#  Marvel analysis of the measured high-resolution rovibrational spectra of
H232S

Katy L. Chubba111email<EMAIL_ADDRESS>Olga Naumenkob Stefan Keelyc,
Sebestiano Bartolottoc, Skye Macdonaldc, Mahmoud Mukhtarc, Andrey Grachovc,
Joe Whitec, Eden Colemanc Anwen Liud, Alexander Z. Fazlievb, Elena R.
Polovtsevab, Veli-Matti Hornemane, Alain Camparguef Tibor Furtenbacherg,
Attila G. Császárg Sergei N. Yurchenkoa, Jonathan Tennysona222email:
<EMAIL_ADDRESS>a Department of Physics and Astronomy, University College
London, London, WC1E 6BT, UK  b V. E. Zuev Institute of Atmospheric Optics,
Siberian Branch Russian Academy of Sciences, Tomsk, Russia  c Highams Park
School, Handsworth Avenue, Highams Park, London, E4 9PJ, UK  d Hefei National
Laboratory for Physical Sciences at Microscale, University of Science and
Technology of China, Hefei 230026, China  e Department of Physics, University
of Oulu, P. O. Box 3000, FIN-90014 University of Oulu, Finland  f Université
Grenoble Alpes, CNRS, LIPhy, F-38000 Grenoble, France  g Institute of
Chemistry, Eötvös Loránd University and MTA-ELTE Complex Chemical Systems
Research Group, H-1518 Budapest 112, P.O. Box 32, Hungary

###### Abstract

44 325 measured and assigned transitions of H232S, the parent isotopologue of
the hydrogen sulfide molecule, are collated from 33 publications into a single
database and reviewed critically. Based on this information, rotation-
vibration energy levels are determined for the ground electronic state using
the Measured Active Rotational-Vibrational Energy Levels (Marvel) technique.
The ortho and para principal components of the measured spectroscopic network
of H232S are considered separately. The verified set of 25 293 ortho- and 18
778 para- H232S transitions determine 3969 ortho and 3467 para energy levels.
The Marvel results are compared with alternative data compilations, including
a theoretical variational linelist.

##  1 Introduction

Hydrogen sulfide, H2S, is a volcanic gas present on earth and other
geologically active planets and moons, such as Io [1], Venus [2], and,
theoretically, hot super-earth "lava" exoplanets [3]. It has recently been
detected above the clouds in the atmosphere of Uranus [4]. On earth, microbial
respiration of seawater sulfate creates a hydrogen sulfide-rich ecosystem [5],
suggesting that H2S is a component for a potential microbial life sustaining
atmosphere on Venus [2], and a potential biomarker for life on exoplanets [6].
H2S influences many physiological processes [7], it is of importance in the
treatment of respiratory diseases [8], and it is used as a measure of the
quality of air near oil refineries [9]. The parent isotopologue, H232S, was
first detected in interstellar space in 1972 [10]. Many of the scientific and
engineering applications mentioned require the detailed knowledge of the
rovibrational energy levels of H232S.

As part of this work, we present the largest compilation of published
experimental rovibrational transition data for H232S. The experimental
database of H232S transitions has been formatted and analysed using the Marvel
(Measured Active Rotational-Vibrational Energy Levels) spectroscopic network
(SN) software [11, 12, 13]. This study builds on the 2012 W@DIS information
system for the hydrogen sulfide molecule [14].

As to the structure of this paper, the next section provides the theory
underlying the present study. Section 3 presents and discusses the
experimental sources used, with results given in Section 4. Section 5
discusses these results and compares the empirical rovibrational energies
presented in this work with corresponding levels previously determined both by
experiment and theory. Finally, Section 6 provides our conclusions. All
transition data and energy levels resulting from this work are included as
supplementary data files.

##  2 Theory

###  2.1 Marvel

The Marvel procedure [11, 12, 13] is based on the theory of spectroscopic
networks [15, 16], with the energy levels represented as nodes and the
transitions between them as edges. The related Marvel code can be used to
critically evaluate and validate experimentally-determined transition
wavenumbers and uncertainties collected from the literature, inverting the
wavenumber information to obtain accurate empirical energy levels with an
associated uncertainty. Marvel has been successfully used to evaluate the
energy levels for molecules such as 12C2 [17], 48Ti16O [18], water vapour [19,
20, 21, 22, 23], H${}_{3}^{+}$ [24], H2D+ and D2H+ [25], 12C2H2 [26], 14NH3
[27, 28], and 12C2H216O [29]. Marvel requires each measured transition to have
an associated uncertainty and for each energy level considered to possess a
unique set of quantum numbers.

###  2.2 Quantum number labelling

The seven quantum numbers that were used for labelling the upper and lower
rovibrational states of H232S are the same as those used in a previous Marvel
investigation to classify water [19]. These quantum numbers are summarised in
Table 1. Normal-mode labelling is used for the vibrations, where
$\nu_{1},\nu_{2}$ and $\nu_{3}$ stand for the symmetric stretch, bend, and
antisymmetric stretch vibrations, respectively. Standard asymmetric-top
quantum numbers are used for the rotations, where $J,K_{a}$, and $K_{c}$ are
the three quantum numbers associated with rotational angular momentum, ${\bf
J}$, and the two projections along the $A$ and $C$ axes. We also provide, as
part of the label, the nuclear spin state (ortho or para), which is deduced by
whether ($v_{3}+K_{a}+K_{c}$) is odd (ortho) or even (para) [26, 30].
Hyperfine coupling associated with nuclear spins has been neglected.

When local-mode notation was used by an experimental source considered in this
work, the quantum numbers were transformed to the normal-mode notation
$(v_{1}v_{2}v_{3})$ to describe a vibrational state. Note that some of the
data sources considered in this study use $J$ and $\tau=K_{a}-K_{c}$ instead
of $J,K_{a}$, and $K_{c}$ [31], where $\tau$ runs from $-J,-J+1,...,J-1,+J$,
with energy increasing from $\tau=-J$. The $+/-$ parity of an asymmetric top
molecule such as H232S is defined by $(-1)^{K_{c}}$ [19]. H232S belongs to the
$C_{2\rm{v}}$(M) MS group [32], which contains irreducible representations A1,
A2, B1, and B2, as given in Table 2.

Table 1: Quantum numbers used to label the upper and lower energy states of H232S. Label | Description
---|---
$v_{1}$ | S-H symmetric stretch ($\sim$ 2614.4cm-1)
$v_{2}$ | Symmetric bending mode ($\sim$ 1182.6cm-1)
$v_{3}$ | S-H antisymmetric stretch ($\sim$ 2628.5cm-1)
$J$ | Rotational angular momentum
$K_{a},K_{c}$ | Projections of rotational angular momentum
ortho/para | Nuclear spin state (see text)
Table 2: Symmetry of the rovibrational states of H232S Symmetry | A1 | A2 | B1 | B2
---|---|---|---|---
Parity | $+$ | $-$ | $-$ | $+$
Nuclear spin state | para | para | ortho | ortho

###  2.3 Selection rules

The rigorous selection rules governing rotation-vibration transitions for a
molecule of the $C_{2\rm{v}}$(M) MS group are given by:

(1) $\displaystyle\Delta J=0,\pm 1,$ (2) $\displaystyle
J^{\prime}+J^{\prime\prime}\neq 0,$ (3) $\displaystyle-{\leftrightarrow}+$

The ortho states of H232S have the nuclear spin statistical weight $g_{\rm
ns}=3$, while for the para states $g_{\rm ns}=1$, thus, ortho transitions have
three times the intensity of para transitions. This is sometimes referred to
as intensity alternation. It is assumed that ortho and para states do not
interconvert. Such transitions are very weakly allowed [33] but have yet to be
observed for H232S.

##  3 Experimental sources

A large number of experimentally determined transition wavenumbers can be
found in the literature for the main isotopologue of hydrogen sulfide, H232S.
We have attempted to conduct a rigorous and comprehensive search for all
useable spectroscopic data. Fortunately, much of the data up to 2012 was
previously collated as part of the 2012 W@DIS information system for hydrogen
sulfide [14], in which some of the authors of this paper were involved. These
data were converted to Marvel format for this work and analysed alongside data
from newly collected sources. This requires the transition wavenumber (in
cm-1) and the associated uncertainty, along with quantum number assignments
for both the upper and lower energy states, and a unique reference label for
each transition. This reference indicates the data source the transition
originates from. The data source tag is based on the notation employed by an
IUPAC Task Group on water spectroscopy [19]. An extract of the input file in
the required format is given in Table 3; the full file can be found in the
supplementary data for this work.

33 sources of experimental data were used in the final data set. The data from
more recent papers are generally provided in digital format, but some of the
older papers had to be processed through digitalisation software, or even
manually entered in the worst cases. After digitalisation the data were
converted to Marvel format, as described above.

Table 3: Extract from the Marvel input file for the ortho transitions for H232S. The full file is supplied as part of the supplementary information to this paper. All energy term values and uncertainties are in units of cm-1. The meaning of the upper and lower state assignments can be found in Table 1.  Transition | Uncertainty | Upper state assignment | Lower state assignment | Reference
---|---|---|---|---
33.12631 | 0.00015 | 0 0 0 5 2 3 ortho | 0 0 0 5 1 4 ortho | 94YaKl_1
33.12631 | 0.00018 | 0 0 0 3 0 3 ortho | 0 0 0 2 1 2 ortho | 94YaKl_2
34.00529 | 0.00010 | 0 0 0 8 7 2 ortho | 0 0 0 8 6 3 ortho | 94YaKl_5
34.15779 | 0.00010 | 0 0 0 11 9 2 ortho | 0 0 0 11 8 3 ortho | 94YaKl_7
34.21980 | 0.00010 | 0 0 0 6 4 3 ortho | 0 0 0 6 3 4 ortho | 94YaKl_8
34.24062 | 0.00012 | 0 0 0 4 2 3 ortho | 0 0 0 4 1 4 ortho | 94YaKl_9
35.73512 | 0.00010 | 0 0 0 8 6 3 ortho | 0 0 0 8 5 4 ortho | 94YaKl_13

Table LABEL:t:sources gives a summary of all the data sources used in this
work, along with the energy range, number of transitions (assigned (A) and
verified (V)), and comments, which are detailed in Section 3.1. Table
LABEL:t:notused lists those data sources which were considered but not used,
with comments on the reasons for their exclusion from the analysis. The IUPAC
reference tag given in these tables matches those used in the unique labels in
the Marvel input files, given in the supplementary data and illustrated in
Table 3.

As transitions have never been observed between ortho and para states of
H232S, they form two separate principal components (PCs) of the experimental
spectroscopic network. All input and output files supplied in the
supplementary data to this work are split into either ortho or para.

Table 4: Data sources used in this study with frequency range, numbers of transitions (A/V for assigned/verified), and comments, which are detailed in Section 3.1. As far as we are aware all experiments were conducted at room temperature.  Tag | Ref. |  Range (cm-1) | A/V | Comments |
---|---|---|---|---|---
72HeCoLu | [34] | 1.17–25.55 | 37/35 | (4a) |
95BeYaWiPo | [35] | 4.39–85.41 | 112/84 | (4b) |
68CuKeGa | [36] | 5.63–14.15 | 6/6 |  |
71Huiszoon | [37] | 5.63–7.23 | 2/2 |  |
14CaPu | [38] | 7.23–53.17 | 70/70 |  |
85BuFeMeSh | [39] | 10.02–20.9 | 6/6 |  |
94YaKl | [40] | 33.13–259.76 | 366/366 |  |
13CaPu | [41] | 33.97–37.45 | 4/4 |  |
13AzYuTeMa | [42] | 45.25–359.79 | 1158/1139 | (4c) |
83FlCaJo | [43] | 50.77–307.51 | 426/387 | (4d) |
18UlBeGr | [44] | 729.78–1735.41 | 2267/2267 |  |
82LaEdGiBo | [45] | 1003.46–1495.28 | 397/396 | (4e) |
83Strow | [46] | 1082.03–1257.07 | 123/123 |  |
96UlMaKoAl | [47] | 1178.05–1359.78 | 41/41 |  |
98BrCrCrNa | [48] | 2141.30–4249.85 | 7473/7473 |  |
18Horneman | [49] | 2180.35–4220.46 | 4460/4460 |  |
84LeFlCaJo | [50] | 2180.36–2945.81 | 2113/2111 | (4f) |
81GiEd | [51] | 2192.48–2823.11 | 715/704 | (4g) |
96UlOnKoAl | [52] | 3614.70–3887.66 | 106/106 |  |
05UlLiBeGr | [53] | 4000.59–6653.79 | 2347/2347 |  |
97BrCrCrNa | [54] | 4500.88–5595.01 | 5221/5219 | (4h) |
18Liu | [55] | 4514.79–5555.58 | 3337/3335 | (4i) |
04BrNaPoSi_c | [56] | 5688.27–6676.71 | 3178/3178 |  |
04BrNaPoSi_a | [57] | 7169.19–7898.97 | 2878/2876 | (4j) |
04UlLiBeGr_b | [58] | 7226.81–7994.09 | 1855/1855 |  |
04UlLiBeGr_a | [59] | 8405.91–8905.28 | 589/589 |  |
04BrNaPoSi_b | [60] | 8412.73–8906.11 | 1179/1175 | (4k) |
03DiNaHuZh | [61] | 9541.01–10000.71 | 1736/1728 | (4l) |
01NaCa_a | [62] | 10787.33–11297.99 | 1105/1097 | (4m) |
94GrRaStDe | [63] | 11948.91–12246.28 | 227/145 | (4n) |
97VaBiCaFl | [64] | 12324.55–12670.68 | 399/387 | (4o) |
99CaFl | [65] | 13060.51–13357.14 | 219/206 | (4p) |
01NaCa_b | [66] | 16186.25–16436.57 | 173/154 | (4q) |
Total |  | 1.17–16436.57 | 44 325/44 071 |  |
Table 5: Data sources considered but not used in this work.  Tag | Ref. | Comments
---|---|---
02CoRoTy | [67] | Data taken from other sources
94WaKuSu | [68] | Data taken from other sources
96SuMeKr | [69] | Data taken from other sources
97SuMeKr | [70] | Data taken from other sources
97Sumpf | [71] | Data taken from other sources
98PiPoCoDe | [72] | Data from http://spec.jpl.nasa.gov. Data does not appear to be experimental.
73HeDeKi | [73] |  Data taken from other sources, [36, 34, 37]
02KiSuKrTi | [74] |  Data appears to be taken from the 1996 edition of HITRAN [75]
06Polovtseva | N/A |  Thesis, no citation. Referenced in [14]
87LeFlCaAr | [76] | Energy levels
13Azzam | [77] |  All results are given in 13AzYuTeMa [42]
94KoJe | [78] | Energy levels
95FlGrRa | [79] | Energy levels
85LaEdGiBo | [80] |  Calculated values, compared to experimental values from 83FlCaJo [43]
94ByNaSmSi | [81] | Energy Levels
01TyTaSc | [82] | Vibrational energy Levels
69MiLeHa | [83] |  Could not be validated with the other sources to a reasonable accuracy (see Section 3.2)
56AlPl | [31] |  Could not be validated with the other sources to a reasonable accuracy (see Section 3.2)
69SnEd | [84] |  Could not be validated with the other sources to a reasonable accuracy (see Section 3.2)

###  3.1 Comments on the experimental sources of Table LABEL:t:sources

(4a) 72HeCoLu [34] contains 2 lines which have been cited as taken from other
experimental sources, for which the original data are already in our dataset.
These duplicates were removed.
(4b) 95BeYaWiPo [35] contains 28 lines which have been cited as from other
experimental sources, for which the original data are already in our dataset.
These duplicates were removed.
(4c) 13AzYuTeMa [42] gives data for rotational lines recorded at room
temperature. The equipment used provides spectra with very high sensitivity
and this enabled transitions between highly rotationally excited states to be
recorded, even at room temperature. These high rotational excitations may not
be so accurately assigned as those of lower energy, as there are no data from
other sources in the same region to confirm these measurements. The
supplementary data from the original paper does not contain the original
experimental transitions; this was provided by the corresponding author. There
were 19 lines which could not be validated against other data and so were
removed from our dataset (see Section 3.2).
(4d) 83FlCaJo [43] contains 39 lines which have been cited as from other
sources which are already present in our dataset. The duplicates were removed.
(4e) 82LaEdGiBo [45] contains one line which could not be validated and so has
been removed from our dataset.
(4f) 84LeFlCaJo [50] contains 2 lines which could not be validated against
other more recent data and were removed from our dataset. All data from this
source were recalibrated; the calibration factor employed was 0.999 999 746
below 2450 cm-1 and 1.000 000 0714 above 2450 cm-1.
(4g) 81GiEd [51] contains 11 lines which could not be validated against more
recent data and thus were removed from our dataset.
(4h) 97BrCrCrNa [54] contains 2 lines with no corresponding energy in the AYT2
linelist (see Section 5) and so were removed from our dataset.
(4i) 18Liu [55] contains 2 lines with no corresponding energy in the AYT2
linelist and so were removed from our dataset.
(4j) 04BrNaPoSi_a [57] contains 2 lines with no corresponding energy in the
AYT2 linelist and so were removed from our dataset.
(4k) 04BrNaPoSi_b [60] contains 4 lines which could not be validated and were
removed from our dataset.
(4l) 03DiNaHuZh [61] contains one blended line which has been removed from our
dataset. A further 7 lines were found to include a level with no corresponding
energy in the AYT2 linelist and so were removed from our dataset.
(4m) 01NaCa_a [62] contains 8 transitions which were found to have no
corresponding levels in the AYT2 linelist and so were removed from our
dataset.
(4n) 94GrRaStDe [63] tentatively assigned the vibrational band in their data
as ($v_{1},v_{2},v_{3}$)=(2,2,2); however, in 95FlGrRaSt [79], a paper
published later with the same authors, the revised assignment is
($v_{1},v_{2},v_{3}$)=(3,0,2). We adopt the latter vibrational label in our
dataset. No other data sources probe this vibrational band. The frequency is
only given to 3 decimal places, so the uncertainty was altered to match. The
original dataset has 82 transitions either labelled with an asterisk to
indicate lines from the H234S isotopologue, or labelled with a dagger to
indicate lines which have been perturbed due to those from H234S [63]. We have
commented these out by adding a minus sign to the wavenumber and added the
label $\\_pt$ to the end.
(4o) 97VaBiCaFl [64] contains 4 lines which could not be validated and so were
removed from our dataset. A further 8 lines were found to contain a level with
no corresponding energy in the AYT2 linelist and so were removed from our
dataset.
(4p) The assignment for 99CaFl_185_na_ct from 99CaFl [65] leads to different
spin states (ortho/para) for the upper and lower state, which is forbidden. We
have removed this transition from our dataset. 5 other lines could not be
validated and so were removed from our dataset. 7 lines were found to have no
variational counterparts in the AYT2 linelist and so were removed.
(4q) 01NaCa_b [66] contains one line which could not be validated and so was
removed from our dataset. A further 18 lines were found to contain levels with
no corresponding energies in the AYT2 linelist and so were removed from our
dataset.

###  3.2 General comments

All transitions which were considered but not processed in the final dataset
have a minus sign in front of the transition wavenumber (indicating that
Marvel will ignore it) in the input files provided as supplementary material
and are labelled with a comment at the end of the reference (see the comments
of Section 3.1). We used a cut-off of 0.035 cm-1 as the largest acceptable
uncertainty and removed any transitions with an uncertainty greater than this
this, with a note in Section 3.1 to indicate how many transitions from a
particular data source could not be validated to within this accuracy.

The publications 97BrCrCrNa [54], 04BrNaPoSi_c [56], 04BrNaPoSi_a [57], and
04BrNaPoSi_b [60] include just a short description of the assignment and
modelling of the Fourier Rransform (FT) spectra between 4500 and 8900cm-1.
Detailed information on the theoretical treatment of these spectra will be
published separately in due course. Some of these transitions, between
4400–8000 cm-1, were previously reported in the HITRAN-2012 [85] and
HITRAN-2016 [86] databases, while the second decade region, 8400–8900cm-1, is
presented for the first time. It is also worth mentioning that the Marvel
dataset includes a considerable number of transitions which include new,
unpublished energy levels. As only the upper energy levels of the transitions
were reported in 05UlLiBeGr [53], 04UlLiBeGr_b [58], and 04UlLiBeGr_a [59],
the corresponding transition wavenumbers were recovered from A.-W. Liu (18Liu
[55]) using the energy levels obtained from 97BrCrCrNa [54], 04BrNaPoSi_c
[56], 04BrNaPoSi_a [57], and 04BrNaPoSi_b [60]. For this reason, the
transitions in the Marvel dataset which are referred to as from 05UlLiBeGr
[53], 04UlLiBeGr_b [58], and 04UlLiBeGr_a [59] contain a larger number of
assignments than reported in the original publications. The data of the first
hexad region between 4500–5600cm-1 were totally assigned based on the energy
levels reported in 97BrCrCrNa [54]. The highly accurate (an accuracy of
0.0005cm-1) set of H232S transitions recorded using a FT spectrometer between
2200–4250cm-1 was provided by V.-M. Horneman (18Horneman [49]) and assigned
using the energy levels reported in 98BrCrCrNa [48].

###  3.3 Sources from the HITRAN database

H232S has been included in the HITRAN database [87, 88, 89, 90, 85, 86] since
1991 [87]. The following are sources for the line positions and energy levels
of the H232S data in the HITRAN database, up to the 2016 release: 96UlMaKoAl
[47], 83FlCaJo [43], 98BrCrCrNa [48], 13AzYuTeMa [42], 94YaKl [40], 95BeYaWiPo
[35], 05UlLiBeGr [53], 04UlLiBeGr [58], 03DiNaHuZh [61], 01NaCa_b [66],
84LeFlCaJo [50], 82LaEdGiBo [45], 85LaEdGiBo [80], 94ByNaSmSi [81]. The
variational ExoMol AYT2 linelist [91] was used to assign transitions in the
$\nu_{2}$ excited vibrational state in the 2012 release [85].

##  4 Results

The Marvel website (http://kkrk.chem.elte.hu/marvelonline) has a version of
Marvel which can be run online. The variable NQN (number of quantum numbers)
is 7 in the case of hydrogen sulfide, as illustrated in Table 3 which shows an
extract of the ortho input file to Marvel.

Marvel automatically assigns the lowest energy state in a particular component
of the spectroscopic network to 0. The ground rovibrational state is included
in the para set of energy levels, however there needs to be a “magic number”,
corresponding to the energy of the lowest ortho state, which is added to all
the Marvel ortho-symmetry energies. Here, this was taken as the ground
vibrational state $(v_{1}v_{2}v_{3})=(000)$ with the lowest rotational energy
(see Section 2.2), $J=1,K_{a}=0,K_{c}=1$, of 94KoJe [78], who determined the
value of 13.74631 cm-1. The output for the ortho energies in the supplementary
data, and the extract in Table 6, all have this magic number added for the
principal spectroscopic network. There are a small number (4 para) of energy
levels which are not joined to either of the principal components. If more
experimental transitions became available in the future it would be possible
to link these to the principal network.

Table 6: Extract from the Marvel output file for the ortho energy levels of H232S. The full file is supplied as part of the supplementary information to this paper. All energies and uncertainties are in units of cm-1. NumTrans gives the number of transitions which are linked to that particular energy level. Assignments are as given by Table 1. Assignment | Energy | Uncertainty | NumTrans | Sym
---|---|---|---|---
0 | 0 | 0 | 1 | 0 | 1 | ortho | 13.746310 | 0.000001 | 177 | B1
0 | 0 | 0 | 1 | 1 | 0 | ortho | 19.375630 | 0.000001 | 153 | B2
0 | 0 | 0 | 2 | 1 | 2 | ortho | 38.297765 | 0.000002 | 271 | B2
0 | 0 | 0 | 2 | 2 | 1 | ortho | 55.161605 | 0.000002 | 265 | B1
0 | 0 | 0 | 3 | 0 | 3 | ortho | 71.424233 | 0.000002 | 282 | B1
0 | 0 | 0 | 3 | 1 | 2 | ortho | 95.056325 | 0.000001 | 344 | B2
0 | 0 | 0 | 3 | 2 | 1 | ortho | 107.368228 | 0.000001 | 331 | B1
0 | 0 | 0 | 4 | 1 | 4 | ortho | 114.177613 | 0.000001 | 282 | B2
0 | 0 | 0 | 3 | 3 | 0 | ortho | 117.392015 | 0.000001 | 265 | B2
0 | 0 | 0 | 4 | 2 | 3 | ortho | 148.418340 | 0.000001 | 379 | B1

We collated and considered a total of 44 325 transitions from 33 experimental
sources (25 474 ortho and 18 851 para). Of those 254 were found to be
inconsistent (could not be validated to within 0.035 cm-1) with others and so
removed from the final data set, leaving a total of 44 071 transitions used as
input into Marvel (25 293 ortho and 18 778 para).

Figure 1 gives a visual representation of NumTrans which link the states of
the Marvel ortho-H232S network, against corresponding upper state energies.
Those upper states which are dark blue in colour, linked by only 1 transition,
should be considered less reliable than those in red, which are supported by
hundreds of different transitions, up to 404. The values of NumTrans for each
level are given in the energy level files in the supplementary data.

Figure 1: The lower-state energies of the experimental ortho-H232S transitions
used in this work, against corresponding upper-state energies. The vertical
bars along the horizontal-axis show the lower-state energies, while the
horizontal bars along the vertical-axis give the upper-state energies. Each
circle represents a particular transition, with the size proportional to the
log of NumTrans, the number of transitions supporting the upper state. This
value ranges from 1 (dark blue) to 404 (red).

##  5 Comparison to other derived energy levels

We used a number of sources to compare energy level values against those
determined in this work. 94KoJe [78] contains a list of energy levels, which
are taken or calculated from other sources [43, 80, 50, 84, 31, 76]; see Table
LABEL:t:96KoJe for a breakdown by vibrational band. AYT2 [91] is a theoretical
variational linelist for H232S calculated as part of the ExoMol project [92,
93], a database of theoretical linelists for molecules of astrophysical
importance. AYT2 is appropriate up to temperatures of around 2000 K and
designed for use in characterising the atmospheres of cool stars and
exoplanets. The states file, available from www.exomol.com, gives the
calculated energy levels, which can be compared against our empirical energy
levels. For more information on the format of the ExoMol data files, please
refer to [93]. 95FlGrRaSt [79] contains rovibrational energy levels from the
($v_{1},v_{2},v_{3}$)=(2,0,3) and (3,0,2) vibrational bands. The authors
outline an experimental procedure used to deduce these levels but they do not
provide the experimental transition data so we could not include the data from
this source in our dataset. 87LeFlCaAr [76] has a table of rovibrational
energy levels for the ($v_{1},v_{2},v_{3}$)=(2,1,0), (1,1,1) and (0,1,2)
vibrational bands. 96UlMaKoAl [47] contains rovibrational energy levels from
the ($v_{1},v_{2},v_{3}$)=(0,1,0) vibrational band.

Table 7: Original sources of energy level data from 94KoJe [78]. Tag | Ref. | Vibrational bands | Comments
---|---|---|---
83FlCaJo | [43] | (0,0,0) | Experimental source used in this work
85LaEdGiBo | [80] | (0,1,0) |  Calculated values, compared to experimental values from 83FlCaJo [43]
84LeFlCaJo | [50] | (0,2,0) (1,0,0) (0,0,1) | Experimental source used in this work
69SnEd | [84] | (1,1,0) (0,1,1) | Experimental source used in this work
56AlPl | [31] |  | (0,2,1) (2,0,0) (1,0,1)
---
(3,0,0) (2,0,1) (1,0,2)
(0,0,3) (2,1,1) (3,0,1)
(2,0,2) (1,0,3) (3,1,1)
Data source not considered reliable. All $J=0$ levels only.
87LeFlCaAr | [76] | (2,1,0) (1,1,1) |  |  94KoJe [78] only contains levels from this source up to $J=5$.
---
See text and Figure 2 for a full comparison.

We compared the pure rotational levels determined in this work both against
those given in 94KoJe [78] and the calculated values from the AYT2 linelist
[91], up to $J$=5. These data are given in Table LABEL:t:compare. The pure
rotational levels are closer to those from the experimentally determined
sources of 94KoJe than to those of variationally calculated AYT2, as would be
expected.

Table 8: Comparison of pure rotational levels from this work with those of 94KoJe [78] (see Table LABEL:t:96KoJe) and AYT2 [91] up to $J$=5. NumTrans gives the number of transitions linking a particular state. All energies and uncertainties are in cm-1. The uncertainty refers to this work.  $J$ $K_{a}$ $K_{c}$ | State | Sym | NumTrans | Uncertainty | This work | 94KoJe | Difference | AYT2 | Difference
---|---|---|---|---|---|---|---|---|---
0 0 0 | para |  A1 | 58 | 0.000098 | 0 | 0 | 0 | 0 | 0
1 0 1 | ortho |  B1 | 170 | 0.000001 | 13.746310 | 13.74631 | 0.00000 | 13.746340 | 0.000030
1 1 1 | para |  A2 | 157 | 0.000002 | 15.090119 | 15.09011 |  $-$0.00001 | 15.090151 | 0.000032
1 1 0 | ortho |  B2 | 147 | 0.000001 | 19.375630 | 19.37563 | 0.00000 | 19.375563 |  $-$0.000067
2 0 2 | para |  A1 | 223 | 0.000001 | 38.016095 | 38.01607 |  $-$0.00003 | 38.016264 | 0.000169
2 1 2 | ortho |  B2 | 258 | 0.000002 | 38.297765 | 38.29775 |  $-$0.00001 | 38.297961 | 0.000196
2 1 1 | para |  A2 | 211 | 0.000001 | 51.140188 | 51.14016 |  $-$0.00003 | 51.140060 |  $-$0.000128
2 2 1 | ortho |  B1 | 253 | 0.000002 | 55.161605 | 55.16158 |  $-$0.00003 | 55.161460 |  $-$0.000145
2 2 0 | para |  A1 | 181 | 0.000001 | 58.368870 | 58.36884 |  $-$0.00003 | 58.368621 |  $-$0.000249
3 0 3 | ortho |  B1 | 265 | 0.000002 | 71.424221 | 71.42426 | 0.00004 | 71.424760 | 0.000539
3 1 3 | para |  A2 | 213 | 0.000001 | 71.465192 | 71.46515 |  $-$0.00004 | 71.465653 | 0.000461
3 1 2 | ortho |  B2 | 329 | 0.000001 | 95.056313 | 95.05630 |  $-$0.00001 | 95.056267 |  $-$0.000046
3 2 2 | para |  A1 | 272 | 0.000001 | 96.392529 | 96.39246 |  $-$0.00007 | 96.392428 |  $-$0.000101
3 2 1 | ortho |  B1 | 313 | 0.000001 | 107.368216 | 107.36820 |  $-$0.00002 | 107.367927 |  $-$0.000289
4 0 4 | para |  A1 | 205 | 0.000001 | 114.172246 | 114.17217 |  $-$0.00008 | 114.173085 | 0.000839
4 1 4 | ortho |  B2 | 270 | 0.000001 | 114.177601 | 114.17758 |  $-$0.00002 | 114.178495 | 0.000894
3 3 1 | para |  A2 | 250 | 0.000002 | 115.340656 | 115.34059 |  $-$0.00007 | 115.340038 |  $-$0.000618
3 3 0 | ortho |  B2 | 247 | 0.000001 | 117.392003 | 117.39199 |  $-$0.00001 | 117.391407 |  $-$0.000596
4 1 3 | para |  A2 | 306 | 0.000001 | 148.140653 | 148.14058 |  $-$0.00007 | 148.140740 | 0.000087
4 2 3 | ortho |  B1 | 362 | 0.000001 | 148.418328 | 148.41831 |  $-$0.00002 | 148.418470 | 0.000142
5 0 5 | ortho |  B1 | 264 | 0.000045 | 166.343488 | 166.34345 |  $-$0.00004 | 166.344887 | 0.001399
5 1 5 | para |  A2 | 196 | 0.000077 | 166.344005 | 166.34417 | 0.00016 | 166.345606 | 0.001601
4 2 2 | para |  A1 | 308 | 0.000001 | 170.335799 | 170.33574 |  $-$0.00006 | 170.335471 |  $-$0.000328
4 3 2 | ortho |  B2 | 383 | 0.000002 | 173.967266 | 173.96726 |  $-$0.00001 | 173.966794 |  $-$0.000472
4 3 1 | para |  A2 | 261 | 0.000001 | 182.648548 | 182.64849 |  $-$0.00006 | 182.647978 |  $-$0.000570
4 4 1 | ortho |  B1 | 290 | 0.000002 | 195.661414 | 195.66142 | 0.00001 | 195.659965 |  $-$0.001449
4 4 0 | para |  A1 | 219 | 0.000003 | 196.802189 | 196.80212 |  $-$0.00007 | 196.800713 |  $-$0.001476
5 1 4 | ortho |  B2 | 356 | 0.000037 | 210.217262 | 210.21727 | 0.00001 | 210.217731 | 0.000469
5 2 4 | para |  A1 | 270 | 0.000001 | 210.264821 | 210.26477 |  $-$0.00005 | 210.265229 | 0.000408
5 2 3 | ortho |  B1 | 387 | 0.000001 | 243.343434 | 243.34344 | 0.00001 | 243.343211 |  $-$0.000223
5 3 3 | para |  A2 | 333 | 0.000001 | 244.392517 | 244.39250 |  $-$0.00002 | 244.392149 |  $-$0.000368
5 3 2 | ortho |  B2 | 363 | 0.000001 | 263.738931 | 263.73895 | 0.00002 | 263.738560 |  $-$0.000371
5 4 2 | para |  A1 | 298 | 0.000001 | 271.106067 | 271.10604 |  $-$0.00003 | 271.104809 |  $-$0.001258
5 4 1 | ortho |  B1 | 316 | 0.000002 | 277.337562 | 277.33758 | 0.00002 | 277.336633 |  $-$0.000929
5 5 1 | para |  A2 | 229 | 0.000002 | 296.104442 | 296.10442 |  $-$0.00002 | 296.101282 |  $-$0.003160
5 5 0 | ortho |  B2 | 278 | 0.000001 | 296.677577 | 296.67760 | 0.00002 | 296.674560 |  $-$0.003017

The pure vibrational levels from this work are also compared with those from
94KoJe [78] and AYT2 [91], see Table LABEL:t:compare_vib. It should be noted
here that there are some differences in labelling between these data sources
and those used in this work. For example, the pure vibrational levels of
94KoJe [78] which are labelled ($v_{1},v_{2},v_{3}$)=(2,0,2), (1,0,2) and
(3,0,0), match the normal mode labelling of (4,0,0), (3,0,0) and (1,0,2),
respectively, according to the labelling of 01TyTaSc [82] and the experimental
sources used in this work.

Table 9: Comparison of pure vibrational levels ($J=0$) from this work with those of 94KoJe [78] (see Table LABEL:t:96KoJe) and AYT2 [91]. NT stands for NumTrans, the number of transitions linking a particular state. All energies and uncertainties are in cm-1. The uncertainty refers to this work. State is from this work and Sym is the corresponding symmetry label as used in AYT2 [91] (see Section 2.2).  $\nu_{1}$ $\nu_{2}$ $\nu_{3}$ | State | Sym | NT | Uncertainty | This work | 94KoJe | Difference | AYT2 | Difference
---|---|---|---|---|---|---|---|---|---
0 0 0 | para |  A1 | 58 | 0.000098 | 0 |  |  | 0 | 0
0 1 0 | para |  A1 | 5 | 0.000477 | 1182.576991 | 1182.5742 |  $-$0.0028 | 1182.569618 |  $-$0.007373
0 2 0 | para |  A1 | 4 | 0.000629 | 2353.964679 | 2353.9655 | 0.0008 | 2353.907317 |  $-$0.057362
1 0 0 | para |  A1 | 3 | 0.000663 | 2614.407743 | 2614.4074 |  $-$0.0003 | 2614.394829 |  $-$0.012914
0 0 1 | ortho |  B2 | 2 | 0.000669 | 2628.454821 | 2628.4552 | 0.0004 | 2628.463320 | 0.008499
0 3 0 | para |  A1 | 3 | 0.000621 | 3513.789974 |  |  | 3513.705072 |  $-$0.084902
1 1 0 | para |  A1 | 3 | 0.000390 | 3779.166566 | 3779.1710 | 0.0044 | 3779.189348 | 0.022782
0 1 1 | ortho |  B2 | 2 | 0.000625 | 3789.269211 | 3789.2720 | 0.0028 | 3789.269878 | 0.000667
0 4 0 | para |  A1 | 1 | 0.001000 | 4661.672219 |  |  | 4661.605794 |  $-$0.066425
1 2 0 | para |  A1 | 2 | 0.000707 | 4932.699369 |  |  | 4932.688937 |  $-$0.010432
0 2 1 | ortho |  B2 | 2 | 0.000707 | 4939.104010 | 4939.2300 | 0.1260 | 4939.129851 | 0.025841
2 0 0 | para |  A1 | 2 | 0.000707 | 5144.986319 | 5145.1200 | 0.1337 | 5145.031868 | 0.045549
1 0 1 | ortho |  B2 | 2 | 0.000707 | 5147.220560 | 5147.3600 | 0.1394 | 5147.166622 |  $-$0.053938
0 0 2 | para |  A1 | 2 | 0.000707 | 5243.101919 |  |  | 5243.158956 | 0.057037
1 3 0 | para |  A1 | 2 | 0.000928 | 6074.581067 |  |  | 6074.566059 |  $-$0.015008
0 3 1 | ortho |  B2 | 2 | 0.000707 | 6077.594560 |  |  | 6077.626636 | 0.032076
2 1 0 | para |  A1 | 2 | 0.000707 | 6288.146119 | 6288.1428 |  $-$0.0033 | 6288.134723 |  $-$0.011396
1 1 1 | ortho |  B2 | 3 | 0.000637 | 6289.172875 | 6289.1739 | 0.0010 | 6289.128284 |  $-$0.044591
1 2 1 | ortho |  B2 | 4 | 0.000632 | 7420.092707 |  |  | 7420.077786 |  $-$0.014921
1 0 2 | para |  A1 | 1 | 0.002000 | 7576.381719 | 7576.3000 |  $-$0.0817 | 7576.413281 | 0.031562
2 0 1 | ortho |  B2 | 2 | 0.001414 | 7576.544710 | 7576.3000 |  $-$0.2447 | 7576.596211 | 0.051501
3 0 0 | para |  A1 | 2 | 0.001414 | 7752.263219 | 7751.9000 |  $-$0.3632 | 7752.343205 | 0.079986
0 0 3 | ortho |  B2 | 2 | 0.001414 | 7779.321260 | 7779.2000 |  $-$0.1213 | 7779.352004 | 0.030744
1 3 1 | ortho |  B2 | 1 | 0.002000 | 8539.561310 |  |  | 8539.565999 | 0.004689
1 1 2 | para |  A1 | 2 | 0.001414 | 8697.141469 |  |  | 8697.133905 |  $-$0.007564
2 1 1 | ortho |  B2 | 2 | 0.001481 | 8697.154984 | 8697.3000 | 0.1450 | 8697.179341 | 0.024357
1 4 1 | ortho |  B2 | 1 | 0.005000 | 9647.167310 |  |  | 9647.098855 |  $-$0.068455
2 2 1 | ortho |  B2 | 1 | 0.005000 | 9806.667310 |  |  | 9806.712978 | 0.045668
1 2 2 | para |  A1 | 1 | 0.005000 | 9806.733119 |  |  | 9806.748170 | 0.015051
2 0 2 | para |  B2 | 1 | 0.005000 | 9911.023119 | 9911.0500 | 0.0269 | 9911.102285 | 0.079166
3 0 1 | ortho |  A1 | 1 | 0.005000 | 9911.023310 | 9911.0500 | 0.0267 | 9911.112478 | 0.089168
3 1 1 | ortho |  A1 | 1 | 0.005000 | 11008.695310 | 11008.7800 | 0.0847 | 11008.774494 | 0.079184
3 0 2 | para |  A1 | 1 | 0.001000 | 12149.460119 |  |  | 12149.552318 | 0.092199
1 0 4 | para |  A1 | 1 | 0.015000 | 12524.637119 |  |  | 12524.834491 | 0.197372
4 0 1 | ortho |  B2 | 1 | 0.015000 | 12525.214310 |  |  | 12525.346292 | 0.131982
3 1 2 | para |  B2 | 1 | 0.015000 | 13222.762119 |  |  | 13222.790827 | 0.028708
2 1 3 | ortho |  A1 | 1 | 0.015000 | 13222.772310 |  |  | 13222.798559 | 0.026249

We compared all the rovibrational energy levels given in 94KoJe [78],
95FlGrRaSt [79], 96UlMaKoAl [47], and 87LeFlCaAr [76] against our set of
Marvel energy levels, taking the aforementioned labelling differences into
account for 94KoJe [78] (see Table LABEL:t:96KoJe). These comparisons are
illustrated in Figure 2. Most levels are in good agreement, with only a few
levels with differences between 0.1–0.4 cm-1. These levels with the largest
differences are from 87LeFlCaAr [76] and 56AlPl [31]. All the levels which are
compared to those of 87LeFlCaAr [76] have several transitions which link them
to other energy levels in our Marvel dataset, and originate from different
sources, which increases their reliability. This indicates that the lack of
accuracy for these particular energies originates from 87LeFlCaAr [76] and not
the Marvel dataset. The transition data from 56AlPl [31] was considered for
use in the current study, but the data could not be validated with that from
newer sources (see Table LABEL:t:notused), and so the energy levels are also
assumed to be unreliable.

Figure 2: Deviations in cm-1 between energy levels from this work and: 94KoJe
[78] (see Table LABEL:t:96KoJe), 95FlGrRaSt [79], 87LeFlCaAr [76], 96UlMaKoAl
[47]. Note the logarithmic vertical–axis.

As mentioned above, AYT2 [91] is a theoretical variational linelist for H232S
calculated as part of the ExoMol project [92, 93]. Highly accurate
experimental energy levels provide essential input for testing and improving
theoretically calculated line positions such as those within AYT2. The ExoMol
data format convention is to have a ‘states’ and a ‘trans’ file. The states
file gives the calculated rovibrational energy levels, which can be compared
against our empirical energy levels, and the trans file contains the
Einstein–$A$ coefficients for transitions between these states [93]. This
format allows calculated linelists to be retrospectively updated using more
reliable experimental energy levels in order to improve their accuracy, see
[94] for an example.

To compare our energy levels with those from the AYT2 [91] linelist, a
comparable states file was made; we labelled our states with the same A1, A2,
B1, B2 symmetry labels which are given in the AYT2 states file, see Table 2.
We then matched all states with this same symmetry, $v_{2}$, $v_{1}+v_{3}$,
and $J$, and searched for the closest value within these given parameters.
Figure 3 gives the result of this comparison.

Work is underway to update the existing ExoMol AYT2 states file and linelist
based on the energy levels obtained in this work, using only those levels
based on the transitions we are most sure about. Care should be taken in
general when using Marvel energy levels with a low value of NumTrans (see
Figure 1). This, along with the uncertainty, gives an indication of the
reliability of a particular energy level.

Figure 3: Deviations, in cm-1, between the MARVEL energy levels from this work
and the variationally calculated linelist AYT2 [91]. Different colors
represent different rovibrational symmetries (see Table 2 in Section 2.2).

##  6 Conclusions

A total of 44 325 measured experimental rovibrational transitions of H232S
from 31 publications have been considered in this work. From this set, 3969
ortho\- and 3467 para-H232S energy levels have been determined using the
Measured Active Rotational-Vibrational Energy Levels (Marvel) technique. These
results have been carefully compared with alternative compilations of energy
levels.

A variational high-temperature linelist for H232S has been computed as part of
the ExoMol project, AYT2 [91]. Our new Marvel energy levels will be used to
improve the accuracy of this theoretical linelist.

A significant part of this work was performed by pupils from Highams Park
School in London, as part of a project known as ORBYTS (Original Research By
Young Twinkle Students) [95]. The Marvel studies of 12C2H2 [26] and 48Ti16O
[18] were also performed as part of the ORBYTS project and further studies on
other key molecules will be published in due course, involving high-school
students from the UK, Australia, and Hungary. A paper discussing our
experiences of performing original research in collaboration with school
children can be found elsewhere [96].

##  7 Supplementary Data

Please refer to the web version of this work for access to the supplementary
data. There are four files provided, as listed in Table 10. The column
definitions are given in Table 11 for files 1 and 2 (Marvel input files) and
Table 11 for files 3 and 4 (Marvel output files).

Table 10: Supplied supplementary data files. File | Name
---|---
1 | H2S_ortho_input_transitions.txt
2 | H2S_para_input_transitions.txt
3 | H2S_ortho_energylevels_output.txt
4 | H2S_para_energylevels_output.txt
Table 11: Definition of columns in files 1 and 2. Column | Label | Description
---|---|---
1 |  Energy (cm-1) | Transition wavenumber
2 |  Uncertainty (cm-1) | Associated uncertainity
| Upper assignments: |
3 | $v_{1}$ | S-H symmetric stretch
4 | $v_{2}$ | Symmetric bending mode
5 | $v_{3}$ | S-H antisymmetric stretch
6 | $J$ | Rotational angular momentum
7 | $K_{a}$ | Projection of rotational angular momentum
8 | $K_{c}$ | Projection of rotational angular momentum
9 |  ortho/para | Nuclear spin state
| Lower assignments: |
10 | $v_{1}$ | S-H symmetric stretch
11 | $v_{2}$ | Symmetric bending mode
12 | $v_{3}$ | S-H antisymmetric stretch
13 | $J$ | Rotational angular momentum
14 | $K_{a}$ | Projection of rotational angular momentum
15 | $K_{c}$ | Projection of rotational angular momentum
16 |  ortho/para | Nuclear spin state
17 | Ref | Unique reference label
Table 12: Definition of columns in files 3 and 4. Column | Label | Description
---|---|---
1 | $v_{1}$ | S-H symmetric stretch
2 | $v_{2}$ | Symmetric bending mode
3 | $v_{3}$ | S-H antisymmetric stretch
4 | $J$ | Rotational angular momentum
5 | $K_{a}$ | Projection of rotational angular momentum
6 | $K_{c}$ | Projection of rotational angular momentum
7 |  ortho/para | Nuclear spin state
8 |  Energy (cm-1) |  Marvel energy assignment
9 |  Uncertainty (cm-1) |  Marvel uncertainty
10 | Num Trans | The number of transitions in the dataset which link to this state

##  8 Acknowledgements

We would like to thank Jon Barker, Fawad Sheikh, and Sheila Smith from Highams
Park School for continued support and enthusiasm, along with Highams Park
students Jack Franklin and Samuel Sheppard for their assistance in this
project. We are grateful to Laura McKemmish for helpful discussions and
advice, Clara Sousa-Silva for setting up the ORBYTS education project, and to
Anita Heward, Will Dunn, Marcell Tessenyi, and the rest of the Twinkle team
for their ongoing support. We give thanks to Shuiming Hu for contributions to
the data used in this work. This work was supported by STFC Project
ST/J002925, the RSF (grant 17-12-01204), the ERC under Advanced Investigator
Project 267219, NKFIH (grant K119658), and the COST action MOLIM: Molecules in
Motion (CM1405).

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# Design of a Smooth Landing Trajectory Tracking System for a Fixed-wing
Aircraft

Solomon Gudeta and Ali Karimoddini S. G. Gudeta and A. Karimmodini are with
the Department of Electrical and Computer Engineering, North Carolina
Agricultural and Technical State University, Greensboro, North Carolina,
USA.Corresponding author: A. Karimoddini, Tel: +13362853313,
<EMAIL_ADDRESS>
###### Abstract

This paper presents a landing controller for a fixed-wing aircraft during the
landing phase, ensuring the aircraft reaches the touchdown point smoothly. The
landing problem is converted to a finite-time linear quadratic tracking (LQT)
problem in which an aircraft needs to track desired landing path in the
longitudinal-vertical plane while satisfying performance requirements and
flight constraints. First, we design a smooth trajectory that meets flight
performance requirements and constraints. Then, an optimal controller is
designed to minimize the tracking error, while landing the aircraft within the
desired time-frame. For this purpose, a linearized model of an aircraft
developed used under the assumption of a constant approach airspeed, is used.
The resulting Differential Riccati equation is solved backward in time using
the Dormand Prince algorithm. Simulation results show a good tracking
performance and the finite-time convergence of tracking errors for different
initial conditions of the flare-out phase of landing.

##  I Introduction

One of the most difficult tasks in aircraft control is achieving a safe and
comfortable landing. A review of accident statistics indicates that over 45
percent of all general aviation accidents occur during the approach and
landing phases of a flight [1]. A closer look shows that the cause of over 90
percent of those cases was pilot related. Loss of control was also a major
contributing factor in 33 percent of these cases. In many cases, the accidents
have happened due to not landing within a targeted landing time [2]. Aircraft
landing has fives phases: the base leg, the glide slope, the flare-out, the
touchdown, and the after-landing roll. The glide slope and the flare-out are
the most crucial phases. In the glide slope phase the aircraft descends along
a predefined straight line at a steady state descent angle; whereas, in the
flare-out phase the aircraft gradually raises its nose while landing. There
are several performance requirements and constraints associated with aircraft
landing. The aircraft needs to follow the desired trajectory and land smoothly
within a targeted landing time. In addition, the aircraft landing controller
should be robust to wind turbulence, measurement noise, and actuator failures.

Many researchers have applied various control techniques to design an
automatic aircraft landing control system. Different control methods such as
classical control [3, 4], optimal control [2, 5, 6, 7], adaptive control [8,
9], nonlinear control [10, 11], and intelligent control [12, 13, 14] have been
used to solve the automatic aircraft landing problem. In [15], the flare-out
maneuver control was successfully implemented using the classical feedback
control method. [3] implemented a PID controller combined with an evolutionary
technique to tune control gains. Generally, classical controllers often lack
optimality in achieving desired performances. [2] applied an optimization
technique called the parametric expansion method to synthesize a linear time-
varying aircraft landing controller. In [16, 13, 17, 5, 18] issues such as
robustness to measurement noise, actuator faults, wind disturbances, and
flight uncertainties were investigated and $H_{2},H_{\infty}$, LQR/LQG, mixed
$H_{2}/H_{\infty}$ control techniques were employed to design optimal and
robust aircraft landing controllers. Recently, concepts from adaptive control
and intelligent control such as adaptive synthesis based on dynamic inversion
theory [13, 19, 10], neural networks [14], quantitative feedback theory [9],
sliding mode control [8], structured singular value $\mu$-synthesis, or fuzzy
techniques [20] were used to design automatic aircraft landing controllers. In
[11, 18, 8] nonlinear control techniques were applied to improve the control
performance of the landing controller. Most of the aforementioned techniques
are based on the asymptotic convergence of tracking errors. This does not
guarantee a safe, smooth, and comfortable landing, which often requires the
aircraft to accurately track a particular landing profile while landing within
a given period of time and satisfying performance requirements and flight
constraints.

The contribution of this paper is, therefore, designing an optimal controller
that guarantees the finite-time convergence of the landing trajectory tracking
errors. For this purpose, we first design the desired landing path which
satisfies performance requirements and flight constraints. The flare-out
maneuver phase is designed as an exponential path, which connects end point of
glide slope phase and the touchdown point on the runway. Furthermore, the
touchdown point is captured as a boundary condition. In addition, performance
requirements and constraints on control input and output signals are
determined. Subsequently, we formulate the tracking problem by designing a
performance index which should be minimized in a targeted landing time. We
then have converted the aircraft landing problem to a Linear Quadratic
Tracking (LQT) problem where we minimize the designed cost function for a
linear model of the aircraft respecting boundary conditions. Control gains are
found by solving differential Riccati equations backward in time using the
Dormand Prince algorithm. The simulation results are analyzed to show that the
developed trajectory tracking system can handle a range of angle of attack and
headwind disturbances.

In the remainder of this paper, in Section II, the aircraft landing problem is
formulated. In Section III, the desired trajectory and performance index are
designed by incorporating the performance requirements and flight constraints.
An LQT controller is developed for aircraft landing by minimizing the designed
performance index to guide the aircraft to follow the desired landing
trajectory. In Section IV, the proposed flight controller is simulated and the
results are discussed. Finally, Section V concludes this paper.

##  II Problem formulation

The aircraft landing process is described in Figure 1. The glide slope phase
and flare-out phase are the most critical steps. After the base leg, the
aircraft should descend towards the runway center-line with the flight path
angle $\nu$ between 2.5 to 3.5 deg. As the aircraft descends to a certain
altitude above the ground, the flare-out phase begins. In the flare-out phase
of landing, the pitch angle $\theta$ of the airplane must be gradually
adjusted to have a value between 0 to 10 deg, and the aircraft should approach
the touchdown point smoothly. The flare-out phase is the most difficult part
of the landing maneuver, as the run-way is not in the pilot’s field of view
and the touchdown point has to be accurately reached within a targeted time
smoothly, following the landing trajectory. In this paper, we assume that the
lateral and longitudinal dynamics of the aircraft are decoupled, and the
lateral controller [21] takes care of the lateral deviations from the center
line of the runway. The focus of this paper, therefore, is on the design of a
longitudinal controller to track the desired landing trajectory.

Figure 1: An aircraft landing phases: the base leg, the glide slope, the
flare-out, the touchdown, and the after-landing roll.

The longitudinal motion of the aircraft is governed entirely by the elevator
deflection angle $\delta_{e}(t)$. The longitudinal dynamics of an aircraft is
given by the so-called short period equation of motion [2] of the aircraft:

$\dot{\theta}(s)=\frac{K_{s}(T_{s}s+1)}{(\frac{s^{2}}{\omega_{s}^{2}}+\frac{2\zeta
s}{\omega_{s}}+1)}\delta_{e}(s)$ (1)

where $\dot{\theta}$ is pitch rate, $K_{s}$ is short period gain, $T_{s}$ is
path time constant, $\omega_{s}$ is short period resonant frequency, and
$\zeta$ is short period damping factor. Assuming the velocity $V$ of the
aircraft to be constant, pitch angle rate $\dot{\theta}$ and altitude $h$ are
related [2] through vertical acceleration $\ddot{h}$ as:

$\ddot{h}(s)=\frac{V}{T_{s}s+1}\dot{\theta}(s)$ (2)

Combining (1) and (2), a fourth-order linear differential equation is found:

$h^{(4)}(t)+2\zeta\omega_{s}h^{(3)}(t)+\omega_{s}^{2}\ddot{h}(t)=K_{s}V\omega_{s}^{2}\delta_{e}(t)$
(3)

Substituting

$\displaystyle\ddot{h}(t)=\frac{V}{T_{s}}\theta(t)-\frac{1}{T_{s}}\dot{h}(t),h^{(3)}(t))=\frac{V}{T_{s}}\dot{\theta}(t)-\frac{1}{T_{s}}\ddot{h}(t)$
(4a) $\displaystyle
h^{(4)}(t)=\frac{V}{T_{s}}\ddot{\theta}(t)-\frac{1}{T_{s}}h^{(3)}(t)$ (4b)

in (3), the new aircraft equation of motion becomes

$\begin{split}\frac{V}{T_{s}}\ddot{\theta}(t)-(\frac{V}{T_{s}^{2}}-2\zeta\omega_{s}\frac{V}{T_{s}})\dot{\theta}(t)-(2\zeta\omega_{s}\frac{V}{T_{s}^{2}}-\omega_{s}^{2}\frac{V}{T_{s}^{2}}\\\
-\frac{V}{T_{s}^{3}})\theta(t)-(\frac{1}{T_{s}^{3}}-2\zeta\omega_{s}\frac{1}{T_{s}^{2}}+\frac{\omega_{s}^{2}}{T_{s}})\dot{h}(t)=K_{s}V\omega_{s}^{2}\delta_{e}(t)\end{split}$
(5)

from which one can derive, the state space representation of the aircraft’s
model as:

$\displaystyle\dot{x}(t)=Ax(t)+B\delta_{e}(t)$ (6a) $\displaystyle y=Cx(t)$
(6b)

where
$x(t)=\big{[}\begin{smallmatrix}h(t)&\dot{h}(t)&\theta(t)&\dot{\theta}(t)\end{smallmatrix}\big{]}^{T}$,
$B=\big{[}\begin{smallmatrix}0&0&0&b_{4}\end{smallmatrix}\big{]}^{T}$,
$A=\bigg{[}\begin{smallmatrix}0&1&0&0\\\ 0&a_{22}&a_{23}&0\\\ 0&0&0&1\\\
0&a_{42}&a_{43}&a_{44}\end{smallmatrix}\bigg{]}$, $a_{22}=\frac{-1}{T_{s}}$,
$a_{23}=\frac{V}{T_{s}}$,
$a_{42}=\frac{1}{vT_{s}^{2}}-2\frac{\zeta\omega_{s}}{vT_{s}}+\frac{\omega_{s}^{2}}{v}$,
$a_{43}=2\frac{\zeta\omega_{s}}{T_{s}}-\omega_{s}^{2}-\frac{1}{T_{s}^{2}}$,
$a_{44}=\frac{1}{T_{s}}-2\zeta\omega_{s}$, $b_{4}=\omega_{s}^{2}k_{s}T_{s}$
and C is the identity matrix.

Figure 2: Definition of aircraft coordinates and angles [2]. $\alpha$ is the
angle of attack, $\theta$ is the pitch angle, $\nu$ is the flight path angle.

Our aim is to design a controller for the aircraft to follow the landing
trajectory with the minimum tracking error $e(t)$ given as:

$e(t)=x(t)-r(t)$ (7)

where $r(t)$ $\in$ $\mathbb{R}^{4}$ is the desired landing trajectory over the
landing time interval $[t_{0}$, $t_{f}]$. Then, given an aircraft model in (6)
and boundary conditions as $x(t_{0})=x_{0}$ and $x(t_{f})=x_{f}$, we address
the following problems for the aircraft landing:

###### Problem 1

Design a desired landing flight trajectory $r(t)$ which captures performance
requirements and flight constraints.

###### Problem 2

Design a performance index $J$ to capture performance requirements and
constraints

###### Problem 3

Design a controller that tracks the desired trajectory $r(t)$ and satisfies
the given performance requirements and constraints.

##  III Landing trajectory tracking

###  III-A Trajectory design

The aircraft landing trajectory is the desired aircraft landing path, which
the aircraft follows over a specified period of time $[t_{0},t_{f}]$. Landing
trajectory must ensure safe and comfortable landing. The desired landing path
is expressed as a desired altitude which is a function of forward distance.
Its important phases are the glide slope phase and flare-out phase as shown in
Figure 3. The glide slope phase is the first part of a landing path. It is
modeled as a straight line path whose slope is given in terms of the flight
path angle $\nu$. The other part is the flare-out path which is initiated when
the aircraft reaches to a certain height ${h_{f}}_{0}$ above the landing
surface. Flare-out path is an exponential path with a near zero flight path
angle at touchdown. In Figure 3, the desired landing path is shown in forward
distance (X)-altitude (h)-plane.

Figure 3: An aircraft landing path where $({X_{g}}_{0},{h_{g}}_{0})$ and
$({X_{g}}_{f},{h_{g}}_{f})$ are the beginning and the endpoint of the glide
slope path; $({X_{f}}_{0},{h_{f}}_{0})$ and $(X_{t},0)$ are the beginning and
the endpoint of the flare-out path; $(0,0)$ is the origin of X-h plane, and
$(X_{\infty},h_{c})$ is the final point on the flare-out path usually defined
below the ground. The origin of the X-h plane is assumed to be at the
beginning of the runway and the aircraft must land within one-third of the
runway.

In the glide slope phase the aircraft tracks a straight line path pointing
toward the beginning of the runway. The desired altitude, $h^{d}$, is computed
based on a given flight path angle $\nu$:

$h^{d}=-tan(\nu^{d})(X-{X_{g}}_{0})+{h_{g}}_{0}$ (8)

As soon as the flare-out phase starts, the desired altitude $h^{d}$ is
computed as:

$\begin{split}h^{d}=-h_{c}+({h_{f}}_{0}+h_{c})e^{-K_{x}(X-{X_{f}}_{0})}\end{split}$
(9)

where $K_{x}$ is a constant which defines the curvature of flare-out maneuver
path, depending on the distance between the origin of X-h plane and the
touchdown point.

The objective of a good, stabilized final approach is to descend at an angle
and airspeed that permits the airplane to reach the desired touchdown point
with minimum floating; in essence, a semi-stalled condition [1]. To accomplish
this, it is essential that both the descent angle and the airspeed be
accurately controlled. Since on a normal approach the power setting is not
fixed as in a power-off approach, the power and pitch attitude are adjusted
simultaneously as necessary to control the airspeed and the descent angle, or
to attain the desired altitudes along the approach path. In this paper, we
focus on the flare-out phase to achieve a safe and comfortable landing. In
(9), ${X_{f}}_{0}$, $h_{c}$ and $K_{x}$ are unknowns. We can solve for these
unknowns from $tan(\nu^{d})$, ${h_{f}}_{0}$, and $X_{t}$ under the following
constraints:

  * •

Slope continuity constraint: For a smooth phase change, the slope at the
beginning of the flare-out path should be equal to the slope of the glide
slope path. Thus, the slope of glide slope path can be calculated as

$\dot{h^{d}}=-\tan(\nu^{d})$ (10)

Similarly, the slope at the beginning of flare-out path is

$\dot{h^{d}}={-K_{x}}({h_{f}}_{0}+h_{c})e^{-K_{x}(X-{X_{f}}_{0})}|_{X={X_{f}}_{0}}$
(11)

Using (10) and (11), $K_{x}$ is computed as

$K_{x}=\frac{\tan(\nu^{d})}{{h_{f}}_{0}+h_{c}}$ (12)

  * •

Path continuity constraint: This constraint guarantees the continuity of the
path when transiting from the glide slope phase to the flare-out phase. By
design, we have ${X_{f}}_{0}={X_{g}}_{f}$. We then need to ensure that $h^{d}$
at ${X_{g}}_{f}$ given in (8) is the same as $h^{d}$ at ${X_{f}}_{0}$ given in
(9) resulting in

${X_{f}}_{0}=\frac{{h_{g}}_{0}-{h_{f}}_{0}}{tan(\nu^{d})}+{X_{g}}_{0}\\\ $
(13)

  * •

Touchdown constraint: The landing path intersects the ground at the touchdown
point $X_{t}$. Thus, at touchdown point we have

$-h_{c}+({h_{f}}_{0}+h_{c})e^{-K_{x}(X_{t}-{X_{f}}_{0})}=0$ (14)

Finally, $K_{x}$ and $h_{c}$ are found by solving (12), (13) and (14).

Under the assumption of constant aircraft forward velocity $\dot{X}$, at any
time moment $t>t_{0}$, the forward distance $X$ on the flare-out curve will be
approximately

$X={X_{f}}_{0}+\dot{X}(t-t_{0})$ (15)

Substituting $X-{X_{f}}_{0}$ from (15) in (9), the landing trajectory in
flare-out phase is governed by

$h^{d}=-h_{c}+({h_{f}}_{0}+h_{c})e^{-K(t-t_{0})}$ (16)

where $K=K_{x}\dot{X}$. By differentiating (16), the desired rate of descent
in flare-out phase is:

$\dot{h_{d}}=-K({h_{f}}_{0}+h_{c})e^{-K(t-t_{0})}$ (17)

The value of the pitch angle is crucial only during the last few sec of
landing before the touchdown point. In flare-out maneuver phase, the desired
pitch angle, $\theta_{d}$, is::

$\theta_{d}=0$ (18)

Also, for a safe and comfortable landing, the pitch rate $\dot{\theta_{d}}$
should be:

$\dot{\theta_{d}}=0$ (19)

###  III-B Performance requirements

The following landing requirements and flight constraints are considered for
the aircraft landing in this paper:

  1. C1:

The landing path should be an exponential path such that it ensures a safe and
comfortable landing.

  2. C2:

The rate of descent $\dot{h}$ should be non-zero in order to avoid
overshooting. In addition, the rate of descent at touchdown should be a small
negative value in order to prevent over stressing of landing gear.
Requirements to avoid undesirable behaviors such as landing gear over stress,
aircraft floating down runway and aircraft overshooting runway are captured by
magnitude of rate of descent at touchdown point. Typically based on type of
the aircraft, a value between 60 to 180 ft/min is considered as an ideal rate
of descent at touchdown point.

  3. C3:

The lower limit on the pitch angle $\theta$ should be $0^{\circ}$ in order to
prevent the nose wheel of an aircraft from touching down first, and the upper
limit on the pitch angle $\theta$ should be $10^{\circ}$ to prevent the tail
gear from touching down first:

  4. C4:

During landing, in the flare-out maneuver phase, the angle of attack must
remain below 80 percent of the stall value. The stall value is assumed to be
$18^{\circ}$ [2]. Hence, the rate of change of angle attack is restricted to a
value less than 20 percent of the stall value:

  5. C5:

The longitudinal motion control is mainly executed by the elevator. To avoid
saturation, the elevator is not permitted to operate against the mechanical
stops during the landing process. Thus, the deflection of the elevator is
restricted between $-35^{\circ}$ and $+15^{\circ}$.

In addition, the aircraft should follow the designed landing trajectory and
reduce the tracking error within the targeted landing time $t_{f}$ with
minimum control effort $\delta_{e}(t)$. This can be captured by the following
performance index:

$\begin{split}J=[Cx(t_{f})-r(t_{f})]^{T}P[Cx(t_{f})-r(t_{f})]+\\\
\int_{t_{0}}^{t_{f}}\\{[Cx(\tau)-r(\tau)]^{T}Q[Cx(\tau)-r(\tau)]+\delta_{e}^{T}(\tau)R\delta_{e}(\tau)\\}d\tau\end{split}$
(20)

where $P\geq 0$, $Q\geq 0$, $R>0$, are all symmetric weighting parameters. The
terms outside the integral ensure that the states are close to the desired
values at the touchdown point, avoiding an early or a late landing. On the
other hand, the terms inside the integral push the aircraft to track the
desired landing trajectory with minimum control effort in the landing time
$[t_{0},t_{f}]$.

###  III-C Controller design based on linear quadratic tracking

Given the aircraft model in (6) and the boundary conditions
$x(t_{0})=x_{0}=\big{[}\begin{smallmatrix}h(t_{0})={h_{f}}_{0}&\dot{h}(t_{0})&\theta(t_{0})&\dot{\theta}(t_{0})\end{smallmatrix}\big{]}^{T}$
and
$x(t_{f})=x_{f}=\big{[}\begin{smallmatrix}0&0&0&0\end{smallmatrix}\big{]}^{T}$,
our aim is to find an optimal control $\delta^{*}_{e}(t)$ that guides the
aircraft to follow the desired landing trajectory $r(t)$ over the time
interval $[t_{0},t_{f}]$, minimizing the performance index in (20), as
described below:

$\min\limits_{\forall t\in[t_{0}\,\,t_{f}]}J$ (21)

subject to

$\begin{cases}\dot{x}(t)=Ax(t)+B\delta_{e}(t)\\\ x(t_{0})=x_{0}\\\
x(t_{f})=x_{f}\end{cases}$

Defining the co-state vector $\lambda(t)=S(t)x(t)-v(t)$, where
$S(t)=C^{T}P(t)C(t)$ and $v(t)=C^{T}Pr(t)$, and defining the Hamiltonian
function
$H=\frac{1}{2}[Cx(t)-r(t)]^{T}Q[Cx(t)-r(t)]+\frac{1}{2}\delta_{e}^{T}(t)R\delta_{e}(t)+\lambda^{T}(Ax(t)+B\delta_{e}(t))$,
the constrained optimization in (21) will be converted to

$\begin{split}\min\limits_{\forall
t\in[t_{0}\,\,t_{f}]}\bar{J}=[Cx(t_{f})-r(t_{f})]^{T}P[Cx(t_{f})-r(t_{f})]+\\\
\int_{t_{0}}^{t_{f}}(H-\lambda^{T}\dot{x}(t))d\tau\end{split}$ (22)

whose solution can be found from $\delta J(x(t))=0$, where $\delta J$ is the
variation of the functional $J$. The optimal control function will be

$\displaystyle\delta^{*}_{e}(t)=-K(t)x(t)+R^{-1}B^{T}v(t)$ (23a)
$\displaystyle K(t)=R^{-1}B^{T}S(t)$ (23b)

in which $S(t)$ and $v(t)$ can be found by solving the following differential
equations in a backward way with a given boundary conditions:

$\displaystyle-\dot{S}=A^{T}S+SA-SBR^{-1}B^{T}S+C^{T}QC$ (24a)
$\displaystyle-\dot{v}=(A-BK)^{T}v+C^{T}Qr$ (24b) $\displaystyle
S(t_{f})=C^{T}PC$ (24c) $\displaystyle v(t_{f})=C^{T}Pr(t_{f})$ (24d)

##  IV Simulation

A simulation environment is set up by integrating the MATLAB and FlightGear
simulation Software in which we simulated an aircraft landing from the
altitude of 95 ft to the runway 10L of San Francisco airport. The information
on the approach plate (see Table IV) of runway 10L of the airport is used to
design a safe and comfortable landing trajectory for an aircraft with model
parameters given in Table I.

$N_{0}$ |  Symbol |  Parameter value
---|---|---
1 |  $K_{s}$ |  -0.95 $sec^{-1}$
2 |  $T_{s}$ |  40 $sec$
3 |  $\omega_{s}$ |  1 $rad/sec$
4 |  $\zeta$ |  0.5
TABLE I: The aircraft model parameters

$\displaystyle\begin{split}P=\bigg{[}\begin{smallmatrix}0.9&0&0&0\\\
0&0.01&0&0\\\ 0&0&1&0\\\ 0&0&0&1\end{smallmatrix}\bigg{]}\\\
Q=\bigg{[}\begin{smallmatrix}0.00067&0&0&0\\\ 0&0.0265&0&0\\\ 0&0&150&0\\\
0&0&0&65\end{smallmatrix}\bigg{]}\\\ R=1\end{split}$ (25)

Table IV summarizes parameters extracted from the approach plate. Similarly,
assuming this particular aircraft has a nominal initial altitude ${h_{f}}_{0}$
of 100 ft, the desired flight path angle $\nu$ of 3 deg, and the pitch angle
$\theta$ of 0 deg, then using (12) to (15), we found the desired landing
trajectory parameters ${X_{f}}_{0}$, $K_{x}$, $h_{c}$, $K$, as shown in Table
IV.

The designed desired landing trajectory is plotted in Figure 4 in solid blue.
The desired altitude $h^{d}(t)$ and the desired altitude rate $\dot{h}^{d}(t)$
are exponential functions while the desired pitch angle $\theta_{d}$ and the
desired pitch rate $\dot{\theta}_{d}$ are constant functions. Both exponential
functions and constant functions have derivatives of all orders. Therefore,
the designed landing trajectory is a smooth landing trajectory.

$N_{0}$ |  Name |  Parameter value
---|---|---
1 |  ${X_{g}}_{0}$ |  -34346 $ft$
2 |  ${h_{g}}_{0}$ |  1800 $ft$
3 |  $X_{t}$ |  3957 $ft$

TABLE II:  Parameters extracted from the approach plate

$N_{0}$ |  Name |  Parameter value
---|---|---
1 |  $K_{x}$ |  0.00049
2 |  $\dot{X}$ |  256 $ft/sec$
3 |  $K$ |  0.1385
4 |  $h_{c}$ |  6.68 $ft$
5 |  ${X_{f}}_{0}$ |  -1908 $ft$

TABLE III: The designed landing trajectory parameters

$N_{0}$ |  Parameter Name |  Parameter value
---|---|---
1 |  $t_{0}$ |  0 sec
2 |  $t_{f}$ |  20 sec
3 |  $h_{0}$ |  95 ft
4 |  $\dot{h}_{0}$ |  -14 ft/sec
5 |  $\theta_{0}$ |  -0.05 rad
6 |  $\dot{\theta}_{0}$ |  0 rad/sec
7 |  $h_{f}$ |  0 ft
8 |  $\dot{h}_{f}$ |  0 ft/sec
9 |  $\theta_{f}$ |  0 rad
10 |  $\dot{\theta}_{f}$ |  0 rad/sec
TABLE IV: Simulation parameters: Initial and final conditions for case I

Figure 4: The desired smooth landing trajectories (solid blue) and the
simulated aircraft landing trajectories (solid red).

###  IV-A Case I: Simulation results for the initial condition given in Table
IV

For the given initial conditions in Table IV, the simulated aircraft landing
trajectory is shown in Figure 4 in red. The controller gain is found solving
(24d) in a backward way using the Dormand-Prince algorithm. To respect the
Constrains C1-C5 in Section III-B, we have tuned the performance index
parameters $R$, $Q$, and $P$ to values in (25). Comparing the desired altitude
$h^{d}(t)$, the desired altitude rate $\dot{h}^{d}(t)$, and their respective
simulation values in Figure 4, we can see that the aircraft has accurately
tracked the desired landing trajectory in a desired landing time of 20 sec
satisfying Constraints C1-C5 in Section III-B. This comparison is summarized
in Table V.

Figure 5: The trajectory tracking error for the flight simulation results in Fig. 4 Figure 6: Optimal controller gains for the flight simulation results in Fig. 4 $N_{0}$ |  Constraint |  Constraint validation based on simulated flight data
---|---|---
C1 |  Exponential (smooth) trajectory |  Satisfied by design in (9)
C2 |  $60\leq\lVert\dot{h}^{d}(t_{f})\rVert\leq 180ft/min$ |  $\lVert\dot{h}^{d}(t_{f})\rVert$ = 62.7 ft/min
C3 |  $0^{\circ}\leq\theta(t_{f})\leq+10^{\circ}$ |  $\theta(t_{f})\approx 0$
C4 |  $\alpha(t)<18^{\circ}$ $\Delta\alpha(t)<3.6^{\circ}$ |  Satisfied through $\theta\approx 0$ and $\dot{\theta}\approx 0$
C5 |  $-35^{\circ}\leq\delta_{e}(t)\leq+15^{\circ}$ |  $-22.3^{\circ}\leq\delta_{e}(t)\leq+2.4^{\circ}$
TABLE V: Landing constraints C1-C5 validation for the given initial conditions
in Table IV

###  IV-B Case II: Simulation results for different initial conditions

Head wind and deviation of a desired flight path angle from a desired value of
3 deg introduces a deviation in the gliding distance and the pitch angle which
in turn give rise to a deviation in the initial altitude of the flare-out
phase as shown in Figure 7.

Figure 7: The deviations in the gliding distance and the pitch angle due to
head wind disturbances cause deviations in the initial altitude of the flare-
out phase.

However, the designed controller can handle these deviations in the initial
altitude $\Delta{h_{f}}_{0}$ and pitch angle $\Delta\theta$ by adjusting
control input in order to maintain the rate of descent and the desired
approach airspeed, while avoiding the saturation region of the elevator
actuator (Constraint C5) .

Figure 8: Trajectory tracking performance for different initial conditions of
the flare-out phase.

Acceptable deviations in the initial altitude and pitch angle around the
desired landing trajectory is determined by varying them correlationally until
the control limiting values are attained instantaneously. Accordingly,
$\Delta{h_{f}}_{0}$ and $\Delta\theta$ are found to be 20 ft and 1 deg
respectively, and the corresponding region of operation is shown in Figure 8
in green.

Figure 9: The control input signal for the flight simulation results for case
I (left) and case II (right).

##  V conclusion

In this paper, a smooth trajectory tracking system was designed for a fixed-
wing aircraft landing. The aircraft landing problem was converted to a finite-
time LQT problem. Information on the approach plate of the runway was employed
to design a desired smooth landing trajectory. We considered multiple
performance requirements and constraints to design an optimal controller which
minimizes the trajectory tracking error. The designed controller gives an
accurate tracking performance for the flare-out phase of aircraft landing. The
simulation result demonstrated the finite-time convergence of the trajectory
tracking error. Moreover, the robustness of the designed controller was
evaluated against different initial conditions due to the headwind
disturbances. This work can be extended by employing a nonlinear model of an
aircraft in the presence of turbulent crosswinds and system uncertainties.

## Acknowledgment

This research is supported by Air Force Research Laboratory and OSD under
agreement number FA8750-15-2-0116.

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