Page 116 - Neural Network Modeling and Identification of Dynamical Systems

3.4 ANN-BASED CONTROL OF DYNAMICAL SYSTEMS 105
system (see, for example, [63–66]), has the form locity of the pitch q andthepitchangle θ can be
carried out using the assumption V = const. In
m(V x − V z q) = X, this case, equations for V x and V z become equiv-
˙
˙
˙
˙
m(V z + V x q) = Z, alent to the equation θ = q, from which it follows
˙
that we can use the system of two equations, i.e.,
I y ˙q = M y , (3.21)
theequationfor q and any of the above equiva-
θ = q, lent equations.
˙
˙
H = V sinθ, Here we choose the system of equations
˙
where X, Z are the projections of all forces acting m(V z + V x q) = Z,
(3.22)
on the aircraft on the Ox-axis and the Oz-axis, I y ˙q = M y .
respectively; M y is the projection of all the mo-
ments acting on the aircraft onto the Oy-axis; q The system of equations (3.22) is closed, since
is angular velocity of pitch; m is the mass of the the angle of attack α entering into the expres-
aircraft; I y is the moment of inertia of the aircraft sions for Z and M will be equal in the case un-
der consideration to the pitch angle θ,which is
relative to the Oy-axis; V is the airspeed; V x , V z
are the projections of the airspeed on the Ox-axis related to V z by the following kinematic depen-
and the Oz-axis, respectively; H is the altitude of dence:
flight.
V y =−V sinθ.
The system of equations (3.21) can be sim-
plified, based on the choice of the trajectory of Thus, the system of equations (3.22) describes
motion and some physical features inherent in the transient processes concerning the angular
the aircraft. Let us first consider the steady hori- velocity and the pitch angle, which occur imme-
zontal flight of an airplane that occurs at a given diately after breaking the balance corresponding
altitude H with a given airspeed V .Asiswell to the steady horizontal flight.
known [63–66], in this case, from the solution of Let us reduce the system of equations (3.22)
the system of equations to the Cauchy normal form, i.e.,

X(α,V,H,T,δ e ) = 0, dV z Z
= − V x q,
Z(α,V,H,T,δ e ) = 0, dt m (3.23)
dq M y
M y (α,V,H,T,δ e ) = 0, = .
dt I y
we can find the angle of attack α 0 ,the thrust of
In (3.23), the value of the pitch moment M y is
the engine T 0 , and the angle of deflection of the
(0) a function of the control variable. This variable
elevator (all-moving stabilizer) δ e , necessary is the deflection angle of the elevator (or all-turn
for this flight. Suppose that at the time t 0 ,thede- stabilizer), that is, M y = M y (δ e ).
flection angle of the stabilizer (or the value of the So, in the particular case under consideration,
corresponding command signal) has changed by the composition of the state and control vari-
the value δ e . The change in the position of the
ablesisasfollows:
stabilizer disturbs the balance of the moments
acting on the aircraft, as a result of which its an- x =[V z q] , u =[δ e ]. (3.24)
T
gular position in space will change before this
affects the change in the value of the aircraft ve- As noted above, the analysis uses an indirect
locity vector. This means that the study of tran- approach to estimating the dynamic properties
sient processes with respect to the angular ve- of the plant based on the nonlinear reference