=Paper= {{Paper |id=Vol-2590/short3 |storemode=property |title=Nonlinear Oscillations Prevention in Unmanned Aerial Vehicle |pdfUrl=https://ceur-ws.org/Vol-2590/short3.pdf |volume=Vol-2590 |authors=Iuliia Zaitceva |dblpUrl=https://dblp.org/rec/conf/micsecs/Zaitceva19 }} ==Nonlinear Oscillations Prevention in Unmanned Aerial Vehicle== https://ceur-ws.org/Vol-2590/short3.pdf

Nonlinear Oscillations Prevention in Unmanned
Aerial Vehicle

Iuliia Zaitceva[0000−0001−9957−022X]

ITMO University, 49 Kronverksky pr., 197101 St-Petersburg, Russia
juliazaytsev@gmail.com

Abstract. The paper considers the prevention of nonlinear oscillations
of a small unmanned aerial vehicle of a fixed wing (UAV) in a closed loop.
This phenomenon is a rapidly developing aircraft oscillations accompa-
nied by increasing amplitude in angular velocities and angles. The pilot
can overcome the process of small amplitude oscillations by disconnect-
ing from the control loop. However, since the oscillations begin suddenly
and develop so quickly that the pilot does not have time to react. Of-
ten these events lead to loss of control and stability of the aircraft. In
manned aircraft, this phenomenon has been known since the inception
of large aviation. In literature it is called the pilot induced oscillations
(PIO) or the aircraft pilot coupling (APC). Oscillations prevention is
proposed to be implemented by introducing into the control loop a non-
linear corrective device that provides the control system in question with
the desired phase stability margin. Simulation results of pitch angle are
presented with active pilot control and the maximum possible time delay
and amplitude of input signal.

Keywords: Nonlinear correction · UAV · Oscillations · PIO · Aircraft ·
Time delay· Phase shift· Saturation.

1 Introduction

Despite its prescription, the issue of PIO still exists and the last accident in
commercial aviation dates from 2014 [7]. At the same time, UAVs became the
successors of this problem. In recently years UAVs have become widely used in
various fields of human activity. In this research we consider the UAV called
”Phastball” which can operate both offline and/or remotely with the participa-
tion of a human operator. It was found from flight experiments that nonlinear
oscillations occur when the UAV control modes switch from manual to control
mode from a ground station [15]. After switching, the time delay of the trans-
mitted input signal increases, and there is also the actuator rate limit, which
Copyright c 2019 for this paper by its authors. Use permitted under Creative
Commons License Attribution 4.0 International (CC BY 4.0)
2 Iuliia Zaitceva

affects the stability of the entire flight control system. Typical trigger of PIO
are limit of control surface actuator, high pilot gain and time delay of signals.
Its appear generally under conditions of demanding maneuvering, dramatically
changing flight control modes and weather conditions. Oscillations can be severe
irreversible, leading to loss of control of the aircraft. Thus, the task of oscillation
prevention relates to the field of flight safety.
All existing methods are not suitable for all types of aircraft; any of them
requires intensive modeling and testing. The emergence of PIO is not always
associated with external factors, but in any case, the prevention of PIO is as-
sociated with ensuring the stability of the system. Currently, there are several
approaches for prevention PIO. Inverse dynamics method [13,17]. The disadvan-
tage of this method is the occurrence of errors in the transition to the original
model, which is overcome in adaptive method [19] including neural network. In
the paper [9] the authors propose to transfer full control to the electronic unit in
dangerous moments of the flight. This method has the disadvantage that no fault
tolerant avionics exist and the control system structure loses its mechanical re-
serve. The control allocation PIO method containing optimal control is proposed
in [1]. In [8] H∞ optimization and anti-windup compensator were designed for
actuator loss in aircraft control system. Also such simplicity of implementation
approaches are known as the phase compensation [3, 14, 18] using prefilters of
various configurations. All of them are aimed at suppressing a high-frequency
spectrum in the control loop and compensating for the phase lag between the
input and output signal. Thus, in this work, a nonlinear prefilter easy to imple-
ment for preventing PIO in UAV is intended, characterized in that it expands
the capabilities of the control system without loss of control accuracy at any
level of the input signal compared to previously proposed ones.
The rest of the paper is organized as follows. The mathematical description of
the system under study is given in Sec. 2. Representation of suggested nonlinear
corrective device is given in Sec. 3. Simulation and concluding remarks are given
in Sec. 4.

2 UAV-Pilot Model
Consider UAV-pilot model consisting of 3 components: pilot, nonlinear actuator
and UAV as shown in Fig. 1. ϑ∗ is the pitch reference signal, ϑ is the pitch
output signal.

2.1 Pilot Model
One of the most important classes of piloting tasks is compensatory tracking
tasks in which the pilot acts on the displayed error between a desired command
input and the comparable vehicle output motion to produce a control action.
The pilot model corresponding to the flight mission is taken in the form [16]
with the parameters obtained in [15]:
TL s + 1 −τ s
Wp (s) = Kp e , (1)
TI s + 1
Nonlinear Oscillations Prevention in Unmanned Aerial Vehicle 3

𝜗∗ Pilot 1 1 𝜗
𝑘" UAV
model 𝑠 𝑠
- - -
Saturation

𝑘#

Fig. 1. Structure of UAV-pilot model

where Kp is the pilot static gain, which expresses the response to control
error ratio. High level of Kp means that the pilot in active control. TL and TI is
the lead and lag time constants respectively, τ is effective time delay, including
transport delays and high frequency neuromuscular lags.

2.2 Actuator Model

A second-order nonlinear actuator model contained the rate limit component is
described by the equations [4]:
(
0, if (δ ≤ δ̄ + ) ∪ (vδ > 0))|(δ ≤ δ̄ − ) ∪ (vδ < 0)
δ̇(t) =
v, other,
(2)
v̇(t) = satv̄ (w(t)),
w(t) = K1 (δ ∗ (t) − δ(t)) − K2 v(t),

where δ̄ + , δ̄ − – the upper and lower boundary of the steering wheel devi-
ation, v̄(·) is the saturation function, v̄ – rate limit, the coefficients K1√, K2
set the characteristic
√ frequency ωa and the damping coefficient ξa : ωa = K1 ,
ξ = K2 /2 K1 .
We take the characteristic of the rate limit nonlinearity as odd-symmetric
and it is represented as describing function [12] depending on input amplitude
A only:
2B h sin(2ψ) i
J(A) = ψ+ , (3)
πb 2
where B/b – slope of the linear part of the saturation function characteristic,
ψ = arcsin(b/A) is the phase shift between input and actuator signal. As shown
in paper [6], if the magnitude of the phase shift exceeds the phase stability
margin of the system, then the system loses stability. This can lead to both
oscillations of unlimited amplitude and self-oscillations. Also in paper [2], it is
argued that actuator rate limit does not always lead to PIO, from which it can
be concluded that the PIO phenomenon only occurs under certain conditions
and a wide range system parameters.
4 Iuliia Zaitceva

2.3 UAV Model

The UAV model is presented in the form of a transfer function from pitch angle
to elevator deflection obtained in [15]:

29.11s2 + 115.5s − 49.29
Wplant (s) = (4)
s4 + 7s3 + 22.6s2 − 10.64s + 0.3
For simulation transfer function (4) was reduced through the same channels
to the third order. Also due to its instability using the method of root-locus
curve [5] the pitch rate feedback was introduced.
The simulation result of noncorrective flight control system with pulse gen-
erator reference signal is shown in Fig. 2.

Pitch angle, ϑ*, ϑ, deg
500
ϑ*
400 ϑ

300

200

100

−100

−200

−300

−400
0 20 40 60 80 100 120 140 160 180 200
t, s

Fig. 2. Time history of pitch angle under Kp = 1, τ = 0.5 s and θ∗ = 40 deg

Fig. 2 shows that that an increase in the amplitude of the input signal to 40
degrees leads to oscillatory processes, the system lose their stability.

3 Nonlinear Corrective Device

Based on results from Sec. 2, the most desirable corrective device should be such
that its frequency characteristics have the properties of attenuation gain, accom-
panied by an increase in phase advance with increasing frequency. The inclusion
of such a correction device improves the relative stability of the flight control
system, i.e. increased of phase and modulus margin. Consider the corrective de-
vice in the form of a nonlinear filter containing separate amplitude and phase
Nonlinear Oscillations Prevention in Unmanned Aerial Vehicle 5

channels [10, 11]:
y = |x1 |sign(σ),
|x1 | = x · WA (s), (5)
σ = x · Wp (s),
where y is the vector of state output, x is the vector of state input, WA (s),
Wp (s) are the low-pass filters of amplitude and phase formation, respectively,
are selected in the following form:
k
WA (s) = , (6)
T2 s + 1
T1 T s + 1
Wp (s) = · , (7)
T T1 s + 1
where 0 < T1 < T2 are time constants. Coefficient k is chosen less than 1
to attenuate the amplitude. In this study, it became necessary to introduce an
additional low-pass filter WA compared with study [3]. It introduce the negative
phase shift, but information about it is not saved using the module capture unit.
The main positive phase shift is introduced by the filter (7):

ωT (1 − T1 /T )
φp (ω) = arctan >0 (8)
1 + ω 2 T 2 T1 /T

And the sign function isn’t stored information about the amplitude of the
signal passing through the filter (7). Thus, from one branch of the correction
device, we get the value of the desired amplitude, from the other – the desired
phase. A distinctive feature of the correction device is the independence of the
frequency characteristics from the input amplitude, and an increase in phase
margin with increasing frequency. This structure (5) is universal, but there is no
general methodology for choosing its parameters due to the variety of nonlinear
systems. Nyquist plot for (5) is shown at Figs. 3, 4.

Fig. 3. Amplitude frequency response
6 Iuliia Zaitceva

Fig. 4. Phase-frequency characteristic

Simulation results are carried out at a fixed value of T = 0.01 and several
attitude ratio values of T1 /T . The maximum attenuation of the amplitude at
Fig. 3 corresponds to the maximum positive phase shift at 4. To correct the
system of interest, a filter configuration with a phase margin of 100 degrees was
chosen.

4 The Results of Nonlinear Correction and Conclution

The corrective device described by (5) is embedded into the structure in Fig. 1
between pilot and actuator. The simulation results of corrective flight control
system are shown in Fig. 5.

*
Pitch angle, ϑ , ϑ, deg
45
ϑ*
40 ϑ

−5
0 20 40 60 80 100 120 140 160 180 200
t, s

Fig. 5. Time history of pitch angle
Nonlinear Oscillations Prevention in Unmanned Aerial Vehicle 7

The parameters of the corrected system are selected as follows: Kp = 8,
τ = 1.5 s and θ∗ = 40 deg. These are extremely valid parameters for the such
kind of system. Thus the introduction of the nonlinear corrective device system
allows to increase the amplitude of the input signal and to introduce the time
delay, together with empowering pilot. Last-mentioned is an important property
when fulfilling demanding flight conditions and maneuverability of UAV.
In conclusion, in this work, the UAV flight control system with the participa-
tion of a human operator was investigated. To prevent the nonlinear oscillations
the nonlinear corrective device with set parameters was introduced, which al-
lows to form independently of the amplitude of the input signal and separately
desired phase and amplitude frequency characteristics of the system. Allowable
numerical boundaries of the parameters of the adjusted system were established.
An additional low-pass filter has been introduced to attenuate the amplitude of
the control signal. Thus, the introduction of non-linear correction increases the
maneuverability of the drone in conditions of vigorous control and actuator rate
limit.

References

1. Acosta, D.M., Yildiz, Y., Craun, R.W., Beard, S.D., Leonard, M.W., Hardy, G.H.,
Weinstein, M.: Piloted evaluation of a control allocation technique to recover from
Pilot-Induced Oscillations. J. of Aircraft 52(1), 130–140 (Jan 2015)
2. Amato, F., Iervolino, S., Pandit, M., Scala, S., Verde, L.: Analysis of pilot-in-the-
loop oscillations due to position and rate saturations. Proc. of the 39th IEEE Conf.
on Decision and Control 4, 3564–3569 (2000)
3. Andrievsky, B., Kuznetsov, N., Kuznetsova, O., Leonov, G., Seledzhi, S.: Nonlin-
ear phase shift compensator for Pilot-Induced Oscillation prevention. In: Prepr.
9th IEEE Europ. Modeling Symp. on Mathematical Modeling and Computer
Simulation (EMS 2015). pp. 225–231. Madrid, Spain (6 – 8 October 2015),
http://uksim.info/ems2015/start.pdf
4. Andrievsky, B., Kuznetsov, N., Kuznetsova, O., Leonov, G., Mokaev, T.: Localiza-
tion of hidden oscillations in flight control systems. SPIIRAS Proceedings 6(49),
5–31 (2016). https://doi.org/10.15622/sp.49.1
5. Besekerskij, V., Popov, E.: Theory of automatic control systems (Teoriya sistem
avtomaticheskogo regulirovaniya). Nauka publ., Moscow. (1975)
6. Chechurin, L., Chechurin, S.: Physical fundamentals of oscillations. Frequency
analysis of periodic motion stability. Springer (2017)
7. Federal Aviation Administration: Pilot induced oscillations, https://www.
skybrary.aero/index.php/Pilot Induced Oscillation
8. Guilhem, P., Jean-Marc, B.: Application of robust antiwindup design to the longi-
tudinal aircraft control to cover actuator loss. 19th IFAC Symposium on Automatic
Control in Aerospace 46, 506–511 (2013)
9. Itoh, E., Suzuki, S.: A new approach to automation that takes account of adaptive
nature of pilot maneuver. In: Automation Congress, 2006. WAC ’06. World. pp. 1–8
(July 2006). https://doi.org/10.1109/WAC.2006.375735
10. Khlypalo, E.I.: Nonlinear correction devices in automatic systems. ”Energiya”
(1973)
8 Iuliia Zaitceva

11. Khlypalo, E.: Consideration of dynamic nonlinearity of magnetic amplifiers in de-
signing automatic systems. Automation and Remote Control 24(11), 1394–1401
(1963)
12. Krylov, N., Bogoliubov, N.: Introduction to nonlinear mechanics. In: Annals of
Mathematics Studies, vol. 11. Princeton University Press, Princeton, N. J. (1947)
13. Lane, S., Stengel, R.: Flight control design using non-linear inverse dynamics. Au-
tomatica 24(4), 471–483 (1988)
14. Liebst, B.S., Chapa, M.J., Leggett, D.B.: Nonlinear prefilter to prevent pilot-
induced oscillations due to actuator rate limiting. AIAA J. of Guidance, Control
and Dynamics 25(4), 740–747 (2002)
15. Mandal, T., Gu, Y., Chao, H., Rhudy, M.B.: Flight data analysis of pilot-induced-
oscillations of a remotely controlled aircraft. In: Proc. AIAA Guidance Naviga-
tion and Control Conf. (GNC 2013), Boston, MA. pp. 1–15. AIAA (Aug 2013).
https://doi.org/10.2514/6.2013-5010
16. McRuer, D., Jex, H.: A review of quasi-linear pilot models. IEEE Transactions on
Human Factors in Electronics HFE-8(3), 231–249 (1967)
17. Meyer, G., Su, R., Hunt, L.: Application of nonlinear transformations to automatic
flight control. Automatica 20(1), 103–107 (1984)
18. Rundqwist, L., Stahl-Gunnarsson, K.: Phase compensation of rate limiters in un-
stable aircraft. In: Proc. Int. Conf. Control Applications (CCA’96). pp. 19–24 (Sep
1996). https://doi.org/10.1109/CCA.1996.558586
19. Rysdyk, R., Calise, A.: Robust nonlinear adaptive flight control for consistent
handling qualities. IEEE Trans. Control Syst. Technol. 13(6), 896–910 (2005)