Page 30 - Neural Network Modeling and Identification of Dynamical Systems

18 1. THE MODELING PROBLEM FOR CONTROLLED MOTION OF NONLINEAR DYNAMICAL SYSTEMS

path angle; V is airspeed; W = mg is the weight where α is the angle of attack, deg; θ is angle of
of the aircraft. pitch, deg; q is the angular velocity of the pitch,
In the climb problem, the state of the aircraft deg/sec; δ e is the deflection angle of the con-
as a dynamical system is described by two vari- trolled stabilizer, deg; C L is the lift coefficient;
ables: airspeed V and the flight path angle γ . C m is pitch moment coefficient; m is mass of the
2
Accordingly, the state vector x ∈ X in this case aircraft, kg; V is the airspeed, m/sec; q p = ρV /2
has the form is the dynamic air pressure, kg·m −1 sec −2 ; ρ is
3
air density, kg/m ; g is the acceleration of grav-
2
2
x = (x 1 ,x 2 ) = (V,γ ). ity, m/sec ; S is the wing area, m ; ¯c is mean
aerodynamic chord of the wing, m; I y is the mo-
Example 2 (Curved aircraft flight in the horizon- ment of inertia of the aircraft relative to the lat-
2
tal plane). The system of equations of motion for eral axis, kg·m ; the dimensionless coefficients
an airplane that performs a turn with roll and C L and C m are nonlinear functions of their argu-
sideslip can be written in the form [28–34] ments; T , ζ are the time constant and the rela-
is
tive damping coefficient of the actuator; δ e act
dV the command signal to the actuator of the all-
m = T − D,
dt turn controllable stabilizer (limited to ±25 deg).
d ˙
mV =−Y cosφ − Lsinφ + T sinβ cosφ, In the model (1.14), the variables α, q, δ e ,and δ e
dt are the states of the controlled object, the vari-
0 = Lcosφ − W. able δ e act is the control. Here, g(H) and ρ(H)
(1.13) are the variables describing the state of the envi-
ronment (gravitational field and atmosphere, re-
Here D is the aerodynamic drag force; L is the spectively), where H is the altitude of the flight;
lift; Y is the lateral force; T is thrust of the power m, S, ¯, I z , T , ζ are constant parameters of the
c
plant; is the yaw angle; V is the airspeed; β is simulated object, C L and C m are variable param-
the angle of sideslip; φ is the roll angle; W = mg eters of the simulated object.
is the weight of the aircraft.
We usually impose some limitations R X on
In thetaskofperformingaturnwithrolland
the suitable combinations of state values of the
sideslip, the state of the aircraft as a dynamical
system S,aswellasconstraints R U on appropri-
system is described by such variables as the air-
ate combinations of control values. As a general
speed V and the yaw angle . Accordingly, the
rule, we also have certain restrictions on the val-
state vector x ∈ X in this case has the form
ues of a combination of the state vector x and the
control vector u, i.e.,
x = (x 1 ,x 2 ) = (V, ).
x,u ∈ R XU ⊂ X × U
Example 3 (Longitudinal angular motion of an
aircraft). A system of equations describing the = X 1 × ··· × X n × U 1 × ··· × U p . (1.15)
longitudinal short-period motion of an aircraft
can be written in the form [28–34] Taking into account the above definitions, the
system S can be represented in the following
¯ qS g general form:
˙ α = q − C L (α,q,δ e ) + cosθ,
mV V
q p S ¯c (1.14) S = {U, ,Z},{F,G},{X,Y},T , (1.16)
˙ q = C m (α,q,δ e ),
I y where U are controllable influences on S; and
2 Z are uncontrollable influences on the states and
T ¨ δ e =−2Tζ ˙ δ e − δ e + δ e act ,