Page 105 - Neural Network Modeling and Identification of Dynamical Systems

94 3. NEURAL NETWORK BLACK BOX APPROACH TO THE MODELING AND CONTROL OF DYNAMICAL SYSTEMS

Bearing in mind the definition (3.1), we can 3.2.1 Main Types of Models
distinguish the following main classes of prob-
There are two main approaches to the rep-
lems related to dynamical systems: resentation (description) of dynamical systems
[12–14]:
1. U,P,Y is behavior analysis for a dynamical
system (for U and P find Y); • a representation of the dynamical system in
2. U,P,Y is control synthesis for a dynamical the state space (state space representation);
• a representation of the dynamical system in
system (for P and Y find U);
terms of input–output relationships (input–
3. U,P,Y is identification for a dynamical sys-
output representation).
tem (for U and Y find P).
To simplify the description of approaches to
Problems 2 and 3 belong to the class of in- the modeling of dynamical systems, we will as-
verse problems. Problem 3 is related to the pro- sume that the system under consideration has
cess of creating a model of some dynamical sys- a single output. The obtained results are gener-
tem, while problems 1 and 2 associate with us- alized to dynamical systems with vector-valued
output without any difficulties.
ing previously developed models.
For the case of discrete time (most important
for ANN modeling), we say about the model
that it is a representation of a dynamical system
3.2 NEURAL NETWORK BLACK
in the state space if this model has the following
BOX APPROACH TO SOLVING form:
PROBLEMS ASSOCIATED WITH
x(k) = f(x(k − 1),u(k − 1),ξ 1 (k − 1)),
DYNAMICAL SYSTEMS (3.2)
y(k) = g(x(k),ξ 2 (k)),
Traditionally, differential equations (for con- where the vector x(k) is the state vector (also
tinuous time systems) or difference equations called phase vector) of the dynamical system
(for discrete time systems) are used as models whose components are variables describing the
of dynamical systems. As noted above, in some state of the object at time instant t k ; the vec-
cases such models do not meet certain require- tor u(k) contains the input control variables of
the dynamical system as its components; vectors
ments, in particular, the requirement of adapt-
ξ 1 (k) and ξ 2 (k) describe disturbances that affect
ability, which is necessary in case the model is
the dynamical system; the scalar variable y(k)
supposed to be applied in onboard control sys-
is the output of the dynamical system; f(·) and
tems. An alternative approach is to use ANN g(·) are a nonlinear vector-valued function and a
models that are well suited for application of scalar-valued function, respectively. The dimen-
various adaptation algorithms. sion of the state vector (that is, the number of
In this section, we consider ANN models of state variables in this vector) is usually called the
order of the model. State variables can be either
the traditional empirical type, i.e., models of the
available for observation and measurement of
black box type [1–11] for dynamical systems.
their values, or unobservable. As a special case,
In Chapter 5, we will extend these models to
dynamical system output may be equal to one
semiempirical (gray box) ones by embedding of its state variables. The disturbances ξ 1 (k) and
the available theoretical knowledge about the ξ 2 (k) can affect the values of the dynamical sys-
simulated system into the model. tem outputs and/or its states. In contrast to the