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1. 52 ACTUATOR keH ea HE a 1 Kd o ale ale a ora Fig 4 The model of the non linear actuator This signal is introduced in the reference model as an additional input 3 one compares it with y 7 Non linear adaptive system for the command of the helicopters pitch s angle inside the reference model and after integration it leads to the modify of the signals y and y The block diagram of the subsystem formed by 52 and actuator is presented in fig 4 R Reference model r 2 NEURAL NETWORK r 2 Forming subsystem Calculus subsystem Calculus subsystem eps r 2 for vector delta r 2 Wx beta Vz r 2 6 4 1 6 8 deg 2 Time sec Time sec Bi 10 4 came 0 5 o 0 5 Time sec Time sec Fig 6 Time characteristics in the case of linear actuator s use Fig 7 Time characteristics in the case of non linear actuator s use 5 Time sec Time sec In the case of non linear actuator for the case of the longitudinal movement of the helicopter equation 33 the system from fig 3 includes the model of non linear actuator fig 4 in which x 6 the block of calculus for 32 is replaced with the subsystem from fig 4 One chooses T 0 03sec and the control limits in
2. l Av 43 M e M V M B M V 44 The component v has the form l a v F b5 45 bo and the control laws 6 and 26 become 1 we A h e nx vek y k y one v V Y bok y Doky bav byv s 46 0 the equation of the closed loop system the equation of the error s dynamic is B 7 Af aA 7 N bova bo 47 The characteristic equation of this system is s bokas bok 0 48 Using the notations bok 209 bok 5 setting 0 7 and 10rad s one obtains k k4 b 1 p DTE One considers E e 5 CECE 1 o z y e and 2 E the observer state 19 The gain matrix L is calculated so that matrix A A LC is stable A is the matrix of system from equation Romulus LUNGU Mihai LUNGU Constantin ROTARU 6 47 The component v has the form va WolV n 49 with o of form 23 with a 1 0 9 0 8 0 7 0 6 0 5 0 4 n of form 28 n n 5 d 0 05 W and V are the solutions of equations 22 no 1 vo vt d vt 2d v t 3d y t y t a 50 T 23 0 12 5 k 0 115 P and P are calculated with 24 One obtains v using 7 where k 0 8 k 0 7 Z 50 The values of the coefficients from 33 are X 0 0553 X 1 413 X 32 1731 X 19 9033 X 0 0039 X 11 2579 M 0 2373 M 6 9424 M 68 2896 M 0 002 M 38 6267 Z 0 0027 Z 0 0236 Z 0 2358 OD Z 0 1233 Z 0 5727
3. 11 bo The compensator may be described by the state equations G AG b e Vpa CS de 12 where has at least the dimension r 1 e c e le s etd c 100 0 13 The state equation of the linear subsystem with the input v e and the output y from fig 1 is x Ax b v e v v a Va 7 14 010 0 0 001 0 T VA opel 15 000 1 i 000 0 rxl The stable state x x V E 0 verifies the equation Ax 0 and taking into account equation 14 leads to the equation of the error vector e x X x Ae bv blv 7 8 16 Introducing in the block diagram from fig 1 a linear dynamic compensator a reference model and a non linear adaptive controller with neural network one obtains the block diagram from fig 2 equivalent with the one from 5 E E pray y T A y REFERENCE h y Sml MODEL LINEAR u s gt COMMAND gt INVERSE A cea DYNAMIC s MODEL 2 COMPENASTOR NON LINEAR ADAPTIVE CONTROLLER Fig 2 Automatic control system with non linear adaptive controller With notations Bel asl oO pel e el 17 G b c A 0 Ol where Z is the identity matrix one obtains AE b v 7 z CE 18 A b C d from 12 are calculated so that A is a Hurwitz matrix For the estimation of the vector the paper s authors propose the introducing in the linear dynamic compe
4. the input equation u amp the output equation y 0 the system s state equation is obtained v Xe X Xe Xs X v TX M M 0 M M Jo M l 0 0 99 0 o o 6 1a O18 33 B 1 0 B O Bp B V Z Z Z Zg Z V LZ 5 Non linear adaptive system for the command of the helicopters pitch s angle From 33 one yields y 0 0 9 M V M 0 M B M V_ M5 34 The second equation 34 must be completed with a non linear term h au h 8 y h ru h lt 35 The relative degree of the system being r 2 in 8 the output of the reference model has the form y Oro 7 Yer So 0 7 10 rad s 36 s 2 amp 98 0 From the analysis of equations 34 one notices that H s from fig 1 has the terms s and s in the denominator and the term b in the numerator Choosing b one gets the transfer function of the system with output y HO 37 Equation 10 becomes y Ayytbvts 38 By elimination of 6 between the equation 0 M V M 0 M B M V M 5 39 and the equation 38 one yields M V M 0 M B M V M5 M bo E 40 From this one identifies v h x 5 and x7 lo l one gets bv 7 a J M6 h x 8 41 For the calculus of 6 and g equation 33 may be written Z X Xg X Ve Xe X Xs 0 B B B 0 B 0 1 B i 42 V Z Ze MN Ve Ze Z Zs 8 One obtains 8 bov M 2
5. B 0 0101 B 2 1633 B 4 2184 Z 0 0698 M 0 5M M 2M The block diagram of the system for automatic command of the pitch angle is presented in fig 3 This structure of the system from fig 3 and its project represents one of the authors contributions in this paper a NONLINEAR Eq 44 8 ACTUATOR z agaca 32 43 y gt REFERENCE E MODEL 7 6 36 OBSERVER 19 ee et ee ee al NEURAL NETWORK Fig 3 Block diagram of the system for automatic command of the pitch angle Actuators characteristics time delays nonlinearities with saturation zone lead to neural network s training difficulties This is why a block PCH is introduced it limits the adaptive pseudo control v and v by the mean of one component which represents an estimation of the actuator s dynamic PCH Pseudo control Hedging PCH moves back the reference model introducing a correction of the reference model s response it depends on actuator s position 3 15 Because the dependence between and 6 is expressed by a non linear function h one yields h x h x 8 it results a difference between two functions h x 5 h x 8 Taking into account that h x 8 h x h x v y function v becomes v v h x
6. I M KASHKONEI A J Robust Adaptive Control Systems Using Neural Networks The International Journal of Control vol 3 2006 7 pp 3 JOHNSON E N CALISE A J Adaptive Guidance and Control for Autonomous Launch Vehicles IEEE Aerospace Conference Biy Ykg MT April 2001 13 pp 4 LUNGU M Sisteme de conducere a zborului Editura Sitech Craiova 2008 329 pp 5 CALISE A J HOVAKYMYAN N IDAN M Adaptive Output Control of Nonlinear Systems Using Neural Networks Automatica 37 8 August 2001 pp 1201 1211 6 GREGORY L P Adaptive Inverse Control of Plants with Disturbances Stanford University 2000 7 HOVAKIMYAN N NARDI F KIM N CALISE A J Adaptive Output Feedback Control of Uncertain Systems Using Single Hidden Layer Neural Networks IEEE Transactions on Neural Networks 13 6 2002 22 pp 8 SHARMA M CALISE A J Adaptive Trajectory Control for Autonomous Helicopters ALAA Guidance Navigation and Control Conference 14 17 August 2001 Montreal Canada 9 pp 9 TRIVAILO P M CARN C L The inverse determination of aerodynamic loading from structural response data using neural networks Inverse Problems in Science and Engineering vol 14 Issue 4 June 2006 pp 379 395 10 ELTANTAWIE M A Aplication of neuro fuzzy reduced order observer in magnetic bearing systems Proceedings of the International MultiConference on Engineers and Computer Scientists vol H Hong Kong 2010 11 BALE
7. NON LINEAR ADAPTIVE SYSTEM FOR THE COMMAND OF THE HELICOPTERS PITCH S ANGLE Romulus LUNGU Mihai LUNGU Constantin ROTARU University of Craiova Faculty of Electrical Engineering Avionics Department Blv Decebal No 107 Craiova Romania Military Technical Academy Department of Aviation Integrated Systems George Cosbuc Blv no 81 83 Bucharest Romania Corresponding author Mihai LUNGU E mail Lma1312 yahoo com mlungu elth ucv ro The paper presents a new complex adaptive non linear system with one input and one output SISO which is based on dynamic inversion The system consists of a dynamic compensator an adaptive controller and a reference model Linear dynamic compensator makes the stabilization command of the linearised system using as input the difference between closed loop system s output and the reference model s output The state vector of the linear dynamic compensator the output and other state variables of the control system are used for adaptive control law s obtaining this law is modeled by a neural network The aim of the adaptive command is to compensate the dynamic inversion error Thus the command law has two components the command given by the linear dynamic compensator and the adaptive command given by the neural network As control system one chooses the non linear model of helicopter s dynamics in longitudinal plain The reference model is linear One obtains the structure of the adaptive con
8. STRASSI P P POPOVA E PAIVA A P MARANGON LIMA J W Design of experiments on neural network s training for nonlinear time series forecasting Elsevier Journal Neurocomputing 72 2009 1160 1178 12 CHEN X GAOFENG W WEI Z SHENG C SHILEI S Efficient sigmoid function for neural networks based FPGA design Springer Publisher 2006 13 FERRARI S Smooth function approximation using neural networks IEEE Transactions on Neural Networks vol 16 no 1 January 2005 14 SHAO L WANG J SHAO S Study on the fitting ways of artificial neural networks Journal of Coal Science and engineering vol 14 no 2 June 2008 15 HOVAKIMYAN N KIM N CALISE A J PARASAD J V R Adaptive Output Feedback for High Bandwidth Control of an Unmanned Helicopter ALAA Guidance Navigation and Control Conference 6 9 August 2008 Montreal Canada Received March 5 2010
9. amics in longitudinal plain The reference model is linear One obtains the structure of the adaptive control system of the pitch angle and Matlab Simulink models of the adaptive command system s subsystems Using these some characteristics families are obtained these describe the adaptive command system s dynamics with linear or non linear actuator The authors contributions in this paper are 1 the structural block diagram of the adaptive command system from fig 1 with the linear part described by equations 8 10 2 the block diagram from fig 2 equivalent with the one from 5 where the linear dynamic model has the structure from fig 1 the structural block diagram from fig 3 and its project 4 the Matlab Simulink models of the subsystems of the structure from fig 6 the graphical characteristics from fig 6 for the linear actuator case and from fig 7 non linear actuator case with the model from fig 5 which describe time evolution of the helicopter s pitch angle time evolution of the command law s components and offer information regarding the quality and the stability of the non linear model s dynamic processes for the adaptive command system of the helicopters pitch angle REFERENCES 1 CALISE A J Flight Evaluation of an Adaptive Velocity Command System for Unmanned Helicopters AJAA Guidance Navigation and Control Conference and Exhibit vol 2 11 14 August 2003 Austin Texas 2 HOSEINI S M FARROKH
10. f and h unknown non linear functions u and y measurable One projects an adaptive control law v in rapport with the output using a neural network NN NN models a function that depends on the values of input and output of the system A at different time moments so that y t follows the bordered signal x t The feedback s linearization may be made by transformation 5 A v h y u 2 where v is the pseudo command signal and h y u the best approximation of h x u h x y u The Romulus LUNGU Mihai LUNGU Constantin ROTARU 2 equation 2 is equivalent with the following one u hy yv 3 If h h one yields y v otherwise i h y v 8 4 e e x u h x u h y u 5 is the approximation of function h inversion s error Assessing y to follow y v has the form 5 6 7 VS yy v Ya tY 6 where v 4 is the output of the dynamic linear compensator for stabilization used for the liniarised dynamic 4 with 0 v the adaptive command that must compensate s and y has for example the form 8 9 al Z lal xZ r FI with k k gt 0 gain constants Z the Frobenius norm of matrix Z the ideal matrix of the neural network and E EPB with P B matrices and E vector The derivative y is introduced for the conditioning of the dynamic error y y y This derivative is given by a reference model command filter 5 y may be cum
11. ical calculus of function 23 and for the solutions obtaining of the equations 22 and 24 are presented in 4 Second output of the compensator is used for obtai ning of an error signal that is useful for training of the neural network From 4 and 6 one yields yO O v Ve YE es 25 yes ye 26 Error may be approximated with the output of a linear neural network NN 5 14 e W7 n p n lt K 27 where W is the weights matrix for the connections between the hidden layer and the output layer NN has 2 layers and one hidden layer uln the reconstruction error of the function and y the input vector of NN n i aO sof 28 vi t wa v t d vt n r 1d 7 97 yO y t d yt n 1d 7 29 with n 2 n and d gt 0 with W is the estimation of W v is projected so that v WT n 30 3 ADAPTIVE SYSTEM FOR THE COMMAND OF HELICOPTERS PITCH ANGLE Lets consider the case of non linear dynamic of an experimental helicopter R 50 with one input and one output SISO its dynamic is given by the equation 1 with x V OBV u 8 y 8 31 where V V_ are the advance velocity respectively the vertical velocity and the pitch angle and the pitch angular velocity B the longitudinal control angle of the main rotor 6 the cyclic longitudinal input Choosing the linearised model of helicopter 15 and annexing the actuator s equation 6 6 32
12. nsator s structure of a linear state observer of order 2r 1 described by equations see fig 3 A Lz 2 2 C 19 with the gain matrix L calculated so that matrix A a LC is stable Considering w the sensor s error y the measured value of y then y y y w and the compensator s equations become AE b v Y Gw z CE Aw 20 with H i 0 G7 bd b If the state of the compensator is known one uses a reduced order observer for estimation of the vector e 10 Romulus LUNGU Mihai LUNGU Constantin ROTARU 4 A L z 2 z c 21 The gain matrix L is obtained so that the matrix A a 7 L c is stable With vectors and the vector T l is obtained The signal E TPD is used for the neural network s training the weights W and V are obtained with equations 11 W ty 2 6 6 V 7 PB KW W V r bn t PBs kV V J 22 where the role of B is played by b In 22 o is for example the sigmoid function 12 13 o z I e 23 a is the Jacobian of vector 6 W and V the initial values of weights W V Ty Ty gt O l Z0 is x p eal ki kyo PBlyy k PB Pal P and P the solutions of Liapunov equations ATP PA ATP PA O 24 P from the signal used for the neural network s training is the solution of first equation 24 with A A d bc The programs for the numer
13. position and speed of the actuators 5deg respectively 50deg sec 15 In fig 5 the Matlab Simulink model for the structure from fig 3 is presented one has chosen 5grd Each subsystem of the system from fig 5 represents a complex Matlab Simulink model In fig 6 the functions t A t E t Va t 5 t SA and v t 0 6 with blue color continuous line and 6 6 with red color dashed line are presented If the actuator is a linear one 0 gt 0 v gt the adaptive component of the command compensates the approximation s error h 5 gt 5 and v gt 0 If the actuator is non linear one obtains the time characteristics from fig 7 additionally the characteristics v t Romulus LUNGU Mihai LUNGU Constantin ROTARU 8 and e appear When v 0O the actuator is in the saturation state and it works in the linear zone when v 0 The characteristic 6 6 phase portrait of the system shows that the non linear system tends to a stable limit cycle The project of the structure from fig 5 and of its subsystems and the obtained graphical time characteristics represent contributions of this paper s authors 4 CONCLUSIONS The aim of the adaptive command is to compensate the dynamic inversion error Thus the command law has two components the command given by the linear dynamic compensator and the adaptive command given by the neural network As control system one chooses the non linear model of helicopter s dyn
14. trol system of the pitch angle and Matlab Simulink models of the adaptive command system s subsystems Thus characteristics that describe the adaptive command system s dynamics with linear or non linear actuator are obtained Key words dynamic inversion neural network adaptive command helicopter pitch angle 1 INTRODUCTION The complexity and incertitude that appear in the non linear and instable phenomena are the main reasons that require the projecting of non linear adaptive structures for control and stabilization in these cases the linear models are far from a good description of the flying objects dynamic Another reason is the non linear character of the actuators The observers must be easily adaptable and their project algorithms must allow the state s estimation of the flying object even in the case of their failure or no use of the damaged sensors signals In these situations it s good to use the real time adaptive control based on neural networks and dynamic inversion of the unknown or partial known nonlinearities from the dynamic model of the flying object 1 The neural network s training is based on the signals from state observers these observers get information about the control system s error 2 3 4 2 ADAPTIVE COMMAND BASED ON DYNAMIC INVERSION Let s consider the dynamic system A with single input and single output described by the equations i f xu y h x 1 with x n x 1 n known
15. ulated with other signals and it results the component v of form 11 Let s consider H s the transfer function of the linear subsystem of A flying object with the input u and the output y having to the numerator a p order polynomial and at the denominator a r order polynomial p lt r 1 For this system the paper s authors propose the command structure from fig 1 with the linear part described by the equations 8 10 A 1 vA s Y z y E ee gt h x u G hyv pj A d wv p b b by v INVERSE PE 3 I l i MODEL NON f pf pm ppm pmpa te g Ls yo ag h y u h x u i Fig 1 The block diagram of the adaptive command system based on dynamic inversion Considering YT ly 9 yO ZT o var ag ay Aa DT bo bi 8 8 with b i 0 p r ad Or 1L the coefficients of the transfer function s numerator and denominator for the system with input uw and output y the linear system with input v and output y is described by equation yO NY b Z e 9 If p 0 then Z v b b and the previous equation becomes E shee 10 In the particular case y y one obtains 3 Non linear adaptive system for the command of the helicopters pitch s angle vy 5 4977

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