The paper considers the prevention of nonlinear oscillations of a small unmanned aerial vehicle of a xed wing (UAV) in a closed loop. This phenomenon is a rapidly developing aircraft oscillations accompanied by increasing amplitude in angular velocities and angles. The pilot can overcome the process of small amplitude oscillations by disconnecting from the control loop. However, since the oscillations begin suddenly and develop so quickly that the pilot does not have time to react. Often 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 nonlinear 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.
Despite its prescription, the issue of PIO still exists and the last accident in
commercial aviation dates from 2014 [
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
associated with ensuring the stability of the system. Currently, there are several
approaches for prevention PIO. Inverse dynamics method [
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. # is the pitch reference signal, # is the pitch output signal. 2.1
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 ight mission is taken in the form [
Wp(s) = Kp ∗
Pilot model
Saturation " 1 # 1
UAV
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, is e ective time delay, including
transport delays and high frequency neuromuscular lags.
A second-order nonlinear actuator model contained the rate limit component is
described by the equations [
K2v(t); +) [ (v > 0))j( ) [ (v < 0) where +, { the upper and lower boundary of the steering wheel deviation, v( ) is the saturation function, v { rate limit, the coe cients K1, K2 set the characteristic frequency !a and the damping coe cient a: !a = pK1, = K2=2pK1.
We take the characteristic of the rate limit nonlinearity as odd-symmetric
and it is represented as describing function [
J (A) =
2B h
b
+
sin(2 ) i
2
;
where B=b { slope of the linear part of the saturation function characteristic,
= arcsin(b=A) is the phase shift between input and actuator signal. As shown
in paper [
The UAV model is presented in the form of a transfer function from pitch angle
to elevator de ection obtained in [
29:11s2 + 115:5s Wplant(s) = s4 + 7s3 + 22:6s2
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 [
The simulation result of noncorrective ight control system with pulse generator reference signal is shown in Fig. 2.
Pitch angle, ϑ*, ϑ, deg ϑ* ϑ 500 400 300 200 100
0 −100 −200 −300 −4000 20 40 60 80 100 120 140 160 180
200 t, s Based on results from Sec. 2, the most desirable corrective device should be such that its frequency characteristics have the properties of attenuation gain, accompanied by an increase in phase advance with increasing frequency. The inclusion of such a correction device improves the relative stability of the ight control system, i.e. increased of phase and modulus margin. Consider the corrective device in the form of a nonlinear lter containing separate amplitude and phase where y is the vector of state output, x is the vector of state input, WA(s), Wp(s) are the low-pass lters of amplitude and phase formation, respectively, are selected in the following form:
WA(s) = Wp(s) = k
; T1 T T2s + 1
T s + 1
T1s + 1
;
where 0 < T1 < T2 are time constants. Coe cient k is chosen less than 1
to attenuate the amplitude. In this study, it became necessary to introduce an
additional low-pass lter WA compared with study [
And the sign function isn't stored information about the amplitude of the
signal passing through the lter (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.
channels [
= x Wp(s); (5) (6) (7) (8)
Simulation results are carried out at a xed 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 lter con guration 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 ight control system are shown in Fig. 5.
45 40 35 30 25 20 15 10 5 0 −50
Pitch angle, ϑ*, ϑ, deg ϑ* ϑ 20 40 60 80 100 120 140 160 180
200 t, s
Fig. 5. Time history of pitch angle
The parameters of the corrected system are selected as follows: Kp = 8, = 1:5 s and = 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 ful lling demanding ight conditions and maneuverability of UAV.
In conclusion, in this work, the UAV ight control system with the participation of a human operator was investigated. To prevent the nonlinear oscillations the nonlinear corrective device with set parameters was introduced, which allows 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 lter 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.