S}$ by projection. Of
course, if $\|\phi(t-d+1)\|$ is close to zero, numerical problems
may occur, so it is the norm in the literature (e.g. [4] and [3])
to add a constant to the denominator[
An exception is [1] where
the ideal algorithm (<ref>) is used and
Lyapunov stability is proven,
but a convolution bound on the exogenous inputs is not proven, and the high-frequency gain is assumed to be known.
however as pointed out in our earlier work
[13], [14] and [16],
this can lead to the loss of exponential
stability and a loss of a bounded gain on the noise.
As proposed in [13], [14] and [16],
we turn off the estimation if it is clear that the noise
is swamping the estimation error. To this end,
with $\delta\in(0,\infty] $, we turn off the estimator if the
update is
larger than $2\|{\cal S}\| + \delta$ in magnitude;

\begin{equation}
\rho(t)
1 & & \text{if } | e(t+1)|< (2\Vert {\cal S}\Vert +\delta)\|\phi(t-d+1)\|\\
0 & & \text{otherwise};
\end{matrix}\right.
\nonumber
\end{equation}

given the initial condition of $\hat\theta(t_0 )
=\theta_0 \in \R^{m+n+d}  $,
for $t\geq t_0 $ we define[If $\delta=\infty $, then we adopt the understanding that $\infty \times 0 = 0$, in which case this formula in (<ref>) collapses into
the original version (<ref>).]

Proj_S { θ̌(t+1) }.

Analyzing the closed-loop system requires a careful examination of the estimation algorithm.
First define the parameter error
by $\tilde\theta(t) := \hat\theta(t)-\theta^* $.
The following result lists properties which are
equivalent to those
of Proposition 1 in [16]
for the
$d $-step ahead adaptive control setup.

For every
$t_0\in\Z$, $ x_0\in\R^{n+m+3d-2}$, $
\theta_0 \in{\cal S}$, $ \theta\in{\cal S}_{ab}
, w\in\ellb_\infty
and $\delta\in(0,\infty] $,
when the estimator (<ref>) is applied to the plant (<ref>),
the following holds:

|e(t+1) |/ϕ(t-d+1) ,

θ̃(t) ^2
≤θ̃(τ) ^2
- 1/2e(j+1) ^2 / ϕ(j-d+1) ^2

2w(j-d+1)^2 / ϕ(j-d+1) ^2

§.§ The Control Law

With the natural partitioning

\begin{equation}
\hat\theta(t)=:
\begin{bmatrix}
\hat\alpha_0(t)
% & \hat\alpha_1(t)
& \cdots & \hat\alpha_{n-1}(t) &
\hat\beta_0(t)
% & \hat\beta_1(t)
& \cdots & \hat\beta_{m+d-1}(t)
\end{bmatrix}^\top,
\nonumber
\end{equation}

the model reference adaptive control law
(based on the Certainty Equivalence principle)

\begin{equation}
\overline y^*(t+d) = \phi(t)^\top \hat\theta(t);
\nonumber
\end{equation}

solving this for $u(t)$
and using the reference model (<ref>),
we have

-∑_i=0^n-1 α̂_i(t) y(t-i)
-∑_i=1^m+d-1 β̂_i(t) u(t-i) +

∑_i=0^n'-d h_i r(t-i)

It is convenient for analysis to
define an auxiliary
tracking error:

it is easy to show that

-ϕ(t-d)^⊤θ̃(t-d) + w(t-d),

e(t)   =
-ϕ(t-d)^⊤θ̃(t-1) + w(t-d),

as well as

\begin{equation} \label{error1}
\overline\varepsilon(t)=e(t)+ \phi(t-d)^\top \left[\hat\theta(t-1)-\hat\theta(t-d)\right],
\quad t\geq t_0+d.
\end{equation}

Observe that we can compute
$\overline \varepsilon (t),\, t\in\{t_0,t_0+1,\ldots,t_0+d-1 \} $, from $x_0,w$ and $y^* $.

In the next section we develop several models used in the analysis, after which we state and prove
our result.
The approach borrows ideas from our previous work
on the $d$-step ahead setup [16]
and extends them to the
Model Reference Adaptive Control (MRAC) case.

§ THE ANALYSIS

In the pole-placement adaptive control setup of our earlier work [14],
a key closed-loop model consists of an update equation for $\phi(t) $, with the state
matrix consisting of controller and plant estimates;
this was effective because the characteristic polynomial of this matrix is time-invariant
and has all roots in the open unit disk.
If we were to apply the same idea in our case here, then the characteristic polynomial
would have roots which are time-varying, with some at zero and the rest at the roots
of the corresponding naturally defined polynomial $\hat{\boldsymbol \beta}(t,z^{-1} ) $,
which is time varying, and it may not have roots in the open unit disk.
On the other hand, in the $d$-step ahead adaptive control setup of our earlier work [15] and [16],
these difficulties
were dealt with by constructing
three different models for use in the analysis:
a model that does not use parameter estimates but is driven by the tracking error,
a crude model to bound the size of growth of $\phi(t) $,
and a crucial model which is driven by perturbed versions of the present and past values of $\phi(\cdot) $.
Here in this paper, which deals with the more general
MRAC problem,
we construct similar, though not identical,
but they need more careful analysis than ones in the $d $-step ahead control case.

§.§ A Good Model

Here we first obtain an equation which avoids using
parameter estimates, though it is driven by the weighted sum of the tracking error $\overline\varepsilon(\cdot) $.
By extending
the idea from [15] and [16],
using the definition of $\varepsilon $ we obtain
a formula for $y(t+1) $,
using the plant equation (<ref>) we obtain
a formula for $u(t+1) $;
it is easy to see that there exists a matrix $A_g\in\R^{(n+m+d)\times(n+m+d)} $
(which depends implicitly on $\theta \in {\cal S}_{ab} $) so that the following holds:

A_g ϕ(t) +
ε(t+1) +

∑_i=0^d a_d-i ε(t+1+i)
_1 y^*(t+1)

∑_i=0^d a_d-i y^*(t+1+i) - w(t+d+1)

The characteristic polynomial of $A_g $
is $\tfrac{1}{b_0}z^{n+m+d}{\mathbf B}(z^{-1}) $,
so all of its roots are in the open unit disk.

The model in (<ref>) is similar to the good model obtained in
the analysis in the $d $-step ahead control case in [15] and [16]
where it is driven by the tracking error $\varepsilon $.
However in the case considered here we would like to obtain
a model which is, instead, driven by $\overline\varepsilon $;
this will turn out to be crucial in analyzing the closed-loop behavior.
To this end,
from (<ref>) and the definitions of $\overline y $ and $\overline y^* $,
it is easy to see that

\begin{equation}
{\cal E}(z)=\frac{1}{{\mathbf L}(z^{-1})} \overline{\cal E}(z)
\label{tfL}
\end{equation}

we can represent $\varepsilon(t) $ as the output of an
$n' $th-order system
driven by
$\overline \varepsilon $ as follows:
\zeta(t)
\begin{bmatrix}
\varepsilon(t) & \varepsilon(t-1) & \cdots & \varepsilon(t-n'+1)
\end{bmatrix}^\top,

$A_l\in\R^{n'\times n' } $ defined by

\[
\begin{smallmatrix}
-l_1 & -l_2 & \cdots & -l_{n'-1} & -l_{n'}
\\
1 & 0 & \cdots & 0 & 0
\\
0 & 1 & \cdots & 0 & 0
\\
\vdots & & \ddots & \ddots & \vdots
\\
0 & 0 & \cdots & 1 & 0
\end{smallmatrix}
\right],
\]

we have

A_l ζ(t) +

_1 ε(t+1)

Note that (<ref>) is driven on the RHS by $d+1 $ terms of $\varepsilon(\cdot) $;
but from (<ref>) we have

\begin{equation}
\varepsilon(t+1+j)=\e_1^\top \zeta(t+1+j ),\quad j=0,1,\ldots,d.
\label{barerr2}
\end{equation}

With this in mind,
we construct the following $(n'(d+1)) $th-order system
driven by $\overline \varepsilon(\cdot) $:

\begin{equation}
\left[
\begin{smallmatrix}
\zeta(t+d+2)
\\
\zeta(t+d+1)
\\
\vdots
\\
\vdots
\\
\zeta(t+2)
\end{smallmatrix}
\right]
% &
\underbrace{
\left[
\begin{smallmatrix}
A_l &   &   &   &
\\
I_{n'} &  &  &  &
\\
& I_{n'} &   &  &
\\
& & \ddots &  &
\\
&  &  & I_{n'} & 0
\end{smallmatrix}
\right]
{=:\widetilde A_l }
% x'(t)
\underbrace{
\left[
\begin{smallmatrix}
\zeta(t+d+1)
\\
\zeta(t+d)
\\
\vdots
\\
\vdots
\\
\zeta(t+1)
\end{smallmatrix}
\right]
\e_1
\overline \varepsilon(t+d+2)
\label{xi_sys_no2}
\end{equation}

At this point we can combine the models
(<ref>) and (<ref>)
together with the linking equation
to obtain a model driven by the exogenous inputs
and $\overline \varepsilon $
(rather than $\varepsilon $):

\begin{equation}
\eta(t)
% &
\frac{1}{b_0}
\e_{n+1}
\biggl[
\sum_{i=0}^{d} a_{d-i} y^*(t+1+i) - w(t+d+1)
\biggr]
\e_1 y^*(t+1),
\label{eta_2}
\end{equation}

if follows that there exists
a matrix
$\tilde B\in\R^{(n+m+d)\times(n'(d+1))}  $, which depends continuously on $\theta \in {\cal S}_{ab}  $,
to obtain
the following $(n+m+d+n'(d+1)) $th-order system:

\begin{equation}
\begin{bmatrix}
\phi(t+1)
\\
\overline\zeta(t+1)
\end{bmatrix}
\underbrace{
\begin{bmatrix}
A_g & \tilde B
\\
& \tilde A_l
\end{bmatrix}
}_{=: \widetilde A_g }
\underbrace{
\begin{bmatrix}
\phi(t)
\\
\overline\zeta(t)
\end{bmatrix}
}_{=: \overline\phi(t) }
+ \eta(t)
\e_{n+m+d+1}
\overline\varepsilon(t+d+2).
\label{goodmodel3}
\end{equation}

Before presenting an even
better model suitable for analysis
we need to analyze a couple of crude models of the closed-loop behavior.

§.§ Crude Models

At times, we will need to use crude models to bound the size of the growth of
$\phi(t) $
the size of the growth of
$\overline\phi(t) $
in terms of the exogenous inputs. Following a similar analysis
of the crude model in [15] and [16],
we now use (<ref>) to describe $y(t+1)$,
and use the control law (<ref>)
with the obtained equation for $y(t+1) $
to describe $u(t+1)$;
so, we can appropriately define matrices $A_{1}(t)$, $B_{1}(t)$ and $B_{2}(t)$ in terms of $\theta\in{\cal S}_{ab} $
and $\hat\theta(t+1)\in{\cal S} $
so that
we have the following crude model of the behavior of $\phi(\cdot) $:

B_1(t) y^*(t+d+1) +B_2(t) w(t+1),

we combine (<ref>) and (<ref>)
to obtain an equation for $\overline\phi(t) $:
we can appropriately define matrices $B_3(t),B_4(t) $ so that

+B_3(t) y^*(t+d+1) +

B_4(t) w(t+1)
_n+m+d+1 ε(t+d+2),

Now we want to find a representation for $\overline\varepsilon(t+d+2)$ on the RHS above
in terms of $\phi(t) $:
from (<ref>)
we have $\overline \varepsilon(t+d+2)=-\tilde\theta(t+2)^\top\phi(t+2)+\overline w(t+2)   $,
so we use (<ref>)
to find a representation of $\phi(t+2) $ in terms of $\phi(t) $
into (<ref>); then we can appropriately
define matrices $A_2(t), B_5(t), B_6(t), B_7(t), B_8(t) $ so that the following crude model of the behavior of $\overline\phi(\cdot) $ is obtained:

+B_5(t) y^*(t+d+1) +

B_6(t) w(t+1)
B_7(t) y^*(t+d+2) +

B_8(t) w(t+2)
_n+m+d+1 w(t+2),

Due to the compactness of ${\cal S}_{ab} $, ${\cal S}_{\alpha\beta} $ and ${\cal S} $,
we can obtain the following immediately.

There exists a constant $c_1\geq 1 $ such that
for every $t_0\in\Z $, $x_0\in\R^{n+m+3d-2} $, $\theta_0 \in{\cal S} $,
$\theta\in{\cal S}_{ab} $, $r,w\in\ellb_\infty $, and $\delta\in(0,\infty] $,
when the adaptive controller (<ref>) and (<ref>) is applied to the plant (<ref>),
the following holds:

A_1(t) ≤c_1,     A_2(t) ≤c_1,
B_1(t) ≤c_1,

B_2(t) ≤c_1,
B_5(t) ≤c_1,
B_6(t) ≤c_1,

B_7(t) ≤c_1,
B_8(t) ≤c_1,

§.§ A Better Model

The good closed-loop model (<ref>) is driven by
a future value of $\overline \varepsilon(\cdot) $.
We now combine it with the crude model (<ref>) to obtain
a new model which is driven by perturbed version of $\overline\phi(t) $,
with weights associated with the parameter estimation updates.
Before proceeding,
motivated by the form of the term in the
parameter estimator,
we define

ϕ(t-d+1)/‖ϕ(t-d+1)‖^2 e(t+1).

There exists a constant $c_2 $ so that for every $t_0\in\Z$, $ x_0\in\R^{n+m+3d-2}$, $
\theta_0\in{\cal S}$, $ \theta\in{\cal S}_{ab}$, $ r,w\in\ellb_\infty $,
and $\delta\in(0,\infty] $,
when the adaptive controller (<ref>) and (<ref>)
is applied to the plant (<ref>),
the following holds:

[A_g + Δ(t) ]
t≥t_0 ,

ν(t+j) ,

See the Appendix.

The result in Proposition <ref>
looks very similar, though not identical,
to the analysis leading up to the main result
of [15] and [16]
on the $d$-step ahead adaptive control problem.
Notice that the matrix $\widetilde A_g $
is a function of $\theta\in{\cal S}_{ab} $ and
the coefficients of ${\mathbf L}(z^{-1}) $;
it lies in a corresponding compact set ${\cal A} \subset \R^{(n+m+d+n'(d+1))\times(n+m+d+n'(d+1)) } $.
Furthermore, the eigenvalues of $\widetilde A_g $ are
at the origin, the roots of ${\mathbf L}(z^{-1}) $, and the roots of ${\mathbf B}(z^{-1}) $,
so they are all in the open unit disk;
we can use classical arguments to prove that
for the desired reference model
there exists constants $\gamma $ and $\sigma\in(0 ,1) $
so that for all $\theta\in{\cal S}_{ab} $, we have

\begin{equation}
\bigl\|  \widetilde A_g^k  \bigr\| \leq \gamma \sigma^k,\quad k\geq0.
\label{ATV1}
\end{equation}

Indeed, we can choose any $\sigma $ larger than

\[
\underline\lambda:=
\max_{\theta\in{\cal S}_{ab} }
\bigl\{|\lambda|: \lambda\in\C, {\mathbf B}(\lambda^{-1})=0 \text{ and } {\mathbf L}(\lambda^{-1})=0  \bigr\}.
\]

Equations of the form given in (<ref>) appear
in classical adaptive control approaches.
While we can view (<ref>) as a linear time-varying system,
we have to keep in mind that $\Delta(t) $ and $\bar\eta(t) $ are implicit nonlinear functions
of $\theta $, $\theta_0 $, $x_0$, $r $ and $w $.
However, this linear time-varying interpretation is very convenient for analysis;
to this end, let $\boldsymbol\Phi_A $ denote the state transition matrix of a general time-varying
square matrix $A$.
The following result is useful in analyzing our closed-loop system.

With $\sigma\in(\underline\lambda,1 ) $,
suppose that $\gamma\geq 1 $ is such that (<ref>) is satisfied for
every $\widetilde A_g \in {\cal A} $.
For every $\mu\in(\sigma,1) $, $g_0\geq0 $, $g_1\geq0 $, and

$g_2 \in \bigl[0, \frac{\mu-\sigma}{\gamma} \bigr),$

there exists a constant $\bar\gamma\geq1 $ so that for every
$\widetilde A_g\in{\cal A} $ and
$\Delta \in {\mathbb S}\bigl(\R^{(n+m+d+n'(d+1))\times(n+m+d+n'(d+1)) } \bigr) $ satisfying

\[
\sum_{j=\tau}^{t-1}
\|\Delta(j) \|
\leq
g_0 + g_1 (t-\tau)^{\frac{1}{2}}+g_2(t-\tau), \;\;
\bar t \geq t>\tau \geq \underline t,
\]

we have

$\|{\boldsymbol\Phi}_{\widetilde A_g + \Delta }(t,\tau) \|
\leq
\bar\gamma \mu^{t-\tau},
\quad \bar t \geq t>\tau \geq \underline t.$

we present the main result proving that the closed-loop system
enjoys very desirable linear-like behavior.

§ THE MAIN RESULT

For every $\delta\in(0,\infty] $ and $\lambda\in(\underline\lambda,1 ) $,
there exists a constant $c >0 $ so that
for every $t_0\in \Z $, $\theta \in {\cal S}_{ab}  $, $r,w \in \ellb_\infty $,
$ \theta_0 \in {\cal S} $, and plant initial condition
$x_0 $,
when the adaptive controller (<ref>) and
is applied to the plant (<ref>),
the following bound holds:

\begin{equation}
\|\phi(t)\|
% &
\leq
c \lambda^{t-t_0} \|x_0 \|
\sum_{j=t_0}^t c \lambda^{t-j }
(|r(j) | + |w(j)| ), \quad t\geq t_0.
\label{th_bd1}
\end{equation}

Furthermore, if $w=0 $, then

≤c (x_0 ^2 + r _∞^2 ).

The above result shows that the closed-loop
system experiences linear-like behavior. There is a uniform
exponential decay bound on the effect of the initial condition,
and a convolution bound on the effect of the exogenous inputs.
This implies that the system has a bounded gain (from $w$ and
to $y$) in every $p$-norm. For example, for $p = \infty $, we see
from the above bound that

\begin{equation}
\|\phi(t) \|
\leq
\tfrac{c}{1-\lambda} \left(
\lambda^{t-t_0} \|x_0 \|
+ \|w\|_\infty+ \|r\|_\infty,
\right)
\quad
t\geq t_0.
\nonumber
\end{equation}

In the absence of noise, most
adaptive controllers
that the tracking error is
square summable, e.g. see [4].
Here we prove a stronger result, namely,
an upper bound on the $2$-norm
in terms of the size of $x_0 $ and $r$.

Fix $\delta\in(0,\infty] $ and $\lambda\in(\underline\lambda,1) $.
Let $t_0\in \Z $, $\theta \in {\cal S}_{ab}  $, $r,w \in \ellb_\infty $,
$\theta_0 \in {\cal S} $, and $x_0\in\R^{n+m+3d-2} $ be arbitrary.
Now choose $\sigma\in(\underline\lambda,\lambda ) $.
We will analyze (<ref>) to obtain
a bound on $\overline\phi(t) $.

Before proceeding, we see that there exists $\gamma_1 $ so that for every
$\widetilde A_g \in {\cal A} $,

$\|\widetilde A_g^k \| \leq \gamma_1 \sigma^k, k\geq0.$

Also, we need to compute a bound on the sum
of $\|\Delta(\cdot ) \| $; since there are $d$ terms
on the RHS of (<ref>),
by the Cauchy-Schwarz inequality
we obtain

∑_j=τ^t-1 Δ(j)
≤d c_2
∑_j=τ+1 ^t+d-1

≤d^2 c_2
∑_j=τ+1 ^t+d-1
ν(j+1) ^2
(t-τ+d-1 )^1/2,

but $(t_2-t_1+d-1)^\frac{1}{2} \leq d(t_2-t_1)^\frac{1}{2}, t_2>t_1 $, so
incorporating this and the definition of $\nu(\cdot) $ we have

∑_j=τ^t-1 Δ(j)
≤d^3 c_2
∑_j=τ+1 ^t+d-1
ν(j+1) ^2

d^3 c_2
∑_j=τ+2 ^t+d
|e(j+1)|^2/ϕ(j-d+1) ^2

Also for ease of notation, let us define

\begin{equation}
\tilde w(t)
% &
\sum_{j=1}^{\max\{3,d+1\}}
|\overline y^*(t+d+j)|
|w(t+j)|+|\overline w(t+j)|).
\nonumber
\end{equation}

Now we consider the closed-loop system behavior. To proceed,
we partition the
timeline into two parts: one in which the noise $\overline w(\cdot) $ is small versus
$\phi(\cdot) $
and one where it is not.
To this end,
with $\mathfrak{v}>0 $ to be chosen shortly,

\[
S_{\text{\sf good}}:=
\left\{
j\geq t_0 : \phi(j) \neq0 \text{ and } \tfrac{|\overline w(j) |^2 }{\|\phi(j) \|^2 } < \mathfrak{v}
\right\},
\]

\[
S_{\text{\sf bad}}:=
\left\{
j\geq t_0 : \phi(j) =0 \text{ or } \tfrac{|\overline w(j) |^2 }{\|\phi(j) \|^2 } \geq \mathfrak{v}
\right\};
\]

clearly $\{j\in\Z : j\geq t_0 \} =S_{\text{\sf good}} \cup S_{\text{\sf bad}}$.[If the noise is zero, the $S_{\text{\sf good}}$ may be the whole timeline $[t_0,\infty) $.]
Observe that this partition implicitly
depends on $\theta\in{\cal S}_{ab} $, as well as the initial
conditions. We will easily obtain bounds on the closed-loop system behavior on $S_{\text{\sf bad}}$;
we will apply Proposition <ref> to analyze the behavior on $S_{\text{\sf good}}$.
Before proceeding, we partition the timeline
into intervals which oscillate between $S_{\text{\sf good}}$
and $S_{\text{\sf bad}}$. To this end, it is easy to see
that we can define a (possibly infinite) sequence of intervals of the form $[k_i,k_{i+1}) $ satisfying:
(i) $k_0=t_0 $; (ii) $[k_i,k_{i+1}) $ either belongs to
$S_{\text{\sf good}}$ or $S_{\text{\sf bad}}$; and
(iii) if $k_{i+1}\neq \infty $ and $[k_i,k_{i+1}) $ belongs
to $S_{\text{\sf good}}$ (respectively, $S_{\text{\sf bad}}$), then the interval $[k_{i+1},k_{i+2}) $ must belong to
$S_{\text{\sf bad}}$ (respectively, $S_{\text{\sf good}}$).

Now we analyze the closed-loop behavior on each interval.

Case 1:
The behavior on $S_{\text{\sf bad}}$.

Let $j\in [k_i,k_{i+1})\subset S_{\text{\sf bad}} $ be arbitrary. In this case,
we have
either $\phi(j)=0 $ or $\tfrac{|\overline w(j) |^2}{\|\phi(j) \|^2 } \geq \mathfrak{v} $.
In either case, we have

ϕ(j) ≤1√(𝔳)
|w(j) |,
j∈[k_i,k_i+1 );

then from the crude model (<ref>)
and Proposition <ref>,
we have

\begin{equation}
\|\phi(j+1) \|
% &
\leq
\tfrac{c_1}{\sqrt{\mathfrak{v}} }
|\overline w(j) |
c_1 |\overline y^*(j+d+1) |
% \nonumber
% \\
% &\quad
c_1 |w(j+1) |,
\;\;
\nonumber
\end{equation}

combining this with (<ref>) yields:

\begin{equation}
\|\phi(j) \|
\leq
\left\lbrace
\begin{matrix*}[l]
\tfrac{1}{\sqrt{\mathfrak{v}} }
|\overline w(j) |,
& j=k_i
\\
c_1 \left(\tfrac{1}{\sqrt{\mathfrak{v}} }+1\right)
[|\overline w(j-1) |
\\
\quad
|\overline y^*(j+d) |
% \nonumber
|w(j) |],
\end{matrix*}
\right.
\nonumber
\end{equation}

Case 2:
The behavior on $S_{\text{\sf good}}$.

Suppose that $[k_i,k_{i+1} ) $ lies in $S_{\text{\sf good}}$;
notice that the bound on $\|\Delta(t) \| $ in (<ref>)
occasionally extends outside $S_{\text{\sf good}}$;
so we handle the first $d+1$ and last $d+1$ times steps separately.

To this end, first suppose that $k_{i+1}-k_i\leq 2(d+1) $;
then using the crude model on $\phi $ in (<ref>) and Proposition <ref>, it is easy to show that if we define $\gamma_2:= \left(\frac{c_1}{\lambda}\right)^{2d+2} $, then we have

ϕ(t)  ≤γ_2 λ^t-k_i ϕ(k_i)

γ_2 λ^t-j-1
(|y^*(j+d+1)|+|w(j+1)| ),

Now suppose that $k_{i+1}-k_i> 2(d+1) $.
Define $\overline k_i:= k_i+d+1 $ and $\underline k_i:= k_{i+1}-d-1 $.
By the second part of Proposition <ref> and using the facts
that $\|\tilde \theta(t) \|\leq 2\|{\cal S} \|  $
and that
$\frac{|\overline w(j)|^2 }{\|\phi(j) \|^2}< \mathfrak{v}   $ for $j\in[k_i,k_{i+1} ) $,
we obtain from (<ref>):

8S ^2+
(t-τ)^1/2 ,

k_i+1 ≥t>τ≥k_i.

If we restrict $\mathfrak{v}\leq1 $,
and define $\gamma_3:= d^3c_2\bigl(
\bigl[
8\|{\cal S} \|^2+
\bigr]^\frac{1}{2} + 2
\bigr)
then we obtain

\begin{equation}
\sum_{j=\tau}^{t-1}
\|\Delta(j)\|
% &
\leq
\gamma_3
\gamma_3
\mathfrak{v}^\frac{1}{2}
\quad \underline k_{i+1} \geq t>\tau \geq \overline k_i.
\nonumber
\end{equation}

We now apply Proposition <ref>:
set $g_0=0, g_1=\gamma_3, g_2=\gamma_3\mathfrak{v}^\frac{1}{2}, \mu=\lambda, \gamma=\gamma_1 $;
we need $\gamma_3\mathfrak{v}^\frac{1}{2}< \tfrac{\lambda-\sigma}{\gamma_1} $,
so if we set $\mathfrak{v}:= \min \left\{1,
\tfrac{1}{2} \bigl(\tfrac{\lambda-\sigma}{\gamma_3\gamma_1}\bigr)^2 \right\} $,
then from Proposition <ref>
we see that there exists a constant $\gamma_4 $ so that
the state transition
matrix $\boldsymbol\Phi_{\widetilde A_g+\Delta }(t,\tau)  $ satisfies

\begin{equation}
\|\boldsymbol\Phi_{\widetilde A_g+\Delta }(t,\tau)\|
\leq
\gamma_4 \lambda^{t-\tau},
\quad \underline k_{i+1} \geq t>\tau \geq \overline k_i.
\label{state_bd1}
\end{equation}

Before solving (<ref>), we obtain a bound on $\bar\eta(t) $;
from Proposition <ref>,
we see that
$\|\nu(t)\| \leq \sqrt{8\|{\cal S} \|^2+4\mathfrak{v} }
,\underline k_{i+1} \geq t \geq \overline k_i  $,
so there exists a constant $\gamma_5 $ so that
$\|\bar\eta(t) \|
\leq
\gamma_5
\tilde w(t),
% \quad
\;
\underline k_{i+1} \geq t \geq \overline k_i.$
Then, using the bound in (<ref>) to
solve (<ref>) we see that there exists a constant $\gamma_6 $
so that

\begin{equation}
\|\overline \phi(t) \|
% &
\leq
\gamma_6 \lambda^{t-k_i} \|\overline \phi(\overline k_i) \|
% \nonumber
% \\
% &\qquad
\sum_{j=k_i}^{t-1}
\gamma_6 \lambda^{t-j-1}
\tilde w(j),
\quad
t\in[\overline k_i,\underline k_{i+1}].
\label{good_bd5}
\end{equation}

We want to have a bound on the whole interval $[k_i,k_{i+1} ) $,
and we would like it to be in terms of $\phi $ instead of $\overline\phi $.
First, we
use the crude model on $\overline\phi(\cdot) $ in (<ref>)
and Proposition <ref> to find bounds
on $\|\overline\phi(t) \| $ for $t\in[k_i, \overline k_i  ] $ and
for $t\in[\underline k_{i+1}, k_{i+1}  ] $,
and on $\|\overline\phi(\overline k_i) \|$ in terms of $\|\overline\phi(k_i) \|$
and combine them
with (<ref>)
to see that there exists a constant $\gamma_7 $
so that

\begin{equation}
\|\overline \phi(t) \|
% &
\leq
\gamma_7 \lambda^{t-k_i} \|\overline \phi(k_i) \|
% \nonumber
% \\
% &\qquad
\sum_{j=k_i}^{t-1}
\gamma_7 \lambda^{t-j-1}
\tilde w(j),
\quad
t\in[ k_i, k_{i+1}].
\label{good_bd6}
\end{equation}

Next we obtain a bound in terms of $\phi$.
It is obvious that $ \|\phi(t) \|\leq\|\overline\phi(t) \|,t\in[ k_i, k_{i+1}]  $.
Then from the definitions of $\overline\phi(\cdot),\overline\zeta(\cdot)  $ and
$\zeta(\cdot) $, it is easy to see that there exists a constant $c_3 $ such that

$\|\overline \phi(k_i) \|
\leq
\sum_{j=1}^{d+1}
\|\phi(k_i+j) \|
\sum_{j=-n'+2}^{d+1} |y^*(k_i+j)|;$

use the crude model on $\phi(\cdot) $ in (<ref>) and Proposition <ref>
to obtain bounds on $\|\phi(k_i+j) \|,j=1,2,\ldots,d+1 $,
in terms of $\|\phi(k_i) \|$.
Incorporating all of the above into (<ref>) and
after simplification, we
see that there exists a constant $\gamma_8 $ so that

≤γ_8 λ^t-k_i  ϕ(k_i)

γ_8 λ^t-j-1
∑_q=1^n'-2 |y^*(k_i-q)|,
t∈[ k_i, k_i+1],

which we combine with (<ref>)
to conclude Case 2.

We now glue together the bounds obtained on $S_{\text{\sf good}}$ and $S_{\text{\sf bad}}$
obtain a bound which holds on all of $[t_0,\infty )$
using the identical argument used in gluing together similar
bounds in the proof of Theorem 1 of [16].
Last of all, we simplify the resulting
quantity by using causality arguments
to remove extraneous terms, and end up with
the bound in (<ref>).

Finally we prove asymptotic tracking.
Suppose that $w=0$; then we see from the estimation algorithm
that in this case, $\rho(t)=1 \Leftrightarrow \|\phi(t-d) \|\neq0 $.
So from (<ref>), the first part of Proposition <ref>, and the Cauchy-Schwarz inequality,
we have

\[
\rho(t-1)
\frac{\overline \varepsilon(t)^2}{\|\phi(t-d) \|^2}
\leq
d \sum_{j=0,\|\phi(t-j-d) \|\neq0}^{d-1}
\frac{e(t-j)^2}{\|\phi(t-j-d) \|^2};
\]

so by the second part of Proposition <ref>
we can see that

∑_t=t_0+d,ϕ(t-d) ≠0 ^∞ε(t)^2/ϕ(t-d) ^2

Then it is easy to see
by the boundedness of $\phi $ proven in (<ref>)
and by the fact that $\overline \varepsilon(t)=0 $ when $\phi(t-d)=0 $,

\begin{equation}
\sum_{t=t_0+d}^\infty
\overline \varepsilon(t)^2
\leq
8d^2\|{\cal S}\|^2\max_{j\geq t_0-d} \|\phi(j) \|^2
\leq
\left(\tfrac{4d\|{\cal S} \|c}{1-\lambda}\right)^2
[ \|x_0 \|^2+ \|r\|_\infty^2].
\nonumber
\end{equation}

But $\varepsilon $ and $\overline\varepsilon $
are related by a stable transfer function
(see (<ref>)),
so if we apply Parseval's Theorem then we
obtain a bound on $\varepsilon $ of the form
given in (<ref>).

§ ROBUSTNESS

It turns out that the convolution bounds
proven in Theorem <ref> will guarantee robustness to
a degree of time-variations and unmodelled dynamics.
We have shown that
the corresponding model reference adaptive controller provides
a convolution bound with gain $c$
and decay rate $\lambda$ when
applied to the time-invariant nominal plant; we can apply
Theorems 1 and 2 of [23] to
show that, in the presence of a degree of time-variation
(slow enough parameter time-variations and/or occasional
jumps) and small enough unmodelled dynamics,
the controller still provides linear-like properties.
Furthermore, we can also obtain
bounds on the average tracking error both
in the case of no noise under slow time-variations,
as well as in the noisy case, by adapting
arguments used
in the proofs of Theorems 4 and 5 of [16].

§ A SIMULATION EXAMPLE

We now provide a simulation example to illustrate the
results of this paper. Consider the time-varying plant

\begin{multline*}
y(t + 1) =
-a_1(t)y(t) - a_2(t)y(t - 1)
\\
b_0(t)u(t) + b_1(t)u(t - 1) + w(t),
\end{multline*}

with $a_1(t) \in [-2, 2], a_2(t) \in [-2, 2], b_0(t) \in [\tfrac{3}{2}, 5]$ and
$b_1(t) \in [-1, 1]$.
Note here that the delay $d=1$.
We want to apply an adaptive controller such that the closed-loop
system follows the behavior of a reference model (<ref>) with $n'=2 $;
following the discussion at the beginning of Section <ref>,
we transform the plant into the predictor form by way of long division:
we see that

$ \alpha_1(t)=a_2(t)-l_2$, $\beta_0(t)=b_0(t)$, and $\beta_1(t)=b_1(t)$.

We choose a reference model
represented by

${\mathbf L}(z^{-1}):=1-\frac{1}{2}z^{-2},
$ and $
{\mathbf H}(z^{-1}):=\frac{1}{2},$
which has poles in the open unit disk
as required;
then we can set

\begin{multline*}
{\cal S}:=
{\cal S}_{\alpha\beta}=
\biggl\{
\left[
\begin{smallmatrix}
\alpha_0\\\alpha_1\\\beta_0\\\beta_1
\end{smallmatrix}
\right] \in\R^4
\alpha_0\in[-2,2], \alpha_1\in[-\tfrac{5}{2},\tfrac{3}{2}],
\\
\beta_0\in [\tfrac{3}{2}, 5], \beta_1\in [-1, 1]
\biggr\}.
\end{multline*}

We apply the adaptive controller (<ref>) and (<ref>)
(with $\delta =\infty $)
to this plant
with the plant parameters given by:

% &
= 2 \cos(\tfrac{1}{100}t)$,
$a_2(t) = -2 \sin(\tfrac{1}{300}t)$,

% &
= \tfrac{13}{4} - \tfrac{7}{4} \cos(\tfrac{1}{125}t)$,

$b_1(t) = -\cos(\tfrac{1}{50}t)$,

and the disturbance given by:

w(t) =
\left\{
\begin{smallmatrix*}[l]
\tfrac{1}{10} \cos(10t), & 200 < t \leq 500
\\
0, & \text{otherwise}.
\end{smallmatrix*}
\right.

We set
$r(t) $ to be a unit square wave with period of $200$ steps.
We set $y(-1) = y(0) = -1, u(-1) = 0$, and the initial parameter
estimates to the midpoint of the respective intervals.
Figures <ref> and <ref> show the results.
The controller does
a good job of tracking when there is no disturbance;
the tracking degrades when the disturbance enters the system
tracking performance improves when the disturbance returns
to zero.
You can also see that the estimator tracks the
time-varying parameters fairly well.

The first plot shows both $y(t)$ (solid) and $y^*(t) $ (dashed);
the second plot shows the control input $u(t)$.

The plots show the parameter estimates $\hat\theta(t) $ (solid)
as well as the actual parameters $\theta^* $ (dashed).

§ CONCLUSION

In this paper,
we show that
a model reference adaptive controller
using a parameter estimator based on the original
projection algorithm
provides desirable linear-like closed-loop properties:
exponential stability, a bounded gain on the noise in every $p$-norm,
and a convolution bound on the exogenous inputs;
this is never found in the literature,
except in our earlier work.
This can be leveraged to directly
prove tolerance to a degree of parameter time-variations and
a degree of unmodelled dynamics.
Also, in the noise-free case, asymptotic tracking is achieved
with an explicit upper bound on the square sum of the tracking error.

We would like to extend these linear-like results
to the case when the sign of the high-frequency gain is unknown
by using multiple estimators along the lines
of [22] and [24].
At present, our proof does not extend to that case
since the key transition matrix is not deadbeat,
but we are working on an alternative proof approach.

First, define a square matrix
of size $(n+m+d+n'(d+1))$,
\left[
\begin{matrix}
I_{n+m+d} &
\\
& {\boldsymbol 0}
\end{matrix}
\right];$

then we have

\begin{equation}
\overline\phi(t)^\top P=\begin{bmatrix}
\phi(t)^\top & {\boldsymbol 0} \end{bmatrix},
\label{Peq1}
\end{equation}

$\overline\phi(t)^\top P \overline\phi(t) = \phi(t)^\top \phi(t)= \|\phi(t)\|^2 .$

Now define

\begin{equation}
\tilde\Delta(t):=\rho(t-1)
\frac{\overline\varepsilon(t)}{\Vert\phi(t-d)\Vert^2}
% \tilde B_j
\e_{n+m+d+1}
\overline \phi(t-d)^\top P
% \\
t\geq t_0+1,
\label{deltaj_def}
\end{equation}

and $\eta_0(t):= [1-\rho(t-1)]\overline \varepsilon(t), t\geq t_0 +1$;
then by using these definitions, we can represent the
in the RHS of (<ref>)

_n+m+d+1 ε(t+d+2)
Δ̃(t+d+2) ϕ(t+2)

_n+m+d+1 η_0(t+d+2) .

We now use (<ref>) (which is valid for $t\geq t_0 $)
to represent $\overline\phi(t+2) $
in the RHS of (<ref>)
in terms $\overline\phi(t) $;
so if we do this and incorporate the result into (<ref>),
then we are now able to rewrite the model (<ref>)
in the desired form of (<ref>)
by defining

and by grouping
the remaining terms (containing exogenous signals) and

_n+m+d+1 η_0(t+d+2)

A_2(t+1) _n+m+d+1 w(t+2)+

_n+m+d+1 w(t+3)

concluding the first part of the proof.

We now prove the desired bound on $\Delta(t) $.
By (<ref>)
we obtain from (<ref>):
$\|\tilde\Delta(t) \| \leq \rho(t-1)
\frac{|\overline\varepsilon(t)|}{\Vert\phi(t-d)\Vert},
\; t\geq t_0+1; $
from (<ref>) and the first part of Proposition <ref>, we can see that

Δ̃(t)  ≤ρ(t-1)

Then from (<ref>),
using the bound in (<ref>) and Proposition 2,
we can easily show that there exist a constant so that (<ref>) is proven.

Finally, we prove the bound on $\bar\eta $.
From (<ref>) and the definition of $\rho(\cdot) $,
it is easy to show that

\leq
\bigl(
1+ \tfrac{4\|{\cal S}\| }{\delta}
\bigr)
|\overline w(t-d) |,
% \quad
\;
t\geq t_0 +d ;$

by incorporating this bound
and using the definition of $\eta(t) $ in (<ref>)
along with the bound in (<ref>)
into (<ref>),
it is easy to see that there exists a constant
so that we obtain the desired bound (<ref>).

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# Classification-Aware Neural Topic Model Combined With Interpretable Analysis
- For Conflict Classification

Tianyu Liang1, Yida Mu 1, Soonho Kim2, Darline Larissa Kengne Kuate2, Julie
Lang2, Rob Vos2, Xingyi Song1
1Department of Computer Science, University of Sheffield, Sheffield, United
Kingdom
2International Food Policy Research Institute, Washington, DC, USA
{tliang9, y.mu<EMAIL_ADDRESS>

###### Abstract

A large number of conflict events are affecting the world all the time. In
order to analyse such conflict events effectively, this paper presents a
Classification-Aware Neural Topic Model (CANTM-IA) for Conflict Information
Classification and Topic Discovery. The model provides a reliable
interpretation of classification results and discovered topics by introducing
interpretability analysis. At the same time, interpretation is introduced into
the model architecture to improve the classification performance of the model
and to allow interpretation to focus further on the details of the data.
Finally, the model architecture is optimised to reduce the complexity of the
model.

##  1 Introduction

Hundreds of conflicts break out every day around the world, many of which have
a major impact on the world’s political and economic situation. A recent
example is Ukraine Crisis, which has caused energy scarcity in Europe, a
reduction in world food production and many other repercussions. For
governments and institutions such as the IFPRI, the impact of conflict events
can be greatly reduced if they are classified, analysed and responded to in
the shortest possible time.

Our goal is to develop a deep learning model suitable for the classification
of conflict information. This model should be able to classify conflict
categories and discover category-related topics. Most importantly, the model
must have high reliability as the consequences of conflicting information are
often very serious. We therefore want to combine text classification, topic
modelling and interpretable analysis to solve the problem.

Text classification assigns category labels to different texts for the purpose
of distinguishing textual information. Recurrent neural networks (RNNs),
convolutional neural networks (CNNs) and graph neural networks (GNNs) have all
been applied to text classification tasks (Tai et al., 2015; Zhu et al., 2015;
Cheng et al., 2016; Kalchbrenner et al., 2014; Kim, 2014; Johnson and Zhang,
2015; Peng et al., 2018). More recently, Sun et al. (2019) provides a fine-
tuned BERT-based pre-training model (Devlin et al., 2019) for text
classification tasks generic solution with new state-of-the-art results on
eight extensively studied text classification datasets.

The topic model is designed to automatically find a range of topics and topic
words from a collection of documents. One of the most classic topic models is
latent dirichlet allocation (LDA) (Blei et al., 2003), which is an
unsupervised, non-hierarchical model. Many subsequent research has been based
on LDA, such as the Hierarchical Latent Dirichlet Allocation (HLDA) proposed
by Griffiths et al. (2003). In 2016, Miao et al. (2016) proposed a generative
neural variational document model (NVDM), which models the likelihood of
documents using a Variational Auto-Encoder (VAE) (Kingma and Welling, 2013).
In order to purposefully uncover topic words related to the target (e.g.
sentiment), many researchers have also proposed alternative approaches. For
example, Ding et al. (2018) added topic consistency to the training as part of
the loss as well, thus making the latent variables dependent on the topic
target as well.

Neural network-based deep learning can be described as a black box, and humans
are not yet able to fully explain or peer into the entire deep learning
process. So the question arises whether humans can be trusted with the
decision-making mechanisms of such data-driven AI systems. The lack of
interpretability leads to a reduction in the reliability of deep learning,
hence the importance of interpretable analysis. In an earlier study, Koh and
Liang (2017) hoped to find parts of the training data/training points that
could be used as a basis for interpretation by introducing the influence
function. Some researchers, on the other hand, have tried to find explanations
for the prediction results from the test data itself. Such explanations can be
found by perturbing the data (Li et al., 2016), extracting attention weights
(Wiegreffe and Pinter, 2019) or calculating the saliency scores of the input
sequences (Jain et al., 2020), etc. Lei et al. (2016); Jain et al. (2020) used
a combination of generators and encoders to extract rationales.

Figure 1: The architecture of the CANTM.

##  2 Preliminary Works

Text classification and topic modelling have been important areas of research
in natural language processing. These two areas are extremely interrelated,
but few studies have effectively integrated them into a unified system. One
successful example is the CANTM model proposed by Song et al. (2021) on topic
modelling of online text messages during the Covid-19 epidemic, which is able
to effectively identify disinformation related to Covid-19 and simultaneously
classify the information, helping to address issues such as citizens’ distrust
of government and healthcare.

The architecture of CANTM is shown in Figure 1. The model is divided into
three parts, BERT embedding, the classifier-regularised VAE (M1) and the
classifier-aware VAE (M2), where the VAE architectures are used as topic
models. The model first uses a BERT pre-trained model to extract segment
embeddings $h$ from the input text sequence $x$. In the encoder part of M1,
$h$ is transformed into the parameters $\mu$ and $\sigma$ of the Gaussian
distribution via the linear layers $linear_{\mu}$ and $linear_{\sigma}$
respectively. The aim of the M1 encoder is to generate the latent variable
$z$, which can be considered as hidden topics. The M1 decoder part uses the
latent variable $z$ as input to reconstruct the bag of words of the input
text. The M1 classifier also uses the latent variable $z$ as input, and
generates classification probabilities after passing through a fully connected
layer containing a softmax activation function. Note that since the classifier
uses hidden topics as the basis for classification, it has not seen real data,
which can reduce the overfitting of the model. The architecture of M2 is
similar to M1, except that it takes the classification probabilities $\hat{y}$
output from M1 as input as well, in order to generate hidden topics $z_{s}$
guided by the classification information. Furthermore, the M2 classifier is
not used to output the final classification, but only to compute joint loss
during training. The joint loss function of CANTM is a combination of the loss
functions of its subcomponents and is calculated as

$\displaystyle\mathcal{L}=\lambda\mathcal{L}_{cls}-ELBO_{x_{bow}}-ELBO_{x_{bow},\hat{y}}$
(1) $\displaystyle-\mathbb{E}_{\hat{y}}[log\ p(x_{bow}|\hat{y})]$

CANTM has good classification and topic discovery capabilities, but it is not
fully suitable for conflict information. Firstly, it does not introduce
interpretability analysis to demonstrate the reliability of the model.
Secondly, the topics discovered by CANTM are to some extent disturbed by a
large number of neutral words present in the input text, thus making the
relevance of the discovered topic words to the category information reduced.
Moreover, the CANTM architecture has redundant parts, which affects its
computational efficiency.

##  3 Methodology

Our model is based on an improvement of CANTM, which we call Classification-
Aware Neural Topic Model Combined With Interpretable Analysis (CANTM-IA).
CANTM is used as the base model because it combines text classification and
topic modelling, which aligns with our goals. Secondly, the stacked VAE
architecture of CANTM effectively allows us to discover the hidden topics of
the target categories. In addition, topics can also be seen as an
interpretation of the classification model, which facilitates our
interpretability analysis and improvement of the model in conjunction with
rationale.

We introduced interpretability analysis specifically by calculating the
attention weights of the last layer in the BERT pre-trained model
corresponding to the CLS labels and averaging them into the saliency score of
the corresponding word piece. The magnitude of the saliency score is used as a
visual representation of the importance of different parts of the original
sample, and the parts with high saliency scores are used as the rationales of
the sample. BERT parameters are frozen during training and only the last
transformer encoding layer weights are unlocked for fine-tuning.

Afterwards, we use the saliency scores of the rationales instead of the bag of
words of the entire input sequence as the reconstruction target in the VAE
architecture. This has several advantages. First, using rationales (the part
of the input sequence with high contribution) as the reconstruction target
allows the topic model to focus more on the important information of the input
sequence, which can reduce the interference of category-irrelevant words by
the topic words and indirectly improve the classification performance of the
model. Second, since the decoder uses rationales to guide the discovery of
hidden topics and the classifier uses hidden topics for classification, it can
be argued that these rationales explain both the hidden topics and the
classification results.

In addition, there is a redundant structure in the M2 decoder part of the
CANTM model. As shown in Figure 1, $m$, as the variable that combines the
input $h$ with the classification result $\hat{y}$, already introduces
classification information for the rest of M2. That is, the process of
generating the variable $z_{s}$ has been guided by the classification
information, which generates the class-aware topics. Therefore, there is no
need to reintroduce the classification result $\hat{y}$ in the decoder part of
M2, and the purpose can be achieved by directly reconstructing the target
using $z_{s}$ as the hidden topic variable.

Combining the above optimisation methods, the modified CANTM-IA model is shown
in Figure 2.

Figure 2: The architecture of the CANTM-IA.

##  4 Experiments

###  4.1 Dataset

We use The Armed Conflict Location & Event Data Project (ACLED), a
disaggregated data collection, analysis, and crisis mapping project, as our
source dataset Raleigh et al. (2010). The ACLED dataset collects six types of
events. We use data spanning a full 3 years between 25 June 2019 and 24 June
2022 as experimental data. Of these, the volume of data for the conflict
category Protests is 415,588, which far exceeds the volume of data for the
other categories. In order to ensure a balanced dataset, a quarter of the
data, i.e. 103,897 items, are randomly selected as the data of category
Protests for the experiment. In addition, 50,000 texts from WMT News Crawl
Dataset 111http://www.statmt.org/wmt13/training-monolingual-news-2012.tgz are
used as the out-of-domain data. The details of the experimental dataset are
shown in table 1, with 90.43% of the ACLED data and 9.57% of the regular news
data. The training set, validation set and test set are sampled from the
original dataset in a 7:1:2 ratio

| Type of
---
conflict
| Battles
---
| Explosions/
---
Remote violence
| Protests
---
| Riots
---
| Strategic
---
developments
| Violence agai-
---
nst civilians
| Out of
---
domain
| Total
---
Train | 76202 | 61222 | 72727 | 34097 | 30294 | 56329 | 35000 | 365871
Valid | 10887 | 8747 | 10390 | 4872 | 4328 | 8047 | 5000 | 52271
Test | 21772 | 17493 | 20780 | 9744 | 8656 | 16094 | 10000 | 104539
Total | 108861 | 87462 | 103897 | 48713 | 43278 | 80470 | 50000 | 522681
Table 1: Information of the experimental data set.

###  4.2 Experimental Setup

We compare our CANTM-IA model with two strong baseline models: BERT and CANTM.
For BERT model, a linear layer of dimension 300 is connected to BERT [CLS]
Token output and uses a fully-connected layer with a softmax activation
function as a classifier to output the classification results. For CANTM
model, we using a bag of words of size 500 and a hidden topic variable of
dimension 100.

Three sets of experiments are conducted to compare the choice of parameters
and the impact on CANTM-IA. The first set uses rationales with a ratio of 10%
of the number of tokens in the input text as the reconstruction target,
denoted as CANTM-IA (ratio 0.1). In the second set, this proportion is 50% and
is denoted as CANTM-IA (ratio 0.5). In addition, a fine-tuning experiment is
carried out to fine-tune the model parameters using the CANTM-IA architecture
on the trained CANTM model for only 1 epoch. The rationales used for the fine-
tuning experiment are scaled to 50% and the model is denoted as CANTM-IA
(fine-tune). Other model parameters are kept consistent with the CANTM
baseline system.

We use BERT-base-uncased in experiments, only the last transformer encoding
layer is unlocked for fine-tuning, and remaining BERT parameters are frozen
during training.

###  4.3 Results

The overall classification results are shown in Table 2. BERT is a strong
baseline with a solid classification accuracy (0.9738). On this basis, CANTM
and CANTM-IA still obtained better classification performance by using hidden
topics as the basis for classification. The best performing CANTM-IA (ratio
0.5) model achieved an accuracy of 0.9780 and an F1 score of 0.9791, which
demonstrates the effectiveness of using hidden topics as a basis for
classification. Furthermore, the classification performance of the CANTM-IA
(fine-tune) is improved over the CANTM model, even after only 1 fine-tuning.
This suggests a positive contribution of the topic model guided by rationale
to the effectiveness of text classification. The F1 scores for each sub-
category in the dataset are given in Table 3.

| Model
---
| Accuracy
---
| F-1
---
BERT (baseline) | 0.9738 | 0.9749
CANTM (baseline) | 0.9751 | 0.9760
CANTM-IA (fine-tune) | 0.9766 | 0.9775
CANTM-IA (ratio 0.1) | 0.9774 | 0.9787
CANTM-IA (ratio 0.5) | 0.9780 | 0.9791
Table 2: Comparison of the classification performance. | Model
---
| Battles
---
| Explosions/
---
Remote violence
| Protests
---
| Riots
---
| Strategic
---
developments
| Violence agai-
---
nst civilians
| Out of
---
domain
| BERT (baseline)
---
0.9583 | 0.9817 | 0.9904 | 0.9754 | 0.9693 | 0.9501 | 0.9994
| CANTM (baseline)
---
0.9628 | 0.9836 | 0.9879 | 0.9707 | 0.9736 | 0.9540 | 0.9998
| CANTM-IA (fine-tune)
---
0.9646 | 0.9854 | 0.9885 | 0.9704 | 0.9769 | 0.9575 | 0.9996
| CANTM-IA (ratio 0.1)
---
0.9633 | 0.9849 | 0.9910 | 0.9769 | 0.9777 | 0.9570 | 0.9997
| CANTM-IA (ratio 0.5)
---
0.9655 | 0.9847 | 0.9911 | 0.9772 | 0.9774 | 0.9584 | 0.9995
Table 3: F1 scores of the models for different categories of classification
results.

It should be noted that since the ACLED data is cleaned by a professional data
agency, the content of the data is to a large extent highly normative and
accurate. As a result, classification performance can be extremely good even
for the baseline model. This makes it appear that the improved model cannot
outperform the baseline model by much in terms of experimental results.
However, in this case, due to the large amount of data in the dataset, even a
subtle advantage is evident in the face of the number of accurate predictions.

Type of conflict |  Topic words in CANTM |  Topic words in CANTM-IA (ratio 0.5)
---|---|---
| Battles
---
| forces military fatalities killed positions
---
clashed militants taliban coded azerbaijan
| clashed killed clashes fire attacked
---
clash fired attack small militants
| Explosions/
---
Remote violence
| shelled forces fatalities injuries unknown
---
positions artillery smm osce airstrikes
| shelled casualties fired targeted fatalities
---
total airstrikes artillery killed carried
| Protests
---
| report protest people city government
---
members held workers gathered protested
| protest protested demonstrated held protesters
---
gathered staged demonstration workers demanding
| Riots
---
| report police rioters demonstrators demon-
---
stration clashed group people stones injured
| rioters demonstrators clashed demonstration
---
set attacked beaten beat burning fire
| Strategic
---
developments
| property destruction forces military arrested
---
township district seized movement security
| arrested set destroyed looted fire seized
---
military destruction burned forces
| Violence
---
against civilians
| killed shot man men fatality armed
---
found colonia unidentified body
| killed shot man attacked armed
---
people found beat abducted dead
Table 4: Top 10 topic words for each conflict type.

We show the top 10 topic words for each conflict category for the CANTM and
CANTM-IA (ratio 0.5) models in table 4. As can be seen, the category-related
topic words extracted by CANTM already provide a good overview for each
conflict category. However, there are still many neutral words such as
’report’, ’city’ and ’unknown’ in the CANTM results. This is due to the fact
that CANTM uses a bag of words from the complete input sequence for topic
reconstruction, which makes a large number of neutral words that appear in the
conflict text influential in the reconstruction process. The weights of the
reconstruction matrix with respect to this token is strengthened during
training, and the relevance of this word to the relevant category is
increased. In contrast, CANTM-IA cleverly reduces the influence of such
neutral words. Because CANTM-IA uses rationales as the reconstruction target,
this allows the model to focus more on the conflict-related information itself
and thus ignore irrelevant neutral words. This results in a greater
concentration of topic words that are relevant to the classification results
and more representative of the categories. It also demonstrates the
effectiveness of category-related topic words extraction and ensures the
possibility of subsequent analysis of the model.

Model |  Note of event (with highlighted rationales)
---|---
| BERT
---
(baseline)
|  Property destruction: Around 13 May 2022
---
(as reported), in Ta Tang Ku village of Pin-
laung township (coded as Pinlaung) (Pa-O
Self-Administered Zone, Shan-South state),
the PNO soldiers destroyed the Catholic
statue of the Virgin Mary and a Catholic
church. The statue is well respected.
| CANTM
---
(baseline)
| Property destruction: Around 13 May 2022
---
(as reported), in Ta Tang Ku village of Pin-
laung township (coded as Pinlaung) (Pa-O
Self-Administered Zone, Shan-South state),
the PNO soldiers destroyed the Catholic
statue of the Virgin Mary and a Catholic
church. The statue is well respected.
| CANTM-IA
---
(ratio 0.5)
| Property destruction: Around 13 May 2022
---
(as reported), in Ta Tang Ku village of Pin-
laung township (coded as Pinlaung) (Pa-O
Self-Administered Zone, Shan-South state),
the PNO soldiers destroyed the Catholic
statue of the Virgin Mary and a Catholic
church. The statue is well respected.
Table 5: Comparison of rationales extracted from the same sample. Words in red
are rationales.

We show rationale examples in table 5 that are extracted by the different
models from the same sample. It can be observed that while the rationales
extracted by CANTM from can focus on conflict-related information (”soldiers”,
”destroyed”, ”church”), there is also some irrelevant information that is
focused on (”as”, ”reported”, ”of”, ”zone”). CANTM-IA (ratio 0.5), on the
other hand, focuses precisely and intently on the conflict information itself
(”the”, ”soldiers”, ”destroyed”, ”a”, ”church”). Note that although the words
”the” and ”a” appear to be meaningless and category-independent words on their
own. However, the model incorporates contextual information. Therefore, it can
be argued that the CANTM-IA model also combines and pays some attention to
coherent semantics, which makes the rationales extracted by CANTM-IA more
coherent than previous rationales, and allows for better interpretation of the
model’s classification decisions and topic selection. The rationales
comparison experiment shows that the rationales extracted by CANTM-IA focuses
on the conflict information itself and can reasonably and effectively explain
the model’s conflict type classification results. This ensures the reliability
of the model’s classification decisions and allows CANTM-IA to provide humans
with reliable results for further analysis of conflict information to a
certain extent.

##  5 Conclusion

We proposed a Classification-Aware Neural Topic Model (CANTM-IA) for Conflict
Information Classification and Topic Discovery in this paper. The
classification results and topic models of CANTM-IA can be reliably
interpreted using rationales. Also, rationales are introduced into the topic
model to improve model performance. Finally, the model architecture has been
optimised. Compared to the baseline systems, CANTM-IA has improved predictive
performance, reliability and efficiency. Our future work will be to adapt the
model to other types of data and to refine the way in which interpretable
analysis is introduced.

##  6 Ethics and Broader Impact Statement

###  6.1 Ethics

Only publicly available
dataset222https://www.prio.org/misc/Download.aspx?file=%2fcscw%2frd%2fReplication+Data%2fReplication+data_Raleigh+et+al+47(5).zip
is used in this paper Raleigh et al. (2010). No ethical approval is required
for this work.

###  6.2 Implications

Our work has several potential practical implications:

  * •

Our model outperforms two strong baselines, BERT and CANTM, in terms of
predictive performance. It can also serve as a competitive baseline for future
research.

  * •

Our explainable neural topic model, CANTM-IA, can be utilized for other NLP
downstream tasks, such as stance detection (Mu et al., 2023) and rumor
verification (Derczynski et al., 2017), providing interpretable predictions.

## Acknowledgments

This research is partially supported by an European Union Horizon 2020 Project
(Agreement no.871042 under the scheme “INFRAIA-01-2018-2019 – Integrating
Activities for Advanced Communities”: “SoBigData++: European Integrated
Infrastructure for Social Mining and Big Data Analytics”
(http://www.sobigdata.eu)).

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