Page 34 - Neural Network Modeling and Identification of Dynamical Systems

22 1. THE MODELING PROBLEM FOR CONTROLLED MOTION OF NONLINEAR DYNAMICAL SYSTEMS

1.2.1 Types of Adaptation 1.2.1.2 Structural Adaptation
Following [35], we distinguish the types (hi- However, sometimes the required behavior
erarchical levels) of adaptation as: variability for the system S cannot be achieved
by varying only the values of the system pa-
• parametric adaptation; rameters ϑ(t i ) ∈
. The next hierarchical level of
• structural adaptation; adaptive systems are systems capable of struc-
• adaptation of object; tural adaptation, i.e., systems capable to change
• adaptation of control goals.
their structure (a set of elements of the system S,
as well as the relations between these elements)
1.2.1.1 Parametric Adaptation
according to the changing situation λ(t i ) ∈ S
Parametric adaptation is performed by ad- S
and the goal γ (t i ) ∈
.
justing the value of the tuning parameters ϑ(t i ) ∈ In the simplest case the system S includes a

of the system S, which are a subset of system set of structurally alternative variants of the rule
parameters w; see page 17 (such parameters can S S P
={ p } , p = 1,...,N . At the time instant
be, for example, the controller gains). t i , a rule with a certain index p is applied. The
In this case, we consider that the rule S
specific value of this index is determined by the
depends not only on x, u, ξ, t,aswasindi- S S
switching rule = (λ,γ ,t). A more com-
cated earlier, but also on ϑ(t i ) ∈
. This means plicated, but more exciting example is the sys-
S
S
that = (ϑ) is a parametric family of func- tem S that undergoes evolutionary changes in
tions. If we specify some constant value of the
the structure under the influence of the environ-
vector ϑ(t i ) ∈
, we thereby select a function
S
S
= (x,u,ξ,t) from this family. The rule ment E (and, possibly, some other factors).
S
S
= (λ,γ ,t) defines values ϑ(t i ) transferred In biology, a mechanism of this kind is called
S
S
to = (ϑ(t i )), which leads to a change in the adaptation (which means an irreversible evo-
nature of the response of the system S to the ef- lutionary change in the genotype of the sys-
fects of the environment E, i.e., to a change in tem). On the other hand, the accommodation
behavior of this system. discussed in Section 1.2.1.1 is a reversible adjust-
Possible mechanisms for changing the values ment of parameters.
of the vector of the parameters ϑ(t i ) ∈
for the S
1.2.1.3 Adaptation of Object
system are not discussed here; we will consider
them in the following sections. It is quite possible that no variations of
The values of the vector of the parameters the system S parameters ϑ(t i ) ∈
or structure
ϑ(t i ) ∈
for the system S can be piecewise con- would allow us to achieve certain goals. This
stant, i.e., their values will remain constant not situation is quite natural, because the potential
capabilities of any system are limited, and these
just for a single ordered pair λ(t i ),γ (t i ) ∈ ×

, but for the entire subdomain i ×
i ⊂ ×
limits are caused by the “design” of the system.
of such a domain. This approach is rather widely If a case of this kind arises, then the next level of
used in control systems. We refer to it as Gain adaptation may be involved, namely an adapta-
Scheduling. tion of the object.
The adjustment might also be continuous, so In Section 1.1.1.1, we formulated the thesis
that each pair λ(t i ),γ (t i ) ∈ ×
corresponds that there is a particular system, which is the ob-
to some value ϑ(t i ) ∈
. ject of our study, and there is all the rest that
This concept of parametric adaptation corre- we did not include in this system. We call this
sponds to the concept of accommodation in biol- “all the rest” as the (external) environment. The
ogy. adaptation of an object involves a revision of the