Page 111 - Neural Network Modeling and Identification of Dynamical Systems

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

and Hammerstein–Wiener type (Fig. 3.2)[39–50].
These systems are sets of blocks of the type
“static nonlinearity” (realized as a nonlinear
function) and “linear dynamical system” (real-
ized as a system of linear differential equations
or as a linear recurrent network). The Wiener
model (Fig. 3.2A) contains a combination of one
nonlinear block (N) of the first type and the
next linear block (L) of the second type (struc-
ture of the form N–L), the Hammerstein model
(Fig. 3.2B) is characterized by a structure of the
type L–N. Combined variants of these structures
consist of three blocks: two of the first type and
one of the second type in the Hammerstein–
Wiener model (structure of the type N–L–N,
FIGURE 3.2 Block-oriented models of controllable dy- Fig. 3.2C) and two of the second type and one of
namic systems. (A) Wiener model (N–L). (B) Hammerstein
model (L–N). (C) Hammerstein–Wiener model (N–L–N). the first type in the Wiener–Hammerstein model
(D) Wiener–Hammerstein model (L–N–L). Here F(·), F 1 (·), (a structure of the form L–N–L, Fig. 3.2D). This
and F 2 (·) are static nonlinearities (nonlinear functions); L(·), block-oriented approach is suitable for systems
F 1 (·),and F 2 (·) are linear dynamical systems (differential of classes SISO, MISO, and MIMO. Some works
equations or linear RNN).
[39–50] show ANN implementations of models
of these kinds.
Using the ANN approach to implement mod-
dynamic ANN models. This approach, as well
as some others, based on the use of feedforward els of the abovementioned types in comparison
with traditional approaches provides the follow-
neural networks, finds some application in solv-
ing advantages:
ing problems of modeling and identification of
dynamical systems [30–38]. • static nonlinearity (nonlinear function) can be
The second situation is related to the block- of almost any complexity; in particular, it can
oriented approach to system modeling. With be multidimensional (function of several vari-
this approach, the dynamical system is repre- ables);
sented as a set of interrelated and interacting •the F(·) transformations required to imple-
blocks. Some of these blocks will represent the ment the block-based approach, which is of-
realization of some functions that are nonlinear ten used to solve various applied problems,
in the general case. These nonlinear functions are formed by training on experimental data
can be realized in various ways, including in characterizing the behavior of the dynami-
the form of a feedforward neural network. The cal system under consideration, i.e., there is
value of the neural network approach in this no need for a laborious process of forming
case is that a particular kind of these ANN func- such relationships before the beginning of the
tions can be “recovered” on the basis of exper- modeling process;
imental data on the simulated system by using • a characteristic feature of ANN models is
appropriate learning algorithms. their “inherent adaptability,” realized through
A typical example of such a block-oriented the learning processes of networks, which
approach are the nonlinear controlled systems provides, under certain conditions, the pos-
of Wiener, Hammerstein, Wiener–Hammerstein, sibility of on-line adjustment of the model