Page 107 - Neural Network Modeling and Identification of Dynamical Systems

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

FIGURE 3.1 General structure of the NARX-model. (A) Model with parallel architecture. (B) Model with series-parallel
architecture.

where y p (k) is the observed (measured) output implements the following mapping:
of the process described by the dynamical sys-
tem [12–14]. g(k) = ϕ NN (y p (k − 1),...,y p (k − n), (3.5)
We assume that the output of the dynami- u(k − 1),...,u(k − m),w),
cal system is affected by additive noise, and the
where w is a vector of parameters and ϕ NN (·) is
point of summation of the output signal and
noise precedes the point from which the feed- a function implemented by a feedforward net-
back signal comes. In this case, the output of the work.
system at time step k will be affected by the noise Suppose that the values of parameters w for
signal both at time step k and also at n previous the network are computed by training it in such
a way that ϕ NN (·) = ϕ(·), i.e., this network accu-
time steps.
rately reproduces the outputs of the simulated
In the case the function ψ is represented by a
dynamical system. In this case, for all time in-
feedforward neural network, the representation
(3.4) corresponds to a NARX-type model (Non- stants k,the relation
linear Auto-Regressive network with eXoge- y p (k) − g(k) = ξ(k), ∀k ∈{0,N},
neous inputs) [15–23], i.e., nonlinear autoregres-
sion with external inputs, in its series-parallel will be satisfied, i.e., the simulation error is equal
version (see Fig. 3.1B). to the noise affecting the output of the dynami-
As noted above, we consider the case when cal system. This model can be called ideal in the
the additive noise affecting the output of the dy- sense that it accurately reflects the determinis-
namical system influences outputs not only di- tic components of the dynamical system process
rectly at current time step k, but also via the out- and does not reproduce the noise that distorts
puts at n preceding time steps. The requirement the output signal of the system. The inputs of
to take into account previous outputs is imposed this model are the values of the control variables,
because ideally, the simulation error at step k as well as the measured outputs of the process
should be equal to the noise value at the same implemented by the dynamical system. In this
time. Accordingly, when designing a model of case, the ideal model, which is a one-step-ahead
a dynamical system, it is necessary to take into predictor, is trained as a feedforward neural net-
account system outputs at past time instants to work, and not as a recurrent network. Thus, in
compensate for the noise effects that have oc- this case in order to obtain an optimal model, it
curred. The corresponding ideal model can have is advisable to use supervised learning methods
the form of a feedforward neural network that available for static ANN models.