Page 32 - Neural Network Modeling and Identification of Dynamical Systems

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

reality, i.e., about the current situation. The situ- The point x ∈ X ⊆ X in the state space is the
ational awareness concept describes which data state of the system S at some time instant t ∈ T =
are obtained by observations and are available [t 0 ,t N ]. For the continuous time t ∈ T case and a
n
to the system S for generating control decisions. finite-dimensional state vector x ∈ X ⊆ R ,inor-
Awareness of some components of the situation der to specify states at all time instants we need
is usually incomplete (we know their values in- to specify a vector function
accurately) or zero (their values are unknown).
For one part of the components it is provided x(t) = (x 1 (t),x 2 (t),...,x n (t))
T
by direct observation (measurement), while for =[x 1 (t)x 2 (t)...x n (t)] . (1.18)
the other part, awareness is provided algorith-
mically, i.e., by estimation of their values based Taking into account the remarks made above
on the known values of the other components. regarding the transition from the continuous
Thus, when we consider a system S with un- time t ∈[t 0 ,t N ] to the discrete time (1.17), in-
certainties, we actually assume an incomplete sit- stead of continuous phase trajectories (1.18)we
uational awareness for it, i.e., the fact that the val- will consider their discrete representation in the
ues of some part of the internal or external com- form of a set of sequences of the form
ponents of the situation for S are unknown or
known inexactly. x(t i ) ={(x 1 (t i ),x 2 (t i ),...,x n (t i ))},i= 0,1,...,N.
(1.19)
1.1.6 Behavior and Activity of Systems The behavior of the system S is the sequence of
its phase states x(t i ) ∈ X, tied to the correspond-
The current state of the system S is described ing time instants t i ∈ T , i.e.,
by the set (1.9) of variables x i ∈ X i describing it
in the problem being solved. This set is consid- { x(t i ),t i }, t i ∈[t 0 ,t f ]⊆ T, i = 0,1,...,N.
ered to be a tuple of length n, i.e., (1.20)

x = x 1 ,x 2 ,...,x n , x i ∈ X i , i = 1,2,...,n, The activity of the system S is a sequence of its
purposeful actions, each of which is a response
T
or as a column vector x =[x 1 x 2 ...x n ] . Here,
of the form
x ∈ X, X ⊆ X 1 × X 2 × ... × X n .
situation, goal ⇒ action ⇒ result,
The ranges X i of continuous variables x i are usu-
that is,
ally subsets of the set of real numbers R.
Which variables x i areincludedinthe set S
{ λ(t i ),γ (t i ) } −−→ λ(t i+1 ), i = 0,1,2,...,N,
(vector) x that describes the state of the system
S depends on the nature of the given system (1.21)
and the problem being solved. In a context of or, equivalently,
different problems, the same system S may be
described by different sets x that include differ- λ(t i+1 ) = ( λ(t i ),γ (t i )), i = 0,1,2,...,N.
S
ent variables x i . Examples 1 and 2 on page 18
may serve as an illustration of this assertion: the Here λ(t i ) ∈ is the current situation, γ(t i ) ∈
is
S
state of the same aircraft is described by different the current goal, and is the law of evolution
variables for the problem of longitudinal motion of the system S.
during climb phase and for the problem of curvi- All kinds of systems S exhibit this behav-
linear flight in the horizontal plane. ior (1.20), whether they are controllable or not