289 | WISEA J004153.42+402117.6 [IrS] | 10.47 | 40.35 | $0.006\pm 0.002$ | 0.007 | 0.28 | 0.37 | 0.69
311 | WISEA J004424.00+413042.9 [IrS] | 11.10 | 41.51 | $0.01\pm 0.002$ | 0.017 | 0.39 | 0.10 | 0.34
328 | NVSS J004510+410915 [RadioS] | 11.30 | 41.15 | $0.004\pm 0.002$ | 0.152 | 1.65 | -0.72 | 1.08
345 | NVSS J004809+403239 [RadioS] | 12.04 | 40.54 | $0.008\pm 0.002$ | 0.028 | 0.59 | 0.00 | 0.78
351 | SSTSL2 J004044.37+404845.4 [RadioS] | 10.18 | 40.81 | $0.006\pm 0.002$ | 0.002 | -0.25 | 0.72 | 0.49
359 | WISEA J004222.07+401605.6 [RadioS] | 10.59 | 40.27 | $0.006\pm 0.002$ | 0.014 | 0.49 | 0.00 | 0.49
363 | WISEA J003806.84+424410.2 [RadioS | 9.53 | 42.74 | $0.003\pm 0.002$ | 0.007 | 0.31 | 0.31 | 1.05
364 | NVSS J004828+402048 [RadioS] | 12.12 | 40.35 | $0.013\pm 0.002$ | 0.022 | 0.27 | 0.00 | 0.27
394 | SSTSL2 J004418.42+411337.9 [RadioS] | 11.08 | 41.23 | $0.005\pm 0.002$ | 0.014 | 0.43 | -0.03 | 0.56
397 | WISEA J003924.78+423005.0 RadioS] | 9.85 | 42.50 | $0.006\pm 0.002$ | 0.076 | 1.14 | -0.38 | 0.93
399 | B3 0040+410 [RadioS] | 10.90 | 41.34 | $0.008\pm 0.002$ | 0.018 | 0.60 | 0.19 | 0.54
410 | WISEA J003917.87+410300.4 [RadioS] | 9.83 | 41.05 | $0.018\pm 0.002$ | 0.011 | -0.31 | 0.00 | -0.28
448 | NVSS J004437+414511 [RadioS] | 11.16 | 41.75 | $0.003\pm 0.002$ | 0.006 | 0.21 | 0.30 | 0.85
449 | NVSS J004437+414511 [RadioS] | 11.16 | 41.75 | $0.003\pm 0.002$ | 0.006 | 0.21 | 0.30 | 0.85
466 | NVSS J003830+424218 [RadioS] | 9.63 | 42.70 | $0.005\pm 0.002$ | 0.019 | 0.73 | 0.00 | 0.73
468 | WISEA J004444.96+424334.6 [RadioS] | 11.19 | 42.73 | $0.008\pm 0.002$ | 0.007 | 0.05 | 0.32 | 0.28
479 | NVSS J004006+402146 [RadioS] | 10.03 | 40.36 | $0.005\pm 0.002$ | 0.010 | 0.38 | 0.00 | 0.38
518 | NVSS J004424+415421 [RadioS] | 11.10 | 41.91 | $0.006\pm 0.002$ | 0.003 | -0.17 | 0.58 | 0.30
523 | WISEA J003750.66+395253.4 [RadioS] | 9.46 | 39.88 | $0.005\pm 0.002$ | 0.009 | 0.34 | 0.00 | 0.33
545 | NVSS J004612+420410 [XrayS] | 11.55 | 42.07 | $0.004\pm 0.002$ | 0.010 | 0.54 | 0.00 | 0.53
567 | NVSS J004738+422020 [RadioS] | 11.91 | 42.34 | $0.007\pm 0.002$ | 0.008 | 0.01 | 0.00 | 0.01
574 | NVSS J003954+423708 [RadioS] | 9.98 | 42.62 | $0.007\pm 0.002$ | 0.010 | 0.20 | 0.00 | 0.20
599 | NVSS J004608+424558 [RadioS] | 11.54 | 42.77 | $0.012\pm 0.002$ | 0.010 | -0.10 | 0.00 | -0.10
615 | WISEA J003932.05+420424.2 [RadioS] | 9.88 | 42.07 | $0.005\pm 0.002$ | 0.007 | 0.18 | 0.00 | 0.18
619 | SSTSL2 J003926.19+412154.2 [RadioS] | 9.86 | 41.37 | $0.005\pm 0.002$ | 0.006 | 0.06 | 0.00 | 0.06
622 | NVSS J004639+421236 [RadioS] | 11.66 | 42.21 | $0.006\pm 0.002$ | 0.008 | 0.16 | 0.00 | 0.16
625 | SSTSL2 J003939.81+415450.5 [RadioS] | 9.92 | 41.91 | $0.005\pm 0.002$ | 0.008 | 0.66 | 0.00 | 0.09
626 | NVSS J004542+415512 [RadioS] | 11.43 | 41.92 | $0.004\pm 0.002$ | 0.008 | 0.32 | 0.00 | 0.31
629 | SSTSL2 J004607.00+422045.2 [IrS] | 11.53 | 42.35 | $0.003\pm 0.002$ | 0.010 | 0.59 | 0.00 | 0.58
645 | NVSS J004435+412521 [SNR] | 11.15 | 41.42 | $0.011\pm 0.002$ | 0.011 | 0.01 | 0.00 | 0.01


# Exploring the use of deep learning in task-flexible ILC*

Anantha Sai Hariharan Vinjarapu, Yorick Broens, Hans Butler, and Roland Tóth
*This work has received funding from the ECSEL Joint Undertaking under grant
agreement No 875999 and from the European Union within the framework of the
National Laboratory for Autonomous Systems
(RRF-2.3.1-21.2022-00002).A.S.H.Vinjarapu, Y.Broens, R.Tóth and H.Butler are
with the Department of Electrical Engineering, Eindhoven University of
Technology, Eindhoven, The Netherlands. R.Tóth is also affiliated with the
Systems and Control Laboratory, Institute for Computer Science and Control,
Hungary. H.Butler is also affiliated with ASML, Veldhoven, The Netherlands.
Email<EMAIL_ADDRESS>
###### Abstract

Growing demands in today’s industry results in increasingly stringent
performance and throughput specifications. For accurate positioning of high-
precision motion systems, feedforward control plays a crucial role.
Nonetheless, conventional model-based feedforward approaches are no longer
sufficient to satisfy the challenging performance requirements. An attractive
method for systems with repetitive motion tasks is iterative learning control
(ILC) due to its superior performance. However, for systems with non-
repetitive motion tasks, ILC is generally not applicable, despite of some
recent promising advances. In this paper, we aim to explore the use of deep
learning to address the task flexibility constraint of ILC. For this purpose,
a novel Task Analogy based Imitation Learning (TAIL)-ILC approach is
developed. To benchmark the performance of the proposed approach, a simulation
study is presented which compares the TAIL-ILC to classical model-based
feedforward strategies and existing learning-based approaches, such as neural
network based feedforward learning.

##  I Introduction

High-precision positioning systems are essential components in modern
manufacturing machines and scientific equipment, see [1, 2, 3, 4]. To ensure
high-throughput and high-accuracy position tracking, a two-degree-of-freedom
controller structure, consisting of a feedback controller and a feedforward
controller, is commonly utilized, see [5, 6, 7]. The feedback controller
maintains closed-loop stability and disturbance rejection, while the
feedforward controller is primarily responsible for achieving optimal position
tracking performance, see [8]. Nonetheless, with the increasingly stringent
demands in contemporary industry, conventional model-based feedforward
techniques, e.g. [9], are no longer adequate to meet the desired performance
specifications, thus necessitating for alternative feedforward approaches.

_Iterative Learning Control_ (ILC), see [10], has emerged as a viable choice
for feedforward control in motion systems that execute recurring tasks,
enabling accurate position tracking. Despite its advantages, ILC exhibits
significant limitations. Primarily, ILC is dependent on the assumption that
the tracking error recurs from one iteration to the next, limiting its general
applicability. Additionally, conventional ILC performance is constrained to a
single task, see [11].

Several studies have attempted to address the task flexibility limitations of
ILC by drawing on concepts from machine learning and system identification, as
reported in the literature [12, 13, 14]. However, the findings from the
related literature suggest that there exists a trade-off between the
achievable position tracking performance and the degree of deviation from the
core principle of ILC, i.e., direct iterative manipulation of signals. Instead
of compromising local ILC performance to enhance task flexibility, the aim is
to develop a learning-based feedforward strategy that can deliver superior
position tracking performance regardless of the severity of the variation of
the compensatory signal across tasks. Such an ILC variant can be imagined to
make use of imitation learning in order to mimic the behaviour of conventional
ILC policies generalized over multiple trajectories.

This paper introduces a novel approach to ILC, termed Task Analogy based
Imitation Learning (TAIL)-ILC, from a data science perspective. By acquiring
spatial feature analogies of the trajectories and their corresponding control
signals, performance of conventional ILC policies can be replicated. To
facilitate efficient network training, abstract lower-dimensional
representations of signals are utilized. This approach offers numerous
benefits in terms of training and prediction time efficiency, utilization of
large datasets, and high sampling rate handling. The resulting feedforward
controller comprises an encoding policy, a learning policy, and a decoding
policy arranged in a cascade interconnection. Dual principal component
analysis (DPCA), a standard linear dimensionality reduction technique, is
utilized for the integration of the encoding and decoding policies, while a
deep neural network is employed for the learning policy.

The main contributions of this paper are:

  * (C1)

A novel TAIL-ILC approach that tackles the task extension problem of ILC via
learning spatial feature analogies of trajectories and their compensation
signals, enabling direct imitation of ILC policies.

  * (C2)

An efficient implementation strategy is devised for the learning-based
feedforward controller in terms of constructing it through the cascade
interconnection of an encoder, a deep neural network, and a decoder.

This paper is organized as follows. First, the problem formulation is
presented in Section II. Next, Section III presents the proposed novel TAIL-
ILC approach which aims at generalizing ILC performance across various tasks
through imitation learning strategies. Section IV provides a simulation study
of the proposed approach with respect to existing feedforward strategies using
a high-fidelity model of a moving-magnet planar actuator. In Section V,
detailed comparison between the proposed TAIL-ILC approach and neural-network-
based feedforward strategies is presented. Finally, conclusions on the
proposed approach are presented in Section VI.

##  II Problem statement

###  II-A Background

Consider the conventional frequency domain ILC configuration illustrated by
Figure 1, where $P\in\mathcal{R}^{n_{\mathrm{y}}\times n_{\mathrm{u}}}$
corresponds to the proper transfer matrix representation of a _discrete time_
(DT) _linear-time-invariant_ (LTI) _multiple-input multiple-output_ (MIMO)
plant with $\mathcal{R}$ denoting the set of real rational functions in the
complex variable $z\in\mathbb{C}$. Furthermore, the proper
$K\in\mathcal{R}^{n_{\mathrm{u}}\times n_{\mathrm{y}}}$ represents a LTI
stabilizing DT feedback controller, which is typically constructed using
rigid-body decoupling strategies, see [15]. The aim of the conventional
frequency domain ILC framework is to construct an optimal feedforward policy
$f$, which minimizes the position tracking error $e$ in the presence of the
motion trajectory $r$. Under the assumption that the reference trajectory is
trial invariant, the error propagation per trial $k\in\mathbb{N}_{\geq 0}$ is
given by:

$e_{k}=Sr-Jf_{k},$ (1)

where $S=(I+PK)^{-1}$ and $J=(I+PK)^{-1}P$. Generally, the update law for the
feedforward policy is in accordance with the procedure outlined in [16]:

$f_{k+1}=Q\left(Le_{k}+f_{k}\right),$ (2)

where $L\in\mathcal{RL}_{\infty}^{n_{\mathrm{u}}\times n_{\mathrm{y}}}$ is a
learning filter and $Q\in\mathcal{RL}_{\infty}^{n_{\mathrm{u}}\times
n_{\mathrm{u}}}$ denotes a robustness filter with $\mathcal{RL}_{\infty}$
corresponding to set of real rational functions in $z$ that have bounded
singular value on the unit circle
$\mathbb{D}=\\{e^{\mathrm{i}\omega}\mid\omega\in[0,2\pi]\\}$, i.e., finite
$\mathcal{L}_{\infty}(\mathbb{D})$ norm. Both $L$ and $Q$ are required to be
designed for the ILC task at hand. Furthermore, by combining (1) and (2), the
progression of the error and feedforward update is reformulated as:

$\displaystyle e_{k+1}$ $\displaystyle=(I-JQJ^{-1})Sr+JQ(J^{-1}-L)e_{k},$ (3a)
$\displaystyle f_{k+1}$ $\displaystyle=QLSr+Q(I-LJ)f_{k},$ (3b)

which can be reduced to:

$\displaystyle e_{k+1}$ $\displaystyle=(I-Q)Sr+Q(I-JL)e_{k},$ (4a)
$\displaystyle f_{k+1}$ $\displaystyle=QLSr+Q(I-LJ)f_{k},$ (4b)

under the assumption that $Q$ is diagonal and $J$ is approximately diagonal,
which holds in case of rigid-body decoupled systems.

From (4), several observations can be made. First, it can be observed that the
contribution of $r$ to the position tracking error is dependent on the
robustness filter $Q$, which is optimally chosen as identity to negate the
contribution of the reference trajectory towards the tracking error. Secondly,
learning filter $L$ aims to minimize the criterion $\|Q(I-JL)\|_{\infty}<1$,
where $\|\centerdot\|_{\infty}$ stands for the $\mathcal{H}_{\infty}$ norm,
such that the tracking error is steered to zero, which is optimally achieved
when $L=J^{-1}$. Note that these assumptions on $Q$ and $L$ yield the optimal
feedforward update $f_{k+1}=P^{-1}r$, which results in perfect position
tracking. Moreover, when the convergence criterion is satisfied, the limit
policies, i.e. $e_{\infty}=\lim_{k\rightarrow\infty}e_{k}$,
$f_{\infty}=\lim_{k\rightarrow\infty}f_{k}$, correspond to:

$\displaystyle e_{\infty}$
$\displaystyle=\bigl{(}I-J\bigl{(}I-Q(I-LJ)\bigr{)}^{-1}QL\bigr{)}Sr,$ (5a)
$\displaystyle f_{\infty}$ $\displaystyle=\left(I-Q(I-LJ)\right)^{-1}QLSr,$
(5b)

In spite of its simplicity and efficacy, the conventional ILC is hindered by
significant limitations, the most notable of which is its confinement to a
single task. Consequently, its practical utility is restricted to particular
types of machinery.

Figure 1: Control structure with the conventional ILC configuration.

###  II-B Problem formulation

The aim of this paper is to address the challenge of augmenting the task-
flexibility of the conventional ILC by utilizing an imitation learning based
controller. This approach facilitates the generalization of the optimal
feedforward policy, created by the conventional ILC, for a wider range of
motion profiles. The primary objective of this paper is to devise a
feedforward controller that employs a learning-based mechanism, which
satisfies the following requirements:

  * (R1)

The learning-based feedforward approach enables the generalization of the
performance of the conventional ILC across multiple trajectories.

  * (R2)

The scalability of the learning-based feedforward approach is imperative for
its implementation in systems with a high sampling rate.

##  III TAIL-ILC

###  III-A Approach

For a given dynamic system with a proper discrete transfer function
$G\in\mathcal{R}$ under a sampling time $T_{\mathrm{s}}\in\mathbb{R}_{+}$, a
reference trajectory $r$ of duration $T=n_{\mathrm{d}}T_{\mathrm{s}}$ seconds
can be defined as

$r=\begin{bmatrix}r(0)&\cdots&r(n_{\mathrm{d}})\end{bmatrix}^{\top},$ (6)

where $n_{\mathrm{d}}$ corresponds to the length of the signal in DT. This
reference trajectory for example can correspond to a $n^{th}$ order motion
profile. A trajectory class $C\subset\mathbb{R}^{n_{\mathrm{d}}\times
n_{\mathrm{t}}}$ is defined as a collection of reference trajectories such
that each trajectory shares certain prominent spatial features (motion profile
order, constant velocity interval length, etc.) with the others, where
$n_{\mathrm{t}}$ is the number of trajectories:

$C=\\{r_{1},r_{2},r_{3}.....,r_{n_{\mathrm{t}}}\\}.$ (7)

Given a specific combination of the $L$ and $Q$ filters, consider that an ILC
policy $\pi^{*}$ exists which maps a given reference trajectory $r$ to the
optimal feedforward compensation signal $f^{*}$, see (5). This can be formally
expressed as:

$\pi^{*}:r_{i}\rightarrow f^{*}_{i}.$ (8)

Henceforth, $\pi^{*}$ shall be denoted as the expert policy, which is equipped
with learning and robustness filters established through a process model. Our
objective is to formulate an optimal student policy $\pi_{\mathrm{s}}^{*}$
that approximates the performance of the optimal policy $\pi^{*}$ over a set
of trajectories from the pertinent trajectory class. To this end, we endeavor
to determine $\pi_{\mathrm{s}}^{*}$ as a solution to the optimization problem:

$\pi_{\mathrm{s}}^{*}=\underset{\pi_{\mathrm{s}}}{\arg\min}\
\eta(\pi^{*}(r_{i}),\pi_{\mathrm{s}}(r_{i})),\quad\forall
i\in[1,n_{\mathrm{t}}]$ (9)

Figure 2: Offline TAIL-ILC with $r_{i},f_{i}\
\in\mathbb{R}^{n_{\mathrm{d}}\times 1}$.

where $r_{i}\sim C$ and $\eta(\cdot,\cdot)$ is a performance quantification
measure, and $\pi_{\mathrm{s}}$ are parameterized student policy candidates.
The expert policy $\pi^{*}$ is a conventionally designed frequency domain ILC
as described in Section II-A. In TAIL-ILC, the idea is to structure
$\pi_{\mathrm{s}}$ as :

$\pi_{\mathrm{s}}=\pi_{\mathrm{D,Y}}\circ\pi_{\mathrm{C}}\circ\pi_{\mathrm{E,U}}.$
(10)

which is visualised in Figure 2. The TAIL-ILC controller is capable of
generating a feedforward control signal based on a given reference trajectory.
This process is carried out through a series of three sub-policies outlined in
equation (10). The first sub-policy, $\pi_{\mathrm{E,U}}$, projects the
reference trajectory $r_{i}\in\mathbb{R}^{n_{\mathrm{d}}\times 1}$ into a
lower-dimensional space referred to as the latent space. Next, the second sub-
policy, $\pi_{\mathrm{C}}$, predicts a latent space representation of the
feedforward signal, which is then fed into the third sub-policy,
$\pi_{\mathrm{D,Y}}$, to project the latent space feedforward signal back into
the higher-dimensional output space, resulting in
$f_{i}\in\mathbb{R}^{n_{\mathrm{d}}\times 1}$. Notably, the successful
application of TAIL-ILC requires that all reference trajectories share certain
spatial features with each other. The prediction sub-policy,
$\pi_{\mathrm{C}}$, is trained on a set of reference trajectories and their
corresponding feedforward control signals obtained using $\pi^{*}$, which are
projected into the latent space. The use of abstract representations enables
the preservation of the most significant information of the signals while
simultaneously reducing the amount of data used for making predictions,
resulting in several advantages, such as increased training and prediction
time efficiencies. The subsequent sub-section will delve into the development
of each sub-policy in further detail.

###  III-B Student policy $\pi_{\mathrm{s}}$

The three-part student policy $\pi_{\mathrm{s}}:r\rightarrow
f_{\pi_{\mathrm{s}}}$ can be decomposed into three distinct components:

$\displaystyle\pi_{\mathrm{E,U}}$ $\displaystyle:r\rightarrow r_{l}$ (11a)
$\displaystyle\pi_{\mathrm{C}}$ $\displaystyle:r_{l}\rightarrow f_{l}$ (11b)
$\displaystyle\pi_{\mathrm{D,Y}}$ $\displaystyle:f_{l}\rightarrow
f_{\pi_{\mathrm{s}}}$ (11c)

where, $r,f_{\pi_{\mathrm{s}}}\in\mathbb{R}^{n_{\mathrm{d}}\times 1}$ and
$r_{l},f_{l}\in\mathbb{R}^{n_{l}\times 1}$ and $n_{l}$ is the latent space
dimensionality such that $n_{l}\ll n_{\mathrm{d}}$. As mentioned in Section
III-A, the training data for the sub-policy $\pi_{\mathrm{C}}$, namely the
pairs $\\{r_{i,l},f_{i,l}\\}$, are in the latent space. This shows that the
ideal outputs of $\pi_{\mathrm{s}}$ are of the form:

$f_{\pi_{\mathrm{s}}}=\pi_{\mathrm{D,Y}}(f_{l})=f^{\prime}\approx f^{*},$ (12)

where, an approximation error may exist between $f^{\prime}$ and $f^{*}$.
Additionally, we aim at:

$\pi_{\mathrm{C}}(r_{l})=\widehat{f_{l}}\approx f_{l},$ (13)

where, in case of using a deep neural network, $\widehat{f_{l}}$ is the output
of the network and the prediction error $e_{\mathrm{pred}}$ is defined as:

$e_{\mathrm{pred}}=\|f_{l}-\widehat{f_{l}}\|_{2},$ (14)

where, $\|\cdot\|_{2}$ denotes the $\ell_{2}$ norm. Moreover, this implies
that (12) becomes:

$f_{\pi_{\mathrm{s}}}=\pi_{\mathrm{D,Y}}(\widehat{f_{l}})=\widehat{f^{\prime}}.$
(15)

In order to quantify the gap between performance of $\pi^{*}$ and that of
$\pi_{\mathrm{s}}$, a distance measure is used as the performance
quantification measure $\eta$ in (9). This is expressed as:

$\eta(\pi^{*},\pi_{\mathrm{s}})=\frac{1}{n_{\mathrm{t}}}\sum_{i=1}^{n_{\mathrm{t}}}\|f_{i}-\widehat{f^{\prime}_{i}}\|_{2}.$
(16)

Assuming that $\mu$ represents the set of weights and biases of the deep
neural network, improving the performance of $\pi_{\mathrm{s}}$ can be posed
as the following optimization problem:

$\underset{n_{l},\mu}{\arg\min}\ \eta(\pi^{*},\pi_{\mathrm{s}})$ (17)

The proposed approach involves propagating the parameter $\eta$ through the
three sub-policies, with the aim of iteratively optimizing both $n_{l}$ and
$\mu$ via (17). However, given the significant computational burden associated
with this approach, there is a need for a more straightforward alternative or
a reformulation of the problem. With this goal in mind, we introduce the
concepts of the _Expert space_ and _Student space_ to provide alternative
perspectives for addressing the optimization problem at hand.

###### Definition 1

The expert space is defined as the space of all real policies denoted by
superscript$\ {}^{\mathrm{e}}$ having the form

$\pi^{\mathrm{e}}:\mathbb{R}^{n_{\mathrm{x}}\times
1}\rightarrow\mathbb{R}^{n_{\mathrm{d}}\times 1}\quad\forall
n_{\mathrm{x}}\in\mathbb{N}$

Example:

  1. 1.

Expert policy in expert space:

$\pi_{\mathrm{e}}^{\mathrm{e}}:r\rightarrow f^{\prime}$ (18)

where $r,f^{\prime}\in\mathbb{R}^{n_{\mathrm{d}}\times 1}$

  2. 2.

Student policy in expert space:

$\pi_{\mathrm{s}}^{\mathrm{e}}:r\rightarrow\widehat{f^{\prime}}$ (19)

where $r,\widehat{f^{\prime}}\in\mathbb{R}^{n_{\mathrm{d}}\times 1}$

###### Definition 2

The student space is defined as the space of all real policies denoted by
superscript $(\ ^{\mathrm{s}})$ having the form

$\pi^{\mathrm{s}}:\mathbb{R}^{n_{\mathrm{x}}\times
1}\rightarrow\mathbb{R}^{n_{l}\times 1}\ \forall\ n_{\mathrm{x}}\in\mathbb{N}$

Example:

  1. 1.

Expert policy in student space:

$\pi_{\mathrm{e}}^{\mathrm{s}}:r_{l}\rightarrow f_{l}$ (20)

where $r_{l},f_{l}\in\mathbb{R}^{n_{l}\times 1}$

  2. 2.

Student policy in student space:

$\pi_{\mathrm{s}}^{\mathrm{s}}:r_{l}\rightarrow\widehat{f_{l}}$ (21)

where $r_{l},\widehat{f_{l}}\in\mathbb{R}^{n_{l}\times 1}$

Table I summarizes these definitions.

Stated differently, the expert space is comprised of all the decoding
policies, $\pi_{\mathrm{D}}$, which project signals into $n_{\mathrm{d}}$
dimensions, while the student space is composed of all the encoding policies,
$\pi_{\mathrm{E}}$, which project signals into $n_{l}$ dimensions.

TABLE I: Expert and student policies in expert and student spaces | Expert space $(^{\mathrm{e}})$ | Student space $(^{\mathrm{s}})$
---|---|---
Expert policy $(\pi_{\mathrm{e}})$ | $\pi_{\mathrm{e}}^{\mathrm{e}}:r\rightarrow f^{\prime}$ | $\pi_{\mathrm{e}}^{\mathrm{s}}:r_{l}\rightarrow f_{l}$
Student policy $(\pi_{\mathrm{s}})$ | $\pi_{\mathrm{s}}^{\mathrm{e}}:r\rightarrow\widehat{f^{\prime}}$ | $\pi_{\mathrm{s}}^{\mathrm{s}}:r_{l}\rightarrow\widehat{f_{l}}$

Based on the preceding definitions, it is worth noting that our primary
objective is to determine the student policy in the expert space,
$\pi_{\mathrm{s}}^{\mathrm{e}}$. In light of these definitions, the distance
metric specified in (16) can be reformulated as:

$\eta(\pi^{*},\pi_{\mathrm{s}}^{\mathrm{e}})=\eta(\pi^{*},\pi_{\mathrm{e}}^{\mathrm{e}})+\eta(\pi_{\mathrm{e}}^{\mathrm{e}},\pi_{\mathrm{s}}^{\mathrm{e}})$
(22)
$\implies\eta(\pi^{*},\pi_{\mathrm{s}}^{\mathrm{e}})=\frac{1}{n_{\mathrm{t}}}\sum_{i=1}^{n_{\mathrm{t}}}\|f_{i}-f^{\prime}_{i}\|_{2}+\frac{1}{n_{\mathrm{t}}}\sum_{i=1}^{n_{\mathrm{t}}}\|f^{\prime}_{i}-\widehat{f^{\prime}_{i}}\|_{2}$

where, $\eta(\pi^{*},\pi_{\mathrm{e}}^{\mathrm{e}})$ corresponds to the
optimization of $n_{l}$ and
$\eta(\pi_{\mathrm{e}}^{\mathrm{e}},\pi_{\mathrm{s}}^{\mathrm{e}})$
corresponds to the optimization of $\mu$. This separation of the distance
measure (16) allows the optimization problem in (17) to be segmented as:

$\underset{n_{l},\mu}{\arg\min}\
\eta(\pi^{*},\pi_{\mathrm{s}}^{\mathrm{e}})=\underset{n_{l}}{\arg\min}\
\eta(\pi^{*},\pi_{\mathrm{e}}^{\mathrm{e}})+\underset{\mu}{\arg\min}\
\eta(\pi_{\mathrm{e}}^{\mathrm{e}},\pi_{\mathrm{s}}^{\mathrm{e}})$

This segregation allows us to optimize $n_{l}$ independently of $\mu$, thus
simplifying the optimization problem defined by (17).

###  III-C Choice of encoding and decoding sub-policies

The encoding and decoding sub-policies in this work employ DPCA, a well-
established linear dimensionality reduction technique, due to its
computational simplicity. Other commonly-used linear and non-linear
dimensionality reduction methods are also available and have been reviewed in
[17]. DPCA involves the identification of a linear subspace with $n_{l}$
dimensions in an $n_{\mathrm{d}}$ dimensional space, where $n_{l}$ is
significantly smaller than $n_{\mathrm{d}}$. This subspace is defined by a set
of orthonormal bases that maximize the variance of the original data when
projected onto this subspace. The orthonormal bases computed through this
process are commonly referred to as _principal components_.

###### Definition 3

A _data point_ in an arbitrary dataset $H\in\mathbb{R}^{n_{\mathrm{x}}\times
n_{\mathrm{t}}}$ is defined as a vector
$r_{i}\in\mathbb{R}^{n_{\mathrm{x}}\times 1}$ $\forall
i\in[1,n_{\mathrm{t}}]$.

The selection of the principal components for an $n_{l}$ dimensional latent
space for the data points in $C$ involves choosing the right eigenvectors that
correspond to the first $n_{l}$ singular values of $H$. It should be
emphasized that the projection of a data point onto the latent space can be
computed through the following method:

$r_{l}=T_{\mathrm{E}}r,$ (23)

where:

$T_{\mathrm{E}}=\widehat{\Sigma}^{-1}V^{\top}H^{\top}$ (24)

In this context, $r_{l}\in\mathbb{R}^{n_{l}\times 1}$,
$V\in\mathbb{R}^{n_{\mathrm{t}}\times n_{\mathrm{t}}}$ denotes the matrix of
right eigenvectors of $H$ and $\widehat{\Sigma}\in\mathbb{R}^{n_{l}\times
n_{\mathrm{t}}}$ contains the first $n_{l}$ singular values of $H$ along its
diagonal elements. It is worth noting that the value of $n_{l}$ is constrained
by the number of data points in $H$. This feature of DPCA is particularly
advantageous in situations where $n_{\mathrm{d}}>>n_{\mathrm{t}}$. Given the
latent space representation $r_{l}$, a reconstructed data point $r^{\prime}$
can be obtained as:

$r^{\prime}=T_{D}r_{l},\qquad r^{\prime}\in\mathbb{R}^{n_{\mathrm{d}}\times
1},$ (25)

where:

$T_{\mathrm{D}}=HV\widehat{\Sigma}^{-1}.$ (26)

###### Remark 1

The computation of transformations $T_{\mathrm{E}}$ and $T_{\mathrm{D}}$
depend on $n_{l}$. Additionally, considering that we have access to the
dataset $H$, the matrix transformations $\widehat{\Sigma}^{-1}V^{T}H^{T}$ and
$HV\widehat{\Sigma}^{-1}$, the right hand side of (23) and (25), become
constant for a specific problem for a given choice of $n_{l}$.

In the light of Remark 1, for a given dataset
$H\in\mathbb{R}^{n_{\mathrm{d}}\times n_{\mathrm{t}}}$ the encoding
($\pi_{\mathrm{E,U}}$) and decoding ($\pi_{\mathrm{D,Y}}$) sub-policies for
using in the student policy $\pi_{\mathrm{s}}$ can be defined as follows:

$\displaystyle\pi_{\mathrm{E,U}}(r)$ $\displaystyle=T_{\mathrm{E}}r$ (27a)
$\displaystyle\pi_{\mathrm{D,Y}}(\widehat{f_{l}})$
$\displaystyle=T_{\mathrm{D}}\widehat{f_{l}}$ (27b)

##  IV Simulation study

This section presents a comparison study of the TAIL-ILC approach in
comparison with classical ILC, an artificial neural network (ANN) based ILC,
referred to as NN-ILC, see [14], and conventional rigid body feedforward, see
[7, 18], which is obtained by multiplication of the acceleration profile and
the inverted rigid body dynamics of the system:

$C_{\mathrm{FF}}=m\ddot{r_{i}}$ (28)

To facilitate simulation, a high-fidelity model of a moving-magnet planar
actuator (MMPA), depicted in Figure 3, is considered. A detailed description
of a MMPA system is given in [19].

Figure 3: Schematic representation of a MMPA model.

Table II provide a concise overview of the network architecture and training
specifics for sub-policy $\pi_{\mathrm{C}}$ in TAIL-ILC and policy
$\pi_{\mathrm{NN}}$ in NN-ILC, respectively. For the sake of comparability,
the training parameters are kept consistent between the two networks. The
networks are designed and trained using the Deep Learning toolbox in MATLAB
2019b, employing the default random parameter initialization.

The training set consists of 618 trajectories, while the test set includes 42
trajectories, each of which is 2.5 seconds long with a total of 20833 time
samples. Each trajectory corresponds to a fourth-order motion profile,
designed based on the approach presented by [20], and is parameterized with
five parameters in the spatial domain. Individual trajectories are then
generated by sweeping over a grid of values for each of these parameters. The
objective of this study is to evaluate and compare the performance of the
previously mentioned feedforward approaches against the expert ILC policy
$\pi^{*}$, which is the traditional ILC optimized for multiple trajectories of
the same class. The primary aim of ILC in this context is to mitigate any
unaccounted-for residual dynamics in the system and enhance classical model-
based feedforward. Consequently, we also compare the combined performance of
student policies with classical feedforward controllers. We demonstrate the
tracking ability of TAIL-ILC and NN-ILC on two reference trajectories, namely
$r_{1}$ and $r_{2}$, which belong to the same class and are shown in Figure 4.
$r_{1}$ is a randomly chosen trajectory from the training set, while $r_{2}$
is a previously unseen trajectory.

TABLE II: Architecture and training details of the NNs

Parameter | TAIL-ILC | NN-ILC
---|---|---
No. of neurons in the input layer | $618$ | $4$
No. of hidden layers | $3$ | $3$
No. of neurons in hidden layers | $800$ | $6$
Activation | Relu | Relu
No. of neurons in the output layer | $618$ | $1$
Learning rate | $10^{-3}$ | $10^{-3}$
Epochs | $5000$ | $5000$
Optimizer | adam | adam
Minibatch size | $128$ | $128$
Train set |  $618$ trajectories |  $618$ trajectories
Test set |  $42$ trajectories |  $42$ trajectories

Figure 4: x-direction of the references $r_{1}$ and $r_{2}$.

###  IV-A Time domain performance of TAIL-ILC and NN-ILC

A silicon wafer scanning application is considered where the scanning takes
place during the constant velocity interval of the motion profile, see [1]. In
this context, Figure 5 illustrates the position tracking error in
$x$-direction during the constant velocity interval of the reference
trajectories $r_{1}$ and $r_{2}$ respectively. In addition to the performance
of mass feedforward, TAIL-ILC and NN-ILC, the figure also indicates the
performance of the expert ILC policy. This is to facilitate the comparison of
the performance of the two deep learning based ILC variants with the baseline.
As demonstrated in the left Figure, i.e. the performance of the feedforward
controllers on $r_{1}$, the expert ILC policy exhibits the highest overall
performance. Nonetheless, it is noteworthy that the TAIL-ILC policy
outperforms in terms of the peak tracking error achieved compared to the
alternative feedforward approaches, whereas the NN-ILC policy demonstrates a
superior performance in terms of the convergence time of the error.
Nonetheless, when analyzing the right Figure, i.e. the performance of the
feedforward approaches for a previously unseen trajectory $r_{2}$, the expert
ILC policy needs to re-learn the relevant feedforward signal. Conversely, the
TAIL-ILC and NN-ILC policies are capable of achieving similar performance to
the re-learned expert ILC policy without any further training. Additionally,
when combined with a classical mass feedforward controller, both the TAIL-ILC
and NN-ILC policies are observed to yield superior performance in terms of
peak error and settling time compared to the classical mass feedforward
controller alone.

Figure 5: Tracking error for $r_{1}$ (left) and $r_{2}$ (right) during
constant velocity.

###  IV-B TAIL-ILC vs NN-ILC

Table III provides a comparison of the training and prediction properties of
the TAIL-ILC and NN-ILC student policies. Here, we compare the following
parameters:

  1. 1.

$T_{\text{train}}$: Time to train the neural network

  2. 2.

$T_{\text{predict}}$: Time to make predictions for 10 randomly selected test
set trajectories.

  3. 3.

$e_{\text{train}}$: Control signal prediction error averaged over 10 randomly
selected train set trajectories.

  4. 4.

$e_{\text{test}}$: Control signal prediction error averaged over 10 randomly
selected test set trajectories.

  5. 5.

$e_{\text{peak \ tracking}}$: Peak tracking error achieved with the predicted
control signals averaged over 10 randomly selected train set trajectories.

TABLE III: Performance comparison for the $1^{st}$ degree of freedom Criterion | NN-ILC | TAIL-ILC
---|---|---
$T_{\text{train}}$ |  $2.5$ hr |  $20$ min
$T_{\text{predict}}(\text{per sample})$ |  $0.005$ sec |  $0.064$ sec
$T_{\text{predict}}(\text{full signal})$ |  $86$ sec |  $0.064$ sec
$e_{\text{train}}$ |  $0.0055$ N |  $0.0011$ N
$e_{\text{test}}$ |  $0.0013$ N |  $0.0064$ N
$e_{\text{peak\ tracking}}$ |  $1.3\cdot 10^{-7}$ m |  $8.3\cdot 10^{-8}$ m

Here, the average control signal prediction errors of the train and the test
set trajectories are calculated as the values of the performance measure
$\eta$ in (22). As can be seen, though the original signals and trajectories
are extremely high dimensional, the projection of these signals into the
latent space using the proposed TAIL-ILC approach has resulted in significant
improvement in training and prediction time compared to that of the NN-ILC
approach.

Moreover, as observed in Figure 5, the average signal prediction error has
decreased for TAIL-ILC in case of previously seen trajectories whereas the NN-
ILC has improved performance for previously unseen trajectories.

##  V TAIL-ILC vs NN-ILC PERSPECTIVES

In the previous Section, we have seen a comparison of the performance of TAIL-
ILC and NN-ILC controllers for a specific use case. However, it is more
natural to view these controllers as individual instances of two fundamentally
different perspectives of the problem. Hence, it is important to reflect upon
the perspectives that these controllers convey and the consequences for
various aspects of the resulting controllers. This is expected to provide us
with a more generalised reasoning to some of the differences observed in
performances of these two controllers.

###  V-A Time duration of trajectories

The NN-ILC and TAIL-ILC are two approaches of ILC that differ in their
treatment of reference trajectories and feedforward signals. NN-ILC is capable
of handling trajectories of different lengths, as it deals with them sample-
wise. In contrast, TAIL-ILC processes trajectories and signals in their
entirety, making it challenging to manage trajectories of varying durations
due to the fixed input-output dimensionality of neural network learning
models. Additionally, NN-ILC is better equipped to handle instantaneous
changes in reference trajectories compared to TAIL-ILC. A possible solution to
reconcile these perspectives is to use a different class of learning models,
such as a recurrent neural network.

###  V-B Training and prediction time efficiencies

In NN-ILC, the training dataset used for $\pi_{\mathrm{NN}}$ encompasses all
the samples from all the trajectories in the training set, along with their
associated feedforward signals. Conversely, TAIL-ILC employs a training
dataset for $\pi_{\mathrm{C}}$ that solely includes the parameters of the
trajectories and feedforward signals within the latent space, resulting in a
significantly smaller dataset in comparison to the total number of samples.
This characteristic leads to TAIL-ILC presenting shorter training and
prediction times when compared to NN-ILC, as demonstrated by the results
presented in Table III.

###  V-C Generalizability to previously unseen trajectories

Figure 5 demonstrates that NN-ILC outperforms TAIL-ILC in terms of
generalizing performance to previously unobserved trajectories. The improved
performance can be attributed to NN-ILC’s treatment of reference trajectories
as points in an $n$-dimensional space corresponding to $n^{\mathrm{th}}$ order
motion profiles, which allows it to learn a mapping to the corresponding
feedforward signal time samples. As a result, the trained network can more
accurately extrapolate performance to previously unobserved points in the
space of possible reference trajectories. In contrast, TAIL-ILC relies
primarily on analogies between individual tasks on a higher level, which may
result in suboptimal performance when confronted with previously unobserved
trajectories at the sample level.

##  VI CONCLUSION

In this work, we have primarily explored two different perspectives within the
context of deep learning of the task-flexibility constraint of conventional
ILC. While each of the considered approaches has its own advantages and
disadvantages, it has been observed that the use of deep learning techniques
in general could be a useful direction for future research in designing task-
flexible ILC variants.

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# A multiplicative ergodic theoretic characterization of relative equilibrium
states

John Antonioli, Soonjo Hong and Anthony Quas

###### Abstract.

In this article, we continue the structural study of factor maps betweeen
symbolic dynamical systems and the relative thermodynamic formalism. Here, one
is studying a factor map from a shift of finite type $X$ (equipped with a
potential function) to a sofic shift $Z$, equipped with a shift-invariant
measure $\nu$. We study relative equilibrium states, that is shift-invariant
measures on $X$ that push forward under the factor map to $\nu$ which maximize
the relative pressure: the relative entropy plus the integral of $\phi$.

In the non-relative case (where $Z$ is the one point shift and the factor map
is trivial), these measures have a very broad range of application: in
hyperbolic dynamics, information theory, geometry, Teichmüller theory and
elsewhere). Relative equilibrium states have also been shown to arise
naturally in some contexts in geometric measure theory as a description of
measures achieving the Hausdorff dimension in ambient spaces. Previous
articles have identified relative versions of well-known notions of degree
appearing in one-dimensional symbolic settings, and established bounds in
terms of these on the number of ergodic relative equilibrium states.

In this paper, we establish a new connection to multiplicative ergodic theory
by relating these factor triples to a cocycle of Ruelle Perron-Frobenius
operators, and showing that the principal Lyapunov exponent of this cocycle is
the relative pressure; and the dimension of the leading Oseledets space is
equal to the number of measures of relative maximal entropy, counted with a
previously-identified concept of multiplicity.

Let $X\subset A^{\mathbb{Z}}$ be an irreducible shift of finite type and let
$\rho\colon A\to B$ be a surjection of finite sets. Let $\pi\colon X\to
B^{\mathbb{Z}}$ be the one-block map defined by $\pi(x)_{n}=\rho(x_{n})$ and
let $Z=\pi(X)$ be the image of $X$ under the factor map $\pi$, so that $Z$ is
a sofic shift. Write $X^{+}$ for the one-sided version of $X$. Given an
invariant measure $\nu$ on $Z$, and a Hölder continuous function $\phi$ on
$X^{+}$ (which we also view as a function on $X$), recall that a _relative
equilibrium state_ of $\phi$ over $\nu$ is a measure $\mu$ on $X$ such that
$\pi_{*}(\mu)=\nu$ and
$h_{\mu}+\int\phi\,d\mu=\max_{\\{\lambda\colon\pi_{*}(\lambda)=\nu\\}}\big{(}h_{\lambda}+\int\phi\,d\lambda\big{)}$.
We are seeking a description of the relative equilibrium states of $\phi$ over
$\nu$ in terms of the Lyapunov exponents and Oseledets spaces of a cocycle of
Perron-Frobenius operators that we describe below. Briefly, the dimension of
the top Oseledets space will be the number of relative equilibrium states
(counted with multiplicity). For $0<\beta<1$, we introduce a metric $d$ on
$X^{+}$ given by $d(x,x^{\prime})=\beta^{\min\\{n\colon x_{n}\neq
x^{\prime}_{n}\\}}$ and write $C^{\beta}(X^{+})$ for the Lipschitz functions
with respect to this metric. We choose $\beta$ so that $\phi\in
C^{\beta}(X^{+})$.

For $j\in B$, define an operator $\mathcal{L}_{j}$ on $C^{\beta}(X^{+})$ by

$\mathcal{L}_{j}f(x)=\sum_{\\{i\colon ix\in
X^{+};\pi(i)=j\\}}e^{\phi(ix)}f(ix),$

where $ix$ denotes the point in $A^{\mathbb{N}_{0}}$ defined by $(ix)_{0}=i$,
$(ix)_{n}=x_{n-1}$ for $n\geq 1$. We shall study the cocycle over the
dynamical system $\sigma\colon Z\to Z$ where the map corresponding to $z$ is
$\mathcal{L}_{z}\colon=\mathcal{L}_{z_{0}}$. As usual, we define
$\mathcal{L}_{z}^{(n)}=\mathcal{L}_{\sigma^{n-1}z}\circ\ldots\circ\mathcal{L}_{z}$.

An inductive calculation shows that

$\mathcal{L}_{z}^{(n)}f(x)=\sum_{\\{w\colon\pi(w)=z_{0}^{n-1},wx\in
X^{+}\\}}e^{S_{n}\phi(wx)}f(wx),$

where the sum is over words of length $n$, $\pi$ is defined on words by
applying $\rho$ to each symbol and $wx$ is defined by concatenation. As usual,
$S_{n}\phi(wx)$ denotes the sum $\phi(wx)+\ldots+\phi(\sigma^{n-1}(wx))$.

Our main theorem relates, for an ergodic invariant measure $\nu$ on $Z$ and a
Hölder continuous potential, the number of ergodic relative equilibrium states
on $X$ to the multiplicity of the top Lyapunov exponent of the above cocycle.
Some of the terms appearing in the statement will be defined in the next
section.

######  Theorem 1.

Let $\pi\colon X\to Z$ be a factor map from an irreducible two-sided shift of
finite type to a two-sided sofic shift and let $\phi\in C^{\beta}(X^{+})$. Let
$\pi=\pi_{2}\circ\pi_{1}$ be the factorization into a map $\pi_{1}\colon X\to
Y$ of class degree one and a finite–one map $\pi_{2}$ of degree $c_{\pi}$ from
$Y$ to $Z$, where $c_{\pi}$ is the class degree of $\pi\colon X\to Z$.

Let $\nu$ be a fully supported ergodic invariant measure on $Z$ and let
$\nu_{1}$,…,$\nu_{k}$ be the lifts of $\nu$ under $\pi_{2}$ with
multiplicities $m_{1},\ldots,m_{k}$ respectively. These multiplicities satisfy
$m_{1}+\ldots+m_{k}=c_{\pi}$. Let $r_{\pi_{1}}(\nu_{i})$ be the relative
pressure of $f$ over $\nu_{i}$ and $r_{\pi}(\nu)$ be the relative pressure of
$f$ over $\nu$.

Let $(\mathcal{L}_{z}^{(n)})$ be the cocycle of linear operators acting on
$C^{\beta}(X^{+})$ described above. Then the largest Lyapunov exponent of the
cocycle is $r_{\pi}(\nu)$ and the multiplicity of this exponent is

$\sum_{r_{\pi_{1}}(\nu_{i})=r_{\pi}(\nu)}m_{i}.$

##  1\. Background

In this section, we collect a number of theorems and definitions that we will
need for the proof, as well as related setting out a number of related
articles in the literature.

Petersen, Quas and Shin [8] studied the case of measures of relative maximal
entropy, that is the relative equilibrium states in the case where one has a
factor map $\pi\colon X\to Z$ and the potential function $\phi$ is taken to be
0. They established that the number of ergodic measures of relative maximal
entropy (that is relative equilibrium states where the potential is taken to
be 0) over any ergodic invariant measure $\nu$ on $Z$ is bounded above by
$\min_{j\in\mathsf{A}(Z)}|\pi_{b}^{-1}(j)|$, showing that, in particular, the
number of these measures is finite. The bound suffered from a failure to be
invariant under conjugacies. This deficiency was remedied and the bound
improved in the paper [3] of Allahbakshi and Quas, some ideas from which will
play an important role here. For $z\in Z$, if $x,x^{\prime}\in\pi^{-1}z$, we
say that $x$ _transitions to_ $x^{\prime}$, and write $x\to x^{\prime}$, if
for all $n$, there exists $\bar{x}$ such that
$\bar{x}_{-\infty}^{n}=x_{-\infty}^{n}$ and $\bar{x}_{m}=x^{\prime}_{m}$ for
all sufficiently large $m$. We then define an equivalence relation on
$\pi^{-1}z$ by $x\leftrightarrow x^{\prime}$ if $x\to x^{\prime}$ and
$x^{\prime}\to x$. The equivalence classes are called _transition classes_ (a
pigeonhole argument and using the finite type property shows there are
finitely many transition classes). Let $\mathcal{T}(z)$ denote the collection
of transition classes over $z$. There exists a quantity $c_{\pi}$, the _class
degree_ of $\pi$ such that for all right transitive points $z\in Z$,
$\mathcal{T}(z)$ consists of $c_{\pi}$ elements.

######  Theorem 2 (Allahbakhshi and Quas [3]).

Let $X$ be a shift of finite type and $Z$ be a sofic shift. Let $\pi\colon
X\to Z$ be a one-block factor map. There exists a word $W=w_{0}^{n-1}$ in
$\mathsf{L}(Z)$, a position $0\leq l<n$, and a subset $B\subset\mathsf{A}(X)$
whose cardinality is the class degree so that for each element $u_{0}^{n-1}$
of $\pi^{-1}(W)$, there is a word $v_{0}^{n-1}\in\pi^{-1}(W)$ such that
$u_{0}=v_{0}$, $u_{n-1}=v_{n-1}$ and $v_{l}\in B$.

The number of measures of relative maximal entropy over $\nu$ is bounded above
by $c_{\pi}$.

The bound on the number of measures of maximal entropy was extended by
Allahbakhshi, Antonioli and Yoo [1] to the number of relative equilibrium
states of a Hölder continuous (or Bowen) potential function.

In the situation described in the above theorem, $W$ is called a _minimal
transition block_ ; $B$ is a set of _representatives_ and the word $u$ is said
to be _routed through_ $v_{l}$. (The minimality in the name refers to the fact
that the set of representatives is as small as possible). A pair of elements
$x,x^{\prime}$ of $X$ is said to be _mutually separated_ if $x_{n}\neq
x^{\prime}_{n}$ for each $n$. Two subsets $S_{1}$ and $S_{2}$ of $X$ are
mutually separated if for each $x\in S_{1}$ and $x^{\prime}\in S_{2}$, $x$ and
$x^{\prime}$ are mutually separated.

######  Theorem 3 (Allahbakhshi, Hong and Jung [2]).

Let $\pi\colon X\to Z$ be a factor map from an irreducible two-sided shift of
finite type $X$ to a two-sided sofic shift $Z$. If $z\in Z$ is right
transitive, then the elements of $\mathcal{T}(z)$ are mutually separated. In
particular, for each copy of $W$ in $z$, there exists a bijection between
$\mathcal{T}(z)$ and $B$ so that for each $C\in\mathcal{T}(z)$, there exists a
representative $s\in B$ such that each $x\in C$ may be routed through $s$ over
that copy of $W$ and through no other element of $B$.

######  Theorem 4 (Yoo [10]).

Let $X$ be an irreducible two-sided shift of finite type, $Z$ a two-sided
sofic shift, and $\pi\colon X\to Z$ be a continuous factor map. Then there is
a sofic shift $Y$ and factorization of $\pi\colon X\to Z$ as a composition of
factor maps, $\pi_{2}\circ\pi_{1}$ where $\pi_{1}\colon X\to Y$ and
$\pi_{2}\colon Y\to Z$ with the properties that $\pi_{2}$ is finite-to-one of
degree $c_{\pi}$, the class degree of $\pi$ and $\pi_{1}$ is of class degree
1.

######  Theorem 5 (Yoo [11]).

Let $\pi$ be a finite-to-one continuous factor map from a homeomorphism $S$ of
a compact metric space $Y$ to a homeomorphism $T$ of a compact metric space
$Z$. Suppose that $\nu$ is an ergodic $T$-invariant measure. Then there exist:

  * •

a number $d$ such that for $\nu$-a.e. $z\in Z$, $|\pi^{-1}z|=d$;

  * •

ergodic measures $\mu_{1},\ldots,\mu_{k}$ on $Y$ such that
$\pi_{*}\mu_{i}=\nu$;

  * •

multiplicities $m_{1},\ldots,m_{k}$ whose sum is $d$.

In the case where $Y$ is a shift space, there exists a ergodic _degree
joining_ , $\bar{\mu}$, a measure on $Y^{d}$ such that for $\bar{\mu}$-a.e.
$(y^{1},\ldots,y^{d})$, $\pi(y^{1})=\ldots=\pi(y^{d})$; the $y^{i}$ are
mutually separated; and $y^{M_{i}+1},\ldots,y^{M_{i}+m_{i}}$ are generic for
$\mu_{i}$, where $M_{i}=m_{1}+\ldots+m_{i-1}$.

The following theorem gives a criterion for simplicity of the top Lyapunov
exponent of an operator cocycle based on contraction of cones and Birkhoff’s
theorem on contraction of the Hilbert metric. Recall that a _cone_ is a closed
subset $\mathcal{C}$ of a real Banach space $B$ that is closed under addition
and scalar multiplication by a non-negative real number.

For $f,g\in\mathcal{C}$, let $m(f,g)=\sup\\{t\colon f-tg\in\mathcal{C}\\}$ and
let $M(f,g)=\inf\\{s\colon sg-f\in\mathcal{C}\\}$. The projective distance
between two points in the cone is defined as
$\Theta_{\mathcal{C}}(f,g)=\log(M(f,g)/m(f,g))$. (Note that this is not a
metric as it may be infinite; also $\Theta_{\mathcal{C}}(\beta f,\gamma
g)=\Theta_{\mathcal{C}}(f,g)$ for all $\beta,\gamma>0$). The _diameter_ of a
subset $S$ of $\mathcal{C}$ is $\sup_{f,g\in
S\setminus\\{0\\}}\Theta_{\mathcal{C}}(f,g)$. A cone is said to be _$D$
-adapted_ if whenever $f\in B$ and $g\in\mathcal{C}$, then $g\pm
f\in\mathcal{C}$ implies $\|f\|\leq D\|g\|$.

######  Theorem 6 (Horan [5]).

Let $Y$ be a compact Hausdorff topological space and $S\colon Y\to Y$ be a
continuous invertible transformation. Let $\nu$ be an ergodic $S$-invariant
measure on $Y$. Let $B$ be a Banach space and let $\mathcal{C}$ be a
$D$-adapted cone in $B$ such that $\mathcal{C}-\mathcal{C}=B$,
$\mathcal{C}\cap(-\mathcal{C})=\\{0\\}$.

Suppose that for each $y\in Y$, $\mathcal{L}_{y}$ is a linear operator from
$B$ to $B$ such that $y\mapsto\mathcal{L}_{y}$ is continuous (where the linear
operators on $B$ are equipped with the norm topology), that
$\mathcal{L}_{y}(\mathcal{C})\subset\mathcal{C}$ for each $y$ and that there
is a measurable subset $A\subset Y$ with $\nu(A)>0$ and an $n>0$ such that
$\operatorname{diam}(\mathcal{L}^{(n)}_{y}\mathcal{C})<\infty$ for all $y\in
A$.

Then the leading Lyapunov exponent of the cocycle
$(\mathcal{L}^{(n)}_{y})_{y\in Y}$ is simple.

We will use the relative variational principle of Ledrappier and Walters [6].

######  Theorem 7 (Relative Variational Principle).

Let $S\colon Y\to Y$ be a topological factor of a map $T\colon X\to X$ and
suppose that $S$ has an ergodic invariant measure, $\nu$. Then for $\nu$-a.e.
$y$, $P_{T}(\phi,\pi^{-1}y)=r_{\pi}(\nu)$.

##  2\. Proofs

For this section, let $\phi$ be a fixed Hölder continuous function. Given
$\beta<1$, we define a semi-norm on $C^{\beta}(X^{+})$ by
$|f|_{\beta}=\sup_{x\neq
x^{\prime}}|f(x)-f(x^{\prime})|/d_{\beta}(x,x^{\prime})$ (that is the
Lipschitz constant of $f$ with respect to $d_{\beta}$) and a norm by
$\|f\|_{\beta}=\max(\|f\|_{\infty},|f|_{\beta})$. Let $\beta$ be such that
$\|\phi\|_{\beta}<\infty$. This quantity will be fixed from here on.

We define a family of cones, one for each real $a>0$, by

$\mathcal{C}_{a}=\left\\{f\in C^{\beta}(X^{+})\colon f\geq
0;\frac{f(x)}{f(x^{\prime})}\leq e^{ad_{\beta}(x,x^{\prime})}\text{ whenever
$x_{0}=x^{\prime}_{0}$.}\right\\}.$

These cones are widely used in symbolic dynamics and appear, for instance, in
the work of Parry and Pollicott [7], although our usage differs slightly as we
do not impose any condition on $f(x)/f(x^{\prime})$ when $x_{0}\neq
x_{0}^{\prime}$. This is important for us, since some operators that we
consider yield functions that are 0 on part of $X^{+}$.

######  Lemma 8.

Let $a$ be large enough that $b:=\beta(a+|\phi|_{\beta})<a$. Then
$\mathcal{L}_{j}\mathcal{C}_{a}\subset\mathcal{C}_{b}$ for each
$j\in\mathsf{A}(Y)$.

###### Proof.

For each symbol $i\in\mathsf{A}(X)$, set
$\tilde{\mathcal{L}}_{i}f(x)=e^{\phi(ix)}f(ix)$. Suppose $x$ and $x^{\prime}$
agree for $n$ symbols for some $n\geq 1$. Then

$\displaystyle\frac{\tilde{\mathcal{L}}_{i}f(x)}{\tilde{\mathcal{L}}_{i}f(x^{\prime})}$
$\displaystyle=\frac{e^{\phi(ix)}f(ix)}{e^{\phi(ix^{\prime})}f(ix^{\prime})}$
$\displaystyle\leq e^{|\phi|_{\beta}\beta^{n+1}}e^{a\beta^{n+1}}$
$\displaystyle\leq
e^{\beta(|\phi|_{\beta}+a)d_{\beta}(x,x^{\prime})}=e^{bd_{\beta}(x,x^{\prime})}.$

Since for $j\in\mathsf{A}(Y)$,
$\mathcal{L}_{j}=\sum_{i\in\rho^{-1}j}\tilde{\mathcal{L}}_{i}$, the result
follows. ∎

######  Lemma 9.

For $f\in\mathcal{C}_{a}$, $\|f\|_{\beta}\leq\max(3,1+ae^{a})\|f\|_{\infty}$.
It follows that $\mathcal{C}_{a}$ is $D$-adapted with $D=\max(6,2+2ae^{a})$.

###### Proof.

Let $f\in C_{a}$. If $x,x^{\prime}\in X^{+}$ have different initial symbols,
then $|f(x)-f(x^{\prime})|\leq
2\|f\|_{\infty}\leq\max(2,ae^{a})\|f\|_{\infty}d_{\beta}(x,x^{\prime})$. If
they have the same initial symbol, then
$|f(x)-f(x^{\prime})|\leq|f(x)|(e^{ad_{\beta}(x,x^{\prime})}-1)\leq\|f\|_{\infty}ae^{a}d_{\beta}(x,x^{\prime})\leq\max(2,ae^{a})\|f\|_{\infty}d_{\beta}(x,x^{\prime})$,
where we used the mean value theorem for the second inequality. Hence
$|f|_{\beta}\leq\max(2,ae^{a})\|f\|_{\infty}$ so that
$\|f\|_{\beta}\leq\max(3,1+ae^{a})\|f\|_{\infty}$.

For the second statement in the lemma, $g\pm f\in\mathcal{C}_{a}$ implies
$\|f\|_{\infty}\leq\|g\|_{\infty}$, so that $\|g\pm f\|_{\infty}\leq
2\|g\|_{\infty}$ and $\|g\pm
f\|_{\beta}\leq\max(6,2+2ae^{a})\|g\|_{\infty}\leq\max(6,2+2ae^{a})\|g\|_{\beta}$.
Subtracting $g-f$ from $g+f$, we obtain the desired bound. ∎

For these cones, we have the following lemma (which can be seen as a special
case of a result of Andô [4]). Expressing arbitrary Hölder continuous
functions as a difference of elements of the cone will allow us to prove
simplicity of the top Lyapunov exponent.

######  Lemma 10.

For all $f\in C^{\beta}(X^{+})$, there exist $g,h\in\mathcal{C}_{a}$ with
$\|g\|_{\beta},\|h\|_{\beta}\leq(2+\frac{1}{a})\|f\|_{\beta}$ such that
$f=g-h$.

###### Proof.

Let $f\in C^{\beta}(X^{+})$, let $g=f+(1+\frac{1}{a})\|f\|_{\beta}$ and
$h=(1+\frac{1}{a})\|f\|_{\beta}$. Clearly $h\in\mathcal{C}_{a}$. Notice that
$\min g\geq\frac{1}{a}\|f\|_{\beta}$, so that

$\displaystyle\frac{g(x)}{g(x^{\prime})}$
$\displaystyle=1+\frac{g(x)-g(x^{\prime})}{g(x^{\prime})}\leq
1+\|f\|_{\beta}d(x,x^{\prime})/(\|f\|_{\beta}/a)$
$\displaystyle=1+ad(x,x^{\prime})\leq e^{ad(x,x^{\prime})}.$

In particular, $g\in\mathcal{C}_{a}$ and $\|g\|_{\beta}$, $\|h\|_{\beta}$ are
bounded above by $(2+\frac{1}{a})\|f\|_{\beta}$. ∎

######  Lemma 11.

Let $b<a$ and $A\geq 1$. Let $P$ be a subset of the alphabet of $X$. Write
$[P]=\bigcup_{j\in P}[j]$ and

$S_{P}=\left\\{f\colon f(x)>0\text{ iff $x\in[P]$; }f(x)\leq
Af(x^{\prime})\text{ for all $x,x^{\prime}\in[P]$}\right\\}.$

Then there exists $K>0$ such that $\Theta_{\mathcal{C}_{a}}(f,g)\leq K$ for
all $f,g\in S_{P}\cap\mathcal{C}_{b}$.

The conclusion here states that the diameter of the set is finite. This is a
key hypothesis in Birkhoff’s cone contraction argument.

###### Proof.

Let $t$ be chosen sufficiently small to ensure that $\frac{2btA}{1-tA}\leq
a-b$. Let $f,g\in S_{P}\cap\mathcal{C}_{b}$. Using the scale-homogeneity of
$\Theta_{\mathcal{C}_{a}}$, we may scale $f$ and $g$ so that
$\min_{[P]}f=\min_{[P]}g=1$, and hence $\max f,\max g\leq A$.

We claim that $f-tg\in\mathcal{C}_{a}$. Let $x,x^{\prime}\in X^{+}$ have a
common first symbol belonging to $P$ (if $x,x^{\prime}$ have a common first
symbol outside $P$, then $(f-tg)(x)$ is trivially bounded above by
$e^{ad(x,x^{\prime})}(f-tg)(x^{\prime})$ since both of these quantities are
zero). We have

$\displaystyle\frac{f(x^{\prime})-tg(x^{\prime})}{f(x)-tg(x)}$
$\displaystyle\leq\frac{e^{bd(x,x^{\prime})}f(x)-te^{-bd(x,x^{\prime})}g(x)}{f(x)-tg(x)}$
$\displaystyle=e^{bd(x,x^{\prime})}+\frac{t(e^{bd(x,x^{\prime})}-e^{-bd(x,x^{\prime})})g(x)}{f(x)-tg(x)}$
$\displaystyle\leq
e^{bd(x,x^{\prime})}\left(1+\frac{At}{1-At}(1-e^{-2bd(x,x^{\prime})})\right)$
$\displaystyle\leq
e^{bd(x,x^{\prime})}\left(1+\frac{2Atb}{1-At}d(x,x^{\prime})\right)$
$\displaystyle\leq
e^{bd(x,x^{\prime})}e^{(a-b)d(x,x^{\prime})}=e^{ad(x,x^{\prime})},$

so that $f-tg\in\mathcal{C}_{a}$. By symmetry, $g-tf\in\mathcal{C}_{a}$, or
equivalently $(1/t)g-f\in\mathcal{C}_{a}$. Hence
$\Theta_{\mathcal{C}_{a}}(f,g)\leq\log(1/t^{2})$ for all $f,g\in
S_{P}\cap\mathcal{C}_{b}$. ∎

######  Lemma 12.

Let $\pi\colon X\to Y$ be a factor map of class degree 1. Let $W=w_{0}^{n-1}$
be a minimal transition block in $Y$. Then there exists an $A\geq 1$ such that
for any $y\in[W]$, and any $f\in\mathcal{C}_{a}$, $\mathcal{L}^{(n)}_{y}f\in
S_{P}\cup\\{0\\}$, where $S_{P}$ is the set in Lemma 11 and $P$ is
$\\{j\in\mathsf{A}(X)\colon\exists U\in\pi_{b}^{-1}(W)\colon
Uj\in\mathsf{L}(X)\\}$.

###### Proof.

Since $S_{P}\cup\\{0\\}$ is closed under addition and $\mathcal{L}_{y}^{(n)}$
is linear, it suffices to prove the statement for a function $f$ supported on
a single cylinder set. Suppose that $f$ is supported on $[k]$. If there is no
preimage of $W$ whose initial symbol is $k$, we see that
$\mathcal{L}_{y}^{(n)}f=0$ since there are no positive summands. Suppose on
the other hand that $U$ is a preimage of $W$ under $\pi$ starting with a $k$.
Let $j\in P$ and let $V$ be a preimage of $W$ under $\pi$ such that
$Vj\in\mathsf{L}(X)$. Since $W$ is a minimal transition block with a single
representative, there exists a preimage $U^{\prime}$ of $W$ starting with the
first symbol of $U$ and ending with the last symbol of $V$. If $x\in[j]$, We
now calculate

$\mathcal{L}^{(n)}_{y}f(x)\geq e^{S_{n}\phi(U^{\prime}x)}f(U^{\prime}x)\geq
e^{n\min\phi}\|f\|_{\infty}/e^{a}.$

On the other hand, it is clear that
$\mathcal{L}^{(n)}_{y}f(x)\leq|\mathsf{A}(X)|^{n}e^{n\max\phi}\|f\|_{\infty}$
for any $x\in X$. Hence we have demonstrated the hypothesis of Lemma 11 is
satisfied with $A=e^{a+n(\max\phi-\min\phi)}|\mathsf{A}(X)|^{n}$. ∎

We point out that the idea of studying the Ruelle-Perron-Frobenius cocycle
over a factor $Y$ and expressing the operators corresponding to symbols in
$\mathsf{A}(Y)$ as sums of operators indexed by symbols in $\mathsf{A}(X)$, as
well as some of the cones that we study here and the description of $S_{P}$
above, appears in work of Piraino [9].

######  Theorem 13.

Let $X$ be an irreducible shift of finite type, let $\pi\colon X\to Y$ be a
factor map of class degree 1 and let $\phi$ be a Hölder continuous function on
$X$. Suppose $\nu$ is a fully supported invariant measure on $Y$. Then the
cocycle $(\mathcal{L}_{y}^{(n)})$ has a simple top Lyapunov exponent, whose
value is $r_{\pi}(\nu)$.

###### Proof.

Let $\beta$ be such that $\|\phi\|_{\beta}<\infty$ and let $a$ satisfy the
hypothesis in Lemma 8. Let $W$ be a minimal transition block for the factor
map $\pi\colon X\to Y$. By Lemma 10,
$\mathcal{C}_{a}-\mathcal{C}_{a}=C^{\beta}(X^{+})$. By Lemmas 8, 12 and 11, we
see that $\mathcal{L}_{y}^{(|W|)}\mathcal{C}_{a}$ is a finite diameter subset
of $\mathcal{C}_{a}$ for any $y\in[W]$. Since the hypotheses of Theorem 6 are
satisfied (the continuity of $y\mapsto\mathcal{L}_{y}$ is because the map is
piecewise constant and the $D$-adaptedness condition on $\mathcal{C}_{a}$ is
satisfied by the second statement of Lemma 9), the top Lyapunov exponent of
the cocycle $(\mathcal{L}_{y}^{(n)})$ acting on $C^{\beta}(X^{+})$ is simple.

Notice that for $y\in Y$, and $g\in\mathcal{C}_{a}$,

$\mathcal{L}_{y}^{(n)}g(x)=\sum_{W\in\pi^{-1}(y_{0}^{n-1})\colon
Wx_{0}\in\mathsf{L}(X)}e^{S_{n}\phi(Wx)}g(Wx),$

so that $\|\mathcal{L}_{y}^{(n)}g\|_{\infty}$ is bounded above by
$P_{n}(\phi,1,\pi^{-1}y)\|g\|_{\infty}$. By Lemmas 8 and 9,
$\|\mathcal{L}_{y}^{(n)}g\|_{\beta}$ is bounded above by
$\max(3,1+e^{a})P_{n}(\phi,1,\pi^{-1}y)\|g\|_{\infty}$. If $f\in
C^{\beta}(X^{+})$, using Lemma 10, we may write $f$ as the difference $g-h$
with $g,h\in\mathcal{C}_{a}$, each of $\|\cdot\|_{\beta}$ norm at most
$(2+\frac{1}{a})\|f\|_{\beta}$. Hence $\|\mathcal{L}_{y}^{(n)}f\|_{\beta}\leq
2(2+\frac{1}{a})\max(3,1+e^{a})P_{n}(\phi,1,\pi^{-1}y)\|f\|_{\beta}$. Since
shift transformations are expansive with expansiveness constant 1, we have
$\limsup_{n\to\infty}\frac{1}{n}\log
P_{n}(\phi,1,\pi^{-1}y)=P_{T}(\phi,\pi^{-1}y)$, which is $r_{\pi}(\nu)$ by
Theorem 7, so that the top Lyapunov exponent is bounded above by
$r_{\pi}(\nu)$.

For the converse inequality, let $(x^{i})_{i\in\mathsf{A}(X)}$ be a collection
of points in $X^{+}$, where $x^{i}$ starts with the symbol $i$. Now

$\sum_{i\in\mathsf{A}(X)}\mathcal{L}^{(n)}_{y}\mathbf{1}(x^{i})\geq
e^{-c}P_{n}(\phi,1,\pi^{-1}y),$

where $c$ is a constant independent of $n$ where
$|S_{n}\phi(x)-S_{n}\phi(x^{\prime})|\leq c$ whenever
$x_{0}^{n-1}={x^{\prime}}\,_{0}^{n-1}$ (such a $c$ exists since $\phi$ is
Hölder). Then

$\left\|\mathcal{L}^{(n)}_{y}\mathbf{1}\right\|_{\beta}\geq\left\|\mathcal{L}^{(n)}_{y}\mathbf{1}\right\|_{\infty}\geq\frac{e^{-c}}{|\mathsf{A}(X)|}P_{n}(\phi,1,\pi^{-1}y).$

In particular, for $\nu$-a.e. $y$, the limit superior growth rate of
$\|\mathcal{L}^{(n)}_{y}\mathbf{1}\|_{\beta}$ is at least $r_{\pi}(\nu)$, as
required. ∎

###### Proof of Theorem 1.

Using Theorem 4, $\pi:X\to Z$ may be factorized as $\pi_{2}\circ\pi_{1}$,
where $\pi_{1}$ is of class degree 1 from $X$ to a sofic shift $Y$; and
$\pi_{2}$ is finite to 1, and for $\nu$-a.e. point, $\pi_{2}^{-1}(z)$ consists
of $c_{\pi}$ pre-images.

We need a more precise description of the sofic shift $Y$ and the factor code
$\pi_{1}$. The space $Y$ is built from a minimal transition block, $W$ in
$\mathsf{L}(Z)$. Recall the representatives of the transition block are a
subset $B$ of $\mathsf{A}(X)$ of cardinality $c_{\pi}$ such that if
$\pi(x)\in[W]$, then $x$ may be locally modified on the coordinates $(0,n-1)$
to give a point $x^{\prime}\in X$ with $x^{\prime}_{l}\in B$.

The alphabet of $Y$ is then $\mathsf{A}(Z)\times(B\cup\\{\star\\})$. The
factor map $\pi_{1}$ is defined as follows:

$\pi_{1}(x)_{m}=\begin{cases}(\pi(x)_{m},s)&\text{$\pi(x)_{m-l}^{m-l+n-1}=W$,
$x_{m-l}^{m-l+n-1}$ routable through $s$;}\\\
(\pi(x)_{m},\star)&\text{$\pi(x)_{m-l}^{m-l+n-1}\neq W$.}\end{cases}$

That is, $\pi_{1}(x)$ records the image in $Z$, together with the
representatives in $B$ through which the orbit of $x$ may be routed each time
that the orbit passes through a transition block.

We then define an operator cocycle over $Y$. For each $s\in B$, let
$R_{s}\subset\mathsf{A}(X)$ be the collection of symbols in $X$ that a
preimage of $W$ may pass through if it is routable through $s$. By Theorem 3,
these sets are disjoint. Write $[R_{s}]$ for $\bigcup_{i\in R_{s}}[i]$ and let
$q$ be the $l$th symbol of $W$.

The generator of the cocycle is then defined by

$\displaystyle\bar{\mathcal{L}}_{(j,\star)}f(x)$
$\displaystyle=\mathcal{L}_{j}f(x)$
$\displaystyle\bar{\mathcal{L}}_{(q,s)}f(x)$
$\displaystyle=\mathcal{L}_{q}(\mathbf{1}_{[R_{s}]}f)(x).$

That is, each time $\pi(x)$ passes through a transition block, the operator
projects to the part of the function routable through the specified
representative. Note that $(q,\star)\in\mathsf{A}(Y)$, and this appears in the
image of points under $\pi_{1}$ for points in $X$ whose symbol belongs to
$\rho^{-1}(q)$, but where the word $\pi(x)_{m-l}^{(m-l+n-1)}$ is not equal to
$W$.

By Theorem 5, there are finitely many ergodic invariant measures on $Y$
projecting to $\nu$, say $\mu_{1},\ldots,\mu_{k}$; as well as multiplicities
$m_{1},\ldots,m_{k}$ summing to $c_{\pi}$ such that a $\nu$-generic $z\in Z$
has $m_{i}$ $\mu_{i}$-generic pre-images under $\pi_{2}$ for each $i$, with
the whole collection of $c_{\pi}$ pre-images mutually separated. Further,
there exists an ergodic measure $\bar{\mu}$ on $Y^{c_{\pi}}$ where
$\bar{\mu}$-almost every point is supported on the $c_{\pi}$ pre-images of
some point $z\in Z$; the first $m_{1}$ being generic points for the ergodic
measure $\mu_{1}$ on $Y$; the next $m_{2}$ being generic for the measure
$\mu_{2}$ etc. We assume without loss of generality that
$r_{\pi_{1}}(\mu_{1})\geq r_{\pi_{1}}(\mu_{2})\geq\ldots$, where
$r_{\pi_{1}}(\mu_{i})$ is the $\pi_{1}$-relative pressure of $\phi$ over
$\mu_{i}$; and that the maximal value of $r_{\pi_{1}}(\mu_{i})$ is attained
for $i=1,\ldots,p$ (but not for $i=p+1,\ldots,k$).

Since $\pi_{1}\colon X\to Y$ and each $\mu_{i}$ satisfies the conditions of
Theorem 13, we see that there is a simple top exponent $\lambda_{i}$ for the
cocycle $(\bar{\mathcal{L}}_{y}^{(n)})_{y\in Y}$ for each of the measures
$\mu_{i}$. The set of $y\in Y$ for which the exponent $\lambda_{i}$ is
achieved and for which the second Lyapunov exponent is strictly smaller is a
collection of full $\mu_{i}$-measure.

For $\bar{\mu}$-a.e. $(y^{1},\ldots,y^{c_{\pi}})\in Y^{c_{\pi}}$, the simple
top exponent of the cocycle $(\bar{\mathcal{L}}_{y})^{(n)}$ is $\lambda_{i}$
for each $y=y^{M_{i}+k}$ with $k=1,\ldots,m_{i}$ (where
$M_{i}=m_{1}+\ldots+m_{i-1}$ and $M_{1}=0$). In particular, the top exponent
of the cocycle is almost surely simple with exponent
$\lambda_{1}=r_{\pi_{1}}(\mu_{1})$ over each of
$y^{1},\ldots,y^{m_{1}+\ldots+m_{p}}$ and strictly smaller for the other
$y$’s.

We now relate derive a relationship between the cocycle
$(\mathcal{L}^{(n)}_{z})$ over $Z$ and the cocycle
$(\bar{\mathcal{L}}^{(n)}_{y})$ over $Y$. Recall that for $\bar{\mu}$-a.e.
$\bar{y}=(y^{1},\ldots,y^{c_{\pi}})$, one has the equality
$\pi(y^{1})=\ldots=\pi(y^{c_{\pi}})$. Write $\bar{\pi}(\bar{y})$ for this
common value. Next, we claim that for $\bar{\mu}$-a.e. $\bar{y}$,

$\mathcal{L}_{\bar{\pi}(\bar{y})}^{(n)}f(x)=\sum_{i=1}^{c_{\pi}}\bar{\mathcal{L}}^{(n)}_{y^{i}}f(x)$

for all $n$ such that $\bar{\pi}(\bar{y})_{0}^{n-1}$ contains a copy of $W$,
the minimal transition block used in the defintion of $\pi_{1}$.

To see this, notice that if $q=w_{l}$ is the symbol in $W$ over which the
representatives lie, then the following identities hold

$\displaystyle\mathcal{L}_{q}$ $\displaystyle=\sum_{i\in
S}\bar{\mathcal{L}}_{q,i}$ $\displaystyle\mathcal{L}_{j}$
$\displaystyle=\bar{\mathcal{L}}_{j,\star}\text{ for each $j\in\mathsf{A}(Z)$
(including $q$)}.$

So the composition of the $\mathcal{L}_{z_{i}}$ is a composition in which a
number of the terms (those occurring when $z$ contains a copy of $W$) are
replaced by a sum of $\bar{\mathcal{L}}_{q,i}$. Since the $\bar{\mathcal{L}}$
are linear, we may distribute the composition over the sum. Since when $z$ is
right transitive, the transition classes are mutually separated (Theorem 3),
almost all of the terms in the summation vanish; the only ones that survive
are those in which the choices of representative are consistent: the
representative over one instance of $W$ together with the point $z$ determines
the representative over all of the other instances of $W$ by virtue of the
mutual separation of the classes in $\pi^{-1}(z)$.

For $\bar{\mu}$-a.e. $\bar{y}=(y^{1},\ldots,y^{c_{\pi}})$, each of the
$y^{i}$’s is right transitive; and the map $\pi_{1}\colon X\to Y$ is of class
degree 1. The pre-images $\pi_{1}^{-1}(y^{i})$ for $i=1,\ldots,c_{\pi}$ form
the transition classes, $\mathcal{T}(\bar{\pi}(\bar{y}))$ in $X$ over
$\bar{\pi}(\bar{y})\in Z$. By Theorem 13, applied to $\pi_{2}\colon X\to Y$,
for $\nu$-a.e. $\bar{\pi}(\bar{y})$ and each $y^{i}$ with $1\leq i\leq
m_{1}+\ldots+m_{p}$, the cocycle $(\mathcal{L}_{y_{i}}^{(n)})$ has an
equivariant one-dimensional space of functions growing at rate
$r_{\pi_{1}}(\mu_{1})$. By the mutual separation of the transition classes,
these functions are disjointly supported, and hence the dimension of the
Oseledets space with rate $r_{\pi_{1}}(\mu_{1})$ is at least
$m_{1}+\ldots+m_{p}$. Theorem 13 implies that in any two-dimensional space of
functions supported on $\pi_{1}^{-1}(y^{i})$, there is a function whose growth
rate is strictly smaller than $r_{\pi_{1}}(\mu_{1})$.

On the other hand, by Theorem 13 for $i>M_{p}$, the growth rate on
$\pi^{-1}(y^{i})$ is at most $r_{\pi_{1}}(\mu_{p+1})$, which is strictly
smaller. Combining these facts, it follows that the dimension of the fastest
growing space is precisely $m_{1}+\ldots+m_{p}$ as required. ∎

## References

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# Mechanical Properties of Wilberforce Pendulum

###### Abstract

Sanghwa Lee
<EMAIL_ADDRESS>
Korea Science Academy of KAIST, 20-078
ABSTRACT
This paper shows the study of interesting mechanical properties of Wilberforce
pendulum. Analyzing qualitatively of the pendulum, it is able to know how the
phenomenon occurs. By setting of the quantitative model, equation of the
motion is derived. Considering the mass and moment of inertia of the spring,
the experiment was done by changing the moment of inertia as the main
parameter. The results were analyzed by defining oscillation ratio and
conversion factor.

##  1 INTRODUCTION

This study is about the mechanical properties of the Wilberforce pendulum. A
Wilberforce pendulum consists of a mass hanging from a vertically oriented
helical spring. The mass can both move up and down on the spring and rotate
about its vertical axis. The theoretical analysis, which consists of
qualitative and quantitative analysis will be done through mechanical
investigation of the system.

Experiments will be done using the mechanical system, which is called
Wilberforce pendulum, and analysis of the pendulum will be done using various
programs. Investigating the data that is earned through experiments, the
behavior of the pendulum is investigated by comparing both theoretical results
and experimental results. In addition, the motion of Wilberforce pendulum
depends on relevant parameters, so that analysis based on considering relevant
parameters will be done.

##  2 THEORETICAL MODEL

A theoretical model was set up based on the qualitative analysis of the
pendulum. Based on the qualitative analysis and the phenomenon observation,
the theoretical model was set up in two different ways.

###  2.1 Qualitative anlysis

There are two perspectives on the spring. From a perspective of force, the
increase in the vertical displacement leads to the rotational increase of the
angular displacement. Vice versa, the increase of the angular displacement
leads to an increase in vertical displacement. The restoring force acts on the
spring, both vertically and torsionally, which makes sense to the decrease in
vertical displacement and rotational displacement. As a perspective in energy,
periodic transfer between rotational energy and translational energy occurs on
the pendulum as it shows the periodic motion.

In terms of phenomenon observation, beating occurs in the motion of the
pendulum. If the angular frequency of vertical motion and rotational motion is
denoted $\omega_{z}$ and $\omega_{\theta}$ respectively, the beating phenomena
can be explained. When $\omega_{z}>\omega_{\theta}$ satisfies, the energy
conversion is unclear. It is able to observe that the beating occurs through
two motions since the energy is not perfectly transferred. In the same way,
when $\omega_{z}<\omega_{\theta}$ satisfies, the energy conversion is unclear
that the beating occurs. However, for the singularity point that
$\omega_{z}=\omega_{\theta}$, the energy conversion is significant when
observed. The resonance occurs between two oscillations that a large amplitude
is observed. As we define oscillation ratio
$\Omega=\omega_{z}/\omega_{\theta}$, the more $\Omega$ is closer to 1, the
resonance is more clearly shown. The resonance occurs most clearly when
$\Omega=1$.

###  2.2 Reason of phenomenon

As analyzed in qualitative analysis, the restoring force acts on the spring so
that two motions act on each other. This was explained quantitatively by Clive
L. Dym [1]. By Castigliano’s Second Theorem, the complementary energy is
denoted as the following equation.

$\begin{split}U^{*}=&\int_{0}^{L_{c}}\left[\frac{M_{x}^{2}}{2GJ}+\frac{M_{z^{*}}^{2}}{2EI}+\frac{V^{2}}{2GA}+\frac{N^{2}}{2EA}\right]dx\\\
=&\frac{1}{2}f_{11}F^{2}-2f_{12}FT+\frac{1}{2}f_{22}T^{2}\end{split}$ (1)

The first term consisting $f_{11}$ shows the elastic potential energy, while
the last term consisting $f_{22}$ shows the torsional potential energy. The
second term consisting $f_{12}$ shows the coupling energy. This coupling
energy is the source of the beating. It is shown that beating occurs from the
perspective of energy.

###  2.3 Quantitative model

Setting up the coordinate system as a cylindrical coordinate system would be
easier to analyze. The system can be set as Fig. 1.

Figure 1: Coordinate system

Set vertical displacement as $z$, angular displacement as $\theta$, vertical
spring constant as $k$, torsional spring constant as $\kappa$, the mass of the
bob as $m$, and the moment of inertia of the bob as $I$.

As the reason of the phenomenon is discussed at 2.2, considering the coupling
energy term, it is able to predict that coupling spring constant exists other
than vertical spring constant and torsional spring constant. Taking Taylor
expansion of potential energy of the spring,

$\begin{split}&V(z,\theta)\\\ &=V(z_{0},\theta_{0})+\left(z\frac{\partial
V(z_{0},\theta_{0})}{\partial z}+\theta\frac{\partial
V(z_{0},\theta_{0})}{\partial\theta}\right)+\\\
&\frac{1}{2!}\left(z^{2}\frac{\partial^{2}V(z_{0},\theta_{0})}{\partial
z^{2}}+2z\theta\frac{\partial^{2}V(z_{0},\theta_{0})}{\partial
z\partial\theta}+\theta^{2}\frac{\partial^{2}V(z_{0},\theta_{0})}{\partial\theta^{2}}\right)\\\
&+\cdots\end{split}$ (2)

There exist terms $1/2kz^{2}$ and $1/2\kappa\theta^{2}$, and there is one more
term other than those two. The second-power approximation shows the coupling
term $\lambda z\theta$, where $\lambda$ is constant.

###  2.4 Equation of motion

Based on the quantitative model, the force acting on the pendulum can be
interpreted through energy. Using the force-potential relation $F=-\nabla U$,

$\begin{split}m\ddot{z}+kz+\frac{\partial(\lambda z\theta)}{\partial
z}=&m\ddot{z}+kz+\lambda\theta=0\\\
I\ddot{\theta}+\kappa\theta+\frac{\partial(\lambda
z\theta)}{\partial\theta}\rho=&I\ddot{\theta}+\kappa\theta+\lambda
z=0\end{split}$ (3)

Initial conditions are set as the following,

$\begin{split}z(0)=&z_{0}\\\
\dot{z}(0)=\theta(0)=&\dot{\theta}(0)=0\end{split}$ (4)

It is able to substitude the angular frequency as the following.

$\begin{split}\omega_{z}=&\sqrt{\frac{k}{m}}\\\
\omega_{\theta}=&\sqrt{\frac{\kappa}{I}}\end{split}$ (5)

Solving the general solution,

$\begin{split}z(t)=&\frac{z_{0}}{\omega_{1}^{2}-\omega_{2}^{2}}\left((\omega_{z}^{2}-\omega_{z}^{2})\cos{\omega_{1}t}-(\omega_{z}^{2}-\omega_{1}^{2})\cos{\omega_{2}t}\right)\\\
\theta(t)=&\frac{-z_{0}(\omega_{1}^{2}-\omega_{z}^{2})(\omega_{2}^{2}-\omega_{z}^{2})}{\lambda(\omega_{1}^{2}-\omega_{2}^{2})}(\cos{\omega_{1}t}-\cos{\omega_{2}t})\end{split}$
(6)

where

$\begin{split}\omega_{1,2}=\sqrt{\frac{(\omega_{z}^{2}+\omega_{\theta}^{2})\pm\sqrt{(\omega_{z}^{2}+\omega_{\theta}^{2})^{2}-4\left(\omega_{z}^{2}\omega_{\theta}^{2}-\frac{\lambda^{2}}{mI}\right)}}{2}}\end{split}$
(7)

##  3 EXPERIMENT

###  3.1 Experimental setup & measurement

The 3D-Model of the pendulum was first made. The pendulum has a slinky
connected to the bob, and the rod intersects the bob. Two weights are
connected at a distance $d$ from the bob, symmetrically. Based on the model,
the pendulum was made with slinky, rod, and weight. The m

The apparatus of the experiment includes the pendulum with slinky connected to
the ceiling and PASCO Capstone Motion Sensor on the floor. The distance
between the pendulum and sensor was measured 0.62 m. For the rotational motion
of the bob, data was extracted through video analysis using TRACKER.

Measuring the mass and the length properties of the bob, it is able to know
the values $m_{b}$ and $I_{b}$. To be more accurate, the mass and moment of
inertia of the spring should also be considered. When a spring is enlonged by
$x$, consider a piece of spring at distance $\xi$. Then using the following
relationship,

$\begin{split}\dot{\xi}=\frac{\xi}{x}\dot{x}\end{split}$ (8)

Considering kinetic energy of the spring,

$\begin{split}&E_{K}=\int_{0}^{x}\frac{1}{2}\dot{x}^{2}dm\\\
&=\int_{0}^{x}\frac{1}{2}\frac{d\xi}{x}m\frac{\xi^{2}}{x^{2}}\dot{x}^{2}=\frac{1}{2}\left(\frac{1}{3}m\right)\dot{x}^{2}\end{split}$
(9)

Then the effective mass of the whole system $m_{eff}$ can be interpreted with
the mass of the bob $m_{b}$ and the mass of the spring $m_{s}$.

$\begin{split}m_{eff}=m_{b}+\frac{1}{3}m_{s}\end{split}$ (10)

In same way, the effective moment of inertia of the whole system $I_{eff}$ can
be interpreted with moment of inertia of the bob $I_{b}$ and the mass of the
spring $m_{s}$.

$\begin{split}I_{eff}=I_{b}+\frac{1}{3}m_{s}R^{2}\end{split}$ (11)

###  3.2 Experimental results

As shown in Fig. 2 down below, it is able to see the resonance between the
vertical motion (dark blue) and the rotational motion (light blue) clearly.

Figure 2: Resonance graph

The distance between the center of the bob and weight, which is denoted as
$d$, is varied from 2 cm to 9 cm. Fourier analysis was done for the vertical
data through MATLAB. The MATLAB code for FFT is shown in Appendix 1. By
analyzing two modes of the pendulum through Fourier analysis, the spring
constant was calculated by considering it as a fitting parameter. As shown in
Fig. 3, FFT data clearly shows two peaks, which means that the oscillation is
a combination of two different modes beating.

Figure 3: Fast Fourier Transform

Taking the vertical angular frequency as a fitting parameter, the theory and
experiment comparison is shown in the table below. The absolute value of the
error rate between the theoretical vertical angular frequency and the
experimental angular frequency is lower than 4%, which means that the
experiment and theory match well. Other experimental graphs are shown in
Appendix 2.

$d$ (cm) | 2.2 | 4.4 | 5.2 | 6.1 | 8.9
---|---|---|---|---|---
$\omega_{z}$ (rad/s) | 2.82 | 2.90 | 2.82 | 2.89 | 2.89
Error rate (%) | -0.71 | -3.57 | -0.71 | -3.21 | -3.21

Table. 1. Experimental results table

##  4 ANALYSIS

###  4.1 Regression

Since we have varied the distance between the middle of the spring and the
displacement of the bob, we have changed the moment of inertia through the
experiment. It is able to plot the angular frequency of each mode through the
change of the moment of inertia. The following graph was drawn by fitting the
other spring constants, rotational spring constant, and coupling spring
constants. These were set as fitting parameters.

Figure 4: Regression graph

As the regression graph is drawn with DESMOS, which is the graph between the
moment of inertia and angular frequency of the pendulum. The angular frequency
$\omega_{1}$ and $\omega_{2}$ is graphed as Fig. 4. The regression formula is
shown in equation (6). The regression formula is an equation that was derived
in theoretical analysis, and it fits the experimental result well. This
clearly shows that the theoretical result and the experimental result matches
well.

###  4.2 Oscillation ratio

Oscillation ration $\Omega$ was set up to explain the observation of the
phenomenon, on the perspective of energy transfer between translational energy
and rotational energy, which is denoted $\Omega=\omega_{z}/\omega_{\theta}$.
It was also observable that as the oscillation ratio gets closer to 1, the
energy transfer gets closer to 100 percent. It is able to analyze that
information quantitatively. Defining the beat angular frequency $\omega_{b}$,
which is introduced as $\omega_{b}=\omega_{1}-\omega_{2}$, It is able to find
the relationship between $\Omega$. As the main goal is to compare the
difference with $\Omega$ and 1, defining $\Gamma$ as the following will help
the explanation.

$\begin{split}\Gamma=|\Omega-1|=\left|\frac{\omega_{z}}{\omega_{\theta}}-1\right|\end{split}$
(12)

As a heuristic, it is able to predict that $\Gamma$ increases as $\omega_{b}$
increases, and $\Gamma$ decreases as $\omega_{b}$ decreases. Introducing
experimental data as Table. 2,

$d$ (cm) | $\Gamma$ |  $\omega_{b}$ (rad/s)
---|---|---
2.2 | 0.4766 | 2.720
4.4 | 0.1601 | 1.125
5.2 | 0.0507 | 0.9354
6.1 | 0.146 | 1.020
8.9 | 0.370 | 1.277

Table. 2. Oscillation ratio related to beat

###  4.3 Conversion factor

In order to focus on the idea of the energy conversion between translational
energy and rotational energy, it is able to define a conversion factor
$C_{E}$. The conversion factor of energy is defined similarly to the Quality
factor of forced oscillation, which is denoted

$\begin{split}C_{E}=\frac{\Delta E}{E_{max}}\end{split}$ (13)

It shows as Table. 3 as implementing experimental data.

$d$ (cm) | $\Delta E$ | $C_{E}$
---|---|---
2.2 | 0.065 | 0.255
4.4 | 0.171 | 0.074
5.2 | 16.7 | 0.988
6.1 | 0.152 | 0.844
8.9 | 0.076 | 0.295

Table. 3. Conversion factor experimental data

It is clearly shown in an experimental method that the conversion factor is
close to 1 as the motion is close to resonance. This way, the relationship
between the efficiency of energy transfer and beat is explained
experimentally.

##  5 CONCLUSION

The study of mechanical properties of Wilberforce pendulum was done by
theoretical analysis with experimental analysis. The theoretical model was set
up based on the observation of the phenomenon with qualitative analysis.
Analyzing in two different perspectives, force, and energy, it is able to
explain the phenomenon. Especially for the energy perspective, some parameters
were defined and introduced for a quantitative explanation using experimental
data. The reason for the phenomenon was explained using the research of Clive
L. Dym[1], which used Castigliano’s Second Theorem. A quantitative model was
set by designing the coordinate system. The existence of the coupling term was
shown by Taylor expansion of potential energy, and by using it, a mechanical
equation of motion was derived. By solving the differential equation, the
general solution was derived.

Experimental setup and measurement were done by firstly designing the 3D-Model
of the pendulum. The experiment was done several times with PASCO Capstone,
and data was extracted using TRACKER. Using the kinetic energy integration,
mass and moment of inertia of spring were considered as well as the bob.
Experimental results are shown by analyzing experimental data using Fast
Fourier Transform (FFT) and showing the error rate in the source of the table.
Analysis of the experiment was done in three ways, including drawing a
regression graph using DESMOS. The oscillation ratio, which was defined in the
qualitative analysis, was reorganized by defining $\Gamma$, and implementing
experimental data. Also, conversion factor $C_{E}$ is defined, and
experimental data was implemented for analysis.

## Appendix 1

Following is the MATLAB code for FFT.

⬇

y = fft(x);

N = length(y);

Fs = 1/t;

f = (0 : N-1) * Fs / N;

plot(f , abs(y))

## References

  * [1] Clive L. Dym, Consistent Derivations of Spring Rates for Helical Springs. Journal of Mechanical Design (2009)