Methods of the second type are based on the concept of the parametric transfer
function (PTF), which at all steps of investigation uses only finite-dimensional
matrices and determinants.
The basic idea of this method applied to continuous linear nonstationary systems
was already given in [5, 6], where the author introduced the transfer function Ws; tðÞ,
which differs from the ordin ary transfer function WpðÞfor continuous linear
time-invariant (LTI) system by its dependence on the time t as a parameter. The paper
[7] mentions the principal possibility to extend the PTF concept to SD systems.
The further realizations of this idea have been elaborated in [8]. Obviously, the
first detailed presentation of the theory of SISO SD systems based on the PTF is
provided in [9].
Further progress and generalization of the PTF method concerning SISO SD
systems, including the solution of H
2
; H
1
and L
2
optimization problems,
development of corresponding software tools, and solution of applied problems, are
provided in [10, 11]. Hereby, the solution of the optimization problems is based on
the application of the Wiener-Hopf method, connected with the use of separation
and factorization operations . With the results of these works, we are able to state
particular features of the PTF method, which make it useful for the solution of
practical problems. In this regard, we note the follow ing:
1. The PTF method has a universal charact er. Its applicability does not depend on
the structure of the SD system and the entry points of external excitations.
2. The PTF method is based completely on frequency representations and does not
require an intermediate transition to the state space.
3. The PTF method opens the possibility of taking into account the influence of the
structure of SD system on its dynamical characteristics and the results of
optimization.
4. The application of the PTF method allows to find the subset of fixed poles of H2
and L2 optimal systems independently of the type of the stochastic input signal.
The fixed poles are determined only by the properties of the poles of the con-
tinuous elements of the SD system and its structure. It is proved that this set of
fixed poles remains unchange d for SISO SD systems, which are optimal by the
criterion H
1
.
5. The PTF method allows the direct generalization to SD systems, in which the
inputs and outputs of their elements have arbitrary constant delays. In particular,
this property opens the possibility to approximately estimate the effect of
computational delay on the quality of processes in SD systems and results
of their optimization.
6. The existing relation between the PTF and the Laplace transformed output of the
SD system allows to extend the Wiener-Hopf method to the L2 optimization
problem.
7. On the basis of the PTF concept, we can formulate and solve an optimization
problem that guarantees the quality of an SD system for a certain set of external
disturbances.
vi Preface
