Extracting Projectile's Aerodynamic Drag Coefficient. Curve Via High-order Iterative Learning Identification. Yangquan Chen, Changyun Wen. School of EEE ...
TP16 550
Proceedings of the 35th Conference on Decision and Control Kobe, Japan December 1996
Extracting Projectile’s Aerodynamic Drag Coefficient Curve Via High-order Iterative Learning Identification Yangquan Chen, Changyun Wen School of EEE, Nanyang Technological University, Nanyang Avenue, Singapore 639798 Jian-Xin Xu Dept. of Electrical Engineering, National University of Singapore, Singapore 119260 Mingxuan Sun Dept. of Electrical Engineering, Xi’an Institute of Technology, Xi’an 710032,China represents the distance between the tracking radar R’ and the projectile P‘, and u,.(t) be the doppler radar measured velocity data. Then we have u,.(t) = F ( t ) . To formulate our problem, we must transform the projectile’s tangential velocity U’ into a velocity u d in the radial direction of the doppler radar as follows. Let
Abstract A novel application of high-order iterative learning control (ILC) method is presented] which is for identifying or extracting projectile’s optimal fitting drag coefficient curve cdf from radar measured velocity data. ILC method is shown to be effective for such an optimal tracking cont,rol problem (OTCP) with its performance index in a minimax sense and with an arbitrarily chosen initial ‘control’ c d f . The curve identification results from a real flight testing data are compared and discussed for different high-order learning paramet,ers to show the effectiveness of the proposed method.
r y ,TZ1 = [x - x r ] y - Yr1 z - Z r ] ]
+
d m .
+
Then, v d = (u,r, uyry u z 5 ) / F (2) Let e ( t ) = u,.(t) - u d ( t ) which is the trackzng error. The problem is to solve
min
A
eb
Consider a 3-DOF point mass ballistic model. Suppose at. time t , the position of the projectile P’ in earth coordinate system (ECS) is [ ~ ( ty)( t,) , z(t)]* and its relat,ive velocity vector C w.r.t. ECS is [u,(t),uy(t), u Z ( t ) l T The . position of the radar R in ECS is [ z r ( t ) , y ~ ( t ) , z t ( t ) I T , which is known. The 3-DOF point mass trajectory model can be described by iLz(f) = -paSl’(u, - w,)Cdf/2?n0 G y ( t ) = -pasVu,Cdf/2mo - g o (1) G z ( t ) = - p a S l ” 7 ( U ~ - wz)Cdf/2???0
k ( t ) = k . ( t ) , b(t) = U Y ( t ) , t ( t ) = U Z ( t ) where t E [O.T] and T is known, go is the gravitational acceleration, U;,] w, are the wind components in ECS known from meteorological measurements; IT is projectile’s relative velocity 1v.r.t. the wind and 1 - = ( u s - u’,)Z + U : + ( U * - u ; ~ ) ? pa. is the air density, s = n d 2 / 4 is the reference area of the projectile and d is the projectile‘s diameter. ,no is the mass of the projectile. and C d f ( f )is the fitting drag coefficient curve 1v.r.t. the trajectory model (1). n-hich is regarded as the unconstrained control function. Define S ( t ) = [ u z ( t ) :u y ( t ) ,u , ( t ) . z ( t ) .y ( t ) , 2(t)lT and the initial state A .Y(O) is given: I\. is the Mach number. i.e. i\ = \.,/a. where a is the local sonic speed. Let ? ( t ) = R’P’ which I Corresponding author. Tel.:( +6?) 799494T( 0) : Fax: (+63)7912687: E-mail:ecywenOntuvas.ntu.ac.sg
I e(t) I .
= min s u p cdf(t)
cdf(t)
1 An Aerodynamic Identification Problem and High-order Iterative Learning Identification
0-7803-3590-2/96 S5.000 1996 IEEE
=
(3)
tE[O,T]
Difficulties will arise in solving this OTCP formulated by ( l ) ,(2), and (3) where the performance index is a minimax one. When a quadratical index form is used, this O T C P is a singular one [l]. However, this can be easily solved by ‘iterative learning identification’ concept as first investigated in [2]. This paper presents the results of the high-order learning identification scheme given by N (Cdj(t))i+l =x[pk(t)(Cdf(t))l k=l
+ Tk(t)el(t)l
(4)
where i is the ILC iteration number, p k , rk are the time varying learning parameters, N the order of ILC updatA .
+
ing law and I = 2 - k 1. To find a suitable selection of p k , r k , first consider the case N = 1. By referring to ( I ) , (2)and the Theorem 3.1 of [3], it is easy to verify that Pa 5 rTwx r Z w z r v ( t )= -{-v[v, - -- -1). (5) 2m Because w,
