Page 218 - Neural Network Modeling and Identification of Dynamical Systems

6.3 SEMIEMPIRICAL MODELING OF AIRCRAFT THREE-AXIS ROTATIONAL MOTION 209
an aircraft, they can be easily estimated using relative damping coefficients for the actuators of
the appropriate ANN modules obtained during the controlled stabilizer, rudder, and ailerons; D,
¯
¯
the generation of a semiempirical ANN model L, Y are the drag, lift, and side forces; L, M, N are
¯
(see also the end of the previous section). the roll, pitch, and yaw moments; m is the mass
The initial theoretical model of the total an- of the aircraft, kg.
gular motion of the aircraft, used for the devel- The coefficients c 1 ,...,c 9 in (6.6) are defined
opment of the semiempirical ANN model, is a as follows:
system of ODEs, traditional for flight dynamics 2
c 0 = I x I z − I ,
of aircraft [13–19]. This model has the following xz
2
form: c 1 =[(I y − I z )I z − I ]/c 0 ,
xz
⎧ c 2 =[(I x − I y + I z )I xz ]/c 0 ,
¯
¯
˙ p = (c 1 r + c 2 p)q + c 3 L + c 4 N,
⎪
⎨
2 2 c 3 = I z /c 0 ,
¯
˙ q = c 5 pr − c 6 (p − r ) + c 7 M, (6.6)
⎪ c 4 = I xz /c 0 ,
⎩
¯
¯
˙ r = (c 8 p − c 2 r)q + c 4 L + c 9 N,
c 5 = (I z − I x )/I y ,
⎧
˙
⎪ φ = p + q tanθ sinφ + r tanθ cosφ, c 6 = I xz /I y ,
⎪
⎪
⎨
˙ θ = q cosφ − r sinφ, c 7 = 1/I y ,
(6.7)
⎪ 2
⎪ sinφ cosφ c 8 =[I x (I x − I y ) + I ]/c 0 ,
ψ = q
⎩ ˙ + r , xz
⎪
cosθ cosθ c 9 = I x /c 0 ,
⎧
˙ α = q − (p cosα + r sinα)tanβ
⎪
⎪ where I x , I y , I z are moments of inertia of the
⎪
⎪ 1
⎨
+ (−L + mg 3 ), aircraft with respect to the axial, lateral, and nor-
mV cosβ (6.8) 2
⎪ mal axes, kg·m ; I xz , I xy , I yz are centrifugal mo-
⎪
⎪ 1 2
⎪ ments of inertia of the aircraft, kg·m .
⎩ ˙
β = p sinα − r cosα + (Y + mg 2 ),
mV The aerodynamic forces D, L, Y in (6.7)and
¯
⎧ the moments L, M, N in (6.6) are defined by re-
¯
¯
2
˙
⎪ T ¨ δ e =−2T e ζ e δ e − δ e + δ act ,
⎪ e e
⎨ lationships of the following form:
2 act
δ
˙
T ¨ =−2T a ζ a δ a − δ a + δ a , (6.9)
a a
⎪ ⎧ ¯ ¯ ¯
⎪ 2 act ⎪ D =−X cosα cosβ − Y sinβ − Z sinα cosβ,
⎩ ˙
T ¨ δ r =−2T r ζ r δ r − δ r + δ . ⎨
r r
¯
Y =−X cosα sinβ + Y cosβ − Z sinα sinβ,
¯
¯
The following notation is used for this model: ⎪ L = X sinα − Z cosα,
⎩
¯
¯
p, r, q are the roll, yaw, and pitch angular ve- (6.10)
locities, deg/sec; φ, ψ, θ are the roll, yaw, and
pitch angles, deg; α, β are the angles of attack ⎧ ¯
⎪ X = q p SC x (α,β,δ e ,q),
and sideslip, deg; δ e , δ r , δ a are the deflection ⎨
¯
Y = q p SC y (α,β,δ r ,δ a ,p,r), (6.11)
angles of the controlled stabilizer, rudder, and ⎪
⎩ ¯
˙
˙
ailerons, deg; δ e , δ r , δ a are the angular veloci- Z = q p SC z (α,β,δ e ,q),
˙
ties of the deflection of the controlled stabilizer, ⎧ ¯
rudder, and ailerons, deg/sec; V is the airspeed, ⎪ L = q p SbC l (α,β,δ e ,δ r ,δ a ,p,r),
⎨
¯
m/sec; δ act , δ r act , δ a act are the command signals M = q p S ¯cC m (α,β,δ e ,q), (6.12)
e
to the actuators of the controlled stabilizer, rud- ⎪ N = q p SbC n (α,β,δ e ,δ r ,δ a ,p,r).
⎩
¯
der, and ailerons, deg; T e , T r , T a are the time
constants for the actuators of the controlled sta- The variables g 1 , g 2 , g 3 required in (6.8)are
bilizer, rudder, and ailerons, sec; ζ e , ζ r , ζ a are the the projections of the acceleration of gravity on