Page 29 - Neural Network Modeling and Identification of Dynamical Systems

1.1 THE DYNAMICAL SYSTEM AS AN OBJECT OF STUDY 17
The set of variables describing the system S house, or immeasurable, for example, as
includes the following: an atmospheric turbulence);
• measuring noise ζ 1 ,...,ζ r , describing the
• the variables x 1 ,...,x n , describing the state of errors introduced by sensors;
the system S, • variables w 1 ,...,w s (constant or varying), de-
• the variables y 1 ,...,y m , describing the obser- scribing the properties of the system S (their
vation results for the state of system S.
values are directly or indirectly determined
The variables x 1 ,...,x n , describing the state by the decisions made during the design of
of the system S, are combined into a collection S) and influencing the behavior of the system
(vector) x, called the state (state vector) of the through the law of its evolution (including
given system, i.e., both constant and varying parameters of the
law). Examples of constant (permanent) pa-
x = (x 1 ,...,x i ,...,x n ); rameters of the system are aircraft wingspan,
(1.9)
x i ∈ X i ⊂ R; x ∈ R X ⊂ X = X 1 × ··· × X n . wing area, length, etc. Examples of varying
parameters of the system are coefficients of
Here X is the region of all possible states of the aerodynamic forces and moments, which are
23
system S (the space of states ); R X is the domain nonlinear functions of several variables (state
of all admissible states of the system S; R is the variables of the object and the environment,
set of real numbers. as well as control variables).
The variables y 1 ,...,y p , describing the obser- The variables u 1 ,...,u m , describing the con-
vation results for the state of the system S,are trol actions on the state of the system S, are com-
combined into the observation vector, i.e., bined into a collection u, called the control (con-
trol vector) of the given system, i.e.,
y = (y 1 ,...,y j ,...,y p );
(1.10)
y j ∈ Y j ⊂ R; y ∈ R Y ⊂ Y = Y 1 × ··· × Y p . u = (u 1 ,...,u k ,...,u m );
u k ∈ U k ⊂ R; u ∈ R U ⊂ U = U 1 × ··· × U m .
The list of variables that describe the factors
affecting the state of the system S includes the (1.11)
following elements:
Here U is thedomainof all possible values of con-
• the variables describing the effects on S,both trols for the system S; R U is the domain of all
controlled and uncontrolled, including: admissible values of controls for the system S.
• controls u 1 ,...,u m represent controllable
Example 1 (Longitudinal aircraft motion during
influences on S that may be varied in order
climb phase). A system of equations describ-
to meet the control goals of the system;
ing the motion of an aircraft during climb phase
• disturbances ξ 1 ,...,ξ q represent uncon-
for unsteady (V = 0) and nonrectilinear ( ˙ = 0)
γ
˙
trollable influences on S (they can have
flight can be written in the form [28–34]
either known or unknown properties, and
they can be either measurable, for ex- dV
ample, as an outdoor air temperature in m = T − D − W sinγ,
dt
the problem of thermal regulation of the (1.12)
dγ
mV = L − W cosγ.
dt
23 The state space in the theory of dynamical systems is often
also called the phase space, and the states of the system S are Here D is the aerodynamic drag force; L is the
phase states. lift; T is thrust of the power plant; γ is the flight