Page 114 - Neural Network Modeling and Identification of Dynamical Systems

3.4 ANN-BASED CONTROL OF DYNAMICAL SYSTEMS 103
In the following sections, we describe the are characterized by the stability of the motion
principal features of the ANN model for aircraft of the plant and the quality of its transient pro-
motion, and we propose the technology of its cesses.
formation. The next step considers one of the The stability of the plant motion in the vari-
possible applications of ANN models of dynam- able x i ,i = 1,...,n, is determined by its ability
ical systems. This is the synthesis of the control to return over time to some undisturbed value
(0)
neural network (neurocontroller) to adjust the of this variable x (t) after the disturbance dis-
i
dynamic properties of the plant. We form the appears [61].
reference model of the aircraft motion, to the be- The nature of the plant transient processes
havior of which the neurocontroller should try that arise as a response to a stepwise action is
to lead the response of the original plant. We estimated using the appropriate performance in-
build an example of a neurocontroller, which dices (quality indicators), which usually include
produces signals for adjusting the behavior of the following [59,61]: transient time, maximum
the aircraft in a longitudinal short-period mo- deviation in the transient process, overshoot,
tion. This example is primarily based on the re- frequency of free oscillations, time of the first
sults of the ANN simulation of the object. steady-state operation, and number of oscilla-
tions during the transient process.
3.4.1.1 The Problem of Adjusting the Instead of these indices we use the indirect
Dynamic Properties of a Controlled approach based on some reference model to
Object evaluate the dynamic properties of the plant. It
Let the considered controlled object (plant) be can be obtained using the abovementioned qual-
a dynamical system described by a vector differ- ity indicators for the transient processes of the
ential equation of the form [59–61] plant and, possibly, some additional consider-
ations, for example, pilots’ assessments of the
˙ x = ϕ(x,u,t). (3.13) aircraft handling qualities.
Using the reference model, we can estimate
T
n
In Eq. (3.13), x =[x 1 x 2 ... x n ] ∈ R is the vector the dynamic properties of the plant as follows:
T
of state variables of the plant; u =[u 1 u 2 ... u m ] ∈
m
R is the vector of control variables of the plant; ∞ (ref ) 2
n
m
n
R , R are Euclidean spaces of dimension n and I = [x i (t) − x i (t)] dt (3.14)
0
m, respectively; t ∈[t 0 ,t f ] is the time. i=1
In Eq. (3.13), ϕ(·) is a nonlinear vector func- or
tion of the vector arguments x, u and the scalar
n
argument t. It is assumed to be given and be- ∞ (ref ) 2
longs to some class of functions that admits the I = λ i [x i (t) − x i (t)] dt, (3.15)
i=1 0
existence of a solution of Eq. (3.13) for given x(t 0 )
and u(t) in the considered part of the plant state where λ i are the weighting coefficients that es-
space. tablish the relative importance of the change for
The controlled object (3.13) is characterized different state variables.
by a set of its inherent dynamic properties [59, We could use the linear reference model
61]. These properties are usually determined by
the plant response to some typical test action. ˙ x (ref ) = Ax (ref ) + Bu (3.16)
For example, when the plant is an airplane this
action can be some stepwise deflection of its ele- with matrices A and B matched appropriately
vator by a prescribed angle. Dynamic properties (see, for example, [62]), as well as the original