An adaptive controller of the major function for an aerial vehicle''s (AV) longitudinal motion with account for the wing aeroelasticity is discussed in this research. A nonlinear mathematical model of resilient aircraft''s longitudinal motion is designed. Adaptive control system of a resilient AV is built by means of majorizing function and its efficiency is tested under the uncertainty parameter function of AV model.
Trang 1ADAPTIVE CONTROLER OF MAJOR FUNCTION FOR
CONTROLLING ELASTIC AERIAL VEHICLE
Nguyen Duc Thanh*, Nguyen Viet Phuong
Abstract: An adaptive controller of the major function for an aerial vehicle's
(AV) longitudinal motion with account for the wing aeroelasticity is discussed in
this research A nonlinear mathematical model of resilient aircraft's longitudinal
motion is designed Adaptive control system of a resilient AV is built by means of
majorizing function and its efficiency is tested under the uncertainty parameter
function of AV model
Keywords: Adaptive; Controller; Major function; Elastic coupling; Aerial Vehicle.
I INTRODUCTION
Use of unmanned aerial vehicles (UAVs) of light and ultra-light classes rapidly increases They are indispensable in a difficult terrain, extreme conditions of work, the work of the emergency services, in the oil and gas sector and others Being highly maneuverable, high-speed and low altitude, they attract the attention of researchers and developers of control systems as complex, non-linear plants with functional and parametric uncertainty and incomplete state measurements [1, 2] Large number of papers dedicated to design of adaptive control systems based on the plant state vector [4-6] Based on the known approaches in this field, the paper deals with the design and effectiveness research of adaptive control systems for longitudinal motion of the UAV based on the major function
II MATHEMATICAL MODEL OF THE AEROELASTIC WING
Consider a thin unswept wing of the finite span Assume wing contours are symmetrical The nonlinear model of the aeroelastic wing with torsion deformations is shown in Figure 1
Figure 1 The analytical model of the wing aeroelasticity
with torsion deformations
This model is widely used in aeroelasticity research [1] Thrust, pressure and gravity centers are marked as TC, PC and GC correspondingly
The mathematical model of the longitudinal motion of a rigid aerial vehicle without allowance of elasticity can be written in the following form [1, 2]:
a*b
y
x
P.C
G.C
Δ
k Δ T.C
Ya
Mz
b b
Trang 2;
z
z
z z z
(1)
pitch; ω z = dϑ/dt is the angular velocity pitch; M z is the torque about a transverse axis; ay , z
y
a -are aerodynamic coefficients (which depend on the flight velocity
V, the thrust P of the engine, the mass m of the aerial vehicle, the lifting force Y,
the angles of attack and inclination, as well as other factors) [2]; Jz is the total
moment of inertia of the aircraft relative to the transverse axis
The elasticity of the wing can be described in terms of torsional elastic deformations as follows [1, 3]:
;
;
;
(2)
the elastic coefficient of torsion deformations; Jw, Jf are moments of inertia of
wings and fuselage relative to the transverse axis correspondingly; a is the dimensionless factor defining the distance from the wing's center to its PC; b is the
half of the length of the wing
By Introduce the partial derivatives of the longitudinal moment of inertia M z
with respect to angle of attack α, angular velocity pitch ω z and deflection of
elevating rudders δ: M z, z
z
a mathematical model of the longitudinal motion of the aerial vehicle with account for wing aeroelasticity [4]:
1 f
1
;
;
;
.
z
z
z
z
J K
or for convenience of further notes, introduce standard symbols for state variables:
=x 1 ; α = x2; ω z = x 3 ; Δ = x 4 ; ω Δ = x 5 ; y = (inclination angle to be measured)
We obtain:
1 1 2 2 3
3 3 4
5 5 2 6 3 4 4
;
;
;
,
(4)
Trang 3Where a1ay; 2 z;
y
f
K a J
w
K a J
w
M a J
z
z w
M a J
w
M b J
For convenience of taking further equation notes, we introduce vector matrix
symbols for longitudinal motion of a resilient AV
т
( ); ;
x u t y
1 2
3
a a
a
A
0 0 0 0bт;
b xx x x x x1 2 3 4 5т;cт 0k c0 0 0,u t( ) 0u A; 0
0
u
is the programed control, u A is the defined adaptive control, k c is the transfer coefficient
of inclination angle sensor = x 1
III AN ADAPTIVE CONTROL SYSTEM OF THE LONGITUDINAL
MOTION OF AERIAL VEHICLE
An adaptive system for nonlinear objects (4), (5) consists of the following subsystems [3]:
1 A full-order reference model (n ) of the following form: 5
0
м м м мu
x А x b (6)
A M - is the Hurwitz matrix; bм 0 0 0 0kMт,kм- constant coefficient
2 A stationary status identifier (observer):
т
0 0u t( ) (x4 x4 ),
(7)
0 , 0
A b are some obtained constant analysis matrices, for example, through object linearization (5), (6), x
is the evaluation vector of the state value of an
object (5), l is the vector of gain coefficients of observer's feedback (7), x2x2
is the observation error (per control variable)
3 Linear (modal) control:
т
u k x
k - is the numeric (5х1) - vector
4 The adaptive control law has the following form:
1
ˆ
u k diag f xk u (8)
k A is the (5x1) - dimensional vector of adjustable parameters of adaptive law
(8); k b is the adjustable input coefficient, f r* f r*(xr)xr p,p0,1, 2,
are scalar
Trang 4function of the scalar arguments, corresponding to majorizing nonlinear functions
of object with the maximal growth [3] r 1, 5.
5 Regularized algorithms of adaptive law (8) parameter adjustment are expressed
by equations of the form
5
M
;
r
diag f
b Pe
P is a constant symmetric positively definite 5х5 matrix - the solution of
Lyapunov matrix equation in the form of:
т
м м
G is any symmetrical positively definite matrix, e = x xM
is the (5x1) –
dimensional vector of error, γ A , λ A , γ b , λ b are positive gain coefficients of contours' setting algorithms
The choice of majorizing functions is stipulated by the structure of nonlinearities of a controlled object and in case of the given object (4), (5) these are [3, 5]:
1 1; 2 2 , 3 3 , 4 4 , 5 1.
f f x f x f x f
IV SOME RESULTS OF COMPUTERIZED RESEARCH OF ADAPTIVE
CONTROL EFFICIENCY OF RESILIENT AIR VEHICLE
Figure 2 The experimental result: The output signal of the control AV
when using the modal controller
V 3
V 2
V 1
d - wing's angle elastic torsion deformations Δ.
V 1
b- attack angle α.
t(s)
V 3
V 2
V 1
1
deg t(s)
t(s)
c - pitch angle ϑ.
V 3
V 1
V 2
deg
t(s)
a - inclination angle
Trang 5In Matlab Simulink, a program of digital realization was built for the suggested adaptive system (6) - (9) with the following unknown numeric data of the prototype AV:
40 /
k
V m s,ay 5.1194, z 9.24,
y
a a 0.8,b 0.35m,S1.05m2,m k 8kg,
16.7015
z
z
2.82 1 22.1 1315.5 8580 17289 ,
In the course of research parameters flight path velocity of wing's mass center
V k , dimensionless factor defining the frequency of elastic oscillations f y were changed
Figures 2, 3 show two groups with four diagrams of transient processes each:
step responses u 0 = δ 0 B of aircraft angles: a) inclination angle ; b) attack α; c) pitch ϑ; d) wing's angle elastic torsion deformations Δ
Every group of diagrams refers to a set of numeric values of measured
parameters V k,(V1,V2,V3) m/s; f y, Hz: (V1 =20; f y =1,25); (V2=30; f y =1,6); (V3 =
40; f y = 2,1); 1 - reference model process; X-axis on all diagrams renders time (s); Y-axis renders angle (deg.)
Figure 3 The experimental result: The output signal of the control AV
when using adaptive controller of major function
V CONCLUSIONS
The research has revealed that within the given limit adjustment of parameters
V 3
V 2
V 1
d - wing's angle elastic torsion deformations Δ
V 2
V 3
V 1
b- attack angle α
t(s)
V 3
V 2
V 1 1
deg t(s)
t(s)
c - pitch angle ϑ
V 3
V 1
V 2
deg
t(s)
a - inclination angle
Trang 6torsion elastic wings' vibrations (noncontrolled weakly damped) are efficiently suppressed within the simulated adaptive model;
At the same time overload values do not exceed acceptable limits (acceptable overload limits while maintaining flight safety);
Engaging a stationary observer (7) limits adaptive system's (6) - (9) capacity with significant parametric mismatch and nonlinearities, therefore it is probably necessary to search for solutions using an adaptive control approach for output or engage adaptive observers
REFERENCES
[1] Tewari A “Aeroservoelasticity Modeling and Control”// New York: Springer 2015 [2] T Theodorsen “General Theory of Aerodynamic Instability and the
Mechanism of Flutter” // NACA Report 496 (1935)
[3] Путов В.В., Шелудько В.Н “Адаптивные и модальные системы
управления многомассовыми нелинейными упругими механическими
объектами”// СПб.: Элмор, 2007 - 243 с
[4] A Lebedev, L Cherubrovkin “Flight Dynamics of Unmanned Aerial
Vehicles” Moscow, 1962 487 pages
[5] V Putov “Direct and indirect searchless adaptive systems with majorizing
functions and their application to the control of nonlinear resilient mechanical objects”// Publishing House Novie Tehnologii - Mechatronics, automation and
control - Issue #10 - 2007 - pp 4-11
[6] V Putov “Comparative Research of Adaptive Systems on Condition and
Output in Control of Unmanned Aerial Vehicle/ V Putov // Izvestia 'LETI'
2016 Issue # 6 pp 41 -44
TÓM TẮT
BỘ ĐIỀU KHIỂN THÍCH NGHI VỚI HÀM MAJOR ỨNG DỤNG ĐIỀU KHIỂN THIẾT BỊ BAY CÓ YẾU TỐ ĐÀN HỒI
Bài báo nghiên cứu ứng dụng bộ điều khiển thích nghi với hàm major cho kênh chuyển động dọc của thiết bị bay khi tính đến nhiễu động đàn hồi Tác giả đã xây dựng mô hình toán học phi tuyến của thiết bị bay trong kênh chuyển động dọc khi tính đến nhiễu động đàn hồi và tiến hành kiểm tra, tính toán, đánh giá đáp ứng của hệ thống điều khiển thích nghi khi chịu tác động của nhiễu động đàn hồi và ảnh hưởng của các tham số bất định của mô hình
đối tượng bay
Từ khóa: Thích nghi; Điều khiển; Hàm major; Đàn hồi khí động; Thiết bị bay
Author affiliations:
Academy of Military science and technology, Cau Giay, Ha Noi
* Corresponding author: thanhnd37565533@gmail.com.