Linear systems with time-varying coefficients are of a great concern in various fields such as control engineering, network theory, etc.
However, the theoretical treatment is so difficult that very few achievements can be seen so far.
In this paper, the discussions are confined to those classes of systems which can been transformed into linear stationary systems. Those systems are known as the “reducible” systems.
First, a sufficient codition is shown for a system consisting of reducible subsystems to be reducible. As an illustration, a sufficient condition is derived for a general second-order system to be solvable analytically. Next, the class of systems described by
Σ
ni=1ai[
d/
dt+α(
t)]
ix(
t)=Σ
mj=1bi[
d/
dt+α(
t)]
ju(
t)
is shown to be reducible. This particular class of systems is examined in some detail with respect to their equivalent block diagram representation, etc.
Further, discussed are some applied problems such as the compensation of unstable systems, and the optimal control of reducible systems, where the adjoint system of a reducible system is shown to be reducible.
Future problems are suggested at the end.
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