Page 106 - Neural Network Modeling and Identification of Dynamical Systems

3.2 NEURAL NETWORK BLACK BOX APPROACH TO SOLVING PROBLEMS ASSOCIATED WITH DYNAMICAL SYSTEMS 95
input control actions, these disturbances are un- mappings, f(·) and g(·) in (3.2)instead ofasin-
observable. gle mapping h(·) in (3.3).
The design procedure for a state space dy- The choice of the suitable model representa-
namical system model involves finding approx- tion (state space or input–output model) is not
imate representations for the functions f(·) and the only design choice required to take when
g(·), using the available data on the system. In modeling a nonlinear dynamical system. The
the case a model of the black box type is de- choice of method for taking disturbances into
signed, that is, we use no a priori knowledge account also plays an important role. There are
of the nature and features of the simulated sys- three possible options:
tem, such data are represented by sequences of
values of the input and output variables of the • disturbances affect the states of the dynamical
system. system;
A dynamical system model is said to have an • disturbances affect the outputs of the dynam-
input–output representation (a representation of ical system;
the system in terms of its inputs and outputs), if • disturbances affect both the states and the
it has the following form: outputs of the dynamical system.

As shown in [14], the nature of the disturbance
y(k) = h(y(k − 1),...,y(k − n),u(k − 1),...,
effect on the dynamical system significantly in-
u(k − m),ξ(k − 1),...,ξ(k − p)),
fluences the optimal structure of the model be-
(3.3)
ing formed, the type of the required algorithm
for its learning, and the operation mode of the
where h(·) is a nonlinear function, n is the order
generated model. In the next section, we con-
of the model, m and p are positive integer con-
sider these issues in more detail.
stants, u(k) is a vector of input control signals
of the dynamical system, and ξ(k) is the dis-
turbance vector. The input–output representa- 3.2.2 Approaches to Consideration of
tion can be considered a special case of the state Disturbances Acting on a
space representation when all the components Dynamical System
of the state vector are observable and treated as
output signals of the dynamical system. As noted, the way in which we take into ac-
In the case the simulated system is linear and count the influence of disturbances in the model
time invariant, the state space representation significantly influences both the structure of the
and the input–output representation are equiv- model and its training algorithm.
alent [12,13]. Therefore we can choose which of
them is more convenient and efficient from the 3.2.2.1 Input–Output Representation of the
point of view of the problem being solved. In Dynamical System
contrast, if the simulated system is nonlinear, the Let us first consider the case in which distur-
state space representation is more general and bance affects the state of the dynamical system.
at the same time more reasonable in compari- We assume that the required representation of
son with the input–output representation. How- the dynamical system has the following form:
ever, the implementation of the model in the
state space is usually somewhat more difficult
y p (k) = ψ(y p (k − 1),...,y p (k − n),
than the input–output model because it requires (3.4)
u(k − 1),...,u(k − m)) + ξ(k),
to obtain an approximate representation for two