# Limited Attention Allocation in a Stochastic Linear Quadratic System with
Multiplicative Noise

Xiangyu Cui, Jianjun Gao, Lingjie Kong School of Statistics and Management,
Shanghai University of Finance and Economics, Shanghai, ChinaSchool of
Information Management and Engineering and Research Institute for
Interdisciplinary Sciences, Shanghai University of Finance and Economics,
Shanghai, ChinaSchool of Statistics and Management, Shanghai University of
Finance and Economics, Shanghai, China

###### Abstract

This study addresses limited attention allocation in a stochastic linear
quadratic system with multiplicative noise. Our approach enables strategic
resource allocation to enhance noise estimation and improve control decisions.
We provide analytical optimal control and propose a numerical method for
optimal attention allocation. Additionally, we apply our findings to dynamic
mean-variance portfolio selection, showing effective resource allocation
across time periods and factors, providing valuable insights for investors.

Key Words: limited attention allocation; linear quadratic system; factor
model; dynamic mean-variance model

##  I Introduction

The Linear Quadratic (LQ) control problem involving multiplicative noise has
been a prominent research area over the past two decades, with significant
potential applications in various fields, particularly in mathematical
finance. Notably, this problem finds applications in dynamic portfolio
selection and financial derivative pricing, as evidenced in studies such as
[1], [2], [3], [4], and [5]. Researchers have explored diverse aspects of the
LQ control problem with multiplicative noise, investigating formulations with
indefinite penalty matrices on the control and state variables (e.g., [6],
[7], [8]). Additionally, scholars have taken keen interest in exploring this
problem under various constraints, including cardinality constraint (see [9]),
no shorting constraint (see [10], [11]), cone constraint (see [12], [13]),
inequality constraint (see [14]), and linear constraint (see [15]).

While many studies on LQ control problem with multiplicative noise assume
knowledge of the noise distribution, dynamic portfolio selection practices
allow decision makers who employ learning procedures to enhance their
estimation of risky assets’ returns. This learning process incorporates expert
opinions (e.g., [16]), additional data (e.g., [17], [18]), predictive state
variables (e.g., [19]), or other information sources (e.g., analysts’
recommendations as in [20]). Bayesian methods are commonly used to update
these estimations. These dynamic portfolio selection approaches with learning
have considerable relevance to studies on the LQ control problem with
parameter uncertainty, which typically focus on additive noise (e.g., [21],
[22], [23]).

[24] and [25] have further demonstrated that during the learning process,
decision makers utilize specific attention resources, which are inherently
limited. Several works have successfully integrated the concept of limited
attention constraints into dynamic portfolio selection (e.g., [26], [27],
[28], [29]), shedding light on the importance of considering resource
constraints in decision-making. However, there is currently a research gap
regarding the application of these insights to LQ control problem.
Specifically, no existing works have discussed the LQ control problem in the
context of limited attention allocation. This unexplored area presents a
valuable opportunity to investigate the interplay between limited attention
resources and the optimal control policy in stochastic LQ systems. Such
analysis could contribute to a deeper understanding of decision-making
processes under resource constraints and potentially lead to more effective
and realistic control policies.

In this paper, we delve into the limited attention allocation problem within a
stochastic linear quadratic control framework with multiplicative noise. Our
model combines both an attention allocation policy, responsible for utilizing
the limited attention resource to update estimations of the multiplicative
noise, and a system control policy, influencing the states of the system. It
is essential to note that our studied model is not a mere abstraction of those
found in the dynamic portfolio selection literature. While [27], [28], and
[29] assume that decision makers update risky asset returns using a certain
amount of wealth rather than expending attention resources, simplifying the
decision problem, our approach explicitly considers the allocation of limited
attention resources. Moreover, [26] incorporates attention resources for
updating risky asset returns, but the study does not address portfolio
decision-making or the allocation of attention resources across different
periods. Our research significantly contributes to advancing the understanding
of the role of limited attention allocation in decision-making for stochastic
systems. By analyzing the interplay between limited attention resources,
estimation updates, and system control decisions, we aim to provide valuable
insights that can enhance decision-making processes in complex constrained
environments.

The paper is organized as follows. In Section 2, we give the basic setting of
the problem. In Section 3, we solve the problem, and derive the optimal
attention allocation policy and optimal control policy. In Section 4, we
explore the practical application of our findings in the context of dynamic
portfolio selection. In Section 5, we conclude our paper.

##  II Basic Setting

We start our discussion from the following simple LQ control problem,

$\displaystyle(P_{C})~{}~{}\min_{\\{\bm{u}_{t+}\\}_{t=0}^{T-1}}$
$\displaystyle~{}~{}E_{0}\left[\sum_{t=0}^{T-1}\begin{pmatrix}\bm{u}_{t+}&x_{t}\end{pmatrix}\bm{B}_{t}\begin{pmatrix}\bm{u}_{t+}\\\
x_{t}\end{pmatrix}+q_{T}x_{T}^{2}\right]$ s.t.
$\displaystyle~{}~{}x_{t+1}=a_{t+1}x_{t}+\bm{b}_{t+1}^{\prime}\bm{u}_{t+},$
$\displaystyle~{}~{}\bm{u}_{t+}\in\mathbb{R}^{n},$
$\displaystyle~{}~{}\bm{b}_{t+1}=\bm{c}+\bm{D}\bm{f}_{t+1}+\bm{\epsilon}_{t+1},$
$\displaystyle~{}~{}\bm{f}_{t+1}=(\bm{I}-\bm{\varPhi})\bm{f}_{t}+\bm{\eta}_{t+1},$
$\displaystyle~{}~{}x_{0}\mbox{ is given},$

where $\bm{u}_{t+}$ is a $n$-dimensional system control policy,
$\bm{B}_{t}=\begin{pmatrix}\bm{A}_{t}&\bm{p}_{t}\\\
\bm{p}_{t}^{\prime}&q_{t}\end{pmatrix}$, $\bm{A}_{t}\succeq 0$ is a
deterministic, symmetric, and semi-positive definite $n\times n$ matrix,
$\bm{p}_{t}$ is a deterministic $n$-dimensional vector, $q_{t}\geq 0$ is a
deterministic parameter, $q_{T}>0$ is also a deterministic parameter,
$\bm{B}_{t}\succeq 0$ is a deterministic, symmetric, and semi-positive
definite $(n+1)\times(n+1)$ matrix, ′ denotes the transpose operator,
$a_{t+1}>0$ is a deterministic parameter, $\bm{b}_{t+1}$ is a $n$-dimensional
random vector, $\mathbb{R}^{n}$ represents a $n$-dimensional Euclidean space,
$\bm{c}$ is a deterministic $n$-dimensional vector, $\bm{D}$ is a
deterministic $n\times k$ matrix, $\bm{f}_{t+1}$ is a $k$-dimensional random
vector ($k<n$), $\bm{\epsilon}_{t+1}$ follows a $n$-dimensional normal
distribution with zero mean and covariance matrix
$\bm{\Sigma}_{\bm{\epsilon}}$, $\bm{I}$ is the identity matrix, $\bm{\varPhi}$
is a deterministic mean-reverting matrix, $\bm{\eta}_{t+1}$ follows a
$k$-dimensional normal distribution with zero mean and covariance matrix
$\bm{\Sigma}_{\bm{\eta}}$, and the diagonal elements of
$\bm{\Sigma}_{\bm{\eta}}$ are denoted as
$\sigma_{\eta_{1}}^{2},\cdots,\sigma_{\eta_{k}}^{2}$. Different from the
classical linear quadratic control problem in [15], problem $(P_{C})$
introduces a factor structure for the random vector $\bm{b}_{t+1}$ and a mean-
reverting structure for the factor $\bm{f}_{t+1}$ to effectively reduce the
dimensionality of the randomness involved. At time $t$, the decision maker can
utilize the observed factor $\bm{f}_{t}$ to infer the distribution of
$\bm{b}_{t+1}$ and decide the control policy.

Next, we introduce the limited attention allocation structure into problem
$(P_{C})$. At time $t$, the decision maker can utilize not only the observed
factor $\bm{f}_{t}$, but also allocate their attention resources, represented
by the scalar parameter $\Lambda_{t}$, to obtain a noisy signal of
$\bm{f}_{t+1}$, denoted as $\bm{s}_{t+}(\bm{\lambda}_{t})$, according to the
following relationship,

$\displaystyle\bm{s}_{t+}(\bm{\lambda}_{t})=\bm{f}_{t+1}+\bm{v}_{t+1}(\bm{\lambda}_{t}),$

where $\bm{v}_{t+1}(\bm{\lambda}_{t})$ is a white noise with diagonal
covariance matrix

$\displaystyle\bm{\Sigma}_{\bm{v}}(\bm{\lambda}_{t})=\textrm{diag}\left(\frac{\sigma_{\eta_{1}}^{2}}{e^{\lambda_{1,t}\theta_{1}}-1},\dots,\frac{\sigma_{\eta_{k}}^{2}}{e^{\lambda_{k,t}\theta_{k}}-1}\right),$

$\bm{\lambda}_{t}=(\lambda_{1,t},\cdots,\lambda_{k,t})^{\prime}$ denotes the
attention allocation policy, which is the attention resource used for getting
the signal, the deterministic parameter $\theta_{j}$ measures the efficiency
of the decision maker’s information processing of the factor $j$.111The
variance setting for noise $\bm{v}_{t+1}(\bm{\lambda}_{t})$ follows the one
proposed in [26]. Under the setting, the reduction in the entropy of factor
$f_{j,t+1}$ stemming from the knowledge of $s_{j,t+}(\lambda_{j,t})$ is
$\displaystyle
H(f_{j,t+1})-H(f_{j,t+1}|s_{j,t+}(\lambda_{j,t}))=\frac{1}{2}\lambda_{j,t}\theta_{j},$
where $H(\cdot)$ is the entropy function, which further implies that the
variance of the factor $j$’s posterior distribution is
$\sigma_{\eta_{j}}^{2}e^{-\lambda_{j,t}\theta_{j}}$. The larger the
$\theta_{j}$, the higher the efficiency. The attention allocation policy
$\bm{\lambda}_{t}$ should be nonnegative and no larger than the limited
attention resource $\Lambda_{t}$, i.e.,
$\bm{\lambda}_{t}\in\mathbb{R}_{+}^{k},~{}\bm{\lambda}_{t}^{T}\bm{1}\leq\Lambda_{t}$.
By incorporating the limited attention allocation structure into the linear
quadratic control problem, we formulate the linear quadratic control problem
with limited attention allocation as follows,

$\displaystyle(P_{AC})~{}~{}\min_{\\{\bm{\lambda}_{t},\bm{u}_{t+}\\}_{t=0}^{T-1}}$
$\displaystyle~{}~{}E_{0}\left[\sum_{t=0}^{T-1}\begin{pmatrix}\bm{u}_{t+}&x_{t}\end{pmatrix}\bm{B}_{t}\begin{pmatrix}\bm{u}_{t+}\\\
x_{t}\end{pmatrix}+q_{T}x_{T}^{2}\right]$ s.t.
$\displaystyle~{}~{}x_{t+1}=a_{t+1}x_{t}+\bm{b}_{t+1}^{\prime}\bm{u}_{t+},$
$\displaystyle~{}~{}\bm{u}_{t+}\in\mathbb{R}^{n},$
$\displaystyle~{}~{}\bm{b}_{t+1}=\bm{c}+\bm{D}\bm{f}_{t+1}+\bm{\epsilon}_{t+1},$
$\displaystyle~{}~{}\bm{f}_{t+1}=(\bm{I}-\bm{\varPhi})\bm{f}_{t}+\bm{\eta}_{t+1},$
$\displaystyle~{}~{}\Lambda_{t+1}=\Lambda_{t}-\bm{\lambda}_{t}^{\prime}\bm{1},$
$\displaystyle~{}~{}\bm{s}_{t+}(\bm{\lambda}_{t})=\bm{f}_{t+1}+\bm{v}_{t+1}(\bm{\lambda}_{t}),$
$\displaystyle~{}~{}\bm{\lambda}_{t}\in\mathbb{R}_{+}^{k},\quad\bm{\lambda}_{t}^{\prime}\bm{1}\leq\Lambda_{t},$
$\displaystyle~{}~{}x_{0}\mbox{ and }\Lambda_{0}\geq 0\mbox{ are given}.$

The time $t+$ serves as a dummy time slot used to distinguish between the
attention allocation procedure and the system control procedure. It provides a
clear separation between these two stages, allowing the decision maker to
allocate attention resources at time $t$ and then execute the control policy
in the subsequent time slot, $t+$. All the random vectors are defined in a
probability space $(\Omega,\mathcal{F}_{T},P)$. The information set at time
$t$ is denoted as $\mathcal{F}_{t}$, which is the $\sigma$-algebra generated
by
$\\{\bm{f}_{0},\bm{\epsilon}_{1},\cdots,\bm{\epsilon}_{t},\bm{\eta}_{1},\cdots,\bm{\eta}_{t},\bm{s}_{0+}(\bm{\lambda}_{0}),\cdots$,
$\bm{s}_{(t-1)+}(\bm{\lambda}_{t-1})\\}$ and the information set after
attention allocation at time $t$ is denoted as $\mathcal{F}_{t+}$, which is
the $\sigma$-algebra generated by
$\\{\bm{f}_{0},\bm{\epsilon}_{1},\cdots,\bm{\epsilon}_{t},\bm{\eta}_{1},\cdots,\bm{\eta}_{t}$,
$\bm{s}_{0+}(\bm{\lambda}_{0}),\cdots$, $\bm{s}_{t+}(\bm{\lambda}_{t})\\}$.

To be more intuitive, we display both the attention allocation procedure and
system control procedure at period $t$ in Figure 1. During $[t,t+]$, the
decision maker chooses the optimal attention allocation policy, generates the
signal, and updates the information set and the distribution of the factors.
Then, during $[t+,t+1]$, the decision maker chooses the system control policy
based on the updated distribution, and updates the information set based on
observed new realizations of random vectors.

Figure 1: The attention allocation procedure and system control procedure at
period $t$

Problem $(P_{AC})$ is derived from real-world applications, specifically in
the domain of portfolio selection in mathematical finance. In this context,
the components of the problem have concrete interpretations: the random vector
$\bm{b}_{t+1}$ represents the excess return of risky assets, the state $x_{t}$
represents the wealth level at time $t$, and the system control policy
represents the portfolio strategy, the factor structure for $\bm{b}_{t+1}$
follows the well-known factor model, as documented in [30] and [31], and the
factors $\bm{f}_{t+1}$ exhibit mean-reverting behavior, a characteristic
documented by [32]. The investors can acquire knowledge about the returns of
factors using various sources, such as factor valuations (e.g., book-to-market
ratio), macroeconomic indicators, cross-sectional characteristics, and machine
learning techniques. This process of learning factor returns, known as factor
timing, is well-established in the literature (e.g., [33], [34], [35], [36]).
The economic foundation of the attention allocation structure is built upon
this factor timing process.

At time $t$, the decision maker knows that $\bm{f}_{t+1}$ and $\bm{b}_{t+1}$
follow multivariate normal distributions,

$\displaystyle\bm{f}_{t+1}|\mathcal{F}_{t}\sim
N(\bm{\mu}_{\bm{f},t},\bm{\Sigma}_{\bm{f},t}),\quad\bm{b}_{t+1}|\mathcal{F}_{t}\sim
N(\bm{\mu}_{\bm{b},t},\bm{\Sigma}_{\bm{b},t}),$

with

$\displaystyle\bm{\mu}_{\bm{f},t}=(\bm{I}-\bm{\varPhi})\bm{f}_{t},\quad\bm{\Sigma}_{\bm{f},t}=\bm{\Sigma}_{\bm{\eta}},\quad\bm{\mu}_{\bm{b},t}=\bm{c}+\bm{D}(\bm{I}-\bm{\varPhi})\bm{f}_{t},\quad\bm{\Sigma}_{\bm{b},t}=\bm{D}\bm{\Sigma}_{\bm{\eta}}\bm{D}^{\prime}+\bm{\Sigma}_{\bm{\epsilon}}.$

For a given attention allocation strategy $\bm{\lambda}_{t}$, the decision
maker obtains the signal $\bm{s}_{t+}(\bm{\lambda}_{t})$ at time $t+$. Then,
the decision maker can use the signal to update the conditional distributions
of $\bm{f}_{t+1}$ and $\bm{b}_{t+1}$. According to Kalman Filtering theory
(see Corollary E.3.5 in [37]), we have that $\bm{f}_{t+1}$ and $\bm{b}_{t+1}$
at time $t+$ are still multivariate normal distributions,

$\displaystyle\bm{f}_{t+1}|\mathcal{F}_{t+}\sim
N(\bm{\mu}_{\bm{f},t+}(\bm{\lambda}_{t}),\bm{\Sigma}_{\bm{f},t+}(\bm{\lambda}_{t}))\quad\bm{b}_{t+1}|\mathcal{F}_{t+}\sim
N(\bm{\mu}_{\bm{b},t+}(\bm{\lambda}_{t}),\bm{\Sigma}_{\bm{b},t+}(\bm{\lambda}_{t})),$

where

$\displaystyle\bm{\mu}_{\bm{f},t+}(\bm{\lambda}_{t})=(\bm{I}-\bm{\varPhi})\bm{f}_{t}+\bm{\Sigma}_{\bm{\eta}}(\bm{\Sigma}_{\bm{\eta}}+\bm{\Sigma}_{\bm{v}}(\bm{\lambda}_{t}))^{-1}(\bm{s}_{t+}(\bm{\lambda}_{t})-(\bm{I}-\bm{\Phi})\bm{f}_{t}),$
$\displaystyle\bm{\Sigma}_{\bm{f},t+}(\bm{\lambda}_{t})=\bm{\Sigma}_{\bm{\eta}}-\bm{\Sigma}_{\bm{\eta}}(\bm{\Sigma}_{\bm{\eta}}+\bm{\Sigma}_{\bm{v}}(\bm{\lambda}_{t}))^{-1}\bm{\Sigma}_{\bm{\eta}},$
$\displaystyle\bm{\mu}_{\bm{b},t+}(\bm{\lambda}_{t})=\bm{c}+\bm{D}(\bm{I}-\bm{\varPhi})\bm{f}_{t}+\bm{D}\bm{\Sigma}_{\bm{\eta}}(\bm{\Sigma}_{\bm{\eta}}+\bm{\Sigma}_{\bm{v}}(\bm{\lambda}_{t}))^{-1}(\bm{s}_{t+}(\bm{\lambda}_{t})-(\bm{I}-\bm{\Phi})\bm{f}_{t}),$
$\displaystyle\bm{\Sigma}_{\bm{b},t+}(\bm{\lambda}_{t})=\bm{D}[\bm{\Sigma}_{\bm{\eta}}-\bm{\Sigma}_{\bm{\eta}}(\bm{\Sigma}_{\bm{\eta}}+\bm{\Sigma}_{\bm{v}}(\bm{\lambda}_{t}))^{-1}\bm{\Sigma}_{\bm{\eta}}]\bm{D}^{\prime}+\bm{\Sigma}_{\bm{\epsilon}}.$

##  III Optimal Attention Allocation Policy and Optimal Control Policy

To ensure the problem $(P_{AC})$ has unique optimal control policy, we provide
the following assumption.

######  Assumption III.1.

For $t=0,1,\cdots,T-1$, the conditional covariance matrix of $\bm{b}_{t+1}$ at
time $t+$, $\bm{\Sigma}_{\bm{b},t+}(\bm{\lambda}_{t})$, is positive definite,
i.e., $\bm{\Sigma}_{\bm{b},t+}(\bm{\lambda}_{t})\succ 0$.

The following theorem provides the optimal attention allocation policy and the
optimal system control policy for problem $(P_{AC})$.

######  Theorem III.1.

Under Assumption III.1, the optimal control policy and optimal attention
allocation policy for problem $(P_{AC})$ at period $t$ are given by,

$\displaystyle~{}\bm{u}_{t+}^{\star}(x_{t},\Lambda_{t},\bm{f}_{t},\bm{s}_{t+}(\bm{\lambda}_{t}^{\star}))$
$\displaystyle=$
$\displaystyle~{}-(\bm{A}_{t}+E_{t+}[h_{t+1}(\Lambda_{t+1},\bm{f}_{t+1})\bm{b}_{t+1}\bm{b}_{t+1}^{\prime}])^{-1}(\bm{p}_{t}+E_{t+}[h_{t+1}(\Lambda_{t+1},\bm{f}_{t+1})\bm{b}_{t+1}]a_{t+1})x_{t},$
(1) $\displaystyle~{}\bm{\lambda}_{t}^{\star}(\Lambda_{t},\bm{f}_{t})$
$\displaystyle=$
$\displaystyle~{}\mathop{\arg\min}_{\begin{subarray}{c}\bm{\lambda}_{t}^{\prime}\mathbf{1}\leq\Lambda_{t},\bm{\lambda}_{t}\geq\bm{0}\end{subarray}}~{}E_{t}\big{[}E_{t+}[h_{t+1}(\Lambda_{t+1},\bm{f}_{t+1})]a_{t+1}^{2}+q_{t}-(\bm{p}_{t}+E_{t+}[h_{t+1}(\Lambda_{t+1},\bm{f}_{t+1})\bm{b}_{t+1}]a_{t+1})^{\prime}$
$\displaystyle\quad\quad\quad\cdot(\bm{A}_{t}+E_{t+}[h_{t+1}(\Lambda_{t+1},\bm{f}_{t+1})\bm{b}_{t+1}\bm{b}_{t+1}^{\prime}])^{-1}(\bm{p}_{t}+E_{t+}[h_{t+1}(\Lambda_{t+1},\bm{f}_{t+1})\bm{b}_{t+1}]a_{t+1})\big{]},$
(2)

where $h_{t}(\Lambda_{t},\bm{f}_{t})$ represents the optimal objective value
of the optimization problem (2) and $h_{T}(\Lambda_{T},\bm{f}_{T})=q_{T}$,
$E_{t+}[\cdot]=E[\cdot|\mathcal{F}_{t+}]$ and
$E_{t}[\cdot]=E[\cdot|\mathcal{F}_{t}]$. And the optimal objective function
for problem $(P_{AC})$ is given as follows,

$\displaystyle
J_{0}(x_{0},\Lambda_{0},\bm{f}_{0})=h_{0}(\Lambda_{0},\bm{f}_{0})x_{0}^{2}.$

Proof. We solve the problem $(P_{AC})$ by dynamic programming. At time
$(T-1)+$, the decision maker observes the signal
$\bm{s}_{(T-1)+}(\bm{\lambda}_{T-1}^{\star})$ and faces the following one-
period decision problem,

$\displaystyle\min_{\bm{u}_{(T-1)+}}~{}E_{(T-1)+}[\bm{u}_{(T-1)+}^{\prime}\bm{A}_{T-1}\bm{u}_{(T-1)+}+2\bm{u}_{(T-1)+}^{\prime}\bm{p}_{T-1}x_{T-1}+q_{T-1}x_{T-1}^{2}+q_{T}x_{T}^{2}]$
$\displaystyle=$
$\displaystyle\min_{\bm{u}_{(T-1)+}}~{}\bm{u}_{(T-1)+}^{\prime}(\bm{A}_{T-1}+q_{T}E_{(T-1)+}[\bm{b}_{T}\bm{b}_{T}^{\prime}])\bm{u}_{(T-1)+}+2\bm{u}_{(T-1)+}^{\prime}(\bm{p}_{T-1}+q_{T}a_{T}E_{(T-1)+}[\bm{b}_{T}])x_{T-1}$
$\displaystyle\quad\quad\quad+q_{T-1}x_{T-1}^{2}+q_{T}a_{T}^{2}x_{T-1}^{2}.$

Under Assumption III.1, we have
$E_{(T-1)+}[\bm{b}_{T}\bm{b}_{T}^{\prime}]\succ 0$. As $q_{T}>0$ and
$\bm{A}_{T-1}\succeq 0$, we further have the optimal control policy at period
$T-1$ given by,

$\displaystyle\bm{u}_{(T-1)+}^{\star}(x_{T-1},\Lambda_{T-1},\bm{f}_{T-1},\bm{s}_{(T-1)+}(\bm{\lambda}_{T-1}^{\star}))$
$\displaystyle=$
$\displaystyle-(\bm{A}_{T-1}+q_{T}E_{(T-1)+}[\bm{b}_{T}\bm{b}_{T}^{\prime}])^{-1}(\bm{p}_{T-1}+q_{T}a_{T}E_{(T-1)+}[\bm{b}_{T}])x_{T-1},$

where the information set at time $(T-1)+$ contain the signal
$\bm{s}_{(T-1)+}(\bm{\lambda}_{T-1}^{\star})$ and the factor $\bm{f}_{T-1}$.
Substituting the optimal control policy back, we obtain the cost-to-go
function at time $(T-1)+$ as follows,

$\displaystyle
J_{(T-1)+}(x_{T-1},\Lambda_{T-1},\bm{f}_{T-1},\bm{s}_{(T-1)+}(\bm{\lambda}_{t}^{\star}))$
$\displaystyle=$
$\displaystyle~{}[q_{T}a_{T}^{2}+q_{T-1}-(\bm{p}_{T-1}+q_{T}a_{T}E_{(T-1)+}[\bm{b}_{T}])^{\prime}(\bm{A}_{T-1}+q_{T}E_{(T-1)+}[\bm{b}_{T}\bm{b}_{T}^{\prime}])^{-1}$
$\displaystyle\cdot(\bm{p}_{T-1}+q_{T}a_{T}E_{(T-1)+}[\bm{b}_{T}])]x_{T-1}^{2}$
$\displaystyle:=$
$\displaystyle~{}h_{(T-1)+}(\Lambda_{T-1},\bm{\lambda}_{T-1}^{\star},\bm{f}_{T-1})x_{T-1}^{2}.$

As $\bm{\Sigma}_{\bm{b},(T-1)+}(\bm{\lambda}_{T-1}^{\star})\succ 0$ and
$a_{T}>0$, which implies
$\begin{pmatrix}E_{(T-1)+}[\bm{b}_{T}\bm{b}_{T}^{\prime}]&a_{T}E_{(T-1)+}[\bm{b}_{T}]\\\
a_{T}E_{(T-1)+}[\bm{b}_{T}]^{\prime}&a_{T}^{2}\end{pmatrix}\succ 0$ according
to Schur complement. We further have

$\displaystyle\begin{pmatrix}\bm{A}_{T-1}+q_{T}E_{(T-1)+}[\bm{b}_{T}\bm{b}_{T}^{\prime}]&\bm{p}_{T-1}+q_{T}a_{T}E_{(T-1)+}[\bm{b}_{T}]\\\
(\bm{p}_{T-1}+q_{T}a_{T}E_{(T-1)+}[\bm{b}_{T}])^{\prime}&q_{T}a_{T}^{2}+q_{T-1}\end{pmatrix}$
$\displaystyle=$ $\displaystyle
q_{T}\begin{pmatrix}E_{(T-1)+}[\bm{b}_{T}\bm{b}_{T}^{\prime}]&a_{T}E_{(T-1)+}[\bm{b}_{T}]\\\
a_{T}E_{(T-1)+}[\bm{b}_{T}]^{\prime}&a_{T}^{2}\end{pmatrix}+\bm{B}_{T-1}\succ
0$

which implies that $h_{(T-1)+}(\bm{\lambda}_{T-1}^{\star},\bm{f}_{T-1})>0$
according to Schur complement. Then, at time $(T-1)$, the optimal attention
allocation policy is given as,

$\displaystyle\bm{\lambda}_{T-1}^{\star}(\Lambda_{T-1},\bm{f}_{T-1})=$
$\displaystyle\mathop{\arg\min}_{\begin{subarray}{c}\bm{\lambda}_{T-1}^{\prime}\mathbf{1}\leq\Lambda_{T-1},\bm{\lambda}_{T-1}\geq\bm{0}\end{subarray}}~{}\\{E_{T-1}[J_{(T-1)+}(x_{T-1},\Lambda_{T-1},\bm{f}_{T-1},\bm{s}_{(T-1)+}(\bm{\lambda}_{t}))]\\}$
$\displaystyle=$
$\displaystyle\mathop{\arg\min}_{\begin{subarray}{c}\bm{\lambda}_{T-1}^{\prime}\mathbf{1}\leq\Lambda_{T-1},\bm{\lambda}_{T-1}\geq\bm{0}\end{subarray}}~{}~{}\\{E_{T-1}[h_{(T-1)+}(\Lambda_{T-1},\bm{\lambda}_{T-1},\bm{f}_{T-1})]\\}.$

And the cost-to-go function at time $(T-1)$ becomes

$\displaystyle
J_{T-1}(x_{T-1},\Lambda_{T-1},\bm{f}_{T-1})=h_{T-1}(\Lambda_{T-1},\bm{f}_{T-1})x_{T-1}^{2}.$

We assume that the cost-to-go function at time $t+1$ is
$J_{t+1}(x_{t+1},\Lambda_{t+1},\bm{f}_{t+1})=h_{t+1}(\Lambda_{t+1},\bm{f}_{t+1})x_{t+1}^{2}$
and $h_{t+1}(\Lambda_{t+1},\bm{f}_{t+1})>0$. Then, we derive the optimal
control policy and attention allocation policy for period $t$. At time $t+$,
the investors decision maker observes the signal
$\bm{s}_{t+}(\bm{\lambda}_{t}^{\star})$ and face the following one-period
portfolio selection problem according to Bellman’s principal of optimality,

$\displaystyle\min_{\bm{u}_{t+}}~{}E_{t+}[\bm{u}_{t+}^{\prime}\bm{A}_{t}\bm{u}_{t+}+2\bm{u}_{t+}^{\prime}\bm{p}_{t}x_{t}+q_{t}x_{t}^{2}+J_{t+1}(x_{t+1},\Lambda_{t+1},\bm{f}_{t+1})]$
$\displaystyle=$
$\displaystyle\min_{\bm{u}_{t+}}~{}\bm{u}_{t+}^{\prime}(\bm{A}_{t}+E_{t+}[h_{t+1}(\Lambda_{t+1},\bm{f}_{t+1})\bm{b}_{t+1}\bm{b}_{t+1}^{\prime}])\bm{u}_{t+}$
$\displaystyle\quad\quad~{}~{}+2\bm{u}_{t+}^{\prime}(\bm{p}_{t}+E_{t+}[h_{t+1}(\Lambda_{t+1},\bm{f}_{t+1})\bm{b}_{t+1}]a_{t+1})x_{t}+q_{t}x_{t}^{2}+E_{t+}[h_{t+1}(\Lambda_{t+1},\bm{f}_{t+1})]a_{t+1}^{2}x_{t}^{2}.$

We define the new probability measure $Q_{t+}(\bm{\lambda}_{t}^{\star})$ after
observing the signal $\bm{s}_{t+}(\bm{\lambda}_{t}^{\star})$, which is
equivalent to the objective probability measure $P$, as follows,

$\displaystyle\frac{dQ_{t+}(\bm{\lambda}_{t}^{\star})}{dP}=\frac{h_{t+1}(\Lambda_{t+1},\bm{f}_{t+1})}{E_{t+}[h_{t+1}(\Lambda_{t+1},\bm{f}_{t+1})]}.$

As the new probability measure $Q_{t+}(\bm{\lambda}_{t}^{\star})$ is
equivalent to $P$, the covariance matrix of $\bm{b}_{t+1}$ under
$Q_{t+}(\bm{\lambda}_{t}^{\star})$ is also positive definite, which implies
that
$\bm{A}_{t}+E_{t+}[h_{t+1}(\Lambda_{t+1},\bm{f}_{t+1})\bm{b}_{t+1}\bm{b}_{t+1}^{\prime}]$
is positive definite. Thus, the optimal control policy at period $t$ is

$\displaystyle\bm{u}_{t+}^{\star}(x_{t},\Lambda_{t},\bm{f}_{t},\bm{s}_{t+}(\bm{\lambda}_{t}^{\star}))$
$\displaystyle=$
$\displaystyle-(\bm{A}_{t}+E_{t+}[h_{t+1}(\Lambda_{t+1},\bm{f}_{t+1})\bm{b}_{t+1}\bm{b}_{t+1}^{\prime}])^{-1}(\bm{p}_{t}+E_{t+}[h_{t+1}(\Lambda_{t+1},\bm{f}_{t+1})\bm{b}_{t+1}]a_{t+1})x_{t},$

and the cost-to-go function at time $t+$ becomes

$\displaystyle
J_{t+}(x_{t},\Lambda_{t},\bm{f}_{t},\bm{s}_{t+}(\bm{\lambda}_{t}^{\star}))$
$\displaystyle=$
$\displaystyle\\{E_{t+}[h_{t+1}(\Lambda_{t+1},\bm{f}_{t+1})]a_{t+1}^{2}+q_{t}-(\bm{p}_{t}+E_{t+}^{Q_{t+}(\bm{\lambda}_{t}^{\star})}[\bm{b}_{t+1}]E_{t+}[h_{t+1}(\Lambda_{t+1},\bm{f}_{t+1})]a_{t+1})^{\prime}$
$\displaystyle\cdot(\bm{A}_{t}+E_{t+}^{Q_{t+}(\bm{\lambda}_{t}^{\star})}[\bm{b}_{t+1}\bm{b}_{t+1}^{\prime}]E_{t+}[h_{t+1}(\Lambda_{t+1},\bm{f}_{t+1})])^{-1}$
$\displaystyle\cdot(\bm{p}_{t}+E_{t+}^{Q_{t+}(\bm{\lambda}_{t}^{\star})}[\bm{b}_{t+1}]E_{t+}[h_{t+1}(\Lambda_{t+1},\bm{f}_{t+1})]a_{t+1})\\}x_{t}^{2}$
$\displaystyle:=$ $\displaystyle
h_{t+}(\Lambda_{t},\bm{\lambda}_{t}^{\star},\bm{f}_{t})x_{t}^{2},$

where $E_{t+}^{Q_{t+}(\bm{\lambda}_{t}^{\star})}[\cdot]$ denotes the
conditional expectation under probability $Q_{t+}(\bm{\lambda}_{t}^{\star})$.
Following the same argument for $T-1$ period, we have

$\displaystyle\begin{pmatrix}\bm{A}_{t}+E_{t+}^{Q_{t+}(\bm{\lambda}_{t}^{\star})}[\bm{b}_{t+1}\bm{b}_{t+1}^{\prime}]E_{t+}[h_{t+1}(\Lambda_{t+1},\bm{f}_{t+1})]&\bm{p}_{t}+E_{t+}^{Q_{t+}(\bm{\lambda}_{t}^{\star})}[\bm{b}_{t+1}]E_{t+}[h_{t+1}(\Lambda_{t+1},\bm{f}_{t+1})]a_{t+1}\\\
(\bm{p}_{t}+E_{t+}^{Q_{t+}(\bm{\lambda}_{t}^{\star})}[\bm{b}_{t+1}]E_{t+}[h_{t+1}(\Lambda_{t+1},\bm{f}_{t+1})]a_{t+1})^{\prime}&q_{t}+E_{t+}[h_{t+1}(\Lambda_{t+1},\bm{f}_{t+1})]a_{t+1}^{2}\end{pmatrix}$
$\displaystyle=$ $\displaystyle
E_{t+}[h_{t+1}(\Lambda_{t+1},\bm{f}_{t+1})]\begin{pmatrix}E_{t+}^{Q_{t+}(\bm{\lambda}_{t}^{\star})}[\bm{b}_{t+1}\bm{b}_{t+1}^{\prime}]&a_{t+1}E_{t+}^{Q_{t+}(\bm{\lambda}_{t}^{\star})}[\bm{b}_{t+1}]\\\
a_{t+1}E_{t+}^{Q_{t+}(\bm{\lambda}_{t}^{\star})}[\bm{b}_{t+1}^{\prime}]&a_{t+1}^{2}\end{pmatrix}+\bm{B}_{t}\succ
0$

which implies that $h_{t+}(\Lambda_{t},\bm{\lambda}_{t}^{\star},\bm{f}_{t})>0$
according to Schur complement. Then, at time $t$, the optimal attention
allocation strategy is given as,

$\displaystyle\bm{\lambda}_{t}^{\star}(\Lambda_{t},\bm{f}_{t})=$
$\displaystyle\mathop{\arg\min}_{\begin{subarray}{c}\bm{\lambda}_{t}^{\prime}\mathbf{1}\leq\Lambda_{t},\bm{\lambda}_{t}\geq\bm{0}\end{subarray}}~{}\\{E_{t}[J_{t+}(x_{t},\Lambda_{t},\bm{f}_{t},\bm{s}_{t+}(\bm{\lambda}_{t}))]\\}=\mathop{\arg\min}_{\begin{subarray}{c}\bm{\lambda}_{t}^{\prime}\mathbf{1}\leq\Lambda_{t},\bm{\lambda}_{t}\geq\bm{0}\end{subarray}}~{}\\{E_{t}[h_{t+}(\Lambda_{t},\bm{\lambda}_{t},\bm{f}_{t})]\\}.$

And the cost-to-go function at time $t$ becomes

$\displaystyle
J_{t}(x_{t},\Lambda_{t},\bm{f}_{t})=h_{t}(\Lambda_{t},\bm{f}_{t})x_{t}^{2}.$

Therefore, we have prove our main results. Furthermore, the optimal objective
of problem is
$J_{0}(x_{0},\Lambda_{0},\bm{f}_{0})=h_{0}(\Lambda_{0},\bm{f}_{0})x_{0}^{2}$.
$\Box$

According to the theorem, several significant findings emerge. First, the
optimal control policy is a linear function of the current state $x_{t}$,
implying that the decision maker’s system control policy is determined based
on the present state. On the other hand, the optimal attention allocation
policy is independent of the current state $x_{t}$ but is influenced by the
attention resource $\Lambda_{t}$ and the factors $\bm{f}_{t}$. The decision
maker allocates attention resources based on the available attention capacity
and the current factors’ information. Second, the optimization problem used to
determine the optimal attention allocation policy is not convex, indicating
that finding the optimal allocation requires numerical techniques rather than
a straightforward analytical solution. Third, both the optimal control policy
and the optimal attention allocation policy heavily rely on the random
variable $h_{t}(\Lambda_{t},\bm{f}_{t})$. This variable represents the
potential future control opportunity, reflecting the expected impact of future
decisions on current decision-making process. Fourth, when no attention
allocation decisions are made, the optimal control policy reduces to the
policy proposed in Proposition 1 of [15]. This comparison highlights the
significance of incorporating attention allocation decisions to linear
quadratic control problem.

Next, we propose the following algorithm to derive
$h_{t}(\Lambda_{t},\bm{f}_{t})$ and the optimal attention allocation policy
$\bm{\lambda}_{t}^{\star}(\Lambda_{t},\bm{f}_{t})$.

######  Algorithm III.1.

  1. 1.

Choose the constant function, $h_{T}(\Lambda_{T},\bm{f}_{T})=q_{T}$ and set
$t=T-1$. Simulate $L$ realizations of the independent multivariate standard
normal random vectors, $\bm{\varepsilon}_{1}^{(\ell_{1})}$ ($k$-dimensional),
$\bm{\varepsilon}_{2}^{(\ell_{2})}$ ($k$-dimensional),
$\bm{\varepsilon}_{3}^{(\ell_{2})}$ ($n$-dimensional).

  2. 2.

Set up a finite grid consisting of $N$ values of $\Lambda_{t}$ and $M$
realizations of $\bm{f}_{t}$.

  3. 3.

For each realization $\bm{f}_{t}^{(i)}$ and possible $\Lambda_{t}^{(j)}$,
compute $\bm{\lambda}_{t}^{\star}(\Lambda_{t}^{(j)},\bm{f}_{t}^{(i)})$ by
solving the optimization problem numerically,

$\displaystyle\min_{\bm{\lambda}_{t}^{\prime}\bm{1}\leq\Lambda_{t}^{(j)},~{}\bm{\lambda}_{t}\geq\bm{0}}\frac{1}{L}\sum_{\ell_{1}=1}^{L}\bigg{\\{}q_{t}+\frac{a_{t+1}^{2}}{L}\sum_{\ell_{2}=1}^{L}h_{t+1}\big{(}\Lambda_{t+1}^{(j)}(\bm{\lambda}_{t}),\bm{f}_{t+1}^{(\ell_{1},\ell_{2},i)}\big{)}$
$\displaystyle\quad-\bigg{(}\bm{p}_{t}+\frac{a_{t+1}}{L}\sum_{\ell_{2}=1}^{L}h_{t+1}(\Lambda_{t+1}^{(j)}(\bm{\lambda}_{t}),\bm{f}_{t+1}^{(\ell_{1},\ell_{2},i)})\bm{b}_{t+1}^{(\ell_{1},\ell_{2},i)}\bigg{)}^{\prime}$
$\displaystyle\quad\cdot\bigg{(}\bm{A}_{t}+\frac{1}{L}\sum_{\ell_{2}=1}^{L}h_{t+1}\big{(}\Lambda_{t+1}^{(j)}(\bm{\lambda}_{t}),\bm{f}_{t+1}^{(\ell_{1},\ell_{2},i)}\big{)}\bm{b}_{t+1}^{(\ell_{1},\ell_{2},i)}(\bm{b}_{t+1}^{(\ell_{1},\ell_{2},i)})^{\prime}\bigg{)}^{-1}$
$\displaystyle\quad\cdot\bigg{(}\bm{p}_{t}+\frac{a_{t+1}}{L}\sum_{\ell_{2}=1}^{L}h_{t+1}\big{(}\Lambda_{t+1}^{(j)}(\bm{\lambda}_{t}),\bm{f}_{t+1}^{(\ell_{1},\ell_{2},i)}\big{)}\bm{b}_{t+1}^{(\ell_{1},\ell_{2},i)}\bigg{)}\bigg{\\}},$
$\displaystyle:=\min_{\bm{\lambda}_{t}^{\prime}\bm{1}\leq\Lambda_{t}^{(j)},~{}\bm{\lambda}_{t}\geq\bm{0}}\frac{1}{L}\sum_{\ell_{1}=1}^{L}g(\bm{\lambda}_{t};\Lambda_{t}^{(j)},\bm{f}_{t}^{(i)},\bm{s}_{t+}(\bm{\lambda}_{t})^{(\ell_{1},i)}),$

where

$\displaystyle\Lambda_{t+1}^{(j)}(\bm{\lambda}_{t})=\Lambda_{t}^{(j)}-\bm{\lambda}_{t}^{\prime}\bm{1},$
$\displaystyle\bm{f}_{t+1}^{(\ell_{1},\ell_{2},i)}=(\bm{I}-\bm{\varPhi})\bm{f}_{t}^{(i)}+\bm{\Sigma}_{\bm{\eta}}(\bm{\Sigma}_{\bm{\eta}}+\bm{\Sigma}_{\bm{v}}(\bm{\lambda}_{t}))^{-1}(\bm{s}_{t+}(\bm{\lambda}_{t})^{(\ell_{1},i)}-(\bm{I}-\bm{\Phi})\bm{f}_{t}^{(i)})$
$\displaystyle\quad\quad\quad\quad+[\bm{\Sigma}_{\bm{\eta}}-\bm{\Sigma}_{\bm{\eta}}(\bm{\Sigma}_{\bm{\eta}}+\bm{\Sigma}_{\bm{v}}(\bm{\lambda}_{t}))^{-1}\bm{\Sigma}_{\bm{\eta}}]^{1/2}\bm{\varepsilon}_{2}^{(\ell_{2})},$
$\displaystyle\bm{b}_{t+1}^{(\ell_{1},\ell_{2},i)}=\bm{c}+\bm{D}\bm{f}_{t+1}^{(\ell_{1},\ell_{2},i)}+\bm{\Sigma}_{\bm{\epsilon}}^{1/2}\bm{\varepsilon}_{3}^{(\ell_{2})},$
$\displaystyle\bm{s}_{t+}(\bm{\lambda}_{t})^{(\ell_{1},i)}=(\bm{I}-\bm{\varPhi})\bm{f}_{t}^{(i)}+(\bm{\Sigma}_{\bm{\eta}}+\bm{\Sigma}_{\bm{\nu}}(\bm{\lambda}_{t}))\bm{\varepsilon}_{1}^{(\ell_{1})}.$

Denote the optimal value as $h_{t}(\Lambda_{t}^{(j)},\bm{f}_{t}^{(i)})$ and
the optimizer as
$\bm{\lambda}_{t}^{\star}(\Lambda_{t}^{(j)},\bm{f}_{t}^{(i)})$.

  4. 4.

Approximate the functions $h_{t}(\Lambda_{t},\bm{f}_{t})$ and
$\bm{\lambda}_{t}^{\star}(\Lambda_{t},\bm{f}_{t})$ by linear interpolation.
Set $t=t-1$.

  5. 5.

When $t=0$, stop the computation. When $t>0$, go back to step 2.

In step 3 of Algorithm III.1, the objective of the optimization problem is not
a simple summation over all simulated random variables. For a given
realization of $\bm{\varepsilon}_{1}^{(\ell_{1})}$, we can compute the
gradient of function
$g(\bm{\lambda}_{t};\Lambda_{t}^{(j)},\bm{f}_{t}^{(i)},\bm{s}_{t+}(\bm{\lambda}_{t})^{(\ell_{1},i)})$
with respect to $\bm{\lambda}_{t}$ and apply the stochastic gradient decent
method to solve the optimization problem. Please note that the complexity of
Algorithm III.1 does not grow exponentially with respect to period $T$.

##  IV Application in Dynamic Mean-Variance Portfolio Selection

In this section, we explore an application of our findings in the context of
dynamic portfolio selection and consider a discrete-time mean-variance model
with limited attention allocation. In this model, the investor’s objective is
to minimize the variance of the terminal wealth $x_{T}$ while ensuring that
the expected terminal wealth achieves a target value $d\geq\rho_{0}x_{0}$,
where $a_{t+1}>1$ is the total return of a risk-free investment at period $t$
and $\rho_{t}=\prod_{s=t}^{T-1}a_{s+1}$, i.e.,

$\displaystyle\min_{\\{\bm{\lambda}_{t},\bm{u}_{t+}\\}_{t=0}^{T-1}}$
$\displaystyle\mbox{Var}_{0}(x_{T})$ s.t. $\displaystyle E_{0}[x_{T}]=d,$
$\displaystyle(x_{t},\bm{\lambda}_{t},\Lambda_{t},\bm{u}_{t+})\mbox{ obey
constraints of }(P_{AC}),$

where $\mbox{Var}_{0}(x_{T})$ denotes the variance of the terminal wealth.
Following [10], we introduce the Lagrangian multiplier $2\mu$ for the
constraint of expected terminal wealth, define
$y_{t}=x_{t}-(d-\mu)\rho_{t}^{-1}$, and obtain the auxiliary problem as
follows,

$\displaystyle(\mathcal{A}_{\mu})~{}~{}\min_{\\{\bm{\lambda}_{t},\bm{u}_{t+}\\}_{t=0}^{T-1}}$
$\displaystyle~{}E_{0}[y_{T}^{2}]-\mu^{2}$ s.t.
$\displaystyle(y_{t},\bm{\lambda}_{t},\Lambda_{t},\bm{u}_{t+})\mbox{ obey
constraints of }(P_{AC}).$

Then, based on strong duality, we can solve the dynamic mean-variance problem
$(MV)$ by optimizing the objective function of $(\mathcal{A}_{\mu})$ over the
set $\\{\mu|\mu\in\mathbb{R}\\}$.

######  Theorem IV.1.

Under the assumption that $E_{(T-1)+}[\bm{b}_{T}]\neq\bm{0}$, the optimal
portfolio policy and optimal attention allocation policy are given by

$\displaystyle~{}\bm{u}_{t+}^{\star}(x_{t},\Lambda_{t},\bm{f}_{t},\bm{s}_{t+}(\bm{\lambda}_{t}^{\star}))$
$\displaystyle=$
$\displaystyle~{}-E_{t+}[h_{t+1}(\Lambda_{t+1},\bm{f}_{t+1})\bm{b}_{t+1}\bm{b}_{t+1}^{\prime}]^{-1}E_{t+}[h_{t+1}(\Lambda_{t+1},\bm{f}_{t+1})\bm{b}_{t+1}]a_{t+1}(x_{t}-\rho_{t}(d-\mu^{\star})),$
(3) $\displaystyle~{}\bm{\lambda}_{t}^{\star}(\Lambda_{t},\bm{f}_{t})$
$\displaystyle=$
$\displaystyle~{}\mathop{\arg\min}_{\begin{subarray}{c}\bm{\lambda}_{t}^{\prime}\mathbf{1}\leq\Lambda_{t},\bm{\lambda}_{t}\geq\bm{0}\end{subarray}}~{}a_{t+1}^{2}E_{t}\big{[}E_{t+}[h_{t+1}(\Lambda_{t+1},\bm{f}_{t+1})]-E_{t+}[h_{t+1}(\Lambda_{t+1},\bm{f}_{t+1})\bm{b}_{t+1}^{\prime}]$
$\displaystyle\quad\quad\quad\cdot
E_{t+}[h_{t+1}(\Lambda_{t+1},\bm{f}_{t+1})\bm{b}_{t+1}\bm{b}_{t+1}^{\prime}]^{-1}E_{t+}[h_{t+1}(\Lambda_{t+1},\bm{f}_{t+1})\bm{b}_{t+1}]\big{]},$
(4)

where $h_{t}(\Lambda_{t},\bm{f}_{t})$ represents the optimal objective value
of the optimization problem (4) and $h_{T}(\Lambda_{T},\bm{f}_{T})=1$, and the
optimal multiplier
$\mu^{\star}=\frac{h_{0}(\Lambda_{0},\bm{f}_{0})(\rho_{0}x_{0}-d)}{\rho_{0}^{2}-h_{0}(\Lambda_{0},\bm{f}_{0})}$.
The efficient frontier is given as follows,

$\displaystyle\mbox{Var}_{0}(x_{T})=\frac{h_{0}(\Lambda_{0},\bm{f}_{0})(E_{0}[x_{T}]-\rho_{0}x_{0})^{2}}{\rho_{0}^{2}-h_{0}(\Lambda_{0},\bm{f}_{0})},\quad
E_{0}[x_{T}]\geq\rho_{0}x_{0}.$

Proof. The auxiliary problem is a particular linear quadratic control system
with limited attention allocation. With the help of the results in Theorem
III.1, we only need to compute the optimal Lagrangian multiplier $\mu^{\star}$
according to

$\displaystyle\mu^{\star}$
$\displaystyle=\mathop{\arg\max}\limits_{\mu\in\mathbb{R}}~{}~{}\\{h_{0}(\Lambda_{0},\bm{f}_{0})(x_{0}-\rho_{0}^{-1}(d-\mu))^{2}-\mu^{2}\\}.$

As $E_{(T-1)+}[\bm{b}_{T}\bm{b}_{T}^{\prime}]^{-1}\succ 0$ and
$E_{(T-1)+}[\bm{b}_{T}]\neq\bm{0}$, we have
$h_{T-1}(\Lambda_{T-1},\bm{f}_{T-1})<a_{T-1}^{2}$. With the help of
$E_{t+}[h_{t+1}(\Lambda_{t+1},\bm{f}_{t+1})\bm{b}_{t+1}\bm{b}_{t+1}^{\prime}]^{-1}\succ
0$, we further have $h_{t}(\Lambda_{t},\bm{f}_{t})<\rho_{t}^{2}$. Thus, the
optimal Lagrangian multiplier $\mu^{\star}$ is
$\mu^{\star}=\frac{h_{0}(\Lambda_{0},\bm{f}_{0})(\rho_{0}x_{0}-d)}{\rho_{0}^{2}-h_{0}(\Lambda_{0},\bm{f}_{0})}$.
Moreover, the variance of the terminal wealth level achieved by optimal
portfolio policy is

$\displaystyle\mbox{Var}_{0}(x_{T})$
$\displaystyle=h_{0}(\Lambda_{0},\bm{f}_{0})\left(x_{0}-\rho_{0}^{-1}d+\rho_{0}^{-1}\mu^{\star}\right)^{2}-(\mu^{\star})^{2}=\frac{h_{0}(\Lambda_{0},\bm{f}_{0})(\rho_{0}x_{0}-d)^{2}}{\rho_{0}^{2}-h_{0}(\Lambda_{0},\bm{f}_{0})},$

which provides the mean-variance efficient frontier by changing $d$ into
$E_{0}[x_{T}]$. $\Box$

In our empirical analysis, we utilize data from the U.S. market obtained from
Kenneth R. French’s data library website. The dataset comprises monthly
returns of the Fama-French three factors, denoted as
$\bm{f}_{t}=(MKT_{t},SMB_{t},HML_{t})^{\prime}$, and the monthly returns of
U.S. five industry portfolios, including Cnsmr (C), Manuf (Ma), HiTec (Hi),
Hlth (Hl), and Other (O), represented as
$\bm{b}_{t}=(b_{C,t},b_{Ma,t},b_{Hi,t},b_{Hl,t},b_{O,t})^{\prime}$. We set
$a_{t+1}$ equal to the average total return of a risk-free investment,
$1.0036$.

In our analysis, we make the assumption that the efficiency parameters of the
decision maker’s information processing of the Fama-French three factors are
all equal to 0.69, i.e., $\theta_{MKT}=\theta_{SMB}=\theta_{HML}=0.69$.222When
the decision maker uses 1 unit attention resource to learn a factor, the
conditional variance of the factor under $\mathcal{F}_{t+}$ is half of the
conditional variance of the factor under $\mathcal{F}_{t}$. Under the
parameter setting, we consider a three-month mean-variance model with limited
attention allocation. The initial attention resource is $\Lambda_{0}=3$. As
the optimal portfolio policy is analytical, we mainly focus on investigating
the optimal attention allocation policy. Applying Algorithm III.1, we compute
the optimal attention allocation policy,
$\bm{\lambda}_{t}^{\star}(\Lambda_{t},\bm{f}_{t})$ for $t=0,1,2$, with
$13\times 7\times 7\times 7$ grid points.

We first discuss the optimal attention allocation among different periods.
Figure 2 shows the total optimal attention allocation
$\bm{1}^{\prime}\bm{\lambda}_{t}^{\star}$ in different cases of factors and
different periods. Case 1 to case 4 correspond to four different settings for
the factors $\bm{f}_{t}$: $(0.005,0.002,0.003)^{\prime}$,
$(-0.127,0.002,0.003)^{\prime}$, $(-0.127,-0.091,0.003)^{\prime}$,
$(-0.127,-0.091,-0.081)^{\prime}$.333$(0.005,0.002,0.003)^{\prime}$ is the
mean of $\bm{f}_{t}$, and $(-0.127,-0.091,-0.081)^{\prime}$ is the mean minus
three times the standard deviation of $\bm{f}_{t}$ for each factor. The cases
with positive extreme values have the similar pattern, which are not reported
in the paper. Based on the total optimal attention allocation for period 1 in
the four cases of each column, we can observe an interesting pattern. When the
values of the factors are extreme values, particularly in Case 4 where the
market state is three standard deviations below the mean for each factor, the
mean-reverting effect of the factors may generate relatively accurate
estimations of the future factor. As a result, the investors tend to allocate
less attention resource to learn the factors. The total optimal attention
allocation for period 2 in the four cases of each row confirm the finding. In
the last period, the decision maker faces a critical choice as there is no
future period to allocate attention to. As a result, the decision maker may
opt to use all the remaining attention resource available to make the most
informed investment decisions in the current period, which leads to lower
total optimal attention allocation for the left and top cases compared to the
right and bottom cases.

Figure 2: The total optimal attention allocation in different cases of factors
and different periods

Next, we examine optimal attention allocation among factors in Figure 3 when
$\bm{f}_{0}$ and $\bm{f}_{1}$ are both from Case 1. Each curve in the plot
represents the allocation in different factors over three periods. Investors
prioritize attention on MKT and HML factors. Although the first and second
periods face the same factors, their allocations differ due to varying
resource availability and future opportunities. In the last period, with no
future opportunities, investors solely focus on the single-period problem. The
figure reveals how investors strategically allocate attention, considering
market conditions, resource availability, and future opportunities,
influencing their decisions over multiple time periods.

Figure 3: The optimal attention allocation among different factors

Finally, during the out-of-sample period from Jan. 2008 to Dec. 2017, we apply
a rolling window scheme to estimate parameters and conduct three months mean-
variance portfolio selection with limited attention allocation. This results
in 118 terminal wealth values. Without attention allocation, the average
terminal wealth is $1.0648$ (standard deviation: $0.2177$, Sharpe ratio:
$0.2477$). With attention allocation, the average terminal wealth improves to
$1.0848$ (standard deviation: $0.2694$, Sharpe ratio: $0.2744$), a 10.8% boost
in the Sharpe ratio. The strategy’s superior risk-adjusted returns highlight
its ability to make more informed investment decisions.

##  V Conclusion

This paper introduces limited attention allocation into the classical linear
quadratic control problem. The model allows strategic resource allocation,
improving randomness estimations. We derive explicit optimal control and a
numerical method for attention allocation. Empirical analysis extends the
model to dynamic portfolio selection, showcasing effective resource allocation
over time and factors. Attention allocation significantly enhances decision-
making and risk-adjusted performance. The study provides valuable insights
into the importance of limited attention allocation in decision-making
processes.

Our model can be extended for real-world application. We can replace the
abstract white noise setting with concrete signal generation based on factor
timing literature, enhancing applicability. The decision maker can also
allocate resources using optimal weights from pre-given prediction models,
providing flexibility beyond factor attention. These extensions improve the
model’s practical relevance and empower investors to make informed decisions
in diverse scenarios.

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#  On-line Parameter Estimation of the Polarization Curve of a Fuel Cell with
Guaranteed Convergence Properties: Theoretical and Experimental Results

Carlo Beltrán, Alexey Bobtsov, Romeo Ortega, Diego Langarica-Córdoba,
Rafael Cisneros, Luis H. Díaz-Saldierna  C. Beltrán and D. Langarica are with
the Faculty of Sciences, Autonomous University of San Luis PotosÃ (UASLP), San
Luis Potosí, 78295, Mexico (e-mails<EMAIL_ADDRESS>diego.langarica@uaslp.mx). A. Bobstov is with the Department of Control
Systems and Robotics, ITMO University, 197101, Saint Petersburg, Russia
(e-mail: bobtsov@mail.ru). R. Ortega and R. Cisneros are with the Electrical
and Electronic Engineering Department, Instituto Tecnológico Autónomo de
México (ITAM), Mexico City, 01080, Mexico<EMAIL_ADDRESS>rcisneros@itam.mx). L. Díaz is with the Instituto Potosino de Investigación
Científica y Tecnológica, San Luis Potosí, 78216, Mexico (e-mail:
ldiaz@ipicyt.edu.mx).

###### Abstract

In this paper we address the problem of online parameter estimation of a
Proton Exchange Membrane Fuel Cell (PEMFC) polarization curve, that is, the
static relation between the voltage and the current of the PEMFC. The task of
designing this estimator—even off-line—is complicated by the fact that the
uncertain parameters enter the curve in a highly nonlinear fashion, namely in
the form of nonseparable nonlinearities. We consider several scenarios for the
model of the polarization curve, starting from the standard full model and
including several popular simplifications to this complicated mathematical
function. In all cases, we derive separable regression equations—either
linearly or nonlinearly parameterized—which are instrumental for the
implementation of the parameter estimators. We concentrate our attention in
on-line estimation schemes for which, under suitable excitation conditions,
global parameter convergence is ensured. Due to this global convergence
properties the estimators are robust to unavoidable additive noise and
structural uncertainty. Moreover, their on-line nature endow the schemes with
the ability to track (slow) parameter variations, that occur during the
operation of the PEMFC. These two features—unavailable in time-consuming off-
line data-fitting procedures—make the proposed estimators helpful for an on-
line time-saving characterization of a given PEMFC, and the implementation of
fault-detection procedures and model-based adaptive control strategies.
Simulation and experimental results that validate the theoretical claims are
presented.

###### Index Terms:

Proton Exchange Membrane Fuel Cell, Online estimation, Parameter
identification

##  I Introduction

Proton Exchange Membrane Fuel Cells (PEMFCs) have emerged as a promising and
scalable energy conversion and sustainable power generation technology. As a
subset of fuel cell technology, PEMFCs offer a variety of unique features that
make them suitable for diverse applications across various industries such as
electrical transportation and power generation. Among others, energy
efficiency, rapid start-up, compact size and weight, low noise, low emission
and modularity characteristics have widespread this type of fuel cells [1].
However, the determination of the state of health of a functioning Proton
Exchange Membrane Fuel Cell (PEMFC), the optimization of the operating
conditions and its energy management are topics of practical importance [2].
Usually such investigations require complicated experiments and
instrumentation that cannot be applied in field operations; therefore, time-
consuming off-line data-fitting procedures are employed. In recent years
mathematical models that describe the operation of PEMFC under different
operating conditions have been developed. Due to the complexity of the
behavior of the PEMFC these models involve highly nonlinear relations that
depend, again in a nonlinear way, on uncertain parameters. The values of these
parameters, which reflect the actual PEMFC performance, inevitably affect
effectiveness of the developed models in simulation, design, optimal operation
and control. Therefore, it is indispensable to investigate the problem of
parameter estimation of these models. Several unwanted phenomena, such as
catalyst poisoning, flooding or drying of the membrane etc. can be monitored,
and the working parameters of an PEMFC can be quickly regulated with the help
of real time parameter estimation [3]—hence our interest in on-line parameter
estimators. Also, to ensure good performance in the face of uncertainty and
noise, we concentrate our attention on schemes for which we can prove global
convergence properties under some suitable excitation conditions.

A particularly relevant practical model is the one describing the quasi-static
behavior of the PEMFC, which is captured by the so-called polarization curve
[4]. This curve describes the behavior of the cell voltage, which is given by
the equilibrium potential diminished by irreversible losses, often called
polarizations, due to three sources: (i) activation polarization, (ii) ohmic
polarization, and (iii) concentration polarization. An universally accepted
model of the polarization curve is given as [5, Chapter 3]:

$v_{fc}(i_{fc})=v_{rev}-v_{act}(i_{fc})-v_{ohm}(i_{fc})-v_{con}(i_{fc}),$ (1)

where $v_{fc}(t)\in\mathbb{R}$ and $i_{fc}(t)\in\mathbb{R}$ are the PEMFC
voltage and current, respectively. The term $v_{rev}$ represents the
reversible potential, which is not a function of the current and is given by
the classical Nernst equation. Since $v_{rev}$ depends on the PEMFC
temperature and partial pressure, its time scale is significantly slower than
the other losses of electrical nature, and can be assumed to be constant. The
term $v_{act}(i_{fc})$ is the activation polarization described using an
empirical approximation of the Tafel equation. The term $v_{ohm}(i_{fc})$
captures the ohmic losses, and $v_{con}(i_{fc})$ is the concentration
polarization. Classical models for the first two components are given as [5,
Eq. 3.14], [6, Subsection 6.2.1]:

(2a)

while for the concentration term, there are two popular versions reported in
the literature [4, Equation (1)] and [7, Equation (4)]:

$\displaystyle\theta_{5}e^{\theta_{3}i_{fc}}$ (2ca)
$\displaystyle\theta_{5}(i_{fc})^{\theta_{3}}.$ (2cb)

As indicated in [8], the model (LABEL:vact) may be not valid for very small
current densities, i.e., where the influence of the activation polarization is
dominant, and an empirical approximation of the Tafel equation may be
preferred, for instance

$v_{act}(i_{fc})=\theta_{4}\left(1-e^{\theta_{1}i_{fc}}\right).$ (2d)

All the parameters given in these functions—namely
$\theta:=\mbox{col}(\theta_{1},\theta_{2},\theta_{3},\theta_{4},\theta_{5})$—are
positive and uncertain and, in particular, in the concentration term (2c) the
unknown parameter enters in an exponential form, leading to a nonseparable
nonlinear parameterization (NSNLP)—that is, a nonlinear parameterization that
cannot be written as $\phi^{\top}(\cdot){\cal W}(\cdot)$, where the function
$\phi$ depends on measurable signals and is independent of the unknown
parameter, which appear only in the function ${\cal W}$.111Notice that if we
adopt for the activation term the expression (2d) instead of (LABEL:vact), a
second nonseparable parameterization is added. Since dealing with two
nonseparable nonlinearities is still an open problem, this model is not
considered in the paper. The task of estimating on-line the parameters of
systems with NSNLP is essentially open, making the solution of the present
problem unattainable with existing parameter estimation techniques. One of the
contributions of this paper is the development of a procedure to transform the
NSNLP into a separable one, to which we can apply some new parameter
estimation techniques. In particular, the least-squares plus dynamic
regression equation and mixing (LSD) estimator recently reported in [9], which
can deal with separable nonlinear parameterizations (SNLP). In the case, when
the parameterization is linear we can apply simpler least-squares of gradient
algorithms [10, 11]

In the interest of investigating the accuracy of the approximate models we
consider in the paper four descriptions of the polarization curve:

  1. M1

The full model (1), (2) with (2ca) for the concentration term.

  2. M2

Invoking some practical considerations discussed for instance in [7, 12], we
concentrate on $v_{con}(i_{fc})$ and assume that the other terms can be
approximated by a known constant $E_{oc}$, which is the voltage at which the
linearized curve crosses the $y$ axis at the no-current state, leading to the
simplified model

$v_{fc}(i_{fc})=E_{oc}-ae^{bi_{fc}},$ (2e)

where $a>0$ and $b>0$ are some unknown parameters.

  3. M3

Adopting the same arguments invoked above, but taking (2cb) for the
concentration term, leads to

$v_{fc}(i_{fc})=E_{oc}-a\big{(}i_{fc}\big{)}^{b}.$ (2f)

  4. M4

Considering that $\theta_{3}$ in (2c) is a small number—compared with the
other quantities—we assume it is equal to zero and define the function

$v_{fc}(i_{fc})=\theta_{6}+\theta_{1}\ln(i_{fc})+\theta_{2}i_{fc},$ (2g)

where we defined $\theta_{6}:=\theta_{1}+\theta_{5}$.

In models M2 and M3 we take into account the well-known fact that

$\displaystyle ae^{bi_{fc}}$ (2ha) $\displaystyle a(i_{fc})^{b}.$ (2hb)

The design procedure that we follow to solve the parameter estimation problem
consists of four steps.

  1. S1

Proceeding from one of the NSNLP mapping given in M1-M3, construct a new SNLP
of the form

$Y(t)=\phi^{\top}(t){\cal W}(\cdot),$ (2i)

where the functions $Y(t)\in\mathbb{R}$ and $\phi(t)\in\mathbb{R}^{p}$ are
measurable and ${\cal W}:\mathbb{R}^{q}\to\mathbb{R}^{p}$ is a known nonlinear
mapping whose argument is a $q$-dimensional vector containing the unknown
parameters. Notice that for the case of M4 this step can be obviated, since
the parameterization is already separable and, moreover, linear.

  2. S2

Propose a parameter estimator for the SNLP (2i). Due to the more relaxed
signal excitation requirements, we implemented the LSD estimator of [9] in
both simulation and experimental results. However, for a linear regression,
the well-known gradient descent algorithm [10] can be alternatively
implemented.

  3. S3

In the LSD estimator, a condition is imposed on the nonlinear mapping ${\cal
W}$ of the SNLP, namely that it is a monotonizable nonlinearity. As explained
in [13], a necessary and sufficient condition to ensure this property is the
feasibility of an easily verifiable linear matrix inequality (LMI).

  4. S4

Graphically compare the polarization curve resulting from the estimator using
models M2-M4 with the “true” curve. In the simulation results this curve is
that of M1 whereas in the experimental results it corresponds to the curve,
experimentally obtained from the PEMFC system in our test bench.

A key feature of the LSD estimator is that it ensures GEC imposing an interval
excitation (IE) assumption [11] on the regressor $\phi$ of (2i), which is a
considerable milder assumption compared with the classical persistency of
excitation condition [10, 11] imposed in standard gradient or least-squares
estimators. In spite of this fact, we have to point out that the estimation of
the parameters of the full model M1 was unsuccessful. Indeed, in spite of the
selection of “highly exciting” input signals $i_{fc}$ it was not possible to
meet the IE requirement of the LSD estimator for parameter convergence. As
discussed in the paper, this major drawback is due to the fact that some of
the elements of the regressor $\phi$ of the SNLP are linearly dependent.
Interestingly, this phenomenon is absent in the approximate models M2-M4, for
which global exponential convergence (GEC), with all signals remaining
bounded, is indeed ensured.

In spite of our inability to consistently estimate the parameters of the full
model M1, we believe the derivation of an SNLP for this model is an important
contribution to the field. Indeed, to the best of the authors’ knowledge, this
is the first time a GEC on-line estimator of the parameters of realistic
models of the polarization curve is reported. The field being dominated by
standard least squares and extended Kalman filters [14, 15], based on linear
approximations of the nonlinear model, with no guarantee of parameter
convergence. See [16] for a recent survey on the topic.

The remainder of the paper is organized as follows. In Section II we derive
the SNLP for the full model M1 and discuss the issue of lack of excitation of
the associated regressor. Unfortunately, this derivation requires the
measurement of the time derivative of $i_{fc}$. In Section III we consider the
simplified models (2e)-(2g). The details of the proposed LSD and gradient
estimators are given in Section IV. Simulation and experimental results that
illustrate the theoretical developments are given in Section V and VI,
respectively. We wrap-up the paper with concluding remarks and future research
in Section VII. In Appendix A we derive a SNLP for the full model M1 in the
particular case when the current is a sinusoidal signal, which is of interest
in an off-line scenario. Interestingly, in these derivations we obviate the
need for the measurement of the derivative of $i_{fc}$. To simplify the
reading, in Appendix B we give a list of acronyms.

Notation. $I_{n}$ is the $n\times n$ identity matrix For a column vector
$a=\mbox{col}(a_{1},a_{2},\dots,a_{n})\in\mathbb{R}^{n}$, we denote
$|a|^{2}:=a^{\top}a$ and define
$a_{i,j}:=\mbox{col}(a_{i},a_{i+1},\dots,a_{j})$, for $i,j\in\\{1,\dots,n\\}$,
with $n\geq j>i$. The action of a linear time-invariant (LTI) filter ${\cal
F}(p)\in\mathbb{R}(p)$, with $p$ the derivative operator
$p^{n}(w)=:{d^{n}w(t)\over dt^{n}}$, on a signal $w(t)$ is denoted as ${\cal
F}(p)(w)$. To simplify the notation, the arguments of all functions and
mappings are written only when they are first defined and are omitted in the
sequel.

##  II Derivation of a SNLP for the Full Model M1

In this section we derive a SNLP for the full the polarization curve (1), (2)
and (2ca)—called M1 in Section I. Unfortunately, this parameterization does
not contain all of the paramters $\theta$, but only $\theta_{1,4}$. To
estimate the remaining parameter $\theta_{5}$ it is necessary to invoke a
certainty-equivalence argument to compute, using the estimate of the other
parameters, an estimate of it via an elementary algebraic operation, whose
description is also included here. As indicated in Section I, unfortunately,
the regressor of the SNLP derived here contains elements which are linearly
dependent. Consequently, it is not possible to achieve the desired parameter
convergence. This delicate issue is also discussed here. It may be argued that
including a parameterization for which convergence is unattainable is a futile
exercise, however, we believe it is of academic interest to illustrate with an
example the potential appearance of the undesired linear dependence
phenomenon, an issue that is often overlooked in the literature.

###  II-A Derivation of a SNLP for $\theta_{1,4}$

To simplify the notation we introduce the definitions

$y:=v_{fc},\;u:=i_{fc},$ (2j)

and rewrite the polarization curve (1), (2) and (2ca) as

$y=\theta_{4}+\theta_{1}\ln(u)+\theta_{2}u+\theta_{5}e^{\theta_{3}u}.$ (2k)

To derive the SNLP we make the following assumption.

A1 Besides the measurable signals $u$ and $y$ it is possible to reconstruct
the signal $\dot{u}$.

######  Lemma 1.

Consider the model of the fuel cell voltage-current static characteristic
given in (2j), (2k) satisfying Assumption A1. There exists measurable signals
$Y_{M1}(t)\in\mathbb{R},\;\phi_{M1}(t)\in\mathbb{R}^{5}$ such that the
following SNLP holds:

$Y_{M1}(t)=\phi_{M1}^{\top}(t){\cal
W}(\theta_{1,4})+\varepsilon_{t},\quad\forall t\geq 0,$ (2l)

where we defined the nonlinear mapping ${\cal
W}:\mathbb{R}^{4}\to\mathbb{R}^{5}$

${\cal
W}(\theta_{1,4}):=\mbox{col}(\theta_{1},\theta_{3},\theta_{1}\theta_{3},\theta_{2}\theta_{3},\theta_{2}-\theta_{3}\theta_{4}),$
(2m)

and $\varepsilon_{t}$ is a signal exponentially decaying to zero due to the
action of the various filters.222It has been shown in [17, Lemma 1] that the
presence of the exponentially decaying term $\varepsilon_{t}$ does not affect
the validity of our asymptotic behavior claims, hence it is discarded in the
future.

###### Proof.

Differentiating (2k) we get

D2 Notice that the parameter $\theta_{5}$ does not appear in the SNLP. It
turns out that, as explained in Lemma LABEL:lem3 below, the generation of an
estimate is a trivial task with an elementary algebraic computation.

D3 A potential drawback of the result above is the need to reconstruct the
signal $\dot{u}$. In some practical applications the PEMFC current acts as a
control signal [19], and it is often designed including an integral action. In
this case, $\dot{u}$ is known. Another practical solution is to approximate it
using a dirty derivative filter ${p\over\tau p+1}$ that, with small values of
the parameter $\tau>0$, gives a precise reconstruction of the signal
derivative. Moreover, in Appendix A we show that this assumption can be
dispensed when the PEMFC current is of the form $u=A\sin(\omega t)$, with $A$
and $\omega$ unknown, which is a scenario that appears in off-line estimation
tasks.

D4 We underscore the—often overlooked—fact that the filtering action
$z={\lambda p\over p+\lambda}(v)$, which appears in the definition of the
regressor (LABEL:phim1) and in (LABEL:ym1), can be implemented without
differentiation via the realization Figure 1: Simulation result of the linear
dendence of $\phi_{M1}(t)$.

(c) $\mathrm{det}\\{W(\cdot)\\}$. (d) Elements of vector
$\phi_{M1}\in\mathbb{R}^{5}$.

##  III Separable Parameterizations for the Reduced Models M2-M3

In this section we derive separable parameterizations for the approximate
models M2-M3 of the polarization curve, which are given by the NSNLP (2e),
(2f) and (2g), respectively. Interestingly, for the model M2 we will derive a
linear regression equation (LRE) that contains both uncertain parameters $a$
and $b$. Moreover, the construction does not involve any dynamic extension and
is carried out with elementary algebraic operations. The derivation for the
model M3 is also done with simple algebraic manipulations and also leads to an
LRE, but requires a second order dynamic extension and it estimates only $b$.
Consequently, in contrast with the case of M2, it requires the application of
a certainty-equivalent based calculation for the parameter $a$. Finally, the
model M4 is clearly already in LRE form, and the details are given here only
for ease of reference.

###  III-A Reduced Model M2

######  Lemma 4.

Consider the model of the fuel cell voltage-current static characteristic
given in (2e) with known $E_{oc}$ satisfying (2ha). The following LRE holds

$Y_{M2}(t)=\phi_{M2}^{\top}(t)\begin{bmatrix}\ln(a)\\\
b\end{bmatrix},\quad\forall t\geq 0,$ (2s)

where we defined

D6 The proof of Lemma LABEL:lem5 for the model M3 is also extremely simple.
The only difficulty been that we can estimate with the LRE (LABEL:lrem3) only
$b$. However, we can use the relation (2f) to estimate—in a certainty-
equivalent way—the parameter $a$ as follows

${\hat{a}}=(E_{oc}-v_{fc})\big{(}i_{fc}\big{)}^{-\hat{b}}.$

D7 In (LABEL:dotx) we have selected the initial conditions of the state
equations in such a way as to avoid an additive exponentially decaying term
$\varepsilon_{t}$ in the regression (LABEL:lrem3). Although, as discussed in
Footnote 2, this term does not affect the asymptotic behavior of the
estimator, it affects the transient.

##  IV The LSD Estimator and Gradient Estimators

For the sake of completeness in this section we present the LSD estimator of
[9] and the classical gradient scheme [10, 11]. They are given in a generic
style to be applied to NLPRE of the form

$Y=\psi^{\top}{\cal W}(\eta),$ (2z)

with $Y(t)\in\mathbb{R}$, $\psi(t)\in\mathbb{R}^{p}$, $\eta\in\mathbb{R}^{q}$
and ${\cal W}:\mathbb{R}^{q}\to\mathbb{R}^{p}$, or the LRE

$Y=\psi^{\top}\eta,$ (2aa)

with $p=q$.

###  IV-A The LSD parameter estimator

We present in this subsection the LSD estimator for the NLPLRE (2z). The proof
of the proposition below is given in [9, Proposition 1], therefore it is
omitted here.

For the LSD estimator we need to impose the following IE assumption [11] on
the regressor $\phi$.

A2 The regressor vector $\phi$ given in (2z) is IE. That is, there exist
constants $C_{c}>0$ and $t_{c}>0$ such that

$\int_{0}^{t_{c}}\phi(s)\phi^{\top}(s)ds\geq C_{c}I_{p}.$

######  Proposition 1.

Consider the NLPRE (2z), with the regressor vector $\phi(t)$ satisfying
Assumption A2. Define the LSD estimator

A3 The regressor $\psi$ is persistently exciting, that is, there exists $T>0$
and $\rho>0$ such that

$\int_{t}^{t+T}\psi(s)\psi^{\top}(s)ds=\rho I_{q},\;\forall t\geq 0,$

######  Proposition 2.

Consider the LRE (2aa) with $\psi$ verifying Assumption A3, and the gradient
estimator

with $\Gamma\in\mathbb{R}^{q\times q}$ positive definite. Under these
conditions, the parameter estimation errors verify (LABEL:expcon).

##  V Simulation Results

Simulations are conducted to test the estimation algorithms based on the
previously introduced models M2-M4. The simulated system is the Matlab block
corresponding to a $1.26$kW PEMFC stack of the Simscape/Matlab electrical
Library. It consists of 42 cells with an open-circuit voltage of $E_{oc}=42$V.
Also, current and voltage are related in a static manner in this PEMFC block.
Therefore, it does not exhibit the hysteresis behavior usual in a physical
setting. Fig. 2(a) shows the polarization curve of the block. For the
presented simulation tests, the stack is excited with a pulse train varying
from $10$A to $20$A at a frequency of $2$Hz. This yields output voltage
changes from $30.006$V to $28.051$V—see Fig. 2(b).

First, we assessed the performance of the algorithm derived from the LSD
estimator for M2—see Prop. 1 and Lemma LABEL:lem3, respectively. It was
realized with the estimator constants: $\gamma_{0}$=24, $f_{0}=1\times
10^{-5}$, $\gamma_{1}=3\times 10^{4}$, and $\gamma_{2}=3\times 10^{4}$. From
Fig. 2(c), it can be observed the evolution of $\hat{a}$ and $\hat{b}$
obtained during the simulation. Note that $\hat{a}$ converges to $10.3136$
whereas $\hat{b}$ tends to $0.0151$. Fig. 2(f) depicts the estimated
polarization curve plotted from those estimated values. It can be compared
with the polarization curve of the PEMFC system, also included in the figure.

Second, simulation results of the LSD estimator based on M3—see Lemma 4—were
performed. For this test, we set $\lambda=80$, $\lambda_{a}=7$,
$\lambda_{b}=7$, $\gamma_{0}$=24, $f_{0}=1\times 10^{-5}$, and
$\gamma_{1}=600$. To visualize the convergence, $\hat{a}$ and $\hat{b}$ are
plotted as a function of time in Fig. 2(d). At the stop time of simulation
$\hat{a}=7.2641$ and $\hat{b}=0.2178$. The estimated polarization curve
obtained from those values is displayed in Fig. 2(f).

Third, a simulation test was carried out to evaluate the performance of the
LSD estimator with M4—see Lemma LABEL:lem5. For this simulation we take
$\gamma_{0}=24$, $f_{0}=1\times 10^{-6}$, $\gamma_{1}=30$, $\gamma_{2}=30$,
and $\gamma_{3}=30$. Fig. 2(e) shows how the estimations of
$\hat{\theta}_{1}$, $\hat{\theta}_{2}$ and $\hat{\theta}_{6}$ evolve during
the simulation. These converge to the values: $\hat{\theta}_{1}={-1.9271}$,
$\hat{\theta}_{2}={-0.0619}$, and $\hat{\theta}_{6}={35.0619}.$ As previously,
the estimated curve resulting from this test is shown in Fig. 2(f).

As a final remark, it can be observed from 2(f) that M2 represents
appropriately the model for currents within the interval
$u\in[10\mathrm{A},20\mathrm{A}]$, for $u<10$A and $u>20$ the difference is
noticeable and increasing. On the other hand, M3 represents appropriately the
model for the interval $u\in[1\mathrm{A},30\mathrm{A}]$. For currents out of
that range there is a barely noticeable difference. To conclude, Model M4
represents remarkably well the model in the entire operation range and
exhibits the best performance.

(a) Polarization curve of the Matlab’s PEMFC block.

(b) Input current $u$ and output voltage $y$.

(c) Online estimation of $\hat{a}$ and $\hat{b}$ in M2.

(d) Online estimation of $\hat{a}$ and $\hat{b}$ in M3.

(e) Online estimation of $\hat{\theta}_{1}$, $\hat{\theta}_{2}$ and
$\hat{\theta}_{6}$ in M4.

(f) Polarization curve of the simulated PEMFC and the estimated polarization
curves M2-M4.

Figure 2: Simulation results of the online estimation.

##  VI Experimental Results

Experimental tests are carried out for experimental validation of the
previously introduced LSD estimator for models M2-M4 in a fully automated Nexa
PEMFC stack. This PEMFC stack consists of 65 cells, a rated output power of
1.2kW, and a nominal output voltage of 26V. Moreover, the following equipment
is used to perform such tests: an opamp-based data acquisition system for
PEMFC current and voltage measurements, an electronic variable load, and the
DSPACE DS1104 for real-time numerical computations. This test bench is
depicted in Fig. 3.

Figure 3: Test bench using 1.2 $kW$ Nexa PEMFC power module.

The polarization curve of the fuel cell was experimentally obtained by driving
the cell’s current to several operation points and then registering the
corresponding steady-state voltage for each point. The hysteresis phenomenon
causes two different paths in the plot depending on whether the subsequent
current operation points have a rise or fall in their value. One path in the
hysteresis loop of Fig. 5(a) is produced when the current is varied upward
from $0$A to $30$A. The other path is produced when the current is varied
downward from $30$A to $0$A. These two paths have been averaged to obtain a
new curve to which we refer in the sequel as the “polarization curve”—see Fig.
5(a). Moreover, the open-circuit voltage of the stack, measured before
testing, was $E_{oc}=39.8$V.

### Approximation of the polarization curve using curve fitting

As a starting point, we approximate the behavior of the polarization curve in
Fig. 5(a) by means of a curve fitting procedure applied to each one of the
previously introduced models M2-M4. Fig. 4 shows these approximations from
computations using Matlab’s Curve Fitting Toolbox [21]. The obtained fitting
parameters for each model are the following. For M2, $a=4.52$ and $b=0.0463$.
For M3, $a=2.117$ and $b=0.5921$. And for M4, $\theta_{1}=-0.7984$,
$\theta_{2}=-0.3709$, and $\theta_{6}=37.31$.

Figure 4: Polarization curve of the PEMFC system and the approximated
polarization curves using curve fitting applied to M2-M4.

### Experimental results of the online estimation

We now present the results of the online estimation using the three proposed
algorithms. The input current for the three experimental tests is a pulse
train varying from $10\mathrm{A}$ to $20\mathrm{A}$ with a period of 3.3 s—see
Fig. 5(b). To reduce the noise produced when sensing, high frequencies were
filtered out by a low-pass filter from the input current and output voltage
signals. Fig. 5(b) shows each sensed signal and its filtered version used to
run the estimation algorithms. All the tests below were carried out with the
LSD estimator.

The first experimental test consisted in assessing the performance of the
algorithm derived from M2. This result was obtained under the following
estimator gains: ${f_{0}=0.1}$, $\gamma_{0}=5$, and
$\Gamma=\mathrm{diag}(30,~{}30)$. Fig. 5(c) shows $\hat{a}$ and $\hat{b}$
obtained during the simulation time. Note from the figure that $\hat{a}$
converges to $5.3842$ and $\hat{b}$ to $0.0438$. The estimated polarization
curve plotted from those values is presented in Fig. 5(f) together with the
actual polarization curve of the PEMFC system.

Next, results of the estimation based on M3 were performed. Fig. 5(d) displays
signals $\hat{a}$ and $\hat{b}$ obtained as a result of this test. We set
${f_{0}=0.1}$, $\gamma_{0}=2.5$, and $\Gamma=6$. Notice from the figure that
the estimated values are $\hat{a}=2.1532$ and $\hat{b}=0.5936$. The
corresponding estimated polarization curve is displayed in Fig. 5(f).

The final experimental results showing the estimates $\hat{\theta}_{1}$,
$\hat{\theta}_{2}$ and $\hat{\theta}_{6}$ of the LSD estimator for M4 are
shown in Fig. 5(e). We fixed ${f_{0}=6\times 10^{-3}}$, $\gamma_{0}=1.115$,
and $\Gamma=\mathrm{diag}(0.5,0.5,0.5)$. As it can be seen, $\hat{\theta}_{1}$
tends to $-1.2193$ whereas $\hat{\theta}_{2}$ tends to $-0.4233$ and
$\hat{\theta}_{6}$ to $39.6861$. The estimated polarization curve resulting
from this test is shown in Fig. 5(f).

It can be observed from 5(f) that, compared with the other two models, M3 fits
better the actual polarization curve in the entire operation interval. On the
other hand, the reliability of M2 is limited to the interval
$u\in[5\mathrm{A},22\mathrm{A}]$. For current values out of that interval the
mismatch of M2 with respect to the actual polarization curve is evident. On
the other hand, M4 approximate better for current levels above $7$A and for
values close to $0$A, this model is clearly more reliable than M2.

(a) Current-voltage curves of the PEMFC system: the hysteresis loop and the
polarization (average) curve.

(b) Input current $u$ and its output voltage $y$.

(c) Online estimation of $\hat{a}$ and $\hat{b}$ in M2.

(d) Online estimation of $\hat{a}$ and $\hat{b}$ in M3.

(e) Online estimation of $\hat{\theta}_{1}$, $\hat{\theta}_{2}$ and
$\hat{\theta}_{6}$ in M4

(f) Polarization curve of the PEMFC system and the estimated polarization
curves M2-M4.

Figure 5: Experimental results of the online estimation.

##  VII Concluding Remarks and Future Research

We have presented the first solution to the problem of designing a GEC, on-
line algorithm to estimate the parameters of the full model M1 of the
polarization curve of a PEMFC that is described by a NSNLP. The design
proceeds in three steps: (i) the derivation of a SNLP for the parameters
$\theta_{1,4}$; (ii) the application of the LSD estimator to reconstruct these
parameters from this SNLP; (iii) the application of simple algebraic operation
to generate the estimate of $\theta_{5}$. It should be underscored that the
use of the LSD estimator is necessary to deal with the non-linear nature of
the regression equation (without overparameterization). Unfortunately, it is
numerically shown that the elements of the SNLP regression are linearly
dependent. Hence, the vector of unknown parameters is not-identifiable. It
should be underscored that this negative result does not imply that the
original NSNLP (2k) is not identifiable.

Besides the estimator for the full model, we have presented extremely simple
procedures to estimate the parameters of the approximate models M2-M4—which
are widely used for control of PEMFC. In these cases, it is possible to derive
a LRE from which the unknown parameters can be estimated with standard
algorithms.

Simulation and experimental results to evaluate the performance on the
estimator of the full model in the face of parameter variations and noise have
been reported. Also, a study has been carried out to assess the ability of the
reduced models to approximate the behavior of the exact polarization curve.

Current research is under way to use these estimator results in the context of
adaptive control. Also, in the spirit of [19], it is of interest to study the
variations with temperature of the reversible potential, which is assumed
constant in this paper. Some preliminary results on this topic have been
reported in [22].

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##  Appendix A Case of $i_{fc}(t)=A\sin(\omega t)$ with $A$ and $\omega$
Unknown

In this appendix we consider the case when the current takes the form

$u(t)=A\sin(\omega t),$

with $A>0$ and $\omega>0$ unknown, which is of interest for off-line
estimation. We will show below that, in this case, it is possible to remove
the assumption of known $\dot{u}$. Instrumental to obtain this result is the
Swapping Lemma [10, Lemma 3.6.5].

Towards this end, we recall equation (2l), that we rewrite here for ease of
reference:

We concentrate our attention on the terms depending on $\dot{u}$. First,
notice that

${\lambda\over p+\lambda}(\dot{u}\ln(u))={\lambda p\over
p+\lambda}(u\ln(u))-{\lambda p\over p+\lambda}(u).$

Second, note that we can invoke the Swapping Lemma to write

where we have used the fact that $\ddot{u}=-\omega^{2}u$, in the second
identity. Replacing these two identities in the NLPRE above we get

with333Notice the, unlike the map ${\cal W}(\theta_{1,4})$, the new map ${\cal
G}(\theta_{1,4})$ does not contain the term $\theta_{3}$ alone.

${\cal G}(\theta_{1,4}):=\begin{bmatrix}\theta_{1}\\\ \theta_{2}\\\
\theta_{3}\theta_{4}\\\ \theta_{1}\theta_{3}\\\
\theta_{2}\theta_{3}\end{bmatrix}\in\mathbb{R}^{5},\;\xi:=\begin{bmatrix}{\lambda
p\over p+\lambda}(\ln(u))\\\ {\lambda p\over p+\lambda}(u)\\\ -{\lambda p\over
p+\lambda}(u)\\\ -{\lambda p\over p+\lambda}(u\ln(u))+{\lambda p\over
p+\lambda}(u)\\\ -{1\over 2}{\lambda p\over p+\lambda}(u^{2})\\\ {\lambda\over
p+\lambda}\left(u{1\over p+\lambda}(y)\right)\end{bmatrix}\in\mathbb{R}^{6}$
(2ac)

where we defined $\theta_{6}:=\theta_{3}\omega^{2}$. Let us now look at the
term $\dot{u}{\lambda\over p+\lambda}(y)$, to which we apply the filter ${\cal
F}(p)$ and invoke the Swapping Lemma to get

We now proceed to apply the filter ${\cal F}(p)$ to $Y$ above yielding

${\cal
F}(p)(Y)=\theta_{3}\left[\dot{u}{\lambda^{2}\over(p+\lambda)^{2}}(y)\right]+\theta_{6}\left[{\lambda\over
p+\lambda}\left(u{\lambda\over(p+\lambda)^{2}}(y)\right)\right]+\begin{bmatrix}{\cal
G}^{\top}(\theta_{1,4})&\theta_{6}\end{bmatrix}{\cal F}(p)(\xi).$

To eliminate the term that depends on $\dot{u}$ we define the signal

where we defined the vector

$\chi:={\lambda^{2}\over(p+\lambda)^{2}}(y)\xi-{\lambda\over
p+\lambda}(y){\cal F}(p)(\xi)-{\bf e}_{6}{\lambda\over
p+\lambda}\left(u{\lambda\over(p+\lambda)^{2}}(y)\right){\lambda\over
p+\lambda}(y)$

with ${\bf e}_{6}:=\mbox{col}(0,0,0,0,0,1)\in\mathbb{R}^{6}$.

It is easy to show that the map $\begin{bmatrix}{\cal
G}^{\top}(\theta_{1,4})&\theta_{6}\end{bmatrix}:\mathbb{R}^{5}\to\mathbb{R}^{6}$
is monotonizable. Therefore, equipped with the SNLP $Z(t)=\begin{bmatrix}{\cal
G}^{\top}(\theta_{1,4})&\theta_{6}\end{bmatrix}\chi(t)$ it is possible, using
the LSD estimator, to estimate the parameters $\theta_{1,4}$ and $\theta_{6}$.

##  Appendix B List of Acronyms

DREM | Dynamic regressor extension and mixing
---|---
GEC | Global exponential convergence
IE | Interval excitation
LRE | Linear regression equation
LSD | Least-squares plus DREM
LTI | Linear time-invariant
SNLP | Separable onlinear parameterization
NSNLP | Nonseparable nonlinear parameterization
PEMFC | Proton exchange membrane fuel cell