# CONTEMPORARY COSMOLOGY FROM LAKATOS’ VIEWPOINT

J.E. Horvath (IAG-USP, São Paulo, Brazil)

<EMAIL_ADDRESS>
ABSTRACT: An analysis of contemporary Cosmology is presented, with the aim of
identifying the elements present in it according to the scientific program
structure created by I. Lakatos. We look at some modern controversies from
this point of view and clarify the meaning of issues related to them within
this context.

KEYWORDS: Cosmology; Imre Lakatos; General Relativity; Particle physics

##  1 Introduction

The contemporary picture of Cosmology has been slowly developed over the 20th
century and the first decades of the 21st, on a quite successful path which
featured theoretical insight and observational work prompted by increasingly
sophisticated instruments and techniques. This modern phase started with the
construction of relativistic Cosmology, based on General Relativity, and
received input from other areas such as Nuclear and Particle Physics, once the
idea of a hot and dense early Universe (the Big Bang) was put forward and
seriously developed. Within this perspective, we arrived to the present
paradigm, the so-called $\Lambda$CDM (Lambda Cold Dark Matter) Cosmology. For
an outsider, it is hard to understand why cosmologists should be satisfied:
the cold dark matter is an undetected component thought to be fundamental for
structure formation and largely dominant in many scales. On the other hand, a
(constant) energy density $\Lambda$ is the latest ingredient of this model, an
unclustered form of invisible energy making the Universe expansion to
accelerate, and adding up to 70$\%$ of the full matter/energy content of the
Cosmos. These two entities have been indirectly inferred, and there are big
observational/experimental efforts to detect some direct evidence, and also a
huge theoretical activity to go beyond mere ”coincidence Cosmology” and
justify a complete picture of the Universe.

In addition to this disciplinary viewpoint, the philosophical constructions of
the 20th century, championed by Popper, Kuhn, Lakatos, Feyerabend and others
can be invoked to visualize the whole picture and its possible future path. We
have argued before (Horvath 2009) that the acceleration of the Universe
triggered a kind of ”extraordinary science” period, such as suggested by T.
Kuhn (1962). The ”anomaly” evidenced by the acceleration, which is now 20+
years old, may or may not be solved by establishing a Dark Energy or
otherwise. Meanwhile, the other ”anomaly” (the lack of material to match the
observation, leading to the Dark Matter hypothesis) will be a century old soon
(Zwicky 1933), and nothing convincing has been detected to play this role
(Shutt 2013), and also theoretical suggestions are many and never reached any
firm consensus (Oks 2021). Moreover, other controversies (”anomalies” in
Kuhn’s terminology) have arisen and may be important for the future of the
field. Therefore, it is interesting to go back to the 20th century ideas and
revisit the question of how is Cosmology constructed as a research programme,
identifying the main ingredients and putting issues in perspective. This is
precisely the task we will attempt here, using the framework developed by
Lakatos in the last century, with the aim of visualizing a more clear picture
of the whole discipline and perhaps giving some insight on its future
development.

##  2 The hard core and protective belt of modern Cosmology

In Lakatos’ view, the present $\Lambda$CDM Cosmology can be considered a
paradigm (Kuhn 1962), basically a research programme that became hegemonic
(Lakatos 1978). Most cosmologists believe, even with some evidence against it,
that $\Lambda$CDM Cosmology is essentially correct, and will not open any
discussion to confront it with any other model, for example, the now abandoned
Steady State (Kragh 1999). $\Lambda$CDM Cosmology is the last version of 20th
century Cosmology, which evolved from a philosophical/speculative state
towards a real empirical science over the years, mainly after the Hubble
announcement of the expansion of the Universe, in which Slipher, Humason,
Lemaître and others contributed. This discovery shifted the interest from a
static, infinitely old Universe to a dynamical, finite age one. By its very
construction and history, the $\Lambda$CDM Cosmology shows different elements
within, which evolved individually and as a intertwined set. An appraisal of
these constitutive elements follow below.

###  2.1 The hard core

Over the years, and after the consolidation of the Einstenian programme for
the description of gravity, a few fundamental elements constituting the hard
core of the Cosmology research programme can be identified. These elements are
never disputed, and if some problem arises with them, the protective belt can
be modified to save their validity (see below). We suggest that the hard core
of modern Cosmology is constituted by

$\ast$ The Cosmological Principle

A fundamental (meta)postulate in Cosmology has its origins in the
revolutionary work of N. Copernicus around 1500 A.D., which rediscovered and
extended the observations of Aristarchus of Samos (III century B.C.). There
has been a lot of discussion on the true extent of Copernicus’ work, mainly
about the consequences of formulating an heliocentric model that removed the
Earth from the center of the Cosmos.

The basic Copernican Principle is usually stated as there are no ”special”
observers. This sharply contrasts with, for example, Aristotelian Cosmology,
in which the Earth occupies a privileged position in the whole world system.
Sometimes it has been insisted that the inference that we are ”cosmically
ordinary” should be called the principle of Copernican mediocrity (with some
subtle differences not discussed here). And as a deeper scrutinized issue, the
elaborated Genuine Copernican Cosmological Principle which says: The Universe
as observed from any planet looks much the same. The latter form is specially
suitable to link Copernicus with the modern form of the Cosmological
Principle, which has been argued to be substantially stronger than the former
(Beisbart & Jung 2006): Each observer in the Universe, in any position, would
observe the same patterns independently of the direction. A Universe which has
no privileged place (center or differentiated places for the observers), and
which shows the same large-scale pattern in any direction is called
homogeneous and isotropic in Cosmology jargoon. We see that statements based
on the observer’s situation and thus translated into statements about the
properties of the Universe itself. Mathematical models should possess these
properties to comply with the postulated observer status. This is indeed the
case of $Lambda$CDM Cosmology, and challenges that have arisen by observations
which seem at odds with the Cosmological Principle (see below).

$\ast$ Conservation of energy

The conservation of energy has been for a very long time a metapostulate in
Physics, and later related to time-symmetric formal mathematical descriptions
via Noether’s currents (Baez 2020). Ideas suggesting a violation of energy
conservation were suggested from time to time, especially when new branches or
phenomena emerged (Landau 1932), but refuted empirically (Bothe & Geiger
1925), and indeed found not viable by most of the practitioners. This energy
conservation certainly applies to any local process.

It is then easy to envisage why (besides several lines of evidence in favor of
an expanding Cosmos) the Steady State theory was progressively abandoned: to
maintain a steady Universe, a creation field (C-field) had to be introduced,
and related to the Faraday-Lenz phenomenon because the creation of
matter/energy opposes the Universe dynamical state. This C-field can be
interpreted as creating matter/energy from the vacuum, and therefore violating
the simplest form of energy conservation. However, there is an argument
stating that the Universe is not time-invariant (it changes over time
irreversibly), and therefore it does not conserve energy as a whole, since the
latter is the charge associated with time symmetry, although many do not
consider that this is enough to consider a C-field. General Relativity and
most viable alternatives do not rely on any C-field, a feature which is
considered a desirable feature by cosmologists. The community welcomes the
solution of cosmological problems while conserving energy, elevated to the
range of hard core postulate.

It is amusing to consider that the ”modern” form of this conservation law is
actually the last version of Lucretius statement ”nothing can be made from
nothing”, initially expressed by Parmenides (Burnet 2014) many centuries
before him. While ”energy” is the contemporary form of the conserved quantity
(we now think that mass is just a concentrated form of it), the idea is the
basically same. It can be said that this element of the Cosmology hard core
programme is as old as the Western thought itself.

$\ast$ Metric character of gravity

The last suggested element of the hard core is more technical, and related to
the identification of gravitation as a deformation of spacetime, the vision
championed by Einstein. The idea here is that the fundamental equations of
motion describing gravitation relate the sources (matter and energy as a
whole) to the deformation of spacetime, described by a function that
essentially measures distances between spacetime events. This mathematical
object is called a metric, and implies that the mathematical space in which it
lives is endowed with a consistent definite expression to measure distance
between events happening in spacetime (i.e. it is a metric space). General
Relativity is the prime example of such a class, but there is a large family
of metric theories which share this underlying structure, and many
alternatives to General Relativity belong to this class. Different hypothesis
separate the latter alternative theories, for example, Unimodular Gravity is
very similar to General Relativity, but an extra condition on the former can
be interpreted as giving rise to a non-zero cosmological constant. Many other
alternatives have been suggested (Bellucci, Faraoni & Longo, 2022), including
some non-metric ones, but the success of General Relativity in many tests has
created a kind of consensus that, if some alternative is presented, it can
hardly escape a metric character.

An important battery of tests that have not been fully explored yet is
provided by the decade-old precise measurements (Kramer et al. 2021) of the
double pulsar system, in which two pulsating neutron stars have provided a way
to test the Post-Newtonian and some Post-Post-Newtonian corrections deviations
from General Relativity. These deviations have proved to be $\leq\,0.01\%$ at
most, in the strong gravity regime, implying that General Relativity leaves
little room for alternative theories. However, this fact will not be
meaningful for Cosmology until alternative cosmologies (including those based
in non-metric schemes) can be subject to analogue tests and confirm their
status. This is a research programme that has not really started and can be
reveling for the progress in Cosmology. It is important to remark that the
hard core content underlies the formal mathematical structure of $\Lambda$CDM
Cosmology, and suffices for its full formulation. For example, acceptance of
the Cosmological Principle and energy conservation will produce the Friedmann-
Lemaître-Robertson-Walker metric as the only possibility within General
Relativity. Contemporary cosmologists who believe that gravitation has a
dynamical metric character would seek for modified FLRW equations (see below)
and postulate DM/DE added to the matter/energy content or a description of
gravitation that preserves the general feature of a metric. All these
components/modifications will be attached to the protective belt, specifically
to the negative heuristics (see below) in Lakatos’ terminology.

Figure 1: A general scheme of Cosmology within the Lakatos’ proposal. All the
suggested elements are discussed in the text (Section 2)

###  2.2 Heuristics in the protective belt

Lakatos’ view of the structure of research programmes envisaged a protective
belt around the hard core, with a negative and positive heuristics. Even if
his initial definition of ”heuristics” changed over the years (Kiss 2006), the
prevailing late concept can be stated as saying that ”heuristics” studies the
patterns of thinking, (to be later related to methodology). Lakatos suggests
the existence of a negative heuristics, intended to protect the hard core from
being attacked/eroded by the presence of anomalies, redirecting the attention
to auxiliary hypothesis and initial conditions. Changes can be made in the
latter without affecting the inviolable hard core, held as fundamental for the
programme existence. On the other hand, positive heuristics is a set of
recommendations articulated to specify how modifications should be performed,
including new measurements, verification of predictions and related issues.

Modern $\Lambda$CDM Cosmology, as any other research programme, has some
elements which are identified as integrating the negative heuristics. The
first recognition is that Dark Matter and Dark Energy are hypothetical
components postulated to avoid a breakdown of the three fundamental points of
the hard core, as discussed in the last subsection. This qualifies all forms
DM/DE as integrating the negative heuristics, independently of their specific
form. In other words, if we could prove that they do not exist, some other
element would be necessary to square FLRW models with observations, otherwise
it would be necessary to accept that the $\Lambda$CDM framework does not
provide a good description of the whole set of data and must be abandoned.
Much in the same way, modifications to the gravitational theory (for example,
introducing a fundamental length) are mainly elements of the negative
heuristics, because they would allow to retain the hard core if the
modifications fit into a metric framework. Other effects entangled with
$\Lambda$CDM Cosmology may qualify as negative heuristics elements, and even
boost the overall description by ”solving” the DM/DE quandary, for instance
the presence of quantum effects in the equations, generally neglected but are
undoubtedly part of the contemporary physical picture. A full list of
candidate elements in the negative heuristics class would be impossible, but
the above statements suffice to show some important ones and how they qualify
as such.

On the other hand, positive heuristics are devised to improve and move the
programme forward, suggesting new developments and measurements. One of the
most important of this class is the measurement of the expansion rate $H_{0}$,
a source of concern lately (see below). The ”cleanest” suggestion to do so is
by means of the standard siren technique (You et al. 2020), which relies on
the amplitude of gravitational waves detected in events of mergers of neutron
stars and black holes. The novel technique is considered quite free of usual
astronomical problems, since gravitational waves are not subject to them, but
the number of events and the refinement of the fusion events are still short
of providing an accurate ($\sim\,1\%$ or so) determination of $H_{0}$ as
needed. In summary, there are elements belonging to the protective belt in
modern Cosmology which are clearly identified, but an exhaustive list is not
possible because of the variety of matters and the enormous amount of work by
cosmologists.

##  3 Actual problems (anomalies) arising

To begin with the anomaly problem, we have addressed DM and DE in some length
in a previous article (Horvath 2009), and suggested that these are, by far,
the most important anomalies to be solved within $\Lambda$CDM Cosmology. This
is also a prime philosophical issue, since even if detected and confirmed,
this would mean that $\sim\,~{}95\%$ of the content of the Universe is not
what we thought it was. Moreover, in a large class of extra-dimensional
theories (Brane World), there is a solution in which DM and DE are nothing but
projections of the (bulk) extra content onto our (brane) Universe. From the
point of view of the programme, this can be seen as the injection of external
components, which were unpredicted and unexpected, changing the situation
quite radically. On the other hand, if DM/DE arise just from a ”dark sector”
of the Standard Model of fundamental interactions, or even are dark
astronomical objects + zero point energy (the latter somewhat strongly
suppressed respect to the naive calculation, Zel’dovich 1968), the modern
Cosmology programme should go ahead without major problems. This is why
initiatives to determine whether DE varies with time through galaxy counts
(https://www.darkenergysurvey.org/) are under vigorous development.

An increasingly debated anomaly in Cosmology is the so-called Hubble tension.
This issue is simply understood by noting that local determinations of $H_{0}$
differ from large-scale ones (CMBR and related) by $\geq 4\sigma$, i.e. much
more than the uncertainty bars. In other words, even though expected to yield
the same rate $H_{0}$, different methods point out to incompatible results. A
lot of activity has been seen on this serious discrepancy, and its solution
may include new physics, with strong consequences for the whole Cosmology.
However, in the absence of a hint of solution, it is difficult to forecast
what is actually implied for the research programme.

Figure 2: An image from the SDSS survey containing 13 billion galaxies
(Abazajian et al. 2004). The large-scale distribution of galaxies. This image
contains galaxies of all types, which assemble into clusters, superclusters
and filaments easily visible to the naked eye. The survey sees an approximate
fraction of 1/20th of the currently observable Universe.

On the other hand, it is known that far beyond the scale of galaxies, a
variety of filaments, voids and other inhomogeneities were detected in the
largest surveys performed so far (Fig. 2). Some of them are visible to the
naked eye, and their evolution can be studied using numerical simulations.
According to the idea of homogeneity and isotropy, the structure present at
the largest scales must completely “dilute” long before the Hubble scale $\sim
5000\,Mpc$. However, there are works that have discussed the extent to which
this is true in the data, and proposed that the structure does not end, being
self-similar up to the limits of the sample. Many of these are fractal models,
where the structure features scale invariance because it is a product of some
self-organization mechanism. Orthodox cosmologists dismiss a scale-invariant
structure all the way up to the Hubble radius, and insist that homogeneity is
achieved beyond $\sim 300\,Mpc$ or so, although much larger structures seem to
be present in the data.

A prime example of these totally unexpected structures that has been
identified is the Hercules-Corona Borealis ”Great Wall of Quasars” (not to be
confused with the original Great Wall which is a ”little wall” 10 times
smaller in comparison), at scales of $\sim 3000\,Mpc$ (Fig. 3). Such an
enormous structure was not considered possible, and in fact the Cosmological
Principle itself and the treatment of the current Universe as a homogeneous
and isotropic fluid over a length scale may be now questioned, since this type
of structure occupies about half the radius of Hubble. In order to have a
consistent perspective with the discussion, we must also remember that to
present a self-similar structure or periodicity a physical mechanism to
produce perturbations that lead to this structure should exist and be
identified, which is not the case yet.

Figure 3: An image of the Great Wall of Hercules-Corona Borealis, an
association of quasars at a scale comparable to that of the observed Universe
(Horváth et al. 2015). The assumption of isotropy and homogeneity of the
Cosmological Principle is contested based on this type of structure “which
could not exist”.

Another very puzzling feature related to the accepted structure formation has
been recently very provided by observations of very early galaxies with large
masses (Hainline 2023). The standard scenario of fluctuation growth would not
leave enough time to form massive galaxies (with $M\geq 10^{10}\,M_{\odot}$)
present up to a redshift $z\sim 15$, that is, only $sim300-500\,Myr$ after the
Big Bang. On theoretical grounds, much smaller ”galactic blocks” with $M\sim
10^{6}\,M_{\odot}$ were expected. Therefore, it is not clear to what extent
the view of initially small galaxies, growing a factor $10{3}-10^{4}$ by
merging is still tenable or must be revised thoroughly. The list of anomalies
do not stop here: there is a claimed evidence that the dipole asymmetries
measured in the CMBR and the ones inferred from matter distributions (Active
Galactic Nuclei for instance) do not agree (Singal 2023). It is not possible
to make them agree just by adjusting the peculiar motion of the Solar System,
therefore, the possibility that both cosmic reference frames are different
remains. It is not clear what relationship could have with the other anomalies
questioning the Cosmological Principle.

Can these observations, isolated or joint, falsify the Cosmological Principle?
the tentative answer is “yes”, but cosmologists may rescue it by redefining
its validity statistically, for example. Anisotropic models of the Universe
evolution have existed for decades, and therefore this is not a matter of
description, but rather of principles. Thus, the study of structure in the
Universe needs to continue and resolve these and other questions of great
cosmological interest.

##  4 The Multiverse and its philosophical and scientific unresolved issues

We have insisted on the role of the Cosmological Principle as an element of
the hard core of $\Lambda$CDM, but an opposing view (called the Anthropic
Principle) has been discussed to justify our very existence in the late
Universe. However, it has been pointed out, and discussed in length, that the
Anthropic Principle collides with the Cosmological Principle to some extent.
In fact, Carter’s (1974) original motivation in the initial formulation of the
Anthropic Principle was related, in his opinion, to a degree of overstatement
of the power of the latter. Life on Earth and other planets do need special
physical conditions, and a place and time for life emergence which is not
arbitrary. For example, the fluctuations that led to the formation of the
structure can be considered from the point of view of the Anthropic Principle.
This has been emphasized by M. Rees (2001), who observed that if the amplitude
had been a slightly smaller, the primordial gas would never condense into
connected structures, and thus the material enriched in heavy elements by the
stars would be dispersed in space and would not allow a later chemical
evolution, with a sequence of stellar generations. Now, if the amplitude had
been a little higher, regions much larger than clusters of galaxies would have
formed very early in the history of the Universe, and would not have broken up
into stars, but would have formed vast black holes instead. The remaining gas
would be heated to such temperatures that it would emit X-rays and gamma rays,
such that material enriched in heavy elements by stars would be quickly
trapped in black holes (none of this would favor a Universe where life could
flourish to discuss these possibilities today). Therefore, a reasonable
”quite” place within a galaxy, and enough time for the planet to evolve from
pre-biotic Chemistry to the present state are needed.

A further development has attracted the attention of cosmologists lately,
namely the “Anthropic” idea that ours is just a favorable Universe out of a
set of universes integrating a Multiverse. These alternative Universes, with
values of the physical constants and histories that differ from ours, are
considered to be real, not a Gibbs ensemble. It has been recognized that this
kind of idea brings an epistemic shift to what we consider as ”Science”, and
lead to dead ends in the cognition of the Cosmos (Kragh 2017).

The question is whether an epistemic shift would be acceptable, if ultimately
executed. Such attitudes have been seen over the centuries, for example,
Cartesians blamed Newtonians for the introduction of unspecified, mystical
elements, such as a “force” into Natural Philosophy, but our own perception
three centuries later does not see anything wrong with it. We may state that
an epistemic shift happened 300 years ago, and was completed by successive
generations of physicists. Should we ”wait and see”, or rather comply against
the consideration of a physical world which may be unobservable by its very
nature? We may be in the middle of such a process.

Figure 4: Back to square one in Cosmology? A cartoon suggesting the danger of
substantial epistemic shifts

##  5 Conclusions

We have constructed in this paper a picture of the contemporary Cosmology
research program patterned after the work and concepts introduced by Imre
Lakatos (1978). The research program features a hard core, a negative
heuristics and a positive heuristics in a protective belt, identified
according to Lakatos’ criteria as showed in Section 2. Several anomalies, old
and new ones, have been discovered and are under scrutiny at present, and as
in other case studies, it is impossible to forecast what the outcome will be
when solved.

We should remark that to go ahead with this research programme, for many
practitioners it is not just more observations what are needed, but rather
some general principle(s) based of symmetries, which play the role of
justification of cosmological facts. For example, the conservation of angular
momentum (as a consequence of the rotation invariance of the physical problem)
added to the hypothesis that gravitational forces stem from a central field,
were enough to show that planetary orbits should be elliptical, not circular,
in the 17th century. There are no such reasons at hand for contemporary
Cosmology. For example, a very small, but non-zero cosmological constant, far
off the predicted value of Quantum Field Theory (Zel’dovich 1968) does not
have a physical reason behind. Other cosmological issues can be thought along
the same lines, but for now some problems are introduced just as epicycles of
the new picture of the Universe, not necessary features dictated by powerful
reasons.

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# Experimental Benchmarking of
Energy-saving Sub-Optimal Sliding Mode Control

Michael Ruderman Michael Ruderman is with University of Agder, Department of
Engineering Sciences, 4879 Grimstad, Norway.
Email<EMAIL_ADDRESS>research has received no funding.

###### Abstract

The recently introduced _energy-saving_ extension of the _sub-optimal sliding
mode control_ allows for the control-off phases during convergence to the
second-order equilibrium. It enables, this way, a lower energy consumption
comparing to the original sub-optimal sliding mode (SM) algorithm, for the
commutating discontinuous control signal. In this paper, the energy-saving
sub-optimal SM control is experimentally benchmarked against a standard
second-order SM controller which has also discontinuous control action. Here
the so-called terminal second-order SM algorithm is used. The system plant is
affected by the matched bounded disturbances which are unknown, and the output
is subject to the sensing noise. Also, a first-order actuator dynamics can
lead to an additional chattering in SM applications. For a fair comparison,
the same (quadratic) terminal surface is designed when benchmarking both SM
controllers. Moreover, the experimentally compared SM algorithms share the
same amplitude and initial conditions.

##  I Introduction

Sliding mode (SM) control techniques are among the most promising robust
control strategies, as they allow dealing with weakly known and perturbed
dynamic systems. While SM control, as methodology, was introduced to
engineering in [1], followed by several well perceived textbooks, e.g. [2, 3,
4], we refer also to a more recent text [5] that summarizes basic ideas,
approaches, and ongoing issues.

In SM methods, the control authority aims to force an uncertain system
dynamics onto a specified manifold $\sigma$, also called sliding manifold, and
then maintain the system behavior staying on it. In this way, a SM control
provides the reduced-order dynamics and insensitivity both to uncertainties in
the system parameters and to some types of external perturbations. Being in
the (first-order) sliding mode means to ensure $\sigma=0$ for all times
$t>T_{c}$, that is after reaching the sliding manifold at $0<T_{c}<\infty$.
For second-order sliding modes [6], also the time derivative of the sliding
variable is forced to zero, i.e. $\sigma=\dot{\sigma}=0$, thus enabling also
for robust regulation of the systems with relative degree two between the
sliding variable and control value. Here it is worth recalling that the
second-order SM controllers are seen to belong to a more general class of
_higher-order sliding mode_ control, actively developed over the last two and
a half decades for the uncertain systems with relative degree two and higher,
see e.g. [4, 5] and references therein. Applications and experimental case
studies with use of the SM control (and also estimation) techniques are
numerous. Here to mention just a few of the recent, the SM applications can be
found e.g. in the temperature control of chemical fluids for silicon wafer
production [7], in states’ estimation for frequency regulation and economic
dispatch in the power grids [8], in the control of hydro-mechanical drives
[9].

The so called sub-optimal SM control was proposed in [10] as a second-order SM
technique which does not require $\dot{\sigma}$ for a discontinuous control
commutation. Recall that a single commutation at the well-defined time instant
in an unperturbed system renders the time optimal bang-bang control, cf. [11].
In turn, the sub-optimal SM control [10] converges robustly to
$(\sigma,\dot{\sigma})$ origin in presence of the bounded perturbations, while
executing a commutating control sequence $u(t)$ with an increasing frequency.
For a detailed survey of the properties, generalizing extensions, and
applications of the sub-optimal SM control we refer to [12].

Since the sub-optimal SM control $u(t)$ is continuously commutating between
two sign-alternating discrete values $\forall\>t>0$, actually as any
discontinuous SM control does, the corresponding cumulative energy consumption
is proportional to the execution time, or at least convergence time if the
control can be switched off after reaching the origin. Recently, an energy
saving extension of the sub-optimal SM control was introduced in [13], which
allows for the well defined $u=0$ phases during the convergence. The original
sub-optimal SM control was modified by introducing an additional switching
threshold which enables for a mandatory $u=0$ phase between two consecutive
extreme values of the sliding variable, cf. with [12]. This implied the
control value in the finite set $u\in\\{-U,\,0,\,U\\}$, while perturbed
systems with the relative degree two and discrete bounded control action (i.e.
$\max(u)=U$) were under investigation.

In this paper, the energy-saving sub-optimal SM control [13] is experimentally
benchmarked in comparison with a classical terminal second-order SM control.
The latter assumes $u\in\\{-U,\,U\\}$ and provides a finite-time convergence
while compensating robustly for the bounded perturbations $|d(t)|<U$, see e.g.
[14]. Worth noting is also that the energy-saving sub-optimal SM control [13]
is evaluated experimentally for the first time, while the targeted electro-
magneto-mechanical system [15] represents a sufficiently challenging
application case as for uncertainties, disturbances, and noise. The rest of
the paper is as follows. In Section II, we provide the necessary preliminaries
of the energy-saving sub-optimal SM control [13] and the benchmarking terminal
SM control with the corresponding second-order sliding surface. Also the
chattering effects are briefly addressed, as relevant for the real SM and
owing to an additional actuator dynamics, see e.g. [16]. Section III describes
in details the experimental benchmarking system. The experimental results are
shown in Section IV while benchmarking both, convergence of the output value
of interest and the monitored energy consumption. The latter is measured as an
integral of the control absolute value, i.e. proportional to the overall time
when the control is on, i.e. $u\neq 0$. The brief conclusions are drawn in
Section V.

##  II Preliminaries

In this section, after specifying the class of considered systems, the energy-
saving sub-optimal SM control [13], which constitutes an extension of the
original sub-optimal SM control [10] is summarized for convenience of the
reader. Then, the so called terminal SM control [17] is recalled, while the
emphasis is on design of an optimal second-order sliding surface discussed in
[14]. By the end, the appearance and parameters of chattering, due to
additional (parasitic) actuator dynamics, are briefly addressed.

The class of uncertain dynamic systems under consideration is the one which
can be assigned with the well-defined sliding variable

$\sigma=\sigma(\mathbf{x},t)$ (1)

that has the relative degree $2$ with respect to the control input
$u\in\mathbb{R}$. The vector of the system states, assumed as measurable, is
$\mathbf{x}\in\mathbb{R}^{n}$ with $n\in\mathbb{N}$ and $2\leq n<\infty$.
Then, the dynamics of the sliding variable, which is affected by the matched
bounded perturbations $d$, can be written as

$\ddot{\sigma}(t)=u(t)+d(\cdot,t),$ (2)

where $d$ is an unknown exogenous term. The system perturbations are supposed,
however, to satisfy the global boundedness condition

$\bigl{|}d(\cdot,t)\bigr{|}\leq D,$ (3)

where $D>0$ is a known real constant. Another essential boundary of the system
class (1), (2) is

$\max\bigl{(}|u|\bigr{)}=U>D,$ (4)

that is due to a saturated control action.

###  II-A Energy-saving sub-optimal SM control

The sub-optimal second-order SM control, cf. [12],

$\displaystyle u(t)$ $\displaystyle=$
$\displaystyle-\gamma(t)\,U\,\mathrm{sign}(\sigma-\beta\sigma_{M}),$ (5)
$\displaystyle\gamma(t)$ $\displaystyle=$
$\displaystyle\left\\{\begin{array}[]{ll}1,&\hbox{ if
}(\sigma-\beta\sigma_{M})\sigma_{M}\geq 0\\\\[2.84526pt] \gamma^{*},&\hbox{ if
}(\sigma-\beta\sigma_{M})\sigma_{M}<0,\\\ \end{array}\right.$ (8)
$\displaystyle\beta$ $\displaystyle\in$ $\displaystyle[0;\,1).$ (9)

was initially proposed in [10]. Here, $\gamma^{*}\geq 1$ and $\beta$ are the
modulation and anticipation factors, correspondingly. The dynamic state
$\sigma_{M}$ constitutes the last _extreme value_ of $\sigma(t)$ during the
control (5) operates on the sliding variable. The control parameters have to
satisfy, cf. [12],

$\displaystyle U$ $\displaystyle>$ $\displaystyle\frac{D}{K_{m}},$ (10)
$\displaystyle\gamma^{*}$ $\displaystyle\in$
$\displaystyle[1;\infty)\cap\Biggl{(}\frac{2D+(1-\beta)K_{M}U}{(1+\beta)K_{m}U};\infty\Biggr{)}.$
(11)

Recall that for a generic case of the sub-optimal SM controllers [12], the
perturbed dynamics is given by

$\ddot{\sigma}(t)=K(\cdot,t)\,u(t)+d(\cdot,t)\;\;\hbox{ with }\;\;0<K_{m}\leq
K\leq K_{M},$ (12)

cf. with (2), meaning the control input gain can also be uncertain, within the
known boundaries $K_{m}$ and $K_{M}$. Also recall that the condition (10)
represents the control authority that is required to overcome the unknown but
bounded perturbation. The condition (11) ensures the convergence of
$\sigma(t)$ to the origin. Here we note that first when
$(\sigma,\dot{\sigma})=\mathbf{0}$, the second-order SM appears in its proper
sense.

One can recognize that for constrained control action (4) on (2), the
parametric condition (11) vanishes, leading to

$\gamma^{*}=1.$ (13)

The resulted (simplified) sub-optimal SM control

$u(t)=-U\mathrm{sign}\bigl{(}\sigma(t)-\beta\sigma_{M}(t)\bigr{)}$

served as a starting point for the energy-saving sub-optimal SM control
introduced in [13].

The energy-saving sub-optimal SM algorithm [13] is

$\tilde{u}(t)=-\frac{1}{2}U\mathrm{sign}(\sigma-\beta_{1}\sigma_{M})-\frac{1}{2}U\mathrm{sign}(\sigma-\beta_{2}\sigma_{M}).$
(14)

An initializing control action, similar as in the case of the original sub-
optimal SM control [12],

$\tilde{u}_{0}(t)=-U\mathrm{sign}\bigl{(}\sigma(t)-\sigma(0)\bigr{)},\quad\forall\,t\in[0;t_{M_{1}}]$
(15)

is also required for accelerating the reaching of the first extremum at
$t_{M_{1}}$.

Figure 1: Parametric constraints (16)–(18) in $(\beta_{1},\beta_{2})$ plane.

In addition to the disturbance condition $|d|<U$, an appropriate thresholds’
relationship is required, while the conditions for convergence were derived in
[13] as

$\beta_{1}+\beta_{2}>\frac{2D}{U},$ (16) $0\leq\beta_{1}<1,$ (17)
$-1<\beta_{2}<\beta_{1}.$ (18)

The imposed $\beta_{1},\beta_{2}$ constraints (16)–(18) have a well-
interpretable graphical visualization as shown in Fig. 1. The set of
admissible $\beta_{1},\beta_{2}$ values is inside of a color-shadowed
triangle. The $\beta_{1}=\beta_{2}$ edge recovers the original energy-saving
SM control. Obviously, the $D/U$ ratio determines the size of an admissible
$\\{\beta_{1},\beta_{2}\\}$ set. It also determines whether the negative
$\beta_{2}$-values are allowed, while it can lead to a twisting mode during
the convergence, see [13]. Recall that it is on the interval
$(\beta_{2},\,\beta_{1})$ where the control value is switched off, i.e. $u=0$,
and thus operates as energy-saving. This appears at each reaching cycle, i.e.
between two consecutive extremal values of the sliding variable, i.e.
$\sigma_{M_{i}}$ and $\sigma_{M_{i+1}}$, and during the entire convergence
range.

For an energy-saving operation of the introduced sub-optimal SM control (14),
a constrained optimization

$\underset{\beta_{1},\beta_{2}}{\min}\,\Bigl{[}J(\beta_{1},\beta_{2},U,D)-\hat{J}(\beta_{1},U,D)\Bigr{]}$
(19)

was formulated and solved in [13]. This maximizes the overall time difference
of the control-on mode, i.e. $u\neq 0$, between the original sub-optimal SM
control, for which the cost function is denoted by $\hat{J}$, and the energy-
saving SM control (14), for which the cost function is denoted by $J$. Note
that for both cost functions the $\beta_{1}$ parameter is kept fixed, also
under the specified upper bound

$\hat{J}(\beta_{1},U,D)<\hat{J}_{\max}.$ (20)

An explanatory $\hat{J}$ mapping is exemplary shown in Fig. 2. Also we notice
that both cost functions constitute a worst-case multiplicative factor in the
finite time convergence provided in [13].

Figure 2: Convergence time cost function $\hat{J}$ in dependency of
$\beta_{1}$.

The minimization hard constraint is

$J(\beta_{1},\beta_{2},U,D)-\hat{J}(\beta_{1},U,D)<0.$ (21)

while for each $D/U$ realization an optimal $(\beta_{1},\beta_{2})$ pair can
be calculated, see [13]. The results of such constrained optimization
(19)–(21) are exemplary shown in Fig. 3 for the assumed $D/U=0.3$ ratio.

Figure 3: Constrained objective function $(J-\hat{J})$ of the $\beta_{1}$,
$\beta_{2}$ parameters, for exemplary perturbation to control ratio $D/U=0.3$.

For a detailed analysis of the convergence and efficiency of the energy-saving
sub-optimal SM control, an interested reader is further referred to [13].

###  II-B Second-order terminal SM control

For second-order systems, i.e. with the dynamic states $x_{1}(t)$ and
$x_{2}(t)=\dot{x}_{1}(t)$, the so called terminal SM control

$\displaystyle u(x_{1},x_{2})$ $\displaystyle=$ $\displaystyle-
K_{t}\,\mathrm{sign}\bigl{(}\sigma(\cdot)\bigr{)},$ (22)
$\displaystyle\sigma(x_{1},x_{2})$ $\displaystyle=$ $\displaystyle
x_{1}+\delta\,x_{2}^{2}\,\mathrm{sign}(x_{2}),$ (23)

was proposed in [17], while a quadratic sliding surface $\sigma(\cdot)$
similar to (23) was exemplified already before, see [6, Sec. 4]. Remarkable in
(23) is that for an appropriately selected $\delta$, the parabolic sliding
surface coincides with the time-optimal state trajectory of the well-known
Fuller’s problem [11]. This has been further analyzed for the second-order
motion systems in [14] so that for

$\delta=\alpha/U,$ (24)

where $U\equiv\max(u)$ is the maximal control amplitude, the $\alpha=0.5$
value represents the boundary layer of the terminal mode, cf. [14, Fig. 2].
That means for $0<\alpha<0.5$, the control (22), (23) changes to the so-called
twisting mode, see e.g. [4, 5] for details. Both modes, the terminal and
twisting, provide a global finite-time convergence of the $(x_{1},x_{2})$
trajectories to the origin. Also recall that for the control (22), (23)
remains globally finite-time convergent in case of the bounded perturbations
$|d(t)|<D$, the following control-gain condition $K_{t}>D$ must be fulfilled.

With regard to the above basics, the used second-order terminal SM control is
given by

$\displaystyle\bar{u}$ $\displaystyle=$
$\displaystyle-U\,\mathrm{sign}\,(\sigma),$ (25) $\displaystyle\sigma$
$\displaystyle=$ $\displaystyle
x_{1}+U^{-1}\alpha\,x_{2}^{2}\,\mathrm{sign}(x_{2}),$ (26)

while $\alpha>0.5$ must be assigned with respect to $D$ if avoidance of the
twisting mode has to be be guaranteed. Also worth noting is that if the
second-order dynamics is additionally scaled by an inertial term $m$, as in
the experimental system provided in Section III, then the sliding surface (26)
must additionally include $m$, cf. with [14, eq. (13)].

The same (quadratic) terminal sliding surface (26), which provides a finite-
time convergence of the states to origin, can equally be used in combination
with other SM control laws, i.e. differing from (25), as it is done with
energy-saving sub-optimal SM control given in the previous subsection.

###  II-C Chattering due to additional actuator dynamics

The so called chattering, as a well-known and -studied phenomena in SM control
systems, see e.g. in [4, 5], appears due to unmodeled dynamics (often of the
sensors and actuators) and, thus, arises naturally when SM controllers are
implemented. Usually, it results in high-frequency oscillations with a finite
amplitude. Often, such (parasitic) additional dynamics can be captured by a
first-order behavior

$v(s)=(\mu s+1)^{-1}u(s),$ (27)

here written in Laplace domain with the complex variable $s$. The
corresponding time constant $\mu>0$ can approximate well the cumulative first-
order time delay of both, the sensor and actuator in the loop. Then, the
control value $v(t)$ appears as the input channel of the system instead of
$u(t)$, cf. (2), thus increasing relative degree of the sliding variable.

For the energy-saving sub-optimal SM control (14), the analysis of chattering
was provided in [13], following similar lines of development as proposed in
[18]. The solution of the harmonic balance equation, which involves also the
actuator dynamics (27), yields the chattering frequency

$\tilde{\omega}=\mu^{-1}\frac{\beta_{1}+\beta_{2}}{\sqrt{1-\beta_{1}^{2}}+\sqrt{1-\beta_{2}^{2}}}.$
(28)

The corresponding amplitude of steady-state oscillations is

$A_{\sigma}=\frac{\sqrt{\mu^{2}\omega_{c}^{2}+1}}{\omega_{c}^{2}\bigl{(}\mu^{2}\omega_{c}^{2}+1\bigr{)}}.$
(29)

For the terminal SM control, the appearance of chattering is as inherent as
for the conventional first-order SM control with discontinuous control action,
and has an even higher amplitude, see [19]. This appears natural since for the
terminal surface (23) when linearized in vicinity to origin, the gaining
factor of the linearized subsystem in the loop increases largely. This leads,
when using a harmonic balance analysis, see e.g. [16], to an interception of
the corresponding Nyquist plot by the relay’s describing function at already
lower frequencies (comparing to the linear first-order sliding surface) and,
hence, at higher oscillation amplitudes. A more detailed discussion of
chattering in the terminal SM control is omitted here due to the differing
focus of the present work.

##  III Benchmarking system

###  III-A Voice-coil based linear actuator

The benchmarking experimental system in use is a voice-coil based actuator,
see e.g. [20] for basics, with one linear degree of freedom (DOF). The
laboratory view of the actuator is shown in Fig. 4. More technical details,
including the system parameters, can be found in [15]. Most relevant to be
specified is that the sampling rate of the real-time control is relatively
high set to 10 kHz. And the time derivative of the measured output
displacement $x(t)$ is obtained by the discrete time differentiation combined
with a low-pass filter (LPF). The cutoff frequency is 1 kHz. Recall that
$\dot{x}(t)$ is required for constructing the sliding surface $\sigma$, cf.
(26).

Figure 4: Voice-coil based linear actuator with one DOF in $x$ coordinates.

The modeled system dynamics is given by

$\displaystyle m\ddot{x}(t)$ $\displaystyle=$ $\displaystyle v(t)+G+d(t),$
(30) $\displaystyle\mu\dot{v}(t)$ $\displaystyle=$ $\displaystyle Ku(t)-v(t).$
(31)

The overall mass of the actuator mover is $m=0.538$ kg. The input gaining
factor, which takes into account both the electro-magneto-mechanical coupling
constant and the resistance of the anchor coils, is $K=3.28$ NV-1. The
constant gravity term is $G=5.27$ N, while the cumulative unknown bounded
disturbance is $0<\bigl{|}d(t)\bigr{|}<D$. Estimated from various open- and
closed-loop experiments, the upper bound of disturbances is set to $D=1$ N.
The time constant of the electro-magnetic circuit of the actuator $\mu=0.0012$
sec is taken from the manufacturer technical data sheet. The available signals
are the input voltage $u(t)$ (given in V) and the output relative displacement
$x(t)$ (given in m).

###  III-B Noisy system states

The measured output signal is subject to the noise. This is due to a
contactless sensing of the relative displacement by means of the inductive
sensor, while its nominal repeatability is $\pm 12$ micrometer. When the
control signal is off, i.e. $u(t)=0$, the background noise of the output is
still visible. The distribution of an exemplary measured output $x(t)$ at zero
input is shown in Fig. 5 (a). While zero mean-value can be assumed, a nearly
normal distribution can be recognized, and the evaluated standard deviation is
$4.58\times 10^{-5}$ m.

Figure 5: Noisy system states: distribution of the measured output $x(t)$ at
zero input $u(t)=0$ (a), fragment of the measured output $x(t)$ at applied
constant input $u(t)=\mathrm{const}$ (b), fragment of the low-pass filtered
$dx/dt$ obtained by the discrete time differentiation of the measured $x(t)$
(c).

To get an idea about additional process noise, i.e. disturbances, one can
inspect a fragment of the measured output $x(t)$ at the applied constant input
$u(t)=\mathrm{const}$, see Fig. 5 (b). One can recognize a harmonic pattern in
the output relative displacement, caused by a electro-magnetic cogging of the
stator-armature structure. Worth noting is that the disturbance $d(t)$
includes also the mechanical friction effects, see e.g. [21], and the
position-dependent variations of the gaining factor $K$, see e.g. [22]. The
velocity state $\dot{x}(t)$, required for computing the sliding variable
$\sigma(t)$, is obtained by the discrete-time derivative of the measured
output. Subsequently, the second-order LPF is applied for obtaining a smoothed
velocity $w(t)$. The LPF dynamics is given by

$\ddot{w}(t)+2\omega_{c}\,\dot{w}(t)+\omega_{c}^{2}\,w(t)=\omega_{c}^{2}\,dx/dt,$
(32)

where $\omega_{c}=2\pi f_{c}$ is the LPF cut-off frequency. The low-pass
filtered velocity state is exemplary shown in Fig. 5 (c), for a controlled
motion in the time between 0.02 sec and 0.4 sec. Two signal patterns are shown
opposite to each other for the LPF with $f_{c}=\\{100,\,1000\\}$ Hz. The
velocity signal filtered with $f_{c}=100$ Hz is depicted for visualizing both,
a continuous motion affected by the cogging force disturbance and a noisy
velocity in the idle state (i.e. for $t>0.4$ sec). On the contrary, the
velocity signal filtered with $f_{c}=1000$ Hz is used in the control
experiments when calculating the sliding variable $\sigma(t)$. Here it is
worth recalling that the settling time $t_{s}$ of the step response of the
second-order dynamic system (32), i.e.
$|\dot{x}(t)-w(t)|\cdot|\dot{x}(t)|^{-1}<0.01$ for $t>t_{s}$, can be
approximately calculated by

$t_{s}\approx 4.6\,\omega_{c}^{-1}.$

One can recognize that the settling time of the LPF with 100 Hz cut-off
frequency $t_{s,100}=0.0073$ sec is about 6 times larger than the actuator
time constant $\mu$. On the contrary, the settling time of the LPF with 1000
Hz cut-off frequency $t_{s,1000}=0.00073$ sec lies clearly below the actuator
time constant $\mu$ and appears, therefore, as sufficiently low for use of
$w(t)\approx dx/dt$ when calculating the sliding variable $\sigma(t)$.

##  IV Experimental results

The experimental results demonstrated below are obtained by using the control
value

$u(t)=u_{sm}\bigl{(}x,\dot{x}\bigr{)}+u_{gr},$ (33)

where $u_{gr}=1.61$ V is compensating for the constant gravity term, cf. (30),
and the sliding-mode control $u_{sm}=\\{\bar{u},\,\tilde{u}\\}$ is either the
terminal (25) or the energy-saving sub-optimal one (14), both using the same
second-order sliding surface (26). The magnitude of the switching control is
assigned to $U=0.8$ V, that is about 2.5 times higher than the corresponding
upper bound of disturbances, cf. $D$ in section III-A and [13]. The design
parameter of the terminal sliding surface is assigned as $\alpha=1.2$, thus
preserving the terminal algorithm from the twisting mode with respect to the
perturbations upper bound $D$. The threshold parameters of the energy-saving
sub-optimal control are assigned to be $\beta_{1}=0.85$ and $\beta_{2}=0.1$
respectively, cf. [13].

The control experiments were conducted for the reference step $x_{r}=0.015$ m,
starting from the same zero relative position. Note that through a
transformation of coordinates $x^{*}=x-x_{r}$ the step reference control task
becomes equivalent to stabilization in the origin $(x^{*},\dot{x})=(0,0)$.
Below, we will however continue to use $x$ instead of $x^{*}$, this for the
sake of simplicity. For evaluation of the control energy consumption, the
following integral metric is used

$E=\int|u(t)|dt.$ (34)

Recall that for a continuously switching SM control $|u|=U$, while for the
energy-saving sub-optimal SM control $|u|=\\{U,\,0\\}$. Therefore, the metric
(34) reflects the overall time when the control signal is on and, hence, the
energy consumption during the control operation. The measured output response
of both SM controllers under benchmarking is shown Fig. 6 (a).

Figure 6: Measured output response of the SM control $\bar{u}$ and $\tilde{u}$
in (a), and the corresponding integral $E$ of the control absolute control
signal in (b).

Both reach the reference equilibrium in finite time and show the same steady-
state accuracy, while the reaching time of the terminal SM control is slightly
superior comparing to the energy-saving sub-optimal SM control, cf. [13]. This
is not surprising since both controllers maintain the system on the same
sliding manifold (26), while the terminal SM controller is always on. The
benefit of the energy-saving sub-optimal SM control becomes visible when
inspecting the evaluated metric (34) depicted in Fig. 6 (b). Not only during
the reaching phase (i.e. on the time interval bounded approximately by $t<0.4$
sec) but equally after the convergence, the difference in energy consumption
between both SM controllers continue to grow.

Another interesting fact is also related to the sliding-mode chattering, which
can appear due to an additional (parasitic) actuator dynamics, cf. section
II-C. Evaluating the chattering frequency and amplitude by means of (28), (29)
results in

$\tilde{\omega}=520.2\hbox{ rad/sec},\quad A_{x}\approx 3.1\times
10^{-6}\hbox{ m}.$

Note that $A_{x}$ can easily been estimated from $A_{\sigma}$, cf. (29), in
vicinity to origin. Given the sensor resolution and level of the measurement
noise, cf. section III, the above evaluated chattering parameters do not allow
to detect a harmonic oscillations pattern within the measured output $x(t)$.
Similar conclusions can be drawn and observed experimentally for the second-
order terminal SM control in regard to the used actuator with additional
(parasitic) dynamics.

##  V Conclusions

The energy-saving sub-optimal SM control, recently proposed in [13], was for
the first time evaluated experimentally and benchmarked with a conventional
discontinuous SM control, both operating on the same terminal sliding surface.
The latter drives both SM controllers in the second-order sliding mode, while
it appears (in its proper sense) first when reaching the origin in a finite
time. The real experimental system with the measurement and process noise, and
without sensing of the time derivative of the output and of the sliding
variable, is considered. The system is modeled in a simplified way by a
perturbed double integrator. The latter is with the known upper bound of the
matched perturbations and an additional (parasitic) first-order actuator
dynamics. The same switching control magnitude and the designed terminal
sliding surface allow for a fair comparison of both, the conventional
discontinuous and energy-saving sub-optimal SM controllers. The integral of
the absolute control value is evaluated as an adequate energy consumption
metric which is proportional to the overall time when the control with a
constant magnitude was on. The provided experimental case study confirms the
applicability and efficiency of the energy-saving sub-optimal SM control [13].

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