Adaptive controler of Major function for controlling elastic aerial vehicle

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.

Research ADAPTIVE 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 y Ya Mz x T.C Δ P.C kΔ G.C a*b b b Figure 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]: Journal of Military Science and Technology, Special Issue, No.60A, 05 - 2019 Electronics & Automation   ay   ay z  z ;       z ;   J z z  M z ;     ,   (1) θ is the angle to the horizontal, α being angle of attack; ϑ is the angular velocity pitch; ωz = dϑ/dt is the angular velocity pitch; Mz is the torque about a transverse axis; ay , ay z - 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]: J f  z  M y ;      J w   M z  M y ;   M y  K  ; J f  J w  J z  (2) Δ is the wing's angle elastic torsion deformations; My is the elastic torque; K is 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 Mz with respect to angle of attack α, angular velocity pitch ωz and deflection of elevating rudders δ: M z , M z z , M z and by combining equations (1), (3) we obtain a mathematical model of the longitudinal motion of the aerial vehicle with account for wing aeroelasticity [4]:   ay   ay z  z ;       ay    ay z  z ;     z  J f K  ;      ;  z   K   M        J w1  M z   M  z z  z      (3) or for convenience of further notes, introduce standard symbols for state variables:  =x1; α = x2; ωz = x3; Δ = x4; ωΔ = x5; y = (inclination angle  to be measured) We obtain: x1  a1x2  a2 x3 ;   x2  a1 x2  1  a2  x3 ;   x3  a3 x4 ;   x4  x5 ;  x5  a5 x2  a6 x3  a4 x4  b ,  (4) N D Thanh, N V Phuong, “Adaptive controller of major function … elastic aerial vehicle.” Research   z Mz M K M K a  ; a  ; a6  z ; b  z Where a1  a y ; a2  a y ; a3  ; Jw Jw Jw Jw Jf  z For convenience of taking further equation notes, we introduce vector matrix symbols for longitudinal motion of a resilient AV x  Ax  bu (t ); y  c т x; (5) 0 0 a1 a2   0  a1   a2  0  A  0 0 a3    1 0 0 0 a a  a4   т b   0 0 b  ; x   x1 x2 x3 x4 x5 т ; c т  0 kc 00 , u (t )      u A ;   u is the programed control, u A is the defined adaptive control, kc is the transfer coefficient of inclination angle sensor  = x1 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]: A full-order reference model ( n  ) of the following form: x м  А м x м  b м u (6) AM - is the Hurwitz matrix; b м   00 00 kM т , kм - constant coefficient A stationary status identifier (observer):    x  A x  b u (t )  lc т ( x4  x4 ), (7) A , b0 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), x2  x2 is the observation error (per control variable) Linear (modal) control:  u uL  k т x , k - is the numeric (5х1) - vector The adaptive control law has the following form:   т uA  k A diag f r* xˆ  kbu (8) kA is the (5x1) - dimensional vector of adjustable parameters of adaptive law   (8); kb is the adjustable input coefficient, f r*  f r* ( xr )  xrp , p  0,1, 2, are scalar Journal of Military Science and Technology, Special Issue, No.60A, 05 - 2019 Electronics & Automation function of the scalar arguments, corresponding to majorizing nonlinear functions of object with the maximal growth [3] r  1,5 Regularized algorithms of adaptive law (8) parameter adjustment are expressed by equations of the form т т  т т k A   A b M Pex diag  f r 1  A k A ; т  k b   b b M Peu  b kb ,     (9) P is a constant symmetric positively definite 5х5 matrix - the solution of Lyapunov matrix equation in the form of: A мт P  PA м  G   G is any symmetrical positively definite matrix, e = x  xM is the (5x1) – dimensional vector of error, γA, λA, γb, λb 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]: f1*  1; f 2*  x2 , f3*  x3 , f 4*  x43 , f5*  IV SOME RESULTS OF COMPUTERIZED RESEARCH OF ADAPTIVE CONTROL EFFICIENCY OF RESILIENT AIR VEHICLE deg deg V1 V1 V2 V3 V2 V3 t(s) t(s) a - inclination angle  b- attack angle α deg deg V1 V1 V2 V3 V3 V2 t(s) c - pitch angle ϑ t(s) d - wing's angle elastic torsion deformations Δ Figure The experimental result: The output signal of the control AV when using the modal controller N D Thanh, N V Phuong, “Adaptive controller of major function … elastic aerial vehicle.” Research In 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: Vk  40m / s , ay  5.1194 , ay z  9.24, a  0.8 , b  0.35m , S  1.05m2 , mk  8kg , M z  16.7015 , M z  2.2163  z m z  1.6160 ,  k      2.82  22.1  1315.5  8580   17289  , In the course of research parameters flight path velocity of wing's mass center Vk, dimensionless factor defining the frequency of elastic oscillations fy were changed Figures 2, show two groups with four diagrams of transient processes each: step responses u0 = δ0B 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 Vk,(V1,V2,V3) m/s; fy, Hz: (V1 =20; fy =1,25); (V2=30; fy =1,6); (V3 = 40; fy = 2,1); - reference model process; X-axis on all diagrams renders time (s); Y-axis renders angle (deg.) deg deg V3 V3 V2 V2 V1 V1 t(s) t(s) a - inclination angle  b- attack angle α deg deg V2 V3 V2 V1 V1 V3 t(s) c - pitch angle ϑ t(s) d - wing's angle elastic torsion deformations Δ Figure 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 Vk (V1, V2, V3) being flight speed and a being position of PC with reference to GC, Journal of Military Science and Technology, Special Issue, No.60A, 05 - 2019 Electronics & Automation torsion elastic wings' vibrations (noncontrolled weakly damped) are efficiently suppressed within the simulated adaptive model; At the same time overload values 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 # 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 điều khiển thích nghi với hàm major cho kênh chuyển động dọc thiết bị bay tính đến nhiễu động đàn hồi Tác giả xây dựng mơ hình tốn học phi tuyến thiết bị bay kênh chuyển động dọc tính đến nhiễu động đàn hồi tiến hành kiểm tra, tính tốn, đánh giá đáp ứng hệ thống điều khiển thích nghi chịu tác động nhiễu động đàn hồi ảnh hưởng tham số bất định mơ hình đối tượng bay Từ khóa: Thích nghi; Điều khiển; Hàm major; Đàn hồi khí động; Thiết bị bay Received date 15th March, 2019 Revised manuscript 12th April, 2019 Published 15th May, 2019 Author affiliations: Academy of Military science and technology, Cau Giay, Ha Noi * Corresponding author: thanhnd37565533@gmail.com N D Thanh, N V Phuong, “Adaptive controller of major function … elastic aerial vehicle.” ... defined adaptive control, kc is the transfer coefficient of inclination angle sensor  = x1 III AN ADAPTIVE CONTROL SYSTEM OF THE LONGITUDINAL MOTION OF AERIAL VEHICLE An adaptive system for nonlinear... (2) Δ is the wing's angle elastic torsion deformations; My is the elastic torque; K is the elastic coefficient of torsion deformations; Jw, Jf are moments of inertia of wings and fuselage relative... Phuong, Adaptive controller of major function … elastic aerial vehicle. ” Research   z Mz M K M K a  ; a  ; a6  z ; b  z Where a1  a y ; a2  a y ; a3  ; Jw Jw Jw Jw Jf  z For convenience